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univERSITORMCH

THE THEORY OF LIGHT
MACMILLAN AND CO., LIMITED
LoNDON • BOMBAY - CALCUTTA - MADRAS
MELBOURNE
THE MACMILLAN COMPANY
NEW YORK • BOSTON - CHICAGO
DALLAS • SAN FRANCISCO
THE MACMILLAN CO. OF CANADA, LTD.
- TORONTO
THE
THEORY OF LIGHT
BY THE LATE
THOMAS PRESTON, M.A. (DUB.), D.Sc. (R.U.I.), F.R.S.
FELLów of THE ROYAL UNIVERSITY OF IRELAND
PROFESSOR OF NATURAL PHILosophy, UNIVERSITY COLLEGE, DUBLIN
AND INSPECTOR OF SCIENCE SCHOOLS UNDER THE SCIENCE AND ART DEPARTMENT
FIFTH E DITION
EDITED BY
ALFRED W. PORTER, D.Sc., F.R.S.
PROFEssor of PHYSICS IN THE UNIVERSITY OF LONDON
MAC MILLAN AND CO., LIMITED
ST. MARTIN'S STREET, LONDON
1928
COPYRIGHT
First Edition printed 1890. Second Edition 1895.
Third Edition, 1901. Fourth Edition, 1912.
Fifth Edition 1928.
PRINTED IN GREAT BRITAIN
By R. & R. CLARK, LIMITED, EDINBURGH
3°oſºol?-312-to
%.
A. %. %2 tº ‘.
Á sº g? * .2 /
2 # 2, 2 &
PREFACE TO FIFTH EDITION
IN editing a new edition of this work the desire has been to
maintain, as far as possible, the character of the preceding
editions. At the same time it was felt that in some instances
the growth of the subject has required the inclusion of fresh
matter, and has shown that some parts have in the past received
rather undue attention. In the former category may be men-
tioned the treatment of thick lenses and coaxal surfaces in
general (pp. 116-120), an outline of the theory of aberrations
(pp. 134-143), the representation of white light as pulses
(p. 183), extension of the treatment of Haidinger's fringes
(pp. 205-209), and of the analysis of spectral lines by Michelson's
interferometer (pp. 231-233), an extension of the account of
the echelon and Lummer grating (pp. 316-320) and of Talbot's
bands (pp. 320-322). The resolving power of optical instru-
ments has been dealt with much more fully (pp. 322, 328-332),
and in this part the term chromatic resolving power has been
introduced to distinguish the power of separating different wave
lengths by means of prisms, etc., from the type of resolving
power with which microscopes and telescopes are concerned ;
the latter, when necessary, has been termed geometric resolving
power.
Further additions include an account of Michelson's method
of measuring the diameters of stars (pp. 332–333), and a brief
account of recent work on the relative motion of the earth
and the ether (pp. 569-570). The electromagnetic theory of
dispersion is given more fully (pp. 612-624).
V
vi THEORY OF LIGHT
Amongst the parts that have been removed may be men-
tioned a number of the mathematical theorems connected with
the calculations of Fresnel integrals—a few representative ones
only have been retained. All belong more properly to a treatise
on mathematics. Moreover, they seem specially inappropriate
here because, although Fresnel's phenomena are of very great
interest historically in relation to the development of the wave
theory of light, they are of very little practical importance.
The account of MacCullagh's theory has also been much
curtailed.
The whole of the account of diffraction has been rearranged
so as to separate the description of the shadow fringes of
Fresnel from that of the fringes obtained by Fraunhofer in the
focal plane of a lens. In previous editions the order followed
was in the main that adopted by Verdet in his Leçons d'Optique
and by Mascart in his Traité d’Optique. Probably the first
clear separation of the two types of phenomena was made
first by the third Lord Rayleigh in his classical article on the
“Wave Theory of Light,” in the ninth edition of the Encyclopædia
Britannica (Collected Papers, vol. iii). It is hoped that this
change will tend towards clarity.
The principal additions made by previous editors were
distinguished by being included in brackets. It has been felt
to be impossible to carry out the same method, owing to the
breaks in the text which such a method entails. In consequence,
all the brackets have been removed, and an attempt has been
made to blend the previous additions with the main text by
making such changes as were necessary to make the narrative
continuous.
I take the opportunity of expressing my obligations to the
previous editors for the lightening of my task by the valuable
additions they had made to Preston's original work.
Although the principal changes are here indicated, it must
be added that the whole of the book has been revised, and very
numerous small changes have been made in the interests of
clearness of exposition. One example may be given. In dealing
with polarised light a clear distinction has to be drawn between
PREFACE TO FIFTH EDITION vii
the direction of propagation of a wave and the direction in which
the particle in the ether vibrates to and fro. The latter is in
the present edition distinguished by being referred to as the
azimuth of vibration, and in this way confusion is avoided.
I desire to express my thanks to Dr. L. F. Bates, Lecturer
in Physics at University College, for his care in preparing the
index.
ALERED W. PORTER.
April, 1928.
PREFACE TO SECOND EDITION
In this edition the text has been revised throughout, and augmented
by more than one hundred pages of new matter, in conjunction
with which several new diagrams have been introduced. Although
these additions are such as to increase the value of the book con-
siderably, yet I must express my regret that, owing to the pressure
of other engagements, I have been unable for the present to com-
plete my original design, namely, to bring all parts of the work
up to the standard demanded by the present state of science.
Such alterations and additions as I have been able to make
are located chiefly in those portions which relate to the rectilinear
propagation of light, wave reflection and refraction, and the applica-
tion of graphic methods to the solution of diffraction problems.
More detail has been introduced in some places, especially in the
chapter relating to the velocity of light, which now contains an
account of Professor Newcomb's valuable experiments.
To my friend Mr. C. J. Joly, F.T.C.D., I am again indebted for
his great kindness in assisting me with the proofs.
March 1895.
ix b
PREFACE TO FIRST EDITION
THERE is perhaps no greater impediment to the advancement of
Scientific research than the want of an easy channel of communi-
cation with all the most recent discoveries. Many of the most
valuable of these are hidden in the transactions of learned societies,
or Scattered in scientific periodicals, published in several languages,
and in various parts of the world, so as to be practically in-
accessible to many who might otherwise become well qualified to
extend the bounds of natural knowledge.
In no branch of Experimental Physics is the English student
placed at such a disadvantage as in the Theory of Light, for
although we possess some excellent elementary text-books, yet the
field covered by them is so limited that they fall far short of the
requirements of all who wish to know how far investigation has
been carried, or in what directions it remains to be pursued, and
of these which are the most urgent and most likely to be attacked
with success.
Influenced by these considerations, I have been induced to
undertake the present work, with the hope of furnishing the
student with an accurate and connected account of the most im-
portant optical researches from the earliest times up to the most
recent date. I have, however, avoided entering into the more
complicated mathematical theories, yet the mathematical theory, in
its most elementary form, as well as the experiments on which it
is founded, will be found in sufficient detail to enable the student,
furnished with the necessary knowledge of higher mathematics,
to attack at once with profit the original memoirs and theories
recently elaborated by various British and foreign writers.
xi
xii THEORY OF LIGHT
Thus, although a large part of the book is suited to the read-
ing of junior students, yet I hope it will be found sufficiently full
to meet the requirements of those who desire a more special
acquaintance with the subject ; and to render it more really useful
in this respect I have, as far as possible, given reference to
Original memoirs and other sources whence fuller information
may be derived.
The text contains, in addition to the physical theory, a de-
tailed account of the most important experiments and physical
measurements, such as the determination of the velocity of light,
wave lengths, refractive indices, etc.; and in some of the funda-
mental experiments, such as those of Newton on the refrangibility
of light and coloured rings, I have given extracts from the
original accounts, being fully convinced that in power and per-
spicuity they far surpass any Second-hand digest. In this manner
I have endeavoured to direct attention to the great importance
of Newton's work, and to show that in this department of scientific
research also he stands almost without a rival.
Some novelty of treatment will, I hope, be recognised in
the extensive application of graphic methods to the solution of
problems in diffraction. The calculation of the intensity at the
various points of a diffraction pattern by the ordinary methods
presents considerable difficulty and labour, but by the method
employed in the third section of Chapter IX. almost the whole
theory of diffraction is brought within the reach of persons fur-
nished with the most elementary mathematical knowledge, and
it might now reasonably form part of the course of very junior
students,
An account of the recent, and justly celebrated, experiments
of Professor Hertz will be found in the last chapter, together with
the mathematical theory of the electric vibrator and the radiation
of electromagnetic waves. The importance of these experiments
it would be difficult to overestimate, in so far as they teach us to
refer electric and electromagnetic phenomena to the intervention
of the same all-pervading medium, which forms the vehicle by
which energy passes through space from one body to another,
which brings us light and heat from the sun, and to which we
PREFACE TO FIRST EDITION xiii
now look for a knowledge of the process by which one body is
enabled to attract another, as well as for an explanation of the
ultimate constitution of matter itself.
In conclusion, I wish to return my best thanks to Professor
Wm. Booth (Bengal Education Service), who has been good enough
to read through the proofs and make several valuable suggestions.
A considerable portion of the proof has also been read by my
friends Mr. M. W. J. Fry, F.T.C.D., and Mr. C. J. Joly, so that I
trust the work will be found free from any errors or obscurities
of a serious nature. To Professor G. F. FitzGerald, F.R.S., F.T.C.D.,
I am indebted not only for the reading of the proofs, and the
most generous assistance and advice, but also for that teaching to
which I mainly owe my knowledge of Experimental Physics.
22 TRINITY COLLEGE,
DUBLIN, July 1890.
C O N TENTS
CHA PTE R. I
INTRODUCTORY
SECTION I.-Early History . * tº e g * e
SECTION II. —Discovery of the Velocity of Light and the Development of
the Corpuscular Theory º g * wº
SECTION III.—Introduction to the Wave Theory
*
CHAPTER II
PROPAGATION OF WAVES AND COMPOSITION OF WIBRATIONS
Wave motion
Transverse waves
Plane and spherical waves
Simple harmonic motion
Algebraical expression
Relation of colour to pitch &
Average kinetic energy of a vibrating particle
Measure of the intensity of light
Principle of superposition e
Compounding of simple vibrations .
Distribution of the energy . *
Graphic representations of resultant
Rectangular vibrations compounded
Vibration of permanent type
Examples
CHA PTER III
ON THE APPROXIMATE RECTILINEAR, PROPAGATION OF LIGHT
Principle of Huygens—Secondary waves
Poles and half-period elements g wº e
Comparison of two consecutive half-period elements
Effect of a plane wave
Vibration spiral
Effect of a spherical wave
Wave of any form jº g g e
Mathematical presentation of Huygens's principle .
PAGE
1-10
11-23
24-34
35
35
37, 38
39
40
42
44
45
47
48 .
49
51
54-56
56
57-61
XV
xvi. THEORY OF LIGHT
CHAPTER IV
ON TEIE REFLECTION OF LIGHT
Regular and irregular reflection
Laws of reflection e º &
Illustration of reflected and refracted waves
Laws of reflection deduced from the wave theory
Reflection of spherical waves at a plane surface
Measure of curvature ſº g
Reflection of a plane wave at a spherical surface
Reflection of a spherical wave at a spherical surface
Reflection of any wave at any surface
Examples
CHAPTE R. W.
ON THE REFRACTION OF LIGHT
Snell's law g &
Deduction of the law of refraction
Examples . g ſº e g g *
Huygens's construction for the reflected and refracted waves
Total reflection *
The energy equation
The principle of least time
Refraction through a prism . º tº
Relation of colour and velocity e *
Relative retardation in passing through a plate
Refraction of a spherical wave at a plane surface
Refraction of a plane wave at a spherical surface
Refraction of a spherical wave at a spherical surface
Refraction through a lens
PAGE
95
96
97
98
99
100
102
103
104
I06
106
107
109
111
114
120
121
123
129
Examples
Caustics s e
Primary and secondary foci .
Examples •
Newton's experiments
Aberrations of lenses
CHAPTER VI
ON THE DETERMINATION OF REFRACTIVE INDICES
Minimum deviation .
The spectrometer . te tº g †
Measurement of minimum deviation and angle of prism
Formula for the refractive index
Methods of Descartes and Newton
Method of total reflection
Gladstone and Dale's law
Lorenz' and Lorentz' formula
Effect of temperature
134-143
144
146
149
151
152
152
155
156
157
CoNTENTs
xvii
Indices of mixtures .
Indices of gases º &
Molecular and atomic refraction
Influence of pressure
CHA PTER WII
INTERFERENCE FRINGES
Destructive interference
Theory of Young's experiment
Relation of colour and wave length .
Fresnel's mirrors
Fresnel's bi-prism tº ſº
Peculiarities of the bi-prism fringes
Bi-plates e ſe
Lloyd's single-mirror fringes
Fresnel's three-mirror fringes
The optical bench . ſº &
Conditions necessary for interference
Corresponding points of the sources
Limit to the number of fringes º g
Interference under high relative retardation
Nature of white light
Achromatic interference bands
Displacement of fringes
Application to the determination of refractive indices—Billet's split lens.
Abnormal displacement of the central band
Abnormal displacement of the fringes by a prism .
Talbot's bands
Powell’s bands
Examples
CHAPTER VIII
PAGE
158
159
161
162
164
165
166
167
170
173
173
174
175 .
176
177
178
179
180
182
185
185
186
189
191
192
193
194
INTERFERENCE BY ISOTROPIC PLATES
SECTION I.-The Colours of Thin Plates.
Expression for the retardation . g &
Influence of dispersion—Achromatic condition
Example • .
Multiple reflections
Haidinger's fringes *
Effect of silvering the surfaces .
Further considerations .
SECTION II.-Newton's Rings.
Calculation of the radii.
The transmitted rings
Rings with a white centre *
Conditions for large and bright rings
Examination of the rings through a prism
Herschel's fringes
Newton’s observations
196
201
203
204
205
207
208
210
213
214
215
216
217
222
xviii THEORY OF LIGHT
PAGE
SECTION III.-The Colours of Thick Plates.
Brewster's bands # g & tº * ſº g 224
Jamin's interference refractometer tº e e g & 226
Michelson's and Morley's refractometer g 9 g g 229
Fabry and Perot's interferometer g tº tº • e 233
Newton's diffusion rings tº º ſº G . tº 233
The colours of mixed plates . § gº * * & 238
Examples o * g g e tº * ſº 240
Newton's observations . e e e ſº e & 240
CHAPTER IX
DIFFRACTION
SECTION I.-The Elementary Theory.
Introduction . g e & e e * & 242
Straight edge . ty * tº º º & tº 245
Narrow wire . • . gº * & te e 247
Narrow aperture & e gº tº ſº º & 249
Circular aperture e º gº & * g º 250
Zone plates * gº g e º e tº º 252
Opaque disc º g †: tº * g & * 253
Babinet's principle gº & g & º * & 254
Young's eriometer—Coronas . e º * e * 257
Newton’s observations . g * º º e gº 259
SECTION II.-Diffraction Gratings.
Fraunhofer fringes—The diffraction grating . p e e 261
Oblique incidence—Minimum deviation g g * wº 264
Dispersion in the spectra & * g e e & 265
Chromatic resolving power & & sº e * * 265
Purity in the spectrum & g e tº e º 266
Absent spectra . g te e e * e º 267
Reflection gratings g & g e g e t 267
Curved gratings. º tº g g tº te * 268
Rowland's concave gratings . e g † e * 269
Photographs of the spectrum—Choice of a grating . * tº 271
Difficulties of construction º tº e • e º 273
Measurement of wave lengths . º g e 274
SECTION III.-Investigation of intensity in diffraction patterns.
A. General considerations:
Application of the graphic method—Vibration spiral . • tº 279
Method of strip division. e * p gº & 281
B. Fresnel fringes:
Special cases g * º º © * tº 284
General investigation . * º e . * 288
Examples * e g e ge ſº e 290
Second graphic method . * * & 292
Fresnel's integrals tº g © g e g 293
Cornu's spiral . * * * º & e 297
Examples ſº * º º e tº g 298
The colour of skylight . e * º º * 300
CONTENTS xix
PAGE
C. Fraunhofer fringes:
Narrow rectangular aperture . tº § * tº 303
Determination of maxima and minima . ſº * g 306
Two equal rectangular apertures g e e tº 307
Small rectangular aperture g * & tº º 309
Talbot's bands . e . e * g g 310
Diffraction grating fº t e § & t 311
Maxima and minima tº ſº g & tº * 312
Non-equidistant apertures & e tº & tº 315
Échelon spectroscope . g g º * º 316
Lummer-Gehrcke interferometer. e * * ſº 319
Talbot's bands—Fuller investigation . e tº * 320
Resolving power of prisms ſº g & * g 322
Effect of inequalities . g & © ge e 324
Diffraction in optical instruments e & . . 324
Diffraction in telescopes and microscopes te & . 324, 328
Measurement of stellar diameters g & º e 332
Examples e * & e * e e 333
CHA PTER X.
THE POLARISATION OF LIGHT
Transverse vibrations—Plane polarisation . º ſº & e 335
Polarisation by reflection . & & * > te * > & 338
Angle of polarisation * § & e * * & 339
Plane of polarisation º * & & * º & 340
Double refraction . º e & e & g º 340
Polarisation by double refraction . & * te * . 342
Property of tourmaline g º º sº g gº ſe 344
Brewster’s law e e * * e g tº e e 345
Pile of plates & g e e g & & gº 345
Polarisation of the refracted light . * g * e 346
Law of Malus g º * tº º t e º 346
Partial polarisation—Interference of polarised light g tº i. 348
C H A PTER XI
DOUBLE REFRACTION IN UNIAXAL CRYSTALS
Wave surface in uniaxal crystals . e g * e g 350
Huygens's construction jº & tº g p 9 te 352
Verification of Huygens's construction e * * g e 353
Positive and negative crystals º g g * * vº 357
Relation between the two velocities . . . & e * 358
Nicol's prism ſº g § º & * g t 360
Foucault's prism—Rochon's prism . g & * & . 361, 362
Rochon's double image micrometer . e & * > t e 363
Wollaston's prism . g tº ſº g º & . 364
XX THEORY OF LIGHT
CHA PTER XII
DoDBLE REFRACTION (FRESNEL's THEORY)
Wave surface in crystalline media . o º
Wave surface as an envelope—Fresnel's hypotheses
The ellipsoid of elasticity
Azimuths of resolution
The normal velocity surface
Equation of the wave surface
Direction of vibration in the wave front º
Relation of the azimuths of vibration to the optic axes
Wave surface by means of the reciprocal ellipsoid .
Uniaxal crystal º e g
Principal sections of the wave surface
Huygens's construction e º
Optic axes or axes of internal conical refraction
Internal conical refraction—Lloyd's experiment
Axes of single ray velocity
External conical refraction e
Relation between the wave velocities
Relation between the ray velocities
Indices of refraction for biaxal crystals
CHA PTE R XIII
REFLECTION AND REFRACTION OF POLARISED LIGHT
Fundamental principles and hypotheses
Light polarised in the plane of incidence •
Light polarised at right angles to the plane of incidence
Light polarised in any plane to
Elliptically and circularly polarised light
Common light & º º
Total reflection
Fresnel’s rhomb
Newton's rings in polarised light
Theory of Neumann and MacCullagh
Approximation to reflecting power .
CHA PTE R XIV
METALLIC REFLECTION
Partial polarisation by reflection o
Change of phase and elliptic polarisation by metallic reflection
Jamin's experiments
PAGE
366
368
369
371
373
374
375
376
376
377
377
379
379
381
383
384
386
387
388
389
391
393
394
396
398
398
402
403
403
403
405
406
406
CONTENTS
xxi
CHAPTER XV
COLOURS OF THIN CRYSTALLINE PLATES
(1) PARALLEL PLANE-POLARISED LIGHT.
Introduction - s º
Calculation of the intensity . e -
Conditions for interference—Transverse vibrations
Phase difference—Quarter-wave plate
Thick plates e º g e tº
Two plates superposed—Calculation of the intensity
Projection on a screen . e º
(2) CoNVERGENT PLANE-Pol,ARISED LIGHT.
Uniaxal crystal cut at right angles to the axis—Rings and crosses
Isochromatic lines and achromatic lines
Isochromatic surfaces in a uniaxal crystal
Isochromatic surfaces in a biaxal crystal
Achromatic or neutral lines * º *
(3) CIRCULARLY AND ELLIPTICALLY Pola RISED LIGHT.
Parallel circularly polarised light
Convergent circularly polarised light
Elliptically polarised light
(4) DISPERSION OF THE OPTIC AXES
CHAPTER XVI
THE STUDY OF POLARISED LIGHT
Detection of polarisation
Babinet's compensator
Elliptically polarised light .
Quarter-wave plate
Strain compensator . -
General method of detection
Plane polarisation
Circular polarisation.
Elliptic polarisation .
Partial polarisation .
Natural light
Imitation of natural light
Haidinger's brushes .
Polarisation of skylight
CHA PTE R XVII
ROTATORY POLARISATION
Rotation of the plane of polarisation
Biot's laws tº º
Rotation by liquids and vapours
Molecular rotatory power
The tint of passage
PAGE
409
410
414
415
415
416
419
421
423
424
428
430
432
435
436
437
440
441
444
446
447
447
447
448
449
450
450
451
452
454
457
458
458
460
461
xxii THEORY OF LIGHT
Rotatory power and crystalline formſ
The Faraday effect
The Kerr effects
The Zeeman effect § tº e * & ſº *
Division of a plane-polarised ray into two circularly polarised rays—Fresnel's
experiment and theory . º * * e
Oblique transmission through quartz—Airy's investigations
General formula
Nicols crossed • e
General expression for the intensity .
Approximate form of the isochromatic lines
Airy's spirals.
APPLICATIONS
Analysers . º
Jellett's analyser
Laurent's analyser
The biquartz
Poynting's analyser
Soleil’s saccharimeter
De Senarmont's polariscope .
MacCullagh's theory
CHAPTER XVIII
ABSORPTION AND DISPERSION
Selective absorption . § º º
Coefficients of transmission and absorption .
Dichromatism—Critical thickness
Production of colour by absorption .
Absorption lines in the solar spectrum
Radiation from moving molecules g
Anomalous dispersion—Abnormal spectra .
Kundt's law .
TRANSFORMATION OF RADIATIONs
Fluorescence .
Calorescence . e g
Stokes's theory & © * ſº &
Application to the study of the ultra-violet spectrum
Phosphorescence g tº & º
THEORIES OF DISPERSION
Velocity and wave length
Cauchy’s formula
Briot's formula
Anomalous dispersion
PAGE
462
463
466
472
474
478
480
481
483
484
487
489
490
491
492
494
495
497
497
512
513
516
517
517
518
520
520
521
523
523
524
CONTENTS
xxiii
CHA PTER XIX
THE VELOCITY OF LIGHT
Introduction .
Fizeau's method
Cornu's determination e
Brightness of the image at any speed
Experiments of Young and Forbes
Foucault's method
Discussion of the revolving mirror method . - º • .
Michelson's experiments -
Newcomb's experiments e º
Comparison of the velocities in different media . • e º
The wave velocity and the group velocity—Comparison of the velocities
determined by the various methods
RELATIVE MOTION OF MATTER AND ETHER
Effects of relative motion
Aberration e g
Drift produced by moving matter
Fizeau's experiment . & º
Experiments of Michelson and Morle
Experiments by Dayton Miller and Kennedy
Lodge's experiment . & • º
C H APTER XX
THE RAINBOW
Introduction . - tº &
General expression for the deviation
Minimum deviation .
Dispersion . - e
Caustic of the emergent rays
Formation of the bows
Supernumerary bows e
Equation of the caustic—Wave front º
Airy's investigation of the intensity at any point .
Miller's experiments. g te
C HAPTER XXI
ELECTROMAGNETIC RADIATION
Ether demanded by electric phenomena
Polarisation of the ether—Electric waves
Oscillating discharge of a condenser
Hertz's experiments . tº
Radiation of electromagnetic waves.
PAGE
528
529
534
535
537
540
543
545
547
552
554
557
559
561
564
566
569
570
572
573
574
575
575
576
579
580
581
584
585
587
588
591
596
xxiv. THEORY OF LIGHT
Line integral of a vector . tº
Equations of the electromagnetic fiel
Plane wave in isotropic non-conductor
Reflection and refraction º tº
Wave propagation in a crystalline dielectric
Plane wave in isotropic conductor
Dispersion in isotropic non-conductor
Isotropic conductor .
Optically active media
Magnetic rotation . gº
Theory of the Hertzian vibrator
Particular solution g
Field in the equatorial plane
Field at a distance from the vibrator
Field near the vibrator * >
Calculation of the energy radiated .
APPENDIX I.—Indices of refraction of solids
APPENDIX II.-Indices of refraction of liquids .
APPENDIX III.-Standard wave lengths.
INDEX
PAGE
598
599
602
604
607
610
612
619
620
622
624
626
627
630
631
632
635
636
636
637
CHAPTER I
INTRODUCTORY AND HISTORICAL
SECTION I.-EARLY HISTORY
1. Opties—Definition and Division of the Subject.—The science
of Optics is that branch of natural philosophy which treats of the
nature and properties of light and vision. In its domain we meet
with a multitude of experiments of exquisite beauty, and investi-
gations which afford ample scope for all the refinements of modern
mathematical analysis. It also supplies us with instruments of the
highest utility both in the pursuit of scientific inquiry and in the
common enjoyments of life. There is accordingly no department of
science more deserving of our study, whether we consider the beauty
or the multiplicity of its phenomena.
The subject was usually divided by the older writers into Catoptrics
and Dioptrics, which embraced the phenomena arising from reflection
and refraction respectively. These terms have now fallen into disuse,
and two branches of the subject have been developed under the titles
Geometrical Optics and Physical Optics. The former is a purely ideal
structure built on the assumed truth of the laws of reflection and
refraction, with the supposition that light travels through isotropic
substances in right lines or rays. It is consequently a mathematical
development of the two laws by which it assumes the rays to be
controlled, and any inquiry into the physical cause or nature of
light is outside its province. This inquiry comes within the scope of
physical optics, the aim of which is to determine the physical processes
concerned in the production and propagation of light, and to account
for them by dynamical principles. Physiological Optics deals with the
phenomena of vision or the sensation produced by light falling upon
the retina of the eye. -
2. Ancient Use of Metallie Mirrors and Burning Glasses.—The
ancients appear to have been wholly ignorant of the theory of optics,
I |B
2 . EARLY HISTORY CHAP. I
and to have been exceedingly slow in advancing the construction of
optical instruments, almost all the refinements of the subject having
been originated and developed within the past three centuries. Yet it
cannot be doubted that some of the more striking of the fundamental
phenomena were observed and studied in the earliest times of civilisa-
tion. If the physical theories of light and vision were subjects too
profound for their investigation, they could not fail to acquire some
knowledge of the laws of the reflection of light and the formation of
images. The attention of the most careless observer must have been
attracted with wonder to the image of himself depicted in still water:
and the reflected landscape, or the image of a few bushes on the
margin of a lake might have afforded to the humblest inquirer an
assemblage of observations from which the general laws of reflection
could be easily inferred. &
Metallic mirrors, and even glass, seem to have been manufactured
long before any of the speculations of the ancient philosophers were
recorded. They are distinctly mentioned in the Old Testament .
(Exodus and Job). The invention of burning glasses seems to have
speedily followed the art of glass-making. Aristophanes' mentions
them as early as 424 B.C. *
3. Pythagoras, Empedoeles, Plato, and Aristotle.—The sources
from which light is most copiously derived are the sun, stars, and terres-
trial bodies undergoing combustion or heated to incandescence. Such
bodies we say are Self-luminous or simply luminous, while non-luminous
bodies are those which are not visible of themselves, but only when
illuminated, that is, when in the presence of a luminous body. The
former class we say emit light, while the latter do not. We thus
distinguish between luminous and illuminated bodies.
Simple as it may appear to us to regard a luminous body as the
source of some influence, which, acting on the eye, excites the sense
of sight, much doubt appears to have existed among those who first
investigated the subject as to whether objects become visible by means
of something emitted by them, or by means of something issuing from
the eye of the spectator. According to the opinion of Pythagoras
* Comedy of the Clouds, Act II. (performed 424 B.C.) : “Strepsiades. You have
seen at the druggist's that fine transparent stone with which fires are kindled 2
Socrates. You mean glass, do you not ? Strep. Just so. Soc. Well, what will you
do with that ? Strep. When a summons is sent to me I will take this stone, and,
placing myself in the sun, I will, though at a distance, melt all the writing of the
summons.” (The writing was then traced on wax spread over a solid substance.)
Pliny also mentions globes of glass, which, when held to the sun, produced com-
bustion, and Lactantius (A.D. 303) states that a glass globe, filled with water and
held to the sun, could light a fire even in the coldest weather.
ART. 3 PLATO AND ARISTOTLE 3
(died 540–510 B.C.) and his followers, vision was caused by particles
continually projected from the surfaces of objects into the pupil of
the eye; while Empedocles (444 B.C.) and the Platonic school main-
tained that vision was effected by means of something emitted from
the eye itself, which, after meeting something else emanating from the
object, excited the sense of sight. In the theory of Plato' three
elements appear to have been necessary to vision. First a visual
stream of light or divine fire emitted by the eye itself. These visual
rays entered into union with the light of the sun, and the two together,
meeting with a third emanation, from the object seen, completed the
act of vision.
The doctrine of visual rays, and emission theories in general, was
* “. . . and the pure fire which is within us and akin to this they (the gods)
made to flow through the eyes in a single, entire, and smooth substance, at the same
time compressing the centre of the eye so as to retain all the denser element, and
only to allow this to be sifted through pure. When, therefore, the light of day
surrounds the stream of vision, then like falls upon like, and there is a union, and
one body is formed by natural affinity according to the direction of the eyes,
wherever the light that falls from within meets that which comes from an external
object. And, everything being affected by likeness, whatever touches and is touched
by this stream of vision, their motions are diffused over the whole body, and reach
the soul, producing that perception which we call sight. But when the external and
kindred fire passes away in night, then the stream of vision is cut off; for going
forth to the unlike element it is changed and extinguished, being no longer of one
nature with the surrounding atmosphere which is now deprived of fire : the eye no
longer sees, and we go to sleep ; for when the eyelids are closed, which the gods in-
vented as the preservation of the sight, they keep in the eternal fire.
“. . . And now there is no longer any difficulty in understanding the creation
of images in mirrors and in all smooth and bright surfaces. The fires from within
and from without communicate about the smooth surface and form one image which
is variously refracted. All which phenomena arise by reason of the fire or light
about the face combining with the fire or ray of light about the smooth and bright
surfaces. And when the parts of the light within and the light without meet and
touch in a manner contrary to the usual mode of meeting, then the right appears to
be left and the left right ; but the right again appears right and the left left, when
the position of one of the two concurring lights is inverted ; and this happens when
the smooth surface of the mirror, which is convex, repels the right stream of vision
to the left side, and the left to the right” (“The Dialogues of Plato,” vol. ii.
Timaeus, pp. 538, 539, by B. Jowett).
He is speaking of two kinds of mirrors; first the plane, secondly the cylindrical.
Again, p. 561 : “There is a fourth class of sensible things, comprehending many
varieties, which have now to be distinguished. They are called by the general name
of colours, and are a flame which emanates from all bodies, and has particles corre-
sponding to the sense of sight. . . . Of the particles coming from other bodies
which fall upon the sight, some are less, and some are greater, and some are equal to
the parts of the sight itself. Those which are equal are imperceptible, or transparent,
as they are called by us, whereas the smaller dilate, the larger contract the sight,
having a power akin to that of hot and cold bodies on the flesh, or of astringent
bodies on the tongue. . . . Wherefore, we ought to term that white which dilates
the visual ray, and the opposite of this black.”
4 EARLY HISTORY CHAP. I
combated by Aristotle as early as 350 B.C. He maintained that light
is not a material emission from any source, but a mere quality of,
or action (āvépyeva) of a medium which he called the pellucid
(6taſhavés)."
Although the reasoning of Aristotle was very superficial, yet he is
entitled to considerable credit for his sagacious, though vague specula-
tions regarding the nature of light and various optical phenomena.
He may to some extent be regarded as having in a haphazard manner
anticipated the undulatory theory of light, which was established two
thousand years afterwards by the labours of Huygens, Young, and
Fresnel.
4. Knowledge of the Ancients.-The principal phenomena of the
rainbow, halos, etc., had not escaped the notice of the ancients, who
classed all these appearances under the common denomination of
meteors. Aristotle * attributed these phenomena to the reflection of
the sun's rays from drops of rain, and observed that a rainbow may be
made by the spray from an oar, and that in this case it will be visible
* “There is then, let us begin by saying, something which is pellucid. And by
pellucid is meant something which is visible, not visible by itself (to speak without
further qualification), but visible by reason of some foreign colour which affects its
neutral pellucidity. Of this character are air and water, and also many among
solid bodies, water and air being pellucid not in virtue of their qualities as water
or air, but because they both contain the same element as constitutes the everlasting
Empyrean essence. Light is then the action (évépyeta) of this pellucid qua pel-
lucid ; and whenever this pellucidity is present only potentially, there darkness
also is present. . . . Thus we have shown light to be neither fire, nor body gener-
ally, nor even the effluvium or emanation from any body (since even in this case
it would be a body of a kind), but only the presence of fire, or something like it, in
that which is pellucid ; two bodies being unable to exist at one and the same time
within the same space. . . . Darkness in fact is really the removal of such a posi-
tive quality from what is pellucid, so that light must necessarily be its presence. .
Empedocles, therefore, and many others who have followed him, have not described
the phenomenon correctly in speaking of light as moving itself, and as coming some
time or other without our knowing it into existence between the earth and the
surrounding air. . . . And the pellucid itself is also similarly dark, but it is so
not when it is pellucid in actuality, but only so potentially ; for it is one and the
same element which is at one time darkness and at another time light. . . .
“Colour therefore is not visible without the presence of light; this indeed we saw
was the essential character of colour that it is calculated to set the actually pellucid
in movement; and the full play of this pellucid constitutes light. . . . Vision is the
result of some impression made upon the faculty of sense; an impression which
cannot be effected by the colour itself as perceived, and must therefore be due to
the medium which intervenes. An intervening substance then of one kind or
another there must necessarily be ; and were this intervening space made empty,
not only will the object not be seen exactly, but it will not be perceived at all”
(Aristotle's Psychology, book ii. chap. vii., by Edwin Wallace, M.A., Cambridge
University Press Series, 1882).
* Meteor, lib. iii, cap. ii.
ART. 7 EUCLID AND PTOLEMY º 5
to a person who turns his back to the sun in the same manner as in
the case of the natural rainbow.
Notwithstanding the absurdity of the doctrine of ocular beams, as
it was called, the geometers of the Platonic school were acquainted
with two very fundamental points in the science of optics. They
taught first, that light, from whatever source it might be emitted,
travels in straight lines; and secondly, that when it is reflected at
any surface, the angle made with the surface by the incident beam is
equal to that made with the surface by the reflected beam.
We thus find them acquainted with the rectilinear propagation of
light and with the law of reflection—the two facts in the science which
we would naturally expect to have been first discovered.
Epicurus, Lucretius, and the other supporters of the quasi-ten-
tacular theory, although they made few or no experiments, lacked not
fertility in hypotheses to account for the common appearances of nature.
They all had a confused notion that as we may feel bodies at a distance
by means of a rod, so the eye may perceive them by the intervention
of light. It is very remarkable that this strange hypothesis held
ground for many centuries, and little or no progress was made in the
subject till it was established on the authority of Alhazen, an Arabian
astronomer, in the eleventh century A.D. that the cause of vision pro-
ceeds from the object and not from the eye.
5. Euclid.—Shortly after the time of Aristotle the celebrated
geometer Euclid (300 B.C.) drew up a treatise on optics, which has
been handed down with his geometrical works." However, the work
is so imperfect and so inaccurate that some have found it difficult to
attribute it to one whose geometry is characterised by such perspicuity
and accurate reasoning.
6. Ptolemy.—The most celebrated of all the ancient writers on
optics was the Egyptian astronomer Ptolemy, who flourished about the
middle of the second century. He treated of astronomical refraction
and of the increase in the apparent diameters of heavenly bodies when
near the horizon. He also drew up tables of the values which he
found for the angles of incidence and refraction of a beam of light
passing from air into glass and water, but he failed to connect them
by any law, like all the subsequent writers of the next fifteen hundred
years. -
7. Experiment of Cleomedes.—Next to a straight stick appearing
bent when part of it is immersed obliquely in water, the apparent
* (Oxford edition of Euclid's works, 1557.) He endeavoured to refute the
Pythagorean, or emission, theory of light, and investigated the apparent place of
the image formed by reflection at the surface of a polished mirror.
6 EARLY HISTORY CHAP. I
elevation of a coin, or any other object, placed at the bottom of a cup
into which some water is poured, is perhaps one of the oldest experi-
ments depending on refraction. It has been referred to by the oldest
optical writers, and especially by Cleomedes, who (A.D. 50)" also
pointed out that, in the same manner, the air by refraction may
render the sun visible when it is somewhat below the horizon.
8. Previous to Alhazen.—In addition to what has been mentioned,
the ancients had a superficial and fragmentary acquaintance with some
of the properties of the rainbow, mirage, and halos, but limited as it
was, it far exceeded their knowledge of the other branches of physical
science. Their knowledge of the general nature of refraction and of
some of its applications was exhibited in the construction and use of
burning glasses, which were sold as curiosities in the toy shops, and
were probably either glass globes filled with water or balls of glass or
rock crystal. -
9. Alhazen.—After a long interval of inactivity the science of
optics was taken up and cultivated with assiduity in Arabia. The
first real progress in the mathematical theory was made by Alhazen in
the eleventh century. He entered into the anatomy of the eye, and
examined the rôle played by each part of it in the production of vision.
Besides accounting for twilight he showed that by means of the
duration of it the height of the atmosphere might be measured. After
describing the eye, he explains how it happens that with two eyes we
see only one object, and that we see each object, however small, not
by a single ray of light (as was at that time supposed), but by a cone
of rays proceeding from the object to the eye.
Alhazen treated largely of optical deceptions, both in direct vision
and also in vision by reflected and refracted light. In this class of
phenomena he ranks what was known as the horizontal moon ; that is,
the increase in the apparent magnitude of the moon, or any other
celestial object, when near the horizon. In explanation of this pheno-
menon he says that we judge of distance by comparing the angle under
which we see an object with its supposed distance, so that if the angles
under which two objects are seen be nearly equal, and if the distance
of one be conceived greater than that of the other, the more distant
object will be imagined the larger. But the sky near the horizon, he
says, is always imagined farther from us than any other part of the
concave surface, on account of the range of intervening terrestrial
objects by which we judge the distance.”
* Cyclical Theory of Meteors (i.e. stars).
* This account of the horizontal moon Bacon attributes to Ptolemy. As such
it is objected to by B. Porta, De Refractione, pp. 24, 128 (Priestley's History).
ART. 11 ALHAZEN, VITELLIO, BACON 7
Although the work of Alhazen may have been founded on that of
Ptolemy, yet he made such a decided advance in the theory, that for
five hundred years or more he was recognised in Europe as the chief
authority on the subject.
10. Witellio.—In 1270 Vitello or Vitellio, a native of Poland,
drew up a treatise on optics" less prolix and more methodical than
that of Alhazen on which it was founded.” Witellio attributed the
twinkling of the stars to the motion of the air in which the light is
refracted, and he remarked that the twinkling is increased when they
are viewed through water in gentle motion. He also compiled a
table of the angles of incidence and refraction of light at the surface
of water and glass, of much greater accuracy than that previously
given by Ptolemy.
11. Roger Bacon.—Contemporary with Vitellio was our country-
man Roger Bacon, a man of extraordinary genius, who wrote on almost
every branch of science, yet notwithstanding the pains he took with
the subject of optics, he does not appear to have made any advance in
the theory which Alhazen had already laid down before him. Great
as Bacon undoubtedly was, he was far from being free from the
prejudices of his predecessors and contemporaries. Some of the
wildest and most absurd of the speculations of the ancients had
the Sanction of his approbation and authority.
The invention of the magic lantern has been attributed to Bacon,
but it has been much disputed whether he was acquainted with tele-
Scopes. Certainly, if he was unacquainted with spectacles, telescopes,
and microscopes, he anticipated their invention in language more than
prophetic.”
* Published by Risner in 1572, with the work of Alhazen translated from the
Arabic, under the title Thesaurus Opticae, Bas. 1572.
* Witellio is said to have at first denied that he had any knowledge of the works
of Alhazen, but he afterwards retracted this denial and acknowledged himself a
disciple of the Arabian philosopher.
* He says: “If the letters of a book, or any minute object be viewed through
the lesser segment of a glass sphere or crystal, whose plane base is laid upon them,
they will appear far better and larger . . . and therefore this instrument is useful
to old men, and to those who have weak eyes; for they may see the smallest letters
sufficiently magnified.” And again : “Greater things than these may be performed
by refracted vision. For it is easy to understand by the canons above mentioned,
that the greatest things may appear exceedingly small, and on the contrary ; also
that the most remote objects may appear just at hand, and on the contrary. . . .
And thus from an incredible distance we may read the smallest letters. . . . And
thus a boy may appear to be a giant and a man as big as a mountain. . . . So also
the Sun, moon, and stars may be made to descend hither in appearance . . . and
many things of like sort which would astonish unskilful persons" (Opus Majus,
Jebb's edition, p. 377).
8 EARLY HISTORY CHAP. I
12. The Introduction of Telescopes.—Although it would appear
from the writings of Bacon, B. Porta, and others, that the properties
of some form of telescope were known or suspected, yet the construc-
tion and practical applications of the instrument do not appear to have
been known and published prior to the year A.D. 1608. If known
before this date, the instrument was probably the secret possession of
certain individuals who employed it in the demonstration of “natural
magic.”
Like many other discoveries, it is probable that more than one
person had hit upon the idea of the telescope, and had constructed
simple forms of the instrument for their own amusement and “curious
practices” before any public record of the invention was made. For
this reason it is not surprising that the early history of the instrument
should have been the subject of a lively debate, and that the invention
should have been ascribed to different persons and claimed in different
countries. The first person, however, who seems to have independently
constructed a telescope, and who at the same time published his dis-
covery, was Hans Lippershey, a spectacle-maker of Middelburg, in the
year 1608.
No small share of honour in this matter must be ascribed to
Galileo," who, in the following year (1609) (having merely heard that
the Belgian spectacle-maker had constructed an instrument by which
distant objects were made to appear nearer and larger), at once set to
work and independently constructed a telescope for himself. With
such skill and ability did he apply himself in this matter that in 1610
he finished an instrument of such excellence that it revealed the
satellites of Jupiter, and thus broke the dawn of modern astronomy.
It was Kepler (1571–1630), however, who first reduced the theory
of the telescope to its true principles, and laid down the common
rules for finding the focal lengths of simple lenses, and the magnifying
powers of telescopes.
13. B. Porta—Camera Obseura.-At the end of the sixteenth
century John Baptista Porta (1545–1615), a Neapolitan philosopher
and famous collector of mysteries, published his Magia Naturalis. To
him the invention of the camera obscura is due. He remarked that if
light be admitted through a small hole in the shutter of a darkened
room, external objects will be clearly depicted on the white wall, in
their natural colours; and he added that if a convex lens be placed
at the aperture the objects will appear so distinct as to be immediately
recognised. -
14. A. de Dominis—The Rainbow.—In 1611 the true theory of
* Opere, ii. p. 4.
ART. 16 SNELL AND DESCARTES 9
the primary rainbow * was at last arrived at by Antonio de Dominis,
archbishop of Spalatro. He showed that one reflection and two
refractions in the drops of rain were sufficient to bring the rays which
formed the bow to the eye of the spectator. This explanation was
either verified or suggested by viewing a glass globe filled with water
and exposed to the sun's rays under the same circumstances as the
drops of rain. Both the primary and secondary bows were afterwards
explained by Descartes” on mathematical principles.
15. Snell and Deseartes.—The next great step was made by
Willebrod Snellius.* About 1621 he ascertained that when light falls
upon the surface of a refracting medium, such as glass or water, the
sine of the angle of incidence bears a constant ratio to the sine of the
angle of refraction.” He died, however, in 1626 without having
published his discovery. The law of refraction has been consequently
often attributed to Descartes, who first published it in the above form,
but not, as Huygens states, without having previously perused the
papers of Snell. In his investigations concerning the rainbow Descartes
also neglects to mention how far he was indebted to the previous
discoveries of Antonio de Dominis.
The speculations of Descartes on the nature of light bear, some
resemblance to those of Aristotle, and it seems indeed extraordinary
that after the lapse of so many centuries, during which the attention
of many celebrated philosophers was concentrated on the subject, no
real progress had been made in the physical theory of light. Descartes
imagined light to be due to a pressure transmitted instantaneously
through an infinitely elastic medium filling all space, and colours he
attributed to a rotatory motion of the particles of this medium.
16. Newton and Grimaldi.-It was still supposed that every re-
fraction of light actually produced colour, instead of merely separating
* De radiis visus et lucis, 1611.
* Spec. Meteorum, chap. viii. * Professor of Mathematics at Leyden.
* Although tables of the angles of incidence and refraction for glass and water
had been constructed by Ptolemy and Vitellio, the philosophers who studied them
failed to discover the law of refraction which lay hidden in them. Even Kepler
(Paralegomena ad Vitellionem, 1604) laboured unsuccessfully to derive it from the
tables of Vitellio.
Snellius observed that if the refracted ray and the incident ray continued through
the point of incidence be intercepted by any line parallel to the normal to the sur-
face at the point of incidence, the length of the intercepted portion of the refracted
ray is in a constant ratio to the length of the intercepted portion of the incident ray.
The form above, in which it was published by Descartes, is merely the trigonometrical
statement of the law arrived at by Snell.
It is remarkable that the law of refraction in its most improved form was arrived
at independently by our countryman James Gregory. He is therefore entitled to
as much honour in this matter as Descartes.
10 EARLY HISTORY CHAP. I
the colours already existing in ordinary white light, but in 1666
Newton made the important discovery of the actual existence of
colours of all kinds in solar light, which he showed to be no other
than a compound of the various colours, mixed in certain proportions
with each other and capable of being separated by refraction of any
kind.
Whilst Newton was making his earliest experiments on refraction
Grimaldi's treatise on light” appeared, containing an account of many
interesting experiments on the effects of diffraction, which is the name
he gave to a small spreading out of light in every direction upon its
admission into a darkened chamber through a small aperture. This
spreading out (or inflection, as Newton called it) of the light shows
that light does bend round corners and deviate from the rectilinear
path like sound, but to a very small extent, and it forms the subject
of one of the most important branches of physical optics. Grimaldi
observed that in some instances the light from one aperture tended to
extinguish that from another, yet it cannot be admitted, from the
nature of his experiments, that he ever observed any true case of the
interference of light. ---
* Physico-Mathesis de lumine, Bonon., 1665.
ART. 17 WELOCITY OF LIGHT 11
SECTION II.-DISCOVERY OF THE WELOCITY OF LIGHT AND
DEVELOPMENT OF THE CORPUSCULAR THEORY
17. Römer–Finite Velocity of Light.—A new era in the history
of optics was registered by the Danish astronomer Olaf Römer, who
in 1675–6 made one of the greatest discoveries in the history of the
Science—that of the propagation of light in time.” Römer ? was led to
this discovery by a series of careful observations on the eclipses of
Jupiter's Satellites. Each satellite, as it revolves round the planet,
disappears behind Jupiter and is hidden from view, or eclipsed, as
long as the opaque body of the planet is between us and the satellite.
As the periodic time of the satellite is small, its motion is rapid and it
disappears almost suddenly, so that the interval of time between two
successive eclipses can be estimated with tolerable precision. If this
periodic time be known, the dates at which successive eclipses will occur
can be tabulated beforehand; but Römer found that the observed
times of eclipses did not agree with those calculated in this manner,
but that certain inequalities occurred which could be satisfactorily ex-
plained only on the supposition that light travels with a finite velocity.
In order to fix our ideas, let us suppose the earth to be stationary
and that Jupiter is also fixed, and that the satellite under observation
moves round it uniformly with the periodic time T. Under these
circumstances the successive eclipses will follow each other regularly
at equal intervals of time T. On the other hand, if the earth moves
away from Jupiter with a given velocity so that the distance between
them increases uniformly, then the interval of time between two con-
secutive eclipses, as observed from the earth, will be increased from T
to T + 1, where t is the time required by light to traverse the distance
passed over by the earth in the time T + T. So also, if the earth
approaches Jupiter, the time between two eclipses will be diminished
in a similar manner. Now, on account of their motions round the sun,
the distance between the earth and Jupiter increases during one part of
the synodic revolution and diminishes during the remainder. In the
former part the periodic time T will have an apparent increase, and in
the latter a decrease. This increase and decrease was found by Römer to
depend on the rate at which the earth is receding from or approaching
* See Mach, Popular Lectures, p. 52. *
* Hist, et Mém. x. p. 399. Ph. Tr., 1677, xii. p. 893.
12 INTRODUCTION CHAP. I
to Jupiter, and the inevitable conclusion was that light is propagated
with a finite velocity."
The velocity so determined was about 192,000 miles per second ;
while the most recent astronomical observations lead, by this method,
to about 185,000 miles per second for the value of the velocity.
Considering the enormous rate at which light travels, it is not
surprising that Galileo and the Academy del Cimento should have
sought in vain to determine it directly. In recent times, however,
methods of extreme ingenuity have been devised by Fizeau (1849)
and Foucault (1850) for directly measuring the velocity of light in air
or any other transparent medium. These methods will be fully de-
scribed in the sequel (Chap. XIX.), and the results leave no doubt as
to the finite speed of light, and fix it at about 186,000 miles, or
300,000,000 metres per second. -
18. Bradley.—For nearly fifty years after the discovery of Römer
no further evidence was adduced to show that the propagation of light
was not instantaneous, and the results arrived at by the Danish philo-
sopher were doubted, if not denied, in many quarters. However, in
1728 Bradley” discovered what is known as the aberration of light,
which, like many other great discoveries, was made when the author
was in pursuit of another inquiry. Intending to verify some of Dr.
Hooke's observations on the parallax of the fixed stars, he observed
the star y Draconis at Kew in 1725, and found that it was more
southerly than it had appeared before, and on carefully observing it,
and other stars, for a long time he found that they all had an apparent
motion in space. After much speculation as to the cause of this
apparent motion he finally succeeded in solving the difficulty by
taking into account the motion of the earth together with the fact that
light is propagated with a finite velocity, and the result of his calcula-
tions gave a value of this velocity agreeing fairly well with that
arrived at previously by Römer. This showed that the direct light of
the fixed stars travelled with the same velocity as that reflected from
the satellites of Jupiter (see further, Chap. XIX.).
* The time taken by light to travel over the radius of the earth's orbit is about
500 seconds. Römer's estimate was much too high, being eleven minutes. That
the inequalities noticed in the eclipses of Jupiter's satellites might arise from the
finite speed of light was admitted when Römer propounded his views, but never-
theless it was contested that the observed inequalities might be due to want of
uniformity in the motion of the satellite itself. This objection is legitimate, and
the astronomical methods alone do not place the question beyond doubt. All
uncertainty, however, has been removed by the terrestrial methods devised by
Fizeau and Foucault.
* Phil. Thans., 1728, xxxv. p. 637.
.ART. 19 PROPAGATION OF ENERGY 13
19. Energy : its Conservation and Transmission.—When 3,
material particle is in motion we say it possesses a certain store of
energy, which we term kinetic, meaning that this energy is due to the
motion of the body. The particle may give up part or all of its energy
to another, by collision or otherwise, but when any such transference
takes place, the amount of energy gained by one particle is the exact
equivalent of that lost by the other, if the particles are perfectly elastic.
If they are not perfectly elastic, heat energy is developed, and the
kinetic energy gained by one added to the heat energy developed in
the collision is equal to the kinetic energy lost by the other. If
the total energy of any body or system of bodies augments or
diminishes, the energy gained or lost must have been abstracted from,
or given to, some other system. In this respect energy is like matter.
The amount of it in any system can be augmented only at the expense
of some other system. That is, energy, like matter, cannot be created
or destroyed by any machine or process at the disposal of man. All
working engines and animals are mere machines for converting energy
from one form to another, or transferring it from one system to
another. It is in this sense that we speak of the conservation of energy,
or the permanence of energy, just as we speak of the conservation
or indestructibility of matter. This idea of the impossibility of
creating or destroying energy, that is, of its ever disappearing in any
system or form without appearing in equal quantity in some other
system or form, underlies the whole basis of modern physics, and
forms its groundwork, just as the postulated permanence or indestructi-
bility of matter forms the foundation of modern chemistry.*
Now there are two methods by which we may communicate energy
to a body at a distance—take, for example, the case of a ship at sea.
We might fire bullets into it, each bullet carrying a store of energy
which it deposits in the ship when it strikes it. By this means we
might set the ship in motion. But there is another method by which
energy may be communicated to the ship. We may use the water or
medium in which the ship floats. We may spend our energy in
exciting waves in the water. These waves travelling outwards will
break upon the ship and set it in motion, thereby communicating a
part of their energy to it. -
In the former method each bullet acted the part of a messenger
carrying a certain cargo of energy from the person or machine that
projected it to the ship. Here we have a transference not only of the
energy, but also of the matter which carries it. In the case of the
* This subject is treated of more fully in the author's Theory of Heat, chap. i.
section vii.
14 INTRODUCTION CHAP. I
waves, the energy is handed on in succession from one portion of the
water (or medium) to the next, while any element of the water merely
oscillates about its position of rest. We have thus a flow of energy
through the water which affords it a means of transit.
As another example of the transmission of energy through matter,
we may consider the case of a mass attached to one end of a rod or
rope. When the other end is held in the hand and twisted, the
attached body will rotate so as to free the rod from torsion. Here
the energy supplied at one end is transmitted along the rod to the
mass at the other. There is a flow of energy along the rod.
Hence if by any means we obtain energy from a source situated
at a distance, we are forced to seek for the vehicle by which it is
conveyed. Either matter has come to us from the source, carrying the
energy associated with it, as in the case of pellets fired from a gun, or
else the energy has been successively propagated through some medium
existing between us and the source. The probability of the discovery of
other methods of the propagation of energy, or even the possibility of con-
ceiving some new method is perhaps a speculation of a purely visionary
character, and is certainly beyond our grasp at present. It is well,
however, to keep our minds open to the fact that there may be methods
of which we have no direct experience, and of which we may, or may
not, become cognisant as our knowledge of the material universe
increases.
20. Two Modes of Propagating Energy—Two Theories of Light.—
It having been proved that light travels with a finite velocity, and it
being accepted that a luminous body, as such, is the source of some
mechanical influence which we call light, and which is necessary to
vision, and above all that the phenomena of light and heat are mani-
festations of energy, the question arises as to how and where this
energy exists during the interval between the instant it leaves the
luminous body and the instant it reaches the observer. Thus light
(or heat) requires about eight minutes to reach us from the sun; how
and where is this energy stored during the transit, and by what
means is it transmitted from the Sun to us ; Direct action at a dis-
tance is out of the question. We cannot conceive of energy disappear-
ing at the sun and reappearing at the earth after an interval of eight
minutes without having been propagated continuously in the interval
through the intervening space.
In the present state of knowledge we are acquainted with energy
only as associated with matter, so much so indeed that matter
has been defined as the vehicle of energy. Consequently two dis-
tinct and intelligible methods of representing the propagation and
ART. 21 TWO METHODS OF PROPAGATING ENERGY 15
nature of light have been conceived. The first (the emission theory),
which was elaborated by Newton, assumes that a luminous body, as
such, continually emits small particles, or luminous corpuscles, of
extreme minuteness in all directions. These particles are projected
from the body and travel through space with the velocity of light,
carrying with them their kinetic energy; that is, their energy of
motion. This theory accounts at once for such general phenomena
as the rectilinear propagation, and reflection, of light ; but some of its
consequences are quite inconsistent with observed facts. For instance,
the doctrine as ordinarily expounded has led to the conclusion that
light should travel faster in the denser media, like water and glass, than
in the rarer less refracting media, such as air, while experiment proves
the reverse. This and other facts which it has failed * to explain have
been satisfactorily accounted for by the second theory (the wave theory),
which supposes light to be due to a periodic disturbance in a medium
existing between the luminous body and the eye, and permeating all
space. This hypothetical medium is called the ether. We are not directly
cognisant of it by any of our senses, such as touch, taste, or smell,
but nevertheless from the phenomena of light (and electricity) we
cannot but be convinced that such a medium exists, and thanks to the
labours of scientific men, our knowledge of its properties is rapidly
increasing.
According then to the second theory—known as the Wave Theory
—a luminous body is the source of a disturbance in the ether, which
is propagated in waves throughout all space. These waves falling
upon the eye excite the sense of vision. They travel with the velocity
of light, and carry energy from the body which produces them to that
by which they are absorbed. -
Before proceeding to the history and development of this theory,
which is that now universally accepted, we shall first glance at the
emission theory and see how far it will account for the facts.
21. The Corpuseular or Emission Theory.—This hypothesis
assumes that the sensation produced by light is due to a mechanical
action on the retina. It formally states that a luminous body emits
minute particles” which by their impacts on the retina cause the
sensation of vision. Very formidable objections to it are presented at
the outset. For corpuscles moving with such an immense velocity as
* Most of these difficulties may be overcome by introducing suitable hypotheses
concerning the nature of the corpuscles.
* “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” (Newton, Opticks, book
iii. Qu. 29).
16 INTRODUCTION CHAP, I
186,000 miles per second must have an exceedingly small mass,
if their momentum is not appreciable. Now an exceedingly large
number of these particles may be made to act together by concen-
trating them in the focus of a lens or mirror, and the resultant effect
of their impulses might be expected to become visible when subjected
to the test of experiment. This apparently easy test of the materiality
of light was appealed to by many philosophers. The effects they observed
were probably due to extraneous causes, such as draughts caused by
inequalities of temperature, and until recent years it was universally
admitted that no effect of the impulse of light had ever been perceived."
The motion excited in the well-known delicate radiometer of Sir W.
Crookes is attributed to other causes, and it is to be remembered that
in experimenting with this instrument the vane first moves towards
the light, indicating an apparent attraction ; and it is not until
rarefaction is pushed to a certain limit that the motion of the vane
is reversed and exhibits an apparent repulsive action of the light.
However, it was deduced by Maxwell,” as a consequence of the
electromagnetic theory, that a beam of light incident on an absorbing
surface should exert a pressure on it and that the amount of this
pressure, if the beam is normal to the surface, should equal the energy
per unit volume, or energy-density, of the incident beam. The
independent experiments of Lebedev and of Nichols and Hull have
definitely established the existence of this pressure and shown that its
magnitude agrees with the value obtained by Maxwell and not with
that required by the emission theory. According to the latter the
pressure due to such a beam on an absorbing surface should equal
twice this energy-density. For if n be the number of particles per
c.c., v the velocity of each particle and m its mass, nv particles would
strike each sq. cm. of the surface per second ; each of them would
communicate my momentum to that surface and therefore the
momentum communicated per second to each sq. cm., or the pressure
on the surface, would be mny" or twice the energy-density of the beam.
Lebedev * suspended in a glass flask by a fine thread a glass
filament to which were fixed two pair of wings, consisting of two
circular discs. One disc of each pair was polished on each side and
* “The experiments of Mr. Bennet seem to decide this point. A slender straw
was suspended horizontally by means of a single fibre of a spider's thread. To one
end of this delicately suspended lever was attached a small piece of white paper, and
the whole was enclosed in a glass vessel, from which the air was withdrawn by an air-
pump. The Sun's rays were then concentrated by means of a large lens and allowed
to fall on the paper, but without any perceptible effect” (Lloyd's Wave Theory).
* Electricity and Magnetism, vol. ii. § 792.
* Lebedev, Reports, Congrès Intern. de Phys. ii. p. 133, Paris, 1900.
ART. 21 PRESSURE OF LIGHT 17
the other disc was electrolytically covered on each side with platinum
black, the layer on the disc of one wing being much thicker than on
that of the other wing. By means of sets of lenses and a movable
mirror he could reverse the direction of the beam of light falling on
the disc under observation. By thus reversing the direction of the
beam he eliminated the disturbing effects due to convection currents
in the residual gas in the flask, taking the difference of the displace-
ments obtained as independent of this convection. He eliminated
the “radiometer’ effects by means of observations on the discs
covered with different thicknesses of platinum black. These effects
depend upon the difference in temperature of the two surfaces of
the disc, and from the observations on the thick and thin layers he
deduced the value of the pressure on an infinitely thin layer for which
the “radiometer” effect would vanish. The gas pressure was reduced
in the flask to such an extent as to make the disturbing forces as
small as possible. The energy of the incident beam was determined
calorimetrically.
The experiments of Nichols and Hull' eliminated the gas action
still more completely. They employed discs of thin glass silvered on
one side, directing the light alternately on the glass and silver sides.
As the light has passed previously through various pieces of glass, it
will have lost by absorption those rays which would heat the glass,
and the silvered side will be heated in both cases. Further, they
exposed the disc only for a short time to the light, measuring its effect
ballistically, and thus the heating effects were not given sufficient time
to become of importance. They determined the energy of the beam
by measuring thermo-electrically the rise in temperature it produced
when falling on a blackened disc. Later Hull” placed the silvered
side of a thin glass dise in contact with the blackened side of another,
and enclosed the pair in a cell formed of two other thin pieces of glass.
When light is absorbed by the blackened glass, the outer surfaces of
the cell will still be at the same temperature. The results obtained
showed that the disturbing effects had been practically removed.
Poynting * has entirely separated the light pressure from the dis-
turbing gas action, by means of a beam falling obliquely on an absorb-
ing surface. The pressure of such a beam will have a component
tangential to the surface. By arranging a suspended disc so that such
* Nichols and Hull, Am. Acad. Arts and Sciences, xxxviii., 1903, p. 559.
* Hull, Phys. Rev., May 1905. -
* Poynting, Phil. Mag., January 1905, p. 169; cf. also, Poynting, Journ. de
Physique, ix. pp. 657-671, Aug. 1910, where the existence of the back pressure of
light on an emitting surface is also demonstrated. See Poynting and Barlow,
Roy. Soc. Proc. Ser. A., lxxxiv. pp. 534-546, May 11, 1910.
C
18 INTRODUCTION CHAP. I
a tangential force would tend to twist the suspension, while a normal
force would produce no such twist, he observed the effects of the
light pressure and found its value to agree with Maxwell's theory.
This difficulty as to the pressure of light is not the only one
which beset the emission theory at the beginning, for we have seen that
the light of the sun is propagated with the same velocity as that of
the fixed stars, and that which comes directly to us from these bodies
travels at the same rate as that which is reflected by a planet or its
satellite. The speed of propagation would therefore appear to be
independent of the luminous source, as well as of any subsequent
modifications which the light may undergo in the celestial spaces.
Even though the causes producing light in the various bodies of the
universe led to the production of the impulses required to project
material particles with the velocity of light, and though these impulses
are the same in the various bodies, the particles would have their
velocities variously affected by the different attractions of the different
luminous bodies. M. Arago ingeniously escapes this difficulty by
admitting that the molecules may be projected with very different
velocities, but that there is only one velocity which is adapted to
excite the sense of sight.
22. Reflection.—According to the theory of emission each lumin-
ous molecule travels in a right line through a homogeneous isotropic
medium. Let MN (Fig. 1) be the path of one of these molecules, and
let AB be a reflecting surface. As soon as
the molecule comes within a certain very
small distance from the surface, indicated
by the line PQ, it begins to experience the
repulsive or reflecting action of the surface.
The velocity of the molecule at PQ may be
resolved into two components, one parallel
to AB and the other perpendicular to it. The former component
is unaltered by the action of the surface, while the latter is at first
diminished and then reversed, so that the molecule retires from AB
at N with the same speed as it approached it at N. As soon as the per-
pendicular component begins to diminish under the reflecting action of
the surface the path of the molecule (at N) begins to curve, and when this
component is reduced to zero the path of the molecule is parallel to the
surface. After this point the repulsive action of the surface will be the
same as before, and the route of the molecule will be along a curved
path to N’, while at N' it retires with its velocity perpendicular to the
surface reversed, and its velocity parallel to it unaltered. The molecule
therefore emerges at N' free from the influence of the surface in a
Fig. 1.-Reflection of a Light Molecule.

Art, 23 THE EMISSION THEORY 19
direction N'M', making an angle with the surface equal to that made with
it by MN. Thus the theory accounts easily for the law of reflection,
just as it is deduced for the reflection of a perfectly elastic sphere.
23. Refraction.—To deduce the law of refraction from a rare to
a denser medium it is assumed that when the molecule comes within
a very small limiting distance (PQ) (Fig.2)
of the surface of separation AB, it begins
to be attracted towards the surface so that
its component velocity perpendicular to
the surface gradually increases, till it
reaches a limiting distance (PQ) on the
other side of the surface AB. It then
proceeds in the new medium in a right Fig. 2.-Refraction of a Light Molecule.
line N'M', the velocity parallel to the surface remaining the same,
while that perpendicular to the surface is increased by an amount
which is independent of the angle of incidence, but which varies for
different materials. Let the velocity along MN be v and let i (called
the angle of incidence) be the angle which MN makes with the normal
to the surface. Then, if the velocity along N'M' be tº and if r (called
the angle of refraction) be the angle between it and the normal, we
have from the constancy of the velocity parallel to the surface—
v sin i =v' sin r,
sin i p'
or ---.
sln 7" tº
The sine of the angle of incidence therefore bears a constant ratio to
the sine of the angle of refraction." This is the law of refraction;
and the formula shows that if i be greater than r then v is greater
than v. That is, the velocity of light in denser (more refracting)
media is greater than in rare (less refracting) media, for we know by
experience that the ray is bent towards the normal (as in Fig. 2) in
passing from a rare medium, such as air, to a denser medium like
glass or water.
We here reach a crisis in the emission theory, for it has been
proved beyond doubt by direct experiments on its velocity that light
travels faster in a rare medium like air than in a denser (more refract-
ing) medium like water. The emission theory is therefore untenable,
and the wave theory, which has not only successfully explained, but
* If the particle were travelling along M'N' in the more refracting medium, then
in approaching the surface AB it would as before be attracted towards the more re-
fracting medium, so that its component velocity perpendicular to the surface would
be diminished by the attraction, and after traversing the curve N'N, it would
emerge into the second medium in the direction NM, making an angle with the
normal greater than that made by M'N'.

20 INTRODUCTION CHAP. I
even anticipated, the results of experiments, has been universally
adopted.”
24. The prima facie evidence in favour of the emission theory is
very considerable. In the first place it readily accounts for the recti-
linear propagation of light, which at first sight looks more like the
motion of projectiles than the propagation of undulations which have
a tendency to spread out. Then it lends itself at once to the explana-
tion of rays and shadows, while the aberration of light is an immediate
deduction. The so-called rectilinear propagation of light was the great
difficulty which the early supporters of the wave theory had to face,
and the account of it remained in an unsatisfactory state till the time
of Young, a hundred years after the time of Huygens, who sought for
its explanation in certain speculations as to the ultimate constitution
of the ether. That no further progress was made until the time of
Young has been attributed to the great impulse given to the study
of the motion of particles under the action of known forces by the
grand discoveries of Newton, which diverted the attention of men of
science into that channel rather than to the study of the propagation
of undulations.
With regard to the emission theory Sir G. G. Stokes says: ”
“Surely the subject is of more than purely historical interest. It
teaches lessons for our future guidance in the pursuit of truth. It
shows that we are not to expect to evolve the system of nature out
of the depths of our inner consciousness, but to follow the painstaking
inductive method of studying the phenomena presented to us, and be
content to learn new laws and properties of natural objects. It shows
that we are not to be disheartened by some preliminary difficulties
from giving a patient hearing to a hypothesis of fair promise, assuming
of course that those difficulties are not of the nature of contradictions
between the results of observation or experiment, and conclusions
certainly deducible from the hypothesis on trial. It shows that we
are not to attach too great importance to great names, but to in-
vestigate in an unbiassed manner the facts which lie open to our
examination.”
25. Newton’s Theory of Fits of Easy Reflection and Easy Trans-
mission.—The existence of both reflection and refraction at the surface
of a transparent substance presents at first sight a great difficulty in
the emission theory, for it is not easy to conceive how the same
surface may at one time reflect and at another refract an impinging
* This difficulty in the theory may be surmounted by a suitable hypothesis con-
cerning the so-called mass of the luminous corpuscle.
* Burnett Lectures on Light, Lecture I., delivered at Aberdeen, 1883.
ART. 25 THEORY OF FITS 21
molecule. To meet the difficulty Newton was led from his observa-
tions on the coloured rings of thin plates (Chap. VIII.) to endow the
luminous corpuscles with periodic phases or fits, as he terms it, of
easy reflection and easy transmission, so that sometimes they are in
a condition to be reflected, and sometimes in a condition to be re-
fracted at a transparent surface. To communicate these fits to the
luminous corpuscles he imagined all space to be filled with an all-
pervading medium or ether. The luminous corpuscles, on striking a
reflecting or refracting surface, excite waves in this ether which over-
take them at regular intervals, and assist or oppose their motion
periodically, so that at any new surface they are refracted or reflected
according as the wave assists or opposes the corpuscle. The element
of periodicity thus so ingeniously introduced, and which is so funda-
mentally involved in a wave motion, we should naturally expect to
be independent of the angle of incidence. However, to reconcile the
theory with his observations on thin plates, Newton found it necessary
to suppose the length of a fit to vary as the secant of the angle.of
incidence, and it does not appear easy to account for such a law.
Boscovich attributed the fits to a polarity of the luminous mole-
cules, which by rotating presented alternately their different sides to
the reflecting or refracting surface, and Biot expounded the same
theory.”
In conclusion, we may state that we believe an ingenious exponent
of the emission theory, by suitably framing his fundamental postulates,
might fairly meet all the objections that have been raised against it.
It will be found, however, on an examination of the whole, that these
necessary postulates endow the corpuscles with the periodic charac-
teristics of a wave motion, and when this is introduced the corpuscles
themselves may be eliminated, for the wave motion alone sufficiently
explains the phenomena. Hence the one remaining argument against
the supposition of corpuscles is that they are superfluous, for we believe
that no direct test such as has been supposed to be given by the law of
refraction in regard to the velocity of light, or by interference pheno-
mena, can decide between the rival hypotheses.
Eatracts from Newton
The following passages, quoted direct from Newton's writings,
expound his theory in his own words, and show how much more
* Boscovich, Philosophiæ Naturalis Theoria, 1758.
* Biot, Traité de physique, tom. iv. p. 1.
22 INTRODUCTION CHAP. I
closely than is generally supposed it resembles the undulatory theory
now accepted:
“Were I to assume an hypothesis, it should be this, if propounded more gener-
ally, so as not to determine what light is, further than that it is something or other
capable of exciting vibrations in the ether ; for thus it will become so general and
comprehensive of other hypotheses as to leave little room for new ones to be in-
vented ” (Birch, vol. iii. p. 249, December 1675).
Opticks, book ii. part iii. prop. xii. : “Every ray of light in its passage through
any refracting surface is put into a certain transient constitution or state, which in
the progress of the ray returns at equal intervals and disposes the ray at every
return to be easily refracted through the next refracting surface, and between the
returns to be easily reflected by it.
“This is manifest by the 5th, 9th, 12th, and 15th observations (coloured rings).
For by those observations it appears that one and the same sort of rays at equal
angles of incidence on any thin transparent plate is alternately reflected and trans-
mitted for many successions accordingly as the thickness of the plate increases in
arithmetical progression of the numbers, 0, 1, 2, 3, 4, 5, 6, 7, 8, etc., so that if the
first reflection (that which makes the first or innermost of the rings of colour there
described) be made at thickness 1, the rays shall be transmitted at thicknesses 0, 2,
4, 6, 8, 10, 12, etc., and thereby make the central spot and rings of light which
appear by transmission, and be reflected at the thicknesses 1, 3, 5, 7, 9, 11, etc., and
thereby make the rings which appear by reflection. And this alternate reflection
and transmission, as I gather by the 24th observation (viewing them through a
prism), continues far above an hundred vicissitudes, and by the observations in the
next part of this book (colours of thick plates) for many thousands, being propagated
from one surface of a glass plate to the other, though the thickness of the plate be a
quarter of an inch or above ; so that this alternation seems to be propagated from
every refracting surface to all distances without end or limitation. This alternate
reflection and refraction depends on both the surfaces of every thin plate, because it
depends on their distance.
“What kind of action or disposition this is ; whether it consists in a circulating
or a vibrating motion of the ray, or of the medium, or something else, I do not here
inquire. Those that are averse from assenting to any new discoveries but such as
they can explain by a hypothesis, may for the present suppose that as stones by
falling upon water put the water into an undulating motion, and all bodies by per-
cussion excite vibrations in the air, so the rays of light, by impinging on any
refracting or reflecting medium or substance, and by exciting them, agitate the solid
parts of the refracting or reflecting body, and by agitating them, cause the body to
grow warm or hot ; that the vibrations thus excited are propagated in the refracting
or reflecting medium or substance much after the manner that vibrations are propa-
gated in the air for causing sound, and move faster than the rays so as to overtake
them ; and that when any ray is in that part of the vibration which conspires with
its motion, it easily breaks through a refracting surface, but when it is in the con-
trary part of the vibration which impedes its motion, it is easily reflected ; and by
consequence, that every ray is successfully disposed to be easily reflected, or easily
transmitted, by every vibration which overtakes it. But whether this hypothesis
be true or false I do not here consider.”
Opticks, fourth edition, 1750, book iii. Qu. 17: “If a stone be thrown into stag-
nating water, the waves excited thereby continue some time to arise in the place
where the stone fell into the water, and are propagated from thence in concentric
circles upon the surface of the water to great distances. And the vibrations or
ART. 25 EXTRACTS FROM NEWTON 23
tremors incited in the air by percussion continue a little time to move from the place
of percussion in concentric spheres to great distances. And in like manner, when
a ray of light falls upon the surface of any pellucid body, and is there refracted or
reflected, may not waves of vibrations, or tremors, be thereby excited in the refract-
ing or reflecting medium at the point of incidence and continue to arise there, and
to be propagated from thence . . . and are not these vibrations propagated from the
point of incidence to great distances ! And do they not overtake the rays of light,
and by overtaking them successively, do they not put them into the fits of easy
reflexion and easy transmission described above For if the rays endeavour to
recede from the densest part of the vibration, they may be alternately accelerated
and retarded by the vibrations overtaking them.”
And again, Qu. 18: “. . . Is not the heat of the warm room conveyed through
the vacuum by the vibrations of a much subtiler medium than air : . . . And is not
this medium the same with that medium by which light is refracted and reflected,
and by whose vibrations light communicates heat to bodies, and is put into fits of
easy reflexion and easy transmission ?”
In Qu. 19 he employs this ether (as he calls it) to account for gravitation. “Is
not this medium much rarer within the dense bodies of the sun, stars, planets,
and comets, than in the empty celestial spaces between them : And in passing from
them to great distances, doth it not grow denser and denser perpetually, and thereby
cause the gravity of those great bodies towards one another, and of their part towards
the bodies; every body endeavouring to go from the denser parts of the medium
towards the rarer ? . . .”
Qu. 23: “Is not vision performed chiefly by the vibrations of this medium,
excited in the bottom of the eye by the rays of light, and propagated through the
solid, pellucid, and uniform capillamenta of the optick nerves into the place of
sensation ?”
Hearing and animal motion he supposes to be brought about also
by the vibrations of the ether.
24 INTRODUCTION CHAP. I
SECTION III.-INTRODUCTION AND DEVELOPMENT OF THE WAVE
THEORY
26. Early Speculations.—The founding of the wave theory of
light, like the discovery of the laws of reflection and refraction, has
been erroneously attributed to Descartes. In the theory of Descartes
vision was supposed to be excited by a pressure transmitted in-
stantaneously through an infinitely elastic medium filling all space, so
that it contained nothing analogous to the continuous propagation of
waves." The origin of the doctrine might be traced back to the vague
speculations of Aristotle, and some germs of it may be found in the
writings of Leonardo da Vinci,” and in the correspondence of Galileo.
More or less obscure ideas were expressed by Grimaldi and Hooke,”
the latter of whom defined light as “a quick vibratile movement of
extreme shortness”; * but he supposed this movement to be propa-
gated instantaneously in all directions. His theory was consequently
little in advance of the instantaneous pressure of Descartes. However,
it appears that Hooke was quite prepared to admit that light travelled
with a finite velocity (when proved), and that he even anticipated the
proof. -
The founder of a theory is not, however, the author who makes
more or less vague but happy guesses at it, and the credit of discovery
is entirely due to him who demonstrates. Otherwise it would be very
difficult to fix the date at which the undulatory theory of light was
first formulated. *
27. Huygens, Young, Fresnel–The true, founder of the wave
theory is undoubtedly Huygens, who in 1678 first stated it in a
definite form, and in 1690 published a satisfactory explanation of
reflection and refraction on the supposition that light is due to wave
* It is strange that with his ideas as to the nature of heat, which he defines as
“an internal agitation of the particles of a body,” and though this vibratory motion
exists in bodies that are both hot and luminous (i.e. incandescent), and is the cause
of the ‘‘instantaneous pressure" transmitted in all directions, yet there is no state-
ment of a vibration existing in the medium through which the pressure is propagated.
° Libri, Histoire des mathématiques en Italie.
* Micrographia (1665) and Lecture on Light. Posthumous works of Hooke, 1705.
See p. 76, etc.
* Micrographia, p. 15.
ART. 28 HUYGENS, YOUNG, AND FRESNEL 25
motion in the ether.” He also accounted for double refraction in
uniaxal crystals—a phenomenon which had been observed and de-
scribed by Bartholinus” about 1670.
Having failed to account satisfactorily for the rectilinear propaga-
tion of light or the theory of shadows, to which the corpuscular theory
lent itself so easily, the wave theory, so well initiated by Huygens,
fell into disrepute, and remained lifeless for almost a century. It
was then revived by Dr. Young's discovery of the celebrated principle
of interference.
Although Huygens discovered what is known as the polarisation
of light, he was unable to account for it on the wave theory, neither
could Young, for these philosophers supposed the wave disturbance
in the ether to be longitudinal; that is, in the direction of the ray of
light, this being the kind of vibration known to occur in the propaga-
tion of sound. And it was not until Fresnel introduced with brilliant
success a happy guess of Hooke's (1672), viz. that the light vibrations
are transverse—that is, perpendicular to the direction of the ray—
that the great difficulties besetting the theory were removed, and the
known phenomena not only satisfactorily explained, but others not
yet discovered were anticipated. Poggendorff remarks that there
is no other instance in the history of modern physics in which the
truth was so long kept down by authority.
It was the phenomenon of the polarisation of light that led to the
final abandonment of the wave theory by Newton. Having before
his mind the longitudinal or sound vibrations, he could not conceive
how a ray could have different properties on its different sides. He
therefore fell back upon the emission theory, and developed it with a
genius more than human. -
28. Interference—Non-Materiality of Light—Experiments of
Grimaldi and Young.—About 150 years before the time of Young,
Grimaldi” remarked that in certain cases two lights when superposed
can partially destroy each other (and Hooke simultaneously laid claim
* The only author who can be advanced with any show of reason as an antici-
pator of Huygens is the Jesuit Pardis. Huygens mentions the manuscript of Pardis,
and cites him (Traité de la lumière, p. 18) as “one of those who have commenced
to consider the waves of light.” The ideas of Pardis were incorporated in the work
of another Jesuit, C. P. Ango (L’Optique divisée en trois livres, Paris, 1682). It is
here explicitly stated that light is due to waves in the ether, just as sound is due to
Waves in the air.
* Erasmus Bartholinus, Eagerimenta chrystalli Islandici disdiaclastici, Copen-
hagen, 1669; Amsterdam, 1670.
* Prop. xxii. : “Lumen aliquando per Sui communicationem reddit obscuram
Superficiem corporis alicunde ac prius illustratam ” (Physico-Mathesis de lumine, colori-
bus et ride, Bologna, 1665).
26 INTRODUCTION CHAP. I
to the same discovery), but from the manner in which his experiments
were conducted he could not have observed any case of true interfer-
ence. After allowing the sunlight to enter a darkened chamber through
two small holes A and B (Fig. 3) pierced very near each other in the
shutter, he received the diverging cones of light on a screen. Each
depicted a circular spot of light surrounded by a fainter ring. Having
placed the screen at such a distance that these rings partly overlapped,
he observed that the illumination appeared less in the overlapping
Fig. 3.-Grimaldi's Experiment.
portion than in the remainder of the rings. If one of the pencils was
intercepted by an obstacle, this dark portion recovered the brightness
of the rest. Thus darkness, he found, may be produced by adding
one light to another, and on the other hand, the illumination may be
increased by withdrawing a portion of the light. The effect here
observed is, however, probably an optical illusion due to contrast, and
not a true case of interference.
The object of Grimaldi's inquiry being merely to ascertain whether
light was a material or an accident, he prosecuted his research no
further, for he considered the experiment fully proved that light was
not a material substance.
Young, on the other hand, admitted a very small pencil of light
through a narrow slit S in a shutter (Fig. 4). This beam fell upon
a screen perforated by two
small pin-holes A and B, very
near each other. From the
apertures A and B he had
thus two small pencils of
light which he received on a
screen M, and he observed
that at M, where the pencils overlapped each other, instead of uniform
illumination, a series of brilliantly coloured bands appeared (Fig. 5).
When he gradually increased the distance between the pin-holes the
bands gradually diminished in width till they finally disappeared. They
also disappeared when he stopped one of the apertures, or when he
removed the slit S and allowed the sunlight to pass through A and B
Fig. 4.—Young's Experiment.


ART, 28 INTERFERENCE–YOUNG'S EXPERIMENT 27
directly, as in Grimaldi's experiment. This showed that the bands
must be due to the action of the light from A on that from B, and
also that these apertures must
be supplied from the same small
source S.
At any point of a dark band
on the screen, the light coming
to it from one aperture (A) is
apparently destroyed by that
coming from the other (B). In
this case the two lights are said
to interfere destructively. The dark bands are places where the sources
A and B produce opposite effects and neutralise each other, whereas
in the bright bands the two effects are alike and the illumination is very
brilliant. On the whole, however, there is no annihilation of the light.
The deficient illumination of the dark bands is accounted for in the Nodestruc-
excessive brilliancy of the bright bands. The whole quantity of light on ...
the screen is the sum of the quantities which the sources A and B would tion.
furnish separately. The dark bands consequently do not point to the
annihilation of any portion of the light, but merely to a redistribution
of it on the screen; and when we speak of destructive interference
at any point, it must be remembered that the illumination which is
apparently destroyed there exists in equal quantity elsewhere (see
further, Art. 44).
It is sometimes asserted that the mutual interference and con-
sequent production of fringes by two similar sources of light com-
pletely overthrows material or emission theories of light; for, it is said,
we cannot conceive of two substances destroying each other. But it
should be remembered that here we have no destruction of light, the
total quantity remains the same, just as in the case of sand or dust
strewn on a vibrating plate or in a sounding tube. If the dust be
uniformly distributed on the plate before the vibration starts, it will
when the plate is bowed collect along certain lines, leaving the other
parts naked. The total quantity of dust however remains unaltered,
and the same law holds in Young's interference experiment. This
experiment would, consequently, not necessarily overthrow an emission Emission
theory, but rather force it to adopt some new hypothesis concerning *
the corpuscles and their mode of interaction, just as Newton invented
his theory of fits to explain the production of coloured rings by thin
plates.
The principle of interference is one of the most fertile in physical
science, and many beautiful examples of its power will appear in the
Fig. 5.-Interference Bands.

28 INTRODUCTION CHAP. I
sequel. Presently we shall show how it reconciles the apparent recti-
linear propagation of light with the wave theory, and answer the
difficulty suggested by Newton and those who espoused the emission
theory: “If light consists of undulations in an elastic medium, it
should diverge in every direction from each new centre of disturbance,
and so, like sound, bend round interposed obstacles, and obliterate all
shadow.” The reply of the wave theory is that light does bend round
obstacles (as Newton's own experiments prove), but to a very small
extent, on account of the extreme shortness of its wave length, and
shadows exist because the several portions of the laterally diverging
light destroy each other's effect by interference.
Sound is observed to bend round corners very much more than
light, merely because its wave length is vastly greater.
29. A Medium neeessary.--The radiations which we receive from
the sun or from any other luminous body not only affect our sense
of sight, which in itself is an evidence of work done, but also in general
appreciably heat any body on which they fall. Besides the radiations
which affect our sense of sight, and which we term light, a luminous
body in general emits others, which we detect by their thermal or
chemical action. Heat and work being convertible, we may, by
measuring the heating effect of the sun's radiation, calculate the
amount of energy transmitted to us per second. We therefore learn
to regard the sun or any other luminous body as a source from which
energy is emitted in all directions, and the question now arises—by
what means is this energy propagated, and how is it stored while it is
travelling to us from the sun or a distant star, for we know that it is
not transmitted instantaneously, but travels through the interstellar
spaces with a definite velocity, viz. the velocity of light.
Now, with our present experience, it does not seem possible to
conceive of more than two modes by which any body as a source of
a mechanical influence, travelling with a finite velocity, can ultimately
affect and communicate energy to another body situated at a distance.
A mechanical influence implies the intervention of a substance of some
kind, and this substance may be either projected forth from the in-
fluencing to the influenced body, like bullets from a gun, each particle
travelling with a certain velocity and carrying a definite amount of
energy with it ; or it may exist as a continuous medium filling the
space between the two bodies, and being disturbed by one, the dis-
turbance may be propagated from portion to portion of the medium
(each part being agitated by its predecessor, and in turn yielding to
the succeeding element the energy it received) till it finally reaches
the second body. The waves excited by casting a stone into water or
3.
ART. 29 YOUNG'S EXPERIMENT 29
by a sounding bell illustrate the latter method of propagating energy,
and this forms the basis of the wave theory of light.
Young's discovery of the so-called destructive interference of two
lights suggests that light in itself is not a substance emitted by the
luminous body. In addition the emission theory has failed (or requires
interminable patching) to account for the observed facts. Scientists
have consequently been compelled to have recourse to the wave theory,
and to assume that all space is filled with some medium or substance,
if we may so call it, differing in its properties from visible material
bodies, and that a luminous body, as such, is the source of a periodic
disturbance of some kind which is propagated in all directions by
means of this medium, or, as it is called, the ether."
The balance of experimental evidence is in favour of the theory
that all optical and radiation effects are due to rapid periodic changes
of some properties of the ether. Electric, magnetic, and electro-
magnetic effects also appear to be due to the intervention of the same
medium.
We know that sound travels through air, water, glass, and other
material substances with a definite velocity in each, and experiment
proves that the propagation of sound in these substances is due to an
undulatory disturbance or vibration, excited in them by the sounding
body. Sound is not propagated in a vacuum. Its phenomena are
consequences of the vibratory motion of the parts of the material
Substances through which it travels. When a musical note is sounded
energy is transmitted to the air by the sounding body and the air is
thrown into vibration. This energy travels through the air as a wave
motion, and part of it is spent in exciting the tympanum of the
hearer. Light, on the other hand, is propagated with the greatest
facility through the best vacuum we can procure. It traverses the
interstellar spaces where we cannot suppose any material substances
to exist, except perhaps the most excessively attenuated atmospheres
or sporadic groups of meteorites. It is propagated with more or less
facility through transparent substances, but in all cases with a velocity
enormously greater than that of sound. Hence although the presence
of material substances modifies the propagation of light to some
extent (as by refraction or diminution of velocity, etc.), yet they
are by no means necessary to its conveyance from one part of space
to another.
The existence of some medium filling all space as far as the
farthest star becomes therefore a necessity in the rational explanation
of the phenomena of heat and light, and before proceeding to the
* See further, Theory of Heat, pp. 52, 57.
30 INTRODUCTION CHAP, I
development of the wave theory of light, it will be well to consider
some of the fundamental properties of this hypothetical medium—the
ether.
30. The Ether.—The assumption of the existence of a medium
filling all space does not seem to have presented any serious difficulty
to the reception of the wave theory. A far more formidable difficulty
with which the early supporters of the theory had to contend was
presented by the existence of rays and shadows. They could not
explain by the wave theory the apparent rectilinear propagation of
light to which the emission theory lent itself so easily.
Several ethers have been postulated by different philosophers for
different purposes.” Newton supposed that a medium existed in which
his luminous corpuscles travelled, and in which they were capable in
certain cases of exciting undulations. He also attempted to account
for gravitation by the differences of pressure in an ether, but he
published little of his theory, “because he was not able from experi-
ment and observation to give a satisfactory account of the medium
and the manner of its operation in producing the chief phenomena of
nature.”
The only ether which has survived is that conceived by Huygens
to account for the propagation of light. The evidence in favour of
it has accumulated with each discovery of science, and the properties
of it as deduced from the phenomena of radiant light and heat are also
those required to explain the phenomena of electricity and magnetism.
It may be that this same medium forms the vehicle by which gravitation
is maintained between material substances, and in some manner as yet
unknown to us forms the link of connection by which the sun is en-
abled to attract the earth and planets and keep them in their orbits.”
* To Descartes the bare existence of bodies apparently at a distance was a proof
of the existence of a continuous medium between them, for he regarded extension as
the sole essential property of matter, and matter a necessary condition of extension.
“Ethers were invented for the planets to swim in, to constitute electric atmo-
spheres and magnetic effluvia, to convey sensations from one part of our body to
another, till all space was filled several times over with ethers” (J. C. Maxwell).
* “It is true that, notwithstanding the labours of various scientific men, we are
not in a condition to give an explanation of gravitation, but our inability to explain
it by no means proves that it is a primary property of matter, incapable of explana-
tion, or forbids us to suppose that it may in some way be brought about by the
intervention of that same substance which we find it necessary to assume for the
explanation of the phenomena of light on the theory of undulations. . . . Assum-
ing for the moment, as a thing at the present day resting on evidence quite over-
whelming, that light consists of undulations, we cannot fail to be impressed by the
multiplicity of purposes all bearing so intimately on our wellbeing, which, it seems
probable, or not unlikely, are fulfilled by one and the same substance, endowed with
properties which we are only gradually learning” (Stokes's Burnett Lectures).
ART, 30 THE ETHER. 31
The present tendency indeed of physical science is to regard all the
phenomena of nature, and even matter itself, as manifestations of
energy stored in the ether. When we electrify a body a certain
amount of energy is expended, and this is ordinarily regarded as the
energy of the electric charge, and may be recovered at any time by
discharging the body. But where is the energy stored ' We say it
is stored in the ether. So again it may be that the energy spent in
raising a mass from the earth's surface is stored in the ether." Hence
what we call potential energy may be energy stored in the ether, and
if it exists there as motion of the ether, then we may regard all energy
as kinetic. &
To account for the propagation of undulations with a finite velocity
and carrying energy, the ether has been endowed with the two radical
properties of elasticity and density, or rather something corresponding
to elasticity and density.” When sound is propagated through material
substances, rarefactions and condensations are produced, and to the
forces of restitution called into play the propagation of the sound is
due,” while the velocity of the propagation depends on the elasticity
and density of the substance. There is, however, a series of pheno-
mena in light 4 which have no counterpart in the theory of sound,
and which lead to the conclusion that the so-called elasticity of the Transverse
ether is very different from that of the air. They suggest that the ".
vibration of the luminiferous ether must be transverse to the direction
of propagation of the light. Air and fluids cannot transmit transverse
vibrations, for they offer no resistance to distortion, and this is the
property on which the propagation of transverse vibrations depends.
When sound is travelling through air the vibrations of the air are
longitudinal, that is in the direction in which the sound is travelling.
Solids, on the other hand, are capable of transmitting both kinds of
vibrations, but with a velocity enormously less than that of light.
The elasticity of the ether has consequently been assumed to be some- Elastic
what of the nature of that of an elastic solid, but the propagation of tº.
light by it on this hypothesis is encumbered by several difficulties.
The first is the possibility of longitudinal vibrations or undulations
normal to the wave front, as in the case of sound propagation. That
no optical phenomena arise from these has been accounted for by
* See further, Theory of Heat, p. 77.
* On this point see Theory of Heat, p. 53.
* In the case of a vibrating elastic solid the energy is half in the form of kinetic
energy, due to the vibratory motion of the parts of the body, the other half being
potential ; that is, stored up in the distortion of its parts (see Thomson and Tait's
Natural Philosophy, or Love's Theory of Elasticity).
* The polarisation of light.
32 INTRODUCTION onar, i.
supposing the ether incompressible," so that the velocity of propagation
of the longitudinal wave is infinite. Again, the phenomena of polarisa-
tion and double refraction have led to incongruities and artificial
assumptions.
Since the vibrations of transparent bodies travel much too slowly
to allow us for a moment to suppose that the propagation of light
might be due to them, we are forced to conclude that the ether which
Conveys the light is distinct from these transparent media, and inter-
penetrates them all freely (and probably opaque bodies too). It may
be difficult at first to admit that a solid body like glass could possibly
have ether freely pervading it; but we must remember that the ether
is a medium about which our senses give us no direct information.
We cannot see, taste, or smell it. It is only by the intellect that we
become convinced of its existence—by studying the phenomena of
nature, and finding how they may be explained by it. A magnet
attracts a piece of iron even though a plate of glass be interposed
between them, and yet the magnetic influence is one which does not
directly affect our senses, but we must conclude that whatever medium
enables the magnet to put the iron in motion, and communicate kinetic
energy to it, permeates the glass as well as the air and interstellar
space. This medium also propagates light and heat through many
solid substances; it therefore not only interpenetrates them, but is
capable of vibrating within them.”
Now although we suppose the ether to permeate all bodies freely,
yet we must suppose that its vibrations are controlled to some extent
by the matter of these bodies, for we can prove that light travels with
different velocities in different transparent substances (a fact attested
by the refraction or bending of the ray in passing from one transparent
Matter and substance to another), while in opaque bodies it is not propagated at
ether.
all. It is therefore natural to inquire if the free motion of the ether
is influenced by the presence of masses of matter or by matter mole-
cules. Do bodies in motion in this ocean of ether carry with them
the ether they already contain, or do they allow the ether to •pass
through them freely, as water would pass through a net, or, as Young
imagined, like the air through a grove of trees Or does the ether
offer any resistance to the motion of the earth and planets through it,
and do these bodies, by their motion, produce streams and eddies
in it !
* Lord Kelvin suggested (Phil. Mag., Nov. 1888, p. 414) that the ether may be
“contractile,” that is, offer a negative resistance to compression, so as to give a
zero velocity of propagation for the longitudinal wave.
* This free interpenetration follows as a natural consequence of the vortex-atom
theory of matter (Theory of Heat, p. 78, etc.).
ART. 32 SPECIAL FORMS OF WAVE THEORY 33
The whole question of the state of the ether near the earth, and
of its connection with ordinary matter, is still far from being settled
by experiment (see Chap. XIX.).
Whatever difficulties we may have in forming a consistent idea of
the constitution of the ether, it cannot be doubted that the interstellar
spaces are filled with an all-pervading medium in which the ultimate
particles of matter and of our own bodies are continually bathed, and
yet of which our senses afford us no direct cognisance. (And what-
soever other functions appertain to it, one of the chiefest is the con-
veyance from the sun to Nhis system of that energy by which all its
physical life is sustained. }
31. The Vibration.—Although all the phenomena of interference
afford us the strongest evidence that light is propagated as a vibration
of the ether—that is, a rapid periodic change of its state or of some
of its properties—yet the medium being hypothetical, we are almost
wholly ignorant as to what it is that vibrates or how it vibrates. The
direction to which the periodic change of state is related we term the
direction of the vibration, meaning by the vibration that periodic
change, whatever it may be. If it be a periodic displacement, then
the direction of the vibration is the direction of the displacement, but
this does not necessarily coincide with the direction of propagation of
the disturbance. In the case of sound we know well the nature of
the disturbance and the properties of the vehicle. In the case of light
we shall see that at least a component of the vibration is in the wave
front, or transverse to the direction of the propagation.
32. Special Forms of the Wave Theory.—Many of the pheno-
mena of light, such as the colours of thin plates, can be equally
explained by any form of wave theory, and cannot be substantially
modified by a more exact knowledge of the constitution of the ether.
However, in other parts of the subject, such as the theory of crystalline
refraction and reflection, we are compelled to make some hypotheses,
and if on developing the special theory built on these hypotheses we
find a discordance with any observed phenomena, this disagreement
only disproves the special form of theory admitted, but the wave
theory in its general sense remains intact.
The necessity of transverse vibrations, and the incapability of
fluids to propagate them, led to the development of the most celebrated
special form of wave theory—that which regards the ether as an
elastic solid. Here, however, we' need not regard the ether as
possessing all the properties of an elastic solid.; All it requires is
torsional rigidity; that is, resistance, to change of shape, or some
property analogous to the torsional rigidity of elastic solids, to enable
D
{
34 INTRODUCTION CHAP. I
it to transmit the transverse vibrations of light. A substance like a
jelly could transmit either transverse or longitudinal vibrations, and
the velocity of the latter might be very great compared with the
former. To avoid the difficulty introduced by the possible longitudinal
waves, Green and the promoters of the elastic solid theory supposed
the ether to be incompressible, so that the longitudinal disturbance
travelled with an infinite velocity. Sir G. G. Stokes has, however,
remarked that although the ether may act as a perfect fluid for any
finite displacements, yet the displacement occurring in the propagation
of light may be so small and so rapid that for it the medium behaves
as an elastic solid. Cumbered with difficulties which only increase as
it is required to meet the demands of accumulating information, the
elastic solid theory, although in many instances it comes near the
truth, can in optics be regarded only as a first speculation, but never-
theless it must always retain a high historic interest.
The theory which promises most favourably at present is that
which regards the ether as a turbulent fluid, and light as an electro-
magnetic phenomenon arising from very rapidly alternating electric
polarisations or “displacements,” as Maxwell termed them. When a
body is electrified energy has been spent in producing the electrifica-
tion, and this energy is stored in the ether around the body. To
indicate this we may say that the ether is polarised, meaning thereby
that its elements have suffered some directed transformation in pro-
perties, or change of state, by the storing of the energy. Electric
phenomena are manifestations of this energy in transformation, and
when the body is discharged the ether is released from the energy
and the consequent polarisation. Similar remarks apply to an electric
current. Now if a body be rapidly charged and discharged, or a
current be passed rapidly in opposite directions, the ether around will
be as rapidly thrown alternately into opposite states of polarisation,
and when this becomes very rapidly periodic we have the vibration of
the ether spoken of in the wave theory of light. When the rapidity
of these vibrations lies within certain limits (red and violet) they affect
the eye with the sense of sight; below the red we detect them by
their thermal, and above the violet by their chemical action 1 (see
further, Chap. XXI.).
Apart from its probable truth, the electro-magnetic theory of light
shows us how careful we must be to avoid limiting our ideas as to the
nature of the luminous vibration.
* A quasi-rigidity might be conferred on the ether by other motions going on in
it. Thus by filling it with vortices it might become capable of propagating trans-
verse waves and standing electric stress.
CHAPTER II
THE PROPAGATION OF WAVES AND THE COMPOSITION OF
WIBRATIONS
33. Wave Motion.—The essential characteristic of wave motion is
that a periodic disturbance is handed on successively from one portion
of a medium to another. Examples of it are constantly presented to
the notice of every one, as, for instance, when a stone is cast into still
water or when a sounding bell throws the air into vibration.
That which is propagated from one part of the medium to the other
is energy, not matter, for while any element of the medium merely
oscillates about its position of rest, there is a continuous handing on, or
flow, of energy from one part to another. In the case of a projectile
or a current, on the other hand, matter flows from one place to another,
and carries with it its associated energy, so that we have a flow both
of energy and matter. - 2. “
If the medium be homogeneous and isotropic, a disturbance is
propagated with the same velocity in all directions; but if the medium
be not homogeneous the speed may vary from point to point, and if it
be homogeneous but not isotropic the speed may depºd
tion of propagation. & , sº
In general the velocity of propagation depends ºn th; nature of
the medium and the length of the wave, but so fº as fight is con-
cerned the velocity in interstellar spaces seems to 5e the same for all
wave lengths.
34. Transverse Wave.-As an elementary introduction let us
consider the nature of a wave of transverse vibration and its mode of
propagation.
Take a flexible cord AB, one end B being fixed and the other A
free. (A thick piece of india-rubber tubing 3 or 4 yards long answers
very well.) If the free end A be quickly moved from A to A' (Fig.
6, a) and back again, the displacement communicated to A will run
along AB as a wave. Fig. 6 (a) represents this wave travelling along
ºn the direc-



35
36 PROPAGATION OF WAVES CHAP. II
Phase.
AB. If A had been displaced in the opposite direction the wave
would travel as in (8), whereas if A were displaced backwards and
forwards continuously, a continuous series of waves would be pro-
pagated along it as in (y). The experiment (a) may be easily repro-
duced by giving a sudden jerk to one end of a rope lying on the
ground. In all these cases we speak only of what happens as the wave
(a) Solitary wave.
(8) -- --
(y) Periodic waves.
Fig. 6.-Transverse Waves.
travels from A to B, that is, we suppose B very far away, for at
present we are not concerned with what happens after the wave
reaches it.
Here it is evident that each element of the cord merely oscillates
backwards and forwards, like A, through its position of rest, while the
disturbance is propagated from A towards B. There is a flow of energy
along the cord.
Definition.—Two particles such as a and b [Fig. 6 ())] are said to be
in the same phase of motion when their displacements and direction of
motion are the same, and two particles in the same phase are separated
by a complete wave length, or by any whole number of wave lengths.
Two particles such as a and c, whose displacements and directions of
motion are opposite, are separated by half a wave length, or any odd
number of half waves, and are said to be in opposite phases of motion.
The process which takes place in the cord may be illustrated by
supposing AB a row of particles connected by elastic bands. When
A is displaced it drags the adjacent particle after it, which in turn acts
on the third, and so on, the disturbance being handed from one particle
to the next. Thus by the time A has reached its greatest distance A,
(Fig. 7, 8) from its position of rest, the disturbance will have travelled
along the line to some point D, and the particle at D, will be on the
point of beginning to move as A did. In doing so it will pull its
successor after it, and the disturbance will travel on in this way to the
end of the line. But when A begins to return to its initial position it
drags the adjacent particle after it. This in turn reacts on the next
and so on, so that by the time A reaches its initial position the original
disturbance (which was being propagated all the time along the line)

ART, 35 PLANE WAVE 37
has reached a distance D, equal to twice the distance of D, from the
end A of the line. The row of particles being now represented by
Fig. 7 (y), let A, pass through its position of rest to an equal distance
As on the other side. The configuration of the line is now represented
by (6), in which the original disturbance is just about to displace D, at
(a) Connected particles.
(3) Quarter wave.
(y) Half wave.
(6) Three-quarter wave.
(e) Complete wave.
Fig. 7-Transverse Waves.
a distance equal to 3AD, from A. So again when A returns to its
position of rest the line will be represented by Fig. 7 (e), the disturb-
ance having now reached D, while A has made a complete excursion.
AD, represents a complete wave. It is the distance which the dis-
turbance travels along the line while the particle A makes a complete
vibration. Now D, is just about to repeat the oscillation performed
by A, so that if A vibrates continuously D, will be moving in exactly
the same manner and have the same displacement and direction of
motion as A at every instant. A, and D, are in the same phase, and
the length AD, is a wave length. Thus at distances of a wave length,
or any multiple of a wave length apart, the motions of two particles
are exactly similar, while at distances of half a wave length apart, or
any odd number of half waves, the displacements and velocities are
equal but in opposite directions. (This remark is of great importance
in the theory of interference.)
35. Plane Wave.-Instead of a single cord let us imagine an
infinite number of parallel and similar cords with their ends attached
to the same plane. For the sake of clearness let us suppose the plane
to be horizontal and the cords to hang vertically from it. Then if
the plane be moved parallel to itself, so that each point of it describes
a short horizontal line, the ends of the cords, being attached to the
plane, will be displaced horizontally and a transverse disturbance will
run along each. At the end of any time this disturbance will have
travelled to the same distance along each cord, and therefore the locus
of those points which are just about to be disturbed, or the wave front,
will be a horizontal plane. If the plane be caused to oscillate regularly
Wave
length.

38 WAVE MOTION CHAP. II
Wave
front
backwards and forwards a series of similar waves will run along
each cord. - - -
We might now suppose the cords so numerous that they touch
each other, and we are thus furnished with an idea of a continuous
medium, disturbed by a series of plane waves propagated through it.
The end of each cord has been supposed for simplicity to oscillate
along a straight line, but we might equally have supposed it to describe
any curve—thus in the case of waves in water the various particles
generally describe circles or ellipses in vertical planes, parallel to the
direction of propagation.
36. Spherical Wave.—Let us now take the case of an infinite
number of exactly similar elastic cords attached to the same point,
and diverging in all directions from it. Further, let a disturbing force
at the centre, acting in the same manner upon each, cause them all to
undulate alike. The cords being similar in all respects, it is obvious
that the waves propagated along them will be alike and travel with
the same velocity. Thus all points at the same distance from the
centre will be in the same state of disturbance at the same instant ;
that is, the locus of points in the same phase of vibration is a sphere
and the wave fronts or wave surfaces are spherical. If we imagine the
cords to fill all the space around the centre we have the case of a con-
tinuous medium disturbed by a system of spherical waves diverging
from a point.
Definition.—The continuous locus of those points, which are in the
same phase of vibration is called a wave front. The word continuous
is inserted because in oscillatory motion such as we are considering
a system of successive waves are in similar phases and a succession
of similar wave fronts coexist, each a wave length in advance of its
predecessor. In this sense any Surface of equal phase is a wave front.
The wave front might be defined more closely as the continuous
locus of those points which are just on the point of being disturbed,
and in this sense the term is frequently used. This wave front then
marks the limits to which the disturbance has just reached at the
instant considered.
Evamples
1.—Prove that the equation
gy = a sin ºv, — ac)
represents a wave disturbance in which V is the velocity of propagation, N the wave
length, y the displacement of a particle from its position of rest at the time t, and a
the distance from the origin of the same particle.
ARt. 36 SIMPLE HARMONIC MOTION 39
For brevity denote the angle (Vt-w)2T/x by 6 ; then if we change a to 2-EX we
change 6 to 0-F2T, and the sine of the angle remains the same, therefore the value of
y—that is, the displacement of the particle under consideration—is exactly the same
as that of the particle at a distance X from it, or any number of times \from it. Hence
\ is the wave length.
Again let t become t +\/V, then 6 becomes 0+2+, and therefore the value of y
remains the same as before. The value of y for the same particle is therefore periodic
and the periodic time T is N/V-that is, X = WT. But X the wave length is the
distance through which the disturbance is propagated in the periodic time T, con-
sequently the quantity V is the velocity of propagation.
The quantity a is termed the amplitude of the vibration. It is evidently the
greatest displacement of any particle from its position of rest. The amplitude de-
pends upon the nature of the medium and the power of the disturbing centre, and
its magnitude must here be left an open question. The periodic time T can be easily
determined in the case of sound, but in the case of light the vibration is so exceed-
ingly rapid that it can only be determined indirectly from the equation \-VT, by
means of a previous knowledge of V and X.
If we consider a row of particles AB, connected as in Fig. 7 and initially in a right
line, their simultaneous positions when disturbed by an undulatory motion will be
represented by Fig. 6 (Y). This curve is determined by the above equation when we
suppose t to remain constant and as and y to be the abscissa and ordinate of the
particle. But if we confine our attention to a single particle—that is, suppose a con-
stant while t and y vary—then the same curve will determine the displacement y of
the particle at any time t, the time t being now the abscissa of the curve.
Thus the curve which represents the simultaneous displacements of all the
particles also exposes the history of the displacement of a single particle.
Simple Harmonic Motion.—If a particle moves subject to the above equation, it is
said to execute a simple harmonic vibration. The motion here described is that
executed by a particle constrained to move on a right line under the action of a force
varying directly as the distance. Such would be the motion of a particle placed in
a smooth straight tube passing in any direction through the earth.
2. –If a particle moves round the circumference of a circle with uniform velocity,
the foot of the perpendicular from it on any diameter moves backwards and forwards
along that diameter with a simple harmonic motion.
Let the angular velocity of the particle be w, and let the time be reckoned from
Fig. 8.
the instant the particle leaves the extremity A (Fig. 8) of some diameter OA making
an angle a with OX. Then if P be the position of the particle at any time t the
angle AOP=wt and XOP=wt-a. Hence, if OY be at right angles to OX, the
distance y from O of the foot N of the perpendicular from P on the diameter OY is
y=a sin (wt-Ha),
where a is the radius of the circle, and is the amplitude of the vibration of the foot

40 WAVE MOTION CHAP. II
of the perpendicular. If T be the time of a complete revolution we have wiſ-2", so
that the equation may be written
u-a sin ** a
ºſ- ***)
Or substituting from the equation \ = WT we have as before
y=a sin (ºv. **)
If V is the velocity of the particle in the circle, then VT-2ta. Hence if the velocity
in the circle be the velocity of propagation of the wave, and if the time T be the
period of the wave, then the circumference of the circle is equal to the length of the
wave, while its radius represents the excursion of the molecule making simple har-
monic vibrations.
The motion of the foot of the perpendicular is represented in Fig. 8, the variables
being y and t, and the co-ordinates y and awt. Similarly the foot M of the per-
pendicular from P on OX executes the vibration
w-a cos (wt +a)=a cos (***)
37. Algebraical Expression.—Let O be a fixed origin and y the
displacement of P, at any instant, from its position of rest. Then if
a be the distance of P from 0,
measured along OX, the particles
adjacent to P will in their dis-
placed positions lie on a curve, as
indicated on Fig. 9, and the equa-
tion of this curve at the instant
under consideration may be writ-
Fig. 9. ten in the form y = p(x).
Now by the characteristic of wave motion this curve will be propa-
gated forward with a velocity V, and the displacement of any particle
P will be exactly the same as that of P at a time t previously, pro-
vided the difference of the abscissae of P and P is equal to Wt, where t
is the time of propagation from P to P. In fact the wave curve at P'
is merely that at P moved forward through a distance Wł. But at P
we have y = p(x), therefore at P' we have
º/- p(a: - Vt),
for the w of P is equal to the w of P' diminished by Vt.
It is clear in itself that this equation represents a wave motion
* The form y=a sin (*(vº-º-A) was tacitly assumed by Newton for the
equation of motion of the air particles in sound propagation (Principia, lib. ii.
prop. 47).

ART. 37 ALGEBRAIC INVESTIGATION 41
travelling with a velocity V, for y remains unchanged if we replace t
by t-i-T and a by its corresponding value a + WT.
Otherwise thus: If W be the velocity of propagation, the displace-
ments at any two points P and P’ will be the same at the times t and t' if
a' — ac-W(t’ – t) or ac'--a – Wt 4-Wi'.
Hence - -
g = p(a), t) = p(ac', t) = p(a – Wt-i-Wi', t'),
which holds for all values of t', and consequently for t = 0, in which
case we have
gy= q.(Wt – ac).
If another disturbance be running in the opposite direction its equation
will be y = f(Vt + æ), so that if the two be superposed we will have
gy = @(Vt — ac) + f(Wf 4-a),
which represents the general state of disturbance in a finite stretched
wire when we have both direct and reflected waves.
A particular solution of the equation y = p(Vt – a.) is obviously
e (**)
gy= a Sln p g
The value of y will be unaltered if t be increased by 27p/V. The
periodic time is therefore T = 2*p/V, but A = WT = 2+p. Therefore *
v= a sin ºv, – a j.
The general form of y is a periodic function. It has the same
values at distances a, a + \, a + 2\, etc., and at times t, t + T, t + 2T,
etc., and we know by Fourier's theorem (Thomson and Tait, Wat. Phil.)
that any periodic function f(z) of period A can be expanded as a series
in the form
2T2 6trz
f(z)= A1 sin ( X +2) +A2 sin (***) +A5 sin ( X +…) + etc.
Hence we may write
g=A1 sin * (Vt – aft) + A2 sin º (Vt — a 2) + etc.
The complete vibration is therefore made up of a series of superposed
simple harmonic vibrations of wave lengths A, #A, #A, etc.
As we have reason to believe that waves of different periodic times
* Hence p is the radius of a circle of circumference X.
42 WAVE MOTION CHAP. II
Infinite
train of
W&Wes.
produce different impressions on the eye, just as notes of different pitch
produce different effects on the ear, we may therefore restrict our
attention to waves of a definite length, and take
gy= a sin ºv — ac) = a sin (#- •)
OI’
ty= & sin (wt – d.)
as the standard equation of the disturbance in our investigations con-
cerning light of a definite wave length
Definition.—The whole angle 2+(Vt – a WA, or ot — a, is called the
phase of the vibration, and the angle a is sometimes called its epoch.
38. Colour and Frequency or Piteh.-We have seen how a single
wave or a continuous succession of waves (Fig. 6) may travel along a
stretched cord. In a similar manner a continuous system of waves
might be generated in the air or any other medium. Such a system
of waves may or may not affect our sense of hearing. The pheno-
menon of sound is produced by a more or less rapid succession of
waves, but there are major and minor limits to the rapidity of vibra-
tion, outside of which the ear fails to follow, and no sensation of sound
is produced ; that is, the sensibility of the ear is limited, and the pitch
of the note must lie within certain limits in order that it may affect
the sense of hearing. So a wave system in the ether may or may not
affect our sense of sight. The sensations of light and colour are due
to a very rapid succession of waves, but the sense of sight is limited
in range and the ether vibration may be either too slow or too rapid
to affect it. The limits in this case, however, are not nearly so widely
separated as in the case of sound. The rapidity with which the violet
waves succeed each other has been calculated to be less than twice as
fast as that of the red, the latter making about four and the former
about seven hundred millions of millions of vibrations per second.
The sensibility of the eye is thus confined to much narrower limits
than that of the ear, the interval between the red and violet being less
than an octave, while the ear possesses a range of several octaves.
Vibrations too slow to excite the eye are recognised by their thermal
effects, and those which are too quick are generally detected by their
chemical action. The former are most effective in heating our bodies,
while the latter facilitate the prosecution of such arts as photography,"
and are a very important factor in the growth of plants, etc.
Range
limited.
* Captain Sir W. Abney has succeeded in preparing photographic plates with
bromide of silver which are capable of being decomposed not only by the violet end
of the spectrum, but also by the red rays, and by rays of lower refrangibility which
have wave lengths nearly three times that of the red (see “Bakerian Lecture,” Phil.
Trans. Roy. Soc., 1881).
ART, 38 LONG AND SHORT WAVES 43
As the pitch (or musical colour) of a note is determined by the
frequency of its vibrations, so it appears to be the frequency of the
vibration in the luminiferous ether that determines the colour. For
if a wave motion is propagated from one medium to another, the
vibration frequency in the second medium ought to be the same as in
the first, the vibration in the second being excited and forced by that
in the first. This being admitted, it follows that if the velocity of
propagation changes in passing from one medium to another, the wave
length must change in the same proportion, in accordance with the
equation A = WT. Now it has been observed that when light of a
definite wave length passes from one medium to another, as from air
to water, the colour of the light (say sodium light) remains unaltered,
so that the natural inference is that the colour impression on the eye
depends on the vibration frequency rather than on the absolute wave
length.
We have now three methods of detecting solar radiation or wave
motion in the ether. If the frequency of the vibration lies within
certain limits, it affects our sight, and we call it light. Outside these
limits the vibration may be too slow or too rapid to affect our eyes.
In the former case we can detect the wave motion by its heating
effects. It imparts its energy to material substances and to our own
bodies, exciting vibrations in them and producing physical changes,
and we term it radiant heat. It is not to be understood that it is only
the waves too slow to affect our eyes that possess any heating power.
This property is possessed by the luminous waves also, but generally
in a smaller degree, so that those waves which affect our sense of sight
may be called luminous heat waves, and those which do not may be
termed non-luminous, or dark, heat waves. Both the ultra-violet and
infra-red waves may be converted into luminous waves. The con-
version of the former is termed fluorescence and of the latter calorescence
(Arts. 289, 290).
Heat and light are consequently reduced to the same cause, viz.
wave motion in the ether, and any ether wave may be, at the same
time, a light wave and a heat wave. It is not that there are two
classes of waves, heat waves and light waves, but that we have two
senses by which we can detect the same waves. They affect our
bodies with warmth and our eyes with light if their wave period lies
within the range of our sensibility."
* Evidently in Young's opinion the heat rays and the actinic rays differ from the
light rays only in their wave length or period, for he states that “. . . we must
therefore call light an influence capable of entering the eye, and affecting it with a
sense of vision. A body from which this influence appears to originate is called a
44 WAVE MOTION CHAP. II
Long
WaWes.
The waves which are too rapid to affect our sense of sight are
termed actinic waves or rays. Their extreme shortness probably
qualifies them to operate between the molecules of bodies and become
effective in producing chemical changes. Beyond this point we have
no means of detecting the very rapid waves. It is otherwise, however,
in the case of long waves.
We have seen that although the ether vibrations may be too slow
to affect our sight (as air vibrations may be too slow to affect our
sense of hearing), still we are able to detect their presence by their
heating effect. But there is a limit even to this. The ether waves
may be so slow that they will pass through our bodies without giving
up any sensible portion of their energy. How then are we to detect
these long ether waves | The recent brilliant experiments of Professor
Hertz (see Chap. XXI.) have placed the means in our hands. We can
now work with ether waves of any length, from a few inches upwards.
Heretofore we could detect only the radiant heat waves, which are
excessively short, ranging from Toºwo to Toºroo part of an inch.
The Hertzian waves are excited electrically and detected electrically.
They may also be detected by their thermal effects. We have thus,
as it were, acquired another sense, for we have now another means
of investigating the ether, and connecting the various phenomena of
nature.
39. Average Kinetic Energy of a Vibrating Particle.—Let a
particle vibrate according to the equation
gy= a sin (wt – a).
Then its velocity at any instant is given by
Q) =} =aw cos (wt – d.),
and if m be its mass its kinetic energy is #my”. Now the foregoing
expression shows that v varies from zero to ao ; accordingly the mean
energy during a complete vibration will be
1 ſº...a..., mºſ",..., ma?a,” ſº \
#ſºn, dt = 4T ſº cos”(wt — a)dt = 4T l {1 + cos 2(at — a)}dt
ma2002 IT 1 .
=-4T ( +2, sin 20t- •) =#ma*a*,
luminous body. We do not include in this definition of the term light the invisible
influences which occasion heat only, or blacken the salts of silver, although they
both appear to differ from light in no other respects than as one kind of light differs
from another, and they might probably have served the purpose of light if our organs
had been differently constituted” (Young, “On the Theory of Optics,” Lectures on
Natural Philosophy).
ART. 40 INTENSITY OF ILLUMINATION 45
where 0 = 27/T. Hence the average energy is
_ma*a*_mirºa”
- - 4 - T2 . e
Cor.—The mean energy = }mw% = }.}ºmy” where v is the greatest
velocity of the molecule. Therefore the mean kinetic energy is half
the maximum kinetic energy.
40. Intensity of Illumination.—We must now define what we
mean by, and how we measure, the intensity of light. Lights generally
differ in two respects, viz. intensity and colour. In photometry we
say that two lights are equally intense if at the same distance away
they produce the same illumination on a screen. If one candle pro-
duces a certain illumination we say that two such candles together
produce twice as much illumination, or that the intensity of illumination
is twice as great. Similarly the intensity of illumination is directly
proportional to the number of candles, provided they are all of the
same power." Now the average energy of the vibration due to any
source is proportional at each point to the square of the amplitude,
and the average energy due to any number of sources is proportional to
the square of the amplitude of the resultant vibration, and this over any
region is the sum of the average energies due to the sources separately.
This has led to two methods of estimating the intensity of the
illumination in terms of the energy of the vibrating medium. In the
first place, we may take the intensity of illumination as measured by
the energy per unit volume of the medium, and in this case if m stands
for the mass per unit volume the intensity by the foregoing article
will be measured by the formula”
_majºaº mirºa”
4 T2 .
On the other hand, we may measure the intensity by the quantity of
energy transmitted per unit time per unit area across a plane perpen-
dicular to the direction of propagation. In this case the velocity of
propagation will come into account, and the intensity (for a plane
wave will be measured by the product of the velocity and the energy
per unit volume ; that is, by the formula
I = majºa”, mirºa”),
4 TB
* This may be regarded as the definition of an illumination n times as intense as
a given standard. That the intensity varies inversely as the square of the distance
from the source follows as an experimental fact.
* In this expression we have taken into account only the kinetic energy. In the
case of a vibrating elastic solid the whole energy is half kinetic and half potential,
so that if the whole energy be taken as the measure of I, the expressions in the text
ought to be doubled.
Energy
contained,
Energy
trans-
mitted.
46 - WAVE MOTION CHAP. II
In practice, however, we deal only with the relative intensities, and
it follows from either system of measurement that if the periods of
two vibrations be the same, then their intensities are in the ratio of
the squares of their amplitudes, or
I _ a”
I-273,
The relative intensities of two sounds of the same pitch, or of two
lights of the same colour—that is, of the same period or wave length
—are consequently compared by the squares of their amplitudes of
vibration. But we cannot so easily compare the intensities of two
lights of different colour, or two sounds of different pitch, for they
produce dissimilar impressions, and the time (T) of vibration enters
the expression for the mean energy of the motion. In the estimation
of the relative intensities of different sources of light (which forms the
subject of photometry) great difficulty is encountered in the fact that
different sources of light are in general differently coloured."
41. The Intensity varies inversely as the Square of the
Distance.—Let us now consider a luminous point as the source of a
system of spherical waves diverging from it as centre. Consider any
one of these waves of radius r. This wave travels outward, develop-
ing itself and increasing its radius with the velocity of light. It
carries its energy with it, and after a time it is a sphere of radius r".
Now if I denotes the energy per unit time transmitted across unit area
of the first wave surface, and I the energy per unit area of the second,
we have, since the energy of the first is handed on to the second, the
whole energy transmitted per unit time across the wave = 4Tr”I, and
also the whole energy transmitted = 4trºI’.
Consequently
I q.72 f
F=-2 or Irº- I'r”= constant,
or the intensities of the illumination at different distances from a
luminous origin are inversely as the squares of the distances, a relation
which is verified by experiment.
Cor.—Since the intensity is proportional to the square of the
amplitude, it follows that the amplitude of the vibration is inversely as
the distance from the origin, or -
ar=a'r' = constant.
* On the comparison of the relative intensities of the different colours throughout
the spectrum see Colour Measurement and Miature, by Captain Sir W. Abney.
ART. 42 PRINCIPLE OF SUPERPOSITION 47
41a. Intrinsie Brillianey.—The intrinsic brilliancy of an ex-
tended source is defined as follows: Let ds
be an element of area of a source and
ds an element of area of a screen. Let
rays passing between these elements make
angles 6 and 6' with them. Then if I is
the intrinsic brilliancy of the source and
Iſ the intensity of illumination on the screen
due to ds,
I cos 6 cos 6'ds ds'
7.2 º
I'ds’ =
42. The Principle of Superposition.—
When two or more separate disturbances
are simultaneously impressed on the same element of a medium,
the effect may be very complex, but in the case of light the dis-
placements are supposed to be such that the composition of them
by direct superposition is allowable, for calculations made on this
assumption agree with the observed facts, at least to that degree of
accuracy reached by experiment. On this principle is based the whole
doctrine of interference discovered by Young in 1801. It follows
from it that any number of separate disturbances may be propagated
through one another in the same portion of the medium, each emerging
from that portion as if it had not been encountered by the others.
Thus rays of light from objects all around us cross each other's paths
in all sorts of ways, but each travels on as if the others did not exist.
Each portion of the ether is traversed by streams of light from a
multitude of different sources, which are simultaneously propagated
through it. Hence we may conclude that at any instant the dis-
turbance at any point of the ether is that due to the superposition of
all the disturbances which reach it at that instant from the various
parts of the surrounding medium. This is, in a generalised form, the
principle stated by Huygens in 1678. It asserts that the displacement
caused by any source of small vibrations is the same, whether it acts
upon the medium alone or in conjunction with other sources, provided
the displacements considered are very small. The combined effect of
several sources is therefore the geometric resultant of the displacements
which would be produced by them acting separately. -
The nature of this principle may be made more clear by a simple
example. Let a pendulum receive an impulse in any vertical plane
passing through the point of suspension, causing it to vibrate in that
plane. Now when it is at the lowest point of the arc of vibration, let
a second impulse be given horizontally in a plane perpendicular to
FIG. 9a.

48 WIBRATIONS COMPOUNDED CHAP. II
that in which it already vibrates. This impulse, if it had acted on the
pendulum at rest, would have caused it to vibrate in the vertical plane
of the impulse and through an arc depending on the magnitude of it.
Now it is found on trial that the distance of the bob of the pendulum
from either of these vertical planes is the same at any instant as if the
other vibration did not exist, so that each vibration subsists inde-
pendently of the other, although the resultant motion is a compound
elliptic vibration.
The two vibrations are here taken in separate planes in order that
their coexistence may be more easily recognised. When the vibrations
are in the same plane the resultant vibration is also in that plane, and
its amplitude, by the principle of superposition, is the sum of the
amplitudes of the constituents when their directions conspire, and
their difference when they are opposed. In this example the phases
of the two vibrations are supposed to be the same. When they differ,
the principle of superposition asserts that the displacements, not the
amplitudes, are additive.
43. To compound two Simple Vibrations.—Let a particle be
simultaneously impressed by the periodic displacements represented by
the equations"
gh = ah sin (wt – au), and ya-assin (wt – ag),
which agree in periodic time (0 = 27/T), but differ in phase and
amplitude. The resultant displacement at any time t is
v= yj + ye=al sin (at – an) +aq sin (wt – ag)
= sin wi(a cos al-Hag cos dº) – cos wt(a, sin a1 + as sin ag)
= A sin (wt – a),
if * A cos a = a, cos al-Hag cos de
and A sin a = a1 sin al-Haq sin a 2.
Therefore by squaring and adding we find
A?= a,” + a.” + 2a1a2 COS (d.1 * dº),
and by division”
_al sin al-F (to sin dº
tan a =
Q1 COS al-H de COS Cl2
This determines the amplitude A and the phase of – a of the resultant
vibration. If we denote the difference of phase of the component
* Fresnel, CEuvres complètes, tom. i. pp. 288-293; Mémoire cowronné sur la diffrac-
tion, $$ 37-42.
* These results may be obtained geometrically as follows: Let the parallelogram
OABC revolve round the vertex O with uniform angular velocity. Then since
A, B, C describe circles round O with a uniform angular velocity, it follows by
Ex. 2, p. 39, that the feet P, Q, R of the perpendiculars from A, B, C on any
ART. 44 . DISTRIBUTION OF THE ENERGY 49
vibrations by 8 we have 8 = a1 – ag, and the resultant amplitude is given
by the equation
A*= a,”-- a.”--2a1a2 cos 6.
Cor. 1.-If 8 = 0 or 2nt—that is, if one wave is retarded on the
other by any number of wave lengths—we have cos 8 = 1, and
Cor. 2.-If 6 = t or (2n + 1)T, one wave is retarded an odd number
of half wave lengths on the other, and
A2= (al º a2)”.
If in addition a = ag, we have A=0, and the waves in this case destroy
each other. -
Cor. 3.-If 6 = }T or (n + k)t, one wave is retarded a quarter wave
on the other, or (n + k)#A and
A2 - al” + a.”.
44. Distribution of the Energy. — In the case of a medium:
simultaneously disturbed by two sources of similar vibrations, we have
seen that the amplitude of the resultant vibration at any point P is
determined by the equation A*= a,” + a.” + 2a1a, cos 8, where 6 is the
phase difference of the vibrations reaching P, and depends on the
difference of its distances from the sources. If this difference is to
remain fixed, then P may lie anywhere on a hyperboloid of revolution,
having the sources for foci. The whole space around the sources may
right line OX execute simple harmonic vibrations along the line OX. But since AC
is equal and parallel to OB, it follows that PR = OQ, and therefore OR = OP+OQ.
Hence the displacement of R at any
instant is equal to the sum of the
displacements of P and Q ; or, in
other words, the motion of R is the
resultant of the motions of P and
Q superposed. Now R executes a
simple harmonic vibration of ampli-
tude OC and phase XOC ; that is,
if the amplitudes of the vibrations
of P and Q be represented by the
lines OA and OB, while the phases
of the same vibrations are repre-
sented by the angles OA and OB Fig. 10.
make with a fixed line OX, then the amplitude of the resultant of the two will be
denoted by OC, the diagonal of the parallelogram on OA and OB, and the phase of
the resultant will be represented by the angle OC makes with OX. The analytical
method in the text is given in order that the student may become armed at once
with a direct and powerful instrument of attack. [Compare Art. 36, Ex. 2. Figure
10 illustrates the cosine formulae. The sine formulae may, however, be deduced
from the figure by supposing the phase measured clock-wise from the vertical OY.]
E

50 VIBRATIONS COMPOUNDED CHAP. II
be divided up by a system of such surfaces, each determined by a value
of 8 and a corresponding value of A. The average energy of the
vibration of the elements of the medium at any one of these surfaces
is measured by A*, which, from surface to surface, varies from a
maximum (a1 + ag)”, for 8 = 2nt, to (a, - a,)”, for 6 = (2n + 1)T. The
average value, however, of A* throughout the whole medium will be
al” + agº,
or the sum of the average energies of the vibrations excited by the
sources acting separately.
Thus when a medium is disturbed simultaneously by two similar
sources O, and O2, the amplitude of the resultant vibration at any point
is the sum of the amplitudes of the vibrations which the sources would
excite if each acted separately only when the vibrations at this point
are in the same phase. The amplitude at any other point will depend
on the difference of phase, so that in some tracts of the medium there
is a large amount of energy, while in others there is very little. The
interfering action of the two sources causes no destruction of energy,
but merely a redistribution of it in the medium around them.
For example, if a1 = az, that is, if the sources be of equal power,
then at some parts of the medium the energy will be proportional to
(2a)”, or quadrupled, while at others it will be zero, but the average
value throughout the medium will be proportional to 2a”, i.e. it will
be the sum of the average energies of the two equal sources.
Example
Any number of vibrations of a given period compound into one of the same
period. For if
v= yi−F V2 + . . . 4-yn=2a1 sin (wt - qI)
= Q sin wt — P cos wt=A sin (wt – c.).
Where P= 2a1 sin a 1, Q=2a1 cos q1,
tan a = 2a1 sin oil = P
2a1 cos al Q’
and A*=2a1* +22a1a2 cos (al-q2)= Pº-H Q”.
If the epoch angles q1 and az, etc., vary irregularly, it has usually been considered
that the chance is that the term involving cos (an – c.2) will be as much positive as
negative, and will disappear from the result, so that the average value of A* will be
Xa”, or
average intensity = sum of component intensities.
But Lord Rayleigh has shown that this conclusion is incorrect. He finds that
the probability of the resultant being greater than any chosen value r is e-” when
each of n separate amplitudes is equal to unity. This result applies to any one
group of vibrations. It is only when we consider the average value to be expected
for a very large number of groups that the resultant is equal to Xa”. This important
result forms the basis of all methods of photometry.
* Phil. Mag., Aug. 1880; also Collected Papers, vol. i. p. 491.
ART. 45 GRAPHIG REPRESENTATIONS 51
45. Graphic Representations of the Resultant of a System of
Vibrations.—From the equation A* = a,” + a,” + 2a1a, cos 8 it appears
that the resultant amplitude A is the diagonal of a parallelogram of
which the sides are a, and a, and 6 the angle between them.
Again the equation for tan a may be written in the form
a sin (a – d.1)+ as sin (q – q2)=0,
from which it follows that if we draw a line making an angle a with
the diagonal A, it will make angles a, and as with the sides a, and as
Jº
Q
Cºe
P
O ſº, “ -x
Fig. 11.-Resultant of two Fig. 12.-Resultant of a System of
similar Vibrations. similar Vibrations.
of the parallelogram. Thus we have the following construction. Take
any fixed line OX (Fig. 11) and draw OP equal to a, and making an
angle a1 with OX, similarly draw OQ equal to ag, and making
QOX = az, then if we complete the parallelogram OPRQ the diagonal
OR is equal to A, and the angle ROX which it makes with OX is
equal to a, while POQ is equal to the difference of phase 8 or al – ag.
Or, since A sin (ot - a) = a, sin (ot – al) + as sin (ot – ag), it follows that,
if we draw a line making an angle of — a with the side a, of this
parallelogram, and therefore making the angle of – as with the side
as, it will make the angle of — a with the diagonal. For as the
right-hand side of the equation is the sum of the projections on this
line of two sides of the parallelogram, therefore the left-hand side is
the projection on it of the diagonal. If, therefore, OX be so drawn
that POX and QOX represent the phases q, and b, of the component
vibrations, ROX will be the phase 4 of the resultant vibration. The
vibrations, therefore, compound like forces: amplitudes and phases
corresponding respectively to magnitudes and directions.
To compound several vibrations by this method we have only to
draw OP, OP, OPs, etc. (Fig. 12), from O, equal to a, a2, as, etc., the
amplitudes of the vibrations, and making angles P.OX, P.OX, etc.,
equal to phases $1, $2, etc.; then, as in the case of a system of forces
(Minchin's Statics), the resultant amplitude is equal to n. OG, if there
are n amplitudes, and if G is the centre of mean position of the points
P1, P2, Ps, etc. *
It may be more instructive, however, to employ the method of the
polygon of forces. Thus draw OP (Fig. 13) equal to ap and making

52 VIBRATIONS COMPOUNDED CHAP. II
Vibration
CUlrVe.
an angle q, with OX, and from P1 draw PAF, - a, and making ºbs
with OX. Then OP, represents the resultant of the vibrations at
and ag. Similarly for a third draw P2Ps = as, and
making an angle be with OX, and so on for any
number. The line joining O to Pn, the extremity
* -- tº ſº of the line last drawn, represents in magnitude
the amplitude, and its direction makes an angle
3 with OX equal to the phase of the resultant vibra-
Fig. 18.—The Vibration bion.
Polygon.
O –Yº
These graphic methods we shall find of great
utility in the theory of diffraction.
Cor. 1.--When the system of vibrations to be compounded forms a
series in which the phase difference of any two consecutive terms is
infinitesimal, but varying continuously from term to term, then the
angle between any two consecutive sides of the vibration polygon
will be infinitesimal. The polygon will consequently become a con-
tinuous curve, as shown in Fig. 14, such that the element of length at
any point M of the curve represents the amplitude of a corresponding
constituent vibration, and the angle which tangent at this point makes
with OX represents the phase 4 of the same vibration. The line
joining O to M represents the resultant of the system of vibrations
between O and M ; the length of the line OM representing its
amplitude, and the angle MOX its phase. If P be a point on
the curve such that the tangent at P is parallel to OX, then the
arc OMP represents a system of vibrations varying continuously
in phase from zero to tr—that is, over half a period—and OP
represents the amplitude, and XOP the phase of the resultant of the
system.
Cor. 2.-If the system of vibrations forms a series of equal ampli-
tude and equicrescent phase, then the sides of the vibration polygon
P P
2-
%
TM TM
O % X O X
Fig. 14. Fig. 15.
will be equal and equally inclined to each other. In other words, the
vibration polygon will be regular and inscribed in a circle. Hence,
when the number of terms is very great and the constant phase differ-
ence very small, the polygon will, in the limit, coincide with its cir-

ART. 46 GRAPHIC REPRESENTATIONS - 53
cumscribing circle, and it follows that the resultant of an infinite Vibration
number of vibrations of equal amplitude, and varying uniformly in circle.
phase, may be represented in amplitude and phase by the chord of a
circular arc. The general curve of Fig. 14 becomes in this case a
circle, as shown in Fig. 15. The arc OM of this circle represents a
system of vibrations of equal amplitude and phase varying uniformly
from zero to b. The chord OM represents the resultant, and the
amplitude is therefore
OM = 2, sin #4,
while the phase of the resultant is
MOX=#q).
That is, the phase of the resultant is the arithmetic mean of the initial
and final constituents of the series, and it is consequently the same as
the phase of the constituent corresponding to the middle point of the
arc OM; for, in the case of a circle, the tangent at the middle point
of the arc is parallel to the chord of the arc.
The diameter OP of the circle represents the resultant of a system
of vibrations of equal amplitude, and phase varying uniformly between
the limits 0 and tr. The phase of this resultant is #T, and it differs
therefore by a quarter period from the first constituent of the series, or
is the same as the middle term of the series. If the amplitude of each
constituent of the series be a, and if there be n terms in the series,
then if the series be represented by any arc OM of a circle, we have
in the limit for the length of the arc
Arc OM = m.a.
Hence, if r be the radius of the circle and q, the difference of phase of
the first and last terms of the series, we have
rq = ma.
Consequently the amplitude of the resultant is
OM = 27" sin *=nº.
If the phase varies from 0° to T, the arc will be a semicircle and the
amplitude of the resultant will be 2r ; that is,
2770,
T
46. Waves of Different Lengths.-We have good reason to
believe that the velocity of propagation in air is very nearly the same
for light waves of every length, just as we know that sound waves of
different lengths travel with the same velocity. Still whether W be
54 VIBRATIONS COMPOUNDED CHAP. II
Elliptic
vibrations.
constant or not it is generally impossible for two vibrations of different
periods such as
th-an sin (wit --an), and ye=as sin (wº-Faº),
to destroy each other completely. We have seen that when to e os = 0,
any number of such waves combine to form a resultant wave A sin(ot - a)
of the same period as the constituents, and containing no trace of
their distinction. The two waves given above cannot, however, be
so compounded unless their periodic times be the same—that is,
unless we have V1/A1 = V2/A, The consideration, therefore, of waves
of different lengths may be kept perfectly separate, for their ultimate
effect will be the superposition of the separate effects, each possessing
its own individuality.
This point is illustrated by sounding two notes of different pitches
together (for example, a note and its fifth); or by the mixture of two
colours.
47. To compound two Rectangular Vibrations.—If a particle be
subjected simultaneously to two simple harmonic vibrations of the
same period, but in perpendicular directions, the resultant vibration is
in general elliptic.
Fig. 16. Fig. 17.
Let the superposed vibrations along OX and OY (Fig. 16) be given
by the equations
a -a sin 9,
y=b sin (0-6)
where 0 for brevity denotes the phase of one of the vibrations and 6
their difference of phase. Now 6 is a variable containing the time,
and to eliminate it we have
y=b (sin 6 cos 64-cos tº sin 6),
and sin 0–aſa,
!/_2: - _*
therefore 5–2 °0s 5-Hsin 6 V 1- ºr
!/ 2. * ...a _*
Hence (! a 90s •) - S111 *(l ...)
* y” 2.xy
or finally at . . cos º-sinº º

Art. 47 ELLIPTIC VIBRATIONS 55
This equation denotes an ellipse having its centre at O, which is con-
sequently the curve described by the particle under the simultaneous
action of two rectangular vibrations of the same periodic time, of
amplitudes a and b, and difference of phase 6.
If 6 be increased by it the equation becomes
* , y” 22, – e...?
*** aſ cos 3–sin” 8.
The greatest values of a and y are a and b respectively, hence if
tangents AE and BF be drawn parallel to the axes of reference, we
have (Fig. 17)
OA = a, and OB-b.
Also by putting a = 0 and y = 0 successively in the equation of the
curve we find
OC– a sin à, and OD=b sin 6.
Hence
sin 5–90 _OD.
OA. OB
Again if we replace a by its maximum value a, we find y = b cos 8 for
the ordinate of the point E.; and if we replace y by its maximum value
b we find a = a cos 3 for the abscissa of the point F, therefore
cos 5 _AE - BF
APTBP"
Cor. 1.-If the difference of phase 3 be zero or 2nt the equation
of the resultant vibration becomes
º
|-}=0.
This represents a right line passing through O (Fig. 18) and
making an angle with the axis of y, the tangent
of which is a/b.
Cor. 2.-If the difference of phase be 7 or
(2n + 1)n, which corresponds to a retardation
A. -- - - -
#A or (2n+1); the resultant vibration is
Figs...18 and 19.-Phase Differ-
ence a Multiple of m.
represented by
* 14
a ti, 0.
This is the equation of a right line passing through O (Fig. 19) and
inclined to the axis of y on the other side at an angle whose tangent
is a/b.
Thus we see that if two simple rectangular vibrations be com-
pounded, the resultant vibration is a simple rectilinear vibration if
they differ in phase by any number of times 27 (or a retardation of


56 VIBRATIONS COMPOUNDED CHAP. II
any number of times A), and if a difference of phase equal to any odd
multiple of T or a retardation of any odd multiple of half a wave
length be introduced, the resultant vibration is still rectilinear, but its
direction is turned through an angle 26, if 6 be the angle the first
vibration makes with either axis.
This remark is of importance in the theory of polarised light.
Cor. 3.-If 6 = }T, or any odd multiple of T, that is, if there is a
retardation of 4A, we have
º y”
ºtº-l.
Fig. 20.-Phase Difference a Right Angle. Fig. 21. – Circular Vibration.
In this case the axes of the elliptic vibration coincide with the
directions of the component vibrations (Fig. 20).
In the first cases the two rectangular vibrations pass through their
middle points at the same instant, and in this case one passes through
its middle point when the other is at its extremity.
If, in addition, a = b, we have
*4-y” – a”
and the resultant vibration is circular (Fig. 21).
Thus two rectangular vibrations of equal amplitude and periodic
time will compound into a circular vibration if they differ in phase by
any odd multiple of T, or if one be retarded on the other by any odd
multiple of 4A.
48. Vibration of Permanent Type—Polarised Light.—It has
been remarked already that the phenomena of interference lead us to
believe that light is propagated as a periodic disturbance or a periodic
change of some sort in the condition of a medium. This periodic
change is referred to as the vibration; but as to the nature of the change,
whether it is simply a periodic displacement of the elements of the
medium, such as occurs in a vibrating elastic solid, or as to whether
it is something of quite a different nature must remain a matter of
speculation. In representing the vibration by an equation of the form
y=a sin (wt -- a)
no particular assumption need be made as to what it is that y
represents, except that it is the change of condition at the time t. It
is perhaps simplest to regard y as a displacement in the medium such

Art. 48 EXAMPLES 57
as occurs in ordinary wave motion. This is the ordinary notion, but
there is no advantage in imposing such a limitation on the nature of the
disturbance, except for the purposes of distinct conception in the mind.
In representing a periodic change by a simple equation of the fore-
going form it should be remembered, however, that this equation
embraces an infinite succession of similar changes; that is, an infinite
train of waves of invariable type. Light is said to be polarised when
the type of the vibration is maintained invariable. If the ether vibration
consists in a periodic displacement, all the elements of the vibrating
ether in the wave front describe similar, and similarly situated, curves
which remain permanently the same. Thus if all the elements describe
similar and similarly situated ellipses for any time the light is said to
be elliptically polarised during that time. If they describe circles it
is circularly polarised, and if they describe parallel rectilinear paths it
is called plane-polarised light. The essential feature of polarised light
is then that all the elements of ether in any wave front continue to
describe exactly the same kind of orbits, or the nature of the disturb-
ance remains permanently the same.
Examples
1. Any number of simple harmonic vibrations in different directions, differing
in phase but having the same periodic time, compound into an elliptic vibration.
2. Any simple harmonic vibration is equivalent to two opposite circular vibra-
tions. -
[For a;-a cos w!, y=a sin wº,
is a circular vibration, viz. a.”--y”–aº, so also, changing the sign
of a we have
w- a cos wº, y = - a sin wº
as a circular vibration in the opposite direction. Adding the
2-components, and also the y-components, we find for the re-
sultant vibration Fig. 22.
a:- 2a cos w!, y=0.
Consequently the two opposite circular vibrations compound into a simple rectilinear
vibration of double the amplitude.
Conversely, if we start with the simple vibration a -a cos wº, we may write it in
the form
(1) are a cos wº, y = }a sin wº.
(2) are a cos w!, y= – a sin wº.
The first pair represent a circular vibration of amplitude a, and the second pair
represent an opposite circular vibration of amplitude 'a.
This result is of importance in the theory of circularly polarised light.
An inspection of the figure shows at once that a particle moving with simple
harmonic motion along a line OY may be supposed actuated by two equal and


58 WIBRATIONS COMPOUNDED CHAP. II
opposite circular vibrations of half the amplitude, for the velocities perpendicular to
this line destroy each other, while the velocity along it is doubled.]
3. Two simple harmonic vibrations, in rectangular directions, compound into a
parabolic vibration if the periodic time and phase of one is double that of the other.
[The equations of the component vibrations are !
a:= a cos 2(at + C)= a cos 26,
gy=b cos (wt + a)=b cos 6,
therefore eliminating 6, we find
2y” a.
52 Tº + 1.
The above vibration is that obtained by compounding a note and its octave. A
further exercise is that of compounding a note and its fifth (periods in the ratio 2:3),
or any note and another at an octave-H fifth interval from it (periods in ratio 1 : 3).
Diagrams of these cases will be found in treatises on sound under the head “Lis-
Sajous’ figures.”]
4. If a system of vibrations differing in amplitude and periodic time be super-
posed, we have for resultant
* t § t;
y=al sin 2tr (#42)+. sin 2T (#r.) + . . . etc.,
Ti T2
which may be written in the form
*
v= a sin *(***)+a, sin *(***) + . . .
where T = m171=m2T2= etc., m1, m2, etc., being integers. This shows that y is
periodic, for it remains unaltered when for t we substitute t +T. The periodic
time of the resultant vibration is therefore T, the L.C.M. of T1, T2, Ts, etc.
5. The components of an elliptic vibration are
2 = a sin (ot, and y=b sin (wt +6),
find the direction of the axes of the ellipse.
[The directions are given by tan 20=; *. cos 5, where p is the angle made with
the axis of ac.]
6. Two elliptic vibrations, given by the equations
a 1–a1 sin (wt + qi) 22 = a, sin (wt + q2)
Ji = b, sin (wt + 31) Q/2=b2 sin (wt +62) ſ”
are superposed, find the resultant vibration.
[If the components of the resultant vibration be
ac= a sin (wt + a) and y = b sin (wt +8),
we have
a”=a^+a++2a1a2 cos (q1 – ag),
b°=b,”+b,” +2bib, cos (81–82),
a sin a 1+a, sin as tan g=} sin 31 + ba sin 82
tan = tº
a cos an + as cos az’ bicos 314-be cos 32
7. The elliptic vibration
ac= a sin wt, y=b cos wº
is equivalent to the two superposed circular vibrations
a'i = }(a + b) sin wi a 2–3 (a – b) sin wº
g/l = }(a + b) cos wº g/2= – #(a – b) cos wi ſ'
ART. 48 EXAMPLES 59
consequently the elliptic vibration may be regarded as the resultant of two oppositely
directed circular vibrations of amplitudes, #(a +b) and #(a – b) respectively.
8. System of Discrete Particles—Minor Limit to the Periodic Time.—A system of
pellets, each of mass m, is attached to a weightless string at equal intervals a. The
string is stretched with tension T along a smooth horizontal plane, so that the force
of gravity does not enter the question of the horizontal transverse disturbances.
Discuss the motion (P. G. Tait, Ency. Brit., art. “Waves”).
Let the transverse displacement of the nth pellet be y, at the time t. The
equation of its motion is
*/n TVnti-yn tºn-ºn- 1
*::::: wº-mºn T Q, T 0, ſº
Again, if yn = A cos (at – ka!) where a =na, we have, from the above equation,
– maſºcos (wt —ka.) =#eos {wt – k(a + a)} - 2cos (wt — ka:)--cos {ot — k(ac – a)}]
- - * cos (wt — kaº) (1 – cos ka).
Hence - maj”= º sin”;ka.
The greatest possible value of w is consequently 2N/TPma; that is, we must have
w-29/a, if v be the velocity of a disturbance in a cord under the same tension and
of the same mass per unit length. The time of oscillation of a pellet is 2T/w and is
>Traſv or T Nam/T. -
This question is closely connected with Stokes's theory of fluorescence. For if a
disturbing force of shorter period than the limit given above be applied continuously
to one of the pellets, there will be an accumulation of energy in its neighbourhood ;
and this energy, if we suppose the disturbing force to cease, will be transmitted
throughout the system by vibrations of equal or greater period than the limiting
value above, corresponding to light of lower refrangibility than the incident, but
having a definite superior limit of refrangibility.
9. On the Group Velocity of a Train of Waves.—If a disturbance be excited at
any place in a sheet of water, as by throwing a stone into a pond, the disturbance
will be propagated into the still water round and it will be seen that the surface
comes to contain regions of disturbance separated from one another by regions of
comparative freedom from disturbance. If any region of disturbance or group of
waves be observed, it will be seen in this case that individual waves advance through
the group, dying away as they approach its anterior limit; the group advances also,
but not as rapidly as the individual waves. It is obvious that we cannot represent
such a wave-system by a single harmonic term, y=a cos (wt – Kaº), because this
expression implies constancy of amplitude for all values (including negative ones),
of a and t. Sir G. G. Stokes first gave an analytic expression for this case by
regarding the group as formed by the Superposition of two infinite trains of waves,
of equal amplitude, and of nearly equal wave lengths and frequencies, advancing in
the same direction, and the same explanation has been developed by Lord Rayleigh.
In fact, if two such trains be superimposed, there will be at any instant places,
separated by the distance which contains one more complete wave of one train than of
d d
C. r. - Cº.
.* If y, = f(a), then yn H = f(a + a)=e da:f(a), consequently we have yº-i-e *y,
d
- 0– 2 Cº- - Cºl.
and ya–1=e *y. Hence mº; =}( e dº – 2+ e *). which reduces to Tº
when a is very small, so that we recover the equation of vibration of a continuous
cord.
60 s WIBRATIONS COMPOUNDED CHAP. II
the other, where the phases of the waves agree and where therefore there will be
maximum disturbance. The centres of these intervals will be places where one
train has gained half a wave length on the other train, and therefore will be places
of disagreement of phase or of minimum disturbance. As time changes, the separate
trains advance, and, if their velocities be equal, two waves agreeing in phase will
remain in phase. The place of maximum disturbance, and therefore the groups will
advance with the same velocity as that of the separate trains. But, if the train of
shorter wave length advance more rapidly than that of longer wave length, the
waves in front of those agreeing in phase will be coming to be more nearly in phase,
and the group will advance more rapidly than the separate trains; while, if the train
of shorter wave length advance less rapidly than the other, the waves behind those
agreeing in phase will be coming to agree more nearly and the group will advance
less rapidly than the separate trains.
If two infinite trains of waves be represented by cos (wt — kaº) and cos (w't — k'ac)
where k = 27/A and k' = 27/X', their resultant is
cos (wt — kaº) + cos (w't — k'a)
= 2 COS (e. - •º-e) COS (ºrºgºrºerº) -
Now, if k' – k and w' - a be very small, we have a train of waves whose amplitude
varies slowly from one point to another between the limits 0 and 2, forming a series
of groups separated from one another by regions comparatively free from disturbance.
At any time t the position of the middle of that group which was initially at the
origin is given by
(w' – wyt – (k' – k)a, =0.
Hence the velocity of the group is
U =
QC (a)" – (1)
t
or in the limit when the number of waves in each group is infinite, the relation
between the group velocity U and the wave velocity W is
. . /1
__*(i)
– i.-7TN,
a(ſ)
remembering that was 2T/T and that k=2T/\, or, as it may also be written,
U_1 , d log V_, d log V * dV
#=1+ d log k = 1 – &I. W. * U=V-X.
Thus if V oc An
U=(1 — m)W.
Cases—
W cc X, U=0, Reynolds's disconnected pendulums
V ce \}, U =#V, Deep-water gravity waves
W oc A", U =W, Aërial waves, etc.
W ca X - ?, U =#V, Capillary water waves
W oc X - *, U = 2 W, Flexural waves
The theory of capillary water waves (U=#W) has been given by Thomson (Phil.
Mag., November 1871). Their wave length is so small that the force of restitution
due to capillarity largely exceeds that due to gravity. The flexural waves (U=2W) are
those corresponding to the bending of an elastic rod or plate (Rayleigh, Theory of
Sownd, § 191). º
ART. 48 EXAMPLES 61
Professor Osborne Reynolds (Nature, 23rd August 1877; also Brit. Assoc., Ply-
mouth) showed dynamically that a group of deep-water waves will advance with
only half the rapidity of the individual waves. In this case the energy propagated
across any point, when a train of waves is passing, is only one-half of the energy
necessary to supply the waves which pass in the same time, so that if the train of
waves be limited it is impossible that its front can be propagated with the full
velocity of the waves, as this would imply the acquisition of more energy than can
in fact be supplied. Professor Reynolds did not contemplate the cases where more
energy is propagated than corresponds to the waves passing in the same time, but
his argument applied conversely to the results already given shows that such cases
must exist. The ratio of the energy propagated to that of the passing waves is U/W.
Thus the energy propagated per unit time is U/W of that existing in a length W, or U
times that existing in a unit length. Accordingly

Energy propagated per unit time d(a) d(1/T)
Energy (average) per unit length dk T d(1/\)
[Lord Rayleigh, Note on “Progressive Waves” (Proceedings of the London
Mathematical Society, vol. ix. No. 125; also Theory of Sound.)]
CHAPTER III
ON THE APPROXIMATE RECTILINEAR PROPAGATION OF LIGHT
49. The Principle of Huygens — Secondary Waves — Wave
Envelopes.—The first function of any theory of light is to account
satisfactorily for the so-called rectilinear propagation of light in a
homogeneous medium. The commonest observations on the shadows
cast by opaque objects or on the beams of light transmitted through
apertures show roughly that light is propagated in right lines or rays,
and, as we have already mentioned, this apparent rectilinear propaga-
tion would seem at first sight to argue strongly in favour of the
corpuscular theory. When the facts are more closely scrutinised,
however, it is found that the rectilinear propagation is only approxi-
mate, for when the source of light is small, such as a narrow slit, then
the shadow of an opaque obstacle (a wire, for example) is not so wide
as the obstacle, for the light bends round the edges of the obstacle
into the geometrical shadow, just as sound bends round corners, only
in a much smaller degree.
Hence any proposed theory must account, not for an accurate
rectilinear propagation, but for an approximate rectilinear propagation
in which there is a slight bending round corners. The apparent
superiority of the emission theory consequently breaks down, and in
the explanation of this and other phenomena connected with the
passage of light through narrow apertures, and past the edges of
opaque obstacles, the wave theory obtains a decided advantage.
It is well known that large obstacles cast more or less distinct
sound shadows, though in such cases there is considerable bending round
corners, but the question at once arises as to how far the possibility of
observing this bending depends on the wave length. If the wave length
is very small will the amount of observable bending be also very small?
It will be seen from the following considerations that this question must
be answered in the affirmative, and that the approximate rectilinear pro-
pagation of light is a consequence of the minuteness of the wave length.
62
Art. 49 HUYGENS'S PRINCIPLE 63
According to the wave theory each point of a luminous body is a
centre of disturbance, or a source from which waves are propagated in
the ether. It is therefore necessary to form some conception of the
manner in which a wave is propagated.
Let O (Fig. 23) be a centre of disturbance, and let ab be the front
of a spherical wave diverging from it. The radius of the wave
increases with the velocity of light, so that
the disturbance now at ab will an instant
later be at a'b', and the single wave ab in
travelling out will disturb all the elements of
the medium over which it passes. Thus the
disturbance of any one element of the medium
causes a subsequent disturbance of all the
other elements, and we may regard the dis-
placements of the elements of the wave Fig. 23.-Propagation by
front ab as the cause of the subsequent Secondary Waves.
displacement of the elements of ab'. With M, M', etc., as centres,
describe a series of equal small spheres' to represent the wavelets
developed by the centres of disturbance M, M', etc. All these
small spheres touch a sphere * dº having its centre at O, and in
this manner we get a new wave front ab'. In this manner the wave
ab is propagated to a'b' by the small secondary waves arising from the
front of the original wave ab. The envelope of these secondary waves
is the grand wave, and from this point of view of wave envelopes
Huygens regarded a wave as being propagated. The effective part of
each secondary wave in generating the primary wave he supposed con-
fined to that portion of it which touches the envelope.
The energy of ab is thus handed on to a'b', and in the same
manner from a'b' to a "b", etc. Consequently each point of the wave
front in any of its anterior positions may be considered as a centre of
disturbance from which a secondary wave diverges, and the aggregate
effect of these secondary waves at any point the same as that of the
primary wave itself, when passing through that point.
Or, more generally, if round the origin O any ideal closed surface
be described, the whole action at any point outside this surface, of the
waves diverging from O, may be regarded as due to the motion pro-
pagated across the various elements of the surface; that is, each
element of the surface, as a centre of disturbance, sends waves to the
* This assumes the medium to be isotropic.
* The envelope is really a surface of two sheets, one sheet lying behind ab and
theather in front of ab. One of the difficulties of this presentation of the theory
is tºxplain why light is not propagated backwards as well as forwards.

64 RECTILINEAR PROPAGATION CHAP. III
point under consideration, and the disturbance there is the resultant
of their combined action. The general problem, therefore, is to deter-
mine the nature of the disturbance at any point when the full
particulars of the displacement at each point of any surface surrounding
the origin. O are known. It is convenient, however, to choose the wave
front as the surface of resolution, for the vibrations at it at any instant
are all in the same phase. -
Thus if ab be the front of a wave diverging from O, then all points
on the surface of ab are in the same phase of motion at any instant,
and, as time progresses, each point passes through all the phases of the
vibration originated by O. Each point of ab must, therefore, be
regarded as the origin of an infinite train of waves, and in calculating
the displacement at any point outside ab, we must consider all the
waves which reach that point simultaneously from the various points
of ab. These waves will have left the various points of ab at different
times previously, and will, therefore, be in different phases when they
reach the point. The general problem, therefore, is to find the
resultant of a system of vibrations of different amplitudes and phases,
determined by the distances of the various points of ab from the point
in question.
In order to make this clearer let AB (Fig. 25) represent the trace of
a plane parallel to the front of a system of plane waves travelling
towards O. At any instant the various points M1, M2, Ms, etc., of
AB are in the same phase of motion, and this phase changes periodic-
ally as the time progresses. Each point of AB, regarded as a centre
of disturbance, is the origin of an infinite train of waves (which diverge
as spheres around it if the medium be isotropic), so that at any
instant the displacement at O is the resultant of an infinite series of
disturbances propagated from the various points of AB. The phases
of these disturbances are determined by the distances of O from the
various points of AB. Thus, if W be velocity of propagation, the waves
reaching O at any instant are—the wave which left P at the time OP/V.
previously, the wave which left M, at the time OM/V previously, the
wave which left M, at the time OM,ſW previously, etc., etc., so that if
the variation of phase with time be known, then the phases of all the
disturbances reaching O at any instant are known. Their amplitudes
can also be calculated, and so the resultant effect at O can be
estimated," if they depend upon the inclination to the wave front of
Inter-
ference.
* Huygens endeavoured to explain the rectilinear propagation of light by the
principle of wave envelopes alone. Thus if ab (Fig. 23) be an aperture, and Oab a
cone of light falling on it from O, he assumed that the effect of each element of ab
in generating a'b' was confined to the apex of the secondary wave or that part of it
ART. 49 HUYGENS’S PRINCIPLE 65
the line joining O to the corresponding centre of disturbance, and if
the law of this variation be known.
The question now arises as to what is to limit the secondary
waves diverging from each point of the primary wave as origin. In
an isotropic medium the disturbance of each little wave spreads
generally in a spherical form. Are we to suppose the sphere com-
pleted so that each small wavelet is propagated backwards as well as
forwards
We know that a single wave may be transmitted along a stretched
cord, or through the air. In this case the agitation of any point
causes the future agitation of those in advance of it, but of none in
the direction from which it has been propagated. Thus in Fig. 7 the
wave travelling along the cord AB disturbs in turn all those parts of
the cord in advance of it, but has no effect on those behind it. For
this reason it has been supposed by some that each point of the
primary wave sends forward only a hemispherical wave, viz. that half
of the secondary spherical wave which lies in front of the primary
wave. Further, if the disturbance is zero at the base of this hemi-
sphere, it is natural to suppose that the intensity of the disturbance
diminishes gradually from the vertex towards the base. This would
be the case, for example, if the intensity in any direction varied as the
cosine of the obliquity, or inclination to the wave normal.
The law which determines the intensity at each point of a
secondary wave has been investigated by Sir G. G. Stokes." He has Stokes's
shown from the equations of motion of an elastic medium that the *
effect of an elementary wave at an external point varies as (1 + cos ?)
where 6 is the obliquity, or angle between the wave normal and the
line joining the point to the centre of the elementary wave. This
factor will vanish only when 6 = T, that is, for points directly behind the
wave. According to this law the secondary wavelet must be regarded
as a complete sphere, the disturbance varying gradually from a maxi-
mum at its forward apex to zero at the diametrically opposite point in
its rear. The effect produced at any point by a wave element depends
also on the direction of vibration in the element. Sir G. G. Stokes
found it proportional to the sine of the angle between this direction
which touches the envelope. This practically assumed the whole question. Fresnel
(CEuvres, tom. i. p. 174) took into account the secondary waves diverging from
the various points of ab, and showed that these by mutual interference may
neutralise each other in some regions of space, and hence produce darkness and
shadow. te
* On the Dynamical Theory of Diffraction, Math, and Phys. Papers, vol. ii. p.
243; Camb. Phil. Trans. vol. ix. p. 1, 1849. Stokes showed that any function of
1 + cos 0 would satisfy the conditions equally well.
F
66 RECTILINEAR PROPAGATION CHAP. III
and the radius vector to the point, and also that it was necessary to
accelerate the phase of the vibration by a quarter of a period in passing
from the primary to the secondary waves.
It will be sufficient, however, for our present purpose to admit
that the effect produced at any point by a secondary wave depends
upon the obliquity, and diminishes as the obliquity increases.
50. Definition of Poles and Half-period Elements.-The pole
of a wave with respect to any external point O is that point of the
wave which is nearest to 0, or more accurately, that point of the wave
from which the disturbance reaches O in the least time. If the wave
is plane the pole of O is the foot of the perpendicular drawn from it
to the wave front, and if the wave is spherical, the pole is that point
where the wave is intersected by the line joining O to the centre of
the sphere.
In general, if the wave be neither plane nor spherical, we define the
pole (or poles) of a wave as the point (or
points, or continuous locus of points) on the
wave front, from which light is propagated
to O in either a minimum or a maximum
time.
Let P (Fig. 24) be the pole of a wave
with respect to the point O, and denote
OP by b. The disturbance from P reaches 0
sooner than the disturbance from any other
point of the wave. With 0 as centre and a
radius b + . A describe a curve on the wave
front. This curve is the intersection of the
wave front with a sphere of radius b + . A
and centre 0, and will include a small element of area on the wave
front around P which we shall call the first half-period element or zone.
With 0 as centre and a radius b + A describe another sphere meet-
ing the wave front in a second curve which with the first curve will
include a little annulus or strip or area on the wave front. This area
we term the second half-period element. Similarly by describing other
Fig. 24.—Half-period Elements.
-- M -
spheres of radii b + 3. b -- 4. etc., we obtain curves on the wave front
including the 3rd, 4th, etc., half-period elements, the nth half-period
element being intercepted between the spheres of radii b + (n-1)}A
and b + nº A. A half-period element with respect to any point O is
then a narrow strip of the wave front surrounding the pole P of that
point, and such that the difference of the distances of its inner and
outer edges from O is half a wave length.

ART. 5.1 HALF-PERIOD ELEMENTS 67
51. Comparison of two Consecutive Half-period Elements.-Let
us now consider the mutual effect of two consecutive half-period
elements, M.M., and M.M., at the point O (Fig. 24). For this purpose
let the annulus. MIM, be divided into a series of concentric circular
rings of infinitesimal area. This may be supposed to be done by a
system of spheres described round O as centre with radii r, r + 6,
r − 26, r + 36, etc., where 6 is a very small quantity (a submultiple of
|A) and r is the distance of the inner edge of MM, from O. Then,
if MºMº be divided up in the same manner, the first ring of M.M., will
be half a wave length nearer O than the first ring of M,Mg, and being
consequently opposite in phase they will neutralise each other at 0 if
the amplitudes of the vibrations which they transmit are equal. In Corre-
the same way the second ring of M,M, will be opposite in phase to the ""
second ring of M,Mg, the third to the third, etc., so that the consecu- rings.
tive rings of one half-period element tend to neutralise the corre-
sponding rings of the other, and if the amplitudes of the disturbances
due to them are equal they will destroy each other completely.
We have now to examine how far equality in this respect is
realised and on what quantities approximate equality depends. For
this purpose it is necessary to remember that the elementary rings
into which the half-period elements have been divided are such that
they are at uniformly increasing distances r, r + 6, r + 26, etc., from O,
and they consequently send vibrations of
uniformly increasing phase to O. Rings
constructed in this manner will not be
rigorously equal in area, and the ampli-
tudes of the vibrations they transmit
to O may differ accordingly. In order
to find an expression for the area of
any such ring let XX (Fig. 25)
be the width of the annulus, and let
OX' – OX = 3 where 6 is a very small
quantity. Then if a be the mean radius
of the annulus its area is 2TaxX'; but
by similar triangles XX : 6: ; r. 2, where Fig. 25.
r is the mean distance of the strip from O, hence a XX* = rô, and
the expression for the area becomes
2Trö.
This expression shows us that if the rings are so constructed that
6 remains constant, then their areas increase as they recede from the
centre; but the increase of amplitude arising from this increase of
area is exactly counterbalanced by the diminution of amplitude arising


68 RECTILINEAR PROPAGATION CHAP. III
Effect of
obliquity.
from increase of distance from O, if we take as a first assumption that
the amplitude of the vibration transmitted to O by any ring is propor-
tional to the area of the ring and inversely as its mean distance from O
(the energy varying inversely as the square of r and directly as the
square of the amplitude). Since the area of the annulus is 2Trð, this
assumption leads to the conclusion that the amplitude of the vibration
due to any annulus is proportional to
2trö,
and therefore the same for all the rings. The corresponding rings of
two consecutive half-period elements would thus produce equal and
opposite effects at O and would appear to neutralise each other com-
pletely, but as yet we have not introduced any consideration as to how
far the amplitude of the vibration due to any element depends on the
obliquity—that is, upon the inclination of the wave normal to the line
joining O to the element in question. At present we shall merely
assume that the effect of obliquity is to diminish the amplitude so that
the effect produced at O by any element diminishes as the radius of
the ring increases. The result of this assumption is that the element-
ary rings of the half-period element M1M, are more effective at O than
the corresponding rings of M2Ms, and instead of complete neutralisation
we have an outstanding difference for every consecutive pair of zones,
and the resultant of these outstanding differences is the integral effect
of the wave.
It is interesting to notice the effect of this diminution of amplitude
on the resultant of a single half-period element. If such an element
be divided, as above, into a great number of concentric rings of
equicrescent phase, and if there were no diminution of amplitude
arising from increase of obliquity, then, when represented graphically,
the resultant of the whole zone would be represented by the diameter
OP of a circle OMP, as shown in Fig. 15, p. 52, the phase of the
resultant being 90° behind that of the vibration coming from the inner
edge of the zone. The effect of diminution of amplitude, however, is
such that in constructing the curve OMP the elements of length
diminish as we proceed along it by revolving the tangent through
equal increments of angle. This leads to an increase in the curvature
of the amplitude curve corresponding to any given portion of the zone,
and, as a consequence, the phase XOP (Fig. 14) of the resultant is less
than 90° behind that of the inner edge of the zone. The phase of the
resultant of a single half-period element may thus be less than that of
the central element of the zone and may correspond to a ring of the
zone situated between the central ring and the inner edge.
Before concluding the comparison of two consecutive half-period
ART. 52 PLANE WAVE º 69
elements it will be useful to examine the outstanding difference
arising from obliquity in so far as it depends on the wave length. We
have found that the amplitude of the vibration due to any elementary
ring is proportional to 218, when the obliquity is neglected, and if we
denote the obliquity by 6 and the distance OP by p we have cos 0 = p/r,
and consequently any function of 6 may be expressed in terms of r and
the constant p. Hence, if the law of variation of amplitude with
obliquity be known, it may be expressed as a function of r, and the
amplitude of the vibration due to any elementary ring may be written
in the form
A = af(r).
The amplitude of the vibration due to the corresponding ring of the
consecutive zone will therefore be
A’= af(r-i-3A) = af(r) + #axf'(r) + etc.
As these are opposite in phase their joint effect will be
A’ – A =#axf'(r) + etc.,
so that if A is small the outstanding difference will be a small quantity Any law.
proportional to A, viz.—
#axf'(r).
Hence when A is small, the resultant of two consecutive zones is small
compared with either.
For example, if we assume that the amplitude varies simply as the
cosine of the obliquity, we have f(r) = p/r and f'(r)= -p/rº, so that
A — A' =#ap\/r”. Cosime law.
Thus with this law of variation (or with Stokes's law, p. 65) the
outstanding difference varies directly as A and inversely as the square
of the distance ; in other words, as the zones recede from P their
effects become smaller and more nearly equal to each other.
52. Plane Wave.—In order to estimate the whole effect of a
plane wave at any point O we divide it, as in the foregoing article,
into a system of half-period elements with respect to O. Let m,
denote the resultant effect at O of the first half-period element, m,
that of the second, and so on. Then since the corresponding rings of
two consecutive half-period elements are opposite in phase at O, the
whole effect of the plane wave is represented by the series
S = m1 – ma-H ma – ma -- etc.
The only definite knowledge which we have as yet deduced concerning
this series is that the consecutive terms are very nearly equal and
gradually grow less and less in absolute value as they recede from the
70 RECTILINEAR PROPAGATION CHAP. III
Maximum
er l'Ol'.
beginning of the series, and setting out with this information, it may
be easily shown that its sum S approximates in the limit to the value
#m. For since the successive terms decrease continuously according
to some law, from the beginning to the end of the series, it follows
that if ordinates Om, etc. (Fig. 26), be erected at equal distances along
a right line OX, so that the lengths of the ordinates represent the
terms m1, m2, ms, etc., of the series S, then a smooth curve may be
drawn through the extremities of these ordinates which will be either
concave or convex towards the axis OX, according to the law of varia-
. Iſld tion of the constituents
of the series. Now ob-
viously mia = m1 – me,
3723 and mgö = ma – may
b Tºº and consequently S
is the sum of the
intercepts mia, mºb, .
etc., and this sum in-
cludes the alternate
steps only of the stairs leading down from m, to X. But the sum of
all the steps (mi = m.) + (m, -ms) + etc., is obviously m, hence the
Sum of the alternate steps must in the limit be half of the whole—
that is

f 022
Ol,
O - X
S=#m.1.
In deducing this result the only condition that has been introduced is
that the terms gradually diminish to zero, so that the curve gradually
approaches, and ultimately touches the axis OX.”
The same reasoning proves that the sum of all the terms after the
* The series may be written
7ml 7713
s=*[*-m- 7713
5 |+|-gº-ma +º] + the last term,
being #nn, or ºn, 1-mm, according as n is odd or even. Each of the bracketed
terms is small if the values of m diminish slowly ; but since their number is large
we would not be justified for that reason in neglecting them. But assuming, for
example, that the law of variation is such that the effect of each zone is smaller
than the arithmetic mean of the effects of the preceding and following zones, all the
terms in brackets are positive, and therefore
7771 7???,
S > Tº +.
Again, writing the series as .
777, 777 777,
S = m1 – *- *-m-t"; – etc.,
it is shown that, under the same conditions,
711 - Tºn
S < + 2 o
Hence the bracketed terms must be negligible. (Schuster, Theory of Optics.)
ART. 52 PLANE WAWE 71
n” approximates to #mn, and consequently the error introduced by
neglecting all the terms after the m” does not exceed $7mn,
Now, if the wave length be very small, a small area around P will
contain a large number of half-period elements, and the resultant
effect of this portion of the wave will be approximately the same as
that of the complete wave, the error being less than half the value of
the last half-period element, which, when n is large, becomes vanish-
ingly small. We conclude, therefore, that when A is small, the
effective portion of the wave is confined to a small area around the pole
of the wave. If this area be intercepted by an opaque obstacle the
remainder of the wave will have no appreciable effect at O; that is, a
small obstacle at P will screen O almost entirely from the wave, and
this is what we mean when we say that light is propagated in right
lines.
The approximate rectilinear propagation of light is therefore a
consequence of the extreme shortness of the wave length, and is
explained by the principle of interference combined with Huygens's
supposition concerning secondary waves. When the wave length is Bending.
large, as in the case of sound, the bending round corners becomes very
noticeable, but in this case also fairly distinct shadows are cast, and
screening occurs, when the obstacle is large compared with the wave
length. The only difference is that when the wave length is small
the intensity falls off much more rapidly as we recede within the
geometrical shadow, and as the limits of observation are determined
by the intensity, the extent to which bending is observable is enor-
mously less in the case of light than in that of sound.
The foregoing results, deduced from the series S, may be also Graphic
derived in a very convenient and instructive manner by aid of the method.
elegant graphic method of Art. 45. Thus, if we construct the ampli-
tude curve for the first half-period element, it will be represented, as
we have already seen, by an arc Oa'a (Fig. 27), which is very nearly
a semicircle, and in the same way if the construction be extended the
second half-period element will be represented by an arc ab'b, which is
also nearly a semicircle. The resultant of the first half-period is
m1 = Oa,
and the resultant of the second is
- n2=ab,
while the resultant effect of the pair taken together is
- m1 – ma=Ob,
the difference Ob being small compared with either m, or mg. Similarly,
72 RECTILINEAR PROPAGATION CHAP. III
Effect of
screening.
if the construction be extended to the other half-period elements, so
as to embrace the whole wave front, the third half-period will give
rise to the arc bec, the fourth to cd'd, etc., etc. Hence the complete
curve representing the whole wave is a spiral of an infinite number of
Fig. 27. —Vibration Spiral.
nearly circular convolutions of ever-decreasing radius surrounding a
point J on the line Oa, very approximately half-way between 0 and a.
Since OJ represents the amplitude of the vibration excited by the
whole wave, and since Oa = m, it follows that in the limit we have
S=\ma.
So also the figure informs us (as does also the series S) that the effect
of the first two, or any even number of half-period elements, is less
than the effect of the whole wave, while the effect of any odd number
of elements is greater than the effect of the whole wave. In either
case, as the number of elements is increased, the deficit or excess of
effect over OJ gradually diminishes; that is, the lengths Oc, Od, etc.,
become more nearly equal to O.J.
We are consequently led to the conclusion that if the whole wave
is screened off, except the first half-period element, the intensity at
the point O under consideration will be about four times as great as
that produced by the whole wave, whereas, if the aperture be increased
so as to transmit two half-period elements, the intensity at O will be
reduced almost to zero. By increasing the aperture so as to transmit
three half-periods the intensity again rises to a maximum, and by
further increasing it so as to transmit four elements the intensity falls
to a minimum. By still further increasing the aperture the intensity
at 0 passes through a succession of maxima and minima, but these
become less and less pronounced as the number of elements is increased,
so that after the aperture reaches a certain limit further increase will
produce no noticeable effect in the illumination at O. Now a large

Art. 53 SPHERICAL WAVE 73
number of these elements is included in a small area around P, the
pole of O, and we consequently conclude that the effective portion
of the wave is restricted to a small area around P in so far as the
illumination at O is concerned.
On the other hand, if we consider the effect of placing a small
screen, instead of an aperture, at the pole of the wave, we see at once
that when the screen just covers the first zone, the amplitude of the
vibration at O will be represented by a J, for the part Oa'a of the
amplitude curve is cut off while the remainder of the spiral remains
effective. Similarly, if the screen covers two elements, the intensity
at O will be represented by the square of bj, and so on. We con-
clude, therefore, that as the screen increases in size the intensity at
O gradually diminishes to zero without passing through maxima or
minima, and that when the screen is large enough to cover a con-
siderable number of zones the illumination at O falls below the limits
of observation.
These theoretical deductions are fully confirmed by the results
obtained by experiment, as will be seen later on, and it may be well to
remark that there is no complementary relation, such as might at first
sight be imagined, connecting the intensity at 0 when part of the
wave is transmitted through an aperture with that obtained when the
same part is obstructed by a screen, and the remainder transmitted;
for, in the former case, the illumination passes through maxima and
minima as the size of the aperture is varied, whereas, in the latter case,
no such maxima or minima occur.
53. Spherical Wave.—In the same manner we may calculate the
effect of a spherical wave at an external point. Let C (Fig. 28) be
the centre of the wave, P the pole of 0,
and let OM'-OM = 8, where 8 is a very
small quantity. -
The area of the annulus intercepted
on the surface of the wave by two spheres
described round O with radii OM and
OM' is
27ta sin 9. MM'
Fig. 28.-Spherical Wave.
where 6 denotes the angle OCM. For since MM is very small the
annulus is a circle of mean radius a sin 6, viz. the perpendicular from
M on OC, a being the radius CM of the wave.
Now if N be the foot of the perpendicular from M on OM,
NM’ NM’
MM-sº OM'C, , . OM MM-OM sin OM'C–0C sin OCM',

74 RECTILINEAR PROPAGATION CHAP. III
or since the angle MCM’ is very small and NM' = 8, we have
- rö= (a+b) MM’ sin 6,”
consequently the area of the annulus is
(l,
2T) *i;
For a plane wave a = a + b = co, and the area becomes as before
2T1:8. -
Hence if we neglect the variation of amplitude arising from
obliquity and take it to vary directly as the area of the strip and
inversely as the distance r, we find that the amplitude produced at O
by any annulus, such that the difference of the distances of its inner
and outer edges from O is 8, is simply proportional to *
Q,
2rózi,
and consequently all rings for which 6 is the same produce effects of
equal magnitude at O. We find, therefore, as before that when the
effect of obliquity is neglected the consecutive half-period elements
destroy each other at O, but when the obliquity is taken into account
the effects of the various zones at O gradually diminish as they recede
from the pole P, and the whole effect is the sum of the series
S = m1 — me + m3 – mi-H etc.,
in which the terms gradually diminish from left to right. From this
stage all the reasoning of the foregoing article applies together with the
graphic construction, and we conclude that the whole effect is approxi-
mately the same as half that of the first half-period element, or
54. Wave of any Form.—When plane or spherical waves are
reflected or refracted at curved surfaces the wave front in general
becomes of a more complicated character; hence, in order to complete
the problem of the rectilinear propagation of light in isotropic media,
it is necessary to consider the case in which the wave front is a surface
of any form. The first point to be remarked about such a surface is
that it may present several poles with respect to any point O, for it
may be such that the radius vector drawn from 0 to a variable point
on the surface passes through several maxima and minima as the point
traverses the surface. These poles are the points in which the surface
* Or thus: r"=a^+ (a + b)” – 2a(a +b) cos 0, . . rar= a(a + b) sin 6d6, but here
dr = 6 and MM’=ad6, therefore at once rô= (a+b) MM’ sin 6.
* Or if p is the phase difference, the amplitude is proportional to Apa/(a + b).
For a given phase difference the intensity is proportional to X*.
Art. 54 WAVE OF ANY FORM 75
is touched by a sphere of variable radius described round O, and they
may be isolated from each other, or they may in some cases be con-
secutive points on the surface and form a continuous locus or curve of Pole locus.
poles.
Now since the radius vector from 0 to any pole P (Fig. 29) is
either a maximum or a minimum, it follows that the line OP is a
normal to the surface at P. For this
reason the element of surface in the
immediate neighbourhood of P will have
sensibly the same effect at 0 as the cor-
responding element around P taken either
on the tangent plane or on the sphere of
closest contact with the surface at P.
Hence when the wave length is very
small, so that a considerable number of
half-period elements are contained in a small area around P, this
portion of the wave will produce an effect at O which will not be
sensibly increased by increasing the magnitude of the area in question,
or if this portion of the surface be intercepted by a screen, the darkness
at 0 will not be sensibly increased by increasing the size of this screen.
As we recede from P the obliquity increases, and as before we
can see from general considerations that two consecutive half-period
elements at a distance from P very approximately neutralise each
other. For if we consider two consecutive half-period elements
intercepted on the surface by spheres described round O with radii
r – A, r, and r + , A, then, if A be small, any two neighbouring
portions of these elements will be related to each other as the
neighbouring portions of two consecutive half-period elements on a
plane or spherical surface, at a distance from the pole such that the
obliquity is the same. When this obliquity is sensible the outstand-
ing difference of effect between two consecutive half-period elements
will consequently be vanishingly small when A is small, and the whole
effective portion of the wave will be limited to a small area surround-
ing each pole.
In the general case, therefore, the whole illumination at O will
appear to come from the immediate neighbourhood of certain points
on the surface, and this illumination will not be appreciably influenced
by screening off the remainder of the surface. These points may be
isolated and can be treated as separate sources when they are removed
from each other by a large number of half-period elements, the effect
of each being approximately the same as half that of the first half-
period element, or they may be close together and form continuous
Fig. 20.-Wave of any Form.

76 RECTILINEAR PROPAGATION CHAP. III.
loci on the surface, so that the illumination may appear to come to O
from certain curves traced on the surface.
54a. There are two defects in the explanation of the rectilinear
propagation of light as given by Fresnel from considering the primary
wave as equivalent to the secondary waves. It does not adequately
explain why light is not propagated backwards as well as forwards,
nor why it should be necessary to accelerate the phase of the secondary
waves by a quarter of a period in order that their effect at O (see
Fig. 25) should be identical with that of the primary wave reaching
O directly. In Fig. 27, OJ represents the vibration due to all the
secondary waves, but it differs in phase by a quarter period from the
phase of the vibration due directly to the primary wave. Stokes
showed dynamically that, for waves advancing through an elastic
medium, the acceleration of phase by a quarter period was necessary
in order that the secondary waves should be equivalent in effect at
any point to the primary wave reaching that point; but there seems
to be no physical explanation of the necessity. It is interesting to
note, however, that Professor Trouton + showed for Hertzian waves
that a small reflector, that is, one acting approximately like the pole
of a wave, reflects a wave nearly a quarter of a period in advance of
the whole reflected wave. -
Also Gouy” showed experimentally that when convergent light
passes through a focus and becomes divergent, there is an accelera-
tion in phase of half a period, a quarter period before and a
quarter period after passing through the focus. He also showed that,
when a beam of light passes through a very small opening, the waves
passing through are advanced in phase by a quarter of a period. He
attributes this to an anomalous velocity of the waves under such
circumstances, and regarding each point, of the surface locus of the
centres of secondary waves as a small window, he considers the
secondary wave as starting with this anomalous velocity, so that, in
the first instants of its advance, it gains the quarter period in phase.
(Cf. also Art. 329, where the anomalous velocity of the waves leaving
a Hertzian oscillator leads to the gain of a quarter period in the first
quarter period.)
It may be shown mathematically that if a continuous train of
waves is passing through a medium, the disturbance at any point O
at any instant (i) may be calculated by taking the integral, over any
closed surface surrounding the point, of certain disturbances at the
various points of this surface at a previous time ( -:) and that
* Nature, vol. xl. (1889), p. 398.
* Comptes rendus, vol. cx. (1890), p. 1251, and vol. cxi. (1890), pp. 33, 910.
ART. 54a, g HUYGENS'S PRINCIPLE 77
therefore the light disturbance at O may be looked on as due to the
resultant effect of these disturbances being propagated to O with
velocity W.
The proof of this theorem may be developed in the following form.
Green's theorem applied to such a surface for the functions U
and } where r is the distance of any point from O, gives
ſ:#as-ſſſºvule-ſſu;()is ru, (1)
where the normal is drawn out from the space included by the surface,
the volume integrals are taken through the volume included by it,
the surface integrals are taken over its surface, and Uo is the value of
U at O.
Let S be the light disturbance at any point, so that S satisfies the
differential equation for the propagation of a disturbance through a
medium by wave-motion,
d2s
* = V2/V28).
dº? W2(V2s)
Let U be the function S after t – # has been substituted in it for t.
Then if 8 be used for differentiation, so far only as a function
contains the independent variable explicitly, it follows from the above
equation that
d?U_ (; **t 82U
-ā; V'\# * 52 + ...)
Now
dU_3U a 3U
zlº Tºº "7 Tör’
d'U (3 & 3)(3U12' 30
ºf \56,ºr 3r)\82 "Y 3r
_82U , 1 3U aſ 5U. 22 3PU , aſ 32U
~ 5 tº 5: Tº 5’ ºr 3rº"; 5%
therefore
… 1 d?U 2 öU a 320 32U
V*U=ya i +; ; +22; 3:#; ;
d2U mm. 23°U ſº tº - º * r
but dº? T V ga, since U is a function of t V’ -
Therefore
*U 2 öU.9's a 3°U.
V*U=2; +; ; +22,; ;
Substituting in (1) we get
1 du d / 1 " I /2 62U 2 öU 2 *U \,- ,
4-U-ſſ:#as-ſſu;()is-ſ(######X*::)is. ()
78 - RECTILINEAR PROPAGATION § CHAP. III
But *
d / a 3U \_1 3U 22% SU, a / 6 a 5 \6U
#(; ;)=; ;-#. #(; #);
_1 5U 22:26 U, a 3°U a 238U
~72 5, T 73 ºr "25 ºr "73 572
Therefore
#(...) 1 SU a: 82U 1 630
dº ºr ) ºr * 2:35:25; "För'
Substituting in (2) we have
ſ1 d'U d. / 1 d /a: 6U
*U.-Jſ;as -ſuff();s-ſ/ſax;(º)e.
Or if (vr) is the angle between r and the normal to the surface and l,
m, n the direction cosines of this normal
4-U.-ſſ|}}}U cos (vr) — 221% #]as
1 du 1 2 SU
== ſſ [. d, " au cos (vr) – ;cos (vr). #}s.
lauss– 1/s,d\)\ ,s
Jſ;as–ſſ;(zº)is
f ſºl 6U 2, 6U *
=|ſ;astſ ſºor), as
=|ſ; ;ds+. ;cos (ºr). #ds.
Therefore
_ ſ ſº 18U 1 A 1 A 6U
*u-ſ] (###U COS (ºr)-; cos (vr). #)'s
ſ ſ1 SU 5 / U
=|| 7 5, T #(;) • COS (ºr)}ds. • (3)
The light disturbance at 0 at the time may, therefore, be con-
sidered as the effect of disturbances at the previous time t– at
each point of this closed surface surrounding O, which are propagated
to O with velocity V ; and, if the values of 8 and # are known at
each point of this closed surface, the disturbance at O may be calcu-
lated from (3) by integration.
If the light is due to a source C outside s, the disturbance at any
point P distant ri, from C will be
... A t 7:1
s=;cos *(ſ-)
ôU_ SU
5, Târ,
E COS or)( -# COS *(i. - rº) +. º * sin 2-(# * rº) }
cos (vºl.)
ART. 54a, HUYGENS'S PRINCIPLE 79
where the first term may be neglected in comparison with the second,
as A is small compared with r.
Thus, it will be seen from (3) that
80= #ſſ; sin 27t (f _* ;3) (cos (vrì) – cos or)as.
Kirchhoff' and Voigt gave this mathematical presentation of the
secondary waves of Huygens and Fresnel, and found the value
COS wº"
X
at any point. The method of Fresnel's zones may be applied to
demonstrate the usual rectilinear propagation of light, and it follows
that the elementary waves are not propagated backwards. At a point
on the line CO produced (vri)=(vr), and the effect at O of the dis-
turbance vanishes. Moreover, the effect at O is due to disturbances
that are in phase a quarter period ahead of the direct disturbance,
and when these are integrated over the surface they lead to a dis-
turbance at O that agrees in phase with that of the direct wave
from C.
Larmor,” however, points out that this must not be taken as an
exact mathematical demonstration of the validity of Huygens's and
Fresnel's hypothesis, for such would be impossible independently of
the mechanics of the propagation of the wave-train. [See also
Rayleigh, Coll. Papers, iii. No. 148.]
* See Drude's Optics (Mann and Millikan), p. 179; Kirchhoff, Ges. Abh. or Vorles.
iiber math. Optik ; W. Voigt, Kompendium d. theor. Physik, ii. p. 776.
* Proc. Lond. Math. Soc., ser. 2, vol. i. p. 1; see also Love, ibid. p. 37.
, for the factor which determines the amplitude of the disturbance
CHAPTER IV
REFLECTION
55. Reflection, Regular and Irregular.—When light falls upon
the surface of separation of two media, part of it is generally turned
back or reflected. Thus when a pencil of light, admitted into a darkened
room by a hole in the shutter, is allowed to fall upon a polished
metallic mirror, a reflected beam may be seen" leaving the mirror
and travelling along a certain definite path. This portion of the light
is said to be regularly reflected, in contradistinction to another portion
of the light, which, after falling upon the mirror, is scattered at the
surface in all directions—or irregularly reflected. This scattering is
due to the inequalities of the reflecting surface, and it diminishes as
the polish of the surface is made more perfect.”
It is by means of this scattered light that we see most of the bodies
around us which are not self-luminous. Thus if the light were all
regularly reflected from a mirror the eye would be affected only when
placed in the reflected beam, and then a bright image of the sun would
be seen in the mirror. If the eye were elsewhere, no light would enter
it and nothing would be seen. The scattered light, however, is diffused
in all directions from ordinary objects and enters the eye from all parts
of the surface, so that they can be seen in every position of the eye.
In speaking of reflected light in future we mean that light which
is regularly reflected according to the laws which we are about to
enunciate. The so-called irregularly reflected light, when there is any
By means of reflecting dust particles in the air which scatter the light in all
directions. We cannot, of course, see light travelling through space. It only affects
the eye when it enters it, and therefore must either enter it directly from the source
or be reflected into it.
* The expression “irregularly reflected ” is here rather an abuse of terms. There
is no irregularity in the reflection. The irregularity is confined to the reflecting
surface. The ordinary laws of reflection are obeyed in full, and are not departed
from unless the linear dimensions of the reflecting surface, or of the rugosities on it,
are small compared with the wave length.
80 g
ART. 57 ILLUSTRATION OF REFLECTION AND REFRACTION 81
occasion to speak of it, will be specially referred to as scattered or
diffused light. *
56. Laws of Reflection.—The beam of light falling upon the
mirror is termed the incident light, and the angle which its direction
makes with the perpendicular (or, as it is often called, the normal) to
the surface at the point of incidence is named the angle of incidence,
while that part of the light which is reflected is known as the reflected
light, and the angle which its direction makes with the normal to the
surface is the angle of reflection. The relations between the angles of
incidence and reflection have been known from the earliest times, and
are stated in the Laws of Reflection: “The angles of incidence and re-
flection are in the same plane, are on opposite sides of the normal and
are equal.”
The first part of this statement affirms that the reflected ray lies
in the plane containing the incident ray and the normal to the surface
at the point of incidence, while the second expresses the equality of
the angles of incidence and reflection.
The intensity of the reflected light generally increases with the
angle of incidence and with the polish of the surface. It also depends
largely on the nature of the medium from which it is incident, and on
that from which it is reflected. For example, much more light is
reflected, under the same circumstances, from a plate of glass in air
than from the same plate immersed in water. The variation of the
reflecting power of a surface with the angle of incidence is well illus-
trated by comparing water and mercury. At perpendicular incidence
water reflects about the fiftieth part of the incident light, while
mercury reflects about the two-thirds; but at an incidence of 89%.”
they each reflect about 72 per cent of the incident light.
An accurate experimental proof of the laws of reflection is furnished
by observations with such an instrument as the Meridian Circle. Adjust
the telescope to observe a star directly, and then observe the reflection
of the same star seen in the horizontal surface of a basin of mercury.
The telescope in these two observations will be found to make the
same angle, on opposite sides, with the vertical line."
The graduation of such an instrument is the most perfect that can
be accomplished by human skill, and yet the smallest divergence from
the preceding law has never been detected.
57. Illustration of Reflected and Refracted Waves.—If a perfectly
elastic ball impinges directly on another of equal mass at rest, the
* Systematic observations of this kind are made in several observatories. The
minute discordance observed has never been attributed to inaccuracy in the law.
See the annual volume of Greenwich Observations,
G
82 REFLECTION CHAP. IV
second ball will take up exactly the motion of the first, while the first
comes to rest at the spot where it impinged on the other. The whole
process is as if the first ball moved on through the other without
disturbing it or being disturbed itself.
So again, if a number of similar balls be placed in a row (Fig. 30, a),
and if the one at the end (A) be struck in the direction of the row, it
will move forward and impinge on the second and come to rest there,
while the second moves forward to strike the third, and comes to rest
in turn. In this manner the blow
is communicated by each ball to
(8) A* * * * º º º its successor, and the disturbance
travels along the whole line, leaving
Fig. *-Direct and Reflected Waves, all the balls at rest except the last
(B), which moves forward with the velocity and energy initially
communicated to the first (A). This is the case of a compressional
wave travelling in air (as in sound) along a uniform straight tube.
Now let us suppose that after the row AB we have another row
A'B' (Fig. 30, 8) of heavier balls. When the ball B moves forward
it strikes A and rebounds (since its mass is less than that of A').
At the same time A' moves forward and strikes its neighbour,
which in turn performs its part, and the disturbance travels along
the row of larger balls A'B', each coming to rest when it impinges
on its successor, because they are all of the same mass. But the
state of things has now altered in the row AB, for B has rebounded
from A', and, travelling backwards, has struck its neighbour and come
to rest. This disturbance is handed on from ball to ball as a disturb.
ance along the row in the backward direction BA. Hence the dis-
turbance in the first row (AB) has given rise to two other disturbances
—a direct one in the second set A'B', and a reflected one in the back-
ward direction BA in the first set.
In the same manner we may suppose the disturbance to arise in
the row of larger balls. Thus if B be struck the impulse will be
communicated along the line to A', and A' in its turn will move
forward and impinge on B, but as B is of less mass than A', the
ball A will not come to rest, but will follow after B, and therefore
(if the balls be imagined connected with weightless threads) A will
pull its successor after it, which in turn will act upon its neighbour,
and so on. A second disturbance is set up in the row A'B', which
consists of a further motion of the balls in the same direction as the
original disturbance.
We may now liken the two rows of balls to two media separated
by a common surface. The smaller lighter balls will correspond to
(a) is a s = e = * * * *n.

ART. 58 LAWS OF REFLECTION 83
the rarer medium and the heavier balls to the denser. When any
disturbance originates in one of the media it is propagated through Reflected
it, and when it arrives at the surface of separation two new disturb- ...;
ances are set up, one (refracted) in the second medium, and another waves.
(reflected) in the first medium. It therefore follows from the wave
theory that when light, travelling in one medium, comes to the surface
of another, part of it should penetrate into the second medium, or
we should have a refracted wave, and part of it should be propagated
backwards in the first medium in the form of a reflected wave.
Since the phenomena of the reflection and refraction of light exist,
we must admit that the ether is modified in some way by the presence
of matter, and differently in different substances. For example, the
ether in glass cannot be in the same condition for vibration as the
ether in air or water. Its so-called elasticity and density—that is to
say, those properties which enable it to propagate wave motion—are
modified by the substance which it permeates.
58. Deduction of the Two Laws of Reflection.—Let AA (Fig. 31) be
the surface of separation of two media, and AB the front of a plane
wave incident on it. Each successive portion of the surface as soon
as the wave reaches it becomes the centre of two diverging waves,
one (reflected) in the upper medium, and the other (refracted) in the
lower. These waves travel with different velocities, but if the medium
be homogeneous and isotropic the secondary waves will be spherical.
At present we shall confine our attention to the reflected wave.
If BA’ and MP be perpendicular to the wave front AB, then B is
the pole of A and M is the pole of P, so that A' is illuminated by the
element of the wave at B, and P by
the element at M ; or, in other words,
BA’ and MP are in the direction of
what we call the rays of light. When
the light from B reaches A* the light
from M has arrived at P some time
before, and would have reached N if
it had not been obstructed by the -
surface, but on reaching P a reflected Fig. 31. Reflection of a plane wave at a
wave is developed which diverges into Plane Surface.
a sphere of radius PM = PN, and similarly the reflected wave at A
has diverged into a sphere of radius AB = AD.
If the plane of the wave AB and the surface of separation AA'
be perpendicular to the plane of the paper, the line through A per-
pendicular to the plane of the paper is the intersection of the surface
of separation with the wave front at A'. Through this line draw a

84 REFLECTION CHAP. IV
plane to touch the reflected wave diverging from A and let the point
of contact be B'. Now since AB is the radius of this wave, at the
instant the light from B reaches A', it follows that AB = BA’= AD,
since the reflected light travels with the same velocity as the
incident. Hence the triangle AA'B' is equal in all respects to the
triangle AA'B or to AAT). Consequently if from P we let fall a
perpendicular PM" on A'B' we will have PM’-PN, and therefore the
reflected wave diverging from P will touch at M’ the tangent plane
A'B' to the wave from A. Similarly the waves diverging from
every point of the surface will touch the same plane. This plane
is therefore the reflected wave envelope, and A'B' is the trace
of the reflected wave at the instant the light from B reaches the
point A’.
The angle A'AB is the angle between the plane of the incident
wave front and the surface ; it is therefore equal to the angle between
the normal to the wave front, or the ray, and the normal to the surface.
Hence A/AB is equal to the angle of incidence. Similarly AA'B' is
equal to the angle of reflection, but these angles are equal by the
equality of the triangles ABA' and AB'A'. The lines AB, PM’, etc.,
are the normals to the reflected wave front, that is the reflected rays.
Any one of these rays obviously lies in the plane containing the
corresponding incident ray and the normal to the surface. The two
laws of reflection are thus completely accounted for by the wave theory.
Let us now investigate the matter a little more closely. It appears
from what has been already said that the effective portion of the wave
AB in illuminating P is confined to a very small element around M,
the foot of the perpendicular from P on the wave front. So in like
, manner, if A'B' were the incident wave, AB

MI would be the reflected wave, the path being
exactly retraced, and P would be illuminated
by the element of A'B' around M'. We should
4. r ** 4 therefore expect that the element M' of the
Fig. 32. reflected wave is illuminated by the point P
of the surface, or further back still, by the element M of the incident
wave. It is easy to show that this is the case.
On the line AA’ find points P, Pi, P, etc. (Fig. 32), such that the
path MP, M' exceeds the path MPM' by half a wave length, the path
MP.M’ exceeds MPM' by two half wave lengths, etc.; that is,
each path exceeds its predecessor by half a wave length. It is well
known, of course, that the path MPM' is less than any other if the
lines MP and PM’ are equally inclined to the surface. Therefore
the path MPM' is that along which it takes the light the least time to
ART. 59 PLANE SURFACE 85
reach M' from M after reflection from the surface AA’. Therefore in
estimating the illumination at Mſ by the reflection from AA’ we may
consider each point of AA’ as the origin of a disturbance propagated to
M', and find the resultant effect. The surface being divided up into
half-period elements, as indicated above, we can easily show as before
that of these elements those immediately around P are the greatest, and
that the elements diminish rapidly at first and then more slowly till
they become practically equal, and being opposed in effect at M' they
produce no illumination there. Consequently the effective portion of the
surface AA’ in illuminating M' is confined to a very small element of
the surface at P. If M is a single luminous point, an eye placed at M’
will perceive a bright point in the direction M'P. The illumination
which reaches M' from M is propagated in the same time and as if it
came from a point situated at an equal distance on the other side of
the surface. Similarly every point of the reflected wave is illuminated
by a corresponding point of the incident wave as if the light came
from the corresponding point of a line through A parallel to A'B'
(Fig. 31). This line is the reflection of AB in the surface. We see
then that each point of the reflected wave is illuminated by that point
of the incident wave which sends light to it in the least time. This is
an example of “the principle of least time,” which is of very wide
application and fertility in the theory of light.
We have now proved that the disturbance at any point M of the
incident wave is propagated along MP, and after reflection at Pit travels
to M', a corresponding point on the plane A'B'. The plane A'B' is the
locus of the points which are simultaneously disturbed. It is the re-
flected wave, while PM and PM’ are rays obeying the laws of reflection
enunciated above (Art. 56).
Cor.—The time taken by light to travel from any point of the
incident wave to the corresponding point M' of the reflected wave is
the same for all rays and a minimum. For MP + PM’= MN = BA."
59. Reflection of a Spherical Wave at a Plane Surface.—Let us
now consider the case of spherical waves diverging from a centre O
and falling upon a plane reflecting surface AB (Fig. 33). Let the
wave front at the instant under consideration meet the surface at A
and B in the plane of the figure. Each point of the surface between
A and B will have become by this time the centre of a reflected wave.
Thus if the surface had not been present the wave would have pro-
* The decrease in the elements PP, PIP2, etc., is well exhibited by describing a
system of ellipses with M and M' for foci and major diameters equal to 1--3\, l-H \,
!-- #A, etc., where l=MP+ M'P. If the line AA’ is a tangent to the inner ellipse the
intercepts made on it by the other conics are the half-period elements.
86 REFLECTION CHAP. IV
ceeded unobstructed, and would occupy the position ANB. The effect
of the surface, however, is such that M, the foot of the perpendicular
from O on AB, has become the centre of a spherical reflected wave of
radius MN = MN, and any other point P is the centre of a spherical
reflected wave of radius PQ = PQ.
It is clear that all these reflected waves will touch a sphere of
centre O' and radius O'N', where MO = MO. For join O' to P, and
produce the joining line to meet this
sphere at Q'. Then O'Q' = O'N' =
ON=OQ, and OP=OP, therefore
PQ = PQ, or PQ is equal to the
radius of the wavelet diverging from
P, and it is also normal to the sphere
AN'D. Hence the reflected wave
diverging from P touches the sphere
ANT3, and this sphere is therefore
the envelope of the reflected waves,
or the limit to which the reflected disturbance has been propa-
gated when the incident wave meets the surface at A and B. The
reflected wave front is consequently a sphere diverging from O' as
centre or the reflected light appears to diverge from a point O' on the
other side of the surface, and at the same distance from it as O. This
point is termed the image of 0 in the surface.
Before dismissing this case it may be noticed that the effect of a
plane surface in reflecting a spherical wave is simply to reverse its
curvature. Thus the incident wave ANB diverging from O is con-
verted into a wave of equal radius diverging from O'.
60. Measurement of Curvature.—As the effect of reflection, or
refraction, is in general to change the curvature of a wave, it will be
convenient to define the measurement of curvature before dealing with
waves reflected or refracted at curved surfaces. In the case of a
uniformly bent curve, that is a circle, the curvature is measured by
the bend per unit length; that is, by the angle between the tangents
at the extremities of a unit length of the curve. This is numerically
equal to the angle subtended at the centre by a unit length of the
circumference, or the same as the ratio of any angle at the centre to
Fig. 33.
the are subtending it. Thus the general measurement of the curva-
ture of a circle is—
angle 1
Curvature="#"—"
are p
where p is the radius of the circle, and the angle is expressed in
circular measure.

ART. 61 SPHERICAL SURFACES 87
When the curve is not a circle the rate of bending varies from
point to point, but the curvature at any point is still measured by the
limit of the above ratio when the arc and angle are taken very small.
The curvature of a curve at any point is thus the reciprocal of a
length—namely, the radius of curvature, that is, the radius of the
circle osculating the curve at the point in question.
The curvature of a small arc, AB (Fig. 34), may be conveniently The
expressed in terms of the Sagitta PM. Thus we have PM x MQ = MA”, Sagitta.
but when PM is small MQ = 2p approx., and consequently
PM = MA*/2p—that is, for an arc of given chord, the sagitta PM is
directly proportional to the curvature. And this is what would have
been expected, for in the limit PM clearly measures the bulge or
bend of the arc. It follows, therefore, that if two arcs, APB and
prº. A * Z.


4 - 4 f_
\ \

B
Fig. 34. Fig. 35. Fig. 36.
AQB (Fig. 35), have the same chord AB, their curvatures are directly
as their sagittae PM and QM. Or if the arcs touch each other, as in
Fig. 36, and if a tangent be drawn at the point of contact, then the
intercepts PM and QM made by the arcs on a perpendicular to the
tangent are proportional to the curvatures of the arcs. For in the
limit these intercepts clearly measure the amounts of bend of the arcs.
Eac.—If the curvature of a circle be a when its radius is p, prove that when the
radius becomes p + c the curvature becomes
g =="—
1 + co’
[In this case we have a = 1/p and
-ji=#-F#,
p-- c 1 +c/p" l-i-co’
61. Reflection of a Plane Wave at a Spherical Surface.—We
shall now consider the reflection of a plane wave at a spherical surface.
Let AMB (Fig. 37) represent the reflecting sphere, and XABY the
trace of a plane parallel to the front of the incident wave. As the
* This method of treating the problems of reflection and refraction was given in
full by Professor S. P. Thompson in 1889 (Phil. Mag. vol. xxviii. p. 232), and was
partially employed in the first edition of this work.
88 REFLECTION CHA1*. IV
wave approaches the surface (from left to right) it comes into the
position of a tangent plane to the surface, first touching it at some
point M. At this instant M becomes the centre of a reflected wave,
and as the original wave moves farther to the right, each point of the
surface in turn becomes the centre of a reflected wave. Thus when
the incident wave occupies the position XY, the part between A and
B will have been reflected by the surface into the wave AN/E,
such that MIN = MN, for MN is the distance to which the reflected
disturbance travels, while the incident wave travels over the dis-
tance MN.
Now when the arc AB is small, MN is proportional to the curva-
Fig. 37. Fig. 38.
ture of the surface and NN' is proportional to the curvature of the
reflected wave. But
NN'-2NM,
and we conclude that the curvature of the reflected wave is twice that
Impressed of the reflecting surface. The action of the surface in reflecting a
curvature.
Focal
power.
plane wave is therefore to imprint on the reflected wave a curvature
equal to twice the curvature of the surface. In other words, when a
plane wave is reflected at a convex spherical surface, the reflected
wave diverges from a point F half-way between the centre of the
mirror and its surface. This point is called the principal focus of the
mirror, and the distance MF is termed its focal length. The focal
length f of a spherical mirror of radius p is consequently determined
by the equation
f=\p.
Since 1/f measures the curvature impressed on a plane wave by
reflection at a spherical surface, this quantity measures the curvature
producing power of the mirror and is termed the focal power. It is

ART. 62 SPHERICAL SURFACES 89
clear, therefore, that the focal power is equal to twice the curvature of
the mirror.
The case of a concave spherical mirror is shown in Fig. 38, in
which MM represents the reflecting surface and XY is the trace of a
plane parallel to the face of the incident wave, supposed travelling
from left to right. If the reflecting surface had not been present
XY would represent the incident wave in one of its positions, but by
the reflecting action of the surface NN is converted into NN', where
obviously in the limit MIN= MN", and therefore the curvature of the
reflected wave NN' is twice that of the reflecting surface. The centre
F of N'N' is consequently half-way between O and the surface, or the
action of a concave reflecting surface is to convert a plane wave into a
spherical wave, of twice the curvature of the surface, which converges
to a point half-way between the centre of the mirror and its surface.
62. Reflection of a Spherical Wave at a Spherical Surface.—
We have seen that a plane surface in reflecting a spherical wave
simply reverses the curvature of the wave, and that a spherical sur-
Fig. 39. Fig. 40.
face imprints twice its own curvature on a plane wave in reflecting it;
we might therefore suspect that a spherical surface in reflecting a
spherical wave would reverse the curvature of the incident wave, and
in addition impress it with twice the curvature of the reflecting surface.
That this is the case is very easily proved. Thus if we suppose a
spherical wave ANB (Fig. 39), diverging from a source Op to be re-
flected at the surface of a concave spherical mirror AMB, then the
wave which would have occupied the position ANB, if unobstructed,
will be converted into the wave AN'B by the reflecting action of the
mirror, and the relation between the two waves is determined by the
equality MN = MN'.

90 REFLECTION CHAP. IV
From this it follows at once that
DN + DN’=2DM,
or the sum of the curvatures of the incident and reflected waves is
equal to twice the curvature of the mirror.
In the case of a convex reflecting surface, if ANB (Fig. 40) re-
presents the position which the incident wave would have gained if
unobstructed by the reflecting surface, AN'B the reflected wave, then
MN = MN', and consequently
DN’ – DN = 2D \ſ.
But if we remark that in this case the curvature of the incident
wave is opposite in sign to that of the reflected wave and of the
mirror, we may write this equation like the foregoing in the form
DN + DN’=2DM ;
that is, with proper attention to sign we may say in general that the
curvature of the reflected wave is equal to the curvature of the incident
wave reversed added to twice the curvature of the mirror. This may
also be expressed by saying that the curvature of the mirror is the
arithmetical mean of the curvatures of the incident and reflected waves.
Denoting the curvatures of the incident and reflected waves by
or, and or, respectively, and that of the mirror by or, the fundamental
equation for the reflection of a spherical wave at a spherical surface
takes the form
al-H 0.2 = 20.
Hence the sum of the curvatures of the incident and reflected waves is
equal to the focal power of the mirror. i
The interpretation of this equation is that if the incident wave
diverges from a point O, at a distance pi from the mirror, the reflected
wave will converge to (or diverge from) a point O, at a distance p,
from the mirror such that
1 1 2
p, "p, Tp
where p is the radius of curvature of the mirror.
The points O, and O2 are termed conjugate foci with regard to the
mirror, and are such that a wave diverging from either will after
reflection converge to (or diverge from) the other.
Cor.—When O1 is at infinity the wave becomes plane—that is, pi
is infinite, and the incident light forms a parallel beam, so that
or = 1/p = 0; and we have the equation of Art. 61, viz. g., - 20 and
p, = #p=f.
Art. 63 ANY SURFACE 91
63. Wave of any Form reflected at any Surface. — In the
preceding articles we have considered the reflection of plane and
spherical waves, but the theory may be applied to a wave PQ (Fig. 41)
of any form reflected at any surface AB. Consider the light incident
at any point A of the surface. This
point is illuminated by the element of the
wave at P where P is the pole of A, or
that point of the wave from which it
takes the light least time to reach A. If
the curvature of the surface at A is not
infinitely great the light will be reflected
at A as from an element of the tangent
plane at this point, so that the reflected
light travels from A along a direction AP, such that AP and AP
make equal angles with the normal at A, and lie in the same plane
with it according to the ordinary laws of reflection.
It may happen that the wave PQ has more than one pole with
respect to A, so that there may be several points P such that the time
required by the light to reach A from them is either a maximum or a
minimum. In this case the light will appear to come in rays from
each of these points to A, and we will have a corresponding set of
reflected rays. There might be a curve on PQ such that each point
of it is a pole of A, the light then would appear to travel to A from
the whole of this curve. Examples of such cases will appear in the
sequel.
To find the form of the reflected wave take any system of points
A, B, etc., on the reflecting surface and determine their poles P, Q, etc.
With A, B, etc., as centres and radii r, r", etc., such that PA + r =
BQ + r = etc., describe spheres. These spheres touch the reflected
wave P'Q' at the points P, Q'. The reflected wave may therefore be
described either as the envelope of these spheres or as the locus of the
points P, etc., taken on the reflected rays such that PA + AP = QB +
BQ = constant. Hence if a set of rays be drawn perpendicular to any
wave front in a homogeneous isotropic medium, they will after reflec-
tion (or any number of reflections) be perpendicular to the new
wave front," and the length of any ray from wave front to wave
front will be constant and the same for all the rays. A similar pro-
position holds likewise for refraction, if “time taken by * replace
“length of.”
Rays.-From what has been proved we see that it is approximately
legitimate to regard the light emitted from any point as made up of
* Malus, Journal de l'Ecole Polytechnique, cah. xiv. p. 1, 1808.
Fig. 41.

92 REFLECTION CHAP. IV
very narrow pencils or rays, and that after reflection each little beam
or ray is reflected on the other side of the normal to the surface at an
angle equal to the angle of incidence. In dealing with problems in
the reflection of light we may therefore consider the light propagated
in rays if it facilitates the solution. Yet we must carefully bear in
mind that rays have no physical existence, for it is waves that are
propagated and not rays. The following examples are added for the
sake of comparison :
Eacamples
1. If a plane mirror on which a pencil of light is incident be turned through any
angle about an axis perpendicular to the plane of incidence, the reflected light is
deviated through twice that angle.
[The direction of the incident light remains fixed in space, hence if the plane
reflecting surface be turned through any angle 6 the normal will be turned through
the same angle, consequently the angle of incidence i becomes i-E6. The angle of
reflection is also altered by the same quantity, therefore the angle between the
incident and reflected rays is 2i +20, but originally it was 2i, therefore the reflected
ray has been turned through an angle 26.
This theorem is of wide application in practice, for plane mirrors are extensively
used to indicate, by the change in the direction of a reflected ray, the motions of
magnetised or electrified needles, and for many other purposes in physical apparatus.]
2. Show from Ex. 1 that when a plane wave is reflected at a spherical surface
the curvature of the reflected wave is equal to twice that of the surface.
3. Light emanating from a luminous origin O is reflected at a plane surface, prove
from the doctrine of rays that the reflected light appears to come from a point O'
on the other side of the surface, such that the line OO’ is perpendicular to the
surface, and 0 and O' are equally distant from it (Fig. 42).
O
2.
º
O'
Fig. 42.
[The lines joining O and O' to any point of the surface are equally inclined to it,
hence every reflected ray passes through O'. An eye placed in the reflected light
will receive a cone of light of which O' is the vertex. The light will consequently
appear to come from O'. This point is called the image, or reflection, of O in the
surface. The reflected waves are spheres with O' as centre.]
4. Light diverging from a point O is reflected at a concave spherical surface, find
the conjugate focus by the doctrine of rays.
[Let C (Fig. 43) be the centre of the sphere and p its radius, and let P be a point
on the surface at a distance from A small compared with the distance OA or p.
Then if OP be any incident ray and PF a reflected ray, CP is the normal to the
surface, and therefore bisects the angle OPF. Hence
OP : PF : ; OC : CF.


ART. 63 EXAMPLES 93
But approximately OP=OA=p, and FP=FA=p, therefore
pı(p − p)=poſpi - p),
OT e l + 1–2,
p1 p2 p
which is the algebraic statement that the row [AFCO) is harmonic—a property at
once evident, for PA and PC are the bisectors of the angle OPF.
The reflected rays therefore pass through the point F determined by the above
equation, and the reflected waves are spheres round F as centre. Of course this
is only an approximation. When O is infinitely distant the light falls upon the
mirror in a parallel beam, and the point F, to which it converges, is termed the
Principal focus. It lies half-way between C and A, or the principal focal distance f
is equal to #p. This follows from the above equation, for 1/p = 0 and p2=f. For a
plane mirror p=oo, so that p1= – ps.] -
5. The equation of Example 4 may be written in the form
(pi – f')(p3 – f)= f°
where f is the principal focal distance, or 2f=p. f is therefore a geometric mean
between the distances of the conjugate foci O and F from the principal focus.
6. Light diverging from a point O falls upon
a convex spherical mirror, find the position of the
conjugate focus F.
[Here the point O (Fig. 44) and the centre of * *
the mirror lie on opposite sides of the surface. GT Aï C
The radius p is therefore to be reckoned negative. \
Hence if p is positive pa will be negative, and
O and F will lie on opposite sides of the mirror.
The focus F is in this case virtual, that is, the reflected light appears to diverge from
it and the reflected wave is a sphere diverging from F as centre.]
7. If p and q denote the distances of two conjugate foci from the centre of a
spherical mirror, prove that
1 1 2
— — — — = — ”
7) Q p
8. Light diverges from a point F, find the form of the surface" which will accu-
rately reflect it to another point F".
[The surface must evidently be such that the lines from F and F to any point of
it are equally inclined to the tangent plane at that point. Now a fundamental
property of an ellipse is that the lines joining the foci to any point of the curve are
equally inclined to the tangent at that point. Hence light proceeding from one
focus will be reflected to the other, and the surface generated by the revolution of an
ellipse round its major axis will therefore satisfy the conditions of the problem. The
surface is therefore any spheroid having F and F' for foci. If the surface is a hyper-
boloid of revolution then one focus is the virtual image of the other, and if the
surface is a paraboloid of revolution, the second focus being at infinity, the light
proceeding from the first focus will after reflection travel in a parallel beam in the
direction of the axis of the surface.
We arrive at the same conclusion by regarding conjugate foci as two points such
that the time taken by light to travel from one to the other, by reflection at the
* Such a mirror is said to be aplanatic. A spherical surface is aplanatic for rays
diverging from its centre only.

94 REFLECTION CHAP. IV
surface, is constant, or the same for all paths. For if p and p' be the distances of
F and F from any point of the surface we have
p--p' - constant,
but this is the fundamental property of an ellipsoid of revolution. f
If mercury be placed in an elliptic dish and disturbed at one focus the reflected
waves may be seen converging to the other focus.]
9. A luminous point is situated between two plane mirrors inclined at a given
angle 6, find the number and position of the images formed by successive reflections
at the mirrors.
10. Show that when an eye is placed to view any image formed by successive
reflections at two mirrors, the apparent distance of the image from the eye is equal
to the distance actually travelled by the light in coming to the eye from the luminous
point.
11. A luminous point is placed between two plane mirrors inclined at an angle
of 27°. Prove that the number of images is thirteen or fourteen, according as the
angular distance of the point from the nearer mirror is less than or greater than 9°.
12. If the light of the sun be admitted through a small hole an image of the sun
is depicted on a screen placed to receive it, but if it be admitted through a large
aperture we obtain an image of the aperture. Explain this.
[Each small portion of the aperture depicts an image of the sun, and the com-
plete system of these images forms the image of the aperture.]
13. If the fraction of light reflected at the first surface of a parallel plate be a
(there being no regular interference), that transmitted by the first surface, reflected
by the second and again transmitted by the first, is a(1 – a)”. That reflected three
times and transmitted twice is a”(1 - a)”, etc. Hence the whole reflected light is
_ 20.
TIEcº
14. The intensity of the light reflected from a pile of plates has been investi-
gated by Provostaye and Desains (Ann. de Chimie, xxx. p. 159, 1850). If p(m) be
the reflection from m plates the reflection from m + 1 plates, as above, is
©(m+1) = a + (1 - a)*@(m)}{1 + ap(m) + a”(p(m))” + etc.)
a + (1 - 2a) f(m).
R = a + (1 – a)*(a + a” + a” + . . .)
*
1 – a p(m)
But p(1) = a, therefore we find p(2), p(3), etc., and generally
7)? O,
q}(m) = I-F(n − 1)a.
Stokes has extended this to the case in which the plates exercise an absorbing in-
fluence (Proc. Roy. Soc. xi. p. 545, 1862).
15. A system of rays being such that they all cut a given surface orthogonally,
construct a mirror which will reflect the system to a given focus (Sir Wm. R.
Hamilton, Trams. Roy. Irish Academy, vol. xv. p. 85, 1828).
[Take on each ray a point such that the sum (or difference) of its distances from
the orthogonal surface and the given focus is constant. The locus of these points
is the surface of the required mirror.]
16. If rays diverging from a point are reflected at any surface the reflected rays
are cut orthogonally by a system of surfaces, and after reflection at any number of
surfaces the whole length of each ray from the source to any orthogonal surface is
the same for each ray (Hamilton, ibid.).
[The medium is supposed isotropic and the orthogonal surfaces are the wave
surfaces.]
CHAPTER W
REFRACTION
64. Refraction, Snell's Law.—When light is incident on the
surface of a transparent medium a portion is reflected; but another
portion enters the medium, pursuing there in general an altered direc-
tion. This portion is said to be refracted. Generally we may say that
when light is incident at the surface of separation of two media one
portion is reflected back and propagated in the first medium, while
another portion is refracted and transmitted through the second
medium, if it be transparent, but absorbed immediately at the surface,
or within a very small distance from it, if it be opaque.
The angle which the refracted ray makes with the normal to the
surface is called the angle of refraction. If
the first medium is optically rarer than
the second, for example, air and water,
the angle of refraction is less than the
angle of incidence, the refracted ray is
bent towards the normal (Fig. 45); on
the other hand, it is bent or deviated
from the normal if the first medium
is optically denser; that is, more highly refracting than the second.
The relations connecting the angles of incidence and refraction are
known as the Laws of Refraction. These were arrived at by Snell about
1621, but first stated by Descartes in the form: “The incident and
refracted rays are in the same plane with the normal to the surface;
they lie on opposite sides of it, and the sines of their inclinations to it
bear a constant ratio to one another.”
Denoting the angles of incidence and refraction by i and r respect-
ively, the relation between them is stated in the formula
Fig. 45.
sin i
- = Lt.
sln 7"
The constant ratio p is called the inder of refraction. It is in general
95

96 REFRACTION Ch.A.P. V.
greater or less than unity according as the first medium is rarer or
denser than the second." When light passes from vacuum into any
medium the ratio is termed its absolute index of refraction, but when it
passes from one medium to another it is termed the relative index.
65. Deduction of the Laws of Refraction.—The theoretical deduc-
tion of the laws of refraction is in all respects similar to that of the laws
of reflection. Let AB (Fig. 46), as before, be the trace of the incident
plane wave, and AA’ that of the surface
of separation, both planes being per-
pendicular to the plane of the paper.
Let v be the velocity of light in the
first medium and v' that in the second.
Then if t be the time required by the
light to traverse the distance BA", we
have BA = wi = AD, and if the wave
had been unobstructed by the second
medium, it would occupy the position
AND (parallel to AMB) at the end of the time t. However, since
the disturbance travels with a velocity v' in the second medium, the
point A becomes the centre of a spherical wave of radius vºt - AC, or
such that AD : AC : : v : v', for AD = wi. Similarly, at the end of the
time t any point P of the surface will be the centre of a refracted
wave of radius PM’, such that PN: PM’: : " : v'. Therefore
Fig. 46.
AC AD_AA'
PM. T. PNT PA”
and hence if a plane be drawn through Aſ perpendicular to the plane
of the paper to touch the sphere diverging from A, it will also touch
that diverging from P, or from any other point of the surface AA’. The
plane through AC, perpendicular to the plane of the paper, is therefore
the wave envelope or the locus of those points which are just about to
be disturbed at the end of the time t. As in the case of reflection, we
may show that the point M of the refracted wave is illuminated by the
point P of the surface, that is, the disturbance at M is propagated along
the path MPM' from M to M', and we shall show immediately that
this path is that along which it takes light the least time to travel from
M to M', and that the principle of least time is obeyed in refraction as
well as in reflection.
The angle AAT) is the angle of incidence and the angle AAC is
* It is not universally true that the denser media are the more highly refracting,
for example water, of unit density, has a refractive index 1:3336 for yellow light,
whereas oil of turpentine, density 0-885, has an index 1.4744 for the same light.

ART. 65 LAWS OF REFRACTION 97
the angle of refraction; denoting these by i and r respectively, we
have
AD sin i v
AG=sin(z=y+*
Hence we have the law of refraction, viz. “the sine of the angle of
incidence bears a constant ratio to the sine of the angle of refraction,”
while we have also the additional information that this constant ratio,
or the refractive index, is equal to the ratio which the velocity in the
first medium bears to the velocity in the second.
The bending or deviation of a ray of light in passing from one
medium to another is then due to the difference of the velocities of
light in the two media. The greater the change of velocity the greater
the bending. If light travels more slowly in the second medium than
in the first, v is greater than v', i is greater than r, p > 1, and the re-
fracted ray is deviated towards the normal ; the reverse is the case
when v is less than v'.
Now we know by direct observation that the deviation is towards Velocity
the normal when light passes from a rare medium like air to a dense test.
medium such as glass or water. The wave theory therefore indicates
that the velocity of light in air is greater than its velocity in glass or
water in the ratio of their refractive indices, and experiment proves
this to be the case.
The emission theory, on the other hand, points to the opposite
conclusion. According to it sin iſsin r = V/v (Art. 23), so that in those
media where the bending is towards the normal the light travels with
increased velocity. Here then the conclusions of the two theories are
contradictory, and experiment, which alone can decide between them,
supports the wave theory conclusively. The emission theory is conse-
quently untenable in its original form, and requires serious modifica-
tion in its fundamental tenets in order to meet this difficulty.
Ea'amples
1. If the velocities in two media be v, and va, while the velocity in vacuum
is v, the absolute refractive indices of the media are
Al-v/vi, and ſlo-v/v2.
while their relative refractive index, or the index of the second with respect to the
first, is
*12=viſva-Maſui.
The law of refraction may consequently be written in the form
p1 sin i = [12 sin r.
98 REFRACTION CHAP. V.
2. For any number of media we have
112 - 123 * * * *n-1, n=unſui,
or the continued product of the relative refractive indices of n substances is equal to
the ratio of the absolute refractive index of the nth to that of the first.
66. Construction for Reflected and Refracted Waves (Huygens).
—We have now the following construction for the wave fronts of the
two portions into which a beam of parallel light is divided when it is
incident on the surface of separation of two transparent isotropic
media.
Let the plane of the paper be perpendicular to the plane of the
incident wave, and also to the plane surface of separation of the
media.
Let AB (Fig. 47) be the trace of the incident wave on the plane of
Fig. 47.
the paper. The plane of the wave is a plane through AB perpendicular
to the plane of the paper, and the surface of separation is a plane
through AA perpendicular to the paper. With A as centre describe
two spheres of radii vt and ºt, where t is the time of propagation from
B to A', and v and v are the velocities of propagation in the first and
second media respectively.
The radius AB of the upper sphere is equal to AB, and the radius
AC’ of the lower is A'B/u. The ratio of the radii is equal to the
ratio of the velocities in the two media, or equal to the relative
refractive index.
From A draw planes perpendicular to the paper to touch these
spheres, and let A'B' and A'C' be the traces of these tangent planes.
These tangent planes are the limits to which the disturbance is
propagated at the instant the wave from B reaches A. They are the
reflected and refracted wave fronts respectively. The perpendiculars

Art. 67 TOTAL REFLECTION 99
AB' and AC on them from A are the reflected and refracted rays
arising from the ray incident at A.
The media here considered are isotropic, and the waves diverging
from any point are accordingly spherical. This is not the case in all
media, yet the construction for the wave fronts remains the same: viz.
with A as centre describe the waves, whatever shape they be, which
have diverged from it at the instant the wave from B reaches A.
Through A draw tangent planes as before to these waves. These
planes are the reflected and refracted wave fronts.
67. Total Reflection.—If the first medium be rarer (less refracting)
than the second, the radius of the second sphere is less than that of
the first, but the radius AB of the first is equal to BA, which is less
than AA', hence the radius AC of the second sphere is always less
than AA, consequently A lies outside it, and it is always possible to
draw a tangent plane to it from A'. There is then always a refracted
wave. It is otherwise when the second medium is rarer (less refract-
ing) than the first. In this case the
velocity in the second is greater than
in the first, and the radius of the
second sphere is greater than the radius
of the first. It is therefore greater
than AB, and may be greater than
AA’ if the incidence exceeds a certain
limiting value. If the radius of the Fig. 48.
second sphere is greater than AA', the point A will lie inside it, and
it will be impossible to draw a real tangent plane to it from A.
Consequently there is no refracted wave, and the light is all reflected
back into the first medium, as shown in Fig. 48.
The light in this case is said to be totally reflected. This limit is
reached when the radius AC of the lower sphere is equal to AA. In Critical
this case angle.
AB AB v
ATA- AC ºf"
sin i =
so that the limiting angle of incidence at which total reflection occurs
is given by the equation
Sln 1–1.
where p is the refractive index of the rarer medium with respect to the
denser. If, however, p, denotes the relative refractive index of the
denser medium, the limiting angle is given by
sin i = 1/u.
This angle is known as the Critical Angle, and total reflection occurs in

100 REFRACTION CHAP. V.
the case of light passing into a less refracting medium if the angle of
incidence is greater than the critical angle.
The existence of total reflection is frequently taken advantage of
in the construction of optical instruments, and notice of it often comes
within reach of ordinary observation, as when the surface of Water is
viewed in a glass held above the head, the silvery brilliancy of the
surface being due to the total reflection of the light.
For water the critical angle is about g e º § . 48° 27' 40"
For crown glass 9 3 3 x * g ſº e † . 40° 30'
For chromate of lead it is only e g * g t . 19° 28' 20"
Ea'ample
If light is refracted at a plane surface, prove that the deviation—that is, the
difference of the angles i and r—increases as the angle of incidence increases.
[When i is small we have i = ºr, and therefore the deviation is i– 7'-(u - 1)m =
i(u – 1)/u. The deviation conse-
quently increases with the angle
of incidence. When i is not
small it follows at once from the
law of refraction that sin 7 in-
creases more rapidly than sin r,
or that i increases more rapidly
than r ; that is, that i – r in-
creases with i. This may be
shown geometrically as follows:
—Let OP (Fig. 49) be any line
representing the velocity in the
first medium, and on the same
scale let OM represent the velocity in the second. With O as centre and OM as
radius describe a circle. Then if the angle OPM be the angle of refraction the angle
OMN will be the angle of incidence, for we have
OP v1__sin i_sin OMN
ÖMºv, **sinºsin OPM
Fig. 49.
Hence OMN = i, and if OP be parallel to the refracted ray and PM parallel to the
normal then OM will be parallel to the incident ray. The angle POM is consequently
equal to i – r and therefore represents the deviation. Now it is clear that POM
increases as r increases; that is, as i increases, until the line PM becomes a tangent
to the circle. At this point, in the case of light refracted from the second medium
into the first, the limit is reached beyond which total reflection occurs and refraction
ceases.]
68. Relation connecting the Intensities of the Incident, Re-
fleeted, and Refracted Waves.—The Energy Equation.—Since the
incident wave is divided into a reflected wave and a refracted wave,
its energy must be equal to the sum of the energies of the other two.
The reflected and refracted waves derive their energies from the
incident, and it is clear that a pencil of length v of the latter gives
rise to a reflected pencil of length v and a refracted pencil of length v',

ART. 68 THE ENERGY EQUATION 101
and the widths of the pencils are proportional to AB, A'B', and A'C'
respectively (Fig. 47). Hence if a, b, c be the amplitudes of the
corresponding vibrations, the energy per unit volume will be propor-
tional to pa”, pö”, pºcº, where p and p’ are symbols for the two media
representing what we may call the density of the ether or that
property of it which corresponds to the density of ordinary matter,
and by which it possesses energy when in motion. Hence the energy
of an incident beam of length v and width AB will be proportional to
wpa”. AB, and we have the equation
vpa”. AB =vpb°. A'B'--v'p'c”. A'C'.
Hence
vpa” cos i =wpb°cos i + v'p'c” cos r,
or, since w/v' = sin iſsin r, we have
p(a^- b%)_sin 2, .
p'c” Tsin 27
It is not unusual to find it asserted that the square of the ampli-
tude of the incident vibration is equal to the sum of the squares of
the amplitudes of the reflected and refracted vibrations; but this
could be true only if pſp' = sin 2:/sin 2i, a law which might exist if
sin 2:/sin 2i were constant instead of sin iſsin r. We would then
have a refractive index measured by the ratio of the densities of the
ether in the two media. The amplitude of the refracted vibration,
however, depends on the mean energy per unit volume, and this, we
have seen, depends on the density p’ of the ether in the second
medium or on the velocity of propagation.
Cor.—Two suppositions have been made with respect to the
quantities p and p", one by Fresnel, that the velocity of propagation
is inversely as the square root of the ether density, or that the
property of the ether which corresponds to elasticity is the same for
both media. On this hypothesis
p v” sin”
p'Tº Tsing:
by which the energy equation reduces to
a?– b% tan i
cº T tan o'
(Fresnel's energy equation.)
The other supposition, made by MacCullagh, is that p = p, or the
ether density is the same in all substances; we have then
a”—bº sin 2r. (MacCullagh's energy equation.)
cº Tsin 2i.
* This equation may be written down at once by observing that the energy in
the triangle ABA' (Fig. 47) is equal to the sum of the energies in the triangles AB'A'
and AC'A'.
102 REFRACTION CHAP. W
69. The Principle of Least Time or the Law of Fermat.—When
light passes from any point M to another M' (Fig. 31) by reflection at
a surface, we have seen that the rays PM and PM’ are equally inclined
to the surface, and consequently their sum is less than the sum of the
lines joining M and M' to any other point on the surface." The path
MPM' is that which will be traversed in the least time in passing
from M to M' by reflection at the surface.
A similar law governs the refraction of light, viz. if light pass
from any point M (Fig. 50) to any point M' in another medium, the
path MAM’ traversed by the ray is such that the
JMS time occupied in travelling over it is a minimum.
`s For if the time along the path MAM" is a mini-
y mum, the time over this path must be equal to
that occupied in traversing the consecutive (very
near) path MAM'. Hence if AB and A'C be
M’ drawn perpendicular to MA’ and MA respectively,
it follows that the times of travelling over the

distances AC and A'B are equal, since MA = MB and M'A' = M'C.
Hence
Fig. 50.
= ~ ; Ol'
f
Q) Q) q; Q)
A/B AC AA'sin i_AA'sin r ,
that is,
sin i v
sing.”.”
Hence if the time occupied in traversing the path be a minimum, the
ordinary law of refraction is obeyed, and conversely.
Denoting the rectilinear paths in the two media by l and l', the
law of Fermat asserts that
Ž J’ tº º
y + ;=a minimum,
OT
7+ pºl' = a minimum.
If the same ray passes through several media we have X(l/v), that is,
2pl, a minimum, and if the refractive index of the medium changes
from point to point, the path of a ray becomes a continuous curve,
and the length l in the above formula becomes a small element ds of
the curve. The principle of least time, according to which the wave
front is always as far advanced as possible, may, for a medium of
* In reality, the laws of reflection and refraction lead, in some cases, to a maxi-
mum time instead of a minimum. Fermat recognised only the latter case. The
principle is better referred to as that of “unvarying time”—the time differing only
by second or higher powers of small quantities from that which all neighbouring
paths would require.
ART. 70 REFRACTION THROUGH A PRISM 103
variable refractive index, be written in the form
ſº -- ſ --
minimum, Ol' uds= minimum.
We are to observe, therefore, that in all cases of refraction through
prisms, lenses, etc., when light travels from one point to another the
ray pursues that path which requires least time. For example, when
the various rays pass from one focus of a convex lens to its conjugate,
those which travel through the centre of the lens traverse a greater
distance in glass, and a less distance in air, than those which pass near
the edge; but all rays require the same time to travel from one focus
to the other, viz. the minimum time.
70. Refraction of a Plane Wave through a Prism.—A prism of
any material is a wedge-shaped portion of it contained between two
planes called its faces, which intersect in a line termed the edge of the
prism. The angle between the faces is called the angle of the prism.
Let ABC (Fig. 51) be a section of a prism by the plane of the paper,
supposed perpendicular to the edge of the
prism. Consider a plane wave of light incident
on the face AB in the direction PQ, the plane
of incidence being the plane of the paper, and
thus perpendicular to the edge of the prism.
The light being refracted into the prism in
the direction QR, making an angle r with the
normal, will suffer a deviation i = r at the face AB. If the angle of
incidence of QR on the second face be r", and the angle of emergence
along RS be i', the light will suffer a further deviation i – r". The
light will now emerge in the direction RS (if the angle of incidence r"
on the second face be less than the critical angle) and the total devia-
tion 6 from the original course PQ is
6–(i+ iº) – (r. 4-r) = i+ i – A
Fig. 51.
where A is the angle of the prism, which is equal to r + r", since it is
equal to the external angle between the
normals at Q and R to the faces.
A plane wave BM (Fig. 52) incident
on the face AB is refracted into the
prism, and travels through it as a plane
wave AD. It then emerges from the
second face AC as a plane wave CN. Any
point S of the wave CN is illuminated by
a corresponding point P of the incident wave, and the law which governs
the propagation of the disturbance is that the time of propagation along
all paths joining pairs of corresponding points on BM and CN is the
Fig. 52.


104 REFRACTION CHAP. V
Dispersion.
same. For example, the time along MAN = time along PQRS = time
along BC. Hence we should have
MA+ AN = p.BC = u(BD + DC),
OI’
AB sin i + AC sin iſ = p(AB sin r + AC sin r"),
which is true if the law of refraction (sin i = p sin r) is obeyed.
Hence NC is such that the disturbances reach it from MB in the
same time; they are therefore in the same phase, that is, NC is the
wave front after refraction through the prism. The rays which pass
near the edge of the prism traverse a shorter path in glass, but a
longer path in air, than those which pass near the base, the long air
path MAN occupying the same time as the shorter glass path BC, and
the ratio of the lengths of these paths is the refractive index of the
glass.
The deviation (8) is measured by the angle between MA and NA
produced ; but MAB = 90 — i, and NAC = 90 — iſ ; hence 8 = i + i." – A.
Now if p be increased while the angle of incidence i and the angle of
the prism remain the same, the angle of refraction r will be diminished,
for sin i = p sin r. Hence the angle of incidence r" on the second
face will be increased, for 1 +r' = A, and consequently the angle of
emergence i' will also be increased to obey the relation sin i = p sin r".
Hence if, while i and A remain the same, the refractive index be
increased, the angle of emergence i' will be increased, and the total
deviation i + i – A will be increased by the same amount.
The amount of deviation, therefore, depends on the refractive
index, that is, upon the velocity of propagation in the prism. We
should consequently expect that if ordinary solar light contains many
constituent waves which travel with different velocities in the prism,
they should be deviated by different amounts on emerging, and a
beam of parallel light incident on the first face should be dispersed
into several parallel beams on emergence from the second face. That
this is the case was first demonstrated by Newton."
71. Colour and Velocity.—The experiments of Newton prove in
a clear and masterly manner that lights of different colours are refracted
by different amounts in passing through a glass prism or on entering a
new medium, the red light being deviated least, the violet most, and
the intermediate colours (orange, yellow, green, and blue) in continu-
ous transition by intermediate amounts. But we have seen that the
bending of the ray on entering a new medium is a consequence of the
difference of velocity in the two media, and the greater this difference
the greater the deviation.
* See extracts at end of chapter.
ART. 71 COLOUR AND VELOCITY 105
We therefore conclude that the velocity of the violet light is least
and that of the red greatest in the prism, while the intermediate
colours travel with intermediate velocities. We have reason to believe
that all the colours travel with the same velocity in free space, and
with practically the same velocity in air, for if the red travelled faster
than the violet it would follow that a star reappearing after eclipse
should at first appear red, as the red light would reach us first, and
then gradually change tint till it finally became white when all the
colours have had time to arrive. Similarly, when the star is just
disappearing at eclipse it should be violet coloured, as the violet would
be the last to reach us.
Now as no observation of this nature has ever been made it follows
that the violet waves must differ in some respect from the red, and this
difference must exist either in the wave length or the periodic time of
vibration, or both, for if they all had the same periodic time and wave
length there would be nothing left by which we could distinguish the
red waves from the violet. The waves then in air must have different
periodic times and different lengths, for since they travel with the
same velocity, the equation
vT = \
shows that the periodic times are proportional to the wave lengths.
In the case of refraction there is one element which is likely to
remain unaltered, viz. the periodic time of vibration. For the vibration
in the second medium is excited and forced by the vibration in the
first medium, and these will in general be executed in the same time.
The time T then is the same in the incident and refracted wave, so
that if the velocity changes in the second medium the wave length
changes proportionately by the equation
q9"T=X’.
Hence it follows that when refraction occurs
*=A=
y–5-4
and the quantities
*=A=T
remain unaltered.
So far as we have yet gone the theory has not indicated whether
the red waves are longer or shorter than the violet ; all we have
arrived at is that the shorter the wave length the shorter the periodic
time is in a vacuum. Further on experiments will be given which
show that the red waves execute about 395 billion (395 × 10*) vibra-
106 REFRACTION CHAP. V.
tions per second, while the violet vibrate about 763 billion (763 × 10*)
times per second, or nearly twice as fast as the red.
The dispersion of the colours (as it is called) by a prism shows
that although all the waves travel with the same velocity in free space,
yet in dense media, like glass and water, the velocity of propagation
depends upon the wave length or periodic time.
Exercise.—Light incident at an angle i is refracted through a parallel plate of
glass. Waves of lengths \, and N2 are refracted at angles r and re respectively,
find the relative retardation by transmission through the plate.
Let PQ (Fig. 53) be the front of a plane wave incident on the face of the plate
at an angle i, and Q0 the direction of the incident
light. Let PM and PN be the directions of the
refracted rays. The planes OMRI and ONR2,
perpendicular to PM and PN, are the refracted
wave fronts, which on emergence, if the plate be
parallel, will proceed parallel to PQ. Thus the
emergent wave fronts are RIB and R2C, and if the
plate had not been present the original wave front
would have been propagated to OLA. Hence the
effect of the plate is to retard one wave by an
amount AB and the other by A.R. The relative
retardation of one on the other is therefore R&B.
But if e be the thickness of the plate, we have RiD =e cot ri, R.D=e cot rº, hence
R2B = R. Resin i =sin iſ R&D – RID)
=e sin iſcot re-cot ri)
= e(u2 cos re-ui cos ri).
Fig. 53.
This example will be of use hereafter in the consideration of the colours of mixed
plates and of doubly refracting crystals.
Cor.--When i =r-0, the light is incident normally, and we have the retardation
= e(us-pa), as is otherwise directly obvious.
72. Refraction of a Spherical Wave at a Plane Surface.—Let
spherical waves diverging from a centre O (Fig. 54) fall upon the plane
surface AB of a transparent substance. If the wave had not encoun-
tered the second medium it would
at some instant occupy the position
ANB, a sphere of radius OA and
centre O. The velocity in the second
medium is, however, different from
that in the first, so that M has
become the centre of a wave in the
second medium of radius MN, where
MN: MN = n : , or
MN = u . MNº. Fig. 54.
Similarly any other point P will be the centre of a spherical way
of radius equal to PQu, and the refracted wave will be the envelop


ART. 73 PLANE WAVE - 107
of all these spheres. In the figure we have supposed the second
substance more refracting than the first, so that their relative index p.
is greater than unity. The consequence is that the incident wave
ANB is flattened down into another AN'B of less curvature.”
Now MN and MN are proportional to the curvatures of the two
waves. Hence the relation between the curvatures is determined by
the foregoing equation, and may be expressed by saying that the cur-
vature of the incident wave is p times that of the refracted wave.
Denoting these curvatures by or, and as respectively, this relation may
be represented in any one of the following forms:
0.1-902, or 01/02=viſvg, or pagi = A272
where p, and p, are the absolute indices of the first medium and
second medium respectively.
If p and p2 represent the radii of curvature, we have or = 1/pi,
ors = 1/pg, and consequently the relation between the distances of the
points O and O’ from the surface is
p2=Api, or piºi = pºº, or pi/pg=PA/42.
73. Refraction of a Plane Wave at a Spherical Surface.—
The case of a plane wave incident on a spherical refracting surface
is one of extreme simplicity when considered from the point of view
of the wave theory. Thus let AMB (Fig. 55) represent the surface of
a refracting sphere and XABY the trace of a plane parallel to the
front of the incident wave. If the refracting sphere had not been
present XY would represent the incident wave in one of its positions,
but by the action of the sphere the portion ANB of the plane wave is
retarded and takes the form of the curve AN’B, which is approximately
an arc of a circle when AB is small. Now MN and NN’ are propor-
tional to the curvatures of the surface and refracted wave respectively,
and the relation between these is determined by the equation
MN = u MN’=p(MN – NN').
That is,
ANN’ = (u – 1)MN.
Hence if or be taken to represent the curvature of the surface and
o' that of the refracted wave, we have
* If AB is small the arc AN'B will be approximately a circle having its centre at
O'. The true form of the refracted wave is not a sphere, however, but a parallel to
a hyperboloid, so that the curve AN'B is a parallel to a hyperbola (see Ex. 1, p. 123).
108 REFRACTION CHAP. V.
The case of a concave refracting sphere is shown in Fig. 56, where
N'N' is the front of the refracted wave and NN is the corresponding
position of the incident wave. Here again we have MN = p, MN,
from which we derive the same formula as that obtained for a convex
surface. In both cases therefore the action of the refracting sphere is
to imprint on the refracted wave a curvature equal to (p − 1)/p times
Impressed that of the refracting sphere. This impressed curvature or is always
* less than that of the refracting surface ºr, and when p is greater than
unity—that is, when the sphere is more highly refracting than the
medium in which it is immersed—the curvature of the refracted wave
is of the same sign as that of the surface, but of opposite sign when p.
Fig. 55. Fig. 56.
is less than unity. Hence when p is greater than unity the focus F.
to which the refracted wave converges (or from which it appears to
diverge), lies on the same side of the surface as its centre C ; but when
p is less than unity the focus and centre lie on opposite sides of
the surface. The relation between the focal distance f-FM and the
radius p of the refracting surface is, since or = 1f and r = 1/p,
_ !!
ſ=ziº.
If the absolute indices u, and p, be used instead of the relative
index u = u2/un the above formulae become
a’e” -
"lº, and f- *2 p
A42 Alº - All
Focal The curvature ºr impressed on a plane wave by refraction at a spherical
* surface is called the focal power of the surface. It is measured by the

Art. 74 SPHERICAL WAVE 109
reciprocal of f, and means the curvature-impressing power of the
surface.
It has now been proved that the refraction of a spherical
wave at a plane surface changes its curvature from on to re-oriſu,
while the refraction of a plane wave at a spherical surface impresses
a curvature or(u - 1)/u on the refracted wave, and we shall see in the
following article that when a spherical wave is refracted at a spherical
surface the curvature of the refracted wave is the sum of these two
quantities.
74. Refraction of a Spherical Wave at a Spherical Surface.—
Let the surface AB (Fig. 57) of the refracting medium be a sphere
of centre C, and let a spherical
wave diverging from O meet it at A
and B. If the second substance be
more refracting than the first, the
spherical wave, which at any instant
would have occupied the position
ANB, will be flattened into the surface
AN'B. This surface is the envelope
of the sphere described with any point
P of the surface as centre and radius
PQ/p. It is the wave surface, and such
that the disturbances from O reach every point of it in the same time,
viz. the interval required to travel directly from O to A or B. The
refracted wave AN'B is propagated normally in the second medium,
and if ONN' be a common normal to both waves, in any position we
have
Fig. 57.
MN=w MN".
The effect of the refraction is therefore to diminish or increase the
curvature of the wave surface according as the velocity is less or
greater in the second medium.
In the same manner we may construct the refracted wave when
the incident wave and the refracting surface have any form.
If the angle AOB be very small, the refracted wave will be
approximately a sphere diverging from a centre O', and the relation
connecting the curvatures of the two waves with that of the surface
may be easily determined, for the equation MN = p.MN may be
written in the form
DN – DM-u(DN' – DM).
That is
DN = up N'-(u – 1)DM.

110 REFRACTION CHAP. V
Hence if the curvatures of the incident wave, refracted wave, and
surface be denoted by ori, o, and a respectively, we have the relation
a – ori
0 - 0.2
on = u02 - (u – 1)0, or = L.
Therefore or, is determined by the equation
0.1 pi – 1
oro — or, = (M. – 1)0, or oro –– -- – 0.
AlO2 l (9. )0, 2 Al M.
The relation between p1, pg, the distances of the conjugate foci from
the surface, is consequently
A 1 p - 1
p, p. To
If the relative index p be replaced by ps/p, the ratio of the
absolute indices of the two media, the relation between the curvatures
takes the symmetrical form
Pººz - A101=(u2 – 11)a,
and the relation between the conjugate focal distances becomes
... * *2 – P1 = P2 - P4.
p2 Pl p
The same formula holds good for a convex refracting surface if the
sign of the curvature of the surface be reckoned negative. The right-
hand side is a constant and is denoted by 1/f. When the object is
at infinity the image is at p, f, and when the image is at infinity
the object is at – pºlf. These points are called the second and first
principal foci respectively.
If the object and image are measured from the first and second
foci respectively the product of their distances is equal to — plugfº.
The size of the image, in general, is different from that of the object.
The ratio of the transverse linear dimensions of image and object is
called the transverse magnification. Since a ray through the centre of
curvature must go through without change of direction, the ratio
Image Distance of image from centre
ObjectT Distance of object from centre’
the extreme rays being assumed to make only small angles with the
axis, i.e. 4
=*-* (Fig. 57),
I
O pi – p
p
tº p291
OI S1 In Ce - I Lic, -
p=(u2 – 91) A4291 - A4102
1 402.
O Hopi
A positive value of the ratio implies an erect image.
Art. 75 REFRACTION THROUGH A LENS 111
If the distances of object and image from the first and second
principal foci are denoted by U, V (in each case being reckoned
positive when further along the actual rays than the reference point),
the formula for the magnification takes the following forms:
r
M-4-º'- W
Oi U u1.f.
75. Refraction through a Lens.—A lens is a portion of a trans-
parent substance, such as glass, quartz, or rock-salt, bounded by two
surfaces of such a shape that light diverging from a point O (Fig. 58)
and falling on one face, will after transmission converge to, or diverge
from another point O', approximately, at least. That is, the surfaces
of the lens are so shaped that a wave which is spherical before
transmission remains approximately spherical after transmission. The
surfaces of the lens itself, which will effect this accurately, are not
Fig. 58.-Aplanatic Lens.
accurately spherical, but only approximately so. The true curve of
the surface which will accurately refract to a point light diverging from
another point is the Cartesian oval (see Ex. 7, p. 125). A surface
which refracts to a point the light diverging from another point is
called an aplanatic surface, and the points are called conjugate foci with
respect to it. The fundamental relation connecting conjugate foci O
and O’ is that the time of propagation is the same along all the paths
by which the light reaches O' from 0. Thus in Fig. 58, if light
diverging from O is refracted by a double convex lens AB to the
point O', any spherical wave diverging from O emerges from the lens
as a spherical wave converging to O'. Every point of this wave is a
pole with respect to O', and all the disturbances which simultaneously
reach 0 are in the same phase. The illumination at O’ is con-
sequently very intense. At points outside the cone O'AB there is
destructive interference and darkness. -
The relation connecting the conjugate focal distances, or the
curvatures of a wave before and after transmission through a lens,

112 REFRACTION CHAP. V.
follows very simply from the principles of the wave theory. In the
first place, let us consider the case of a concavo-convex lens (Fig. 59)
in which the thickness MN and the aperture AB are small compared
with the focal distances. This is taken as a typical case because the
curvatures of its surfaces are of the same sign, whereas in the double
convex lens (Fig. 58) the curvatures of the faces are of opposite signs.
The relation which we are about to deduce might be obtained im-
mediately by applying the formula of Art. 74 to the two faces of the
lens in succession, but for the sake of illustration we shall approach
the problem directly from fundamental principles. The principle on
which the present investigation is based is that the time required by
light to traverse the path OAO' is the same as that required for the
Fig. 50.
path OMNO’. The air equivalent of the latter path is OM + p.MN + NO”
Hence the fundamental equation is
OA-P O'A=0M+ O'N + wMN.
With centre O and radius OA. describe a circle cutting 00 at P,
and with centre O' and radius O'A describe a circle cutting 00 at Q,
then, writing the foregoing equation in the form
(OA – OM)+(0'A-O'N)=u MN,
we have
– PM-H QN = u MN,
which, expressed in terms of the sagittae, gives at once
DP – DM--DQ+DN=4(DN – DM).
Consequently the relation connecting the curvatures of the waves
entering and emerging from the lens with the curvatures of its faces is
DP+DQ=(u – 1)(DN-DM),
or denoting the curvatures of the waves as before by r, and r, and
the curvatures of the surfaces by or and or', we have
ºn 4-ga-(u − 1)(a' – ºr,

ART. 75 REFRACTION THROUGH A LENS 113
where or and o' are the curvatures of the first and second surfaces,
that on which the light is incident being regarded as the first surface
of the lens.
When expressed in terms of the corresponding radii of curvature
the foregoing equation furnishes the relation between the conjugate
focal distances, viz.
1 1 1 1
; ::=(-1)(;-)
P1 O2
where pi and p2 are the distances of O and O' from the lens, and p and
p' are the radii of curvature of its faces AMB and ANB respectively.
Or, if all distances measured on the same side of the lens as the source O
of a divergent beam falling on it, be regarded as positive, and those on
the opposite side as negative
1 1 (; })
— — — = (p. – 1) ( —; – R.
Ol Po (9. \; p/?
a general formula applicable to a lens of any shape.
If the point O from which the light emanates be infinitely distant
the incident wave will be plane; that is, its curvature ori will be zero,
and the curvature of the transmitted wave will be
a 2–(a – 1)(q – o').
The function of the lens is consequently to change the curvature of a Focal
wave by an amount (p − 1)(a — or'), so that this quantity represents ...” of
the curvature-producing power and is termed the focal power of the
lens. In general, therefore, we may say that the curvature of the
transmitted wave is equal to the algebraic sum of the curvature of the
incident wave and the focal power of the lens; that is, the final curva-
ture of the wave is equal to the algebraic sum of its initial and
impressed curvatures.
The point to which a plane wave or a parallel beam of light is
concentrated is termed the principal focus of the lens, and its distance
from the lens is termed the focal length. Denoting the focal length
by f we have
1–1–1 -t, -i \ſ! -1
-}=-ji=(i. 1)(. })
which expresses the focal power in terms of the refractive index and
the curvatures of the faces of the lens.
In the proof on the preceding page the thickness of the lens has
been supposed to be greatest at the centre. In some lenses, however,
the thickness increases from the centre towards the edge, so that their
thinnest part is at the centre. Lenses are accordingly divided into
two classes according as their greatest or least thickness is at the
I
114 REFRACTION - CHAP. W.
centre. The former are termed convex, or convergent, and the latter Con-
cave or divergent lenses.
Ea'amples
1. Prove that the distance between two conjugate foci with respect to a convex lens
cannot be less than four times its focal length.
[Since the sum of the reciprocals of two conjugate distances is numerically equal
to 1/f, and therefore constant for a given lens, it follows that the product of these
reciprocals is greatest when they are equal, and consequently the sum of the distances
themselves must be least when they are equal. Thus
I (; ...) (; 1 y 4
7, - ( — + — ) = ( — — — ) + —,
fº Pl P2 pl P2 PIP2
which shows that the product of the reciprocals is greatest (i.e. pipe is least) when
q-e 1 1 , 1 pi-H pe
a-e, But j-,+;="; -
Hence pi-i-p, must bear a constant ratio to pipe, and consequently pl+p, is least
when pipe is least, which is when p1=p2=2f. In other words, the least distance
between two conjugate points is 4f.]
2. Find the formulae for the magnification due to a thin lens.
[Since the magnification is the product of the magnifications at the two surfaces,
M=I/O=% *1. Pa *g_P3 Pl,
p1 A/2 p2 M3 P1 A3
Here the suffixes 1, 2, and 3 refer to the object-space, lens, and image-space respect-
ively. (When the extreme media have the same absolute index the magnification
is pa/pi.)
Again, if the lens is at O and the principal foci are at F, and F, while the object
and image are at P and Q, the distances F1P, F20 are called the focal distances of
the object and image ; denote them by U and W,
W := F2%) = F.0 + OQ.
These and all similar equations are true in whatever order on the axis the points
occur, provided that the same direction of measurement is always taken as positive.
Any distance AB (i.e. from a point A to a point B) is positive when in the direction
in which the light goes.
The equations applying to the two surfaces being
P2 4 ºz - Pl
P2 Pl p
*3 P2_P3-P9
ps po p' '
the equation for the lens becomes
*_*, *-* +%-49–
——F--- E - (S8,
p3 plp p' Tf (say),
and
OFI = – pºlf, OF2=ps f:
so that
M_* Ps, 4. Witºſ.
98 p. 93 U - Wif
But
UW = – plus fº,
and . .
— W
M=4ſ--".
U p. |
ART. 75 EXAMPLES 115
3. A lens is employed to cast an image of an object on a screen, show that in
general there are two possible positions of the lens for given positions of the object
and screen, and prove that the length of the object is a geometrical mean between
the lengths of the two images.
[In order that any image may be possible the distance between the object and
screen must be greater than 4f. This being the case, let the lens produce a distinct
image on the screen when placed at a distance pi from the object and p3 from the
screen. Then if Z be the length of the object and Z, the length of the image, it follows
that
! pi
i.e. (1)
Now since conjugate foci are interchangeable it follows that the lens will also
produce a distinct image on the screen when the distance of the lens from the object
is pa and from the screen plº so that if lo be the length of the image in this case we
have -
! pe
- = −t. 2
la Pi (2)
Multiplying (1) and (2) together we have
*=ll, or l- Núl,
Hence if li and l, be measured, the length 2 can be determined when the object is
not accessible for the purpose of direct measurement.
This method of obtaining the length of a luminous object, or the distance between
two luminous points is of practical importance in measurements of the wave length
of light (Art. 91, footnote).]
4. If a lens of absolute index us be situated between two media of absolute indices
ph and u2, prove that the curvatures a 1 and a 2 of the waves entering and emerging
from the lens are connected with the curvatures or and o' of the two faces of the lens
by the equation
*101 - M302=(91-93)0 - (42-93)a'.
[In this case we have, by the principle of least time (Fig. 59),
p1OA + p.2O'A = p.10M + paNN + p.2O'N,
which may be reduced at once, as in Art. 75, observing that, in Fig. 59, a 2 is negative.]
5. Find the relation connecting the curvatures of the incident and emergent
waves in the case of a lens of which the thickness c is not small.
[As before let on be the curvature of the incident wave when it reaches the first
surface or of the lens, then the curvature o', of the wave when it just enters this
surface is given by the equation (Art. 74)
Pigi - 930'2–(91-92)0. (1)
Now the wave when it enters the lens travels a distance c before it meets the second
surface, and consequently its curvature o'", when it meets the second surface is by
the Ex. of Art. 60
// — 0 2
& Oſ *T 14-ca', (2)
But by Art. 74 the curvature gº of the wave emerging from the lens is connected
with o', the curvature of the second surface and with a "2 by the equation
Ago"2-page=(u2-pa)0',
and this by (2) becomes
Wºº's
1 + ca'2
- A102=({2-ph)0'.
116 REFRACTION CHAP. W.
Substituting for usa', from (1) we have at once
Phºlt (As-ºl)g
1 + 4*1914 (u2-91)0% clue
This equation connects the distances, p1=1/q, and pg=1/a, of any pair of conjugate
foci from the surfaces of the lens with the radii of curvature of its faces and the
thickness c. If the incident wave be plane we have a1 =0, and the focal power of the
lens is given by the equation
}=2,- (**) {i + (12 †. – o' }.
This expression shows that the focal power is not the same when the light falls on
the surface o' as when it falls on a, so that the focal power changes when the lens is
reversed with regard to the incident light. When the thickness is neglected this
reduces to the expression of Art. 75, and the focal length remains unaltered when
the lens is reversed.]
6. Determine the focal power of a combination of two thin lenses situated at a
distance c apart.
[Let fi and f, be the focal lengths of the two lenses, then if a plane wave falls
upon the lens f, its curvature when emerging from it is 1/f, and its curvature when
A110.2 = - (Ho-pi)0'.
º ... 1 g
it reaches the second lens is fi + c Hence the curvature of the wave emerging from
I
the second lens is
I l
2–-H F,
Ji-H c fº
which expresses the equivalent focal power of the combination. Denoting this by p
and denoting the focal powers of the lenses by pi and p2 respectively we have
_ $1
*=i #2, tº
The focal power when the combination is reversed is obtained by interchanging pi
and p2 or fi and f, in the foregoing expressions.
If a wave of curvature a falls upon the combination its curvature after trans-
mission will be a + p.] -
7. Determine the conjugate positions of object and image for two thin lenses a
distance d apart.
[Measuring from the first lens of focal length f,
-
Measuring from the second lens of focal length ſº,
+----.
where w is the position of the final image. In these formulae all distances are
reckoned positive when in the direction of the incident light. The distance d is
essentially positive. Eliminating v,
(f1 + f2 — d)ww-f.(fi – d)?! -Hºf (f - d.) w ł.fifed=0,
ww.--aw -- buy + c = 0.
&
OT
This can be written
(u +b)(w 4-a)=ab – c.
When w is infinite was – a ; this determines the second principal focus.
When w is infinite wi- – b ; this determines the first principal focus.
ART. 75 - EXAMPLES 117.
Hence w-i-b and w -- a are the focal distances U and W of the object and the
image respectively, and UW =ab – c.
Inserting the values of a, b, and c,
* fºſ,”
'UW = – 7–2+++*—Ră = constant.
( fitſ... — d.)”
If a second object-image pair be taken,
UW =U.W.,
Ol'
W, Hi–––
wºw, Fu-U-- i.
If further,
––w ––ſiſ,
|U1= Wi-F +j.-d'
then
1 1 —Ji + f2 - d * 1.
WTW-UTU-Tº-F
Hence, if the points W1 and Ul be taken as reference points, and y be the
distance of the image from the point W1, and a that of the image from the point
U1, then
1 1 1
yT 27 F
and the equation is reduced to the same form as for a single thin lens. The points
U, and Wi are known as the first and second principal points of the system. Denote
the points themselves by H, and H2,
The magnification due to the system is the product of the magnifications due to
the separate lenses; i.e.
= --ſi-. *-ſ:
* +f f,
––ſ W – a – f.
T f, U – b +f
Hence, remembering that WU= — Fº,
I' W
M=#= - F.
This is true for all object-image pairs and consequently it is true for the principal
points for which U = – W = F. Hence for the principal points the magnification is
plus unity, and they are consequently often known as the unit-points. An object
placed at H1 would give rise to an erect image of the same size at H2. If Hi is
within the pair of lenses it would, of course, be possible only to place at H1 a virtual
object ; i.e. an image formed by a system through which the rays had previously
passed.
Again, from the two values for the magnification here obtained,
_F--W.
T U – F
But,
y=W – W. = W 4-F,
and
Q: E U * Ul = U *E* F,
therefore M=} and it follows that the object and image subtend equal angles at the
points Hi and H2 respectively. The distances of the principal points from the
lenses are
118 REFRACTION CHA P. V.
Oi H1 = Oi F1 +F1BI = * b +U,
==A*–
Ji-Hºf – d. -
O2H2= OeP2 + F2H2= — Q. -- W,
=-—º-
Ji + f2 – d.
Their distance from one another is
H. Ha-HO,--O,O,--O.H.,
===ſ*—+d-–4–=-- *
fi + f2 – d. Ji-Hſ, -d T fl-ºf-d
For example, take two thin lenses, O, and O2, each of focal length 3g, separated
by a distance 2g, *
o,H,-, *g, *-
3g + 3g – 29
o,H,--- *** - - -3
3g-H 3g – 2g 2
-–tº–=
3g-H 3g – 2g
o,F,--b=-ſº-º--ºg--&g,
fl-Hſ, – d. 4g 4
O.F.--a=+%.
2. j
Hi H2= - 9,
The order of the points in this case is
F1, O1, H2, Hi, O2, F2.
Exceptional case : In one case the foregoing transformations become impracticable.
This occurs whenever fi-Ff. - d =zero ; because in this case the principal points and
principal foci are at infinity. In this case the connection between the positions of
the conjugate points becomes simpler because the term of the second degree vanishes
and we are left with a linear equation
f(fº - d.)w-f:(f - d)?' + fifa = 0,
fi + f2 – di 0,
–fºw--fºw-Hºff,(f1 + f2)=0.
It follows that for two pairs of object-image points
f*(w — wh)=f.”(u – wh).
which owing to
becomes
Since the factors of the variables are squares, if the object moves along the axis the
image always moves the same way. The linear magnification of the image is
—ſ.
• -?
constant and equal t
l
This special case is called a telescopic system, because when a telescope is used to
examine infinitely distant objects, the observer's eye being also adjusted for infinity,
the condition fi-i-f. - d is satisfied. The value given for the linear magnification
should be contrasted with the elementary value for the angular magnification,
- —ſ.
V1z., #l
8. Determine the conjugate positions of the object and image for a single thick
lens when the extreme media are different.
ART. 75 EXAMPLES - 119
[The equations for the refraction at the separate surfaces are
where d is the thickness of the lens. The elimination of v gives
7 d d . . d
fi + 3-#)uw- .( -É)ure (ſ -#)w: fº = 0,
( l A42 Paſaſ Ji A42 iſiſ ſº A42 P.1/aſi. *u.
sº
Or ww.--aw -- buy + c = 0.
Proceed as in Ex. 6 or as follows:
Let A = f; +f, – d, F= Jife 3
- 1 TV 2 Ji-Hjø – d.
d
then 7tw – Al (r * #) + (f -ſi g #)w: Tº = 0.
-> 3 Apiz All A pº A41443 A42
Let - _Paſid - Pºſºd
6 QC F (1, Aus' gy= w -- Aus’
then *g_* =l.
Aſ a F
This is of the same form as for a single surface.
The new origins of co-ordinates, from which a and y are measured, are evidently
AP2 AP2
respectively. These are the principal points, H1 and H2. The first and second
principal foci, F, and F2, are at a = — waſ and y= + usB respectively.
If object and image be measured from F1 and Fº
e UW = — plug Fº.
The magnification is
F W
M=** = – ".
AgP
_u, F+W
U - plaf º
If two points, N2 and N1, are chosen such that
- N20=pûF+W,
NAP= U - 93F,
then the last formula shows that
N20
M = **2.
NIP
These points, N, and N2, are called the nodal points. When the extreme media are
alike they correspond with the principal points, and in that case M =# But
- l
this is not true in general. The definitions of N2 and N1 show that, since H,
and Hi are a possible image and object pair,
N2H2=p1F + F2H2
=(u - Ps)F,
NiH1 = FIBA – As E
- (41 - pa)R,
OT NaN1 – H2H1.
120 REFRACTION CHAP. W.
That is, the separation between the nodal points is always the same as that
between the principal points, The two pairs correspond when p1=ps.
It follows that, in the case of small objects at right angles to the axis, if the
extremity P" of the object is joined to N, and that of the image Q be joined to
N2, the lines so drawn are parallel. This can be expressed otherwise by saying
that the object and image subtend equal angles at the respective nodal points.
There are thus six standard (or cardinal points) defining the lens system : the
two principal foci, Fl and F2 ; the two principal points, H1 and H2 ; and the
two nodal points, Ni and N2. Planes through these points and at right angles to
the axis are known as focal, principal, and nodal planes respectively. A know-
ledge of the positions of F1, F2, H1, and H2, is sufficient to determine position and
magnification of an image of a small object at right angles to the axis.
Since we have shown that the formula for any two co-axal surfaces can be made
to have the same form as for a single surface (by suitably selecting origins), it follows
that combining this equation with that for a third surface we can again reduce the
formula for the new combination to the same form as for one surface ; and so on
for any number of surfaces. The final combination will have principal foci, prin-
cipal points and nodal points which have the same characteristic properties what-
ever the number of components may be. In making this extension it must not be
forgotten that the final image formed by any combination is measured from the
second principal H2 point of that combination. So that, if the next component
surface O, is at a distance du 1 from the last surface Ou_1 of the combination, it is
at a distance du 1 – Ou_i H2 from the last image reference point.]
9. Four thin co-axal lenses of focal lengths fi, f. fs, f, are arranged in order,
the successive intervals being di, d, ds. Show that the reciprocal of the focal
length is -
l Ji-Hſ, - d. fº,
0,
===F | Jº, Jº-Hſs-do, J3,
jº, ſº *ſ, *, *ſ, *.
CAUSTICS
76. Evolute of Wave.—The reflected and refracted waves which
we have so far considered have been either plane or spherical in form,
so that the reflected and refracted rays converge to, or diverge from,
a point. In the case of plane waves this point is infinitely distant.
In general, however, when light diverging from a point is reflected or
refracted at a given surface, the front of the reflected or refracted
wave is neither spherical nor plane, and the rays will not pass through
a single point, but will envelop a surface called the caustic. This
want of convergence of the rays is termed astigmatism.
Now the rays are normals to the wave, and the normals to a curve
(or surface) envelop another curve (or surface) called the evolute."
Consequently the reflected or refracted rays envelop the evolute of the
* In the general case when the wave is not a surface of revolution the caustic
surface consists of two parts, or sheets, generated by the two principal centres of
curvature. The caustic is simply the surface of centres of the wave surface, and
when the surface is one of revolution one of the sheets of the surface of centres
degenerates to the axis of revolution.
ART. 77 CAUSTICS 121
reflected or refracted wave. The evolute of the reflected wave is
therefore the caustic by reflection, and the evolute of the refracted wave
is the caustic by refraction.
It will be sufficient for our present purpose to confine our attention
to surfaces of revolution, the incident light diverging from a point on
the axis of revolution. In this case the caustics will also be surfaces
of revolution about the same axis, and the section of the caustic
surface by any plane through the axis will be the caustic curve of
the generating curve of the surface at which the reflection or refraction
Occurs.
Thus if a surface be generated by the revolution of the curve APQ
(Fig. 60) round the axis
OA, and if O be the origin
of light, then rays OP, OQ
falling upon the surface will
be refracted in directions
PM and QN. The curve
BMN cutting these refracted
rays at right angles, or rather
the surface formed by the
revolution of BMN around
OB, will be the refracted
wave. The evolute of the
curve BMN is the envelope
of the rays refracted at APQ; that is, the caustic surface is formed by
the revolution of the caustic curve around OA.
77. Primary and Secondary Foci...—Let the refracted rays PM and
QN, produced backwards, intersect at F, ; then F is a point on the
caustic curve if the rays PM and QN be considered infinitely near.
Now if the whole figure be revolved round OA through a very small
angle, PQ will describe a little element of the refracting surface, MN
an element of the refracted wave surface, and F, a short line on the
caustic surface perpendicular to the plane of the paper. The element
of the surface formed by the revolution of PQ round OA receives a
cone of light from O. This cone gives rise to a refracted cone which
falls upon the element of the wave surface generated by MN, and the
rays of this latter cone produced backwards pass through a short line
on the caustic surface at F1.
This line is an element of the circle described by F, in revolving
round the axis OA. It is consequently perpendicular to the plane of
the paper; that is, to the primary plane of the refracted cone. (The
plane containing the axis of the cone and the axis of the surface is
Fig. 60.

122 REFRACTION CHAP, V
Focal
lines.
termed the primary plane of the cone, while a plane perpendicular to
this through the axis of the cone is called the Secondary plane.) The
cross section of the refracted cone at F is consequently termed the
primary focal line of the cone. Now in the case of a surface of revolu-
tion it is clear that every normal to the surface meets the axis of
revolution, and the point in which any normal NF, meets the axis
remains fixed while the curve BMN revolves. The point F, is there-
fore the vertex of a cone of rays passing through the circle described
by N about the axis OAB. Similarly every other point on the axis
is the vertex of a right circular cone of rays cutting the wave surface
Orthogonally. Hence the wave normals at two adjacent points M and
N meet the axis of revolution at two adjacent points, embracing be-
tween them an element of the axis which remains the same while the
curve BMN revolves round the axis. The elementary refracted cone
of rays previously considered consequently passes through a second
line at Fº-namely, an element of the axis of revolution. This is the
Secondary focal line of the refracted cone. It lies in the plane of the
paper, and is consequently perpendicular to the primary focal line."
The cross section of the refracted cone at F., by a plane perpendicular
to its axis is not a line, but in general a figure-of-eight-shaped curve.
As the section of the refracted cone degenerates to a line at F, and
again to a line at F, and as these lines are at right angles, it follows
that any cross section between F1 and F, will be an oval curve having
diameters in and perpendicular to the primary plane, which reduce to
zero as the cross section moves to F, or F, respectively. Consequently
Circle of at some place between F1 and F, the diameters of the cross section
least con-
will be equal. The section will here be a circle, or approximately
such. This section is called the circle of least confusion.
The existence of the focal lines and the circle of least confusion
can be easily ascertained by reflecting obliquely from a concave mirror
a cone of light diverging from a bright point, and receiving the re-
flected cone on a white screen or sheet of paper. For one position of
the screen a well-defined line perpendicular to the primary plane is
depicted. This is the primary focal line. As the screen is moved
away farther from the surface the line broadens out into an oval spot,
which in one position of the screen is very nearly a circle—the circle
of least confusion. On moving the screen farther away the oval
* In the general case when the wave surface is not one of revolution the normals
to the surface (i.e. the rays) intersect each other only when taken along the lines of
curvature. These lines form two orthogonal systems on the surface, and the corre-
fusion.
sponding intersections of rays form a surface of two sheets (the caustic surface or
surface of centres). The primary focal line F1 is an element of a line on one sheet
and F, is an element of a line on the other.
ART. 77 EXAMPLES 123
narrows in perpendicular to the primary plane till it again becomes a
line lying in the primary plane. This is the secondary focal line.
When the incidence of a small cone is direct the focal lines and
circle of least confusion all coincide with the geometrical focus C (Fig. Cusp,
60). This is the point where the caustic meets the axis of revolution.
It is a double point or cusp on the caustic curve.
Caustics by reflection may be easily shown by allowing the light
of the sun or lamp to fall upon a narrow riband of polished steel
such as a watch spring. Placed on a sheet of paper and bent into
any desired curve, the spring shows a well-defined bright caustic on
the paper, the part within being brighter than that without the curve.
By varying the form of the spring a variety of caustics with cusps,
contrary flexures, and other singularities may be exhibited. The
plane of the sheet of paper should pass nearly through the sun or
source of light. The bright curve seen upon the surface of a cup of
tea is a familiar example of the caustic of a circle.
Examples
1. Light diverging from a point is refracted at a plane surface, prove that the
caustic curve is the evolute of a conic section.
[Let O (Fig. 61) be the luminous point,
PD the reflecting surface, OP any incident
ray. Draw 0D perpendicular to the sur-
face, and produce it till DO'-DO. De-
scribe a circle about OPO' and produce
the refracted ray PQ backwards to meet
the circle at M. Join OM and O'M.
Then since D is the middle point of
00, the arc OP=0'P or the angle
PMO = PMO’; that is, the refracted ray
bisects the angle M. Now PMO = Fig. 61.
P00'-i, the angle of incidence, and PND=r, the angle of refraction. Hence
sin i_ON O'N
*=sin ºr OM - O'M’
since MN bisects the angle M. Consequently
_ON +O'N
** OMI. O’M’
The locus of M is therefore an ellipse having 0 and O' for foci, and the refracted ray
PQ is a normal to the ellipse at M. The caustic is therefore the evolute of this conic.
The major diameter 2a and the eccentricity e of the conic are given by
2a – 00'ſu, and e=u.
O
or OM - O'M = º = constant.
If the refraction takes place from the rarer to the denser medium, the refracted ray
bisects the angle M externally. M lies between 0 and P, and MO-MO is constant.
The locus of M is a hyperbola, with 0 and O' for foci, major diameter 00'ſu, and
eccentricity u.

124 REFEACTION CHAP. V.
The refracted wave front is got by finding a series of points Q such that
OP+ uPQ=const. The locus of the points Q will be the refracted wave front
cutting the rays PQ normally. The caustic surface is formed by the revolution of
the foregoing evolute round 00'.
The reflected waves in this case are spheres concentric with O', so that the reflected
rays all appear to come from O'.]
2. Parallel rays are reflected from a spherical mirror,
find the caustic.
[Let ACB (Fig. 62) be a section of the mirror, QR
the direction of the incident light, and RP a reflected
ray. With centre O (the centre of the mirror) describe
a circle of radius OF =#0C. Join R0 and upon RT
as diameter describe a circle, centre 0'. Then O'T-
#OT. Let the reflected ray cut this circle at P. Now
the angle POT-2PRT =20RO =2TOF. Therefore in
length the are TF =arc TP.
If therefore the circle RPT rolls on the circle TF
the point P will trace out the epicycloid APFB, the
cusp of which is at F, and since RPT is right, and PT
is the normal to the path of P, it follows that RP is
a tangent to the path. Hence the reflected rays envelop an epicycloid, the cusp of
which is at the principal focus of the mirror.]
3. Light diverging from a point on the circumference of a circle is reflected from
the concave arc, find the caustic.
[The caustic is an epicycloid formed by the revolution of a circle on an equal
circle.]
4. Light diverges from a-luminous point situated outside a sphere, find the
caustic by refraction. -
[A surface of revolution generated by the evolute of a Cartesian oval whose foci
are the luminous point and its inverse with
respect to the sphere.]
5. Find the geometrical focus of a pencil of
light after direct refraction at a plane surface.
[Let the light diverge from O (Fig. 63), and
let OP be any ray nearly parallel to OA, the
perpendicular from O on the surface. Then PA
is small, and OP=0A nearly. If the refracted
ray produced backwards meets OA at F, then
Fig. 62.
FA PF_sin i_ - ! Fig. 63.
OAT PO sin rº", -
so that if OA and FA be denoted by p, and p, respectively, we have
p2=ºpl.
The focus is therefore farther away from or nearer to the surface than the point
O according as u is > or - 1, that is, according as the light passes from the less
to the more refracting medium, or vice versa.
To an eye situated in air objects under water appear nearer the surface than
they really are, while to an eye placed under water objects in the air appear farther
away.]
6. Find the geometrical focus of a pencil of rays after direct refraction at a
spherical surface.
[Let C (Fig. 64) be the centre of the sphere and P any point in it, so that OP


ART. 77 EXAMPLES 125
is nearly parallel to OC. Then if the refracted ray meets OC at F, we have CPO = i
and CPF=r, and
PF_sin O_uCF
PO sin FT CO
Ol'
Ps uſes p) . . . " |
pi (p → p) po p, p
This formula coincides with that which determines the position of the focus by
reflection at a spherical mirror when we put
u- -1, a general substitution which converts
formulae of refraction into the corresponding
formulae of reflection.]
7. Light diverges from a point F, ; to find
the surface which will refract it accurately to
another given point F.
[If p and pº be the distances of F, and F.
from any point P of the surface, then the time
along pi added to the time along pº must be Fig. 64.
constant, or
Pl + P2– constant.
th tº
That is,
plºt ºpe= k.
This is the equation of a Cartesian oval of which F, and F., are the foci.
If the light be parallel, instead of diverging from a point Fu this surface becomes
a spheroid of eccentricity 1/u, and having F., for a focus.
These surfaces were originally studied by Newton and Descartes, and are termed
aplanatic surfaces (see Newton's Principia, book i. § 14, prop. 97).]
* 8. A small oblique pencil is reflected at a spherical
surface to determine the primary and secondary foci.
[Let C (Fig. 65) be the centre of the surface, O the
origin of light, OP and OQ incident rays, and PF, and
QF, the corresponding reflected rays and F, the primary
and F., the secondary focus. Let OP=p, PF=p1,
PF2=p, CP=R, PCA = 0. Then since CP and CQ bisect
the angles at P and Q respectively, we have
2PCQ=POQ+ PFC),
but the perpendicular from P on OQ is equal to p. POQ,
and also to PQ cos i. Therefore P00 = PQ cos iſp. Similarly PF0= PQ cos iſpi,
while PCQ=PQ/R. Therefore
Fig. 65.
*** *
p, "p R cos i
Further to determine pa we have the triangle OPF2=OPC+CPF:, or pH sini +
Rp, sin i =ppa sin 2i. Hence
111_2 cos ‘..]
P2 p R
9. A small pencil is incident obliquely on a plane refracting surface, find the
primary and secondary foci.
[Let OP and 00 (Fig. 66) be two incident rays, PF, and QF, the refracted.
Then if PF = p), PF2=ps, OP=p, we have
sin i_PF2 pg , ,
sin r PO p? or p2=ºp,


126 REFRACTION CHAP. V.
which may be written in the form
Again, if the angles of incidence and refraction for OQ be tº di and r + ºr, PF0=dr,
POQ=di, and we have from the triangles
PF, Q and POQ
º - --
PQ_ ºr , and PQ dº --
pl cos r p cos ?
But sin i = a sin r, therefore cos idi-
a cos rºlr, consequently
ucosºr cos” 0
Fig. 60. Pl p -
ul-
cosº r
P-ºp. ºl
10. A small pencil of light is refracted obliquely at a spherical surface of radius
R, find the foci.
[Denote the angles subtended by PQ at O, Fº and C, by a 3, Y, respectively.
Then, since these angles are very small, we have (Fig. 67)
PQ cost a PQ cosº PQ.
*- : *-*. R
But since the triangles PMO and QMC have equal vertical angles, the sum of the
base angles of one is equal to the sum of
the base angles of the other, or
a + i = y + i +di, ... di = a – Y.
Similarly from the triangles PNF, and
QNC we have
3+ r =y+ r + dr, . . dra B-y.
Substituting for a, 3, Y, we find
- cos 1 'cos r 1
a Poſº ||) ºr-Roſº).
But since sin i = usin r, we have
Fig. 67.
cosidi–w cosºrdr.
cos i (cos i *)-wº *(*-*)
( p R pi R/'
a cosºr cos^i_ucos r-cost
Pl p R
Again, since the area of the triangle OPC is equal to the sum of the areas of OPF,
and CPF, we have
Therefore
ol.
pR sin i =ppa sin (i-r) + Rposin r,
or dividing across by pp.Rsin r, we find
-º-º:
di'
w_1_ucos r-cost
ſº p R


ART. 77 EXAMPLES 127
Hence both results are incorporated in the equations
ucosºr cos”?_ucos r-cost u_1
Pl p R p2 p
11. Determine the foci of a small pencil refracted obliquely through a prism in a
principal plane.
[Using Ex. 9 and neglecting the thickness of the prism we find
_cos” cos” i.
Pi Ticosºcos.7%
p2 = p.
Hence if the prism be in the position of minimum deviation (Art. 78) i = i', and
r= r", and we have
p1 = pa F p,
or the foci coincide. The refracted pencil therefore diverges from a point at the
same distance (p) as the origin from the edge of the prism. This result is of import-
ance in the study of the spectrum.
If the refracting angle of the prism be very small, the angles i, i', ', r", are very
nearly equal, and we have again p1=p2 = p.]
12. Prove that for a prism of angle A
sin #(A + 3) ...,gosł(r-r').
sin A Tº cos (; – d’)
[We have 6 = i + ' – A, and r + r = A, therefore
sin #(A + 3) sin #(t+7).
sin #A T sin #(r--r')
But sin * = p sin r, and sin i' = a sin r", therefore by addition we have
sin #(i+ iº) cosł(; – ?) = p, sin #(ºr + r') cos (r – r"),
and consequently
sin #(i+?) cos #(r-r')
sin (r. 7)7"cos (; – )' etc.
The least value of cosł(r-r')/cos #(i – ') is unity, and when this happens i-i'
and r=r', or the deviation is a minimum. For if i-i' we have i – 7'-' — r", or
* – ? - ?' – r". Hence cos #(? – ?) is less than cos #(, – r").]
13. Show that for a prism . _ _cos r cost.tºos,
?, COS 7" COS 7,
[We have sin i = u sin r and sin i = u sin 7", whence cos idi = p, cos ral, and
cos iſdi' = a cos r dº". But r + r = A, whence dr--dr' = 0, etc. The ratio — . is called
the angular magnification, because, if two incident rays make with each other the
angle di, the emergent rays make with each other the angle – di'. It follows that
the angular magnification is unity for minimum deviation. This magnification is
greater than unity when i > iſ unity.]
14. When two differently inclined rays have the same deviation the value of i for
one is that of i' for the other. Hence if m is the angular magnification for one case,
1/m is the value for the second. An important method, due to Schuster, of adjusting
the collimator of a spectroscope for infinity is founded upon this fact. (See Art. 79.)
15. Prove that if the refractive index of a prism be changed by an amount du,
the deviation 6 will be changed by an amount dò, where
sin A
dô = −1·ſ&p.
cost cost
128 REFRACTION CHAP. V
T . [We have r + r' = A, sin i = p sin r, sin ''-pºsin 7" where A and are constants,
6= i+? – A, and consequently d6=di'.
For the minimum deviation 7'-r' =#A, and the above becomes
2 sin #A
TFERdu,
V1 – u” sin” #A*.
consequently the angle of dispersion of any two colours increases with the angle of
the prism.] -
16. Prove that the position for minimum dispersion is not the same as that for
minimum deviation.
17. If a ray of light pass through a system of prisms of vertical angles A1, A2,
etc., and if a12, a2s, etc., denote the angles between the faces of the consecutive
prisms, prove that the deviation is given by the equation
d6 =
6== i+ iºn 4-2a – 2A
where it is the angle of incidence on the first prism, and i', the angle of emergence
from the last.
[We have 6=2(i+ i - A), and also the relations tº +ia-ana, i'24-is-ags, etc., con-
sequently 2(?--i') — ?) – iºn=20, therefore, etc.]
18. Determine when the deviation produced by a given system of prisms will be
a minimum. -
[By Example 14 the deviation will be a minimum when
dil + di'n = 0.
Now for each prism dr-dr' = 0, cosidi–p cos rar, cost'di' = p cos r'dr', and therefore
tº */
COS 7, , , , COS 7, , ,
--dº -- ;di' = 0,
COS 7" COS 7"
cos r cos ?
cos icos r'
etc., which with the relations di'1+dia- dala-0, etc., become
or, writing =f, we have the series of relations dii-Hſidi'i =0, die-Hjad'2–0,
Hence by multiplication and the relation di, 4-diºn =0 we have
1=Ji.fs.f, . . . ſn.
By Example 11 it appears that the primary focal distance is given by the equation
p1 = f°p, hence for the system of prisms we will have
p1=(ſif.ſ. . . . f.)"p,
which in the case of minimum deviation becomes
p1 = p,
or the system in this case is still aplanatic.]
19. When rays issuing from a luminous origin are reflected at a given surface the
two sheets of the caustic surface have a finite number of points in common. The
intensity of the reflected light is greatest at these points, and each of them is the con-
jugate focus of an ellipsoid of revolution described with the source of light as one
focus and having contact of the second order with the mirror (Hamilton, Trans. Roy.
Irish Academy, vol. xv. p. 107).
20. Along a given ray the intensity varies inversely as the product of the dis-
tances from the two foci of the pencil (ibid.).
21. Prove that, ignoring absorption, scattering, etc., the intrinsic brilliancy
(Art. 41a) of an extended object and of its image are the same.
ART. 77 NEWTON'S EXPERIMENTS 129
[Take the angles as small, and the object and image normal to the axis. The
light received by the area A of the lens is Ids º ; the light received by the image is
... A ds' ds *
I'ds Ž' But F37 º' Therefore I = I’. {_T= {
/
(Clausius has shown that this is true ds 7" 7. / |ds
always if a perfect image be formed. W
The problem is analogous to the corre- Fig. 67a.

sponding principle in heat, according to
which it is impossible to raise the temperature of a body above that of a source
by concentrating heat radiated from the latter upon it.)]
NEWTON'S EXPERIMENTS
“Lights which differ in colour, differ also in degrees of refrangi-
bility” (Newton, Opticks, book i. prop. i. theorem 1).
Eaper. 1.-I took a black oblong stiff paper terminated by parallel
sides, and with a perpendicular right line drawn across from one side
to the other, distinguished it into two equal parts. One of these
parts I painted with a red colour and the other with a blue. The
paper was very black and the colours intense and thickly laid on, that
the phenomenon might be more conspicuous. This paper I viewed
through a prism of solid glass, whose two sides through which the
light passed to the eye were plane and well polished, and contained
an angle of about sixty degrees, which angle I call the refracting angle
of the prism. And whilst I viewed it, I held it and the prism before
a window in such a manner that the sides of the paper were parallel to
the prism, and both those sides and the prism were parallel to the
horizon, and the cross line was also parallel to it; and the light which
fell from the window upon the paper made an angle with the paper,
equal to that angle which was made with the same paper by the light
reflected from it to the eye. Beyond the prism was the wall of the
chamber under the window covered over with black cloth, and the cloth
was involved in darkness that no light might be reflected from thence,
which in passing by the edges of the paper might mingle itself with
the light of the paper and obscure the phenomenon thereof. These
things being thus ordered, I found that if the refracting angle of the
prism be turned upwards, so that the paper may seem to be lifted
upwards by the refraction, its blue half will be lifted higher by the
refraction than its red half. But if the refracting angle be turned
downwards, so that the paper may seem to be carried lower by the
refraction, its blue half will be carried something lower thereby than
its red half. Wherefore in both cases the light which comes from the
blue half of the paper through the prism to the eye does in like circum-
R
130 REFRACTION CHAP. V.
stances suffer a greater refraction than the light which comes from the
red half, and by consequence is more refrangible.
Exper, 2–About the aforesaid paper, whose two halves were painted
over with red and blue . . . I lapped several times a slender thread
of very black silk in such a manner that the several parts of the
thread might appear upon the colours like so many black lines drawn
over them, or like
long and slender
dark shadows cast
upon them. I might
have drawn black
lines with a pen,
but the threads
Fig. 68. were smaller and
better defined. . . . I placed a candle to illuminate the paper
strongly, for the experiment was tried in the night. . . . Then at a
distance of six feet and one or two inches from the paper upon the
floor I erected a glass lens four inches and a quarter broad, which
might collect the rays coming from the several points of the paper,
and make them converge towards so many other points at the same
distance of six feet and one or two inches on the other side of the
lens, and so form the image of the coloured paper upon a white
paper placed there. . . . The aforesaid white paper . . . I moved
sometimes towards the lens and sometimes from it, to find the places
where the images of the blue and red parts of the coloured paper
appeared most distinct. Those places I easily knew by the images
of the black lines which I had made by winding the silk about the
paper. For the images of those fine and slender lines (which by
reason of their blackness were like shadows on the colours) were
confused and scarce visible, unless when the colours on either side of
each line were terminated most distinctly. Noting, therefore, as
diligently as I could, the places where the red and blue images of the
Dispersion coloured paper appeared most distinct, I found that where the red
of foci half of the paper appeared distinct, the blue half appeared confused,
so that the black lines drawn upon it could scarce be seen; and, on
the contrary, where the blue half appeared most distinct the red half
appeared confused, so that the black lines drawn upon it were scarce
visible. . . . The distance of the white paper from the lens when the
image of the red half of the coloured paper appeared most distinct being
greater by an inch and a half than the distance of the same white
paper from the lens when the image of the blue half appeared most
distinct. In like incidences, therefore, of the blue and red upon the

ARt. 77 NEWTON'S EXPERIMENTS 131
lens, the blue was refracted more by the lens than the red, so as to
converge sooner by an inch and a half, and therefore is more
refrangible.
“The light of the sun consists of rays differently refrangible”
(Newton, Opticks, book i prop. ii, theorem 2).
In a very dark chamber, at a round hole about one-third part of
an inch broad, made in the shut of a window, I placed a glass
prism, whereby the beam of the sun's light, which came in at that
hole, might be refracted
upwards towards the op-
posite wall of the chamber,
and there form a coloured
image of the sun. The
axis of the prism (that is,
the line passing through
the middle of the prism
from one end of it to the
other end parallel to the
edge of the refracting angle) Fig. 69.
was in this and the following experiments perpendicular to the
incident rays. About this axis I turned the prism slowly, and saw
the refracted light on the wall, or coloured image of the sun, first
to descend and then to ascend. Between the descent and ascent Minimum
when the image seemed stationary, I stopped the prism and fixed "**
it in that posture that it should be moved no more. For in that
posture the refractions of the light at the two sides of the refract-
ing angle—that is, at the entrance of the rays into the prism and
at their going out of it—were equal to one another. . . . The prism
therefore being placed in this posture, I let the refracted light fall
perpendicularly upon a sheet of white paper at the opposite wall of
the chamber, and observed the figure and dimensions of the solar
image formed on the paper by that light. This image was oblong,
and not oval, but terminated with two rectilinear and parallel sides,
and two semicircular ends. On its sides it was bounded pretty
distinctly, but on its ends very confusedly and indistinctly, the
light thus decaying and vanishing by degrees. The breadth of this
image answered to the sun's diameter, and was about 2% inches
but the length of the image was about 10} inches, and the
length of the rectilinear sides about 8 inches, and the refracting angle
of the prism, whereby so great a length was made, was 64 degrees.
With a less angle the length of the image was less, the breadth
remaining the same. If the prism was turned about its axis that way

132 REFRACTION CHAP. V.
which made the rays emerge more obliquely out of the second
refracting surface of the prism, the image soon became an inch or two
longer, or more; and if the prism was turned about the contrary way,
so as to make the rays fall more obliquely on the first refracting
surface, the image soon became an inch or two shorter. And, there-
fore, in trying the experiment I was as curious as I could be in
placing the prism by the above-mentioned rule exactly in such a
posture that the refractions of the rays at their emergence out of the
prism might be equal to that at their incidence on it.
Now the different magnitude of the hole in the window-shut and
the different thickness of the prism where the rays passed through it,
and the different inclinations of the prism to the horizon, made no
sensible changes in the length of the image. Neither did the different
matter of the prisms make any ; for in a vessel made of polished
plates of glass, cemented together in the shape of a prism and filled
with water, there is a like success of the experiment according to the
quantity of the refraction.
This image of the spectrum was coloured, being red at its least
refracted end and violet at its most refracted end, and yellow and
blue in the inter-
mediate spaces. .
I then placed a
second prism im-
mediately after the
first in a cross posi-
tion to it, that it
might again refract
the beam of the
sun's light which
came to it through the first prism. In the first prism the beam
was refracted upwards, and in the second sideways. And I found
that by the refraction of the second prism, the breadth of the
image was not increased, but its superior part, which in the first prism
suffered the greater refraction, and appeared violet and blue, did again
in the second prism suffer a greater refraction than its inferior part,
which appeared red and yellow, and this without any dilatation of the
image in breadth.
Exper, 6.-In the middle of two thin boards I made round holes
a third of an inch in diameter, and in the window-shut a much
broader hole being made to let into my darkened chamber a large
beam of the sun's light, I placed a prism behind the shut in that
beam to refract it towards the opposite wall, and close behind the
Fig. 70.

ARt. 77 NEWTON'S ExPERIMENTS 133
prism I fixed one of the boards in such a manner that the middle of
the refracted light might pass through the hole made in it and the
rest be intercepted by the board. Then at a distance of about twelve
feet from the first board I fixed the other board in such manner that
the middle of the refracted light which came through the hole in the first
board and fell upon the opposite wall might pass through the hole in
this other board, and the rest being intercepted by the board might
paint upon it the coloured spectrum of the sun. And close behind
this board I fixed another prism to refract the light which came through
the hole. Then I returned speedily to the first prism, and by turning
it slowly to and fro about its axis, I caused the image which fell upon
Fig. 71.
the second board to move up and down upon that board that all its
parts might successively pass through the hole in that board and fall
upon the prism behind it. And in the meantime I noted the places
on the opposite wall to which that light after its refraction in the
second prism did pass; and by the difference of the places I found
that the light which being most refracted in the first prism did go to
the blue end of the image, was again more refracted in the second
prism than the light which went to the red end of that image, which
proves as well the first proposition as the second. And this happened
when the axes of the two prisms were parallel or inclined to one
another, and to the horizon in any given angles.
“Homogeneal light is refracted regularly without any dilatation
splitting or shattering of the rays, and the confused vision of objects
seen through refracting bodies by heterogeneal light arises from the
different refrangibility of several sorts of rays” (Opticks, book i. prop.
v. theorem 4).
Erper. 12.-In the middle of a black paper I made a round hole
about a fifth or sixth part of an inch in diameter. Upon this paper I
caused the spectrum of homogeneal light described in the former
proposition so to fall that some part of the light might pass through
the hole of the paper. This transmitted part of the light I refracted

134 1&EFRACTION - CHAP. W.
with a prism placed behind the paper, and letting this refracted light
fall perpendicularly upon a white paper two or three feet distant
from the prism, I found that the spectrum formed on the paper by this
light was not oblong, as when it is made (in the third experiment) by
refracting the sun's compound light, but was (so far as I could judge
by my eye) perfectly circular, the length being no greater than the
breadth, which shows that this light is refracted regularly without any
dilatation of the rays.
Note.—In finding the focal length of a lens by an image cast on a screen,
observe that when the screen is too near the edge of the image is red, and
when too far the edge is blue ; so by this means the screen can be placed
very accurately.
ABERRATIONS OF LENSES
We shall now briefly indicate the investigation of several defects
an image possesses in addition to it being represented by focal lines.
(a) Central Spherical Aberration at a Single Refracting Spherical Surface.
—Referring to the figure where P is the object, Q the point where
a refracted ray cuts the axis, C
the centre of curvature, and A
the point of incidence, we have
the following equations:
CQ_sin r CP_sin i
o C Q P AQTsin P APTsin P
Fig. 71a. . . CQ. AP =*0.
AQ CQ pi
Let v = OQ and u = OP, then the equation becomes
v– R AP up
v-R AQT up
OI’
p1(v – R) poſu - R)
JºRWEREFLEE ºf Jºãº
This equation is exact, and there is one object-image pair which satisfies
it exactly. For if -
w – R v – R. and Boſw-R)_A(v-R)
af2 Toy2 ” 20 Q)
2
the equation is satisfied. Solving these two equations simultaneously
pºll = pºv = (p, + p.)R. These conjugate points are called aplanatic
points. In other cases the position of the image is different for
different values of W. We shall suppose the angle p so small that
powers of it beyond the second can be neglected.
Write the quantities under the square root sign in the form
• Vi-ºwn)
Q}

ART. 77 ABERRATIONS OF LENSES 135
and put sin”(/;) = |*/A, and expand. The equation becomes
1 1 (v–R)*W* ... / 1 1 \ , up(w-R)*p”
*|†-; +**=p(-i) *:::::--
Q)
Now when p = 0, the image is at v, where
| 11–, ſl 1.| •
All #-.]-ºſ, w
hence by subtraction
All | * * ; – *{ *; r)*_ *: ºr) g
On the right hand the approximate value of w (viz. v.) may be used
and we may use the equation
, (w- R}_*(vo — R).
*0 a. vo
*[. -º-º-º:*(;-ſº \.
'v vo º pºw ºvoſ
whence finally
The value of vo may be eliminated by the elementary equation for
images, and it is more simple to deal with reciprocals.
Let - -
a = 1/w, 8-1/vo, p=1/R, and *-*=de.
Q Vo
then
de-ºº-ºo-º-1)*
where y =Rſ. This value is the small difference between reciprocal
distances. But -
1 l . 1
d (...) — — º * jº,
whence dv, the longitudinal aberration, is easily calculated. Repeating
Q,
N
I Fig. 71b.
the operation for the second surface of a thin lens, the aberration for
such a lens can be obtained.
(b) Focal Lines Due to Central Oblique Passage through a Thin Lens.—
Let i be the angle between the axis and the chief ray of the thin
pencil (Fig. 71b). The retardation for the ray that goes straight

136 REFRACTION CHAP. V.
through is (u cos r – cos ?) e where e is the thickness and r the
refraction angle in the lens. An extreme ray PAQ, in the plane
of the diagram, owing to its extra length of path, is retarded by
2 -
#( +}) • For the focus Q, these retardations are equal. Putting
$) = B cos ? we obtain
I 1 ., 26
; Fiji (a cos r-cos ”Bººcos2?"
But when i = 0
1, 1 (1 − 1)2e 1 1
**** B2 =(p. p(#).
*-*.
Jo
Hence
* ...}\ …— * ... / 1 1
(+.) cos” i- (a cos r – cos i) (***)
This gives the position Q, of the primary focal line.
If we take an extreme ray at right angles to the plane of the
diagram the only change is that for y we must write B. Hence, if
OQ, = va,
+}=(acoºr-coºd (#)
This gives the secondary focal line Q, The distances are taken
as positive when as on the diagram. If we measure in the direction of
the incident light as positive, the signs of u and R, must be changed.
(c) Curvature of the Image.—The image formed in the previous case
is twofold. But in general it has the additional defect that neither of
the two images, in general, is plane, even if the object is plane. This
property is known as the curvature of the image. We will consider
- , that the object PP' also
is curved (Fig. 71c);
and, as we will only
deal with a small value
of i, that they may be
replaced by their oscu-
latory spheres.
1
If we put k = 1 + 5.
Fig. 71c.
I for a primary focal sur-
face, and equal 2. for a secondary focal surface, both may be treated
simultaneously.
The equations required are
l, 1-lit Kë
v 'w f :
and
1 1 1
a- *
vo 100 f

ART. 77 ABERRATIONS OF LENSES 137
If we divide the former by cosi (i.e. 1 – ?/.) and subtract it from
the latter, we have
QN PM _(k+3)? &
ONOO− OMOP-TTFT
In the limit for small angles ON. OQ = (ON)” and OM. OP = (OM)”;
also if QQ' and PP' are approximately circles [QN]*=QN. 2p nearly,
and (PM)” = PM. 2po nearly where p and po are their radii of curvature.
Hence
1 ſ Q'N \*__1_ſ PM \*_ _ (k++)*
2p \ON 2p, U OMJ T f :
and finally, because each of the quantities in brackets equals sin” =?”
1 1 2k + 1
;------
Inserting the appropriate value of k gives one or other of the
curvatures. The locus of the circles of least confusion is between
these two surfaces.
(d) Helmholtz's Equation.—Let the object and the image formed
by an optical system be at P and Q on the axis (Fig. 71d). If PP' is
a small object line at right
/ //
angles to the axis its image is p’ p" Q Q
a similar line QQ'. Now PP'
may be regarded as a wave
front (when the curvature of JP Q

this can be neglected) and QQ' Fig. 71d.
as the wave front in a later
position. Let PP" and QQ" represent another pair of wave fronts
making a small angle with the former pair. Then the time to
pass from P' to Q' must be the same as from P to Q, and the latter
must be the same as from P” to Q"; hence
p.P'P” – aſ Q'Q”
where p, and pſ are the indices for the extreme media.
Let PP"|PP' = a and Q'Q"/00 = a,
then
gaPP'-p'a'QQ'.
This is known as Helmholtz's equation. It gives the linear magnifi-
cation QQ'ſ PP' in terms of the angular magnification aſſa. When the
extreme media are alike these magnifications are the reciprocal of each
other. An example of this law has already been given in connection
with telescopic systems at the end of Art. 75, Ex. 7.
(e) Abbe's Sine Condition.—Let PP and QQ' (Fig. 71e) be a small
object and image at right angles to the axis. Let PS and P'S be
two incident rays from the extremities of the object, and let QS', Q'S'
138 REFBACTION CHAP. W.
be the corresponding emergent rays. The points S and S are there-
fore possible object and image points and the optical lengths of the
distances between them are the same along each path. The optical
length from P to Q is (if p and p" are the refractive indices of the ex-
Fig. 71e.
treme media) pr’S +[SS'] – p'QS', and the optical length from P' to Q' is
AP'S +[SS']— u'Q'S'.
Moreover, each of these optical lengths is a constant for a small
range of 6, provided that the system gives a point-image of a point.
Hence
u(PS–PS) – u'(QS' – u'. Q'S')= constant,
or in terms of 6 and 6'
p.PP" sin 6 – p". QQ' sin 6' = constant.
But 6 and 6' vanish together, and therefore the constant is zero.
Hence
p. PP' sin 6 = p.'QQ' sin 6'.
Now if the magnification is to be the same for all such small pencils,
QQ'ſ PP = constant, and therefore
p' sin 6"/(u sin 6) = constant
for all the rays of the wide angle pencil by which the image is formed.”
If this condition is not satisfied the image of a small body on the
axis (such as PP') formed by the rays passing through different zones
of the system will be different. In other words, although Q is by
f
/
|
t
—w
P O
Fig. 71.f.
supposition an exact image of P, Q will be a blurred image of P'
when formed by a wide angle beam.
This important theory can also be proved in an alternative way.
Charles Hoskin, Jr., J. Royal Micros. Soc, vol. iv. p. 337 (1881). See also
J. D. Everett, Phil. Mag., 6th ser. vol. iv. p. 166 (1902).





ART. 77 - ABERRATION OF LENSES 139
Let P and Q represent a small object, PP', and its image, QQ',
both object and image being at right angle to the plane of the
diagram. Let PM, PN be two incident rays in a plane perpendicular
to the diagram, and M'Q, N'Q the corresponding emergent rays.
Then from the axis being an axis of revolution the angle MON = angle
M’O'N' = d.d. (say).
Then
MPN=A=\;=\9:3°–
PN *E*=sin 6. dq,
and similarly
M'QN' = sin 6'dq).
But by Helmholtz's formula
p.a. PP' = p/a'. QQ',
p and p' being the first and last indices.
Hence
p sin 6. PP' =p'sin 6'QQ',
which is the same result as before.
(f) Stops and Pupils.-The character of a lens system is different
according as wide or only narrow pencils succeed in passing through it.
Any diaphragm introduced to cut off marginal rays is known as a stop.
e-"
sº- > L^T hºmºmº
_2~<
_-_-

Fig. 719.
In the case of a single meniscus lens as shown, if a small stop is
suitably placed on the object side the rays that fall on the lens are
limited to narrow pencils each coming from one point of the object.
These narrow pencils are imaged as focal lines with a circle of least
confusion (where the beam is narrowest) between these lines. By
suitably placing the stop it is found possible to bring all these circles
approximately to one plane, and thus do away with curvature of
the field. When no actual diaphragm is used in a lens system the
boundaries of the lenses act as stops.
Let the image of each stop be calculated as formed by all the lenses
that precede the particular stop. It is clear that the widest angle
pencil that can pass through the system from the axial point of the
object, is determined by the particular stop-image which subtends the
1 40 REFRACTION CHAP. W.
smallest angle at that axial point. This stop-image is called the
entrance pupil. The image of the entrance pupil formed by the entire
system is called the eſcit pupil; all the rays that pass through the
entrance pupil must pass through the exit pupil. The exit pupil is
referred to by older writers as the Ramsden's circle or bright spot.
The position of the entrance pupil determines the rays which the
object can send through the system. If it is very near the object
itself wide-angle pencils can fall on the system, though in the case of
a small entrance pupil they would fall symmetrically. A small
entrance pupil remote from the object will transmit small-angle
pencils only ; these may come from parts of object that are not near
the axis, and fall obliquely on the lens but will pass centrally
through it. Attention must be paid to these pupils in designing a
system so as to be as free as possible from defects. In using an
instrument the entrance pupil of the eye should be made to coincide
with the exit pupil of the instrument.
(g) Windows.—Referring again to the images of the various stops in
the parts of the system anterior to each. The stop-image which sub-
tends the smallest solid angle at the centre of the entrance pupil is
called by von Rohr the entrance-window; it determines the extent of
the object which will be seen through the system i.e. the field of view.”
(h) Chromatic Aberration.—Even when a lens system has been
corrected for other defects the final image is still imperfect when white
light is used, owing to its position being different for different colours.
This defect is known as chromatic aberration. We will consider only
the case of a narrow axial pencil.
For a single thin lens the elementary formula gives
q = (u – 1)C
where q is the reciprocal of the focal length (i.e. the focal power) and
C is the algebraic sum of the curvatures of the two faces. Hence
dq = Cdu. If two such lenses of different glass are placed co-axally
and in contact
and
dº?= Chdui + Cºdple
— a dia due
=ºi +*i
du,
pºl –
Representing it by Al and Ag for the two lenses, we can make the
The quantity l is called the dispersive power of the material.
* For further information see Whittaker, The Theory of Optical Instruments
(Camb. Tracts on Mathematics and Mathematical Physics).
ART. 77 ABERRATION OF LENSES 141
focus the same over the range du by making
$141 + $3A2=0 ;
*.e. by taking lenses such that *= –. --→ *
1
For transparent substances A is always positive; hence the focal
powers of the lenses must be of opposite sign. If the first is con-
vergent the second must be divergent. If this compensation is made,
A
*-*(1-#)
The values of A are always considerably less than unity. To avoid
fractional values the reciprocal is usually taken instead of A itself,
and represented by v (the new value). The equation is then written
*-*(·-·).
The value of v varies between about 60 for crown glasses and 30 for
flint glasses. One of the achievements of modern technics has been
to make glasses with a small v and a Small refractive index. The
accompanying table gives some typical cases of glasses, with fluor-spar
(which is used in combinations) for comparison.

Commercial Name. Description. (u D-1)104. !/.
O 218.8 Boro-silicate crown 5013 65.9
O 198 - Densest flint 7782 26.5
O 378 Extra light flint 5473 45.9
O 2164 Crown 5102 58.4
Fluor-spar CaF2 4.338 97.3
The value of v for any one glass is different in different parts of the
spectrum : the result is that when achromatism has been effected in
one part (say for the middle region) there is still a residual aberration
which is known as the Secondary spectrum. By properly selecting the
two kinds of glass this residual aberration can be made very small.
When two co-axal thin lenses are separated by the distance A, as in
many kinds of eye-piece, the focal power of the combination is given by
‘P= @1 + q2 – A$1%2.
If the focal power is independent of the colour
0= dºl-H dº - A (pid#2+ $23%),
o-º-º-A(***).
vi v
2 P2 V1
Or
If both components are of the same glass, vi = v, and
©1 + $2 – 2A$1%2 = 0,
fl-Hja-2A.
Ol'
142 & REFFACTION CHAP. V.
This condition is satisfied in the case of various eye-pieces; e.g.
Kellner (f=f, = A), Huygens (f, = 3f, A = 2f ). Huygens designed
his eye-piece to diminish spherical aberration, the bending of the ray
being equally distributed between the lenses. -
This condition, however, only provides for equality in the focal
powers for different colours and not for identity in the positions of
the second principal focus. The position of the second principal plane
(from which the focal length is measured) also depends upon the
colour. If we write the quadratic expression governing the relation
between object and image in the form (p. 116) .
ww.--aw - buy + c = 0,
the distance O2H, from the second lens to the second principle plane
is seen to be
Agh -
– O.H,- — —tº-
21, 12 $1 + q2 - Aq1952
(p. 118). To make Oaha invariable is equivalent to making invari-
able its reciprocal
(b1-H ºbº → A$192.
A@l 5
that is,
d(baſºl) == Adºbo,
OT
+- _ $2
(i. •)*=#ae.
OT
1. – A\?, ?, ?,
(; a) vs pl" vi’
OT
Aºi = (91 – vº)/v1.
If the lenses are of the same kind of glass v1 = va, and either A must
be zero (i.e. the lenses must be in contact) or else bi must be zero,
which is a useless case. It is impossible, therefore, to provide
achromatism of O.H., for separated lenses of the same kind of glass.
But with different kinds it is possible. For example, let v = 2ws, then
A must be fi/2, and then P = {1 + ba/2. But we must also satisfy
the condition found for F to be invariable, and this provides an
equation from which be can be found in terms of $1; but this con-
dition makes the focal power of the combination zero. This can be
; whence
seen most easily if the reciprocal of O.H., be written :
* I
1T dºp qā
ART. 77 ABERRATION OF LENSES 143
But if dip = 0 it follows that dº, must also be zero; i.e. the separate
lens must be achromatic. Hence it is impossible to obtain complete
achromatism with non-achromatic separated lenses. The above con-
siderations apply only to images formed by narrow pencils lying close
to the axis. The general problem of securing achromatism over the
whole field of view of a microscope is a much more complicated one.
It is now being realised that an instrument (e.g. a microscope)
should be designed as a whole. It is accordingly on this scheme not
attempted to make each part achromatic. Chromatic aberration is
permitted—even encouraged—in the objective and then compensated
for by the eye-piece (which is called a compensating eye-piece). This
scheme gives a greater facility in diminishing the aberrations of all
kinds. This is specially applied in what are called apochromatic
systems in which, by a selection of appropriate materials (of which
fluor-spar is one), corrections can be made which were impossible with
old-fashioned glasses.
CHAPTER WI
ON THE DETERMINATION OF REFRACTIVE INDICES
78. Minimum Deviation.—Newton, when investigating the solar
spectrum, always placed the refracting prism so that the incident and
emergent rays were equally inclined to the faces of the prism. In
other words, the pencil of light passed symmetrically through the
prism so that i = i and r = r.
In this case we may easily show that the deviation is a minimum.
With any centre O (Fig. 72) describe two circles CD and AB with
- radii proportional to the velo-
cities of light in the two media
(air and glass); that is, the
ratio of the radii is equal to
the refractive index of the
prism. Draw OA parallel to
the incident ray, and at A
Fig 12. draw CAN parallel to the
normal to the first face of the prism. Then OAN = i, and since
sin º-, OC_sin OAN sin İ
sin 77*TOATsin OCA ºf sin OCA'
it follows that OCA = r. Consequently OC is the direction of the ray
within the prism. If, therefore, CN’ be drawn parallel to the normal
to the second face the angle OCB = 7", and since OC/OB = p, it follows
as before that OBN’ = i, and therefore OB is parallel to the emergent
ray. Hence the angle AOB is equal to the whole deviation, while
AOC and BOC are the deviations at the first and second faces respect-
ively, and ACB = A the angle of the prism. The angle ACB is
therefore constant, and the problem is now reduced to finding when
the arc AB is a minimum, for AB measures the angle AOB which is
the total deviation.
We shall now show that the arc AB, intercepted on the inner circle

144
ART. 78 MINIMUM DEVIATION 145
by the lines CA and CB inclined at a constant angle A, is least when
CA = CB, that is, when the lines are equally inclined to CO. For let
CA and CB be this position, and let CA and CB' be a consecutive
position, that is, such that the angle ACA’ = BCB = 0 where 6 is very
small. Then
AA' sin à, and BB sin "..
AC Tsin A' BC sin B'
Hence if AC = BC we have
AA' sin B'.
BB’Tsin A'
But obviously sin B'- sin B and sin A'< sin A, B' being an exterior
and Aſ an interior angle, and B = A, therefore sin Bº sin A' or
AA'- BB'. Consequently A'B'- AB or the arc AB is less than its
consecutive value on either side, and is therefore a minimum * (see
also Ex. 12, p. 127).
In practice the position of minimum deviation is easily determined.
Newton, who worked with a small pencil of solar light from an aper-
ture in the shutter, placed the prism edge downwards. Consequently
the refracted pencil was bent upwards, and the spot on the screen was
displaced towards the ceiling as well as converted into a coloured
spectrum. He found that by gently turning the prism round its edge
the spectrum moved down the screen, and by rotating it in the opposite
direction the spectrum moved up so as to increase the deviation. By
rotating it slowly in the former direction so that the spectrum moved
down, or the deviation diminished, he found that as he turned the
prism the spectrum moved down the screen to a certain position, where
it came to a standstill, and then commenced to move upwards again,
no matter which way he rotated the prism. This position is that of
* Or again, let the circle centre C radius CB' cut the circle BA again in A".
Then AA'ſ AA"=CA'/CA" because CA bisects A"CA'. Therefore BB' = AA"<AA’.
Or thus: since 6= i+?' – A,
we have for 6 a max. or min. 0=di + di',
also, since r + r = A, 0 = dr H-dr'.
Again, sin i = p, sin r and sin i' = u sin r",
therefore cos idi = p, cos rar and cos i'di' = a cos r'dr'.
Hence by division we have
cos i cos r . cos” i cos” r
cos 7Tcos.7’ cos277 cosºr”
Or
1 — pºsin” r 1 — sin” r sin” r
Iºsinº TI- sinº FTsinº,”
the third expression being obtained from the first and second by taking the ratio of
the difference of their numerators to the difference of their denominators. Hence
*=r', and consequently i = '. It is easily seen that this corresponds to a minimum
rather than a maximum value of 6.
L
146 DETERMINATION OF REFRACTIVE INDICES CHAP. VI
the minimum deviation, and is, as Newton observed, that in which the
ray passes symmetrically through the prism.
The same principle is used at present, although the method of
procedure is different, the spectrum being now viewed through a
telescope instead of being projected on a screen.
An instrument fitted to observe a spectrum is called a spectroscope,
or a spectrometer if, in addition, it be graduated to measure the devia-
tion of the refracted rays.
79. The Spectrometer.—Fig. 73 represents the outline of a spectro-
meter. The source of light is placed before a narrow slit S in the
end of a telescope tube, or if greater illumination be desired the light
Fig. 73.-The Spectrometer.
of the sun or electric arc is concentrated at S by means of a condensing
lens. At the other end of the tube an achromatic lens L is placed and
focussed on S, so that the light emerges from L in a parallel beam.
This part of the apparatus is called the collimator, and its function is
to procure a parallel beam of light. In practice this is effected by
focussing the observing telescope on a distant object, so that the image
and the cross-wires may be viewed together without parallax, and
therefore the image is formed in the focal plane of the telescope. The
collimator is then viewed directly through the telescope and adjusted
till the image of the slit and the cross wires are viewed together
without parallax. A better method, due to Schuster, may be employed
instead (p. 127). The telescope is turned round to correspond with
a deviation considerably greater than the minimum value; two
positions of the prism can, in general, be found which will deviate the
rays from the collimator so as to pass through the telescope. Select

ART. 79 THE SPECTROMETER 147
the position of the prism for which the angle of incidence is the greater,
and in consequence the angular magnification is less than unity. If
the telescope is now focussed on the image of the slit it will thereby
be adjusted for rays more nearly parallel than those issuing from the
collimator. Turn the prism into the second position, the focus is
thereby upset, but it can be restored by adjusting the distance of the -
collimator slit. This adjustment necessarily brings about greater
parallelism of the rays from the collimator, because in this position
the prism increases the divergence of rays. By going back to the
former position and readjusting the telescope, and then to the second
and readjusting the collimator, and so on, the rays from the collimator
can very quickly be made as parallel as may be desired. In observing
dispersion, the light emerging from the collimator falls upon the face
AB of the prism ABC. The prism is placed with its refracting edge
(A) vertical on a horizontal table, at the centre of the instrument,
which can be levelled by means of three adjusting screws. In passing
through the prism the light suffers dispersion, and there emerges a
parallel beam of red light and also a parallel beam of violet, with
beams of the other colours situated between them. The pencil of
red light is brought to a focus R and the violet light to a focus V
by the object glass M of the observing telescope. . The other colours
are focussed at points lying between V and R, so that a real spectrum
is depicted in the focal plane of the telescope. This spectrum is then
viewed through a suitable eye-piece N.
The collimator and telescope are attached to the graduated circle,
their axes are parallel to its plane, and their directions meet above its
centre. The telescope is generally fitted with cross wires and a
Ramsden's eye-piece, and turns in a plane parallel to the graduated
circle about its centre while the collimator is fixed. The position of
the telescope with reference to the circle is determined by a vernier.
The table which supports the prism turns round a vertical axis and
carries an arm furnished with a vernier, which slides on the graduated
circle and determines its position.
Since the property of a lens is to concentrate a system of parallel
rays to a single focus, we see that the use of the collimator is of great
importance in producing a pure spectrum. For since the red light
emerges from the prism in a parallel beam, it will form a red image of
the slit at R, similarly there will be a violet line at V, and the other
coloured images will be spread out between R and W without over-
lapping if the slit be sufficiently narrow. The dispersion, as the
separation of the colours is called, depends on the nature of the prism
as well as its angle (see Ex. 15, p. 127), and can be greatly increased
148 DETERMINATION OF REFRACTIVE INDICES CHAP. VI
by allowing the light to pass through several prisms in succession so
that a very long spectrum may be obtained.
If the collimator is not adjusted for infinity the rays from the slit
will be incident at various angles on the prism and the image will
consist of focal lines. If the telescope is focussed for the primary
focal line the image of a vertical slit (the refracting edge of the prism
being supposed vertical) will appear as a vertical line scarcely inferior
in sharpness to that obtained with the collimator adjusted for infinity.
In the position of minimum deviation these focal lines will coalesce,
provided that the angle of divergence of the rays is sufficiently small.
In order that the image of the slit as seen in the telescope should
be as fine as possible, care should be taken about the illumination of
the slit. This is specially the case when the indices for two neigh-
bouring wave lengths are being determined. It will be shown later
(Art. 162c) that the image differs from the one which would be
calculated by the method of geometric optics, and that to obtain it as
narrow as possible it is necessary to employ a wide emergent beam
from the prism. There are different ways available for securing this.
The first arrangement is as shown in Fig. 73a. If the source is itself
broad—a sodium flame, for example
—it must be placed so close to the
sºil slit as to subtend at least as great an
angle at it as the collimator lens
does. If this would bring the
source so near the slit that there
would be danger of damaging it, an
auxiliary lens or condenser must be employed. There are two ways of
using it. The source AA', supposed to be fairly broad, is placed in
the first principal focal plane of the condenser (Fig. 73a). Any one
point of the slit is now illuminated by many points of the flame. If
A is one extreme point of the source which sends light through the
slit, the inclination 61 of the ray is the same as that of the secondary
axis AC through A. The adjustment must be made so that 61 is at
least as great as the semi- .
angle subtended by the col- "|→T =l–
limator lens at the slit. 7
The other method is to I-T-
produce a diminished image º
º Fig. 73b.
of the source on the slit.
The angle 61 must, by choosing the distance of the source, be as
great as in the other cases (Fig. 73b).
When very narrow slits are in use it is necessary for 61 to be
Fig. 73a.

ART. 80 MEASUREMENT OF THE ANGLE OF A PRISM 149
considerably greater than the limit specified above so as to gain
additional light by means of beams that are diffracted by the slit and
would not otherwise fall upon the collimator lens.
To test whether or not full illumination has been secured, the slit
should be widened and a piece of thin paper or ground glass placed in
contact with the collimator lens; it should be illuminated throughout
the whole extent of the lens. The back of the prism can be tested in the
same way. An alternative way is to remove the eye-piece of the tele-
scope. On looking down the tube the field should appear “full of light.”
80. Measurement of Minimum Deviation and Angle of Prism.—
Since the deviation depends on the wave frequency (or colour) of the
light, the prism can be placed in the position of minimum deviation
only for some selected colour at once. To measure the minimum
deviation for any ray the telescope is turned to view the slit of the
collimator directly, and its reading is then taken on the graduated
circle. The prism of the substance of which we wish to determine the
refractive index is now fixed on the table of the spectroscope, and the
telescope is turned to view the spectrum produced by the prism. The
prism is then rotated (by turning the table which supports it) through
a small angle, and this will generally either increase or diminish the
deviation. Turning it in one direction will increase, and in the opposite
will diminish, the deviation. This latter direction being determined,
the prism is slowly turned, and the spectrum is followed by the tele-
scope till the spectrum comes to a standstill, and any further turning
of the prism in either direction will increase the deviation. The
reading of the telescope is now taken, and the difference between this
and the former reading is the minimum deviation of the ray under
consideration.
To determine A, the angle of the prism, turn the table of the
spectroscope so that the edge of the prism
(Fig. 74) is towards the collimator, and the
light from it falls upon and is reflected
from both faces. The telescope M is now
turned until the light reflected from the
face AB enters it, and an image of the slit
is seen by reflection in the field of view
coinciding with the cross wires.
Now turn the telescope into the position
Mſ, so that an image of the slit is seen
by reflection in the face AC. The differ-
ence between the readings of the telescope in these two positions—
that is, the angle through which it has been turned—is twice the
Fig. 74.

150 DETERMINATION OF REFRACTIVE INDICES CHAP. W I
angle of the prism, for this is only an illustration of the
principle that if a plane reflector is turned through any angle the
reflected ray is turned through twice that angle (Chap. IV. Ex. 1). Or
thus: the angle between the reflected ray AM and the incident ray
produced is equal to twice the angle between AB and the incident
light, so also the angle between AM" and the incident ray produced is
twice the angle between the face AC and the incident light. There-
fore the angle between AM and AM", or the angle through which the
telescope has been turned, is equal to twice the angle of the prism.
The prism should be placed on the table, so that the light from
the central portion of the collimating lens falls upon its edge, and
does not pass by it. At the same time it is to be so adjusted that it
is possible to see the slit by reflection from both faces, as described
above. These conditions are generally secured by placing the prism
with its edge a little in front of the centre of the supporting table.
This method of determining A supposes the faces of the prism to
extend right up to its edge, and that near the edge the faces exhibit
no curvature. If any doubt exists as to the faces being truly plane
at the edge we may employ another method of determining A, which
depends upon keeping the telescope fixed and rotating the table.
This method may be used when the table is furnished with a vernier
arm to determine its position.
Clamp the telescope M in any position, so that its axis is inclined
to the axis of the collimator, and turn the table till the image of the
slit is seen in the telescope by reflection from the face AB of the prism.
Read the vernier and then turn the table until the slit is seen by re-
flection from the other face AC. Read the table vernier again. The
difference between the two readings is easily seen to be tr – A.
Having measured both A and 8, we determine p by the formula of
Art. 81. A Bunsen burner with a bead of salt, or a piece of pumice
stone saturated with salt solution, in it gives a yellow sodium light
which is very homogeneous. By using it we determine the refractive
index for yellow light and avoid complications. To ensure accuracy the
determinations of A and 8 should in all cases be repeated several times.
In some modern spectroscopes, e.g. the autocollimation spectro-
scope," the use of a separate collimator is dispensed with by using a
prism with one face plane-polished and silvered.
The slit is fixed in the focal plane of the objective of the observing
telescope and the upper and lower halves of it are illuminated by light
admitted through two windows in the sides of the telescope and
* “Úber eine neue Spekstroskop-Konstruction,” Dr. C. Pulfrich, Zeitschrift für
Instrumentenkwºnde, xiv. p. 354, 1894.
Art. 81 METHOD OF DESCARTES 151
directed on the slit by two reflecting prisms. Parallel beams of light
leave the objective, are refracted at the first surface of a prism, fall
normally (for one wave length in a given position of the prism) upon
its posterior face, are there reflected and, returning in the reverse
direction, are brought to a focus in the focal plane of the objective.
Increased dispersion is obtained as the light goes twice through
the prism ; and the spectra of two sources may be formed in close
proximity to one another for purposes of comparison.
The instrument" may be converted into a spectrograph for photo-
graphing both the visible and ultraviolet spectrum by using the telescope
as a collimator, substituting a transmitting prism for the reflecting
one and attaching a camera. For observing the ultraviolet spectrum,
objectives and prisms of quartz or quartz-fluorite are employed.
81. Formula for the Refractive Index.-The prism being placed
in the position of minimum deviation for any particular ray of light
we have
6=2i-A, or i=}(A+6),
and A =2n, or r=#A,
but sin i = a sin r,
therefore sin (A +6)=usin A,
_sin (A +3)
and T sin A
a formula which determines p, when A and 8 have been meas-
ured.”
Cor—If the prism be very thin, so that A and 6 are very small, the
sines of the angles may be replaced by their circular measures, and the
above formula becomes
5-(u – 1)A.
Ez. 1. The minimum deviation 3 varies with the colour or wave length of the
* “Ein neuer Spektrograph für sichtbares und ultraviolettes Licht,” F. Löwe,
Zeitschr. f. Instrumentenkunde, xxvi. 1906.
* Method of Descartes.—The method employed by Descartes to determine the
refractive index of a substance is of consider-
able interest. Transmitting a horizontal
pencil of light through an aperture O (Fig.
75) in a vertical screen AB, it fell upon the
vertical face of a prism placed against an
aperture O' in the vertical screen A'B'. In
this case
* =0, re-O, r'EA, and 5–1 – A,
and therefore
sin (A +6)=usin A. Fig. 75.
The angle 6 which the refracted pencil makes with the horizon is determined by the
equation O'A' = AP tan 6 (Dioptrices, caput x, p. 140).

152 DETERMINATION OF REFRACTIVE INDICES CHAP. VI
light. The change du of the index and the corresponding change d6 of the minimum
deviation are connected by the equation
cos (A + 6
du– 3 º *A ),
Ev. 2. A prism of silicate of lead having a refracting angle of 21° 12 produced
a minimum deviation of 24° 46' in homogeneous red light; find the refractive index.
[L sin (A +5)=L sin (22° 59')=9:59158,
L sin A = L sin (10° 36')=9-26470.
Therefore logue 0-32688 and u-2-123].
82. Liquids.-The refractive index of a liquid may be determined by
enclosing it in a hollow prism the faces of which are plates of parallel
glass. If the glass sides of the hollow prism be uniform plates they
will not cause any dispersion or deviation of the light. The whole pro-
cedure is consequently the same as in the case of transparent solids."
83. Method of Total Reflection.—By measuring the critical angle
(a) of a substance we at once determine its relative index of refraction
by the formula
sin a = 1/u.
The advantage of this method is that it does not require the con-
struction of a prism, and may therefore be applied to a solid substance
which it is not convenient to cut. Wollaston * applied the method
to liquids, and in this application it has been perfected by Terquem
and Trannin.” Two plates of parallel glass (Fig. 77) cemented together
at their edges by a little gum or Canada balsam, so as to enclose a
thin layer of air, are immersed in a vessel containing the liquid under
consideration. The vessel, which is a square box with parallel glass
sides, is first fixed on the table of the spectrometer and the telescope
is adjusted to view the collimator slit, the light passing normally
through the sides of the box. The prepared glass plates are now
Liquids.
* Newton's Method.-An ingenious method of measuring the refractive index of a
- liquid was employed by Newton. The liquid was placed
in a rectangular vessel with a flat glass bottom M (Fig.
76). This vessel is attached to a frame AB, which moves
freely round a horizontal axis O, carried by a vertical
support OP. A pencil of solar light falls upon the sur-
face of the liquid, and the carrier AB is turned round 0
till the pencil emerging from the base of the vessel is
parallel to AB. In this position the refracted beam is
perpendicular to the base M of the vessel. The angle of
refraction is consequently equal to the inclination of AB
to the vertical. This is determined by a graduated circle
attached to OP. The angle of incidence is the inclination
of the incident ray to the vertical, and this may be
determined by repeating the experiment when there is
no liquid in the vessel.
* Wollaston, Phil. Trans. p. 365, 1820.
* Terquem and Trannin, Journal de Physique, first series, tom. iv. p. 232, 1875.
Fig. 76.

Aitt. 83 METHOD OF TOTAL REFLECTION 153
placed in the liquid with their plane faces vertical (the table of the
spectroscope being horizontal), and so inclined to the light from the
collimator that total
reflection just occurs.
When this happens
the field of the tele-
scope becomes quite
dark. It is clear that
in this case the angle
of incidence on the glass plates is the angle of total reflection of the
liquid and air. By turning the plates round the vertical in the opposite
direction, the light reappears in the telescope, and then, on continued
rotation of the plates, again vanishes. The angle through which the
plates have been turned is twice the critical angle a, and
sin a = 1/u.
In a similar manner the index of a thin plate of a solid may be Solids.
determined by immersing it in a liquid of known index po greater
than that of the solid. Here we have
sin a = uſuo:
This method has been adopted by Kohlrausch for measuring the
indices of crystals.”
It is important that the light should fall upon the plates containing
the air film in a parallel beam, for the whole pencil will then suffer
total reflection at the same instant, and the field of the telescope will
become dark suddenly. Care should also be taken to place the
plates vertically. This being secured, with monochromatic (sodium)
light, the disappearance of the image is almost instantaneous, and
Terquem and Trannin estimate the experimental error at less than
fifteen seconds. When white light is used, the image passes through
shades of yellow, orange, and finally the pure red of the extreme
spectrum. The accuracy of the method is shown by the following
table, where the indices are compared with the determinations of
Fraunhofer, and of Gladstone and Dale, the slight differences being
attributable to the state of purity of the liquid :
Fig. 77.
Ray. | Temp. 2a. index.
- –
18° 97° 20' 30" | 1.3317 | 1.33171 Fraunhofer.
-- 18° 97° 9' 50" | 1.3336 | 1.33171 --
Benzine 19°.5 | 84° 41' 20" | 1.4846 | 1.4860 Gladstone and Dale,
C
D
. . A
Glycerine . . . A 18° 85° 55' 20" | 1.4672 | 1.4664 -- --
Amyllic Alcohol . . A 18° 91° 10' 0" | 1.4000 | 1.3990 -- --
A
Water .
5
Sulphide of Carbon 20° 76° 55' 0" | 1.6078 1-6076 -- --
| Wied. Ann. iv. 1, 1878.

154 DETERMINATION OF REFRACTIVE INDICES CHAP. VI
When the liquid can be procured in small quantities only, or when
it is imperfectly transparent or pasty (such, for example, as the
crystalline lens of the eye), Wollaston's original method may be used.
A small drop D (Fig. 78) of the liquid is attached to the lower surface
of a right-angled prism ABC, resting on a horizontal table AB fur-
nished with a small hole to receive the drop. The drop is viewed
through the face AC of the prism by means of a telescope T pivoted
at the centre of a graduated circle, which slides on a vertical rod fixed
to the table. The telescope is
first placed at such a height that
the table, or any mark below the
drop, may be seen through it, BC
being turned towards the light
of the clear sky. As the gradu-
ated circle is lowered, the tele-
scope being continually directed
towards the drop, a position is
reached for which the ray MD makes the critical angle with the
normal ND. Rays making an equal angle with the normal on the
other side are thus totally reflected, and the surface at D therefore
becomes suddenly brighter, and the mark below the drop disappears.
The telescope is now fixed, and its inclination (6) to the vertical is
observed. In this case NDM is the critical angle a, and NMD =
90 – a, while TMO = 90 – 6. But the refractive index of the glass
is given by
Fig. 78.
_sin TMO cost.
*Tsin NMDT cosa
Hence
008 a “*”. and sin a _l Vº cosº.
A. L.
But if the refractive index of the drop be denoted by p', we have
u’-usin a = Nuº-cos”6.
An objection to the method is the difficulty of making the angle of
the prism exactly right. To avoid this, Malus worked with an acute-
angled prism, of which the index and angle were accurately determined.
Examples
1. When the angle A of the prism is acute, prove that
a' =sin(0–A) cos A+sin A VLA-sin” (9 - A)
where 6, as before, is the angle the telescope makes with the vertical.

ART. 84 LAW OF GLADSTONE AND DALE 155
[The angle of incidence of the ray DM on the face AC is r-A-a, and the angle
of emergence is i = A – 6. But sin i = p, sin r, therefore sin (A - 0) = p sin r, and
p'= p sin q = p sin (A-r)|= — p. sin 7' cos A+ p sin A cos r, etc.]
2. Determine the refractive index of the prism by the same method.
[Since the medium below the prism is air, we have p' = 1, and
p = cosec A V1+sin” (3-A)-2cos Asin (0– A).]
84. Gladstone and Dale's Law—Variation of Refractive Index
with Density.—When a substance is compressed, or its temperature
varied, the density (p) alters, and this is accompanied by a corre-
sponding variation in the refractive index (p). Gladstone 1 and Dale
found that these two quantities were related by the equation
**=constant
p + -
The physical interpretation of this law is not far to seek if we
admit that the refracting substance consists of refracting molecules of
constant index distributed through the ether. The quantity p – 1
represents the excess of the refraction or path retardation due to the
presence of the molecules, and will be proportional to their number
per unit volume, that is, to the density. Thus if p be the index of a
plate of the substance and e its thickness, the corresponding path in
free space of a ray traversing the plate will be pſe, but if the thickness
absolutely occupied by the molecules of the substance be e and the
constant index” of a molecule be m, then a thickness e – e of the plate
is occupied by ether, and the corresponding free space path for the
plate will be e – e 4 ms. Hence
|weae – e –- me,
OI’
u-1={(m-1).
62
But eſe measures the relative volume occupied by the molecules,
and is therefore proportional to the density of the body. Conse-
quently
º = constant.
Since the density of a solid or liquid can be changed only very
slightly, experiments on the law are necessarily confined within ex-
tremely narrow limits, and we can only accept it as an approximation
to the truth in absence of other support.
. In the case of gases, however, the density can be varied consider-
* Gladstone and Dale, Phil. Trans, p. 887, 1858, and p. 317, 1863.
* The quantity e is the thickness of the plate of the same face area that the
molecules of the plate e would form if arranged with no free spaces between them,
and the m is the index of this derived plate. -
156 DETERMINATION OF REFRACTIVE INDICES CHAP. VI
ably, but on the other hand p – 1 is here a very small quantity, and
therefore experiments which sufficiently verify the formula -
A - 1
---
K
will also verify the formula
Q
pi* – 1
= 2x,
for u + 1 = 2 very nearly, if p exceeds unity by a small amount.
The emission theory indicates that for all bodies the quantity p” – 1
should be proportional to the density, and consequently it has been
sought to establish the latter formula. Thus, if v be the velocity in
a given substance and v the velocity in a vacuum, the work done
on a luminous corpuscle in passing into the substance from vacuum will
be proportional to the density of the substance, but the work done on
the corpuscle is proportional to the change in the square of its velocity,
therefore
v” – vº– kp.
That is,
p” – 1 = kp/vº.
This shows that the index is independent of the angle of incidence, and
that p” – 1 is directly proportional to the density of the substance.
From his experiments on the variation of the index of water under
compression, M. Jamin concluded that (p.” - 1)/p was constant, but M.
Mascart found that the results of M. Jamin were more accurately
represented by the formula of Gladstone and Dale, and this conclusion
has been verified by Quincke."
This formula (.*= constant) also follows from the Sellmeier
formula of dispersion (see Art. 297; cf. also the Ketteler-Helmholtz
formula of dispersion) on the assumptions that the natural frequencies
of the molecules are independent of the density and that the constants
An are proportional to the density. -
Lorenz” and Lorentz” have deduced theoretically the formula
2
pº — 1 1
p3+2 ;= Constant,
and later Planck derived the same formula from a different theory.
This formula has been found by various observers to agree well with
* Ketteler proposes the empirical formula
2 – 1
*=d-a-gº-ºº: . . . )=constant.
The ratio (Hº- 1)/(u"–1) for the different colours he finds to be sensibly
constant so that a, 8, Y, etc., do not depend on the wave length. In the case of
water below 8° they appear, however, to depend on the temperature (Wied, Ann.
xxx. p. 299, 1887; Journal de Physique, second series, tom. vii. 1888).
* Lorenz, Wied. Ann. xi. p. 70, 1880. * Lorentz, ibid. ix. p. 641, 1880.
ART. 84 LAW OF GLADSTONE AND DALE 157
experimental results for liquids and gases for wide variations of
pressure and temperature, and has been found to hold for many
substances through the liquid into the gaseous state. - -
Pockels" examined the influence of mechanical stress on the
refractive index of glass, determining experimentally the change
in refractive index with density, and found the Gladstone-Dale
formula to agree best with his results, the Lorenz-Lorentz formula
not being applicable. *
Effect of Variation of Temperature.—If the temperature varies the
molecular index may also vary. If this be so it follows from the
equation
p. – 1 = v(m – 1),
where v is the volume of the molecules in a unit volume of the sub-
stance, by differentiating logarithmically with regard to the tempera-
ture 6, that
1 du 1 dv 1 dºm
*-* - -
urid㺠Jä" miº
But the density of the medium is proportional to v. Hence
1 du 1 dp_ 1 dom
*-* * - - - -m-m-me ---
or, more conveniently for calculation, if V be the volume at 6 of a
unit mass of the substance, then W = 1/p, and
_l dº. 1.1 &W = 1 &m
p — I d6 W d6 m – l d6
Gladstone's law, which assumes m to be constant, requires the
second member to vanish. M. Dufet” found it always negative and
very nearly constant for liquids, but greater in absolute value than
indicated by theory when m is a variable. In the case of solids the
sign was positive and the magnitude less than that deduced from theory,
whereas in the case of solutions when concentrated they behaved as
solids, but when dilute they behaved as liquids.
Or in all liquids the molecular index decreases as the temperature
rises, and in all solids the molecular index increases with the tempera-
ture, while the quantity ### remains very nearly constant.
The effect of temperature on the refractive index is shown in the
following table after Gladstone and Dale. The indices are for the D
line, except that of phosphorus, which is for the C line :
* Phys. Zeitschr. ii. pp. 693-695, August 31, 1901.
* “Sur la loi de Gladstone et la variation de l’index moléculaire,” par M. H.
Dufet, Journal de Physique, November 1885. See also September 1885.
158 DETERMINATION OF REFRACTIVE INDICES CHAP. vi
T - - Sulphide of Absolut Methyl PhoSDh
*...* | *;"| Water | Ether | }. ºš |*ś"
0° 1:6442 1 3330 * * * tº º e Hº & &
10° | . I-6346 | 1.3327 | 1.3592 1-3658 || 1:3379
20° | 1.6261 || 1 °3320 || 1 3545 | 1.3615 e tº $
30° | 1.6182 1-3309 || 1:3495 || 1:3578 - © tº 2:07:41
40° 1-6103 || 1:3297 tº e is 1°3536 | 1.3297 || 2:0677
50° * * * 1°3280 1 °3491 tº gº º 2°0603
60° l'3259 1 3437 2:05.15
85. Indices of Mixtures and Solutions.—The law of Gladstone
may be applied to calculate the refractive index of a mixture of two
substances which have no chemical action on each other.
Let V1 and V, be the volumes, p, and p, the densities, p1 and ps
the indices of the components of the mixture, and let V, p, p. be the
corresponding quantities for the mixture, we have
pW = Wip, + W202,
and since each of the components occupies a volume V in the mixture
we can say that Wipſ V and V209/W are the densities of the components
in the mixture. Consequently by Gladstone's law
* - 1 º' - 1 (P1-1) V.
Pl p 1 Pl Wi
9 – I f * l p - V
- P2 p 2 P2 V2
where p'i and p', are the new indices of the components in the mixture
according to the law. The index p of the mixture will therefore be
given by (p − 1) = (p/1 – 1) + (p/2 − 1), or by the foregoing equations
2-1-0-1)}+(4-1)}
or, finally, if M and M, be the masses of the components,
(M,4M,)tº 1 =M,” + Mºrº.
p p1 P2
Experiment shows that this formula is approximately true for
mixtures of liquids and mixtures of isomorphous crystallised salts, and
it ceases to hold for mixtures of Saline solutions.
Effect of Change of Temperature.—If the temperature (6) varies, we
have
3 (4-1) M diſºl - 1 d /ue – 1
(M. Mºż(**)=M%(**)+M;(**)
2
But by the equation (n − 1) = 0(m = 1) it follows that
#(**)=#–º l_ dºm
d6\ p / Tp d6 T p m – 1 d6'
ART. 86 INDICES OF GASES 159
Hence
86. Gases.—The index of refraction of a gas may be determined,
like that of a liquid, by enclosing it in a hollow prism with faces of
parallel plate glass. The prism is placed on the table of the spectro-
meter, and the gas under consideration is admitted through drying
tubes, if necessary, its temperature and pressure being noted. The
telescope, as usual, is first adjusted to observe the collimator slit directly,
and if the plate sides of the prism are of truly parallel glass, the prism,
when placed on the table of the spectroscope, will produce no deviation
when it contains free air. This being ensured, any future deviation is
due to the refractive difference between the gas contained by the prism
and the outside air. Any such deviation will in general be very
small, so that the table of the instrument (with the prism attached)
may be turned through 180°. This will throw the deviation to the
opposite side (as the edge of the prism is now turned in the opposite
direction), so that the difference between the two readings of the tele-
scope will be 26, and by means of the ordinary formula we obtain the
relative index of the gas.
To obtain its absolute index we require first the index of air at
zero and 760 mm. This is determined by first pumping as much
as possible of the air out of the prism. Let the pressure of the
residual air be h', its index pſ, and the density p, then by the law of
Gladstone and Dale
* -l ºo-1,
p' P0.
tº ’ — * p' *º * –*—
if 6 be the temperature.
But if the exterior air be at a pressure h, density p, and index u,
we have also
* - 1 Po-l,
p P0 -
OT a-1+(n-1)}=1+(4,-1)++r
P0 760(1 + a 6)
Experiment gives the relative index of the air in the prism with
respect to the exterior air; if this be m we have
, º' 160(1+d6)+(90-1)!'.
Tu T760(1 + aff)+(uo – 1)h
_j , (m. – 1)760(1+d6).
p0=1 + h' — m/,
Hence
To determine the refractive index vo of any gas at zero and 760
160 DETERMINATION OF REFRACTIVE INDICES CHAP. VI
(the pressure being h"), we have as before, by Gladstone's law,
'— //
"=1+(0-1)/Häjää
where v' is the index at h’ and 6.
If the outside air be at h and 6, we have for it, as above,
h.
*=1+(8–1)THº
Consequently if m, be the relative index of the gas with respect to
the air, we have
_w'_(1+d6)760+(vo- 1)h',
p. (1 + q6)760+(uo – 1)h
from which we obtain vo.
Dulong" worked with a constant deviation. He introduced the
different gases successively into a hollow prism, but varied the
pressure so that each produced the same deviation. Thus for air
at h' and 6 in the prism, while the exterior air is at h and 6, we have
for the relative index
'ml
(1+d6)760 + (go – 1)h'
(I-Faà)760+(u, -1)}. '
772, E
while for a gas at h", and 61, with relative index m, we have
_(1+d61)760+(v0–1)h',
T(I-Faº)760+(a, -i)h,”
hence if the pressure be adjusted to give a constant deviation, we
have m = m, which determines vo as a function of po a known quantity.
If the exterior pressure and temperature have remained constant
during the experiment, we have 6, = 0 and h, → h, and therefore
(90–1)h' = (v0 – 1)h'i,
or the refractive powers (u – 1) of two gases are inversely as the
pressures required to produce the same deviation. ſº
The refractive index of a gas may also be determined by delicate
methods depending upon interference phenomena as mentioned in
Arts. 122, 123.
* Annales de Chimie et de Physique, second series, tom. xxxi. p. 154, 1826.
[TABLE
ART. 86 INDICES OF GASES 16.1
INDICES OF REFRACTION OF GASEs
Index. Density. Index. Density.
Air tº 1.000294 | 1.000 || Ethylene 1'000678 || 0-978
Oxygen . 1°000273 || 1:106 || Marsh Gas . . 1°000443 || 0 °555
Hydrogen 1.000138 || 0-069 || Chloride of Ethyl . 1-001095 || 2:234
Nitrogen 1.000300 || 0-971 || Hydrocyanic Acid . 1-000451 || 0-944
| Chlorine 1-000772 || 2:470 || Ammonia . 1°000385 0'596
Nitrous Oxide 1.000503 || 1:520 || Phosgene tº 1*001159 || 3 °442
Nitric Oxide . . 1.000303 || 1:039 || Sulphydric Acid 1*000644 || 1 "I'91
Hydrochloric Acid 1-000449 || 1:247 || Sulphurous Acid 1'000665 || 2:234
Carbonic Oxide 1-000340 | 1957 || Ether . g . 1-001530 2'580
Carbonic Anhydride | 1.000449 || 1:524 || Sulphide of Carbon | 1.001500 || 2:644
Cyanogen . . 1'000834 || 1:806 || Phosphuretted Hyd. | 1.000789 || 1:214

Kayser and Runge using a Rowland grating and a prism of com-
pressed air found the following indices corresponding to various wave
lengths:
INDICES FOR DRY AIR FOR 0° AND 760 MM.

Wave Length. Index. Wave Length. Index.
5630 1.0002927 2860 1°0003088
4430 2955 | 2850 3094
4200 2967 : 2550 3158
3250 3035 2360 32.19
These results they find agree very well with Cauchy's formula (see
Art. 292), and if A is expressed in millionths of a millimetre
p = 1.00028817-1-1-316X-2+ 31600X-4
(Astronomy and Astrophysics, vol. xii. p. 426.
ib. p. 455).
The additive nature of the formula for the refractive index of a
mixture and the fact that the formula may be extended so as to
apply to many cases of chemical combination, point to refraction
being an approximately additive atomic property. Thus the formula
of mixtures applied to the molecule gives that the molecular refrac-
tion (or the product of the molecular weight of the substance and
its refraction constant) is equal to the sum of the products obtained
by multiplying the number of atoms of each element in the molecule
by the atomic refraction of that element (i.e. by the product of the
refraction constant of the element by its atomic weight). Using
the Lorenz-Lorentz formula, the atomic refraction of an element is
2 — e * tº * g
º e * for light of a definite wave length, A being the atomic weight
and u the refractive index of the element of density p.
See also Hasselberg,
Similarly
M
162 DETERMINATION OF REFRACTIVE INDICES CHAP. VI
g e ſº 2 — e º
the molecular refraction for a substance is ; © * where M is its
molecular weight.
For example, Brühl found p = 1°471 for carbon tetrachloride of
density 1-632. Its molecular refraction is therefore
g 2 — e
#; ;=262s.
Brühl's determinations of the atomic refraction of carbon and chlorine
were 2.365 and 6:014 respectively, giving 2.365 + 4 x 6:014 = 26.42
for the molecular refraction of the carbon tetrachloride, in good
agreement with the value 26-28 determined directly.
However, for some substances, other observers have found that
the determination of the atomic refraction by the Gladstone and Dale
formula [*] gives results more in accord with their observa-
tions.
Influence of Pressure upon Refractivity of Gases
At moderate pressures the variation of the refractive index of a
gas with pressure is sufficiently well represented either by Gladstone
and Dale's or by the Lorenz-Lorentz formula.
For pressures below atmospheric the following values were deter-
mined by Posejpal" for A = 5462 tenth-metres, (p − 1) = Kp(1 + Bp),
p being the pressure in mm, of mercury.

(u – 1)106 y *
Atmospheric Pressure. K × 106. BX 108.
Air 292-63 + 0.04 0.36269 + 0.00015 357+ 39
CO2 447 '96 -- 0:28 0.55191 + 0.00026 1063 + 55
At high pressures the only thorough investigation has been made by
P. Phillips” on carbon-dioxide. By experimenting with a gas just
above its critical temperature large continuous changes of density
may be produced, and for this reason Phillips' determinations were
made at 34° C. The measurements were made by aid of the fringes
obtained when light was transmitted through a Fabry and Perot
étalon, which was enclosed in a strong chamber into which the gas
could be compressed. The étalon produced the usual circular system
of bands, but a slit was adjusted so as to cut off all but a narrow
horizontal strip across the system of circles. The volume of the
vessel was determined, and thence the density of the compressed gas,
1 Bull. Inter. Acad. Sci., Bohemia, 1918-20.
* Proc. Roy. Soc., A, xcvii., 1920.
ART. 86 REFRACTIVITY OF GASES 163
by allowing the gas to escape into a measuring cylinder and measuring
it at about atmospheric pressure.
The following table shows typical values which were obtained
from three different wave lengths:
Refractivity p, — 1.
Density p.
A ==5790. A = 5461. A = 4358.
0 71235 0 °1659 0 °1663 0 1686
0.62777 0° 1460 0 - 1462 0.1483
0 °54'300 0 - 1260 0 1262 0-1279
0 °41'550 0 °09590 0 °09610 0 °097.28
0 °33133 0.07623 0.07637 0.0773]
0 °20114 0 °04606 0-04617 0 04677
0 '11558 0 °02645 0.02646 0.02687
From these numbers the Lorentz constant can be calculated for each
wave length. The reciprocals of the values calculated follow very
precisely a linear law of variation with the square of the density, the
total variation being about 1 per cent. For wave length 5461 Å.U.
the relation between the refractive index and the density is
Au” + 2
pi* – 1
p=6'581+0.1130p”.
An explanation of this constant is given later (Art. 336).
CHAPTER VII
INTERFERENCE FRINGES
87. Destructive Interference.—So far we have been engaged in
considering the mode of propagation of a luminous wave and the
modifications which it undergoes when it encounters the surface of
a new medium. We shall now proceed to inquire into the effects
produced when two series of waves are propagated simultaneously in
the ether from two small luminous origins close together.
When two waves arrive simultaneously at the same point of
space, the ether there will be thrown into vibration by both, and we
have already shown (Chap. II.) that in this compound motion each
vibration may be regarded as acting independently of the other. If
the constituent vibrations are in the same direction, the effects are
added, and the amplitude of the resultant vibrations will be equal to
the sum of the amplitudes of the constituents, but if they are opposed,
the resultant amplitude is equal to the difference of the amplitudes of
the constituents. In this latter case, if the vibrations are of equal
amplitude, they should completely destroy each other. This is usually
spoken of as destructive interference.
We know from experience that two sets of sound waves may
neutralise each other and produce silence," so also two sets of water
waves when superposed many produce a dead level. If then there is
any truth in the undulatory theory of light, something of the same
kind should take place with two sets of light waves. That this actually
occurs is proved abundantly by the following experiments.
In all cases of interference, however, it is to be carefully re-
membered that light (regarded as energy) is never annihilated. The
Energy
redis-
tributed.
* Two organ-pipes tuned in unison and mounted close together produce only
a very faint sound at external points. They start in opposite phases, and the effect
which would be produced by one is just neutralised by the other. Other examples of
interference occur in the beats of two notes nearly in unison, in the nodes of organ-
pipes, and vibrating strings, etc.
- 164
ART. 88 THEORY OF YOUNG'S EXPERIMENT 165
distribution alone is altered, so that the illumination, instead of being
diffused regularly, is concentrated in some places at the expense of
others.
88. Two Small Apertures—Theory of Young's Experiment.—
Let us suppose that two sets of waves always exactly alike start from
two near luminous origins A and B (Fig. 79). If the directions of the
disturbances transmitted to any point P by the two sources conspire,
the amplitude of the disturbance at
P will be doubled, but if the com-
ponent vibrations be opposed at P,
they will destroy each other, and no
effect will be produced at this point.
In the former case the illumination at
P is four times that produced by
either of the sources acting singly ;
in the latter case the illumination is zero. The illumination sent by
one source is swept away by that contributed by the other'—a result
observed in the justly celebrated experiment of Dr. Young, and
apparently opposed to the idea of the materiality of light (Art. 28).
Let us now examine the theory of Young's experiment a little
more closely. The direction of the vibrations sent by A and B to any
point P of a screen will conspire, and the amplitude of the disturbance
will be doubled, when they arrive at P in the same phase. Now we
suppose the waves to set out from A and B in the same phase, so that
they will arrive at P in the same phase if the path AP is equal to BP,
or differs from it by any number of complete wave lengths. But the
vibrations will be opposed at P if the waves arrive there in opposite
phases, which will be the case if AP differs from BP by one half-wave
length, or any odd number of half-wave lengths. We therefore con-
clude that if
Fig. 79.
AP– BP-n}, -
the point P on the screen will be very bright, or dark, according as
n is even or odd. If then O be the middle point of AB, and if OM be
perpendicular to AB, the distances AM and BM will be equal, therefore
the lights should be in accordance at M, and it should be very bright.
At M, there will be darkness if the difference of the distances of this
point from A and B is half a wave. At M, there will be brightness
again if the difference of its distances from A and B is two half-wave
* It is usually said that the light from one source is destroyed at these places by
that from the other. This is misleading, however, as there is no real destruction,
for the light is merely taken away from some places and heaped up at others.

166 INTERFERENCE. CHAP. W II
lengths. Similarly, darkness will occur again at M, brightness at
M, and so a series of bright and dark points occur alternately.
The point M, is such that the difference of its distances from A
and B is n; ; if then this difference remains constant the point Mn
may lie anywhere on a hyperbola (as far as the plane of the paper is
concerned), having A and B for foci.
In space the locus of Mn is obviously an hyperboloid of revolu-
tion, viz. that generated by the revolution of the foregoing hyperbola
round the line AB as axis. §
On the screen then we have not a series of bright and dark points,
but a series of alternately bright and dark lines, or bands perpendicular
to the plane of the paper. These lines are the intersection of the
screen with the hyperboloid loci just mentioned, which are so little
curved as to coincide sensibly with their asymptotes.
The distance of any band from the central point M is very easily
calculated. For if P corresponds to a retardation of n half-wave
lengths, the distance MP is small. Denote MP by a , MO by a, and
with P as centre describe an arc of a circle BC. This arc is approxi-
mately a straight line perpendicular to OP, and AB is perpendicular
to OM, therefore the angle ABC is equal to the angle POM. And
hence their circular measures are equal, or ->
PM. A.C. or 2 º'
OMT BC’ a T c
where c represents the distance AB. Hence
a \
QC = — • ?? -
c “2’
and the point P is bright or dark according as n is even or odd.
Ea'amples
1. The apparent angular distance from the centre M of any fringe of order n
as seen from the point O (Fig. 79) is independent of the distance of the screen MP.
[For if 0, be the angle POM subtended by the central and nth bands at O we
have
an º 8
a T2C Tc '
ºn -
which is independent of the distance a..] -
2. The distance an of the nth fringe from the centre M is proportional to the
wave length and to the order (n) of the fringe and inversely as the apparent angle
of the two sources as seen from M.
X
[For wn-ºn 2 and : measures the apparent angle of the sources.]
89. Colour and Wave Length.-This formula shows that the
ART. 90 FRESNEL'S MIRRORS 167
distance of any fringe from the central one M depends on the wave
length being in direct proportion to it. Hence, if composite light be
used, we should expect to find rainbow-coloured bands" instead of
merely bright and dark lines. This is what is actually observed, and,
moreover, the inner edge of each band is violet, while its outer edge is
red, showing that the violet wave lengths are shorter than the red.
Having accurately determined the magnitudes of ſº, a, n, and c, the
formula gives the wave length of any particular kind of light. By this
method the length of the red waves is found to be about 0000266 in.,
the violet about "0000167 in., and the mean wave about 00002 or
Hoºvo in, or stºo mm.
90. Fresnel's Mirrors.--When Young first published his experi-
ments, scientific men were by no means inclined to admit that the
phenomena observed were due to interference in the manner conceived
by their illustrious discoverer. It was known that the image of a
small luminous origin, formed by the light admitted through a very
small hole, was surrounded by coloured bands, and that light suffered
a similar modification in passing near the edge of an opaque obstacle
(see Newton's Observations, § 134). The bands observed by Young
might then be attributed to this modification (or diffraction). They
might be a variety of diffraction bands.” Objections were therefore
raised, and to remove them it was necessary to devise some method of
obtaining two small sources of light close together wholly independent
of apertures or edges of opaque obstacles. This was first contrived by
Fresnel, whose experiments are justly ranked amongst the most im-
portant and instructive in the whole range of physical optics.
In his first experiment * Fresnel used two plane mirrors inclined at
an angle of nearly 180°, so that they almost lie in the same plane. A
beam of light diverging from the focus of a lens or from a very narrow
slit is allowed to fall upon them. Each mirror reflects the light which
* If the colour depends on the wave length. The existence of these bands con-
sequently indicates that each coloured light has a definite wave length and pariod,
just like each musical note.
* Diffraction bands are also due to interference and the principle of their produc-
tion is essentially the same as that underlying interference bands. The latter are
produced by the interfering action of two distinct waves, while the former are pro-
duced by interference between the elements of a single wave as explained in Art. 126.
Since it may equally be said that both cases are cases of diffraction, it will be
realised that the question is one of terminology. When the solution to a problem
is obtained by resolving a wave into elements in the manner introduced by Fresnel,
it is the diffraction method that is employed. When this refined analysis is un-
necessary, and we obtain a solution by considering the superposition of entire waves
—as was done by Young—we are solving by the interference method.
* CEuvres, tom. i. pp. 150, 186, 268,
168 INTERFERENCE CHAP. VII
falls upon it, and we have therefore two reflected beams whose direc-
tions are inclined at a very small angle.
If S (Fig. 80) be the source of light, OM and ON the two mirrors,
the cone of light reflected from OM appears to come from a vertex
A, the reflection of S in OM; similarly the light reflected from ON
appears to come from B, the reflection of S in ON. If, therefore, the
mirrors are inclined at a very obtuse angle, the points A and B will
be very close together, and the reflected beams should give inter-
ference phenomena on a screen placed across any part of the region
where they overlap, similar to those which would be produced if A and
B were two small apertures. Here now we have two sources of light
close together without the aid of edges or apertures, and the result is
conclusive in favour of Young's theory. A brilliant system of fringes
is produced, similar to those anticipated by the theory. In order to
Fig. 80.-Fresnel's Mirrors.
satisfy ourselves that these bands are really produced by the mutual
action of the two beams, we have only to intercept one of them by
covering the corresponding mirror with lampblack and the whole
system instantly vanishes. They also vanish when the mirrors are
parallel.
Suppose the point of light S to lie in the plane of the paper, and
let the line of intersection of the mirrors be perpendicular to it and
meet it at O. Now SA is perpendicular to the mirror OM, and Am
=Sm. Similarly BS is perpendicular to ON, and meets it produced
at a point n such that Bn = Sn, while the angle ASB is equal to the
external angle between the mirrors, since it is the angle between the
perpendiculars to them from S. Also since OS = OA = OB the points
S, A, B lie on a circle having its centre at O, therefore the angle ASB
at its circumference is half the angle AOB at its centre, or the angle
AOB = 20, if a denote the external inclination of the mirrors. Hence
if DOC be perpendicular to both AB and the screen, D is the middle

ART. 90 FRESNEL'S MIRRORS 169
point of AB and C is the centre of the fringes. If we denote OD by
a and OC by b we have AB = 2a sin (0, since OB is approximately equal
to OD (or sin 0 = tan o), and the formula w-ºn. for the distance of
the nth fringe from the centre becomes
a- (“tº mºeºtºn?,
2a sin w 2 20:00 2
since 0 is a very small angle.
For the success of this experiment very careful adjustment is
necessary. The polished surfaces of the mirrors should extend right
up to the line of intersection of the two faces. If the mirrors are
made of glass it should be black, or else silvered on the first face,
otherwise the reflections from the second surface of the glass destroy
the effect. Polished black glass is commonly used instead of polished
metallic mirrors.
The mirrors M and N are usually attached to a plate, which can be
fixed to one of the uprights of the optical bench. The mirror N can
turn round the axis O ; this axis is fixed to the plate and three screws
pass through the plate to adjust the level of the mirror M. Another
screw is furnished with a spiral spring which keeps the mirror pressed
against the three screws. By means of these the mirror M is adjusted
till its edge is parallel to the axis O. When this is arrived at, a screw
enables the mirror N, by turning round O, to vary the angle between
the mirrors. The other mirror can be screwed forward parallel to
itself. This motion displaces A along the line SA, and the result is
that the central band with the whole fringe system is displaced across
the screen. The complete system of fringes may in this manner be
caused to pass in succession over any desired part of the screen.
In order to obtain the fringes, the mirror M should be adjusted,
so that its plane is parallel to that of N, as tested by looking at the
image of a straight line reflected in both mirrors and seeing that it
keeps straight however the plate be moved. Then fixing the plate to
its upright on the bench, the slit and its image in the two mirrors may
be focussed by a lens in the focal plane of the eye-piece, the slit adjusted
to be vertical and the axis O made vertical, as tested by the images in
the mirrors being made parallel to the image of the slit. If the planes
of the mirrors are only parallel, they may be brought into coincidence,
so as to make the two images formed by them coincide, by turning
the screw, which moves M. parallel to itself.
After removing the lens it is well to push the eye-piece near to the
mirrors, when a small movement of the screw which rotates N should
cause fringes to appear, where the light from the two images overlaps.
170 INTERFERENCE CHAP. VII
These may be made as perfect as possible by narrowing the slit and by
slight further adjustments of the slit or the axis O for parallelism, and
then the eye-piece may be drawn back in order to separate the fringes
far enough for measurement. The plate carrying the mirrors and the
eye-piece should have been put in such positions, by rotation round a
vertical axis and motion perpendicular to the line of the bench, that
this movement of the eye-piece does not cause the fringes to move
across the field of view.
After the fringes have been measured, the distance between the
two sources may be determined by the method described in the note,
p. 171.
If the screw which rotates N be standardised, the angle between
the mirrors may be calculated from counting the number of turns of
the screw given to it in rotating N from its position, when the planes
of the mirrors coincide, to its position, when the measured fringes are
obtained. -
The distances a and b can be measured on the scale of the optical
bench ; the distance a by the micrometer motion of the cross wires in
the eye-piece, and n can be counted. We thus arrive at a determina-
tion of the wave length of any particular kind of light which we may
choose to fix on.
In practice a narrow line of light (an illuminated slit) is used.
In Fig. 80 the mirrors are planes perpendicular to the paper through
OM and ON, while the slit is perpendicular to the paper through S.
91. Fresnel's Bi-prism.–In the above experiment a pencil of
light was divided by reflection into two others inclined at a small angle,
Fig. 81.-Fresnel's Bi-prism.
and these produced the phenomena of interference. It is possible to
procure the same result by refraction, and this is the basis of Fresnel's
second experiment."
Let CDE (Fig. 81) represent a glass prism with a very obtuse
angle E, and let light from O fall perpendicularly on the opposite
face CD. The whole prism is as if made up of two prisms CE and
* This experiment has been sometimes wrongly attributed to Pouillet. It was
first described by Fresnel (CEuvres, tom, i. p. 330).

ART, 91 FRESNEL’S BI-PRISM d 171
DE of very small angle (at C and D) placed base to base at E, and
hence the name bi-prism.
The light which falls upon the upper half of the prism is bent
downwards, and appears after emergence to diverge from a point B,
while that which falls upon the lower half is bent upwards, and appears
to diverge from A. The less the angles C and D the nearer together
will be the points A and B, so that by diminishing these angles suf-
ficiently the emergent cones of light will be similar to those coming
from two very near origins, and interference effects will be presented
as before.” -
The distances from the prism of the virtual foci A and B are
very approximately the same as that of the luminous origin O. For
since the refracting angles C and D are very small, the focal lines
of each refracted cone coincide, and p = pg=p by Ex. 11, p. 127.
In practice a narrow strip of light, from a slit, is used. By this
means very much increased brightness is obtained without loss of
definition, as the various parts of the slit, if it be very narrow, give
rise to coincident systems of bands. The length of the slit is carefully
adjusted parallel to the edge of the prism, and the fringes are parallel
to their common direction. The field is usually bordered with other
systems of bands. These arise from diffraction, and will be explained
farther on. º -
The distance of the nth band from the centre of the system is
usually derived as follows. If a denote the distance of the origin O
from the prism, and b the distance of the prism from the screen, we have
AE = BE = a very approximately, and consequently if c denote the
distance AB between the virtual foci" and 8 the angle BEO or the
1 An elegant method of determining c is given by Glazebrook (Physical
Optics). Introduce a lens between the prism and eye-piece. This lens will form
images of A and B in the focal plane if it is properly placed. Now in general two
such positions of the lens can be found, and if di and d2, di' and dº be the dis-
tances of the lens from the focal plane and from the points AB in the two positions
respectively we have #
1 1 1 1
But di-H da–d,’ 4-dº, therefore di = d2" and d2=d,'. Hence if cI and cº be the cor-
responding distances between the images of A and B in the focal plane of the eye-
piece we have, c being the distance AB,
C1 C C2 C
+ = +, and # = +..
e - di dº' di' do'
Therefore * cº-cº, or c=Vee,
e, and c, are measured by a micrometer eye-piece. The lens must be chosen of such
a focal length that both positions lie on the screen side of the prism.
172 - INTERFERENCE CHAP. Wii
deviation produced by the thin prisms, we have, if a = angle of prism,
c=2a sin 6=2a(u – 1)0,
for since the angle of the prism is small sin 6 = 8 – (u – 1)a. Hence
, a + b ... A a + b X
* - 5
~ a "3-3. II)." 2
which shows that the bi-prism 1 is equivalent to a pair of mirrors
inclined at an angle (p − 1)a. -
It is easy to see, however, that this simple proof is defective.
It is based on the assumption that the virtual foci A and B are in
fixed positions in spite of the variation of the inclinations of the rays
proceeding from them. If this were indeed the case, moving the
prism slightly downwards (i.e. keeping the plane through CD un-
changed) would not shift the fringes. In reality such a movement
shifts them very nearly (a + b)/a times the movement itself. It is
more satisfactory to start by taking the deviations of the rays that
meet on the screen as being equal—at least to a first approximation,
which is all that is necessary. It then follows that the actual lengths
of the paths of the rays starting from the real source O are equal
(neglecting third-order quantities) when the thickness of the prism is
neglected. The relative retardation on reaching the screen is entirely
due, to a first approximation, to the different thicknesses of glass
traversed. Distinguish the data for the two rays by suffixes. Let
the angles which the incident and emergent rays make with the axis
be D and b, and the angle of the prism at C be a. Then
ô= 61 -- pi = 02+ p2=(u – 1)a,
and
a8= b% +a; ;
or eliminating p
(a + b)61, =bö--ac, and (a + b)62- bb — a.
The fringes (bright and dark) are given by -
*=(-1)(.-e)
since this is the relative retardation, where e, and e, are the thick-
nesses of glass traversed. But
e2 - €1 = a (61 – 92)a (1st approx.).
Hence, finally,
º- -1)** _ 2aa.
3 =\!. a + b T a H-bº
* If the prisms CE and DE were placed edge to edge instead of base to base, A
and B would be interchanged and the emergent cones would have no common part.
Interference bands could, however, be obtained by interposing a lens in the paths of
the cones bringing them to real foci A' and B' images of A and B. The light diverg-
ing from A' and B' will produce fringes.
ART. 93 BI-PLATES 173
This is the same result as before. The compensation which enables
the simple method to give the right result in the case of the fixed
prism arises from the fact that not only is the virtual source
different for different inclinations of the emergent ray, but a difference
of phase must at the same time be attributed to it.
92. Peculiarities of the Bi-prism Fringes.—The fringes produced
by a bi-prism differ in some respects from those obtained by reflection.
For, on account of the dispersion in the glass, the foci A and B will
be different for the different colours. Thus the violet light will appear
to come (on account of its greater refrangibility) from two points A,
and B, a little farther apart than the two from which the red appears
to come. Denoting these distances by c, and c, the formula for the
distance of the nth red band from the centre is (by Art. 88)
º _**", \r
"T c, "2"
and for the violet
º _***,\!.
"T c, " 2
The difference between a, and a, is consequently greater than if there
was no dispersion by the prism. The iris-coloured bands are therefore
broadened by the dispersion, and the overlapping is proportionately
increased. The fringes of the bi-prism are bright, for the prism allows
a great quantity of the light which falls upon it to pass through.
These fringes are then bright, and very easily procured—the apparatus
requiring very little trouble in setting up.
93. Bi-plates.—A beam of light may be subdivided by refraction
through two plates, of the same nature and equal thickness, placed at
Fig. 82.-Bi-plates.
an angle as indicated in Fig. 82. Two pieces M and N of parallel
glass are cut from the same plate to ensure equality in thickness, and
placed at an angle. On the bisector of the angle between them is
placed the luminous origin O. The light which falls upon the plate
N passes through it in a direction CD, and emerges parallel to its
original direction, appearing to diverge approximately from a point A.
Similarly the light which emerges from the plate M diverges from a
virtual focus B. The emerging cones are received by a lens L which

174 INTERFERENCE CHAP. W 11
brings them to real foci A and B. After diverging from A and B
the beams will overlap and produce fringes.
The lateral displacement of a ray in passing through a parallel
plate of thickness e is easily seen to be e sin (; – r)/cos r, and conse-
quently the distance c between the virtual foci A and B is very
approximately
c=2esin (i-r)/cos r,
which, if 20 denote the angle between the plates, may be put at once
in the form
c=2e costſ1-(u2+(49–1) cotºl,
since i−90° - 6 very nearly.
94. Lloyd's Single-mirror Fringes.—A convenient method of dis-
playing interference bands caused by the mutual action of direct and re-
flected light was devised by Dr. Lloyd of Trinity College, Dublin. A
polished mirror of metal or of black glass is placed so that the rays
from a luminous origin B (Fig. 83) are reflected from it at nearly
grazing incidence. The reflected rays diverge from a virtual focus A
which is the image of the
origin B, so that a point
P on a screen placed be-
yond the mirror receives
light directly from B, and
also by reflection from the
mirror, or, regarding A as
the source of the reflected light, P is supplied by the origins A and B,
which may be brought close together by making the angle of incidence
nearly 90°. -
Since the reflected light is confined to the upper side of the mirror,
less than half the complete system is formed, and it might be imagined
that under no circumstances could more than one-half the system be
obtained. However, by interposing a thin transparent plate in the
path of the direct beam, or by holding the magnifier through which
they are examined somewhat excentrically, the bands may be displaced
(see Art. 102) so as to detach themselves from the mirror until the
complete system is seen, as in Fresnel's experiments. The adjust-
Fig. 83.-Lloyd's Single Mirror.
* “A New Case of Interference of Rays of Light” (Lloyd, Trans. Roy. Irish
Academy, vol. xvii.; read 27th January 1834).
Dr. Lloyd's method seems to have been anticipated by Professor Powell in the
following passage: “Beautiful sets of colours (the theory of which is evidently de-
pendent on interference) are seen on viewing a candle, or line of light, by very oblique
reflection from any moderately polished surface, as ivory, ebony, etc., held close to
the eye” (Rev. B. Powell, Phil. Mag, and Ann., January 1832). These bands may,
however, have been due to diffraction.

Ant. 95 FRESNEL’S THREE-MIRROR EXPERIMENT 175
ments in this experiment are easily made ; it requires no special
apparatus, and the bands are bright and well marked.
Dr. Lloyd states that the centre of the system does not correspond Displaced
to the line of intersection of the mirror and screen, but that the bands."
are all displaced through half the interval of a band width from the
mirror edge. This, he suggests, indicates that the reflected light has
been accelerated by half a wave length, or that its phase has been
increased by T at reflection. The experiment is, however, a delicate
one ; moreover, the effect is more complicated than the simple
interference method implies. The observation must therefore not be
considered as settling the question of the change of phase on reflection.
95. Fresnel's Three-Mirror Experiment. – Fringes produced by
the use of three plane mirrors have also been obtained by Fresnel."
*
------------
-
-
Fig. 84.—Fresnel's Three Mirrors.
Of the two pencils which produce the bands, one is reflected from the
mirror M (Fig. 84) and the other successively from the mirrors L and
N. The planes of the mirrors L and N intersect at O on the surface
of M. Let to and to be the angles they make with it respectively.
Now if the line OS makes an angle 0 with the mirror L, the light
reflected from L will appear to come from a point Sº where the arc
SS = 29, and if this light falls upon N its reflected beam will diverge
from A, where the arc SA=2(0) + 0 – 6), for the mirror N makes an
angle, to + 0–0 with OS'. Hence
SA=2(w-w').
So also the light reflected from M will diverge from B where
SB =2(w--0).
Consequently the arc
AB =2(w' – 9).
Hence if we denote OS by a we have OA = OB = a, and the chord
AB = 2a sin (w'-6).
* Fresnel, CEuvres, tom. i. p. 703.

176 INTERFERENCE CHAP. VII
The system is consequently equivalent to two mirrors inclined at an
angle @' – 6.
Since the mirror M is shaded by the others, the interfering pencils
will be incomplete and unsymmetrical with respect to the central line
OC. The central fringe may therefore be found near one extremity
of the system, or it may lie entirely outside the visible fringes or
common part of the interfering pencils. By increasing the path of the
doubly reflected pencil, or diminishing that of the other, the fringes
may be displaced on the screen so that the whole system may be
viewed. This is readily done by screwing forward the central mirror
M in the direction of the normal to its plane.
The angles to and to may have any value up to 45°, for it is only
necessary that o' – 6 should be small. Fresnel worked with a = 0,
and varied the angle between 7° 30', and 40°.
If a change T of phase accompanies each reflection the twice re-
flected beam will be accelerated by half a wave length on that reflected
from the central mirror, and therefore, as in Lloyd's bands, the central
band or that corresponding to equal distances CA and CB, will be
black. This is confirmed by the experiment. It should also be
further remarked that, as in case of a single mirror, the right side
of A corresponds to the left of B and the left to the right, as if B
were the reflection of A with respect to the central line OC (see
Art. 98).
96. The Optical Bench.-Interference experiments are usually
made on an optical bench. This apparatus usually consists of a
horizontal bar, accurately graduated, along which three vertical up-
rights can slide freely. Attached to each upright is a vernier, so that
the distance between them can be determined accurately by the scale
on which they slide. Each is also furnished with a head-piece, which
can be raised or lowered or turned round a vertical axis at will.
The first upright is furnished so as to hold a metal piece, which
carries a slit capable of being adjusted to any convenient width by
means of a screw. A fine adjusting motion of the head allows the
slit to be brought accurately parallel to any desired direction. The
second upright carries a frame in which can be placed a metal ring,
which holds the plate containing the two small apertures, or any other
apparatus for producing interference or diffraction, such as the bi-prism,
a fine wire, an opaque edge, a diffraction grating, etc. The third
carries a micrometer eye-piece furnished with a cross wire. The axis
of this eye-piece is horizontal; that is, parallel to the bench on which
all the pieces slide. The head of the second upright can be moved by
means of a horizontal screw perpendicular to the length of the scale
ART. 97 CONDITIONS NECESSARY FOR INTERFERENCE 177
so that each piece which it carries can be brought into the line joining
the slit and observing telescope.
The use of a lens is legitimate in experiments on interference, for
light brought to a focus by a lens is concentrated there without any
relative change of phase in its components; since all the rays brought
to that focus travel over paths which require the same time.
The eye-piece being furnished with a cross wire and micrometer
screw, the distance a of any band from the centre can be measured,
and the value of A can be calculated accordingly.
97. Conditions neeessary for Interference.—When commencing
these investigations we assumed the waves emitted by A and B at any
instant to be always exactly the same, and the theory indicates that
this is necessary in order to have interference fringes or points at
which the effects destroy each other continually. If the phases of the
waves from A varied irregularly a great number of times per second
with respect to those from B, we should have at any given point P
neutralisation and co-operation succeeding each other so rapidly that
nothing but a mean effect would be perceived, and this would be
merely the sum of the mean effects of each source taken separately.
Now if the disturbances come from two independent sources, such as
the two different parts of a flame, the relative phases of the two would
be purely casual, and no fixed and permanent neutralisation could be
expected. Observation shows that no interference effects are mani-
fested unless the two interfering streams of light come originally
from the same source, and subsequently traverse slightly different
paths, and this is what the theory anticipates.
In the experiments of Grimaldi the apertures were illuminated
directly by the Sun, and consequently no interference phenomena
could possibly have been observed. This point was particularly
noticed by Young, who allowed the sunlight to pass first through a
narrow slit, and then through the two small apertures. He re-
marked that the fringes disappeared when one of the apertures was
stopped, and also when the slit was removed, so that the two apertures
were illuminated directly by the sun. In this case each point of the
sun produces a distinct set of fringes, but the multitude of sets
become so superposed and interspersed that all visible effect is
obliterated.
An essential condition then is that the two apertures be supplied
from the same source, so that the waves diverging from A and B at
-any instant may be exactly alike. To effect this in practice, a narrow
slit is usually placed symmetrically near the apertures with its length
perpendicular to the line joining them. The light from a lamp or
N
178 INTERFERENCE CHAP. VII
other source falls first upon the slit, or is focussed on it by means of a
condensing lens, and after diverging from it reaches the two apertures.
The slit, being very narrow, is like a single line of light, each point
of which is symmetrically situated with respect to A and B, and sends
waves to each which are exactly alike, so that the whole resultant
wave emitted by A is the same as that emitted by B. These waves,
on arriving at any point P of the screen, will produce the phenomena
of permanent interference." The bi-prism and other apparatus for
producing interference bands are therefore to be regarded as contriv-
ances to procure two similar origins of light in close proximity.
From this it will be easily understood how it is that two lamps, or
two candles, can never be expected to destroy each other's effects
anywhere when placed close together like two organ-pipes tuned in
unison. Two lamp flames have no permanent phase relation. The
waves sent out by one are not necessarily similar to, or in any way
related to, the waves sent out by the other. Each point of a flame is
an independent source of light, and the waves emitted by it continually
vary in character; while for two sources to produce darkness at any
point the necessary condition is that they should continually send
waves to this point opposite in phase, but in other respects exactly
alike. - §
98. The Corresponding Points of the Sources.—In Young's
experiment the two apertures are supplied by the same small source,
and in the case of Fresnel's mirrors and bi-prism the interfering
origins are images of the same source, and are therefore similar—the
right-hand side of one corresponding to the right-hand side of the
other, and the left to the left. With Lloyd's single mirror it is some-
what different, for here the two interfering sources are the luminous
origin and its image, but the right side of the image corresponds to
the left side of the origin, and vice versa. Now continuous interference
can be expected to occur only between rays issuing from corresponding
points of the interfering sources, for the waves emitted by the various
points of the source (slit) have necessarily no fixed phase relation
when it is supplied directly by a flame. Hence with the bi-prism and
mirrors it is the right side of one image that interferes with the right
of the other, and the left with the left, but in Lloyd's experiment the
right side of the source interferes with the left of the image, and
vice versa. In the latter case, when the slit has any sensible width
* The fringes are only produced distinctly when the source is very narrow. The
width of the aperture should be so small that the displacement of the centre of the
system, incurred by using in turn the two edges of the slit as linear apertures,
would be small compared with the width of an interference band.
ART, 99 LIMIT TO THE NUMBER OF FEINGES 179
the centre of symmetry must be the same for the bands produced by
all corresponding points, but the distance c between the corresponding
points is variable. In Fresnel's experiments, on the other hand, the
distance c is constant, while the centre of symmetry varies. In Lloyd's
experiment then the central bands are exactly superposed for all the
groups of corresponding points, and the width of the slit does not
interfere with the achromatism of the central line. The widths of
the bands produced by different pairs of corresponding points will,
however, be different (since the band width varies inversely as c), and
this results in a confusion which increases from the centre outwards."
In Fresnel's experiments the band width is the same for all pairs of
corresponding points, and the width of the slit merely leads to a
lateral displacement of the central line of the various systems, so that
the condition for distinctness is that the width of the slit be narrow
compared with the width of a band, and this limiting width of the slit
is independent of the order of the bands.
99. Limit to the Number of Fringes.—The formula (p. 165) for
the distance of any bright band from the centre of the system shows
that the width of any band, measured from darkness to darkness, is
w-a)/c.
The band width is therefore directly proportional to the wave length
of the light employed. If the light could be procured absolutely
homogeneous—that is, of a single wave length A–then theoretically
the screen should be covered with an infinite number of similar bands,
having nothing to distinguish one from another.
With ordinary light the case is very different. Each colour gives
rise to a system of bands, and of these the red bands are broadest
and the violet narrowest, the width of the former being about twice
that of the latter. Hence it happens that after a few alterations the
nth red band coincides with the (n + 1)th violet, or perhaps the dark
spaces of one system are filled up with the bright bands of another, so
that overlapping and superposition of the multitude of systems from
* The measure of the confusion arising from the variation of c in the finite width
of the slit in Lloyd's experiment is easily found. Thus the distance of the nth
band of any system from the centre is a = max/c, and therefore the interval 30, be-
tween two corresponding bands when c varies is
63; = — *s.
C
or if w be the band width = a \ſc we have
ða; Öc
it — 71, -— 3
20 C
therefore the slit must be made narrower as ºv increases if the distinctness of the
bands is to be preserved.
180 INTERFERENCE CHAP. VII
the red to the violet takes place, and this leads to the final obliteration
of all visible effect at a short distance from the centre (Fig. 85).
What occurs then is that a few
(ten or twelve) bright rainbow-
coloured bands are seen which
become less and less distinctly
marked, finally merging into
one another and fading into
uniform illumination at a
short distance from the central
line. This line is white, as it is a bright line for all wave lengths.
Theoretically there is not a single place of complete darkness, for this
would entail at that place a complete discordance of phase for all
wave lengths, whereas any point at a distance a from the centre will
be bright, for the wave length determined by na) = ca. The very
existence of any visible bands with white light depends on the limited
sensibility of the eye, which is confined to about one “octave,” and
on its capability of making chromatic distinctions.
Want of purity in the light is therefore detrimental to the pro-
duction of a large number of visible bands.
100. Interference under high Relative Retardation.—It follows
from the foregoing considerations that the more homogeneous the light
the greater the number of observable interference bands, and if per-
fectly homogeneous light—that is, light of a single wave length—could
be obtained, the screen should be covered with an infinite system of
similar bands. In practice, however, it has not as yet been possible to
obtain strictly homogeneous light. An approximation to it may be
obtained by casting the spectrum of any source of light on a screen,
furnished with a very narrow slit, in such a way that a narrow strip
of the spectrum is transmitted through the slit. This slit may be used
as the source of light in an interference experiment, and when so
employed, the number of bands directly observable is vastly greater
than that given by a source of ordinary white light. Still, in this
case also, the light is heterogeneous, for the slit transmits a narrow
band of the spectrum, and this band contains a group of waves varying
in length by an amount depending on the width of the slit, and, as
the band width will be different for the different constituents of the
group, there will be overlapping, and ultimately obliteration of the bands.
A very convenient source of approximately homogeneous light is
that of the sodium flame." This consists of two narrow bands very
Fig. 85.-Overlapping of the Fringes.
* Proposed by Brewster, Annales de Chimie et de Physique, second series, tome
xxxvii. p. 437, 1828.

ART, 100 LARGE RELATIVE RETARDATION 181
close together in the yellow part of the spectrum, so that it contains a
group of waves of different lengths, and overlapping ultimately occurs.
With this source of light Fizeau observed 50,000 bands, and more
recently this number has been largely increased by Professors A. A.
Michelson and E. W. Morley." Using the light of incandescent sodium
vapour (in an exhausted tube provided with aluminium electrodes)
interference was observed with a retardation of over 200,000 wave
lengths, i.e. over 4 inches. This number was still further increased by
using the light from Plücker tubes containing vapour of mercury or
thallium chloride which gave interference with a difference of path of
540,000 and 340,000 wave lengths respectively. In these experiments
a special form of apparatus, called an interference refractometer, was em-
ployed (Art. 123), by which any desirable difference of path could be
easily introduced between the two interfering beams. Experiment
consequently proves that the number of bands directly observable
increases with the purity of the light.
It may also be shown that interference takes place in the regions
beyond the limits of the visible fringes where overlapping exists to such
an extent that the field appears to be uniformly illuminated. In these
regions any point is a place of brightness for certain wave lengths and
of darkness for others, and consequently the light there is a mixture
in which only certain constituents of the original light are represented.
Hence, if a slit be opened in any part of this region so that the light
falling on it may be transmitted and examined in a spectroscope, the
spectrum will exhibit certain dark bands corresponding to those waves
which are destroyed by interference at the slit. This method was
employed by Fizeau and Foucault.” If a narrow slit be opened * at
the centre M of the fringe system the light which is transmitted
through it will give a complete spectrum, since the central fringe is a
place of brightness for all colours. On the other hand, if a slit be
Spectro-
Scopic
analysis.
opened at any other part of the fringe system the transmitted light .
will give a spectrum exhibiting those colours for which the slit is a
place of brightness, and consequently crossed by dark bands showing
the absence of those colours which are destroyed by interference at the
slit. If a be the distance of the slit from the centre of the system,
the wave lengths of the dark bands satisfy the equation
* A. A. Michelson and E. W. Morley, Journal of the Association of Engineering
Societies, May 1888, and Address before the American Association, Cleveland Meet-
ing, August 1888, by A. A. Michelson.
* Fizeau and Foucault, Annales de Chimie et de Physique, third series, tome xxvi.
p. 138, 1849; Comptes rendus, 24th November 1845.
* The width of the slit should not be more than a moderate fraction of that of a
band.
182 INTERFERENCE CHAP. VII
O, A
w=;(2n+1);
In the experiment of Fizeau and Foucault a slit was opened at
the centre of the system produced by Fresnel's mirrors, and by screwing
forward one of the mirrors parallel to itself, the system of fringes was
displaced gradually across the screen, so that they passed in succession
over the slit. As the first dark band comes on the slit, a dark band is
seen in the spectrum which passes from the violet to the red as the
mirror is gradually displaced, then a second succeeds it, and then
two, three, or more dark bands simultaneously appear crossing the
spectrum, till finally they become so numerous and narrow that to
separate and distinguish them the resolving power of the spectroscope
requires to be increased. Hence the extent to which interference can
be observed is limited only by the resolving power of the spectroscope.
It has been usually held that even if the number of observable
fringes were not limited by overlapping, yet a major limit to this
number would be determined by certain irregular changes taking place
in the source of light. Thus if we consider a point P so far from the
central line of the fringes, that AP – BP is a large number of wave
lengths, then the light reaching P at any instant from B is a large
number of wave periods in advance of that which reaches it from A.
The former wave was emitted from the source some time before the
latter, and if the nature of the waves emitted by the source has
changed in the meantime, the waves reaching P at any instant from
A and B will be dissimilar, and have no constant phase relation.
For example, in the case of a monochromatic source, if the vibra-
tion be represented by a simple equation of the form a = a sin (ot + a),
we deal with an infinite train of waves propagated in a perfectly
regular manner. If this train be supposed broken up by sudden
changes of phase originating in the source, then we have to deal
with a system of groups of waves, and the conditions of propagation
will be altered, so that the state of affairs at any point ceases to be
represented by the foregoing simple equation, and the problem becomes
much more complicated.
No conclusion as to the nature of the vibrations causing white
light can be drawn from the fact that with it interference bands may
be observed under high relative retardation. Gouy and Rayleigh
have shown that if the disturbance in the medium were due to quite
irregular impulses from the source, the action of a prism would be to
analyse the complex disturbance at any point into its constituents of
definite wave length, in the same way that a complex periodic function
is analysed mathematically into its simple harmonic components in a
ART. 100 LARGE RELATIVE RETARDATION 183
Fourier series. The limits of the retardation within which interference
will occur depend entirely upon the resolving power of the spectro-
Scope and are not concerned with the nature of the vibrations in the
Source. In M. Gouy's opinion the nature of white light may be best
understood by assimilating it to a disturbance originated by a sequence
of entirely irregular impulses, though Lord Rayleigh points out that
the impulses themselves must not be supposed to be arbitrary, as in
that case there would be no room for distinguishing the radiations of
various temperatures.
Professor Schuster has shown that, whether we consider white
light as due to the vibrations of a large number of systems each
sending out homogeneous vibrations of a definite wave length, or that
it is due to separate instantaneous impulses, there is a definite limit
of retardation, depending on the resolving power of the spectroscope,
beyond which interference cannot take place. He points out that the
disturbance produced by a single impulse is spread out by a grating
into a disturbance lasting over a finite time, which increases with the
number of lines in the grating, and he holds that a prism produces a
similar effect, owing to the group velocity in the prism being less than
the wave velocity. His conclusion is thus the same as Rayleigh's,
viz. that “there is no distinction between regular and irregular light
beyond that which is brought out by the distribution of energy in the
spectrum. If the intensity vanishes except for a definite wave length,
we must call the vibration completely regular ; if the intensity is the
same for all vibrations, it will appear that we must call it completely
irregular.” He explains the production of fringes without the use of
a spectroscope as a resonance effect in the eye or photographic plate.
The effect of an impulse is spread out over a certain time and, if this
lasts until the effect due to the disturbance travelling by a different
path begins, interference may result. (See further, M. Gouy, Journal
de Physique, 2*, tom. v. p. 354, 1886; Lord Rayleigh, Phil. Mag.
vol. xxvii. p. 460, 1889; Arthur Schuster, Phil. Mag. vol. xxxvii.
p. 509, 1894.)
It is easy to calculate a form of pulse which will correspond, for
example, to the distribution of energy in “white ” light; meaning by
this term the radiation emitted by a full radiator (i.e. a perfectly
black body), Assuming Wien's law of distribution, the form of pulse
is given by the integral
° km. 1 '5
$/= - 2 {}
6 70, cos madm
w 0
-()*() ºr sº (;)
184 INTERFERENCE CHAP. VII
where tan / =# and T is the gamma function. A little more
than half of this pulse is shown in Fig. 85a by a continuous curve.
The full curve is symmetrical with respect to the vertical through the
circle. Since the phases of the components are of no importance, in
regard to the distribution of energy, an infinite variety of forms
exists. If we had taken sin no instead of cos na; the pulse would
be given by the dotted curve in the same figure. It is diagonally
symmetrical with respect to the same vertical; i.e. it is as much below
the axis on the left as it is above it on the right. Planck's equation
gives very similar pulses because the amounts of energy at the lower
values of n (where Wien's equation fails) are so small. Planck's
Fig. 85a.
equation can be developed into an infinite series of Wien equations
with different values for k, and each can be integrated separately.
If two dark bands appear in determinate parts of the spectrum
corresponding to wave lengths A and A', then by the above equation
(p. 182)
(2n + 1)A=(2n' + 1)A’=26,
where n and n' are two whole numbers, and 6 is the difference of the
distances of the interfering origins from the slit in the screen. Be-
tween these two bands a number N of other dark bands may occur
which can be counted. Then we have
—, , –2n+1 \-X.
N + 1 = 'n' — m = 2- -N-
2X’ 2X
OI’ – ------ f - *--------a-º, tº
2n + 1 (N+1);ºn 2n' + 1 (N+1), ºv
Knowing n or n' we can calculate 8, the relative retardation of the
pencils when they reach the slit. -

ART. 102 ACHROMATIC FRINGES 185
101. Achromatic Interference Bands.—The interference bands
ordinarily obtained are highly coloured, and this happens because, in
the formula,
X
Q} *n
! E: — -3
C 2
the distance c is the same for all colours while A is variable." If, how-
ever, by some means c be made different for the different colours so
as to be directly proportional to A, we will have Aſc = const., and
the bands will be of the same width for all colours. The fringes will
therefore be achromatic, and the want of homogeneity of the light will
offer no obstacle to the production of a large number of fringes. This
may be easily arranged by using Lloyd's mirror and a diffraction
grating (Art. 139, etc.) with which to form a spectrum. White
light from a narrow slit falls in succession upon a grating and an
achromatic lens, so as to form diffraction spectra in the focal plane of
the lens. One of these spectra’ is used as the proximate source of
light in the interference experiment, and since the deviation of any
colour in the diffraction spectrum varies as A, it is only necessary to
arrange the mirror so that its plane passes through the white central
image in order to realise the conditions for achromatic bands. When
the adjustments are carefully made the whole field is filled with fine
bands, which become coloured only at the edges of the field.
With less perfection the diffraction spectrum may be replaced by
a prismatic one so arranged that Aſc is constant for the most luminous
rays. “The bands are then achromatic in the same sense that an ordi-
nary telescope is so. In this case there is no objection to a merely
virtual spectrum, and the experiment may be very simply executed
by Lloyd's mirror and a prism of about 20° held just in front of it.”
“It is interesting to observe the effect of coloured glasses upon the
distinctness of the bands. If the achromatism be in the green, a red
or orange glass, so far from acting as an aid to distinctness, obliterates
all the bands after the first few. On the other hand, a green glass,
absorbing rays for which the bands are already confused, confers
additional sharpness. With the aid of a red glass a large number of
bands are seen distinctly, if the adjustment be made for this part of
the spectrum.” “
102. Displacement of the Fringes—Diminished Speed in Denser
Media.-In the deduction of the law of refraction the wave theory
pointed out that the velocity of light should be less in the denser or
* In the case of the bi-prism c varies, being least for the red and greatest for the
violet, and this exaggerates the overlapping and increases the colouring.
* Lord Rayleigh used the second.
* Lord Rayleigh, Phil. Mag. vol. xxviii. pp. 77, 189, 1889.
186 INTERFERENCE CHAP. VII
more refracting media, and we mentioned in passing that the emission
theory pointed to an opposite conclusion. This point can now be
decided by means of the phenomena of interference. It is obvious
that the central fringe is situated in that place to which it takes the
light the same time to travel from the two interfering origins. If now
a thin plate of glass (or other transparent substance) be interposed
in the path of one of the beams, the
light of that beam will be retarded
or accelerated according as it travels
slower or faster in the glass than in
air. The point then at which the two
beams will arrive in the same time will
Fig. 88-Displacement of the Central Band, be displaced on the screen. The central
band will be moved towards the path of the beam in which the plate
(Fig. 86) is interposed if the light travels slower in the glass, but to
the opposite side if it travels quicker in it. The result is decisively
in favour of the diminished velocity of light in the more refracting
media, and the difference of velocity may be measured by the amount
by which the fringe is displaced.
It is easy to calculate the relation between the displacement and
the refractive index of the interposed plate. Let P be the position
of the central fringe when the plate is interposed in the path of
the ray BP. The time of travelling over AP is the same as that
of travelling over BP; hence if v and v' denote the velocities in
air and in the plate respectively, and e the thickness of the plate,
then
BP – e e AP,
tº - d.
Ot
BP-e-Hue-AP.
Hence
e(u – 1)=AP– BP=n.
if the central fringe is displaced through the distance occupied by m
fringes, a result which was obvious, for the retardation introduced by
the passage through the plate is (, – 1)e.
103. Application to the Determination of Refractive Indices.—
The amount of displacement in the foregoing important experiment
furnishes us with a method of determining the refractive index of a
substance when the displacement of the central fringe and the thick-
ness of the plate are known.' Fresnel and Arago” applied it to the
* It also determinese when we know u and the displacement.
* By observing the position of the fringes formed by two rays, one of which
passed throngh vacuum and the other through air.

ART. 103 CIRCULAR FRINGES 187
determination of the refractive indices of gases. It is susceptible of
great accuracy, the minutest change in the index of refraction of air
being observed—such, for instance, as the change due to a rise of ºr
of a degree in temperature.
By the same method it was ascertained that the refractive index
of dry air is about one-millionth greater than that of air saturated
with vapour. Arago also pointed out that the scintillation of the
stars is due to interference arising from the changes in the refracting
powers of portions of the atmosphere through which the different
portions of light reach the eye.
M. Billet" has devised a very convenient method of producing
interference fringes, and showing the effect of a plate interposed in
the path of the interfering pencils in displacing the fringes. It
consists of a lens cut into two halves, L and L. (Fig. 87), which can be
separated or brought close together at will by means of a micrometer
Fig. 87-Billet's Split Lens.
screw. Their sections are brought parallel by turning one of them
round a fixed axis by means of another screw. The luminous origin
O produces two images A and B situated on the optic axes OA and
OB of the two halves, the motion of which allows the images A and B
to be brought as close together as desired. The light diverging from
these images produces fringes on a screen placed at any part of their
common path, and it is easy to interpose plates of any transparent
substance in the path of either or of both simultaneously.
With the bi-prism the thin plate may be placed over one-half of
the prism, or if two plates are to be compared, one may be placed
over each half. These experiments were first executed by Fresnel
and Arago,” and gave rise to the construction of interference
refractometers.
An ingenious modification of Billet's experiment has been recently
suggested by M. G. Meslin,” in which fringes of a circular form are
* Ann, de Chimie et de Phys, third series, tome lxiv. p. 385.
* (Euvres de Fresnel, tome i. pp. 125, 691, and several memoirs of Arago (CEuvres,
tome x. pp. 298, 312).
* G. Meslin, Comptes rendus, 1893; and Journal de Physique, 3me, tome ii.
p. 205, 1893.

188 INTERFERENCE CHAP. VII
Circular
fringes.
obtained. Thus in all the forms of experiment so far described the
line AB joining the interfering sources is at right angles to the
direction of propagation of the light, and for this reason the fringes
obtained on the screen on which the light falls are approximately right
lines perpendicular to AB. This is the case because the surfaces of
constant retardation are hyperboloids of revolution round the line AB,
having A and B for foci. But if by any means the line AB is turned
so as to be parallel to the direction of propagation of the light—that is,
if it is turned so as to pass through O—then the axis of revolution of
the surfaces of interference will be perpendicular to the screen, and
their cross sections on the screen will be a system of concentric circles.
In Billet's experiment AB is at right angles to the central line, because
L and L are separated by displacement at right angles to this line, but
Fig. 88.
if the displacement were made parallel to this line, then the line AB
would pass through O.
This is shown in Fig. 88, in which the lower half of the lens is
displaced through an interval CC’ parallel to the central line. With
this arrangement the light which passes through L is brought to a
focus A on the line joining 0 to its optic centre C, while the light
passing through L' is brought to a focus B on the same line, and if a
screen PQ be placed anywhere between A and B, interference fringes
will be depicted on it in the region of the overlapping beams of light.
The surfaces of constant retardation in this case are not hyperboloids
but ellipsoids of revolution round the line AB, and having A and B
for foci. For if we consider any point X on the screen which receives
light from both L and L', the light reaching it from L will have
travelled over a path 6 +AX, where 6 represents the equivalent path
from 0 to A and is the same for all rays passing through L. Similarly
the light reaching X from L will have travelled over a path 8 - BX,
where 6 is the equivalent path from 0 to B through L. The differ-
ence of path at X for the two constituents is consequently
(6+AX)-(6'-BX)=(6 – 6')--AX + BX.
But 3’- 6– AB = const.,

ART, 104 EXAMPLE 189
consequently the path retardation at X is
AX + BX – AB,
and if this is to remain constant we must have
AX + BX = const.
That is, X must lie on the surface of an ellipsoid of revolution, having
A and B for foci. The cross section of any one of these ellipsoids by
the plane PQ is a circle, and bright and dark bands are accordingly a
system of concentric circular rings. These circles are not, however,
complete, for the beams of light overlap only on one side of the
central line, and at most only half of the complete system, i.e.
semicircles, can be obtained. .
Eacample
For a given position of the screen prove that the radii of the consecutive rings
are proportional to the square roots of the whole numbers 1, 2, 3, etc.
[Take the middle point of AB as origin and the line AB as axis of a, so that the
equation of any one of the ellipses may be written in the form
3:2 m/?
3/
:*#=1,
where AX+BX=2a. Hence if AB = 2C the relative retardation at X is
6= AX+BX – AB=2(q – c).
But 8°– a”— cº, and consequently, since 6 is small compared with either a or c, we
have approximately
8°– (a — c)(a +c)=#ö(a +c) = cö ;
since a = c + #6 is nearly equal to c. Hence, in the equation of the ellipse,
6 2 2
7/ = - Cº,” — Q.
y=. 3.
we may write a=c and 3– Nc3, so that we have
== 6 12 — a 2
3/ Vº o:2).
But y is the radius of the ring corresponding to the retardation 6, and for the
bright and dark rings 6=#n\, according as n is even or odd.
When the screen passes through the middle point of AB the rings reach their
maximum size, when a = c they degenerate to zero.]
104. Abnormal Displacement of the Central Band.—It has been
pointed out by Sir G. G. Stokes' that the method of determining the
refractive index of a plate by the displacement of a system of inter-
ference fringes is subject to a theoretical error depending on the
1 Brit. Assoc. Report, 1850; also Math. and Phys. Papers, vol. ii. p. 361.
190 INTERFERENCE CHAP. W II
dispersive power of the plate. In the absence of dispersion the
retardation (8) introduced by the plate would be independent of
Å, and would therefore be completely compensated at the point
a = a&ſc. But when there is dispersion the retardation 6 depends on
A, and the different colours are unequally retarded by the plate.
The violet fringe system will consequently be most displaced, and the
red least. If u be the linear displacement of the fringe system of
wave length A, we have u = a&ſc and 6-(p − 1)e = f(A) suppose, con-
sequently -
w-ºx). (1)
The centre of the complete displaced system is therefore not neces-
sarily at the point reached by the two pencils in the same time, but is
determined by the coincidence of bright bands of the most brilliant
parts of the spectrum. Measured from the original centre, the position
of the nth bright band of wave length. A will be
•=ºu-ºo ſo). (2)
When this quantity is as independent of A as possible, the best coin-
cidence of the various bright bands will occur, and the position will
correspond to the centre, or, as Cornu terms it, the achromatic band
of the displaced system. This will happen when da/d\ = 0, or when
n is the nearest integer to
w--f'(x)= -$ du.
& d’A
Substituting this value of n in (2) we find for the displacement of
the central white band
•={{ſ0)-xfo)) (3)
where f(X)=(u – 1)e.
This when expressed in terms of u gives by (1)
dw •
QC E QM, - ^ix. (4)
and when expressed in terms of the band width w = ax/c, we have
Airy's formula *
dw p.
(5)
Q = 20 - ?U—s
dw
The final term on the right-hand side of each of these equations
is inherently negative, for since the refrangibility increases as the wave
length decreases, it follows that the displacement of the bands corre-
* “Remarks on Mr. Potter's Experiment on Interference” (G. B. Airy, Phil.
Mag., March 1833).
ART. 105 POTTER'S EXPERIMENT 191
sponding to a given wave length must increase as the wave length (or
band width) decreases. Hence du and dA (or dw) must have opposite
signs. The final term in the foregoing equations (which represents
the abnormal displacement of the central white band caused by disper-
sion) will consequently be additive, and will increase the normal effect
of the interposed plate.
Exercise.—Assuming the truth of Cauchy's formula,
B C
A = A +...+x+
the relative error is
^ſ(\}_ _ _i (234.4%
}}=-º;(; X4 ' ' ' .)
2 C
=-jº (a-Arº).
105. Abnormal Displacement of the Fringes by a Prism—
Potter's Experiment. — Repeating an experiment of Professor
Powell's," Mr. Potter” found that a prism interposed in the pencils
of light reflected from Fresnel's mirrors, arranged to produce inter-
ference bands, systematically deviated the fringes towards the thick
end of the prism by an amount greater than that of the calculated
centre of the interfering pencils. This he considered inconsistent
with both the wave and emission theories, but Sir G. B. Airy &
showed at once that this phenomenon followed as an immediate con-
sequence of the wave theory. The investigation is that which we
have already applied to determine the position of the central or
achromatic band when the fringes are displaced by a plate inter-
posed in the path of one of the pencils. If u be the linear displacement
of the fringes corresponding to a wave length A and w the width of a
band, the displacement of the achromatic band is
2, wº
dºw
* Powell's experiment was a simple variation of that of Art. 102. The interposi-
tion of the thin plate is attended by difficulties on account of the extreme proximity
of the interfering pencils. Powell suggested the use of a thin prism of 4° or 5° re-
fracting angle, the edge of the prism being parallel to the line of intersection of the
mirrors. The two pencils then pass through different thicknesses of the prism.
Powell says, “The whole set of stripes are seen in the deviated image, unaltered,
except by a trifling degree of colour and a slight shifting towards the more refrangible
end of the spectrum, obviously due to prismatic refraction ” (Rev. Baden Powell,
Phil. Mag. and Ann., January 1832).
* The prism used by Potter was an ordinary glass prism of 43° angle (R. Potter,
jun., Phil. Mag., February 1833).
- * “Remarks on Mr. Potter's Experiment on Interference " (G. B. Airy, Phil.
Mag., March 1833).
192 INTERFERENCE CHAP. VII
as before. The quantity du/dw being negative, the abnormal effect is
added to the regular deviation produced by the prism.
The abnormal displacement of the central band is therefore a con-
sequence of the heterogeneity of the light. In fact, with perfectly
homogeneous light we would have nothing whereby to distinguish the
central band, for the fringe system would consist of a set of perfectly
similar bands. The effect of the prism is to displace the apparent
centre of the system. The nth band is rendered achromatic, but
the system is no more achromatic than before, for the widths of the
component bands and the overlapping remain unaltered.
If a diffraction grating be used instead of a prism, the deviation
will vary as the wave length—that is, u varies as w, and consequently
u – wiluſdw = 0.
106. Talbot's Bands.-A remarkable system of bands was dis-
covered by H. F. Talbot," and their complete explanation was first
given by Airy,” whose calculation is very complicated, but his final
result may be obtained from very elementary considerations, which
are given in the theory of diffraction (see Art. 158). At present we
shall merely give Talbot's general account. He describes his experi-
ment as follows: “Make a circular hole in a piece of card of the size
of the pupil of the eye. Cover one-half of this opening with an
extremely thin film of glass (probably mica would answer the purpose
as well, or better). Then view through this aperture a perfect spec-
trum formed by a prism of moderate dispersive power, and the spectrum
will appear covered throughout its entire length with parallel obscure
bands resembling the absorption produced by iodine vapour.”
In Fig. 89 the thin plate is represented in three different positions.
At L it is situated directly between the eye of the observer and the
- eye-piece of the telescope,
at M it is placed between
the prism and the object
glass, and at N between
the prism and the colli-
mator.
Fig. 89.-Talbot's Bands. An imperfect explan-
ation of the bands was given by Talbot on the principle of simple
interference. Thus if 8 be the retardation suffered by any ray,
of wave length A, in passing through the plate, then one half of
the light passing through the aperture will be retarded relatively
to that which passes through the other, and if the quotient 8/A is a
* H. F. Talbot, Phil. Mag., 1837, part i. p. 364.
* Airy, Phil. Trans., 1840, part ii. p. 225, and 1841, p. 1.

Art. 107 POWELL’S BANDS 193
whole number, the two halves will agree in phase, but if 8 is equal to
(2n + 1)}\, they will be opposite in phase and destroy each other.
Now 8/A varies from one colour to another, so that agreement and
opposition in phase will recur alternately as we pass from one end of
the spectrum to the other. It is consequently traversed by a system of
dark bands. This explanation would, however, indicate bands on which-
ever side of the field the thin plate was placed. All that can rightly
be inferred is that if bands occur their spacing must be as stated.
A similar experiment is that of Brewster," by which he imagined
he had discovered what he termed a new polarity of light. “While
examining the solar spectrum formed in the focus of an achromatic
telescope after the manner of Fraunhofer, he placed a thin plate of
glass before his eye in such a manner as to intercept and retard one-
half of the pencil which was entering one-half of the pupil. He was
then surprised to find that when the edge of the retarding glass plate
was turned towards the red end of the spectrum, intensely black lines
made their appearance . . . but upon turning the plate of glass half
round (still keeping its plane perpendicular to the axis of the eye), so
as to present the edge past which the rays entered the eye to the
violet end of the spectrum, the dark bands disappeared.” In inter-
mediate positions the bands appeared more or less distinct, according
as the edge was more presented to the red or to the violet end. The
thinner the glass the more distinct the lines, and they were formed in
any part of the spectrum. Brewster remarked that “An examination
of these lines affords the very best means of determining the dispersive
powers of substances; for their distance from one another increases
or diminishes exactly as the entire length of the spectrum is increased
or diminished, and the number of them in the same part of two spectra
of different lengths is always the same.”
107. Powell's Bands.”—In a hollow glass prism (Fig. 90) con-
Fig. 90.-Powell's Bands. Fig. 91.
taining some highly refractive liquid, such as oil of sassafras, anise, or
cassia, a plate of glass is inserted with its lower edge parallel to the
“On a New Polarity of Light” (Sir D. Brewster, Brit. Assoc. Report, 1837).
* “On a new Case of Interference of Light” (Baden Powell, Phil, Trams., 1848).
O

194 INTERFERENCE CHAP. VII
edge of the prism, and so that its plane nearly bisects the angle of the
prism, while it extends only through the upper half of the liquid,
leaving the lower or thinner part clear. The light from a slit being
transmitted through it in the usual manner, the spectrum thus formed
is crossed by a number of dark bands parallel to the slit or edge of
the prism.
With some liquids and plates the bands are sensibly equidistant;
in others they increase in number and fineness towards one end of the
spectrum. In most cases they extend throughout, but in some they
are deficient at one part of the spectrum.
If the thickness of the plate exceed a certain limit the bands
become too numerous and too fine to be seen ; if less than a certain
limit they become too few, broad and faint; while for some inter-
mediate thickness they appear most vivid and distinct.
When the plate is inclined either way, even to being in contact
with the side of the prism, the bands are still seen, but they suffer a
slight displacement downwards as the plate is inclined.
Some combinations of liquid and plate, such as glass with oil of
turpentine, or water, give no bands with this arrangement. Stokes
pointed out that in this case bands are produced by placing the plate in
the thinner part of the prism, leaving the wider part clear (Fig. 91).
Stokes has varied the experiment by inserting the glass plate in
a vessel with parallel sides, and allowing the light from a prism to
fall upon it (Fig. 92).
With plates of crystallised substances,
such as Iceland spar, two sets of bands, one
finer than the other, are presented. On
applying a Nicol's prism (Chap. XI.) each set
disappears alternately, leaving the other
visible at each quarter of a revolution of the analyser, showing that
they are due to the two oppositely polarised pencils. The finer bands
belong to the extraordinary, and the broader to the ordinary ray.
The explanation of the general' formation of the bands is afforded
by the simple interference theory on the same lines as those due
to Talbot.
Fig. 92.-Stokes's Modification.
Examples
1. Assuming” the truth of Cauchy's law of dispersion, determine to what degree
of approximation \ſc can be made independent of X by means of a prism in the
experiment of Art. 101.
* The complete investigation has been given by Stokes (Phil. Trans. p. 227,
1848; Math, and Phys. Papers, vol. ii. p. 14).
* These Examples are taken from Lord Rayleigh's article, loc. cit.

ART. 107 EXAMPLES - 195
[According to Cauchy's law we may take c-A-B/\*, and therefore
N__N°
c TAX2 – B
Hence if A/c is to be stationary when X has a prescribed value No, we must have
d(\/c)=0, that is,
The proportional deviation of X/c from the prescribed value \oſco is consequently
* = *.x=-&-
Nº. 3W,” Xixº']
2. If the achromatism is to hold 'good in the neighbourhood of the D line
(N,+ 58890), find the proportional variation of Aſc for the C line (N = 65618).
[Using the formula of Ex. 1 we have No- '58890 and N = 65618, therefore
X/c
=-|- = 1°0155.
No/eo |
3. In Ex. 2 determine the order (m) of the band at which the C-system is dis-
placed through half a band width relatively to the D-system.
[The relative displacement at the nth bright band is 62 = ma (N/c-Xoſco)=}^\oſco,
if the displacement is half a band width.
Hence
. Using Ex. 2 this reduces to n=32—that is, after 32 complete periods the bright
bands of one system coincide with the dark bands of the other.]
4. If the prism be not employed, prove that the bright bands of one of the
systems of Ex. 3 will coincide with the dark bands of the other when m = 4:2.
[When the prism is not employed c is constant and the formula for m in Ex. 3
becomes
5. If the two systems differ in wave length by a small amount 5X, prove that
the formula for m in Ex. 3 becomes approximately
XAN 2 ÖA
= 1 | tº 7. |.
$0. *(*) [i+:
[This formula gives the order of the band at which complete discrepance first
occurs between the systems No and No.4-6\, and it shows that when 5X is small the
order of the band is inversely proportional to the square of 6X.
The corresponding effect will occur without a prism at the band
T
º +\ +\
70, E _X^0_ =3:9,
so that the effect of the prism is to increase the number of bands in the ratio
2\o : 36X.]
CHAPTER VIII
INTERFERENCE BY ISOTROPIC PLATES
SECTION I.--THE COLOURS OF THIN PLATES
108. General Statement of the Phenomena.-The examples of
interference which we are now about to discuss are noteworthy on
account of the peculiarities which they present and their frequent
occurrence to ordinary observation. Here it is no longer necessary to
have a very narrow source of light.
When ordinary white light falls upon a thin film of a transparent
substance, such as a soap bubble or a film of oil spread on the surface
of water, brilliant colours are generally observed, and sometimes all the
tints of the rainbow are exhibited. These colours were first observed
by Boyle and Hooke, and the latter succeeded in blowing glass
sufficiently thin to exhibit them distinctly. They are often developed
in mica and other minerals which possess a lamellar structure, but
the most familiar instance of their exhibition is in the froth of liquids
and films of oil. The colours vary with the thickness of the film,
and disappear altogether when it exceeds certain limits. This is well
exhibited by dipping the mouth of a wine glass into soap water. The
viscid aqueous film which adheres to it after immersion displays the
whole succession of these phenomena. When the film covering the
mouth of the glass is held in a vertical plane it appears at first uni-
formly white, but as it grows thinner by the gradual descent of the
fluid, colours begin to be exhibited at the top, where it is thinnest.
These colours form horizontal bands which become more and more
brilliant as the thickness diminishes, but when the thickness is reduced
to a certain limit at the upper part the film becomes quite black, and
it has at this place arrived at such a stage of tenuity that it is no
longer able to support its own weight, and the film bursts.
Permanent films capable of showing all these effects can be
obtained by dropping on water dilute solutions of celluloid in acetate
196
ART. 109 RETARDATION IN THIN PLATE 197
of amyl. The solution spreads over the water, and when the solvent
evaporates the solid celluloid is left behind.
Every one is familiar with the fact that polished steel becomes
coloured in various shades when exposed to the air. These colours
are due to a thin film of the oxide of the metal which is gradually
formed on the surface."
The same appearances are displayed in a still more striking
manner when two plates of glass (which contain a thin film of air
between them) are pressed together, or when a convex lens is laid on
a plate of glass. Around the point of nearest approach successions of
coloured rings of great brilliancy are presented, which dilate as the
pressure is increased so as to diminish the thickness of the included
air film.
109. Thin Plates.—In the cases cited above, the thickness of the
film usually varies from point to point. For simplicity let us first
consider the case of homogeneous light falling upon a thin uniform
plate or film of a transparent isotropic substance, for example, a film
of air enclosed between two parallel plates of glass. The fringes so
obtained are known as Haidinger's fringes.
The light issuing from one point of a source and converted into a
parallel beam by means of a lens incident in the direction A, B, (Fig. 93)
on the first surface of the film is divided
there into two portions, one reflected parallel
to BC and the other transmitted parallel to
B.C. This latter portion is further divided
at the second surface, one part being trans-
mitted and the other reflected back along
CB to suffer refraction at B and emerge in
part from the plate again in the direction BC,
making an angle with the normal equal to
the angle of incidence. The direction BC is therefore the same as that
of the light which is directly reflected at the first surface. In this
direction we have therefore two streams of light, one coming from the
first surface by the reflection there of the incident light, and the other
emerging from the first surface after it has been refracted into the plate
and reflected at the second surface. This latter stream of light, having
traversed the plate, will be retarded relatively to the light which is
reflected directly at the first surface. The two streams will reinforce
Fig. 93.-Thin Plate, Retardation.
* This oxide is formed rapidly when the temperature is high, and the thickness
of the film depends so invariably on the temperature that artists are in the habit
of estimating the temperature by the colour developed. Thus steel in the process
of tempering is spoken of as having received a yellow heat or blue heat, etc.

198 THE COLOURS OF THIN PLATES CHAP. VIII
or weaken each other according as they are in the same or in opposite
phases."
The calculation of the difference of phase of these two streams of
light is very simple. Draw B,D and BD, perpendicular to AB and
B,C, respectively. Thus B,D and BD, are the fronts of the incident
and refracted waves respectively. At the instant the light from D
reaches B the disturbance from B, has travelled to D, in the second
medium. Accordingly the first pencil of light is about to leave the
surface at B along BC at the same instant as the refracted ray is
leaving D, to traverse D.C., B. The retardation is therefore
6= C,D, +C, B.
Produce B, C, to meet at E the normal drawn to the plate at B.
Then if r be the angle of refraction into the plate it is obvious that
BEC, a r, C, B = C,E, and therefore
ô= D, E = BE cos r=20 cos r
where e is the thickness of the film.
So far the theory seems to indicate that the brightness will be
greatest when the difference of path 2e cos r is an even number of half
wave lengths, and least when this difference is an odd number of half
waves, but the results of observation show that the conditions of light
and shade are exactly reversed.
Now if the difference of phase depends only on the difference of
path traversed by the pencils, when this difference vanishes (which is
the case when the thickness of the film is infinitesimally small) the two
pencils should be in the same phase and the illumination should be a
maximum. But we know that if at any point the thickness of the
plate is zero there will not be any light reflected there, and the point
will appear dark; the light passes straight through, Hence we have
* “This mode of explaining the phenomena of thin plates was pointed out by
Hooke in a remarkable passage in his Micrographia, some years before the subject
was taken up by Newton. In this passage he very clearly describes the manner in
which the rings of successive orders depend on the interval of retardation of the
second ‘pulse' or wave with respect to the first, and therefore on the thickness of
the plate. But he does not seem to have had any distinct idea of the principle of
interference itself, and his conception of the mode in which the colours resulted from
this “duplicated pulse’ is entirely erroneous. Euler was the next who attempted
to connect the phenomena of thin plates with the wave theory of light, but the
attempt, like all the physical speculations of this great mathematician, was signally
unsuccessful, and the subject remained in this unsettled state until the principle of
interference was discovered by Young. When this principle was combined with the
suggestion of Hooke the whole mystery vanished. The application was made by
Young himself, and all the principal laws of the phenomena were readily and simply
explained ” (Lloyd, Elementary Treatise on the Wave Theory of Light, third edition,
p. 138).
ART, 109 CALCULATION OF THE RETARDATION 199
arrived at two opposite conclusions, and of these two the latter is
undoubtedly correct. The difference of phase therefore must depend
on something else as well as on the difference of hath traversed by the
pencils. This second element is not far to seek.
It has been shown (Art. 57) that when a pulse travels along a row
of balls (or a cord) and passes into another row of different density, it
is subdivided into two pulses, one traversing the second system and
the other reflected in the first system. Let us confine our attention
to the reflected wave. Suppose the first pulse to be propagated as a
forward displacement, then the reflected pulse will be propagated as a
displacement in the direction of propagation or the reverse, according
as the first system is less or more dense than the second. In the first
case the pulse is reflected without change of sign, in the second it is
reflected with change of sign. The displacement in one case is in the
opposite direction to that in the other, so that if the two were super-
posed they would neutralise each other. This is expressed by saying
that the waves are in opposite phases (see further, Art. 112). e
Now in the case of the thin plate, if the light at the upper face
is reflected in passing from a dense medium to a rare, then at the
second surface it is reflected in passing from rare to dense. The two
portions are reflected under opposite conditions, so that if one is
reflected without change of sign, the other is reflected with change of
sign, and the two reflected waves are in opposite phases. Accordingly
the act of reflection under the opposite conditions introduces half a
period difference of phase. Hence the whole retardation is
2e cos r + 3\
measured in the medium of which the plate is composed. If p be the
refractive index of the material of the plate, the air distance which
corresponds to this is
2ple cos r + 3\
where A is now the wave length in air. The illumination of the plate
will consequently be a maximum if 2e cos r = an odd number of half
waves (measured in the plate), a minimum if 2e cos r = an even
number of half wave lengths. If e = 0, or the thickness of the plate
is infinitely small, the retardation is half a wave and the illumination
is a minimum, which agrees with experiment. If the wave length be
measured in air the brightness will be greatest when 2ple cos r is an
odd number of half wave lengths, and least when the same quantity is
an even number. - -
In the foregoing we have been considering homogeneous light ;
that is, light of a definite wave length A. If, however, the incident
Change of
sign.
200 THE COLOURS OF THIN PLATES CHAP. VIII
White
light.
Over-
lapping.
light is heterogeneous, and contains many wave lengths for which A, p.,
and r are different, it will happen that the condition for brightness
will be satisfied by some of the constituent waves, while the condition
for darkness is satisfied by others. It follows, therefore, that in the
reflected beam only some of the constituents of the original light will
be represented. -
Thus when ordinary solar light is incident on a thin film the
light which comes from any point of it to the eye will not include
any of the wave length satisfying the equation (A being the wave
length in air)
2ple cos r = m\
where n is any whole number. The light from this point will be
accordingly coloured, the colour at any point depending both on
the thickness of the film and on the angle of incidence. If the
angle of incidence or the thickness varies from point to point of
the film, corresponding variations of colour will occur, but if the
incidence and thickness be constant, the colour will be uniform. If
the light returning from any point of the film be analysed in a spec-
troscope the spectrum will consequently be crossed by certain dark
bands corresponding to those waves which have been neutralised by
interference. The number and closeness of these dark bands will
increase with the thickness of the plate, and when the thickness
reaches a certain limit they become so fine and close that the
resolving power of the spectroscope may fail to separate them.
This means that interference has ceased to be observable by reason of
excessive overlapping. This overlapping increases with the thickness
of the plate, and leads to the obliteration of the bands, but it should
be remembered that interference takes place in thick plates just as in
thin, in the same way as interference exists in Fresnel's experiment
in the regions distant from the central line. The thinner the plate
the less the overlapping, and the more observable the phenomenon.
This overlapping will be diminished, and the limiting thickness of the
plate at which interference can be observed will be increased by
increasing the homogeneity of the light.
A simple case is that in which the film is a thin wedge bounded
by two planes inclined at a small angle. The lines of equal thickness
are parallel to the edge of the film, consequently with monochromatic
light it will appear crossed by a system of parallel bright and dark
bars, and with white light these are replaced by a system of parallel
coloured bands.
In general, however, it is not necessary to resort to any particular
contrivance in order to obtain interference fringes by reflection from
ART. 110 CONDITION FOR ACHROMATISM * 201
thin films. If a film of uniform thickness could be procured it would
appear uniformly coloured when a beam of parallel light is reflected
from it. In practice, however, it is impossible to secure perfectly
plane surfaces, and the film of air enclosed between two so-called
parallel plates of glass is not of uniform thickness throughout. As a
consequence the film, instead of being uniformly coloured, generally
exhibits coloured fringes forming curves which encircle the points of
nearest approach of the plates. We are thus furnished with an
exceedingly delicate optical test of the planeness of the surfaces. On
Fringes.
pressing the plates closer together so as to reduce the thickness of the
film the bands dilate, showing how the colour at any point depends
upon the thickness of the film.
1 10. Influence of Dispersion on the Colour of a Film—Con-
dition for Achromatism.—The order of the colours in fringes pre-
sented by thin films (or the series of tints passed through by a film of
uniform thickness as the thickness is varied) may differ considerably
from that exhibited in the interference bands produced by the mirrors
of Fresnel and Lloyd. This arises from the fact that in the latter
case the path retardation (6) at any given point is the same for all
wave lengths, and consequently the phase retardation 8/A varies from
colour to colour by reason of the variation of A alone. In the case of
a film, however, there is a second agency through which the phase
retardation may be altered, namely, the dispersion within the film,
and the change arising from this cause may be either of the same or
of opposite sign to that arising from the variation of A when there is
no dispersion, so that the colour effect may be either increased or
diminished by it.
Thus, let us consider a definite case in which a uniform film,
enclosed between two infinite media A and B, is viewed by an eye
situated in A, and let a parallel beam of white light traversing the
medium. A fall upon the film so as to be reflected to the eye in ques-
tion. In this case the angle of incidence is the same for all wave
lengths, but the angle of refraction into the film is different for
the different colours. As a consequence the path retardation 2e cos r
varies from colour to colour, and the expression for the phase retarda-
tion for a given colour (namely, 2e cos r|A) varies both in its numerator
and denominator. When the variation of cos r is the same as that of
A for all values of A, the phase retardation will be the same for all
colours; that is, all the colours will be equally affected by interference,
and the dispersion in the film will have produced achromatism. This
will happen when
COS )*
= constant, :
X
202 THE COLOURS OF THIN PLATES CHAP. VIII
and the corresponding angle of incidence is determined by this
condition, where A is the wave length measured in the film.
It can be easily seen that achromatism may be produced in this
manner when the film is less refracting than the media between which
it is enclosed. For in this case the angle of refraction into the film
increases with the refrangibility of the light ; that is, r increases, or
cos r diminishes, as A diminishes, and consequently there may be
some angle of incidence for which the variation of cos r is the same
as that of A from colour to colour, and cos r|A may be the same
for all. When the film is more highly refracting than the enclosing
media, on the other hand, the value of cos r diminishes as A in-
creases, and then the effect of dispersion in the film is to exaggerate
the colour effect which would be produced by the variation of A
alone.
At nearly perpendicular incidence the variation of cos r for the
different colours is very small, and the effect of dispersion on the
colour of the plate is not very sensible, but as the angle of incidence
approaches the angle of total reflection from the film, the angle of
refraction approaches 90°, and the variation of cos r with A becomes
much more considerable, and may even pass the limit at which colour
compensation is produced. If perfect compensation should occur, the
bands produced by a film of variable thickness will be simply black
and white, and a uniform film will pass through alternations of black
and white as its thickness is varied.
In order that perfect achromatism may be produced it is clear that
the refractive index of the film must be related to the wave length in
some particular manner; that is, there must be a particular law of dis-
persion in the film. This law is contained in the achromatic condition
cos r|A = const. Thus, if we write cos r = k \, we have
sin?r=1 — k”X”,
and consequently if i be the angle of incidence (which in the foregoing
case is the same for all colours) at which achromatism occurs, we have
sin T = p sin i, where p is the refractive index of the film with respect
to the medium A, and consequently the law of dispersion required is
p” sin” i = 1 – k”X”,
which may be written under the more general form
pi*= a +b)\*.
It should be observed that the common case of a film of air enclosed
between two parallel plates of glass does not satisfy the conditions
demanded by the foregoing investigation unless some special contriv-
ART. 110 CONDITION FOR ACHROMATISM 203
ance be adopted to cause the light to enter the glass so as to fall in a
parallel beam on the film." If no such precautions be taken a parallel
‘beam of white light falling upon the first glass plate will be dispersed
within the plate, so that the angle of incidence on the film will vary
from colour to colour, while the angle of refraction into the film will
be the same for all colours, and equal to the incidence on the glass
plate. In this case the light falling upon the film is not one parallel
beam of white light, but a dispersed beam, consisting of parallel beams
in different directions for the different colours, and the dispersion
which occurs in the glass is corrected by an equal and opposite
dispersion in the film, so that the light within the film is a parallel
beam of white light. The path retardation 2e cos r is consequently
the same for all colours, and the phase retardation 26 cos r/A varies to
the full extent from colour to colour without compensation of any
sort. The series of tints passed through by such a film, as its thick-
ness is varied, should therefore be the same as that presented in the
interference fringes produced by Fresnel's mirrors.
Ea'ample
If the refractive index pº be related to the wave length X by the equation
b
*=a+5. y (1)
prove that achromatism will be most nearly approached when *
cotºr =
(2)
2(p. – a)
—-
[Colour compensation will take place when cos r|X is stationary ; that is, when
X sin ºrdr--cos ral) = 0. (3)
But p sin i =sin r, therefore since i is the same for all colours, we have sin rſp =
constant and consequently t
p, cos rar – sin rdu = 0. (4)
Combining (3) and (4) we obtain
cotºr=-“ - (5)
pſ d'A'
Now by equation (1) we find
du 2b 2
... = -j- -, (4-6)
consequently (5) reduces at once to the required relation (2).
* This may be easily arranged by using a prism and a plate (as in Fig. 99) instead
of two plates. A parallel beam of white light incident normally on one of the faces
of the prism will fall on the film as a parallel beam of white light, and it may be so
arranged that the reflected light falls normally on the second face of the prism so as
to emerge to an eye in air in the condition in which it leaves the film.
* Lord Rayleigh, Phil. Mag. vol. xxviii. p. 192, 1889; and Ency. Brit., Art.
“Wave Theory.”
204 THE COLOURS OF THIN PLATES CHAP. VIII
In the case of Chance's “extra-dense flint" glass
b= .984 × 10−", and for the sodium lines w=1:65,
X=5-89 × 10−", consequently achromatism occurs when
r=79° 30'.]
111. More Complete Investigation—Multiple Reflections.—The
theory of thin plates as it came from the hands of Young laboured
under an imperfection which, however, was soon removed. Thus it is
easily seen that the intensities of the two portions of light reflected
from the two surfaces of the plate are not equal." These two portions
therefore can never wholly destroy one another, and the intensity of
the light in the dark rings can never entirely vanish, as it appears to
do when homogeneous light is employed. Poisson was the first to
point out and to remedy this defect in the theory. It is evident,
in fact, that there must be an infinite number of partial reflections
within the plate, at each of which a portion is transmitted, and it is
the sum of all these portions that must be taken into account.
In the foregoing discussion we assumed that the only light which
emerged from the plate along BC (Fig. 94) is that ray which after
refraction at B, is reflected at C1. It
is obvious, however, that there is a
multitude of other rays which also
emerge in this direction. For if we
take BB = BB, a B.B., etc., it is clear
that a ray incident at B, will after re-
fraction pass along B.C.,B,C, B and
part of it will emerge at B along BC.
Similarly a ray incident at B, will, after alternate reflections at the
two surfaces of the plate, emerge in part along BC. It is thus
evident that the complete stream of light which issues along BC
from the interior of the plate consists of several parts, the first of
which is by far the most powerful and the others diminish rapidly to
zero. The foregoing calculation is therefore only approximate, and it
becomes necessary to calculate each of the components and to sum
their effects. If the retardation suffered by the ray A,B,C, in the
plate is 8, we have 6 = 2e cos r, and it is obvious that the retardations
of the consecutive rays incident at B, Bº B, etc., are 26, 38, 48, etc.,
respectively. We thus know the phases of the components as they
arrive at B, but to calculate their joint effect it is necessary also to
know their amplitudes. For this purpose it is necessary to examine
the relation connecting the reflection and refraction coefficients. It
Fig. 94.—Multiple Reflections.
* The condition for the equality of these two portions would violate the condition
of Art. 112.

ART. 112 CALCULATION OF INTENSITIES 205
must be added that, in order that the rays incident at BB,B, ... may
interfere with one another, they must all have come from a single
point of the source, otherwise there will be no permanent phase
relationship between them. This will be the case if the source is at
the first principal focus of a lens which transmits the rays to the film.
Alternatively, we might consider all the arrows on Fig. 94 to be
reversed. The line CB then represents a single ray issuing from the
source, which is split up by multiple reflections into a number of rays
BA, B, A, etc. If these are now collected to a focus by a lens,
interference takes place at this focus. This latter method corresponds
most nearly with usual practice, for light is taken from a bright cloud
or other source without the intervention of a lens, and the reflected
rays are collected by a lens. If the
eye alone serves as this lens the
fringes are formed on the retina.
112. Haidinger's Fringes—Cal-
eulation of Intensities.—We will
adopt the second method and take
the single incident ray as of ampli-
tude unity. The rays which form
the reflected system have amplitudes
a, bed, beºd, be"d, etc. ; bd
and those which form the transmitted
system have amplitudes
bd bdc?, bdc", etc. Fig. 95.
The factors in these expressions have the following meanings:
At each outside reflection the amplitude is changed in the ratio a
2 3 entering refraction 3 y 7 5 3 3 b
2 3 inside reflection 3 3 } } 2 3 C
3 3 leaving refraction 5 y 3 2 3 3 d
Each ray is retarded by a phase angle 8 with respect to the previous
member in the bundle, and the first emergent ray is retarded 8/2 with
respect to the first reflected ray.
Summing the vibrations in the two bundles we get
reflected y = a cos wi-H bed cos (wt – 5) + bo'd cos (wſ – 26), etc.,
and
transmitted y=bd cos (w * #) +bdc” cos (w - ; - •) + balcº cos (w - #- 2.) etc.
These two series can each be summed if we substitute the real part
of eº for cos 9 where 6 is any angle. The series for y reflected
becomes

206 THE COLOURS OF THIN PLATES CHAP. VIII
$/R = * clot(a + bede-tê + bºde -2.64 to infinity)
R. P. bcde – tº
-: t *º-
since all terms but the first form a geometric series; and
e 8
t – ºr
3/T * bda” *(1+*-***-*to infinity)
Kot-3)
_R.P. bale
T of 1 – cºg-wó ‘
. Now when the thickness of the film is zero, ö is also zero. In this
Case yn = 0, y = 1, because an infinitely thin film can reflect no light
and must transmit all; hence
bcd
a +1 º–0.
and
bd
3. – ---5 — 5
1 — cº
and therefore
a = – c and ba- 1 – a”.
Thus the coefficient of reflection of amplitude at the internal surface
is equal and opposite to the value at the external surface. Since a
reversal of amplitude is equivalent to a change of phase of 180° this
is equivalent to a shift of half a wave length ; it is in fact the A/2
referred to on p. 199. Previously the occurrence of this phase
change was based upon an analogy derived from the behaviour of
ivory balls. We now see that it is demanded by the fact that an
infinitely thin film must reflect no light and transmit the whole.
No information is here obtained as to the phase change at each
reflection, it is only the difference at the two surfaces that is found.
The effect at each surface can only be found when the true theory of
the mechanism of light transmission is ascertained.
We can now write
wº-": a-ſi -º-º-"; as "(H+)
R. P (ot-3)
$/T = of (1 – a”) e
1 — a”e – w8
Now write as an abbreviation
$/R= Fº et wi (A -- th)
= (A cos wt – B sin wt)
= VA3+B” cos (wl – p).
The intensity is therefore A* + Bº.
But this can most easily be obtained by multiplying e” (A + i B) by
ART. 113 EFFECT of SILVERING THE SURFACES 207
its conjugate complex quantity e-" (A – B); for this gives the value
A” + Bºat once. Similarly for yr Hence
© e 1 – c – w8 1 – c + 16
== - t ſ - ------ - Lot ( - l-
IR = Intensity of reflected beam = acto ( --- jº) X (0.6 - Loo ( —a’e + 3)
_ a”(2–2 cos 6)
1 + a” – 26% cos 6
Aa2 sin”?
2
= 3.
. c. 6
(1 – a”)” +4a” sin”;
8 8
o t(ot-3) (1-62), “‘ā)
e * I 1. - .. (1 – a”e
IT = Intensity of transmitted beam = Tøø – 5 1 – a”e + vö
(1 – a”)?
8
tº
(1–62)2+4a” sin”;
It should be observed that a is the ratio of the reflected amplitude
to the incident, and therefore a” is the ratio of the respective in-
tensities; i.e. it is the reflecting power in the ordinary sense of the
expression. Its value varies both with the nature of the media, which
are separated by the reflecting surface and also with the angle of
incidence (§§ 209, 210). Its value determines the character of the
fringes which are seen.
113. Effect of Silvering the Surfaces.--This can be illustrated
by examining the effect to be expected when the surfaces of the film
are thinly silvered. The reflecting power for crown glass, at nearly
normal incidence, is about 0:04, while a very thin coat of silver may
increase it to 0-80 or more.
With the plain glass we may write approximately
. 26
I n = 16 sin”. H '08(1-cos 3)
8
IT = 1 – 16 sin”.
The transmitted beam shows therefore exceedingly little contrast :
while the reflected intensity varies from zero to 0-16. The reflected
system is therefore fairly conspicuous while the transmitted system is
insignificant. Moreover, for the maxima of IR, the variation follows
a cosine law, so that the parts which are bright above the average are
as broad as those for which the brightness is below the average.
Contrast this case with the thinly silvered case in which a’ = 0-8.
We now have
IR = 1 nearly,
- - 4 •04
I = --—#8.
T = '04 -- 4 sin”;
208 THE COLOURS OF THIN PLATES CHAP. VIII
The fluctuations in the reflected system are now insignificant, while
those of the transmitted system show marked contrast; varying in fact
from unity to 0-01 nearly. Moreover, the character of the fringes is
different. Taking the transmitted set, which alone are of any im-
portance, It remains nearly zero except for values of 8/2 very nearly
equal to 0, T, 24r, 3T, etc. In other words, the bright fringes are very
narrow and are separated by broad patches which are of very weak
intensity. This is of immense importance in practice. The fringes
correspond to different directions for different colours; and those due
to two lights of nearly equal wave length are more easily separated
from one another when the bright fringes are narrow. The property
that is called the chromatic resolving power is increased by the silver
coat. Application is made of this fact in the Fabry and Perot
interferometer (§ 123).
Since, as indicated in Fig. 95, the eye has to bring a group of
parallel rays to a focus it is clear that the eye must be focussed for
infinity. Since one point of the source gives rise to an illumination
at only one point of the retina it is clear that an extended source must
be used if the whole of the field of view is to be illuminated. The
fringes will then be seen as circles—because the retardation will
be the same provided r is the same—with their centre corresponding
to the normal to the film. An easy way to obtain them is to take a
bright source—such as a mercury arc-containing only a limited
number of colours; interpose a lens between the arc and the eye in
such a position that the lens appears “full of light °; and then also
interpose, close to the eye, a thin cleave of mica. A very beautiful
set of coloured fringes is thus obtained. [In reality it is a double set,
very nearly superimposed, because there are two refractive indices
concerned since mica has crystalline properties; however, this com-
plication need not be attended to at first.] Or a piece of white paper
may be placed near the arc so as to be brightly illuminated by it, and
this paper may then be examined through the mica placed, as before,
close to the eye. Microscope cover-slips are usually not uniform
enough to produce the fringes well.
114. Haidinger's Fringes — Further Considerations.— In the
preceding section it has been assumed that the transmitted system is
due to the superposition of an infinite number of transmitted rays.
When the film is thick and the incidence is not nearly along the
normal it is clear that from any film of finite area only a few multiple
reflections can take place. Let there be N components in the trans-
mitted system. The expression for YT must be summed to N terms
instead of to infinity. The result is
ART. 114 HAIDINGER's FRINGEs 209
(1–62N)2+4a2N sin (;)
IT= (1 – a”)? e
(1—a”)2+4a.” sin"(...)
It is to be noticed that the frequency of fluctuation of the numerator
is N times as great as that of the denominator. The result is that
between two successive maxima given by the denominator there will
be N – 1 minimum values controlled by the numerator. The appear-
ance will be sufficiently well represented by Fig. 140 for the case of
N = 6. The main maxima are known as principal maxima, the smaller
ones as secondary maxima. The space between a principal maximum
and the adjoining minimum may be taken as indicating the resolving
power for two different wave lengths.
For suppose that a principal maximum for the wave length A + dA
just coincides with the first minimum next the principal maximum for
wave length A and for which the retardation is PA, it may be taken
that the two principal maxima will be seen as a distinct double fringe.
Then since the first minimum occurs §th of the distance between the
maxima of order P and P + 1 we have
X+d), P+.
→---F-
Ol'
dN_ _1_
A TNP'
If d) is in reality the least difference of A that can be detected, à is
called the chromatic resolving power, which is consequently equal to
NP. Hence greater resolving power is obtained both by increasing
the thickness of the film and by increasing the value of N. The
effect of half-silverising the surfaces can be explained by the increased
number (N) of transmitted beams, of sensible intensity, which help to
form the transmitted system.
Evample.
Find the relation between the reflection and refraction coefficients by means of
the principle of retraceability of rays.
[An incident amplitude unity is reflected as a and refracted as b. Now retrace
the rays. The amplitude a is in part reflected as a” and in part refracted as ab ;
while the amplitude b is in part reflected internally as bc and in part refracted as bd.
These retraced rays must reconstitute the original rays reversed ; i.e. in the first
medium
1 =ba + a”,
0 =bc + ab,
and in the second medium
Ol'
c= — a, and b& = 1 – a”.]
This was first shown by Stokes. (On the perfect blackness of the central spot in
Newton's rays, Camb. and Dublin Math. Jour., 1849, vol. iv. p. 1, or Stokes' Math.
and Phys. Papers, ii. p. 89.)
P
210 NEWTON'S RINGS CHA P. VIII
-
SECTION II. THE ColourED RINGs of THIN PLATES
115. Newton's Rings.--It has been already mentioned that when
two pieces of ordinary plane glass are pressed together the thin film
of air enclosed between them generally
exhibits a series of highly coloured
fringes running in curves round the
point of nearest approach of the glasses.
When one of the pieces of glass has a
spherical surface while the other is
plane, as shown in Fig. 96, the layers
of equal thickness in the film form a
system of concentric circles around the
point of nearest approach, and, when
viewed in ordinary daylight, a system
of highly coloured rings is seen encircling the central spot. These
appearances are known as Newton's rings," and they form one of the
most beautiful and easily produced examples of interference.
There is an important difference between these fringes and those
due to Haidinger. It is a familiar fact that Newton's fringes (as well
as those of a soap bubble or thin oil film) appear to be painted—to
use Newton's own phrase—on the film itself and are not at infinity.
In general the problem is somewhat complicated; but we can get
some idea of it by confining examination to the neighbourhood of the
normal and to thin films. The P
two chief rays into which the
incident ray bifurcates, diverge 3/
from (or converge to) one another;
the eye must be focussed for the
point of divergence in order that
the effects due to these two rays
may be focussed upon the retina. It is easy to see that for small
angles and a wedge-shaped film, if the region of incidence is distant
b (Fig. 960) from the edge E and if r2 is the angle of incidence
Fig. 96.-Newton's Rings.
Fig. 960.
Newton, in studying the formation of these rings, “took two object-glasses, the
one a plane-convex for a fourteen-foot telescope, and the other a large double convex
for one of about fifty foot; and upon this laying the other with its plane side down-
wards I pressed them slowly together to make the colours successively emerge in
the middle of the circles” (Opticks, book ii. p. 172).

ART. 115 NEWTON'S RINGS - 211
on the second surfaces, then the two chief rays will intersect at
a point P in front if the incidence is (as on the diagram) toward
the edge E, or behind it if the incidence is away from E.; and
the distance AP = º approximately. For a parallel film b is infinite,
and so is AP; when b is small, then rºb is very small and P is very
near the film.
The laws according to which these rings are formed are very easily
deduced by remarking that the thickness of the film varies approxi-
mately as the square of the distance from the point of contact. Thus
if OQM be the spherical surface (of radius R) of which the lens is a
part, and O its point of contact with the plate of glass, then
OP2 = PQ X PM = 2Re,
since PM is very approximately. equal to 2R, the diameter of the
sphere, and PQ = e the thickness of the film at P. Now the film con-
sists of circular rings of uniform thickness. Thus at all points of the
circle of radius OP around the point of contact the thickness of the
film will be the same, and equal to PQ. Denoting the corresponding
radius OP by p we have the general relation
p?=2Re,
consequently if e satisfies the equation
26 cos r = nº
the ring will appear bright or dark according as n is odd or even.
The radii of the bright rings are therefore given by
p= WR secr. (2n+1)}\,
and the radii of the dark rings by
p= s/Rsecr. m).
when m is any whole number, and A is the wave length in the film.
If n = 0, then p = 0 and the centre of the system is dark, as we should
have expected, since we have supposed that the thickness of the film
is zero at this point. The radii of the successive bright and dark
rings are proportional to the square roots of the consecutive numbers,
the bright rings corresponding to the odd and the dark rings to the
even number. -
Since the thickness of the film at any point varies as the square
of the distance from the centre, it follows that the thicknesses which
correspond to the successive rings are proportional to the natural
numbers, at the dark rings to the even numbers, and at the bright
rings to the odd.
212 ** NEWTON'S RINGS CHAP. VIII
These laws were arrived at with great accuracy by Newton him-
Wave self, but he did not stop here. He found that in his experiments the
.º absolute thicknesses of the film corresponding to the dark rings were
++ ºroro, +rgºod in., etc., when the angle of incidence was 4°. From
this we find A = ++}ro inch approximately, which corresponds to the
most luminous part of the spectrum in the neighbourhood of the
yellow. These measurements are the first from which the wave lengths
of light might have been determined, and Newton made use of them
for the purpose of ascertaining the length of a fit, his attention being
concentrated on the development of the emission theory.”
Velocity If water instead of air be placed between the glasses the radii of
test. the rings are observed to be much smaller. This then is a proof that
light travels slower in water than in air, for here the thickness of
water required to retard one component on the other by a definite
amount is less than the thickness of air, which produces the same
result. The formula which determines the radii of the rings also
points to the same conclusion, for the wave length in any medium is
proportional to the velocity, and we infer that the radius of any ring
varies approximately as the square root of the velocity in the film
when the incidence is nearly normal. Thus, from the contraction
exhibited by the rings when water or any other fluid replaces air
between the lens and plate, it is possible to compare the velocities of
light in these media.
When violet light is used the rings are smaller” than with red
light, and the theory points out that the ratio of the radii are as the
square roots of the wave lengths; hence we have again arrived at the
conclusion that the violet waves are shorter than the red, and we can
again not only compare their magnitudes, but absolutely determine
their lengths in any given substance.
When ordinary solar light is used a series of iris-coloured rings
are exhibited, violet at the inner and red at the outer edge. The
order of succession of the colours laid down by Newton 4 from the
* Newton, Opticks, book ii.
* “If the rays which paint the colour in the confines of the yellow and orange
pass perpendicularly out of any medium into air, the interval of their fits of easy
reflection are the swºrth part of an inch. And of the same length are the intervals
of their fits of easy transmission” (Newton, Opticks, book ii. part iii. prop. 18).
For the thicknesses at the dark rings were rrºw, rrºwn, etc. This obviously
corresponds to half a wave length, so that we have for yellow light the first deter-
mination of A=41%try in.
* This may be observed (following Newton) by illuminating the glasses with
light from different parts of the spectrum, or more simply, by looking at the rings
formed by ordinary light, through differently coloured glasses.
* Opticks, book ii. obs. 4; see also table, p. 228.
ART. 1.16 THE TRANSMITTED RINGS 213
centre outwards for the successive rings was : (1) black, blue, white,
yellow, red; (2) violet, blue, green, yellow, red; (3) purple, blue,
green, yellow, red ; (4) green, red; (5) greenish-blue, red; (6) greenish-
blue, pale red; (7) greenish-blue, reddish-white. This list is generally
referred to as “Newton's Scale of Colours.”
Thus we read of the red or blue of the “third order,” meaning
thereby that red or blue which is seen in the third rainbow-coloured
ring which encircles the central dark spot.
With white light the alternations are few, for the coloured rings
soon became superposed and overlapped, so as to obliterate all traces
of interference and colour, and the rings fade gradually into uniform
illumination.
116. The Transmitted Rings.-The rings we have spoken of so
far are produced by the interference of the streams of light reflected
from the two surfaces of the thin film. It is obvious that the light
transmitted through the film should also exhibit interference pheno-
mena, but of a complementary character, the maxima and minima
of one system corresponding to the minima and maxima of the
other. Thus when the film is looked at from the other side a system of
rings is observed, complementary in character to those observed by
reflection, and consequently encircling a white centre. These rings
are much paler than the reflected system, for on account of the great
difference in the intensities of the interfering pencils," there are no
points of absolute darkness, so that the bright rings do not stand out
so prominently as in the reflected system.”
If the two ring systems are viewed at once it follows that uniform
illumination should be the result. This has been verified by Arago.”
Placing a glass plate and lens in contact in a vertical position over a
* When light is incident perpendicularly on glass about 4 per cent is reflected ;
the intensity of the first reflected beam is therefore about 3's of the incident beam.
The rest enters the glass and loses 4 per cent again by reflection at the second
surface, so that ('96)”, or '9216 of the original light passes through. The light trans-
mitted after being twice reflected inside the plate is I('96)” (ºr)”, or about $ per cent.
That four times reflected inside will only be gº of this, and so on. The difficulty
then is to understand how such a small quantity of light when superposed on the
strong direct beam should produce any perceptible rings at all. However the
intensity of the weaker beam is (#)”, if the intensity of the stronger is unity, hence
the amplitude of the weaker is ºr, and the maximum and minimum intensities in
the transmitted beam will be
(1+%)*=1+3* g.p.,
so that the difference is as much as ºr of the stronger beam. (For the complete
calculation see Arts. 112, 113, 114.)
* Noticed by Newton, Opticks, book ii. obs. 9.
* Arago, CEuvres complètes, tome x. p. 16 (note).
214 NEWTON'S RINGS CHAP. VIII
horizontal sheet of uniformly illuminated white paper, an eye situated
at E (Fig. 97) will receive light from B by reflection at C, and also
light from A by transmission. Both the reflected and transmitted
systems are in the field of view, but being
exactly complementary, the result is uni-
form illumination.
Specimens of ancient glass sometimes
show transmitted colours of great brilliancy.
Brewster's explanation is that owing to
superficial decomposition of the glass we
have here to deal with a series of thin plates of nearly equal thick-
nesses. With such a series the transmitted colours should be much
purer, and the reflected much brighter than is usual with a single plate."
117. Rings with a White Centre.—The system of rings pro-
duced by the transmitted light as we have observed starts from a
white centre, but the feature of the reflected system is that their
central spot is black. The theory accounts for the black spot by
showing that the two interfering streams of light are reflected under
different conditions, one in passing from dense to rare, and the other
in passing from rare to dense, the result being that a difference of
phase of half a period is introduced between the two beams.
Let us now consider the case of two plates enclosing a film of a
refractive index intermediate between those of the plates themselves.
Thus suppose the film to be denser (more refracting) than the first
plate, and less refracting than the second plate, then the reflection at
the first surface of the film will take place when the light is passing
into a more refracting medium, and the same will be the case at
the second surface also. The light is therefore reflected under
similar circumstances at both faces, and no difference of phase is
introduced. It follows, then, that in the system of rings formed
under these circumstances the central spot should be white.
Young verified this anticipation of the theory by enclosing oil of
sassafras between two lenses, one of which was of flint glass, and the
other of crown glass. By this experiment Young justified the hypo-
thesis of the loss of half an undulation. …
The oil of sassafras may be replaced by a mixture of essence of
cloves and essence of laurels. If the film be of higher index than the
object-glasses between which it lies, the centre of the rings should still
be black. Arago verified this by using oil of cassia, of which the
index is superior to that of flint glass.
Fig. 97.-Arago's Experiment.
* The analytical investigations of Stokes for a pile of plates (Proc. Roy. Soc, vol. xi.
p. 545, 1860) may be applied to this question.

Art. 118 CONDITIONS FOR LARGE AND BRIGHT RINGS 215
When the third medium differs from the first, the theory of thin
plates becomes more complicated. In one case, however, no colours at
all should be exhibited, viz, when the film is backed by a perfect
reflector. In this case the waves are reflected in toto, so that the
reflected and transmitted systems become superposed. A calculation,
similar to that in Art. 113, shows that, if the intensity of the light
reflected at C.C.C. (see Fig. 94) be unchanged by these reflections, the
intensity of the light finally reflected at B is equal to that of the
incident light and independent of 6. However interference fringes
have been obtained from a polished metallic surface covered with a
film of gelatine." In fact if any light is absorbed by either of the
reflecting surfaces, the energy of the finally reflected beam cannot be
the same as that of the incident beam and light of certain wave lengths
may be absent.
118. Conditions for Large and Bright Rings.-The formula of
Art. 115 shows that the diameter of the nth ring increases with
the radius of curvature of the lens; that is, with the tenuity of the
film. Hence, in order to obtain wide rings, a lens of very small
curvature should be employed, but with a given piece of apparatus
there is still another factor to be considered in estimating the magni-
tude of the rings, viz. the angle of refraction into the film. The
diameters of the rings depend on the secant of this angle, and they
therefore increase with it. With a simple piece of apparatus, such as
that shown in Fig. 96, when the angle of incidence is increased, there
is great loss of light by reflection at the first or upper surface of the
glass, and only a small fraction
of the incident beam reaches
the film, so that a large angle
of incidence is detrimental to
brightness, and the rings ob-
tained will be very faint. This
difficulty may be avoided by
using a prism and a lens, as
shown in Fig. 98, instead of the Fig. 98.
plate and lens employed by Newton. One face of the prism is placed
on the curved surface of the lens so that light falling nearly
perpendicularly on one of the other faces will enter the prism in large
quantity, and in such a direction that the angle of refraction into the
film is also large. Viewed through the third face of the prism the
rings obtained in this manner are both bright and large. The same
* See Wood, Phil. Mag. 6, vii. p. 376, April 1904, and cf. chapter on Metallic
Reflection, infra.

216 - NEWTON'S RINGS CHAP. VIII
Achro-
matising
effect of
dispersion.
result is obtained with a glass plate and a prism having one face
polished into a spherical form of small curvature.
The chief peculiarity of the rings obtained in this manner is
not so much the brilliancy, or the diminished influence of the
thickness of the film, as the more or less perfect achromatism
produced by dispersion, especially in the neighbourhood of total
deflection, so that the bright rings are nearly white instead of being
highly coloured. This happens because the more refrangible rays
are more deviated by the prism, and therefore enter the film at
a greater angle, so that for a given thickness of film they have a
smaller path retardation, as explained in Art. 110. This means
that the dispersion increases the diameters of the rings corresponding
to the more refrangible rays, and when the diameters of the rings of a
given order are the same for all wave lengths there is perfect achro-
matism, and the rings are black and white. This compensating effect
of dispersion diminishes the confusion which arises from overlapping,
and increases the number of visible rings. As the incidence augments
the violet rings may become larger than the red, and a reversal of
colour occurs. This takes place near total reflection, which happens
first for the violet light, and is attended by a rapid change in the value
of cos 1 (see further, Art. 120). -
119. Examination of Newton's Rings through a Prism.—When
Newton's rings are examined through a prism some remarkable pheno-
mena are exhibited. They are described in his twenty-fourth observa-
tion, Opticks (book ii.): -
“When the two object-glasses are laid upon one another so as to make the rings
of the colours appear, though with my naked eye I could not discern above eight or
nine of those rings, yet by viewing them through a prism I could see a far greater
multitude, insomuch that I could number more than forty . . . and I believe that
the experiment may be improved to the discovery of far greater numbers. . . . But
it was on but one side of these rings, namely, that towards which the refraction was
made, which by the refraction was rendered distinct, and the other side became more
confused than when viewed with the naked eye. . . .
“The arcs where they seem distinctest were only black and white successively,
without any other colours intermixed. .
“I have sometimes so laid one object-glass upon the other that to the naked eye
they have all over seemed uniformly white without the least appearance of any of the
coloured rings; and yet by viewing them through a prism great multitudes of those
rings have discovered themselves. And in like manner plates of Muscovy glass and
bubbles of glass blown at a lamp furnace, which were not so thin as to exhibit any
colours to the naked eye, have through the prism exhibited a great variety of them,
ranged irregularly up and down in the form of waves. And so bubbles of water,
before they began to exhibit their colours to the naked eye of a bystander, have
appeared through a prism, girded about with many parallel and horizontal rings; to
produce which effect it was necessary to hold the prism parallel, or very nearly parallel,
to the horizon, and to dispose it so that the rays might be refracted upwards.”
Ant. 120 HERSCHEL’S FRINGES 217
Newton attributes these “odd circumstances” to the dispersing
power of the prism. The blue being more refracted than the red, it
is possible that the nth blue ring may be so displaced relatively to
the nth red ring that, at part of the circumference, the displacement
may compensate for the difference of diameters. A white strip
may thus be formed in a situation where without the prism the
mixture of colours would be complete, so far as could be judged by
the eye.
A simple case is that in which the thin film is a wedge bounded
by plane surfaces inclined at a small angle. If the edge of the prism
is parallel to the intersection of the faces of the plate, by drawing back
the prism it will be possible to adjust the effective dispersing power
so as to bring the nth bars to coincide for any two assigned colours,
and therefore approximately for the entire spectrum. The formation
of this achromatic band depends upon the same principles as the
fictitious shifting of the centre of a system of Fresnel's bands when
viewed through a prism (Art. 105).
In order to explain the increased number of the bands observed by
Newton, Rayleigh showed that it is necessary to take account of the
curved nature of the surfaces producing the rings. “The width for
each colour varies at different parts of the plate and it is possible that
the blue bands from one part, when seen through the prism, may fit
the red bands from another part of the plate.”
120. Herschel's Fringes.—A very simple and effective method of
obtaining coloured fringes by reflection from a thin plate of air was
first pointed out by Sir
William Herschel.” On
a perfectly plane piece
of glass or a metallic
mirror, before an open
window, place an equi-
lateral prism. The light
falling upon the exposed
face of the prism is re-
flected at the base and
emerges from the other
face (Fig. 99). To an observer looking in through the latter face the
field appears divided into two parts, one brightly illuminated, which
arises from the occurrence of total reflection at the base of the prism, and
the other comparatively dark, the light there being partly transmitted.
-
W. Wºº
Fig. 99.
Rayleigh, Phil. Mag. vol. xxviii. p. 202, 1889.
* Sir William Herschel, Phil. Trans., 1809, p. 274.

218 Newton’s RINGs CHAP. VIII
The line of separation of the two parts would be circular to an eye
situated in the prism, but the apparent shape of the curve seen by the
eye outside is the distorted form of a circle seen by refraction through
the prism. Since total reflection occurs at slightly different incidences
for the different colours, it follows that the curve of separation will be
iris-coloured. Inside this coloured band, and running parallel to it,
we have, in addition, a system of beautifully coloured fringes, the
breadth and number of which vary with every change of pressure.
These bands do not require for their formation a perfect polish
in the lower surface nor extreme thinness in the air film, for they
may be seen very well when the prism is separated from the lower
plate by the thickness of thin tissue-paper or a fibre of cotton-wool.
That excessive thinness of the air film is not necessary to the
production of tolerably broad fringes when the angle of incidence
approaches the angle of total reflection may be inferred at once from
the expression 22 cos r, which gives the path retardation of the
interfering pencils in the case of a film of thickness e. Near the
critical angle r is nearly 90°, and cos r is very small, so that the
retardation may be small even with a sensible thickness of film."
When the prism and plate combined are held up to the light a
transmitted iris is seen, lined with a similar system of fringes, on
looking through the plate and the
base of the prism.
The experiment may also be
conducted by merely looking
through two prisms placed in con-
tact (Fig. 100). In this form it
was repeated by H. F. Talbot.”
If the prisms be equal, isosceles,
and right-angled, then when
placed with their hypothenuses
in contact their ends will form
squares. Looking through the
combined prisms at the sky, a system of bands is seen, and looking at
the interface so as to see the light reflected there, another system is
observed, the latter being complementary to the former.
Fox Talbot describes a modification of the experiment as follows:
Fig. 100.
* These fringes may also be obtained very conveniently from a film of air enclosed
between two plane glass plates which are separated by two fragments of the same
thread of platinum wire (about ſº mm. in diameter), cemented around their edges,
and plunged vertically in water contained in a rectangular glass vessel in the manner
described in Art. 83.
* H. F. Talbot, Phil. Mag, 1836, p. 401.


ART, 120 TALBOT's MODIFICATION 2.19
“But the beauty of the appearances may be surprisingly increased by
transporting the apparatus into a dark chamber and suffering a
pencil of the brightest solar light to pass through the prism, or to
be reflected from the face AC. If then a sheet of white paper be
held up, at any distance from the prism, the coloured bands are
depicted upon it with the greatest vivacity and distinctness. The
transmitted bands have altogether a different character from the
reflected ones, so that it is impossible to mistake one for the other,
even without reference to the path of the ray.
“The coloured bands are not, as has been supposed, isochromatic
lines. The deviation is sometimes very marked, so that a band in the
course of its progress acquires very different tints from those which it
possessed originally. This fact may be considered of some importance
with respect to the theory. It takes place when the prisms are in
close contact and the bands few in number. But the following is still
more deserving of attention. When the contact of the prisms is
diminished by interposing a hair between them (still pressing them
together), the coloured bands depicted upon the paper become more
numerous, narrow, and crowded. Frequently they alternate a great
number of times with two complementary colours. This appeared to
me so remarkable that I repeated the experiment with additional
care. The radiant point of solar light was made smaller by trans-
mitting the ray through a lens of short focus, and the position of the
combined prisms was slowly altered by turning them round their
centre. The appearance of the bands on the paper was all the time
carefully noted. I soon found a position of the prisms in which
the remarkable phenomenon occurred of a complete compensation of
colour—that is to say, that the bands were black and white. At the
same time they were become exceedingly narrow and numerous. . . .
They resembled more than anything else the closely ruled parallel
lines by which Shadows are produced in some kinds of engraving, and
which are often employed in maps to represent the sea.
“Now it requires in ordinary circumstances the employment of
very homogeneous light, in order to produce bands anything like these
in number and distinctness. In the present instance, on the contrary,
common solar light was employed. . . . These bands are best seen in
the light reflected from the face AC.”
The achromatism here referred to is accounted for by the disper-
sion which takes place in the glass, and which becomes highly effective
when the light falls upon the film at an angle nearly equal to the
critical angle, so that the angle of refraction into the film is nearly 90°,
as explained in Art. 110.
220 * NEWTON'S RINGS CHAP. VIII
In the first place, it may be remarked that when the prism is
isosceles, so that the angle A is equal to the angle B, then a ray enter-
ing the first face AC (Fig. 99) at any angle will leave the second face
BC at the same angle, and consequently a beam of white light entering
the face AC will leave the face BC without dispersion. In other
words, the dispersion produced in the angle A will be compensated in
the angle B. The function of the prism is therefore to vary the angle
of refraction r into the film in such a way that cos r|A is approxi-
mately the same for all the colours. When A is not equal to B there
will be a further displacement of the fringes such as would be produced
by regarding them through a prism of angle A — B, as described in
Arts. 105 and 119.
Let the angle of incidence on the first face of the prism be i, and
the corresponding angle of refraction ri. Then if i and r be the angles
of incidence and refraction for the film, we have
sin 71=A, sin rì, (1)
sin r=1, sin i, (2)
7+ r1=A (3)
where p is the refractive index of the prism, and the film is supposed
to be air. When the film is other than air, equation (2) can be
modified accordingly. These equations combined with the achromatic
condition GOS 7°
→-- Const. (4)
determine the angle of incidence at which achromatism takes place,
when the law of dispersion in the glass is known.
Thus in Talbot's form of the experiment a parallel beam of white
light was used, so that it is the same for all wave lengths; hence by
differentiating the foregoing equations we obtain
p cos ridri + sin ridu = 0, (1')
p. CoS idº-F sin idu =cos rair, (2’)
di + dri = 0, (3')
A sin rdr--cos ral} = 0. (4!)
Combining these equations we obtain at once M. Mascart's relation 1
colºr=-ji.* X;
which may be written in the equivalent form *
cotºr = — ——º:--sin A_X dº.
sin icos rip, d\
The foregoing applies to Talbot's form of the experiment in which
the angle of incidence i, is constant, and all the light of a given colour
* M. Mascart, Traité d'optique, tome i. p. 449.
* Lord Rayleigh, Phil. Mag. vol. xxviii. p. 196, 1889.
ART. 120 ACHROMATIC CONDITIONS 221
is refracted at the same angle, so that the bands arise from variations
of the thickness of the film, such as when a hair is placed between the
prisms to produce a wedged-shaped film.
In Herschel's form of the experiment the prism is placed before
the open sky, so that the angle of incidence is variable, and light of a
given wave length is not all refracted at the same angle. These bands
are broad and richly coloured, if the air layer is thin, becoming finer
and less coloured as the thickness increases. They are produced near
the limit of total reflection by the variation of the angle of incidence
when the thickness of the film is constant. Of course bands of an
intermediate character are produced when the angle of incidence and
the thickness of the film both vary.
The theoretical condition for constant thickness is better satisfied,
if, after Mascart, we place the layer of air in the focus of a small
radiant point (electric arc). In this case the area concerned may be
so small that the thickness in operation can scarcely vary, and the
ideal Herschel's bands are seen depicted on a screen held in the path
of the reflected light. It will, of course, be understood that bands
may be observed of an intermediate character in the formation of
which both thickness and incidence vary. Herschel's relate to one
particular case—that of constant thickness; Talbot's to the other
especially simple case of constant angle of incidence.
From the present point of view there is one very important dis-
tinction, because one is achromatic and the other is not. To under-
stand this, we follow Herschel's bands in greater detail.
In the same notation as before
6=2e cos r = m\,
and the question to be investigated is the relation of i, to n. The
band of zero order (n = 0) occurs when r = 90°, that is at the critical
angle. For two successive bands we have
mX=2e cos r, (n + 1)A=2e cos (r-H dr),
therefore
Also
X = – 26 sin rdr.
cos indii = p, cos ridri,
so using previous results we find
- d;, & cos”, - cost —X mX” cos ri
*Toos iſ a cos i 2e sin ºr Tag” cosi, cost sin F
Near total reflection sin r = 1, (q.p.) and the factors cos ri, cos i, cos i
vary but slowly with the order of the band and also with the wave
length. Hence the width of the band is approximately proportional
to the order, the square of the wave length, and the inverse square
of the thickness.
222 - NEWTON'S RINGS CHAP. VIII
When the light is white, the centre of the system will be where
there is coincidence of bands of the order n in spite of the variation of A.
About the achromatic centre thus determined the visible bands will
be grouped. At the central band n is the same for the various colours,
consequently the widths of the various systems are at this place
approximately proportional to A*. Hence these bands are less achro-
matic than ordinary bands or Newton's rings, in which the width is
proportional to A, and this theoretical conclusion is in harmony with
observation (see Lord Rayleigh's paper, Phil. Mag., Sept. 1889).
Newton's Observations on the Coloured Rings of Thin Plates
Opticks, book ii. part i. obs. 4: “I took two object-glasses, the one a plane-
convex for a fourteen-foot telescope, and the other a large double convex for one
of about fifty foot; and upon this laying the other with its plane side downwards I
pressed them slowly together, to make the colours successively emerge in the middle
of the circles, and then slowly lifted the upper glass from the lower to make them
successively vanish again in the same place. The colour, which by pressing the
glasses together emerged last in the middle of the other colours, would upon its first
appearance look like a circle of a colour almost uniform from its circumference to its
centre, and by compressing the glasses still more, grew continually broader till a new
colour emerged at its centre, and thereby it became a ring encompassing that new
colour. And by compressing the glasses still more the diameter of this ring would
increase, and the breadth of its orbit or perimeter decrease until a new colour emerged
in the centre of the last ; and so on until a third, a fourth, a fifth, and other follow-
ing new colours successively emerged there, and became rings encompassing the
innermost colour, the last of which was the black spot. And, on the contrary, by
lifting up the upper glass from the lower, the diameter of the rings would decrease,
and the breadth of their orbit increase, until their colours reached successively to the
centre ; and then by being of a considerable breadth, I could more easily discern and
distinguish their species than before. And by this means I observed their succession
and quantity to be as followeth.
“Next to the pellucid central spot made by the contact of the glasses succeeded
blue, white, yellow, and red. The blue was so little in quantity that I could not
discern it in the circles made by the prism, nor could I well distinguish any violet
in it, but the yellow and red were pretty copious, and seemed about as much in
extent as the white, and four or five times more than the blue. The next circuit in
order of colours immediately encompassing these were violet, blue-green, yellow, and
red ; and these were all of them copious and vivid, excepting the green, which was
very little in quantity, and seemed much more faint and dilute than the other
colours. Of the other four the violet was the least in extent, and the blue less than
the yellow and red. The third circuit or order was purple, blue, green, yellow, and
red; in which the purple seemed more reddish than the violet in the former circuit,
and the green was much more conspicuous, being as brisk and copious as any of the
other colours except the yellow ; but the red began to be a little faded inclining very
much to purple. After this succeeded the fourth circuit of green and red. The
green was very copious and lively, inclining on the one side to blue and on the other
side to yellow. But in this fourth circuit there was neither violet, blue, nor yellow,
and the red was very imperfect and dirty. Also the other colours became more and
more imperfect and dilute, till after three or four revolutions they ended in perfect
whiteness.”
ART, 120 NEWTON's OBSERVATIONS 223
“Obs. 5.--To determine the interval of the glasses, or thickness of the inter-
jacent air, by which each colour was produced, I measured the diameters of the
first six rings at the most lucid part of their orbits, and squaring them, I found
their squares to be in the arithmetical progression of the odd numbers, 1, 3, 5, 7, 9,
11. And since one of these glasses was plane and the other spherical, their intervals
at those rings must be in the same progression. I measured also the diameters of
the dark or faint rings between the more lucid colours, and found their squares to
be in the arithmetical progression of the even numbers, 2, 4, 6, 8, 10, 12. And it
being very nice and difficult to take these measures exactly ; I repeated them divers
times at divers parts of the glasses, that by their agreement I might be confirmed in
them.”
“Obs. 10.-Wetting the object-glasses a little at their edges, the water crept in
slowly between them, and the circles thereby became less and the colours more ſaint,
insomuch that as the water crept along, one half of them at which it first
arrived would appear broken off from the other half, and contracted into a
less room. By measuring them I found the proportions of their diameters to the
diameters of the like circles made by air to be about seven to eight, and consequently
the intervals of the glasses at like circles, caused by those two mediums water
and air, are as about three to four. Perhaps it may be a general rule, that if any
other medium more or less dense than water be compressed between the glasses,
their intervals at the rings caused thereby will be to their intervals caused by
interjacent air, as the sines are which measure the refraction made out of that
medium into air.” -
224 THICK PLATES CHAP. VIII
SECTION III. THE Colours of Thick PLATES
121. Brewster's Bands.--When a pencil of light falls in succes-
sion upon two transparent plates which are not very thin,' some of
the many portions into which it is divided by partial reflections at
their bounding surfaces are frequently in a condition to interfere and
produce coloured bands. These fringes were observed by Sir D.
Brewster” in 1815. The apparatus employed in the experiment con-
sisted of a straight tube (Fig. 101) blackened on
the inside and closed at one end by a disc con-
taining a small aperture O. Two uniform * glass
plates of equal thickness were placed near each
Fig. 101. other at the other end of the tube, one of them
being at right angles to the axis of the tube, and the other inclined to
it at a very small angle. This angle could be varied by means of a
micrometer screw.
In order to understand the formation of these fringes it is only
necessary to notice that when the aperture O is illuminated and
viewed through the plates the light which reaches the eye consists of
many distinct components arising from successive reflections within
and between the plates, and a corresponding series of images of the
source is consequently presented.
This series of images may be divided into a system of groups.
Those of the first group correspond to light that has traversed the
space between the plates once, so that they are formed by light that
has not been reflected from one plate to the other, but which may
have suffered reflection within either plate. The first image of this
* A thick plate in optics means one of thickness large compared with the length
of a wave of light.
* Brewster, Edinb. Trans, vol. vii. p. 435, 1815.
* A plate of perfectly uniform thickness cannot be procured in practice, and it is
difficult to obtain plates sufficiently uniform to give good fringes by this method.
A plate of glass generally has its faces inclined to each other so as to form a prism of
very small angle. The lines of constant thickness in such a plate are approximately
rectilinear and parallel, and can be seen when the plate is viewed by reflection in
monochromatic light. The interference fringes follow the lines of constant thick-
ness, and if the plate be cut in two along a line perpendicular to the direction of
the fringes, the two parts when superposed by folding them round the line of
section will readily exhibit Brewster's bands, for in this case the corresponding rays
of the interfering pencils traverse the plates at places of equal thickness.

ART, 121 BREWSTER’s BANDS 225
group may be called the principal image, and it is formed by light
that has been directly transmitted through the plates, such as AA in
Fig. 102. Behind this there is an image formed by light that has
suffered two internal reflections in one of the plates, such as the rays
BB and CC, and so on for multiple internal reflections. Now the
image formed by BB will coincide with that formed by CC when
the plates are parallel and of equal thickness; but when the plates are
slightly inclined the thickness traversed by the light in one of them
will differ by a small amount from that traversed in the other, and
there will be a small relative path retardation introduced between the
pencils BB and CC, so that interference bands will be produced.
Regarding the image as a source of light we may say, then, that the
image formed by BB interferes with the image formed by CC, and
Fig. 102.
produces an image crossed by fringes. The image formed by AA
presents no fringes, for there is no other pencil of light of approxi-
mately the same path. The other images of this group are also
crossed by bands; for example, the light that has suffered four
reflections in the plate M interferes with that which is four times
reflected within the plate N, etc. The second group consists of light
that has traversed the space between the plates three times. The
first image of this group is formed by light that has traversed each
plate only once, such as the ray DD, and, like the first image of the
first group, it presents no fringes. A little behind this, however,
comes the image formed by EE, and this is interfered with by FF, so
that fringes are presented. Other images are formed by rays, such as
GG, that have suffered four or more reflections, and they also present
bands which are explained in the same manner. The third group is
formed by light that has traversed the space between the plates five
times, and so on.
The relative retardation of the pencils forming the interference
Q

226 s THICK PLATES CHAP. W III
bands on the second image of any group is easily expressed in terms
of the angles of refraction into the plates. For if e be the common
thickness of the plates it is clear that the ray EE suffers a relative
retardation 2e cos r in the plate M, while FF suffers a relative
retardation 26 cos r" in N. The two transmitted pencils have
consequently a relative path retardation of
ô=2e (cos 1 — cos r").
This vanishes when r = 7'-that is, when the light is incident on
the two plates at the same angle, or is parallel to the plane bisecting
the obtuse angle between the plates. The light incident in this
direction consequently determines the central fringe of the system.
Brewster describes the experiment as follows:
“In order to observe the phenomenon to the greatest advantage, let the light of a
circular image subtending an angle of 1° or 2° be incident perpendicularly, or nearly
so, upon two plates of parallel glass placed at a distance of one-tenth of an inch, and
let one of the plates be gently inclined to the other, till one or more of the reflected
images be distinctly separated from the bright image formed by transmitted light
and received upon the eye placed behind the plates. Under these circumstances the
reflected image will be crossed with about 15 or 16 beautiful parallel fringes . . .
the direction of the fringes is always parallel to the common section of the four
reflecting surfaces.
“All the preceding experiments were made with plates which were cut out of
the same piece of glass, and had therefore the same thickness. I now tried plates of
different thicknesses, both when ground parallel and when cut from a common plate
of glass; but I could never render the coloured fringes visible, unless when the glass
was parallel, and exactly of the same thickness in both plates.”
122. Jamin’s Interference Refractometer. —The interference
bands obtained by Brewster's method have been turned to account
by M. Jamin' in the construction of a very delicate refractometer.
Two plates of parallel glass as nearly as possible of equal thickness
(about 1 cm.) are silvered on their backs and supported (on an
optical bench or otherwise) so that the distance between them can be
altered at will. Their plane faces can be placed in the vertical, which
is supposed perpendicular to the plane of the paper in Fig. 103.
The first plate AB is fixed, and the light of the sun or any other
large source falls upon it. After reflection it is received on the second
plate CD, which can be turned round a horizontal axis by means of
a screw situated at its back, and round a vertical axis by means of
the screw Q. The displacement of this plate round the vertical is
measured by the movement of an arm on a graduated arc at W. Part
of the light incident on the first face AB is reflected there, and
after penetrating the second plate is reflected at its second surface
1 Jamin, Ann. de Chim, et de Phys., third series, tome lii. p. 163, 1858.
ART, 122 JAMIN'S REFRACTOMETER 227
and emerges from the plate. A second part of the light penetrates
the first plate, and after reflection at its second surface it is reflected
from the first surface of the second plate. Thus of the two beams
one is reflected at the front of the first plate and the back of the
second, while the other is reflected at the back of the first plate and
the front of the second." Hence if the plates be parallel, the two
pencils will traverse equal and similar paths, and they will therefore
be in the same phase on leaving the second plate ; but if the second
plate be turned through a small angle the ray which is refracted in it
will pursue a path slightly different from that traversed by the ray
refracted in the first plate ; there will therefore be a difference of
S
Fig. 103.
phase between the beams as they leave the second plate, and the
phenomena of interference will be presented. .
If we start with the two plates parallel and the plane of incidence
horizontal, then by turning the screw at the back of CD the plates
will become inclined to each other and their line of intersection will
be horizontal. A system of horizontal fringes will consequently
appear in the field, increasing in width as the inclination of the
plates is diminished. When the plates are rigorously parallel the
* There is of course a whole system of images formed by successive reflections
within the plates and at their bounding surfaces. Thus an image is formed by light
reflected directly from the face AB and then from CD. Behind this there is the
image considered in the text, and which is crossed by bands because it consists of two
nearly coincident images—namely, that formed by reflection from AB and C/D’
combined with that formed by reflection from A'B' and CD. The next image is
bright and may be considered the principal image of the system, and is formed by
the light reflected from the metallic surfaces A'B' and C'D'. This is followed by
other fainter images produced by light that has suffered two or more internal
reflections within the plates. These are also crossed by interference bands, and the
whole system is situated sensibly on the perpendicular to the plates drawn through
the source.

228 - THICK PLATES CHAP. VIII .
bands should disappear entirely and give place to a uniform illumina-
tion. As this stage is approached, however, the bands generally
become deformed and form undulating curves indicating want of
homogeneity in the glass or imperfections in the planeness of the
surfaces.
The central fringe corresponds to zero retardation and, as in
Brewster's experiment, is formed by those rays of the interfering
pencils which make the same angle with the two plates. These rays
are consequently parallel to the plane bisecting the obtuse angle
between the plates. The effect of moving the screw Q is to displace
the central band vertically and raise or lower the whole fringe system
in the field of view.
This may be easily seen by considering the images s, and sº formed
of a source by the first and second surfaces respectively of the first
plate. If the second plate is parallel to the first, the image s, formed
of s, by its second surface will coincide with the image 8, formed of
s, by its first surface. If, however, the second plate be turned slightly
round a horizontal axis, s, and Ss will lie on the normals drawn through
s, and sº to the second plate at nearly the same distance from the plate
as before ; that is, the images will be separated vertically and will form
two sources in a vertical line that will produce horizontal fringes.
The width of these fringes will depend on the distance apart of these
two sources; i.e. on the rotation of the second plate round the horizontal
axis. A rotation of the second plate round a vertical axis separates
the two sources horizontally; that is, it turns the perpendicular, bisecting
the line joining them, in a vertical plane through the line of sight,
and therefore raises or lowers the fringes in the field of view.
When a slender pencil of light (obtained by placing a suitable
diaphragm in the path of the incident light) is used, tubes containing
gases or thin plates of different substances may be placed in the paths
of the pencils between the plates, and the relative speeds of light in
these substances determined by the corresponding displacement of the
fringes. The apparatus may thus be used as a refractometer.
The retardation produced by the passage of one of the pencils
through a thin plate of any substance, or through a tube of gas, of
which it is desired to measure the refractive index, is determined by
means of a compensator K. Thus if a tube of gas or a thin plate be in
the path of one pencil, a thin plate of parallel glass may be intro-
duced across the path of the other, and if this latter be of the proper
thickness, there will be no resultant displacement of the fringes. The
one will compensate the other. The proper thickness of the parallel
glass plate may be adjusted by constructing it of two slender prisms
ART, 123 JAMIN'S COMPENSATOR 229
of equal angle, placed on each other with their edges in opposite
directions. Placed thus they form a parallel plate, the thickness of
which may be varied at will by sliding one prism upon the other. This
may be done conveniently by keeping one fixed, and displacing the
other by means of a screw. When the instrument is once standard-
ised we know immediately the retardation produced by the substance
in the path of the first pencil.
The compensator used by M. Jamin consisted of two glass plates
(shown Fig. 104), fixed to a common axis, and inclined to each other
at a small constant angle. One pencil passes through one plate, and
the other through the other plate. The retardation produced by a
plate depends on the angle of incidence, and is least when the ray
passes through it normally. When the rays fall on the plates
perpendicularly to the plane bisecting the angle between them, they
introduce equal retardations, and therefore no displacement of the
fringes. In any other position the plates displace the fringes by an
amount depending on their thickness and the angles of incidence; so
that, by rotating the axis to which they are fixed, any desired
displacement of the fringes can be produced, and any previous
displacement compensated. The sensi-
tiveness of this compensator can be
varied at will by altering the angle
between the plates, and the retardation
introduced by it may be easily calculated,
but it is best to graduate it by experi-
ment, which shows that the relative
retardation is very nearly proportional
to the angle through which it is turned
from the position of zero displacement.
M. Jamin by this method has determined
the index of refraction of several gases,
and compared those of dry and moist air. By the same process he also
investigated the effect of compression and change of temperature on
the refractive index of water."
123. Michelson and Morley's Interferometer.—An interference
refractometer has been described by Professors Michelson and Morley”
which readily permits of the introduction of any relative retarda-
tion between the interfering pencils, and consequently allows of the
Fig. 104.—The Compensator.
| Jamin, Ann. de Chim, et de Phys., third series, tome lii. p. 163, 1858; and tome
lxi. p. 385, 1861.
* Professors A. A. Michelson and E. W. Morley, Journal of the Association of
Engineering Societies, May 1888.

230 THICK PLATES CHAP. VIII
observation of interference bands corresponding to a large difference
of path. This apparatus is the same as that employed in their ex-
periments on the relative motion of the earth and the ether (Art. 315).
By means of a collimating lens, light from a source of light (e.g. a
sodium flame) S (Fig. 105) is made to fall in a nearly parallel beam
upon a plane glass plate A inclined to it at any angle (usually 45°).
Part of this light is transmitted in the direction AB and part is
reflected in the direction AD. Both of these beams are received
perpendicularly on plane mirrors C and D, so that they return to the
plate A along their original paths and pass in part into the observing
telescope T. Now the pencil reflected from D traverses the plate A
three times before it reaches T, and in order to compensate for this a
Fig. 105.
similar plate of glass B is introduced into the path of the pencil
reflected from C. Hence if AC = AD, and if the plates are parallel,
the two pencils will have traversed paths of equal length, but it is to
be observed that they have both suffered reflection at the same face of
A, one internally and the other externally, and consequently when
the adjustment of the mirrors is exact the whole field will be dark.
In this case the image of the mirror C in A coincides with D. When
the adjustment is altered there will be a path retardation between the
pencils equivalent to that of an air film of thickness AC – AD, and
whose angle is equal to that between D and the image of C in A.
One of the mirrors is furnished with a micrometer screw by which
it can be moved parallel to itself so as to introduce any difference of
path desired.
If the image of C in A is parallel to D, a circular set of interference
fringes will be seen in the telescope. Michelson has discussed the
* Michelson, Phil. Mag., April 1882, and Sept. 1892.

ART. 123 MICHELSON AND MORLEY'S INTERFEROMETER 23]
theory of these fringes and shown that the clearness of the fringes
changes with variation in the difference of the paths of the interfering
beams, in a manner which depends upon the structure of the light in
the source.
If a monochromatic beam is split into two equal parts, each of
amplitude a, and a phase difference is set up between them, we have
gy=aſcos wi-H cos (wt – 2%)],
and the intensity is 2a”(1 + cos 24).
In the case of a heterogeneous beam—i.e. one consisting of several
wave lengths and in which the energy is distributed according to the
law f(\)—the sum of the intensities is
ſ (1 + cos 2q,)f(\)dX.
There is here no question of interference, because the intensities
corresponding to different frequencies are independent of one another
and they are additive. In this expression 2q, = *R where R is the
space retardation: so that q, is also a function of the wave length.
The practical use which has been made of the interferometer is to
determine what the value of f(X) is for a single spectrum line. Such
a line is not a mathematical line, but consists of a small range of
wave lengths. If the range about the mean wave length Ao is
sufficiently small we can write
I
;
where a is small ; and
20–£(1+2)R.
No
Now, on the assumption that the range is small, we can take R as a
constant throughout the integration, for the variation in q is mainly
due to the change in A. Writing R = PA,
I = ſ + COS (#6 +opx) }*@*
where p(a)da replaces f(X)dX.
Hence
I = ſ q.(a)da: +cos 2TPſcos 2TPa. . p(a)dae – sin 2+Pſsin 2T Pac. p(a)da:
=T+ G cos 2TP – F sin 2+P.
This gives the intensity, due to the narrow range of wave length, for
any one value of P; i.e. for any given position of the movable mirror.
Next, consider what happens when the mirror is shifted. This results
in a change in the value of P during which T does not change, and F
and G change very little compared with the change in the cosine and
232 THICK PLATES CHAP. VIII
sine factors. If we treat them as constant and differentiate with
respect to P we get
G sin 2+P.+ F cos 2TP=0,
which, together with the expression for I, gives
I=T-E VG24-F4.
The upper sign clearly corresponds to a maximum value of I, and the
negative to a minimum value.
Michelson made an eye-estimate of the values of the intensities
of the maxima and minima near the centre of the field, and calculated
what he called the “visibility” defined by
I — I
IIl{i,x. min.
V=ſ*H
IIl{LX. min.’
It is clear that the visibility is a measure of the contrast between the
brightest and the darkest parts of the field. When the mirror is
advanced the visibility of the fringes changes, and visibility curves
for a given source were obtained by plotting V against the positions
of the mirror.
Unfortunately, although the visibility curve is different for
different spectral lines, there is no direct way of determining the
structure; i.e. the values of the elementary vibrations of the source
from these curves. It is necessary to assume various structures, to
calculate the visibility curves that correspond, and find, by comparison,
to which of them the experimental visibility curve most nearly
corresponds. As an example, assume that the structure is given by
q(x)=e-k”
where k is constant.
Then
CAE) --- Tr2P2
G = ſ cos 2TPa; . -*a*- : , -º
~ CO
F=O,
CO ºmmºn
t-ſ e-kºda = w/º:
– CO
and thence
- 2p2
-º-,-º:
Thus V plotted against the order of interference, P, should in this
case be a curve of the same type as h(a) against a ; i.e. a “probability”
curve. The cadmium red line corresponds admirably to this case.
ART. 124 NEWTON'S DIFFUSION RINGS 233
In other cases more complicated assumptions must be made. The
cadmium green line can be represented by two such exponential
groups of unequal maxima, and the blue line by two still further
separated. The orange-red line of oxygen is fitted by three such
groups.
The method is a very ingenious one, and although it has been
superseded by more direct methods of analysis it is of importance
historically, because it was the first method of successfully analysing
into components a spectral line which with an ordinary spectroscope
appeared single.
MM. Fabry and Perot have devised an interferometer in which,
with a monochromatic source, very sharp maxima of illumination are
obtained, separated by comparatively wide bands of darkness. It
consists of two parallel plane plates of glass, separated by an air layer,
which may be very exactly adjusted for parallelism and one of which
may be moved perpendicularly to its plane. The opposing surfaces
of the glass are silvered in the same silvering bath, so that light falling
on them may be partially reflected and partially transmitted, to the
same extent at each surface. The theory of this interferometer has
already been given (Arts. 112-114). If a parallel beam of light
from an extended source pass nearly normally through the plates,
and be examined through a telescope focussed for infinity, a set of
Haidinger fringes will be seen ; i.e. a series of circles whose centre
corresponds to the beam which falls precisely along the normal.
Owing to the high reflecting power of the partially silvered surfaces
the maxima will consist of very narrow bands (Art. 114). The light
examined may be considered as made up of lights issuing from the
original source, and from all the virtual sources corresponding to
successive reflections. These will interfere regularly because the
phases will form an arithmetic progression.
If the light examined consist of light of two different wave lengths,
two sets of narrow fringes will be formed, the interval between the
fringes being different for light of each wave length. Thus the fringes
will coincide in places and these coincidences will be separated by
regions in which the fringes are separated. The smaller the difference
between the wave lengths, the greater will be the interval between two
coincidences; and as high orders of interference are obtained with the
apparatus, the resolving power of the apparatus is very great and very
close lines may be separated.
124. Newton's Diffusion Rings.-The preceding cases of inter-
ference may be produced with tolerably thick plates, and are known as
the interference colours of thick plates. Coloured rings, with thick
234 THICK PLATES chap. viii.
plates, of a distinct kind were discovered and described by Newton,"
and are known as diffusion rings.
Newton allowed a pencil of sunlight to fall perpendicularly upon a
glass mirror, ground concave on one side and convex on the other, to
a sphere of nearly six feet radius, and silvered on the convex surface.
Holding at the centre of the sphere a screen of white paper, with a
small hole at its centre to allow the beam of light to pass and repass,
he observed four or five rainbow-coloured rings on the paper, encircling
the aperture through which the light poured. These were similar to
the transmitted rings of thin plates, the squares of the radii of the
bright rings being proportional to the even numbers, while those of
the dark rings were in the ratios of the odd numbers.
When the mirror is inclined a little so as to throw the reflected
image slightly to one side of the aperture the rings are formed as
before, but their centre is at the middle point of the line joining the
aperture and its image. This central spot changes its appearance in a
remarkable manner as the image recedes from the aperture, being
alternately bright and dark when homogeneous light is used, but with
white light it assumes every gradation of colour.
Newton endeavoured to account for these rings, like those of thin
plates, by the emission theory; but it is to Young we owe the
application of the wave theory to their explanation. His work was
afterwards completed by Herschel, Stokes, and Schäfli.
The rings are much better marked when the surface of the mirror
is slightly dimmed, as, for example, by coating it over with a weak
mixture of milk and water. This
suggests that the phenomena are due
in some way to the light which is
scattered at the surface of the mirror,
for we know that the effect of dim-
ming the surface is to increase the
quantity of light scattered or irre-
gularly reflected at it. We will there-
fore attempt on this supposition to
explain the phenomena in the more
general case when the light comes
from a point L (Fig. 106) of the screen not far from the axis of the
mirror, the plane of the screen passing through the centre O of the
mirror and being perpendicular to its axis ON. Denote the distance
LO by u, the radius OA of the first surface of the mirror by a, and
the radius ON of the second by a + e, where e is the thickness of
Fig. 105.
* Newton, Opticks, book ii. part iv.

ART. 124 NEWTON'S DIFFUSION RINGS 235
the mirror. Consider the illumination at any point Q (not in the
plane of the paper) on the screen. -
The rings must be due to the interference of the light scattered
or diffracted by the same particle of dust. It has been shown by
Stokes that no regular interference is to be expected between portions
of light diffracted by different particles of dust, for the diffusion is
accompanied by a difference of path which varies from point to point
(Camb. Trans. ix. p. 147, 1851). In his memoir there is a complete
discussion of the whole case.
A ray of light LM, after incidence at M, will be refracted in part
along MN, and after suffering reflection at N it will arrive at P, where
a portion will be regularly refracted and some of it will be scattered.
Let PQ be the scattered ray which reaches Q. Now another stream
of light will also reach Q. For consider the ray incident at P, part of
it will be scattered on entering the plate, and some one PR of these
irregular rays will after regular reflection and refraction at R and S
respectively reach Q along SQ. It will thus have traversed a path
LPRSQ differing little from that traversed by LMNPQ. The two
pencils will therefore be in a condition to produce interference effects.
It may be observed that both the rays have suffered the scattering at
the same point P of the surface, the first at emergence and the second
at incidence. The two have therefore passed over similar paths and
suffered similar transformations. They are consequently beams of the
same nature, and may interfere.
It only remains now to calculate the difference of the paths tra-
versed by the rays. We have *
-º-º- w?\} w?
LM-Jº-a(1+...) =a+3. (approx.),
since u is very small compared with a, Again, if i be the angle of
incidence (LMO) of the ray LM, its sine is approximately equal to its
tangent or circular measure, and consequently
sin i = OL w
* OMT,”
and therefore
e sin i w
SlT1 7" E — E — ”
A. AL(t
But r = MNA, therefore
- % MA2 w?
— / 22 2\2 — –--- * *T Tº || 3
MN = (e?--MA”) = (i tº )=c(l **)
since
*=sin MNA=sin r=#.
6 A.0%
Similarly
NP = MN and PQ=a+. º
236 - THICK PLATES CHAP. VIII
But the path MN + NP is traversed in glass, so that the equivalent path
in air will be p times as great ; the whole equivalent air path is then
LM + p.(MN +NP)+ PQ,
OT
a tº 12 € 1+2+. +a+*.
2, #4 2p*a? 20,
From this the value of the path LPRSQ may be written down at once
by observing that the first ray travels regularly till it emerges at P,
where it is scattered. But if the second path be traversed in the
reverse direction we see that, setting out from Q along QS, refraction
and reflection take place regularly at S and R till emergence at P,
where scattering takes place. Traversing the second path in the direc-
tion QSRPL is then the same as the first path in the direction LMNPQ
with this difference, that the point Q is at a distance v from O, whereas
the distance of L from it is u. Consequently the second path may be
written down from the first by interchanging u and v. It is therefore
q2 q;2 no.2
at:19a (1+3,...) + °t 2, Q
The difference of the path is consequently
6
- º – wº).
Ö
When this difference is an even number of half wave lengths the two
beams are in the same phase, and all points at the distance v from O
are bright. If the difference is an odd number of half waves this
circle is dark.
There is therefore a series of alternately bright and dark rings
encircling the point O. The radii of the rings are the values of p,
which satisfy the equation
€ (p” *)=n.
wº"? 20*) = 2’
the bright rings corresponding to even values of n and the dark rings
to the odd values.
If u = 0, we have then
a formula which embraces the laws laid down by Newton for the case
in which the origin of light is at the centre of the mirror.
Thus the diameter of any ring varies inversely as the square root
of the thickness of the mirror, while, as in the transmitted rings of
thin plates, the diameters of the bright rings are proportional to the
square roots of the even numbers, and the diameters of the dark rings
to the square roots of the odd numbers. Finally, the squares of the
ART. 124 NEWTON's DIFFUSION RINGS 237
radii vary directly as the wave length, so that with white light the
circles are rainbow-coloured bands, changing from violet at the inner
to red at the outer edge.
When u = w the retardation is zero for all wave lengths, conse-
quently the circle with centre O, and passing through the luminous
point L, will be bright for all colours, and will therefore be white, and
(opposite to L) at the other extremity of the diameter of this circle
there will be an image of the point L formed by regular reflection at
the surface of the mirror." This white circle is surrounded by another
system of coloured rings. The difference of the squares of the radii
of any two consecutive rings is constant, and the area of the annulus
between any consecutive pair is therefore the same for all pairs and
is equal to
trula” A
rº-pº)=* . g.
The successive circles therefore approach each other more and more
closely the farther we go out from the centre, so that at a short distance
from the centre the appearances become so confused that all effect is
obliterated. These rings may be also observed either with a plane
glass mirror or with a convex mirror, but they are more faint. With
a convex mirror it is necessary to use a convergent pencil of light in
order to obtain a real image of the source.
They were obtained by the Duc de Chaulnes” with a concave
metallic reflector, in front of which he placed a very thin plate of glass
or mica dimmed with a mixture of milk and water. The metallic
reflecting surface plays the part of the silvered back of Newton's
mirror, while the plate of glass or mica acts as its first face, and the
air space between the plate and the reflector corresponds to the glass
of the mirror.
The rings may also be well observed without a screen, as suggested
by Stokes.* All that is required is to place a small flame at the
centre of curvature of the mirror, so that it coincides with its image.
The rings are then seen surrounding the flame.
Dr. Whewell 4 observed a similar system of coloured bands formed
when the image of a candle, held near the eye, is viewed by reflection
in a plane glass mirror placed at a distance of some feet. This
* With regard to these Newton observes: “The incident and reflected beams
of light always fell upon the opposite parts of this white ring, illuminating its peri-
meter like two mock suns in the opposite parts of an iris" (Opticks, book ii. part
iv. obs. 10).
* De Chaulnes, Mém. de l'Acad. des Sc. p. 136, 1755.
* Stokes, Phil. Mag. p. 419, 1851.
* Stokes, Camb. Phil. Trans. pp. 148, 149, 1851.
238 THICK PLATES CHAP. VIII
observation was communicated to M. Quetelet," by whom it was
published. In repeating the experiment together they found it
essential that the mirror should not be perfectly bright, and to ensure
the production of the bands it was sufficient to breathe gently on the
surface of a cool mirror. Instead of moisture, which quickly evaporates,
M. Quetelet recommended a tarnish of grease.
125. The Colours of Mixed Plates.—When the space between two
glass plates is filled with a mixture of two substances in a finely divided
state—such as water and air, or water and oil—light will in general
traverse the different parts of the mixture in different times, and the
interval of retardation will depend upon the difference of the velocities
of light in the two media, and upon the varying arrangement of the
media between the two plates. Portions of the transmitted light will
therefore be in a condition to interfere, and coloured rings are seen
when a luminous object is viewed through the glasses.
The colours of mixed plates were first observed by Dr. Thomas
Young and described in the Philosophical Transactions for 1802.” He
produced them by interposing small globules of water, or butter,
between two glass plates, or two object-glasses, pressed together so
as to give the ordinary colours of thin plates. In this way little
cavities of air were surrounded with water or butter, and on
looking through the combination he saw fringes or coloured rings
several times larger than those of thin plates which would have been
produced had air alone been contained between the glasses. The
rings were seen by the direct light of a candle, and began from a
white centre like those produced by transmission through an air
film. On the dark space next the edges of the plate he observed
another system of fringes complementary to the first and beginning
from a dark centre like those produced by reflection. This latter
system was always brighter than the former. Brewster * in repeating
the experiments “tried transparent soap and whipped cream, which
gave tolerably good results; but I obtained the best effect by using
the white of an egg beat up into froth. To obtain a proper film of
this substance I place a small quantity between the two glasses, and
having pressed it out into a film I separate the glasses, and by holding
them near the fire I drive off a little of the superfluous moisture. The
two glasses are again placed in contact, and pressed together so as to
produce the coloured fringes or rings, they are then kept in their
* Quetelet, Corresp. Phys. et Math, tom. v. p. 394, 1829.
* Also republished in 1807, Elements of Natural Philosophy, vol. i. pp. 470, 787;
vol. ii. pp. 635, 680.
* Sir D. Brewster, “On the Colours of Mixed Plates,” Phil. Trans. p. 73, 1838.
ART. 125 MIXED PLATES 239
place either by screws or by wax, and may be preserved for any
length of time.” Brewster considered the colours as due to diffraction,
regarding mainly the case of light diffracted at a straight edge.
Verdet objected to this, because the phenomenon does not depend on
the form of the globules, but Wood+ points out that the colours
absent in the transmitted system do not appear in the reflected light,
and that, therefore, some of the energy of the incident light remains
unaccounted for on the interference explanation. He completes the
explanation by means of the diffraction effects due to a thin transparent
lamina, the light passing close by the edge of the lamina and that
passing through the lamina near to the edge giving diffraction fringes.
If a dark object be behind the glasses and if the incident light be
somewhat oblique, the rings change their character and°resemble the
ordinary reflected system of Newton. One of the portions of the
interfering light in this case suffers reflection, and accordingly half a
period difference of phase is introduced. These rings contract as the
obliquity of the light is increased. The opposite occurs in the case of
Newton's rings.
The colours of mixed plates were attributed by Young to interfer-
ence, on the supposition that part of the transmitted light passes
through one of the constituents of the mixture and part through the
other, and the explanation from this point of view has been followed
up by Verdet.” Thus if part of the incident light be refracted through
the plate at an angle ri, and another part at an angle rº, then by Ex.
Art. 71 the relative retardation between these parts is
6= e(p, cos ri – pie cos re),
and when this is an even or an odd number of half wave lengths there
will be a corresponding increase or diminution of intensity. For
normal incidence 8 = e(p, — us), and the bright and dark rings will
correspond to
26(91-92)=m\,
according as n is even or odd.
In the case of the coloured rings of thin plates seen by transmitted
light the bright and dark rings correspond to thicknesses
2e' = m\}\,
according as n is even or odd.
Hence the thicknesses e and e', at which rings of the same order
(n) occur in the two experiments, are in the ratio
A
6
a - #(91-92).
* Wood, Phil. Mag., ser. 6, vol. 7, p. 379, April 1904.
* Verdet, Optique physique, tome i. p. 155.
240 - THICK PLATES CHAP. W III
Eacamples
1. If the constituents of the mixed plate be water and air, we have approximately
p1=# and pig- 1, therefore
e'
;
But the squares of the diameters of the rings are proportional to the thicknesses,
consequently the diameters of corresponding rings of the two systems are in the
ratio 1 : V6. *
2. The expression pl cos ri – pº cos re increases with the angle of incidence, and
the rings consequently contract in diameter. For
dö– e(us sin redra-pa sin ridri)=e sin (drº – dri),
but pi sin rì = p, sin r2, therefore dra/drl = p, cos riſug cos rº, therefore dra is greater
than dri, therefore d6 is positive and 6 increases with i.
Newton's Observations on the Diffusion Rings of Thick Plates
Opticks, book ii. part iv.: “There is no glass or speculum how wellsoever polished,
but, besides the light which it refracts or reflects regularly, scatters every way
irregularly a faint light, by means of which the polish'd surface, when illuminated
in a dark room by a beam of the sun's light, may be easily seen in all positions of the
eye. There are certain phenomena of this scattered light which, when I first observed
them, seem’d very strange and surprising to me. My observations were as follows.
“Obs. 1.-The sun shining into my darken'd chamber through a hole one-third
of an inch wide, I let the intromitted beam of light fall perpendicularly upon a
glass speculum ground concave on one side and convex on the other, to a sphere of
five feet and eleven inches radius, and quick-silvered over on the convex side. And
holding a white opake chart, or a quire of paper at the centre of the spheres to which
the speculum was ground—that is, at a distance of five feet and eleven inches from
the speculum—in such a manner, that the beam of light might pass through a little
hole in the middle of the chart to the speculum, and thence be reflected back to the
same hole : I observed upon the chart four or five concentric irises or rings of colours,
like rainbows, encompassing the hole much after the manner that those which (in
the fourth and following observations of the first part of this third book) appear'd
between the object-glasses, encompassed the black spot, but yet larger and fainter
than those. . . . When the sun shone very clear there appeared faint lineaments of a
sixth and seventh. If the distance of the chart from the speculum was much greater
or much less than that of six feet, the rings became dilute and vanished. And if
the distance of the speculum from the window was much greater than that of six
feet, the reflected beam of light would be so broad at the distance of six feet from
the speculum where the rings appeared, as to obscure one or two of the innermost
rings. And therefore I usually placed the speculum at about six feet from the
window ; so that its focus might fall in with the centre of its concavity at the rings
upon the chart.”
‘‘Obs. 3.-Measuring the diameters of these rings as accurately as I could, I
found . . . the squares of the diameters of the (bright) rings in the progression, 0,
1, 2, 3, 4, etc. I measured also the diameters of the dark circles between these
luminous ones, and found their squares in the progression of the numbers #, 13, 2%,
3}, etc.” (that is 1, 3, 5, 7, etc.).
“Obs. 5.— . . If the speculum was illuminated with red (light), the rings
were totally red with dark intervals, if with blue they were totally blue, and so
ART. 125 NEWTON's observations 241
of the other colours. . . . In the red they were largest, and the indigo and violet
least, and in the intermediate colours, yellow, green, and blue, they were of several
intermediate bignesses.”
“Obs. 7.—By analogy . . . it seemed to me that these colours were produced
by this thick plate of glass much after the manner that those were produced by
very thin plates. For upon trial I found that if the quicksilver were rubbed off
the backside of the speculum, the glass alone would cause the same rings of colours,
but much more faint than before ; and therefore the phenomenon depends not upon
the quicksilver, unless so far as the quicksilver by increasing the reflexion of the
backside of the glass increases the light of the rings of colour. I found also that a
speculum of metal without glass . . . produced none of these rings, and thence I
understood that these rings arise not from one specular surface alone, but depend
upon the two surfaces of the plate of glass whereof the speculum was made, and upon
the thickness of the glass between them.”
Experimenting with specula of different thicknesses, Newton found that the
squares of the radii of the rings varied inversely as the thickness of the glass (see
Observation 9).
“Obs. 12. —When the colours of the prism were cast successively on the specu-
lum, that ring which in the two last observations was white, was of the same big-
mess in all the colours, but the rings without it were greater in the green than in the
blue, and still greater in the yellow and greatest in the red. And, on the contrary,
the rings within that white circle were less in the green than in the blue and less in
the yellow and least in the red.”
CHAPTER IX
DIFFRACTION
SECTION I.—THE ELEMENTARY THEORY
126. Diffraction—Introductory.—By far the greatest difficulty
encountered at the outset of the wave theory was the explanation of
the rectilinear propagation of light. This difficulty confronted the
early founders of the theory, and those who feared to encounter it
became partisans of the emission theory from which the principle
flowed as a natural consequence. A closer examination of the facts,
however, shows that light suffers some deviation from the rectilinear
course in passing by the edge of an opaque obstacle; that it does bend
round corners as required by the wave theory, but, as the wave length
is excessively small, the intensity falls off rapidly within the geometri-
cal shadow, and the amount of bending observable is so slight that
careful examination is required to detect it. When light passes
through a very small aperture, the dimensions of which are comparable
with the wave length, it is not propagated through the aperture as a
definite ray or pencil, but diverges in all directions just as sound does
when passing through an aperture of a few feet in diameter.
On the other hand, well-marked sound shadows are formed by
huge obstacles, such as mountains or large buildings, and the existence
of these may be often noticed by the most casual observers.
The phenomena which occur when light passes through a very
narrow aperture or close to the edge of an opaque obstacle, and which
arise from the light deviating from the rectilinear path, are classified
and studied under the title of Diffraction. These appearances are
depicted at the boundaries of the geometrical shadow, and while all
other theories have failed to account for them, the wave theory
explains and even predicts the phenomena truly.
The first observations on diffraction were made by Grimaldi,” about
Bending.
* Physico-mathesis de lumine, coloribus et iride, Bononiae, 1665.
- ... . . 242
ART. 126 INTRODUCTION 243
the middle of the seventeenth century. Having admitted a cone of Grimaldi.
light into a darkened chamber through a very small aperture, he
found that when a small opaque obstacle was placed in the cone
its shadow on a screen was much larger than its geometric projection,
so that the light suffered some deviation from the rectilinear course
in passing the edge of the obstacle. On observing the shadows
attentively he found that they were bordered by three iris-coloured
fringes, running parallel to the edge of the shadow, and decreasing
in width and intensity as their distance from it increased. Similar
fringes may also be observed, under favourable conditions, within the
shadows of narrow obstacles such as a fine wire or hair.
The phenomena of diffraction were subsequently examined by
Hooke 1 and Newton.” The experiment of Grimaldi was varied Newton.
by Newton, who transmitted light through a very narrow aperture
between two knife edges and observed the image it cast upon a
screen. The image was bordered by three iris-coloured bands, in
which the colours succeeded each other as in the rings of thin
plates—violet nearest the shadow and red farthest removed from
it. He observed the same appearances in the exterior of the
shadow of many obstacles, but he does not mention the brilliant
fringes which occur in the interior of the shadows of narrow obstacles,
although Grimaldi had observed the crested fringes at the angles of
shadows.
The first application of the wave theory to the explanation of
diffraction phenomena was made by Dr. Young,” who attributed the Young.
fringes to the interference of the direct light which passes very close
to the edge with the light reflected by the edge at grazing incidence.
That the effects are not produced by the interference of two pencils of
light, such as Young imagined, is proved by the fact that they do not
depend on the degree of polish or sharpness of the edge. Fresnel
observed that whether light passed over the back or edge of a razor
the fringes produced were the same, and this he confirmed by exact
experiments.” If the fringes depended in any way on the light
reflected at the edge of the obstacle, they should vary in intensity or
position when the degree of polish or the material of the edge is
altered.
To ascertain whether the form of the edge had any effect on
* Hooke, Posthumous Works, p. 190, London, 1705.
* Newton, Opticks, book iii. . . . .
* Young, “On the Theory of Light and Colours,” Phil. Trans. p. 21, 1802;
Lectures on Natural Philosophy, pp. 342, 365, 1807.
* Fresnel, CEuvres complètes, tom. i. pp. 148, 280.
244 ELEMENTARY THEORY OF DIFFRACTION CHAP. l X
Fresnel.
the fringes, Fresnel took two plates of steel, the edge of each
being rounded off in one half of its length, but sharp in the
remaining half. He placed the rounded portion of each opposite
the sharp part of the other. If the position of the fringes
depended on the sharpness of the edge the effect would here
be doubled, and the fringes would appear broken in the midst.
On the contrary, they were found perfectly straight throughout
their entire length. Young's explanation is therefore incorrect, and
it is to Fresnel that we owe the true solution of the problem, and the
deduction of general expressions for the effect of a wave at any point.
According to Fresnel the phenomena of diffraction are to be
attributed to the mutual interference of the secondary wavelets which
diverge from the wave front. Each element of the primary wave,
when it reaches the obstacle, is considered as the centre of a diverging
wavelet, and the resultant of all the secondary waves he expressed by
means of two integrals taken within limits determined by the parti-
cular nature of the problem under consideration. The problem of
diffraction was thus solved by the principle of Huygens combined with
the principle of interference.” -
Diffraction phenomena are, therefore, due to the mutual interfer-
ence of the disturbances propagated from the various elements of a
single wave, just as the interference phenomena described in the fore-
going chapters are due to the mutual interference of two trains of
waves. The bright focus of a lens exists because the disturbances
propagated there by the wave passing through the lens agree in phase
and produce an intense effect. At other points destructive interference
of the wavelets occurs, and there is no illumination.
In the hands of Fresnel 4 the theory was developed to such
perfection that little room was left for addition, and by the exact
agreement of the results of careful observation with the anticipations
of analysis, the evidence furnished in its favour is so overwhelming that,
to those who impartially examine it, little doubt is left as to its truth.
In this section we shall confine ourselves to a general explanation
of the phenomena by elementary methods, so that we may become
acquainted with the general facts of some important cases before
entering upon the more intricate investigations. It should be observed
* Marian (Mém. de l'Académie des Sciences, p. 53, 1738) and Dutour (Mém. de
l'Académie des Sciences, tome v. p. 365, 1768) conceived that the fringes depended on
the refraction of the light by a thin layer of condensed air on the edge of the obstacle,
but that this is not the case is proved by the fact that diffraction takes place in
vacuum exactly as in air. º
* [It appears from a note of Prof. Preston's that his opinion of Fresnel's work on
diffraction was considerably modified after the publication of the second edition.]
ART. 127 STRAIGHT EDGE 245
that the fringes which are known as Fresnel's are obtained without the
use of a lens. They are formed at any distance from the source, and
are of the nature of shadows and not images. Needless to say, when
they are formed they can be examined by means of an eye-piece with
the object of seeing them on a larger scale.
127. A Straight Edge.—The first case of diffraction which we
shall consider is that presented when light, diverging from a luminous
point (such as the image of the sun produced by a lens of short focal
length), passes by the straight edge of an opaque screen. Let O
(Fig. 107) be a luminous point emitting spherical waves, AB an
opaque obstacle perpendicular to the plane
of the paper, then if the light be propagated
accurately in right lines from Othere should
be uniform illumination above the line OAM,
and complete darkness below it on a screen
PQ placed to receive the light after it passes
over the obstacle AB. It is observed, how-
ever, that the illumination does not become
zero immediately below M, but that it fades
away continuously and rapidly, and there is complete darkness at a
small distance below M. Immediately above M, on the other hand,
the illumination is not uniform, but passes through a series of maxima
and minima, giving rise to a series of brilliant fringes parallel to the
edge of the obstacle AB. These fringes become less distinctly marked
the further we recede from M, the boundary of the geometrical shadow,
till at length they are wholly obliterated and merge into uniform
illumination at a short distance from M.
The shadow is thus not distinctly marked by the line OAM, as
the geometrical theory of optics would indicate, but the light fades
away gradually on one side, and passes through many alternate suc-
cessions of brightness and darkness, forming fringes, on the other.
The appearance of these fringes is independent of the distances of the
two screens from the luminous point, the scale merely varying accord-
ing to circumstances, and from their constant character they may be
readily recognised in experiments in which they occur associated with
other fringes.
Let us now consider the illumination at any point P on the screen
outside the geometrical shadow. We have already found (Chap. III.)
that the effect of the wave SA at P is confined to a limited number
of half-period elements around the pole R. If therefore P is so far
removed from M that none of the effective elements of the wave at P
are intercepted by the screen AB, then the illumination at P will be
Fig. 107.

246 ELEMENTARY THEORY OF DIFFRACTION CHAP. IX
affected in no way by the obstacle AB. But if P be so near M that
the arc RA contains only a portion of the effective elements of the
lower half of the wave, the other part being intercepted by the
obstacle, we may roughly consider the illumination at P as consisting
of two portions, one the entire half wave RS and the other the part
Fringes RA consisting of a few half-period elements. Now if RA contains an
outside, even number of these elements they will mutually interfere in pairs
and should have little effect at P. Whereas if RA contains an odd
number the effect should be much greater at P. We are led to infer
then that the illumination at P is a maximum or a minimum according
as the arc RA contains an odd or an even number of half-period
elements. That is, if
AP–RP=(2n + 1% (maximum brightness),
and if
AP - RP = 2n} (minimum brightness),
the difference of the distances AP and RP remaining constant, the
point P will move along a hyperbola having A and O for foci, for
OP – AP=OR – (AP–RP),
and OR is constant, therefore the difference of the distances of P
from the fixed points O and A is constant. P therefore describes a
hyberbola, but its curvature is so small that it almost coincides with
its asymptotes.
Denoting OA by a and AM by bwe can easily calculate the distance
a = PM of any bright or dark band from M ; for " ,
3:2 a;2
op=a+b+giº and AP=b+.
Therefore *
o:2/1 1
AP-RP=}(;-zii),
OI’
* -º-º-º: for a bright or dark band
2 b(a+b).T “2 (for a bright or dark band).
Hence for maximum brightness
= x/ *2n+1),
1 Or thus: the angle PAM is very small and equal to twice the angle ARD,
where D is the point in which a circle, centre P, radius PR, cuts PA, therefore
2: ...AD 6 6 ſc b(a + b)
+ = 2-3 = 2 -——-s: = 2----—-r 02–2 \*— 14 .. 6.
b *AR a . A.AOR *a*ary Or º – 2 C!, 6
ART. 128 NARROW WIRE 247
and for minimum brightness
- b(a + /
º'- Vº . 2m).
When the screen PM is moved nearer to or farther from the
obstacle AB, the distance a remains constant while b varies; the corre-
sponding values of a are the ordinates of the hyperbola mentioned
above.
To determine the illumination at any point Pinside the geometrical
shadow, we must observe that the part of the wave which propagates
light to P is only a fraction of half a wave. The point R (Fig. 108)
is now below the edge of the obstacle, so
that some of the powerful elements are
intercepted by the screen. The light from
the point A is, however (of the effective
part of the wave), that which requires least
time to reach P. If therefore we divide
the part AS of the wave into half-period
elements with respect to P beginning at A,
the first element AM, will be the most
powerful, and the others will be smaller and become rapidly
equal to one another, so as to destroy each other's effect at P.
The resultant effect at P is therefore confined to a few half-
periods near the edge A of the obstacle, and these will give a
resultant illumination at P. However, as P sinks farther into the
shadow, the obliquity of the line PA to the wave front will increase,
and the consecutive half-period elements AM1, M2, Ms, etc., with respect
to P will gradually become smaller and more nearly equal in effect, till
finally, when P is a small distance below M, the whole effective portion
of the wave is cut off and the resultant at P is zero.
Consequently we conclude that the light falls off continuously but No fringes
rapidly within the geometrical shadow, and alternations of brightness *
and darkness do not occur.
It is not difficult to see that diffraction fringes can be exhibited
only when the angular diameter of the source of light is small. For
if the luminous origin subtends any considerable angle at the eye, each
point of it will give rise to a corresponding set of fringes, and the
multitude of sets will be so superposed and intermixed as to obliterate
all visible effect.
In practice a strip of light from a narrow slit is used, and the fringes
are viewed through an eye-piece mounted on an optical bench.
128. Narrow Wire.-Let us now consider the case of a very
narrow opaque obstacle, such as a hair or a fine wire. Let the opaque
Fig. 108.

248 ELEMENTARY THEORY OF DIFFRACTION CHAP. IX
screen AB of the preceding article be limited on the lower side, so
that it becomes a narrow strip intercepting the light from O. The
shadow on the screen MN (Fig. 109) will be bounded externally on
each side by a system of fringes similar to those just described, and
accordingly attributable to the same cause. The system on either side
is produced by the light which passed that side of the obstacle acting
nearly independently of that which passed
the other side. The upper system is due
mainly to the diffraction of the light from
O over the upper side A of the obstacle,
and the lower system by the diffraction of
the light at the lower side B.
In addition to these two systems of
fringes, however, there is another set of
brilliant bands situated inside the geo-
metrical shadow (MN) of the obstacle (if it be sufficiently narrow).
This system consists of finer lines than the others, and, unlike them,
of equal width throughout. It remains therefore to account for this
internal system by the theory."
In the preceding article we have seen that the effect of the portion
AS of the wave at any point P inside the geometrical shadow is due
entirely to a few half-period elements at A, the others mutually de-
stroying each other. The portion AS of the wave may therefore be
replaced by a small luminous source near A, so far as its effect in
illuminating P is concerned. Similarly the lower portion BT of the
wave, in illuminating P, is equivalent to a small luminous source near
B. Thus any point P within the shadow is illuminated by both sources
at the same time; these sources will therefore, like two small near
apertures, produce the phenomena of interference inside the shadow,
and the fringes which occur there are accordingly accounted for.
If the obstacle AB is not very narrow, then there will be no internal
fringes, but a gradual fading away of the light at each side of its
geometrical shadow and the usual system of external diffraction fringes
on each side. When the obstacle is narrow, however, the illumination
inside M overlaps that inside N, and interferes with it. Or we might"put
Fig. 109.
* Diffraction fringes may be exhibited on a minute scale by candle light, with no
other apparatus than a small lens having a fine wire stretched across in contact with
its surface. Holding the other surface next the eye, if we look through the lens at
the flame of a candle at some distance, or, what is better still, at its light admitted
through a narrow slit, the wire being parallel to the slit, the dark image of the wire
will be seen edged by the external fringes, and the shadow marked by the internal
fringes in a remarkably beautiful and distinct manner (see B. Powell, Phil. Mag.
and Ann., January 1832).

ART. 129 NARROW RECTANGULAR A PERTURE 249
it thus: with the straight edge any point inside the shadow is illuminated
by a small source at A. If the obstacle be narrow this point is also
illuminated by a small source at B, and if the distance AB between
these sources is small enough, they will interfere and produce fringes
in the interior of the geometrical shadow. It is clear that these fringes
are given by the equations
AP – BP=m, } and *=ºn}
where n is even for the bright bands and odd for the dark ones.
That the internal fringes are due to the interference of the two
portions of light which pass over the edges of the narrow obstacle was
proved conclusively by Dr. Young. He showed that when the light
from one side was intercepted by an opaque screen either before or
after it reached the obstacle, the whole system of internal fringes dis-
appeared, and the ordinary external diffraction fringes alone remained
on that side over which the light was allowed to pass. It is clear
therefore that the internal fringes are due to the joint working of
the light which passes both sides of the obstacle, whilst the external
fringes on the upper and lower sides of the shadow are due to the
practically independent action of the portion of the light which passes
on these sides respectively.
129. Narrow Rectangular Aperture.—Let us now turn to the
quasi-complementary case—that in which the light from a source O is
admitted to a screen through a very narrow slit or rectangular aper-
ture. First consider the illumination at any point P (Fig. 110) of the
screen outside the boundary of the geo-
metrical image of the aperture. As before,
we divide AB into a series of half-period
elements with respect to P, beginning at A,
as the light reaches P from this point first.
Then the point P will be the centre of a
bright or dark band according as the differ-
ence BP – AP is an odd or an even number
of half wave lengths. If the difference is equal to an even number of
half waves there will be an even number of half-period elements in
AB, which will mutually interfere at P, and the effect will be less than
if there is an odd number of half-period elements in AB. There will
thus be two systems of fringes, one on each side of the geometric image
of the aperture, the bright and dark bands of which correspond to odd
and even values of n respectively. This is just the reverse of what
takes place in the fringes formed in the interior of the shadow of an
Fig. 110.

250 ELEMENTARY THEORY OF DIFFRACTION CHAP. IX
opaque obstacle. The position of these fringes is given as usual by
the equations
X a \
BP- AP="3 and * . "2
where n is even for the dark and odd for the bright bands.
Now if the screen is so remote from the aperture that AM – BM is
less than half a wave, then the first band will lie outside the edge of
the image, and the systems of fringes already mentioned will represent
the complete phenomena. But if the screen is so near the aperture
that the difference of the distances of M (or N) from A and B is a
number of half waves, fringes are visible within the projection of the
aperture also. The illumination at any point Q of this image is due
to the two portions into which OQ divides the wave AB. These por-
tions are sensibly different in magnitude as well as obliquity, and their
joint effect at Q requires a more complete investigation (Art. 154).
130. Circular Aperture.-Among the most striking of the pheno-
mena of diffraction are those produced when light, diverging from
a luminous origin, passes through a small circular aperture such as a
pin-hole in a sheet of lead. When the aperture is viewed through a
lens it appears as a brilliant spot surrounded by a series of vivid rings,
and as the distance between the aperture and the eye is altered these
rings vary in the most beautiful manner. The central white spot con-
tracts to a point and vanishes as the eye approaches the aperture while
the rings close in upon it in succession, and thé centre passes in
succession through a series of most beautiful hues similar to those
presented in the colours of thin plates.
The points of maximum and minimum intensity on the central
line are easily determined. Thus let O be the luminous origin (Fig.
111), AB a section of the aperture,
and OMP a line through its centre M.
The illumination at P is found by divid-
ing the wave into half-period elements
as it diverges through AB. Thus let
M1, M2, Mg, etc., be a series of points
on the wave front marking the half-
period elements. Then, as shown in
Art. 53, the consecutive zones approxi-
mately destroy each other, and if the
aperture transmits an even number of them, the illumination at P will
be very feeble whereas if it transmits an odd number the illumination
will be largely increased. Hence as P travels along the axis of the
aperture the intensity of the illumination at it passes through a
Fig. 111.

ART. 130 CIRCULAR APERTURE 251
succession of maxima and minima. The distance of any point of
maximum or minimum intensity from the aperture is easily calculated.
For if we denote OM and PM by a and b respectively it follows at
Once from the expression of Art. 53 that the area of each half-period
element is very approximately equal to
Tab),
a + b
since the radius of the aperture is small compared with either a or b.
Denoting this radius by r, the area of the aperture will be Tr”, and if
this contains n half-period elements we have approximately
o TabnX
Tr?” = - #
a + b
The positions of the points of maximum and minimum intensity
along the axis are consequently given approximately by the equation 4
& __a^*_
T wax – rº
where r is the radius of the aperture. The dark points correspond to
the even values of n, and the brightest points to odd values.
Since the distance of any bright point from M depends on A, it
follows that when white light is used points of maximum brightness Dispersion
for the different colours will be situated at different distances from M, **
the red being nearest M and the violet farthest away; there will not,
therefore, be any points of complete darkness, but the centre of the
image will pass through a succession of richly coloured hues, following
each other nearly in the order of Newton's scale.
The dark points correspond to the even values of n, but a closer
calculation shows that in this case, as in the case of a rectangular
aperture, the points of maximum brightness are not exactly half-way
* This result may be easily deduced directly by expressing the path retardation
of the ray PAO, which passes the edge of the aperture, relatively to the ray PMO
which passes through the centre. Thus since r is small we have
7.2
7.2
, and PA = b + + .
OA=a+; 2b
Consequently the retardation is
7.2/1 1
ô= (OA --PA) - (a+b)=(+}) tº
That is,
2, 2abó
7** = —-
a + b
where 6=$nX when the aperture contains a whole number of half-period elements.
This expression also gives Tab)/(a+b) as the approximate area of each half-period
element.
252 ELEMENTARY THEORY OF DIFFRACTION CHAP. IX
between the points of darkness. Both these questions will therefore
be resumed in the next chapter, and be more fully dealt with.
131. Zone Plates.—We have seen that the consecutive annuli
into which the circular aperture AB is divided approximately destroy
each other in pairs at P. If then the alternate annuli—e.g. the 2nd,
4th, 6th, etc.—be covered with some opaque substance, the others
would be left free to have their full effect, and we should expect P to
be brightly illuminated. The theory here is in complete accordance
with the results of observation. Such plates, with alternately opaque
and transparent annuli, may be obtained by photography, and it is
found that, although there is an important difference, they resemble
a lens in bringing light from any origin O to the same point P as focus.
The difference is that the light which passes through the second trans-
parent annulus arrives at P a complete period later than the light
from the first. Similarly the path of the light from the third is a
wave length greater than the path of the light from the second, and
so on. Whereas in the case of a lens the characteristic is that all the
light which reaches the focus from O arrives there in the same time ;
the phases of all the waves which reach the focus are the same
because the times occupied by them in travelling there from O are
the same for all. A zone plate has therefore the property of a
condensing lens.
To construct a zone plate it is only necessary to describe on a
slip of white paper a series of concentric circles of radii proportional to
the square roots of the natural numbers, i.e. 1, w/2, x/3, V4, etc.
The areas of these circles are directly as the natural numbers, and the
area included between any pair of consecutive circles is constant,
hence if the alternate annuli be blackened over, the others remaining
white, we will have a sketch of a zone plate, but on much too large a
scale. A miniature photograph of it may now be made on a thin plate
of glass, and it is found that when this plate is interposed in the path
of a beam of light it produces the effect described. The light is
brought to a focus at a point P such that the rings on the plate are
half-period elements with respect to it. The focus of the red light is
of course, as before, nearer the plate than the focus of the violet light.
The reverse holds in the case of a lens, for the violet, being most
refrangible, comes to a focus nearest the lens.
If it be possible, instead of blackening out every alternate zone,
to alter the phase due to these zones by half a period, a fourfold
intensity at P will be obtained compared with that due to a zone
plate. This result has actually been brought about by R. W. Wood,
who makes what he calls a phase-reversal plate. A drawing is
ART. 132 OPAQUE CIRCULAR DISC 253
prepared as before with alternate zones blacked out. This is photo-
graphed on a sensitive plate, consisting of a gelatin film impregnated
with a solution of bichromate of potash. The parts on which light
falls are made less soluble by the light. If the plate be now
“developed” by soaking in slightly warmed water, the parts which
have not been illuminated dissolve more quickly than the alternate
zones. They are fixed by being simply allowed to dry. If a number
of such plates be developed for different times the expectation is that
one or more will approximately give the desired result. This follows
because any change of phase so brought about less than half a wave
length will be better than none at all.
132. Opaque Circular Dise.—In applying the theory to the case
of diffraction by an opaque circular disc, Poisson was led to the
startling conclusion that the illumination at the centre of the shadow
should be the same as when the disc is removed, and Arago showed
that this was verified by experiment." Without entering into a com-
plete investigation we can see that the illumination along the axis of
the disc should be nearly uniform and approximately the same as
when the disc is removed except close to the disc. For, take any
point on the axis of the disc, and divide the wave into half-period
elements with respect to it, beginning at the edge of the disc. The
first half-period element, which immediately surrounds the edge of
the disc, is the most powerful, and plays the part Ordinarily taken
by the centre zone of the wave. As in Chap. III., it follows that the
resultant effect is approximately equal to half that of the first existing
zone, and when the obliquity is very small this will be very nearly
the same as #m (Art. 52), which represents the effect of the whole
wave. The illumination, however, must fall off gradually as the
point under consideration approaches the disc, for when the point
is near the disc the obliquity of the first zone passing the edge is
greater than when the point is farther away. In other words, the
effective portion of the wave is to some extent cut off when the
point is near the disc.
This result is easily understood by remembering the bending, or
* Verdet asserts (Leçons d'optique physique, t. 1, p. 250) that Delisle was the
first to observe the existence of a bright spot at the centre of the shadow of an
opaque screen, of circular form and of very small dimensions. This assertion is
repeated in several text-books. The reference given by Verdet is to a paper by
Delisle (Mém. Acad, des Sciences, 1715, p. 166). This paper, however, deals entirely
with the easternal fringes, the innermost of which is declared to be the brightest of
the half-dozen that were seen. There is no reference to internal fringes or to a
bright spot. Nor is there any such reference in Delisle's Memoirs published in
1738 at St. Petersburg, where he had been appointed Professor of Astronomy. *
Uniform
intensity
along axis.
254 ELEMENTARY THEORY OF DIFFRACTION CHAP. IX
diffraction, of the light which takes place into the geometrical shadow.
When the obstacle is a small circular disc this diffracted light overlaps
from all sides at each point on the axis. At any one of these points
all the diffracted light is in the same phase, and there is consequently
no destructive interference at any point on the axis. At a point not
on the axis the components of the diffracted light differ in phase,
and there is interference, so that, when white light is used, what
is observed is a system of rings surrounding a white centre, the in-
tensity at the centre being practically the same as if the disc were
removed.
The experiment is not very difficult to perform. It is necessary
that the disc should have a very smooth edge; otherwise each
inequality gives rise to extra fringes
which mix and cross so as to introduce
great confusion. It can be suspended
by fine fibres or mounted on an optic-
ally finished glass plate or on a selected
sheet of mica. The size that can be
used depends upon the brightness of
the source, which is usually a strongly
illuminated pin-hole–quarter of an
inch diameter works very well. The
source does not need to be precisely
on the axis; and in consequence the
bright spot obtained is an image of the source when an extended
one is used. The accompanying figure (Fig. 111a) shows the effect
for a triangular source."
133. Babinet's Principle.—When light is transmitted through a
very small aperture we have seen that there will be illumination
at points considerably outside its geometrical image. But when the
aperture is of sensible magnitude, an image of it is depicted on the
screen, and at the borders of this image the illumination falls off
gradually. Now at any point outside the image, or at any point of
the pattern where the illumination is zero, the effect produced by any
part or parts of the aperture must be exactly equal and opposite to
the effect of the remainder of the aperture. Thus if S be the area of
any portion or any number of portions, isolated or continuous, of the
* Porter, Phil. Mag., April 1914. We thus have the possibility of obtaining
sharp images of small objects, using an opaque disc instead of a lens. The image
consists of numerous bright spots partly overlapping their neighbours. The side of
this triangle is more than forty times the diameter of the true bright spot in length.
When a pin-hole is employed as source the spot obtained has always a very abrupt
edge, whereas for the true spot the boundary should die away gradually.
Fig. 111a.


ARt. 133 BABINET’S PRINCIPLE 255
aperture, S, the area of the remainder, and S the whole area, we have
S=Si-H S2,
and if the vibration produced by S. at P, a point outside the image, be
A = a sin b, then the vibration produced by S, will be y, = – a sin (b,
for the resultant is zero. Hence it follows that if any part or parts S,
of the aperture be supposed opaque, the remainder S, being left trans-
parent, so that diffraction may occur when light is transmitted through
the transparent parts, the vibration and illumination at P will be
determined by the equations -
yi = a sin p, and Ii = a”,
while if the foregoing opaque portion S, be supposed transparent, and
the transparent portions S. opaque, the illumination at P when the
light is transmitted through S, will be determined by
Aſ2= – a sin p, and I2 = a”,
that is, I = I2 = a”, or the illumination is unaltered when we suppose
the transparent parts of the aperture to become opaque and the
opaque parts transparent.
This principle is due to Babinet," and applies to any point at
which the illumination due to the whole aperture is zero.
If the point P be illuminated, then the whole aperture S transmits
a vibration represented by the equation
gy=A sin (wt + C),
while any selected portion S, of it produces the vibration
vi = ah sin (wt + qi),
and the remainder of the aperture produces
!/2=ao sin (wt + q2).
The vibrations due to the portions S, and Sº, in general, will differ
in phase and amplitude, but they will be connected with the vibration
due to the complete aperture S by the equation (Art. 43)
A*=&º-Faº-H2a1a2 cos (an – ag),
or denoting the corresponding illuminations by I, II, and I, we have
I=I, +I, +28/II, cos (al-qa).
If al – as = }T, then I = I, +Ig. It should be remarked therefore that
we have the equation I, +I, = I only in the special case when the vibra-
tions from S, and S, differ in phase by a quarter period. Attention is
* Babinet, Comptes rendus, tome iv. p. 638, 1837.
256 ELEMENTARY THEORY OF DIFFRACTION CHAP. IX
directed to this point, for it is not unusual to find it assumed that
I = I, +I, universally, or that the illumination in one case is exactly
complementary to that in the other."
When we know the vibration y produced by the whole aperture,
and the vibration y, produced by any part, we can calculate by the
above equations the vibration ye, and illumination I, produced by the
remainder. Thus if the aperture S be supposed very large, so that
practically the whole wave may reach any point P on the screen, then
if y be the vibration excited at P by the complete wave, y, that when
any area S, of the wave is stopped by an opaque obstacle, y, that of
the remainder, viz. the vibration due to that part of the wave which
passes the obstacle, then we have -
v=v, Fya.
The vibration y will be practically uniform all over the screen, so that
we have
$/1+ ya- Constant,
but I, +I, is not constant, for we have seen that I, +I, is not equal to
I except in a very special case. If y, be the vibration at any point
when light passes through a narrow rectangular aperture, and y, the
vibration at the same point when the aperture is replaced by a wire of
the same width, than y1 + y, = y, which is constant over the screen, but
we cannot say that the illuminations at any point are complementary
in the two cases. The same remark applies to the case of a circular
aperture and an opaque disc of the same dimensions.”
Such screens are called complementary Screens, but the term must
not be understood to refer to any complementary relation between the
Quasi-com- illuminations; it merely signifies that the transparent portions of one
* screen are replaced by opaque parts in the other, and vice versá. In
the case of an opaque disc we have seen that the illumination is
approximately uniform along the central line, while in the case of the
circular aperture it passes through a series of maxima and minima. So
again it is the outside of the image of a large obstacle that is bordered
with diffraction fringes, whereas in the case of the corresponding aper-
ture the fringes lie within the geometrical image and the patterns are
not complementary.
* [These results are well illustrated by Fig. 10 on p. 49 if we suppose the circle
C deleted and the angle AOB variable. OP, OQ, OR represent yi, y, and y = y, + y,
respectively, and OA, OB, OC are proportional to VI, VI, and VI. If I=I, +I,
the angle AOB (the phase difference) must be right. If NI, and VI are known as
well as the difference AOC of the corresponding phases, WI, is found as being
proportional to CA or OB.]
* If I corresponds to a circular aperture and I2 to a disc, then (Art. 132) I2 = I,
and II+2&/III, cos 3–0, or cos 6= – 3 MI/I.
ART. 134 YOUNG'S ERIOMETER 257
134. Coronas—Young's Eriometer.—We have already seen that
when light, diverging from a luminous point, passes by the edges of
an opaque obstacle, systems of coloured fringes are formed parallel
to the edges of the shadow. In the case of a circular disc, or a
circular aperture, the fringes form a system of concentric circular
rings. Instead of a single aperture if we have a large number of
irregularly distributed small equal apertures (or of a large number of
equal circular discs) in an opaque screen it may be shown (see Art. 158)
that the diffraction pattern is the same as that produced by a single
aperture multiplied in intensity by the number of apertures.
Instead of opaque discs we might equally have small regular
globules of condensed vapour, as in a cloud, and it is to the diffraction
by these globules that the coloured rings seen around the sun and
moon, when observed through a thin cloud, are due. These rings are
observed close to the surface of the sun and moon in hazy weather,
and must not be confused with the larger rings or halos formed at
some distance away. The halos are often seen in northern latitudes,
and are due to ice crystals floating in the atmosphere, the angular
radius of the first being from 22° to 23°.
Newton observed coloured rings around both the sun and moon,
and he was the first to attribute them to the action of water
globules in the air. He describes them as follows (Opticks, book ii.
part iv.):
“For in June 1692 I saw by reflexion in a vessel of stagnating water three halos
crowns, or rings, or colours about the sun, like three little rainbows, concentrick to
his body. The colours of the first or innermost crown were blue next the sun, red
without, and white in the middle between the blue and red . . . these crowns
enclosed one another immediately, so that their colours proceeded in this continual
order from the sun outward. . . . The like crowns appear sometimes about the
moon, for in the beginning of the year 1664, February 19th, at night, I saw two such
crowns about her. The diameter of the first or innermost was about three degrees,
and that of the second about five degrees and a half. . . . At the same time there
appeared a halo about 22° 35' distant from the centre of the moon. It was elliptical,
and its long diameter was perpendicular to the horizon, verging below farthest from
the moon. I am told that the moon has sometimes three or more concentric crowns
of colours encompassing her next about her body. The more equal the globules of
water or ice are to one another, the more crowns of colours will appear, and the
colours will be the more lively. The halo at the distance of 22% from the moon is
of another sort. By its being oval and remoter from the moon below than above, I
conclude that it was made by refraction in some sort of hail or snow floating in the
air in a horizontal posture, the refracting angle being about 58° or 60°.”
Fraunhofer confirmed Newton's views by showing that these
coloured rings may be produced artificially by looking at a source of
light through a plate of glass covered with fine globules of condensed
vapour or with lycopodium dust. The condition necessary for the
- S
Coronas.
Halos.
258 ELEMENTARY THEORY OF DIFFRACTION CHAP. IX
success of the experiment is that the globules should be of sensibly
uniform size. He also obtained them with a large number of small
metallic discs of equal size placed between two plates of glass, and he
found that the diameters of the rings varied directly as the wave
length, and inversely as the diameters of the discs.
M. Verdet' reproduced an allied phenomenon by covering the
object-glass of a telescope with a copper plate containing a large
number of small circular holes distributed irregularly. On observing
a distant source of light, he saw at the focus of the telescope a system
of rings similar to the coronae. These, however, belong to the
category of Fraunhofer fringes.
These appearances were also observed by Young,” who in a very
ingenious manner contrived to apply them to the measurement of the
diameter of fine fibres, or small particles of any kind.
The apparatus invented by Young consisted of a metal plate per-
forated with a small hole of about ºr inch (0-5 mm.) in diameter.
Around this aperture was a circle of smaller holes nearly half an inch
in radius. The flame of a lamp was placed immediately behind the
aperture, and the plate viewed through the substance under examina-
tion. The central aperture is seen surrounded by a ring, which can
be brought to coincide with the circle of small holes in the plate by
moving the substance backwards or forwards along a graduated scale.
The distance between the substance and aperture is read off on the
scale, and this varies inversely as the diameter of the ring, but theory
shows that the diameters of the rings, produced by different sized
particles, vary inversely as the diameters of the particles (see Art.
163). Consequently it follows that the diameters of the particles are
directly proportional to the distances between the substance and aper-
ture when the ring appears to coincide with the circle of small holes
perforated in the plate. An experiment is therefore made with
particles of a known diameter, and this gives the constant of the
instrument from which the diameters of any other small particles may
be determined. Thus if d be the distance between the plate and
substance containing particles of known radius r, when the ring and
halo appear to coincide, and if 8 and p be the corresponding quantities
for any other substance, then
or p=}.
.
_ 7"
- Tº 3
d
* Verdet, CEuvres, tom. v. p. 314.
* For an account of Young's Eriometer see Art. “Chromatics,” vol. iii. of Supple-
ment to Ency. Brit. 1817; see also Phil, Mag. March 1881, and vol. xx. p. 354,
1885.
ART. 134 NEWTON'S OBSERVATIONS 259
Newton's Observations
Opticks, book iii. part i. fourth edition, 1730.—“Grimaldo has informed us that
if a beam of the sun's light be let into a dark room through a very small hole, the
shadows of things in this light will be larger than they ought to be if the rays went
on by the bodies in strait lines, and that these shadows have three parallel fringes,
bands, or ranks of color'd light adjacent to them. But if the hole be enlarged the
fringes grow broad and run into one another, so that they cannot be distinguished.
These broad shadows have been reckon’d by some to proceed from the ordinary
refraction of the air, but without due examination of the matter. For the circum-
stances of the phenomenon, so far as I have observed them, are as follows.
“Obs. 1.-I made in a piece of lead a small hole with a pin, whose breadth was
the 42nd part of an inch. For 21 of those pins laid together took up the breadth
of half an inch. Through this hole I let into my darkened chamber a beam of the
sun's light, and found that the shadows of hairs, thread, pins, straws, and such like
slender substances placed in this beam of light were considerably broader than they
ought to be if the rays of light passed on by these bodies in right lines . . . a hair
of a man's head, whose breadth was but the 280th part of an inch, being held in
this light, at the distance of about twelve feet from the hole, did cast a shadow which
at the distance of four inches from the hair was the sixtieth part of an inch broad—
that is, above four times broader than the hair, and at a distance of two feet from
the hair was about the 28th part of an inch broad—that is, ten times broader than
the hair, and at the distance of ten feet was the 8th part of an inch broad—that is,
35 times broader.
“Nor is it material whether the hair be encompassed with air or with any other
pellucid substance. For I wetted a polished plate of glass and laid the hair in the
water upon the glass, and then laying another polished plate of glass upon it, so
that the water might fill up the space between the glasses, I held them in the afore-
said beam of light, so that the light might pass through them perpendicularly, and
the shadow of the hair was at the same distances as big as before. . . . Therefore
the great breadth of these shadows proceeds from some other cause than the refrac-
tion of the air.”
He further observed that “The shadows of all bodies (metals, stones, glass,
wood, horn, ice, etc.) in this light were border'd with three parallel fringes or bands
of coloured light. . . . The colours proceeded in this order from the shadow : violet,
indigo, pale blue, green, yellow, red; blue, yellow, red; pale blue, pale yellow
and red.”
“Obs. 8.-I caused the edges of two knives to be ground truly strait, and
pricking their points into a board so that their edges might look towards one
another, and meeting near their points contain a rectilinear angle, I fastened their
handles together with pitch to make this angle invariable. The distance of the
edges of the knives from one another at the distance of four inches from the angular
point, where the edges of the knives met, was the eighth part of an inch ; and there-
fore the angle contained by the edges was about one degree 54'. The knives thus fixed
I placed in a beam of the sun's light, let into my darken'd chamber through a hole
the 42d part of an inch wide, at a distance of 10 or 15 feet from the hole, and let
the light which passed between their edges fall very obliquely upon a smooth white
ruler at a distance of half an inch or an inch from the knives, and there saw the
fringes from the two edges of the knives run along the edges of the shadows of the
knives in lines parallel to those edges without growing sensibly broader, till they
met in angles equal to the angle contained by the edges of the knives, and where
they met and joined they ended without crossing one another. But if the ruler was
held at a much greater distance from the knives, the fringes where they were farther
260 ELEMENTARY THEORY OF DIFFRACTION CHAP. Ix
from their place of their meeting were a little narrower, and became something
broader and broader as they approach'd nearer and nearer to one another, and after
they met they cross'd one another, and then became much broader than before.
“Whence I gather that the distances at which the fringes pass by the knives are
not increased nor alter'd by the approach of the knives, but the angles in which the
rays are there bent are much increased by that approach, and that the knife which
is nearest any ray determines which way the ray shall be bent, and the other knife
increases the bent.”
“Obs. 10,-When the fringes of the shadows of the knives fell perpendicularly
upon a paper at a great distance from the knives, they were in the form of hyper-
Fig. 112.
bolas. . . . Of these hyperbolas one asymptote is the line DE, and their other
asymptotes are parallel to the lines CA and CB."
(In Fig. 112 CA and CB are parallel to the edges of the knives, and DE is the
bisector of the external angle between them.)
Allowing the light from the small hole to pass through a prism and form a
spectrum on the opposite wall, he found that the shadows of objects placed in the
light between the prism and the wall were bordered with fringes of the colour of that
light in which they were held. “And comparing the fringes made in the several
colour'd lights, I found that those made in the red light were largest, those made in
the violet light were least, and those made in the green were of a middle bigness.”

ART. 135 THE DIFFRACTION GRATING 261
SECTION II.-DIFFRACTION GRATINGS
135. Fraunhofer Fringes—The Diffraction Grating.—We now
proceed to the elementary explanation of the appearances presented
when a distant source of light is viewed through a system of very
narrow, equal, and equidistant, rectangular apertures. Such a system
is named a grating or a diffraction grating. Gratings are ordinarily
formed by tracing a number of parallel equidistant lines on a glass
plate with a fine diamond point. These lines act like a system of
Fig. 113.
fine opaque wires, in that the light incident on them is reflected back
in all directions and refused transmission, while it passes freely through
the transparent spaces between the lines. Such gratings frequently
contain as many as 20,000 or even 40,000 lines to the inch, the
ruling being so fine that the striae are invisible except under a powerful
microscope. -
When a luminous origin is looked at through a grating a central
or direct image is seen, and on either side of it there are several
spectral images richly coloured with all the tints of the rainbow.
These spectra increase in breadth and diminish in brilliancy as they
recede from the centre, and they may be seen by merely viewing an
ordinary candle flame or gas jet through the eyelashes, or through a
pocket-handkerchief, or through a piece of ordinary wire gauze. The
peculiarity of these fringes or spectra is that a lens is employed for
their production, and they are formed in the focal plane of this lens.

262 DIFFRACTION GRATINGS CHAP. Ix
The name of Fraunhofer fringes is given to them. They are very
much more important than those of Fresnel, which have already been
described.
To observe the spectra to advantage a telescope should be first
focussed on the luminous origin. The grating being then placed before
its object-glass, the spectra are formed in its focal plane, and are
viewed with all the advantages of amplification and distinctness
through the eye-piece. In the following elementary explanation we
shall therefore suppose the light after passing through the grating
BM (Fig. 113) to fall upon a lens, which we may regard as the
object-glass of a telescope. Owing to their mode of production they
constitute a true image of the source.
Let M.N., M.N., MANs, etc. (Fig. 113), be the apertures—that is,
the transparent portions—of the grating, supposed perpendicular
to the plane of the paper, and let the width of each of the
apertures be a while the width of each ruling is b, and for simplicity
suppose the light to be incident perpendicularly to the grating and
the plane MB to be a wave front.
Now consider the light which falls upon the lens in any direction
OP (Fig. 113) where 0 is the optic centre of the lens. Streams of
light fall upon the lens in this direction from the apertures and are
brought to a focus at P, consequently the disturbance at P will be the
resultant of all the disturbances sent to it by the various streams
from the apertures. Draw M.D. (Fig. 114) perpendicular to the
direction OP. Each stream is propagated from
this line to P in the same time; such a line may
be referred to as a standard plane. But the light
is incident in the same phase at every point of
the grating, since we have supposed the incident
wave front parallel to it, consequently the light
which reaches P from the second aperture will be
retarded on that which reaches it from the first
and interference will occur. Thus the difference
of path between the first element of the first
aperture and the first element of the second is
equal to M.D., and the same difference exists between all the corre-
sponding pairs of elements of these apertures, while the same remark
applies to every other consecutive pair of apertures. If, there-
fore, M.D., is an even number of half-wave lengths, the light
from all the apertures will arrive at P in the same phase, and will
reinforce each other, and the illumination at P will be very great;
but if M.D., is an odd number of half-wave lengths, the light from
Fig. 114.

ART. 135 THE DIFFRACTION GRATING 263
the first aperture will be destroyed by that from the second, the light
from the third by that from the fourth, and so on. Hence in this case
the illumination at P will be zero.
Now M.D., -(a + b) sin 6 if the direction OP makes an angle 6
with the normal to the grating; hence P will be very bright if
(a +b) sin 6=2n}\,
and dark if
(a + b) sin 6 = (2n+1)}\.
Let us now suppose 6 to increase from zero while m takes the consecu-
tive values, 0, 1, 2, 3, etc. The value 0 = 0 corresponds to n = 0, so
that there is no retardation, and the light from all the apertures
arrives in the same phase at M on the central line OM (Fig. 113).
This then is a bright point for all wave lengths, and with ordinary
light will be white. An interval of darkness (see Art. 137) now occurs
as 6 increases from zero to the value 0, given by the equation
sin 61 = \ſ(a+b).
If OP, be drawn in this direction, meeting the focal plane at Pl, then
P1 is a very bright point. Another interval of darkness occurs till 6
reaches the value determined by
sin 62=2\ſ(a+b),
which gives another bright point P, and so on. We have therefore
a succession of bright places P1, P2, Ps, etc., with dark intervals
between them, and like appearances also on the lower side of the
central line OM. The direction from O to the nth bright point is
given by the equation
sin 6, = m\ſ(a+b).
What has been said so far applies to light of a definite wave
length. When white light is used a brilliant rainbow-coloured band White
or spectrum appears at each of the points Pi, P., Ps, etc. For the light.
angle 6 corresponding to any bright point increases with the wave
length, consequently the points of maximum illumination for the
red light are farther removed from the centre than the corresponding
points for the violet light. What were bright points at P1, P2,
etc., with monochromatic light, are now drawn out into exquisitely
coloured spectra, violet at the inner and red at the outer edge.
Several spectra are formed on each side of the central image; the
first pair being separated from the second pair, and from the central
image, by completely dark bands. Overlapping of the spectra will Over-
occur when the deviation of the violet of any order is less than that *
of the preceding red. This will take place between the spectra of
264 DIFFRACTION GRATINGS CHAP. IX
order higher than the second, for since the deviation is proportional to
the wave length, and since the wave length of the red is approxi-
mately twice that of the violet, it follows that the deviation of the red
of the second spectrum will be approximately the same as that of the
violet of the third spectrum, while the red of the third will be more
deviated than the violet of the fourth, and so on. The separation of
the superposed parts of the spectra at any place may be effected by
means of a prism.
Examples
1. Explain how the lens gives only one bright point on the screen when the
opaque parts of the grating are made transparent.
2. Show by Babinet's principle (Art. 133) that when a lens is used the bright-
ness of the lateral spectra remains the same when the opaque and transparent
parts of the grating are interchanged.
136. Light incident obliquely – Minimum Deviation. – If the
incident light be not perpendicular to the plane of the grating,
but falls upon it at an angle i with the normal, the retardation
is given by the equation
6–(a ; b)(sin 9-º-sin i).
For if MD and MD (Fig. 115) be drawn perpendicular to the inci-
dent and transmitted beams respectively, the
retardation will obviously be ND + ND'; but
ND = (a+b) sin i and ND = (a+b) sin 6. There-
fore, etc.
The position of the nth spectrum in this case
| is determined by the equation
(a + b)(sin ºn--sin i)=m\.
The angle of incidence i may be determined
by measuring the angle 2: between the direct
light and that reflected regularly from the face of the grating.
Diffraction spectra, like refraction spectra, exhibit a minimum
deviation. The deviation suffered by the light of the nth spectrum is
given by the equation
But
Fig. 115.
D= i+0m.
(a + b)(sin i +sin ºn)=m\,
hence the deviation of the nth spectrum will be a minimum when
* For a maximum or minimum value of D we have
dID=di + d6a–0,
and from the second equation, when X and n are given, we have
cos idi-H cos Badºn–0.
Therefore cost-cos ºn-that is i-9, for each is less than 90°.

ART. I38 RESOLVING POWER 265
* = &n-that is, when D = 2i, or when the angle of incidence is equal
to the angle of diffraction. We have then
2(a + b) sin # D = m\.
In the position of minimum deviation the definition of the spectrum
is considerably augmented. It has consequently been used by M.
Mascart” in his determination of A. In modern spectroscopy,
however, a photographic method is always used, and the grating is in
a fixed position for the whole spectrum.
187. The Dispersion in the Spectrum.—The nth spectrum of
any monochromatic source is given by sin i + sin 6 = n\ſ(a + b). The
smaller the grating interval, a + b, the larger will be the value of 6.
If we consider a change of A equal to dx we have cos (d0=
70,
a-Fö
tional to cos 0. If 6 is not greater than 6°, cos 6 is a constant within
about 1 part in 200, so that d6 is nearly proportional to dx. The
spectrum is then called a “normal” spectrum. The quantity d6/dx
is the dispersion. So far as the dispersion is “normal” the spectra
formed by different gratings are similar to one another. This is never
the case with prisms; for these d6/dx may be much greater in the
violet than in the red; in some cases it is even negative. This is
known as the irrationality of the spectrum. -
138. Chromatie Resolving Power.—We may show that each of
the spectra for a given monochromatic beam is a very small patch or
narrow line according to whether the source is a point or a line. For
let P be a line of maximum brightness; at it, all the diffracted beams
meet in the same phase (or rather differing by whole multiples of 21,
which comes to the same thing). Now consider a point differing in
position from P by such an amount that successive beams differ in
phase by 2n(n+); the first and last of the N beams differ there-
dA ; hence the corresponding change in 6 is inversely propor-
fore by 2ş. N = 2it. The second set of N/2 beams and the first set
of N/2 beams differ by #2T =T and therefore destroy one another.
This point is therefore one at which there is no illumination. But
since n increases by unity in passing from one spectrum to the next,
and only by 1/N in passing from a maximum to this point of Zero
illumination, it follows that all the light in a single spectrum is con-
centrated in a very narrow region when N is big. Consider now a
Second wave length X + dA such that its nth order spectrum coincides
* Mascart, Ann. de l’École norm. tom. i. et iv.; Comptes Rendus, tom. lvi. p. 138;
tom. lviii. p. 1111.
266 - DIFFRACTION GRATINGS CHAP. IX
with the point of zero illumination. Then
m(A+d)\)= (*#).
Or
iX
dNT
The spectra for the two wave lengths should be distinguishable
from one another when no closer than this. The quantity A/dx=nN
is called the chromatic resolving power. In reality the image will
appear a double one even if the components are some ten or twenty
per cent closer. The above limit is, however, a convenient standard
by which different gratings may be compared.
139. Purity of the Spectrum.—In the above discussion it has been
assumed that the slit source is infinitely narrow. This of course
cannot be the case in practice. The slit subtends an angle Ai (say)
at the collimator lens, and consequently the issuing rays fall at angles
differing at most by Ai upon the grating. The angles of diffraction
that correspond will vary through a range A6, and the general equation
7.N.
(sin i + sin 0)=#; shows, by differentiation, that for a given wave
length the two ranges are connected by cos iAi + cos 6A6= 0. The
image is therefore more complicated than we have supposed. Each
line of the slit gives rise to a diffraction patch of breadth d6 as
calculated in the last section; the centres of these patches are spread
out over a range A6. To allow for this we will suppose that the
duplicity of the image will just be distinguishable when the separation
between their centres is
#A0+d6+}Ad=a+A9
instead of d6. This convention corresponds to the case in which the
inner edges of the two images of the slit satisfy the condition imposed
in Art. 138. -
Now d6. (a +b) cos 0 = \dim where an=#
and cos 0. A6= — cos iAi ;
also Ai-slf where s is the actual width of the slit and f the focal
length of the collimator lens. Whence for resolution
dN__1_ d6+ A6 1 [...}.sº eij,
X T N n " Tº GTT Nº. " f X
Inspection of a figure will show that N(a + b) cos i is the total breadth
of the beam received by the grating. The fraction A/dx is called the
purity of the spectrum. It gives a practical measure of the chromatic
resolving power taking the spectroscope as a whole. Although the
ART. 141 REFLECTION GRATINGS 267
expression obtained above has no pretensions to accuracy, it corresponds
well with experimental determinations except in extreme cases. Any
attempt at obtaining greater precision would be out of place, because
actual gratings differ in many respects from the ideal grating which
has been taken as the basis of our considerations.
It should be observed that if the angle of incidence, i, is increased
so that cos i tends toward zero the purity is increased. This is
easily understood if it be recalled that when i is large the angular
magnification is less than unity. In such a case the geometric image
of the slit has less angular width than the rays issuing from the slit
itself—even a wide slit (e.g. 2 mm. wide) gives spectral lines so sharp
that the sodium orange lines are clearly separated by a single flint-
glass prism. The ideal resolving power is not altered by this rotation
of the grating, provided that the grating remains illuminated through-
out its whole extent (i.e. N = const.).
140. Absent Spectra.-The general conclusion which we have
drawn is that if (a + b) sin 6 or M.D., is any even number of half-wave
lengths—that is, any whole number of wave lengths—then the illumi-
nation at P is a maximum, and a spectrum is formed there. We will
now show that it may happen that M.D., is an even number of half-
wave lengths and yet there is no illumination at P, in fact, the
spectrum is wanting. This will happen when the direction OP is
such that there is an even number of half-period elements in each
aperture. Each aperture will then produce zero effect at P, and there
will be no illumination at that point. Now suppose that a and 3 are
the two smallest whole numbers which measure the ratio of a to b, then
a = ka and B = kb, so that if M.D., - (a + 8).A. we must have ND1 = a \,
since M.D., ; NID1 = a + b : a = a + 8: a. Hence there is an even number
of half-wave periods in the aperture M.N., and therefore in every
aperture, consequently each aperture produces no effect at P, and the
result is darkness at that point. Hence if we say that M.D., a m).
corresponds to the nth spectrum, we may say that the
(a + 3)th, 2(a + 8)th, etc., m (a +8)th
Spectra are wanting.
141. Reflection Gratings.-Spectra similar to the preceding may
also be obtained by reflection, and first-class gratings may be formed by
ruling very fine parallel grooves on a polished metallic surface. The
streams of light regularly reflected from the polished intervals between
the rulings proceed from a virtual image of the source as if they came
through the intervals from behind the surface. If the surface be
plane the case will be analogous to that of the transparent grating
268 DIFFRACTION GRATINGS CHAP. IX
just considered, and the expression for the retardation, when
the light is incident at an angle i and diffracted at an angle 0.
becomes
5–(a+b)(sin i-Esin 0).
Appearances of the same nature and attributable to the same cause
are often observed when a metallic surface has been polished with
a rather coarse powder. The powder leaves minute striae which
affect the light as described above. A simple way of producing
a similar result is by passing the finger over the surface of a
piece of glass moistened with the breath. The exquisite colour of
mother-of-pearl and other striated substances (formed of a vast number
of very thin layers) are natural instances of the same phenomena.
142. Curved Gratings.-Let the surface on which the lines are
ruled be not plane, but have any curved section AB (Fig. 116) per-
pendicular to the lines of the grating. If
light from a source S fall upon it at an
angle i with the normal, and be diffracted
at an angle 6, the retardation will be
PQ(sin i + sin 6).
Consequently for brightness we have as
before, if PQ = (a+b),
m\ = (a+b)(sin i + sin (),
Fig. 110.
the negative sign being taken in the case
of reflection, if S and S. lie on opposite sides of the normal (as in
Fig. 116).
To obtain the image S. of S, virtual or real, let us consider two
rays SPS and SQS incident at angles i and i + di, and diffracted at 0
and 0 + d6 respectively, and let the normals to the curve at P and Q
meet at C. Denote the small angles at S, C, and S by a, B, and y
respectively. Then clearly we have
a + i = 3-i-H di-supplement of angle at M,
3+0=y+0+d6 =supplement of angle at N.
Consequently di = a – 8, and d6 = 8 – y.
But if S is a focus, 8 must be stationary, and this condition gives
sin i-sin 6 =const.,
Ot
cos idi – cos tºdt} = 0.
Hence, substituting for di and d6, we obtain
(a - 3) cos i-(3-y) cos 0–0. (1)

Art. 143 ROWLAND'S CONCAVE GRATINGS 269
Now if we write
PS=p, PC = R, PS'-p', PQ=e,
we have
pa-ecos i, R3–e, p'y–e cost.
Therefore (1) becomes
cos 1 1 cos tº
| | | " - " " - ----- 2
cos ( p |) cos *(i. p' ) 0, (2)
from which we find at once
2
* , , "ºº", , (3)
p(cost-cosi) – R cosº.'
Here we may regard p and i as the polar co-ordinates of S, and pſ
and 6 those of S’; hence if S describes any curve, S will describe
another, the focal curve, defined by the above equation.
Cor. 1.-If p = R cos i, then p = R cos 0–that is, if S describes a Rowland's
circle on R as diameter, S will move on the same circle. Hence if the “”
grating curve be a circle of radius R, a source situated on a circle
described on R as diameter will give spectra situated on the same circle.
This important deduction has been utilised in a masterly manner
by Professor Rowland in the construction of his concave gratings.
Cor. 2–When i = + 6, if p = R cos i (as in Fig. 117) we have
p = p. Hence in the position of minimum deviation, as for prisms, the
source S and its image S are equidistant from the grating. Con-
sequently the diffracted rays returning to S form an image superposed
on the source.
Cor. 3.-If the grating be plane R is infinite, and
p'--p cos”0/cos^i.
143. Rowland's Concave Gratings.-Professor Rowland has suc-
cessfully ruled fine gratings on a concave spherical surface of polished
speculum metal. The lines are the inter-
sections of the surface with a series of
parallel equidistant planes, one of which
(the central one) passes through the centre
of the sphere.
Let PM (Fig. 117) be the surface of
the grating, C its centre, and M the
middle point of the ruled surface. On
CM as diameter describe a circle. Then,
as shown above (Cor. 1), a source S on
this circle will give an image at S on the same circle, defined by the
equation.
Fig. 117.
m\ = (a+b)(sin i-sin 6),
the negative sign occurring here because we have taken S and S on
opposite sides of the normal MC.

270 DIFFRACTION GRATINGS CHAP. IX
Real diffraction spectra will consequently be formed on the circum-
ference of the circle SCS', having their lines parallel to the lines of the
grating. If three arms of equal length be hinged at F, the principal
focus of the spherical surface, one to carry the grating G, another the
slit or luminous origin S, and the third an observing telescope or screen,
by rotating this latter all the spectra may be successively observed.
The length of each arm is half the radius of curvature of the grating.
In the third arm, instead of the screen or photographic camera, a
sensitive radiometer may be substituted, and the heating effects of the
various parts of the spectrum studied, as has been done by Professor
S. P. Langley." -
In order that a large part of the field of view may be in focus at
once Professor Rowland places the eye-piece at C, so that 0 = 0, and
the value of i for the nth spectrum is then given by
(a +b) sin i = n\.
This arrangement is secured mechanically by placing the slit at S
(Fig. 118), the intersection of two arms SG and SC set at right angles.
At the extremities G and C of these
arms the grating G and the camera
or eye-piece C are placed. This
arrangement is specially advan-
tageous for photographing the
spectrum. Rails are placed on
SG and SC for the locomotion of
the grating and camera-box.
SG and SC are heavy wooden
beams of which SG is fixed, while
SC has a slight freedom of rotation
about S, controlled by screws at C.
The rails for the grating-holder and camera-box are of iron and fastened
to these beams by screws which admit of adjustment, so that the rails
may be straightened if the beams warp. GC is a tubular wrought-
iron girder pivoted at its ends directly over the rails, on two iron
carriages. Its length is approximately equal to the radius of the
grating, and has a range of adjustment of about 6 inches. The
carriages have wheels resting on the iron ways, and these enable
the girder to be easily moved from one position to another. The
camera-box and grating are themselves movable along GC, and have
freedom of motion, but can be finally clamped in place.
The slit, which is generally open not more than 001 inch, can be
Fig. 118.-Concave Grating.
| S. P. Langley, Phil. Mag, vol. xxi. p. 394, May 1886.

ART. 144 ROWLAND'S CONCAVE GRATINGS 271
adjusted parallel to the lines of the grating. This adjustment is one
of the last to be made in mounting the grating, and is executed by
turning the slit until the definition is the best possible, a condition
very important in photographing the spectrum. If the slit be out
0°-5 the definition is spoiled.
Stops can be placed at the top and bottom of the slit, thus causing
the grating to be illuminated only by the centre of the solar image;
otherwise the definition may be endangered by the rotation of the
sun. The image of the Sun on the slit should consequently be large.
With the apparatus in the Johns Hopkins University it is 1.2 cm., and
this is reduced one-half by the stops. For solar work the light is
thrown on the slit by means of a condensing lens and a totally reflect-
ing prism." Between the lens and the prism absorbing solutions can
be placed. For ordinary purposes a 10,000 grating is sufficient, but
for photographing in the ultra-violet it is best to have a 20,000 grating
with a ruled space of 5% inches on a 6-inch polished surface. The radius
of curvature is generally 21:5 feet. The photographic plates are 19
inches long, 2 inches wide, and fºr inch thick, which allows them to be
bent to the required radius without breaking.
In the Astrophysical Journal for 1910 A. Eagle describes an
alternative way of mounting a grating, which is easier to set up and
has other advantages which are detailed in his paper. In this mode
of mounting the grating is turned so that the diffracted rays return
practically along the path of the incident beam.
144. Spectrum Photographs—Choice of a Grating.—Mostgratings
give a brighter spectrum at one side than at the other; and so before
placing the grating in the holder it must be examined to see which
side should be used. Every grating has spectra of different brightness
on the two sides; and one should be used which is bright in the par-
ticular spectrum desired for observation. The red of one spectrum may
be bright and its violet faint. Further, the various parts of the grating,
especially if it be concave, may give spectra of varying brightness.
For instance, the second spectrum may be uniformly bright for all parts
of the grating, while one end of the grating may give a bright third
spectrum and the other a faint one. Having selected the grating
which we wish to use, it is mounted in its holder and the colli.
mating eye-piece is put in place. The focus is then carefully adjusted
by altering the length of p till the cross-hairs are exactly at the centre
of curvature of the grating. On moving the bar the whole series of
* In Fig. 118 the light is thrown on the slit by means of a totally reflecting prism.
This of course is necessary only when the aspect is such that the light cannot be
thrown on the slit directly. *
272 DIFFRACTION GRATINGS CHAP. IX
spectra are seen in exact focus. The rail SC on which the carriage
moves is graduated to equal divisions representing wave lengths, since
the wave length is proportional to the distance SC. The instrument
may thus be set to any particular wave length we desire to study, or
the wave length may be obtained by a simple reading. By having a
variety of scales, one for each spectrum, we can immediately see what
lines are superimposed on each other, and identify them when we are
measuring their relative wave lengths. Replacing the eye-piece by a
camera, the spectrum may be photographed with the greatest ease.
“We put in the sensitive plate either wet or dry, and move to the part
we wish to photograph. Having exposed that part we move to another
position and expose once more. We have no thought for the focus,
for that remains perfect, but simply refer to the table giving the proper
exposure for that portion of the spectrum, and so have a perfect plate.
Thus we can photograph the whole spectrum on one plate in a few
minutes from the F line to the extreme violet, in several strips each
20 inches long, and we may photograph to the red rays by prolonged
exposure. Thus the work of days with any other apparatus becomes
the work of hours with this. Furthermore, each plate is to scale, an
inch on any one of the strips representing exactly so much difference of
wave length. The scales of the different orders of spectra are exactly
proportional to the order. Of course the superposition of the spectra
gives the relative wave lengths. To get the superposition, of course,
photography is the best.” "
Between the slit and the camera-box no lens is interposed. Besides
the saving of light and cost, there are no corrections necessary for
spherical aberration, imperfection of lenses, etc. The concave
grating is astigmatic ; i.e. a point of light, as the source, is brought to
focus, not in a point, but in a line. By the astigmatism a small
spark of light at the slit is broadened out into a wide spectrum,
greater accuracy in comparing solar and metallic lines is afforded,
and a spectrum is obtained which is broad enough to stand
enlarging.
The spectrum is normal at C, 6 being small (see Art. 137). Further,
in this case nA = (a+b) sin i. But SC = p sin ice A, thus if an absolute
wave length be marked on SC and the instrument is in perfect adjust-
ment, we can mark on the arm SC a scale of wave lengths for each
spectrum, and the absolute wave length of any one line is known at once.
It is important to notice that this scale on the beam is identical with
the scale on the photographic plate (as follows from the values of SC/A
and d6/dx), and that all the spectra are in focus at C at the same time,
* Professor H. A. Rowland, Phil. Mag. p. 197, September 1883.
ART. 145 DIFFICULTIES OF CONSTRUCTION 273
and stay in focus, however C moves along SC, it being rigidly attached
to G. The concave grating, besides, is the only spectroscope suitable
for the ultra-violet and infra-red. Much longer photographic plates
can be used than with any other instrument, since they can be easily
bent, so as to be throughout in focus.
A 10,000 grating has on the whole better definition than a
20,000 one, and is of course much cheaper. It is only for work with
the camera in the ultra-violet part of the spectrum that it becomes
necessary to use a 20,000 grating. This is due to the fact that the
same dispersion and resolving power are obtained with a lower order
of spectrum, where there are fewer overlapping spectra, with the
20,000 than the 10,000 grating.
A spectroscope is used for two purposes—to measure the lines in
the solar or metallic spectra, or to establish coincidences simply.
For both these purposes the concave grating is far superior to any
other on account of the overlapping spectra. *
Owing to the astigmatism of the grating it is not possible to adopt
the usual method of illuminating part of the slit with the solar image
and part with the spark or arc, and so a different and far better plan
is adopted. A compound photograph of the two spectra is taken.
The solar spectrum is photographed along the middle of the sensitive
plate; the sunlight is then turned off and the metallic spectrum
is allowed to fall upon the upper and lower part of the plate.
Then, finally, the sunlight is turned on again along the middle of the
plate. If there has been any gradual displacement of the camera
during the operation the error is eliminated by this process if the two
times of exposure to the solar spectrum are the same. A record of the
barometer and thermometer readings must be kept, for corrections due
to variations of temperature and pressure may be considerable.
145. Diffleulties of Construction.—The difficulties attending the
construction of a grating are described by Mr. J. S. Ames : * “It takes
months to make a perfect screw for the ruling engine, but a year may
easily be spent in search of a suitable diamond point. . . . Most points
make more than one ‘furrow’ at a time, thus giving a great deal of
diffused light. Moreover, few diamond points rule with equal ease
and accuracy up hill and down. This defect of unequal ruling is
especially noticeable in small gratings, which should not be used for
accurate work. Again, a grating never gives symmetrical spectra;
and often one or two particular spectra take all the light. This is, of
course, desirable if these bright spectra are to be used. Generally it
is not so. . . . It is not easy to tell when a good ruling point is
* J. S. Ames, Phil. Mag., May 1889.
T
274 - DIFFRACTION GRATINGS chap. ix
found ; for a “scratchy’ grating is often a good one ; and a bright
ruling point always gives a “scratchy’ grating. When all goes well it
takes five days and nights to rule a 6-inch grating having 20,000
lines to the inch. Comparatively no difficulty is found in ruling
14,000 lines to the inch. It is much harder to rule a glass grating
than a metallic one ; for to all of the above difficulties is added the
one that the diamond point is continually breaking down.”
146. Measurement of Wave Lengths.-The introduction of the
diffraction grating provided an exceedingly accurate method of de-
termining the absolute wave length corresponding to any part of the
spectrum. This method involves the accurate measurement of the angle
of deviation of the ray under consideration, and also the measurement of
the absolute length of the grating or grating space. The latter is diffi-
cult to determine accurately. Metallic gratings are much larger than
glass gratings, and consequently an error in measuring them is of less
importance in the result. However, it requires several days to rule a
large grating, and as the coefficient of expansion of speculum metal is
more than twice that of glass, changes of temperature give rise to
greater irregularities in ruling, but this advantage of the glass grating
is more than counterbalanced by the great difficulty in ruling one
free from flaws occasioned by the breaking down of the diamond point
on the hard material.
The interference methods of determining A usually require the
exact determination of some very small length; they are therefore
much inferior to the diffraction method, which lends itself more
readily to linear measurement besides affording very pure spectra.
Transmission gratings may be used in two ways: (1) with the
incident light perpendicular to the plane of the grating, in which case
mX = (a+b) sin 6,
and (2) in the position of minimum deviation, when the equation
which determines A is
mX=2(a +b) sin #ID.
The former method was used by Mr. Louis Bell" as offering fewer
experimental difficulties; but with either method the accuracy with
which the angular deviation can be determined far surpasses that of
the measurement of the grating space (a + b).
The spectrometer and grating being placed in exact adjustment,
readings may be taken on the D1 line in the spectra on both sides of
the slit, and the angle measured five or six times in succession.
Mr. Bell worked with the third spectrum, as in it the definition
* Louis Bell, Phil. Mag. p. 265, March 1887.
ART. 146 MEASUREMENT OF WAVE LENGTHS 275
was particularly good, and being of the highest order that could be
conveniently observed, an error in the angle could produce little effect
in the result. No correction was considered necessary for the effect
of the velocity of the apparatus through space (see Chap. XIX.).
. The accurate determination of wave lengths was first rendered
possible by Fraunhofer's researches in the solar spectrum. The dis-
covery of dark lines in the spectrum gave a definite standard of refer-
ence, and Fraunhofer" himself, with a wire grating, necessarily very
defective, obtained very fair determinations of the wave length of the
D line.
His mean result was 0005888 mm., which is remarkably accurate,
considering his gratings and the fact that most of his angles of devia-
tion were less than 1°.
These determinations were not improved on till the importance of
spectroscopic work was established by the great researches of Bunsen
and Kirchhoff, and the art of ruling gratings was much improved by
Nobert. Mascart employed four or five of Nobert's gratings, and
worked with them in the position of minimum deviation ; that is, so
that the incident and diffracted rays made equal angles with the plane
of the grating. This method avoids the necessity of placing the
grating perpendicular to the axis of either telescope, but it is rather
more difficult in the experimental work, and is perhaps of questionable
utility.
In the same year (1868) Ångström's researches appeared, and for
long remained the standard of reference in all questions of wave
length. He used Nobert's gratings, and in spite of the fact that these
were small and inaccurately ruled, giving imperfect definition and
showing numerous “ghosts,” his results would have been very nearly
exact if his standards of length had been correct.
In all the earlier determinations of the wave length insufficient
attention was paid to the measurement of the grating spaces, and this
of course requires an accurate standard of length. Thalén,” who
assisted Ångström in his work, corrected it afterwards for the error in
the assumed length of the Upsala metre.
Ten years after Ångström's research Mr. C. S. Peirce" again
attacked the problem with Rutherford gratings far superior to any
previously used.
* “Neue Modifikation des Lichtes durch gegenseitige Einwirkung und Beugung
der Strahlen, und Gesetze derselben,” presented to the Munich Academy in 1821.
See Bell, “On the Absolute Wave Length of Light,” Phil. Mag. p. 246, April
1888. '
* Thalén, Sur le spectre du fer, Upsala, 1885.
* American Journal of Science, third series, xviii. p. 51, 1879.
276 g DIFFRACTION GRATINGS CHAP. IX
The grating space (a + b) is never perfectly uniform throughout
the whole extent of the ruled surface. Regular or periodic variations
produce “ghosts” and differences in focus of the spectra on opposite
sides of the centre. There are other variations of an irregular
character, such as the displacement or omission of one or more lines,
or, what is far worse, the more or less sudden change in the grating
space, forming a part having a grating space peculiar to itself. This
latter is by far the most formidable type of error. The other irreg-
ularities are harmless, and occur in most gratings. If the abnormal
portion of the grating be confined to a few hundred lines, they will
merely diffuse a certain amount of light without producing false lines
or sensibly injuring the definition. They will, however, lead to an
incorrect result if we determine the grating space by measuring
the total length of the grating, and dividing by the total number
of lines.
Mr. Bell" describes an experiment illustrating the effect of these
errors of ruling: “Place a rather bad grating—unfortunately only too
easily obtained—on the spectrometer, and, setting the cross-hairs care-
fully on a prominent line, gradually cover the grating with a bit of
paper, slowly moving it along from one end. In very few cases
will the line stay upon the cross-hairs. A typical succession of
changes in the spectrum is as follows: Perhaps no change is observed
until two-thirds of the grating has been covered. Then a faint
shading appears on one side of the line, grows stronger as more and
more of the grating is covered, and finally is terminated by a faint
line. Then this line grows stronger till the original line appears
double and finally disappears, leaving the displaced line due to the
abnormal grating space.”
The effect of an abnormal portion of the grating is, therefore, to
cause a displacement of the lines of the spectrum and lead to error in
the evaluation of the deviation, as well as in the calculation of
the grating interval. The abnormal spacing generally occurs at
the end of the grating where the ruling was begun, for the
engine after starting requires some little time to settle down to a
uniform state.
To detect and evaluate the errors of irregular spacing Mr, Bell”
proposes the calibration of the grating. In this process the grating is
examined under a microscope from end to end. This gives the varia-
tions in the lengths of the spaces in different parts of the grating.
Bell's mean result for D, with four corrected gratings, two on glass and
* Bell, Phil. Mag. p. 362, May 1888.
* Louis Bell, Phil. Mag. p. 363, May 1889.
ART. 146 EFFECTS OF RULING ERRORS 277
two on speculum metal, gave at 20° C. and 760 mm. pressure in air
Xp1=5896-18 tenth metres,
or in vacuo
Xp1=5897-90 tenth metres,
which, as far as errors of observation go, he considers should be cor-
rect to within one part in half a million. The wave lengths of the
other lines in the spectrum derived from this are in air at 20° and
760 mm.
A (line between “head" and “tail” El (line between “head” and “tail”
of group) º tº . 7621-31 | of group). e º . 5270°52
2 3 5 5 . 6884 °11 E2 3 y 2 3 . 5269 °84
C 3 3 3 3 . 6563'07 Cl 3 3 3 2 . 5183 '82
{; 5 5 5 5 . 5896-18 F 5 5 y 5 . 4861 °51
D2 3 3 3 3 5890'22
Mr. Bell has given with the above figures the chief results pre
viously obtained for D, as follows:
Mascart . e o * . 5894 '3 Angström corrected by Thalén 5895.
Van der Willigen , tº . 5898 '6 Müller and Kempf. te . 5896 °2
Ångström . e g . 5895'13 || Macé de Lépinay . †: . 5896'0'
Ditscheiner . & 9 . 5897 '4 Kurlbaum . e { } . 5895-9
Peirce . e e ſº . 5896-27 | Bell e t e wº . 5896-18
Every method of determining wave lengths must necessarily involve
the uncertainties of the standards of length used. The experimental
difficulties involve small but troublesome corrections, such as the effect
of moisture in the atmosphere, changes of pressure, uncertainty as to
the true temperature of the grating, and variations of the grating
Space.
Runge (Astronomy and Astrophysics, vol. xii. p. 426) gives an
example on the necessity of allowing for the dispersion of air in deter-
mining wave lengths by the method of coincidences: “Suppose at a
temperature of 20° Celsius and a pressure of 760 mm. two rays have
the wave lengths 6000 and 2000. They would then exactly coincide
if one was observed in the first and the other in the third order.
Now reduce both rays to vacuo. They would no longer coincide,
the ray in the third order lying to the less refrangible side by
0.635 × 3 – 1:633 = 0.272. Since the corrections are proportional to
the density of the air a barometric pressure of say 770 mm. and a
temperature of 12° Celsius would make the distance between the two
lines 0-011, an amount which is larger than the probable error that
may be reached with well-defined lines.”
He gives the following table for reducing to vacuo Rowland's
278 DIFFRACTION GRATINGs CHAP. IX
standard wave lengths and all wave lengths that are determined from
them by interpolation :
t

Wave Lengths. º Wave Lengths. º
8000 2164 Ångström 4000 1:109 Ångström
7000 1.893 , , 3000 0.857 ,,
6000 1:633 ,, 2500 0.739 , , ,
5000 1.369 , , 2000 0.635 ,,
Compare the table on p. 161 containing Kayser and Runge's
indices for dry air. - -
ART. 147 GRAPHIC METHOD 279
SECTION III.-INVESTIGATION OF THE INTENSITY IN
DIFFRACTION PATTERNS
A. GENERAL CONSIDERATIONS
147. Application of the Graphie Method.—In the foregoing
sections we have considered merely the general character of the effects
produced by diffraction when light, diverging from a luminous point,
falls upon a narrow aperture or passes by the edge of an opaque obstacle.
The actual calculation of the intensity of the illumination at any point
of a diffraction pattern (that is, the fringe system produced by diffrac-
tion) is generally a problem of some difficulty when attacked directly
by the analytical method, but in many cases the solution may be
effected with great simplicity by means of the elegant graphic method
introduced in Art. 45. We shall therefore recapitulate briefly this
method of representing the resultant of
a system of vibrations of the same
period, but of different amplitudes and
phases.
Thus it has been shown that if a
polygon be constructed so that the
lengths of its sides (Fig. 119) represent
the amplitudes an as . . . an of a system
of vibrations simultaneously superposed
on a particle, while the angles which
these sides make with a given line OX
represent the phases of the corresponding vibrations at any instant,
then the closing side OP of the polygon represents the amplitude, and
the angle XOP which it makes with OX represents the phase of the
resultant vibration. In this figure the phases of the successive vibra-
tions are taken in ascending order of magnitude, and are such that there
is an abrupt change in passing from each to those adjacent to it. If,
however, the change of phase be not abrupt, but varies continuously in
passing from each to the next, and if the amplitudes be very small,
then the sides of the polygon will be very short, and the angles which
they make with each other will be very small, so that in the limit the
figure becomes a continuous curve, as shown in Fig. 120. This is
what happens when we attempt to represent the effect of a complete
wave, or of any part of it, at an external point.
The shape of such a curve depends on the manner in which the
Fig. 119.

280 GRAPHIC METHOD CHAP. IX
Curvature.
amplitudes and phases of the vibrations change in passing from each
to the next of the system, the curvature at any point being measured
by the rate of variation of phase with ampli-
tude. Thus if the element of length of the
curve be denoted by ds, and if the angle
between two consecutive elements be dºb, then
dº will be the difference of phase between
two consecutive vibrations, and the radius of
curvature at the corresponding point of the
vibration curve will be ds/dº. The form of
Fig. 120. the curve representing the effect of any part
of a wave will consequently depend on the manner in which the wave
is subdivided into elements of area.
In the particular case in which the vibrations form a system of
equal amplitude and uniformly increasing phase the curvature is the
same at all points, and the vibration curve (as already noticed in Art.
45) is a circle. This is the case discussed in Arts. 51 and 52, in which
the influence of obliquity is neglected, and where the wave front is
divided into ring elements corresponding to equal increments of phase.
The areas of these elements are not equal, but vary directly as their
distances from the point O, at which their joint effect is to be calculated,
and it follows therefore that when the influence of distance only is
considered, the amplitudes of the vibrations produced at O by the
The ring various rings are the same. When the influence of obliquity is taken
method.
into account, on the other hand, the amplitudes form a diminishing
Fig. 121.-Vibration Spiral.
series, and therefore in the vibration curve ds continually diminishes
as q, uniformly increases, so that the curvature gradually increases as
we proceed along the curve, and consequently the effect of the wave,
instead of being represented by a circle, is represented by a spiral


Art. 148 METHOD OF STRIP DIVISION 281
curve encircling a point J. (Fig. 121) with convolutions of ever-dimin-
ishing radius.
This method of dividing the wave front into elements of equally
increasing phase is undoubtedly the most desirable when it can be
applied, but there are many cases in which it ceases to be convenient,
and other methods of subdivision have to be adopted. For example,
the ring method applies at once if we wish to calculate the intensity
at any point on the axis of a circular aperture, or the effect of any
circular or annular portion of a wave at a point on its axis. When
the aperture is rectangular, or when the light is diffracted over a simple
straight edge, it is best to divide the wave into elementary strips
parallel to the edge of the obstacle. We shall consequently consider
this method of strip division in some detail before applying it to The strip
particular problems. In the ring method the surface of the wave is "**
divided into elements of area by means of a system of spheres having
a common centre, while in the strip method the surface is intersected
and divided into strip areas by means of a system of planes having a
common edge.
148. The Method of Strip Division.—Let two parallel right lines
AB and CD (Fig. 122) be drawn
in the front of a plane wave so
as to include a very narrow strip
of the wave, and let it be re-
quired to determine the effect of
this strip at any external point
O. Taking the strip to be very
narrow we may represent it by
the right line AB (Fig. 123),
and divide it into elements of
length, PM, MIM, etc., by
points taken on it at equally in-
creasing distances from O. These Fig. 122.
elements are such that
OM - OP=OM, OM, -etc. =OX'-OX=6.
They are elements of uniformly increasing phase, but are not of equal
length. To express the length of any one of them, XX for example,
we have by similar triangles
(OX'-OX) : XX’: ; PX' : OX'.
Denoting OX' – OX by 6, and writing r and a for OX and PX-that

282 - GRAPHIC METHOD - CHAP. IX
Single
strip.
is, for the distances of the element from O and P respectively—we
have 1
_rö.
Tac
XX’
Hence, if we neglect the influence of obliquity, the amplitude of the
vibration contributed by the element XX will be proportional to
XX_3
7" 90s
We conclude therefore that when the strip is divided into elements
corresponding to uniformly increasing phases their effects diminish as
they recede from the pole P, the diminution of effect being inversely
as the distance of the element from the pole of the strip. When the
influence of obliquity is taken into account this rate of decrease in the
effects of the elements, as they recede from the pole, is made still more
rapid. Hence when we compare two consecutive half-period elements
of the strip, as in Art. 51, it follows that when they are near the pole
they differ considerably in effect, but when they are far away from the
pole they approximately neutralise each other.
Denoting the effects of the consecutive half-period elements by ml,
mo, ms, etc., and taking into account the two halves of the strip, viz.
AP and BP, the whole effect of the strip may be written in the form
S=2(m) — me + m3 – ma-H etc.).
In this series the terms diminish rapidly at first, so that they soon
become very small, and ultimately equal and opposite. We may
therefore conclude that when the wave length is small the whole
effective portion of the strip is confined to a small region in the
neighbourhood of the pole. In the same way every other strip may
be reduced to a small effective portion in the neighbourhood of its
pole with respect to O, and the whole wave may be replaced by a
narrow equatorial band QQ (Fig. 122) passing through P in a direction
at right angles to the strips.
The calculation of the effect of the whole wave at O is conse-
Equatorial quently reduced to that of the equatorial band, QQ. Now this band
band.
is cut into elements by the strips ABCD, etc., which correspond to
equal increments of phase, and as the effects of these at O vary
inversely as their distances from P (when the effect of obliquity is
neglected), it follows that the successive elements produce effects at O
which rapidly diminish as they recede from the pole. This diminution
is rendered still more rapid by the influence of obliquity, which also
causes a falling off in amplitude as we recede from the pole. The
* This merely expresses that rār = ada.
ARt. 149 METHOD OF STRIP DIVISION 283
result is that the effective portion of the equatorial band is confined
to a small area in the neighbourhood of P, and, as before, the effect of
the whole wave may be limited to a small
area surrounding the pole.
Now if the effect of a single strip,
such as AB (Fig. 123), be represented
graphically in the manner already ex-
plained, it is clear that (since the ampli-
tudes of the vibrations contributed by the
successive elements of the strip rapidly
diminish as they recede from the pole) the
curvature of the vibration curve will in-
crease rapidly at first, and then more
slowly, so that the curve will be a spiral
similar to that shown in Fig. 120, with
convolutions of ever-decreasing radius en-
circling a point J. This spiral represents one half of the strip, and
the other half will be represented by the same spiral repeated. The
effect of the whole strip will consequently be represented in
amplitude by 20.J., and in phase by the angle XO.J. This angle
measures the difference in phase between the resultant vibration con-
tributed by the whole strip, and that contributed by the central point
or pole of the strip.
Having determined the amplitude and phase of the vibration con-
tributed by each strip—that is, by each element of the equatorial band
QQ—a spiral may be drawn in the same way to represent the effect of
the complete wave. Now since the phase of the vibration contributed
- by the whole wave differs from that arriving whole
from the pole by 90° (Art. 52), it follows ".
that the spiral representing the equatorial
band must encircle a point J on the axis OY
(Fig. 124), and must start from 0, not as a
tangent to OX, but, making an angle with it,
determined by the phase difference between
the resultant vibration of a whole strip
and that contributed by its pole. This
spiral represents one half of the equatorial
band, and the other half is represented
by the same spiral repeated, so that the whole amplitude is
measured by 20.J.
In the case of a spherical wave the surface may be divided into a spherical
series of strips by a system of planes passing through a diameter of ****
Fig. 123.
Fig. 124.


284 GRAPHIC METHOD CHAP. IX
Spherical
wave.
Cylindrical need consider is the cylindrical wave, such as
Wave.
the sphere. Thus if APB (Fig. 125) represents the front of a spherical
wave diverging from C, and if it is required to calculate the effect at
O of the whole, or some portion of it, the
surface of the wave may be divided into
a system of strip areas by a system of
planes passing through a diameter AB
of the wave drawn at right angles to OC.
These planes, which may be called mer-
idian planes, intersect the surface in great
circles, and divide it into elementary
strips, which are lunes of small area.
Each of these strips may be reduced in
effect to a small area in the neighbour-
Fig. 125. hood of its pole, and may be represented
by a spiral curve after the manner of Fig. 120. It follows therefore
that, as in the case of a plane wave, the spherical wave may be
replaced in effect by an equatorial band lying along the great circle
PMN, which cuts the meridian planes orthogonally. This band in
turn may be represented graphically by a spiral curve after the manner
of Fig. 124, and its effective portion is restricted to a small area in
the neighbourhood of P. When represented in this manner the ampli-
tude of the resultant vibration is 20.J. (Fig. 124), -
and its phase is 90° in advance of that of the
vibration arriving from the pole of the wave.
The only other form of wave which we
**
diverges from a long narrow slit. In this case
the surface of the wave may be divided into a
series of rectilinear strips parallel to the length
of the cylinder by a system of planes drawn |
through its axis, as shown in Fig. 126. Each
of these strips may be reduced to a small Fig. 126.
effective portion, and the whole wave as before may be replaced
by an equatorial band PMN. This in turn reduces to a small
portion around P, and may be represented graphically, as in the
case of a plane wave.
B. FRESNEL FRINGES
149. Fresnel Fringes.—Before making any calculation of the
actual intensity at the various points of a diffraction pattern, we shall


ART. 149 FRESNEL FRINGES 285
consider in a general manner the fluctuations of intensity in some of
the elementary cases already noticed (Sec. I.).
For this purpose, when the strip method of division is employed, each
half of the wave gives rise to a spiral curve, and these spirals are
identical in shape and position. In-
stead of superposing them, however, it
will often be found convenient, for the
purposes of representation, to draw one
of them above the line OX, and the
other below it, as shown in Fig. 127.
The chord joining any two points on
the upper spiral will represent the re-
sultant effect of a corresponding por-
tion of one half of the wave, and the
chord joining two points on the lower
spiral will represent a portion of the
other half of the wave. Thus OJ re-
presents the whole effect of the upper half of the wave and OM
represents the effect of a portion of the lower, whereas JM is the
resultant of these two (viz. OJ and OM) taken together. This applies
to the case of light diffracted over a straight edge.
Straight Edge.—It has been already indicated that when light
passes by the edge of an opaque screen a system of fringes exists just
outside the geometrical shadow, but that inside the shadow the light
fades away gradually without passing through any alternations of
brightness and darkness. The illumination at any point outside the
shadow is contributed by a complete half wave RS (Fig. 107) and a
fraction RA. The effect of the half wave will be represented by OJ
and the fraction by OM (Fig. 127), where M is some point on the
other half of the spiral. The resultant effect will therefore be repre-
sented by JM. Hence as M moves along the spiral JM passes through
a series of maxima and minima. There is consequently a series of
alternations of brightness and darkness outside the geometrical shadow.
For as the point on the screen moves outwards from the shadow, the
point M moves round on the spiral towards J', and JM passes through
a maximum and a minimum every convolution. The least value is
JO, that due to half a wave. The intensity then rises and falls, the
values of the maxima being greater than JJ', that due to the complete
wave, and the values of the minima greater than JO, that due to half
Fig. 127.
This application was published by M. Cornu as a “Méthode nouvelle pour la
discussion des problèmes de diffraction dans le cas d'une onde cylindrique” (Journal
de Physique, tom, iii. 1874).

286 GRAPHIC METHOD CHAP. IX
a wave. At the edge of the geometrical shadow the amplitude is O.J.,
and the intensity is consequently one-fourth of that produced by the
whole wave.
For the illumination at any point within the geometrical shadow
we have only to deal with part of the spiral OJ
(Fig. 128), as only a fraction of half a wave is
now in action. As Precedes within the shadow
the tracing point M recedes from O along the
spiral OJ, moving round towards the point J.
The line JM, which represents the resultant
vibration, continually decreases towards zero,
without passing through any maximum or mini-
mum values. The illumination therefore falls off gradually to zero
within the shadow.
Narrow Aperture.—Let us now consider the illumination produced
at any point by a very narrow rectangular slit. Since the element of
arc of the spiral measures the amplitude of vibration of a corresponding
element of the wave, it follows that the length of the complete arc
of the spiral which represents the wave from the slit is simply pro-
portional to the width of the slit, consequently the amplitude of the
resultant vibration at any point on the screen will be measured by the
right line joining the extremities of a constant length of the curve,
viz, an arc proportional to the width of the slit. Inside the geometrical
projection the arc in question passes
through O and belongs partly to one
half of the spiral and partly to the
other. Outside the projection the arc
is situated altogether in one half of
the spiral. In all cases it is clear
that there will be generally fluctuations
of intensity at different points of the
Screen.
If the slit be very narrow, so that
the corresponding arc of the spiral is
small, then inside the projection the
intensity will remain constant over a
considerable range, and be very nearly proportional to the square
of the width of the aperture, since here the arc will nearly agree
with its chord.
Narrow Wire.—The case of a narrow wire can be deduced from that
of a narrow rectangular aperture of the same dimensions. Inside the
shadow of the wire the effect of that part of the wave which passes one
Fig. 128.
Fig. 129.


ART. 149 FRESNEL FRINGES 287
side of the wire is represented by JM (Fig. 129) where M is some point
on the spiral OJ, and the effect of the part passing the other side of the
wire is represented by J'M' where M' is some point in the spiral OJ."
and the arc MM is proportional to the width of the wire. If the wire
subtends a considerable number of half-period elements at the screen,
the arcs OM and OM will contain several convolutions of the spirals
and the lines JM and J'M' will be nearly equal, so that there will
be destructive interference when JM and J'M' are in opposite
directions, and maximum illuminations when they are in the same
direction. In the interior of the shadow we have thus a system of
interference bands. As we approach the borders of the shadow, the
point M, suppose, moves towards O on the spiral and the point M’
moves towards J', since the arc MM must be of constant length. The
lines JM and J'M' in this case differ considerably in magnitude, so
that when they are parallel and in opposite -
directions there will still be some resultant
illumination and the minima will not be
places of complete darkness. Outside the
shadow we have a complete half wave and a
fraction from one side of the wire. These
will be represented by OJ and OM' respec-
tively, while from the other side we have a
portion represented by JM", where M" is
some point on the spiral near J'. The are
MM" is absent and of a constant length pro-
portional to the width of the wire. Thus if from J' we draw J'N
(Fig. 130) parallel and equal to the chord of the arc MM", then since
JJ represents the effect of the whole wave, and since J'N represents
the effect of the intercepted portion, it follows that JN will represent
the transmitted portion. JN therefore represents the effect at a point
outside the shadow, and as J'N revolves round J-that is, as the point
recedes from the edge of the shadow—the line JN will pass through
a series of maxima and minima, which represent the external fringes.
In the case of the narrow aperture it is JN that is intercepted and
J’N transmitted.
Two Narrow Rectangular Apertures.—In the case of two equal nar-
row apertures we have to consider the resultant of two arcs of the
spiral of lengths proportional to the widths of the apertures. These
arcs are separated by an arc of constant length, proportional to the
distance between the apertures. The resultant is therefore the vector
sum of the chords of the two arcs; that is, the diagonal of the
parallelogram having its adjacent sides parallel and equal to the chords.
Fig. 130.

288 GRAPHIC METHOD CHAP. IX
This diagonal will pass through a maximum or minimum value accord-
ing as the chords of the arcs are parallel and in the same or opposite
directions. t
Two narrow wires give a corresponding system of fringes, and the
diffraction patterns afforded by other arrangements may be investigated
in a similar manner.
Circular Aperture.—The intensity at any point on the axis of
a circular aperture may be easily expressed in the same manner.
Thus we have already seen (Art. 53) that when the aperture is divided
into circular annuli, corresponding to equal differences of phase, the
amplitudes of the vibrations produced by these elements are equal, and
therefore (when the influence of obliquity is neglected) the vibration
curve is a circle. Hence, as in Art. 154, if s be the length of the arc
of the circle which represents the resultant vibration, we have
OM = 2R sin p, and s-2Rq).
Therefore the intensity may be expressed in the form
1-alº
where 24, is the phase difference of the vibrations from the centre and
the circumference of the aperture. 3.
When the radius of the aperture is small compared with the other
distances involved we have approximately, as in Art. 130,
20–º-*(a++,+,-,-)=tº
where r is the radius of the aperture, a its distance from the luminous
origin, and b its distance from the screen. Now by Art. 53 the arc
S is proportional to 27taô/(a + b), and this by the foregoing reduces to
Tr”/b. Hence the expression for I becomes
1-(+) sin”?
b q.”
where
q = Tr”(a + b)/2ab)\. -
150. General Investigation.—The components of the resultant
vibration at any point and the intensity of the illumination may be
very easily expressed by means of the vibration spiral. Thus if a. and
y be the co-ordinates of any point on the spiral, and q, the inclination
of the tangent at that point to the axis of a, then p is the phase of
the vibration from the corresponding point of the wave.
Now in any curve we have, if ds be the element of arc,
da;=cos pds, and dy=sin ºpds.
Therefore
Ø Fiſ cos pds, and y=iſ sin pds,
ART. 150 GENERAL INVESTIGATION 289
and consequently the radius vector is given by the equation
rº–294-yº–Iſcos *ds?--Iſsin ºds?.
But the resultant amplitude is given by the radius vector r of the
spiral, consequently the intensity of illumination is measured by
I=[ſcos wds?--Iſsin qds]”.
The phase ºbo of the resultant vibration is given by the inclination
of r to the axis of a, and we have consequently
–9– ſ sin ºpds
tan poº-º-º:
2C ſcos ºds'
the integrals being taken between limits which are determined by the
problem under investigation.
- Now the element of arc ds of the spiral is taken proportional to
the amplitude of the vibration contributed by an element of area dS
of the wave front, and this amplitude is taken proportional to the area
and inversely as the distance (r) of the element from the point under
consideration. Further, the amplitude will depend on the inclination
of r to the wave normal; that is, on the obliquity, as well as on the
direction of the vibration in the wave front. The full expression for
ds is consequently of the form
dS
ds= ... f.
where f is some function of the co-ordinates, which diminishes as the
element recedes from the pole of the wave.
The general expressions for the resultant consequently take the
form
*=ſ for tº v= | f in º.
The intensity of the illumination is consequently
1=[ſ for wºrſſ f in ºf
while the phase of the resultant is
q ſy sin •º
tan *-*=* -;
Q} dS'
ſ f cos p 7.
In the foregoing investigation the amplitude of the vibration con-
tributed by an element dS of the wave front, situated at a distance r,
has been represented by the expression fds/r, and it is consequently
- IJ
290 GRAPHIC METHOD CHAP. IX
desirable to investigate the constitution of the symbol f. Now if A
be taken to represent the amplitude of the vibration in the wave
front, then the amplitude produced by an element dS at a distance
r may be written in the form •
AdS
a===biº
where p is some function of the angles which r makes with the wave
normal and the direction of vibration, and b is a quantity which has
yet to be determined. Now a and A are of the same dimensions,
being both amplitudes, whereas iſ is a function of angles and is of zero
dimensions, consequently b must be the inverse of a length. But the
only other quantities that can have any reference to a are v and A, and
of these v is eliminated by the fact that it is a function of the time ;
hence the only length that can enter into the quantity b is the wave
length A. We conclude, therefore, that b is of the form k|A where k
is a constant of zero dimensions. Using this notation the complete
expression for the amplitude is of the form
kAdS
7-X, Y
Comparing this with the form făS/r, we find that f is of the form
f=*V.
where k may be taken as unity. We have thus reached the important
result that the factor f contains the reciprocal of the wave length as a
constituent besides its dependence on the obliquity and the amplitude
of vibration in the original wave front. -
Ea'amples
1. If the radius vector r, from an external point O, to any element of a plane
wave makes an angle 6 with the wave normal, the components of the vibrations
at O are
w=}ſcos (2tbuſX). f. du,
y=kſ sin (2TrbuſN). f. du,
where we sec 6–1, and k is a constant.
[Divide the wave up into circular elements around the pole P as centre (Fig. 24).
Let p be the radius of one of these elements and r its distance from O. Then if
OP=b, we have p-b tan 0 and r-b sec 6= b(1+u),
2T 2Trb 2tröw,
*=#(r- b)= *(see 6 – 1)= -X →
dS=2irpdp=2irb% tan 6 secº 6.d6 =2|rb°(1+u)du.
Therefore dS/r=2irbdu, etc.]
ART. 150 |FXAMPLES 291
2. If f=constant, show that the vibration curve is a circle.
[Here we have
a;=k| "cos (2irbuſN)du = a sin (2tbuſX),
g =kſin (2tbuſN)du = a (1 – cos (2TrbuſN)}
where a is a constant.
Hence
až--(y – a)*= a”,
which shows that the amplitude curve is a circle, of radius a, passing through the
origin and having its centre situated on the axis OY (cf. Art. 154).] *
3. In the same case prove that the vibration excited at O by the complete wave
is in phase a quarter period behind that which reaches it from the pole P.
[For the complete wave we have, if we assume that cos oo =sin oo = 0,
CO
*=eſ cos (2tbuſX)du = 0,
gy=kſ sin (2Tbw/N)du =k.
Hence if q be the phase of the resultant vibration, we have
tan p-y/a:= co,
therefore
q = $tr.]
4. If the effective portion of a wave be confined to a small portion around the
pole, which is sensibly plane, and for which f is constant, the amplitude curve will
be a circle.
[We have
dS=2irpdp= trap”,
and 2T p” • p”
6–5. 35 since r=b+;
Hence
dS=bXdó,
and therefore, since f is supposed constant,
8 -
w=aſ cos 3d6= a sin 6,
J 0
8 g
v=eſ sin Ödö = a(1 — cos 6),
O
and
a’--(y – a)*= a”, etc.]
5. Determine the radius of curvature of the amplitude curve, Ex. 1.
[The amplitude of the vibration excited by the element dS is proportional to
Zºë–2-y ... dw.
Hence the element ds of the amplitude curve is proportional to flu, and if p be the
angle the tangent to it makes with OX, we have dºp-2Trbdu/\, Hence the radius
of curvature p is proportional to
ds
p=#=x f.
292 SECOND GRAPHIC METHOD CHAP. IX
Hence if f=constant the amplitude curve is a circle, and if f diminishes gradually
from the pole of the wave, the curve is a spiral of ever-decreasing radius of curvature
as we move along it, setting out from O.]
6. If
f=1+cos *—º.
we have for a plane wave (Ex. 1)
a:-/:ſcos (2+bu/\) #au,
- y=k | sin (2itbuſX) #ºu.
151. Second Graphic Method.—A second method of representing
the resultant amplitude of a system of superposed vibrations has been
given in Art. 45. This method is based on a second graphic representa-
tion of the resultant of a system of forces, and it follows that if a
system of lines OP, OP, etc. (Fig. 12), be drawn from any point O
such that the lengths OP, OP, etc., represent the amplitudes of the
vibrations, and the angles they make with a fixed line OX their phases,
the resultant will be represented by n times the line OG, joining O to
the centre of mean position (G) of the points Pi, P, etc.
If in this manner we plot down the curve for a complete wave,
beginning at the pole of the wave, we see at once that we have a
spiral curve surrounding O with many con."
volutions of ever-decreasing radius (Fig. 131).
With this spiral the resultant effect of any
portion of the wave is not represented by
the chord joining 0 to a point on the curve,
but by that joining it to the centre of
gravity of the corresponding arc of the curve.
This obviously passes through fluctuations
as in the case of the other spiral.
This method may be applied with the
greatest facility to the calculation of the intensity of the illumination
at any point of a diffraction pattern when the incident light is parallel.
In this case the curve becomes a circle, and the problem is reduced at
once to the finding of the centre of gravity of an arc of a circle, lead-
ing to all the results we have already arrived at. The deduction of
these results by this method will form a simple and useful exercise.
The equation of the second spiral, and the integrals giving the
general expression for the intensity, may be derived very simply. For
if a. and y be the co-ordinates of the centre of mean position of the
extremities of the radii vectores of the spiral, we have
Fig. 131.
nº->w, and my–Ny.

ART. 152 FRESNEL's INTEGRALS 293
Hence -
(m. OG)*=(nº)*-i- (my)*=(S,v)” (Sy)”.
Now any radius vector p of the spiral represents one of the
constituent vibrations in amplitude and phase. The length of p is con-
sequently (as in Art. 150) given by the equation
dS
p="../
But if a. and y be the co-ordinates of the extremity of p we have
-
2–p cos p, y=p sin ºp,
where q is the phase of the corresponding vibration. Hence the
expression for the intensity of the illumination
(m. OG)*=(>2)-4-(>y)*=(>pcos p)*--(>p sin ºp)*
I– | cos *] + [ſºn *].
152. Fresnel's Integrals.-In the foregoing articles it has been
proved that the intensity of the illumination at any point of a diffrac-
tion pattern may be expressed in general as the sum of the squares of
two integrals taken between limits defined by the nature of the problem.
By making certain assumptions and approximations Fresnel deduced
special forms of these integrals which we shall now consider. Let us
take the case of a cylindrical wave and let it be divided into a system
of narrow strips by lines
drawn on the surface of
the wave parallel to the
axis of the cylinder, as
indicated in Art. 148.
Further, let each of these
strips be replaced by its
central effective portion,
so that the whole wave is
reduced to its equatorial band. By this process the calculation of
the effect of the wave is reduced to finding that of a circular band,
and we know that of the whole band it is only a small portion in
the neighbourhood of the pole that is effective. The whole effect
is therefore equivalent to that of a small arc of a circle. Let a be
the radius of this circle—that is, the radius of the cylindrical wave–
and let b be the distance of the point O (Fig. 132) at which the effect
is sought, from the pole P of the wave. Then if OM = r and if PM =s
we have
gives as before
Fig. 132.
r”– (a+b)++ a”-2a(a+b) coss/a,

294 FRESNEL'S INTEGRALS CHAP. IX
consequently when s is small we have approximately
That is, the relative path retardation of the vibration from M is
proportional to the square of the arc PM. The phase difference is
consequently y
_2T * — _T(a +b)
*= (r-t)=*.*.*.
Hence if we assume that the amplitude of the vibration produced at
O by an element ds of the circle is simply proportional to the length
of the element, the quantity fas/r in the preceding integrals must be
replaced by ds, and the expression for the intensity becomes
I = |ſco. was] + [ ſsin was]'.
or, writing q in the form #Tv” where v = S w/2(a + b)/abX, we have ds
proportional to do, and therefore I may be expressed in the form
I = ſſCOS Arvia) + [ ſsin irwayſ e
This is the formula deduced by Fresnel, and the two integrals which
appear in it are known as Fresnel's integrals.
It should not be lost sight of that in expressing I in this form
certain assumptions and approximations have been made, and that it
is on account of these assumptions and approximations that integrals
of this form appear. Thus it is practically assumed that the effective
portion of the equatorial belt is so small that q is sensibly propor-
tional to sº, and that r is approximately constant and equal to b.
Further, the amplitude of the vibration contributed by any element of
area is taken proportional to the area, and this amounts to assuming
either that fºr is constant, or that the function f is constant as well as r.
Now in examining the effect of a plane or a spherical wave at any
point, we have seen (Arts. 52 and 53) that the only factor left
by which the approximate rectilinear propagation may be explained
is that the function f is not constant, but is such that it decreases
as the obliquity increases; that is, as r increases. Consequently it is
not legitimate to assume either that f is constant or that fºr is con-
stant, for it is on the variations of f and r (however small) that
the whole outstanding effect depends in the case of a complete wave.
The final result obtained by Fresnel amounts to saying at once
that the effective portion of the wave front is confined to a small arc
which is so short that throughout it q, may be taken proportional to
s”, and this is what stamps Fresnel's integrals with their peculiar form.
ART. 152 LAW OF OBLIQUITY 295
This assumption confines the effective portion of the wave to a small
portion around the pole, and therefore virtually introduces a law of
diminution of effect with obliquity; or, in other words, a form of the
function f.
To determine the law which Fresnel introduced inadvertently by
these assumptions let us suppose that the wave is divided into ring Fortuitous
elements, as in Art. 53, so that dS is proportional to rdºp. Then lº."
fdS/r = kfaq, where k is a constant, and f cos ºdS/r = kf cos ºdd. Now
in order that this may take the form cos 2°dz it is only necessary to
suppose that
1
f oc - -
Vº Vr-b
Thus the law inadvertently introduced is that f varies inversely as the
square root of the phase retardation, and consequently its rate of decrease
as we recede from the pole is very rapid, and the effect of a small area
surrounding the pole must be approximately the same as that of the
whole wave. We should therefore expect that the value of either of
Fresnel's integrals when taken between the limits 0 and v should be
sensibly the same whether we take v fairly large or infinitely great.
In fact, as v increases from zero the value of each integral fluctuates
and passes through a series of maxima and minima, but these fluctua-
tions soon become less and less pronounced, the maxima decreasing
while the minima increase until they become sensibly equal, and the
value of the integral remains sensibly constant as w increases in-
definitely.
This may be seen geometrically by plotting the curve y = cos wº, in
which the value of the ordinate varies periodically between the limits
+ 1 as a increases (Fig. 133). The -
area of this curve is ſydºr = ſcosa”da,
and may therefore be taken to re-
present the value of one of Fres-
nel's integrals. Now while y varies
between constant limits + 1, it is
to be remarked that the distance
between two consecutive points of
intersection of the curve with the axis of a continually diminishes
as a increases. In fact, if a. and a + h_be the abscissae of two consecutive
points of intersection of the curve with the axis of a, we have
Fig. 133.
*=(2n+1)T/2, and (z+h)*=(2n +3)7/2,
so that by subtraction we obtain
h”--2aºh – m,

296 FRESNEL'S INTEGRALS CHAP. IX
and, therefore, as a, increases h must diminish. The total area of the
curve is thus the sum of a system of loops which are alternately
positive and negative and which decrease indefinitely in absolute
value. Hence the integral, after passing through a series of maxima
and minima, rapidly attains a stationary value, as shown in the
following table after Gilbert :
TABLE OF FIRESNEL's INTEGRALs (Gilbert)*

Q). J."cos #mrv?dv. f: sin ºrvºdv. Q). ſ. cos Amv2dv. ſ. sin #m-vºdv.
0.0 0.0000 0.0000 2.6 0.3389 0.5500
0.1 0.1000 0.0005 2.7 0.3926 0.4529
0-2 0. 1999 0.0042 2.8 0.4675 9.3915
0.3 0.2994 0.0141 2.9 0.5624 0.4102
0-4 0.3975 0.0334 3.0 0.6057 0.4963
0.5 0.4923 0.0647 3.1 0.5616 0.5818
0.6 0.58||11 0. 1105 3.2 0.4663 0.5933
0.7 0.6597 0.1721 3.3 0.4057 0.519.3
0.8 0.7230 0.2493 3.4 0.4385 0.4297
0.9 0.7648 0.3398 3.5 0. 5326 0.4153
1.0 0.7799 0.4383 3.6 0.5880 O. 4923
1. 1 0.7638 0.5365 3.7 0.5419 0.5750
1.2 0.7154 0.6234 | 3.8 0.4481 0.5656
1.3 0.6386 0.6863 | 3.9 0.4223 0.4752
1-4 0.5431 0.7135 4.0 0.4984 0.4205
1.5 0.4453 0.6975 4-1 0.5737 0.4758
1.6 0.3655 0.6383 4.2 0.5417 0. 5632
1.7 0.3238 0. 5492 4-3 0.4494 0.5540
1.8 0.3337 0.4509 4-4 0.4383 0.4623
1.9 0.3945 0.3734 4-5 0.5258 0.4342
2.0 0.4883 0.3434 4-6 0.5672 0. 5162
2. 1 0. 5814 0.3743 4.7 0.4914 0.5669
• 2.2 0.6362 0.4556 4.8 0.4338 0.4968
2.3 0.6268 0. 5525 4-9 0.5002 0.4351
2-4 0.5550 0.6197 5.0 0.5636 0.4992
2.5 0.4574 0.6192 OO 0. 5000 0.5000
When the values of the integrals expressing the intensity are known,
then if one of them be denoted by a , and the corresponding value of
the other by y, it is clear that when a curve is constructed with a and
y as co-ordinates, the radius vector r drawn from the origin to any
point on the curve will represent the resultant vibration arising from
the corresponding portion of the wave. For we have r" = a + y”, and
consequently 7° represents the intensity of the illumination, or r
represents the amplitude of the resultant vibration and the angle it
* Gilbert, Mém, cowronnés de l'Acad, de Bruxelles, tome xxxi. p. 1, 1863. The
values of the cosine integral for v = 0°1 and 1-8 have been corrected. A more
extended table for values of :* for intervals of 0-1 from 0 to 20 is given in British
Association Report (Oxford Meeting, 1926).
ART. 152 CORNU'S SPIRAL 297
makes with the axis of a represents its phase. The curve constructed
in this manner is, in fact, the vibration spiral. Using the values of
Fresnel's integrals given in the foregoing table, M. Cornu constructed
the spiral shown in Fig. 134. The branch OM, MJ refers to one
half of the wave, and the branch OJ" to the other. The part OM,
represents the first half-period element, M.M., the second, M.M., the
third, and so on, the successive convolutions winding with ever-
increasing curvature round a central point J. This point is situated
on the line bisecting the angle between the axis, for its co-ordinates are
*=ſº cos trººdoº, and v=ſ sin Tºdw-?,
consequently at J and Jº we have ºr = y; that is, these points lie on the
Fig. 134.—Cornu's Spiral.
line bisecting the angle XOY. The phrase of the resultant would thus
appear to be only 45° in advance of that which arrives from the
central element. It must be remembered, however, that in evaluating
I by this method the wave surface has been reduced by the strip
method to an equatorial band, so that each element of this band is
the resultant of a strip. The phase of each element of the band is
thereby advanced by 45° relatively to that of the vibration from the
pole of the strip. The whole phase of the resultant is consequently
90° in advance of that arriving from the pole of the wave surface.
This rotates the spiral through 45° and throws the points J and Jº on
the axis OY, as in Fig. 127.

298 FRESNEL's INTEGRALs CHAP. IX
Methods of evaluating Fresnel's integrals have been given by
Fresnel, Gilbert, Cauchy, and Knochenhauer. These methods are
noticed in the following examples and will be found at length in
Verdet's CEuvres, tom. v. p. 328, etc.; Optique. Physique, tom. i.
f
Ea'amples *
1. At what distance from a slit of width 2c is it necessary to place a screen so that
the central fringe may be of minimum intensity ?, Determine this intensity (Cornu).
Here we must determine the points of the two branches of the spiral which are
nearest. Since in this case they are symmetrically situated, the line joining them
passes through 0, and with a compass the minimum distance is found to correspond
to v = +1875. Since s-c we have * . . . º
*(***).2–11.87Re .
*::::: *=(1875).
Again the intensity is measured by the square of the chord of the curve, and this
is (1.08)?=1:17 where the intensity of the light due to the whole wave is JJ’”=2.
The relative intensity of the central band is therefore 0.585.
2. Prove that
{+1t
ſ cos #Tv°dv =; sin #tr(?” +2iu) — sin irº,
i
i-Hw 1 e sº
ſ sin #7tvºdva #| — cos $tr(i.” +2iw)+cos *]
i -
(Fresnel, CEuvres, tom. i. p. 319),
where w is a small quantity and i given.
[Replacing v by i-H w we have
cos $try” – cos #1 (i+w)*=cos #tr(i.”--2iw)
neglecting wº. Hence
--w t&
ſ cos #m vºdvi- | cos #tr(i.” +2iu)dw
3. o
l& tº
=cos $tri”| cos triudu — sin #tri”ſ sim triudw,
O O
which is integrable at once. -
By this method Fresnel calculated the values of the integrals, taking was 0 - 1 and
i successively equal to 0, 0 1, 0 2, 0 3, etc.]
3. Prove that
J 2.7a, – 2_ T20,5 TrAvº *
ſº ºw-cosirº (; Tā āt T3. ET; g- . .
tº 2 Trv3 Tr3.ji + Töyll * )
+sin #Tº(H-Tă. H +T3TETS Ti-. . .
(Kmochenhauer, Die Undulationstheorie des Lichtes, p. 36).
[Integrate by parts.] -
4. Writing the result of Ex. 3 in the form
ſ cos #Trvºdva M cos #twº 4-N sin #7tvº,
AlèT. 152 EXAMPLES 299
show that
w)
ſ sin #trvºdva-M sin #7tv°–N cos $try”.
O -
Hence - -
U 2 lº 2
|ſ COS arvae +|ſ sin #tvºdv | = M24-N2,
O 0. -
and -
dM &N.
;=1 * TrvīN, dºv == TrvM.
5. Prove that - -
CO o/ 1 1. 3 . 5 1 . 3. 5. 7. 9
2,7as — *! —t – a-º.
| cos #trºdva cos $try (* rºſt T6Ayll tº tº ..)
º 1 1 . 3 1 .. 3. 5. 7
* *! ---------------.
sin #7tv G. #5; +–F, F. . .)
(Cauchy, Comptes rendus, tom. xv. pp. 534, 573).
[Integrate by parts.]
6. Writing the equation of Ex. 5 in the form
CO
| cos $trºdva P cos $trº” – Q sin #tw”,
ty
show that
ſ CO sin #try”dv= P sin #tw”--Q cos #tw”,
and also 1)
# =rvo — 1, }= — TrvP.
7. If axes of reference be taken in the wave front at P, or parallel to it through
O, we have
p”=a^+ y”, and dS=da dy.
i. º
Hence if f=constant and rab, that is, if the effective portion is limited to a small
area around the pole P we have 6-trpº/b} (Ex. 4, p. 291), and the intensity of the
illumination is proportional to
[ſcos #Tc(a^+ y”)da!dy]*-i- [ſsin #Trc(a^+ y”)da dy]?
where c is a constant (2/b)\) in the case of a plane wave, and it represents 2(a +b)/abX
for a spherical wave of radius a.
Denoting these integrals by M and N respectively, we have by expansion
M =iſCOS #rcºdaſcos #Trcy”dy–Jsin #Trca”daſsin #Trcy”dy,
N =iſsin #Trcºdaſcos #tcy”dy--ſCOS #rcºdaſsin #Trcy”dy.
Replacing ca” or cy” by vº, the calculation of these integrals is reduced to the calcula-
tion of the integrals (Fresnel’s)
C =iſcos #Trvºdv, S =iſsin #Trv°dv.
Particular Cases
(a) Complete Wave.—If the wave is unobstructed, we have
+co +oo
c=ſ COS $trºdva-1, s=| sin #Trv°dv= 1.
- CO - CO
300 FRESNEL'S INTEGRALS CHAP. IX
Hence.
M = 0, and N=}
Therefore
M*N*=;
and
tan ºp-N/M = co, . . q =#T.
(b) Straight Edge.—If the wave passes over the edge of an opaque obstacle, we
integrate with respect to y between the limits + oo and – co, so that
M =iſcos $7tv°dv –ſ sin #7tvºdv,
N =iſ cos #trºdv-Hiſsin #irvºdv,
M2+ N2= 2[ſcos #Trvºdvī’ + 2[ſsin #Trv°dv]”.
Outside the shadow the integration extends over one half of the wave and part
of the other half, thus
CO †y t)
| cos try”dv-- | cos #Trv°dv=% + | cos $7tw”dv.
O O ()
U 2 U 2
I = | 3 + ſ COS Arvae -H ſ 3. +ſ sin arvae º
But inside the shadow the integration extends over part of half a wave, from v to
co, and
Hence outside
CO CO ty º
| cos #Trv°dv= ſ cos $7twºdy – ſ cos $7tvºdvº-3 – | cos #Tv°dv.
ty - 0 O O
Hence inside the shadow
2 {j 2
I = |: * | cos irº -- | ** | sin invas.
The cases of a narrow wire and narrow slit are treated in the same manner.
153. On the Seattering Action of very Small Particles, and
the Colour of Skylight.—In applying the wave theory to deduce the
ordinary laws of reflection of light (Chap. IV.) the linear dimensions
of the obstacle at which reflection occurs have been supposed enor-
mously great compared with the length of the reflected waves. For
this reason the ordinary laws of reflection involve no reference to the
wave length, and the screening action of the obstacle is perfect except
for a narrow band around the edge of the geometrical shadow. The
width of this band, however, and the intensity at any point of it,
depend on the wave length (being greater for the longer waves than
for the shorter), and consequently a small obstacle will not be so
effective as a screen to the longer waves as to the shorter.
Thus we have seen (Chap. III.) that an obstacle will screen a point
from the influence of a wave when it is wide enough to cover a large
number of half-period elements of the wave front, but the width of
any half-period element increases with the wave length, and conse-
ART, 153 ACTION OF SMALL PARTICLES 301
quently a given obstacle will act most effectively as a barrier to the
waves of shortest length. We are consequently prepared to admit
that when ordinary white light passes through a medium (such as
the atmosphere) in which very small particles are suspended, the
waves of greater length will be more freely transmitted than those
of higher refrangibility. After passing through a certain thickness
of such a medium the light from a white source should consequently
appear yellowish (like the snow on a distant mountain), and as the
thickness increases the tint should become reddish.
To say that the longer waves are most freely transmitted is
equivalent to saying that the shorter are most copiously reflected,
and we should therefore expect the light reflected (or scattered) in
any direction by such particles of matter to be rich in the rays of
higher refrangibility. Now the light which reaches us from the open
sky is that part of the light of the sun that has been reflected, or
scattered, by the fine particles of matter suspended in the atmosphere,
and if these particles are small compared with the wave length of
light, the tint of sky should belong to the blue end of the spectrum
rather than to the red.
In estimating the quality of this light, however, it must be re-
membered that it has suffered from the modifying action of trans-
mission as well as scattering. For it is clear that the light which
reaches the eye after scattering in a certain locality is merely the
residuum of the solar light which has survived after transmission
through a certain thickness of the air, as well as the scattering
action of the particles. Thus the light first passes through a certain
thickness of air and is modified by transmission. This modified
light is then reflected or scattered by certain particles, and the
scattered light is subsequently transmitted through some thickness
of air to the eye of the observer. Now we have seen that transmis-
sion is detrimental to the rays of higher refrangibility, while the
scattering cuts off those of lower refrangibility, and the light which
survives and reaches the eye will therefore be weak in both ends of
the spectrum. In other words, it will be composed chiefly of the
waves of intermediate length from the region of the blue or green.
The exact colour of course will depend on the size and plenitude of
the particles; for example, when the particles are relatively large
they will affect all wave lengths of light in practically the same
degree, and the scattered light will be white.
This explanation of the blue colour of the sky was proposed by
Lord Rayleigh' in 1871, and the law according to which the scattering
* The Hon. J. W. Strutt, Phil. Mag. vol. xli. pp. 107,275.
302 - ACTION OF SMALL PARTICLES OHAP. IX
takes place for waves of different lengths may be easily deduced from
elementary considerations. Thus we have seen (Art. 150) that if
the amplitude of the vibration contributed by an element of a wave
surface to any point be taken to vary directly as the element of area,
and inversely as the distance, then the complete expression for the
amplitude must contain AT", so that the intensity must vary inversely
as the square of the wave length. In the same way, if we consider
the light scattered in any portion of space laden with very small
particles, and if we assume that the amplitude of the vibration con-
tributed by any element of volume d'V of this space is proportional
to that element of volume, and inversely as the distance, then in the
expression and reasoning of Art. 150, dS must be replaced by dV,
and the complete expression for the amplitude must contain A-", so
that the intensity of the scattered light will vary inversely as the
fourth power of A; that is, when scattering alone is considered,
I
sº Xi'
I
Now the light which reaches the eye is scattered, and also transmitted
(both before and after scattering) through some thickness 2 of the
medium. Hence its composition will be determined by finding the
effect of transmission on the quantity Is. This problem is equivalent
to that of finding the intensity of a pencil of light after transmission
through an absorbing medium when the absorption varies inversely as
the fourth power of the wave length. Hence, as in Art. 282, we
have
, , ; } . . dI kdo:
T X4
where k is a constant, and dI the change of intensity in passing through
a layer of thickness do. The intensity of the light reaching the eye
after suffering scattering as well as transmission through a total thick-
ness aſ of the medium is therefore
I= I e-kw'A',
and substituting for I, we have finally
I = #. -kºlaº.
This expression exhibits the joint effects of scattering and transmission,
and shows how I diminishes for the large values of A as well as for
the small. The maximum value of I corresponds to some intermediate
wave length Am given by the equation
At =ka.
771, 2
ART. 154 NARROW RECTANGULAR APERTURE 303
and the corresponding maximum value of I is
In = A^n, *e-1 = Aſeka,
while the intensity I, corresponding to any wave length A, is related
to Im by the equation
- + = yel-V where y =[Am/\]*.
In order to test this theory Lord Rayleigh compared the blue
light of the sky taken from the neighbourhood of the zenith with
sunlight diffused through white paper. The results of this comparison
are contained in the following table for the fixed lines C, D, bs, F of
the spectrum, and, considering the difficulties and uncertainties of
the comparison, the observed and calculated values appear to agree
tolerably well. It must of course be observed that the comparison is
based on the assumption that the light diffused through white paper is
similar to the scattered light which illuminates the sky.
C. D. bº. F.
25 40 63 80 (calculated)
25 41 70 90 (observed)
It appears therefore that the skylight, when compared with that
diffused through white paper, was bluer than that required by the
theory; and this, Lord Rayleigh suggests, may arise from the possible
yellowness of the paper, or from the yellowness of the sunlight when
it reaches us compared with that in the upper regions of the atmo-
sphere where it is diffused. * . *
In a later paper Lord Rayleigh discusses the interesting question
as to what the particles are which cause the scattering of light in the
atmosphere. That small particles of saline or other matter (including
organic germs) play a part cannot be doubted, and to them may be
attributed much of the bluish haze by which the moderately distant
landscape is often suffused. . “But it seems certain that the very
molecules of the air themselves are competent to scatter a blue light
not very greatly inferior to what we receive.” Early estimates were
based upon a value for the number of molecules of air per cubic
centimetre, which has since been shown to be too small. Schustei
has pointed out that the modernly accepted number almost exactly
accounts for the degree of atmospheric transparency observed at high
elevations in the United States, apparently justifying to the full the
inference that the normal blue of the sky is due to molecular scattering.
C. FRAUNIHOFER FRINGES
154. Narrow Rectangular Aperture.—It is very easy to examine
by the graphic method the case in which the incident light is a parallel
304 FRAUNHOFER FRINGES CHAP. IX
beam, or, in other words, a plane wave. We shall first take the case
of a narrow rectangular aperture of width a, and at present we shall
omit all consideration of the length of the aperture
and investigate only those phenomena which arise
from the narrowness of its width.
Let AB (Fig. 135) be a cross section of the
aperture by a plane drawn at right angles to its
length, and let the incident light make an angle
90° – i with the width AB; it is required to deter-
mine the intensity of the illumination at any point
on the other side of the aperture when a lens is
Fig. 135. placed before the aperture, as in Fig. 113, so that
all the light which leaves the aperture parallel to a given direction
is focused at a single point. For this purpose let us take the beam
of diffracted light which leaves the slit in any given direction AX,
making an angle 90° - 6 with AB. Then it follows, as in Art. 136,
that the relative path retardation of the extreme rays AX and BY is
5–a (sin i + sin 0),
and their difference of phase is found by multiplying this by 27/A.
Now let the aperture be divided into a very great number of exceed-
ingly narrow strips of equal width, the lengths of the strips being
parallel to the length of the aperture. This is equivalent to dividing
AB into a great number of small ele-
ments of equal length, and since the light
is parallel it is clear that each of these
elements produces vibrations of equal
amplitude and corresponds to equal in-
crements of phase.” The problem there-
fore reduces to the calculation of the re-
sultant of a number of vibrations of
equal amplitudes and uniformly increas:
ing phases. Such a system, we have
already seen (Art. 45), gives a vibration curve of uniform curvature-
that is, a circle—and the resultant is consequently represented in
amplitude and phase by the length and direction of a chord OM (Fig.
136) of a circle. The intensity of the diffracted light consequently
Fig. 136.
1 The angles are taken this way in order to embrace the case in which the plane
of incidence is not the same as the plane of diffraction. Here i and 0 are the angles
which the rays make with a plane perpendicular to AB.
* Each term ought in strictness to be multiplied by a factor depending on the
obliquity, such as Stokes' factor (1+cos 6). When comparing the intensities for
different values of 6 it cannot correctly be omitted.


ART. 154 NARROW RECTANGULAR APERTURE 305
depends on the final position of the tracing-point M ; that is, upon
the phase difference of the extreme rays AX and BY. For example,
when this phase difference is an even multiple of T the intensity is
zero, for then the tracing-point coincides with O, and OM is zero.
Now if we denote the phase difference" of the extreme rays by 24, we
have 2% = OCM = 2+8/A, and therefore -
TO,
*=5 (sin i + sin 6).
But we also have
OM=2R sin #0CM=2R sin (p
where R is the radius of the circle. Further, if s be the length of the
arc OM, then by the manner in which the curve is plotted S must be
proportional to the width of the aperture. *
So that if we take the constant of proportionality to be unity we
may write
S = 0.
But s = 2Rºb, therefore 2R = aſh, and the expression for OM becomes
OM = asin q,
y
and the resultant intensity is measured by
- I = a” º
Cor. 1.-Since the line OM makes with OX an angle MOX = }MCO,
it follows that the phase of the resultant vibration is the same as that
of the vibration contributed by the middle strip of the aperture.
Hence if the vibration from B be represented by y = sin ot, that from
A will be y = sin(ot + 2b), and the equation of the resultant vibration
is
sin q,
3/= 6–4. sin (wt + q2).
Cor. 2.-If the light be incident perpendicularly on the aperture,
the intensity at any point of the diffraction pattern is proportional to
* It is to be remembered that the tracing-point may have described the circle
several times, and finally settled in the position M. The angle 24, is the whole
angle through which the radius CM has revolved, and the arcs referred to in the
text is the whole length of arc described by M.
It is also worthy of note that R varies from point to point of the screen. It is
* or the width of the aperture that remains constant, and it is for this reason that
the maxima are not determined by p=an odd multiple of T/2, as shown in Art. 155.
X
306 FRAUNHOFER FRINGES CHAP. IX
155. Determination of the Maxima and Minima.-The expression
for the intensity at any point, being a function of the angle 6 of dif-
fraction, will vary from point to point on the screen, and pass through
a series of maximum and minimum values. This has already been indi-
cated by the elementary examination of Art. 129, but we are now in
a position to inquire into the phenomena more accurately.
For brevity we have written * = (sin i + sin 0), and we have
found the intensity measured by a sinº/dº, hence as the angle 0 varies,
the intensity passes through a series of maximum and minimum values
as follows.
Minima.-I = 0 when q = nºt, excluding the value n = 0, which
corresponds to a maximum. Hence at points in the direction tº de-
termined by the equation
sin i + sin 6-m)/a.
there is complete darkness when n
is any integer other than zero.
Marima.-Equating to zero the
first derived of sin ºbſº, we find
that the values of q, which make I
a maximum satisfy the equation
p=tan p."
To solve this equation graphically,
plot first the curves
(1) y=a, and (2) y=tan a.
The first represents a line bisecting
the angle between the axes of a and
y (Fig. 137). The second consists
of an infinite number of branches
of amplitude T. The first branch AA passes through the origin O
and touches at infinity the lines w = + , T, which are its asymptotes.
The second branch BB cuts the axis of w at the point r = + and touches
at infinity the lines w = }T and z = }T. The maximum values of I
correspond to the values of y which satisfy both these equations, since
Fig. 137.
: .2 - -
1 The first derivative of *-*.*.*(* **) The condition
cos ºp-0 does not lead to either maxima or minima, because tan q is then infinite.
It is better to factorise the derivative thus: *(o: * ...) The conditions
sinºp
then are p” =0 for minima, and q cot q=1 for the maxima.

ART. 156 TWO EQUAL RECTILINEAR APERTURES 307
then we have a = tan a. The maxima are therefore determined by the
intersection of the line OC with the curves AA, BB, CC', etc. The
corresponding values of q are less than the odd multiples of T, but
as n increases the value of q approaches more and more nearly to
(2n + 1)}T. The values of q corresponding to the maxima values of
the illumination have been given by Schwerd' as follows:
po-0 In=1
ºple 1:4303r In-sinºpſºp,”
pº-2'45.90m- Ig-sin"pºſtpº
pa-3:47.09m. Is–sinºs/pº
p1=4:47.74m. I, esinºp/q,”
ps-5°48'18"r Is–sinºps/pº
pº- 6:4844T Id=sinºps/pº
prº-7-4865t In-sin"p:/pº
being approximately in the ratios of the quantities 1, (...)
37.
Thus, while ºn becomes more nearly (2n + 1)}T as n increases, the
corresponding values of I (the maxima illuminations) decrease rapidly,
(...)
etc., which correspond to ºf being equal to odd
multiples of T. If the intensity of the first
maximum be taken as unity, the values of the
second, third, and fourth will be approxi-
mately ºn, ºr, and Ho respectively. Fig.
138 represents the variations of intensity, its
abscissae being the angle q, and its ordinates
the corresponding intensities. The first maxi-
mum is very much greater than the others,
and these again diminish very rapidly. With
white light we have a series of rainbow-
coloured fringes, violet at their inner and red
at their outer edges. The spectra formed Fig. 138.
by a single narrow aperture Fraunhofer terms spectra of the first class.
156. Two Equal Rectilinear Apertures. – The same graphic
method may be applied with facility to the case of two very narrow
apertures, each of width a separated by an opaque interval of width b.
Let the apertures be OA and BC (Fig. 139). Then, as in Art. 154,
the effect of each aperture may be represented by an arc of a circle
of magnitude 2a, where
a-". (sin i +sin (),
and these arcs (Fig. 139) will be separated by an arc AB of magnitude
2/3 given by the equation
* Schwerd, Beugungserscheinungen (Mannheim, 1835).

308 FRAUNHOFER FRINGES CHAP. IX
8–' (sin i +sin 6).
The resultant amplitude due to each will be measured by the chord
OA or BC—that is, by a sin aſa-but as the resultant vibrations due
to OA and BC differ in phase by an amount
2(a + 3), namely, theangular distance between
their middle points, it follows by Art. 43
that the resultant intensity is measured by
…?
I– (2)”.” cos” (a +3).
The equation of the resultant vibration is
therefore
y=2a. *...* cos (a + 8) sin (wt +2a+8),
Fig. 139.
and this corresponds in phase to the middle strip of the opaque interval.
For the resultant vibration transmitted by OA is
1/1 = 0. ** sin (wt -- a),
and that transmitted by BC is
wea". “sin (w/ -- 3a-H 28).
But y = y + y, therefore, etc.
The intensity depends on two variable factors, one sin aſa, which
gives the fringes of a single aperture, and the other cos (a +/3), which
gives a system of fringes corresponding to the interference of the lights
from the two apertures. This factor vanishes when
(a +B)=(2n+1).
that is, when
(a + b)(sin i +sin 6)=(2n+ 1. -
In this case the light from the second aperture is an odd number of
half-period elements behind that from the first, and the two destroy
each other by interference. But if
a +3=nt,
Ol'
(a +b)(sin i +sin 0) = n\,
then the two are concordant, and the illumination is a maximum.
These are termed maxima and minima of the second kind, or spectra of
the second class.
We may therefore consider the phenomena observed as the super-

ART. 157 SMALL RECTANGULAR APERTURE - 309
position of these two systems of fringes; that due to the first factor, a
diffraction system, and that due to the second factor, an interference
system. The intensity is zero when either factor vanishes.
The dispersion of the second system being inversely as a + b is less
than that of the first system, which is inversely as a, it follows that,
when the apertures are not very close, the second system is nearly all
contained within the first two bands of the first system.
The points of maximum intensity consequently do not in general
coincide either with those of the first system or with those of the second.
157. Small Rectangular Aperture.—In the calculation of Art.
154 the diffraction pattern is considered only in so far as it depends
on the width of the aperture. In the case of a long narrow slit the
pattern consists of a system of rectilinear bands parallel to the length
of the slit, and these bands arise from the narrowness of the aperture.
The length of the slit, in fact, is so great that all diffraction effects in
this dimension are lost, for the whole effective portion of a strip of the
wave taken parallel to the length of the slit is transmitted when the
aperture is long. *
On the other hand, when the aperture is short as well as narrow,
so as to have the shape of a small rectangle of length a and width b,
then the limited length comes into operation and produces diffraction
effects. The whole effective portion of a wave strip taken parallel
to the length is not transmitted, but partly obstructed, by the
aperture. The pattern consists in fact of a system of bands parallel
to the length of the aperture, and also a system of bands parallel to
the width. The former arise from the limited width of the aperture,
and the latter from its limited length. In order to determine the
intensity, let the aperture be divided into a great number of very small
strips parallel to its length. Each of these strips will be of length a,
and will give rise to a vibration, at any point under consideration, of
amplitude (Art. 154)
sin c. TQ, , . . . .
A=aº. , where a=5. (sin i + sin 6), *
and the incident light makes an angle 90° – i with the length of the
slit, while the diffracted light makes an angle 90° - 6. We have now
to find the resultant of a great number of vibrations of amplitudes
A = a sin aſa and varying in phase from of to ot + 28, where
6–ºsin '--sin 6’),
90° – i' and 90° - 6' being the angles which the incident and dif-
fracted light make with the direction of the width of the slit. But
310 TALBOT'S BANDS CHAP. Ix
as before, the resultant amplitude of these will be All sin (3/8. Hence
the intensity of illumination will be measured by
o, sin” a sin”
I =a^*. .*
The illumination at any point therefore depends on two variable
factors, one of which gives rise to a series of bands parallel to the side
a of the aperture, while the other gives
a series of bands parallel to the side b of
the aperture. These lines enclose a
system of rectangles (Fig. 140) similar
to the aperture turned through 90°.
The greater the length of a side the
narrower are the bands perpendicular
to that side. It is thus we have only
one system of parallel fringes with a
long narrow slit, for the width of the slit is so small that the
bands parallel to the length are fairly broad, while those parallel to
the width are invisible on account of the length of the slit.
158. Talbot's Bands. The system of bands which are seen
crossing a tolerably pure spectrum, when it is viewed through a small
hole half covered with a thin transparent plate, has been mentioned
already in Art. 106. We are now in a position to deduce easily
the expression for the illumination at any point, the aperture being
supposed rectangular.
Let AB (Fig. 141) represent the plate covering half the aperture
AC. The effect of the illumination from AB will be represented by
Fig. 140.
Fig. 141.
Fig. 142.
an arc OA = 2a (Fig. 142) of a circle, but as the plate produces a
retardation, it follows that the ray from B (Fig. 141) which does not
traverse the plate will be accelerated relatively to that which passes
through the plate by some amount 26. Hence the effect of the free



ART. 159 DIFFRACTION GRATING 3.11
part BC of the aperture will be represented by the arc BC = 2a (Fig.
142) of the circle, where AB = 26, the retardation in the plate. It
follows easily that the inclination of the chords OA and BC is 2(a – 8),
consequently if they are each equal to p, their resultant is
2p cos(a-6). -
But by the preceding article p = absin a sin Bag, where ab is half
the area of the aperture, consequently the resultant illumination is
proportional to
sin” a sin”8
a 32
which is the formula deduced analytically by Airy." A table of
the values of this expression for various values of a and 3 was
constructed by Airy, and he also plotted curves showing the fluctuations
of intensity. This accounts for production of bands with mono-
chromatic light, but does not indicate why, in order that bands may
be formed, the slip must be held on a particular side of the opening.
We return to this question later (Art. 162b).
159. The Diffraction Grating—Any Number of Parallel, Equal,
and Equidistant Narrow Rectangular Apertures.—In the case of a
system of n very narrow equal apertures,
separated by equal opaque intervals of width
l, we have to find the resultant of a system
of amplitudes represented by the chords of
n arcs of a circle each of magnitude 2a,
and separated from each other by arcs each
equal to 2/3 (Fig. 143).
If we take any axes of reference OX and
OY, and if any one of the chords makes an
angle m with OX, the consecutive chord
will make an angle m + 2a + 2/3 = n + y sup-
pose, and the other chords will make angles n + 2y, m + 3), . . .
74 (n − 1)) respectively with OX. Hence if X denotes the sum of the
projections of all the chords on the axis OX, and if p be the length of
each chord, we have
X=p(cosm-i-cos (m+y)+cos(m+ 2y)+. . . cos (m+(n-1)}]
cos” (a – 6),
Fig. 143.
In the case of a long narrow aperture (Art. 150), such as we are now
considering, p is given by the equation
* Airy, Phil. Trans. p. 1, 1841.

312 DIFFRACTION GRATING CHAP, IX
Similarly if Y denotes the sum of the projections on OY, we have ,
Y=p(sin m + sin (m+y)+sin (m+2))+. . . sin {n+(n-1)-y}]
_sin {n++(n-1))}sin #7ty
=p sin #7 e
Hence 1
2, v2_a sinº #my
x+x =&#.
But X* + Y” is the square of the resultant amplitude. Consequently
the resultant intensity is measured by
_a sin” n(a+8)
T" sin” (a +8)
where p” is the intensity produced by a single aperture (Arts. 149-157).
Cor.—The phase d of the resultant vibration is given by the
equation
tan •=}=tan {n++(n-1))}
=tan {n+(n − 1)(a +8)};
it is consequently the same as the phase of the vibration from the
middle point of the grating. Hence if the equation of the vibration
from the first aperture of the grating be y = p sin ot, the equation of
the resultant vibration will be
sin m(a +3)
sin (a +8)
sin {at +(n-1)(a +8)}.
160. Determination of the Maxima and Minima Intensities.—
The expression for the intensity of the illumination produced at any
* Otherwise thus denoting V-1 by i, we have, since cos 0+i sin 0–9.
g & g º, º
x+y=pſe"+ jor),0+2), e º º l=P* "(1 -e”)
& 1–e"
Similarly -
º * - ?, – ?/m,
x-sy-aſe "+2". tº ſº tº j-pe "d-2")
- 1 — e – vy
Multiplying we find
iny —iny
x, ye-ſº-º-º-º-º: Q=ºos º-, sinº
(2– .” —eſ”) * 1 — cos y sin” #0
So also by addition we find
x=0 * {n+}(n-1)-y} sin #7)
=p sin #7 ’
and by subtraction
Y = sin {n+}(n-1))} sin #7 y
=p sin #7.
2
ART. 160 MAXIMA AND MINIMA INTENSITIES 3.13
point by a grating is the product of two variable factors; one pº
corresponds to the diffraction produced by a single aperture, and has
been already discussed. The other factor,
F, sin”(a + 8)
* sin” (a +8).”
also produces a series of maxima and minima corresponding to the
interference of the light from the various apertures. For brevity let
us write a = a + 8, then the bright and dark bands are determined by
equating to zero the first derived of sin” maſsin”/a:—that is, by
2 sin mac e g
: (n sin a cos na! — cos a sin na)=0
Slſh*Q}
; v.2
S] n°770;
=2...-(n cot na — cot a,
Sl]]*Q}
which is satisfied by
(1) (sin na)/(sin a =0, (2) m cot na =cot a..”
Minima.—In the first case the amplitude clearly vanishes and we
have a series of minima of zero value. They occur when sin na = 0
and sin a = 0. -
Principal Maxima.--—If, however, a = mir, both the numerator and
denominator of the expression sin na / sin a will vanish. Its true value,
however, will be m, so that the intensity will be a maximum and pro-
portional to n*. This corresponds to
a +3= mir, or (a+b)(sin i+sin 6)=m\.
These maxima are very intense and are termed principal maxima.
There are obviously n – 1 minima between two principal maxima.
Secondary Maajima.—The roots of the equation
m, cot mac- cot ac
other than a = mir (which correspond to the principal maxima) give
rise to another set of maxima termed secondary maxima, much less
intense than the principal maxima. From this equation we find for
any of the maxima
sin” na; m?
sinº, Ti-F(m?–1) sin” a.”
which shows that the ratio of these secondary maxima to the principal
maxima (n”) is
1
I+(n3–I) sinº'
* The usual expression is tan ma;=ntana: ; but this takes no account of the
& e p. Tr 3 Tr 5 * º
maxima which occur when *=3 −. º, etc., when n is odd, since for these
2
l
angles tan nº-n tan as and not n tan ac,
314 DIFFRACTION GRATING CHAP. IX
These secondary maxima are strongest when they are near the
-- - - 2n + 1
principal maxima, and they become zero or unity when w = ** – T
->
according as m is even or odd. Hence when n is large and the
principal maxima have intensity m” the secondary maxima nearly
midway are quite negligible. But this is not altogether the case for
the secondary maxima that are next-door neighbours of a principal
one. The following gives the ratio of intensities, and also the posi-
tions for the case of various numbers of openings:
Ratio of Intensities of
m. nºr. Principal to Adjacent
Secondary.
3 minºr 4-270° 9
4 ,, H-263° 37' 13-5
5 ,, .4261° 12' 16-0
15 l, 42.57° 50' 20-6
Infinite º, H257° 27' 21-2
Thus even with an infinite number of openings the secondary intensity
is nearly one-twentieth of the principal. It is at first surprising that
they are not seen except when n is small. But it must be remembered
that the intensities of actual spectral lines are themselves only feeble,
and one-twentieth their in-
tensity is below visibility.
The distribution for six
openings, as calculated by
Schwerd, is shown in Fig.
144.
The secondary maxima
may be determined by the intersections of the curves.
Fig. 144.
(1) y=n cot ma", and (2) y=cot w,
in a manner analogous to that employed in the case of a single
aperture (Art. 155).
The first equation represents a curve asymptotic to the line r = mir
while the second is a similar curve, or a set of similar curves
asymptotic to nº = mr where m is an integer.
The equation
(n+1) sin (n − 1)2 =(n-1) sin (n+1)2
also determines the maxima and is preferable, because it lends itself
more readily to graphical treatment.

ART. 161 IRREGULAR SPACING 315
The symmetry of a shows that the éffect of this factor remains un-
altered if the opaque portions be made transparent and the transparent
portions opaque, as this merely amounts to interchanging a and 3.
The resultant illumination at any point being determined by the
product of the two factors sin” aſa” and sin” maſsin” a, to obtain it we
must multiply the ordinates of the curve (Fig. 144) by the corresponding
ordinates of the curve relative to a single aperture (Fig. 138). The
variations of the latter are very feeble compared with those of the prin-
cipal maxima, so that they scarcely affect the position of a maximum.
If it should happen, however, that a zero value of sin aſa should
correspond to a principal maxi-
mum of the other curve, then
this maximum will be absent.
This will happen when

I
a(sin i + sin 6)= m\,
and also (a + b)(sin i+sin 6) = m/\;
Or 0, 790,
bT m' – 'm
where m and m' are whole
numbers, and the corresponding
principal maxima, or spectra,
are absent (see Art. 144). -
In Fig. 145 1S shown the | L/TN
actual distribution of intensity 6 T 27- 37T 3:
in the case of three openings Fig. 145.
separated by spaces equal to the width of the openings. It will be
seen that the spectra of even order are absent.
161. Any Number of Narrow Rectangular Apertures, Parallel
and Equal, but not Equidistant.—The foregoing calculation is based
on the supposition that the opaque intervals are of equal width, or, in
other words, that the grating has been ruled uniformly ; the investi-
gation may, however, be applied with the greatest facility to the case
in which the length b is variable, which is that of a system of equal
apertures placed at random distances apart.
Each aperture will be represented by an are 2g of a circle, and these
arcs will be arranged at random round the circumference of the circle :
that is, separated by variable intervals. Let the chords of the arcs 24
make angles m, m3, . . . m with a fixed axis OX, and let each chord
be p as before then, the sum of the projections of the chords on OX is
X=p(cos mi + cos ma--cos ma-F . . . cos mn),
316 IRREGULAR SPACING onar. IX
and the sum of their projections on the perpendicular axis OY is
Y=p(sin mi +sin m3+sin ma-H . . . sin mn).
Hence the square of the resultant amplitude is
X*+Y”=p^{n+2X cos (mi = m2)].
Now the sum X cos (mi = m2) embraces all combinations of the angles
mi, m2, . . . mm, and the values of its constituents vary irregularly,
taking random values between + 1 and — 1. Admitting that their
sum is negligible or sensibly zero (see Example, p. 50), we have
X?-- Y?=mp”,
or the intensity is n times that produced by a single aperture. Sub-
stituting for p from Art. 157, we have
I = ma?b? sin” a sin” 8 e
q.” 32
162. The Échelon Spectroscope.—We have seen (Art. 138) that
the dispersion of a grating varies directly as the order (n) of the spec-
trum and inversely as the distance (a + b) between consecutive rulings.
Consequently to increase the dispersion the rulings must be made closer
or else a spectrum of higher order must be observed. On account of
the extreme faintness of high order spectra the second alternative
afforded no promise of improvement in the construction of gratings
until Michelson investigated the subject.
He pointed out the conditions, depending
on the shape of the grooves made in
ruling, under which a reflecting grating
would concentrate most of the light
into particular group of spectra, and he
constructed a most ingenious and valuable
transmission grating by means of which
the same effect is produced. Michelson
took a number of plane-parallel plates of
optical glass of exactly the same thick-
ness (e) and built them up in the form of a stair, each step being of
the same depth (f). To ensure equality of thickness it is necessary
to cut them from a single plate whose faces have been made as
nearly as possible parallel to one another.
The échelon may be regarded as a grating in which there is a
series of openings each of width f, through which light is transmitted.
The light that is so transmitted in any one direction, 6, has a phase
:
P
:
R

--
A
Bl
Al
:
A>.
'o,
Fig. 145a.
* A. A. Michelson, The Astrophysical Journal, vol. viii. p. 37 (June 1898).
ART. 162 THE ÉCHELON SPECTROSCOPE 317
(in the focal plane of an objective) depending partly upon the thick-
ness of glass passed through, and partly on the value of 6. Let
PACQ and P.B.A.Q, represent rays of light incident normally upon
the plates and diffracted at corresponding points A and A1 on con-
secutive steps. Draw A/C at right angles to AQ. The retardation of
the second ray on the first is pl?,A1 – AC if p is the refractive index
of the glass. Now AC is equal to the sum of the projections of
AB, BIA, and A1C on its direction; that is, AC = – AB, sin 6 + B1A1
cos 0, or – f sin 6 + e cos 6. The phase difference is therefore
*. (pe +f sin 6 – e cos 6). This corresponds to 22 in Art. 160. Hence
sin”ma:
sin”a:
where n is the number of plates. This must be multiplied by the
the factor F, arising from the distribution of the openings is
factor due to each opening. If 20 = º • f sin 6 the final value for the
intensity is
I=f?. sin”a sin°ma:
*/ Tº sinº.
We may conclude therefore that principal maxima will be obtained
whenever a is an integral times it unless sin”aſa” vanishes; i.e. the m”
spectrum will be given by
m\=ple +f sin 6 – e cos 6
=(u – 1)e-Hºfb approximately
for small angles, which alone have any importance. Now a for a
f
single opening is ; sin 6 = . 6 approximately.
Thus
a=; [m) — (p − 1)2].
Let us suppose that e, p, and 6 are such that at the m” maximum a = 0,
then this spectrum will be as bright as possible. If we now examine
the (m + 1)” maximum, it will be found that a = T, and therefore the
value of sin a is necessarily zero, so that the principal maximum will
be absent (Arts. 140 and 160). This is true also for all the spectra
except the m”. Thus the light is concentrated into a single spectrum
as in the case of prisms. However, the condition that a = 0 for the
m” spectrum may not be quite satisfied. It is obvious, however, that
sin aſa has no great magnitude except, say, for a between + # • If the
m” maximum of F, corresponds to a =#, then the (m-1)” will
T g te tº
correspond to a = ; in this case these are the only two maxima
T 4 »
which have any considerable magnitude, and there cannot be more
3.18 ÉCHELON SPECTROSCOPE CHAP. IX
than two. We have considered the incidence to be normal ; when it
is not so, sin i + sin 6 (to a first approximation") must replace sin 6. It
is clear that we can bring about the satisfaction of either of the above
values of a by slightly inclining the grating ; i.e. altering the angle i.
The importance of this grating arises from its very high chromatic
resolving power. Since the equation sin”aſa”. sin”naſsin” is of the
same form as for the ordinary grating, the resolving power is also
given by the same formula as in that case if p can be taken as inde-
pendent of A; i.e. R.P. = nm. When 0 = 0, m = (p − 1)e/A. Let p = 1%,
e = 0.5 cm., A = 5 x 10−9 cms. ; then m = 5000.
If there are 20 plates, the resolving power is 100,000. A Rowland
grating, having 14,000 lines to the inch and ruled over 6 inches,
would only have a resolving power in the first order of 84,000.
Messrs. Adam Hilger have made échelons containing 50 plates.
The difference between the two types of grating can be displayed
by indicating that the Rowland produces comparatively large devia-
tion for even very low order spectra, but comparatively small
dispersion. In the case of the échelon several orders are visible at
once in the field of the telescope, while the dispersion for different
colours is very great. This is seen easily by examining the equation:
for the retardation. The value of m changes by unity for a change of
angle do = } (this gives the separation of two adjacent orders of
f
spectrum) while a change of wave-length dA changes the angle for a
given order by
7mdX e du
Ad=== - ? ;dX,
and the ratio
, .4%
A6 ºn-eſs -
— = &\.
d 6 \
According to the usual convention, resolution takes place when A0
is d6|n; hence the resolving power is
N Op.
#=n(m -*. $).
The term involving the variation of p is negligible in practice.
The overlapping of the spectra is overcome by isolating the spectral
line under examination by means of an ordinary spectroscope. The
eye-piece is removed, and a slit placed in the focal plane of the
1 The exact expression for a when the incident angle is i is TſN}e(p, cos r-cos ?)
+f (sin i +sin 6)} where r is the internal angle in each plate. It is doubtful, how-
ever, whether anything can be gained by estimating this to more than the first order
of small quantities owing to other complications which have been neglected.
ART. 1620, LUMMER-GEHRCKE INTERFEROMETER 319
objective allows only one such line to pass through a collimator to the
échelon grating. A slit is necessary because, as with the ordinary
grating, the coloured images are images of the source.
162a. Lummer - Gehreke Interferometer.—In 1903* Lummer
and Gehrcke introduced an apparatus which may be regarded as an
alternative to a Fabry and Perot interferometer. In the Fabry
instrument (with two parallel plates) the increased reflecting power
which is necessary to bring about high resolving power is obtained
by half-silvering the surfaces. The new apparatus consists of a single
plate with a right-angle prism cemented to it at one end. Light

–Hº-
→=
admitted at right-angles to the hypotenuse of the prism is multiply
reflected at the two surfaces of the plate, part of it emerging into the
air at each incidence upon each surface. The prism serves to prevent
the large loss of light which would occur if the light were incident
very obliquely. Two groups of parallel beams are thus obtained, to
each of which the equation for a transmitted system (Art. 114)
applies, viz.:
(1-a2N)2+ 4a2N sin? (*)
IT = (1 – a”)” º
(1 – a”)?-- 4a” sin";
The rays will emerge at a nearly grazing angle if the internal angle
of incidence, r, is little less than the critical angle. But it can be
shown that the reflecting power, a”, in such a case approaches the
value unity (see Chap. XIII.), and it is on this fact that the success
of the device depends. The result that was otherwise gained by
silvering is here gained by altering the angle of incidence. For a
given angle i the reflecting power depends upon how the light is
polarised; it takes either the value sin”(; – r)/sin”(i + 1) or the value
tan”(; – r)/tan”(; + 1) according to whether the plane of polarisation of
the incident light is parallel or perpendicular to the plane of incidence.
It is somewhat greater in the former case.
* Ann, d. Phys, w. Chem., 4th ser. vol. x. 457.
320 TALBOT'S BANDS -- CHAP. IX
th.
If we assume resolution will occur when the m” order maximum
for a second colour is at the (m+.
grating," the chromatic resolving power will be given as for a grating
C.R.P. = Nºm.
Now N is approximately l/(2e tan r) and m is approximately
(2pe cos r)/A; whence
h ſº
) for the first, as in the case of a
l 1 — sin” r
C.R.P. =X*-in-.
If we take r as nearly the critical angle sin r = 1/p, whence
- , l
C.R. P. #(º- 1),
which is independent of the thickness. For l = 10 cms, p = 1.5, and
A = 5 x 10-5 cms. C.R.P. = 225,000. This is an upper limit for this
length of plate.
Needless to say, the plate must be very accurately a parallel plate.
With the first ones made, a number of supernumerary images
(“ghosts”) appeared, and special means had to be employed in order to
distinguish true from false. Plates made by Hilger do not show these
defects. It is used in the same way as an échelon for the resolution
of isolated spectral lines. But no narrow slit is necessary; indeed, if
one were employed, only a
series of point images would

| | | be obtained. To obtain a
X X X X X system of lines light must
- .2 – 1 5/60-75 -- . 1 + 2 85
: º be allowed to fall from
various points of the source,
as in the case of Haidinger's fringes, such as are produced in a Fabry
and Perot's interferometer. With the plate placed horizontally, and
a vertical slit, the spectral maxima will be horizontal lines the sharp-
ness of which does not depend on the slit ; the lengths of these
maxima will be proportional to the width of the slit. In Fig. 145c
is shown the resolution of the mercury green line into components,
the length of the lines indicating their relative intensities.
162b. Talbot's Bands.-We are now in a position to examine
further the formation of Talbot's bands. This becomes easy when it
is realised that the arrangement is in reality an échelon grating of
two openings, and therefore the distribution of intensity in the focal
plane of a lens is
sin” a sin”. 2a:
I = 7-2 . —F-
q.” sin”a:
* This would follow at once if the minima were zeros: they become more and
more nearly so as a” approaches unity.
ART. 162b TALBOT'S BANDS 321
where, for small angles,
a={r(i+0), and •=#| || * 1)+r(4)]=s+a.
If for any particular wave length 8 = mr where m is an integer
and a = 0, I will be the greatest maximum for that wave length. For
a somewhat longer wave length,
A + dA, the value of 8 will be differ.
ent. The value of 8 varies chiefly
with A ; there is in addition a small
variation due to change of p, which
we will neglect. The value of 8+ a
may still be maintained unchanged if
the angle of incidence, i, be changed
according to the equation
da_ _dB
tº NT ºx" Fig. 145a.
Owing to 6 being a fixed direction and i + 9 sen sibly zero, this equation is
practically
dź (9 - 1)6
2XT TXr '
which is a positive quantity. Thus, in order for the chief maximum
for each of the two wave lengths to be in juxtaposition, the value
of i must increase with the wave length. Such a change in i can be
brought about by appropriately placing a prism in advance of the
grating. If it is placed in the position shown in the diagram it is at
once seen that i will be greater for red light than for blue. If it were
inverted so that the refracting edge was toward the point A the
condition could not be satisfied ; the prism would in fact tend to
separate these maxima, and the resultant effect would be uniform
distribution of light; i.e. there would be no bands.
For a prism of the same kind of glass as the slip and of angle A
the deviation is g
d; dD du (u-1)e
D=(u-)a, and K--...--A;=*.
to satisfy the required condition. Hence for a prism of given angle
a definite proportion must exist between e and r.
Even then perfect blackness will not in general be obtained, so far
as the range of wave length, dA, is concerned, for this would require
that the interval between two successive maxima should be constant
for the range. Following the method of examination which we have
previously employed, since
Y

322 TALBOT'S BANDS OHAP. IX
n\ = (p. * 1)e,
and
(m+1)\=(u – 1)e + r(61 + i)
where 6, is position of the (m + 1)” spectrum for A ;
therefore -
X = r(91+ i).
If the (m + 1)” spectrum for A + dA is in the same position, 61,
X+d) = r(91+ i + di),
Or
ax=rdi-ºx.
This requires that m = 1; so that a very thin slip would be required
to make the adjacent maxima correspond.
1626. Resolving Power of Prisms.—From the point of view of the
general theory of diffraction a prism must be regarded as a diffraction
grating having a single opening. The distribution of light in the
focal plane of a lens when a parallel beam
is incident is therefore rºsin”uſu” where
u = [T/A]{(i+ 6)r — (p − 1)e}.
All angles are here assumed to be small.
For the wave length, A, the chief maximum
is at 6 where i + 6 = (u – 1)e/r. Since eſr
is the angle of the prism, this position corre-
sponds to that calculated for the image by the
methods of geometric optics. On the diffrac-
tion theory, however, this image is accom-
panied by lateral spectra (on each side). The first zero is when
w = ºr, and its separation from the centre of the chief image is there-
fore A9 = A/r. The chief image for a neighbouring wave length,
X + d \, is given by (i + 0 + d6)r = (p − 1)é;
whence
Fig. 145e.
_*-*. *
d6 = —- e=;du.
On equating d6 to — A6, in accordance with the usual convention
in regard to chromatic resolving power, we arrive at the following
equation :
Chromatic resolving power =X/d\= - §
The same result is true when none of the angles are small. We
have proved (p. 127, Ex. 15) that
cos r cos ?
is the general formula for the angular dispersion due to a prism. If

ART. 1626 RESOLVING POWER OF PRISMS 323
the difference of length of the extreme paths through the prism be
e and the breadth of the emerging beam be B it is easily shown that
the value of di'ldu given above equals eſb. The diffraction effects will
be those due to an opening of breadth B when the light is incident
normally upon it; consequently A6 = A/B instead of as before.
Finally, putting di' = – A6 the same value for the C.R.P. is obtained.
Lord Rayleigh has also obtained the same result by another
method. If B, and B are two positions of the wave front for light
of wave length. A before and after refraction, then every ray takes the
same time in passing from one to the other ; or if we reckon the
total optical path for each ray (i.e. each length of path multiplied by
Fig. 145f.
the refractive index) the sum represents this time and will be the
same for each ray. Hence
PA+AQ=PM 4-pe--NQ'.
Now calculate the optical length from P to Q for a wave length
of refractive index u + du. We do not need to determine the
precise path that this ray will take. Owing to Fermat's principle
the time taken by it is a minimum (when comparison is made with
neighbouring paths), and therefore we can calculate it for the neigh-
bouring path PAQ without introducing an error of more than the
second order. This path is therefore sensibly the same as for A. The
beam will be inclined to the first, one, and the new wave front
through Q will take up such a position as QQ", and we have
P'M+(u, + du)e +NQ” = PA + AQ=P'M+pe 4-NQ’;
whence - S. -
Ai Q” - Q = sº edu, •-->
and the angle turned through is
Q” - Q' ** &
*—B--- du.
The remainder of the proof is the same as before. When the
beam does not extend to the refracting edge of the prism, e equals

324 & DIFFRACTION - CHAP. IX
the difference between the thicknesses of glass passed through by the
extreme rays. - $
As an example, take a heavy silica flint glass the refractive index
for A 4861 ÅU being 1771 and for A 5893 being 1751. To separate
the sodium orange lines, for which A/d\ = 1000, would require a prism
with a base of 0-5 cm. In practice, when allowance is made for the
slit-width even this width of base would be insufficient.
A number, N, of prisms placed in line with one another would
constitute a grating which would have N times the resolving power of
a single one for which again the light would be concentrated, in the
main, in one spectrum as in the case of the échelon.
162d. Effect of Inequalities.—The values found for the various
cases are ideal limits; many causes will tend to reduce the resolving
power obtained. In the case of gratings any inequality in the ruling,
such as is brought about by variations of temperature during the ruling
process, tends to broaden each spectral image. In bad cases this
even gives rise to supernumerary lines. For example, if half the
grating has a different spacing from the other half all the spectral
lines will be duplicated. -
In use, any inequality in illumination acts, in general, in the same
direction. For example, if the amplitude of the incident wave changes
according to the cosine law from unity at one edge to zero at the
other, the integral for the case of one opening becomes of the form
Y = ſ"cos ma; cos (wt — q2 sin 6)da, [where 7)0.7° = º
-> ;ſ "cos [wt – (g + m)assin 6]da;+ ;|"cos [wt – (q – m)a sin 6]da:
- =}{*cos (wt – wi)+
sin wo
101 Qſ,
cos (wt-ntº) }
2 **
where
2u, - (q+m}r sin 6, and 2wo- (q - m)r sin 6.
The intensity is therefore
} *******-*)
instead of the first term only. In the case supposed [mr = T/4] the
third term is zero and each spectral line would be doubled.
163. Diffraction in Optical Instruments—Circular Aperture.—
In many optical instruments the aperture through which the light
enters is limited by a circular stop, and for this reason the study of
diffraction in the case of a circular aperture has attracted special
attention.
(a) Telescopes—For example, when a point source of light, such as
Art. 163 CIRCULAR APERTURE 325
a distant star, is viewed through a telescope, the wave falling upon the
object glass is limited by a circular aperture, and instead of having a
point image of the source (as the geometrical theory would lead us to
expect with an aplanatic lens) in the focal plane of the object glass, there
is a diffraction pattern' which consists of a bright central spot surrounded
by a series of rings alternately bright and dark. With white light the
central spot is approximately white and the rings are coloured. The
brilliancy of the rings diminishes rapidly from the centre outwards.
For the exhibition of several rings a strong source of light is required,
and with a faint source only the central spot can be observed. The
diameter of the central spot in all cases varies inversely as the diameter
of the aperture.
Let the incident light be parallel and fall normally on the aperture,
and let the diffracted pencil make an
angle 0 with the normal to its plane.
Let C be the central point of the
aperture, CN a normal to its plane,
and CO the direction of the diffracted
beam, then OCN = 0, and the plane of
diffraction at C–that is, the plane of
CO and CN–meets the plane of the
aperture in a line A.B. Let P be any
point of the aperture, and let the
radius vector CP(= p) make an angle q, with AC. Then the element
of area at P is
Fig. 146.
dS=pdqdp,
and the phase retardation 8 of the vibration from this element, relative
to that from A, is the same as for that from D, where PD is perpen-
dicular to AB, and therefore **
5– AD sin * * sin 6. (r-p cos ºp)=h(r-p cos p)
2m . - -
where h = º sin (), and r is the radius of the aperture. Hence the
vibration excited by the element dS at P is proportional to
sin (wt +h(r-p cos p)}pdºpdp,
and the resultant vibration for the complete aperture will be
2m ºr
| ſ p sin (wt +hr-hp cos p)dºpºp
--> -- 2m ºr 2m fºr
=sin (wt +hr) ſ | p cos (hp cos p)dºpºp-cos (wt +hr) ſ | p sin (hp cos ºp) dºpºp.
- - o -
The second double integral on the right is zero, for the elements of
* Noticed by W. Herschel in 1782, Phil. Trans. Roy. Soc. p. 52, 1805.

326 DIFFRACTION OHAP. IX
it which arise from any two points situated at equal distances on
opposite sides of C are of opposite signs and destroy each other.
Hence the resultant vibration is -
2T 7.
sin (wt + hºr) | | p cos (hp cos ºp)dºdp,
and the intensity in the direction CO is
I = [ | 27r|º cos (hp cos oned
The integration with respect to p is obtained at once by parts,
*:: *
* thus— h
7. _|_P º aſ 1 |r .
ſ p cos (hp cos *-kāraºp sin (hp cos *| h cosº Jo sin (hp cos p)dp
G
sin (hºr cos p) + wºrst cos (hºr cos p) — r)
–––.
Th cos p
sin (hºr cos b) issiºr cos p)
= 7-2 •--------- e
(hr cos ºp) (#hr cos ºp)”
- tº . e. 2Trr .
Hence, writing 2m = hr = ** sin 6, we have
2T sin (2m cos p) o ſ2m. sin'on cos p), , Tl2
— I a.2 ºm- - *-sº-
1-[aſºae-isſºl.
The further integration, with respect to q, may be obtained in
series, for we have -
sin as 3:2 a.4 0.6
Tag T = 1 - B + 5 * 7 + . . . .
sin” a 1-cos 22 2*a*, 2*a*_2'a"
# - Tº T- H 6 s.
Therefore -
2tr (2m cos (p)”, (2m, cos p)*
—,2 ſ” ſ 1 –
WI=r o {l 3. + |5. e e is ..}dº
- 2m. 2°(m cos (p)*, 2*(m cos p)*
*ſ {1- |4 + |6. - * tº e }de.
But we have
5. . . (2n − 1)
. 4. 6 . . . 27.
2tr,
therefore finally
- 1 / m\? 1/m2\2 1/2n3\2 1/m3\? -
Ji-rr'ſ 1-#(#) +(;) -(e) +}(;) tº e e }.
Or denoting the series within the bracket by S, we have
º I=(irrº)2S3,
which is the result obtained by Airy.
* G. B. Airy, Camb. Phil. Trans. p. 283, 1834. The series s—ſº where
J1(2m) is Bessel's function of order unity.
ART. 163 CIRCULAR APERTURE - 327
The series S is convergent for all values of m and passes alternately
through positive and negative values as m increases from zero. The
intensity consequently presents maximum and zero values, correspond-
ing to # = 0, and S = 0, respectively; and when the corresponding
values of m have been found, the deviation 6 is given by the equation
Trr sin 6 -> ^n\
777, - —, or sin 6 = −.
X T7"
Hence the deviation corresponding to any bright or dark ring,
being small, is proportional to A and inversely as the radius of the
aperture. The intensity is proportional to the square of the area of
the opening.
The subjoined table taken from Verdet's Optique Physique contains
the values of m/T corresponding to the first few maxima and minima.
The table shows the rapidity with which the maxima decrease, and
also that the difference between two consecutive values of m tends to
become constant and equal to #t.

"m/tr. Intensity S2. m/m. intº
1st max. 0 1 1st min. 0 °610 0
2nd max. 0 '819 0 °01745 2nd Imin. 1 * 116 0
3rd max. 1-333 0-00415 3rd min. 1 *619 0
4th max. 1 ‘847 0 001.65 4th min. 2:120 0
5th max. 2-361 O'00078 5th min. 2’621 0
In the foregoing equation 6 is obviously the angular width of a
ring as seen from the optical centre, and as this is very small we may
write 6 for sin 6. Hence the angular width of the first dark ring is
given by the equation
0–0.61%.
When two very close point-sources of light are viewed through a
telescope their diffraction patterns will overlap, and they cannot be
distinguished as distinct sources if the overlapping of the central spots
of their images exceeds a certain limit. Now the intensity falls from
unity at the centre of the spot to 0°37 at a distance from the centre
equal to half the radius of the first dark ring. Consequently if the
distance between the centres of the two central spots is equal to the
radius of the first dark ring the intensity should be 0.74 half-way
between the two centres and unity at each centre. This variation of
intensity should be easily observable, and a double star should there-
fore be resolved by a telescope when the angular separation of the
328 DIFFRACTION CHAP. IX
components is 0-61 A/r. In reality, observations on stars through
telescopes show that a double star can be recognised as double when
the stars are separated about 20 per cent. less than the above
conventional limit. *
(b) Microscopes.—The consideration of the resolving power of
microscopes is based on the same principles. The essential differences
are that the stars are at great unknown distances, and it is their
angular separation that can be dealt with ; further, the objective
subtends a small angle at a star. In the microscope problem the
object is “accessible,” and its actual size is the important datum ;
moreover, the lens subtends a large angle at it. The particular
approximations that are made in the case of the telescope no longer
apply. The following method of attacking the problem is due to
Lord Rayleigh : If PP' is small object the rays from P meet in an
Fig. 146a.
image-point at Q. At this image-point the various rays that meet
are isophasal. From the point P' rays proceed and are diffracted to
Q, the extreme rays having traversed the paths P'AQ and P'A'Q.
Their paths differ therefore by 2PP' sin 0, as is easily seen if perpen-
diculars are dropped on the dotted lines from P' and P. If the whole
difference is equal to A one half of the beam will cancel the other half
very nearly and the effect at Q will be zero, or nearly so. According
to the usual convention we may consider that the points PP' will then
be clearly seen separate in the image-plane. Hence the smallest
distance apart of points for which they will be recognisable as distinct
is PP =; i. 3 or thereabouts. (This can be compared with the value
for a telescope if we write PP' = ua and AA’= 2w sin 9.) If the
mediums in the object space have a refractive index p the same
expression holds good, but A is the wave length in that space.
No
Writing the standard value \o we have PP = ºrg.
2p sin 6
The geometrical

ART. 163 CIRCULAR APERTURE 329
resolving power is taken as the reciprocal of this and is equal to
2p sin 6
No
The resolving power is increased by increasing it, and by diminishing
the wave length of the light employed. The increase due to p is
one of the objects gained by employing an immersion lens—oil of
cedar-wood or bromonaphthalene being placed so as to fill up the gap
between the objective and the cover-glass.
It should be noted that the points P and P' have been treated as
though they were independent sources (like two stars). A microscope
object is, however, not self-luminous, and it is necessary to consider
how it is illuminated. If the
condenser focuses a perfect
image of the actualsource(flame,
etc.) upon the object, each
point on the object receives
light only from one point of
the source, and so also for
other points. In this case the
light from P has no per-
manent phase relationship
with that from P'. The case
is analogous to that of two
stars or two candles—the in-
tensities of their contributions
at Q must be simply added
together (Art. 44, Example). Fig. 146b.
If the source shone directly
upon the object each point of the source would illuminate a wide
extent of the object, and there is consequently a permanent phase
relation between the contributions which the lens receives from neigh-
bouring points. Fig. 146b is drawn to illustrate the differences. The
dotted lines show the images from two neighbouring points, themaximum
of each being at the same place as the zero of the other. If the lights
are independent, the sum of the ordinates gives the resultant intensity
(Curve A). It will be seen that there is a pronounced minimum at A
—the points should be seen as distinct points. If a permanent rela-
tion existed at P and P' the resultant at Q would have been as shown
by Curve B, and there is no indication of duplicity. Thus a condenser,
properly used, increases the resolving power—nearly doubling it in fact.
Abbe developed an alternative way of regarding the problem
which is very instructive. He took as his representative object a
The term p sin 9 is called the numerical aperture (N.A.).

330 DIFFRACTION OHAP. IX
grating, and investigated the spectra which it would produce in the
back focal plane of the objective. These spectra are Fraunhofer
spectra. They make angles with the axis given by PA = e sin 6 when
the light is incident normally on the grating. If due to a slight
obliquity only one spectrum was formed and sent light to the eye-
piece, uniform illumination would be seen ; i.e. there would be no
indication of the object having a structure. If two spectra are
formed and the light from them enters the eye-piece, a set of fringes
such as are formed by Fresnel's mirror will be seen ; the image has
now a structure similar in its periodical rulings to the object itself.
In other words, we have the first beginnings of an image. It is very
imperfect, however, and has no definite localisation; the eye-piece may
be withdrawn considerable distances without sensibly changing them :
except in size there is in fact no focus. If three or more spectra contri-
bute, localisation of the fringes is introduced, and we now have an image
if we focus for a particular plane; moreover, it is more nearly a perfect
image. Abbe rightly considered that in order to get a perfect image
all the diffracted rays would have to be gathered together in appro-
priate phases. Indeed the principle of reversibility requires that this
should be so.
The distance from the objective of the plane in which the fringes
are localised is connected with the spacing of the fringes by the Ordinary
Tules of geometric optics, with the spacing and distance of the grating-
object.
Abbe's condition for the limit of resolution, as quoted above, is
twice that of the value obtained by Rayleigh's method ; i.e. the
resolving power is half of Rayleigh's value. This is a consequence
of his grating-object not behaving as a self-luminous object.
When a given resolving power has been obtained it does not
follow that the detail resolved will actually be seen. It is the “thing
seen * with which we are concerned, and this depends upon who sees.
The human element enters, and in consequence it is not possible to
make exact statements. All that it is useful to get is a rough
estimate by which the quality of optical instruments can be com-
pared. Now the eye does not distinguish detail finer than that
which subtends 1 or 2 minutes of arc ; the division lines on a scale,
for example, are not detectable unless the distance between two
neighbours subtends this angle. If it does not, the angle must be
magnified by means of a lens. The permissible magnification may be
considerably greater than that which gives the above angle. We do
not place a scale just at the limit at which the divisions are seen ;
advantage is gained often by bringing them much nearer. In the
ART. 163 - CIRCULAR APERTURE 331
case of a microscope a magnification of 1000 for N.A. = 1, or 1500 for
N.A. 1-5 is permissible. But it scarcely needs to be said that no
amount of magnification will reveal the detail if, owing to insufficiency
of resolving power, the detail is not there.
The attainment of as close an approach as possible to perfect
images is limited by the extent of the elimination of all the aberra- "
tions calculable by the methods of geometrical optics. Professor
A. E. Conrady emphasises the fact that extension of numerical
aperture has surpassed the value warranted by the existing design
and construction of lenses. The same may be said of condensers
even more emphatically. Moreover, the action of a condenser is
modified by the presence of a “slide” of very imperfect optical quality
through which the light must pass. Again, with biological specimens
the objective “focuses” a thin layer only. If this happens to be near
the top surface of the specimen the light from the condenser is scat-
tered by the layers beneath it ; if it is near the bottom, the same
statement applies to the emergent light. . In either case there must
be diminution of resolution.
(c) Ultramicroscopy.—The above considerations give no indication
of the visibility of isolated particles, but only of the possibility of
detecting detail in them ; for example, their shape. If each gives
sufficient light (either by self-luminosity or by illumination by means
of a powerful beam athwart the line of vision), the particle will be
seen. This latter mode of illumination is employed in what is known
as ultramicroscopy. The visible disc seen is certainly much larger
than the geometric image of the particle: it is a diffraction image
from which comparatively little can be learned owing to difficulties
of interpretation. Similarly, an isolated luminous line gives an image
much wider than its geometric image. A dark line on a bright back-
ground (the light of which is all of one phase) may well remain
visible when the width of the line is only one thirty-second part of
the minimum distance required for resolution.
Rayleigh describes the following experiment : “In front of the
naked eye was held a piece of copper foil perforated by a fine needle
hole. Observed through this, the structure of some gauze just dis-
appeared at a distance from the eye equal to 17 inches, the gauze
containing 46 meshes to the inch. On the other hand a single wire
'034 in. in diameter remained fairly visible up to a distance of 20
feet. The ratio between the angles subtended by the periodic struc-
ture of the gauze and the diameter of the wire was thus
•022 240 9.1.”
O34 T7 T
332 DIFFRACTION CHAP. IX
This experiment illustrates the need of caution in the interpre-
tation of (e.g.) fine lines of pearlite on a dark background formed by
the rest of a metallurgical specimen. -
163a. Measurement of Stellar Diameters.-Lord Rayleigh has
shown that the resolving power of a telescope objective would be
increased by stopping out all but a peripheral zone of it. The
central diffraction disc is thereby made smaller; i.e. the radius of the
first dark ring is reduced. Unfortunately, at the same time, the
relative intensity of the first bright ring becomes brighter, and hence
more confusion results between the overlapping diffraction figures
formed by the light from neighbouring stars.
Michelson has adapted this restriction of aperture to the measure-
ment of the angular diameter of stars. Instead of employing the
whole of such a peripheral zone, let the whole of the lens be stopped
out except two small patches at opposite ends of a horizontal
diameter; as a particular case, let these patches be rectangles of
horizontal breadth a, and let b be the distance between their centres.
This is equivalent to a diffraction grating of two openings. Consider
the variation of the intensity along a horizontal diameter of the image
in the principal focal plane. For any point source at infinity on the
axis of the telescope the intensity is given by &
sin”u, sin”2v
I = a2 nº sinºv
where 2u = qasin 6 and 2v = qb sin 6, and where 6 is the angular
distance from the principal focus.
If a second luminous point exists so that the two subtend an
angle i at the objective, the intensity due to it is
, .2 sin”v' sin”2v'
I'=*.*.*.
where 2u =qa (sin 6 + sini) and 2v'-qb (sin 6 + sin i).
The two sources being independent of each other, the resulting
intensity is
I+ I’= a” sin”, sinºv, sin?w' sin”2v'
w?" sin?” 202' sin”v'
Since a is taken small compared with b we may neglect the difference
between u and uſ, and the intensity for different values of 9 is
proportional to
sin”2ny sin”2^)/
sinºv " sinºy’
i.e. to
2+ cos 2w + cos 2w'.
There will thus, in general, be variations of intensity along the
ART. 1630, ASTRONOMICAL APPLICATIONS 333
horizontal diameter. These fluctuations will, however, disappear if
cos 20/= – cos 20 and be replaced by uniform illumination proportional
to 2 ; because this condition requires that w" – v, i.e. qb sin i, shall be
an odd multiple of a right angle, and ql sin i depends only upon b and
i and is independent of 6 (which defines any particular point on the
diameter); so that when the intensity is 2 at one point of the
diameter it is the same value at all. Variation of b can bring about
the satisfaction of this condition; in all other cases fringes appear.
Thus for a given value of b the fringes will first disappear when
sin i–A.
4
This allows two stars to be judged as being two when about ten
times as close as the limit found for the same objective without any
diaphragm. It should be observed that the inference is only an
indirect one—the stars themselves are not seen separate.
Similar observations enable even the angular diameter of a single
star to be estimated ; but to effect the estimation it is necessary to
integrate the effects arising from all the elements into which the
complete disc of a star may be divided.
Michelson has still further increased the sensitiveness by placing
two mirrors at 45°, one in front of each of the apertures, and two
other mirrors, also at 45°, one on each side, so as to catch light from
the star and reflect it to the former pair. The extreme rays so
collected make with one another the angle that the distant pair
subtends at the star. The vanishing of the fringes depends upon the
distance between the distant pair, although the angular distance
between the fringes (when they appear) still depends upon the distance
between the apertures. Examining a Orionis Michelson found that
the fringes vanished with the outside mirrors 121 inches apart. From
this he inferred that the angular diameter of that star is about 047
seconds of arc.
Eaſamples
1. Taking AB (Fig. 146) and a perpendicular to it as axes of a and y respectively,
show that with an aperture of any form, the intensity at O in the direction CO
making an angle 6 with the wave normal is
2 2
I=[ſy sin(** sin º). +[ſues (** sin •)a.j e
[Divide the aperture into narrow strips perpendicular to the axis of 2. The area
of one of these strips is yda, the length of the strip being y and its width day. Also
if the phase of the vibration from the origin of co-ordinates be wt, the phase retarda-
tion of the vibration from the element yda, will be ** sin 6, and hence it will be
334 - DIFFRACTION CHAP. IX
represented by
3/ sin(**** sin º).
The resultant of the whole aperture will be
ſº sin(**** sin •),
=sin otſu cos(** sin •), too. otſu sin(*. sin 0).
Therefore, etc.] -
2. The sine of the deviation (sin 6) corresponding to a maximum or minimum
intensity (Ex. 1) is proportional to the wave length A. Hence the linear dimensions
of the patterns produced by differently coloured homogeneous lights are proportional
to the wave lengths.
3. With similar apertures the sines of the deviations (sin 6) corresponding to a
maximum or minimum intensity of a given order are inversely as the linear dimen-
sions of the apertures.
[For let m be the ratio of the corresponding lines of two apertures, and the
expression for the intensity given by one aperture be that of Ex. 1, the expression for
the other will be derived by substituting ma;, and my for a, and y, viz.
-12 2
[ſº sin (ºn, sin o)as -- [ ſnºw cos(*n. sin •), tº
Hence if 9, makes the first a maximum or minimum, the second will be made a
maximum or minimum by 62, if m sin 62=sin 61, that is, if
sin 61 m
sin 92. T 1
4. The intensities transmitted to corresponding points by two similar and
similarly situated apertures are in a constant ratio.
[This follows from the expression of Ex. 3, the constant ratio being mº.]
5. If one aperture can be obtained from another by displacing parallel to them-
selves its ordinates (y) without altering their lengths, the intensities in the plane az
are the same for the same directions.
6. The intensity at any point due to a system of equal, similar, and similarly
situated apertures is equal to the intensity produced by a single aperture multi-
plied by that produced by a system of points similarly situated on the apertures,
one on each.
[Consider a point situated on one of the apertures and take a point situated simi-
larly on each of the other apertures. The vibration produced by this system of
points will be of the form. A sin (wt +6). As the chosen point moves over the first
aperture, the corresponding points will move over the other apertures and obviously
A will remain constant.
Hence if the intensity produced by a single aperture, or by a system of similarly
situated points on the system of apertures, be zero, the intensity produced by the
whole system will be zero. In this case then there will be in general two series
of minima.]
7. Show that the pattern produced by diffraction through an elliptic aperture
may be determined by reduction to the case of a circular aperture.
8. Determine the character of the diffraction pattern produced by a square
aperture.
9. Determine the character of the pattern produced by diffraction through a
triangular aperture.
CHAPTER X
ON THE POLARISATION OF LIGHT BY REFLECTION AND DOUBLE
REFRACTION
164. Transverse Vibration—Plane Polarisation.—In the study
of the phenomena with which we have been hitherto engaged, we have
deduced no evidence whatsoever as to the nature of the vibrations in
the luminous waves. Throughout our investigations of the phenomena
of interference we have only supposed the vibrations of the interfering
rays to be similar to each other, but as to how they are directed in
space, or as to the relation of this direction to the direction of propa-
gation of the ray, we have made no assumption, nor yet dealt with
any phenomena calculated to attract our attention to it. From time
to time our knowledge of the theory of sound has proved of consider-
able assistance in the study of analogous phenomena in the theory of
light, but we now approach a class of optical phenomena which have
no analogue in the theory of sound. These have consequently been
supposed to arise from the difference in the nature of the vibrations of
the ether which constitute light and those of the atmosphere which
produce sound.
In the case of sound we know that the vibrations of the atmosphere
are longitudinal—that is, in the direction in which the sound is being
propagated ; the same is the case when rods and strings vibrate longi-
tudinally, or in the direction of their length. However, in the case of
a sounding fiddle-string (or the cord of Art. 34) the vibrations of the
string are perpendicular to its length, or transverse. A cord or rod
is thus capable of two distinct kinds of vibrations, longitudinal and
transverse, and these are propagated along the cord with different
velocities. So, in general, if a disturbance is being propagated by the
vibration of an elastic medium in any direction, we may resolve the
vibration into two others, one longitudinal, and the other transverse to
that direction, and if this applies to the ether, we should have the two
species of waves in it.
335
336 POLARISATION OF LIGHT CHAP. X
The question now at issue is whether the luminiferous vibrations
(supposing them to be periodic displacements) are longitudinal or
transverse, or if both species exist in the ether and affect our sense of
sight. This question can only be answered by experiment, but before
adducing any evidence on the point, it will be well to return to the
consideration of a vibrating cord (Art. 34).
Let AB (Fig. 6) represent a stretched cord, such as a fiddle-string.
If this cord be rubbed in the direction of its length, it will start to
vibrate in that direction, i.e. longitudinally, and emit a piercing note.
But if a bow be drawn across it at right angles to its length, it will
oscillate transversely and emit quite a different note. In the former
case the appearance of the string is the same on all sides, or in all
planes drawn through its length, but in the latter case the string
vibrates in a definite plane, and its appearance is not the same in all
planes drawn through its length. It has definite sides on it with
regard to the space around it. It looks flat, like a ribbon; when
vibrating it has acquired sides and may be said to be polarised.
Now suppose the string AB to pass freely through a narrow rect-
angular slit a little wider than the diameter of the cord. It is clear
that the longitudinal vibrations will not be interfered with, but will
continue unmodified, no matter how the slit is turned round the
vibrating string. The case is quite different when AB is vibrating
transversely, for when the length of the slit is in the direction of the
vibration of the string—that is, parallel to the plane of vibration—the
string is free to oscillate, and its vibrations will not be disturbed.
However, as the slit is turned round the cord so as to be at right
angles to the plane of vibration, the cord is no longer free to oscillate,
and its motion is interrupted by the sides of the slit. Thus, by
turning the slit round the cord, we detect the transverse vibrations
which give a two-sidedness or polarity to the vibrating string, and even
though it were invisible we should learn that it does not present the
same aspect on all sides."
It occurs to us now to endeavour to determine by some analogous
method the nature of the luminiferous vibrations, and for this purpose
we may, with profit, first operate with a plate of tourmaline cut
parallel to the axis of the crystal. When a beam of light is allowed
to fall perpendicularly upon such a plate part of it is transmitted, and
we shall see now that this transmitted portion has the peculiar two-
sided property of the transversely vibrating string. To the eye the
transmitted light appears to have suffered only a slight colouring due
* This might also be detected by placing a smooth plane surface against the
string.
ART. 164 DETECTION OF TRANSWERSE WIBRATION 337
to the natural tint of the crystal. But if we allow the light which
passes through the plate to fall upon another similar plate, we find
that the light from the first passes freely through the second when
the two are parallel; that is, similarly placed with respect to the ray
as A and B (Fig. 147). As the second plate is rotated (AT3) round
the beam as axis, its plane being always perpendicular to the ray, the
light transmitted by it is
observed to fade gradually,
and when it has been turned
through a right angle (A"B")
the light is completely ex-
tinguished. It is immaterial
which plate is turned, but
the bodily turning of both at the same time is without effect. If the
rotation be continued beyond a right angle, the light reappears and
increases in intensity till a second right angle has been completed,
when it is as bright as at the outset. When the plates are at right
angles to each other, i.e. crossed, the light from the first is refused
transmission through the second, just as the transverse vibrations of
the string were stopped by turning the slit.
Now the beam transmitted through a single plate of tourmaline
remains unaltered in intensity as the plate is rotated, we therefore
detect no two-sidedness in ordinary or natural light; but it is quite
otherwise with the transmitted beam, for we see that the amount of
it which the second plate allows to pass depends on the orientation of
that plate with respect to the first, and in one position it entirely
refuses a passage to the light. This beam, then, which has passed
through the plate of tourmaline has acquired a two-sidedness. It is for
this reason said to be plane-polarised, and for this reason it has been
supposed that its vibrations are transverse, and all in one plane, like
the vibrations of a string plucked aside. .
It is very important to remark that there is one position of the
second plate which entirely cuts off the beam transmitted through the
first, and this has been taken to show that the light vibrations cannot
be longitudinal, for if they were, the orientation of the second plate
would not be expected to affect them, just as the turning round of the
slit did not affect the string vibrating longitudinally. The fact that
no light gets through the second plate in one position appears to
show that the longitudinal vibrations, if they exist in the ether, are
not propagated as light by themselves, but it does not prove that a
longitudinal component does not exist in the vibration as a necessary
part, for the extinction of the transverse constituent might entail that
Z
Fig. 147.
y -
Longi-
tudinal
component.

338 POLARISATION OF LIGHT CHAP. X
of the longitudinal as well. In fact, it is not easy to see how waves
of transverse vibration can be propagated in a medium such as an elastic
solid without being attended by more or less of the longitudinal
element. Further evidence on this point will appear in what follows.
165. Polarisation by Reflection—Biot's Polariscope.—The polari-
sation of light was first noticed by Huygens when studying the refraction
of light through a crystal of Iceland spar, but it remained an isolated
fact in science for more than a century afterwards. About 1808 Malus
discovered, accidentally, that light when reflected from the surface of
glass acquires properties similar to those possessed by the light trans-
mitted through a plate of tourmaline—that, in fact, light may be
polarised by reflection—and pursuing the inquiry further, he found
that the same occurs when light is reflected from water and other
transparent substances. Hence the two-sidedness of the light which
has passed through a tourmaline plate may be detected by allowing
it to fall upon a plane glass plate. By turning the plate round the
beam the reflected light is seen to vary in intensity, and in one
position of the plate the reflected light vanishes altogether. By
keeping the glass plate stationary and rotating the tourmaline we
may obtain the same results.
Similarly the beam may fall first upon the glass and afterwards
be transmitted through the tourmaline with the same effect. In one
case we polarise the light by transmission through the tourmaline and
analyse it (that is, detect its polarisation) by reflection from the mirror,
and in the other case it is polarised by reflection
from the mirror and analysed by transmission
through the tourmaline. It is clear, therefore,
that the tourmaline may be dispensed with
altogether and replaced by a plane glass mirror,
since the mirror can act the part either of a
polariser or an analyser. On this principle
instruments termed polariscopes have been con-
structed. One of the first of these was designed
by Biot," but it has long since been superseded
by more commodious forms. It is represented
in Fig. 148. At each end of a tube T plane
mirrors of polished black glass are placed. Each
mirror is capable of two motions—one round a
diameter of the tube; that is, round an axis
perpendicular to the axis of the tube. The amount of this rotation is
measured by the graduated circles M and N. The second motion
Fig. 148.
* Biot, Traité de physique, tome iv. livre sixième, chap. i. p. 255.

ART. 166 ANGLE OF POLARISATION 339
is round the axis of the tube. This is obtained by the mirrors
being attached to rings G, H, which are graduated and movable on
the tube. * ,
The mirrors can thus be inclined at any angle to the incident light
and to each other.
166. Angle of Polarisation.—When the planes of the two mirrors
are placed parallel, the light which is reflected from the first and falls
upon the second is there partially reflected. By rotating either mirror
round the axis of the tube, the amount of reflected light will diminish
and become a minimum when the mirror has been rotated through
90°, so that the mirrors are crossed. The amount of light reflected
in this position will depend upon the angle of incidence, and for one
particular value of this angle the reflected light will vanish entirely."
We consequently infer that there is one particular angle of incid-
ence at which light is completely polarised by reflection from glass, and
this is termed the angle of polarisation, or the polarising angle.
When light is reflected from glass, the reflected beam in general is
only partially polarised. It cannot be completely extinguished by a Partial.
* According to Fresnel's theory (see Art. 210), light, incident at the polarising
angle, should not be reflected at all, if polarised perpendicularly to the plane of in-
cidence. Lord Rayleigh draws attention (Phil. Mag. vol. xvi. pp. 444-449, Sept. 1908)
to uncertainties arising from the nature of the reflecting surfaces. These may be
covered with films of moisture, grease, or even condensed air, and may be still
further complicated by residues of the polishing material. Thus he found for water
(Phil. Mag. (5), vol. xxxiii. p. 1, 1892) that, with proper precautions to remove grease,
when the light is polarised perpendicularly to the plane of incidence, much less re-
flection may be obtained than that observed by Jamin, or, in other words, the polarisa-
tion of light incident at the polarising angle is much more complete for water than
Jamin supposed. Also Drude (Wied. Ann, vol. xxxvi. p. 532, 1889) found that the
cleavage surfaces of transparent crystals, when fresh, gave complete polarisation
within the limits of error of his experiments. However, Rayleigh finds for glass that,
while with careful polishing it may be brought to give zero elliptic polarisation, with
further polishing it again gives elliptic polarisation, but with the sign reversed of
the component vibration in the plane of incidence. Such a polished surface is,
however, soon contaminated under ordinary conditions; part of this contamination
may be removed by breathing on the glass and then wiping it with a cloth, but for
its complete removal repolishing is necessary. It has also been found that the
elliptic polarisation is affected by pressure on the reflecting surface (Volke, Amm. d.
Phys. (31), vol. iii. pp. 609-632, March 1, 1910). It is to be noted that Fresnel's theory
is based upon the occurrence of a completely discontinuous change occurring at the
surface of separation of the two media, though it is unlikely that such an abrupt
change actually takes place. Apart from contaminating films, probably a rapid but
continuous transition occurs from one medium to the other, due to interpenetration of
the two media (see Wheeler, Phil. Mag. vol. xxii. pp. 229-245, August 1911). Drude
has determined the effect of the film or layer, if very thin, and shown that ellipticity
of different sign may be expected according as the film is of greater refractive index
than either medium, or of refractive index intermediate between those of the
two media. *
340 POLARISATION OF LIGHT CHAP. X
Maximum.
Direction
of vibra-
tion.
tourmaline plate or by reflection from another mirror at any incidence.
The amount of polarisation depends upon the angle of incidence, and
at one particular angle the polarisation becomes complete.
We must not hastily infer that for every substance there is an angle
of complete polarisation. In fact, it is proved by experiment that
as the angle of incidence increases, the polarisation in general also
increases to a maximum, and then decreases after passing through the
angle of maximum polarisation. For each substance there is an angle
of incidence which gives a maximum of polarisation, and this angle is
termed the polarising angle of the substance.
M. Jamin, who investigated this subject, found that only a few
substances, of refractive index about 1:46, polarise light completely
by reflection. For all other substances the polarising angle is merely
the angle of maximum polarisation. For glass the polarising angle is
about 57°, and for pure water 53° 11'.
167. Plane of Polarisation.—When plane-polarised light falls at
the polarising angle upon a glass mirror, the intensity of the reflected
light depends upon the position of the plane of incidence with regard
to the ray. Thus as the mirror is rotated round the ray, keeping
the angle of incidence constant the intensity of the reflected pencil
varies, and that particular plane of incidence in which the light is
most copiously reflected is called the plane of polarisation. When the
polarised light has been obtained by reflection, it is obvious that the
plane of polarisation, as defined above, is the plane of reflection of the
light, for this beam would be most copiously reflected by a second
surface when it is parallel to the polarising surface.
According to the theory of Fresnel, the vibrations of plane-
polarised light are perpendicular to the plane of polarisation. Thus
the direction of vibration in light, polarised by reflection, is perpen-
dicular to the plane of reflection; that is, parallel to the reflecting
surface. The relation of the direction of vibration to the plane of
polarisation has been a subject of much dispute, and we postpone its
consideration for the present. It will be well to bear in mind, how-
ever, that something may be going on both in and perpendicular to the
plane of polarisation, and that the vibration may not be a simple
displacement in the wave front (Chap. XXI.).
168. Double Refraction.—Hitherto we have assumed that when
light is incident on the surface of a transparent medium, the refracted
portion pursues in all cases a single definite direction. That this is
not always the case was discovered by Erasmus Bartholinus, a Danish
philosopher, about the year 1669. Experimenting with a cleave of
Iceland spar (carbonate of calcium), he found that a beam of light on
A
ART. 168 DOUBLE REFRACTION 341
refraction at its surface travels through the cleave in two determinate
pencils, one of which traverses it according to the known laws of
refraction, while the other is bent according to a new and extraordinary
law not previously noticed."
A few years after the discovery of double refraction, Huygens.”
gave a satisfactory explanation of it in uniaxal crystals on the principles
of the wave theory, and whilst repeating the observations of Bartho-
linus, he was led to the discovery of the “wonderful phenomenon" of
the polarisation of light.
The property of producing two refracted beams is called double
refraction, and is possessed by all crystallised minerals except those
whose fundamental form is the cube. It belongs also to animal and
vegetable substances possessing a regular arrangement of parts, and
to all bodies whose parts are in a state of unequal compression or
dilatation.
The angular separation of the two refracted pencils varies with the
direction of the incident ray with reference to the natural figure of the
crystal. In every doubly refracting crystal there is at least one direc- optic axis.
tion, and in many two, in which no such separation occurs. These
directions are called optic aces of the crystal. The refracted rays
are most widely separated when the incident beam is perpendicular to
the optic axis.
In Iceland spar, the substance in which it was first observed, the
phenomenon of double refraction is very strikingly exhibited. This
mineral crystallises in many forms, all of which may be reduced by
cleavage to therhombohedron, which is accordingly thefundamental form.
It is also found in considerable masses of great purity and transparency.
The rhombohedron is bounded by six
parallelograms, the angles of which are
101° 55' and 78° 5' respectively. Two of
the solid angles, a and b (Fig. 149),
diametrically opposite, are contained by
three obtuse angles, while each of the
remaining four is bounded by one obtuse
and two acute angles. The direction
making equal angles with the faces at the summits of the obtuse
solid angles is called the aris of the crystal. Thus if the edges of the
rhomb be equal to each other, the line ab adjoining the obtuse solid
Fig. 149.
An account of these experiments was published at Copenhagen in 1669 under
the title Experimenta Crystalli Islandici dis-diaclastici, quibus miraet insolita refractio
detegitur. A full account of them is given in the Edinburgh Philosophical Journal,
vol. i. p. 271.
* Huygens, Traité de la lumière, “De l'estrange réfraction du Cristal d’Islande."

342 POLARISATION OF LIGHT CHAP. X
Both
beams
plane-
polarised.
angles, or any parallel line, is the crystallographic axis. The angles
at which the faces themselves are inclined are 105° 5' and 74° 55'.
When a transparent cleave of Iceland spar is placed over a small
black dot on a sheet of white paper, two images of the dot are seen on
looking through the crystal, and if the eye be held perpendicularly
over the face of the crystal while it is rotated over the dot, one of the
images remains stationary, while the other rotates round it. The
fixed image appears a little nearer than the movable one, and the line
joining them is parallel to the shorter diagonal of the rhombic face
through which they are observed when the sides are made equal.
169. Polarisation by Double Refraction.—The properties of light,
which has suffered double refraction, may be best examined by allow-
ing a pencil from a small aperture to pass through a large cleave so
as to receive the two emerging beams on a screen.
Of the two portions into which the refracted light is divided by a
uniaxal crystal, that which obeys the ordinary laws of refraction is
called the ordinary ray, and this gives an image on the screen called
the ordinary image. The other refracted ray does not obey the
ordinary laws of refraction. It is called the eſciraordinary ray and
gives an “extraordinary image.” These rays and images may be denoted
by the letters O and E. By placing a plate of tourmaline in the
path of either of the rays produced by double refraction, it is found
on rotating the tourmaline that in one position it refuses to transmit
the ray, while in other positions the ray is transmitted with more or
less freedom. Both the rays then are two-sided or plane-polarised.
It must be realised that it is the ray in the rhomb that follows an
“extraordinary” law; there is nothing extraordinary about the ray
when it has emerged from the spar. If owing to the position of the
spar the ordinary ray is polarised in the vertical plane, the extra-
ordinary is found by the above tests to be polarised in the horizontal
plane. This latter ray, when it falls upon a second cleave of spar,
will follow either the extraordinary or the ordinary law according to
the orientation of the second spar. The term “extraordinary image"
must therefore be considered only as an abbreviation for “image
formed by the extraordinary ray.”
This behaviour of rays may be shown by allowing the pencils to
pass through a second crystal of calcspar. When the second crystal is
parallel to the first, the two behave just like one of their joint thick-
ness. In this position the ordinary and extraordinary rays of the first
crystal traverse the second as ordinary and extraordinary rays, but
when the second is rotated through an angle, each of the rays O and
E is doubly refracted in it, so that four rays emerge from the second
ART. 169 POLARISATION BY DOUBLE REFRACTION 343
crystal and four images are depicted on the screen. The ordinary ray
of the first rhomb gives rise to an ordinary O, and an extraordinary
Oe, while the extraordinary gives rise to an ordinary E, and an extra-
ordinary E. As the rhomb is turned, the pair Oe and Eo, which are
faint at first, gradually increase in brightness, while the other pair Oo
and E, diminish in intensity and finally vanish when the rhomb has
been rotated through 90°. We have now again only two images, viz.
O, and Eo. On rotating the rhomb farther, these images grow fainter,
while the other pair O, and Ee reappear and increase in intensity till
the rhomb is rotated through 180° from its first position. Here we
have only two images O., and E, as at starting. Thus in one position
the second rhomb allows the pencils to pass freely through—the
ordinary as ordinary, and the extraordinary as extraordinary—while
in a position at right angles to this it refracts the ordinary ray entirely
as an extraordinary, and the extraordinary entirely as an ordinary,
and in any intermediate position it refracts both doubly.
By utilising the persistence of visual impressions, this form of the
experiment is susceptible of a very elegant modification. Let one of
the pencils emerging from the first rhomb fall upon the second. This
beam will in general be divided into two others by the second, and the
refracted pencils will depict two spots on a screen placed to receive them.
By placing the screen at a proper distance the spots may be made to
overlap partly. Now let the second rhomb be rotated rapidly around
the direction of the incident pencil of light. The spots on the screen
will describe circular luminous bands on the screen which partly over-
lap. Three concentric rings are consequently presented. The middle
ring is due to the overlapping of the two images, and is uniformly
bright all round, for the overlapping images are complementary in all
positions. This central ring is bordered on either side by other rings,
one due to the ordinary image and the other to the extraordinary.
These do not appear uniformly bright. The former will be brightest
in the plane of polarisation, and darkest in the perpendicular plane.
The reverse is the case in the extraordinary ring.
The ordinary and extraordinary rays emerging from a rhomb of
Iceland spar or any other doubly refracting crystal are consequently
in a condition singularly different from that of common light, for while
a beam of natural light is always divided into two of equal intensity
on entering the crystal (except when it passes in one particular direc-
tion called the optic axis), the subdivision of the ordinary or extra-
ordinary ray by a second rhomb depends on the orientation of the
second crystal with respect to the first. -
It was in this form that the polarisation of light was first noticed
344 - POLARISATION OF LIGHT chap. x
Double
absorption,
by Huygens. On analysing his observations, he determined that a
beam of solar light is always divided into two (except when it traverses
the crystal in a direction called the optic axis), and that each of the
resulting beams will be singly or doubly refracted by a second crystal
of spar according to the relative position of the principal Sections; these
are planes drawn through the optic axes of the crystals and per-
pendicular to the refracting faces. If the principal planes of two
crystals be either parallel or at right angles to each other, then the .
rays which emerge from the first are not doubly refracted by trans-
mission through the second. If the principal planes are parallel,
the ordinary ray from the first traverses the second as an ordinary
ray, and the extraordinary as extraordinary ; but if these sections
are at right angles, the ordinary ray from the first is refracted as an
extraordinary ray in the second, and the extraordinary ray as an
ordinary. -
The ordinary wave is found to be polarised in the principal plane,
and the extraordinary wave is polarised perpendicularly to the principal
plane, the plane of polarisation being defined as in Art. 167.
A beam of plane-polarised light possesses the following character-
istics: ‘.
I. It is not divided into two others by a doubly refracting crystal
in two positions of the principal section with respect to the ray, while
in other positions it is divided into two pencils which vary in intensity,
and are complementary as the crystal is rotated.
II. It is not reflected at the surface of a transparent substance
when the plane of incidence is perpendicular to the plane of polarisa-
tion, provided that the angle of incidence has a certain value depending
on the nature of the substance. *
170. Property of Tourmaline.—In our preliminary experiment we
obtained polarised light by transmitting a pencil of ordinary light
through a plate of tourmaline. Now tourmaline is really a doubly
refracting crystal and divides the intromitted beam into two parts—an
ordinary and an extraordinary—yet it is only the extraordinary beam
that emerges from the plate, for the ordinary pencil is rapidly absorbed
by the crystal, so that a plate of small thickness (1 or 2 mm.) is almost
impervious to it. It is this property that renders tourmaline plates so
readily adapted to the operations of either producing polarised light or
of analysing it when already obtained.
The wave velocity is then not the only property affected by crystal-
line structure. In many crystals the two polarised rays suffer different
rates of absorption, and it is this property that qualifies a single plate
of tourmaline to act the part either of a polariser or an analyser. The
ART. 171 BREWSTER'S LAW 345
property of double absorption is without doubt intimately related to
that of double refraction. -
Tourmaline, however, is not a very transparent mineral for either
ray, and strong beams of polarised light cannot well be obtained with
it. Hence it is quite unfitted for work when the illumination is
faint, and consequently other forms of polarisers and analysers—that
is, apparatus for producing and studying polarised light—have been
invented. The most important of these are Nicol's and Foucault's
prisms (see Arts. 182, 183). -
171. Brewster's Law.—About the year 1811 Sir David Brewster
commenced an extensive series of experiments with the object of deter-
mining the polarising angles of different media, and of connecting
them by a law. The outcome of his investigations was the simple and
remarkable law, “The indea of refraction of the substance is the tangent of
the polarising angle.” This law, which is expressed by the formula
tan i = p,
informs us that the polarising angle ranges in different substances
from 45° upwards, and, when the refractive index is known, the angle
of polarisation is inferred.
Since the refractive index is different for differently coloured lights,
it follows that the angle of maximum polarisation is different for the
different rays of the spectrum, and consequently if a beam of Solar
light be reflected successively from two glass plates whose planes of
reflection are at right angles, the reflected beam will never be wholly Colour at
extinguished, but will be reduced to a residuum coloured with a red rºws
or blue tinge, according as the angle of incidence is the polarising
angle of the more or less refrangible rays. When the angle of incidence
is that of polarisation of the most luminous part of the spectrum, the
reflected light is of a purplish tint, formed by the mixture of the
remaining rays in different proportions. These effects are best marked
in highly dispersive substances.
Ö The geometrical interpretation of Brewster's law is that when a
pencil of light falls upon a transparent substance at the polarising
angle, the reflected and refracted rays are at right angles. For if
tan i = p, then; – #.
OI’
cos i =sin r and i + r = 90°,
therefore i and r are complementary, or the reflected and refracted
rays are at right angles.
172. Pile of Plates.—The law of Brewster applies to light reflected
from the surface of the rarer as well as from the surface of the denser .
346 POLARISATION OF LIGHT CHAP. x
medium, and since the refractive index in the former case is the
reciprocal of that in the latter, it follows that the angles of polarisation
in the two cases are complementary. From this it follows that when
a beam of ordinary light falls upon a parallel plate of any transparent
substance at the polarising angle, the refracted portion will meet the
second surface at the polarising angle, and if it still contains an un-
polarised part this will be wholly or partially polarised by reflection
there. Hence if several plates of glass be arranged parallel to one
another, a pencil of light incident on the first at the polarising angle
will, after refraction, meet all the succeeding surfaces at their polarising
angles also. So that all the light reflected from these surfaces will be
plane-polarised. Such an arrangement is termed a pile of plates, and
is very useful as a polariser when the light is not incident in a
parallel beam as in the case of skylight, for the reflected beam is much
more intense than that obtained from a single plate.
173. Polarisation of the Refracted Light.—So far we have only
considered the modification which the reflected pencil has suffered.
However, when the refracted pencil is examined, it is found to contain
a quantity of polarised light. The relation between this polarised
light and that of the reflected beam is very intimate. It was discovered
by Arago and stated thus: “When an unpolarised ray is partly reflected
at, and partly transmitted through, a transparent surface, the reflected and
transmitted portions contain equal quantities of polarised light, and their
planes of polarisation are at right angles to each other.”
The operation of the plate is purely selective, for the polarised
component which is missing in the reflected light, is represented in
the transmitted light. Each beam, in general, also contains an amount
of common or unpolarised light.
If the transmitted light be received upon a second parallel plate,
the portion of common light which it contains undergoes a further
subdivision, and so on for any number of plates. Hence, when the
number of plates is sufficiently great, the transmitted light will be com-
pletely polarised, and consequently if a beam of light be incident on a
pile of plates at the polarising angle the transmitted light, after traversing
a certain number of the plates, will suffer no further diminution in in-
tensity except by actual absorption in the plates. For the refracted light
will become wholly polarised in a plane perpendicular to the plane of in-
cidence, and no portion of it will be reflected by the succeeding plates.
174. Law of Malus.-These results were established by Malus
who also conjecturally assumed that when a pencil, polarised by reflec-
tion at one plane surface at the polarising angle, is allowed to fall
upon a second at the polarising angle, the intensity of the twice-reflected
ART. 174 LAW OF MALUS - 347
beam varies as the square of the cosine of the angle between the two planes
of reflection. The truth of this law was afterwards established by the
experiments of Arago. -
Thus if we assume with Fresnel that the direction of vibration is
perpendicular to the plane of polarisation, then if the incident light be
polarised in a plane making an angle 6 with the plane of incidence,
the incident vibration, of amplitude a, may be resolved into two com-
ponents, one a cos 6 perpendicular to the plane of incidence and the
other a sin 6 parallel to it. The former is polarised in the plane of
incidence, and is reflected; the latter is polarised at right angles to
the plane of incidence, and is transmitted. The reflected light is thus
in all cases polarised in the plane of reflection, and its intensity is
proportional to cos”6. The law of Malus is thus simply accounted for
by the principle of resolution of vibrations. -
Common light, we know, is broken up by a crystal into two parts
polarised at right angles to each other, and conversely, the preceding
law of Malus enables us to regard common light as consisting of two
equal plane-polarised rays polarised in planes at right angles. For
consider two such rays, and let a and 90° – a denote the angles which
their planes of polarisation make with the plane of reflection; then if
I denotes the intensity of each of the rays, the intensities of the
reflected rays are I cos”a and I sin”a respectively, and the sum of these,
I cos°o. + I sin”q = I,
is the resultant intensity of the reflected. It is therefore constant and in-
dependent of the position of the plane of reflection with respect to theray;
but this is the distinctive characteristic of ordinary unpolarised light.
Hence in ordinary light the direction of vibration is supposed to
be quite irregular, but when it falls upon a doubly refracting medium
the vibrations behave as though they were resolved in two definite
directions, constituting two equally intense rays polarised in per-
pendicular planes and differently refracted by the medium.
If a pencil of common light of intensity I falls upon a crystal of calc-
spar, it is divided into ordinary and extraordinary rays, each of intensity
#I, and if these fall upon a second crystal, they will be again subdivided
according to the law of Malus as indicated in the following table:

Incident Light. In First Crystal. In Second Crystal. P ãº, Sum.
Oo-3; I cos” a 0
I |. Oe=#I sin” a 90 #I I
Eo-#I sin” o, 0
E=#I Ee= }I cos” a 90 #1]
348 POLARISATION OF LIGHT Ch.A.P. x
175. Partially Polarised Reflected Light.—When light is incident
on a reflecting surface at an angle either greater or less than the polar-
ising angle, it was observed by Malus that the reflected light possesses
only in part the properties of polarised light. Neither of the two pencils
into which it is divided by a rhomb of Iceland spar ever completely
vanishes, but each varies in intensity as the rhomb is rotated. He
consequently concluded that the reflected light consisted of two parts,
one perfectly polarised, while the residue remains in the state of
ordinary light. Partially polarised light is then, according to Malus,
a mixture of ordinary light with a part wholly polarised, and in this
hypothesis he has been followed by most subsequent philosophers,
for the light possesses all the properties of such a mixture.
If this partially polarised light be reflected from a second surface
at the same angle, the reflected pencil is found to contain an increased
quantity of polarised light, and by augmenting the number of reflec-
tions the light may be almost wholly polarised. This was first noticed
Fig. 150.
by Sir David Brewster, and he found that light may be polarised at
any incidence by a sufficient number of reflections, the number of re-
flections required increasing as the angle of incidence is more removed
from the polarising angle. Hence the utility of a pile of plates.
176. Interference of Polarised Light.—We have seen that light
suffers an important modification by transmission through doubly
refracting crystals and also by reflection, and it has also been found
that this modified or polarised light, as it has been called, obeys the
ordinary laws of reflection, refraction and dispersion. It is important
therefore to determine if the phenomena of interference are produced
by it under the same circumstances as by ordinary light. For this
purpose the refractometer of M. Billet, already described (Art. 103),
will be found convenient. A beam of plane-polarised light produced
by a tourmaline T, or otherwise, falls upon a narrow slit S (Fig. 150).
The light diverging from the slit falls upon the segments L and L of
a divided lens, which bring the rays to foci A and B. At these
points thin plates of Iceland spar, or tourmaline, or other doubly
refracting crystals may be placed. If tourmalines be at A and B there
is a single polarised beam from each, and it is found that when their

ART, 176 INTERFERENCE OF POLARISED LIGHT 349
principal planes are parallel we have interference fringes on the screen,
as in the case of ordinary light, but when one of them is rotated so
that they are crossed the fringes disappear entirely. This is what we
should have expected, for we have already learned (Art. 47) that two
rectangular vibrations differing in phase in general compound into an
elliptic vibration.
If thin plates of Iceland spar be placed at A and B with their
principal sections parallel, and if the incident light be polarised in or
perpendicular to their principal sections, it will traverse the crystals
without double refraction, either as ordinary or as extraordinary rays.
The planes of polarisation of the emerging beams will be parallel and
interference fringes will be formed on the screen. If however the
incident light is polarised in any other plane, double refraction will
occur in the plates at A and B, each giving rise to an ordinary and
to an extraordinary pencil. The two ordinary beams interfere and pro-
duce a system of fringes. Superposed on these fringes we have another
system produced by the extraordinary beams, but no destructive
interference occurs between the ordinary and extraordinary parts.
If now one of the plates A and B be turned through 90°, so that
they are crossed, the ordinary ray from A will be polarised parallel to
the extraordinary ray from B, and these will interfere and produce
a system of fringes. The centre of the system will however be dis-
placed towards A, for the ordinary ray travels more slowly in the
crystal than the extraordinary, as we shall see immediately. Similarly
the extraordinary ray from A will interfere with the ordinary from B
and produce a system displaced towards B.
Fresnel and Arago, who investigated directly the interference of Conditions
polarised light, summarised their conclusions as follows:" .
(1) Two rays of light polarised at right angles do not interfere de-
structively under the same circumstances as two rays of ordinary light.
(2) Two rays of light polarised in the same plane interfere like two
rays of ordinary light.
(3) Two rays polarised at right angles may be brought to the same
plane of polarisation without thereby acquiring the quality of being
able to interfere with each other.
(4) Two rays polarised at right angles, and afterwards brought to
the same plane of polarisation, interfere like ordinary light if they
originally belonged to the same beam of polarised light.
The fact that rays polarised in perpendicular planes cannot inter-
fere destructively is in itself an indication that the direction of vibration
is transverse to the direction of propagation.
* Fresnel, CEuvres, tome i. p. 521.
CHAPTER XI
DOUBLE REFRACTION IN UNIAXAL CRYSTALS
177. Wave Surface in Uniaxal Crystals.-Before proceeding to
the general theory of double refraction and the more complicated
phenomena arising from refraction in biaxal crystals, it will be well to
first give a general statement of the phenomena and laws of refraction
in uniaxal crystals, and it will be interesting afterwards to see how
these may be deduced as particular cases of the general theory when
we suppose the two optic axes of a biaxal crystal to coincide.
It has been already stated that double refraction was discovered
in Iceland spar by Erasmus Bartholinus. Soon after its discovery
Huygens,” who had already unfolded the wave theory of light and
accounted for ordinary refraction and reflection, was naturally anxious
to reconcile the new properties of light discovered by Bartholinus with
the same theory, and in his desire to assimilate the two classes of
refraction he was happily led to assign the true law of ea traordinary
refraction in uniaxal crystals. He had already shown that the form of
the wave of light propagated in glass and isotropic substances was a
sphere, and as one of the rays in Iceland spar was found to obey the
ordinary laws of refraction he assumed that the corresponding wave
was also a sphere. The law which governed the other ray, though not
so simple, he imagined to be next in order of simplicity, and he
assumed the extraordinary wave to be a spheroid—that is, an ellipsoid
of revolution. a
The velocity of the extraordinary ray in any direction is conse-
quently given by the following construction: “Let an ellipsoid of
revolution be described round the optic axis having its centre at the
point of incidence ; and let the greater axis of the generating ellipse
be to the lesser in the ratio of the greatest to the least index of refrac-
tion; then the velocity of any ray will be represented by the radius
vector of the ellipsoid which coincides with it in direction.”
* Huygens, Traité de la lumière, chap. v.
350
ARt. 177 WAVE SURFACE IN UNLAXAL CRYSTALS 351
The surface described here is the wave surface of Huygens. The
law correctly describes how the velocity in any direction of propagation
can be represented; but it gives no hint as to what determines
the velocity. In reality it is not the direction of propagation which
determines it but the direction in which the vibrations take place, as
will be shown later. To distinguish these directions from one another,
and thus to avoid confusion, we shall henceforth speak of the direction
of propagation and of the azimuth of vibration which is at right angles
to the direction in which the wave goes. For any one direction of
propagation there are an infinite number of azimuths of vibration,
for any one of which the velocity along the ray is different (in general)
from that for any other.
The law of Huygens was found to apply to many crystals besides
Iceland spar, but in all of these there was only one optic axis, or one
direction along which a ray of light passed without division. The re-
searches of Brewster, however, made known a class of crystals having
two axes or two directions of no separation of the ray." Huygens's
law was then found not to be general, and in this state the problem was
taken up by Fresnel, who proposed a theory which met not only all the
requirements of the ascertained facts of the refraction in biaxal crystals,
but which even outran the existing knowledge and predicted results of
the highest consequence, afterwards verified by direct observation.
According to the construction of Huygens the wave surface or the
secondary wave in a uniaxal crystal consists of two portions or sheets, Two sheets.
one a sphere which gives rise to the ordinary ray, and the other a
4-
Fig. 151. –Negative Crystals.
Fig. 152.-Positive Crystals.
spheroid giving rise to the extraordinary ray. These two surfaces,
the sphere and spheroid, touch at two points, and the line joining
these points is the optic axis.
In the case of Iceland spar and all negative crystals (Art. 180)
* They are different from the optic axes as will be seen later (Art. 202).


352 DOUBLE REFRACTION IN UNLAXAL CRYSTALS CHAP. XI
the sphere is entirely within the spheroid, and in the case of quartz
and positive crystals the spheroid is within the sphere. Thus if the
spheroid is generated by the revolution of the ellipse
* y”
nºt jail
round its minor axis b, and the sphere by the revolution of the circle
*4 y” – bº
round the same axis, any section of the wave surface through the
optic axis (b) is an ellipse (axes a and b) and a concentric circle of
radius b touching it at the extremities of the axis minor (Fig. 151).
In positive crystals the wave surface is generated by the revolution
of the ellipse round its axis major and a concentric circle of radius a
touching it at the extremities of that axis (Fig. 152).
A section of the complete wave surface by a plane perpendicular
to the optic axis consists of two circles, one of radius a and the other
of radius b.
Huygens verified his theory by well-contrived experiments, but
much more accurate measurements were necessary to prove the extra-
ordinary wave to be truly an ellipsoid of revolution. These measure-
ments were made in 1802 by Wollaston, and afterwards by Stokes,
Mascart, and Glazebrook with the most perfect optical instruments,
resulting in the complete verification of Huygens's hypothesis.
178. Huygens's Construction.—Let us now seek the directions of
the two refracted rays in a crystal of
calcspar when a plane wave falls
upon it from air. Let IA (Fig. 153)
be the direction of the light incident
on the face of the crystal, and let
AB be the trace on the plane of the
paper of the incident wave front,
and AA’ the trace of the face of the
Fig. 153. crystal. The incident wave front is
a plane through AB perpendicular to the paper, and the face of the
crystal a plane through AAſ, also perpendicular to the plane of the
paper.
When the disturbance reaches A this point becomes the centre of
a spherical wave reflected back into the air, and also the centre of a
double wave propagated in the crystal. The plane of the paper will
cut this wave in two curves, the sphere in a circle and the spheroid
generally in an ellipse. The diagram represents the particular case
in which the optic axis lies in the plane of the paper, and consequently


ART. 179 VERIFICATION OF HUYGENS's WAVE SURFACE 353
the circle and ellipse touch each other. If the disturbance from B
reaches Aſ at the instant the wave in the crystal is just developed to
the extent represented in Fig. 153, then through a perpendicular
drawn at A* to the plane of the paper—that is, through the line in
which the incident wave meets the surface—draw a tangent plane to
the sphere. This plane will be the ordinary wave front. It will
touch the sphere at C in the plane of the paper, and AC will be the
ordinary refracted ray. Through the same line draw a tangent plane
to the spheroid. This plane will be the front of the extraordinary
wave, for all the wave surfaces diverging from the various points of
AA’ at the same instant will touch it. If this plane touches the
spheroid at a point Cº, then C will not in general lie in the plane of
the paper, unless the optic axis lies in (or is perpendicular to) that
plane ; AC will be the extraordinary ray, and in general it will not
be in the plane of incidence or obey the ordinary laws of refraction.
If the optic axis lies in the plane of the paper—that is, the plane of
incidence, as in the diagram—the point C will also lie in that plane, so
that the extraordinary ray AC" will lie in the plane of incidence and
will thus obey one of the laws of refraction.
There is one case, however, in which the extraordinary ray should
obey both laws of refraction, viz. when the optic axis is perpendicular
to the plane of incidence. In this case the section of the spheroid by
the plane of incidence is equatorial, and is therefore a circle, so that
the extraordinary ray AC" is not only in the plane of the paper, but
the sine of the angle of incidence bears a constant ratio to the sine
of the angle of refraction. This ratio is called the extraordinary indew,
of refraction. If the velocity in air be denoted by unity, and the
velocities of the ordinary and extraordinary rays by b and a, the radii
of these circles, then
po- 1/b = ordinary index,
pe-1/a = extraordinary index.
The extraordinary index of refraction is thus the least value of the
index of refraction of the extraordinary ray in negative crystals.
It is necessary to add that on Fresnel's theory the azimuth of the
vibration of which AC is the ray is at right angles to the plane of the
paper; whereas for the extraordinary ray AC’ it is parallel to C'A',
and it is thus not in general at right angles to the ray.
179. Verification of Huygens's Construction.—The simplest and
most exact method of showing that one of the rays in Iceland spar
obeys the ordinary laws of refraction, no matter in what direction it
traverses the crystal, and consequently that its wave is spherical, is to
cut several slices in different directions from a rhomb of spar and
2 A
354 DOUBLE REFRACTION IN UNLAXAL CRYSTALS CHAP. x1
to cement them together and then to cut the whole into a prism,
having its edges perpendicular to the planes of junction (Fig. 154).
Examining through this prism the light from the slit
of a spectroscope, the extraordinary spectra furnished
by the different slices are observed to be differently
deviated, but there is only one ordinary spectrum, or
the ordinary spectra furnished by the different slices all
coincide; that is, the refractive index of the ordinary
ray is independent of the direction in which it traverses
** the crystal. Its wave surface is therefore spherical;
and its refractive index po is measured in the same manner as that of
any uncrystallised substance.
To verify the construction for the extraordinary wave, we examine
the following cases.
(1) Refracting Face parallel to the Optic Awis, and the Plane of Incidence
perpendicular to the Azis.-Let the face of the crystal be a plane
through AA' (Fig. 155) perpendicular to the plane of the paper (which
is supposed to be the plane of incidence). Then in this case the optic
axis at A is a line through A perpendicular to the plane of the
diagram. The section of the sphere is a circle of radius b, and the
section of the spheroid, being its
equatorial section, is a circle of
radius a. The tangent planes from
a line through A perpendicular to
the plane of the paper, to the
sphere and spheroid, touch them at
C and C in the plane of incidence.
Taking the velocity of light in air
as unity, the velocity of the ordinary
ray will be b, the velocity of the
extraordinary in this case will be a, and the refractive index for the
extraordinary ray is
Fig. 155.
sin i_1_
sin r a T’”
It therefore obeys both laws of refraction.
Cutting a prism of Iceland spar with its refracting edge parallel
to the optic axis, two spectra are obtained; the light of one is
polarised in the principal plane and the other in the perpendicular
plane. By interposing a plate of tourmaline in the path of either
it can be cut off and the other examined. The indices p, and p.
can thus be calculated for the several rays of the spectrum. The
results of experiment are in complete accordance with the foregoing


ART. 179 VERIFICATION OF HUYGENS's WAVE SURFACE 35
5
theory, and the section of the extraordinary wave perpendicular to the
optic axis is therefore a circle of radius a = 1/u.
This wave is then a surface of revolution round the optic axis.
To determine the form of the generat-
ing curve, we shall study the refrac-
tion in a plane passing through the
optic axis.
(2) Optic Awis parallel to the Face of
the Crystal and to the Plane of Incidence.—
When the refracting surface contains
the axis of the crystal and the plane of
incidence passes through that axis, the
section of the spheroid by the plane of
incidence will be an ellipse whose lesser axis (the optic axis) lies in
the surface (Fig. 156). The section of the sphere will be a circle
touching the ellipse at the extremities of the minor axis. The
tangent planes from A', as before, will touch the sphere and spheroid
in C and C respectively in the plane of incidence, and since A"
is on the minor axis it follows that the line CC' will meet AA’ at
right angles. For the polar of any point on the chord of contact of a
circle and ellipse having double contact is the same with regard to
both curves. Hence CC" must be the polar of A with regard to both
Fig. 156.
Tº
-º-
ſ -
ſº
In -
DI
Fig. 157.-Verification of Huygens's Wave Surface.
the circle and ellipse, and is consequently perpendicular to AA. We
have therefore -
tan r DC a
* * * *_*.
tan r" DC b a.
This remarkable relation has been verified by Malus' as follows: Two
scales AC and BC (Fig. 157) engraved on a plate of polished steel
are inclined to each other at a small angle and divided into small
* “Théorie dela double réfraction" (Mémoires des Savants trangers, tome ii. p. 303).


356 DOUBLE REFRACTION IN UNLAXAL CRYSTALS CHAP. XI
equal parts. A thick plate of the crystal having its faces parallel to
the optic axis is laid on the scale, so that its principal section is
perpendicular to AC, and viewed through a telescope LM mounted on
a graduated vertical circle in a plane parallel to the optic axis of the
crystal. The scale and rhomb are supported on a horizontal circle, and
the horizontality of the upper face of the rhomb is ensured by turning
the platform round and observing that the image, by reflection from
it, of a distant point is not displaced. The proper orientation of the
crystal is secured by means of fine lines marked on the scale per-
pendicular to AC, the crystal being turned round till both images of
these lines are coincident.
Two images of each scale are seen, and if these be denoted by ac,
a'c', be, bºc', there will in general be some point of be coinciding with
some point of a 'c'. Let this point be h. Then h is the image of some
point D of the scale AC, and also of some point E of the scale BC. If
the axis of the telescope is directed to view this point, it will cut the
surface of the rhomb at H and the position of H can be determined
with reference to the scales; or the divisions at E and D which appear
to coincide can be read off and the distance ED determined by actual
measurement.
If e be the thickness of the crystal HP we have
ED = EP – DP=eſtan r" – tan r).
But tan r is known, for the angle of incidence is equal to the inclina-
tion of HM to the vertical, and is consequently equal to the angle made
with the vertical by the axis of the telescope, and sin i = p, sin r:
therefore r is known and r may be determined by the above formula.
If this value of r agrees with its value as determined by the formula,
tan r u.
tan r u,'
the experiment will have proved that the section of the extraordinary
wave by a plane through the optic
axis is an ellipse, and consequently,
since the wave front is a surface of
revolution, it must be a spheroid
of axes a and b. The experi-
mental results are here in accord-
ance with the theory; but the
method is not extremely sensitive,
and indirect methods are much more sensitive and verify the
theory with greater precision.
(3) Optic Awis perpendicular to the Refracting Surface.—When the
Fig. 158.

ART. 180 NEGATIVE AND POSITIVE CRYSTALS 357
optic axis is perpendicular to the face of the rhomb it must be parallel
to the plane of incidence, and the section of the wave surface should
be a circle of radius b and an ellipse of axes a and b, as in Fig. 158. If
a circle of radius a be described with centre A it will touch the
ellipse at the extremities PQ of its axis. A tangent from A to this
circle will touch it at C", and if the tangent to the ellipse touch it at
C’ then C'C' will be perpendicular to the axis PQ of the ellipse. Con-
sequently if the angle OAC" (= AAC") be denoted by p, we have
tan p_DC 9 Pe.
tan r"TDC"Ta Tuo
But sin p = AC"/AA' = sin iſ us, hence
tan "=" tan p==*==–#.
b b WI–a4 sin” i u,vu,3–sinº
This relation, like the preceding, has been verified by Malus, and
it therefore affords additional evidence that the surface of the extra-
ordinary wave is an ellipsoid of revolution.
180. Negative and Positive Crystals.-All uniaxal crystals may
be divided into two classes. In the first class, to which Iceland spar
belongs, the wave surface consists of a spheroid with a sphere interior
to it touching it at the extremities of the axis minor. The radius of
the sphere is consequently always less than the corresponding radius
of the spheroid, and the ordinary velocity is less than the extra-
ordinary. The ordinary index is consequently greater than the extra-
ordinary and the latter ray is less bent towards the normal than the
former. For this reason such crystals are called negative or repulsive
Crystals.
In the second class, to which quartz belongs, the reverse is the
case, the spheroid is inside the sphere and the ordinary velocity is
therefore greater than the extraordinary. The index of refraction
of the extraordinary ray is consequently greater than that of the
ordinary ray. The former ray is more bent towards the normal
than the latter. These are therefore called positive or attractive
Crystals.
Since the sine of the angle of incidence at total reflection is the Critical
reciprocal of the refractive index, it follows that the critical angle is angle.
always less for the ordinary than for the extraordinary ray in negative
crystals, while the reverse is the case in positive crystals.
The following tables contain the values of the ordinary and
extraordinary indices for a few crystals:
358 DOUBLE REFRACTION IN UNIAXAL CRYSTALS CHAP. XI
The ray.
POSITIVE CRYSTALS

Rºo. Rºe.
Quartz e g e e w; * e 1 544 1 *553
Sulphate of Potash . * ge º 1 °493 1 : 502
Dioptase . te e * e e 1.667 1:723
Ice . º tº tº tº g g 1 "306 1 "307
Zircon tº te e º g ſº I '92 to 1 '96 1.97 to 2-1
NEGATIVE CRYSTALs
Rºo. Mºe,
Iceland Spar, º g º º 1 -658 1486
Tourmaline . e º & tº 1 -637 to 1 '644 1.619 to 1 °622
Beryl . © & g º º 1°584 to 1 '577 1:578 to 1 '572
Apatite g e º * * 1 *646 1 *642
Nitrate of Soda . te ſº g I ‘5854 1 °3369

181. Wave Velocity and Ray Velocity.—In the case of ordinary
refraction, such as occurs in isotropic media, the wave diverging from
any point is a sphere, and, as shown in Art. 65, the refracted ray is.
perpendicular to the front of the refracted wave. For this reason the
direction of the ray is the same as the direction of propagation of the
wave, and the ray velocity is the same as the wave velocity.
The direction of the ray in all cases is found by joining the centre
of disturbance A (Fig. 46) to the point C, in which the secondary
wave diverging from A touches the wave envelope A/C. When the
secondary wave is spherical the ray is perpendicular to the wave
envelope, but when the secondary wave has any other shape the ray
in general will not be normal to the wave envelope, and the direction
of the ray will not be the same as the direction of propagation of the
wave. Thus in Fig. 153 A'C is the front of the ordinary wave, and
AC is the ordinary ray emanating from A. So also A'C' is the front
of the extraordinary wave; that is, the extraordinary wave envelope,
and AC" is the extraordinary ray from A. The extraordinary ray is
therefore in general not at right angles to the front of the extraordinary
Wà,Ve. -
Now as the secondary wave (that is, the sphere and spheroid)
diverges from A the point C and C' move along the lines AC and
AC’ with certain definite velocities, termed the ray velocities; and it
is clear that the ray velocities are proportional to the radii AC and
AC’ respectively. At the same time the planes A/C and A'C' (that
ART. 181 WAVE VELOCITY AND RAY VELOCITY 359
is, the wave fronts) move forward with certain definite velocities,
termed the wave velocities. The direction of wave propagation is normal
to the wave front, thus the ordinary wave front moves in the direction
of the normal to AC, while the extraordinary wave moves in a direction
perpendicular to A'C'. It is clear, therefore, that the direction of
propagation and velocity of the extraordinary wave are in general not
the same as that of the extraordinary ray, for while the ray velocity is
proportional to the radius vector AC the wave velocity is proportional
to the perpendicular from A on A'C'. It is to the ray velocity that
the statement of Huygens's law in Section 177 applies. The relations
connecting the wave velocities and the ray velocities may be very
easily found as follows:
Relation between the Ordinary and Eatraordinary Wave Velocities.—Since
the wave is propagated in the direction of the normal, it is clear
that the ordinary wave velocity is measured by AC, the perpendicular
from A on A/C, while the velocity of the extraordinary wave is
measured by the perpendicular p from A on A'C'. Now since A'C' is
a tangent to the ellipse ºſa” + y”/b% = 1, it follows that the perpen-
dicular p from A on A'C' is given by the equation
p?= a” sin” a + b% cos” a
where a is the angle which p makes with the axis minor of the ellipse ;
that is, the angle between the wave normal and the optic axis.
From this equation we obtain at once the relation
p” – b%–(a”— b”) sin” a.
Now b is the velocity of the ordinary wave, and this equation proves
that the difference of the squares of the two wave velocities is propor-
tional to the square of the sine of the angle between the optic axis
and the direction of wave propagation, or normal to the front of the
extraordinary wave.
Relation between the Ordinary and Eatraordinary Ray Velocities.—Since
the ray velocities are measured by the radii AC and AC", then if we
denote AC’ by r and the angle which it makes with the optic axis
by 6, we have at once from the equation of the ellipse
1 _sin” 6 cos? 6
2- aſ Tā
3.
which gives at once the relation
1 1 1 1 \ .
;3 - 53– :-)sinº.
Hence the difference between the squares of the reciprocals of the
360 DOUBLE REFRACTION IN UNIAXAL CRYSTALS CHAP. xi
ordinary and extraordinary ray velocities bears a constant ratio to the
square of the sine of the angle between the extraordinary ray and the
optic axis. 4
This proposition is due to Biot, and similar relations of a more
general nature will be found to hold for biaxal crystals (see Arts. 206,
207).
182. Nicol's Prism.—The most effective and convenient method
of procuring a strong beam of plane-polarised light is by double
refraction. When ordinary light is transmitted through a crystal of
Iceland spar two réfracted beams arise, and these we have seen are
both plane-polarised and their planes of polarisation are at right angles.
Hence if one of the beams is intercepted by any means the other will
furnish a pencil of plane - polarised light. An attempt might be
made to stop one of the two refracted beams by placing an opaque
diaphragm on the second face of the crystal, but it may be easily
seen that this method would present difficulties, for unless the source
of light is very small or the rhomb very long the refracted beams
will overlap.
The former condition entails a great reduction of the illumination,
and the latter requires large specimens of Iceland spar of sufficient
purity, which are costly. The difficulty might, however, be evaded
by receiving the light on a lens placed in contact with the first face of
the crystal. After transmission through the lens and crystal the light
will converge to two foci—the ordinary rays to one and the extra-
ordinary to the other. One of these foci may now be covered by an
opaque diaphragm, and the rays diverging from the other received by
a second lens and reduced to parallelism if necessary. The first lens
may be plano-convex and may be cemented to the face of the crystal
with Canada balsam.
The most convenient method, however, is to stop one of the pencils
by total reflection inside the crystal. This is the method adopted in
Nicol's prism. A long lozenge-shaped rhomb of calcspar is formed by
cleavage from a crystal, so that its length AC (Fig. 159) is about
three times its width AD. This rhomb is cut in two by a plane
passing through the obtuse angles A and A’ and parallel to the longer
diagonal BD of the end ; that is, perpendicular to the principal plane,
ACA'C', which contains the optic axis and is normal to ABCD.
The cut faces are then polished and cemented together in their original
position by a thin film of Canada balsam. The refractive index of the
* Iceland spar is rather friable, and in practice it is found easier to grind away
half of the rhomb instead of cutting it as generally described. The remaining halves
of two rhombs thus ground are then cemented together.
ART. 183 FOUCAULT.'s PRISM 361
balsam is greater than that of the extraordinary ray" in the spar and
less than that of the ordinary. Now total reflection occurs only in
passing from a more to a less refracting medium. It follows, therefore,
that the extraordinary ray falling on the balsam will be always trans-
mitted, but if the incidence be sufficiently oblique the ordinary ray will
be totally reflected at the surface of the balsam and refused transmission
through the crystal (Fig. 160). The extraordinary ray alone is there-
Fig. 159. Nicol's Prism. Fig. 160.
fore transmitted, and the light emerging from the prism is plane-
polarised at right angles to the principal plane.
The plane of division of the crystal must be drawn so that the
light incident nearly normally on the end of the rhomb may fall upon
the Canada balsam at an angle not less than the critical angle for the
ordinary ray. This angle is easily calculated, for the ordinary index
of the spar is about 1:65, while the index of the balsam is nearly
1:55. Hence the index of refraction of the ordinary ray from the
crystal to the balsam is 1-55/I-65 = 939, and the sine of the angle of
incidence for total reflection must have this number for its minor limit.
Hence if the angle of incidence of the ordinary ray on the balsam is
equal to or greater than 69° 30', total reflection will occur.
183. Foucault's Prism.—The Canada balsam in Nicol's prism
might be replaced by any substance of less refractive index than
calcspar for either ray, or both. It is clear that the less the index of
the substance between the two segments of the prism the less the
critical angle, and the less the critical angle the shorter the rhomb
required to construct a prism of given width.
* The value of the extraordinary ray is not the value to be found in tables.
This latter value is that for a ray vibrating parallel to the axis. The value here
concerned depends on the angle of incidence and lies in between the two extreme
refractive indices. -

362 DOUBLE REFRACTION IN UNIAXAL CRYSTALS CHAP. XI
For this reason Foucault dispensed with the balsam, or cement,
altogether, and in the prism which bears his name there is a film of
air between the two segments. The critical angles for the ordinary
and extraordinary rays are about 37° 14′ and 42° 23'. Hence if the
angle of incidence on the film of air is intermediate between the
critical angles of the ordinary and extraordinary rays, the former will
be totally reflected and the latter transmitted. Although the use of
the air film permits of a considerable shortening and consequent reduc-
tion in price of the rhomb, yet there is more loss of illumination by
reflection from the film. With Nicol's prism the index of the balsam
sis so near that of the extraordinary ray that the last-named is trans-
mitted almost in its entirety. -
184. Other Prisms.--A great many varieties of Nicol prism have
been devised. The ordinary variety deviates the ray to one side,
though the emergent ray is parallel to the incident. This is obviated
in the Prazmowski prism, through which the light goes at right angles
to the end faces which are not cleavage planes but are ground down so
as to suit normal incidence. Reference should be made to an article
by the late Prof. S. P. Thompson in the Transactions of the Optical
Convention for 1905.
185. Rochon's Prism.—If a small pencil of light be transmitted
through a parallel plate of a doubly refracting substance both the
emergent pencils will be parallel to the incident beam and therefore
parallel to each other, while their interval of separation will be pro-
portional to the thickness of the plate for a given angle of incidence.
But if the faces of the plate be inclined at an angle, so as to form
a prism, the emergent beams will be inclined to each other and their
separation will increase as they recede from the prism.
Such a separation of the rays is useful in many investigations, and
in order to render the divergence as wide as possible the prism should
be cut with its refracting edge parallel to the optic axis, so that the
incident light may be perpendicular to that axis, for in this case the
difference of the ordinary and extraordinary indices is greatest. Such
a doubly refracting prism may be achromatised by means of a prism of
glass with its refracting edge turned in the opposite direction.
A better arrangement, however, is that employed by Rochon. Two
prisms of calcspar, or quartz, of the same angle are cut so that the
refracting edge in one is parallel to the optic axis and in
the other perpendicular to it (Fig. 161). They are then cemented
together with their edges in opposite directions so as to form a
parallelopiped. Thus in a normal cross section of the united prisms
the section ABD of one contains the axis, its direction being per-
A Rt. 186 ROCHON'S DOUBLE IMAGE MICROMETER 363
pendicular to the face AD, while the section BCD of the other is
perpendicular to the axis.
A ray incident normally on the face AD of the first prism travels
through it without division, as its direction is parallel to the optic
axis; on arriving at the interface double
refraction occurs, but the ordinary ray
proceeds undeviated. The extraordinary,
however, is deviated towards the edge or
towards the base of the prism ADB accord-
ing as the crystal is positive or negative.
In the first prism the two rays travel with a Fig. 101. Rochon's Prism. .
common velocity, namely, that of the ordinary ray, and in negative
crystals the ordinary velocity is least, while in positive crystals it is
the greatest. If A denotes the angle of the prism and 6 the devia-
tion, then since the angle of incidence on the interface is A, the
angle of refraction is A + 8, and
sin (A + 3) tº a
sin A v, b
or, as 8 is small, we have approximately
1 + 6 cot A=alb,
from which we find
a – b
6– 5 tan A.
Again, if r be the angle of emergence from the other face of the
prism, then, since the angle of incidence there is 6, we have
sin 5
:--te-d',
slin nº
if we take the velocity in air as unity. Consequently 3 – a sin r, and
we have, by the previous equation,
- 1 1
sln 7" – (; - ...) tan A = (un-ue) tan A,
which gives the angular separation (r) of the ordinary and extraordinary
rays, since the ordinary ray traverses the system normally without
deviation.
Rochon's prism is ordinarily constructed of quartz, so that the
deviation is in the opposite direction; that is, towards the base of the
second prism, since in quartz po is less than pe.
186. Rochon's Double Image Micrometer.—If a Rochon's prism
be placed in an ordinary telescope (Fig. 162) between the object glass
and its principal focus, two images, an ordinary and an extraordinary,
will be formed there. The distance between the images will depend

364 DOUBLE REFRACTION IN UNLAXAL CRYSTALS CHAP. xi
on the distance of the prism from the focus; accordingly, by moving
the prism suitably, the images may be brought into contact. For this
purpose the prism is movable within the telescope, and when the
images appear in contact the distance of the prism from the focus can
be read off on a graduated scale. By this means Arago determined
the apparent diameters of the planets with great accuracy.
Let f be the focal length OF of the object glass and a the distance
between the prism and the focus F. Then, if a be the apparent
angular diameter of the planet,
FQ=ftan a-a: tan 5,
in which, if we know f, ºr, and 6 the deviation, we can determine a.
For a given instrument the quantities f and 6 will be constant, and
could be determined by direct observation on an object of known
Fig. 162. – Double Image Micrometer.
diameter placed at a known distance. The quantity tan 6'ſ being
determined may be regarded as the constant of the instrument.
Again, if an object of height h be at a distance d, we have
h =dtan a,
from which we find either h or d if the other be known.
187. Wollaston's Prism.—This prism differs from that of Rochon
only in that the optic axis of the first prism ABD (Fig. 163) is parallel
to the face AB, so that it is merely Rochon's prism turned through a
right angle. A ray incident normally on the face AB travels along
the normal in the crystal as an ordinary ray with velocity v, and also
as an extraordinary ray with velocity v. On reaching the interface
the ordinary ray is refracted extraordinarily, for the principal planes of
the two prisms are at right angles, hence the angle of emergence of
the ordinary ray is, as in Rochon's prism, given by the equation,
sin r=(un-u.) tan A.
The extraordinary ray in the first prism ABD traverses it with a
velocity we, but it traverses the second prism as an ordinary ray with

Art. 187 WOLLASTON'S PRISM 365
a velocity vo. Hence, exactly as before, the angle of emergence from
the second face CD is also given by the above equation. The two
emerging rays are therefore equally deviated
on opposite sides of the normal and their
angular separation is doubled. By this
duplication the feeble double refraction of
quartz is rendered very sensible.
It should be remarked, however, that
the deviation is different for the different Fig. 168,-Wollaston's Prism.
colours, and the images are coloured. In Rochon's prism the ordinary
image has suffered no deviation and is therefore uncoloured.
If all four external faces of the prism are polished, it is both a
Rochon and Wollaston prism—a Wollaston if used as shown, a Rochon
if the light goes between the faces BC and AD.

CHAPTER XII
DOUBLE REFRACTION (FRESNEL’S THEORY)
188. The Wave Surface in Crystalline Media. — In homo-
geneous isotropic substances, such as glass, the physical qualities are
the same in all directions around any point, and if any element be
displaced the forces of restitution called into play will be opposite and
parallel to the displacement. The disturbance will travel with the
same velocity in all directions, and the wave surface will be spherical.
The optical properties of such substances are also alike in all directions,
and the elementary ether waves are spherical.
It is otherwise in the case of crystalline substances. Here the
physical properties, such as hardness, elasticity, compressibility, and
conductivity for heat and electricity are different in different directions
around any point. If any element of such a medium be displaced the
forces of restitution will not in general be parallel to the displacement,
but may be inclined to it, and will not in general tend to pull the
element directly back into its original position. A disturbance will
not travel with the same velocity in all directions, and the wave
surface will not be spherical.
It is consequently to be expected that the optical character of a
crystalline substance will also depend upon its structure, and that the
velocity of propagation of the ether waves will be different in different
directions, so that the wave diverging from any point will not be a
sphere, but a surface of some shape determined by the state of the
ether within the crystal and its relation to the molecules of the crystal.
In the case of a uniaxal crystal we have already seen that the wave
surface consists of two sheets, one a sphere and the other a spheroid.
Thus if we assume that the ether within a crystal possesses the
properties of a homogeneous elastic solid, the solution of the problem
of double refraction is afforded by the theory of elasticity, and as such
it has been developed by Green, MacCullagh, Neumann, and Cauchy.
This is known as the elastic solid theory, and is based on the supposi-
366
ART. 189 WAWE SURFACE AS AN ENVELOPE 367
tion that the ether consists of distinct particles, regarded as material
points (at least as far as Fresnel is concerned), which exert forces on
each other in the directions of the lines joining them and varying as
some function of the distance. In crystalline media the arrangement
of the ether particles is supposed to be different in different directions,
but symmetrical with respect to three mutually rectangular planes.
The ether is thus modified in its arrangement.or properties by the
presence of the crystalline matter. According to Fresnel the density
is modified, while according to Neumann and MacCullagh it is the
elasticity which undergoes modification. In all cases the ether is
supposed homogeneous, and its axes of symmetry are parallel to those
of the crystal.
We might, however (very reasonably), assume the ether to be
isotropic everywhere, and the same in all bodies as in free space,
and then proceed to explain reflection, refraction, dispersion, etc., as
the effects of the momentum communicated to the matter particles by
the motion of the ether. When light traverses a transparent substance
the matter particles may be set in vibration, thus absorbing some of
the energy of ether waves and impeding their progress; and further,
the amount of this may depend upon the direction of propagation in
crystalline media, but it will be the same in all directions in isotropic
substances. Within a crystal the velocity of propagation and the
absorption of energy in any direction may also depend upon the relation
of that direction to the azimuth of vibration. It is on this basis of
the interaction of the ether and the matter particles that Boussinesq,
Voigt, Sellmeier, Helmholtz, Lommel, and Thomson have built their
theories.* Great difficulties arise, however, in connection with the de-
velopment of the elastic solid theory, especially in connection with the
conditions that relate to the boundaries between two media. Fresnel's
theory is given here as an illustration, in terms of easily pictured
phenomena, of the general way in which polarisation may arise.
189. The Wave Surface as an Envelope.—Whatever be the nature
of the assumed dynamical conditions, everything is finally reduced to
the determination of the velocity of propagation of a plane wave, and
the mode of vibration which must exist in such a wave in order that
it may be propagated with a determinate velocity. The problem
before us is therefore to determine the law according to which a
plane wave travels in any assigned direction through a crystal in the
most general case. When this is known the deduction of the form
and properties of the wave surface becomes merely a matter of
geometry.
* See Glazebrook’s “Report on Optical Theories,” Brit. Assoc. Report, 1885.
Isotropic
ether.
368 FRESNEL’S THEORY OF DOUBLE REFRACTION CHAP. XII
Fresnel arrived at the form of the wave surface by considering it as
the envelope of a system of plane waves. Thus if a system of plane
waves starts from any point in various directions at the same instant,
each wave travelling in the direction of its normal with a velocity
depending on its direction of propagation, then after any given interval
all these plane waves will touch the wave surface, for the surface
touching all these planes will be the limit to which a disturbance
travelling out in all directions from this centre will have reached in
the given time. The construction for the wave surface considered as
an envelope is therefore as follows: On the radii drawn out from a
point measure lengths proportional to the velocities of normal propaga-
tion in their directions, and at each point thus determined draw a
plane perpendicular to the corresponding radius. The envelope of
these planes will be the wave surface. -
The principle involved in the preceding, viz. that the plane wave
moves parallel to itself, is very fundamental. For this wave preserves
not only its direction of motion but also the identity of its vibration,
and, for these to be preserved, it is necessary that the elastic forces called into
play by the displacement should be parallel to the azimuth of the displacement.
190. Fresnel's Hypotheses.—The hypotheses on which Fresnel
founded his theory may be summarised as follows: -
(1) The vibrations of polarised light are at right angles to the
plane of polarisation.*
(2) In all cases the elastic forces called into play by the propaga-
tion of a train of plane waves (the vibrations being rectilinear and
transverse) bear a constant ratio to the elastic forces developed by
the displacement of a single molecule, the others remaining at rest.
(3) When a plane wave is propagated in any homogeneous medium,
it is only the component of the elastic force parallel to the wave front
which is effective in the propagation of the wave.
(4) The velocity of propagation of a plane wave, of permanent type,
in any homogeneous medium, is proportional to the square root of the
effective component of the elastic force developed by the vibrations.
Little can be said in support of the second hypothesis, and Fresnel
himself was conscious of its weakness. Nevertheless the close agree-
ment of experiment with the results of Fresnel's theory must always
entitle it to favourable consideration. The fourth hypothesis is intro-
duced on account of a vague analogy between the transverse vibrations
of the ether and those of a stretched string, while the difficulty of the
third is removed by the supposition that the ether is incompressible,
* In the final form of Fresnel's theory, this is rather a deduction than a funda-
mental hypothesis, as pointed out by Walker, Nature, vol. lxxiii. p. 319, Feb. 1, 1906.
ART, 191 ELLIPSOID OF ELASTICITY 869
so that the velocity of propagation of the longitudinal vibrations is
infinite. Although dynamically unsound Fresnel's theory of double
refraction will ever possess a high historic interest, and we shall accord-
ingly detail its leading features and discuss the geometry of the wave
surface deduced from it.
191. The Ellipsoid of Elastieity.—Fresnel assumed the ether to
consist of particles mutually attracting each other, and which, when
disturbed from their positions of rest, vibrate under the influence of
their mutual attraction.
Assuming the ether to be molecular, and that each molecule is in
stable equilibrium under the influence of the others, we can show that
the potential energy of displacement of a single molecule is a quadratic
function of the three components of the displacement parallel to any
set of mutually rectangular axes.
Let V be the potential of the molecule at any point a, y, z, due to
the whole system of particles. The components parallel to the axes
of reference, of the force on the particle at this point, will be each zero,
since it is in equilibrium initially, and therefore we have
# =0, #=0. #=0. (1)
Now let the particle at a, y, z be displaced to a near point a + š, y + m,
2 + (, while the others remain at rest. The potential at this point will
be V + dW, or, since {, m, Č are supposed very small,
GW. GW G.W d°W ad”V, ad”V, o, dºw
#####(ºn d; ** # **śr etc.). (2)
The components of the force on the molecule are found by differentiating
(2) with respect to £, m, { respectively. Hence, remembering the
equations (1), we have
x=tºr *Y., day
Tº dº ""dºdy" ‘dºdº
d2V 02W G2V
G2V d2W 62V
— t--- -
**śāst"; +&i. |
or, writing
d”W_ A (72V B d2V Č2V 62V G2V
dº? dºñº Ž-9, 37; ", i.e", dºº"
We have, for the forces parallel to the axes of reference,
X = A; +H") + §).
Y= H$4-B7 + Fº
Z= G# + Fn + Cº.
(4)
Hence if we construct the quadric
2
B
370 FRESNEL's THEORY OF DOUBLE REFRACTION CHAP. XII
A£2-i-Bº-i-Cº-1-2Fm; + 2Cº+2H$n = 1, (5)
we have, denoting the left-hand member of this equation by 2S,
dS v_dS _dS - -
X=#| ||Y=; Z=#. (6)
Now & m, { are proportional to the direction cosines a, B, y, of the
displacement, and if the displacement be taken as unity, Ś, m, Č will be
numerically equal to a, 9, y respectively; but the resultant force on
the molecule will not be in general in the same direction as the dis-
placement, for the direction cosines of the resultant force are propor-
tional to X, Y, Z, and these by (6) are proportional to the direction
cosines of the normal to the quadric (5) at the point 3, m, Č. The
resultant force on the molecule is consequently not parallel to the
direction of displacement, viz. along the radius vector to £, m, Č, but is
along the normal to the quadric. *
There are, however, three directions at any point along which if a
molecule be displaced the resultant force will be parallel to the dis-
placement, and tend to restore the particle to its original position.
These directions are the axes of the quadric (5), and if we take them
for axes of co-ordinates the equation of the quadric may be written in
the form * -
a?aſº-H bºy” + c222–1. (7)
In this case the expression (4) for the components of the force on a
molecule displaced through a distance p = W& Fºº & will become
X = a”#, Y=b°m, Z= c^{ 8
Or X=pa'a, Y=pb°3, Z=pcºy J’ (8)
while the restoring force along the line a, B, y of the displacement will be
F=Xa + Y94-Zy=p(a’a”--b%3°4-c”y”),
OI’
F=} x- (9)
where r is the radius vector of the quadric (7) drawn in the direc-
tion a, B, Y.
Fresnel then assumed that the elastic forces due to the relative dis-
placements occurring during the propagation of a wave are proportional
to the elastic forces, determined as above, for absolute displacements,
if the displacement is in each case in the same direction. Further,
likening the medium to a collection of stretched cords (cf. Art. 35), and
considering the stationary waves formed by interference between an
advancing train of waves and the waves reflected by a plane reflecting
surface, he assumed that, as in the case of a cord the length of the
cord that will vibrate with a certain frequency is proportional to the
ART. 192 AZIMUTHS OF RESOLUTION 371
square root of the tension, or to the square root of the effective restoring
force on the displaced elements, so in the case of the medium the
distance between two consecutive nodes and therefore the velocity of
propagation of the wave is proportional to the square root of the
effective restoring force on the displaced elements. He considered the
ether as incompressible and the component F, along the line of displace-
ment of the particle as the effective restoring force, and it followed
that the velocity of propagation of a plane wave, in which the displace-
ment of the molecules is parallel to a given direction, is inversely
proportional to the radius vector r of the ellipsoid of elasticity drawn
in that direction. *
Cor.—If v be the velocity of propagation of a plane wave whose
vibrations are in the direction a, 8, y, then, since v c 1/f, we may
write equation (7) in the form,
v2–v,”a” + v.”3°4-vºy”, * ` (10)
Or - -
(v2 – v.2)a?--(v2–v,”)3°4-(v*-vº)Y*=0 (11)
where ve v, v, are now the velocities of propagation of waves vibrating
parallel to the axes of elasticity. If pºp ps, ps be the principal refractive
indices we may write, if we take the velocity outside the crystal to be
unity,
1 1 I
*-* === - - ..) ºr --- 12
A-, *, *s-, (12)
and (10) takes the form
- 2 2 2
w2= *, + Bºr Y. (13)
All* Alo" / 3"
192. Azimuths of Resolution.—The essential condition, for the
propagation of a plane wave without alteration, is that the effective
component of the elastic force developed by the displacement should be
parallel to the displacement. This condition is only satisfied for two
azimuths in any plane, viz. the axes of the conic in which that plane
cuts the preceding quadric.
When the elements in the front of a plane wave are displaced
parallel to a given direction in the wave front, we have seen that the
force of restitution on each element will not in general be in the
direction of the displacement, nor even in the plane, but will be normal
to the ellipsoid of elasticity; that is, perpendicular to the central
section of the ellipsoid which is conjugate to the direction of displace-
ment. Thus if AB (Fig. 164) be the section of the ellipsoid of
elasticity by the plane of the wave front, and OA the azimuth of
displacement of the elements in the front of the wave, OB the radius
of the section conjugate to OA, and OC the radius of the ellipsoid
37.2 FRESNEL's THEORY OF DOUBLE REFRACTION CHAP. XII
Circular
sections.
Polarisa-
tion.
Optic
axes.
conjugate to the section AB, then the force of restitution will be
parallel to ON the normal to the plane BOC. Now if the projection
of ON on the plane AOB coincides
with OA, the plane of ON and OA
must be perpendicular to the plane
AOB. But ON is perpendicular to
OB, therefore OB will be perpen-
dicular to OA; that is, OA and OB
are the axes of the section AOB.
Thus the resultant force on any par-
ticle may be resolved into two com-
Fig. 164. ponents, one in the wave front and
the other perpendicular to it. If the displacement is along either
of the axes of the elliptic section AOB, the force component
in the wave front will also be along that axis, and these axes
are the only azimuths in the wave front possessing this property.
These two azimuths are termed the singular directions or azimuths of
resolution in the plane of the wave.
If the plane be parallel to either of the circular sections of the
quadric, every direction is a singular direction, and a vibration in any
direction in this plane will be propagated without alteration and there
will consequently be no double refraction.
Thus when a displacement occurs in any azimuth in the front of a
wave it is only its components parallel to the singular directions that
are propagated as permanent waves, and these are propagated with
different velocities (except when the plane is parallel to a circular
section). Consequently the bifurcation of the ray on entering a
crystal is accounted for, as is also the polarisation of the two rays, and
the fact that their planes of polarisation are at right angles.
We have now arrived at the fundamental law of double refraction—
In one and the same direction of propagation two systems of plane waves
are propagated, having their vibrations parallel to the aires of the section of the
ellipsoid of elasticity by a diametral plane perpendicular to the direction, and
the velocities of propagation of the two systems are inversely proportional to
the lengths of these aires.
We see also that there are in general two directions of propagation
along which there will be no double refraction, and these directions
are perpendicular to the circular sections of the ellipsoid of elasticity.
They are termed the axes of single wave velocity or the optic awes of
the crystal. For waves propagated in these directions the velocity is
the same whatever be the azimuth of vibration. The wave fronts
will coincide, but there may be a separation of the rays (Art. 203).

ART, 193 PROBLEM 373
193. Problem.—To a variable plane drawn through the centre of
the ellipsoid
a perpendicular is drawn at the centre of the section, and portions
ON, and ON, (Fig. 164) are taken on it which, measured from the
centre, are equal to the axes of the section, to find the locus of their
extremities.
If r be the length of either axis, and l, m, n the direction cosines
of the normal to the plane, we have *
02/2 2m2 c2m2
----s---º-rº, + → = 0.
r2 – 0.2 ŽIE **I.
(See Salmon's Geometry of Three Dimensions, Art. 101.)
In this equation r is the length of the radius vector of the
required locus, and l, m, n are its direction cosines; the equation of
the surface is therefore
(232 bºy” cº”
7.2 — a 2 7.2 *g b? 72 - c2T
0.
194. The Normal Velocity Surface.—Around any point O con-
struct the ellipsoid of elasticity, and consider a system of plane waves
passing through O in all directions at the same instant. Let any one
of the planes cut the ellipsoid in the section AOB (Fig. 164) of which
the axes are OA and OB. Draw a normal at O to the plane of the
section and measure off ON1 and ON, on it inversely proportional to
the axes OA and OB, then if planes be drawn through N, and N,
parallel to the plane of the section, they will represent simultaneous
positions of the waves which will be propagated with their fronts
normal to ONN, the one having its vibrations parallel to OA and
the other parallel to OB, for we have already seen that the velocities
* If aſ, b’ be the axes of any central section, l, m, n, the direction cosines of the
normal to its plane, and R the intercept made on it by the surface, we have
1 1 1 1 1 1
aſt 5% + R2+ 2 + 5 tº
therefore -
1 1 1 1 1 /2 m2 m2
--> * ===-- *** - sºme “-º º
But
1 p” =#1 ***.
a'25'2Ta2bºcº Tb2S2 cºa? ' a 252
Whence the quadratic for 1, either semi-axis, is
1 1 / 1 – 72 1 – on” 1 — m? + £ on 2 m2 0
which may be written in the form above, by remembering that 1 = lº--m” + n°.
374 FRESNEL’S THEORY OF DOUBLE REFRACTION CHAP. XII
of propagation of these waves are inversely proportional to OA and
OB respectively. These planes therefore envelop the wave surface.
If the plane AOB be supposed to turn round O, the points N, and
N, determined as above will each trace out a sheet of a surface, and
the locus of both will therefore be a surface of two sheets, termed the
surface of normal velocities, since any radius vector of it determines the
two normal velocities of the plane waves propagated in that direction.
The equation of this surface is easily found, for it is described as
the locus of points on the normal (at the centre) to a variable central
section of the quadric
a’aº-bºy” + c22°–1,
the distances from the centre to the points being inversely propor-
tional to the axes of the section. The locus is therefore found from
the equation of the preceding article, by changing a, b, c, r into
their reciprocals, and we thus obtain
3.2 ſ? 22
Q
t + º - + = 0.
7.2 — a 2 '72 – b2 '72 – c.2
Cor. 1.-If v be the velocity of propagation in the direction l, m, n,
we have r proportional to v and ſº, y, z, proportional to l, m, n, therefore
72 m? + 7.2
v”— v,” v°– v.” vº-vº
= 0.
Cor. 2.—The direction cosines of the vibration being a, 6, y, we
have (Art. 191, Cor.)
(v*— v.”)a?--(v*— v.”)(3°4-(v*— vº))”–0,
and, since the vibration is perpendicular to the wave normal, we have
also
la + m3+ my=0.
Combining these equations with that of Cor. 1 we find
***, *-ºs-º-º:
in P- - , ).
195. Equation of the Wave Surface.—If v be the velocity of
wave propagation in the direction l, m, n, the wave surface is the
envelope of the plane
la + my + m2 = v (1)
where v is related to l, m, n by the equation
72 m? m?
wizatiºtº-9. (2)
and in addition we have always
!?-- m” + m2 = 1. - (3)
ART. 196 DIRECTION OF WIBRATION 375
Differentiating (1), (2), (3), regarding l, m, n as variables, we obtain 4
Xdl+ Yolm -- Zdn – dv- 0,
!d! ... main ... ndº { *******) do-0
w-º-º-º- \ºtº-ºy"Gººſ”,
ldl + malm + nām = 0
where X, Y, Z are the co-ordinates of the point where the plane touches
the wave surface. From which by the use of indeterminate multipliers
A and B, we have
X = Al + Blſ(v* – v,”), (4)
Y = Am + Bºm/(v* – v.”), * (5)
Z=An -H Bn/(v* – vº”), (6)
72 m” m* \
*(ºtºtº-1 (7)
Multiplying (4), (5), (6) by l, m, n respectively and adding we find
v= A, * (8)
while the same equations, squared and added, give with the aid of (7)
X*4-Y24-Z2–A24. B/v,
or, writing R* = X* + Y” + Z” and using (8) we obtain
B= v(R” – vº). (9)
Substituting these values of A and B in (4) we have
- R*-*_ R*-viº.
x=ioriº-º- *Evº Q
Therefore
y_*-** g X &
R” – v,” v
Similarly
m=º. X,
R” – woº v
and
a-º-º. 4.
R” – v.” v
Hence by substituting these values of l, m, n in (1), we have finally
2 - as 2 2 - nº.2 2 — nº.2 2 V 2 72
x4: "...yº.º. Zºº. 22-2 (.... 1%
R” – v.” R” – v.” “R” – vº y
Rºt R. "R2
OI’
v.2X2 v.”Yº v.27% -
H2 *g v,” R2 - v.” R” * va”
0,
which is the equation of the wave surface.
196. Direction of Vibration at any Point of the Wave Front.
—We have seen that in each direction in the crystal two waves can in
general be propagated with different velocities, and these are plane-
polarised in planes at right angles. The azimuths of the vibrations
* Archibald Smith, Phil. Mag., 1838, p. 335.
376 FRESNEL’S THEORY OF DOUBLE REFRACTION CHAP. XII
in these waves are parallel to the axes of the elliptic section of the
elasticity ellipsoid. We shall now show that this azimuth in each
case is parallel to the projection of the radius vector of the wave
surface on the tangent plane at the corresponding point. For by
Cor. 2, Art. 194, we have l, m, n related to a, B, ) by equations of
the form
7 - ºn. k? ºn.
---- nº -o-, -k o-, - Kºy.
* – tº "" vº-vº "º" vº-wº y
Therefore the equations (4), (5), (6) of the preceding Article become
X=Al-H BKa, Y=Am+ Bx3, Z=An + Bxy,
which show that the direction of vibration a, B, y lies in the plane
containing the radius vector to X, Y, Z and the perpendicular l, m, n on
the corresponding tangent plane to the wave surface. But the vibra-
tion takes place in the wave front, and is therefore parallel to the line
joining X, Y, Z to the foot of the perpendicular on the tangent plane,
or the direction of vibration in any ray is parallel to the projection of
the ray on the corresponding tangent plane.
197. Relation of the Azimuths of Vibration to the Optic Axes.
—The planes of polarisation of the two rays which correspond to any
given plane wave front are very simply
related to the optic axes. Let ON (Fig.
165) be the normal to the plane of the
wave, OM and OM the optic axes. Then
the planes of polarisation of the two rays
are the planes which pass through ON,
and the axes (OA and OB) of the section
in which the ellipsoid of elasticity is cut
by the plane through O to which ON is
Fig. 165. normal. Now the circular sections of the
ellipsoid are perpendicular to OM and OM', they will therefore meet
the elliptic section ARB in radii which are equal to each other and
perpendicular to the planes MON and MON respectively. That is,
the radii of the section which are perpendicular to the radii OR and
OR are equal, OR and OR are therefore also equal, and they are
consequently equally inclined to the axes OA and OB of the section,
and hence the planes containing ON and these axes will bisect the
angles between the planes MON and MON. That is, the azimuths
of vibration of the two rays bisect the angles between the planes con-
taining the wave normal and the optic axes.
198. Equation of the Wave Surface by Means of the Recip-
rocal Ellipsoid.—We have already deduced the wave surface as the

ART, 199 UNIAXAL CRYSTAL 377
*
envelope of a variable plane. It occurred to MacCullagh that the
wave surface might also be described as the locus of a point by using
the reciprocal quadric
a;2 iſ? 22
* .####1.
If any plane be drawn through the centre of this quadric cutting it in
a section of axes aſ and b', and if along the normal at its centre por-
tions be measured off equal to the axes a' and b respectively, the locus
of the extremities of these portions will be the wave surface. Its
equation is thus found at once to be #
a23:2 + bºy? + C22 = 0
Developed in terms of the co-ordinates, a, y, z, the equation becomes
(º-Fy” +2)(a^* + l2/2+ c22) — a”(b2+ c2)a?–52(c2+ a”),” – c’(a2+b^)2+ a”lºc?=0.
The surface is consequently of the fourth degree, and consists of two
sheets. We shall suppose a > b > c.
199. Uniaxal Crystal.—If two of the principal velocities are equal,
b = c suppose, the above equation may be written in this form
(*****-*)[aº-Hºº-ºº-oºl=0.
The surface consequently breaks up into the sphere
a?--y24-2°–b”,
and the spheroid
a2a3+b^{y^+2)=a^b°.
In this case the sphere and ellipsoid of revolution touch each other
where the axis b meets them, and this is the optic axis of the crystal.
The sphere lies entirely within the spheroid, so that the ordinary index
of refraction is greater than the extraordinary, and the crystal is
negative like Iceland spar when b is less than a. g
If, on the other hand, b is equal to a, then the radius of the
sphere is equal to a, and the spheroid lies entirely within the sphere,
the two touching where a meets them; a is the optic axis and the
crystal is positive or attractive, the ordinary index being less than the
extraordinary.
Finally, if a = b = c the wave surface reduces to the sphere
(3°4-y?4-2'-a')?=0, 7 w
which represents the case of isotropic media, or crystals belonging to
the cubic system. •
200. Principal Sections of the Wave Surface. —If in the
I MacCullagh, “On the Double Refraction of Light in a Crystallised Medium
according to the Principles of Fresnel” (Trans. Royal Irish Academy, June 1830).
378 FRESNEL's THEORY OF DOUBLE REFRACTION CHAP. XII
equation of the wave surface we make successively a = 0, y = 0,
2 = 0 we obtain the equations of the curves of section of the wave
surface by the planes of yº, ºr, and ry respectively. Each of these
Fig. 166. Fig. 167.
sections consists of two curves, namely, a circle and an ellipse having
the same centre.
(1) Thus making a = 0, we find
(y”-----a”)(bºy” + cºº-bºº")=0,
therefore the section of the surface by the plane /* consists of the
circle of radius a,
y”--2°–a4,
and the ellipse
bºy” + cº-lºcº
of axes b and c, which consequently lies entirely within the circle, as
shown in Fig. 166.
(2) Now suppose y = 0, and we find the section by the plane &
to consist of the circle
***=l,”,
and the ellipse -
a” + cº-a”,
the radius of the circle being b and the axes of the ellipse being
a and c it follows that the circle meets the ellipse in four points, as
represented in Fig. 167.
(3) Again, making z = 0, we find the circle
*-i-yº-cº,
and the ellipse
a” ºr bºy”-aºbº.
The circle therefore lies entirely within the ellipse, as shown in
Fig. 168.
Fig. 169 represents a segment of the surface of the wave by the

ART, 201 CONSTRUCTION OF HUY GENS 379
principal planes. It intercepts segments b and c on the axis of a,
and a on the axis of y, and a and b on the axis of z. It presents four
conical points in the plane zº, where the circle of radius b meets the
ellipse of axes a and c. The lines OP, OP to these points are such
that only one ray will be propagated in their direction, and they are
consequently called the aires of single ray relocity.
Models of the wave surface may be procured, and an examination
of one of them will assist the ideas of the student with reference to
the nature of the surface.
201. Construction of Huygens.—The form of the wave surface
being known, the directions of the refracted rays are determined by
Fig. 168.
Fig. 169.
tangent planes drawn to the two sheets of the surface according to
the construction of Huygens (Art. 66). Thus when a plane wave is
incident on the face of a biaxal crystal each point of the surface is a
centre from which elementary wave surfaces diverge; each wavelet
consists of two sheets, and the envelope consists of two planes, one
touching all the interior sheets, the other all the exterior. To
determine the directions of these planes it is only necessary to con-
struct one wave surface, and draw tangent planes to both its sheets
through the trace of the incident wave on the face of the crystal, as has
been already indicated in Arts. 66 and 178. The line joining the
centre of any wavelet to its point of contact with the tangent plane
gives the direction of the refracted ray.
It may happen that no real tangent plane can be drawn to one or
both of the sheets of the wave surface under the prescribed conditions,
and total reflection will then occur as in the case of isotropic media.
202. The Optic Axes or Axes of Single Wave Velocity–Axes
of Internal Conical Refraction.—The form of the section of the






380 FRESNEL'S THEORY OF DOUBLE REFRACTION CHAP. XII
wave surface by each of the principal planes has been arrived at in
Art, 199. Each principal plane cuts the surface in a circle and an
ellipse having the same centre, but in only one case, that of the plane
zº, do the circle and ellipse intersect.
Here the radius of the circle is b and
the axes of the ellipse are a and c.
The curves consequently intersect in
four points P, P., P., P1, and have
four common tangents MN, MANI,
M’N, MTN'i (Fig. 170). -
Now the planes passing through
these common tangents and per-
pendicular to the plane of the sec-
tion are tangent planes to the wave
surface. Moreover, they do not,
like ordinary tangent planes to a
surface, merely touch it at one point,
or even at two points, M and N, etc. The points P, Pe etc., are what
are termed conical points on the surface; they are little pits or dimples,
and the tangent planes MN, etc., cover them over and touch the
surface, as Sir William Hamilton proved, all round the perimeter of a
circle of contact."
The lines OM, OM are perpendicular to these planes, and they are
therefore such directions of the wave normal that only one wave front
exists, for the plane MN touches both sheets of the wave surface.
* The points on the surface S at which the tangent plane is parallel to the axis of
y satisfy the condition º–0. Applying this to the wave surface we find
y{b^{2^+y^+3*) – bºſcº-Faº)+a” + bºy” +cº –0.
The factor y=0 corresponds to points situated in the plane az, which obviously
satisfy the conditions of the problem. The second factor represents an ellipsoid
which clearly possesses the same planes of circular section as the ellipsoid of
elasticity a” + bºy” +cº–1. Hence a plane wave parallel to a circular section of
the ellipsoid of elasticity—that is, perpendicular to either optic axis—will meet the
above quadric in a circle, at every point of which it touches the wave surface. For
on eliminating y” between this quadric equation and that of the wave surface, the
result breaks up into factors
- a 2 – 52 a2 cº a2 bº v/ a 2– º)
- * * + b || -- * * + b
( + ºr Vº Vº)( * Vº b2 – c.
- a 2– bº /a2–gºſ. a2 – 52 v/. - º)
- * * – b - . 1 - - - -
(*** V.E. Vº)( * Vº l, b2 – c.2
which are the four tangent planes at the conical points.

ART. 203 INTERNAL CONICAL REFRACTION 381
The directions OM and OM’ are for this reason termed the axes
of single wave velocity. For these directions there is only one plane
wave, for other directions there are two ; they are therefore the optic
awes of the crystal.
The angle between the optic axes may be easily expressed in terms
of the principal velocities, for since OM(= b) is a perpendicular to the
tangent MN to the ellipse ºſcº + 2*|a” = 1 (Art. 200), its length is
given by the equation
b?– c’ cos”q + a” sin” p
where q is the angle OM makes with the axis of w—that is, half the
angle MOM'.
Consequently
; , , / D3-cº = . /*-*. tan º- Vº
sin q = Vº cos p = Vº an ºp až– 52'
or, in terms of the principal indices of refraction,
I.T.T., - ăTº I.T. 2
sin 4–% ^/ **, *, cos 4–% w/ º, tan p-ºl w/ **, *,*.
P2 'N' pliº – ps P2 (Y p.1% – ps P3 ‘Y pl" — u,
Since tan q = 2/3 it follows that the equations of the optic axes
OM and OM’ are
*== x º and y=0,
which shows that they are normal to the circular sections of the quadric
a’aº 4-5°y?--cºz”— 1.
In general a, b, c are functions of the wave length, and consequently
the angle between the optic axes varies with the colour of the light.
203. Internal Conical Refraction—Lloyd’s Experiment.—The
direction of a refracted ray is given by the line joining the centre of
disturbance to the point of contact of the wave surface with the wave
envelope, as determined by Huygens's construction.
Hence if O (Fig. 170) be the centre of disturbance (say a point on
the face of a crystal on which a plane wave is incident), and if MN be
the direction of the front of the refracted plane wave, it follows that
any line from O at any point of the circle of contact of MN with the
wave surface is a possible direction for the ray in the crystal. We
should expect then that a ray incident on the face of the crystal in
such a direction that the refracted wave in the crystal is parallel to
MN (that is, the plane wave in the crystal travels in the direction of
the optic axis) should on entering the crystal be divided not into two
rays but into a cone of rays, viz. the cone joining O to the circle of
contact of MN with the wave surface.
This result was predicted by Sir William Hamilton, and at his
38.2 FRESNEL’S THEORY OF DOUBLE REFRACTION CHAP. x 11
request the experiment was undertaken by Dr. Lloyd," who found the
anticipations of the theory verified in a most remarkable manner.
A plate of aragonite was used, having its faces perpendicular to
the bisectors of the angle between the optic axes, which, in the crystal
submitted to experiment, was about 20°. One of these lines is parallel
to OM (Fig. 171), and its direction was determined beforehand by
means of the phenomena of the colours of crystalline plates. A slender
pencil of light, SO, limited by
two screens, one of which, CD,
was at some distance from the
plate, and the other, which
was a thin leaf of metal, was
pierced by a small hole and
placed on the face of the plate.
The emergent rays were re-
ceived on a screen of silver
paper, EF. The minuteness of this phenomenon and the perfect
accuracy required in the incidence rendered it very difficult to observe.
The crystal was moved with extreme slowness so as to vary the direc-
tion of incidence very gradually, and when the required position was
obtained the two images on the screen EF suddenly spread out into a
continuous ring of light. No sensible enlargement of this ring could be
observed as the distance of the screen EF from the plate was altered,
showing that the emergent beam was cylindrical, and that consequently
the path of the light in the crystal was the cone OMN. The angle of
this cone was found to be 1° 50', and its magnitude as indicated by
the theory was 1° 55', so that the observed and theoretical values
agreed very closely.
To measure the angle of incidence Lloyd received the pencil
reflected at 0 on a screen, and marked the point K where it fell.
He then removed the plate and arranged a theodolite so that its axis
of rotation passed through O. He was then able to measure the angle
SOK, which is double the angle of incidence. The observed angle
of incidence was 15° 40', and its value as indicated by theory 15° 19.
The agreement of the theory and experiment is thus exceedingly com-
plete. The diameter of the ring on the screen EF determines the
angle of the refracted cone.
The existence of conical refraction has been regarded as one of the
most striking proofs of the general correctness of Fresnel's theory of
double refraction, but Stokes” has pointed out that it is not competent
Fig. 171.
* Lloyd, Trans. Roy. frish Acad, vol. xvii. p. 145, 1833.
* Stokes, Brit. Assoc. Report, 1862.

Art. 204 AXES OF SINGLE RAY VELOCITY 383
to decide between the several theories which lead to Fresnel's wave
surface as a near approximation. Internal conical refraction depends
upon the existence of a tangent plane to the wave surface which touches
it along a plane curve, and this property would be possessed by the
wave surface arising from any reasonable hypothesis. Other forms of
the wave theory, based on very different assumptions, lead to Fresnel's
wave surface exactly. The existence of conical refraction cannot
therefore be regarded as deciding in favour of Fresnel's particular
doctrine.
Azimuth of Vibration.—It is easy to determine the azimuth of the
vibrations in each of the rays which constitute the cone OMN, for the
azimuth of vibration of any refracted ray is found by projecting the
ray on the corresponding tangent plane to the wave surface (Art. 196).
Now M is a point on the circle in which the
tangent plane touches the wave, and OM is
perpendicular to this plane, therefore if ON (Fig.
172) is any ray of the cone its projection on the
plane of the circle will pass through M and con-
sequently be the chord MN of the circle. Hence
the azimuths of the vibrations of the different
rays of the cone are parallel to the chords of the
circle of contact drawn from M to the points Fig. 172.
where the rays meet this circle. It follows therefore that two rays
meeting this circle at diametrically opposite points are such that
their vibrations are at right angles; they are therefore polarised
at right angles. To verify this it is only necessary to receive the
emergent cylinder of light on a tourmaline plate or a Nicol's prism,
and of the two extremities of the same diameter of the ring one will
be completely dark and the other brightest, the illumination gradually
fading round the ring from the latter point to the former.
204. Axes of Single Ray Velocity.—We have seen that the wave
surface presents four singular points in the plane zº. These points P,
P', P, Pºp are common to both sheets of the surface, and are such that
at any one of them P an infinite number of tangent planes can be
drawn to the surface, and not merely two, as Fresnel appears to have
imagined, viz. –one to the circle and one to the ellipse (Fig. 170).
This system of tangent planes forms a tangent cone to the surface at
the conical point P. Now if a ray travels in any direction in the
crystal, the velocity of the ray is measured by the radius vector of the
wave surface drawn in its direction. Consequently in any direction
we have in general two ray velocities, since the radius vector has in
general two values, one given by each sheet of the wave. But if a

384 FRESNEL's THEORY OF DOUBLE REFRACTION CHAP. XII
ray travels in the direction OP there is only one value of the radius
vector, and consequently only one velocity of the ray. . Consequently
both rays travel in the direction OP (or OP') with the same velocity,
and these directions are called the awes of Single ray velocity. They are
generally very close to the optic axes, or axes of single wave velocity,
but they are not on that account to be confounded with them. -
The angle between the axes of single ray velocity may be easily
expressed in terms of a, b, c. For the co-ordinates a and 2 of P are
common to the circle *
a 24-2°– U2,
and to the ellipse
a’aº 4-cº–a’e”;
therefore
----- —a--
b? — cº
a”
2 -
(l, º, ands-avſ
2
a” — c
*-ºv/
– c’
Hence if the angle POM (Fig. 170) be denoted by p, we have tan p = 2/a,
and therefore
_c / a7 – 5% :... . . . .” 5–3 _d ÉTº
cº-Nº, in *-*Nº, tan *** V =.
The right lines joining O to P and P. are consequently given by
the equations
__La /ö” – c’
2 F +. ^/ 2.3T B2
. 2, and y=0;
they are therefore perpendicular to the circular sections of the
reciprocal ellipsoid
2
o:2 aſ? .
###=1.
C
º, tř
Cor.—The above values of the co-ordinates of P show that it is a
singular point on the surface, for they satisfy the equations
dS dS dS
# =0, #-0. #–0
where S denotes the equation of the wave surface. There is con-
sequently at P a tangent cone to the surface.
The axes of single ray velocity are called the awes of external conical
refraction.
205. External Conieal Refraction.—The direction pursued by a
refracted ray, after emerging from the crystal, is determined by the
position of the tangent plane to the wave surface at the point where
the ray meets it. But at any one of the conical points P there is
an infinite number of tangent planes enveloping a cone, consequently
the ray which traverses the crystal in the direction OP (or OP) may
on emergence pursue the direction determined by any one of these
Ant. 205 EXTERNAL CONICAL REFRACTION 385
tangent planes. The emergent beam should therefore be of a conical
form. -
Dr. Lloyd found that this was fully verified by experiment.
Taking the plate of aragonite already mentioned, he placed on each face
of it a thin plate of metal, perforated by a very minute aperture, as
shown in Fig. 173. These plates were so adjusted that the line
connecting the two apertures coincided with the direction of the axis
of single ray velocity. A flame of a lamp was then brought near the
aperture O in such a manner that the central part of the convergent
beam should have an incidence
of 15° or 16°. When the adjust-
ment was completed a brilliant
annulus of light was seen on
looking through the aperture P
in the second plate. Whenever
the second plate was ever so
slightly moved, so that the line.
OP connecting the apertures no
longer coincided with the axis of Fig. 173.
single ray velocity, the phenomenon rapidly changed and the annulus
resolved itself into two separate images.
The incident light was also brought to a focus on the surface of
the plate by means of a lens of short focal length. In this case the
upper plate was dispensed with and the lamp removed to a distance.
The rays of the sun were also used and the emergent light received
on a screen. Each ray of the incident cone of light is doubly refracted
at 0, but for all the rays on a certain conical shell of the incident
cone of light one of the refracted rays will travel along OP, and
emerge as a ray of the external cone. If the aperture in the second
plate is not very small, some of the rays which do not travel exactly
along OP will be allowed through, and considerable discrepancy will
occur between the results of observation and theory. However,
when the necessary correction is applied, the agreement, between the
theoretical and observed magnitude of the angle of the cone was
found to be nearly complete, the observed angle being 2° 59' and the
calculated angle 3° 0' 58".
Dr. Lloyd also determined that when external conical refraction is
exhibited the ray OP is parallel to the axis of single ray velocity.
In order to do this he observed the angle of incidence of the axis of
the convergent beam on the first face, and he found it 15° 58' when
the comical refraction occurred. He also calculated by theory the
angle at which the axis of the incident cone should meet the first
2 C.

386 FRESNEL’S THEORY OF DOUBLE REFRACTION CHAP. XII
face in order that the refracted ray should be parallel to the axis of
single ray velocity, and he found it to be 15° 25' 8".
It was also found by experiment that “the angle between the planes
of polarisation of any two rays of the come is half the angle between the planes
containing the rays themselves and the avis.” This remarkable law is also
in complete accordance with the theory as in the case of internal
conical refraction (Art. 203).
206. Relation between the Velocities of Propagation of a Plane
Wave and the Position of the Wave Normal With respect to the
Optic Axes.—The velocities of the two waves travelling in any direc-
tion are given by the radii in that direction of the surface
of normal velocities; they are consequently determined by the
equation
72 on? m?
jºtziºt; 2–0
where l, m, n are the direction cosines of r. Hence
r — [(b2+ c2)!?--(c2+ a”)m°4-(a” + 5°)n?]??-->5%”!?=0,
or, denoting the roots of this equation by r" and r*, we have
r” + r"?=x(52+ c2)??,
and
7'27'2–Xb2C2/2.
Now if r makes angles 6' and 6" with the optic axes, and, using the
notation of Art. 202, if the direction angles of the optic axes be q,
#T, #7 – b, and T – $, $1, $1 – 4 respectively, we have
cos 6' =l cos q + m, sin (p,
cos 0"— – l cos p + n sin ºp.
Therefore
cos 6' — cos 6" / a? – c.” A
= ~~~~~ =# (cos 6' — cos 6" -F-35 rt. 202.
2 2 cos p # (cos 6' — cos 6") ^/ 2 T2 ( )
_cos 0' +cos 0" / f/ / a2 – c.2
T 2 sin q - =# (cos 6' + cos 6") ^/ ; I2 5 §
* A remarky ble variation of the phenomena took place on substituting a narrow
linear aperture for the small circular one in the plate next the lamp, the line being
so adjusted that the plane passing through it and the aperture on the second face
should coincide with the plane of the optic axes. In this case, according to the
hitherto received views, all the rays transmitted through the second aperture should
be refracted doubly in the plane of the optic axes so that no part of the line should
appear enlarged in breadth in looking through this aperture ; while according to
Sir William Hamilton the ray which proceeds in the direction OM should be
refracted in every plane. The latter was found to be the case ; in the neighbour-
hood of each of the optic axes the luminous line was bent, on either side of the
plane of the axes, into an oval curve. This curve, it is easy to show, is the comchoid
of Nicomedes, whose asymptote is the line on the first surface (Lloyd, Wave Theory
of Light, p. 212).
ART. 206 RELATION CONNECTING RAY. VELOCITIES 387
and
Hence
r^2+y^2=a2+ c2 – 4 (cos 6' – cos 0")*(a” – c’)++ (cos 0' + cos 0")*(a”—cº)
=a^+ c2+ (a”— c”) cos 6' cos 0",
and
r’?,’” – a "cº-- 4 (a”—cº)*(cos” 0' + cos” 0")++(a+-cº) cos 6' cos 0".
From which we have
(r'2–r”)?=(r” +r”)?–4r”,”–(68–69)” sin” 0' sin” 0",
or, finally, since r" and r" measure the velocities, we have
w” – w”=r” – r"?=(a?– c’) sin 9' sin 0",
and
v°4-v"?=a^+ c2+(a” – c’ cos 6' cos 0",
which establishes a relation between the velocities of the two plane
waves which are propagated in any direction and the angles which
this direction makes with the optic axes.
Cor.—The normal velocities of the two waves propagated in the
same direction are
v% =#(a2+ c2)++(a” – c’) cos (6'-6"),
v”=#(a^+ c2)++(a” – c’) cos (6'-6").
207. Relation connecting the Ray Velocities in a given Direction,
and the Angles made with the Axes of Single Ray Velocity.—Writing
the equation of the wave surface in the form
rºXa?!?– r?»a”(b?--cº)!?--a%2c% = 0,
and denoting its roots by r" and r", we have
I 1–sſ 1. l 2–1. l 2/1 - 1 aſ 1 I
#}=-(+) =;4; +? (i. #)+n (*-ā)
and
l 72
—HR,+ 2. —
7'2, "2 b2C2
Now if the radius vector r makes angles 6 and 6" with the axes of
single ray velocity, and if the direction angles of these lines be iſ, #7,
#T – p, and T – p, #7, #1 — ſº, respectively, then
cos 6' =l cos p + m, sin p,
cos 6"— – l cos p + m, sin p.
Therefore
*-*-º-º-º:
cos 6' — cos 6" / ab w/ a” – c.2
t==== Goºd - COS 6 }: 22 Tº
and
* cos 6' + cos 6" b a? – c.2
= tº:-- *|† — =# (cos 6' + cos 6")- * *
# (cos ?: cosº"; Vº
388 º
FRESNEL's THEORY OF DOUBLE REFRACTION CHAP. XII
while m” is determined by the equation
on?= 1 – 7% – m”.
Hence
1 1 1 , 1 1 1 f f/
zāt ºf at at (#-3) cos 6' cos 0",
and
& 1 \2
Whence
I 1 l l & / *
-a – †, = | * – H \sin 6' sin 6".
7'2 r”2 a? cº
The difference of the squares of the reciprocals of the ray velocities is
consequently proportional to the product of the sines of the angles
which the common direction of the rays makes with the axes of single
ray velocity. Denoting the velocities of the rays by v' and v" we may
write this relation in the form g
e v'–2– w”-2–(a-?– c-2) sin 9' sin 9".
Cor. 1–Since the velocities are inversely proportional to the
refractive indices, we have
p”— u"*=(u,” – us”) sin 6' sin 6",
or approximately
p' – u"=(ul – us) sin 6' sin 6".
Cor. 2.--Since the relative retardation introduced by a plate of
thickness e is e(u' – p") for normal incidence, we have
- 6= e(u' – u")= e(ul – ps) sin 6' sin 6".
Cor. 3.—The ray velocities are given by the equations
v'-2–4(c-4--a-”) — (c-º- a--)cos(6'-6"),
v"–9–$(c-44-a-”) –$(c-4-a-”) cos (6'4-6").
BIAXAL CRYSTALS
PRINCIPAL INDICES FOR SODIUM LIGHT

Least (...) Mean (#). Greatest (#). Temp. Observer.
Aragonite e 1°53013 I'68157 1 °68589 ... Rudberg.
Borax . g 1 *4463 I “4682 1 °4712 23° Kohlrausch.
Mica g º 1 °5609 1°5941 l'5997 23° 33
Nitre . e 1 °3346 1 °5056 1 ‘5064 16° Schrauf.
Selenite . . 152082 1.52287 1 °53048 17° | W. Lang.
Sulphur (prism) 1°9505 2'0383 2*2405 16° Schrauf.
Topaz . I '61161 1 '6] 375 1 *62109 ... Rudberg.
C HAPTER XIII
REFLECTION AND REFRACTION OF POLARISED LIGHT
208. Fundamental Principles and Hypotheses. – The first
attempt to determine the relation between the intensities of the
incident, reflected, and refracted pencils when light falls upon the
surface of separation of transparent media was made by Young,” but
he confined his investigations to the particular case of perpendicular
incidence. The amplitude, form, and phase of the incident vibra-
tion being given, the problem before us is to determine the
amplitudes, forms, and phases of the reflected and refracted vibrations.
Thus if the simple vibration
Aſ = a sin (wt + a)
gives birth to the simple reflected and refracted vibrations
- $/l = b sin (wt + 3), and ye=c sin (wt + y),
which differ from the incident in phase and amplitude, we require to
determine b and c, B and y in terms of the known quantities.
In approaching this subject hypotheses must be made in two
departments, one with respect to the nature of the vibration, and the
other with respect to the nature of the difference in the properties
of the ether in different media, and as to whether the change of con-
dition is sudden or gradual at their surface of separation.
The hypotheses adopted by Fresnel lead to formulae which are in
very close agreement with the results of experiment. He founded his
theory (in 1821–23) on the following principles.”
(1) The Principle of the Conservation of Energy, from which it follows
that the energy of the incident wave is equal to the sum of the
energies of the reflected and refracted waves which arise from it.
Denoting the amplitudes of the corresponding vibrations by a, b, c
respectively, we have the energy equation (Art. 68)
p(a” — b”) sin 2i =p'c” sin 2, (energy equation)
* Art. “Chromatics,” Ency. Brit. Supplement.
* Fresnel, CEuvres, tom. i. pp. 441-799.
889
390 REFLECTION AND REFRACTION OF POLARISED LIGHT ch. xiii
connecting the amplitudes of the incident, reflected, and refracted
vibrations. *
This equation of course is deduced on the supposition that the
Second medium absorbs no part of the refracted pencil. In general,
however, the second medium absorbs a portion of the refracted light,
and the corresponding energy appears as heat in the elevation of
temperature of the substance. The calculations and formulae founded
on this equation are consequently limited by this supposition, and they
therefore apply only to the case of waves which are transmitted without
absorption. It is further assumed that the transverse vibrations of
the incident light excite only transverse (or light) vibrations in the
second medium, or that the entire energy of the incident light appears
again as light in the reflected and refracted pencils. In the case of
elastic solids, however, both longitudinal and transverse vibrations are
produced by reflection and refraction, so that we have in general two
reflected and two refracted waves, one transverse and the other
longitudinal, and these are propagated with different velocities.
(2) Hypothesis of Uniform Elasticity of the Ether.—Some hypothesis
must now be made concerning the symbols p and p" which are called
the densities of the ether in the two media. Fresnel assumes that
the velocity of propagation in any medium varies inversely as the
square root of the ether density in that medium, so that
Nº'-v sini
Jºy Tsinº
Now the velocity of propagation of waves in elastic matter is
measured by the square root of the elasticity divided by the density;
this assumption is consequently analogous to saying that the elasticity
of the ether, or that property of it which corresponds to the elasticity
of ordinary matter, is the same in all media. Introducing this assump-
tion into the energy equation we have Fresnel's modified form
*E*- . (Fresnel's energy equation.)
(3) Continuity of the Displacement.—To determine the ratios of b
and c to a we require another equation. This is obtained by suppos-
ing that the displacement remains the same in crossing the surface
of separation. Thus if planes be drawn parallel to the interface and
very near it, one in each medium, the velocities and displacements of
the elements in these planes can only differ by an infinitely small
fraction of their own value. If the ethers in the two media be treated
as two portions of different elastic substances (like jellies for example)
in contact, then at the interface they must always remain in contact ;
ART. 209 POLARISED ANGLES-PLANE OF INCIDENCE 391
that is, during the motion there is no slipping of one on the other
parallel to the surface, and also there should be no separation or
relative motion perpendicular to the surface. The displacement at
the common surface must be the same in the two media, and this
must include the longitudinal displacement, or pressural wave, as well
as the transverse vibrations which are supposed to constitute light.
According to the elastic-solid theory the ether belonging to any
medium always remains in that medium, never crossing the interface
or changing its density. If, however, we look upon the ether in the
two media as being continuous but differing in density, a portion of
the ether in either may cross the interface into the other, and a thin
layer of the ether at the surface might suffer rapid periodic changes of
density. However, if we admit Fresnel's assumption that there is no
change of phase in crossing the surface, this layer of variable density
must be infinitely thin compared with the length of a wave, so thin,
in fact, that the phases of the vibrations on each side of it may be
considered the same." -
Fresnel did not consider the component of the displacement per-
pendicular to the surface, he merely secures continuity parallel to the
interface, so that there is no tangential slipping of the ether in one
medium on that in the other. >
MacCullagh, on the other hand, worked on the supposition that
the vibrations in the two contiguous media are equivalent ; that is, the
resultant of the incident and reflected vibrations is the same, both in
magnitude and direction, as the resultant refracted vibration. This
hypothesis he termed the principle of equivalent vibrations.”
209. Light Polarised in the Plane of Ineidence.—According to
the theory of Fresnel the direction of the vibration is perpendicular to
the plane of polarisation, so that for light polarised in the plane of
incidence the vibration is parallel to the surface of separation. Again,
since no change of phase is supposed to accompany the reflection and
refraction, the extreme displacements a, b, c will be attained at the
same instant. Hence the complete displacement in the first medium
is their algebraic sum a + b (where b may be inherently positive or
negative with respect to a), and by the third principle at the interface
this must be equal to c, the displacement in the second. Hence to
determine b and c we have the equation of continuity
a + b = c, (1)
* See further Glazebrook’s “Report on Optical Theories,” Brit. Assoc. Report,
1885, p. 186.
* MacCullagh, “On the Laws of Crystalline Reflection and Refraction,” Trans.
Roy. Irish Acad. vol. xviii., January 1837.
392 REFLECTION AND REFRACTION OF POLARISED LIGHT ch. xiii
together with Fresnel's energy equation,
a” — b%=cº tan & cot r. . (2)
Dividing (2) by (1) we obtain
a – b = c tam & cot r. (3)
Combining (1) and (3) we find at once
sin (? – r)
sin (i+r)' (reflected)
and
tº sºme 2a cos ? sin r
T Tsin (; Er) Tº (refracted)
Thus the sign of b is opposite to or the same as that of a according as
? is greater or less than r ; that is, according as the second medium is
more or less refracting than the first.
It should be remarked that the relative intensities of the incident
reflected and refracted rays are not measured merely by a”, b”, cº, but
by the rates at which energy is propagated by the corresponding
waves (Art. 68); that is, according to Fresnel's theory, by
a?: b”: c’ tan i cot r,
or by
2 . .2sin” (i-r). ...sin 2; sin 2r
sin” (i+r) “sin” (i+r)
Cor. 1.-The expression for b may be written in the form
sin (i-1) p cos r-cos ? p. — cos iſ cos r
— Q. --—- : = - 0, -— = – Q —H-
sin (7+ r.) A cos r + cos ? p-H cos iſ cos r
Hence if i = 1 = 0° we have cos i = cos r = 1, and therefore at perpen-
dicular incidence
pi – 1
= — C, --
a-FI’
which is the expression arrived at by Young.
As i increases from 0° to 90° the value of b increases numerically
from Young's value to — a, because
cos ? { 1 — sin” . }
cos rT U1 – (sin” i)/u?/ ?
which diminishes continuously from unity to zero as i increases from
0° to 90°.
Similarly
_2a cos i sin r_ 2a cos i
sin (i+ r.) Tº cos r +cos i
Therefore at perpendicular incidence
2a.
C E . —-->
a-Fi'
ART. 210 POLARISED ANGLES-PLANE OF INCIDENCE 393
consequently c decreases from this value to zero as i increases from
zero to #T. The intensities at perpendicular incidence are propor-
tional to
• *(iii) “º
These formulae have been verified photometrically by Arago, while
Provostaye and DeSains, by means of the thermopile, have confirmed
their accuracy for heat radiations.
210. Light Polarised Perpendicularly to the Plane of Incidence.
—In the case of light polarised perpendicularly to the plane of incid-
ence the vibration, according to Fresnel's theory, is in the plane of
incidence ; but the vibration must be in the wave front, it is therefore
along the azimuth AB (Fig. 47) and makes an angle i with the sur-
face of separation. In the reflected and refracted waves it is along
A'B' and A'C', making angles i and r respectively with the surface.
Hence the algebraic * sum of the displacements in the upper medium is
(a + b) cos i parallel to the surface, and in the lower medium c cos r,
consequently for no slipping at the 'surface of separation we have
(a + b) cos i = c cos r.
Combining this with the energy equation
a” — b%= c” tan icot º'
we have, by division, -
a – b = c sin iſsin r.
Therefore
tan (i – r)
T“tan (i+ry (reflected)
and
2a cos i sin r
*Tsin (; E7) cos(?–7). (refracted)
Hence, if i + r < 90°, we see that if i be greater than 7 then a and
b will have opposite signs; but, on the other hand, when the
first medium is more refracting than the second, a and b have the
same sign. The relative intensities are in the ratios a”: b%: c’ tan i
cot r ; that is,
a stan” (: - ?) 2 sin 27 sin 27' &
* * tan? (i+r)' sin” (i+r) cos” (i – r)
Cor. 1.-Writing the expression for b in the form
A cos r – cos i cos (? --r)
g 2
b = — a ... • &
A cos r +cos i cos(i-1)
1 b is here taken as positive when its component along the surface is in the same
direction as the component of the incident vibration along the surface, i.e. (Fig. 47),
if AB is the positive direction of the incident vibration, B'A' is the positive direction
for the reflected vibration.
394 REFLECTION AND REFRACTION OF POLARISED LIGHT CH. XIII
obtained from the above value by merely dividing the numerator and
denominator by sin r, we see that when i = 1 = 0 we have
p. – 1
* Ti,
-* –
the square of which measures the intensity of the light reflected
normally. This expression is the same as that which determined the
normally reflected beam when the incident light is polarised in the
plane of incidence, as it obviously should be, for in both cases when
the light is incident normally, the vibration is parallel to the surface,
and the two should be reflected according to the same law.
Similarly when i = r = 0 the expression for c becomes the same
in both cases, and the reflected and refracted intensities are com-
plementary.
Cor. 2.-If i + r = 90°, which is the general condition at the angle
of maximum polarisation, we have
b=0, and c= a/u.
The light is therefore all refracted, and its intensity, measured by
c” tan icot r, becomes at once cºp.” (since i + r = 90°), or a”, which is the
measure of the intensity of the incident light.
Hence the amplitude of the reflected pencil decreases from Young's
value to zero, as the angle of incidence increases from zero to the
polarising angle (tan i = p). It then changes sign (or the phase changes
by T) as i increases, and attains the value a at grazing incidence.
211. Light Polarised in any Plane—Change of the Plane of
Polarisation by Reflection.—Let us now suppose the incident light
to be plane-polarised in a plane inclined at any angle a to the plane
of incidence. The direction of the vibration now makes an angle
90° – a with the plane of incidence (following Fresnel), and we can
therefore resolve it into two components
*
a cos a, and a sin a,
perpendicular to, and in, the plane of incidence respectively. The
former component may be regarded as a beam of light polarised in
the plane of incidence, and it will give rise to reflected and refracted
rays determined by Art. 209, in which a is to be replaced by a cos a.
The latter component is a pencil polarised perpendicularly to the plane
of incidence, and also gives rise to reflected and refracted rays deter-
mined by Art. 210. -
Hence the reflected light is the resultant of two portions, one
polarised in the plane of incidence and the other polarised perpendicu-
larly to it, the amplitudes of these portions are
ART. 211 CHANGE OF PLANE OF POLARISATION BY REFLECTION 395
- sin (? - ?) , º, tan (? – r) t
a cos"sin (i+r)' and – a sin a tan (i+r) (reflected)
respectively, while the refracted light consists of
cos ? sin r g cos i sin r
2a cos 0. sin (i+7)' and 20 sin a sin (i+ r) cos (; — r)' (r effected)
polarised in and perpendicular to the plane of incidence respectively.
If then the reflected light be plane-polarised in a plane making an
angle 6 with the plane of reflection, its components perpendicular to
and in this plane are b cos B and b sin 6 respectively, where b is the
amplitude of the reflected vibration. So also if c be the amplitude of
the refracted vibration, and y the angle its plane of polarisation makes
with the plane of refraction, its components perpendicular to and in
that plane are c cos y and c sin y respectively, consequently we have
b cos 3– – a cos asin (?-?)
2– sin (i+ r.)
* . . . tan (? – ?) | * (reflected)
b sin 8= — a sin *tan(TF)
cos i sin º'
ecosy-2acosa; l
c sin y=2a sin a cos i sin º' | tº (refracted)
sln y = sin (7+ r) cos (; — r)
From these equations we can determine the positions of the planes
of polarisation of the reflected and refracted pencils, for we have at once
i cos (7+?'
tän 3=tan *#}
and
tan y = tan a sec (; — r).
Hence 1
tan 8-tan y cos (i+r).
So also for the amplitudes of the vibrations, we have
2. sin” (i- ºr)\ *
tan” (; — r) + COS* Cl
sin” (i+r).J.'
b2 = 0.2 { sin” atan” (; + r.)
and
_4a” cos”: sin” r ſ sin” a
sin” (i+1) \cos” (i-r)
4a2cos”, sin?” tº 2 //;
=#( +sin” a tan (i-r)}.
2
+ cos” *}
while the intensities of the incident, reflected, and refracted beams are
* The relation connecting a, 8, and y may be stated in the form : “The vibrations
in the incident and reflected waves coincide with the projections of the refracted
vibration on those waves, or, the planes of vibration of the three waves intersect in
a line which is the direction of the refracted vibration " (MacCullagh, Trans. Roy.
Irish Academy, vol. xviii. 1837). - *
396 REFLECTION AND REFRACTION OF POLARISED LIGHT CH. XIII
respectively proportional to
- a”, b”, cºtan icot "'.
Cor. 1.-If i + r = 90°, we have 3 = 0; that is, if the light is incident
at the angle of maximum polarisation, the reflected light is polarised
in the plane of reflection. - *
Cor. 2.—From the expression for tan £8, it is clear that while i + r is
less than 90°, we have cos (; + r)<cos (; – ), and therefore tan 83 tan a,
while as i increases (a remaining constant) {3 passes through Zero as
i+r passes through 90°, and becomes negative when i + r increases
beyond 90°. The numerical value of tan (8 is therefore always less
than that of tan a, or the effect of the reflection is to bring the plane
of polarisation nearer to the plane of incidence, and at the angle of
maximum polarisation the two coincide.
Cor. 3.-If the first medium is more refracting than the second,
total reflection will occur when r = 90°, and in this case tan B will be
equal to – tan a, so that a and 6 are supplementary. -
Cor. 4.—If the same pencil of plane-polarised light be reflected m
times at the same incidence i, and in the same plane we have for the
azimuth of the resultant vibration after the m reflections
cos” (i+ r.)
tan = tan c. *
8m cos" (i −7)'
while if the refracted beam suffers n refractions by passing through
parallel plates of the two media, we have
tan yn = tan a secº (? – r).
For by refraction into a plate we have tan yi = tan a sec (; – ), and by
the refraction out at the second surface we have tan y, a tan yi sec (; — ),
OT
tan Y2–tan a secº (i − r).
In passing through n parallel plates the ray suffers 2n refractions, and
tan y = tan a secºn (i – r).
The general effect of reflection is therefore to make the plane of polar-
isation more and more nearly that of incidence, while for refraction it
is more and more nearly at right angles thereto. It is as though the
azimuths of vibration rotated in jumps; but the total change can never
be as great as ninety degrees.
212. Elliptically and Circularly Polarised Light.—We have seen
already (Art. 47) that two rectangular vibrations differing in amplitude
and phase compound into an elliptic vibration, and conversely we may
AIRT. 212 ELLIPTICALLY AND CIRCULARLY POLARISED LIGHT 397
decompose an elliptic vibration into two rectangular vibrations of the
form
a:= a sin wā, and y = b sin (wt + 6).
Consequently, if the incident light be elliptically polarised, we may
resolve it into two components, one polarised in the plane of incidence
and the other polarised perpendicularly to it, and apply the foregoing
formulae to determine the nature and intensity of the reflected and
refracted rays. The general expressions for these elliptic vibrations can
be formed without difficulty, but as their discussion presents no particu-
lar interest, we shall only consider the case of circularly polarised light.
If the incident light be circularly polarised, its component vibrations
may be represented by
o:= a cos wi, and y = a sin wi.
The components of the reflected and refracted vibrations may be
written down from the formulae of Arts. 209, 210; thus for the
reflected vibration we have
sin (; — r)
sin (i+ r.)
-* - atan (? – r)
T *tan (i+ r.)
Cos wt = A cos w!,
sin wt = B sin wt,
so that
.2 2
or the reflected light is elliptically polarised, the axes of the ellipse
being respectively in, and perpendicular to, the plane of reflection.
If the light is incident perpendicularly, we have i = r = 0 and
A = B = – aſp. - 1)/(u + 1),
so that the reflected light is circularly polarised and its intensity is
— 1 \2
•(#).
It is to be remarked, however, that the sense of the circular vibration in
the reflected light is opposite to that in the incident light, for the a com-
ponent of the vibration is affected with the negative sign.” This theoret-
ical deduction of Earnshaw has been verified experimentally by Powell.”
The character of the refracted light may be investigated in a
similar manner. It is to be observed in all cases that the y component
of the reflected vibration vanishes when i + r = 90°, or the reflected
light is always plane-polarised in the plane of reflection at the angle
of maximum polarisation. *
* The sign of y is also negative, but the positive direction of b with regard to
the direction of propagation of the light is opposite to that of a. See Note, Art. 210.
* Powell, Phil. Mag. (3), vol. xxii. pp. 92, 262.
398 REFLECTION AND REFRACTION OF POLARISED LIGHT CH. XIII
213. Reflection and Refraction of Common Light.—The charac-
teristic of ordinary light is that on transmission through a doubly
refracting crystal, such as Iceland spar, it is divided into two pencils
of equal intensity. If the crystal be also doubly absorbing, the
ordinary and extraordinary rays will be of unequal intensity, and (as
in the case of tourmaline) only one of them may be transmitted. On
entering the crystal, however, the light is divided into two equal beams
vibrating in two rectangular planes, and this whatever be the orienta-
tion of the crystal. Hence if ordinary light be resolved into any two
rectangular components they will be equal, and therefore the com-
ponents of the incident beam in and at right angles to the plane of
incidence will be equal in intensity, and each half that of the
incident light. Consequently the intensity of the reflected light will
be (Arts. 209, 210) : ºn 2 /A tan” (i.
I=leſ:#;+#|
and the intensity of the refracted light will be
I'- *[. º sin 2r. º sin 27 sin 2. e
sin” (i+r) ' sin” (i+r) cos” (, – r)
In the case of the reflected light the second term within the bracket
represents the light polarised perpendicularly to the plane of incidence.
If the two terms were equal, then the reflected light would be like the
common light of the incident beam. The second term, however, is
always less than the first, except under nearly grazing incidence, con-
sequently there will in general be an excess of light polarised in the
plane of incidence; that is, the reflected light will be partially
polarised. -
At the particular incidence i + r = }T the second term vanishes
entirely, and the reflected light is wholly polarised in the plane of
incidence.
Again the first term in the expression for the intensity of the
refracted light is less than the second, and this indicates that there is
an excess of the refracted light polarised perpendicularly to the plane
of incidence, but the whole refracted light is not completely polarised
in this plane at the polarising angle. At this incidence the reflected
light alone is completely plane-polarised, and the two beams contain
equal quantities of polarised light.
214. Total Reflection.—When the first medium is more highly
refracting than the second, total reflection occurs when the angle of in-
cidence reaches a certain critical value determined by p, sin i = 1. The
corresponding value of r is 90°, and if i be increased beyond its critical
value a value of sin r greater than unity is required, so that the angle
ART. 214 TOTAL REFLECTION 399
r becomes imaginary, and the formulae which determine the amplitudes
and intensities of the reflected and refracted pencils are no longer
applicable.
Experiment proves that when i exceeds its critical value the
refracted ray ceases to exist, and the reflected ray is equal in intensity
to the incident. Theory confirms this result, for the construction of
Huygens shows that in the second medium no real wave envelope can
be drawn, and the conservation of energy then requires that the
intensity of the reflected ray should be equal to that of the incident.
Taking the case of light polarised in the plane of incidence, we
have for the amplitude of the reflected ray
sing-r-, sinicosi-sinisſiº
e g & fº † * e -—5—5–; ;
sin (; +r) usin icos i-sin i Vl-ºsinº
by writing sin r = p sin i, u, being the index of refraction from the less
to the more refracting medium. Hence when i increases beyond the
critical value we have p sin i-1, and the expression for b becomes
imaginary. This imaginary form has been interpreted by Fresnel in
the following manner, which is undoubtedly ingenious, but which
must be regarded rather as an interesting curiosity than as a rigorous
demonstration.
Dividing the numerator and denominator of the expression for b
by sin i and multiplying each by the conjugate form of the denomin-
ator, it becomes
a * cos' it (1 Asinº)-29 cos; Vl-Asimº.
p” cos” – (1 – u% sin” i)
The denominator of this expression is simply (p.” - 1), so that it
reduces to
2 –L 1 – 9, 12 sin? . & ----- *
aſº. *J. singi-i N/Ti } = a(P – Q V-1)
p?– 1 p.” – 1
where
2 2 cº, 2 ..." e
pº-H 1 - 2p* sin” . 2pſ, cos ? /−
P=== and Q = *Nº,
and therefore
P2+ Q2=1.
Hence if we take P = cos 8 we will have Q = sin 8, and the expression
for b becomes
4 b= a(cos 6 – V-1 sin 3).
Hence if the equation of the incident vibration be y = a sin ot, the
equation of the reflected vibration will be -
y=a(cos 3 - V-1 sin 6) sin ot.
Fresnel suspected that since the occurrence of V-1 in geometry
400 REFLECTION AND REFRACTION OF POLARISED LIGHT CH, XIII
indicates a rotation of 90° in the position of the line whose length
is multiplied by it, so it is probable that here the imaginary quantity
V - I denotes a change of ºr in the phase of the vibration to which
it is attached. Proceeding on this assumption the equation of the
reflected vibration becomes
g=a {cos 6 sin wi – sin à sin (wt +90°)}
= a(cos 6 sin wt — sin à cos w!)
= a sin (wt – 6).
The interpretation, therefore, according to Fresnel is that the phase
of the reflected vibration has been altered at reflection by an amount
6 given by the equation
tan -}=*#####!
while the amplitude of the reflected vibration is equal to that of the
incident.
In the same manner we may treat the case of light polarised at
right angles to the plane of incidence, and the corresponding quantities
P', Q', 8' will be found to be as follows:
__{i}+1 – (u"+1) sin” i
Tſº-I) (ºr I) sinº-I}’
Q’ = 21, cos i Vu% sin” i – 1
T(2–1) {(u2+1) sin” i-1}'
29 cost Wººsin”? - 1.
(u” + 1)–(94-1-1) sin” i
— P’
tan 6' =
When the light is polarised in any azimuth, we may resolve it into
two components, one polarised in the plane of incidence and the other
polarised perpendicularly to it, and when total reflection occurs the
former will suffer a change of phase 8 and the latter a change 8', deter-
mined as above. The difference 8 – 8 is all that concerns us experi-
mentally, and we have
-- 1-(ºil) sinºsinº,
cos (6 – 6')= (u%+ 1) sin” i – 1
the reflected beam should therefore be elliptically polarised, the phase
difference of the two components being 8 – 8.
Cor. 1.-If 6 – 8 = 0; that is, if the change of phase produced by
the reflection is the same for the component in the plane of incidence as
for that perpendicular to it, the reflected ray will be plane-polarised.
In this case cos (8–6) = 1, and we have, therefore,
1 – (u%+1) sin” i +2p* sin" is (u%+ 1) sin” i – 1,
OI’
p” sin” . — (p.” + 1) sin” i + 1 = 0.
Hence
ART. 214 TOTAL REFLECTION 401
2 H / 1,2 —
sing; (ºf 9+.+1)+(u”- 1)
2p.”
= 1 or 1/u.”.
The first value of sin i corresponds to grazing incidence, and the second
to the limiting incidence for total reflection. At the two limits of total
reflection, therefore, the reflected light is totally plane-polarised.
Cor. 2.-The difference of phase 8 – 8 passes through a maximum
value at an angle of incidence determined by the equation
(u” + 1) sin” i-2,
and the corresponding maximum is given by
A 80°– (u” +1)*_4°–(9 – 1)*
cos (6-8)==#H#==#H#
Cor. 3.−If cos (8 – 8') = 0, the axes of the reflected elliptic vibration
are in, and perpendicular to, the plane of incidence, and the correspond-
ing incidence is given by
29% sin” i - (u%+ 1) sin” i + 1 = 0.
If at this incidence the plane of polarisation of the incident light
makes an angle of 45° with the plane of incidence, the reflected light
should be circularly polarised. This affords an experimental means of
verifying the foregoing theoretical results, but in order that the values
of i, determined by the foregoing quadratic, should be real, it is neces-
sary to have
p?-3+ V8,
consequently, a substance of refractive index nearly equal to 3 would
be required. -
The required difference of phase may, however, be produced by two
or more total reflections from a substance such as glass with a smaller
index of refraction. Thus if the azimuth of the plane of polarisation
of the incident light be 45°, and if a difference of phase of 45° be intro-
duced at each reflection, the light twice reflected will be circularly
polarised, or if a difference of phase of T/2n be produced at each reflec-
tion, the light n times totally reflected will be circularly polarised. In
the first case the angle of incidence is determined by the equation
cos (6–6) = 1/V2,
Or
40° sin" i – (2+ V2) (º-F1) sin” i +2+ V2=0.
This gives real values of i for glass and media whose index of refraction
lies between 1:4 and 1-6.
We may at once write down the corresponding formulae for three,
or four, total reflections; that is, for 8 – 8 = T/6, T/8, etc.
2 D
402 REFLECTION AND REFRACTION OF POLARISED LIGHT CH. XIII
215. Fresnel's Rhomb. —In verification of the foregoing con-
clusions Fresnel constructed a parallelopiped of glass such that a ray
of light AB (Fig. 174) falling normally on the
end suffers total reflection internally at B, where
it falls upon the face at an incidence of 55°, and
again at C, and then emerges normally at the
other end of the rhomb. At B a difference of
phase of 45° is introduced—that is, A retarda-
tion—and the same difference is again produced
at C, so that in all 90° difference of phase is in-
troduced, or a retardation of A, and thus if
the incident light be polarised at an angle of
45° to the plane of incidence ABC, its com-
ponents in and perpendicular to the plane of
incidence will be equal, and the emergent light is found to be
circularly polarised.
Conversely, if the incident light be circularly polarised, the rhomb
introduces a further difference of phase of 90°, so that the emergent
light is plane-polarised. Hence, generally, if a ray of light originally
plane-polarised in an azimuth of 45° to the plane of incidence be passed
through any number (n) of Fresnel's rhombs, the emergent light will
be circularly or plane-polarised according as n is odd or even. With
such rhombs we may therefore test between ordinary light and cir-
cularly polarised light.
By means of this rhomb we may also convert elliptically polarised
light into plane-polarised light. For if the axes of the elliptic vibra-
tion be in and perpendicular to the plane of incidence ABC, the two
internal reflections will introduce a further difference of phase of 90°
between its components, and the emergent light should be plane-
polarised. This light will then be extinguished by a Nicol's prism, and
we can therefore test between elliptically polarised light and partially
polarised light.
Fresnel's rhomb consequently possesses all the properties of a
quarter-wave plate (Art. 223), and is preferable to it in working with
white light, since the difference of the velocities of the different colours
in glass is inconsiderable. However, the emergent beam of light
changes its position when the rhomb is rotated, so that there is trouble
in following it with the other parts of the apparatus. It is best there-
fore to use a quarter-wave plate in working with monochromatic light.
216. Theory of Neumann and MacCullagh.-The principle of
the continuity of the ether adopted by Fresnel demands that the dis-
placements parallel to the surface of separation should be the same in
Fig. 174.

ART. 216 THEORY OF NEUMANN AND MACCULLAGH 403
both media. But it also requires the displacements perpendicular to
the surface to be the same, consequently if we have the equation
(a +b) cos i =c cos r (no slipping),
we should also have
* (a – b) sin i = c sin r (no separation).
Hence by multiplication we obtain
(a”— b%) sin icos i = cºsin r cos r. • " (i)
But by the conservation of energy we have
p(a”— b”) sin icos i =p'c” sin rcos r, (2)
and therefore the equation derived from the principle of continuity
will be in accordance with that derived from the conservation of
energy if
p=p'.
Hence Fresnel's assumption that the density of the ether is different
in different media is inconsistent with the continuity of the ether
at right angles to the surface,
Adopting the above equations, Neumann and MacCullagh postulate
that the density of the ether is the same in all media, but that its
elasticity is different. On solution they find expressions for the
amplitude of the reflected rays which differ from those arrived at by
Fresnel only in that the expression for light vibrating in the plane of
incidence is that which Fresnel arrived at for light vibrating perpen-
dicularly to the plane of incidence, so that according to their theory
the direction of the vibration lies in the plane of polarisation and not
perpendicular to it as in the theory of Fresnel.
According to either theory the reflecting power for ordinary
light is
*{#. (? – r), tan” (, – r) e
* \sin” (i+r) ' tan” (i+ r.)
If this expression is expanded in powers of i, remembering that
sin i = p sin r, the first term in the brackets gives
(#) (1+ #, i. (3+ 12p. – p?), etc.).
the second gives
– 1\ 2 222 24
(#) (1-º-; (9 – 12p. -- 5p.”), etc.),
and the reflecting power for ordinary light is
— 1 2 º t?
(i. }) (1 T2a: (1 – 4p, + A*), etc. }.
404 REFLECTION AND REFRACTION OF POLARISED LIGHT CH. xIII
tº:
This shows that the reflecting power remains sensibly constant if the
fourth power term can be neglected. Moreover, when i increases from
zero, the reflecting power will increase or diminish, for small angles,
according as 1 – 4p, + p.” is less or greater than zero. Thus, for increase,
p = 2 + V3-P where P is a positive quantity; but if p > 3.732 . . .
or less than 0.268 . . . the reflecting power would decrease at first
with increase in i. No transparent substance has a value of p, either
as great or as low as these limits, and consequently an increase is to be
expected in all cases whether the reflection takes place externally
or internally.
CHAPTER XIV
METALLIC REFLECTION OF POLARISED LIGHT (FRESNEL’S THEORY)
217. Partial Polarisation by Reflection in General.-Malus, who
discovered the polarisation of light by reflection from glass, observed
that natural light is never completely polarised by reflection from
a metallic surface. The laws deduced from the theory of Fresnel
are therefore not applicable to metals or highly refracting substances.
M. Jamin, who investigated the question, found that only a few sub-
stances completely polarised light by reflection, that the angle of
incidence at which this occurred was tant*(u), and for these sub-
stances u = 1'46. For all other substances there is an angle of
maafimum polarisation determined by the equation
tan t = p,
instead of an angle of total polarisation, and this angle is termed the
polarising angle.
Malus was of opinion that common light is never polarised by
reflection from metals, but in 1813 Brewster corrected this error, and
showed that the reflected light was partially polarised, the amount of
polarisation depending on the incidence and passing through a maxi-
mum at a certain angle.
Biot verified the observations of Brewster, and remarked that if
light is partially polarised by one reflection at a metallic surface, it
ought to be completely polarised by a sufficient number of reflections
taking place at the same angle of incidence. In 1830 Brewster 1
found that when plane-polarised light is reflected from a metallic
surface it remains plane-polarised if the incident ray is polarised in or
perpendicular to the plane of incidence, but in any other case the
light is partially “depolarised" (that is, becomes in part elliptically
polarised) by the reflection.
* Brewster, Phil. Trans., 1830, p. 287.
405
406 METALLIC REFLECTION ohar. xiv
218. Change of Phase and Elliptic Polarisation by Metallie
Reflection.—When the incident light is polarised in any azimuth a, it
may be replaced by two components, one parallel to and the other at
right angles to the plane of incidence. Now these components may
become altered in two respects by reflection. In the first place, their
amplitudes may be changed, and, secondly, their phases may be altered. .
The change of amplitude alone merely alters the plane of polarisation
(i.e. rotates it through a certain angle), the reflected light remaining
plane-polarised ; hence if the reflected light is not plane-polarised,
the phases of the component vibrations must have been changed by
the reflection, and changed by different amounts. If this view be cor-
rect the depolarisation observed by Brewster is none other than elliptic
polarisation arising from the change of phase introduced by reflection
between the components parallel and perpendicular to the plane
of incidence, and that this is so is supported by many experiments.
Thus when plane-polarised light suffers reflection at a metallic
surface it experiences in general a change of phase by the reflection,
but the change is less for light polarised at right angles to, than
for light polarised in, the plane of incidence. Consequently if the
incident light be polarised in or at right angles to the plane of
incidence, the reflected light will remain plane-polarised in the same
plane; but if it be polarised in any other plane the reflected com-
ponents in and perpendicular to the plane of reflection will differ in
phase, and the reflected light will in general be elliptically polarised.
This difference of phase increases from zero at normal to ºr at grazing
incidence. e
In the case of light polarised in the plane of incidence the reflected
light increases in intensity from normal to grazing incidence when it
is all reflected; but when the incident light is polarised at right
angles to the plane of incidence the reflected ray diminishes in
intensity from normal incidence to a certain angle at which it becomes
a minimum, it then increases again to grazing incidence. This mini-
mum is little marked for silver, but is very decided in the case of steel
and certain metallic oxides.
219. M. Jamin’s Experiments.-A method of observation, sus-
ceptible of great accuracy, has been employed by M. Jamin to
measure the intensity of the light reflected from metals. The
principle of his method is the comparison of the light reflected from
the metal with that reflected from glass. He employed a plane
mirror, one half of which was metal and the other half glass. Light
polarised by a Nicol's prism fell upon this mirror, and the reflected
* Annales de Chimie et de Physique, third series, tom. xix. p. 296.
ART. 219 JAMIN'S EXPERIMENTS 407
light was examined by a doubly refracting analyser. Two images are
formed by the analyser, each of which consists of two parts, one
formed by reflection from the glass, the other from the metal. The
latter half is generally coloured, so that it is easily distinguished.
If the incident light be polarised in the plane of incidence, and
if the principal plane of the analyser make an angle a with the
plane of incidence, the intensities of the ordinary and extraordinary
images will be measured by
* Ordinary. Extraordinary.
Glass . e g b°cos” a, b°sin” a,
Metal . tº ſº an” cos” a, m” sin” a,
{º * > sin (? – ?'
where b, according to Fresnel's formulae, is equal to — “sin #: and m
is a coefficient for the metal.
To determine m we might seek the case when the two halves of
the images are of equal intensity, and we would then have
in 2 / o –
nº-tº-aº r)
sin” (i+ r.)
But as this incidence does not exist, and since the ordinary image of
one part of the mirror and the extraordinary of the other vary from
zero to a maximum in inverse senses, there will always be two values
of a, which make the ordinary image of one half equal to the extra-
ordinary of the other, and we will have *
m” cos” an=b” sin” al, or m” sin” as = b% cos” as,
from which we obtain
on*=b% tan” al-b% cot” aa,
the angles an and as are therefore complementary. In practice both
a, and as are determined, and one observation corrects the other.
The colouring in the image of the metal renders it difficult to say
when the intensities of two images compared are equal, and besides
this the images are not very close to each other. #
For light polarised at right angles to the plane of incidence we
have similarly
m”=b°tan” al-b” cot” aſ,
where
yo getan (?-?).
tan” (i+r)
To calculate m it is necessary to know the index of refraction of the
glass. This M. Jamin determined by the change of the plane of
polarisation caused by reflection. If the azimuth of the incident
light be a, and that of the reflected 6, we have
cos (i+ r.)
tan 8–tan a cos (; – r)
408 METALLIC REFLECTION CHAP. XIV
Consequently if a = 45° we have
cos (i+r)_1 - tan i tan º'
tan 8–0. (i-r)|T1--tan i tan r"
OI’
& 1 – tan 8 ©
tan tan "Ti Itan #-tan (45°– 8),
Or
tan r=tan (45°–3) cot i.
This equation determines the angle r, and the refractive index is
found from the equation sin i = p sin T.
Jamin applied the same method also to transparent substances
and his experiments show that, for them, the difference of phase is
small from normal incidence to an angle a little less than the polarising
angle; it then increases rapidly, reaches 90° at the pólarising angle,
and becomes very nearly equal to it at an angle a little greater than
the polarising angle. The change of phase therefore does not occur
suddenly at the polarising angle, but takes place continuously and
rapidly in the neighbourhood of this angle. According to Fresnel's
theory (Art. 211) we should say that the difference of phase between
the two reflected components is zero up to the angle of maximum
polarisation, that there it suddenly changes to T, and remains so up
to grazing incidence. -
A verification of the elliptic polarisation of light by metallic
reflection was observed by Brewster. He found that if a ray of
plane-polarised light be reflected twice at the same angle, but in
perpendicular planes, from two similar metallic surfaces, the ray will
be again plane-polarised after the two reflections. De Sénarmont
interpreted this experiment mathematically as follows:
Let the reflection change the component parallel to the plane of
incidence by altering its amplitude in the ratio m : 1 and its phase
by 8, and let the corresponding quantities for the other component
be m' and 8. Then the incident and reflected vibrations in Brewster's
experiment, in and perpendicular to the plane of incidence respectively,
3.I’6.
Incident. Once reflected. Twice reflected.
a sin wit, am sin (wt +6), amm'sin (wt + 6+ 6’),
a' sin wi, a'm' sin (wt +6’), a'mºn" sin (at 4-6-- 6').
Hence the phases of the twice reflected components are the same, and
the ratio of their amplitudes is the same as originally. The reflected
beam is therefore plane-polarised in the primitive plane.
CHAPTER XV
INTERFERENCE OF POLARISED LIGHT COLOURS OF THIN
CRYSTALLINE PLATES
1. Parallel Plane-Polarised Light
220. Introductory Statement.—We now proceed to the study of
the phenomena which occur when polarised light is transmitted through
thin plates of doubly refracting substances. The first discoveries in
this region were made by Arago" in 1811. Placing by chance a thin
plate of mica in the path of a pencil of plane-polarised light (the blue
light of the sky) and examining it through a doubly refracting prism
(Iceland spar), he observed that both the ordinary and the extraordinary
images were richly coloured. In general, when plane-polarised light is
transmitted through a thin plate of any doubly refracting substance and
then examined by means of a doubly refracting analyser, both images
are richly coloured and, if they overlap, their common portion appears
white, which shows that the colours of the images are complementary.
If plane-polarised light be received by a Nicol's prism, or other
analyser, we know that in one position of the Nicol the light is refused
transmission. The Nicol being set in this position, if a thin plate of a
crystal be introduced across the path of the light, the capability of
transmission through the Nicol is suddenly restored, and a portion of
the light is transmitted which depends on the position of the interposed
crystal. For this reason the light was said to be “depolarised" by the
crystal, and by means of this property the doubly refracting structure
of many substances was detected by Malus, where the separation of
the ray was too small to be observed directly. The effect is due to
light becoming elliptically polarised in passing through the crystal.
The colours appear to be painted on the crystal slice as in the case
of the colours of soap films. They can be projected on a screen if the
lens is so placed that the slice and the screen are in conjugate planes.
* Arago, CEuvres complètes, tom. x. p. 36.
409
410 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV:
A medium originally isotropic may acquire the power of double
refraction when subjected to strain, and if the strain be homogeneous
the optical properties of the substance are similar to those of a natural
crystal, the principal axes of the wave surface coinciding with those of
the strain. A feeble doubly refracting power is conferred on glass by
bending or straining it. Unequal heating produces the same effect, as
it is accompanied by expansion, which gives rise to internal strain.
This double refraction may be detected by examining the glass between
crossed Nicols, and so delicate is this test that it is difficult to find
large pieces of glass so free from internal strain as to show no revival
of light when so examined.
221. Intensity of Illumination at any Point.—Let us now
see how far the physical theory accounts for these appearances.
In the first place, there are three essential conditions for their
production.
(1) The polarisation of the incident light.
(2) The interposition of a thin crystalline plate.
(3) The action of an analyser on the light after passing through
the plate.
Let the principal plane of the polarising Nicol be parallel to OP
(Fig. 175), and let the principal plane of the analysing Nicol be parallel
to OA. Then since it is the extraordinary ray that is transmitted
through the Nicol, and since by Fresnel's
hypothesis the vibrations of this ray are in the
principal plane, it follows that the incident
vibration at O will be parallel to OP. On
entering the plate it becomes broken up into
two others, polarised at right angles to each
other, one parallel to OX and the other parallel
to OY, where OX and OY are two determinate
rectangular directions in the crystal. Hence if the incident vibration
be y = a sin of it gives rise to
Fig. 175.
a cos a sin wº, and a sin a sin wº
along OX and OY where POX = a.
As these waves travel through the plate with different velocities
they will be unequally retarded, and consequently on emergence they
will differ in phase by an amount 8. The transmitted vibrations
therefore take the form
a cos a sin wº, and a sin a sin (wt 4-5).
On reaching the analyser these vibrations become resolved parallel to
its principal plane OA. If therefore AOX = 3 we have two vibra-

ART, 221 EXAMINATION OF THE INTENSITY 411
tions parallel to the principal plane of the analyser : the component
a cos a sin of along OX gives a cos a cos Á sin of along OA, and the
component a sin a sin (ot + 8) gives a sin a sin B sin (ot + 8) along OA,
and these compound into a resultant vibration
gy= a cos a cos 3 sin (ot + a sin a sin 8 sin (wt + 6),
parallel to OA. The intensity of the resultant vibration is therefore
given by (Art. 43, Chap. II.) the equation
I=a^{cos” a cos” 8+sin” a sin” B+2 sin a cosa sin B cos & cos 3}
=a^{cos” a cos” 8+sin” a sin” 8+2 sin a cos a sin 8 cos 3 (1 – 2 sin” #6)}
=a^{(cos a cos 3+sin a sin 8)” – 4 sin a cos a sin 8 cos 3 sin” #6}.
Hence, finally,
I=a^{cos” (a – 8) – sin 20 sin 28 sin” #6}
where a - 8 is the angle between the principal planes of the polariser
and analyser.
If the analyser be merely a doubly refracting rhomb two images
will be presented. The above expression refers to the extraordinary
image, and it is seen in the same manner that the intensity of the
ordinary image is
Io=a^{sin” (a – 8)+sin 20 sin 28 sin” $6},
which shows that it is complementary to the extraordinary. We shall
confine our attention to the extraordinary, as it is that furnished by a
single image analyser such as Nicol's prism. g
In the case of white light 8 will be different for the various wave
lengths," and if a. also varies with the wave length, the general expres-
sion for the intensity will be
I=cos” (q – 8)Xa”— sin 20 sin 282a” sin” #6.
The first term being independent of 8 will have no effect in producing
colour in the image, but in the second term 8 will depend on the wave
length, and consequently the different colours will enter it in different
amounts. If the incident light be white the transmitted light will in
general consist of two parts, one of which is white, depending on the
first term, and the other more or less coloured, arising from the second.
With a given plate the combined rotation of the Nicols, or the rotation
of the plate round its normal, will affect all the colours in the same
proportion, and consequently the tint of the second term will remain
* In some crystals the dispersion sensibly modifies the relative retardation as
dependent on the wave length. Herschel (Art. “Light,” Ency. Metropolitana, § 915)
observed that the rings exhibited by a common variety of uniaxal apophyllite were
approximately achromatic, indicating that 6 was almost independent of A, and under
these circumstances a very great number of rings may become visible. It must be
added that no allowance has been made for loss of light in reflections.
The colour
term.
412 COLOURS OF THIN CRYSTALLINE PLATES ("HAP, XV
Comple-
mentary
relation.
the same, but its intensity will vary as sin 20 sin 28 varies. Now
sin 20 sin 28 may be either positive or negative, and for this reason the
resultant colour of the plate may be either of two different tints. For
example, when sin 20 sin 28 is positive (which will hold as the plate
is rotated from a = 0° to 8 = 90°), the resultant light will consist of a
certain quantity of white light, from which a varying amount of light
of a given colour is subtracted, and when sin 20 sin 28 is negative
(which will hold from 8 = 90° to a = 90°), the resultant will consist of
a given quantity of white light, to which a varying quantity of light of
a given colour is added. In each case the resultant tint remains
unaltered as the plate is rotated (until sin 20 sin 2/3 changes sign),
except in so far as it becomes more or less diluted by the greater or
less admixture of white light arising from the first term of I.
The colouring, depending on the second term, will be most marked
when a – 8 = 90°, and least marked when a – 3 = 0, the corresponding
values of I being
I= 2a:” sin” 20 sin” #6 (colour most marked),
I= 2a"(1 — sin” 20 sin” #6) (colour least marked).
In the former case the field of the analyser would be dark if the plate
were removed. In both cases the appearances are most marked when
sin 20 = 1, or a = 45°; that is, when the principal planes of the polariser
and analyser bisect the angles between the principal planes of the
plate.
When the light falling on the thin plate is not polarised there is
no exhibition of colour in the field of the analyser, and the images are
white if the incident light be ordinary white light. Let the common
light be resolved into two components—one vibrating vertically, the
other horizontally. The former will be treated by the slice and
analyser as though it had been obtained from a Nicol with its principal
azimuth vertical; the other, as if by a Nicol with its principal azimuth
horizontal. But these effects are complementary. Hence they add
up to white light.
Cor. 1.-The intensity in any position 6 of the analyser is comple-
mentary to that in the perpendicular position 8 + 90°. For if I, and I,
denote these two intensities, we have
Ii =a^{cos” (a — 8) – sin 20 sin 28 sin” #6},
and changing 3 into 90°,+ 3, we have
I2 = a”{sin” (a – 3)+sin 20 sin 28 sin” #6},
therefore
Il + Ig == a?,
or the sum of the two intensities is equal to that of the incident light.
ART. 221 EXAMINATION OF THE INTENSITY 4 13
The effect of rotating the analyser through 90° is therefore to
change the intensity, and also the tint, into its complementary. If
the analyser be merely a doubly refracting rhomb, the two images
will exhibit these complementary tints, and if the analysing rhomb be
removed altogether these complementary images will become super-
posed, and we have a single image without colour.
Cor. 2.-For a given relative position of the Nicols—that is, for Achro-
a – B = constant = y—the intensity of the coloured component will vary º
as the plate is rotated round the normal to its plane, and the image
will be, uncoloured when
sin 20 sin 28 = 0;
that is, when a = 0° or 90°, and when B = 0° or 90°. There are con-
sequently four positions in which the image is achromatic, viz. when
the principal section of the plate is parallel or perpendicular to the
principal plane of the polariser or analyser, and the intensity of the
achromatic image is
I= a” cos” y.
This will be a maximum when y = 0°–that is, when a = 8, or when the
polariser and analyser are parallel, and zero when y = 90°–that is, when
the Nicols are crossed. In both these cases the four positions giving
an achromatic image obviously reduce to two.
Cor. 3.−If a = 3, or the principal planes of the Nicols are parallel,
we have for the intensity of the transmitted light
I = a”(1 — sin” 20 sin” #6).
Cor. 4.—If in addition a = 0° or 90°, we have
I = a”,
or the transmitted light is equal to the incident. The plate has here
no effect, as is easily understood, for if a. is equal to 0° or 90°, the
direction of vibration of the incident light is parallel to one of the
possible directions in the crystal. It therefore passes through unaltered
to the analyser.
If a = 8 = 45°, I = a” cos” #8. Hence if 8 = (2n + 1)" and a = 8 = 45°,
the transmitted light is zero. * -
Cor. 5.-For any given positions of the Nicols and plate the intensity
is a maximum when sin #8 = 0 and a minimum when sin #8 = + 1
if sin 20 sin 28 is positive, or
I= a” cos” (a - 3), 6 = 2ntr, (max.)
I=6%cos” (a – 8) – sin 20 sin 28}, 6=(2n + 1)7. (min.)
But if sin 20 sin 2/3 is negative, the case is reversed. The difference
414 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
of phase 8 is determined by the thickness of the plate traversed and by
the wave length of the light under investigation. º
Cor. 6.—If a - 8 = 90°, so that the Nicols are crossed, we have
sin 28 = — sin 20, and
I = a” sin” 20 sin” #6.
So that for a given value of a the intensity I will be a maximum when
8 = (2n + 1)" and zero when 8 = 2nt. In the former case the intensity
will be the greatest possible when a = 45°, for then I = a”, and the
intensity will be zero if a = 0 or 90°.
Cor. 7.—To determine the phase of the resultant vibration
v= a (cos a cos 3 sin wt +sin a sin 8 sin (ot-H 6)}
= A sin (wt + p.),
we have
A sin p- a sin a sin 3 sin 6,
A cosp=q{cos a cos 3+sin a sin 3 cos 3},
sin a sin 3 sin 6
cos a cos B+sin a sing cos 3
tan p-
Hence if a. and 8 are the semi-sides, and 6 the included angle, of a
spherical triangle, p is half the spherical excess.
Cor. 8.-The light emerging from the thin plate is in general
elliptically polarised, the equation of the vibration being (Art. 47) .
22 g/* 22:y cos 6
ZººZº. Tº T ºf = 0.2 sin” 6.
COS* O. SlT1*. CT, SII) Cº., COS C,
222. Conditions for Interference — Transverse Vibrations.—
The phenomenon of double refraction shows that in the crystal the
light is divided into two waves travelling with different velocities.
On emerging therefore from a thin plate one will be retarded on the
other, and in general a difference of phase will exist. Hence if light
undulations were purely longitudinal, like those of sound, the emerging
waves should interfere, and the thin plate alone should be sufficient to
produce all the phenomena of interference and colour without either
polariser or analyser. Such, however, is not the case, and it was
found by Fresnel and Arago, who investigated the subject of the
interference of polarised light experimentally, that two plane-polarised
rays interfere and produce fringes as ordinary light only when they are
polarised in the same plane and originally belonged to the same plane-
polarised pencil (Art. 176).
Two rays polarised in different planes in general combine into
a resultant elliptic vibration, and this is an immediate consequence of
the theory of transverse vibrations. The office of the analyser in these
experiments is to transmit one component only of each of the two
waves from the crystal; these components having the same plane of
ART. 223 QUARTER-WAVE PLATE - 415
polarisation. They can then interfere with the production of colours.
The other component of each is stopped; if transmitted, these also would
interfere with one another, and, as we have seen, would produce the com-
plementary effect. If both pairs act together they produce only white
light. All the phenomena are therefore in confirmation of the theory
which supposes that the vibrations of the ether which constitute
light involve a component which is transverse to the direction of
propagation.
223. Calculation of the Difference of Phase—Quarter-Wave
Plate.—So far we have only considered the nature of the vibration
which reaches the eye from a single point O of the thin plate, or from
a uniform plate transmitting parallel light. If the incident light be
not parallel, and if the thickness of the plate varies from point to point,
the retardation 8 of one component on the other will also vary from
point to point of the plate, so that the calculation of the appearance of
the plate requires the determination of 6 at each point.
A ray incident at an angle i gives rise to two refracted rays, and
the relative retardation introduced by a plate of thickness e we have
already found (Ex..., p. 106) to be
6–e sin i (cot re-cot ri)= e(p, cos re-p1 cos 7'1),
and on the calculation of the quantity 8 the determination of the
character of the pattern presented in the field will depend.
For normal incidence the retardation becomes e(p, — pi), and if the
thickness of the plate is so adjusted that
e(u2 -- Pi) – #X,
the plate will be a quarter-wave plate for the wave length A. Such
plates are of importance in the study of circularly polarised light.
224. Thick Plates—Colours produced by Superposition.—The
phenomenon of colour we have seen to be due to the difference of
phase introduced by the thin crystalline plate, and if the plate be not
thin this difference may be a great number of wave lengths, so that,
as in the case of Newton's rings, the colours of different orders may
come to be superposed, and the resultant light will be white.
Thus for a given position of the Nicols and plate (a and 8 given)
the intensity will be a maximum or a minimum according as sin” #8 = 0
or 1, assuming sin 20 sin 28 positive. The maximum or minimum over.
intensity for any colour will consequently correspond to retardations lapping.
mA and (2n + 1); A respectively. But if the plate is thick n will be
very great, so that if A and Aſ are two near wave lengths, we may have
2n , 3Å = (2n + 1)}\",
416 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
and thus if the plate is dark for A'it will be bright for an adjacent wave
length A. It will therefore be bright for many wave lengths along
the whole range of the spectrum, and will consequently appear white.
When examined under a spectroscope it should, nevertheless, exhibit a
spectrum crossed by many dark bands.
The tints may, however, be produced in thick plates by superposing
two of them in such a manner that the ray which has the greater
velocity in the first shall have the lesser velocity in the second. Thus
if the crystals be uniaxal and both positive (or both negative) their prin-
cipal sections should be placed at right angles, whereas these sections
should be placed parallel if the crystals are of opposite denominations.
225. Superposition of two Crystalline Plates.—Let us now examine
the appearances presented when plane-polarised light passes successively
through two superposed thin crystalline plates, before being received
by the analyser.
Using the notation of Art. 221, we shall take the direction of the
vibration incident on the first plate to be parallel to OP, and this
according to Fresnel's theory is parallel to the principal plane of the
polariser when the light is polarised by transmission through a Nicol's
prism. Then if OX and OY be the two directions of vibration in the
first plate, the components of the light emerging from it may be
written in the form
a cos a sin (wt + 61), and a sin a sin (wt + 6'1)
where 8, and 8', are the phase retardations introduced by the first plate.
When these vibrations reach the second plate they are each split up
into two components, except in the particular cases in which the
principal planes of the plates are parallel to each other or crossed.
For the sake of simplicity we shall first consider the case in which
Principal the principal planes of the plates are parallel to each other. In this
łºń. case the foregoing vibrations traverse the second plate without sub-
division, and on emerging from it they may be written in the form
a cos a sin (wt + 3 + 62), and a sin a sin (wt + 6'1+ 6'2)
where 8, and 8', are the phase retardations introduced by the second
plate and correspond to 8, and 81 respectively in the first. The whole
retardation of one ray is 8, + 8, while that of the other is 81 + 8, ; hence
falling upon the analyser we have two rectangular vibrations differing
in phase by an amount (6, + 82) — (81 + 82), and the intensity of the light
vibrating in the direction of the principal plane of the analyser is
consequently found from the expression of Art. 221 by merely
replacing the quantity 8 by the quantity (61 + 82) — (81 + 82). This is
ART. 225 TWO CRYSTALLINE PLATES SUPERPOSED 4 1 7
the extraordinary image, and therefore its intensity is given by the
expression
I, = a”{cos” (a — 8) – sin 20 sin 28 sin” #(61 + 62 – 6'1 – 6'2)}.
Similarly the intensity of the ordinary image, if transmitted by the
analyser, will be
Io =a^{sin” (a — 8)+sin 20 sin 28 sin” #(61--62 – 6'1 – 6'2)}.
Thus when the principal planes of the plates are parallel, the combina-
tion acts as a single plate of thickness equivalent to the joint thicknesses
of the two. When the plates are crossed the phase retardation of one Plates
ray is obviously 8, + 8, while that of the other is 8', 4-8, ; consequently *
the difference of phase on reaching the analyser is (8, + 6.) – (6', + 8,),
and the intensity of the extraordinary image is
I, =a^{cos” (a — 8) -- sin 20 sin 28 sin” #(6, 4-6'2 – 6'1–62)},
while the intensity of the ordinary image is given by the complementary
expression
I, =a^{sin” (a – 8)+sin 20 sin” 28 sin #(6, 4-6'2–61–62)}.
In the same way the expression for the intensity may be written down
at once when any number of plates are superposed with their principal
planes either parallel or crossed, or when some of them are parallel
while others are crossed, for we have merely to replace the quantity 8
in the formulae of Art. 221 by a corresponding quantity 28 – 28, where
28 is the whole phase retardation of one of the rays emerging from
the system of plates, and 26 the whole phase retardation of the
other.
In the more general case in which the principal planes of the plates Principal
are inclined to each other at an angle, the calculation is more tedious plane; in-
but presents no serious difficulty. Thus, using the same notation, if the ined.
vibrations emerging from the first plate parallel to OX and OY are
a cos on sin (wt + 61), and a sin a 1 sin (wt + 6'1);
then if the directions of vibration OX’ and OY in the second crystal
be inclined at an angle as to those in the first, viz. OX and OY, the
foregoing vibrations give for the vibration parallel to OX’
X = a cos on cos og sin (wt +6, + 62) + a sin a sin a sin (wt + 6'1+62),
while parallel to OYZ we have
Y= a sin all cos as sin (wt + 6'1+ 6'2) – a cos an sin as sin (wt + 3 + 6'2).
2 E
a-
418 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
Hence if the principal plane OP of the analyser makes an angle as
with OX', the component vibråtion parallel to OP is
v=X cos as +Y sin ag.
Substituting for X and Y and collecting the coefficients of cos of and
sin ot we find --
gy = P cos wi-i-Q sin wt.
where
P|a = cos all cos de cos as sin (6, 4-62) + sin at sin a 2 cos as sin (6'1 + 52)
+sin al cos og sin as sin (6'1+ 6'2) — cos du sin ag sin as sin (6, 4-62),
and
Q/a = cos at cos as cos as cos (6, 4-62) + sin a sin as cos as cos (6'1+ 32)
+ sin al cos az sin as cos (6'1+ 6'2)-cos al sin a 2 sin as cos (6, 4-6'2).
Consequently for the intensity of the light vibrating parallel to the
principal plane of the analyser—that is, for the extraordinary image—
we have g
I, + P2+ Q”.
Hence by taking the sum of the squares P and Q, and noticing that
the coefficients of the quantities sin (81 + 82), etc., in the expressions
for P and Q are exactly the terms which occur in the expansion of
cos (a1 – as – as), we obtain
I.-a”{cos” (al-as-as) – sin 20, sin 20, cos 2as sin” #(3,-6)
– sin 2011 cos” as sin 203 sin” #(6, 4-62 – 6'1 – 6'2) -
+cos 2q1 sin 2012 sin 2013 sin” #(62 – 6'2)
+sin 2011 sin” as sin 203 sin” #(6, 4-6'2 – 6'1–62)}.
In the same way for the intensity of the ordinary image
Io = a”{sin” (an – as – ag)+sin 2011 sin 2012 cos 2013 sin” #(61 – 6'1)
+sin 2011 cos” as sin 2013 sin” #(614-62 – 6'1 – 6'2)
– cos 2q1 sin 2012 sin 203 sin” #(62 – 6'2)
– sin 2011 sin” as sin 203 sin” #(6, 4- 6'2 – 6'1 – 62)}.
By adding these expressions together we find that the two images are
complementary, for we have
Io-H I, = a”.
The tint of the image in either case will vary with a 1, a2, and as ; that
is, when the analyser or either of the plates is rotated.
Cor.—Making a, = 0, we obtain the expression for two plates with
their principal sections parallel, and making as = 90°, we obtain the
formulae for two plates crossed.
The foregoing enables us, in a simple manner, to determine
ART. 226 PROJECTION ON A SOREEN 419
whether a crystal is positive or negative. For this purpose take a sign
thin plate of the crystal and observe the tints it produces in polarised"
light. Now superpose on it a plate of quartz or some other crystal of
known sign so that the principal sections of the two plates may be
parallel. The two crystals will be of the same or contrary signs
according as the new tints presented in the analyser are higher or
lower in Newton's scale of colours than those afforded before the inter-
position of the quartz plate.
If the principal sections of the two superposed plates are parallel
or perpendicular the images presented by the analyser are not altered
by interchanging the positions of the plates, but this is not generally
the case when the principal sections make any other angle with each
other, for the above formulae show that if the plates be interchanged
the values of I, and I, also change except when a 2–0° or 90°.
226. Projection on a Screen.—The phenomena indicated by our
theory may be observed by simply looking at the sky through two
Fig. 176.
Nicol's prisms separated by a thin crystalline plate. But by pro-
jecting the images on a screen as follows they may be observed on a
larger scale and exhibited to an audience.
Let the light of the sun, reflected from a heliostat if necessary, be
transmitted through a polarising Nicolor Foucault prism P (Fig. 176).
The polarised light from P falls upon a system of two highly con-
verging lenses L and L', having a common focus at S. It is clear that
L produces an image of the sun at S and the light leaves L. in the
same condition as it enters L. If another converging lens L' be placed
in the path of the light it will again be brought to a focus at S', where
we shall have a second image of the sun from which the light will
diverge. It is at S', where the pencil is very narrow, that the analysing
prism is placed.
Now if a diaphragm be placed at FF we may regard each point of
it as the origin of the small conical pencil of 16 aperture which it
receives from the sun. Any one of these pencils diverging from F
will be refracted at L and travel along LS in a parallel beam and then

420 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
come to a real focus F" from which it diverges, and falling upon L" is
brought to a focus F" upon a screen F"F". -
If we wish to observe the tints produced by a crystalline plate
when the incident light is parallel, it is placed at FF or F"F", the
pair of lenses L and L' in this case having obviously no effect. The
incident light being polarised in an azimuth a to the principal section
of the plate will be divided by it into two parts polarised at right
angles and differing in phase. Both these parts are concentrated by
L” at S', where the analyser is placed, and are again reduced by it to
a definite polarised pencil, which paints an image on the screen F"F".
If the analyser is merely a doubly refracting prism we shall have two
images on the screen which are complementary in colour, as is verified
by the fact that if they are partially superposed the overlapping por-
tion is always white, no matter how the plate is changed. By inclining
the crystalline plate to the direction of the light, by turning it round
a line in its plane, we alter the thickness of the plate traversed and
change the difference of phase, and therefore vary the tints of the
images. -
For the success of these experiments it is necessary to work with
strong light, on account of the magnitude of the image on the screen.
The solar light may be replaced by electric light, cast by a lens on the
polarising Nicol in a parallel beam.
2. Convergent or Divergent Plane-Polarised Light
227. The chief object of the lenses L and L' is to study the
phenomena produced in convergent light. For this purpose the
crystalline plate is placed at S. From such a point as F proceeds a
come of rays whose angle depends upon the original aperture through
which the light comes. This cone becomes an inclined parallel beam
after passing L. Every ray in this beam goes through the crystal at
the same angle, and is divided by it into two rays which are relatively
retarded the same amount for each original ray. But F is imaged at
a point F" on the screen, and no other ray reaches F". Hence the
interference colour at F" is the same for all the rays meeting there.
Any other point in the plane FF gives rise to a parallel beam through
the crystal passing at a different angle. It is finally represented by
an image-point on the screen. The colour at this point will depend
upon the retardation which is common to every ray of this second
beam. Hence the colours are completely sorted out on the screen.
We shall now consider various special cases in detail. It must be
observed that the crystal is not imaged on the screen and that the
Art. 228 UNIAXAL CRYSTAL 421
effect at any point of the screen does not arise from one point of the
crystal only, but from the whole area of it which is traversed by rays
having one definite angle of inclination.
228. Uniaxal Crystal–Plate cut at Right Angles to Optic Axis.-
Let a plate of a uniaxal crystal, cut perpendicularly to its axis, be placed
at S so that the axis of the beam is perpendicular to the face of
the plate, and consequently passes through it in the direction of the
optic axis.
Consider a circle through FF (at right angles to the axis). All Rings.
the rays which pass through points of this circle meet on the screen
at the corresponding points of the image of the circle on the screen.
If O is the centre of the circle FF (Fig. 177), and X is a point on it,
then the rays from it will all be
resolved in the crystal into two com-
ponents—one vibrating in the principal
section, and therefore parallel to OX,
and the other at right angles to that
azimuth. These rays experience a re-
lative retardation, the same for all that
proceed through X, and the colour
on the corresponding point on the
screen depends upon this retardation
and upon no other factor. Now the Fig. 177.
retardation will be the same for all such points X which lie on the
circle. If, however, we consider a larger or smaller circle the inclination
of the rays through the crystal will be different, and so also will the
retardation and the ultimate colour. Hence the appearance on the
screen will be a sequence of coloured rings. Any one of these, however,
is not equally bright all round. Comparing Figs. 177 and 175 we see
that if OP (Fig. 177) is parallel to the principal section of the polariser,
and OA to that of the analyser, then the angle XOP = a and XOA = 3.
Further, applying the considerations of Art. 221, the intensity is
given by
I– *{ cos”(a - 3) – sin 20 sin 23 sinº),
for homogeneous light.
If we denote the variable angle XOP by 0, and the angle AOP by
y—that is, if we write tº for a and y for a – 8–the expression for the
intensity becomes
I =a^{cos' y—sin 20 sin 2(9-7) sin” .3}.
Now if the point X be anywhere on the line OP or on its perpen-

422 COLOURS OF THIN CRYSTALLINE PLATES CHAP. xv.
Crosses.
dicular OP', the incident vibration, being parallel to OP, will be in the
principal plane in the former case and perpendicular to it in the latter,
so that it will pass through the plate without decomposition until it is
finally resolved in the analyser. The lines OP and OP' should there-
fore be each of uniform illumination, and should exhibit no colour. Our
formula points to the same conclusion, for if X is on OP or OP' we have
a = 0° or 90°, the term on which the colour depends vanishes, and in
both cases
I=a” cos” (a – 3)=a” cos”)
where y is the angle between the principal planes of the polariser and
analyser. We have therefore a rectangular cross of uniform illumina-
tion a” cos”y, having one arm parallel to and the other perpendicular
to the principal plane of the polariser.
Fig. 178. –(a – 8–0°). Fig. 179.-(a – 8–90").
Similarly, if B = 0° or 90°, we have the same value of I, viz.
I = a” cos”), and hence another uniform cross exists in the field having
its arms respectively parallel and perpendicular to the principal plane
of the analyser.
In general, then, two rectangular crosses are seen in the field, and
these are of the same uniform illumination, a” cos”), and uncoloured.
If y = 0°–that is, if the principal planes of the polariser and
analyser are parallel—the two crosses coincide, and we have one white
cross of uniform illumination a”, viz. that of the incident light
(Fig. 178).
If y = 90°–that is, if the Nicols are crossed—the two crosses again
coincide. But in this case the single cross is black, for cos y = 0 and
the intensity is zero (Fig. 179).
If sin 20 sin 2/3 is positive, the points of maximum intensity along
a given radius between the brushes are determined by
sin” .6–0, or 6–2mm,

ART. 229 GENERAL CONSIDERATIONS 423
and the minima by *
sin” #6=1, or 6= (2n + 1)"r.
But if sin 20 sin 28 be negative the intensity will be least when 8 = 2nt,
and greatest when 8 = (2n + 1)T.
The curves of equal intensity for a given position of the Nicols are
determined by the equation
cos” y – sin 20 sin 2(9 – y) sin” #6= constant.
The effect of superposing on a uniaxal plate, cut perpendicularly to Sign test.
the axis, another uniaxal plate cut in a similar manner is the same as
an increase or a decrease in the thickness of the first plate according as
they are of the same or of opposite signs. If they are of the same
sign the rings contract, and if they are opposite signs the rings
dilate. We are thus afforded with a ready and simple means of
determining the sign of a crystal, by comparison with another uniaxal
crystal of known sign.
229. General Considerations.—When slices of crystal, uniaxal
or biaxal, are examined in this way the figure on the screen is, in
general, traversed by two systems of lines. If white light is used
one of the systems of lines is brilliantly coloured, and these are termed
the isochromatic lines or simply the fringes. Their form depends on the
nature of the section of the crystal—that is, on the direction in which
it has been cut with reference to the optic axis or axes—and their
brightness depends on the position of the analyser, the colour being
most distinctly marked when the polariser and analyser are crossed
(a – 8 = 90°), as indicated by the equation
I=a^{cos” (q – 3) – sin 20 sin 28 sin” $6}.
The lines of the second system are not coloured. They intersect the
fringes and are termed the achromatic or neutral lines, and, as the above
equation shows, they are determined by the equation
sin 2a sin 28 = 0;
that is, a = 0° or 90°, and 8 = 0° or 90°, while the points of maximum
intensity are determined by
sin” #6= 0, or 6=2ntr,
and the minimum points by
sin” #6=1, or 6=(2n + 1)T, *
if the product sin 20 sin 28 be positive; but if this product be negative
the maximum points will be determined by
sin” $6 = 1, or 6– (2n+1)"r,
424 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
and the minimum points by
sin” #8 = 0, or 6=2mir.
The maximum lines in one case, therefore, correspond to the minimum
lines in the other.
Now it is easily seen that sin 20 sin 28 generally changes sign in
passing across a neutral line. For such a line arises from sin 20 = 0,
or sin 28 = 0, or both ; but when any function passes through a zero
value its sign in general changes. Thus sin 20 passes through zero
and changes sign as a passes through the value #1ſ. Hence in crossing
a neutral line corresponding to sin 20 = 0 the quantity sin 20 sin 28
will change from a positive to a negative value or vice versa, and
consequently the bright fringes at one side of the neutral line will
correspond to the dark bands at the other. If, however, sin 20 = 0 and
sin 28 = 0 simultaneously, the quantity sin 20 sin 28 will not change
sign, and the fringes will preserve the same tint in crossing the
line.
Hence in crossing a neutral line the isochromatic fringes in general
change to the complementary tint, except when sin 20 and sin 2/3
vanish simultaneously; an example of this latter case occurs when
we have a uniaxal plate cut perpendicularly to the axis and a - 3 = 0°
or 90°. Here the neutral line is a rectangular cross; sin 20 and sin 28
vanish simultaneously, and the rings in each quadrant are the same as
those in the adjacent quadrants.
We shall now consider the isochromatic lines, or fringes, which are
found to be as follows:
(1) In uniaxal crystals the fringes are—
Circles for a section perpendicular to the axis.
Hyperbolas for a section parallel to the axis.
Elliptic or hyperbolic arcs for an oblique section.
(2) In biaxal crystals they are— *
Closed rings for a section perpendicular to an axis.
Hyperbolas for a section parallel to the plane of the axes.
Lemniscates for a section perpendicular to the bisector of
the angle between the axes.
230. Isochromatic Surface in Uniaxal Crystals.-The fringes
presented in the case of a uniaxal crystal cut perpendicularly to the
optic axis have been already discussed (Arts. 228, 229). We shall
now investigate the general case, in which the thin plate may be cut
from the crystal in any direction."
Change of
tint.
* M. Bertin, “Mémoire sur la surface isochromatique,” Annales de Chimie et de
Physique, third series, tome lxiii. p. 57, 1861.
ART. 230 ISOCHROMATIC SURFACES 425
The characteristic of a fringe is that the retardation 6 is the same
at every point of it, and the locus of points in space for which 8 has
a given constant value will be an isochromatic surface. To every value
of 6 there will be a corresponding surface, and if with the radiant
point as origin a system of these surfaces be described corresponding to
retardations of 1, 2, 3, 4, etc., half-wave lengths, their intersections with a
plane drawn parallel to the face of the crystal will determine the corre-
sponding isochromatic curves or fringes exhibited in the field of view.
In general a ray will be divided into two rays within the crystal,
travelling with different velocities, but as the crystal is thin, and the
difference of index is small, we may, in the approximate calculation,
suppose both rays to travel along the same inclined line, one with
the ordinary velocity vo and the other with a velocity v. The times
occupied in traversing the oblique distance p through the slice is
to = p/vo and t = p/v respectively, so that the time retardation is
- 1 1
6-t-p(-i).
and the path retardation is
6= (po- p.) p.
For the calculation of the isochromatic surfaces we may take an
origin at any point O (Fig. 181), and draw lines in all directions to
represent various rays through the plate, and of lengths such that the
values of 8 would be the same. The surface on which the free ends
of these lines lie is then an isochromatic surface.
Now the wave surface in a uniaxal crystal consists of a sphere of
radius b or 1/uo (in a negative crystal like Iceland spar), and a spheroid
of which the generating curve is
poºr” + p.”y°= 1. (1)
If r denotes any radius vector of this ellipse we have p inversely pro-
portional to r, and hence for the retardation in traversing a thickness
p we have
B b (1) 6=p(uo-pº). (2)
ut by
p?= u,” cos” 0 + p.” sin” 6, (3)
and by (2) 6
*= ( : — go )*, 4
*=(-1) (4)
Therefore, by combining (3) and (4) we find
6 2 &
(. * *) =u,” cos”04-u,” sin” 0.
Hence
(6 – puo)*=p,” + p.”y”,
426 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
which, since p” = aſ + y”, gives
{(u.”—u,”) y”4-3%-49.23°(º-Fy”),
the generating curve of the isochromatic surface.
The isochromatic surface is formed by the revolution of this curve
round the line representing the axis of the crystal. Its general form is
represented in Fig. 180. In the neighbour-
hood of the axis of y the curve resembles
an hyperbola, and the surface an hyper-
boloid of revolution." By assigning various
values to 8 we determine corresponding
surfaces, and the intersections of these with
a plane parallel to the face of the crystal
give the isochromatic curves or fringes.
Thus if 8 = T, which corresponds to a retardation #A, the correspond-
ing surface intersects the plane in the first dark fringe, etc.
(1) Section Perpendicular to the Optic Azis.-If the thickness of the
plate be e the section of the isochromatic surface by the face of the
plate cut perpendicular to the axis is found by putting a = 6. The
curve is obviously a circle of which the radius is r = y determined by
the equation -
Fig. 180.
(u,” – pº)” – 26°(u,” + p.”)r” – 4p,”6°e?-- 64 = 0,
OT
,a_*(***)+29.3°/ſºfſº-º.
(u.” - p,”)?
Since 8 is small compared with e, the expression for r reduces to the
approximate form
,2__24.0%
T 1, 2 sº
po” – gº
or since pie + po = 2po nearly, we have the approximate expression
Öe
q2 = ©
Mo - Me
The consecutive rings are determined by making n equal to the con-
secutive whole numbers in the equation 8 = n}\, the dark rings corre-
sponding to the odd values of n and the bright rings to the even values.’
(2) Section Parallel to the Aais.-As the equation of the surface is
{(u.”—p,”) (y” +?)+6%2=49.23°(a 24-y” +2°),
we obtain the equation of a section parallel to the plane of a. and y
* Neglecting y”.

ART. 230 ISOCHROMATIC SURFACES 427
by putting z = e. Using the approximate form, the equation of the
surface is
-- o u,5°
u,(y”--zº) - *=ºL. }*
and the equation of the section by the plane z = e is
52
*-*=p (ººº-º)
Me!/*-1. Me (a, - ºº
6
= 2 *(*º-1) approximately.
Alet e(un-u.) PI ately
Corresponding to various values of 8 we have a system of hyperbolae,
and the figure (Fig. 181) represents the curves corresponding to 8 = }n}\,
m being any whole number. If
the absolute term vanishes the
corresponding hyperbolae re-
duce to a pair of lines, and
the appropriate value of 6 is
3 = (po-po)e. The equation of
the lines is
uci/*-uº-0,
and as pºe is nearly equal to po
the lines are very close to the
bisectors of the angles between
the axis of w and y, and are
therefore nearly at right angles. If (a, -po)e is equal to a whole
number of half-wave lengths, these lines are comprised in the system
of fringes. -
If the retardation is less than (, – p.)e the hyperbola cuts the axis
of a, while, if the retardation is greater than (po-po)e, it cuts the axis
of y, and the lines corresponding to 6 = (po-po)e separate the two sets
of curves. The fringes are most brilliant when the polariser and
analyser are parallel, or crossed, and make an angle of 45° with the
principal plane of the plate, but when the principal plane is parallel or
perpendicular to the principal section of the analyser the fringes
disappear entirely.
(3) Oblique Section.—The section of an isochromatic surface by a
plane inclined to the axis is a curve of the fourth degree which approxi-
mates to a circle when the plane is nearly perpendicular to the axis
and to an hyperbola when the plane is nearly parallel to the axis.
For an obliquity of 45° it is only the portions of the fringes near their
vertices that are seen, and these are right lines perpendicular to the
principal section of the plate.
Fig. 181.

428 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
231. Isochromatic Surface in Biaxal Crystals.-In the case of
biaxal crystals the relative retardation of the rays in traversing a
distance p may, as before, be written in the form -
6= p(uſ * pa")
where p' and p" are the roots of the equation
72 - m? m2
+ 5 + F
p? - pº a? - ug” p? * * º
0,
in which l, m, n are the direction cosines of p, and p is a radius vector
of the isochromatic surface when 8 is constant (see Art. 206).
Now from this equation we have
p” + a”= X(u,”-- pº)”,
and
p”u”=>uºus”.
But since
6
;=p' * p",
we have
/2 *2 62 "-4 /2, ſ/2
pºº + p. 2) - At “A.”.
Hence
ſ 2 . . 2, 2 *\* 2u.2/2
(2(º-tº- p? = 42pºus”!”,
or in Cartesian co-ordinates, the equation of the isochromatic surface is
{(u,” + agº)a?--(ug” + p.1°),”-- (u,”-- p.”)2°– 6*}”
=4(a!” + y^+ 2*)(u,”ugº”--pºu,”y”-- piºu..”).
All the surfaces are obtained by giving various values to 6. They are
similarly situated, and their intersections with
the face of the crystal give the isochromatic
curves, each corresponding to a certain constant
difference of phase.
The form of these surfaces is represented
in Fig. 182. The section by the plane az which
contains the optic axes is found by making y
zero in the above equation. It consists of two
branches, one having asymptotes parallel to the
optic axes, the other interior to it is closed and parasitic, not being
related to the question in hand. It is obvious that in the directions
of the optic axes the values of p should be infinite, for since these
Fig. 182.
* M. Bertin remarks that “la surface isochromatique des cristaux à deux axes
ressemble à une croix de Saint-André dont les bras seraient cylindriques et dirigés
suivant les axes optiques du cristal.”

A Rt. 231 ISOCHROMATIC SURFACES 429
are the optic axes v H tº or p = p", and therefore p is infinite for a
given retardation.
The sections made by the other co-ordinate planes are closed curves
of the fourth degree.
(1) Section Parallel to the Plane of the Ares.—The sections of the
isochromatic surface by a plane parallel to that containing the optic
axes differ little near the centre from hyperbolas. On substituting
y = e and making 6 = (p., -pº)e, the equation of the section of the
surface reduces to
0 = pºº-p12°4 terms of the 4th degree with small coefficients.
That is, the section nearly coincides with the lines
2- +a; /u. -
- ^ All
If we substitute
5
1. –
*/ 13-141
- e.
where e is a small quantity, in the equation of the surface, we find for
the indicatrix
*-*-*.
showing the nature of the hyperbolic curves with the above lines for
asymptotes. These fringes are like those of a uniaxal crystal cut
parallel to its axis (Fig. 181).
(2) Section Perpendicular to a Bisector of the Angle between the Optic
Awes.—The sections of the isochromatic surfaces by a plane perpen-
z=506
Fig. 183.
dicular to the internal or external bisector of the angle between the
optic axes—that is, parallel to a y—resemble the several forms of
lemniscates.
The sections of the surface corresponding to a given value of 8 by
planes drawn at right angles to the plane of the optic axes, and at
distances 2 = 206, 2 = 506, z = 606 respectively from the centre, are
represented in Fig. 183.

430 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
In this case we can determine the approximate form of the fringes
directly. For if OM and OM (Fig. 184) be the optic axes, p" and p"
the indices for the rays travelling in any
direction OP which makes angles 6 and 6.
with the axes, then
6=OP(u' – u").
But by Art. 207, Cor. 1, we have the approxi-
mate formula
Fig. 184.
u' – u"-(a-pa) sin 6 sin 9',
and approximately
sin º-º: sin º'-. OP=e.
Therefore
6 - - PM. PM’
:=(W-4")=(1-4) (OM)
that is,
PM. PM’-constant,
which is the polar equation of a lemniscate having M and M.
for foci.
(3) Section Perpendicular to an Optic Azis.-In this case the iso-
chromatic lines will be a system of closed curves resembling
deformed lemniscates encircling the axis to which the section is per-
pendicular.
232. Achromatic or Neutral Lines.—The uncoloured lines are
determined by the equation
sin 20 sin 28–0.
They are consequently the locus of a point on the face of the crystal
such that the planes of polarisation of the rays in the crystal are
at that point either parallel or perpendicular to the principal plane of
the polariser or analyser.
(1) Section Perpendicular to the Bisector of the Angle between the Optic
Awes.—The planes of polarisation of the rays travelling in any direc-
tion OP (Fig. 184) are the bisectors of the angles between the
planes OPM and OPM’, which contain the ray OP and the optic axes
OM and OM respectively. The traces of these planes on the face of
the crystal will, for a small angle of incidence, approximately coincide
with the bisectors of the angle MPM'. Hence if the plane of the
paper be taken parallel to the face of the crystal, and if the optic axes
meet it at M and M' (Fig. 185), the point P will describe a neutral
line if the bisector of the angle MPM is parallel or perpendicular to
the principal plane of the polariser or analyser. Let OX be parallel
to the principal plane of the polariser, and take OX and the perpen-

Art. 232 ACHROMATIC CURWES 431
dicular line OY as axes of reference, O being the middle point of MM’.
The bisector of the angle MPM' must be perpendicular to OX, there-
fore the triangle APB is isosceles. Hence
if the co-ordinates of P be a, y, and those
of M be aſ, y', we have
got pho-º-º, and got Pao-ºtº,
3/- 1/ y -- y
equating these expressions we find at once
zy–a'y'.
The locus of P is consequently a rect-
angular hyperbola, passing through M.
and M', of which the asymptotes are OX
and OY, lines parallel and perpendicular
to the trace of the principal plane of the polariser.
In the same manner we find a second rectangular hyperbola cor-
responding to 3 = 0° or 90°, also passing through M and M', and
having for asymptotes lines parallel and perpendicular to the trace of
the principal plane of the analyser.
The uncoloured lines therefore consist of four hyperbolic branches
passing two and two through M and M', the apparent extremities of
the optic axes, the asymptotes of one curve being a parallel and a per-
pendicular to the trace of the principal plane of the polariser, while
those of the other bear a corresponding relation to the principal plane
of the analyser.
If the principal plane of the polariser is parallel to MM’-that
is, if MM" coincides with OX-the hyperbola reduces to its asym-
Fig. 185.
Fig. 186. Fig. 187.
ptotes w/ = 0, and becomes a rectangular cross. If in addition
the optic axes coincide, the other hyperbola will also become a rect-
angular cross, as we have already seen to be the case for a uniaxal
crystal.


4.32 COLOURS OF THIN CRYSTALLINE PLATES CHAP. xv.
If the polariser and analyser are either parallel or crossed, the two
hyperbolas coincide, and we have a single rectangular hyperbola of
which one branch passes through M and the other through M', as
shown in Fig. 186, while if in addition the plane of polarisation be
parallel or perpendicular to MM’ the curve reduces to a rectangular
cross as shown in Fig. 187. When the polariser and analyser are
parallel we have a - 3 = 0, I = a”, and the brush is bright, but if the
Nicols be crossed a — 8 = 90°, and the brush is dark.
The neutral hyperbolic bands are made to pass through all possible
forms by rotating the plate between the Nicols, and, as in the case of
uniaxal crystals, the coloured fringes change to the complementary
tint in crossing the neutral lines (Art. 229).
(2) Section Parallel to the Plane of the Ares.—In this case the optic
axes OM and OM (Fig. 188) at any point O in the face of the crystal
lie in the plane of the face, and for a ray incident
nearly normally at 0, the planes of polarisation of
the two refracted rays will bisect the angle MOM’
internally and externally. Hence if the angle of
incidence be small the bisectors OX and OY will
have the same direction at every point, and therefore
if they are parallel or perpendicular to the principal
plane of the polariser or analyser at one point they
will be so at all, and the whole field will be uncoloured. There are
consequently no neutral lines, but in four positions the whole field is
uncoloured.
(3) Section Perpendicular to an Aris–When the plate is cut at
right angles to one of the optic axes the fringes form a system of con-
centric rings, and these are crossed by two neutral bars which in
general are not rectangular. When the Nicols are parallel or crossed
the two bars coincide and the neutral line is sensibly straight, and
either white or black.
Fig. 188.
3. Circularly and Elliptically Polarised Light
233. Parallel Circularly Polarised Light.—The phenomena
which are presented when circularly or elliptically polarised light is
transmitted through a thin crystalline plate were investigated by Airy.”
The characteristic of circularly polarised light is that it results from
two equal plane-polarised parts, polarised at right angles and differ-
ing in phase by a quarter of a period. Its components in any
* Wide examples to Chap. II. p. 57.
* Airy, Camb. Trans, vol. iv.

ART. 233 - CIRCULARLY POLARISED LIGHT 433
two rectangular directions may therefore be represented by the
equations
* a = a cos wi, and y=a sin wt,
the intensity of the original beam being 2a”. It may be obtained by
transmitting plane-polarised light through a quarter-wave plate placed
with its principal sections at an angle of 45° to the principal section of
the polariser (or a Fresnel's rhomb suitably placed), the action of these
instruments being to divide the incident plane ray into two others
vibrating in perpendicular directions, and differing in phase by a
quarter period. The determination of the phenomena which arise
when circularly polarised light is transmitted through a crystalline
plate may therefore be obtained from Art. 225 by introducing the
condition that the first plate introduces a phase difference of a quarter
period. The direct investigation of the intensity in the field of the
analyser is, however, very simple. Let the components of the circular
vibration falling on the plate be
2=a cos ot, and y = a sin wi,
parallel to the two azimuths of vibration in the plate as in Art. 2] 2.
The passage through the plate introduces a further relative difference
of phase 8, so that the emerging vibrations may be written in the form
a = a cos wi, and y=a sin (wt +6).
These, resolved parallel to the principal plane of the analyser, give
the vibration
& Cos a cos wt + a sin a sin (wt + 3) = a(cos a + sin a sin 6) cos wit -- a sin a cos 6 sin wi.
The intensity is therefore
I=a^{(cos a + sin a sin à)*+ (sin a cos 3)*}
= a”(1+sin 20 sin 6),
which is independent of the orientation of the polariser.
If the analyser be merely a doubly refracting rhomb we shall have
in the same manner for the ordinary image
I' = a”(1 — sin 20 sin 6),
and therefore
I+ I’=2a”,
so that the images are complementary.
Had we started on the supposition that the a-component of the
vibration is retarded relatively to the y-component, we would have
had
a;= a cos (wt + 6), and y=a sin wt,
2
F
434 COLOURS OF THIN CRYSTALLINE PLATES CHAP. XV
Colour
term.
Light
circularly
analysed.
and for the intensity in the field of the analyser we would have the
complementary expressions
I= a”(1 — sin 20 sin 6),
I’= a”(1 + sin 20 sin 6).
Hence if the sense of the retardation be reversed the two images
in the field of a doubly refracting analyser will interchange tints.
The formula for I shows that the image possesses a certain colour
depending on the term sin 20 sin 8, and that there are two positions
in which the image is achromatic, given by the equation
sin 20 = 0;
that is, when the analyser is parallel or perpendicular to the principal
section of the thin plate, and in both these positions the intensity is
the same and equal to half that of the incident light, for we have
I = I’= a2.
The tint, depending on 8, will vary with the thickness of the plate,
but here it will not follow the law of colour in Newton's rings which
holds in the case of plane-polarised light, for the term on which the
colour depends is here proportional to sin 8 and not to sin” #8.
If a second quarter-wave plate or a Fresnel's rhomb be introduced
between the crystal and the analysing Nicol in such a manner that light
passing through the Nicol would be circularly polarised by the plate
(that is, so that the principal plane of the Nicol makes an angle of
45° with the principal plane of the plate), then the light falling on the
crystal will be circularly polarised and the light emerging from it will
be circularly analysed. In this case the vibrations emerging from the
crystal are, as before, of the form
a cos wi, and a sin (wt + 6).
Hence if the principal plane of the quarter-wave plate be inclined at
an angle a to that of the crystal the two components emerging from
the plate are
a cos a cos wi + a sin c. sim (wt + 6),
and
a sin a sin wit -- a cos a cos (wt + 6).
These resolved parallel to the principal plane of the analysing Nicol,
which is at 45°, give an expression which is simply their sum multiplied
by 1/x/2 ; that is,
#(co. (wt — a) + cos (wt + 6– a)}.
^
ART. 234 CIRCULARLY POLARISED LIGHT 435
Hence
I=2aº cos” #6.
There are consequently no neutral lines, and the isochromatic lines are
the same as when the light is plane polarised and analysed.
234. Convergent Circularly Polarised Light.—If the incident
light be convergent, achromatic lines and coloured fringes are presented
as in the case of plane-polarised light, but here they are of a simpler
character. The uncoloured lines are determined by the equation
sin 20 = 0,
therefore they are only half as numerous as in the case of plane-
polarised light, for in the latter case the neutral lines are determined
by the equation sin 20 sin 28 = 0.
Thus with a uniaxal plate cut perpendicularly to the optic axis
we have with plane-polarised light in general two rectangular crosses
of uniform intensity and neutral tint, but with circularly polarised
light we have a single cross, the arms of which are respectively parallel
and perpendicular to the trace of the principal section of the analyser.
The intensity of the neutral lines is uniform and equal to a”, being
independent of the angle between the principal planes of the polariser
and analyser.
For a given position of the Nicols, if sin 20 is positive, the points
of maximum intensity correspond to
sin 6 = 1, or ö– (4m + 1)}tr;
that is,
6=", 57r 97ſ
5' 2 T2 . etc., (max.)
and the points of minimum intensity to
sin 6= – 1, or 6–(4m – 1)}tr;
that is,
37ſ 77r 11tr $º
3=#, T2 - 2 etc. (min.)
In the case of plane-polarised light we have seen that the bright
and dark fringes are determined respectively by
6=2ntr=0, 2tr, 4tr, etc.,
and
6=(2n + 1)T = T, 37ſ, 57r, etc.
Hence the bands correspond to the even multiples of #7 in the latter
case and to the odd multiples in the former,
436 COLOURS OF THIN CRYSTALLINE PLATES CHAP. xv.
Sign
test.
The effect of the circular polarisation has been to displace the
fringes through a quarter of an order. In one pair of opposite quad-
rants they are pulled in towards the centre by this amount, and in the
other pair they are pushed out by the same amount, for the bright
rings in any quadrant correspond to the dark rings in the adjacent
quadrants, each fringe changing to its complementary in crossing the
neutral lines (Figs. 189, 190).
This result affords a convenient method of determining the sign of
a crystal, for if two crystals have opposite signs then sin 6 will be posi-
tive for one and negative for the other, consequently the bright rings
afforded by one will correspond to the dark rings of the other. But
we have seen that the bright rings in any quadrant correspond to the
dark rings in the adjacent quadrant, therefore for a given position of
Fig. 189. Fig. 190.
the Nicols the rings afforded by one crystal will be similar to those of
the other turned through a right angle.
If the thin plate be cut from a biaxal crystal perpendicular to the
mean line, the isochromatic curves are lemniscates, and crossing these
we have the two branches of a rectangular hyperbola passing through
the apparent extremities of the optic axes, and having for asymptotes
a parallel and a perpendicular to the trace of the principal plane of
the analyser. The fringes change to the complementary tint where they
cross the neutral lines; that is, the bright bands in any quadrant
correspond to the dark bands in the neighbouring quadrants.
235. Elliptieally Polarised Light.—If the light incident on the
plate be elliptically polarised, its components parallel to the principal
directions in the plate will differ in amplitude and phase. The effect
of the plate is to alter this difference of phase, so that emerging from
the plate the vibrations may be written in the form
a; = & cos w!, and y = b cos (wt + 6).

ART. 236 DISPERSION OF THE OPTIC AXES 487
Resolving these parallel to the principal plane of the analyser we have
finally the vibration
a cos a cos wi-b sin a cos (ot + 3)=(a cos a + b sin a cos 3) cos wi-b sin a sin 6 sin ot.
The intensity is consequently given by
I = (a cos a + b sin a cos 6)*--b” sin” a sin” 6
= a” cos” a + b% sin” a + absin 20 cos 6.
The uncoloured lines are therefore again given by the equation
sin 20 = 0.
The forms of the fringes have been investigated by Airy in some of
the simpler cases, particularly that of a uniaxal crystal cut perpendicu-
larly to the optic axis.
4. Dispersion of the Optic Aases
236. Dispersion of the Axes.—The optic axis of a uniaxal crystal
is the same for light of all colours, but in the case of biaxal crystals
the angle between the optic axes depends on the principal indices
ph, pg, ps, and as these quantities are functions of the wave length, it
follows that in general the angle between the optic axes will be
different for differently coloured lights. Thus if 26, and 26, denote the
angle between the optic axes for the violet and the red rays respectively,
we have to determine them in terms of the corresponding principal
velocities (Art. 202)
2 Tj, 2
tan 6, = w/ º
a) a,” - b,”
to 3
* — c,”
tan 6, = b2 - c.2’
* T Lº-
and as a, b, c do not, in general, vary proportionally in passing from
one colour to another, it follows that 6 will vary with the wave length.
This dispersion of the optic axes, as it is termed, depends on the
character of the crystal, presenting different peculiarities according as
the crystalline axes are rectangular or oblique.
(1) Orthorhombic System.—Crystals of the orthorhombic (or trimetric)
system possess three rectangular planes of symmetry. The crystallo-
graphic axes are unequal but mutually rectangular.
Now the optic axes are situated in the plane (~2) of the greatest
and least axes of elasticity. Consequently if we have a > b >c for all
wave lengths the optic axes for all colours will be situated in the same
plane (ac), but if it should happen that for some colours we have
a > b >c and for others b > a i-c, the optic axes of the former will lie in the
plane (ac), and those of the latter in the perpendicular plane (bc). So
4.38 COLOURS OF THIN CRYSTALLINE PLATES CHAP, XV
again if for any colour we have a > c->b, the corresponding optic axes
will be situated in the plane (ab). The pairs of optic axes situated in
any plane have, however, a common bisector of the angle between them,
viz. the axis of elasticity which lies in that plane; they are therefore
symmetrically situated with respect to this line. When the variations
of a, b, c from colour to colour are small, compared with the differences
of the quantities themselves, then the optic axes of the various colours
(as in the case of aragonite) will all lie in the same plane, and be
symmetrically situated with respect to the same line. The bisector of
the acute angle between the optic axes is called the acute bisector or
first median line.
Thus the lemniscate fringes presented by a plate cut perpendicu-
larly to the greatest or least axes of elasticity—that is, perpendicularly
to either bisector of the angle between the optic axes—will possess the
same centre but will have different foci for the different colours, the
distance between them increasing or decreasing from the red to the
violet, according to the nature of the crystal ; but in some cases the
angle between the axes attains a maximum value for some colours
between the red and violet, and then decreases from this towards both
ends of the spectrum. (Normal dispersion of the aſces.)
(2) Monoclinic System.–In the monoclinic system of crystals two
of the crystallographic axes are inclined at an oblique angle, while the
third is perpendicular to their plane. Thus of the angles between the
axes two are right and one oblique. There is consequently only a
single plane of symmetry, viz. that containing the oblique axes. The
third axis, which is perpendicular to this plane, preserves the same
direction for all colours, but the directions of the other two axes may
vary in the plane of symmetry, so that great complication may occur
in the fringes. Two distinct cases arise :
(a) If for any colour the bisector of the angle between the optic
axes coincides with that crystallographic axis which is perpendicular
to the plane of the other two, this line will bisect the angle between
the optic axes for all the other colours, but since the oblique axes in
the plane of symmetry vary with the wave length, it follows that the
plane containing the optic axes may be situated in any azimuth.
Borax presents this mode of dispersion, the optic axes for the various
colours being situated in different planes but possessing a common
bisector. Thus the isochromatic fringes afforded by a plate cut per-
pendicularly to the bisector of the acute angle between the optic axes
(the greatest axis of elasticity in negative, and the least in positive
crystals) possess the same centre, but their axes may be in any
plane through it. (Crossed dispersion of the awes.)
ART. 236 DISPERSION OF THE OPTIC AXES 439
(3) If the optic axes for any colour are in the plane of symmetry—
that is, the plane containing the oblique axes of the crystal—then the
optic axes for all colours will be situated in the same plane, but as their
directions may vary in any manner, the angles between them will not
have a common bisector and the isochromatic curves will not have a
common centre, but their centres will be situated on a right line with
respect to which the curves are symmetrical. This species of dispersion
is exhibited in gypsum. (Inclined dispersion of the aſces.)
(3) Triclinic System.—In the triclinic system of crystals the three
crystallographic axes are inclined at oblique angles, and the position of
the axes of optical elasticity may vary in any manner from colour to
colour. We may consequently be presented simultaneously with the
modes (a) and (8); that is, the optic axes for the different colours may
be situated in different planes and at the same time they may not have
a common bisector.
Effect of a Change of Temperature.—An elevation of temperature
generally diminishes the refractive index of a transparent substance.
Hence in general a variation in the temperature of a crystal causes a
corresponding variation in the quantities a, b, c, so that the angle
between the optic axes for any given colour will in general change
with the temperature of the crystal. And further, if at one tempera-
ture the optic axes for all colours lie in the same plane, then at
another temperature some may remain in that plane while others are
displaced to the perpendicular plane. This was observed by Brewster 1
in sulphate of sodium. In this substance the optic axes at ordinary
temperatures are all situated in the same plane, but they exhibit great
dispersion and the fringes are greatly confused. Operating with mono-
chromatic light of various colours, the angle between the optic axes
was found to increase from the violet to the red, being small for the
former and considerable for the latter. On gradually raising the
temperature the angle was seen to diminish for all the colours, the
violet axes shrinking closer and closer till they finally coalesced, and
the crystal for these rays then behaved as if uniaxal. On continuing
the elevation of temperature the violet axes separated again, but now
in the perpendicular plane, and at 60° the angle between them attained
a considerable magnitude. In operating with white light the pheno-
mena become extremely confused and indistinct.
In the case of the oblique systems of crystals an elevation of
temperature may displace one axis notably more than the other, so
that the mean line corresponding to this colour becomes displaced in
direction.
* Brewster, Phil. Mag. (3), vol. i. p. 417.
CHAPTER XVI
ON THE STUDY OF POLARISED LIGHT
237. Detection of Polarised Light.—The fringes and colours
exhibited when polarised light is transmitted through a thin crystal-
line plate placed before an analyser afford a most delicate test of the
presence of polarisation in a beam of light, or of doubly refracting
structure in a transparent substance. It is to be expected, therefore,
that instruments depending in principle upon the exhibition of these
fringes should have been early invented and applied to the study of
polarised light. In general any instrument which acts as a polariser
may also be used as an analyser. Thus a Nicol's prism may be used
either to produce a pencil of plane-polarised light or to detect polarisa-
tion in any other pencil. If light be transmitted through a Nicol
the intensity of a transmitted beam will vary as the Nicol is rotated
if the incident beam is polarised or partially polarised. But this
method of detecting the presence of polarisation depends on the
variation of the intensity of an image, which will be very minute, and
consequently escape observation if the quantity of polarised light in
the incident beam is small.
Instruments depending on the production of colours or fringes are
much more delicate, and detect the presence of small traces of polar-
isation. We shall describe those of Savart and Babinet.
238. Savart's Polariscope.—Savart's polariscope is a simple and
delicate contrivance for the detection of plane-polarised light.
A thin plate of a uniaxal crystal is divided into two halves in
order to secure two similar thin plates of equal thickness. The two
portions are now superposed so as to form a plate of double the thick-
ness, and one of them is rotated through 90°, so that their principal
sections are at right angles. The plates, so placed, are mounted in a
small tube before a Nicol's prism, or any other analyser, of which the
principal section is turned parallel to the bisector of the angle between
the principal sections of the plates.
440
Art. 239 BABINET'S COMPENSATOR 441
If a parallel beam of plane-polarised light falls normally on the
plates, the phase difference between its two components, parallel and
perpendicular to the principal section, introduced by one plate is
exactly removed by the second plate ; but when a beam of convergent
or divergent plane-polarised light is allowed to fall upon the plates the
field of the analyser is crossed by a system of coloured rectilinear
fringes parallel to the bisector of the angle between the principal
planes of the plates. The sensitiveness of the instrument increases
as the plane of polarisation of the incident light approaches the direc-
tion of the fringes—that is, the bisector of the angle between the
principal sections—and it is sufficiently delicate easily to detect the
polarisation of the light of the sky.
239. Babinet's Compensator.-The compensator devised by M.
Babinet admits of the study of polarised light by means of coloured
fringes, even though the incident light be parallel. It is an exceed-
ingly sensitive polariscope, and has been successfully adapted, in a
modified form, by M. Jamin to the study of elliptically polarised
light.
It consists of two slender right-angled prisms or wedges of quartz
ABD and BCD (Fig. 191), placed together with their hypothenuses
in contact so as to form a thin plate of
rectangular cross section ABCD. In the
prism ABD the optic axis is parallel to
the face through AB, and to the plane of
the section ABCD, while in the prism
BCD the axis is parallel to the face
through CD, but perpendicular to the
plane of the section. Thus in both
prisms the axes are parallel to the faces
of the plate, but perpendicular to each other, as in Wollaston's prism
(Art. 187), but on account of the extreme smallness of the angles of
the wedges in this apparatus the separation of the rays during trans-
mission is negligible.
Hence if plane-polarised light falls normally on the face AB it will
be broken up into two parts, one vibrating parallel to AB, and the
other perpendicular to it, and these vibrations on entering the second
prism will retain their directions of vibration but interchange their
velocities of propagation, for the vibration parallel to AB is parallel to
the optic axis in the first prism but perpendicular to it in the second.
It consequently follows that the ordinary ray in the first prism
becomes an extraordinary ray in the second, and rice versa. Thus if
any ray PQ traverses a thickness e in the first prism, the relative
Fig. 191.

442 STUDY OF POLARISED LIGHT CHAP. XVI
retardation of the two vibrations will be e(p, -po), and for a thickness
e' in the second prism will be – e'(ue – po), since the ray which travels
fastest in the first travels slowest in the second, and vice versa. Hence
the whole relative path retardation of the vibrations produced by
transmission through the plate is
6=(e– e')(ge-po).
The ray MN which passes through the centre of the plate (e = e()
suffers no relative retardation in its component vibrations, and remains
polarised in the same plane as at incidence. As we move from N
towards D the component vibrations at emergence will differ in phase,
but at a certain distance a from N the phase difference will be equal to
T, and the retardation will be #A. At a distance 20, it will be A, and
so on. Consequently on the line CD we will have a system of equally
spaced points at which the phase difference will be a multiple of tr, and
the emergent light will be plane-polarised, and on the face of the
crystal there will be a corresponding system of right lines, at a com-
mon distance a from each other. At a centre N, and at points on
CD, distant any multiple of 2d. from it, the phase difference will be a
multiple of 21, and the transmitted light will be polarised in the same
plane as the incident, but at the intermediate points, viz. those distant
from N by an odd multiple of a, the phase difference will be an odd
multiple of T, and the transmitted light will be plane-polarised also,
but in this case the plane of polarisation will be inclined at an angle
2a to the plane of polarisation of the incident light, where a is the
angle between the primitive plane of polarisation and the principal
plane of the face AB (see Art. 47, Cor. 1). The points at which
the plane polarisation exists are determined by the equation
(e – e')(ue — Ho) = m. §).
At a point any distance a from N the path retardation is 8 where
ð ac Q}
#X * a' e tº 6 - #A.
and we have the relation
f 2.
(e-e')(4,-4)=. tº
2
. At points where aſa is not a whole number, the phase difference will
be other than a multiple of T, and the transmitted light will in general
be elliptically * polarised.
* At points half-way between those where the plane polarisation occurs, the phase
difference will be 90° or an odd multiple of 90°, hence if the incident light be
polarised at an angle of 45° to the axis of the quartz, the component vibrations will
be of equal amplitude, and when they differ in phase by 90° the light will be circu-
ART. 239 BABINET'S COMPENSATOR 443
Hence if the compensator be viewed through a Nicol's prism, turned
so as to extinguish the plane-polarised light of the central line N, it
will also extinguish the light from a system of parallel lines on either
side of N, at a distance 2a from each other. Between these dark lines
there will be a certain amount of illumination. At points half-way
between them—that is, on a system of lines distant from N by odd
multiples of a-the light is also plane-polarised, but at an angle 2a
to that at the central line. The Nicol will not extinguish this plane-
polarised light but will transmit more or less of it according to the
value of a, and (if a = 45°) the light from these lines will be completely
transmitted by the Nicol, so that they will be very bright and the
fringes will then be best marked, for between the dark lines corre-
sponding to the w = 2nd we shall have the spaces the brightest possible.
Hence if a = 45°–that is, if the incident light is polarised at an angle
of 45° to the principal plane of the face—the fringes will be most
distinct; the dark bands being given by
a;=2na,
and the brightest lines by
a;= (2n + 1)a.
Thus in general we have two systems of lines along which the light
is plane-polarised, one corresponding to retardations equal to even
multiples of $A and the other to odd multiples, and the planes of
polarisation of the two systems are inclined at an angle 2a. If the
principal plane of the analyser be perpendicular to the primitive plane
of polarisation—that is, if the Nicols be crossed—a system of dark lines
corresponding to the former will appear in the field, and if the analyser
be rotated through an angle 20, a system of dark lines corresponding
to the latter will be presented, while for intermediate positions of the
analyser there will be no perfectly black bands, but merely lines of
maximum and minimum intensity.
The distance a may be measured by fixing a very fine cross wire,
furnished with a micrometer screw, so as to coincide with the central
dark band. By turning the screw the wire is displaced to the next
dark line, and the distance 2a through which it has been displaced
may be found; or the wire may be displaced to the nth dark line
larly polarised. There will therefore in this case be a system of lines along the face
of the plate at which the transmitted light is circularly polarised, and these lie half-
way between the lines of plane polarisation. They may be detected by introducing
a quarter-wave plate or Fresnel's rhomb, which will reduce the circularly polarised
light to plane-polarised light; this may be quenched by a Nicol's prism, and a
corresponding system of dark lines will be presented in the field.
444 STUDY OF POLARISED LIGHT CHAP. XVI
from the centre, the distance 2na read off, and a calculated. The retard-
ation at any point depends on the wave length, and consequently if
white light be used the bands will be rainbow-coloured and the central
line alone will be quite black.
Cor.—If the angle ABD of one of the prisms be i we have
(ple - po) tan i-X/4a,
for
tan i-*: , and (e – e')(ue — *)=}}
which by multiplication give the above result. Knowing a and i, this
gives a method of determining either A or pe – po, when one of them is
known otherwise. -
240. Elliptieally Polarised Light.—M. Jamin applied Babinet's
compensator, in a modified form, to the determination of the constants
which characterise an elliptically polarised ray. In the foregoing we
have supposed the cross wire movable while the compensator remains
fixed, but in M. Jamin's apparatus the cross wire remains fixed and
one of the wedges forming the compensator is moved parallel to itself
by means of a micrometer screw, the other wedge remaining fixed.
The effect of this is to diminish or increase the difference of thickness
e – e' under the wire according to the direction of motion. The
motion of the wedge consequently displaces the whole system of fringes
across the field, and any particular band may be brought under the
cross wire. It is clear that the distance through which the wedge
must be displaced in order to displace the system through the width
of a band is twice as great as the corresponding displacement 2a of
the fibre in the first form of the apparatus. For here the thickness of
one wedge under the wire remains constant while the other varies, but
in the case of a movable cross wire the thickness of one increases and
that of the other simultaneously diminishes by the same amount, so
that to produce the same difference of thickness under the fibre it is
only necessary to move it through half the amount. Hence if 2b be
the distance through which it is necessary to displace the wedge in
order to displace the fringes under the cross wire by the width of a
band, we have for the retardation at a distance a from the central
band
º
Ö a: A .
XT 25 Ol 6=; 2
and therefore
b = 2a.
The instrument may now be applied to the determination of the
characteristics of elliptically polarised light.
ART. 240 ELLIPTICALLY POLARISED LIGHT 445
Determination of the Phase Difference.—If the incident light be ellip-
tically polarised we may resolve the vibration at each point into two
components parallel to the axes of the quartz wedges. These com-
ponents will differ in phase and amplitude, and are of the general form
a = A cos (wt + a), y= B cos (wt +8).
The effect of transmission through the plate is to change the phase
difference a - B by an amount
6 = *(e – e')(ue - po),
which will vary from point to point. It is clear then that we will
have a system of parallel lines for which the whole phase difference
a – 8 + 8 will be equal to multiples of T, and the transmitted light
along these lines will be plane-polarised, consequently when the com-
pensator is viewed through a Nicol, properly placed, a system of bright
and dark bands will be exhibited in the field, and under the central
band the total phase difference will be zero.
Now let us suppose that the apparatus is so adjusted that with
plane-polarised light the central band is under the fibre. Then, when
the incident light is elliptically polarised, the central band will not be
under the fibre but will be displaced across the field. At this band
the original phase difference existing between the component vibrations
has been neutralised by that introduced by the plate. This phase
difference is consequently determined by turning the micrometer screw
till the central dark band is brought under the wire. If the displace-
ment be a we have therefore
**=} or a -8-1.
Position of the Aales.—The azimuths of the axes of the elliptic
vibration may also be determined by means of the compensator. We
know that the phase difference of the component vibrations taken
along the axes is 90° (see Art. 47). Hence set the compensator so
that the cross wire is over the central line N–that is, over the central
dark band when the incident light is plane-polarised—and turn the
screw by an amount #b, so that the line under the fibre will correspond
to a retardation of #A or a phase difference of 90°. Now allow the
elliptically polarised light to fall on the compensator, and if the central
black band be not under the fibre turn the compensator round the
direction of the incident light, and so cause the fringes to move across
the field, till the central band is under the fibre. In this position
446 STUDY OF POLARISED LIGHT CHAP. XVI
the axes of the elliptic vibration are parallel to those of the quartz
wedges.
Ratio of the Aaces.—When two rectangular vibrations compound
into another rectilinear vibration, the tangent of the angle which the
direction of the resultant makes with one of the components is measured
by the ratio of the amplitudes of the components (Art. 47, Cor. 1).
Now an elliptic vibration is divided into two components parallel to
the axes of the compensator, and at the black bands their phase differ-
ence is a multiple of T so that they compound into a rectilinear vibra-
tion, which, since it is extinguished, must be perpendicular to the
principal plane of the analyser; consequently the principal plane of
the analyser makes with one of the principal sections of the compen-
sator an angle the tangent of which is equal to the ratio of the amplitudes
of the components of the elliptic vibration. Hence if the compensator
be set so that its axes are parallel to the axes of the elliptic vibration,
then the tangent of the angle between one of its principal planes and
the principal plane of the analyser will be equal to the ratio of the
axes of the elliptic vibration.
241. Applieation of the Quarter-wave Plate.—A simple but
much less precise method of analysing elliptically polarised light is to
reduce it to plane-polarised light by transmission through a quarter-
wave plate. The property of such a plate is to introduce a phase
difference of a quarter period (90°) between the component vibrations
which are transmitted through it. Hence if the light be received
normally on the plate, placed so that the azimuths of vibration in it
are parallel to the axes of the ellipse, the component vibrations in the
transmitted light will differ in phase by 180°, and it will therefore be
plane-polarised, and capable of being extinguished by a Nicol's prism.
Hence the plate and Nicol are adjusted by trial till the light is
extinguished, and in this position we know that the principal sections
of the plate are parallel to the axes of the elliptic vibration, and further,
the principal plane of the analyser being perpendicular to the azimuth
of vibration of the plane-polarised light emerging from the plate will
make an angle with one of the principal sections of the plate, the
tangent of which is equal to the ratio of the axes of the elliptic
vibration.
Since the retardation in passing through a plate depends on the
wave length, it follows that a quarter-wave plate will only act as such
for some particular wave length. Fresnel's rhomb, on the other hand,
is almost free from this objection, and introduces a phase difference
of 90° very approximately for all values of A. It is consequently
better adapted for work with white light. It suffers from the defect
ART. 242 STRAIN COMPENSATOR 447
that it displaces the beam sideways so that the analyser requires to
be similarly displaced. This is overcome in a modification of the
rhomb devised by A. E. Oxley." * *
242. The Strain Compensator.—A slip of glass when stretched
(or compressed) acquires the doubly refracting properties of a uniaxal
crystal of which the optic axis is parallel to the direction of stretching.
When interposed in the path of a pencil of plane-polarised light the
glass plate will therefore in general convert it into elliptically polarised
light, and conversely it may, with proper stretching, reduce to zero, or
T, the phase difference already existing in an elliptic vibration, and so
convert it into plane-polarised light. In this application it was em-
ployed by Dr. Kerr’ in his celebrated experiments on the reflection of
plane-polarised light from the pole of a magnet (Art. 259).
Methods of producing and identifying Polarised Light
243. General Method of Detection.—It may be useful here to re-
capitulate the methods by which the various kinds of polarised light may
be obtained, and to point out the special characteristic qualities by which
they may be distinguished from each other, and from common light.
Polarisation of any kind may be detected by means of the colours
produced by transmission through a thin crystalline plate and analyser,
and a knowledge of the character of the fringes afforded by the
different kinds of polarised light would enable us by this method alone,
not only to detect the presence of the polarisation, but also to determine
its class. Each kind may, however, be studied separately as follows.
244. Plane Polarisation.—Plane-polarised light was first obtained
by double refraction and afterwards by reflection from glass. In all
cases of double refraction the refracted pencils are both plane-polarised,
and their planes of polarisation are at right angles. An obvious method
of obtaining a beam of plane-polarised light is to transmit ordinary
light through a doubly refracting crystal and to intercept one of the
refracted pencils. This may be done in three ways. If the crystal be
large, so as to afford considerable separation of the refracted rays, one
of them may be stopped by an opaque diaphragm, while the other is
allowed to pass for use or examination. If the separation of the rays
be small, so as not to permit of the use of a diaphragm, one of the
pencils may be absorbed and the other transmitted. This happens
in some natural crystals, such as tourmaline, which possesses a high
coefficient of absorption for the ordinary ray, but freely transmits the
* Oxley, Brit. Assoc. Report, xxi. p. 517, and Phil. Mag. (1910).
* Kerr, Phil. Mag., November 1875.
Methods of
producing.
448 STUDY OF POLARISED LIGHT CHAP. XVI
extraordinary. Finally we possess the method commonly used in
practice, viz. that of intercepting one of the beams by total reflection
while the other is transmitted, and on this principle such instruments
as Nicol's and Foucault's prisms are constructed.
The second method of obtaining plane-polarised light, viz. by
reflection at the polarising angle from some transparent substance,
such as glass, is very convenient, requiring no special apparatus, but it
is less perfect than the method of double refraction, for it is found by
experiment that very few substances completely plane-polarise light by
reflection, and that these have a refractive index of about 1'46.
It is to be remembered that any instrument which acts the part of
a polariser may also be used as an analyser to examine the properties
Methods of of a polarised beam. Examined by any analyser, such as Nicol's prism,
detecting.
Produced.
the characteristic of plane-polarised light is that it may be completely
extinguished by the analyser placed in a certain azimuth, viz. Such
that the principal plane of the analyser is parallel to the plane of
polarisation of the incident light. Examined by transmission through
a doubly refracting rhomb, we obtain in general two refracted pencils
of different intensities, and in two positions of the rhomb one of the
pencils vanishes so that the incident light is transmitted in a single
beam. We consequently possess the means of detecting it, and dis-
tinguishing it from all other kinds of polarised light, and from
ordinary light.
245. Circular Polarisation.—A circular vibration arises when
two rectangular plane-polarised vibrations of the same period are
compounded, which differ in phase by 90° and are of equal amplitude.
The equations of such a vibration may be written in the form
a = a cos w!, y=a sin wt
(see Fig. 21), and the vibration is said to be left-handed or counter-
clockwise. Or the equations may be
a:= a cos (ot, y = — a sin wt,
and the vibration is said to be right-handed or clockwise.
Circular polarisation is therefore produced when a plane-polarised
ray is divided into two others, vibrating at right angles, equal in
amplitude and differing in phase by 90°. This is obtained by trans-
mitting plane-polarised light through a quarter-wave plate, the axes of
the plate being inclined at 45° to the plane of polarisation of the inci-
dent light. The same effect is produced with much less colouring by
transmission through Fresnel's rhomb, the plane of reflection being
inclined at 45° to the plane of polarisation.
When circularly polarised light is examined through a Nicol the
ART, 246 ELLIPTIC POLARISATION 449
illumination of the field is independent of the orientation of the Nicol.
In this respect it resembles ordinary light, but the two may be easily Detected.
distinguished. For if in the path of a circularly polarised beam we
interpose a quarter-wave plate or a Fresnel's rhomb, a further quarter-
period difference of phase will be introduced between the component
vibrations, and this will either add to or destroy the original phase
difference. We have consequently emerging from the quarter-wave
plate two rectangular vibrations, of which the phase difference is either
zero or T, and these compound into a rectilinear vibration, so that the
light is plane-polarised. This may be completely extinguished by a
Nicol's prism, and we have the means of deciding whether the original
light was circularly polarised, or merely a beam of ordinary light; for
the latter on transmission through the quarter-wave plate will still
retain the character of ordinary light and will never be quenched by
the Nicol.
The quarter-wave plate and the Fresnel's rhomb consequently not
only act as producers of circular polarisation, but also assist in detect.
ing it where it already exists.
246. Elliptic Polarisation.—The rectangular components of an
elliptic vibration differ both in phase and amplitude. The equations
of the vibration may be written in the general form
a:= a cos wit, y= b cos (wt + 3),
in which case the motion in the elliptical path is right-handed, provided
b is positive. This follows from the fact that both y and a diminish
between of + – 8 and ot = T – 8 while in the rest of the path they
both increase algebraically. If b is negative the motion is left-
handed or counter-clockwise. An elliptically polarised ray may there-
fore be regarded as the resultant of two plane-polarised rays vibrating
at right angles, and differing both in phase and amplitude. It con-
sequently includes under it, as particular cases, both plane and circular
polarisation.
It is clear, then, that any process which splits a beam into two plane- Produced.
polarised beams, polarised at right angles, and differing in phase and
amplitude, will in general afford elliptically polarised light. This is
effected by transmitting plane-polarised light through a crystalline plate,
and in general by reflection from metallic or crystalline surfaces.
When angular separation takes place between the two components, as
is especially the case when the incidence is oblique, it must be borne in
mind that interference can only take place at the point of divergence,
real or virtual, of these components.
Elliptically polarised light when examined through a Nicol will Detected.
2 G
450 STUDY OF POLARISED LIGHT CHAP. XVI
give an illumination which will vary as the Nicol is rotated, being
greatest when the principal plane of the analyser is parallel to one
axis of the ellipse and least when parallel to the other. It is thus
distinguished from ordinary light, but the same property is possessed
by a mixture of ordinary and plane-polarised light. It therefore
remains to distinguish between elliptic and partial polarisation. This
is effected by the help of a quarter-wave plate or a Fresnel's rhomb
properly placed. Thus if the light be transmitted through a quarter-
wave plate, and if the axes of the plate be placed parallel to those of
the elliptic vibration, the phase difference of the components will
be reduced to zero, or increased to T, and the transmitted light will
be plane-polarised. This may be extinguished by a Nicol, and
the elliptic polarisation is completely distinguished. The further
study of the elliptic vibration is effected by means of Babinet's
compensator.
247. Partial Polarisation.—The term partially polarised light is
generally applied to a mixture of ordinary and plane-polarised light.
When examined through a Nicol the illumination of the field will vary
as the Nicol is rotated, and this arises from the suppression of the
plane-polarised component of the light. At first sight it might there-
fore be suspected that such a beam possessed elliptic polarisation, but
as already indicated, a quarter-wave plate assists us to distinguish
between the two.
248. Common Light.—It now becomes a matter of interest and
importance to inquire into the nature of the vibration in common or
unpolarised light. The characteristic of common light is that, on
transmission through a doubly refracting rhomb, it affords two pencils
of light which do not change in intensity as the rhomb is rotated
round the direction of the incident pencil. Circularly polarised light
also possesses the same quality, but is, in addition, capable of afford-
ing coloured fringes when transmitted through a thin crystalline plate
and subsequently analysed. Plane and elliptically polarised light also
afford such interference fringes, but in the case of common light they
are wholly absent.
When transmitted through a thin crystalline plate ordinary light
gives two rays polarised at right angles, and each of these gives an
image in the analyser ; but the two images are complementary in
intensity and superposed. The result is that when the analyser is
rotated the illumination remains uniform and equal to half that of the
incident light. Hence if the vibration of common light be elliptic,
circular, or rectilinear, the nature of the vibration cannot remain per-
manent, for the light would then possess the qualities of polarised
ART. 249 IMITATION OF NATURAL LIGHT 451
light and exhibit fringes when transmitted through a thin crystalline
plate.
This characteristic quality of common light has led to the supposi-
tion that in it the vibration frequently changes its character, or that
it consists of successive series of elliptic vibrations (including circular
and plane as particular forms), all the vibrations of any series being
similar to each other, but those of one series having no necessary
similarity in azimuth or form to those of any other. We know that
many thousand interference bands may be presented when common
light is reflected from Fresnel's mirrors (Art. 100), and that a limit to
the observable number of these bands is imposed by the limited
resolving power of our spectroscopes. Now interference can only be
expected between two sources when they emit similar vibrations. Con-
sequently if common light affords many thousand fringes the vibrations
emitted by the source must remain similar for as many periods at least
(see p. 182).
Hence if we suppose the vibration to change suddenly at fre-
quent intervals, the vibrations of any series between two consecutive
changes must be similar to each other, and there must be many
thousand vibrations in each series, say a quarter of a million. Now the
orange light of the spectrum executes about 500 billion—500 x 10%—
vibrations per second, consequently the form of the vibration might
change 500 times per second and still have a billion similar vibrations
in each set. This would permit of the exhibition of millions of inter-
ference bands (a number quite beyond our power of observation), and
yet would sufficiently account for the absence of fringes by trans-
mission through a thin plate. For if we have a number of similar
vibrations in any azimuth, these will give all the phenomena of inter-
ference rings and colours as long as the vibrations remain similar, and
if the vibration changes to the perpendicular azimuth we will again
have coloured fringes, but this time of the complementary character.
Hence if the azimuth or nature of the vibration changes many times
per second the several systems will be produced in such rapid succes-
sion that only their combined effect will be perceived, and the field
will appear uniformly illuminated.
249. Imitation of Natural Light by the Rotation of Polarised
Light.—If ordinary light be transmitted through a Nicol's prism the
emergent light will be plane-polarised, and the vibration will take
place in a certain definite azimuth determined by the position of the
principal plane of the prism. If the Nicol be now rotated round an
axis parallel to the direction of the incident light, the plane of polarisa-
tion will rotate at the same rate, and the transmitted light when
452 STUDY OF POLARISED LIGHT CHAP, XVI
doubly refracted will possess, like ordinary light, the property of
giving two beams of constant intensity, and uniform illumination will
be produced instead of colours and rings, when transmitted through a
thin crystalline plate. It would be possible, however, to distinguish
between a rapidly rotating plane-polarised pencil and ordinary light.
For if the rotating beam be examined through a Nicol which is
rotated round an axis parallel to the direction of the incident light at
a rate equal to that of the polarised beam, then if the principal plane
of the analyser be parallel to the plane of polarisation of the ray
in one position it will remain so permanently, and the light will be
completely quenched.
If a thin plate of mica be attached to the polariser the light emerg-
ing from the mica plate will in general be elliptically polarised, and
the combined rotation of the Nicol and plate will give an elliptically
polarised beam, of which the vibrations are similar but rotating rapidly,
so that the axes of the ellipses are directed in all azimuths. Such a
beam was found to produce the same polarisation phenomena (such
as coloured rings, etc.) as circularly polarised light, but if the Nicol
be kept fixed and the mica plate rotated, so that the sense of the
elliptic vibration is reversed every half-revolution, the transmitted
pencil behaved like common light. -
It has been shown (Art. 48, Ex. 2) that a plane-polarised ray may
be considered as the resultant of two opposite circularly polarised rays
Rotating of the same period. If the circular vibrations be of different periods
the direction of the resultant vibration will rotate uniformly, and there-
fore a rotating plane-polarised ray, such as we have considered above,
may be regarded as the resultant of two opposite circularly polarised
rays of different periods. This difference in period means a difference
in wave length and refrangibility, and Airy” has consequently remarked
that a gradual change in the vibration, such as the uniform rotation
of its azimuth, is not an admissible representation of the nature of
common light, for we may regard a uniformly rotating vibration as
the equivalent of two others of different periods, and it should give
rise to two polarised rays of different colours. The velocity of rotation
"required to produce any such separation would, however, be far in
excess of anything we could possibly attain.
250. Haidinger's Brushes.—It was discovered by Haidinger that
plane-polarised light may be detected by the naked eye, and its plane
of polarisation may also be ascertained. The phenomenon observed is
the appearance of a pale yellow patch or brush bounded on either side
ray.
* Dove, Pogg. Amm. vol. lxxi. p. 97. See Verdet, CEuvres, tom. vi. p. 88,
* Airy, Undulatory Theory of Optics, art. 183.
ART. 250 HAIDINGER'S BRUSHES 453
by curved arcs in form like the branches of a hyperbola. The axis of
this brush lies in the plane of polarisation, and on each side of its neck
there is a violet or bluish patch.
These brushes are supposed to arise from the polarising structure of
the eye itself, and by some persons they are observed even when the light
is only partially polarised. Most persons, however, can see them only
when the light is completely polarised, and even then with difficulty.
They are best observed by looking at a bright cloud through a Nicol's
prism, the Nicol being revolved round its axis so as to alter the plane
of polarisation. They gradually disappear when the eye is directed
towards them in the same position, being visible for only a couple of
seconds after the polarised light is first received by the eye; but when
the position of the plane of polarisation is changed by rotating the
Nicol the brushes continue in the field and are seen to revolve with
the Nicol. They may be seen by looking for a few moments at one of
the images afforded by a rhomb of Iceland spar and then at the other,
and so on alternately, or by looking at the sky, alternately towards the
horizon and the zenith, in a plane at right angles to the line joining the
observer to the sun when the sun is near the horizon.
M. Jamin attributes this phenomenon to the refracting coats of
the eye. Thus if plane-polarised light be transmitted through a pile
of plates the intensity of the transmitted beam will depend on the
relation of the plane of incidence to the plane of polarisation. Now
the crystalline lens of the eye consists of curved layers of unequal
refracting powers, and, on account of the curvature, the relation of the
plane of polarisation of a given pencil of polarised light to the plane
of incidence at each point will vary from point to point, with the
result that the intensity of the light transmitted will vary from point
to point on the retina, and the general appearance of the brushes would
be accounted for. Helmholtz," however, took objection to this explana-
tion, and found that in working with homogeneous light” the blue was
the only colour with which the brushes were visible, and that the extent
of the brushes was limited to the yellow spot of the eye. To account
for the brushes it therefore suffices to suppose that the yellow spot
is doubly refracting to a slight extent, and that it absorbs the
* Physiological Optics.
* The office of the different colours in the production of these brushes was first
investigated by Sir G. G. Stokes, who found that in the red and yellow light of the
spectrum no trace of them could be observed. The brushes began to be visible in
the green, were more distinct on passing into the blue, were particularly strong
about the line F, could be traced as far as the line G, and when they were no longer
visible the cause appeared to be merely the feebleness of the light, not the in-
capacity of the greater part of the violet to produce them (Stokes, Brit. Assoc.
Report, 1850; Collected Papers, vol. ii. p. 362).
454 STUDY OF POLARISED LIGHT CHAP. XVI
extraordinary ray more strongly than the ordinary in the case of
blue light.
251. The Polarisation of Skylight.—It has been already noticed
that, in general, polarisation is produced to some extent by reflection
and refraction, and it is consequently not surprising that the light of
the rainbow, and even the light of the open sky, should be more or
less polarised. The polarisation of the blue light of the sky was
noticed by Arago as early as 1811. When the open sky is viewed
through a Nicol's prism variations of brightness are observed as the
Nicol is rotated on its axis. This change of brightness is most notice-
able when the sky is viewed in a direction at right angles to the
rays of the sun, and observation proves that the light of the sky is
polarised, and that the direction of most perfect polarisation is at right
angles to the direction of propagation of the solar rays. This is some-
times expressed by saying that the place of maximum polarisation in
the sky occurs at an angular distance of 90° from the sun."
The blue light of the sky has been imitated, and artificial skies
produced in a most elegant manner by the late Professor Tyndall. If
the peculiarities of skylight are due to the scattering action of very
Small particles suspended in the air, the object of a test experiment
must be to procure a region of space in which very fine particles are
known to be suspended and to allow a beam of light to play upon it.
This was secured by passing a pencil of light through a tube contain-
ing a small quantity of vapour of iodide of allyl.” By the action of
the light the vapour is gradually decomposed, and a very fine cloud
forms within the tube. At first the precipitated particles are exceed-
ingly small, and the colour observed is a delicate blue, but as the
experiment progresses the particles gradually increase in size, and the
blue gradually brightens, “still maintaining its blueness, until at
length a whitish tinge mingles with the pure azure, announcing that
the particles are now no longer of that infinitesimal size which scatters
only the shortest waves.” ” Further, when the incipient cloud was
viewed transversely through a Nicol's prism, the scattered light was
found to be completely polarised in the plane passing through the axis
of the tube and the eye of the observer—that is, the plane containing
the incident and scattered ray—just as the blue light of the sky is
* In the vertical plane through the sun neutral points have been observed by
Arago, Babinet (Comptes rendus, tom. xi. p. 618, 1840), and Brewster (Brit. Assoc.
Report, 1842, part ii. p. 13). At these points there is no polarisation, and in the
intervals between them the light is polarised in rectangular planes.
* Several other substances will also produce the desired effect, as nitrite of amyl,
bisulphide of carbon, benzole, benzoic ether, etc.
* Tyndall, Heat a Mode of Motion, p. 491.
ART. 251 POLARISATION OF SKYLIGHT 455
polarised in a plane passing through the solar rays and the eye of the
observer.
That the preponderance of blue in the light of the sky is due to
the scattering action of very small particles suspended in the air seems
to be placed beyond doubt by these experiments. Now we have seen
that this preponderance becomes at once intelligible when we consider
the action of particles of linear dimensions small compared with the
wave length of light (Art. 153), and Lord Rayleigh" has further
shown that the polarisation of the scattered light may be explained
by supposing that the particles load the ether so as virtually to Loaded
increase its inertia, or that property of it which corresponds to inertia”
(Art. 308).
This loading action of the particles might be counterbalanced by
supposing suitable forces to act at all points of the ether where the
inertia is altered. If these forces were applied the loading would be
neutralised, and the waves would pass unbroken just as if the particles
were not present, so that there would be no scattering of the light.
These applied forces must have the same period and direction as the
undisturbed luminous vibrations, and the light actually scattered is
the same as would be caused by the action of forces exactly the
opposite of the foregoing acting on the medium otherwise free.
Now on account of the smallness of the particles we may take the
forces acting throughout the volume of any one of them as a whole.
The periodic disturbance arising from the action of this force will be
symmetrical round the direction of the force, and the transverse vibra-
tions which it originates will be such that the azimuth of vibration in
any ray lies in the plane containing the ray and the axis of symmetry;
in other words, the azimuth of vibration in the diffracted ray makes
the least possible angle with the azimuth of vibration in the incident
ray. Further, the intensity of the scattered light should vanish in the
direction of the axis of symmetry, for the transversal to a ray propa-
gated in this direction becomes indeterminate. This being conceded,
let us suppose that a solar ray is regarded transversely, so that the
line of vision is at right angles to the ray. The vibration in the ray
may in this case be divided into two components, one along the line
of vision and the other at right angles to it, both of these components
being in the wave front. Now the component along the line of vision
will give rise to no scattered light in this direction, for this is its axis
of symmetry, and consequently the scattered light received by the eye
arises entirely from the component at right angles to the line of vision.
It is therefore completely polarised, and the plane at right angles to
* Hon. J. W. Strutt, Phil. Mag. vol. xli. p. 107, 1871.
456 STUDY OF POLARISED LIGHT CHAP, XVI
the azimuth of vibration—that is, the plane of polarisation in Fresnel's
theory—contains the line of vision and the incident ray.
When the angle between the line of vision and the incident ray is
other than 90° the scattered light is a mixture arising from both the
foregoing components, and the polarisation becomes less and less com-
plete as the line of vision approaches the direction of the ray, and in
this direction it altogether disappears.
CHA PTER XVII
ROTATORY POLARISATION
252. Rotation of the Plane of Polarisation—Arago’s Discovery.
—When plane-polarised light is reflected at the surface of a transparent
substance, the planes of polarisation of the reflected and refracted
pencils do not in general coincide with that of the incident light (Art.
211). This change of plane of vibration is an abrupt one occurring
in the act of reflection or refraction, and is due to the two components
being unequally reflected (or refracted), so that the resultant vibra-
tion is differently orientated. The change of plane is never as great
as 90°. It must not be confounded with another phenomenon discovered
by Arago, which will now be described.
In 1811 Arago discovered that a continuous rotation of the
plane of polarisation occurs when plane-polarised light is transmitted
through quartz in the direction of its optic axis, and this property is
also possessed by many other substances. The rotation is here due
to the action of the medium through which the light passes, and its
amount is directly proportional to the thickness traversed. When
plane-polarised light passes through a uniaxal crystal, such as Iceland
spar, in the direction of the axis, the plane of polarisation of the
transmitted pencil coincides with that of the incident light, but it is
different in the case of rock-crystal (quartz). Here the plane of Right and
polarisation at emergence is not the same as at incidence, but is".
inclined to it at an angle depending on the thickness of the quartz
plate. Further, some specimens of quartz rotate the plane of polarisa-
tion to the right (as seen by an observer receiving the light) and some
to the left. The former are called right-handed or dextrogyrate, and
the latter left-handed or levogyrate. It is found, further, that if the
slice of quartz (e.g.) is reversed (back to front) it still rotates the
plane in the same sense as before ; in this it is analogous to a screw ;
* Mém. de la première Classe de l'Institut, tom. xii. p. 93. CEuvres complètes, tom.
x. p. 35, 1812.
457
458 ROTATORY POLARISATION CHAP, XVII
a right-handed screw is right-handed from whichever end it is
observed.
253. Biot's Laws.—This remarkable phenomenon was investigated
with great care and success by Biot," who deduced the following laws:
(1) The amount of rotation is proportional to the thickness tra-
versed by the ray. -
(2) The rotation effected by two plates is the algebraic sum of the
rotations produced by each separately.
(3) The rotation augments with the refrangibility of the light, and
is approximately proportional to the inverse square of the wave length.
In support of the third law, M. Broch gives the following numbers
for quartz. The first line gives the rotation p produced by a quartz
plate (one millimetre thick) on the different rays, and the second line
gives the product of p by the square of the corresponding wave
length :
Ray B C D E F. G
p 15° 30' 17°24' | 21° 67’ 27° 46' | 32° 50' | 42° 20'
pX” 7238 7429 751.1 7596 7622 7841
For essence of turpentine Wiedemann found
10° 9' 14° 5' | 18° 7' 23° 2' | 32° 75'
4690 4871 5184 54.71 6044
º
These tables show that the rotation is not strictly in the inverse
ratio of the square of the wave length, but that the product p}\” in-
creases with the refrangibility of the light. Biot's Law is accordingly
only an approximation to the truth (see Art. 338).
254. Rotation produced by Liquids and Vapours.--It was soon
discovered that many liquids and solutions possess, like quartz, the
power of rotating the plane of polarisation of a ray transmitted
through them, but in a much less degree. Thus a plate of quartz
1 mm, thick rotates the plane of polarisation of the red rays about
18°, while an equal thickness of turpentine only produces a rotation
of a quarter of a degree. -
The rotatory power of liquids and vapours is studied by enclosing
them in a tube through which the polarised light is transmitted. It
is found that liquids and solutions do not lose their rotatory power
when diluted with other liquids not possessing this property, and they
retain it even when in the state of vapour. Biot consequently con-
cluded that the power possessed by liquids of rotating the plane of
* Biot, Mém. de la première Classe de l’Institut, tom. xiii. p. 218, 1813; Amm.
de Chimie et de Physique, 1815, etc. This definition of right-handed rotation is that
of Biot, which is now universally adopted. The opposite convention introduced by
Herschel would be quite appropriate from the physical standpoint, but it is not so
convenient in practice.
ART. 254 ROTATION PRODUCED BY LIQUIDS AND WAPOURS 459
polarisation is inherent in the ultimate molecule. In this respect they
differ from quartz, which loses its rotatory power when it loses its
crystalline arrangement. Thus Sir J. Herschel found that quartz held
in solution by potash (liquor of flints) was destitute of rotatory power,
and the same was observed by Sir D. Brewster with respect to fused
quartz.
It is, perhaps, not surprising that crystalline substances should, on
account of some special molecular arrangement, possess rotatory power
and affect the propagation of light within the mass in a manner
depending on the direction of transmission. The loss of this power
when the crystalline structure is destroyed, as when quartz is fused,
is consequently an event which would be naturally expected, but the
possession of it in all directions by fluids and solutions, in which there
cannot be any special internal arrangement of the mass of the nature
of a crystalline structure, is not a thing which one would have been
led to expect beforehand. To Faraday it appeared to be a matter of
no ordinary difficulty. It must be pointed out that a large number of
slices of quartz, cut perpendicularly to their axes and suspended in a
fluid (of about the same index, so as to diminish reflections), and with
their axes directed at random, should rotate the plane of polarisation.
This follows from the fact (Art. 252) that reversal of the axis does not
reverse the sign of rotation. It is just possible that the light in
traversing a solution in which the molecules are free to move may, on
account of some peculiarity of structure, cause the molecules to take
up some special arrangement, so that the fluid becomes, as it were,
polarised by the transmission of the light, in a manner somewhat
analogous to that in which a fluid dielectric is polarised in a field
of electrostatic force. Experiments made to test this point on a
column of sugar 165 cms. long were made by Paris and Porter," using
(a) a steady source of light; (b) a very short-lived spark. The
difference in the mean values for the two cases did not amount to
more than one part in 13,000. The duration of the spark was not
more than 3 × 107" seconds. If there was any alteration of position
of the molecules, it must therefore have taken place in less than this
short interval.
When two or more liquids are mixed together their combined
rotation is often equal to the algebraic sum of the rotations which
the constituents would produce separately when taken in thicknesses
proportional to the volumes they occupy in the mixture. This pro-
perty has been applied to the analysis of compounds containing a
substance which possesses rotatory power and is combined with others
* Paris and Porter, Phil. Mag., Jan. 1914.
460 ROTATORY POLARISATION CHAP. XVII
which are neutral. This important application has been found of
much industrial value in the analysis of saccharine solutions, and for
this purpose instruments termed Saccharimeters have been invented
(Art. 273, etc.).
- Calling right-handed rotations positive and left-handed negative,
the rotations produced by a thickness of one decimetre of each of the
following substances, in the case of red light, are 4
Essence of Turpentine – 29°-6 Essence of Seville Orange + 78°-94
5 3 Citron + 55°-3 3 3 Mint – 16°-14
y 3 Aniseed — 0°-7 1, Sassafras + 3°-53
2 3 Lavender + 2**02 $ 3 Rosemary + 3°:29
3 3 Fennel + 13°-16 7 5 Carraway + 65°.79
Solution of Sugar (cane), 50 % in water + 33°64
5 5 Quinine 6 % in alcohol – 30°
5 § Strychnine 14% in alcohol — 6°-6
5 3 Brucine . 5 % in alcohol — 12° to - 16°
* 3 Morphine 5 % in soda – 11°-5
For yellow light (D) and one millimetre thickness we have
Quartz 21°-67 Hyposulphate of Potash 8° 39'
Cinnabar 3.25° 3 y Lead 5° 53'
Chlorate of Soda 3°-67 3 3 Strontium 1° 64'
Bromate , , 2°'80 5 y Calcium 2° 9’
255. Molecular Rotatory Power.—When a substance possessing
rotatory power is held in solution by a liquid which is inactive, it is
found that the rotation of the plane of polarisation is proportional to
the quantity of the active substance traversed by the ray; that is, the
rotation is proportional to the number of molecules of the active sub-
stance per unit volume and to the length of the path of the ray.
This has given rise to the opinion that the rotatory power is inherent
in the ultimate molecule, and hence the term molecular rotatory power.
If unit weight of the solution contains a weight w of an active
substance, and a weight 1 – w of the inactive solvent, and if the
specific gravity of the solution be or, the volume of unit weight will be
v = 1/q, and the density of the active substance in the solution will be
a: = += wo.
Q)
The rotation produced by a thickness e will therefore be
p = Mewa *
where M is a constant which measures the rotation produced by unit
thickness of a solution containing the active substance in unit density.
The constant M is called the molecular rotatory power of the Substance.
If two active substances be mixed, their weights being in the ratio
* See Werdet, CEuvres, tom. vi. p. 270.
ART. 256 DISPERSION OF THE PLANES OF POLARISATION 461
w: (1 - w), the rotation produced by the second will be
p' = M'e(1 — wya.
where M' is its molecular constant. Hence their combined rotation will be
... p-Ep’ - e.g. {Mw:EM'(1–w)}.
The quantity M is found to be not absolutely constant. In general
it augments a little as the proportion of the solvent is increased, and
in the case of the organic alkaloids and their salts marked deviations
are exhibited.
Temperature Effect.—The effect of a change of temperature is gener-
ally to change the molecular rotatory power. In some substances, such
as essence of oranges and essence of turpentine, the rotatory power is
diminished by a rise of temperature, while in others, such as quartz
and chlorate of sodium, it increases with the temperature.
M. Gernez' expresses the molecular rotatory power in the form
M = a – bf – ct”
where a, b, c are constants for any given wave length. Thus for the
essence of oranges and essence of turpentine M. Gernez finds for the Dray
Essence of orange M = 115-91 – 0-1237t- 0:000016:2,
,, turpentine M- 36-61 – 0-004437t.
(>
In the case of quartz the molecular rotatory power may be repre-
sented between — 20° and 100° by the formula
Quartz M-Mo(I+0-000146324t +0.0000000329t’)
where Mo-21-658.
256. Dispersion of the Planes of Polarisation—Tint of Passage.
—Since by Biot's third law the rotation is different for the different
wave lengths, it follows that when white light is used the planes of
polarisation of the various constituent colours in the transmitted pencil
will have suffered different amounts of rotation, and consequently, if
this light be examined by means of a Nicol's prism, only one of its
constituents will be quenched, viz. that colour for which the plane of
polarisation is parallel to the principal plane of the Nicol. The other
colours will be more or less transmitted according to the position of
their planes of polarisation. The field will consequently appear
coloured and the colour will vary as the Nicol is rotated. If the thick-
ness of the quartz plate does not exceed 5 mm., then for a certain
position of the Nicol the field possesses a greyish-violet colour, called
the tint of passage (named by Biot the teinte Sensible). If the Nicol
be slightly turned from this position the field becomes red, and
for a rotation in the opposite direction it appears blue. On passing
* M. Gernez, Ann. de l'École Normale, tom. i. p. 1.
462 ROTATORY POLARISATION CHAP. XVII
through this position the transition from red to blue is very rapid.
It is accordingly a position attainable with considerable accuracy.
When the tint of passage has been obtained for a quartz plate, the
principal plane of the Nicol is parallel to the plane of polarisation of
the greenish-yellow rays, and these rays are consetluently quenched by
the Nicol and absent from the emergent light. This point may be
tested by submitting the transmitted light to spectroscopic analysis.
The spectrum will be found to be crossed by a dark band correspond-
ing to the missing rays.
When the analyser is placed so as to produce the tint of passage,
the illumination of the field is a minimum, for the rays corresponding
to the brightest part of the spectrum are absent and the illumination
of the field is produced by the less intense parts, viz. the red and violet
ends of the spectrum. A general expression for the intensity corre-
sponding to any position of the analyser is easily obtained. Thus if
the intensities of the various colours in the incident light be I, I, I,
etc., for the red, orange, yellow, etc., and if the primitive plane of
polarisation of the light make an angle a with the principal plane of
the Nicol, then after passing through the quartz they will make angles
a + ar, a + ao, etc., where ar, ao, etc., are the angles through which the
" planes of polarisation of the corresponding colours (red, orange, etc.)
are rotated by the quartz. Hence the intensity of the red in the field
of the analyser will be I, sin” (a + ar) with corresponding expressions
(see Art. 174) for the other colours. The complete expression for the
illumination will consequently be
I = I, sin” (a + ar) + Io sin” (a + ao) + . . . 4- I, sin” (a + ay).
257. Relation of Rotatory Power to Crystalline Form.—The
curious fact that some specimens of quartz
rotate the plane of polarisation to the
right, while others rotate it to the left,
was found by Sir J. Herschel to be in-
timately connected with a difference of
crystalline form. The ordinary form of a
quartz crystal is a six-sided prism topped
by a six-sided pyramid. The solid angles
at the junction of the prism and pyramid
are often absent, and replaced by facets,
or small secondary planes, which are ob-
Fig. 192—Right-handed Crystal liquely inclined to the other faces of
of Quartz. º
the crystal. The figure shows a right-
handed quartz crystal : the left-handed variety is like the image of
this one as seen in a vertical mirror. The two varieties are said to

ART. 258 THE FARADAY EFFECT , 463
be enantiomorphous." In the right-handed variety the facets lean
over to the right, as indicated by the dotted lines.
258. Rotation produced in a Magnetic Field—The Faraday Effect.
—In 1845 Faraday” made the remarkable discovery that isotropic
substances, more especially those possessing a high refractive power,
such as heavy glass, may also acquire the property of rotating the
plane of polarisation when under the influence of powerful magnetic
force. In Faraday's experiment the soft iron pole-pieces of an electro-
magnet were each pierced with a cylindrical hole, and a pencil of
plane-polarised light was transmitted through them, the pole-pieces
being placed so that the axes of the holes were in the same straight
line. The direction of this line and the direction of the transmitted
pencil of light were consequently parallel to the lines of magnetic force
when the magnet was excited. Between the poles, and in the path of
the light, a block of dense glass (silicated borate of lead) was placed,
and the light was received by an analyser, adjusted so that the field
was dark.” On allowing a current to pass in the coils of the magnet,
a powerful magnetic field was produced and the light reappeared in
the analyser, but it could be again extinguished by properly turning
the analyser. This showed that the pencil emerging from the glass
was still plane-polarised, but that the plane of polarisation was rotated
by the influence of the magnetic field. When the current was
reversed the direction of magnetisation was changed, and it was found
that the direction of rotation of the plane of polarisation was also
reversed.
The direction in which the rotation of the plane of polarisation takes
place is determined by a rule similar to Ampère's law connecting the
direction of the lines of magnetic force encircling an electric current with
* Sir David Brewster subsequently discovered that the amethyst, or violet quartz,
is made up of alternate layers of right-handed and left-handed quartz. This remark-
able structure may be traced in the fracture of the mineral, for the edges of the
layers crop out, and give to the fracture the undulating appearance which is peculiar
to the mineral. But the structure in question is displayed in the most beautiful
manner when we expose a plate of this substance to polarised light. The colours
exhibited in polarised light likewise reveal the existence of crystals of quartz pene-
trating others in various directions, when no striae, or other external appearances,
indicate their presence (Lloyd, Wave Theory of Light, p. 239).
* Faraday, Eaſperimental Researches (xixth series), vol. iii. p. 1; Phil. Trans.,
1846, p. 1.
* It is difficult to secure blocks of glass free from internal stress, and, as already
remarked (Art. 220), glass in this condition plays in some degree the part of a doubly
refracting crystal and “depolarises” the light. It is accordingly important that
the block of glass used in this experiment should be well annealed or else corrected
in some way by a compensating plate. The silicated borate of lead glass softens
at a temperature below that of boiling oil and can be well annealed. Flint glass
possesses a more feeble rotatory power, and crown glass is more feeble still.
464 ROTATORY POLARISATION CHAP. XVII
the direction of the current. Thus, as Faraday states it, if a watch be
placed between the poles of the magnet, so that its face is towards
the north pole and its back towards the south, then the direction of
motion of the hands of the watch is the direction of rotation of the
plane of polarisation; in other words, to a person looking along a line
of magnetic force—that is, from north to South, or from places of higher
to places of lower magnetic potential—the rotation is from left to
right. Or, again, if we regard a line of magnetic force as due to an
electric current encircling that line, then the direction of rotation is
the same as the direction in which the current circulates. A rotation
in this direction is consequently referred to as a positive rotation.
When the light passes through the field in a direction parallel to the
lines of magnetic force the effect is greatest, and when the direction
of the light is perpendicular to the lines of force there is no rotation
of the plane of polarisation.
Faraday's experiments extended over a vast variety of substances
which included solids, liquids, and gases, but in the latter he was not
able to observe any decided effect. In many of these experiments
the electromagnet was dispensed with, and the magnetic field produced
within a long helix of wire, carrying an electric current, was employed
instead. By this means the light could be transmitted through a
great thickness of any transparent substance, placed as a core within
the helix, and feebly rotating substances could be examined with more
effect. In all cases the magnetic field of the solenoid behaved as that
of the electromagnet, except that with the former the light in the
field of the analyser was suddenly restored when the current was
switched on, whereas with the latter the restoration of the light was
more or less gradual.
In the case of substances in which the induced rotatory power was
feeble, it was found best to work with the Nicols turned a little from
the position of complete extinction, so that with no current there was
a little illumination in the field of the analyser. When the current
was turned on in one direction this illumination was increased, whereas
in the other direction it was diminished, and this change of intensity in
opposite directions doubled the effect to be observed.”
Rule.
* Faraday noticed that the rotation produced by a substance (water) was ap-
parently unaffected by placing an iron bar within the helix, and that it was the
same whether it was placed along the axis or near the side of the spiral. It was
also the same whether the substance was enclosed in an iron or a brass tube. When
water was placed in an iron tube # inch thick, and this tube placed within another
of the same thickness but a little wider diameter, the rotation was increased. On
placing these two within a third iron tube the effect diminished, but was still very
considerable.
ART. 258 THE FARADAY EFFECT 465
The property thus induced by the magnetic field is similar to that
possessed by saccharine solutions and other rotatory substances, in so
far that a rotation of the plane of polarisation is effected, but there
is one important difference. In the case of rotatory substances the
direction of rotation is the same whatever be the direction of trans- Distinction.
mission ; that is, if a ray be rotated to the right in passing from one
end A to the other end B of a crystal of quartz, it will also be rotated
to the right in passing from B to A.” The result is, that if a ray pass
from A to B, and be then reflected directly back so as to pass from
B to A, the rotations will neutralise each other, and the ray after the
double transmission will return polarised in the primitive plane of
polarisation. This is easily conceived, for if the observer be supposed
looking in the direction AB, and if the incident ray in passing from
A to B be rotated to his left (that is, to the right of an observer look-
ing in the direction BA who would receive the emergent ray), then
the reflected ray in returning from B to A will be rotated by an equal
amount to his right, so that the two will neutralise each other. In
the case of the magnetic field, however, the direction of the rotation
depends on the direction of transmission, as well as on the nature of
the medium. If the rotation is to the right in passing from a place
of higher to a place of lower potential, then it will take place to the
left when passing from lower to higher. It follows, therefore, that if
a ray pass from A to B and be then reflected back so as to pass from
B to A, the rotation of the plane of polarisation will be doubled. In
the magnetic field the rotation is in the same direction whether we
conceive the ray as moving from us or towards us, but with a quartz
crystal, if the rotation be to the right for a ray advancing towards us,
it will be to the left for one travelling from us.
This difference in character, of the rotation produced by a quartz
crystal and that effected by a substance placed in a strong magnetic
field, points to a difference in the nature of the causes which primarily
induce the rotation.” The fact that the rotation is removed by re-
traversing a piece of quartz, or any natural rotatory substance, indi-
cates something analogous to a twisted structure in the substance, but
the doubled effect of the magnetic field seems to indicate that some-
thing of the nature of a rotation is going on in the field. This
rotation must be an affection of, or at least depends on, the matter
occupying the field, and does not necessarily lead to the conclusion
that the free ether in a magnetic field is in rotatory motion, for the
* That is, to an observer receiving the light in both cases.
* If the coil, still carrying the same current, is inverted as well as the direction
of the ray the difference between the two cases disappears.
2 H
466 ROTATORY POLARISATION CHA P. XVII
Faraday effect is in one direction in some substances and in the
opposite direction in others. It is consequently induced by the action
of the magnetic field on the matter molecules occupying it, the effect
of the magnetic action being to originate something analogous to a
rotation in the matter molecules, and the direction of rotation being
determined by the nature of the matter. -
The rotation is found to be proportional to the difference of mag-
netic potential between the points where the ray enters and leaves the
medium. Thus if V1 and V, be the values of these potentials, we have
for the rotation
p = c(V1 – V2)
where c is a constant (known as Verdet's constant) depending on the
nature of the medium. It follows, therefore, that the rotation will be
greatest when the ray travels in the direction of the lines of force, and
will vanish in directions perpendicular to these lines, while for oblique
directions it will vary as the cosine of the angle of inclination.
Since the time of Faraday this rotatory power has been found
to be impressed on most diamagnetic substances, and it has been
observed in gases as well as in solids and liquids, by Becquerel,
Kundt, and others. By reflecting the pencil of light backwards and
forwards several times through the field, a feeble rotation may be
much amplified and rendered sensible, for if the ray traverses the
field n times the rotation will be n times that produced by a single
passage.
The results of these experiments prove that most isotropic sub-
stances (including solids, liquids, and gases), when subject to magnetic
force, rotate the plane of polarisation of the transmitted light in the
positive direction—the positive direction of rotation being taken, as
already defined, to be the direction of the Ampèrean current producing
the field. Further, a positive rotation occurs when light is transmitted
through thin films of magnetic substances such as iron, nickel, and
cobalt. It is positive also in the case of oxygen (which is magnetic),
but in the case of a concentrated solution of ferric chloride the rotation
is negative. The negative rotation of other magnetic salts may be
recognised by the diminution of the positive rotation of the solvent."
259. Kerr's Experiments.--(1) Electrostatic Effect.—After dis-
covering the magnetic rotation of the plane of polarisation Faraday
sought for a corresponding effect in substances subjected to electro-
static stress, but without success. This effect was first observed by
Dr. Kerr 9 of Glasgow, who found that a dielectric under electric
Results.
* A. Kundt, Phil. Mag. vol. xviii. p. 327, 1884.
* J. Kerr, Phil. Mag. vol. 1. pp. 337-446, 1875.
ART. 259 THE KERR EFFECTS 467
stress acquires the double refracting properties of a uniaxal crystal of
which the optic axis is in the direction of the lines of electric force,
some substances behaving like negative and some like positive crystals.
Glass and olive oil and quartz belong to the former class, while bisul-
phide of carbon, paraffin, oil of turpentine, and resin belong to the
latter.
In glass the effect was well marked, and the experiment was con-
ducted by drilling two holes in opposite ends of a block of the
material. Metallic terminals were inserted in these holes and con-
nected to the electrodes of an induction coil, which possessed a
sparking distance of from 20 to 25 cm., and this distance could be
varied at pleasure. The thickness of glass between the terminals was
about a quarter of an inch clear, and across this space, at right angles
to the lines of electric force, a pencil of plane-polarised light was
transmitted. When the induction coil was out of action the transmitted
light was plane-polarised and could be extinguished" by a Nicol pro-
perly adjusted. This position being accurately secured it was found
that when the coil was thrown into action, so that the glass was under
electric stress, the light reappeared in the field of the analysing Nicol;
nor could it be again quenched by any rotation of the Nicol in either
direction. The light thus becomes elliptically polarised by the dielectric,
and the character of the elliptic vibration may be examined by means
of a Babinet's compensator (Art. 239). *
The effect is regular and best marked when the light crosses the
lines of force at right angles, and the plane of polarisation is inclined
to them at 45°; but when the plane of polarisation. is either parallel
or perpendicular to the lines of force there is no effect. The optical
effect does not attain its full intensity at once, but increases gradually
from zero to its final state in as much as thirty seconds after the
electric field is excited, and in the same manner the optical effect
fades away gradually when the electric stress is removed.
This effect may be completely compensated by a slip of glass com-
pressed, or stretched, in the direction of the lines of force (see Art.
242). From this it is inferred that the effect of the electric action
is optically equivalent to compressing the glass in the direction of the
electric force, the glass acquiring the properties of a negative uniaxal
crystal having its axis parallel to the lines of force. The result is as
if the molecules of the glass assumed a regular crystalline arrangement
* In order that extinction should be produced it was found necessary to introduce
a plate of glass in front of the analyser to neutralise the “depolarising” effect of
the experimental block containing the terminals. This was called the neutralising
block and was adjusted by trial.
468 ROTATORY POLARISATION CHAP. XVII
Elliptic
polarisa-
tion.
in the electric field, and the effect is as good with the alternating
electrifications produced by an induction coil as with a continued
electrification of the same strength in one direction produced by a
frictional machine; the alternating electrifications produce contrary
electric polarisations, but their actions conspire in arranging the
molecules.
If 6 be the relative retardation introduced between the components
of the elliptic vibration per unit thickness of the dielectric, Kerr
found 6 to be proportional to the square of the electric force F;
that is,
6 = }cF2
where k is a constant depending on the nature of the dielectric, and
may be either positive or negative. This law contains a result found
experimentally—namely, that the optical effect is independent of the
direction of the electric force ; that is, it remains the same when the
direction of the force is reversed.
(2) Magnetic Effect.—Shortly after the discovery of the modifica-
tions produced in the optical properties of a dielectric when subject to
the action of electric force, Dr. Kerr 2 was led to examine the effects
produced when plane-polarised light is reflected from the surface of a
magnetised body, such as the polished pole-piece of an electromagnet.
In approaching this subject we must remember that when plane-
polarised light is reflected from the surface of a polished metal the
reflected light is in general elliptically polarised (Art. 218). There
are two cases, however, in which the reflected light remains plane-
polarised—namely, when the light falling upon the surface is polarised
either in or at right angles to the plane of incidence—and in these two
cases the reflected light can be extinguished by the analysing Nicol.
Let us suppose, therefore, that light polarised in the plane of incidence
falls upon the polished pole-piece of an electromagnet (which, for the
present, is unmagnetised), and that the reflected pencil is received by
an analyser placed in the position of complete extinction. If the mag-
netising current be now turned on the illumination is observed to
reappear in the field of the analyser, and this cannot be extinguished
by rotating the analyser in either direction. The reflected light is
consequently not plane-polarised. Extinction, however, may be pro-
duced by introducing a compensating strip of strained glass into the
path of the light, and this shows that the reflected beam is elliptically
polarised. -
This is the new fact discovered by Dr. Kerr, and is described by
* J. Kerr, Phil. Mag. vol. ix. p. 157, 1880.
* J. Kerr, Phil. Mag. vol. iii. p. 321, 1877; and vol. v. p. 161, 1878.
ARt. 259 THE KERR EFFECTS 469
him as a rotation of the plane of polarisation, thus: “When plane.
polarised light is reflected regularly from either pole of an electro-
magnet of iron, the plane of polarisation is turned through a sensible
angle in a direction contrary to the nominal direction of the magnetis.
ing current; so that a true south pole' of polished iron, acting as a
reflector, turns the plane of polarisation right-handedly.”
The arrangement of the apparatus is shown in Fig. 193. The
source of light was a paraffin flame S, placed at a distance of one foot,
or less, from the polished polar surface. Close to the flame came the
polarising Nicol P, through which the light passed in a horizontal
direction and fell upon the pole of the magnet, inclined so that the
angle of incidence could be varied at pleasure. At a few inches from
the pole the reflected light was received by the analyser A, the Nicols
being adjusted so that complete extinction was secured when the pole
was unmagnetised. In
order to obtain any
optical effect it was
found necessary to
secure intense con-
centration of magnetic
force upon the reflect-
ing surface at the
point of incidence of
the light. For this purpose a block of soft iron, W (one of the several
pole-pieces of the magnet), about two inches square and three inches
long, was planed off at one end into a blunt wedge with a well-rounded
edge. This wedge, called the submagnet, was placed over the centre
of the reflecting pole, resting on two splinters of hard wood so as to
leave a narrow chink through which the light passed, the width of the
chink being about ºn of an inch. With this arrangement an intense
concentration of magnetic force is produced in that part of the field
which is utilised optically, the lines of force being sensibly perpen-
dicular to the reflecting surface.
In the greater part of his experiments Dr. Kerr (as recommended
by Faraday) first adjusted the Nicols to perfect extinction, and then
turned the analyser through a very small angle, so that the light was
faintly restored in its field. The magnetising current was then switched
on, and the illumination in the field of the analyser was observed to
increase or decrease according to the direction of the current. Within
the limits of his experiments, which included incidences varying from
0° to 80°, Dr. Kerr found that the rotatory effect of reflection from
Fig. 193.
* By a true south pole is here meant the north-seeking pole of a magnet.
The sub-
magnet.

470 ROTATORY POLARISATION CHAP. XVII
Kundt's
experi-
ments.
Normal
incidence.
magnetised iron was negative; that is, in a direction opposite to that
of the magnetising current.
Subsequently Professor Kundt extended the experiments to
incidences ranging between 0° and 90°, and he found that the
direction and amount of the rotatory effect depended not merely on
the angle of incidence, but also on whether the plane of polarisation
was parallel or perpendicular to the plane of incidence. When the
light was polarised in the plane of incidence the direction of rotation
was found to be the same for all angles of incidence, and opposed to
the Ampèrean molecular currents of the reflecting magnet. On the
other hand, when the light was polarised at right angles to the plane
of incidence, then the direction of rotation was the same as that of the
Ampèrean current for angles of incidence between 0° and 80°, but in
the opposite direction for incidences between 80° and 90°. In both
cases the rotation reached a maximum at an incidence of about 65°.
Further, it was found that iron exhibited an anomalous rotational
dispersion, the red rays suffering a larger rotation than the blue by
reflection from a magnet as well as by transmission through a
magnetised film of iron.
In order to examine the case of normal incidence Dr. Kerr found
it necessary to modify the apparatus, as shown in Fig. 194. The
wedge-shaped submagnet of Fig.
193 was dispensed with and
replaced by a block of soft iron
rounded at one end into the
frustum of a cone. A small
conical hole was drilled through
the centre of this block, and it
was then placed directly over
the centre of the polar surface,
with its axis vertical, as shown
in Fig. 197, the block being
separated from the pole of the
magnet by a wide ring of writ-
ing-paper. Above the block a
thin plate of glass C was placed, on which the horizontal beam of light
from the polariser fell at an angle of 45°, and was reflected vertically
downwards through the hole in the iron block. This beam, after
suffering direct reflection at the surface of the pole, returned vertically,
and passed in part through the plate C into the analyser A placed in
Fig. 194.
* Kundt, Berlin, Sitzungsberichte, July 10, 1884; Phil. Mag. vol. xviii. p.
308, 1884.

ART. 259 THE KERR EFFECTS 471
position to receive it. The results obtained with this form of apparatus
are complicated by the introduction of the glass plate C, for if any Glass plate.
rotation occurs the light returning from the magnet to the plate will
experience a further rotation during refraction through the plate, as
explained in Art. 211. This disturbing effect of the plate seems to
have been first recognised and allowed for by Professor Kundt.
In the foregoing experiments the lines of force are perpendicular
to the reflecting surface, and to complete the investigation Dr. Kerr
examined also the effect when the reflector is magnetised parallel to Force
its surface. For this purpose the electromagnet was placed upright º to
upon a table, and a rectangular bar of soft iron, one of whose sides was
carefully planed and polished, was placed across the poles of the electro-
magnet, with the plane of its polished face vertical (the plane of the
paper being horizontal), as shown in Fig. 195.
With this arrangement, in which the magnetic force is approxi-
mately parallel to the reflecting surface, it was found that when the
light was polarised in
the plane of incidence
the magnetic rotation
was negative—that is,
opposite in direction to
the Ampèrean current,
which would produce
the magnetisation—for
all angles of incidence;
while for light polarised at right angles to the plane of incidence the
rotation was negative for incidences between 90° and 75°, vanished
at 75", changed sign there, and remained positive for all angles of
incidence between 75° and zero.
The particular case of perpendicular incidence on the side face was
examined, and it was found that no optical effect was produced whether
the plane of polarisation coincided with the direction of magnetisation
or made an angle with it. From this it appears that there is no
magneto-optic effect when the plane of the wave front is parallel to
the lines of magnetic force.
The optical effect produced when light is reflected at the surface of
a magnetised substance may be brought into harmony with that pro-
duced by transmission through substances placed in a magnetic field,
by the supposition that the light during reflection penetrates a thin
film of the reflecting pole, and suffers rotation within the substance
while traversing the film. This point of view has been worked out by
Professor Kundt, who examined the light reflected from the second
Fig. 195.

472 ROTATORY POLARISATION CHAP. XVII
face of a glass plate. The faces of the plate were not parallel, but
inclined at an angle, so that the light reflected from the first face was
well separated from that which penetrated the plate, and, after suffering
reflection at the second face, emerged again through the first. This
twice refracted and once reflected beam was examined when the block
of glass was placed upon the poles of an electromagnet with the lines
of force parallel to the reflecting face, and it was found that when the
incident light was polarised in the plane of incidence the emergent
light was rotated in the positive direction for all angles of incidence,
but that when the incident light was polarised at right angles to the
plane of incidence the rotation was negative from normal incidence
up to the polarising angle (56°4), and was positive from the polarising
angle to grazing incidence.
That a change of sign should occur at the polarising angle in the
case of light polarised at right angles to the plane of incidence, and
not in the case of light polarised in the plane of incidence, is indicated
theoretically by the formulae of Art. 211. -
The mathematical investigation, and the explanation of the elliptic
polarisation in these experiments on the basis of Maxwell's theory, have
been given by FitzGerald" and others. The rotatory effect discovered
by Faraday when light passes along the lines of magnetic force, and
the doubly refracting effect discovered by Kerr when light is trans-
mitted across the lines of electric force, agree in this respect, that they
both depend upon the presence of matter in the field; they both vary
in sign according to the nature of the substance occupying the field,
and they do not occur in the free ether.
260. Zeeman Effect.—The influence of a magnetic field upon the
frequency of light vibrations was first demonstrated by Zeeman * in
1896. He observed that when sodium light was subjected to a powerful
magnetic force the spectroscopic lines appeared to broaden, returning
at once to their original width when the magnetic force was removed.
Lorentz pointed out that, on his theory of light vibrations being due to
the vibrations of electric charges or small electrically charged particles,
the light should be circularly polarised at its edges when looked at
along the lines of force ; while, if the light were viewed across the
lines of force, the edges would be plane-polarised in a plane parallel to
the lines of force, and the centre of the line would be plane-polarised
in a plane perpendicular to the lines of force. Zeeman found that
these conclusions were verified by his experiments. Thus he placed
* G. F. FitzGerald, Phil. Trans., 1880, pt. ii. p. 691.
* Phil. Mag. (5), vol. xliii. p. 226, March 1897; vol. xliv, p. 55, July 1897;
and vol. xlv. p. 179, February 1898.
ART, 260 THE ZEEMAN EFFECT 473
an electromagnet, with bored pole-pieces, with the axis of the holes in
line with the centre of a concave grating. Between the grating and
the eye-piece he placed a quarter-wave plate and a Nicol so as to
extinguish right-handed circularly polarised light and found that, on
viewing the spectrum of a sodium flame placed between the poles of
the electromagnet, a reversal of the current of the electromagnet caused
the sodium line to move in the focal plane of the eye-piece: the visible
edge of the line for one direction of current becoming the quenched
edge for the reversed current.
Later Zeeman observed that when the blue cadmium line, obtained
by sparking between cadmium electrodes in a strong magnetic field,
was observed along the lines of forces, it appeared as a double line,
the two lines being circularly polarised in opposite directions. When
it was looked at in a direction at right angles to the lines of force,
through a Nicol, it appeared as a double line with a central dark line,
or as a narrow bright line according as the Nicol was turned so as to
extinguish light plane-polarised perpendicularly to the lines of force,
or in the horizontal plane through the lines of force. He obtained
similar results for other lines, and measured the displacements due to
the magnetic field by photographing their spectra.
Preston * did much valuable work in this connection, and was the
first to observe and photograph the distinct tripling of the lines when
viewed perpendicularly to the lines of force. He also showed that
more complicated effects are produced by a magnetic field upon spectral
lines, some lines appearing as triplets, others as quartets or doublets,
others again appearing as six or even nine distinct lines. He also
indicated the division of the spectral lines of a substance into groups
according to their behaviour in a magnetic field, and found that these
groups corresponded with the series of Kayser and Runge. His
measurements led him to the conclusion that the corresponding lines
of the natural groups into which a given spectrum resolves itself
possess the same value of 6A/A*, A being the wave length and 6A the
difference in wave length of the side lines due to a given field, and
that further this value is the same for corresponding lines in homo-
logous spectra of different substances.
Lorentz supposed that light vibrations are due to the vibrations
of electrically charged bodies of definite mass, called electrons. These
vibrations may be resolved along and perpendicular to the lines of
force. The component along the line of force will be unaffected by
the magnetic force, and will give rise to plane-polarised light when
* Preston, Phil. Mag. (5), vol. xlv. p. 325, April 1898; Sci. Trans. Roy. Dubl.
Soc. vol. vi. (Series II.), p. 385, vol. vii. (Series II.), p. 7.
474 ROTATORY POLARISATION CHAP. XVII
observed across the lines of force, but will not produce any light effect
in the direction of the lines of force. The components in the plane
perpendicular to the lines of force may be regarded as two circular
vibrations of the same period in opposite directions, but the periods
of the electrical charges revolving in circles in planes perpendicular
to the magnetic force will be altered by this force, one being retarded
and the other accelerated. A spectroscope of sufficient resolving power
should therefore give two lines of light, circularly polarised in opposite
directions, when the light is observed along the lines of force, but
when the light is observed perpendicularly to the lines of force, these
circular vibrations are seen end on, and give two plane-polarised lines,
the component parallel to the lines of forces giving the centre line of
the triplet plane-polarised in a perpendicular plane to the plane of
polarisation of the outside pair.
This elementary presentation of the theory suffices to account for
what may be called the normal doublet and triplet produced by the
magnetic field. In 1891 G. Johnstone Stoney" had published a theory
to account for double lines in the spectra of gases from the considera-
tion of the orbital motions of electrons being subject to perturbations.
The magnetic field furnishes a cause for such perturbations, and Preston
applied the consideration of these perturbations to explain the more
complicated types of multiple lines produced by the magnetic field.
FitzGerald” drew attention to the connection between the Faraday
rotation of the plane of polarisation of light and the Zeeman effect.
The magnetic field causes the frequency of vibration of the electron to
be different according to the sense in which it is rotating ; that is,
according to the character of the circularly polarised light which it is
producing. It follows that the absorption of molecules in a magnetic
field will be different for light circularly polarised in opposite senses.
Hence, according to modern theories of dispersion, the dispersion of
the medium will be different, and therefore the velocities of beams of
oppositely circularly polarised light through the medium will be
different, and hence a beam of plane-polarised light will have its plane
of polarisation rotated by a substance in a magnetic field.
261. Division of a Plane-Polarised Ray into two Circularly
Polarised Rays.-The first theoretical interpretation of the phenomena
of rotatory polarisation was given by Fresnel on the hypothesis that
a rectilinear vibration may be regarded as the resultant of two opposite
circular vibrations (see Art. 48, Ex. 2). According to this theory a
* Trans. Roy. Dubl. Soc. vol. iv. p. 563. See also Larmor, Phil. Mag. (5), vol.
xliv. p. 503, December 1897.
* FitzGerald, Proc. Roy. Soc. vol. lxiii. March 1898; Collected Papers, p. 464.
Art. 262 CIRCULAR DECOMPOSITION 475
plane-polarised ray may be regarded as the resultant of two opposite
circularly polarised rays. If this be so a plane-polarised ray in passing
through any substance may be decomposed into two others circularly
polarised in opposite senses, which may or may not be propagated
with the same velocity, according to the nature of the substance. If
they travel with the same velocity then at emergence they will com-
bine again into a plane-polarised ray, polarised in the same plane as
before transmission, but if the opposite circular vibrations travel with
different velocities then at emergence one will have suffered a retarda-
tion relatively to the other with the effect, as we shall presently show,
that the plane of polarisation of the emergent beam will be inclined
to the primitive plane of polarisation, and the rotation will be
accounted for.
To test this point, and place it beyond doubt that this decomposi-
tion in reality occurs, and is not a result of the mere manipulation of
formulae, Fresnel devised the following experiment. Several prisms of
quartz, alternately right-handed and left-handed, were arranged in
the shape of a parallelopiped so as to form an achromatic combina-
tion. Thus in Fig. 196 the prisms ABC and BED are right-handed
while BCD and DEF are left-handed, and
in all the axis is parallel to the faces AE
and CF. Now if a plane-polarised ray
passing through quartz in the direction of
the axis is really divided into two opposite
circular vibrations which travel with
different velocities, then on reaching
the interface BC the quicker ray in ABC will be the slower in
BCD and vice versa, and consequently at the face BC one ray will
be deviated from the normal and the other towards it. There
will thus be a separation of the rays which will be augmented at
the faces BD and DE, so that by sufficiently increasing the number
of prisms the separation should be rendered visible. Fresnel found
that this was the case, that two emergent pencils were presented,
and that both were circularly polarised."
262. Rotation of the Plane of Polarisation by Retardation of a
Circular Component.—Having shown that a plane-polarised ray may
be decomposed into two opposite circularly polarised rays, it is easy
to show that the effect of introducing a relative retardation between
the circular vibrations is to rotate the plane of polarisation. Thus
Fig. 196.
* The same results follow with ordinary light. In fact, the experiment was
first devised to show the double refraction of unpolarised light (Fresnel, CEuvres,
tom. i. p. 743, etc.).

476 ROTATORY POLARISATION CHAP. XVII
if two particles M and M' (Fig. 197) describe the same circle in opposite
directions starting from the same point Y, it is clear that their velocities
parallel to OY will be equal and in the same direction, while their
velocities perpendicular to OY will be equal but oppositely directed.
Hence if the two motions be simultaneously impressed on the same
molecule its velocity parallel to OY, at any instant, will be double that
of M or M', and its velocity perpendicular to OY will be zero, so that
it will simply vibrate backwards and forwards along the line OY with
an amplitude of excursion equal to twice the radius of the circle.
Now let us suppose that one of the circular vibrations is, by any
cause, relatively retarded so that when M travels round the circle to
Y the other has not yet reached it, but has attained a position Mº
(Fig. 198) at an angle 8 from M. Draw OP bisecting the angle MOM'.
If the two circular vibrations now move with the same velocity (which
Fig. 197. Fig. 198.
will be the case when the light emerges from the active substance) the
two particles will cross at P, and their combined motions impressed on
a single molecule will cause it to vibrate along the line OP. Thus if
the original vibration in the incident plane-polarised light be parallel
to OY, and if one of its circular components be retarded relatively to
the other, by transmission through quartz, or otherwise, the plane of
polarisation of the emergent light will be inclined to that of the in-
cident light at an angle p given by the equation
p=#6
where 6 is the relative phase retardation.
The same result may be arrived at very easily by means of the
equations of the vibration. The incident vibration y = 2a sin ot is
equivalent to the two opposite circular vibrations
*1 = a cos w!, y = a sin wº, (left-handed)
a 2–- a cos w!, ye-a sin wº, (right-handed)
and if the latter pair be accelerated in phase with respect to the

AItT. 262 CIRCULAR DECOMPOSITION 477
former by an amount 8 the equations of the transmitted vibrations
may be written in the form
a'i = a cos wt, vi = a sin ot,
aca– – a cos (at +6), */2= a sin (wt + 6).
Consequently we have for the transmitted vibration
a = a cos wt — a cos (wt + 6)=2a sin #8 sin (wt + #6),
y = a sin wb -- a sin (wt + 3) = 2a cos #6 sin (wt + #6).
These are two perpendicular vibrations of the same phase, which there-
fore compound into a resultant rectilinear vibration, making an angle
p with the axis OY, which is determined by the equation
tan p = 0:/y=tan #6.
Therefore, as before, p = #8. *
An expression for the rotation p may be easily found in terms of
the wave lengths of the circular components. For if A, and A, be the
wave lengths of the circular components in the quartz, then if e be
the thickness of the crystalline plate we have
e=nº}\, = m2\g
where n, and n, are the numbers of vibrations executed by the two
components during transmission. The relative phase acceleration of
the two components on emerging from the plate is consequently
1 1
6=2tr01 -n}=2~(, smº #).
Hence the expression for the rotation is
=re (+--
p = tr \l N2 tº
In the case of Iceland spar and other uniaxal crystals both rays travel
with the same velocity in the direction of the axis. In such substances
we have A = As, and no rotation of the plane of polarisation is pro-
duced. In quartz, on the other hand, the wave surface consists, as is wave
usually assumed, of a sphere and a concentric spheroid, but they do *
not touch. The radius of the sphere is greater than the major axis of quartz.
the spheroid, so that the latter is entirely enclosed within the former
and Al is not equal to Ag: This difference in wave length is much
smaller than the difference between the corresponding indices and the
extraordinary index, so that for many purposes the crystal may be
considered as having two chief indices only.
Experiment shows that p is approximately proportional" to I/A*;
we therefore conclude that 1/A1 – 1/A2 is approximately proportional
* In fact, if \g-Xi + 6N1, 1/\l – 1/A2=6)1/A1*.
478 ROTATORY POLARISATION CHAP. XVII
to 1/A*. The formula for the rotation may then be written in the form
K6
p=x,
where e is the thickness of the substance passed through, A the wave
length of the light in air, and k a constant depending on the nature of
the substance.
263. Decomposition of a Rectilinear Vibration into two opposite
Elliptic Vibrations—Oblique Transmission through Quartz. It has
been seen in the foregoing articles that two circular vibrations of the
same amplitude and opposite senses compound into a rectilinear vibra-
tion, and conversely, it has been proved by experiment that a pencil
of plane-polarised light is actually divided into two opposite circularly
polarised beams by transmission through quartz in a direction parallel
to the optic axis of the crystal.
From a mathematical point of view, however, the circular com-
ponents have no particular claim to be regarded as the only legitimate
constituents of a plane-polarised ray. A rectilinear vibration may be
the resultant of a variety of systems of constituents, which may be
rectilinear, or elliptic, or other than elliptic, instead of circular. Thus
the rectilinear vibration
a; = & COS w!
is mathematically equivalent to the two opposite elliptic vibrations
a 1 = (a — k) cos wi, $/l = b sin (wt -- 6),
a 2 = k cos wt, Q/2= – b sin (wt + 6),
for these, when added together, yield the original vibration. The
choice of components must therefore be settled by experiment, and
Fresnel's experiment (Art. 261) marks the circular components as the
proper constituents when light is transmitted through quartz in the
direction of the axis. -
In directions inclined to the axis of the crystal, however, it was shown
in 1831 by Sir G. B. Airy" that the decomposition must be regarded as
elliptic, the ellipse becoming a circle in the particular case of transmission
parallel to the optic axis. Thus when a plane-polarised ray is transmitted
through rock-crystal in a direction inclined to the axis, it is divided
into two others which are elliptically polarised ; the elliptic vibrations
in the two being in opposite directions, and their greater axes coin-
ciding in direction with the principal plane and the perpendicular plane
respectively. Thus the major axis of one elliptic vibration coincides
in direction with the minor axis of the other, as in Fig. 199. The
* G. B. Airy, Trans. Cambridge Phil. Soc. vol. iv. pp. 77, 199, 1833.
A Rt. 263 ELLIPTIC DECOMPOSITION 479
ratio of the axes is the same in both, but varies with the inclination
of the ray to the optic axis, being equal to unity parallel to the
optic axis, and to zero for directions perpendicular to it – the
elliptic vibration including the circle and right line as particular
cases. Thus a circularly polarised ray may pass through quartz
in the direction of the optic axis
without alteration, a plane-polarised
ray may do so in the perpendicular
direction, and a given elliptic ray in
any intermediate direction.
The phenomena presented by oblique
transmission through quartz have all
been explained satisfactorily by Airy Fig. 199.
on the assumption of this special decomposition, together with the
supposition that one of the components is retarded relatively to the
other during transmission.
In order to express this conveniently we shall take the initial
rectilinear vibration to be of the form
w-(1+k”) cos w!.
It is evident that this may be replaced by the components
arl-cos w!, y = k sin wº,
re-/* cos wº, y2 = -k sin wº,
and these components are of the form required by Airy's theory, for
they are elliptic, of opposite signs, and such that the major axis of
one coincides with the minor axis of the other while the ratio of the
axes is the same in both.
Now if the components (, y) be retarded by an amount 3
relatively to the components (ºr, y), the equations of the emergent
vibration will be
*1 =cos w!, y = k sin wº,
are-k” cos (wt 4-6), ya- -k sin (wt #6).
Therefore for their resultant we have
w-cos w = k” cos (wt +6)=A cos (w/ -- p)
where
/* sin 5
2- - -4 - -
A*=1 + 2}ºcos 54 k", and tan *-ijącos 5
Similarly
y=k sin wº -k sin (wt H 6) = B cos (wt 4 p.)
where
B*=4/* sin” .5, and tan p =tan #6.
If k = 1 the vibrations are circular, and we have q = p = }6. This
applies to the case of propagation parallel to the axis.

480 ROTATORY POLARISATION CHAP. xv. 11
264. General Formula.--When convergent plane-polarised light
is transmitted through a thin plate of quartz cut perpendicularly to
the axis, and then examined by means of an analyser, the isochromatic
curves obtained exhibit some remarkable peculiarities which are
not presented in the case of crystals which do not
possess rotatory power. To determine these pecu-
liarities it becomes necessary to obtain an expression
for the intensity in the field of the analyser when
the light traverses the quartz plate in a direction
oblique to the axis.
- Let the incident light be plane-polarised, and
Fig. 200. let OP (Fig. 200) be parallel to the principal plane
of the polariser, OX and OY to the directions of vibration in the
crystal. Then if the angle POX = a, the incident vibration, being
parallel to OP by Fresnel's theory, will give components of ampli-
tudes a cos a and a sin a parallel to OX and OY respectively.
Writing a = 1 + k”, the vibration a = (1 + k”) cos a cos of parallel to
OX may be replaced by the two opposite elliptic vibrations
a 1-cos a cos wº, y1=k cos a sin wº }
a 2-k” cos a cos w!, ye=-k cos a sin wiſ'
(1)
in which the ratio of the axes of one is equal to the ratio of the axes
of the other, but the minor axis of one is in the direction of the major
axis of the other, as indicated in Fig. 199. So also the vibration
y = (1 + k”) sin a cos of parallel to OY may be replaced by the elliptic
vibrations
a' = k sin a sin wº, y’i-sin a cos w \
wº- -k sin a sin ot, y'a-kº sin a cos w ſº
(2)
In passing through the crystal one (the left-handed, say) of the
elliptic vibrations is retarded on the other by an amount 8, so that on
emergence the vibrations (1) are of the form
(3)
*1 =cos a cos w!, !/1=k cos a sin wº }
a 2-k” cos a cos (wt - 6), ya- - k cos a sin (wt – 5) J'
and equations (2) become
(4)
a'i-k sin a sin (wt - 6), y' =sin a cos (wt – 5) }
a'2– – k sin a sin wº, y'a-k” sin a cos w! -
Hence if the principal plane of the analyser OA makes an angle B
with OX, we have for the vibration in that plane
Y=(21-22-21--a'a) cos 3+(y-º-yº-yº-yº) sin 3,
from which the general expression for the intensity in the field of the
analyser may be obtained. Let us first consider the case in which the
Nicols are crossed.

ART. 265 NICOLS CROSSED 481
265. Nicols Crossed.—In the particular case in which the polariser
and analyser are crossed we have 6 – a = 90°, and the calculation be-
comes much simplified. Thus the vibration in the principal plane of
the analyser is
Y= (y1-ya-yı --yº) cos a -(r) + æs--a'-a'a) sin a
=2% sin #3 cos (wt – #6)+(1-kº) sin 20 sin #6 sin (wt – 5).
Hence the intensity is given by the equation
I={4}*--(1-kº)" sin” 2a) sin” .3. (1)
When the incident light is a parallel beam, and the thickness of
the plate uniform, the directions of vibration OX and OY will be the
same at all points of the plate, so that a remains constant as well as 6.
The colour and intensity will therefore be the same at all points unless
the plate possesses irregular structure.
With a convergent beam on the other hand 6 will increase as we
proceed along any radius vector drawn from the centre of the field.
Further, the two directions of vibration within the plate for a ray
incident at any point X will be along and perpendicular to OX, as
explained in Art. 228, so that these directions vary as X rotates
around the centre O. The angle a will consequently not be constant,
and in the foregoing formula for the intensity it will play the part of
the direction angle to the radius vector drawn to the point X, at which
the intensity is expressed. Hence, in general, as X moves over the
face of the crystal the intensity will vary by reason of the variations
of a as well as those of 8. For direc-
tions slightly inclined to the axis,
k=1, and we have approximately
I=4 sin” .5. (2)
Consequently in the neighbourhood
of the optic axis the intensity de-
pends only on the value of 8, and as
8 will be different for the different
colours, it follows that when white
light is used the central spot will be
coloured and will not be crossed by
any uncoloured lines.
In directions considerably inclined to the axis, we have approxi-
mately k = 0, and the expression for the intensity becomes
Fig. 201.
I =sinº 2a sin” .5. (3)
The term sin 20 gives rise to a dark rectangular cross (Fig. 201) Cross and
Central
spot.
corresponding to a = 0° and a = 90°, while the term sin” ſº produces *
2 I

482 ROTATORY POLARISATION CHAP. XVII
a system of concentric circular rings, as in the case of an ordinary
uniaxal crystal, alternately dark and bright when monochromatic
light is used, but with white light they will be iris-coloured. The
pattern presented in the field consequently consists of a cross, together
with a system of coloured rings surrounding a central spot of tint
depending on the thickness of the plate.
It should be observed that the intensity, as expressed by formula
(1), will vanish only when 8 is zero, or a multiple of $71, and for this
reason the dark cross of the pattern is not black. The illumination is
Rotating
and non-
rotating
crystals
compared.
simply least along the directions a = 0 and a = }T, etc., and greatest
along the lines a = }T and a = }T, etc. Near the centre of the field k is
sensibly equal to unity, and the intensity given by equation (2) is the
same in all directions around the centre. The dark cross consequently
does not begin to appear until we recede to some distance from the
centre, and its arms do not completely interrupt the rings in any
region. They merely approach blackness at a considerable distance
from the centre where k is sensibly equal to zero, and where the
intensity is given by equation (3).
In the case of uniaxal non-rotating crystals the cross is black and
traverses the centre of the field, as shown in Fig. 179. This happens
because the ordinary and extraordinary waves travel with the same
velocity in the direction of the optic axis, and consequently 8 is zero
at the centre of the field. In the case of rotating crystals, however,
the two waves travel with different velocities, even in the direction of
the optic axis. According to Airy's theory the two sheets of the wave
surface do not touch each other in a rotating crystal. The radius of
the sphere in quartz is greater than the greatest axis of the spheroid,
so that the latter is contained within the former, and as a consequence
8 is not zero at the centre of the field. Now in the case of a non-
rotating crystal the expression for 6 at a distance r from the centre is
(p. 426) of the form 8 = mr^ where m is a constant depending on the
nature of the plate. Hence in the case of a rotating plate the retarda-
tion is of the form
ô = mr^+ 60,
8, being the retardation at the centre of the field where r = 0. By
Art. 253 it follows that 8, varies inversely as A*, so that 6 is a
function of the wave length and will not vanish simultaneously for
the various colours. For this reason when white light is used the
central spot is never black but coloured, with a tint which depends
upon the thickness of the plate. As in the case of a non-rotating
plate the difference between the squares of the radii of two con-
secutive rings is constant for light of a given wave length, but
ART. 266 GENERAL EXPRESSION FOR THE INTENSITY 483
the squares of the radii are no longer proportional to the natural
numbers.
When the principal plane of the polariser is parallel to that of the
analyser, the complementary pattern is exhibited. In this case 6 = a,
and the vibration is
Y=(21+ æs--a'1+2'2) cos a + (yì 4-ye-Fºſ' 4-1/2) sin a.
Hence we find
I'-(1+ k”)?– {4k?--(1 – k”)” sin” 20}sin” #6,
an expression which might have been written down from the value of
I with the consideration that I and I' must be complementary, and that
the amplitude of the incident light was taken equal to (1 + k”).
If a doubly refracting rhomb be used as analyser, I and I repre-
sent the intensities of the extraordinary and ordinary images re-
spectively.
To obtain these effects experimentally the apparatus previously
described in Fig. 176 should be employed and the slice placed at S.
266, General Expression for the Intensity."—When the principal
planes of the Nicols are inclined to each other, the calculation of the
intensity becomes more complicated.
Substituting from equations (3) and (4) of Art. 264 we have
Y=cos 34cos wi (cos a + k” cos a cos 3 – k sin a sin 6)
+sin wt (k” cos a sin 6 – k sin a + k sin a cos 3)}
+sin 8(cos w! (k cos a sin 6+ k” sin a + sin a cos 3)
* +sin wit (k cos a -k cos a cos 6+sin a sin Ö)}.
Writing this in the form
Y = A cos w! + B sin wit,
the intensity is given by the equation
I = A2 + B2
where
A= cos 3 (cos a + k” cos a cos 3 – k sin a sin 6)
+sin 3 (kcos a sin 6 + k” sin a + sin a cos 6)
=k* (sin a sin 8+ cos a cos 3 cos 6) – k sin (a — 8) sin 6
+cos a cos 3+ sin a sin 8 cos 6,
and
B=cos 3 (k” cos a sin à – 2k sin a sin” #6)
+sin 8 (sin a sin à --2k cos a sin” #6)
=k* cos a cos 3 sin 6 – 2k sin (a – 8) sin”36--sin a sin 3 sin 6.
Squaring and adding these expressions, we find that I may be written
in the form
I = Lk" – Mk3 + Nk? — Mk + L.
* See note by G. Quesneville, Comptes rendus, cxxi. pp. 522-524, 1885.
484 ROTATORY POLARISATION CHAP. XVII
The coefficient of k+ is
L=(sin a sing+cos a cos 3 cos 3)2+cosº a cos”g sinº
=sin” a sin” 8+ cos” a cos” 8+2 sin a cos a sin 3 cos & cos 6
= cos” (q – 3) – sin 20 sin 23 sin” #6,
and the term independent of k is at once seen to have the same value.
Seeking the coefficient of k” we have -
M = 2 (sin a sin 8+ cos a cos 3 cos 6) sin (a — 8) sin 6
+4 cos a cos B sin (a – 3) sin à sin” #6
=2 (sin a sin 8+ cos a cos & cos 5-H2 cos a cos 3 sin” #5) sin (a — 8) sin 6
=2 sin (a – 3) cos (a – 8) sin 6
= sin 2 (a – 8) sin 6,
and we obtain the same expression for the coefficient of k.
Finally the coefficient of k” is
N = 2 (sin a sin 8+ cos a cos 3 cos 6) (cos a cos 3+sin a sin 3 cos 3)
+sin” (a — 8) sin” 6+2 sin a cos a sin 8 cos & sin” 6+ 4 sin” (a — 8) sin"33
=2 cos 6 (sin” a sin” 8+ cos” a cos”8)+sin 20 sin 28+4 sin” (a – 8) sin” #6
=2 cos” (a — 8) – 4 sin” #6 (sin” a sin” 8+cos” a cos” 8– sin” (a – 8)}
=2 cos” (a — 8) – sin” # 644 cos 2 (a — 8) – 2 sin 20 sin 28}.
Hence
I=(k++ 1){cos” (a – 8) – sin 20 sin 28 sin” #6} – k(1+k”) sin 2 (a – 3) sin 6
+k”[2 cos” (a — 8) – {4 cos 2 (a – 8) – 2 sin 20 sin 28}sin” ºbj
=(1+k”)” cos” (a – 3) — k(1+k”) sin 2 (a — 8) sin 6 -
– {4k” cos 2 (a – 8) + (1 – k”)” sin 20 sin 28}sin°46.
Which may also be written in the form
I=[(1+k”) cos (a – 8) cos #6–2k sin (a – 8) sin #6]*-ī- (1 – k”)” cos” (a +8) sin” #6.
Cor. 1.-If k = 0, we have
I= cos” (a — 8) -- sin 20 sin 28 sin” #6,
which is the ordinary formula for uniaxal crystals devoid of rotatory
power.
Cor. 2.-If k = 1—that is, if the ray passes along the axis—
we have &
I= 4{cos (a – 3) cos #6 – sin (a – 3) sin #6}*=4 cos” (a – 8+$6).
This expression is a maximum when 6 – a = #8, and zero when a – 8
= }(T – 6). This indicates that the light emerging from the quartz
plate is plane-polarised, and that the plane of polarisation has been
rotated through an angle 6 – a = #8.
Cor. 3.--When the Nicols are crossed a — 8 = 90°, and, as in Art.
268, we have
I= {4}^+ (1 – k”)” sin” 20}sin” $5.
267. Approximate Form of the Isochromatic Lines. – When
the incident light is parallel the angles a and B have the same values
Aft'. 267 APPROXIMATE FORM OF ISOCHROMATIC LINES 485
at every point of the plate, and the intensity is uniform. With a
convergent beam, however, the angle a is variable, and, as already
remarked, is the direction angle of the radius vector to the point under
consideration. For given positions of the Nicols the angle a - 3 will
be constant, viz. the angle between their principal planes, and denoting
it by y, the expression for the intensity may be written in the form
I= {(1+k”) cos y cos $6–2k sin y sin #6}*+ (1 – k”)” cos” (20 – y) sin” #6.
In this expression 8 and k vary from point to point, and at any given
point 8 is a function of the wave length, so that the investigation of
the isochromatic lines becomes very complicated. An idea of their
approximate form may, however, be obtained by finding the shape of
the bright and dark lines for the case of monochromatic light, on the
supposition that along any one of these lines k is sensibly constant ;
that is, that the different points of a bright or dark line are not at
very different distances from the centre. In accordance with this
supposition we have merely to express that I is a maximum or a
minimum when 3 is regarded as the variable. For this purpose Airy
made use of a subsidiary angle X, determined by the equation
2/c
tan X = I-E k? tan y.
When this is substituted in the first term of the expression for I it
becomes
(1+k”)” cos” y secºx cos” (x+$6),
or, replacing secº X, the expression for I takes the form
I= {(1+k”)” cos” y + 4% sin” y} cos” (x+$6)+(1 – k”)” cos” (20 – y) sin” $6.
The points of maximum and minimum intensity along the radius
vector corresponding to a given value of a are found from the
equation dI/d6 = 0. Writing the expression for I in the form
I = P cos” (x + $6) + Q sin” #6,
and differentiating with regard to 8, we have
Psin (2x + 6) = Q sin 6.
From this equation it follows immediately that
sin (2x+ 5) + sin à P+Q
sin (2x+6) – sin 3TT P-Q’
Hence
tan (x + 6) = – tan X #}
- vº.
or substituting for P and Q we have finally
486 ROTATORY POLARISATION CHAP. XVII
(1+/* cosº y +4% sin” y + (1 – kº)” cos” (20-y)
tan (x + 6)= -tan *(1+/ºcosº y Hººsinºy (1 Kºcosº (2a–7)
Writing – tan Q for the right-hand member of this equation we
have
tan (x+5)=-tan Q,
and consequently
x+6=T – Q,
where Q is an angle which varies with a as the point under considera-
tion moves along a bright or dark line. The polar equation of any
such line may be expressed by substituting crº 4-6, for 6 in the fore-
going (as in Art. 265), and we then have
crº=(T-Q – X) - 80
where x is a constant, when y and k are given, and Q varies with a
Fig. 202. Fig. 203.
only. Hence, as the point moves along a bright or dark band, the
radius vector r will increase or decrease according as Q decreases or
increases. The rings will consequently deviate from the circular form
by amounts depending on the variations of Q. Now Q is greatest
when 2a – y = nºr; that is, in the four rectangular positions of the
radius vector corresponding to
a = Y, a = }(y-º-T), a = }(y-º-2T), a = (y-º-3T).
But y = a – 8, and consequently the equation a = }) is the same as
a = - B, and this means that the radius vector of a bright or dark
band will be least along a pair of rectangular lines such that at each
point of them the angle y between the principal planes of the polariser
and analyser is bisected by the principal planes of the plate. The
greatest values of r correspond to the least values of Q, and these
occur when 2a – ) = (n + k)7; that is, in the four rectangular positions
a = }(y- *T), a = }(y-º-ººr), etc.

ART. 268 CONVERGENT CIRCULARLY POLARISED LIGHT 487
Hence the positions of the maxima differ from those of the minima by
45°, and the general shape of a bright or dark band is a square-shaped
curve such as that shown in Fig. 202, the greatest values of the radius
vector corresponding to the diagonals of the square. Close to the
centre k is nearly equal to unity, and the expression for I will vanish
when x + #8 = (n + $)T. But at the centre x = y, consequently when
3 = (2n + 1)7 – 2), there will be a black spot at the centre. Now, for
points at a given small distance from the centre 8 is constant
and I will be greatest when 2a – y = nºr, and least in the directions
2a – y = (n + k)T ; that is, along the diagonals of the square curve.
The spot at the centre will consequently be a cross whose arms are
along the diagonals of the square. With white light this central cross
will be coloured. At a considerable distance from the centre k
approaches zero, and Q becomes very small, and remains sensibly
constant along any band, so that in the outer regions the bands
become sensibly circular, as shown in Fig. 203.
268. Convergent Circularly Polarised Light — Caleulation of
the Intensity.—When the incident light is circularly polarised we
may represent the component vibrations parallel to the principal
sections of the thin plate by
ac=(1+k”) cos wi, y= — (1 + k”) sin wt,
if the amplitude of the incident light be (1 + k) M2. These com-
ponents become respectively, after transmission through the plate,
21 = COS aſt, $/l = k sin wt \
a 2 =k* cos (wt – 6), q/2 = — k sin (wt – 6) ſ”
ac'1=k cos (wt – 6), Aſ'1'= — sin (wt – 6) \
a'2– – k cos wt, gy's = – k” sin wt ſ'
Hence for the vibration parallel to the principal plane of the analyser
we have
Y=cos &#cos wi(1+%” cos 6+k cos 3 – k)+sin wt(k” sin 6+k sin 6)}
+sin 8(cos wi(k sin 6+sin 6)+sin wt(k - k cos 6-cos 6- 6°)}.
Denoting the coefficient of cos of by A and that of sin ot by B we
have I = A* + B2 where
A = k” cos & cos 6+k (cos Á cos 6+sin 8 sin 6 – cos 3)+cos B+ sin 3 sin 6,
B=k*(cos 3 sin 6 — sin 3)+k(cos 3 sin 6 – sin 8 cos 6+sin 8) – sin 8 cós 6,
and the expression for I becomes
I=(1+k”)” – 4k(1 – k”) cos 28 sin” $6+ (1 – k") sin 28 sin 6.
Cor. 1.-If k = 0, we have, as in Art. 233,
I = 1 + sin 28 sin 6.
488 ROTATORY POLARISATION CHAP. XVII
Cor. 2.-If k = 1, we have
I =4;
that is, the light is transmitted parallel to the axis without alteration.
269. Isochromatic Lines—Airy's Spirals.-In order to determine
the form of the bright and dark lines in the case of circularly polarised
light the general expression for I obtained in the preceding article may
be transformed, as in Art. 267, by using a subsidiary angle x related
to B by the equation
-2
tan x= *...* an 23.
Substituting this in the expression for I it becomes
I = (1+k”)*-ī-2(1-k") sin 23 cosec x sin #8 sin(x-5)
=(1+k”)*--(1-kº) sin 23 cosec × {cos(x-3)-cos x}.
Now if we consider a radius vector corresponding to a given value of
Fig. 204. Fig. 205.
B, and if we regard k as sensibly constant in passing along this radius
from a bright band to the adjacent dark band, then X will also be
sensibly constant within this range, and I will pass from a maximum
to a minimum when x – 8 passes from 2nt to (2n + 1)7. Hence,
regarding x as a function of 3 as we pass along a dark band, we shall
have
x - 6–(2n+1)"r,
or writing crº 4-8, for 6 as before, we have for the equation of the band
crº=x-(2n+1)"r-60.
For a given value of n this represents a spiral curve in which r
continually increases as x increases positively. When n is increased
by unity a second spiral is obtained which is the same as the first,
rotated through two right angles, and when n is changed into n + 2

Art. 270 & ANALYSERS 489
the first spiral is repeated, and so on. The complete figure conse-
quently consists of two similar spiral curves, mutually enwrapping
each other, their positions differing by 180°, as shown in Fig. 204.
Close to the centre k is sensibly equal to unity, and the intensity
is approximately equal to (1 + k”)”, so that the central tint is sensibly
white. The spirals consequently will not appear very near the centre,
and at a distance from the centre they will be faint in the directions
parallel and perpendicular to the principal plane of the analyser ; in
other words, the outer regions will be traversed in these directions by
a dark cross.
When the quartz plate is right-handed and the incident light
right-handed the spirals enwrap each other from right to left, and in
the reverse manner when the sign of the plate or that of the incident
light is changed." t
Eacamples
1. Plane-polarised light passes in succession through two quartz plates of equal
thickness and opposite rotations; show that the intensity of the illumination in the
analyser, when the Nicols are crossed, is
2k. 2 .
- - .2 2 --- 2 g & - N s * \ $ 2 & e
I = 4 (1 – K2) (Hºcos a sin #6+ sin 2. COS *} sin” º
2. The curves of zero intensity in Ex. 1 are given by the equations
sin #6= 0,
2k tan #6+ (1 + k”) tan 20 = 0.
The former gives a system of concentric circular rings, and the latter represents
four distinct spirals passing through the points where the circles meet the axes OX
and OY (Fig. 205).
APPLICATIONS
270. Analysers.-The peculiar property which some substances
possess of rotating the plane of polarisation of light may be used as a
means of detecting their presence in solutions when mixed with other
substances which are inactive ; and not only does it afford a qualitative
test, but also an exact determination of the amount of active substance
present, when we know the amount by which the plane of polarisation
is rotated. The application of this property to the analysis of solu-
tions consequently involves the exact measurement of the rotation
of the plane of polarisation, and this involves primarily the accurate
determination of the position of the plane of polarisation. This deter-
mination cannot be made with sufficient precision by means of an
analyser such as a Nicol's prism alone, for with such an analyser the
* Coloured drawings of these figures are given in Airy's Memoir.
490 ROTATORY POLARISATION CHAP. XVII
position of the plane of polarisation of a beam is determined by placing
the Nicol so that the light is completely quenched, or else transmitted
most copiously. Either of these positions can be attained with only
moderate certainty, as it is difficult to ascertain when the field is
exactly brightest or darkest, for near the position of maximum
or minimum illumination the change of intensity, as the Nicol is
rotated, is small, and is scarcely appreciable for a small displacement
in either direction.
To remedy this defect analysers of special construction have been
invented which involve the use of an additional piece of apparatus in
conjunction with the analysing Nicol. This piece is so designed that
the position of the plane of polarisation is determined by comparing
the equality of two lights seen simultaneously and in juxtaposition.
The superiority of this method depends upon the fact that we can say
with much more certainty when two parts of the field are equally
illuminated than when any part of it has attained its least or greatest
brightness.
271. Jellett's Analyser and Saccharimeter. — The analyser
devised by Jellett enables us to determine the position of the plane of:
Fig. 200. Fig. 207.
polarisation by comparing the equality of two lights seen at the same
time and in juxtaposition. To construct the additional piece a rhomb
of Iceland spar, about two inches in length, is taken, and its ends are
sawn off at right angles to its length, so as to form a right prism ABCD
(Fig. 206). This prism is divided by a plane through PQ parallel to
its edges, and making a small angle with the longer diagonal AC of
the base ABCD. One of the two halves, PQCD, is now reversed, and
the two are cemented together with their surfaces of section in contact
as represented in Fig. 207, which is obtained from Fig. 206 by

AT&T. 272 ANALYSERS AND SACCHARIMETERS 491
rotating PQCD through two right angles, keeping the two pieces in
contact at their common surface of section. -
Let us now suppose that a pencil of plane-polarised light falls
normally on the face ABCD. The ordinary ray passes through
undeviated and polarised in a plane perpendicular to AC–that is,
according to Fresnel's hypothesis, vibrating parallel to AC–while the
extraordinary ray is deviated to one side, and, if the rhomb be
sufficiently long, the two will be separated at emergence. Hence in
the Jellett prism ABQD'C' (Fig. 207), when light is transmitted
through it normally, the vibration of the ordinary ray will be parallel
to AO in the part AOQB and parallel to CO in the part C'OQD'.
Consequently, if a cylindrical pencil of light traverse the prism so as to
be equally divided by the plane of section OQ, the extraordinary rays
will be thrown off in opposite directions in the two halves, and may be
stopped by a diaphragm suitably placed. The ordinary rays, on the
other hand, will be transmitted without deviation, and will constitute
a cylindrical pencil equally divided by the plane of section, but polar-
ised in planes inclined at an angle 2a = AOC, where a = AOP, the
angle which the plane of section makes with the diagonal AC (Fig.
206).
Now if the incident light be plane-polarised, the component
parallel to AO is transmitted by one half, and the component parallel
to CO by the other, and therefore if we look through the analyser,
the intensities in the two halves of the field will be different, except
when the plane of the section is parallel to the primitive plane of
polarisation, or perpendicular to it. For one position of the analyser
one half of the field will appear dark and for another position the
other half will be dark, and the angle between these positions is 20.
while for a position midway between them the two halves are equally
illuminated. In this position the primitive plane of polarisation is
parallel to PQ (Fig. 207). The experiment consequently consists in
equalising the intensities by rotating the analyser, and as the position
of equal intensity can be detected by the eye with much accuracy, the
instrument affords an exact determination of the position of the primi-
tive plane of polarisation.
Jellett employed this analyser in his saccharimeter to determine
the rotation of the plane of polarisation produced by saccharine solu-
tions, and he estimated 1 the probable error in the result of a single
observation with such an instrument to be only 0:02 of a grain in a
cubic inch of the solution.
272. Laurent’s Analyser and Saceharimeter. — In Laurent's
* Lloyd's Wave Theory of Light, p. 247.
492 ROTATORY POLARISATION CHAP. XVII
analyser' one half of the field of view is occupied by a plate of quartz,
or gypsum, cut parallel to the axis, and of such a thickness that it
introduces a retardation of . A between the ordinary and extraordinary
rays. The other half of the field is empty or covered by a glass plate
of sufficient thickness to absorb the waves of length. A to the same
extent as the crystalline plate.
Let APB (Fig. 208) be a semicircular plate of glass, AQB the plate
of quartz, and AB the direction of its optic axis.
Then if the plane of polarisation of the incident light be parallel to
OP, it will be transmitted without alteration through the glass and
emerge from it still plane-polarised parallel to OP; but in the crystal-
line plate it will be divided into two components, one parallel to OA
and the other perpendicular to it, and one of
these will be retarded relatively to the other
by a half-wave length. The result is that the
right emerges plane-polarised from the crystal-
line plate, but the plane of polarisation will be
inclined to 0A on the other side at an angle
AOQ = AOP (Art. 47, Cor. 2). The planes of
polarisation of the rays emerging from the two
halves of the compound plate will therefore be
inclined at an angle 2a, where a = AOP. Con-
sequently if a Nicol be fixed before the plate the two halves of the
field will in general appear unequally illuminated, but if the principal
plane of the Nicol be parallel to AB the illuminations of the two
halves will be the same.
Since the thickness of a plate required to produce a retardation A
depends on the wave length, it follows that Laurent's analyser can
be used only with monochromatic light. Jellett's analyser, on the other
hand, is not subject to this restriction, but may be used with white light.
In Laurent's saccharimeter the light is sifted through a plate of
bichromate of potash, which allows only the yellow rays to pass. It then
falls upon the polarising Nicol, to the second face of which the quartz
plate is fixed. After transmission through the plate it passes through
a tube containing the solution under investigation, and thence through
the analysing Nicol to an eye-piece where it is received by the observer.
273. The Biquartz.-A simple and fairly accurate means of
determining the plane of polarisation is afforded by the biquartz.
This consists of two semicircular plates of quartz placed in juxta-
position, each being cut at right angles to the axis. In one the
rotation is right-handed, and in the other it is left-handed. The two
* Journal de Physique, 1874 and 1879.
Fig. 208.

ART. 273 THE BIQUARTZ 493
plates are of the same thickness, so that they produce equal rotations
of a given ray, but in opposite directions. Thus if QOP (Fig.
209) be the direction of the incident vibration, it will be rotated
to the right in the plate APB, and by an equal amount to the left in
the plate AQB. Now if the incident light
be white the various simple colours will be
rotated through different angles, and it
follows that if the emergent light be
analysed by a Nicol, waves of different
refrangibilities will be quenched in the two
halves of the field, and they will con-
sequently appear of different colours.
There is one position of the Nicol,
however, in which the two halves of the
field may have the same colour. For consider the ray of which
the plane of polarisation has been turned through a right angle.
In one plate it will be turned to the right, and the vibration
of this ray will be parallel to OQ'. In the other plate it is turned
through 90° to the left, and the vibration will be parallel to OP’.
Hence the vibrations of this ray will be parallel to the same direction
in both halves of the field, viz. to a line P'OQ' perpendicular to POQ.
Thus if the principal plane of the Nicol be parallel to POQ the light
of the same wave length will be quenched in both halves of the field,
and the other colours will be present in the same proportion in each
half, so that they will exhibit the same tint.
By suitably adjusting the thickness of the quartz plates the ray
which is rotated through a right angle may be made any one we please.
If it be that corresponding to the brightest part of the spectrum (the
yellow) this light will be absent and the common colour of both halves
will be the tint of passage (see Art. 255). A minute rotation of the
Nicol in one direction will render one half of the field blue and the
other half red, and a displacement in the opposite direction will render
the former half red and the latter blue. The thickness of the biquartz
which exhibits the tint of passage is about 3.75 mm. When this is
secured the plane of polarisation of the incident light is perpendicular
to the principal plane of the analyser.
When the tint of passage is attained the substance under investi-
gation is placed either between the polariser and the biquartz, or
between the biquartz and the analyser. If the substance rotates the
plane of polarisation the tint of passage will disappear from the field
of the analyser, and to restore it the Nicol must be rotated in the
direction in which the plane has been rotated and through an equal
Fig. 209.

494 ROTATORY POLARISATION CHAP. XVII
angle. In practice the light should fall normally on the biquartz, and
the axis of rotation of the analysing Nicol should be parallel to the
direction of the incident light. The greater part of the errors arising
from these conditions not being accurately fulfilled may be corrected
by rotating the Nicol through 180° and taking a second reading.
A plate formed of four quadrantal pieces of quartz, alternately
right- and left-handed, is also used in the same way.
A method of greater accuracy is to subject the light to spectro-
scopic analysis. For this purpose it is thrown on the slit of a spectro-
scope by a lens so adjusted as to form a real image of the biquartz on
the slit. Two spectra are seen in the field, one from each half of the
biquartz and situated one above the other. Each of these is crossed
by a dark band corresponding to that colour extinguished by the
analysing Nicol. By rotating the Nicol the bands move across the
spectra in opposite directions, so that the Nicol may be so adjusted
that they are one above the other. When this is attained the light of
the same wave length has been quenched in each half of the field, and
the principal plane of the Nicol is parallel to the direction of vibration
of the incident light. When the spectroscope is employed a single
quartz plate may be used instead of the biquartz. For when solar
light is used, and a spectrum formed which exhibits the Fraunhofer
lines, the dark band arising from the quartz plate may be brought to
coincide with any one of the dark lines of the spectrum. On intro-
ducing a rotatory substance the dark band will be displaced across the
field, and to restore it to its original position the analysing Nicol must be
turned through an angle equal to the rotation produced by the substance."
274. Poynting's Analyser.—The leading idea in the design of
analysers for the purposes of Saccharimetry (as already illustrated in
the foregoing examples) is to construct a piece of apparatus such that
when a beam of plane-polarised light is transmitted through it, the
light transmitted in one half of the field shall be polarised in a plane
which is inclined at an angle to the plane of polarisation of the light
transmitted through the other half. This, in general, is obtained by
covering the two halves of the field with two pieces possessing rotatory
power such that the plane of polarisation is rotated through different
angles in passing through them.
The most obvious method of effecting this is to cover the two
halves of the field with layers of different thickness of some rotatory
substance, for example, with two plates of quartz of different thickness
cut perpendicularly to the axis, or with strata of a rotatory liquid
of different thickness. This is the method employed by Professor
* M. Broch, Amm. de Chimie et de Phys. (3), tome xxxiv. p. 119, 1852.
ART. 275 SOLEIL'S COMPENSATOR AND SACCHARIMETER 495
Poynting." In the first form of apparatus a uniform circular plate of
quartz, cut at right angles to the axis, was divided along a diameter
into two halves. One half was slightly reduced in thickness, and the
two were then reunited so as to form a circular plate as before, with
this difference, that one half of the plate now exceeded the other in
thickness. When a beam of plane-polarised light is transmitted through
such a plate the plane of polarisation of the light issuing from one half
will be inclined to that of the light issuing from the other by an angle
depending on the difference of thickness of the two halves of the plate.”
In the second form of apparatus a rotatory liquid is employed, and
the arrangement is exceedingly simple. The liquid, say a solution of
sugar, is contained in a cell through which the beam of polarised light
is transmitted. A piece of plate glass several mm. thick is immersed
in this cell with its faces at right angles to the beam of light, and it is
so arranged that one half of the beam passes through the plate while the
other half passes by its edge. The former half of the beam, since it passes
through the glass, obviously passes through a less thickness of the rota-
tory liquid than the latter, and is consequently less rotated. Hence the
light emerging from the two halves of the field is polarised in planes
which are inclined to each other. The angle between the planesof polaris-
ation can be varied by varying the thickness of the interposed glass plate.
275. Soleil's Compensator and Saccharimeter.—In the saccha-
rimeter of M. Soleil the rotation produced by any substance under
Fig. 210.
examination is compensated by means of a plate of quartz of which
the thickness can be varied at will. This part of the apparatus is
called the compensator, and is similar to Babinet's compensator (Art.
240) only in so far as it consists of two quartz wedges abc and def
(shown exaggerated in Fig. 210), of which one can slide on the
other along their common face so that their joint thickness may be
varied. In this compensator, however, both wedges are cut with
J. H. Poynting, Phil. Mag. vol. x. p. 18, 1880.
* In order to vary this angle Professor Poynting suggests that one half of the
plate might be made up of two wedges, as in Babinet's compensator (Art. 240), so
that its thickness could be altered at pleasure by sliding one of the wedges over the
other.

496 ROTATORY POLARISATION CHAP. XVII
their faces ac and ef perpendicular to the axis, and they are both left-
handed crystals. The light traverses both wedges in the direction of
the optic axis, and the action of the compensator is consequently to
introduce any desired amount of left-handed rotation. In conjunction
with this pair of left-handed wedges a right-handed quartz plate Q is
used, so that when the thickness of abcd is varied, the combined effect
of Q and abcd is to introduce any desired amount of rotation, which
may be either right-handed or left-handed according to the thickness of
abcd. Thus when the thickness of abcd is properly adjusted, the right-
handed rotation produced by Q will be exactly neutralised. For a
greater thickness of abcd the combined effect of Q and abcd will be a
left-handed rotation, while for less thicknesses the resultant effect will
be right-handed. ---
In the saccharimeter the light is polarised by a Nicol's prism PN.
It then falls upon a biquartz B, and afterwards traverses the sub-
stance S under examination. After emerging from the active substance
it traverses a right-handed quartz plate Q and then falls upon the com-
pensator abcde. Finally the apparatus is terminated by the analysing
Nicol AN and a small Galilean telescope T, focussed on the biquartz B.
In making an experiment the tube S is filled with water,” and the
telescope is focussed on the biquartz. The micrometer screw by which
one of the wedges of the compensator is moved is now turned to the
zero position, and the analysing Nicol is rotated till the tint of passage
is obtained in the field. The water is now removed from the tube S
and the rotatory substance is introduced. If the substance is active
the plane of polarisation is rotated and the tint of passage disappears.
It is, however, restored by turning the micrometer screw of the com-
pensator so as to alter its thickness in such a manner that the rotation
produced by the substance S is neutralised. If the substance is right-
handed the thickness of the compensator must be increased, and if
left-handed, diminished. The angle through which the micrometer has
been turned in order to restore the tint of passage determines the
amount of rotation produced by the substance.
This method requires the light to be white and the liquid colour-
less in order to obtain the tint of passage. To apply the instrument
to cases in which the light or the liquid is coloured, M. Soleil added
an extra piece which he called the producer of the tint of passage. This
part is placed in front of the saccharimeter, and consists of a Nicol N'
and a quartz plate Q'. The incident light is polarised by the Nicol
* This avoids the necessity of readjusting the focus of the telescope when the
rotatory liquid is introduced, for if the telescope were focussed on the biquartz when
the tube S is empty, it would not be in focus when the tube is filled with a liquid.
ART. 276 DE SENARMONT'S POLARISCOPE 497
N", and after traversing the rest of the apparatus, the field of the
analyser will in general be coloured. By turning the Nicol Nº a
position can generally be found in which the plate Q gives a tint com-
plementary to the colour of the liquid for the light employed. The
colour effect is thus neutralised and the tint of passage can be
obtained.
276. De Senarmont's Polariseope.—In Soleil's compensator, just
described, the characteristic piece consists of two wedges of left-handed
quartz placed so as to form a plate of variable thickness. If, however,
the wedges are of opposite sign so that one of them is right-handed
while the other is left-handed, the plate will be such that along the
central line the plane of polarisation will be rotated as much to the
right in one prism as to the left in the other, and the resultant rota-
tion along this line will be zero. Hence when the Nicols are crossed
this line will be black. On one side of this line the resultant rotation
will be to the right, and on the other side it will be to the left, according
to the sign of the wedge of predominating thickness at the place in
question, so that on either side of the central dark band coloured bands
are symmetrically situated analogous to the bands observed in Babinet's
compensator. When the analysing Nicol is turned the central band is
displaced to one side or the other by an amount depending on the
rotation of the Nicol.
This apparatus is known as De Senarmont's* polariscope, and it
affords a very accurate means of determining the position of the plane
of polarisation of any source of light which is partially or wholly
polarised.
277. MacCullagh's Theory of the . Double Refraction of
Quartz.-The singular laws of the double refraction of quartz,
and its peculiar property of rotating the plane of polarisation,
were first connected in a general theory by MacCullagh.”
In a common uniaxal crystal two waves are propagated in any
direction, one according to the ordinary laws of refraction, and the
other in a manner depending on the inclination of the wave normal to
the optic axis. These waves are polarised at right angles and travel
with different velocities. Consider a plane wave passing through a
crystal in a direction making an angle 6 with the optic axis. Take
the wave normal for the axis of 2 and the axes of a; and y in the wave
front, the former being perpendicular to the optic axis. The axes of
a; and y will be parallel to the azimuths of vibration in the ordi-
nary and extraordinary waves respectively (according to Fresnel's
* De Senarmont, Ann. de Chim. (3), tome xxviii. p. 279, 1850.
* MacCullagh, Trams. Roy. Irish Academy, vol. xvii. 1836.
2 K
498 ROTATORY POLARISATION CHAP. XVII
hypothesis), and if £ and m be the displacements the equations of the
two vibrations are - -
ºš A3%, dº Bºn - (1)
º
--> -
ŻT *dz2, …}}T *dº
where, in a positive crystal (a ->b), we have
A= a”, and B-a” cos” 64-b% sin” 9. (2)
To account for the rotatory polarisation, and represent the twisted
structure of quartz (p. 465), MacCullagh introduced differential co-
efficients of the third order into the equation of the vibrations, and
assumed
d”; d”; , ºdºm
# = A;+0. (3)
dºm pºdºm odºš
These equations are satisfied by
#= p cos º: (vt – ?), m = q sin #(ot-2)
if, writing q|p = k, we have the equalities
vº–A–2tr0kſ!,
w?= B-2TCſki,
that is, if
2TrC 1
A-B-º(*-ī) = 0,
which by the formulae (2) becomes
! sin” 6
*-*.*@-tº-1=0,
a quadratic in k, of which one root is the negative reciprocal of the
other. t
Denoting the roots by k and – 1/k, we see that there are two
elliptic vibrations as derived by Airy. Thus we have
271-
2 e
$1=p cos #(ºt — z), mi–pk sin !
(vić - 2),
and
v,”– A – 2+Ck|l.
This is an elliptic vibration propagated with velocity v, and having its
axes in the ratio 1 : k.
The second value (– 1/k) gives the vibration
2 . 27ſ
$2 =p cosé (og - 2), 72= -#sin +(vº - 2),
where
vº- A + 2tr0/kl.
ART. 277 MACCULLAGH’S THEORY 499
The axes of this vibration are in the ratio — k : 1, and it is propagated
with a velocity vs. Consequently we have in general two elliptic
vibrations travelling with different velocities and having the same
eccentricity, but the major axis of one coincides with the minor axis
of the other.
When the light passes in the direction of the axis of the quartz we
have k = + 1, which shows that there are two rays circularly polarised
in opposite senses. Their velocities of propagation are
vº–a4–2TC/l, vº–a4+2+C/l,
TrC TC
•=a( -:) *=(l +;).
The ordinary velocity a is consequently their arithmetic mean.
Let the quartz be a parallel plate of thickness e. The time retard-
ation of one ray relatively to the other in passing through the plate
will be
€ 6 63 TrC\-1 e TCY-1 27teC
r=#-º-;(1-#)"-(1+...)"—º.
Taking the velocity in air as unity, and the wave length \, correspond-
ing to l in the crystal, we have l ; A = a + 1, and the phase difference
on emergence will be
or approximately
6 = 2irt 4trºeC &
- NTT Taº?
But the rotation is half the phase retardation, therefore
_21%C
P=-aº'
which contains the experimental law of M. Biot, viz. that the rota-
tion is inversely proportional to the square of the wave length. The
sign of C determines whether the rotation is right-handed or left-
handed, and the divergence from Biot's law exhibited by some sub-
stances might be accounted for by supposing that in these substances
C is a function of the wave length.
Although the additional terms introduced by MacCullagh are
quite empirical, yet the investigation based on them is instructive as
showing one form of equation leading to results consistent with
experiment. It does not follow that they are the only terms which
would be equally consistent. Much work is being done at the present
time in determining the precise arrangement of the atoms in a crystal
of quartz (by the employment of X-rays), and it is probable that a less
empirical solution of the problem may be based upon it.
500 ROTATORY POLARISATION CHAP. xvil
Eacamples
1. If the assumed equations of motion be
d?: Ad”; , , dºm
dźT*::::: + C. y
dºm – dºn ,dº;
#=Bā-C;
show that k is determined by the quadratic
! sin” 6
k? — 2TrC
C’
2 -*
(a” – bº)k — G- 0.
This equation indicates that the ratio of the axes is not the same for both the elliptic
vibrations, and explains the difference between the ratios observed by Airy.
2. Given - -
a = 0.64859, b=0.64481, X = 0000258 in.,
e=0.03937 in., p=0.333,
show that the quadratic for k is
k°–258 k sin”9 – 1 = 0.
CHAPTER XVIII
ABSORPTION AND DISPERSION
278. Selective Absorption.—When light is being transmitted
through any substance part of it is generally taken up by the medium
or absorbed, and this absorption is different in general for rays of
different refrangibility. It is on this account that white light generally
becomes coloured after traversing a sufficient thickness of any substance.
There is no body perfectly transparent—that is, transparent to rays of
every refrangibility—and, on the other hand, there is no substance, in
the same sense, perfectly opaque. Air" in sufficient thickness colours
the light of the sun, at first yellow and afterwards red, and pure water
produces the same effect in a more decided manner, while gold in a
thin leaf transmits a faint greenish light. Metallic silver appears to
allow the actinic rays to pass, and an ordinary stone wall is transparent
to the long waves from a Hertzian vibrator (Chap. XXI.).
The character of the absorption which takes place in any substance
may be analysed by submitting the transmitted light to examination
in a spectroscope. The spectrum formed by the transmitted light will
be crossed by dark bands corresponding to the colours which have been
most rapidly absorbed. Some substances exercise marked absorption
in one or more parts of the visible spectrum, while others transmit
freely the luminous rays, but absorb certain of the non-luminous waves.
Thus the spectrum of light transmitted through a solution of chloro-
phyll (the colouring matter of the leaves of plants) exhibits dark bands
in the red, yellow, green, and violet, and human blood produces
marked absorption bands in the yellow and green portions of the Absorption
spectrum, which are visible even when the quantity of blood is so *
small as scarcely to affect the colour of the solution. Permanganate
* The transmitted light is the residue which passes through when absorption
proper, and internal reflection arising from irregular structure of small suspended
particles, have played their part. The relation of absorption to internal reflection
deserves consideration.
501
502 ABSORPTION AND DISPERSION CHAP. xv.111
of potash (Condy's fluid) exhibits several dark bands in the green part
of the spectrum, when the solution is dilute, and these are character-
istic of the permanganates. With stronger solutions the dark bands
broaden out, and ultimately unite to form one broad dark band, the
Gases.
Analogy.
whole region being now absorbed. And again, Brewster discovered
that a solution of oxalate of chromium and potash produces a single
narrow absorption band, which is so well defined that he suggested its
use in the measurement of refractive indices with artificial light, when
sunlight cannot be obtained.
For the most part, however, the light which has passed through an
absorbing medium exhibits a single maximum of absorption. The rays
in a certain region of the spectrum are most effectively stopped ; but
in the case of gases the phenomena are very different. The rays
stopped by them are characteristic and well defined, so that the
spectrum of the transmitted light exhibits a certain number of fine dark
lines, distinctly separated from each other. Gases accordingly allow
almost all the light to pass except a few rays of definite refrangibility.
When light of any wave length is intercepted by an absorbing
substance the energy of the corresponding ether vibrations is imparted
to the matter, and in general reappears as heat, as is manifested by the
rise in temperature of the absorbing substance. The vibrations of the
ether are taken up by the matter molecules, and these in turn vibrate
and become centres of disturbance. What is absorbed then is energy.
The matter molecules are intimately bound up with the ether, and the
energy of the ether vibration is converted (possibly) into energy of
motion in the matter molecules.
Now if we consider any system of matter molecules it will be
easily inferred that some ether vibrations will be much more powerfully
absorbed than others, viz. those most competent to excite that vibration
which the matter molecules execute freely. It is well known that a
series of slight taps may excite a considerable oscillation in a common
pendulum, the condition being that the taps be timed to the period of
swing of the pendulum. The same point is illustrated in the resonance
of organ pipes which are excited by vibrations of their own free period.
So, again, if a number of ships at sea be disturbed by waves, each ship
will have its own free period, and if for some of them this happens to
be the same as that of the waves, these particular ships will be thrown
into a state of violent oscillation. They will thus act as absorbers of
the energy of the waves. If the periods are not synchronous the
motion caused by one wave will be destroyed by another, so that the
energy absorbed from one wave will be given out to some other, and
there will be no accumulation of energy in the ship.
ART. 278 SELECTIVE ABSORPTION - 503
It consequently follows that of the multitude of waves in a pencil
of white light those which will be most freely absorbed by any sub-
stance will be those which synchronise in period with the free vibrations
of the matter molecules. The molecules absorb waves of their own Selection.
period of vibration, and these again are obviously the waves which the
molecules will excite when they become centres of disturbance." It is
consequently to be expected that different kinds of matter should
absorb waves of different periods or rays of different refrangibility.
That they should, in fact, select certain rays in preference to others,
and this is what is known as selective absorption.
In the case of sounding bodies we know that some can take up
and resound to vibrations of almost any period, while others confine
their vibrations to some fundamental tone and its harmonics. The
former class is illustrated in the sounding board of a piano or the body
of a violin, while examples of the latter are an organ pipe or a stretched
string. So it is in the case of absorbing substances. Some, such as
black opaque bodies, absorb the luminous waves of every length, and
others, such as gases, absorb only waves of certain definite periods.
When an opaque substance, like a piece of iron, is heated, at low
temperatures it emits the longer or dark heat waves. As the tempera-
ture is raised the vibrations become more intense, the molecular
agitation more excited, and waves of shorter period are emitted, so
that at a certain stage the waves of the red end of the spectrum
appear and the iron mass assumes a dull red heat. Carrying the
elevation of temperature still further, vibrations of shorter and shorter
periods are superadded, till finally all the waves of the visible spectrum
are emitted and the mass has reached a white heat.
In the case of a gas, on the other hand, the molecules have free
paths; they are at distances from each other which are large compared
with the dimensions of the molecule, and each, in making its excursions
between its successive impacts with the others, will vibrate in its own Gases.
free period of oscillation. The waves absorbed by a gas will therefore
be well defined and of certain definite periods, viz. those of the free
vibration peculiar to the molecule. And these, again, will be of the
same period as those emitted by the gas when it becomes incandescent ;
that is, the spectrum of an incandescent gas will consist of one or
more well-defined bright lines, images of the slit, which correspond
exactly to the dark absorption lines which appear in the spectrum of
white light which has passed through the gas when cold. Thus the
* Tourmaline rapidly absorbs the ordinary wave ; that is, it absorbs light which
is plane-polarised in a certain direction, and when heated, so as to become in turn a
source of light, it radiates plane-polarised light.
504 ABSORPTION AND DISPERSION CHAP. XVIII
spectrum of incandescent sodium vapour contains a bright double
yellow line, and the spectrum of white light which has passed through
cold sodium vapour is crossed by a dark double line in the yellow
corresponding exactly to the bright yellow line given by the vapour
when hot. The spectrum of a given gas may, however, vary consider-
ably with changes of temperature and pressure.
Thus hydrogen enclosed in a vacuum tube at atmospheric pressure
exhibits, with a certain amount of continuous spectrum, four bright
lines, three of them situated in the red, blue, and violet respectively,
and the fourth, a faint one, in the extreme violet. As the pressure is
decreased they grow sharper and more distinct, while the continuous
part of the spectrum grows fainter and dies away. If, however, the
exhaustion be pushed further, the red and violet lines fade out and
the green alone is left. On the other hand, if more hydrogen be
compressed into the tube, the bright lines will be found to broaden
out and grow fainter, while the continuous spectrum becomes more
brilliant. So, again, oxygen at high temperatures—that is, with a
strong spark—shows a number of bright lines chiefly in the violet, but
if the temperature be lowered only four lines are presented—one in
the red, two in the green, and one in the blue.
Increase of temperature in general introduces new lines, and
increase of pressure causes them to grow wider and broaden out into
a continuous spectrum. This is not contrary to expectation, for
increase of temperature corresponds to increased molecular agitation,
and increase of pressure brings the molecules closer together, so that
collision becomes more frequent. During collision the vibrations are
constrained and the emitted waves are altered. The denser the gas
the greater is the proportion of time spent in collision and the less
the free path of the molecule, so that the free or characteristic
vibrations of the molecule grow less and less predominant in the
emitted light. The result is that when the gas is compressed to near
its liquid volume the irregular vibrations have gained the field, and
the characteristic lines of the gas have broadened out into a continuous
spectrum like that of a solid or liquid.
279. Coefficients of Transmission and Absorption.—When a
beam of light of a given wave length is transmitted through a layer
of an absorbing substance a certain fraction of it is absorbed, and it is
assumed that this fraction is independent of the intensity of the
incident beam. It follows from this that the amount of light which
passes through a number of equal layers diminishes in geometrical
progression as the number of layers increases in arithmetical pro-
gression. Thus if I denotes the intensity of the incident light, Ia
ARt. 280 CRITICAL THICKNESS 505
will be the intensity after transmission through unit thickness, where
o, is a proper fraction and depends upon the nature of the substance
and the refrangibility of the light employed. For a given wave length
a will be different for different substances, and for a given substance
a will vary with the wave length. The intensity of the beam incident
on the second unit of thickness being Ia, it follows that the intensity
of the beam transmitted through it will be Ia . a = Ia”. Similarly the
intensity after transmission through three, four, etc., units will be
Ia", Ia", etc., respectively, and the intensity after transmission through
a thickness a will be
I’ = Ici”.
The quantity a is termed the coefficient of transmission. If the in-
cident pencil be not monochromatic, but be composed of waves of
various lengths, of which the intensities are I, I, Is, etc., with co-
efficients of transmission ap ag, as, etc., the intensity of the transmitted
beam will be
I’= IIdiº + Iza.” + Isagº + . . . =XIa”.
The quantity, and the quality, of the transmitted light will con-
sequently depend upon the primitive composition of the beam—that
is, upon the nature of the source ; also upon the nature of the ab-
sorbing substance—that is, upon the various coefficients a1, ag, etc.;
and finally upon the thickness traversed.
If, on the other hand, we deal with the absorption so that a beam
of intensity I is changed by an amount – dſ, when transmitted through
a layer of thickness do, we assume that dI for light of a given wave
length is proportional to do, and also to I, so that we have
— dI=8Ida:
where 6 is a constant depending on the nature of the substance, and
on the wave length of the light. This constant is termed the
coefficient of absorption of the substance for the light in question.
Integrating this equation we have at once
log. (I/Io) = -8a.
where Io is the intensity of the incident beam, and I the intensity after
it has traversed a thickness a of the substance. Writing this equation
in the form
I= I0e-8%,
We see that the coefficients of absorption and transmission are con-
nected by the relation a = e−8.
280. Dichromatism—Change of Tint—Critical Thiekness.—
The coefficients of transmission being in general different for the
506 ABSORPTION AND DISPERSION CHAP. xviii.
different colours, it follows that the emergent light will be coloured,
the colour being generally the same for all thicknesses traversed, and
the tint merely becoming deeper as the thickness is increased. There
are, however, some media in which the colour of the transmitted light
varies with the thickness. Thus if I, and I, be the initial intensities
of the two colours which predominate in the transmitted light, an and
as their coefficients of transmission, then their intensities in the trans-
mitted beam will be Ila,” and Isa,”. Consequently, if we have I1-I,
and ag-al, it will follow that for small thicknesses Ila,” will be greater
than Iga,”, but as the thickness increases these quantities will become
more nearly equal, till at last exact equality will be obtained, and for
greater thicknesses Iga,” will be greater than Ila,”. The thickness at
which equality is reached is determined by
Ilai"= Isaº,
or, taking logarithms of both sides, we have
a-ºº: Iz – log. II,
log, al- log. do
which determines the thickness at which the change of colour takes
place. For thickness less than this I, predominates, but for greater
thicknesses I, predominates. For example, cobalt glass transmits both
blue and red light, but the blue to a less extent than the red. For
this reason, when the thickness is considerable, the blue rays of the
spectrum are almost entirely absorbed, and the colour of the trans-
mitted light is red. The glass appears blue, on the other hand, when
the thickness is small.
The more ordinary case is due to the absorption being mainly in a
single region of the spectrum while the remaining parts of the spectrum
are but slightly and nearly equally absorbed.
281. Colour produced by Absorption.—From the foregoing con-
siderations it is clear that absorption must be a prime agent in the
production of colours in natural bodies. When white light passes
through any substance some of its components are wholly or partially
absorbed, and the transmitted beam is more or less coloured in con-
sequence. The tint depends on the nature of the missing rays, and is
the resultant of the rays which have been allowed to pass. Ordinarily
bodies are seen by light which has been scattered at their surfaces by
rugosities or inequalities. Structural inequalities in the interior also
scatter the light which has penetrated beneath the surface, so that it
is reflected back and reaches the eye, after having traversed some
thickness of the material, weakened and robbed of some of its con-
stituents by absorption.
ART. 281 COLOUR PRODUCED BY ABSORPTION 507
An excellent illustration of the effect of interior scattering is pre-
sented in the striking dissimilarity in appearance between a liquid and
its froth. Pure colourless water gives a white foam, clear glass and
ice may be powdered into white dust, and the white froth on beer
scarcely exhibits a trace of the deep colour possessed by the liquid.
In all these cases the incident light suffers partial reflection at the
surface of each bubble or particle on which it falls. The intromitted
rays are almost all returned by partial reflections at a multitude of
surfaces, and emerge again from the substance in all directions. The
consequence is that such accumulations of particles or films are highly
opaque to light, and if they exercise no particular absorbing effect
on any of the simple colours they will reflect all colours equally and
appear intensely white ; such, for example, is a cloud. If, however,
the material absorbs some of the simple colours more powerfully than
others the emergent light will be coloured, and this we designate
the colour of the body. It may, however, happen that the substance
absorbs some of the rays more readily than others, and yet in a
finely divided state it appears white because nearly all the light is
reflected very close to the surface, and does not pass through a
sufficient thickness of the material to tinge sensibly the scattered
light. This happens in the case of the froth of beer just mentioned,
or when blue or red glass is finely powdered, the resultant powder
being nearly white. Some substances, however, absorb particular rays
so strongly that passage through a very thin film may tinge the
scattered light in a decided manner.
It is interesting to note the variations in appearance produced
when a coloured body is placed successively in the different parts of
a pure spectrum. A white lily placed in the red looks red, in the
blue it appears blue, and placed in the green it exhibits a green colour.
The white lily contains no body which absorbs any particular colour
more readily than any other. It affects all wave lengths alike, so that
it assumes the colour of the light that falls upon it, and in white light
it appears white:
A coloured flower, on the other hand, when held in the different
parts of the spectrum, will appear of the same colour as that part in
which it is held, but it will vary much in brightness as it is moved
from one part to another. Placed in a colour which it does not absorb
it will appear bright, for all the light that falls upon it will be reflected
and traverse the flower cells without sensible diminution, but in a
colour which it absorbs freely it will appear almost black, for nearly
all the incident light will be absorbed. Thus a red poppy will show
a brilliant red when placed in the red part of the spectrum, but it
508 ABSORPTION AND DISPERSION CHAP. XVIII
will appear dark in the green or blue. Its cells are filled with a fluid
which absorbs the green and blue, but which freely transmits the red.
The effect of internal reflection, and subsequent absorption, in pro-
ducing colour is well shown by placing a carefully filtered coloured
liquid in a basin. The light falling on the surface of the liquid pene-
trates the interior, and after reflection at the sides of the basin it
emerges and reaches the eye, having suffered absorption, and the
coloured liquid is seen. If, however, the interior of the basin be
painted black, no light will be reflected at its sides, and consequently
none will reach the eye from the interior, and the liquid will appear
black. The upper surface of the liquid reflects all the rays in the
same proportion. Any selective reflection of the surface could be
detected by observing in it the image of a white object. This will
appear coloured if the rays are reflected in different proportions. By
sprinkling a little flour or powdered chalk in the solution, the full
colour may now be restored, for the light which enters the liquid will
be reflected from the white particles, and, emerging, will reach the
eye, robbed of those rays which are absorbed most copiously by the
solution. t
Reflection then is the proximate cause of the colour of these
bodies, inasmuch as without reflection no light would reach the
eye, but absorption is the ultimate cause, for it is thus that the
reflected light is deprived of some of its constituents and becomes
coloured.
At this point it will be easy to understand the colour displayed
by a mixture of pigments. From what has been said the mixture
should present a tint resulting from the combined effect of the colours
transmitted by all the pigments. Thus a mixture of blue and yellow
paints appears green, not because green is situated between blue and
yellow in the spectrum, but because green is the only colour which is
freely transmitted by both. The blue pigment absorbs the red, orange,
and yellow rays, while the yellow paint absorbs the violet, indigo, and
blue. Hence the green alone is suffered to pass through both and
emerge to the eye after internal reflection.
Many bodies, however, possess surface colour, and seem to select
some rays for reflection in preference to others. Thus many of the
aniline dyes appear one colour when viewed by reflected light and
another when viewed by transmitted light. To observe this a small
quantity of a solution in alcohol may be spread on a glass plate and
allowed to evaporate. A thin film of the aniline is thus deposited
on the plate, and this will present one colour when looked at and
another when looked through. A similar selection appears to take
ART, 281 COLOUR PRODUCED BY ABSORPTION 509
place in reflection from metals, gold reflecting yellow and copper red.
A large number of substances have been found to possess this surface
colour, reflecting a very large percentage of certain rays and a very
small percentage of other rays. Thus Wood * shows for cyanine, which
has an absorption band near the D lines, that the spectrum of the light
reflected from it has a very dark band in the green, and that its surface
colour arises from the absence of this green light, and therefore is not
yellow but a deep plum colour. It follows from the electromagnetic
theory that the intensity of the reflected light depends upon the
coefficient of absorption of the body for the light, and upon the index
of refraction, and as the refractive index is large on the red side of the
absorption band (see Art. 284), but low on the green side, the intensity
of the reflected light will be great near the absorption band on the red
side, but diminished on the green side. Wood traces, by means of
curves, the influence of both these quantities for cyanine, and explains
the position in the green of the band of minimum reflection, showing
that scarcely 2 per cent of the light of corresponding wave length is
reflected at normal incidence.
The light transmitted through such a substance will consequently
be deficient from two causes. It will be largely deprived of certain
rays by reflection at the surface and of certain others by absorption in
the interior, and the constitution of this doubly sifted beam will
determine the colour by transmission, or the body colour of the
substance.
On the other hand, the light reaching the eye by reflection con-
sists of two parts, one consisting of light near the absorption band,
which has been abnormally reflected, and the other of light which has
been reflected in the ordinary way. This mixture determines the
colour as seen by reflection, and when examined through a Nicol's
prism shows a marked change in character. The ordinary reflected
part may be plane-polarised at a proper angle of incidence, and can be
extinguished by the Nicol, but the anomalous part of the beam, as in
the case of metallic reflection, and for similar reasons, is never plane-
polarised, and will therefore be always visible through the Nicol. The
result is that the colour of the surface will change as the analyser is
turned. For fuchsine, under such circumstances, elliptic polarisation
is found in the yellow and to a greater extent in the green and the
* This property of reflecting a large percentage of certain wave lengths has been
applied to the isolation of waves of definite wave length in the infra-red by suc-
cessive reflections from several surfaces of an absorbing substance, such as fluor-spar
sylvine, or quartz. See Preston's Heat, 2nd ed. p. 606.
* Wood Phys. Rev. vol. xiv. pp. 315-318, May-June 1902. See also Wood. Phil.
Mag. vol. iii. pp. 607-632, June 1902.
510 - ABSORPTION AND DISPERSION CHAP. XVIII
blue; it appears a rose colour by transmitted and green by reflected
light, but when examined through a Nicol it appears a peacock blue.
282. Absorption Lines in the Solar Spectrum.—When a pure
spectrum of the sun's light is obtained it is found to be crossed by an
enormous number of very fine dark lines. These lines were first
noticed by Dr. Wollaston, but they were subsequently rediscovered and
investigated with great skill by Fraunhofer, after whom they are known
as “Fraunhofer's lines.” It is easy to understand how this want of
continuity in solar light escaped notice till great precautions were
taken to secure a comparatively pure spectrum. For if the waves
emitted by the sun were all of one period—that is, if its light were
perfectly homogeneous—then on transmission through a prism there
would be no separation or dispersion, and in the spectroscope a narrow
image of the slit would be seen ; while if the light contained only
waves of two definite lengths, say red and blue, then two narrow
images of the slit would be seen, one red and the other blue, and
these would be separated by a dark space, the width of which will
depend upon the dispersive power of the prism or the resolving power
of the spectroscope.
With light composed of many colours we shall have as many
coloured images of the slit arranged parallel to each other, and
separated by dark spaces corresponding to the waves which are
not represented in the original beam. When the constituent waves
of the composite beam become sufficiently numerous the various
images of the slit will be in close proximity, and a stage will be
arrived at when they will unite to form a continuous band or even
overlap each other. This stage may obviously be postponed by
increasing the dispersion—that is, by employing several prisms—and
also by diminishing the width of the slit.
. The number of different waves in a composite beam may, how-
ever, be so great as to defy resolution by any apparatus which we
can construct, and the spectrum will then appear quite continuous;
but we are not, therefore, to conclude that it comprises light of all
possible wave lengths, but merely that its constituents are too numer-
ous to be separated by our apparatus. Consequently, whether a
spectrum appears continuous or not will ultimately depend upon the
resolving power of the instrument employed.
The existence of dark lines crossing the solar spectrum shows
us that the corresponding waves are either wholly or largely absent
from the light of the sun when it reaches us, and the inference
is that they have either been absorbed by some medium between us
and the sun, or else they have not been emitted at all.
AIRT. 282 ABSORPTION LINES IN THE SPECTRUM 511
Now the spectra of incandescent solids, such as iron, charcoal, etc.,
do not exhibit the numerous dark lines seen in the solar spectrum, and
if the sun be an incandescent mass we infer that he emits the missing
rays, that they are intercepted by some absorbing medium, and that
this medium is the atmosphere of cooler gases surrounding the inner
nucleus of the sun." Thus the dark D lines in the yellow part of the
solar spectrum indicate that the light from the inner part of the sun
passes through a stratum of sodium vapour in the outer atmosphere,
for it is found that these lines correspond exactly to the absorption
bands of sodium vapour, and that the corresponding rays are emitted
by the vapour when incandescent.
By comparing the other dark lines with the absorption lines
produced in continuous spectra by other vapours, or with the
bright lines afforded by them when incandescent, the existence
of many other substances has been detected in the sun, and the
examination of the constitution of the heavenly bodies has been
rendered possible. The spectrum of iron vapour consists of a
large number of bright lines, and of these over four hundred have
been identified with dark lines in the solar spectrum, and correspond-
ing coincidences of the dark lines with the bright lines given by many
other substances have been observed. Thus Angström and Thalén
identified the following:”
Iron tº . 450 Manganese . 57 Hydrogen 4
Calcium . . 75. Chromium . 18 Aluminium 2
Barium . . 11 Nickel tº . 33 Zinc 2
Magnesium . 4 Cobalt . . 19 Copper 7
Besides the examination of the resultant light from the sun as a
whole, that emitted by any selected portion of it may also be examined.
An image of the sun an inch or two in diameter may be formed by
means of a lens of considerable focal length, and the slit of the spectro-
scope may be adjusted to any desired part of the image, and the
radiations from that part may be separately studied. This has been
done by Janssen, Lockyer, and many others, with the result that a re-
markable variation of appearance occurs, especially in the neighbourhood
* If all the dark lines were the effect of absorption while passing through the
earth's atmosphere or through some substance existing between the earth and
the sun, then the spectra of all the stars should exhibit absorption bands at least as
extensive as those of the solar spectrum. Lines produced by absorption in the
earth's atmosphere should be common to all spectra, and should vary in intensity
according to the thickness traversed by the light ; that is, at Sunrise and mid-day.
* For the results of Rowland’s great work on the solar spectrum see “Pre-
liminary Table of Solar Wave - Lengths,” Astrophysical Journal, vol. i. p. 29
(1895), and vols. following. *
512 ABSORPTION AND DISPERSION CHAP. XVIII
of spots and faculae.” Sometimes some of the lines corresponding to
an element are seen bright and straight, while others are faint or
crooked.
283. Width of the Spectral Lines—Radiation from Moving
Molecules.—In discussing the radiation from an incandescent gas, we
have so far neglected the motions of translation of the individual
molecules. We have supposed that the molecules emit waves of
certain definite periods depending upon their own period of free
vibration. In this case the spectrum of an incandescent gas should
consist of certain definite lines, each of purely monochromatic light,
and the absorption bands produced in a continuous spectrum should
be of infinitesimal width. Taking into account the motions of transla-
tion of the molecules while describing their free paths, and applying
Doppler's principle, it will follow immediately that the spectral lines
exhibited by an incandescent gas as well as the absorption bands in
the solar spectrum should be of definite width. This application has
been discussed by Ebert,” and the whole investigation with reference
to interference has been taken up and extended by Lord Rayleigh.”
Thus if v be the velocity of light and u the velocity of the moving
molecule, 6 the angle which its direction of motion makes with the line
of sight, then the natural wave frequency n is changed into n' where
n’=—*—.
^) – 'u coS 6
The new wave length. Aſ will be vſn', and will consequently be con-
nected with the old wave length A = vſm, by the equation
X’=X(1 – cos 6)
where u is small compared with v.
As a first approximation u may be supposed to be the same for
every molecule, and the limiting values of the frequencies will be
*(l ++) and n(1 -:)
Q) Q)
the result being that the spectral line which would otherwise be
* [At the Exeter Meeting of the British Association (1869) Janssen described a
method of studying the distribution of any substance in the sun which produces
a brilliant line in the spectrum. A large image of the sun is formed on a movable
slit and a spectrum is produced by a grating. A second slit, moving in correlation
with the first, allows the bright line alone to act on a photographic plate imme-
diately behind it. In this way Mr. George Hale, and more recently M. Deslandre,
have been most successful in obtaining photographs of the sun produced solely by
the K line (see Hale, Astronomy and Astrophysics, vol. xii. (1893), pp. 241, 450).]
* Ebert, Wied. Ann. xxxvi. p. 466, 1889.
* Lord Rayleigh, Phil. Mag. vol. xxvii. pp. 299, 484, 1889.
ART. 284 ANOMALOUS SPECTRA 513
infinitesimally narrow, and correspond to a single wave frequency m,
will now be broadened out into a uniformly illuminated band, having
the above limits for the frequencies at its edges. Thus by the motion of
the molecules the emitted light is rendered to some extent heterogeneous.
The waves are not of a single period, but vary over a certain range
depending on the velocity of motion of the molecules, and conversely, the
light absorbed by a gas will not be of a single definite wave length, but
will consist of a group or groups of waves lying between certain limits.
ANOMALOUS DISPERSION
284. Anomalous Spectra.-Intimately connected with the dark
absorption bands is the remarkable phenomenon of anomalous or
abnormal dispersion. Hitherto we have always spoken of the spectrum
as presenting an invariable succession of colours in the order violet,
blue, green, yellow, orange, and red. We have incidentally mentioned
that spectra obtained by prisms of different substances present peculi-
arities characteristic of the material of the prism, and that spectra
produced by different prisms cannot well be compared on account of
the irrationality of dispersion, as the irregularity in the spreading out
of the colours is termed.
- It has been found that refraction spectra are not complicated by
the irrationality of dispersion alone, but that in many cases the order
of the colours is entirely changed, and that sometimes the spectrum
may not even present a continuous appearance, but may be broken
into parts isolated from each other by broad dark bands.
Although anomalous dispersion may appear at first sight to be a
strange and unexpected phenomenon, yet, on duly considering the
mode of propagation of light in refracting media, we are led to expect
it. For the refractive index of any medium, for waves of a given
frequency, is determined by the velocity of propagation, and in general
the velocity of propagation will vary with the wave length in a manner
depending on the nature of the medium. Thus if the long waves
happen to travel faster than the short waves in some media, we are
not justified in concluding that there are not media in which the
reverse may be the case. The velocity of waves of a given frequency is
influenced by the internal structure of the medium, and the constitu-
tion of one medium might be such that the violet waves travel faster
than the red, while the reverse may be the case in another medium,
just as in one medium the red may be more strongly absorbed than
the violet, while in others the violet is more strongly absorbed than
the red. The so-called irrationality of dispersion, and anomalous
2 L
514 ANOMALOUS DISPERSION CHAP. XVIII
dispersion, are consequently phenomena which might to be expected
just as we should expect the different colours to be absorbed in
different proportions by a given medium, and differently in different
media.
The existence of anomalous dispersion" seems to have been first dis-
covered by Fox Talbot” about 1840, but the discovery does not seem
to have been followed up. In 1860 M. Le Roux * discovered that
iodine vapour possessed a very remarkable absorbing power. He
found that it transmitted only the red and violet rays, and that of
these the red are the more refracted, contrary to what takes place in
the ordinary cases of refraction. At a temperature of 700° C. the
indices for iodine vapour of the red and violet rays were found by
M. Hurion 4 to be
ur-.1-0205, u, -1-019.
In 1870 Christiansen” detected the existence of anomalous disper-
sion in alcoholic solutions of fuchsine (one of the aniline dyes). This
solution gives a marked absorption band in the green, so that this
colour is entirely absent from the spectrum of the transmitted light.
Of the remaining colours the red, orange, and yellow occur in their
natural order; that
is, the red is less
refracted than the
orange, and the
Fig. 211. orange less than the
yellow. The violet, however, suffers a very peculiar deviation. It is
less refracted than the red, and separated from it by a dark interval.
The arrangement and relative intensities are shown in Fig. 211, the
spectrum is elongated to an extraordinary extent, the green is absent,
the violet is least refracted, and is separated from the other colours by
a dark space. The prism used by Christiansen consisted of two glass
plates inclined at an angle of about 1°. The dotted lines show the
position and length of the spectrum obtained with the same prism
when filled with pure alcohol."
The indices for the various rays were as follows:
| Anomalous equals av and ºuaxós, that is, uneven's etymologically it does not
mean ‘contrary to law.”
2 Tait, Light, $196, and Proc. Roy. Soc. Edinburgh, 1870-71.
* Le Roux, Ann. de Chimie et de Physique, third series, tome lxi. p. 285, 1861.
* M. Hurion, Journal de Physique, first series, tome vii. p. 181.
* Christiansen, Pogg. Ann., 1870-72.
* See also papers by A. Pfüger, who used prisms of the solids of very small angle
40” to 140", in Wied. Ann. vol. lvi. p. 412; vol. lviii. p. 670; vol. lxv, pp. 171,
214.


ART. 284 ANOMALOUS SPECTRA 515
A. B C D E. F G. H
- -
Fuchsine Solution |
(18 per cent) 1:450 1:50.2 1-561 - - 1-312 || 1:258 1-312
Pure Alcohol . 1-3628 not measured 1-3654 || 1:3675 || 1:3696 || 1:3733 1-3761
In determining the indices by the method of total reflection, it is found
at normal incidence that the reflected light is strongly coloured green.
As the incidence is increased the spectrum of the reflected light shows
first (besides the green) blue, then violet, red, orange, and yellow
enter in succession. We conclude, therefore, that the green is re-
flected at all incidences, and that of the other colours the blue is the
least refracted and the yellow most.
In working with a hollow prism containing a solution, the action
of the solvent may be eliminated by enclosing the prism in a vessel
having parallel glass sides and filled with the solvent liquid.
Kundtl has made an extensive series of observations on anomalous
dispersion, and has shown that it appertains to all bodies which possess
what is known as surface colour, that is, whose colour by reflection is
different from the colour by transmission (§ 281). Thus fuchsine
colours the transmitted light red, but the light reflected from it is
green. The solutions, even though very dilute, of all bodies of this
class show marked absorption bands, and they reflect in a marked
manner, light of certain frequency. Among the substances examined
by Kundt were the blue, violet, and green anilines, solution of indigo
in strong sulphuric acid, carmine, permanganate of potash, and cyanine.
The greater part of the observations were made by the method of
crossed prisms. This method has been employed by Newton in his
investigations on the refrangibility of solar light (see Chap. V. p. 132).
If two prisms of a substance, such as glass, which exhibits regular
dispersion, be crossed, the spectrum afforded by the first will be
displaced more or less by the second, but the displacement will affect
all the colours in a continu-
ous manner, the violet most
and the red least. If, how-
ever, one of the prisms ex-
hibits anomalous dispersion,
the displacements may take
place in opposite directions, Fig. 212.
and be very different for the various colours. The displaced spectrum
will then consist of coloured patches more or less isolated from
Kundt, Pogg. Ann., 1871-72.

516 ANOMALOUS DISPERSION CHAP, XVIII
each other. Fig. 212 shows a spectrum of permanganate of potash
obtained by the method of crossed prisms. It exhibits five absorption
bands in the green, and is inflected in contrary directions at the two
extremities."
285. Kundt's Law.—As a result of his experiments M. Kundt con-
cluded that in going up the spectrum, from the red towards the violet,
the deviation is abnormally increased below an absorption band, while
above the band the deviation is abnormally diminished by the absorption.
Or in going up the spectrum the refractive index is increased where the
coefficient of absorption increases rapidly, and diminished where the
absorption rapidly diminishes.
Thus on either side of an absorption band there is an abnormal
change of refrangibility of such a kind that the refraction is increased
below, that is on the red side, and diminished above the band. An
analogous interaction frequently occurs in vibrating systems of nearly
the same period. Thus if to the bob of a pendulum P, executing
horizontal vibrations, another pendulum p be attached, the effect will
be to increase or diminish the period of P according as the period of
p is shorter or longer than that of P; that is, the effect of p is to
increase or diminish the virtual inertia of P according as the natural
period of the former is shorter or longer than that of the latter. If
P tends to vibrate more rapidly than p, then the effect of p is to make
it vibrate more rapidly still, and vice versa. Below an absorption band
the ether vibrations are slower than those of the matter molecules.
The effect of the matter is consequently to increase (abnormally) the
virtual inertia of the ether, and therefore the refrangibility. Above
1 Professor H. Becquerel (Comptes rendus, vol. cxxvii. p. 399, December 5,
1898; and vol. cxxviii. p. 145, January 16, 1899) has exhibited the anomalous
dispersion of sodium vapour near the D lines. He projected the image of the
crater of an arc light on a horizontal slit, placed in the focus of a collimator lens.
The parallel beam then passed through a sodium flame in the form of a prism
with its refracting edge horizontal, and was focussed by a lens into an image of
the horizontal slit upon the vertical slit of a highly dispersive spectroscope.
On either side of the two dark D lines the spectrum was sharply curved so that
for wave lengths slightly exceeding those of Di and D2 the vapour possessed
an index of refraction rapidly increasing near the lines; and for wave lengths
slightly less than D, or D, the index decreased when approaching the lines (see
Lord Kelvin, Phil. Mag. March 1899, and W. H. Julius, Proc. of the Royal Acad.
of Sciences, Amsterdam, or Astrophysical Journal, vol. xii. p. 185, October 1900).
In his interesting paper Julius suggests that the extraordinary curvature of certain
solar lines may be due to anomalous dispersion in portions of the sun's atmo-
sphere, and that the “flash” spectrum seen at totality in an eclipse may be caused
by abnormal refraction of the light of the photosphere occurring in the atmosphere
of metallic vapours surrounding the sun. Wood and Ebert have, independently,
imitated the solar conditions in this case, and obtained a bright line spectrum by
abnormally refracting white light through sodium vapour of varying density.
ART. 286 CALORESCENCE 517
an absorption band the period of the matter is longer than that of the
ether, and the effect is an abnormal diminution of the virtual inertia
of the ether, and therefore of the refraction.”
TRANSFORMATION OF RADIATIONS
286. Fluorescence.—In discussing the phenomena of absorption it
has been pointed out that of the many constituents of a beam of solar
light some are absorbed more particularly by one body and some by
another. In general the energy of an ether wave will be taken up by
the matter if it is of the proper period to excite the matter molecules
to vibration, yet some bodies appear competent to take up and absorb
the energy of waves of all periods, at least of those lying within the
limits of the visible spectrum. The general effect of absorption is to
raise the temperature of the body. Thus light is taken up and heat
is emitted instead—that is, waves of a short period are absorbed—and
in their place the longer infra-red or heat rays are emitted. This is
obviously a case of the transformation of radiations of a high period
into others of a lower period. Energy is absorbed in the form of short
waves and is emitted in long waves, and this species of degradation is
always in operation when light falls upon and heats material bodies.
We shall see immediately that light waves of a short period, as those
from the violet end of the spectrum, are absorbed by some substances
and emitted again as light waves of a lower refrangibility. For
example, the violet, or even the ultra-violet, may be specially absorbed
and emitted again as green or red rays. This is degradation of the
same sort as the former, and is termed fluorescence. When light is
absorbed by lamp-black and emitted as heat radiations, we detect the
transformation by our sense of heat, but in the case of fluorescence
the degradation is seen by the eye; the change in period has not been
sufficiently great to place the vibration outside the limits of the visible
Spectrum.
287. Calorescence.—The converse transformation may also be
effected, viz. the conversion of waves of long period into those of
shorter period. The long heat waves may be converted into waves
sufficiently short to affect the eye, and may thus be rendered visible.
This converse transformation, the rendering visible of the infra-red
waves, is termed calorescence. It may be demonstrated by concentrat-
ing the non-luminous radiations by means of a condensing lens at the
focus of which a piece of platinum foil is placed. Professor Tyndall
* This principle was used by Lord Rayleig has early as 1872. See Phil. Mag.,
1872, “On the Reflection and Refraction of Light by intensely opaque Matter.”
518 TRANSFORMATION OF RADIATION CHAP. XVIII
first exhibited it in this manner by focussing the radiation of an
electric arc. on a slip of platinised platinum. The beam was sifted
of all the luminous radiations by transmission through a solution
of iodine in bisulphide of carbon, so that only the obscure heat radia-
tions fell upon the platinum. In this manner the slip was raised
to incandescence and emitted light; that is, the visible infra-red
radiations of long period were converted, in part at least, into the
higher luminous radiations. In this experiment, however, the platinum
can never be heated to as high a temperature as that of the source at
which the light has originated ; so the light emitted cannot be as great
in intensity as that from a black body at the temperature of the
source, and in practice it is very much less.
288. Fluorescent Substanees — Stokes's Theory.—The pheno-
menon of fluorescence was first observed by Sir David Brewster" in
an alcoholic solution of chlorophyll, and was termed by him internal
dispersion. He found that when a pencil of solar light passes through
a green chlorophyll solution, the path of the beam is marked by a
brilliant red light. It was then noticed by Herschel’ that when the
sun's rays fall upon a dilute solution of sulphate of quinine the surface
of the liquid, where the light falls, exhibits a bright blue colour, which
penetrates to a small distance within the liquid. Herschel found
that this blue light was confined to the surface stratum on which
the light fell, and that light which had once produced this effect was
incapable of developing it again in other solutions of the same
substance. -
Thus if the pencil of solar light, which passes through a cell contain-
ing sulphate of quinine, be transmitted through a second cell of the
same solution, there will be no development of the blue shimmer at the
surface of the second cell where the light enters it. That part of the
solar beam which developed the blue colour in the first solution has
been absorbed there, and the transmitted pencil contains no rays
competent to excite it in the second. If the light be concentrated, by
means of a lens, to a point in the interior of the liquid, the blue
shimmer may mark the whole path of the beam, and be not merely
restricted to a thin surface layer. The same appearances are ex-
hibited in the greenish surface colour of canary glass (coloured with
oxide of uranium), and in the bluish surface tint so frequently
observed in some kinds of ordinary paraffin oil. Fluorescence is
also finely developed in solutions made from the bark of the horse
chestnut.
* Brewster, Eighth Report, Brit. Assoc. and Phil. Mag., 1848.
* Herschel, loc. cit.
ART. 288 - FLUORESCENCE 519
Sir G. G. Stokes,” to whom the whole explanation of this pheno-
menon is due, has shown that it is exhibited in a greater or less
degree by many substances, including white paper, bone, ivory, cotton-
wool, etc. In reflecting on the possible explanation of Herschel's
observations it occurred to him that the blue light dispersed in the
solution of quinine might not be the blue rays of the incident beam,
but might be due to the degradation of the rays of higher refrangibility,
which are mostly invisible, and the results of experiment completely
verified the speculation.” -
To find what rays are most effective in producing fluorescence in
any substance, it is only necessary to place a piece of it in the various
parts of a pure spectrum of solar (or electric) light. The spectrum
should be formed with prisms and lenses of quartz, as this material
transmits the ultra-violet radiations very freely. When the substance
is moved from the red end of the spectrum towards the violet it is
found that at a certain point it begins to glow with colour, always of
a refrangibility lower than that which falls upon it. Thus when a
slip of uranium glass is so treated there is scarcely a trace of colour
exhibited until it is moved up to the blue part of the spectrum ; here
it begins to glow with its characteristic yellowish light, and the effect
persists as it is moved through all the higher colours, and even to a
considerable distance beyond the limits of the violet. The ultra-violet
waves, which are too short to affect the sense of sight, are absorbed by
the uranium glass, and afterwards emitted as rays of lower refrangibility,
sufficiently slow to be detected by the eye. *
Solutions of chlorophyll, sulphate of quinine, and other liquids
may be examined in the same manner by enclosing them in a small
test-tube. When a test-tube containing sulphate of quinine is held in
the red end of the spectrum nothing strange is observed. In the red
it looks red, and in the green it appears green, but in the blue and
violet a marked change occurs in the appearance. The pale blue
shimmer exhibited in the sun's rays begins to show itself, and
increases as the solution is moved towards the end of the spectrum,
remaining visible even beyond the limits of the violet. The red and
green rays are inactive ; it is the rays of higher refrangibility from the
violet extremity of the spectrum that operate on the sulphate of quinine.
If the blue fluorescent light emitted by the quinine be examined
in a spectroscope it will be found to contain rays from various parts of
the spectrum. It is not a homogeneous blue light. All its constituents,
however, are of lower refrangibility than the violet or ultra-violet which
excited them.
* Stokes, Phil. Trams., 1852-53. * Compare Ex. 8, Art. 48.
520 TRANSFORMATION OF RADIATION CHAP. XVIII
Wood" and Moore have examined the spectrum of the fluorescent
light emitted by sodium vapour, and found it to coincide absolutely
with the spectrum of the light absorbed by the same vapour under
similar conditions.
In chlorophyll the greater part of the fluorescent effect is produced
by the visible light of the spectrum, and not so much by the ultra-
violet, as in the case of sulphate of quinine. Held in the red end of
the spectrum, the chlorophyll glows with a deep red. It has already
been stated that the spectrum of solar light transmitted through a
solution of chlorophyll exhibits a marked absorption band in the red
between the lines B and C. The fluorescent light emitted by the solu-
tion when held between these lines is found, on analysis, to be of
lower refrangibility than the light absorbed. As the test-tube is
moved up the spectrum the red glow grows fainter, but again rises in
intensity as each absorption band is crossed. In the blue and violet
the red glow is more continuous, and at the end of the violet it
assumes a brownish tint, due to the presence of some green in the
fluorescent light.
289. Applieation—Study of the Ultra-violet Spectrum.—One
important application of fluorescent substances arises in the means they
afford us of mapping the solar spectrum beyond the limits of the
violet. This remarkable prolongation of the solar spectrum beyond
the violet was first noticed by Sir John Herschel in throwing the
spectrum on turmeric paper. Herschel attributed this prolongation,
which was yellow, to a peculiar reflecting power of the paper.
Casting the complete solar spectrum on the surface of a transparent
fluorescent substance, various effects of the different colours may be
observed simultaneously. At the red end there is in general no
fluorescent effect. Towards the blue and violet, and for considerable
distances beyond the confines of the violet, the fluorescent shimmer is
emitted from the surface layer, and dark lines are visible in some parts,
indicating absorption bands in the solar spectrum beyond the violet
similar to the Fraunhofer lines which are presented in the coloured
part. The waves at these parts do not exist in the solar beam when
it reaches us. They have been absorbed in the solar or terrestrial
atmosphere, or else they have not been emitted by the sun at all.
290. Phosphoreseenee.—The fluorescent light which we have been
discussing is emitted only while the substance producing it is illumin-
* R. W. Wood and J. H. Moore, Phil. Mag. vol. vi. pp. 362-374, September 1903.
Wood has also observed the presence of polarisation in the light emitted by the
vapours of sodium and potassium, when fluorescing, Phil. Mag. vol. xvi. pp. 184-189,
July 1908. -
ART. 291 & VELOCITY AND WAVE LENGTH 521
ated. On intercepting the light which falls upon a solution of sulphate
of quinine the blue shimmer from its surface layer disappears at once.
Some substances, however, especially the sulphides of barium, strontium,
and calcium, persist in emitting light even after the incident beam has
been cut off; that is, they shine in the dark for some time after ex-
posure to the light. This phenomenon is called phosphorescence. It
has, however, nothing to do with the luminosity of phosphorus, which
is due to slow oxidation. It is merely fluorescence lasting after the
exciting cause has been removed. The light absorbed during exposure
is emitted as light of lower refrangibility, and continues to be emitted
by many substances for long or short intervals after they are removed
from the light and brought into a darkened chamber. The duration
of the phosphorescence after the incident light has been cut off is, in
most substances, so short that it is impossible to detect it without
some specially contrived method of investigation. To pursue this
inquiry, Becquerel invented an ingenious phosphoroscope, by means of
which we can determine with considerable accuracy the duration of
the phenomenon after the direct light has been cut off. With this
apparatus the existence of phosphorescence to a greater or less degree
has been detected in most substances. -
THEORIES OF DISPERSION
291. Dependence of the Welocity on the Wave Length.-We owe
to Newton the important discovery that the light of the sun is com-
posite, that is, it consists of a system of simple colours, or, as we say
now, of a great variety of waves of different periods. The separation
of solar light into its constituent simple colours was effected by Newton
with the aid of a prism of glass. The different colours are deviated by
different amounts in passing through the prism, and the transmitted
light, when received on a white screen, paints on it a coloured band or
spectrum. The property which transparent substances possess of thus
separating the various constituents of white light is called dispersion,
the inequality in the refraction of the various colours leading to their
separation or dispersion from one another.
Now the teaching of the wave theory, and of the emission theory
also, is that rays which are refracted by different amounts on entering
any medium must travel through that medium with different velocities,
the relation between the absolute refractive index and the velocity for
any ray being, according to the wave theory,
Q)
— ‘0
* = .
522 THEORIES OF DISPERSION CHAP. XVIII
where v, is the velocity in free space—that is, in the ether. This
velocity is, as far as we know, the same for light of all colours. A
condition which may therefore be introduced into any theory of
dispersion is that it does not exist in free space, and is a phenomenon
arising from the interaction of the ether and matter.
The constancy, or approximate constancy, of to for all waves ranging
from the red to the violet at least, is inferred from considerations
such as the following. Certain stars exhibit rapid changes in bright-
ness and are called variable stars. One of these, Algol, passes in
three and a half hours from one of the second to one of the fourth
magnitude in brightness. Whatever be the cause of these changes,
whether it be due to eclipse or otherwise, they ought to be accom-
panied by corresponding exhibitions of colour, if the various colours
travel through space with different velocities. Thus if the red light
travels faster than the violet in interstellar space, as it certainly does
in glass and common transparent substances, then when the star is
growing faint it should be coloured blue or violet, and when it is grow-
ing bright it should appear red. No trace of a coloured tint has ever
been observed, although light requires several years to reach us from
this star. If the difference in the velocities of the red and violet
amounted to the one hundred thousandth part of the value of either—
that is, a variation of 001 per cent—an interval of at least an hour
would elapse between their times of arrival and the corresponding
changes in tint should be observed. We may therefore conclude that
the waves corresponding to the various colours of the spectrum traverse
the free ether with the same velocity, but it is not to be assumed that
waves of every length pass with the same velocity. The foregoing
observations indicate only an equality, or an approximate equality,
between the velocities of the very limited sets of waves which affect
the eye. The long waves far below the limits of the red may travel
with velocities very different from that with which the short ultra-
violet waves are propagated.
In the case of sound notes of all pitches travel with equal.
velocities, and the common velocity deduced by theory is a result of
the supposition that the length of the wave is vastly greater than the
displacements, or sphere of action, of the vibrating molecules. When
the wave length is very small, as in the case of light, the conditions
are very different, and we cannot assume that the velocity of propaga-
tion will be independent of the time of vibration. Even in the case of
a stretched string, vibrating transversely, the velocity of propagation
depends upon the wave length owing to the stiffness of the string, and
the dependence becomes more pronounced in the case of the transverse
ART. 292 CAUCHY'S FORMULA 523
vibrations of a bar. Hence the phenomenon of dispersion is not
peculiar to light.
292. Cauchy’s Formula.-Starting from these principles, Cauchy"
showed that the velocity will in general be a function of the wave
length. By proceeding to a higher order of approximation he arrived
at the relation
c d
Xi+N+ tº
which expresses the refractive index in the form
2– b
Q} =a+x+
C
Xi+ . . .
where A, B, C, etc., are constants depending on the nature of the medium
and diminishing rapidly in magnitude as we proceed to the higher terms.
The formula shows that waves of short period are more highly
refracted than the waves of longer period, and that the latter approach
a limited index p = A. Within the limits of the visible spectrum it
represents, when taken to three or four terms, the results of experiment
very accurately, but fails to représent the facts when applied to the
dark radiations beyond the red, as shown by the experiments of
Prof. Langley.” In media, however, which exhibit anomalous dispersion
it is not true that shorter waves possess the higher refractive indices,
and the formula takes no account of this class of phenomena, which
point to an intimate connection between the production of dispersion
and the existence of absorption.
In Cauchy's method of investigation the assumption is that the
distances between the molecules are comparable with the wave
length, but in the modern methods which consider the direct inter-
action of the ether and matter, it is the periodic time of the wave
which is supposed to agree with, or approach, the period of free vibra-
tion of the molecule, when strong absorption takes place.
293. Briot's Formula.-The work of Cauchy was taken up and
treated in a more general manner by Briot.” Taking into account
more directly the interaction of the ether and the matter molecules, he
obtained the formula
e-A tºt
}=&A ºr e tº e
which consists of two parts, one a series similar to that of Cauchy, and
the other the term kA”, which depends on the direct action between
the ether and the matter.
The matter is supposed to affect the ether in two ways. Firstly,
* Cauchy, Nowveawa, exercices de mathématique, 1835.
* S. P. Langley, Ann. de Chimie et de Physique, tome ix. p. 496, 1886.
* Briot, Essai swr la théorie mathématique de la lumière, Paris, 1864.
524 THEORIES OF DISPERSION CHAP. XVIII
by modifying its distribution in the body, and this gives rise to the
Cauchy series; secondly, by exercising a direct action on its motion
when vibrating, and this introduces the term k\*. The matter is
supposed not to be disturbed sensibly by the luminous vibrations, but
to exert on the ether a reaction towards its equilibrium position and
proportional to its displacement. This formula represents to a high
degree of accuracy the dispersion of the luminous rays, and also
agrees fairly with Langley's investigations in the infra-red."
294. Anomalous Dispersion.—To account for the existence of
anomalous dispersion, theories, based on the mutual reaction between
the ether and matter, have been developed by Boussinesq, Sellmeier,
Helmholtz, Ketteler, Lommel, etc.
Cauchy regarded the matter and ether as moving together, while
Briot assumed the matter to be sensibly undisturbed ; on the other
hand, Boussinesq supposed the ether of constant density, but that the
matter is partially displaced, this displacement leading to a reaction
between the matter and the ether, but being so small that the elastic
forces brought into play in the matter may be neglected. Sellmeier
further supposed that the reaction between the matter and the ether
is proportional to their relative displacement (§ – £1), obtaining as the
equations of motion of the ether and matter respectively,
32: 32
p;=\; -ºs-à),
62: (1)
p;#=k(k-à).
The matter molecules or atoms are to be regarded as capable of vibrat-
ing with certain natural periods of vibration, forced vibrations of period
T being set up in them by the light waves. The second of equations
(1) corresponds to a set of molecules with natural period T, where
4tr?
PiTº =k.
A particular solution of these equations is of the form
t piz
& — - - -
# = b cos 2-(# #)
t 2.
$1= b, cos *(# - º )
. being the wave length in the matter.
(2)
* Ketteler finds that the formula
2
p?= — k\*-i- a”-- sº» sº
represents the facts very accurately. From Langley's experiments on sel gemme
he determines
k=0.000858, a”=2:32883, D= 1 1410, X,2=0.01621
(Journal de Physique, 29, tome vii., 1888).
ART. 294 ANOMALOUS DISPERSION 525
Substituting the solution (2) in the second of equations (1), we
find
b
b1__X*
(3)
where A1 is the wave length in ether of light of the same frequency as
the natural frequency of the matter molecules, i.e. where A*/A*=T,”/T”.
The first of equations (1) gives, on substitution of (2), for the
index of refraction for light of wave length X,
A1X”
*=1+xº. (4)
If the substance contains sets of molecules with different natural
periods of vibration, a new equation must be added to equations (1)
for each set of molecules, and a corresponding term added to the first
of the equations. The resulting formula for p? will become
Am N*
X2 -- Amº
where the summation must be taken to include all the natural periods
of vibration of the molecules of the substance.
If A1 is small compared with A–that is, if the substance is trans-
parent to light in the visible spectrum, but has an absorption band in
the ultra-violet—p, obviously diminishes as A increases. The formula
(4) may be written
p?= 1 + X (5)
X,2\-l
*=1+A (i-x)", (6)
which reduces to Cauchy's form.
If, in addition, there be an absorption band in the ultra-red, for
which Āg is large compared with A, (5) gives
9. A X2\–1 X,2\–1 Asy
*=1-#"(1-3) +A.(1-3) (7)
A,X* X,2 X,4
=i-º-A(i+...+.)
neglecting higher powers, which includes Briot's form.
The Sellmeier formula represents well the dispersion of transparent
bodies, and accounts for anomalous dispersion. Thus, if AI corresponds
to an absorption band, (6) shows that p becomes abnormally large for
wave lengths a little greater than A1, while expansion in the form of
(7) shows that p is at first imaginary for wave lengths a little less
than A1, but that it becomes real again, though of abnormally small
magnitude, as A diminishes. If the refractive index as ordinate be
plotted against the wave length as abscissa, a curve is obtained convex
to the wave-length axis as the wave length diminishes towards the
absorption band, but concave to that axis as the wave length diminishes
526 THEORIES OF DISPERSION CHAP. XVIII
below the absorption band. For sodium vapour, e.g., it gives results
in accordance with Becquerel's" photographs, and Wood” has traced its
agreement with the dispersion on each side of the sodium lines.
Kelvin * interprets the imaginary values of p as indicating selective
reflection; i.e. that no light of such wave lengths can enter the medium.
However, the formula does not apply through an absorption band,
and indeed the theory does not really account for absorption or
for the transformation of light energy that takes place in absorption.
The assumed particular solution (2) of the equations (1) gives for the
ratio of the amplitudes (biſb) of the vibrations in the matter and the
ether the value A*/(A* – A,”), which becomes infinite for A = A1. The
particular solution does not then apply at the absorption band, as it is
not concordant with the initial and limiting conditions. To make
it concordant the complementary function needs to be added ; and
under the assumed conditions it would take an infinite time for it
to become negligible.”
Helmholtz introduced the conception of the vibrations of the
atoms being subject to frictional forces, thus accounting for the trans-
formation of light energy that occurs in absorption. He expressed
by means of differential equations the motion of the ether and that of
the matter, taking into account in the former case the elastic reactions
of the ether and the mutual forces arising from the interaction of the
ether and the matter; in the latter case, besides these mutual forces,
he considered the elastic reactions of the matter and the frictional
forces arising from the adjacent matter. Solving these differential
equations, he obtained formulae connecting the refractive index and
absorption coefficient with the wave length, viz.
© k2 X2 * 2 sº X*(X2 * \m.”
*-ī; -1=-Pºt?Anº
and Ak\_ 2A a)\º
Tº "[X3-X,Z), Hoº
where k is the damping factor of vibrations passing through the
medium, and a depends upon the friction.
When the coefficient of absorption is small, so that its square may
be neglected, the formula becomes for one absorption band
2 = 1 — PX2 QX'
p?=1 — PX +xº
* Becquerel, Comptes rendus, December 5, 1898, and January 16, 1899.
* Wood, Phil. Mag., September 1904, pp. 293-324.
* Kelvin, Phil. Mag., March 1899, pp. 302-308.
* Carvallo, Reports, Congrès Intern. de Phys. ii. p. 193, Paris, 1900.
ART. 294 ANOMALOUS DISPERSION 527
as Wüllner has pointed out. If P and Q are equal, as experiment
shows for water and some other substances, this formula reduces to
the Sellmeier formula. It applies fairly satisfactorily in the visible
region to a substance with an absorption band in the ultra-violet,
but transparent through the visible spectrum.
Ketteler + obtained a modification of Helmholtz' formula, involving
a term dependent upon the refractive index of the medium for infinitely
long waves. The formula reduces to the form
*=º twº-sº
for wave lengths not close to the absorption bands in a transparent
medium, with an absorption band in the ultra-violet and another in
the infra-red.
Paschen” has found that this formula represents the dispersion of
fluorine far down into the infra-red. Rubens* and Trowbridge have
verified it for rock-salt and sylvine for wave lengths as great as
0-018 mm., while Rubens 4 and Nichols extended the observations up to
A = 0225 mm. For quartz it was found necessary to take account of
two absorption bands in the infra-red in addition to one in the ultra-
violet, and the existence of these two absorption bands was verified.
These equations fit in so well with the complicated phenomena of
refraction and dispersion that no doubt can be felt that they are based
on sound physical principles. But modern discovery has shown that
the vibrators concerned must include the unitary electric charges—
the electrons—which play such an important part in physical processes.
We shall develop the analogous equations in terms of electromagnetic
theory in the concluding chapter (Chap. XXI).
* Ketteler, Wied. Ann. tom. liii. p. 268, 1894.
* Paschen, Wied. Ann. tom. liii. p. 334, 1894.
* Rubens and Trowbridge, Wied. Ann. tom. lx. p. 724, 1897.
* Rubens and Nichols, Wied. Ann. tom. lx. p. 418, 1897.
CHAPTER XIX
EXPERIMENTAL DETERMINATIONS OF THE WELOCITY OF LIGHT
295. Introduction.—The first attempt to determine the velocity
of propagation of light was made by Galileo. The principle of the
method employed was as follows. Two observers, A and B, are
situated at a distance, and each is furnished with a lamp which can
be quickly screened or uncovered. If A uncovers his lamp it will
be seen after a certain interval by B, and this interval will measure
the time occupied by the light in travelling from A to B. If B now
uncovers his lamp it will be seen after an equal interval by A. Hence
if B uncovers his lamp at the instant he perceives that of A, then A.
will perceive that of B at a time after he uncovered his own, equal to
twice the time required by light to traverse the distance between A
and B, together with a small interval of time depending on B,
namely, the time required by B to perceive the flash of A's lantern
and uncover his own. This personal interval becomes of such weight
(when compared with the very small time required by light to
traverse such distances as those which could be employed in an
experiment of this character) that if any definite result had been
obtained it would have been altogether worthless. The fundamental
principle of this method, however, is correct, and is the same as that
on which one of the most celebrated of modern methods is based,
namely, Fizeau's, or that which may in general be termed the eclipse
method.
It is interesting to notice how the academicians might have
obtained an estimate of the velocity of light by attending to the
conditions on which the delicacy of their method depended. In the
first place, having determined that the velocity of light, if not infinite,
was certainly very great, it becomes a matter of prime importance to
eliminate the personal interval arising from the slowness of perception
and movement of the second observer B. At the instant the light
from the first station reaches the second, the conditions of the problem
528
ART. 296 FIZEAU'S METHOD 529
require that a return beam of light shall leave the second station and
travel to the first. The most obvious way of securing this is to
replace the second observer B by a reflecting surface, such as a
polished mirror, so that when A uncovers his lantern the light issuing
from it may fall upon the mirror and be reflected back at the second
station without loss of time. Thus when A uncovers his own lantern
he may observe its image by reflection in the mirror at the second
station, and the interval between the uncovering of the light and the
perception of the image is the quantity to be estimated. The
personality of B is hereby eliminated ; but this is not enough, for the
personality of A remains.
The whole problem is now reduced to devising an accurate method
of estimating this interval. One obvious source of error is a personal
one attending the observer, and arises in the uncertain estimate of the
interval between deciding to move the screen and actually moving it ;
and also of the interval between the eye receiving the return flash and
the perception of the flash by the brain.
To eliminate this the screen used to uncover the light should in its
motion be employed to cut off the return rays. For example, let the
lantern be covered, and let the eye of the observer be placed close to it
and directed so as to look into the distant mirror. Then if the screen
be suddenly moved so as to uncover the lantern and cover the eye of
the observer, the return flash will not be received by the eye if the
time occupied in moving the screen from the lantern to the eye is less
than the time required by light to travel to the distant mirror and
return.
The problem is now reduced to the devising of some mechanical
method of moving a screen, or a train of screens, across the field
of view so that the source of light and the eye of the observer
shall be covered and uncovered in succession. This was first done
in 1849 by Fizeau by using a toothed wheel, which rotated in
the field of view in such a way that the opaque teeth passed before
the eye and intercepted successively the light emerging from the
Source and that returning to the eye after reflection in the distant
mirror. This method is explained in the following article.
FIZEAU's METHOD
296. General Principles of the Method.-In the method adopted
by M. Fizeau' a beam of light from a source S (Fig. 213) is in-
troduced through a collimator fixed to the side of a telescope, and
* H. Fizeau, Comptes rendus, tome xxix. p. 90, 1849.
2 M
530 THE VELOCITY OF LIGHT CHAP. xix
after reflection at the surface of a plate of parallel glass inclined at
45° to the axis of the tube, the pencil comes to a focus at a point F,
which is the principal focus of the object-glass of the telescope. It
follows therefore that the light diverging from F will emerge from the
telescope in a parallel beam. This beam, after traversing a distance
of three or four miles, falls upon a lens L, and is thereby focussed on
a reflector R, which is part of the surface of a sphere having its centre
of curvature at the centre of the lens L.
On account of this arrangement it follows that the beam of light,
as a whole, after reflection at the surface of the mirror, emerges from
the lens L in a parallel pencil, and, retracing its former path,” is again
brought to a focus at the point F. It then diverges from F and falls
upon the inclined glass plate, where it is in part reflected and in part
transmitted. The transmitted portion is received by an eye-piece E,
Fig. 213.
and enters the eye of the observer, so that an image of the source S
is seen in the field of view.
The eyepiece E combined with the object-glass at the observing
station constitutes a telescope, which may be referred to as the “observ-
ing” telescope, and similarly the apparatus RL at the distant station
may be referred to as the reflecting collimator. When the apparatus is
properly adjusted these two pieces should be directed towards each other
in such a way that an image of the object-glass of each is formed in
the principal focus of the other. This might be effected before the
mirror R is placed in position by attaching an eye-piece to the tube
RL so as to form a true telescope, whose axis can be directed as
required towards the observing station. This being done the mirror
R has to be placed in position. In order to do this with precision
the following device was resorted to by Messrs. Young and Forbes in
the experiments described subsequently (Art. 299). The mirror was
attached to a cap which screwed on to the end of the tube RL, and
* It is to be observed that a ray entering one half of the lens L is returned after
reflection at R through the other half, so that the path of the reflected ray does not
coincide with that of the incident.

ART. 296 FIZEAU'S METHOD 531
an exactly similar cap was fitted with a piece of ground glass, so that
the glass in one cap occupied the same position as the mirror in the
other. The ground glass cap was first screwed on to the tube RL, so
that the image was received upon the glass in proper focus, the number
of turns of the screw required to effect this being noted. The glass
cap was then removed, and the mirror cap screwed on through the
same number of turns. The centre of the mirror must now be brought
into coincidence with the centre of the lens L, and this was effected
by means of three screws at the back of the mirror. To test this
adjustment a tube about one foot long was placed on the collimator so
as to project in front of the object-glass, and a small ring was placed
at the end of this tube, the ring being supported at the centre of the
tube by three strips of metal. On looking through this ring the
observer ought to see an image of his eye when the adjustment is
perfect, and the screws at the back of the mirror were altered until
this was the case.
To direct the reflecting collimator RL so that its axis shall point
to the observing telescope, it was found most convenient to look
through its object-glass L at the mirror R (the head of the observer
being kept out of the path of the light as far as possible). An
image of the distant station was then seen in the mirror, and the
direction of the axis was changed until the light coming from the
observing telescope was seen in the centre of the mirror.
A toothed wheel, connected to clockwork driven by weights, so
that it could be set in uniform and rapid rotation round an axis parallel
to the axis of the telescope, is placed with its edge at F, so that, as it
rotates the light is alternately intercepted and allowed to pass between
its teeth. Let us suppose the wheel to be at rest, and that F is situated
in the space between two teeth. In this case the beam is allowed to
pass, and a bright image is seen in the field of view. But if F falls
upon a tooth, the light reaching it from S will be reflected directly
back into the eye. To avoid this the teeth may be blackened, or
bevelled so as to reflect the light against the sides of the telescope
which are also blackened. If this is secured there will be no illumina-
tion in the field except when the light passes through a tooth space
and returns after reflection at the distant mirror.
Now if the wheel rotates very slowly the image in the field of
view will appear and disappear successively as the spaces and teeth
pass before F, but if the speed be increased so that several teeth pass
per second, the succession of brightness and darkness will be so rapid
that, owing to the persistence of the visual impression, a permanent
image will be seen. Hence if the angular width of a space be a and
532 THE VELOCITY OF LIGHT CHAP. XIX
the width of a tooth be B, so that the combined width of a tooth
and space is a + 8, it will follow that if the intensity of the image
seen when the wheel is at rest be Io, its intensity when the wheel is
rotating slowly (but rapidly enough to cause a continuous impression)
will be
O,
I = ais"
If the velocity of light is infinite the illumination of the image
will remain constant for all speeds, but if it is propagated in time
then I will depend upon the speed of rotation. Some of the light trans-
mitted through a space will in returning fall upon the adjacent tooth and
be intercepted, and if the speed be great enough, so that when the light
returns a tooth has moved into the position previously occupied by the
space, then all the returning light will be intercepted, provided the teeth
be at least as wide as the spaces, and complete extinction will be effected.
What occurs, therefore, is that at first a bright image is observed,
which diminishes in brightness as the speed of rotation is increased,
and is finally extinguished if the width of a tooth be equal to or greater
than that of a space. If B is greater than a the light will remain
eclipsed until the speed is sufficiently increased to remove the obstruct-
ing teeth and bring the spaces into position for the returning light.
The image now reappears and gradually grows in brightness as the
speed is raised. It reaches a maximum and then fades away again
into darkness, and so on in succession for higher and higher speeds.
If a = 8, or the teeth and spaces are equal in width, then for certain
particular speeds the light will be completely eclipsed, but will reappear
for speeds either less or greater. If B is less than a the light will
never be wholly eclipsed, but will merely fall to a minimum and rise
again to a maximum in alternate succession.
In Fizeau's experiments the teeth and spaces were of equal width,
each being a quarter of a degree, so that the wheel possessed 720 teeth.
In this case, if D be the distance between the toothed wheel and the
reflector, v the velocity of light, and T the time occupied in traversing
the distance 2D, we have
T = ~~~.
Q)
In this time the wheel will have turned through an angle 07' = 2+n.T,
if n is the number of revolutions per second. Consequently, if a be
the angular width of a space, the first eclipse will occur when
a = 2+NAT=4TN, D/v
where N, is the speed of the revolution when the first eclipse occurs.
ART. 296 FIZEAU'S METHOD 533
If the speed increases the image will reappear and grow in intensity
till it reaches maximum brightness at the speed
2a = 4trN2D/v.
As the speed increases further the illumination will wane, and a second
eclipse will occur when
30. = 4trNaD/v.
Similarly the pth eclipse will occur at the speed Nº-1 where
(2p – 1)a=4trNep-1B/v,
from which we have
(2p – 1)a. 2p – 1
***
where m = rſa is the number of teeth contained in the wheel. Reckon-
ing the first maximum brightness as that which occurs after the first
eclipse, it is clear that the pth maximum will occur at the speed Neo
given by the equation
2pa = 4trNaply/v,
from which we have a corresponding formula for v.
Fizeau endeavoured to determine the speeds at which the suc-
cessive eclipses occurred, but this was a matter of extreme difficulty
and uncertainty. In the first place, the intensity of the light return-
ing to the telescope is greatly weakened by transmission through the
apparatus and by reflection at the glass plate G, so that the image seen
is necessarily faint even when at its maximum brightness. It is again
rendered less distinct by the extraneous illumination in the field of the
telescope, caused by reflection from the teeth of the wheel. For when
the wheel rotates, the light when not passing between the teeth is
reflected from them back into the field of view, and causes a general
illumination of the whole field. It is therefore very desirable to sup-
press this reflected light as much as possible, and to this end Messrs.
Young and Forbes, in repeating the experiment, bevelled the teeth so
that the light reflected from them fell upon the blackened sides of the
telescope. They also smoked the wheel itself, so as to diminish its
reflecting power as much as possible.
But even when the distinctness of the image is secured it is very
difficult to decide at what instant it is completely extinguished, it being
much more easy to say when two images seen simultaneously are
equally intense than when any image of varying intensity has reached
its maximum or minimum brightness. Determinations deduced from
direct observations of the speed at maximum or minimum brightness
are consequently attended with considerable uncertainty.
Working with a distance D = 8633 metres, Fizeau found that the
534 THE WELOCITY OF LIGHT CHAP. XIX
first eclipse occurred when the speed of revolution was 12-6 turns per
second, and the final result for the velocity was put down at 70,948
leagues of 25 to the degree. This is taken to represent a velocity of
about 315,000 kilometres per second.
297. Cornu's Experiments.—In 1874 M. Cornu" made a deter-
mination of the velocity of light by Fizeau's method with greatly im-
proved apparatus. Accuracy is very difficult to attain in the method
of observation adopted by Fizeau, since it is almost impossible to
determine when the image is exactly eclipsed. To evade this difficulty
M. Cornu placed the mechanism of the toothed wheel in electrical
connection with a chronograph, so as to mark every hundred revolu-
tions. At the same time a clock marked seconds, and tenths of seconds
were recorded by means of a vibrating spring. The observer had also
under his control a key by means of which he could record any instant
at which he wished to know the velocity. From the chronographic
record the speed and rate of change at every instant could be obtained.
Allowing the speed to increase, the illumination will sink to a certain
value and the speed is then signalled. After complete extinction the
image will reappear, and when it attains the former brightness the
speed is again signalled. The speed corresponding to zero brightness
is the mean of these two, and is found from the chronographic record.
In M. Cornu's experiments the distance D between the two stations
was nearly 23 kilometres, so that he was enabled to observe eclipses up
to the thirtieth order; that is, to make 15 teeth of the wheel pass before
the flash from the distant mirror returned.
The method, however, is not very desirable, in that it is not capable
of delicate measurement in regard to the quantity directly estimated—
namely, the brightness of the image. For the eclipses are not sudden
phenomena occurring at well-marked speeds, but are so gradual that it
is difficult to say precisely when they occur, and even in the method
adopted by M. Cornu to evade this difficulty there must still remain
considerable uncertainty. His investigations, however, surpass in re-
liability and comprehensiveness anything which had been previously
attempted.
The final value obtained for the velocity of light was 300,330
kilometres per second in air, and this corresponds to 300,400 kilometres
in vacuo. This result is, however, somewhat too high, for recent
experiments have established that the velocity is undoubtedly less
than 300,000,000 metres per second.”
* Cornu, Annales de l'Observatoire de Paris (Mémoires, tome xiii. 1876).
* Assuming that the velocity determined by the toothed-wheel method ought
to be the same as that determined by the revolving mirror. [Recently a new
Art, 298 BRIGHTNESS OF THE IMAGE 535
298. Brightness of the Image at any Speed.—If the angular
width of the spaces and teeth be a and 6 respectively, and if the intensity
of the image when the wheel is at rest be Io, its intensity when the
wheel is rotating slowly will bear to Io the ratio a (a +/3), for this
is the ratio of the quantity of light allowed to pass per second when
the wheel is rotating slowly to the quantity when it is at rest or alto-
gether removed. But if the wheel rotates through an angle e while the
light travels over the double journey between the two stations, the space
available for the transmission of the returning light will be reduced
from a to a - e, and the intensity of the image will be
-- e.
I - * Isº
where e = 0T = 2+n.T, when the wheel is making n revolutions per
second.
If the number of teeth be m we have m(a +/3) = 27, and therefore
e = mn.T(a + 3), so that if a/(a + 3)=k we obtain
I =(k - mn.T)In.
Hence the intensities are represented by the ordinates, and the
speeds by the corresponding abscissae of the
right line,
y=(k-mTw).In,
which makes an intercept OB = k10 on the
axis OY (Fig. 214) corresponding to zero
(very slow) speed" and maximum intensity,
and an intercept OA = k/mT, which corre-
sponds to the first extinction. Three cases Fig. 214.
are presented according as a is equal to, less than, or greater than 3.
If a =/3–that is, if the teeth and spaces are equal in width—the
initial intensity (for a low speed) is klo – Io. As the speed increases
the intensity falls in proportion, and complete extinction occurs at the
speed
k 1 º
- ---
As the speed increases beyond this value the intensity rises propor-
tionately, and again attains the maximum value kI, a I, at the speed
determination of the velocity of light has been made at Nice Observatory by M.
Perrotin using Fizeau's original apparatus as modified and used by Cornu (Comptes
rendus, tome crxxi. pp. 731-734). The preliminary results of 1500 measures indicate
a velocity of 299,900+ 80 km., which agrees far better with the value derived from
the revolving mirror method than with Cornu's determination.]
* That is, only fast enough to produce persistence of the visual impression.

536 THE VELOCITY OF LIGHT CHAP. XIx
Nº = 2N1. A further increase of speed diminishes the intensity, and a
second extinction occurs at the speed Na = 3N, and so on in succession.
The variation of the intensity with the speed is represented by the
broken line BA, B, A, etc. (Fig. 215).
If a B or k < , then the first extinction occurs at the speed e = a,
or OM = k/mT = km/2m D (Fig. 216). The intensity will now remain
zero till e = 3, which
occurs at the speed
ON = (1-k)|2m.D. The
brightness again rises to
a maximum klo at B,
corresponding to a speed
e = a + 8, or N = n/2m D.
Fig. 215. The arithmetic mean of
the speeds OM and ON, corresponding to the first extinction and
first reappearance, is equal to the speed at which the first eclipse
would occur if the teeth and spaces were equal, and each (a + (8)
in angular width. This speed, corresponding to A1 (Fig. 216), the
first central extinction, is (OM + ON) = p/4m.D=N1. The speed
corresponding to B, the first maximum brightness, is 2N1; that at
the second central extinction, Ag, is 3N1, and in general the maximum
brightnesses occur at the even multiples of Ni, and the central
extinctions at the odd multiples.
Since mT = 2m D/v = 1/2N, the expression for the intensity at any
speed a before the first
eclipse is
º
1-(-; ).
and since the intensity
must be the same when
the speed is increased
from a to n=2p N, ++, it follows that between the pth and (p + 1)th
eclipses we have (+ r = 2pN1-m)
- 2px1-m) r +( º)
1-(+ 2N, ),={* 12 2N, },
the positive sign applying to the case in which the brightness is
decreasing and the negative sign to that in which it is increasing:
Thus between the pth and (p + 1)th eclipses, when the brightness is
increasing (m-2p N1),
Fig. 216.
I = I0ſ k-p+ º) (brightness increasing),
º 2N,


ART. 299 EXPERIMENTS OF YOUNG AND FORBES 537
and when the brightness is decreasing (n-2p N1)
I– I,(: +p- º) (brightness decreasing).
I
When a -3, or k > , the image is never totally eclipsed but sinks
from a maximum klo to a minimum determined by the speed e = 3, and
the corresponding mini-
mum intensity is given
by the equation
1–. Leº – 1) Id.
The intensity remains
stationary, as represented
in Fig. 217, till the speed
attains the value e = a, from which it will increase and attain a
maximum when e = a +/3, and so on. The variations of the intensity
between two consecutive minima are subject to the same laws as in the
foregoing cases.
299. Experiments of Young and Forbes.—An elaborate series
of experiments on the velocity of light was executed by Dr. J. Young
and Professor G. Forbes," using a modified form of Fizeau's apparatus.
The chief novelty of their method is the introduction of a second
reflecting telescope R'L' (Fig. 213) situated behind the first and nearly
in the same line, so that two images were seen close together (not r}o
of an inch apart) at the edge of the revolving wheel. The ratio of the
distances of the two reflectors from the toothed wheel was 12:13.
Denoting these distances by D and D', and the speeds at which the
corresponding images are first eclipsed by N, and Nº.1, it is clear that if
D′ be greater than D, then N, will be greater than NT, so that the
image from the farther reflector will be first eclipsed and also the first
to reappear. What happens then is that as the speed of the wheel is
increased from rest, one image I’ fades more rapidly than the other I,
till the former is extinguished and the latter remains alone in the field,
if the width of the spaces is less than that of the teeth. One of two
things may now occur. The image I may reappear before the other
I is completely extinguished, or the latter may be eclipsed before the
former reappears, so that both images are extinguished. In the former
case I on reappearance will increase in brightness with the speed,
and I approaching eclipse will continually diminish, so that a speed
will be reached at which the two are equal in brightness; we have
then I = I’, or
Fig. 217.
* Young and Forbes, Phil. Trans, pt. i. 1882.

538 THE WELOCITY OF LIGHT CHAP. XIX
70, / 72
The speed being increased, I will increase and I diminish to zero, and
after eclipse will reappear while I' passes through its maximum, and
begins to diminish. As I diminishes I will increase, and equality will
again be reached, when we have
72, f *
I,(*-irº)=1 (*11-sº)
and in general, if equality occurs between the pth and (p + 1)th eclipses
of both, we have
70, / / 7. * 70,
I,(*-pº)-1-(º-sº)
If equality be again established after the next eclipse of I’—that is,
after the pth eclipse of I and the (p + 1)th eclipse of I'—since I is
now decreasing and I' increasing, if the speed be n', we have
m’ — Tº m/ \
Let I'o - plo and Nº = gnu, then we have g = D/D', and the fore.
going equations become
70, 72,
*-nº-(º-sº)
'n' 'n'
**p-º-º-º-ti)+;
}.
which by subtraction give
****)=sºn) ( +).
Now if the distances D and D′ be so chosen that their ratio g is equal
to 2p}(2p + 1), we have
Q g-H p_{n+n') g +p
-— ,
29 g 2N, 9
OI’
N-" # 3.
which gives N1 in terms of observed quantities, and hence we have the
velocity of light in the form z
v=4mDN,-"Pºtº2.
In the experiments of Young, and Forbes the observations were made
at the 12th equality, and g was almost exactly equal to 12/13. The
value of v deduced was 301,382,000 metres per second. The source
ART. 299 EXPERIMENTS OF YOUNG AND FORBES * 539
of light employed was a Siemens electric lamp, and the speed was
registered electrically. The distances D and D' were 3-18845 and
3:44928 miles respectively.
Any want of achromatism in the lenses will impart a correspond-
ing colour to the image, and atmospheric absorption will produce a
similar effect. Messrs. Young and Forbes observed that one of the
images was in general red and the other blue, but closer examination
proved that when the brightness of either was increasing its colour Coloured
was red, and when its brightness was fading the colour was blue. This "*
they concluded to indicate a difference in speed of the rays of higher
and lower refrangibility. Thus if the blue rays travel faster than the
red, then the red rays will be eclipsed at a lower speed than the blue,
so that as the speed of the wheel is increased, the rays from the red
end of the spectrum will be first eclipsed, and the fading image will
appear blue. On the other hand, when the image is reappearing after
eclipse, the rays which travel slowest will be the first to gain admis-
sion through the adjacent tooth space, and the growing image will
appear red.
To test this point experiments were made with the red light and
blue light from the spectrum (of the electric arc) formed by a prism,
and from the average of the results the experimenters concluded that
the blue rays travel about 1.8 per cent faster than the red. This
difference is so great that, in the absence of other support, the effects
observed have not been generally accepted as due to a difference in
the velocities of the various rays, but it is surmised that the colouring
is rather due to some extraneous cause not yet fully determined.
{Cornu draws attention to the diffraction effects arising (1) from
the waves of light on their way to the distant telescope grazing the
nearer one, and (2) from the use of a mirror with a central hole instead
of a glass plate, in order to increase the brightness of the image.
From this it results that the telescopes receive diffracted pencils
from the edge of the central hole and send back waves diffracted by
the edges of their objectives. To these diffraction effects he attributes
the high value of the velocity obtained by Young and Forbes and the
difference in the observed velocities of the blue and red rays.)
It is clear that such a great difference as 1:8 per cent in the
velocities of the red and blue rays should be detected in the other
methods of estimating the velocity of light. For example, in Foucault's
method, to be presently described, the image of the slit should be
drawn out into an elongated spectrum, but no such colouring or
elongation has ever been observed.
* Cornu, Reports, Congrès Inter, de Phys, vol. ii. p. 229, Paris, 1900.
540 THE VELOCITY OF LIGHT CHAP. XIX
FoucAULT.'s METHOD
300. Foucault's Experiments.-As early as 1834. Wheatstone *
had employed a rotating mirror to determine the velocity of electricity
and the duration of the electric spark. He further conceived that the
same method might be used to determine the velocity of light, and
test between the rival theories as to whether the speed of light was
greater in the more refracting or less refracting media. The suggestion
was taken up by Arago,” but it was not until 1850 that the experi-
ment was designed in a form capable of giving accurate results by the
ingenuity of M. L. Foucault.” The principle of the method is as
follows: -
Solar light, transmitted through a rectangular aperture S (Fig.
218), falls upon an achromatic lens L, and afterwards upon a plane
Fig. 218,
mirror R, which can be made to rotate rapidly round a vertical axis,
the plane of the paper being supposed horizontal. A concave mirror
M is fixed at a distance. The surface of this fixed mirror is spherical
and its radius is equal to the distance RM, while its spherical centre
is at R on the axis of rotation of the moving mirror. Let us first
suppose the mirror R at rest, and so placed that the light reflected
from it comes to a focus upon the fixed mirror M, and forms there a
real image of the slit S. The pencil reflected from M returns along
its former path, is reflected from R, traverses the lens a second time,
and comes to a focus at S, forming an image superposed on the slit.
For the convenience of observation a plate of parallel glass is placed
near S in the path of the beam of light, and inclined to it at an angle
of 45°. The pencil reflected from M when returning to S meets the
1 Wheatstone, Phil. Trans. p. 583, 1834.
* Arago, Annuaire du Bureau des Longitudes pour 1842, p. 287.
* Foucault, Comptes rendus, tome xxx. p. 551, 1850; tome ly, pp. 501, 792, 1862.

ART, 300 FOUCAULT'S METHOD 541
plate where it is in part reflected, and forms an image of S at a, which
is observed through an eye-piece. A fine wire may be placed across
the centre of the slit parallel to its length, that is vertically, so that
the image at a is crossed by a dark vertical line, over which the
fibre of the eye-piece can be accurately placed in making the
measurements.
Let us now suppose that the mirror R is caused to rotate. As
long as the light from R falls on M there will appear an illuminated
image at a, and it is important to remark that on account of the curva-
ture of M and its arrangement, as already described, the image a remains
fixed though R has changed its position, but when R turns round so
that the reflected beam does not fall upon M there will be no illumina-
tion at a. If the rotation be very slow, brightness and darkness
will alternately succeed each other at a, and when the rotation is
sufficiently rapid the persistence of the visual impression leads to
a permanent image appearing at a. The brightness of this image Brightness.
will obviously be much less than when the mirror is at rest, the ratio
of the two being that of the arc M to a whole circumference. Hence
by increasing the magnitude of M the brightness will be increased
in the same ratio, and if in addition the revolving mirror be polished
on both sides the illumination will be doubled.
Let T be the time required by the light to traverse and return
along the distance RM = D, then vT = 2D. But during this interval
the mirror R has turned through an angle (0ſ, if its angular velocity
be 0 = 2nt, where n is the number of turns per second. The axis of the
pencil returning through the lens to a will consequently be rotated
through an angle 20'ſ, viz. twice the rotation of the mirror. The
image a will consequently be displaced to aſ, and the image of S to S',
where SS' = aa' = a suppose. The distance a is measured by means of
the micrometer attached to the eye-piece.
Now the light returning from M is reflected from R and appears
to come from a point situated at an equal distance behind R, so that
the pencils forming the images at S and S appear to come from
sources S, and S', behind R, wherefore RS, - RS' = D, and the lines
joining S and S to the optic centre of the lens pass through S, and
S', respectively. Denote the distances of the lens from the slit and
revolving mirror by a and b respectively. Then since the angle SLS'
is very small, we have its circular measure
SS-5SA, and SS,-2D6,
ſt b + D
where 6 is the small angle between the two positions of the mirror.
542 THE WELOCITY OF LIGHT CHAP. XIX
The speed.
tº
Therefore
Q: F 2a D6 F 2a Day'T == 4awD? 5
b+D T b + D v(b+D)
Ol'
Q) = 8trºmaD?
a:(b+ D)
which expresses v in terms of quantities which can be measured.
In the final experiments of Foucault a beam of solar light was
reflected horizontally from a heliostat through the aperture S. The
sight used was not a fine wire stretched across the aperture, but a
microscope scale which consisted of fine lines traced on a piece of
silvered glass at a distance of ſº mm. from each other. This scale
was placed at S, so that the light passed through it and an image of it
was viewed at a in the field of the observing microscope. What was
observed therefore was not the displacement of the image of the slit
but the displacement of the image of this scale. The revolving mirror
was a piece of glass silvered and polished on one face. This was sup-
ported in a strong ring frame, and its diameter was 14 mm. The
radius of curvature of the fixed mirror M was 4 metres, so that with a
single fixed mirror, as in Fig. 218, the distance D in the foregoing
formula would be 4 metres. In the actual experiment, however, this
was increased to 20 metres by using five fixed mirrors instead of one.
For this purpose M was turned a little to one side, so that the light
reaching it from the revolving mirror was not reflected back directly to
R as already described, but to another fixed mirror of equal radius. From
this it was reflected to a third, and then to a fourth, and finally to a
fifth, which received it normally, and returned it along its previous
path to the revolving mirror, and thence to the field of the observing
microscope. The lens L, which had a focal length of 1.9 metres, was
placed between the revolving mirror and the first fixed mirror (Fig. 219),
and not, as in Fig. 218, between the revolving mirror and the aperture.
In order to determine the speed of the revolving mirror, and to
control it during an observation, a most ingenious device was resorted
to. A finely divided toothed wheel was placed between the observing
microscope and the reflecting glass plate, so that the image of its
toothed edge appeared in the field of view. This wheel was driven
by clockwork at a uniform speed, which could be accurately deter-
mined. Now the beam of light entering the field of view is not con-
tinuous, but intermittent. It consists, in fact, of a succession of
flashes, each flash corresponding to a complete turn of the revolving
mirror R. If the beam of light were continuous, the teeth of the
revolving disc would be seen rapidly crossing the field at a speed
ART. 301 REVOLVING MIRROR METHOD 543
depending only on the rate at which it is driven, and at any consider-
able speed they could not be distinguished in passing. With the
intermittent beam, however, the teeth are illuminated for a very short
time once during each revolution of the turning mirror R, and the
result is that if the toothed wheel turns through an angle correspond-
ing to any whole number of teeth in the time between two consecutive
flashes, the position of the teeth in the field of view will always appear
to be the same when they are illuminated, and the teeth will conse-
quently appear to be stationary. If the speed of the toothed wheel
be greater or less than this the teeth will appear to have a slow for-
ward or backward motion in the field of view.
The revolving mirror was driven by an air turbine so that its
speed could be controlled, and during an observation this was so regu-
lated that the image of the toothed wheel appeared stationary in the
field of view. The speed of the toothed wheel being known, the
number of teeth passing between two consecutive flashes could be
determined, and from this the speed of the rotating mirror was easily
found. §
The observed displacement of the scale was 0.7 mm., and the final
result for the velocity of light was 298,000,000 metres per second.
301. Discussion of the Revolving Mirror Method.—In Foucault's
investigations the distance D was so small (being only 20 metres with
five fixed mirrors) that a large angular deviation of the image was out
of the question, and by reason of the many reflections there was
necessarily a serious loss of light. In order to obtain a large deflec-
tion with a given speed it is necessary to work with a large distance
between the two mirrors, and as there is always light lost by reflec-
tion and absorption in passing over a great distance it is necessary
to attend to the conditions which render the image most brilliant.
When the displacement of the image is large the reflecting glass
plate is unnecessary and may be dispensed with. The image formed
by the returning light can then be observed directly without being
weakened by the reflections attending the use of the glass plate. Now
the angular deviation of the return image, for a given speed of the
revolving mirror, increases with the distance D, and for a given
angular deviation the displacement of the image is proportional to the
distance between the source and the revolving mirror, or, as it is
called, the radius. Hence for a large displacement of the image the
distance between the mirrors, the radius, and the speed should each
be made as large as possible. The second condition is obviously in
conflict with the first, for the slit and the fixed mirror must be
situated in the conjugate foci of the lens L.
544 THE VELOCITY OF LIGHT CHAP. XIX
When the lens is placed between the revolving mirror and the slit,
as in Fig. 218, the quantity of light returned by M to R varies
inversely as the distance D. Thus with a concave mirror of one
decimetre diameter placed at a distance of one kilometre the light
returned to the revolving mirror would not be as much as wºrn
part of the light reflected from it. This quantity is further reduced
by atmospheric vibration, diffusion, and absorption, and with large
distances it is almost impossible to construct a mirror of such uniform
curvature that the rays will fall normally on every part of its surface.
In general only a part of its surface will satisfy this condition, so that
a portion only of it is effective, and this leads to a further decrease in
the brilliancy of the return image. Hence with a concave mirror of
one decimetre diameter for each kilometre of distance only a small
fraction of the grºwn part of the light reflected from the revolving
mirror is really utilised when the apparatus is arranged as in Fig. 218.
On the other hand, when the lens is placed between the revolving
mirror and the fixed mirror, as in Fig. 219, it is easily seen that if R
and M are in conjugate foci of L, then the light reflected from R will
fall upon M as long as
the axis of the reflected
beam falls upon the lens,
however great the dis-
tance D may be. This
arrangement, however,
cannot be made, for it is
Fig. 219. the slit S and not the
mirror R that must be in the conjugate focus of M ; nevertheless, it
may be approximated to by bringing the slit close to the revolving
mirror, and the brilliancy of the return image will be increased
accordingly ; that is, approximately in the ratio of the angular
diameter of the lens, subtended at the centre of motion, to that of the
mirror M at the same point. -
The advantages derived from increasing the distance D are
attended by serious defects in the return image caused by atmo.
spheric diffusion and vibration. For the light which forms the image,
instead of travelling accurately along a definite line ML, between the
fixed mirror and the lens, is scattered through a certain angle. This
leads to an error in position of some parts of the image, which, when
expressed in linear measure, will be proportional to the focal length
of the lens. In other words, a limit is soon reached, beyond which
the angular accuracy with which a micrometer wire can be set on the
image of a star is not increased by increasing the length of the

ART. 302 MICHELSON'S EXPERIMENTS 545
telescope. The result is that the brilliancy of the image cannot be in-
creased many fold by increasing the focal length of the lens without
at the same time increasing too much this source of error.
Another possible drawback to the use of the lens in the position
shown in Fig. 219 is that an image of it, or of some part of it, will
flash through the field of view with every revolution of the mirror,
and this will lead to a certain amount of illumination in the field. It
is obviously desirable that this, as well as all other extraneous illumi-
nation, should be excluded from the field of view, in which a very
faint image is to be observed.
... 302. Michelson's Experiments.-The chief objection to Foucault's
experiments is that the deflection was too small to be measured with
sufficient accuracy, and to remedy this defect Professor Michelson 4
modified the arrangement of the apparatus in such a way that the
return image was displaced through 133 mm., or about 200 times
that obtained by Foucault. With a large displacement such as this
the inclined glass plate could be dispensed with, and the return image
was observed directly through a micrometer eye-piece placed on the
same stand as the slit through which the light was transmitted. The
eye-piece consisted of a single achromatic lens of about 2 inches focal
length. In the focus of this, and nearly in the same vertical plane as
the face of the slit, a single vertical silk fibre was stretched. In
measuring the deflection the eye-piece was so placed that the fibre
bisected the slit, and it was then moved till the fibre bisected the
deflected image of the slit.”
The revolving mirror was a disc of plane glass about 14 inch in
diameter and 0-2 inch thick. It was silvered on the front surface, so
that reflection took place from one surface only. The lens was placed
between the two mirrors, as in Fig 219, so as to secure greater bright-
ness of the return image, and its focal length was 150 feet. The
revolving mirror was placed 15 feet inside the principal focus of the
lens, and the distance between the two mirrors was about 2000 feet.
In the first set of experiments the fixed mirror was plane, being about
* A. A. Michelson, Astronomical Papers for the American Ephemeris and Nautical
Almanac, vol. i. part iii. p. 117, 1880.
* It may be observed that in the experiments of Michelson and Newcomb the
image of the slit was worked with rather than the image of a wire, or scale, placed
across the slit, as in the experiments of Foucault. This was necessary on account
of the great distance between the mirrors, for an accurate image of a line could not
be formed when the distance was considerable. Further, when the distance is large
the transmission through the atmosphere renders the image very unsteady, especi-
ally about the middle of the day. It was only during the hour after sunrise, or the
hour before sunset, that a sufficiently steady image of the slit could be obtained in
these experiments.
2 N
546 - THE WELOCITY OF LIGHT CHAP. XIX
7 inches in diameter, and a small telescope was attached to it (with
the line of collimation at right angles to the surface of the mirror) for
the purposes of adjustment.” The lens was 8 inches in diameter and
was not achromatic, but on account of its great focal length, as com-
pared with its aperture, the want of achromatism was not appreciable.
The “radius,” or the distance between the slit and the revolving
mirror, was about 28 feet. In making an experiment it was found
necessary to incline the axis of the revolving mirror slightly to the
right or left, so that the light falling directly on it from the slit
should not be reflected into the eye-piece but should pass either above
or below it. Without this precaution the light of the slit would be
flashed into the field of view at every revolution of the mirror and
would overpower that of the image to be observed.
The revolving mirror was driven by an air turbine controlled by a
cord leading from its valve to the observer's table. To measure the
speed of rotation a tuning-fork, bearing on one prong a steel mirror,
was used. This was kept in vibration by an electric current from
five “gravity” cells. The fork was so placed that the light from the
revolving mirror fell upon it and was reflected to a piece of plane
glass (placed in front of the eye-piece of the micrometer), inclined
at 45°, and thence to the eye. When the fork and the revolving
mirror are both at rest, an image of the revolving mirror, is seen.
When the fork vibrates this image is drawn out into a band of light.
When the mirror revolves this band breaks up into a number of
moving images of the mirror, and finally, when the mirror makes as
many turns as the fork makes vibrations, these images are reduced to
one, which is stationary. This is also the case when the number of
turns is a submultiple of the number of vibrations. When it is a
multiple (or simple ratio), the only difference is that there are more
images. Hence to make the mirror execute a certain number of
turns, it is simply necessary to pull the cord attached to the valve to
the right or to the left, until the image of the revolving mirror comes
to rest. In this way it was possible to keep the mirror at a constant
speed for three or four seconds at a time, and this was sufficient for
an observation. In a large number of the expériments the speed was
approximately 258 revolutions per second, and the mean result for
the velocity of light in vacuo was put down at
v=299,910+50 kilometres per second.
In a subsequent series of supplementary measures Professor
* The particulars of adjustment are described in Professor Michelson's paper,
loc. cit.
ART. 303 .* NEWCOMB'S EXPERIMENTS 547
*
Michelson determined the velocity of white and coloured light in air,
water, and bisulphide of carbon. In these experiments the arrange-
ment of the apparatus was the same as before ; the fixed mirror,
however, was slightly concave, and had a diameter of 15 inches.
Particular attention was paid to the appearance of the return image,
in order to detect if it indicated any difference in the velocities of the
different colours in air, such as was supposed to have been observed
by Young and Forbes. The actual width of the slit was 0.19 mm.
and the width of the return image was only 0.25 mm. The colour of
the central portion of this image was always yellowish, and occasion-
ally both borders were observed to have a pale violet tinge. There
was consequently no indication of a spectral drawing out of the image
such as would result if the different colours travelled with different
velocities. A difference of velocity such as that obtained by Young
and Forbes should have yielded a spectral image of about 10 mm. in
width. Finally, experiments were made in which a plate of red glass
covered one half of the slit so that one half of the return image was
white while the other was red. The two halves of the image were
found to be exactly in line, and showed no break or displacement
'such as would attend a difference of velocity in the different colours.
The weighted mean of 318 observations on sunlight gave for the
velocity in a vacuum
v= 299,865 kilometres,
while the weighted mean of 267 observations on electric light gave
v=299,835 kilometres.
303. Newcomb's Experiments.-The most recent investigation of
the velocity of light by the revolving mirror method was made at
Washington in the years 1880-82 by Professor Simon Newcomb,”
and this determination is perhaps one of the most reliable and complete
that has yet been recorded here. -
In these experiments the quantity directly measured was not the
linear displacement of the return image but its angular deviation. The
plan of the apparatus is shown in Fig. 220, and it will be seen that
the method followed by Foucault and Michelson of placing the lens
between the two mirrors was not adopted by Newcomb. After due
consideration the method of Fig. 218 was employed, the lens being
placed between the slit and the revolving mirror. The general
* A. A. Michelson, Astronomical Papers for the American Ephemeris and Nautical
Almanac, vol. ii. part iv. p. 237, 1885.
* S. Newcomb, Astronomical Papers for the American Ephemeris and Nautical
Almanac, vol. iii. part iii. p. 113, 1885.
548 THE VELOCITY OF LIGHT CHAP. XIX
feature of the apparatus is that two telescopes were disposed with
their axes at right angles to each other, one, F, termed the sending
telescope, being used to cast a pencil of light on the revolving mirror,
and the other, L, the observing or receiving telescope, being employed
to receive the return beam.
The light of the sun, thrown from a heliostat, entered the slit S of
the sending telescope, and after passing along the tube F was reflected
N º º
º | lº-
______________
Fig. 220.
by a plane mirror at the elbow C through the object-glass J. It then
fell upon the revolving mirror contained in the box m, and was there
reflected along the line Z to the distant fixed mirror. The object-
glass of the receiving felescope was immediately below J, and its tube L
was mounted in adjustable Y's on a frame NN, which moved hori-
zontally around a vertical axis coinciding with the axis of rotation of
the mirror. The farther end of this telescope was fitted with a pair of
microscopes, p and h, for reading the divisions of the graduated arc below.
With the apparatus disposed in this manner it was necessary to










ART. 303 NEWCOMB'S EXPERIMENTS 549
elongate the revolving mirror to such an extent that the light, when
cast upon the upper part of it by the upper or sending telescope F,
should, after reflection at the fixed mirror, return, not to the same
part of the revolving mirror, but to a part somewhat lower down,
so as to enter the lower or observing telescope L. By this means
the light, irregularly reflected at the surface of the revolving mirror
where the incident beam falls upon it, is prevented from entering the
observing telescope, and greater darkness of the field is thereby secured.
With this arrangement almost all the extraneous light can be shut
out of the field, and a very faint image of the slit can be observed.
Further, by causing the mirror to revolve first in one direction, and
then in the other, deviations on opposite sides of the zero can be
observed, and in this manner the zero error can be eliminated.
In order to strengthen the illumination of the return image, and to
avoid loss from any slight displacement of a single mirror, two fixed
mirrors were used side by side. These were concave, and each had
an aperture of about 40 cm., and a radius of curvature of about 3000
metres. -
The revolving mirror, shown in Fig. 221, formed one of the
chief novelties of the apparatus. It consisted
of a square steel prism, the vertical height of
which was 85 mm., while the horizontal cross
section was a square of 37.5 mm. side. The
four vertical faces of the prism were nickel-
plated, and each face in turn acted as reflector
during the revolution, so that the brightness
of the image was quadrupled. A pair of
circular plates C was fastened to the top of the
mirror, and another pair D to the bottom, each
pair holding a set of twelve fans. These con-
stituted the fan wheels on which the air blast
impinged, and set the mirror in rotation.
Either set could be used separately, so that the
mirror could be driven in either direction, or Fig. 221. The Revolving
the two could be placed in action simultane- Mirror.
ously in such a way that one counteracted and controlled the other.
The speed was determined directly by a wheelwork system geared
into a small pinion fixed to the axis of the mirror. This pinion geared
into a larger wheel," on the axis of which a second pinion was fixed.
* It was found that the best metal wheels were worn out almost at once under the
high speeds at which the first wheel was forced to move. For this reason it was neces-
sary to resort to raw-hide as the material for the first wheel, and this proved successful.

550 THE WELOCITY OF LIGHT CHAP. XIX
This in turn geared into a second wheel, which made one revolu-
tion for every twenty-eight revolutions of the mirror. This wheel
broke an electric circuit at each revolution, and by this means
every twenty-eighth turn of the revolving mirror was recorded
on a chronograph along with the beats of a sidereal break circuit
chronometer. g -
In making an experiment the observing telescope was first set in
a fixed position corresponding to some desirable deflection of the
return image, and the speed of the revolving mirror was then so
adjusted that the return image entered the field of view and came to
rest upon the micrometer wires of the eye-piece. In order that the
observer at the eye-piece should be able to regulate the speed of the
mirror, one of the valves T could be controlled by means of an end-
less cord X, which passed through pulleys around the side of the
instrument to the front of the observer's table. This valve being shut
the other was opened and the mirror set in motion. When the speed
reached the proper limit the image was seen entering the field, and as
it approached the cross wires the chronograph was started. The cord
X was then moved so as to open slightly the valve to which it was
attached. This allowed a slight counterblast to act upon the other
fan wheel, and by this means the speed could be regulated with the
greatest delicacy. When the image was steadily adjusted on the wires
it was kept there for about two minutes, and the corresponding record
on the chronograph furnished the speed of rotation. It was remarked
that the higher the speed the greater the delicacy with which the
image could be adjusted to the cross wires. The observing tele.
scope was thén set on some division on the other side of the zero, and
the mirror was made to rotate in the opposite direction, and a run
taken as before. -
The final conclusion from these experiments was that the velocity
of light in air is 299,728 kilometres per second, and in vacuo 299,810
km., with a probable error estimated at 40 or 50 km.
Using only the results of those determinations which were sup-
posed to be free from any constant error, the concluded velocity was
In vacuo, v=299,860+30 kilometres.
Cornu" considers that the limits of error given by Michelson and
Newcomb for their determinations must be considerably enlarged,
partly owing to diffraction at the edges of the reflecting surface, and
partly owing to the unknown results accruing from the great velocity
* Cornu, Reports, Congrès Intern. de Phys, ii. p. 240, Paris, 1900.
ART. 303 NEW COMB'S EXPERIMENTS 551
with which the image of the source sweeps across the reflecting surface
of the fixed mirror. This velocity is comparable with that of light,
and yet it is assumed that reflection takes place according to the same
laws as if no such transverse velocity existed. H. Lorentz," however,
maintains that diffraction will only alter the sharpness of the image
without displacing it, and that the velocity with which the beam of
light traverses the fixed mirror would not affect the directions of the
reflected rays, even though it were of much greater magnitude than it
actually is in the experiments. Cornu considers the most probable
value of the velocity of light in a vacuum to be 300,130 + 270 kilo-
metres per second of mean solar time.
The latest determinations have been made by Michelson (assisted
by F. Pearson)" replacing the square-sectioned mirror by mirrors
with 8, 12, and 16 faces respectively. These were driven by air-
blasts and the speed determined stroboscopically by comparison with
a valve-controlled fork, the rate of which was compared with a free
pendulum furnished by the United States Coast and Geodetic Survey.
The range of path was between Mount Wilson and Mount San Antonio
(twenty-two miles). The octagonal mirror, making 528 turns per
second, rotates through one-eighth of a turn during the time of passage
of light from the revolving mirror to the distant station and back,
thus presenting the succeeding face of the mirror to the returning
beam at (very nearly) the same angle as at rest. Definite observations
were begun in June 1926 and continued till the following September.
The following results were obtained, each of which is the mean of
about twelve determinations:
—-
- Mirror. Year. Velocity in vacuo. Weighted Mean.
Glass (8 sided) . e 1925 299802
* > * 2 3 1925 756
j 3 5 * 1926 813
Steel (8 sided) . e 1926 795 gº
Glass (12 sided) . . 1926 796 29.9796 -- 4
Steel (12 sided). º 1926 796
Glass (16 sided). * 1926 803
2 3 5 y 1926 789
It is proposed to extend the investigation by employing a range of
eighty-two miles between Mount Wilson and Mount San Jacinto.
The following table gives a summary of the chief determinations
that have been made since Foucault.
* Lorentz, Arch. Néer. 6, pp. 303-318, 1901.
* Astrophys, Journ. lxv. p. 1 (1927).
552 THE WELOCITY OF LIGHT CHAP. XIX
RESULTs obTAINED FOR THE VELOCITY OF LIGHT IN WACUo
* Velocity i
Observer. Kij
Foucault, at Paris, in 1862 . º & º e - º 298000
Cornu, 3 3 1872 . e e e - º e 29.8500
$ 3 1874 . & g * * * w te 300.400
Foregoing, as discussed by Helmert . º e • 2.99990
Young and Forbes, 1880-81 . º e º - e e 301.382
Michelson, at Naval Academy, 1879 . e - - e 2999 10
3 y Cleveland, 1882 . t º º º © 29.9853
Newcomb, at Washington, 1882—
(a) Using only results supposed free from constant error . 299860
Perrotin, between Nice Observatory and Mont Vinaigre . 299901
Michelson and Pearson, 1925-1926 . . - e º 29.9796+ 4
J’ 304. Comparison of the Velocities in Different Media.—The first
application of the revolving mirror method was not directed so much
to a determination of the absolute velocity of light in air or any other
medium as to the comparison of the velocities in different transparent
media. The emission theory demanded a greater velocity for light in
the more highly refracting media, while the wave theory required the
reverse, and the burning problem of the time was to obtain a direct
experimental test. For this reason the earlier experiments of Foucault
were directed to determine whether the velocity of light in air is
greater or less than in water. If light travels slower in water than in
air, then it is clear that if a tube of water be placed between the
revolving mirror and the fixed mirror, so as to be traversed by the
reflected beam of light, a longer time will be spent in the double
journey between the mirrors, and the deflection of the return image
will be increased. On the other hand, the deflection will be diminished
when the water is interposed if the velocity in it is greater than in air.
To test this point, therefore, it is only necessary to interpose a column
of water and observe how much, and in what direction, the deflection
is changed at any given speed of the mirror.
ART. 304 DIFFERENT MEDIA 553
The difficulty of comparing the two deflections at the same
speed is eliminated by arranging the experiment so that the two
return images are seen in the field of view simultaneously. This
was the plan adopted by Foucault, and was effected by placing
the tube of water T (Fig. 222) between the revolving mirror R
and a second fixed mirror Mſ of the same curvature as M, and
having its centre at R. When the revolving mirror is set in
motion, a return image will appear as before in the field of view by
reflection from M. This may be termed the air image. For the same
reason there is another return image arising from the light cast upon
M’ and reflected there. This may be called the water image. For
small speeds these two images will be superposed at a, but when the
speed is increased they separate, the water image a' being more dis-
placed than the air image a', showing that light travels faster in air
than in water.
The image observed was that of a fine vertical wire stretched across
the slit parallel to its length, and in order that the two images should
be simultaneously in focus, it was necessary to place a slightly con-
verging lens L'in front of the tube T. This correction was necessary
because the lens L is so placed that S and M are in conjugate foci
when the space between the mirrors is filled with air, consequently S
and M cannot be in conjugate foci of L when the water tube is
interposed. -
In this manner Foucault proved that light travels faster in air than

554 THE WELOCITY OF LIGHT CHAP. XIX
in water, but he made no estimate of the ratio of the velocities. This
was done in 1883 by Professor A. A. Michelson," the apparatus being
disposed in the manner devised by Foucault. The tube T contained
distilled water and was about 3 metres long, the distance between the
mirrors was about 5 metres, and the “radius” 10 metres. The speed
was 256 revolutions per second, and the ratio of the velocity of light
in air to that in water was found to be 1:330. The refractive index
of water for yellow light is 1.333, and the difference between these
two numbers was considered within the limits of experimental
error.
In the case of carbon bisulphide the refractive index for the mean
yellow rays is 1:64, and the ratio of the velocities was found to be
1758. This result is about 7 per cent too high, and the discrepancy
could not be regarded as within the limits of experimental error (see
Art. 307). -
Experiments were also made by Professor Michelson on the
velocities of differently coloured lights in carbon bisulphide. For this
purpose the solar light was passed through a direct vision spectroscope
before it fell upon the slit, and by turning the prism through a small
angle, either end of the spectrum could be observed. The results
indicate that the orange-red light travels 1 or 2 per cent faster
in carbon bisulphide than the greenish-blue light. -
WAVE VELOCITY AND GROUP VELOCITY
305. The Velocity determined by the Methods of Römer and
Fizeau.-Shortly after the announcement of the results of Messrs.
Young and Forbes, Lord Rayleigh * raised and discussed the question
as to what it is that is really determined in observations on the velocity
of light such as we have described. - If we could deal with a single
wave and observe its progress, we could determine its speed ; that is,
the wave velocity. In the case of light, however, we cannot follow
the motion of a single wave. Here we deal with a succession of
flashes or groups of waves, and measure the velocity with which some
impressed peculiarity travels. Thus in determinations by the eclipses
of Jupiter's satellites, or by Fizeau's method, the light is rendered
intermittent, and the velocity observed is that of a limited train of
waves; that is, the group'velocity.
The relation of the group velocity U to the wave velocity V will
* A. A. Michelson, loc. cit.
* Lord Rayleigh, Nature, 25th August 1881 and 17th November 1881;
Collected Papers, vol. 1, p. 537.
ART. 306 ABERRATION METHOD - 555
be found in a note appended to Chap. II. (p. 59). If k = 27/A and
w = 27/T we have -- . . -->
da, GW
U=::=V-Yº,
so that U is identical with V only when V is independent of k—that is,
of the wave length, and this relation is generally supposed to hold for
light traversing free space. The equation shows that a complete
knowledge of V completely determines U, but a complete knowledge of
U does not determine W without the aid of some auxiliary assumption.
The usual assumption is that W is independent of the wave length, in
which case U will also be independent of the wave length. If, how-
ever, W is not independent of A, we may substitute some dispersion
formula, such as
W = A + BK* + CK++ . . .
Using theformula W = A + BK”, and taking the wavelengths of the orange-
red and green-blue lights to be in the ratio 6 : 5, Lord Rayleigh finds
that, if the results obtained by Young and Forbes were correct (viz.
that blue light in space travels 1.8 per cent faster than red light),
the wave velocity for red light would be nearly 3 per cent less than
the velocity determined by the eclipse method, the relation between U
and V being
W =U(1 * 0.0273).
To illustrate the difference between U and W we will calculate
the value for air. From Kayser and Runge's formula (Art. 86),
p = 1.000288 + 1.316X-” where A is expressed in millionths of a milli-
metre. -
7W X d
U=V – X*" = r A 4.
Now dX W | + p, d\
X2 |’
U=W(1 – 0.000007) approximately.
9. s
=vſ. _2.63 or for light of A=600,
Thus in this case the difference is inappreciable. On the other
hand, for water it is given by U =VI1 – 0.015] approximately
306. The Velocity determined by the Aberration Method.-In
measurements depending on the aberration of light the velocity deter-
mined does not obviously depend upon the observation of the rate of
propagation of any impressed peculiarity. It is therefore generally
concluded that it is not the group velocity U, and, according to the
usual theory of aberration, must be the wave velocity V. If this be
so, we have no reason a priori to expect that the velocity given by
Bradley's method should be the same as that found by the methods
of Römer and Fizeau, unless the wave velocity be identical with
556 THE WELOCITY OF LIGHT CHAP. XIX
the group velocity, which postulates that the speed is independent
of the wave length. The close agreement of the velocity V found by
observation of the coefficient of aberration and that (U) given by
Fizeau's method leaves little room for the supposition that U is different
from W in air, and points to the conclusion that the velocity is inde-
pendent of the wave length, or that all colours traverse interstellar
spaces with the same speed.
The problem, however, is not so simple as it may seem to be at
first sight. If the source is moving athwart the line of sight, its
distance, with respect to an observer regarded as fixed, is changing
at a continually changing rate, and thus a variable Doppler effect is
introduced. The observer through his telescope receives, therefore,
not a continuous monochromatic beam, but a beam whose dominant
wave length is continually changing. Ehrenfest has shown," and
Havelock” has confirmed his result by an independent method, that
even the aberration method gives the group and not the wave
velocity. The conclusion come to in the previous paragraph is there-
fore unsound.
307. The Velocity determined by the Revolving Mirror Method.
—In this method of determining the velocity of light, the first reflec-
tion of the beam takes place at the surface of a revolving mirror, and
on account of the motion of this mirror, the angle of incidence varies,
while the direction of the incident beam remains constant. It follows,
therefore, that the successive waves are thrown off at different angles,
and consequently the reflected beam is a curved stream of light, in
which the successive wave fronts are inclined to each other at an angle
depending upon the speed of the mirror.
This reflected stream as it travels through space sweeps past the
fixed mirror, and a segment of it is reflected there. By properly
adjusting this mirror the reflected segment is returned in such a
direction that it falls upon the revolving mirror and enters the eye-
piece of the observing microscope. An image is thus depicted in the
field of view, and its illumination arises, not from a continuous stream
of light, but from a succession of flashes, each flash being produced by
a reflected segment, and corresponding to a revolution of the moving
mirror.” The image being produced by a succession of flashes, it
would appear at first sight that the velocity determined by this method
ought to be the same as in Fizeau's experiment, namely, the group
* Ehrenfest, Ann, d. Phys. xxxiii., 571 (1910).
* Havelock, Camb. Tracts in Math, and Phys., No. 17, p. 14; also Rayleigh,
Phil. Mag. xxi. p. 130 (1911).
* It is to be noticed that the stream leaving the revolving mirror is itself inter-
mittent.
ART. 308 EFFECTS OF RELATIVE MOTION 557
velocity U. But Lord Rayleigh 4 has remarked that because the
successive wave fronts are inclined to each other after reflection from
the moving mirror, therefore, as in the Doppler effect, the wave
length is varying from point to point along the wave front in a
direction perpendicular to the axis of rotation, and, if the velocity
depends upon the wave length, the wave front will rotate in the air
during the transit between the two mirrors, so that the velocity
determined by this method may be some function * of V and U.
It has been pointed out, however, by Professor J. Willard Gibbs &
that although the individual waves rotate, yet the wave normal to the
group remains unchanged; or, in other words, if we fix our attention
on a point moving with the group, and ºtherefore with velocity U, the
successive waves all pass through that point with the same orientation.
He shows this by determining the rotation of a wave front in the
interval of time between the arrival of two successive wave fronts at
the moving point, and concludes, therefore, that the velocity determined
by this method is the group velocity U.
Expressing U in the form
|U = d(Vox-1)
T d(ux-1)
where V, is the vacuum velocity, A the vacuum wave length and p, the
index of refraction, and using Verdet's determinations of p, he finds
that on this supposition for carbon bisulphide
#=1745,
for the mean of the D and E lines. The experimental result obtained
by Michelson was 1:76 for light in which the maximum brilliancy was
between D and E, but nearer to D than to E. This agrees best with
the supposition that U was the velocity actually measured and the
same would be true, if values nearer to the line D were used.
RELATIVE MOTION OF MATTER AND THE ETHER
308. Effects of Relative Motion.—When the observer is not at
rest, but is moving through the medium in which a system of waves
is being propagated, or when the source itself is moving through the
medium, or more generally, when the source, medium, and observer
* Lord Rayleigh, Nature, vol. xxv. p. 52, 17th November 1881.
* The value deduced by Lord Rayleigh was V*/U, and the value found by Professor
V2 7. & e
2WTU’ Nature, vol. xxxiii. p. 439, 1886.
* J. Willard Gibbs, Nature, vol. xxxiii. p. 582, 1886.
A. Schuster was
558 RELATIVE MOTION OF MATTER AND ETHER CHAP. XIX
are moving relatively to each other, certain changes may occur in the
observed effects. These we shall now briefly notice.
It is known, for example, that when the observer is moving
through the air towards a source of sound, or when the source is
moving towards the observer, the frequency (or pitch) of the note
appears to the observer to be increased, whereas, when they are moving
away from each other, the reverse occurs. Thus one effect of relative
motion is a change in the observed frequency. This is the well-known
Doppler effect, and if light be a wave motion in a medium, something
of the same sort should occur when the source and observer are moving
relatively to each other. This change of frequency has been already
noticed and calculated in Art. 283, and may be examined spectro-
scopically. -
Again, if the medium itself be in motion, the waves will drift in
the direction of motion, so that the velocity with which they travel
will be greater in the direction in which the medium is moving than
in the opposite direction. A motion of the medium will thus cause a
change in the velocity of propagation. Further, when there is motion
of the source, medium, or observer, the intensity may be different in
different directions, but this may be compensated in some cases by a
change in the radiating power of the source in different directions
imposed by the motion of the source. -
Finally, when the observer is moving relatively to the source a
change may occur in the direction in which the waves appear to travel.
This change in direction is known as aberration, and on it is founded
the method by which Bradley was led to an estimation of the velocity
of light (Art. 18). *
Hence if light be a wave motion, and if its mode of propagation in
the ether be at all similar to that of waves in an ordinary material
medium, then relative motion of the source, medium, and observer
should produce effects similar to those mentioned above, namely,
changes in frequency, velocity, intensity, and direction. We should
bear in mind, however, that in reasoning on this subject we approach
it with ideas derived from the study of material media, and that in
drawing our conclusions we are merely arguing by analogy from
phenomena which occur in media which we can examine and control
to those of a medium of which we are almost wholly ignorant. In
addition we have no knowledge as to the real nature of the periodic
change which occurs in what we term “the vibration,” whether it be a
motion, as ordinarily conceived, or a periodic change of some property
or condition of the ether. We do know, however, that the velocity of
propagation of light is greater in free space than when passing through
ART. 309 ABERRATION 559
a region in which matter is diffused, and that the velocity in a
space occupied by matter is different for waves of different lengths.
As to whether the ether itself is really modified by the presence of
the matter (for example, if there is a real change of density as Fresnel
supposed), or as to whether the change is only virtual, being merely an
affection arising from the influence of the matter on the velocity of
propagation of waves through the ether in a space in which matter
molecules are diffused, is still a subject of speculation. The prevailing
conception seems to have been that the ether within a piece of matter
really differs in some quality from the ether in free space. If there
is a real change of density in the ordinary sense of the term, then
the doctrine of an incompressible ether would present a serious
difficulty. w
It is sufficient, however, to regard the modifications imposed by
matter to be merely virtual, so that the effect of the molecules of
matter distributed through a region of the ether is to influence the
propagation of waves through that region in a manner similar to that
which would be produced by a real change in those properties of the
medium which determine wave propagation through it. Thus if a
system of floats or other bodies be suspended in a certain region of a
fluid, the propagation of waves through this region will differ from
that in the free regions, and this amounts to a virtual change in the
elasticity or density of the fluid in that region. Thus the fluid itself
may be really unaltered by the presence of foreign bodies distributed
through it, but it may be virtually altered as far as wave propagation
is concerned.
The fact before us at present is that the rate of propagation of
waves in the ether is modified by the presence of matter, and the
question which at once arises is as to whether the velocity of propa-
gation of ether waves in a region occupied by matter is influenced by
the motion of the matter through space. This question we shall now
consider in conjunction with the phenomenon of aberration.
309. Aberration.—In order to illustrate the principles of aberra-
tion let us take the case of an observer A moving in the direction AB
(Fig. 223) with a velocity u, while a particle P moves with a velocity
v in the direction PB, and let the lengths AB and PB represent the
magnitudes of u and v respectively, then it is clear that A and P will
reach B at the same instant and will collide there. In other words,
the direction in which P must be fired in order to strike A is not
along the line PA but along a line PB, making a certain angle with
PA. This is illustrated by the well-known fact that in order to hit a
bird flying across the line of fire the gun must be aimed, not directly
560 RELATIVE MOTION OF MATTER AND ETHER CHAP. XIX
at the bird, but at a point somewhat in advance of it. On the other
hand, regarding the subject from the point of view of the observer A,
it is clear that as P moves along PB, and A moves along AB, the line
joining P and A will be always parallel to its initial direction AP.
Hence what A observes is that P approaches him in a direction parallel
to AP and finally strikes him. The direction in which P appears to
Fig. 223. Fig. 224.
approach A is consequently parallel to AP, while the direction in
which it would appear to move if A were at rest would be PB. Thus
the motion of the observer alters the apparent direction of motion of
the particle P, and the angle APB between the apparent and real
directions of motion is termed the aberration.
In order to obtain a clearer view of the matter let us suppose that
a straight tube is laid from A to P, and let this tube be carried by A,
so that the direction of its axis remains fixed; that is, so that it moves
with a velocity of translation u in the direction AB (Fig. 224). Then
it is clear that when A occupies the positions A, A", etc., P will
occupy positions P', P, etc., such that the lines A. P., A"P", are
parallel to AP—that is, if the observer looks along the tube he will
always see P on its axis; or, in other words, P will appear to move
towards him along the axis of the tube. This, then, is the apparent
direction of motion of the particle.
An expression for the aberration e arising from the motion of the
observer may now be written down at once. For if the angle PAB be
denoted by 6, we have
* AB_sine.
º, PB sin tº
therefore
- tº .
sin e--sin ().
º
The quantity uſe is termed the aberration constant, and it is clearly the
sine of the maximum angle of aberration, or the tangent of the aberra-
tion, when the directions of motion are at right angles to each other.
Instead of regarding P as a moving particle, we might regard it as




















ART. 310 DRIFT PRODUCED BY MOWING MATTER, 561
a certain definite or “labelled ” element of a wave front advancing in
the direction PB; and further, if we think of the tube AP as a telescope,
then we see that the direction in which a telescope must be pointed in
order to receive light advancing in the direction PB, will be BP when
the observer is at rest, but will be AP when the observer is moving.
Hence when the motion of the earth around the sun is taken into
account, it should follow that the direction in which a star is seen should
be different at different times of the year. In other words, the motion
of the earth should cause the stars to appear to describe small orbits
around their true positions, these orbits being small ellipses on the
celestial sphere—stars on the ecliptic describing right lines, and stars at
the pole of the ecliptic describing small circles. This is the phenomenon
observed by Bradley, and the explanation put forward is that it arises
from the orbital motion of the earth compounded with the velocity of
light.
The general explanation of the aberration of the stars in this
manner is so simple, and the value of the velocity of light deduced
from it is so close to that obtained by direct experiment (and from
observations on the eclipses of Jupiter's satellites), that the truth of
the explanation can scarcely be doubted. On the emission theory,
indeed, this explanation would appear to be particularly acceptable,
but, when the subject is examined a little more closely, the explanation
is not so simple as it appears at first sight. This arises from the pos-
sible drift which may occur when a wave (or a light corpuscle) is being
propagated through moving matter.
810. The Drift produced by Moving Matter.—In Fig. 224 we
have considered AP as the axis of a tube, and this tube has been sup-
posed empty, so that the velocity of P remains the same whether we
suppose it moving within the tube or outside it. If, however, the
tube be filled with a medium in which P moves with a velocity v', and
if we suppose the motion of P to be unaffected by the motion of the
medium filling the tube, then clearly the inclination of the tube must
be altered to suit the new velocity vſ. In other words, the aberration
will depend on the nature of the substance filling the tube.
In drawing this conclusion, however, we have left out of account
an important consideration, namely, that, on account of the motion of
the substance filling the tube, P may be dragged in the direction in
which the tube is moving, and that by reason of this drag (or drift) a
compensation may occur, so that the inclination of the tube (or the
observed aberration) may remain unaltered. Further, it must also
be taken into account that the direction of motion of P may change, or
refraction may occur at the surface of the medium within the tube, so
2 O
562 RELATIVE MOTION OF MATTER AND ETHER CHAP. XIX
that we cannot say a priori whether the aberration of a star should or
should not be the same when the tube is filled with a given substance
as when it is empty.
As a matter of fact the experiment has been tried by Airy and
Hoek, and the result is that the aberration of the fixed stars is
observed to be the same whether the telescope be filled with water or
air. The problem, therefore, before us at present is to account for
this fact on the supposition that the light waves drift with the matter
through which they are moving, and to determine the law of drift so
that compensation may occur, and the aberration remain independent
of the medium filling the telescope.
Let AP be the direction of the axis of the telescope (Fig. 225)
and PP the direction of the incident light in space. Then if the
telescope were merely an empty tube the element of wave front at P
would move to B' in the original direction PP. But when the tele-
scope is filled with a refracting medium the axis AP is normal to the
refracting surface, and therefore the angle of incidence is APB, so
that the direction of the refracted ray would be a line PQ, making
an angle r with AP when the telescope is supposed stationary.
Now let P'B' represent V the velocity of light in a vacuum, and
let A'B' represent u the velocity of the observer ; then the direction
in which the tube must be
pointed when empty is A(P'.
Hence if the direction of the
telescope is to remain un-
changed when filled with a
refracting substance, we must
have AP parallel to A'P' in
Fig. 225. Consequently, if
this figure be constructed so
that PQ represents (on the
same scale) the velocity of
light in the refracting substance at rest, then, if there is no drift, P
will proceed through the tube as if the medium occupying it were at
rest, and will consequently not reach Q until A has reached a point B
determined by the equality AB = u = A'B'. But if the waves drift, P
will be dragged with the medium so as not to travel along PQ, but
along some path PB in advance of PQ. When PB is such that AB = u,
then the drift is such that complete compensation has taken place. To
express this we may call QB the velocity of drift; that is, the rate at
which P is dragged in the direction of motion. Denoting it by u' we have
AQ = u-u',
Fig. 225.

ART. 310 DRIFT PRODUCED BY MOWING MATTER. 563
and consequently
w-w'_AQ_sin r. (1)
v TPQ Tsin 6
But denoting A'P'B' = APB' by i we have
sin 9 V
sinº Toº
(2)
where W is the vacuum velocity. Combining these equations we have
at Once *
w–w'_sin r u w
To Tsinº V-LV ;
that is,
or, finally,
The law of drift consequently is that the ether waves must be
carried by the moving matter with a velocity uſ in the direction of
motion, and this velocity is less than the full velocity of the matter in
the ratio (u% - 1)/pº.
This is the law deduced by Fresnel on the supposition that the
“ether density’ is different in different substances, and that the
velocity of propagation of light in any substance varies inversely as
the square root of the ether density (Art. 208). Thus if the ether
density in free space be denoted by p, while that in a given piece of
matter is p", then as this piece of matter moves through space it may be
regarded as carrying its contained ether with it as if fixed to it, while
the external ether is dashed away in front and streams round behind,
after the manner of a fluid in which a solid body is moving. Or, on
the other hand, we may regard the piece of matter as moving through
the ether like a network through a liquid, so that to a person moving
with the matter the ether would appear to flow in at the front of the
body and out at the rear, the total quantity of ether within the body
remaining constant. From this point of view the ether outside the
body may be regarded as fixed, while each unit of volume of the body
carries with it as permanently attached to it a quantity p' – p with the
full velocity u of the body. This is equivalent to saying that the whole
ether within the body is not carried forward with the velocity of the body,
but with a velocity uſ less than u, and determined by the equation *
p'w':= (p' – p)w.
* This equation may also be obtained by considering the flux per unit area in
the front of the body as pu, while that within the body is ap', therefore a = up/p'.
Hence the velocity of the ether within the body relatively to that outside is
w' - w – a = (1 – plp')w.
564 RELATIVE MOTION OF MATTER AND ETHER CHAP. XIX
The equation for u' is therefore
*-( -)-(1 - *).
p º
Hence if the ether inside the matter is moving with this velocity (uſ)
relatively to that outside, and if the light waves in traversing this
moving ether are carried with it with the full velocity u/, then the drift
will be such as would render the angle of aberration independent of
the substance with which the telescope is filled.
Whether the drift of the waves is due to a motion of the ether
caused by the moving matter—that is, by the ether being dragged
with the matter—or whether, on the other hand, the ether is to be
regarded as remaining stationary everywhere, while the matter moves
through it, so that the wave drift is caused in some way by the action
of the moving matter on the waves as they are passing through it, is a
speculation on which it is well to keep an open mind. With regard
to this we shall now consider some of the experiments by which it has
been shown that light waves drift when passing through moving matter.
It is to be remarked, however, that these experiments have always been
put forward as proving that the ether itself is carried with the moving
matter, and that the wave drift is the result of the motion of the
ether.
311. Fizeau's Experiment.—The law of drift contained in Fres-
nel's formula has been directly verified by M. Fizeau 1 in a celebrated
Fig. 226,
experiment in which the moving substance was water. A pencil of
light from a narrow slit S (Fig. 226) falls upon a plate of parallel glass
G, from which it is reflected and passes through a lens L, from which
it emerges in a parallel beam and passes through two apertures A and
B. The vessel CD is divided into two chambers by a partition E, and
a current of water can be forced through it entering one chamber and
leaving the other as indicated by the arrows. The ends of the vessel
are closed with plates of parallel glass, and the light from the apertures
A and B, after passing through the vessel, is received by a lens L and
Fizeau, Ann. de Chimie et de Physique, third series, tome lyii. p. 385, 1859.


ART, 311 FIZEAU's EXPERIMENT . 565
focussed on a mirror M. After reflection from M it again traverses
the vessel CD, passes through the apertures, and comes to a focus at
S'. It is to be noticed that the light when it enters the vessel through
A leaves it through B and vice versa, so that the two pencils which are
focussed at S have each traversed similar paths, one from A to M and
back from M to B, while the other passes from B to M and returns
from M to A. If the paths traversed by these beams differ slightly,
interference will take place and the image S will be crossed by a
system of fringes. Now if a current of water be forced through CD,
the pencil which enters at B and returns through A will travel with
the current and the 6ther pencil will travel against the current, so that
if the motion of the water has any effect on the rate of propagation of
the light, the time of passage of one pencil will differ from that of
the other, with the result that an extra phase difference will exist at
S', and the interference fringes will be displaced by a corresponding
amount. -
Let V be the velocity of light in vacuo, v the velocity in water,
and u the velocity of the water, and let the velocity with which
the waves are carried by the water be waſ, then the velocity of the
ray which travels with the current will be v + uſe and the velocity
against the current will be v — ua, so that the difference of the times
required by the two pencils will be
–––––
Q) — M3; Q) + 740,
where l is the length of the water path—that is 2CD. This determines
the phase difference introduced by the motion of the water, and the
corresponding displacement of the fringes at S', when measured, gives
the value of a.
M. Fizeau obtained a sensible displacement of the fringes when the
velocity of the water was 2 metres per second. With a velocity of
7 metres the effect was measurable, and the result supported the
formula proposed by Fresnel, viz. that ether waves travelling in the
interior of transparent media are carried forward, but with a velocity
less than the velocity of the medium in the ratio (p.” - 1)/u”, where p.
is the refractive index.
This result has been further confirmed by the more recent work of
Messrs. Michelson and Morley," and it may be regarded as proved that
the velocity of light is affected by the motion of transparent matter
through which it is passing. However, Fresnel's theory not only
1 Michelson and Morley, American Journal of Science (3), vol. xxxi. p. 377,
1886. º
565 RELATIVE MOTION OF MATTER AND ETHER CHAP. XIX
supposes that the ether waves are carried forward by moving trans-
parent media, but also that the ether outside the moving body is at
rest, or at least that the velocity of the moving body is relative to a
stationary ether around it."
312. Experiments of Michelson and Morley.—The subject of the
relative motion of the earth and the luminiferous ether has also been
examined experimentally by Messrs. Michelson and Morley.” Their
conclusion is that if there be any relative motion between the earth
and the adjacent ether it must be small. Thus the ether in the
neighbourhood of the earth would appear not to be at rest in space,
but to be carried along by the earth. The theory of their experiment
is as follows.
Let a pencil of light SA (Fig. 227), falling upon a piece of plane glass
A, be partly reflected along AB and partly transmitted along AC. If
the reflected and transmitted portions
fall perpendicularly upon mirrors B
and C they will be returned along BA
and CA. Hence, if AB = AC, the
transmitted part of BA and the re-
flected part of CA will interfere along
AD. Let us suppose now that the
ether is at rest, and that the earth
moves in the direction AC, so that
the mirror A is carried to A', while
the reflected light travels to the mirror
B and back again. In this case the
ray SA will be reflected along AB where the angle BAB is equal to
the aberration * a. The reflected ray returns along B'A', and the
Fig. 227.
- 1
* Lorentz has shown, however, that the expression 1 wº ought to be replaced,
in the case of dispersive media, by 1 -º-º:
sodium the change is numerically from 0.438 to 0-451. A very exhaustive repetition
of the investigation by P. Zeeman (Proc. Acad. Amsterdam, 1915) has yielded results
in accordance with this modified formula.
* Michelson and Morley, Phil. Mag. vol. xxiv. p. 449, 1887. -
* {The reflected ray AB makes the angle 45°--a with the normal to the mirror
as may be seen by considering a plane wave falling on the moving mirror and apply-
ing Huygens's construction to determine the reflected wave front, when the motion
of the mirror is taken into account. Thus, if AE are two points on the wave front,
where E is distant EE' in the direction of the wave propagation from the surface of
the mirror when A meets the mirror, if AE is inclined to the moving mirror AE’ at
an angle of 45°, and if the velocity of the mirror in the direction normal to the wave
front is u, the mirror will move through a distance EE", where E'E"/EE”-u/V,
before E reaches the mirror. The reflected wave front will be E"A", where AA" =EE"
For water and the D line of

ART. 312 MICHELSON'S AND MORLEY'S EXPERIMENTS 567
angle AB'A' is equal to 20. The transmitted ray AC returns along
CA', and is reflected at A* along AT), making CAT)' = 90 – a, and
therefore still coinciding in direction with the transmitted portion
of B'A'. The returning rays B'A' and CA’ do not, however, now
meet at exactly the same point A', but this difference is of the second
order and negligible.”
Let v be the velocity of light, u the velocity of the earth, D the
distance AB or AC, T the time occupied by the ray in passing from A
to C, and Tº the time in returning from C to A'. Then, since C is
moving with velocity u, we have vſ = D + uT, so that
P , T = D
y
Q) + M,
and the whole time is therefore
Q)
T+T =2Dža.
The distance traversed in this time is therefore
v2 w?
2Dºº-2D(l +.)
neglecting terms of the fourth and higher orders. The length of the
other path AB'A' is obviously
2 2
2D (1 rº) –2D(l +.) (approx.),
since AA"/AB = 2w/v.
The difference of the paths ACA’ and AB'A' is consequently
Duº/vº. If the whole apparatus be now rotated through 90° the
difference of path will be in the opposite direction, and a displacement
of the interference fringes corresponding to the retardation
6=2Du?/v2
should occur. Taking u to be merely the velocity of the earth in its
orbit, we have u%|v% = 10−5, and measuring D in wave lengths of yellow
light Michelson and Morley found it was equal to 2 x 10" (about 11
and AA"E" is a right angle. The triangles EAE” and A"E"A are equal and if the
angle EAE’’= p, it follows that p = 45°– A"AE", and therefore EAA" = 24.
Also —?—-º'-- *--—“––––.
sin 45°T AE" VEE/2E EE72T V(V-v)2+V2 V2-2a+cº’
C, Cl
therefore •= (1+.)
or 2% = a to the first order.}
* For a more detailed account of the theory, see Larmor, Aether and Matter,
pp. 46-53, Cambridge, 1900.
568 RELATIVE MOTION OF MATTER AND ETHER CHAP. XIX
metres) in their second experiment. Hence, if the ether be at rest—
that is, if the relative motion of the earth with respect to it be u—we
should have a displacement of the fringes equal to
4 x 107 x 10-8-0-4 (of a fringe width).
The actual displacement observed was certainly less than the twentieth
part of this amount and probably less than the fortieth part.
Since the displacement 8 is proportional to the square of the
relative velocity u, the authors of the experiment conclude that the
relative velocity of the earth and the ether is probably less than one-
sixth, and certainly less than one-fourth of the earth's orbital velocity.
They consequently regard it as tolerably certain that if there is any
relative motion between the earth and the ether, it must be small—so
small, in fact, that the ordinary theory of aberration becomes untenable.
In conducting the experiment the chief difficulties encountered
were the distortion of the apparatus produced by rotating it, and
its extreme sensitiveness to vibration. The latter was so great
that the interference fringes could not be observed when working
in the city, except at brief intervals, even at two o'clock in the
morning. . These difficulties were surmounted by placing the apparatus
on a massive stone floating on mercury, placed in a cast-iron trough
which was cemented into a low brick pier. The stone rested
on an annular wooden float, such that there was a clearance of
about 1 centimetre between it and the sides of the iron trough.
A pin kept the float concentric with the trough, and during the
observations the apparatus was kept in slow uniform motion, making
one revolution in about six minutes. This motion was slow enough
to permit readily of the necessary observations, and the strains caused
by bringing the system to rest at each observation were thus avoided.
The negative result obtained in the foregoing experiment, although
being in accordance with the supposition that the ether in the neigh-
bourhood of the earth is at rest relatively to the earth, or that the
ether outside a moving body is dragged along viscously or otherwise
by that body, may, nevertheless, be explained in some other manner.
For example, it has been suggested by FitzGerald that the force
of attraction between two molecules moving through the ether in a
given direction may depend on the angle which the line joining the
molecules makes with the direction of motion." If this be so, the force
will not be the same when the molecules are moving in the direction of
* This idea has been also put forward by Professor H. A. Lorentz, and developed
in a valuable memoir (Versuch einer Theorie der elektrischen und optischen Eh-
scheinwmgen in bewegten Körperm, Leiden, E. J. Brill, 1895).
ART. 312 MICHELSON'S AND MORLEY'S EXPERIMENTS 569
the line joining them, as when they are moving in the perpendicular
direction. By this means it is possible that a body which is spherical
when at rest may become slightly ellipsoidal when moving; or, in other
words, that the length of a given rod may depend upon the angle
which the direction of its length makes with the direction of its motion
through space. This change in the linear dimensions of Michelson and
Morley's apparatus when rotated through a right angle may be such
as to compensate the displacement of the fringes expected from the
motion of the earth through the ether.
The theories of Larmor and Lorentz on the connection between
matter and ether indicate that a contraction in the direction of the
drift should be expected. Various attempts have been made to subject
the FitzGerald-Lorentz hypothesis to the test of experiment. Morley i
and Miller showed that the effect, if existent, does not depend upon
the physical properties of the substance, pine and sandstone being
affected alike. Lord Rayleigh * has suggested that such a deformation
of matter, when moving through the ether, might be accompanied by
sensible double refraction, but his experiments showed that, certainly
for liquids and probably for solids, no double refraction of the order
to be expected exists. His experiments were completely confirmed
and extended by Brace.” On the other hand, Iarmor 4 contends that
this absence of double refraction does not disprove the FitzGerald-
Lorentz hypothesis, but necessitates the additional hypothesis that
matter is electrical, that is, consisting of electrons.
Michelson's experiment has been recently repeated with various
results. Dayton Miller, working at the top of Mount Wilson with an
interference apparatus in which the light paths were 65 metres long,
considered that his experiments indicated a motion of the sun through
the ether with a velocity not less than 200 kilometres per second in
a direction right ascension 262°, declination 65°. It does not appear,
however, that his apparatus was sufficiently screened from temperature
variations. More recently a repetition has been made by R. J.
Kennedy,” using apparatus on a much smaller scale (the light paths
were only 4 metres long) in order to make it easier to obtain a
uniform temperature throughout. The apparatus was enclosed, her-
metically sealed, and filled with helium in order to diminish (by its
small refractivity) the disturbance from such residual inequalities of
density as might occur. Polarised light was employed so to remove
* Morley and Miller, Phil. Mag. vol. ix. May 1905, pp. 680-685.
* Rayleigh, Phil. Mag. vol. iv. Dec. 1902, pp. 678-683.
* Brace, Phil. Mag. vol. vii. April 1904, pp. 317-329.
* Larmor, Phil. Mag. vol. vii. June 1904, pp. 621-625.
* National Acad. of Sc. vol. xii. p. 622 (1926).
570 RELATIVE MOTION OF MATTER AND ETHER CHAP. xix
all light except that due to the nearly-equal-path rays employed.
Extra sensitiveness was gained by silvering the lower half of one of
the mirrors a small fraction of a wave length more thickly than the
upper half, the dividing line between the two levels being straight and
as sharp as possible. Adjustment being made so that the intensity
was the same above and below the dividing line for any one fringe,
any change in the interference would cause a shift of the intensities.
The effect is much the same as in the Lippich half-shade polarimeter.
No fluctuations of relative intensity were observed due to chance
variations of density. On turning through a right angle no shift was
observed depending upon the orientation. The experiment was
carried out at the Norman Bridge Laboratory, Pasadena, and after-
wards at the Mount Wilson Observatory in the 100-inch telescope
building—again with a null result.
313. Lodge's Experiment.—The influence of moving matter on
the velocity of light in its neighbourhood has been examined by Pro-
w fessor O. J. Lodge' in a
series of experiments. The
apparatus consisted of a
pair of circular steel plates
3 feet in diameter. These
plates were mounted with
their planes parallel and 1
inch apart on a vertical
axis, and were then set
spinning at as high a
speed as they would safely
stand without flying to
pieces. If the ether in
the space between the
spinning discs is dragged
..around with them, then a
pencil of light travelling around between the discs in the
direction of motion would be expected to traverse the space with
a greater velocity than a beam travelling in the opposite direction.
To test this a parallel beam of light was divided into two parts by
a semi-transparent mirror (a piece of glass silvered so thinly that
it transmits one-half of the light and reflects the other half), and
the two halves of this split beam were reflected by fixed mirrors in
* O. J. Lodge, “Aberration Problems,” Phil. Trans, vol. clxxxiv. p. 727, 1893.
In this paper several problems connected with aberration and the relative motion
of matter and ether are discussed in detail.
Fig. 228.
---

ART. 314 LODGE'S EXPERIMENT 57.1
such a way that they passed in opposite directions round and round
the space between the spinning discs, as shown in Fig. 228. The
plane of this figure is horizontal, and shows the space between the discs
around which the beams were reflected several times by four fixed
mirrors forming a square. The beams could thus be caused to traverse
a distance of 30 or 40 feet between the discs, and then be allowed to
enter a telescope, where they interfered and produced fringes.
At first a displacement of the fringes was obtained, but this proved
to be spurious, being caused by the disturbing effect of the air-blast
thrown off by the spinning discs. When due precautions were taken
to avoid this spurious displacement it was concluded that there was
nó indication of a shift due to a viscous or other drag of the ether.
Hence the velocity of light in the space just outside moving matter
would appear to be uninfluenced by the motion of the matter—at least
in the case of a small mass such as that used in these experiments.
314.—Although the null effect in these various experiments
(312, 313) can be satisfactorily accounted for on physical lines by
means of the FitzGerald-Lorentz shortening, yet it has at the same
time formed the basis of a much more startling theory, namely, the
theory of relativity. This starts at the opposite end, and lays down
principles from which, without any ad hoc hypotheses in particular
cases, the null effects would follow. It would be out of place to
discuss this theory here. The probability is that both modes of
approach are legitimate. In much the same way as the principle
of energy applied to the case of thermal questions does away with
the necessity of a detailed examination on the lines of Newtonian
mechanics, so the principle of relativity may be a general guide
enabling certain phenomena to be deduced without the need of
knowing the detailed mechanical structure of the system.
CHAPTER XX
THE RAINBOW
315. Introduction of the Geometrical Theory.— Rainbows are
seen when the sun shines upon falling rain or on the spray of a cascade
or fountain. Sometimes only one bow is observed, but often two are
seen, which are arcs of concentric circles, and have their common centre
on the line joining the sun to the eye of the observer. These bows
are seen only when the observer's back is turned towards the sun, and
they have therefore from the earliest times been attributed to the
refraction and reflection of the sun's light by the falling drops of
water." The inner bow, which is called the primary bow, is about 41°
in angular radius. It is very brilliant and presents all the colours of
the solar spectrum, the red being situated at its outer and the violet
at its inner edge. The radius of the outer bow is about 52°, and it is
termed the secondary bow. It is much fainter than the primary bow,
and presents the same succession of colours, but in the reverse order—
being red at the inner and violet at the outer edge. The space
between the two bows is notably darker than the rest of the sky,
which is pervaded by a general faint illumination inside the primary
and outside the secondary bow.
* “This was understood by some of the antients, and of late more fully discussed
and explained by the famous Antonius de Dominis, Archbishop of Spalato, in his
book De Radiis Visits et Lucis, published by his friend Bartolus at Venice in the year
1611, and written above twenty years before. For he teaches there how the interior
bow is made in round drops of rain by two refractions of the sun's light, and one re-
flexion between them, and the exterior by two refractions and two sorts of reflexions
between them in each drop of water, and proves his explications by experiments
made with a phial full of water and with globes of glass filled with water, and placed
in the sun to make the colours of the two bows appear in them. The same explica-
tion. Des-Cartes hath pursued in his Meteors, and mended that of the exterior bow.
But while they understood not the true origin of colours, it is necessary to pursue
it here a little farther ” (Newton, Opticks, book i. part ii. prop. ix.). . . .
One of the “antients” referred to in the above extract was Theodorich, who,
about 1311, anticipated the work of A. de Dominis. His writings were first
published in 1814 by Venturi (see Tait's Light, p. 125).
572
A Rt. 316 DEVIATION PRODUCED BY A SPHERICAL DROP 573
After A. de Dominis the geometrical theory of the bows was taken
up and developed by Descartes, but it was not until Newton had dis-
covered the difference in refrangibility of the several colours, and their
consequent separation by refraction, that the varied colour of the bow
could be explained. The explanation of the colours as it left the
hands of Newton, and the geometrical theory of the phenomena, afford
only a first approximation to the solution of the problem. The
primary and secondary bows, or, as we may call them, the principal
bows, are accompanied by other coloured bands or spurious bows,
close to the inner edge of the primary and the outer edge of the
secondary. These extra bands are called supernumerary or complementary
bows, and their explanation is afforded only by the more rigorous
application of the wave theory.
The complete theory of the rainbow is therefore not to be obtained
from the simple consideration of rays, but must be treated as a pheno-
menon to be explained by the general principles of interference. This
was first pointed out by Young, and the question was afterwards
considered by Potter," the solution being completed by Airy.” It will
be convenient, however, to consider the principles of the geometrical
theory before proceeding to the more complete solution.
316. Deviation of a Ray emerging from a Refracting Sphere
after any number of Internal Reflections.—The geometrical theory
of the rainbow requires a
knowledge of the manner
in which a pencil of par-
allel light is refracted
through a transparent
sphere, and of the form of
the emergent beam. Let
SA (Fig. 229) be a ray
of light falling on a
refracting sphere ABC. Fig. 229.
Let i be the angle of incidence at A, and r the angle of refraction, then
the deviation produced at A is i – r. Again, since OA = OB = OC, it
follows that OBA = OBC = OCB = r, consequently the deviation pro-
duced by reflection at B is T – 2, and the deviation produced by
refraction at C is as before i = r. The complete deviation of the ray
CE emerging after a single internal reflection is therefore
D=2(? – r) + r - 2,
Potter, Trans. Camb. Phil. Soc, vol. vi. p. 141.
* Airy, Trans. Camb. Phil. Soc, vol. vi. p. 379.

574 * THE RAINBOW . CHAP, XX
and it is obvious that, as each reflection introduces a deviation (T – 2r),
the deviation of a ray emerging after n internal reflections will be
D=2(? – r) + n (tr – 27). - (1)
317. Minimum Deviation.—The foregoing general expression for
the deviation shows that it will vary with the angle of incidence.
We shall now prove that there is a certain incidence for which the
deviation introduced is least, and the rays which suffer this least
deviation are those with which we are most concerned in the theory
of the bows. The change of D corresponding to a small variation
of i is by equation (1)
dD=2di – 2(n+1)dr,
and for a maximum or minimum the value of D is stationary, that is
dD = 0, which gives e
di = (n + 1)ar. (2)
But since sin i = p sin r, we have also
cos id: = p, cos rar,
and therefore by (2)
p. cos r = (n + 1) cos i. (3)
From this we obtain at once
• * ~ * — / p”–1.
90s °F AV nº.12,
which determines the angle of incidence at which the deviation is
stationary. It may be easily inferred that this incidence corresponds
to the minimum deviation rather than to the maximum. For in the
case of the ray passing through the centre, the deviation is ntr, which
is obviously greater than that of any other ray. However, by differ-
entiating a second time we find
G2D d?"
But
dr_cos ? dºr (1 - 0°) sin i.
di Tu, cosºr' dº? T p3 cos 3r
This latter is always negative (p = 1), and consequently the second
derived of D is always positive; that is, the corresponding value of D
is a minimum.
Substituting for p in the preceding formula and taking n equal to
1, 2, 3, 4 in succession, the corresponding minimum deviations are
70, Red. Violet.
1. tr — 42°.1 ºr — 40°.22
2 2T – 129°.2 2T – 125°. 48
3 3tr -- 231°.4 37 – 227°.08
4 4tr – 317°.07 47 – 310°. 07
Art. 318 DISPERSION 575
318. Dispersion.—Since the refractive index is greater for the
violet rays than for the red, the deviation of the violet after any
number of refractions will be greater than that of the red, and
we naturally infer that the least deviation of the red light emerging
from a raindrop will be less than that of the violet. The relation
connecting the variation of the minimum deviation with the refractive
index is easily determined, for we have
dD=2di – 2(n+1)dr,
cos idi–w cos rar-i-sin rdu.
Hence using the relation p cos r = (n + 1) cos i, we have
dL 2 sin r_2 ... 2 (n+1)*-*
dº, ... = ***. Vº -
This expression being positive (i = 90°) shows that the minimum devia-
tion increases with the refractive index.
319. The Emergent Rays.-Let ABC (Fig. 230) be a refracting
sphere and consider a beam of parallel rays falling upon it in the
direction SA. Let SABCE be that ray which suffers the least devia-
tion by the two refractions and a single internal reflection. The
deviation being stationary at the minimum, it follows that in the
direction CE there will emerge a pencil of approximately parallel rays,
and in this direction there will be great concentration of illumination
if there is no mutual interference of the rays. Any ray incident
between M and A emerges, after reflection, from the arc MC, and
its deviation is greater than that of EC; so also the rays which
enter the sphere between A and T (the tangent ray) leave it by the
arc T'C, and the emerging rays cut the ray CE, since they suffer a
greater deviation than CE.
Thus all the rays emerging from the arc MT' lie between the lines
OM and CE. Hence the light which emerges after suffering a single

576 THE RAINBOW CHAP. XX
reflection all lies within a cone of about 42°, and the bounding sur-
face of the cone contains the rays which have suffered the least
deviation. Outside this cone the eye will receive no light. In the
direction CE, the edge of the cone, the illumination is very intense,
and inside the cone the rays are divergent and the illumination
feeble.
The rays emerging from the arc CT touch a caustic curve to
which the ray CE is an asymptote and which touches the sphere at T.
The rays emerging from the arc MC touch another branch NP, which
is also asymptotic to EC produced backwards, and this branch meets
the sphere orthogonally at N; that is, it touches the central ray MN.
These properties of the caustic are easily deduced (Art. 322), but we
shall first consider the formation of the bows. Preliminary to this
-
-
-
|
#
.
Fig. 231. – Reflection. Fig. 232. –2 Reflections. Fig. 233.−3 Reflections.
consideration an inspection of diagrams (231, 232, 233), showing the
minimum deviation rays and the emergent pencil for 1, 2, 3, internal
reflections, will be useful.
320. Formation of the Bows.-The general explanation of the
various bows is easily deduced from the foregoing principles. Let
O, O, O, etc. (Fig. 234) be a series of raindrops in the same vertical
line, or successive positions of the same drop, and let the observer's
eye be situated at E. Draw EC parallel to the direction of the inci-
dent light. The line EC produced backwards will pass through the
sun and will be the axis of the bows. Each drop emits light which
has suffered one, two, three, etc., internal reflections, or, as we may
call it, light of the first, second, and third, etc., orders. Let us first
consider the emergent light of the first order. This emerges from each
drop in a cone, of which the semi-vertical angle is about 41° for the
violet and 43° for the red.
Consider the violet first and draw EO, making the angle OEC
equal to the supplement of the least deviation of the violet ; that is,

A RT. 320 FORMATION OF THE BOWS 57.7
about 41°. Then the drop O, will send its least deviated rays to E,
and will consequently appear very luminous. Any drop, such as 0
situated below 0, will also send light
to E, but in this case it will be from
the interior of the cone, the rays
reaching E from O will be divergent,
and the illumination very faint. On
the other hand, drops above 0, will
fail to transmit any light of the first
order to E, for the point E is situated
entirely outside the cones emerging
from them.
If now the line EO, be made to
revolve round EC, keeping the angle
between them constant, it will gener-
ate a right cone of which EC is the
axis, and the point O, will describe
a circle. Every drop on this circle will obviously transmit minimum
deviation light of the first order to E, drops inside the circle will
transmit faint illumination, and drops outside it will send no light to
the eye.
What we have, consequently, is a brilliant circle, the space inside
of which is faintly illuminated and the outside dark. On account of
the sensible diameter of the sun the bright circle will not be merely
a bright line, but will have a sensible width determined by the
apparent diameter of the sun. The least deviation of the red being
about 43", it follows that the drops which transmit the red will be a
little higher up than those which give the violet. The angular radius
of the red circle will be 43°, and the other colours will occupy inter-
mediate positions. We have therefore a spectrum-coloured band about
2° in width, red at the outer and violet at the inner edge.
Since each simple colour gives rise to a band of definite width, on
account of the magnitude of the sun's disc, it follows that overlapping
and considerable mixture of colours is presented in the rainbow. This
may take place to such an extent that all trace of colour may become
obliterated, or nearly so. We have then the phenomenon of the white
rainbow. This may happen when the sun shines on the raindrops
through a thin cloud in the higher regions of the atmosphere. The
sky is then very bright for one or two degrees around the sun, and
the diameter of the effective source of light is very much increased.
Let us now consider the light which suffers two internal reflections.
The inclination to the incident light of the least deviated red ray is
2 P
Fig. 234.

578 THE RAINBOW - CHAP, XX
about 51° and that of the violet about 54° (see Fig. 232). Hence, if
the angle CEO, is equal to 51°, the drop O, will send intense red
light of the second order to the eye, and drops a little above O2 will
transmit the other colours. We will thus have a second bow, pro-
duced by the light of the second order. This bow will be broader and
fainter than the primary, for the light is weakened by the second
interior reflection, and the dispersion is also greater, on account of the
lengthened path within the drop. Drops below O, transmit no second
order light to E, and those above O, illuminate E faintly. It conse-
quently follows that the space between the two bows is dark compared
with the spaces inside the primary and outside the secondary. The
width of the secondary bow is about 3”, its outer edge is violet and
its inner red.
In the same manner we have bows arising from three, four, or
more internal reflections. The third and fourth bows are situated
between the observer and the sun, so that to be seen the rain should
be viewed with the eye directed towards the sun (see Fig. 233).
The brightness of the sun's light is, however, sufficient to render their
faint illumination unobservable. The fifth bow, however, occurs in
the same part of the sky as the first and second, and would be seen
by an observer with his back turned towards the sun but for its
extreme faintness. Its light is so much weakened by the five reflec-
tions, and dispersion, that it is rarely, if ever, detected.
Intersecting rainbows are also sometimes observed. They require
for their production two sources of parallel rays, such as the sun and
its image by reflection in a large sheet of calm water behind the
observer. The usual system of bows is formed by the direct light of
the sun, and the second system, intersecting them, by the reflected
light, which proceeds, as it were, from the image of the sun beneath
the water. The second system is like the ordinary system, but is less
brilliant, and the bows have their common centre as much above the
horizon as that of the direct system is below it.
Lunar bows may also be seen under favourable circumstances.
They are, however, faint and their colours are not easily distinguished,
except the moon be full and other conditions favourable. This,
together with the fact that they occur only at night, renders their
observation rare.
Much foolish discussion seems to have been raised by the question
whether two persons, or even the two eyes of the same person, see the
same rainbow. From what has been said it is plain that the system
of raindrops which transmits rays of minimum deviation to one point
cannot be the same as that which sends them to another. The ring of
ART, 321 SUPERNUMERARY BOWS 579
drops sending light of least deviation to E lies on a cone of which
E is the vertex, EC the axis, and the semi-vertical angle equal to the
supplement of the least deviation. The question might as well be
asked as to whether two persons see the same object by means of the
same rays of light. *
A similar question is, Can a rainbow be seen by reflection ?—that
is, whether a bow apparently reflected in still water is the image of
that seen directly in the sky. In this case the reflected bow is not
the image of that seen in the sky, but is the reflection of that which
would be seen by an eye vertically below that of the observer, and as
much below the surface of the water as his eye is above it.
321. Supernumerary Bows.-The geometrical theory established
by Descartes and Newton was for a long time accepted as affording
a complete and satisfactory explanation of the phenomenon of the rain-
bow. Closer observation, however, showed that in addition to the
principal bows already described, there are other concomitant coloured
bands, or alternations of brightness and darkness, and of these the
elementary theory gives no account. These additional bands are
frequently seen near the inner edge of the primary bow and less fre-
quently at the outer edge of the secondary. They are called Super-
numerary, complementary, or spurious bows, and are best defined near
their summits.
Young" was the first to propose an explanation of these spurious
bows on the theory of interference, and Mr. Potter” afterwards proved
that the complete theory of the rainbow is to be obtained not from
the geometrical theory of rays, but from the principle of interference.
Young remarked that the efficacious rays consisted of two superposed
sets, parallel in direction on emergence, but traversing different paths
in the drop. Thus the rays emerging from the drop near the ray CE
(Fig. 230) consist of two sets, viz. those which escape from the lower
arc CT" and those which escape from the upper arc CM. The former
will obviously be parallel to some of the latter, and as they travel
over different paths in the drop destructive interference is to be
expected when they are in opposite phases. Near the direction of the
minimum deviation EC we should expect not merely a single maximum
of brightness, but a maximum accompanied by alternations of bright
and dark bands.
To complete the solution we require the form of the wave emerging
from the drop. This was effected by Mr. Potter,” who first traced the
* Young, Phil. Trans. p. 8, 1804.
* Potter, Camb. Phil. Trans, vol. vi. p. 141.
* Potter, loc. cit.
580 THE RAINBOW CHA p. xx
caustic enveloped by the emergent rays and then developed the wave
front as its involute. The final investigation and the determination of
the intensity at any point was given by Airy."
322. Caustic of the Emergent Rays.-Let CE (Fig. 235) be one
of the emergent rays, D
its deviation, and P the
point where it inter-
sects the consecutive ray.
Then P is a point on
the caustic. Denote OP
by p, the radius of the
circle by a, and draw
OQ perpendicular to CP.
We have, if OPQ = y,
OQ=a sin i-p sin Y, (1)
Fig. 235.
and since P is the intersection of two consecutive rays, p remains
constant, while i and y change to i + di and y + dy, therefore
a cos idi–p cos yd),
or by means of equation (1)
dy_w cos i-tan Y
diſp cos y tan (2)
But
D=2(i-r) + m (T-21).
Therefore
dL' o ºr of cos i \
º, -2 20 + 1);=2\l (*!). cos rſ’ (3)
since sin i = p sin r.
- - dL) dy -
Now it is clear that . E-4, therefore by (2) and (3)
tan y = -2 tan (1-(n+1). \ (4)
a cos rſ'
and therefore by (1) and (4)
a sin i ----
-- = a sin i , ſ1 cotºy
p Sun Y V +
tº cosº cosºr
- tº …, 2. - -
Vºn ** {{acos r (n+1) cosº.
The value of p becomes infinite when
a cos r- (n+1) cos i =0.
- tº 1
os - /*
cos t ^
n2+2m
That is when
Airy, Camb. Phil. Trans, vol. vi. p. 379.

ART, 323 SUPERNUMERARY BOWS 581
The direction of the least deviated rays is therefore asymptotic to the
caustic. Again, if i = 90°, we have p = a, so that the caustic meets the
drop at the point T', where the light emerges tangentially (Fig. 230).
In the case of a single internal reflection we have n = 1, and if we
take u = }, as is the case for water, we find that p is also equal to a
when i = 0; that is, the caustic meets the drop at the point N where
the light falls normally on the sphere.
The curve for a single internal reflection is shown in Fig. 230.
The rays emerging from the arc CT touch the branch ET, and those
emerging from CM touch the branch NP, while the ray of least devia-
tion CE meets both branches at infinity.
The form of the wave front may be derived from the caustic, for
the rays are normals to the wave front and tangents to the caustic.
The wave front is therefore an involute of the caustic. For the
branch NP this will be a curve OR (Fig. 236), and for the branch ET"
a curve OR. The asymptotic ray CE will be normal to both OR and
OR at 0, and the point O will be a point of inflection, the tangent at
which will be perpendicular to the ray CE.
323. Intensity at any Point. Airy's Investigation.—In order to
determine the intensity at any point we require the equation of the
surface of the emergent wave. Let ROR
(Fig. 236) be a section of the wave front by
the plane of incidence, OX the least deviated
ray, and OY the inflectional tangent to the
curve at O. Taking OX and OY for axes of
da: dºc
reference, we have dy T 0, and dy? Tº 0 at the
origin. Consequently near O the form of the
curve will be sufficiently represented by the
equation
w-Ay",
Fig. 236.
and in calculating the intensity at any point
near the axis OX we need only consider a portion of the wave in the
neighbourhood of the origin.
Let P be any point near the axis OX, so that its ordinate y is
small compared with its abscissa r. Let S be a point on the wave
front, s the arc OS. Then if the vibration propagated to P by
the element at O be ds : sin ot, that transmitted by the element at
S will be ds : sin (of +8) where 6 is the phase difference, and the
complete disturbance at P will be
+ºo
ſ sin (wt +6)dy,
--~

582 THE RAINBOW CHAP. XX.
for near O the curve closely approximates to the axis OY, and ds is
equal to dy. It remains to calculate 8. -
Let the co-ordinates of S be a, and y, those of P, Š and m. Then if
PS = r we have *
r”= (a, -ś)?--(y – m)*=y°–2/m + 7°--A*y°–2Ay”; +8°,
substituting Ay” for a from the equation of the curve. Replacing
& + m” by p’ and neglecting y”, since y is small, we have
r=(p”–2my--y”–2A#y”)*
2 2 3a,3
=2(1-44;-º-º-º: #)
#Tº Tº Tºº Tºp 25
_79_P-
m”
- +++ -i- –5-a-y
p 2p3
m(p” – mº) _A: \ :
p * (ºr p }v.
Now m is small compared with £, so that p = wº-st; and the
expression for r becomes
2 2
r=#+4. ## (#-A),
2: T : "2:
_; , m” my y” :
=s+#-###-Ay,
approximately.
The term involving the second power of the variable may be
removed by writing
l
V=**5As
and we then have, writing c for the absolute term,
7" - C – tº- A28
= C — 4. – Az”,
neglecting 1/3A, which is of the order yº/a, in comparison with 4&n.
The expression for the resultant disturbance is therefore
+co . 2tre 27t/m2
ſº sin(*-*(*A*)}.
and the intensity is proportional to
+o, 2 2, 2 +-co 2
[ſ. COS *(; +A*). +[ſ. sinº (ºr Ae).] e
The second integral is zero, and the first is twice as great as if the
limits were zero and infinity, consequently
smºs * ... 2 r/m2 - A 23 2
i-Iſ. *(;. Ae).]
ART, 323 - SUPERNUMERARY BOWS 583
This integral may be written in the form
° Tr
ſ cos; (w”--mw)dw
0
(M) = 2 *) and m=#2 N \}
T* \ \ } : TX { \4A) "
The values of this integral have been calculated by Airy for successive
values of m by a method of approximation analogous to that adopted
by Fresnel in the evaluation of his integrals.
As m increases, the integral attains a maximum, and then passes
in succession through a series of minima and maxima, the first maxi-
mum being much greater than any of its successors. The zero value
of m, which corresponds to m = 0—that is, for points on the line OX-
does not then give a maximum, but the first maximum occurs at a
point where m/£ has a positive value greater than zero.
It follows therefore that the first bright band—that is, the rainbow
—is not situated exactly on the line of least deviation, but is attained
at a deviation a little greater, or the angular radius of the primary bow
is a little less than that indicated by the geometrical theory, and
inside the bow we have several alternations of brightness and darkness;
that is, supernumerary or spurious bands.
The angular distances between the spurious bands and the prin-
cipal bow vary with the diameters of the raindrops. For the integral
which determines the intensity reaches a maximum or a minimum
for definite values of m, consequently the corresponding ratio m/3
which determines the position of the band varies directly as (A)*.
But two drops of different diameters, a and a', give emergent waves
of which the sections are similar, and if a., y; aſ, y', be homologous
points on them we should have
where
& Ø $/
aſ Tºy Ty
But
& Ay” . tº Aſ dº
a'TA'y” gy”TA (t/2”
and
(#)-(+)
#) * Q, 5
hence m/$ varies inversely as aft—that is, the smaller the drops the
farther apart are the supernumerary bows; and this explains why they
are only well defined near their summits, for as the drops descend
they grow in magnitude, and consequently the lower portions of the
bow deviate less and less from the geometrical position.
584 THE RAINBOW CHAP. XX
Light
polarised.
The same reasoning applied to the second principal bow shows
that it also is accompanied by spurious bands situated at its outer
edge and best defined at its summit. This bow also deviates from
the geometrical position, its radius being somewhat greater than that
assigned by the elementary theory.
324. Miller's Experiments.-Airy's theoretical results have been
confirmed by the experimental investigations of Mr. Miller." A
pencil of sunlight was admitted in a horizontal direction through a
narrow vertical slit, and fell upon a thin vertical jet of water. View-
ing the stream through a telescope (or with the naked eye) portions
of the primary and secondary bows, and a large number of the
spurious bands could be seen forming a series of vertical coloured
fringes arranged side by side. The diameter of the water jet was
about 022 of an inch. The mixture of colours rendered it difficult
to fix upon the brightest parts of the bands. The mean of eight
observations was as follows:
Radius of brightest part of primary bow 41° 32' \
,, first spurious bow 40° 27'ſ '
3 3 2 3
Radius of brightest part of secondary bow 51° 58' \
3 3 } % ,, first spurious bow 53°57'ſ
According to the elementary theory we should have
Radius of brightest part of primary bow 41° 58'9".
2 3 } % ,, secondary bow 51° 12'-9J
In conclusion it may be remarked that the light of the rainbow is
partially polarised. This polarisation was noticed by Biot as early as
1811, and is to be expected as a consequence of the reflection and
refraction suffered by the light in the drops of rain (Art. 173). The
extent to which the light of any bow is polarised may be easily cal-
culated by resolving the incident vibration into two components—one
polarised in the plane of incidence, and the other perpendicular to it,
and then applying the formulae of Arts. 209, 210.
* Miller, Camb. Phil. Trans. vol. vii. p. 277.
CHAPTER XXI
ELECTROMAGNETIC RADIATION
325. Ether demanded by Electric Phenomena—An Electric
Charge a Charge of Energy and an Electrie Current a Flow of
Energy.—To account for the propagation of heat and light—that is,
of radiant energy—we have postulated the existence of a medium
filling all space. But the transference of the energy of radiant heat
and light is not the only evidence we have in favour of the existence
of an ether. Electric, magnetic, and electromagnetic phenomena
(and gravitation itself) point in the same direction.
It is a matter of common observation that attractions and repul-
sions take place between electrified bodies, magnets, and circuits
conveying electric currents. Large masses may be set in motion
in this manner and acquire kinetic energy. If an electric current
be started in any circuit, corresponding induced currents spring up
in all neighbouring conductors ; yet there is no visible connection
between the circuit and the conductors. To originate a current in
any conductor requires the expenditure of energy. How then is
the energy propagated from the circuit to the conductors? If we
believe in the continuity of the propagation of energy—that is, if we
believe that when it disappears at one place and reappears at
another, it must have passed through the intervening space, and
therefore have existed there somehow in the meantime—we are
forced to postulate a vehicle for its conveyance from place to place,
and this vehicle is the ether.
When a body is electrified, what we must first observe is that a
certain amount of energy has been spent ; work has been done, and
the result is the electrified state of the body. . The process of electrify-
ing a conductor is therefore the storing of energy in some way in, or
around, the conductor in some medium (the ether). The work is spent
in altering the state of the medium, and when the body is discharged
the ether returns to its original state, and the store of energy is evolved.
585
586 ELECTROMAGNETIC RADIATION CHAP. XXI
Similarly a supply of energy is required to maintain an electric current,
and the phenomena arising from the current are manifestations of the
presence of this energy in the ether around the circuit. Formerly an
electrified body was supposed to have something called electricity
residing upon it which caused the electrical phenomena, and an electric
current was regarded as a flow of electricity travelling along the wire,
while the energy which appeared at any part of the circuit (if con-
sidered at all) was supposed to have been conveyed along the wire by
the current. The existence of induction, however, and electromag-
netic actions between bodies situated at a distance from each other,
lead us to look upon the medium around the conductors as playing a
very important part in the development of the phenomena. It is, in
fact, the storehouse of the energy.
Upon this basis Maxwell founded his theory of electricity and
magnetism, and determined the distribution of the energy in the
various parts of the field in terms of the electric and magnetic forces.
The conclusions to which Maxwell came were that the ether around
an electrified body is charged with energy, and the electrical phenomena
are manifestations of this energy, and that the imaginary electric fluid
distributed over the conductor was either not there at all or, at any
rate, took only a very subsidiary part in phenomena. When we
speak of the charge of an electrified conductor we refer to the charge
of energy in the ether around it, and when we talk of the electric flow
or current in a circuit we refer to the only flow we know of, viz. the
flow of energy through the electric field in a direction determined
by the circuit.
Maxwell's expressed hope, that electricity (as distinct from the
energy in the surrounding medium) may be dispensed with, has not
been realised. As is now generally known, something can be separated
from matter consisting of very small particles (each of about 1/1800
the mass of a hydrogen atom) which have a negative charge of the
same amount in whatever way they are produced. These are the
electrons. It is true that they might be regarded as a very fine kind
of matter, especially as they are found to have inertia. They con-
tribute, however, an insignificant amount to the total mass of the
matter which has emitted them. What they are is beyond the reach
of science; but for convenience of description they are defined as
pure negative electricity. The main part of the mass of a body is
due to something else—part, at any rate, of this has a positive charge.
When an electric current flows in a metallic wire it is a current of
electrons. But with each electron there moves a disturbance in the
surrounding space (electric and magnetic fields). When the rest of
ART. 326 POLARISATION OF THE ETHER—ELECTRIC WAVES 587
the atom moves it also gives rise to fields of the opposite kind due to
its positive charge. The difference between this and Maxwell's view
is only very slight. It may be summed up by saying that it is still
found convenient to speak of electric charges, and to regard them as
centres from which the surrounding forces can be calculated according
to certain ascertained laws. In regard to details about these centres,
an immense amount of knowledge has been accumulated in recent
years which cannot be described here.
326. Polarisation of the Ether—Electric Waves.—The work
spent in producing the electrification of a conductor is spent on the
ether and stored there, probably as energy of motion. To denote this
we shall say that the ether around the conductor is polarised, this word
being employed to denote that its state or some of its properties have
been altered in some manner by the work done on it; that is, by the
energy stored in it. In the case of a conductor possessing what is
termed a positive charge, the ether around it is polarised in a certain
manner and to a certain extent depending on the intensity of the
charge. If the charge be negative the polarisation is in the opposite
sense, the two being related, perhaps, like right-handed and left-handed
twists or rotations. s
Now consider the case of a body charged alternately, positively and
negatively, in rapid succession. The positive charge means a positive
polarisation of the ether, which begins at the conductor and travels
out through space. When the body is discharged the ether is once
more set free and resumes its former condition. The negative charge
now entails a modification of the ether or polarisation in the opposite
sense. The result of alternate charges of opposite sign is that the
ether at any point becomes polarised alternately in opposite directions,
while waves of opposite polarisations are propagated through space,
each carrying energy derived from the source or agent supplying the
electrification. Here, then, we have a periodic disturbance of some
kind occurring at each point, accompanied by waves of energy travel-
ling outwards from the conductor.
The phenomena of interference lead to the conclusion that light is
the result of a periodic disturbance, or vibration, of the ether, but as
to the nature of the vibration—that is, as to the exact nature of the
periodic change—or what it is that changes, we possess no know-
ledge. From the foregoing we see that alternating electric charges
are accompanied by corresponding changes of state, or vibrations, of
the ether, and if the charge be varied periodically and with sufficient
rapidity, we have a vibration at each point analogous to, and perhaps
identical with, that which occurs in the propagation of light.
588 ELECTROMAGNETIC RADIATION CHAP, XXI
This, then, is the electromagnetic theory of the luminous vibration.
In the older or elastic-solid theory, the light vibrations were supposed
to be actual oscillations of the elements or molecules of the ether about
their positions of rest, such as take place when waves of transverse
disturbance are propagated through an elastic solid. Such a limitation
is, however, unwarranted. All we know is that the change, disturb-
ance, vibration, polarisation, or whatever we wish to term it, is periodic
and transverse to the direction of propagation. The electromagnetic
theory teaches us nothing further as to its nature, but rather asserts
that whatever the change may be, it is the same in kind as that which
occurs in the ether when the charge of an electrified body is altered or
reversed. It reduces light and heat waves to the same category as
waves of electric polarisation; the only quality of the latter required
to constitute the former is sufficient rapidity of alternation. These
speculations have received the strongest confirmation by the recent
important experiments of Professor Hertz. Before describing them
we shall consider the mode of discharge of a condenser. The
theoretical investigation was given by Sir William Thomson’ (Lord
Kelvin) as early as 1853. -
327. Oseillating Discharge.—When any elastic substance is sub-
jected to strain and then set free, one of two things may happen. The
substance may slowly recover from the strain and gradually attain its
natural state, or the elastic recoil may carry it past its position of
equilibrium, and cause it to execute a series of oscillations. Something
of the same sort may also occur when an electrified condenser such as
a Leyden jar is discharged. In ordinary language there may be a
continuous flow of electricity in one direction till the discharge is
completed, or an oscillating discharge may occur; that is, the first
flow may be succeeded by a back-rush, as if the first discharge had
overrun itself and something like recoil had set in. The jar thus
becomes more or less charged again in the opposite sense, and a second
discharge occurs, accompanied by a second back-rush, the oscillation
going on till all the energy is either radiated or used up in heating the
conductors. --
Ilet Q be the charge of the jar at any instant, C its capacity, R the
resistance of the circuit, and L its coefficient of self-induction. Then
if I be the intensity of the current and E the electromotive force, we
have the equation
dI
d
| Sir William Thomson, Phil. Mag., June 1853.
ART, 327 OSCILLATING DISCHARGE 589
In this case E = Q/C, and I = - *3. Therefore
dt
d0 Q_
+R+3=0.
d”9
dº?
The solution of this equation is
Q=Ag"+ Be"
where p, and p" are the roots of the equation
R. 1
2. L ** ss= *
*++,+of+0.
OI’
= -R = , /*_l
g ** - 3I.F M II? - GL"
Writing
* 4L* CL’
we have
p = – R/2L + c, pſ' = – R/2L – a,
and -
_Rt
Q=eT2E(Aeo'4-Be-a')
where A and B are constants determined by the initial conditions, viz.
that initially we have Q = Qo, and I= 0, which give
- - A + B = Q0, and Ap, + Bu'-0,
Or -
R R.
A-2(1H+.) and B=Q(-;) &
Hence at any time we have
_Rt R t R.
Q = Q06 #((+;)* (*-ā)--") *
Consequently the current at any instant is
* d0 Qo _Rt olt - Out
I=-ji=g. Ti(*-e )
Hence if a be real—that is, if we have Rº-4L/C—the quantity Q
will gradually diminish to zero as the time increases.
If, however, we have R*- 4L/C, then a will be imaginary, and
writing
I R2
a' = a W-i = CLTTL2'
the above formulae become at Once
Ré R
== 6. " OT. A g P
Q = Qo #( cos at +3; sin a •)
and Rt.
Q0. e T2L
TCLo.
sin aſt.
590 ELECTROMAGNETIC RADIATION CHAP. XXI
In this case the current starts from zero and rises to a maximum ;
it then falls to zero and becomes reversed, after which it passes
through a series of oscillations. The discharge therefore does not
take place in a single flow from one coating to the other, but a back-
rush sets in, and a series of currents, or oscillations, occur alternately
in opposite directions. -
The current attains its maximum intensity when
tan aſt– 2La'/R (maximum current).
The zero value of the current is reached when
a’t = nºr (zero current),
and consequently the charge at the same time is at its maximum, for
we have I = — d0/dt. Thus the charge oscillates backwards and for-
wards, attaining positive and negative maxima after the lapse of equal
intervals Tſa', the time of a complete oscillation being
2ir
T-7+H:
^/ CLT 4L2
If the resistance be small compared with the reciprocal of the
capacity we may use the approximate formula
T=2ir NCL.
The successive maximum charges occur when I = 0, or at = nºr; they
are therefore
2m-R 37ſ B.
Q0, Q = - Qeº, Q2=Q06 2Laº, Q3= – Q06 2Da’’
The quantities therefore diminish in geometrical progression, and the
energy of the charge diminishes correspondingly on each oscillation,
being lost in heating the circuit.
Whether the discharge is continuous or oscillatory therefore depends
on whether 4L is less or greater than CR”, and an oscillatory discharge
may be obtained either by increasing L or sufficiently diminishing C
and R. f
It will be shown later (§ 328) that the rate of decay of the
oscillations is brought about in part by radiation. This is not taken
account of here. - A
These predictions of analysis have been confirmed, as Thomson
suggested, by examining the spark, during discharge, by means of a
revolving mirror. In Feddersen's experiments the image of the spark
* If the capacity be expressed in electrostatic measure, and the self-induction in
electromagnetic, this expression takes the form 27 NCL/v where v is the velocity of
light, but it is better to keep consistently to one set of units in any equation.
ART. 328 HERTZIAN WIBRATORS 591
in a revolving mirror was viewed through a telescope. When the
resistance of the circuit was high the spark was merely drawn out in
width—that is, at right angles to its length; but when the resistance
was sufficiently reduced, so that the oscillating discharge might occur,
the band was reduced to a broken image consisting of a series of strips,
each strip corresponding to a discharge.”
Paalzow examined the discharge through a vacuum tube in the
same manner. With a high resistance a bluish light showed itself at
one pole, but with a low resistance it was exhibited at both. In the
latter case a magnet held near the tube split the discharge band into
two lines of light, but in the former the magnet effected no separation
of the discharge, showing that the discharge occurred in one direction
only or in both, according as the resistance of the circuit was high
or low.
The time of oscillation of an ordinary Leyden jar is from # to #
millionth of a second, giving rise to electromagnetic waves about 50
to 100 metres long, if we assume their rate of propagation to be the
same, or approximately the same, as that of light. A small thimble-
sized jar would give rise to waves about 1 metre in length, and if
we push the reduction to the limit of the light waves, or 6000 tenth-
metres, we would require a circuit approaching atomic dimensions. This
suggests that the long electromagnetic waves emitted by a condenser
undergoing oscillating discharge are in reality the same in kind as the
short ethereal waves which affect the retina, and that the light and
heat waves are excited by electric oscillations in the atoms or molecules
of the incandescent matter. *
328. Hertz’s Experiments.-The electromagnetic waves radiated
by a condenser undergoing oscillating discharge have been ingeniously
detected by Professor Hertz. The method is analogous in principle to
the use of resonators in the detection of sound waves. It is well
known that a vibrating tuning-fork when held near an open pipe will
throw the air column into vibration and elicit a note from the pipe if
the length of the pipe be so adjusted that its period is the same as
that of the fork. In the same way, if an oscillating discharge be main-
tained in a circuit, it will play the part of the tuning-fork, and if
another circuit be at hand which possesses the same periodic time of
electric oscillation, then it will play the part of the resonator, and will
be thrown into electric vibration. These sympathetic vibrations have
been detected and examined by Hertz as follows.
The vibrator consisted of two brass plates A and B (Fig. 237),
* For an account of the researches made in this department see Wiedemann,
Elektricität, vol. iv. pp. 177, etc.
592 ELECTROMAGNETIC RADIATION CHAP. XXI
to each of which is attached a stiff wire terminated by a brass knob.
The knobs are gilt and placed about 2 or 3 millimetres apart, so
that when the plates are oppositely electrified a spark can easily
cross between them. Electric oscillation then sets up, the charge
passing backwards and forwards from
one plate to the other and diminishing
continually in quantity as its energy
becomes used up in heating the wires,
sparking, and especially by radiation
into space. The time of each oscillation is approximately 27./CL,
and this in the experiment will be about 1/30,000,000 of a second,
if the plates used are about 40 centimetres square.
At each passage of the charge between the plates of the vibrator,
induced currents will occur in neighbouring conductors, and if the
periodic time of an oscillation in one of these should happen to
be the same as that of the vibrator, the oscillations induced in
it will become multiplied and may attain a considerable intensity.
This conductor then acts as a resonator. The
resonator (or receiver) devised by Hertz was
a simple circle of wire (Fig. 238), the ends
terminating in brass knobs which could be
adjusted at a small distance apart. The length p
of the wire 1 being carefully adjusted to suit the
period of oscillation of the vibrator, then, when
the vibrator is in action, the induced current
flows backwards and forwards in the resonating
circle from one knob to the other, and finally attains such a strength
that sparking actually occurs between the knobs.
The charging of the vibrator is effected by connecting its plates
with the terminals of an induction coil. In this way the oscillation is
maintained in the vibrator, and continuous sparking can be obtained
in the resonating circle.
The reflection and interference of electromagnetic waves may now
be observed with facility. Placing a large metal screen (a sheet of
zinc 2 or 3 metres square) immediately behind the resonator, the
sparking is observed to increase in brilliancy, and the distance between
the knobs may be augmented beyond the limiting distance at which
sparking would occur before the screen was brought up. When the
screen is moved back from the resonator to a distance of 2 metres,
say, the sparking ceases, and cannot be obtained even on screwing the

A Fig. 237.—The Vibrator. B

Fig. 238.-The Receiver.
* If the plates of the vibrator are 40 cm. square, the length of the receiver is
about 210 cm. of No. 17 wire.
ART. 328 HERTZIAN WIBRATORS 593
knobs very near each other, but when the screen is moved twice as far
away (4 metres), sparking is again obtained which is more vigorous
than when the screen is entirely removed. Without the screen spark-
ing is obtained at every distance, diminishing only in intensity as the
distance between the oscillation and resonating circuits is increased.
With the screen, on the other hand, sparking is obtained at some places
and no sparking at others, even though these may be nearer the
vibrator.
It should be observed that no sparking is obtained in the reson-
ator when it is placed immediately behind the zinc sheet, and this
shows that the zinc sheet screens off the electromagnetic action of the
vibrator.
The interpretation of these results is that waves of electromagnetic
disturbance are radiated from the vibrator which fall upon the metal
screen and are reflected there. The reflected waves interfere with the


c - T-
º #Diº
-H
Receiver. Fig. 239. Vibrator.
{
direct waves, as in the case of sound. At the distance of a quarter-
wave (2 metres in the above experiment), or any odd multiple of a
quarter-wave from the screen, the two are in opposite phases, and
their combined effect on the resonator is zero, while at distances of
2, 4, 6, etc., quarter-waves from the screen the direct and reflected
waves are in accordance, and vigorous sparking is produced in the
resonating circuit. The corresponding interference of direct and
reflected sound waves is well exhibited in the nodes and loops of
organ pipes.
Having once obtained these electromagnetic waves, and the means
of observing them, Hertz proceeded to inquire into their laws of
reflection and refraction. To this effect he modified the apparatus so
as to render it more sensitive, and concentrated the radiation by means
of zinc reflectors bent into the shape of parabolic cylinders. The
vibrator consists of two brass cylinders (Fig. 239), each about 12 or
13 centimetres long, 3 centimetres in diameter, and rounded or ter-
minated by knobs at the sparking ends. The cylinders are placed in
2 Q
594 ELECTROMAGNETIC RADIATION CHAP. XXI
the focal line of the parabolic reflector, and are connected to the
terminals of an induction coil which excites the oscillations. The
receiver consists of two pieces of thick wire each about 50 centimetres
long. They are also placed in the focal line of a parabolic reflector,
and from the end of each a thin wire passes at right angles through
the reflector to a sparking space at the back, where the induced spark-
ing can be observed without obstructing the waves falling on the
reflector. The total length of the vibrator is nearly half that of the
emitted wave. For success it is necessary that the pole surfaces of the
sparking space should be frequently polished and guarded against the
illumination of simultaneous lateral discharges, for the rays of higher
refrangibility are detrimental to the working of the apparatus.
With this apparatus Hertz detected reflection from the walls and
obstacles around the room, and subjected the electromagnetic radiations
to most of the experiments which it is customary to perform with light
and heat rays. He had a large prism of pitch constructed, and verified
that these waves are refracted in passing through it, just as light and
heat waves are refracted in passing into a new medium. The angle of
the prism was 30°, and a deviation of 22° was observed in the trans-
mitted waves, giving a refractive index of 1.69 for these long electro-
magnetic waves. The refractive index of pitch for light waves varies
from 1-5 to 1-6, but a disagreement in refractive index is not to be
taken as evidence against the similarity of light and electromagnetic
waves, for it would be remarkable, and even contrary to expectation
(see Arts. 292-5, and 336), if the refractive index of waves 1 metre
in length happened to be the same as that of waves 200,000 times
shorter.
From the mode of production of these waves it is clear that they
consist of transverse vibrations and are plane-polarised, for the electric
force is parallel to the vibrator and the lines of magnetic force are
circles round it. The directions of the electric and magnetic forces are
consequently in the wave front and at right angles to each other. If
the receiving mirror be rotated round the direction of the rays coming
from the vibrator the sparking gradually ceases, and when the focal
line of the receiver is horizontal—that is, at right angles to the focal
line of the vibrator—no sparks can be obtained, even though very close
to the vibrator. The two pieces of apparatus thus play the part of
the polariser and analyser in a polariscope. g
The polarisation of the waves may be shown by introducing between
the mirrors a wire screen consisting of wires wound parallel to a given
direction on a light frame, so as to resemble a diffraction grating.
When the direction of the wires is parallel to the oscillator—that is,
ART. 328 HERTZIAN WIBRATORS 595
to the focal lines of the mirrors—the screen is opaque to the radiation,
it refuses to transmit the waves but reflects them copiously. On the
other hand, when the direction of the wires is perpendicular to the
focal lines of the mirrors, the waves are freely transmitted and sparking
is obtained in the receiving circuit. The screen is thus opaque to the
radiation when the direction of the wires is parallel to the electric
force, but transparent when the wires are parallel to the magnetic force.
It thus plays the part of an analyser.
Hertz's experiments have been repeated in this country with
various modifications, and particularly in Dublin by G. F. FitzGerald,
and Dr. F. T. Trouton,” who have further studied the cases of
reflection from non-conducting substances, such as glass and paraffin.
They have also settled the long-disputed Question as to the azimuth
of vibration in relation to the plane of polarisation in plane-polarised
light. Fresnel and his followers required the luminous vibration to
be at right angles to the plane of polarisation, while, according to
MacCullagh and Neumann, it must take place in that plane. On the
other hand, Maxwell's theory indicated that something occurs in both
planes, a magnetic vibration in one and an electric in the other, or
rather a vibration accompanied by magnetic force in one azimuth and
electric force in the perpendicular azimuth. The theory further
indicated that the electric force is perpendicular to, and the magnetic
force in, the plane. of polarisation. This conclusion has been verified
by experiment, for the electromagnetic waves are found not to be
reflected at the polarising angle from the surface of a bad conductor,
such as a wall, when the electric force is parallel to the plane of inci-
dence, but reflection occurs at all angles when the electric force is
perpendicular to the plane of incidence. Plane-polarised light is best
reflected when the plane of polarisation coincides with the plane of
incidence (Art. 167), and it follows, therefore, that for the polarised ray
the electric force is perpendicular to, and the magnetic force in, the
plane of polarisation.
Hertz's experiments have been repeated by Sir Oliver J. Lodge
and Mr. E. J. Dragoumis” at University College, Liverpool, with
a slightly modified form of receiver. In order to render the effect of
the electric oscillations more easily observable in the resonating cir-
cuit, a Geissler's tube was placed in the spark-gap, one electrode of the
* F. T. Trouton, Nature, 22nd August 1889, etc.
* E. J. Dragoumis, Nature, 4th April 1889. An account of the recent work in
this department has been given by Sir Oliver J. Lodge (the work of Hertz and
Some of his successors, being an evening lecture at the Royal Institution, 1st June
1894. Reprinted from The Electrician by the Electrician Printing and Publishing
Company, London).
596 - ELECTROMAGNETIC RADIATION CHAP. XXI
tube being connected with either side of the gap. When the apparatus
is at work the tube lights up and renders the effect of the electric
oscillations visible even at a considerable distance. The detection of
the oscillations by the chemical action of the current on iodide of
potash paper was also tried. -
The sympathetic oscillations in the receiver may also be detected
by means of a delicate galvanometer. One terminal of the galvano-
meter is connected to one side of the spark-gap, and the other to the
other side of the gap. When sparking occurs at the gap a deflection
of the galvanometer occurs, and with a delicate long-coil galvanometer
a very marked effect may be produced. Great delicacy of adjustment
at the gap is necessary, but when properly arranged the apparatus is
very sensitive. By this method FitzGerald has exhibited the effect
to a large audience.
The Hertzian vibrations may also be detected by the method of the
bolometer; that is, by the variation, when heated, of the resistance of
a thin wire placed across the spark-gap,
329. Radiation of Electromagnetic Waves.—When an electric
oscillator is at work a periodic variation, or vibration, is occurring at
each point of the field around it. The distributions of electric force
in the field at the stages t = }T, i = }T, t = {T, t = }T, have been
plotted by Hertz and are represented in Figs. 240-243. If T be
the time of a complete vibration, the state of the field at any instant
during the first half period is just reversed at the corresponding
instant of the second half, so that it is only necessary to trace the
changes in the field during half a complete vibration. Fig. 240 shows
the field at the time t = 0, or t = n}.T. Here the poles of the oscillator
are free from electrification, and no lines of force run out from them.
Electric charges are just about to accumulate on the extremities.
Lines of force are about to spring out, and will continue to spring out
till t = }T. At this time the field of force will be as represented in
Fig. 241. Here it may be observed the lines of force are not drawn
right up to the poles of the oscillator, for the formulae (Art. 339) from
which the curves were plotted regard the oscillator as exceedingly
small, so that in the immediate neighbourhood of a finite vibrator they
do not sufficiently represent the true state of the field. A small
spherical space around the oscillator, to which the formulae cannot be
applied, is consequently left unplotted. The speed at which the dis-
turbance travels out in the initial stage is much greater than the
normal velocity, and when t = }T the electric wave has nearly tra-
versed half a wave length instead of a quarter. There is thus a gain of
* G. F. FitzGerald, Evening Lecture at the Royal Institution, 21st March 1890.
ART. 329 DEVELOPMENT OF THE WAVES 597
a quarter period, and the velocity finally subsides to the normal value
(1/s/Ku) at some distance from the origin. Fig. 243 represents the
field at the time t = }T, and Fig. 242 that when t = {T. In the latter
a singular action is shown as coming into operation. The lines
farthest from the origin are being drawn together by the stress with a
lateral inflection. This inflection approaches nearer and nearer to the
axis of 2 till finally a self-closed surface like a vortex ring detaches
itself from each of the outer lines, and these spread out into space
while the residue sink back into the conductor. The number of
Fig. 240, t=0. Fig. 241, t = T.
_-
|
!
g
(i)||
º/
*F34A
%)
º
i. Nº) -
-342
Fig. 242, t={}T. Fig. 248, t- #T.
)
receding lines is exactly the same as the number of originally expand-
ing lines, but their energy is diminished by that of the detached
portions. These portions carry energy out into space, and in conse-
quence the oscillation must soon subside unless the source be supplied
with energy.
Returning now to Fig. 240 it will represent the state of things
at the time t = }T, the closed surfaces being the detached lines of
force travelling out into space. The oscillator is again without charge,
but charging is about to commence and new lines of force are about to
spring out from the poles. These will compress the detached lines
whose history we have just followed, and the situation at the time



598 ELECTROMAGNETIC RADIATION CHAP. xxi.
t = }T will be as shown in Fig. 241. So also Fig. 243 will represent
the field when t = T, and Fig. 242 when t = gT, whereas Fig. 240
will again show the oscillation about to start a fresh disturbance."
An approximate estimate of the energy radiated in actual working
has been given by Hertz. Two spheres of 15 cm. radius were charged
in opposite senses up to a spark length of about 1 cm., so that their
difference of potential was about 120 C. G. S. electrostatic units. The
charge of such a sphere was accordingly about 900 C. G. S. units, and
the total store of energy possessed was about 60 x 900 = 54,000 ergs
or 55 gramme-centimetres. The length of the oscillator was about 1
metre and the wave length 480 centimetres approximately. The loss
of energy comes out to be 2400 ergs to half a swing, so that after
eleven half-swings one half of the initial energy is radiated. This
rapid damping is made evident by experiment, and could not be
obviated even though the resistance of the oscillator and spark were
negligible. A loss of energy of 2400 ergs in 1:5/100,000,000 of a
second means the performance of work equal to 22 horse-power, and
the oscillator must be supplied at this rate if the vibration is to be
maintained at a constant intensity, in spite of the radiation. During
the first few swings of the vibrator the intensity of the radiation at
about 12 metres distant from the vibrator corresponds to the intensity
of the solar radiation at the surface of the earth (see Art. 344).
Before entering into the mathematical investigation of the pro-
perties of these curves, we shall summarise Maxwell's equations of
the electromagnetic field and give a brief discussion, from the point
of view of the electromagnetic theory, of reflection, refraction, and
dispersion. For this purpose it will be useful, in the first place, to
obtain an expression for the line integral of a vector taken round a
small rectangular circuit.
330. Line Integral of a Vector.—In a field of force the force at
any point may in general be represented in magnitude and direction
by a right line. Such a quantity, possessing
both magnitude and direction, is termed a
directed quantity or a vector. The line integral
of a vector round any curve or circuit may be
defined as / F cos ºds where F represents the
value of the vector at any point of the curve,
ds the corresponding element of the curve,
and ºf the angle between the directions of F and ds. Since F cos q,
is the component of F parallel to ds, it is clear that when F is a
Fig. 244.
* The curves have been recalculated by Professor Pearson and Miss Alice Lea,
making allowance for damping due to radiation (Trans. Roy. Soc., 1900).

ART. 331 FUNDAMENTAL ELECTROMAGNETIC EQUATIONS 599
force, then the line integral is simply the work done in traversing
the circuit.
Let us now consider a small rectangular circuit of sides 8a, and Öy,
and centre O (Fig. 244). Then if the components of F at O be a and
B respectively in the directions of the sides of the rectangle, and if F
varies from point to point of the field, it follows that when the sides
of the rectangle are very small the vector components which appear in
the line integral round the circuit ABCD will be
a – #y along AB, 8+ #º along BC,
CL -- #y along DC, 8– #º along AD.
Hence the line integral taken round the rectangle in the direction
ABCD is -
•-ºw Öa; + 84.4% ôy – at #y 6a: – 8-iº ôy
$/ $/
90
_ (d8, da
* (# dy )way.
The quantity #–% is termed the curl of the vector, and consequently
we have found that the line integral of a vector. taken round a small
rectangular circuit is the product of the area of the circuit by the curl
of the vector, or more generally, the line integral of a vector taken
round the perimeter of any area is equal to the surface integral of the
curl taken over the area.
331. Fundamental Electromagnetic Equations—Isotropic Di-
electric.—We are now in a position to write down the equations con-
necting the electric and magnetic forces in any medium. In the first
place, for simplicity, let the medium be an isotropic non-conductor such
as air or glass.
Then if we consider any circuit placed in such a medium, it is found
by experiment that an induced current is generated in the circuit when
the magnetic field varies, and that the electromotive force of this
current is measured by the rate at which the whole flux of magnetic
induction passing through the circuit changes. That is, the line
integral of the electric force taken round the circuit is equal to the
time rate of change of the surface integral of the magnetic induction
passing through it. Hence if we consider elementary rectangular
circuits of sides, 8y, 82, Öz, &c, &, Sy respectively, having their planes
parallel to the planes of reference, then denoting the components of
the electric force at the centre of the parallelopiped 80, Sy, 62 by P, Q, R,
and the components of the magnetic force by a, B, y respectively, we
600 ELECTROMAGNETIC RADIATION CHAP. XXI
have the line integral of the electric force round the rectangle 8y, 62
equal to
dR d()
(; - #) öyöz,
and if the magnetic permeability of the medium be p, the flux of
magnetic induction through the same rectangle is
• Awačyöz;
hence paying attention to sign we have at once the relation
da_dR d0
T*ā-āy Tú. (1)
similarly
d8 dB dR
" ** Tºjz Tº
_,dy_d0 dB
*d. Tº Tây (3)
(2)
That is, the curl of the electric force is equal to the time rate of change
of the magnetic induction. In the same way the line integral of
magnetic force taken round a circuit is equal to the time rate of
change of the flux of electric displacement passing through the circuit,
so that if the specific inductive capacity of the medium be K we have
the corresponding set of equations"
dP_dy dB
dt T dy dz’
d0 da dy
“dt dz da'
.d.B. d8 do.
iſ = # T dy' (6)
(4)
(5)
* The electric current in a straight wire is related to the magnetic force in the
field around it by the fact that the line integral of the magnetic force taken round a
curve enclosing the wire is equal to the current multiplied by 4tr. Thus if the
current strength be C the magnetic force at a distance r from the wire is 2C/r, and
the integral of this taken around a circle of radius r is 2Trx 20/r=4trC. To apply
this to a dielectric, Maxwell introduced the idea of what he termed “displacement”
currents. Thus if we consider an electrified conductor placed in a dielectric, the
electric density (or displacement) at any point of the conductor being p, the electric
force just outside the surface of the conductor will be 4trp in air and 4Tp/K in any
other medium. This displacement is supposed to exist wherever there is electric
force, and a variation of the displacement corresponds to an electric current, the
current strength being dp/dt. As the displacement at any point of an isotropic
dielectric is a directed quantity, being in the direction of the electric force, it follows
that if its components parallel to the axes be denoted by f, g, h at any point, then
the components of the force at that point are
P=4trf/K, Q = 4rg/K, R=4th/K,
consequently the components of the displacement current are
_dſ_K dB ... dg K d0 _dlv_K dR
*Tºº Ar 27 ° ºf T 4: ... ***, * 1: ...,'
ART. 831 ISOTROPIC DIELECTRIC 601
We have thus six equations connecting the six quantities, a, B, y,
P, Q, R, and from these we may easily deduce a differential equation
for any one of them. Thus if we differentiate (1) with respect to t
and substitute from (5) and (6) for d0/dt and dR/dt we have
Kº-º;(;-#) _d #-#)
*d;2T dy\dy Tāz) Tâz\do, T dz
_dºa dºa dºa d (da, d8, dy
****** :-#(#####)
But since we have supposed the medium to be an isotropic non-con-
ductor in which the magnetic density is zero, or, in other words, that it
contains no source or sink of magnetic force, it follows that the flux
of force taken over the surface of an element of volume is zero ; that is,
do dB., dy_
dºt dºt 3: "".
Consequently if we as usual, denote the operator *****, b v2
Q. y 5 5 p da’ dy?' d2? y
the equation for a becomes
d’a
Pºk;
= V*a,
with similar equations for 3 and y. -
In the same way by differentiating (4) with regard to t and sub-
stituting from (2) and (3) we obtain the equation
d”P_e/2p d /dE d0 dR
ck;=v-P-#(#####).
Hence if we apply the above law connecting the current strength with the line
integral of magnetic force we have
dy d8 GP
dy d2 =4tru-K d;"
Thus the curl of the magnetic force is 4t times the electric current, and, expressing
equations 1, 2, 3 in similar language, we may say that the curl of the electric force
is 4T times the “magnetic current.” Thus KP, KQ, KR are the components of
what may be called the electric flux density, and KdF/dt, Kd2|dt, KdF/dt are the
components of the electric current density. Similarly u0, p8, uy are the components
of the magnetic flux density, and puda/dt, pudg/dt, pudy/dt the components of the mag-
netic current density. When the medium is not a perfect non-conductor the com-
ponents of the conduction current must be taken into consideration, and if the
electric conductivity be k these components will be kP, kQ, kB, so that the equa-
tions become -
d _dy d8
(tre+k}) P=#-; etc.,
d dR d()
* (raiº) **āj-ž. etc.
The factor g which appears in the latter equation has been introduced by Mr.
Oliver Heaviside, and is the magnetic analogue of k ; that is, it is the magnetic
conductivity or the reciprocal of the magnetic resistivity. There is no evidence that
g has any existence.
602 ELECTROMAGNETIC RADIATION CHAP. XXI
But since there is no electricity distributed through the dielectric
the electric density is zero at every point of it ; in other words, there
is no source or sink of electric force, and therefore
dP, d0, d.R.
#####=0,
so that the differential equations for the electric force become
2
AK # =V*P, a K º =V*Q, uk *—vºr.
The components of the electric and magnetic forces thus satisfy the
same differential equation, and this equation is of the same form as
that which determines the vibration of an elastic solid. It follows,
therefore, that a periodic electromagnetic disturbance is propagated
after the manner of a wave motion in an elastic solid, and that the
velocity of propagation is determined by the equation
•===
Wu.K
332. Plane Wave in an Isotropic Non-conductor.—Let us now
consider the simple case of a plane wave propagated through an
isotropic dielectric. Let the plane of the wave front be parallel to
the plane ay, so that the axis of 2 is normal to the wave front; that is,
parallel to the direction of propagation. In this case all the quantities
which determine the character of the wave are functions of 2 and t,
being independent of a and y. Hence the equations (3) and (6) of
Art. 331 become -
dy
tº dB
*:::=0. and Ki-0,
which show that y and R are each independent of t; that is, they are
each zero (or constant) and play no part in the propagation of the
wave. We conclude, therefore, that the electric and magnetic forces
have no component (or at least no periodic component) normal to the
* In the equations of Art. 331, all quantities are measured in terms of the same
kind of units, i.e. electromagnetic units. If the current, electric force, and specific
inductive capacity be expressed in terms of electrostatic units, and magnetic force and
permeability in terms of electromagnetic units, equations (1) to (6) take the form
p da_dR d0 -
tº- od; T dyT dz’ etc.,
4tru _K dP _dy - d6 etc
v. T v di Tay dz’ “
since the electrostatic units of electric force, quantity, and specific inductive capacity
are respectively v, v', v-" times the corresponding electromagnetic units.
The equation at the close of the Art., for the velocity of propagation, applied to
the ether, shows that the ratio of the two units of quantity must be numerically equal
to the velocity of wave propagation in free space.
ART. 332 PLANE WAVE IN AN ISOTROPIC NON-CONDUCTOR 603
wave front ; that is, the disturbance is confined to the plane of the
wave front or is transverse. An electromagnetic wave consequently
possesses the essential property of a light wave in having the vibration
in the wave front, or transverse to the direction of propagation.
Further, it is clear that if the axes of a; and y be so chosen that
the axis of a is parallel to the direction of the resultant magnetic force
in the wave front, then a will represent the whole magnetic force, so
that we have £8 = 0 permanently, and consequently by equation (2),
since R is always zero, it follows that P is always zero. From this
it follows that Q must be the resultant electric force ; that is, if the
resultant magnetic force is parallel to the axis of a, then the resultant
electric force is parallel to the axis of y. In other words, the electric
and magnetic forces are in the wave front, and are at right angles to
each other.
We see, therefore, that according to the electromagnetic theory of
light the vibration is a transverse periodic disturbance attended by
electric force in one direction, and magnetic force in the perpendicular
direction. The former is the direction of vibration postulated by the
theory of Fresnel, and the latter that demanded by MacCullagh. The
war between the rival theories is consequently at an end, for one
speaks of the direction of the electric force while the other speaks of
the magnetic force, so that both are equally right and equally wrong
in dealing with the vibration as a whole.
The general differential equation of propagation becomes simplified
in the case of a plane wave travelling in the direction of the axis of 2.
For since P, Q, R., a, 3, y are independent of a, and y, the equations of
Art. 331 become
d°P_&P
d;2 T dº?’
d?Q_d°Q dºB
plk d;2 T :2’ diz T
p.K 0,
with similar equations for a, 9, y. Hence while R and y are each
zero (or constants), the general expressions for P, Q, a, B are of the
form -
P= p(2 – vt) + p(z+vt)
where v is the velocity of propagation and is equal to 1/Vuk. In
the case of air K is equal to unity if the electrostatic system of units
be employed, and Vp is the reciprocal of a velocity; but if the
electromagnetic system be used p = 1 and VK is the reciprocal of a
velocity. Hence if the light vibration be an electromagnetic dis-
turbance, this velocity should be equal to that of light—that is, of
light waves of the length here considered—and this conclusion is
supported by the results of observation. Again, since for transparent
substances the permeability is found to be practically the same as that
604 ELECTROMAGNETIC RADIATION CHAP. XXI
of the ether, the velocity of propagation in a transparent dielectric is
V1/K times the velocity in free ether; but the index of refraction of
a substance is the ratio of the velocities in free ether and in the
substance, and therefore the square of the index of refraction of a
substance should be equal to its specific inductive capacity (measured
electrostatically). For gases, in which the variation of the index of
refraction with wave length is small, this connection is found to hold.
For dispersive substances, approximate agreement is found, if the
refractive index be taken for the long wave lengths comparable
with those for which the specific inductive capacity can be measured,
but the application of the theory to dispersion will be considered
later.
333. Reflection and Refraction. — In order to investigate
what occurs when the electromagnetic disturbance passes from one
isotropic transparent substance into another, we must consider the
boundary conditions at the surface of separation of the two media. If
this surface be plane and the normal to it be taken as the axis of 2, the
usual conditions that the tangential magnetic and electric forces must
be continuous (which follow from considering the work done in going
round a small rectangular circuit with two sides parallel to the surface
of separation, but one in one medium and the other in the other
medium) lead to the continuity of a, B, P, and Q in passing through
the surface. The same conclusion follows from the presence of the
differential coefficients of a, 3, P, and Q with regard to 2 in the two sets
of equations (1) to (6), Art. 331, since these differential coefficients must,
e tº tº & ŽR d
therefore, be finite. The differential coefficients º, # do not appear,
but if p1; p.2, K1, and K, be the permeabilities and specific inductive
capacities of the two media the third equation of each set shows that
Anyl-plºys and K1R1 = K2R,
where the suffixes indicate the media in which the forces are taken.
As for all transparent media the permeability is not sensibly different
from that of the free ether, p, a p, = 1, and we see that the normal
magnetic force is also continuous, but for media of different specific
inductive capacity the normal electric force is discontinuous at the
surface of separation (considered infinitely thin) and K1R1 = K, Rø.
If we consider a plane wave in the medium Ki falling on the
surface of separation of the two media at an angle i, and take the
electric disturbance in the wave front as of the simple harmonic type,
and further take the plane of incidence as the a, 2 plane, this electric
disturbance may have components
ART. 833 REFLECTION AND REFRACTION 605
271-
A cos X
(la;+ my--nz – Wit),
B cos *(x+ my--nz – Wit),
in the a, 2 and y, z planes respectively, where l, m, n are the direction
cosines of the normal to the wave front.
As l = sin i, m = 0, n = cos i, we have
2 & * o
P=A cost cosº sin 42 cost-Vit),
2tr, . . º
Q=B cosº (a sin it a cost-Vit),
- tº tº 2T . e &
R= — A sin i cos *@ sin i +2 cos ? – Wit),
the positive direction of 2 being taken into the second medium.
From equations (1), (2), (3), Art. 331, we deduce for the magnetic
forces
C, F - B cos ?, cos *(a. sin i+2 cos i – Wit),
Wi X
A 27ſ., . . g
8-y cos-r (a sin i + 2 cos i – Wit),
B . . 2tr, . . • * r
ºy=# sin icos = -(a sin i+2 cos i – Vit).
Wi X
Corresponding equations hold good for the forces in the reflected
and refracted waves. Thus, for the reflected wave, we have
e 2tr, . . -
P’ = A' cos iſ cos #(a sin iſ +2 cos ? – Wit),
2tr, . . . º
Q =B' cos * (a sin iſ +2cos '' – Wit),
tº ge 27r, . . &
R’s — A' sin º' cos #(º sin '+2 cos ?' – Wit),
f 2 -
º T, . . º
a' = — V, cos 'cos V (a sin i' +2 cos ? – Wit),
I -
, A' ... 2 * — —”/ *
8 =v, COS #6. sin º' + 2 cos iſ – Wit),
f
B' . . 2 : " . º
7-#. sin 'cos #(º sin iſ +2 cos ?' – Wit);
and for the refracted wave
w 27r, . r
P2 = A2 cos r cos jø sin r +2 cos r – Waſ),
27r, .
Qa=B2 cos *(* sin r-i-2 cos r–Wot),
2
º 2 g
Ra– – A2 sin r cos sº sin r +2 cos r – W2t),
B 27r, .
ag= – º cos r cos (a sin r +2 cos r–Wºt),
2 W X cº
2 2
8.-: COS **@ sin r +2 cos r – Wat),
W2 X2
Bo . 2T, .
%=# SII) 7" COS *(a. sin r-i-2 cos r – W2%).
2 2
606 ELECTROMAGNETIC RADIATION CHAP. XXI
The boundary conditions must hold for all values of t, a and y at
the plane z = 0, which is only possible, if
sin i_sin iſ sin r
wº-vi-V."
which equations reduce all the forces to dependence on the same
function of a and t. These equations contain the ordinary laws of
reflection and refraction, for they give i = T — i, as the reflected
wave does not, in general, coincide with the incident wave, and also
sin i / sin r = W1/Vs.
Further the boundary conditions require that the total magnetic
force along each axis in the first medium (i.e. the sum of the forces
due to the incident and reflected waves) shall be equal, at the plane
2 = 0, to the magnetic force in the second medium, taken in the same
direction and at the same plane. We thus get
_B cos i B' cos’ B, cost (1)
- V, V, - W2 }
A + A' A2
wº--w,' (2)
B sin i , B' sin i' B, sin r (3)
wº-r-vi---v-
Similarly the components of electric force at the plane z = 0 give
the equations
A cos ?--A' cos ? = A2 cos r, (4)
B+ B' = B2, . (5)
Kı(A+ A*) sin i = K, A2 sin r. (6)
Of these six equations (2) and (6) are identical, as are (3) and (5).
The four independent equations enable us to determine A, B, A, and B,
For we have
p V, cos ?"
B-B-Bºy
B+ B' = Ba,
COS 7°
A — A' = A2+, »
COS 7, ,
therefore
2A = A2ſ +...+...+, ) = A2+*: tº
SLIl 7" COS 7. Sln 7° COS 7,
( sin à cos ..) cos (i+r) sin (i – r)
A2 —r— — ". . ~~ A2 y
Sill 7" COS 7,
sin $100s ..) _A sin (i+r) cos (i-r)
2A'
-:
sin r cos ?
sin à cos r sin (; + r.)
2B = B2ſ H- = D2-T.”
COS 7, Sl. Il 7° COS 7, SIIl 7°
2B'- B,
sin icos r in (i —
1_sin )=-B; (i. w),
cos i sin r cos i sin r
ART. 334 CRYSTALLINE DIELECTRIC 607
_ Atan (i − r)
T“tan (; Er)'
B' = — B
Aſ
sin (; — r)
sin (i+r)'
_ 2A sin r cos i
Tsin (i+r) cos(?–7)”
2
2B cos i sin r
B2=++++
sin (i+ r.)
3.
which are Fresnel's equations for the reflected and refracted waves
(see Arts. 209, 210), attention being paid to the note (Art. 210).
As A vanishes if i + r = 90° (see Cor. 1, Art. 211), we see that
the electric force is perpendicular to the plane of polarisation of a
plane-polarised ray.
For perpendicular incidence the values of A' and B reduce to
A (# ) and – B (#). showing, if p > 1, that both components of the
reflected electrical vibration are of the opposite sign to those of the
incident vibration, as it is to be remembered in the above that
the positive directions in the reflected wave were taken in the same
sense with regard to the wave normal as in the incident wave, but
this wave normal was rotated by the reflection away from the z axis
so as to make with it the angle T — i instead of i.
The components of the magnetic vibration in the reflected wave
prove to be of the same sign as those in the incident wave. Wiener's
experimental results, therefore, that a node is formed photographically
at a reflecting surface, by interference between a beam incident normally
and the reflected beam, show that the electric vibration determines
the photographic effects.
In the above it has been assumed that a discontinuity in K exists
at the surface of separation of the two media. If the change in K be
assumed to take place rapidly, but continuously, in a thin transition
layer between the two media, the boundary conditions are found to
require the introduction of a phase difference between the reflected,
refracted, and incident waves. Thus the elliptic polarisation of light by
reflection may be explained."
334. Crystalline Dielectric.—When the medium occupying the
field is aeolotropic, the specific inductive capacity K, and the magnetic
permeability p, may vary from point to point, and they may also be
different for different directions around the same point. Then, though
the electric displacement satisfies the ordinary differential equation
df / dg dh
#####–0
* See Drude's Theory of Optics, Mann and Millikan, pp. 287-295.
608 ELECTROMAGNETIC RADIATION CHAP. xxi
for an unelectrified medium, the electric force is not proportional to
the electric displacement, as K is not constant, and accordingly the
quantity
dP, dG}, dR
da: ' dy' dz
will not be zero in an unelectrified aeolotropic medium.
If we confine our attention to the case of a homogeneous crystal-
line medium possessing three mutually rectangular axes of electric
symmetry, and if we assume that these axes of electric symmetry are
also axes of magnetic symmetry, then the equations of Art. 339 may
be written in the form
_, do dB d0 dP_dy dB.
*di dy. Tax *dt T dy dz
d6 dB dR r d0 da dy
– º – ; – ...}(i), Kºji=i-j(2),
_º_&_* R&R_38 da
*dt Tala, dy *d; Tax, dy
where K1, K2, Ks are the three principal electric inductive capacities,
and pºp ps, ps, the three principal magnetic permeabilities.
As a first case, if we take the magnetic permeability to be the same
in all directions, so that p1 = p, a pa = p, then by differentiating the
equations (2) with respect to t, and substituting for da/dt, etc., from
the equations (1), we obtain the equations
wº-vºp -- :
ukº-vo - . × 2 (3)
aß= V*R — #|
where
A =#4 º -- º: º
In order to determine how the velocity of propagation of a plane
wave depends on the direction of propagation in the medium, let us
take the case of a plane wave travelling in the direction l, m, n where
l, m, n are the direction cosines of the wave normal, then in general
the electric and magnetic forces at any point, 3, 4, 2 may be expressed
as periodic functions of la + my + m2 – vi where v is the wave velocity.
Hence if we take a simple harmonic disturbance we may write
P= Posin (la + 'my--nz – vt),
ART. 334 CRYSTALLINE DIELECTRIC 609
with similar expressions for Q and R. From this it follows at once
that -
d?P_ d"P_ _,9p &Q —
dº?" sºme 72P, 72T v2P, dºdyT * lm Q, etc.
Consequently, if we write piki = 1/v,”, p.K., - 1/v,”, p.ks = 1/vº” where
vi, we vs are the three principal velocities, then the equations (3)
become
(v*— v,”) P+ liv,”(lP+ m,0+ mR)=0 | -
(v* – v.”) Q+ mºv,”(lp + m (Q+ mR) = 0 }. (4)
(v*— vº) R+mºvº (lP+m,0+ mR) Eðſ
Dividing these equations by vº-v,”, w”— v,”, vº – v.”, and adding them
together, after multiplying by l, m, n respectively, we have
as 2 22, 2 2ns 2
lºv, 'mºve ** = 0
2
1 + § § à *
v?– v.” 'wº – v.” v”—vº
and, remembering that 1 = P + m3+ n°, this may be written in the form
72 m? + m2 = 0 (5)
v?– v.” v2 – v.” v2–vº “
Thus the velocity of propagation of an electromagnetic wave in an
electrically crystalline medium is determined by the same equation as
that deduced by Fresnel for the propagation of a light wave in a
crystal (Art. 194), and, starting from this stage, all the results obtained
in Chap. XII. for doubly refracting crystals may be expressed in terms
of the electromagnetic theory.
Now P, Q, R are proportional to the direction cosines of the electric
force, and in the foregoing equations we see that le + m0 + nP cannot
be zero unless v = v = vs - vs ; that is, unless the medium is isotropic.
But if lp + m0 + n R is not zero the electric force" is not at right angles
to the wave normal, and consequently not in the wave front. It is
easily seen, however, that the electric displacement is in the wave
front. For from equations (4) it is obvious that
* : * : *=–*— - " - –”—.
vº v.” vºTwº-vº vº-vº vº-vº
But P/v,”: Q/v,” : R/v,”= K, P : K.0 : K.R, and these latter are the
components of the electric displacement, and are therefore proportional
to the direction cosines of the displacement. Denoting these by
l', m', n', we have
7 777, 70,
w” – v.” v”— v.” v”— v.”
(6)
7' : m' : n' =
Consequently by equation (5) we have
l!' + mm' + nºn' = 0;
* The electric force is at right angles to the ray, as is easily seen from the
relations of Art. 195.
2 R.
610 ELECTROMAGNETIC RADIATION CHAP. XXI
that is, the electric displacement is in the wave front or at right
angles to the wave normal. This relation may also be deduced
very easily from the equations (2), assuming a, B, y to be functions
of la + my + m2 – vi.
Comparing the relations (6) with those of Art. 196 we see that the
direction of the electric displacement is the same as that of the vibra-
tion considered in Fresnel's theory; that is, the electric displacement
lies in the wave front and is perpendicular to the plane of polarisation,
while the magnetic displacement is in that plane, and is therefore the
vibration considered by MacCullagh.
On the other hand, if we take K to be the same in all directions
while p is variable, we obtain differential equations for a, B, y, which
are exactly the same as those from which we have deduced the pre-
ceding results for P, Q, R. We conclude therefore, that in a medium
which is magnetically aeolotropic, but electrically isotropic, the mag-
netic displacement is in the wave front and at right angles to
the plane of polarisation, while the electric displacement is in the
plane of polarisation, corresponding to the vibration considered by
MacCullagh.
Finally, when the medium is neither electrically nor magnetically
isotropic the equation of the surface becomes
(#####)(;++.)
Kı K2 K3/ \pi plo 'A3
.2 2 2
-i (ºr, I )-;(º)-iſºk) ºr "
91R1 \ºoks Pak2 Makov Maki Pika KSVP1R2 Maki PIP298ki K2K3
This is the surface enveloped by the wave planes. It is of the fourth
degree, and each of the co-ordinate planes intersects it in a pair of
ellipses. In one of these planes the ellipses intersect each other
in four conical points. In all ordinary crystals p1 = p, = pºs= 1 very
approximately, and the foregoing general equation reduces to Fresnel's
well-known form. - - * *
335. Plane Wave in an Isotropic Conductor.—In the above
theory each medium has been assumed to be a structureless non-
conductor, capable only of the electrical currents due to rate of change
of “displacement.” In order that it may afford an explanation of the
dispersion and absorption phenomena that arise when light passes
through matter, the theory must be modified so as to take account of the
electrical and magnetic properties of the matter. As regards the latter,
the molecular theory of magnetism would lead us to expect that all
substances would behave alike for the rapidly alternating magnetic
forces that are produced by the light vibrations, and that, even in the
ART. 335 ABSORPTION IN ISOTROPIC CONDUCTOR 611
case of the magnetic substances for which the permeability would be
much greater than unity for steady or slowly alternating forces, the
molecules would be unable to respond magnetically to the exceedingly
rapid light vibrations. This assumption is fully justified by experiment,
and accordingly p may be taken as equal to unity for all substances so
far as light is concerned. With regard, however, to the electrical forces
and displacements differences will arise that depend upon the nature of
the material medium. Some media will be, at least partially, incapable
of the permanent displacement possible in a dielectric, such displace-
ment being at least in part removed by what may be called a “slip "
occurring in the medium. This removal of displacement, or slip,
corresponds to the conduction current of a conductor and its amount
in relation to the electric force acting depends upon the electric con-
ductivity of the medium. The theory will partly take account of
this property (as explained in note, Art. 331) by equating the
Current to (ºp. # #) where P is the electric force and accordingly
equations (4), (5), (6), Art. 331, will become for an isotropic medium
of conductivity k,
d \p dy dº
(ºrk Kł)P=}-º etc. (1)
These conduction currents possess energy, which reappears as heat,
and accordingly the light energy will be used up in producing them or
absorption will occur when light enters and passes through such a
medium, the intensity of the light diminishing more and more as the
length of its path through the medium increases.
Expressing mathematically that the amplitude of the vibration
diminishes according to the exponential law, as the distance the
wave travels in the medium increases, we may write for a plane wave
in an isotropic absorbing medium
9
gº- #(x+my+m2) 27r g
P= Ae COS *(x + my + m3 – Wit),
with similar expressions for Q and R, k being the coefficient of absorp-
tion and expressing that the amplitude is diminished by the factor
e-* when the wave advances through the distance A, the wave length
in the medium.
We may write P as the real part of
2 2mº,
* **qw my +m2)+ 4.(la-i-my--nz–Wit)
A6
where i = x/-i, when equations (1) become, since dP_ _2TiV,
d; T N
4trkA. dP_dy d6
(#: + K ); * dy * dź” etc,
P,
612 ELECTROMAGNETIC RADIATION GHAP. XXI
It follows from these equations, as in Art. 331, that
2k}\? G2P -
**** -- K \* * – A2 -
( Wi +k); P, etc., (2)
and we have
(*. R) V,”=>/2(1+ix)*=(1+ix)",
l
Ol' KV,”= 1 - k”,
kXVI = k,
or if v is the index of refraction of the medium
KV2 = y2(1 – K?), (3)
kTV2 = y2k (4)
where W is the velocity in the free ether.
These equations are not numerically verified for conducting media
and modification of the theory becomes necessary, but the student
should note that equations (2) are of the same form as the correspond-
ing equations in Art. 331 with a complex dielectric constant substituted
for K and that the coefficients of absorption and refraction of the
wave depend upon the real and imaginary parts of this complex quantity.
336. Dispersion in Isotropic Non-Conductor.—But, besides its
electrical conductivity, and even though this be negligible, the molecular
nature of the material medium must be considered. Various experi-
mental facts—e.g. the facts of electrolysis, the discharge of electricity
through gases, etc.—justify the assumption that matter molecules con-
tain atoms or groups of atoms positively or negatively charged. There
is even reason to believe that the molecules may consist of such positive
and negative charges (electrons, as they are called), either in a fixed
relation to one another or performing definite orbits in the molecule.
We may assume, however, that such electrons are present in the
molecule without either identifying them with the ionic charges of
electrolysis, or making any hypothesis as to the nature of the matter
of the molecule. Such charges will have their positions or movements
affected by the varying electrical forces set up by light vibrations,
and their rate of change of displacement must be taken count of in
estimating the value of the electrical current at a point in the body at
any instant. These electrons," if displaced from their equilibrium
positions, will have natural periods of vibration of their own ; that is,
* The student will observe the similarity between the results obtained on this
hypothesis of vibrating electrons, with certain free periods of vibration, and those
obtained in Art. 294 on the hypothesis that the atoms or molecules themselves are
capable of vibrating with certain free periods of vibration. As indicated in the
present article, the electron theory is able to explain the known facts of dispersion,
but even on the electromagnetic theory it is not a necessary hypothesis for this ;
all that is really required is that the dielectric polarisation shall have certain free
periods of vibration (cf. Whittaker, History of the Theories of the Ether, p. 429).
Art. 336 DISPERSION IN ISOTROPIC NON-CONDUCTOR 613
assuming that no external forces act upon them. Forced vibrations
of them would be set up by the alternating electric forces due to the
light vibrations, and, as in well-known mechanical analogues, if the
period of the applied alternating force agrees nearly with the natural
period of the electron, resonance will occur and the amplitude of the
vibration may be large.
In discussing forced vibrations, Rayleigh 4 shows that, if the
frequency of the forced vibration be a little lower than that of the
natural vibrations, this resonance effect tends to retard the advance of
the forced vibration, leading to an apparent diminution in the elasticity
of the ether or a diminished velocity and therefore an increased
refraction for such vibrations, while, if the frequency of the forced
vibration is a little higher than that of the natural vibrations, an
increased velocity and therefore a diminished refraction will result, as
is found to be the case in bodies exhibiting anomalous dispersion on
each side of an absorption band.
If it be further supposed that a frictional resistance of the nature
of fluid friction is offered to the motion of the electrons, some of the
energy of the light causing these forced vibrations will be degraded
into heat by this frictional resistance and true absorption of light will
occur. This will naturally be most marked when the amplitude of the
vibration is large ; that is, when the period of the applied electrical
force agrees with the natural period of the electron. Electrons of
various natural frequencies will lead to various absorption bands.
Transmission through Dielectrics.-If É be the displacement in the a,
direction of the electron from its equilibrium position, p its frequency
constant (i.e. 21/p equals the period of its free oscillation), m its mass,
e its charge, 2, the resistance coefficient per unit mass, its equation of
motion may be written’
02
mº; =ep — mp”; – 20mr # (1)
where P is the 2 component of the electric force and the frictional
resistance is put proportional to the velocity. As the forced vibrations
maintained by the oscillating force P are alone to be considered, we
may put £-real part of Ae” where to is the frequency-constant for
this force. Inserting this value and cancelling the exponential factor
common to each term gives
– wºmé +p°m; +2rmuwê = eP,
* Rayleigh, Theory of Sound, i. p. 168.
* The first term on the right-hand side expresses the accelerating force, and the
second term expresses the constraint in a form that shows that a vibration of period
2T/p will occur, if the accelerating and frictional forces vanish. s
614 ELECTROMAGNETIC RADIATION CHAP, XXI
OI’
P e2
eš =#|F-ºl. (2)
The component of the current due to the displacement of the electrons
will be #(>Net) where N signifies the number per unit volume of
electrons of any one type and the sign of summation includes all
types of electrons. The idea of a convection current was first introduced
by G. F. FitzGerald (before it was admitted that a current in a metal
is of that type). This convection current is to be added to the
displacement current of Maxwell, viz. #, in order to obtain the
total current which equals * times the line integral of magnetic force
round unit area.
The equations (4), (5), (6) of Art. 331 will then become
4try e? 0P_0y 63
|X== – wº- 2āra) + Ko |; Tây 53° etc., (3)
where the dielectric constant Ko for matter-free ether has been written
for 1/Woº. We are thus led to equations of the same form as the
equations for P, Q, R in Art. 331 with a complex quantity [that in
square brackets] substituted for Ko The quantity in square brackets
is therefore the effective K in the structured medium, and if p has
the same value [as we may quite safely take for granted, with a
reservation in the case of highly magnetic bodies] the velocity of
propagation must vary inversely as the square root of K. It will be
seen therefore that in general this velocity, W, is a complex quantity.
To simplify the exposition we will first suppose that the imaginary
term, 2%ro, is zero, or so small as to be negligible. We then have
2 2
=\;=}=1 + X; (4)
where p is the refractive index of the medium with respect to that of
ether as the standard value. This is an equation of the same form as
was found by Sellmeier and others (Art. 294) from non-electrical
theories. It is found to hold very well for transparent substances—a
few terms of the summation being sufficient to represent experimental
data. Expressed in terms of the frequencies instead of frequency-
constants
Ne2
*=1+XFrºp (5)
f being the frequency of the light and f, the frequency of the free or
natural vibration of one type of the electrons. Any consistent set of
ART, 336 1) ISPERSION IN ISOTROPIC NON-CONDUCTOR 615
units may be used ; if electrostatic ones are chosen Ko = 1, if electro-
magnetic, Ko- 1/(9 × 10°). In at least one respect this discussion
is incomplete. We have supposed any one electron to be acted on as
if it were alone. In reality the effective force due to the light must
be modified by the displacements which the surrounding electrons
undergo. For simplicity we will assume that there is only one type
of electron present. We may take the modification of the field as
proportional to P where this is the sum of the displacements; that
is, P = Neš. Whereas previously (p” – 0°)é = #. we must now write
(p”— ºs-il P+; G Neil
where k is the assumed proportionality, or
4tr, Ne? P
[(2-2)-; *...] =#.
0
Hence, finally, if
4T Ne”
K, ºn T”
2 — 1 — —
M. ***IºIRA’
which can be written
(u°– 1) (p” – wº)= A[1+ k(u” – 1)],
whence
*-1__. A
IIk(º-I)-pº- oft'
H. A. Lorentz found that a good fit could be obtained in some
cases by putting k = 1/3, in which case the equation becomes
*-1-1. A
pº-H2T32 º'
(6)
Since A is proportional to N, the number per unit volume of electrons
concerned, we may take it also as proportional to the density. [This
assumes that no fresh electrons enter into play even for great changes
in density.] We may accordingly write for any one substance
# }= constant. (7)
This is known as Lorentz's constant. The experiments of Phillips
already described (Art. 86) show that the reciprocal of this constant in
the case of CO2 is a linear function of the square of the density.
C. Cuthbertson has shown that for gases at or below atmospheric
pressure one term in the summation is sufficient. The following table
gives the values of the constants in the formula
C_
4-1=jºys'
616 ELECTROMAGNETIC RADIATION CHAP. XXI
the frequencies being expressed in vibrations per second : 1
Substance. C X 10-27. f,2×10-27. #x10".
Helium 1 21238 34992 34'648
Neon . 2'59826 38916 66-767
Argon . 4:71632 17009 275-97
Krypton 5 °3446 12768 418 °59
Xenon. 6'1209 8978 1001
Hydrogen 1 *692 12409 136°35
Oxygen 3.3970 12804 265 '3
Nitrogen 5'0345 17095 294 '5
Fluorine [195 (D line)]
Chlorine 7-3131 96.29 '4 759 °46
Bromine 4 °2838 3919 '2 1093
Assuming that the vibrators are electrons, in each case the number C
must be proportional to the number of these electrons per unit volume;
moreover, if Kepler's law were applicable (a view which is rather
discredited at the present time) f,” would be inversely as the cube
of the radii of the orbits.
For solids, when long ranges of wave length are considered,
several terms must be taken into account. It is customary to write
the equation in the form
My M M
*=K'4xº-Nº-Nº (8)
in which, instead of the frequencies, the corresponding standard
wave lengths (i.e. wave lengths in the ether as standard medium) are
used. The term K' includes terms which arise in the transformation
and which correspond to the value of pº for infinitely long wave lengths.
H. Rubens and E. F. Nichols” have determined the indices of quartz
(SiO2), rock-salt (NaCl), and sylvine (KCl), and applied the above
equation to them. The values of the constants are given below.

Quartz. Rock-salt. Sylvine.
My '01065 "01850 0.150
Mr. 44 °224 8977 10747
Ms 713 55 -- -*.
Nv ‘1031 -127 • 153
\r 8'85 56-12 67:21
\s 20-75 - -
K’ 4 '5788 5-179 4'553
K 4 '55 5'85 4°94
* Cuthbertson, Phil. Mag. S. 6, vol. xxv. p. 592 (1913).
* Wied. Ann. lx. p. 418 (1897).
ART. 336 DISPERSION IN ISOTROPIC NON-CONDUCTOR 617
The terms corresponding to the violet free-frequency are very
small and of little importance except when the light is nearly in
resonance with them. The wave lengths are expressed in terms of the
millionth of a metre (1p), and the constants given require the use
of this unit.
The value of p” when A = Go is K'. This should be the dielectric
constant for long waves. The actual dielectric constants are given in
the final line.
As A = any one of the values Av, As, As the expression makes p.”
infinite. According to the formula for reflection of waves, the reflecting
power of the medium should therefore be infinite. As a fact it is
wery large at these values of A, but not infinite. Use is made of this
fact to isolate light of selected wave lengths. The composite light is
successively reflected from plates of the material. At each reflection
the resonance wave lengths [Av, AR, or Ås, etc.] become more and more
predominant. The final rays are consequently known as reststrahlem, or
remainder rays. *
The fact that the refractive index does not become infinite requires
us to include the neglected frictional term.
It is instructive to look first more closely at the equation
*-i-º-º:
If p” be plotted against 0° the curve obtained is as shown. At
first pº increases with 0°. This agrees with what is obtained in
ordinary cases of dispersion. |
In the neighbourhood of p" a 2 :
it increases very fast, then
becomes negative (i.e. p. be-
comes imaginary and does a |
not exist) and afterwards
becomes positive again, and 1 – ---------4----------------
ends with a value less than
unity, but approaching it
asymptotically. The nega-
tive region is one in which |
light could not be propa- |
gated. It corresponds to an i
“absorption band.” R. W. Fig. 244a.
Wood has obtained curves of precisely this type for sodium vapour in
the neighbourhood of one or other of its two absorption bands, D, and
D. But the same type had been observed previously in the trans-
mission through sundry highly coloured materials like fuchsine, as

618 ELECTROMAGNETIC RADIATION CHAP, XXI
described in Art. 284. For higher frequencies, when p, becomes
positive again, its value is less than unity, and remains so unless there
is a higher resonance frequency. Recent measurements of the refrac-
tion of X-rays through prisms made by Siegbahn demonstrate that for
these rays, which are electromagnetic waves of very high frequency,
the refractive index is less than unity.
When friction is taken into account this dispersion curve is
modified. Schuster has emphasised strongly that real friction is inad-
missible in the treatment of molecular vibrations. But dissipation of
energy occurs by radiation, and therefore there must be some retard-
ing force in phase with the velocity. Hence we may return to
equation (3), where a factor 2r is introduced, merely safeguarding
ourselves by not specifying in detail how the existence of r comes
about. We may now, following the same line as before, write
v°= 1 +
47r Ne?
m Koſp”— w”--2irw] (9)
where vº has been placed instead of p”, because it is obviously a
complex quantity (i.e. it contains an imaginary term, 2%ro), and it is
necessary to enquire into the meaning that must be given to it.
Putting v = a – 18 where a and 8 are the coefficients of its real and
imaginary parts we have
*=w-5-ºf-1+z=#.
A[(p” – wº) – 24rw]
= 1 + s ſº tº
(p” * w?)? + 47°ojº
Hence, separating the real and imaginary parts,
A(p” – wº) )
(p” - o,”)? + 4, 2002
* | (10)
a?-69–1+
aft =
From these expressions a and 8 can be calculated. If there are
other free frequencies they will give rise to corresponding terms.
To find the meanings of a and 8 we write the equations of a
simple wave
v=a cos (wt – ga)
where q = 27/A, and oſq is the velocity of propagation V, and
Vo-Vog.
*=y (w)
The above equation shows that q must have become complex, and
ART, 336 DISPERSION IN ISOTROPIC NON-CONDUCTOR 619
v=Vº = d – uſ?,
(w)
(JO ,Cº.)
q=#-º.
The equation for y therefore becomes (when expressed in real
quantities)
_98
W.” (r)
g/ = ae COS (al * **).
Vo
We see therefore that 3 determines the damping factor, which
indicates that the amplitude falls off exponentially with uniform in-
crease of the distance traversed. At the same time waſvo has taken the
place of q and determines the velocity of change of phase, which is
*Wo Wo.
(JC, C,
There is no longer a quantity which can be spoken of as the velocity
of the wave, because there is no longer any characteristic feature
which travels unchanged. The disturbance changes form as it goes
along. If 3 is small, a becomes the refractive index and the material
refracts according to the sign law.
The value of a” in this case never becomes infinite, though it may
become large ; it becomes unity when 0° = p^; it may become zero.
The value of 8 becomes large in the neighbourhood of 0° = p", and
consequently the term that has been introduced would account for
the strong absorption.
Transmission through Metals.-The foregoing discussion must apply
to conductors as well as to dielectrics, because it is universally recog-
nised at the present day that a steady current in a conductor is a
convection current—the value per unit area being Nev. Nevertheless
the current can still be expressed in the old-fashioned way in terms
of Ohm's Law. Since the potential drop per unit distance is equal to
the electric force (i.e. P in the direction a), it follows by Ohm's Law
that the current per unit area is P/p where p is the specific resist-
ance, or Po where or is the electrical conductivity. If we include the
possibility of Maxwell's dielectric current entering as well, the total
current is
Ko OP
P
4tr 0t + Po,
and 4t times this must replace the quantity in square brackets in
formula (3). We can therefore at once write down that
47ror
wko
w°= 1 +
instead of (9).
620 ELECTROMAGNETIC RADIATION CIH. A.P. XXI
Hence v is complex and, as before, we can write it a - 6, whence
o? g” – 1
ag=}º-º-ºxºv, (11)
0
where T is the time period of the light, and the suffix Zero denotes that
the datum is for the standard medium (the ether). The last expression
is the most convenient if we use electromagnetic units. The value
for copper is a = 1700 and po = 1. Hence aft = Ao x 5.2 x 10*. The
product aſ is very large for optical waves and the first part of (11)
shows that a and 8 must each be large and practically equal. Hence,
very closely indeed,
cº–8*=pûNoa Vo.
Hagen and Rubens tested this formula, determining 6 from the
reflecting power of metals. For wave lengths of 25.5 p at 170° the
agreement was very good (except in the case of bismuth). For
wave lengths 12 p. (1p = 107* cm.) the agreement was poor. They
concluded that for waves of high frequency the self-vibrations of
some of the electrons must be taken into account. -
This investigation is of the highest importance in the discrimina-
tion between rival theories of optical phenomena. In two important
respects the evidence is in favour of the electromagnetic theory. The
fact that the velocity of propagation of light in the ether is the
reciprocal of Vu,K, led to Maxwell's enunciation of this theory. The
fact, just alluded to, that the specific electrical resistance of a metallic
conductor can be calculated for long waves from the optical behaviour
is further evidence which it would be difficult to explain away.
337. Optically Active Media.-The electromagnetic theory may
be applied to explain the behaviour of optically active substances, by
regarding their molecules as unsymmetrical or of definite structure.
Thus Drude, following Goldhammer," extends equation (1), Art. 331,
by the addition to P of a constant x (?-?), thus retaining the
property of the substance which expresses that all co-ordinate
directions in it are alike. For example, he shows that, if the
electrons be restricted by the molecular structure to move in helices
instead of straight lines, such modification of this equation will result.
Making use of these modified equations, he shows by a method similar
to that in Art. 33 I, that the index of refraction of the substance
will be different for circularly polarised light according as it is polarised
right-handedly or left-handedly, and that therefore plane-polarised
light falling perpendicularly on a plate of the substance will have its
* Goldhammer, Jowrmal de Physique (3), i. pp. 205, 345.
ART. 337 OPTICALLY ACTIVE SUBSTANCES 621
plane of polarisation rotated, the direction of the rotation depending
upon whether the index of refraction for the circularly polarised light
is greater for the right-handed or for the left-handed ray. For
crystals, taking account of the dependence of the dielectric constant
upon direction, he deduces the separation of a wave into two elliptically
polarised waves (cf. Art. 263) for a general direction through the
crystal, which become circularly polarised for a direction along the axis.
He deduces the dispersion formula for the rotation 6 of the plane
of polarisation,
km
3-ºxº
where the summation is to be taken so as to include the wave lengths
corresponding to the natural periods of all the active electrons. If
the substance contains electrons whose natural period is small compared
with light, one term in the summation will be * (cf. Biot's formula,
Art. 253).
Thus for quartz, using the wave lengths A1, A2. As for the absorp-
tion band in the ultra-violet, and for the two in the infra-red as
determined in the ordinary dispersion, it is found experimentally that
the corresponding constants ks, ks vanish, indicating that the infra-
red electrons are not optically active, and the resulting formula
s=sºº is found to agree well with the results of observation.
– Al
The following table compares some of the observed values obtained by
Gumlich 4 with the values calculated from the above formula, similar
agreement existing through the whole range of his experiments.

A. 8 obs. 8 calc.
2 : 14 I '60 1.57
1.77 2°28 2 °29
1-08 6-18 6'23
‘59 (D line) 21-72 21-70
*49 31.97 31-92
*34 70-59 70'61
•22 220 72 220.57
True anomalous rotatory dispersion should occur on passing
through an absorption band, and has been observed.” As in ordinary
dispersion this is due to A* – Am” becoming small, as A approaches Am,
and changing sign as A passes through the value Am. Cases have
also been observed 9 in which the rotation at first increases in the .
1 Gumlich, Wied. Amm. vol. lxiv. p. 349, 1898.
* Landolt, Das optische Drehungsvermögen, p. 135.
* Arndtsen, Pogg. Amn. Bd. cv.
622 ELECTROMAGNETIC RADIATION CHAP. XXI
visible spectrum with decreasing wave length, and then diminishes
after attaining a maximum, which may be explained by supposing
the existence of absorption bands in the ultra-violet and infra-red due
to molecules of opposite rotatory power.
It must be observed, however, that the rotatory term has been
introduced ad hoc, and much requires to be done to connect it with
the actual molecular structure of the crystal.
338. Magnetie Rotation.—Two hypotheses have been put forward
to explain, on the electromagnetic theory, the rotation of the plane of
polarisation of light by matter in a magnetic field. On one hypothesis
the substance is supposed to contain electrons in rotation round orbits.
From considering the action of the electric forces set up by light waves
upon such molecular currents, Drude" shows that plane polarised light
will be split up into two beams of light circularly polarised in opposite
directions and travelling with different velocities in the substance,
when in a magnetic field, and that therefore a rotation of the plane
of polarisation will occur. He deduces for the dispersion equation for
this rotation a formula
a' bn
s= (+x+...)
where the wave lengths Am, as before, correspond to the natural
frequencies of the active electrons and are determined from the ordinary
dispersion of the substance.
This equation is found to agree well with experimental results for
certain substances, e.g. bisulphide of carbon and creosote, but the
theory would require that the sign of the rotation should change on
passing through an absorption band, which has not been confirmed by
the experiments of Macaluso and Corbino” for sodium vapour, or by
those of Wood.” On the other hand, Schmauss 4 claims to have
observed anomalous rotation in accordance with the theory for certain
solutions of the rare earths, and Wood * has also obtained confirmatory
results with a solution of praseodymium chloride.
The second theory takes account of the fact that a conductor
carrying a current in a magnetic field is acted on by a force perpen-
* See also FitzGerald, Scientific Writings, p. 57, etc.
* Macaluso and Corbino, Rend. d. R. Accad, d. Lincei, (v.) vol. vii. p. 293 (1898).
Further experiments have been made by Zeeman, who observes the change of sign
in the case of sodium vapour in the region between the components of the doublet
produced by the magnetic field. See for a very extended account of the subject
Zeeman, Magneto-optics (Macmillan and Co).
* Wood, Phil. Mag. vol. x, pp. 408-427, Oct. 1905; and vol. xiv. pp. 145-152,
July 1907.
* Schmauss, Ann, der Phys. vol. x. p. 853 (1903).
* Wood, Phil. Mag. vol. ix. pp. 725-727, May 1905.
ART, 338 MAGNETIC ROTATION 623
dicular to the direction of the current and to the normal component
of the magnetic force. So in a Crookes' tube the kathode rays are
deflected and follow a curved path under the influence of an external
magnetic field, and, if a substance contains electrons in motion, their
direction of motion will be displaced in a magnetic field. Such a result,
known as the Hall effect, has been observed in metals. Drude has
developed this theory to explain the rotation of the plane of polarisa-
tion of light by a magnetic field and deduced the dispersion equation
f /Y 2
•=}{4׺).
This formula agrees nearly as well as the formula obtained on the
other theory with the observed results for bisulphide of carbon and
creosote, and it also agrees with the observations for sodium vapour
of Macaluso and Corbino and of Wood, as it requires the rotation to
be of the same sign on opposite sides of an absorption band.
The Zeeman effect has been briefly described in Art. 260. There
is a remarkable connection, pointed out by H. Becquerel, between it,
the magnetic rotation of the plane of polarisation of light, and the
laws of optical dispersion.
In the Zeeman phenomenon a magnetic field alters the free periods
of vibration of the electrons, splitting each, when the magnetic field is
in the line of light propagation, into two components in the simplest
cases. These altered components are given by
2_, 2–4 (*f;
p rº-3-(*. ).
The symbols p and po stand for altered and original frequency
constants. Since the change is small we will write d(p") instead of
p” – pºo. The two components are circularly polarised.
Now consider the formula for dispersion,
W.2 A.
2— '0 – sº-º-º-º:
pº- wh–14 gº
in the simplest case.
Mathematically
duº duº
dp2T do,”
owing to the form of the equation,
Now the path retardation per unit length is du, and the corre-
sponding phase retardation is *au. But (Art. 262) the rotation of the
plane of polarisation is ; the phase retardation. Therefore the
rotation per unit length is sau, and if we regard the change of p as
being brought about by a change in the value of p (the free frequency)
. 624 ELECTROMAGNETIC RADIATION CHAP. XXI
we have
_ _T duals T2How du
R = — xiàºp T X m. dp?
T2Hea. du.
X ºn dº?"
Now -
r
-*. , SO dw = — *Max,
whence
_HeM du
T2Wºm, d\'
This equation gives the right order for R in the case of CS, and other
substances, but too large. Schuster suggests that the effective mass of
the electron may be larger in the more complicated structures of mole-
cules which give line spectra.
THE HERTZIAN OSCILLATOR
339. The Hertzian Oscillator.—In order to investigate the field
in the neighbourhood of a simple dumb-bell oscillator Professor Hertz”
employed the general equations
_, da_dR_dº dP_dy dB
*d, *z, Tº di Tây dz
d6 dB dR d0 da dy
T * j, fºx. Tº (1), Kºji=i-j. (2),
_, dy_dG}_dP dR_dé da
*dj fºr T ây - dt Tº da; dy
together with the constant flux conditions
do d8, dy_ dP, dG}, dB
######9, and ###= e
Now if the axis of z be taken to coincide with the axis of the
oscillator (Fig. 245), the lines of magnetic force will be circles having
Z their centres on the axis of 2, and their planes
p perpendicular to it. Hence we have y = 0, and the
yº flux condition becomes
s da d8
O X - ãº"dy = 0.
This means that the quantity ady – 3da, is a com-
Fig. 245

plete differential of some function, and if we use
Hertz's notation and represent this function by d/I/dt, we have
* The theory of the electric oscillator was given by Hertz, together with the ex-
periments already referred to, in Wied. Ann., 1888-89. These researches have been
translated into English by Mr. D. E. Jones (Electric Waves, London, Macmillan and
Co., 1893). -
Al-T. 339 THE HERTZIAN OSCILLATOR. 625 -
a º!, g––º = 0
*ājā P- -ā, Y=9.
Substituting these values of a, B, y in the equations (2) we find
dP_d (d”II) ºr d0 d (d’II dR d / d’II d?II
dt T di \ dad: J' di Tài \dyāz) *d, T -#(; #).
Now these equations show that the quantities
_*II, KQ-ºli, KR+(3*I dºn),
da:d: dydz da” dy?
KP
are each independent of t, so that at any point, as far as a periodic
disturbance is concerned, we may take
G2II d?II
KP=; KQ=;
2 2rry
KR = — (#)
da? dy” (3)
and these values of P, Q, R will satisfy the equations (1) provided II
satisfies the equation *
, d2|I
for on substitution we find all the equations satisfied identically. It
may be remarked that the equation determining II is the same in form
as that satisfied by the electric and magnetic forces.
When the distribution is symmetrical round the axis of 2, as in the
case of a simple electric oscillator (Fig. 245), the forces at any point of
the field will depend only on the 2 co-ordinate of the point and on the
distance p = Wºry from the axis of 2. Denoting the electric force
in the direction of p (that is, perpendicular to 2) by E, and the
magnetic force perpendicular to the meridian plane drawn through
the oscillator (which in this case is the direction of the resultant
magnetic force) by H, we have
Q} , A3/ !/ o 33
E = P + (): ; H = a +4 – 8: ,
p o; ;-5,
and by the foregoing equations (3) we have
1 GW 1 GW 1 GW
where
d'II
v=p;
The function V holds an important place in the investigation of the
field. We shall show that the curves in which the surfaces of revolu-
2
* The condition required is that *R*; – V*II shall be independent of a; and y—
that is, at any point it may be a function of 2 and t ; but since the electric and
magnetic forces considered above involve differentiation of II with regard to 2 and y
this function of 2 and t may be taken as zero without affecting the field.
2 S
626 ELECTROMAGNETIC RADIATION CHAP. XXI
tion W = const. are cut by the meridian planes are the lines of electric
force. For if the direction of the resultant electric force at any point
makes an angle q, with the axis of 2 we have
tan ºp-E/R= -: º
by (4), which is the trigonometrical tangent of the angle which the
tangent line to the meridian section of the surface W = const. makes
with the axis of 2. Hence the resultant electric force is tangential to
the surface V, and along its curve of section with the meridian plane.
Again, if two surfaces V and W + dW be taken enclosing a shell, and
if any plane perpendicular to the axis of 2 be drawn cutting the shell,
the flow of electric force across the strip of the plane intercepted by
the shell will be constant. For the area of the strip is 2Tpdp and the
force perpendicular to it is R, therefore the flow is
ſkR . 2Tpdp= -ſºap-ºry, – W2),
similarly the flow across the strip intercepted by the shell on any
surface of revolution round the axis of 2 is constant.
The surfaces V are those whose history and development we have
already traced in Figs. 240-243.
340. Particular Solution.—A particular solution of the equation
d?II 2
is -
el . -
II = ;sin (mºr – ºut) (1)
where 4 e, l, m, n are constant quantities and m and n are taken to
satisfy the relation
770, *-*
7, T WLK :
that is, m = 27/A, n = 27/T, and n/m is the velocity of propagation.
The function W is now easily determined, for by equation (1) we
have f
dII ºſcos (mr – mt) _sin (mºr-ºlt)\ dr
do." T \ 7777° dp'
but dr/dp = sin 6, and pſr = sin 6, therefore
d'II ſ sin g-no) * tº
W = n := - * - smº 2 e &
W =PH, elm{cos (mºr – nt) → jsin 6 (2)
From this value of W we may at once obtain the forces by differentia-
tion. We shall first examine the case of any point situated in the
1 Hertz writes el instead of a single constant A, the signification being, as will
be seen afterwards, that l is the length of the oscillator and e the charge on
either pole.
ART. 341 EQUATORIAL PLANE 627
equatorial plane aſy; that is, the plane through the middle point of the
oscillator perpendicular to its length. -
341. Equatorial Plane—In the equatorial plane, wy, we have
6 = 90°, dz= – rat, p=r, dp=dr.
Therefore by substituting in equation (2) of the foregoing article we
obtain -
rp_1 dV_ 1 dW
KB-ji=-ji=0.
_1 dV_elmºſ cos (mºr – mt)\
Tp de T. r {sin (mr-nb.t==},
_1 dW_elm” cos (mºr — nt) + sin (mr-nb)\ . -
To do T -- U - sin (mºr – mt) — 77,7° m???
H
– KR
The resultant electric force is therefore perpendicular to the equatorial
plane, for since E = 0 the resultant force is R, and is parallel to the
vibrator.
Writing the expression for R in the form
KR=A sin (mt – 6),
we have at once by comparison
e el - -
— A sin Ö == (an” sin mr--mºr cos mr – sin mr),
el e
– A cos 6= 73 (mºr” cos mr – my sin mºr–cos mr).
Hence by squaring and adding we obtain
* —
A = 73 VII mºrº-Hºnºrº.
At small distances from the origin the amplitude A varies inversely as
the cube of r, but for great values of r the amplitude approximates to
the inverse ratio of the distance.
To determine 6 we have
777.7"
1 — m???
2a-2 cº- e tan 707 -
mºr" sin mr--mr cos mr – sin mr
mºr” cos mr-mir sin mr-cos mr" 777,7'
l + 22.2
1 – mºrº
tan 6=
tan m”
Therefore
_ ] 772.7°
- - ô = m^' – tan 'i º
and we have finally
7)??"
el 1--3–E–F–F . -
KR=. VI- mºrº-Fºmºri sin ( 7tt - ????" + tan **) º
7" - 1 – ºn.”
The expression for 8 shows that the phase at a given instant is not
simply proportional to the distance of the point under considera-
tion from the oscillator, but is less by the angle whose tangent is
mr/(1 – mºr”). When r is very large this angle is approximately equal
628 ELECTROMAGNETIC RADIATION CHAP. XXI
to T, hence as we recede from the oscillator the value of 8 becomes
more and more nearly equal to mr – T. Very near the oscillator we
have 8 = 0, since 8 and r vanish together. For small values of r it is
clear that 8 is negative, so that the curve (Fig. 246) representing the
Y relation between 8 and r, at any given instant,
lies below the axis OX near the origin, the dis.

tance r being measured along OX and the angle
__ X 8 along OY. The negative value of 8 will be
O
2’ greatest when döſdr = 0, which corresponds to
2^ mr = x/2; that is, r = A/T v2 = A/(4.4) approxi-
2^ mately. From this point the curve begins to
w Fig. 246. - rise, and as r increases 8 increases almost pro-
portionately, approaching more nearly the value
of mr – T. The curve is consequently asymptotic to the line
8 = mr – T, which meets the axis OY at a distance numerically equal
to T below the origin, and is inclined to the axis OX at an angle of
which the trigonometrical tangent is numerically equal to m.
The velocity of propagation is determined as the speed (dr|di)
with which any surface of equal phase moves forward. Taking
this surface to be that in which R = 0, or the phase a multiple of it,
we have then
mt – mr-i-tan" Hºa–0, or Nºr.
Hence
- 777.7°
tan (m? -n!)=i-jºº
a result which follows also directly from the original equation for R.
Therefore
_dr n 1-m”+mºr.
Tdt Tºm m°r” (mºr?–2)
This expression shows that the value of v is infinite both at the origin
and at the distance r = x/2/m = A/(4.4). This is the point where the
tangent to the curve (Fig. 246) is parallel to OX, the interpretation
being that at this point a small distance is traversed without change
of phase. At points between this point and the origin v is negative,
and for greater values of r the velocity is positive. Hence the wave
spreads out from this point in both directions. It is here that they
are thrown off into space, part being radiated and part receding again
into the oscillator, as indicated in Hertz's diagrams (Figs. 240-243).
For large values of r the velocity approaches more nearly the normal
velocity v = n/m = A/T.
ART. 341 EQUATORIAL PLANE 629
The magnetic force may be treated in the same manner. Thus, if
we write
H=A' sin (nt – 6’),
we have
e elm *
A sin 3’- — ; (mrsin my 4-cos mr),
7
eln, , .
A' cos 6' = º (sin m?" – my cos mr).
Hence
6/77, 1–F–F
Aſ = } VIE m”,
7-
and -
1
g tan m^* + —
, mºr sln mºr-H cos mº" 777,7"
tan 6' = * -: 3
7???" COS 772.7° — SlT1 772.7° 1
- 1 — — tan myr
777.7°
OI’
^ l
ô' = mr-i-tan-4--,
7)?,?"
and we have finally
elºn — I
H == VT-Fºmºrºsin (u — ????" — unº).
At the origin we have r = 0, and therefore 8 = #ir, so that initially
the electric and magnetic components differ in phase by a quarter
period. At great distances, however, we have 8 = mr, so that the
phase of the magnetic component does not increase proportionately to
r, but there is a loss of #1, with the result that at a distance from
the origin the electric and magnetic components differ in phase by T.
The curve (Fig. 247) indicating the relation between 8' and r will
therefore meet the axis OY at a distance numeric- Y
ally equal to #T, and will finally approach
asymptotically the line 8 = mr. This line passes
through the origin and is inclined to the axis 41
OX at an angle whose tangent is numerically
equal to m. This curve and that of Fig. 246 -
are therefore asymptotic to lines which are 6 - X
parallel, and the distance between these lines, Fig. 247.
measured parallel to OY, is numerically equal to T, showing that the
phases of the electric and magnetic components differ by T.
The velocity of propagation of the magnetic component may be
found by considering the wave front where the magnetic force is
zero. We have then

1
tan (mt – mº') = −.
7)7,7°
Therefore
_dr m 1 + mºr”
*
T dº Tºm ºn 272
630 ELECTROMAGNETIC RADIATION CHAP. XXI
When r = 0—that is, at the vibrator—the velocity is infinite, but on
receding from the vibrator the velocity rapidly approximates to the
normal value n/m = A/T.
For the time rate of change of the magnetic force we have
dH_elmnº { sin (mºr – mt)
* \
dt 7" 7777° cos (mºr — mi) ſ
Throwing this expression into the form
A" sin (nt – 6"),
we find at once
& g e/m2 e
A” sin 6" = º (mºr cos mr – sin mºr),
A/ "— elm? . c.3 &
A” cos 6” = — -a- (my sin mr-i-cos m?").
Therefore e
, elm.” -------
A" = 7.2 Wi + m”,
tan 6” = sin mr-mºr cos mr tan mr-mºr
T cos mr-i-mºr sin mr 1 + mºrtan my
OT
6" = mr – tan-1 mr.
At the origin 8" is zero, and at great distances 3" = mr- 3. so that
Y the curve representing the variation of 8" with r

passes through the origin and finally approaches
asymptotically the line 8" = mr – 3. which meets
_2^ the axis OY at a distance ; below the origin,
Ol * and the tangent of whose inclination to the
axis OX is m. It consequently lies half-way
between the asymptotic lines of Figs. 246 and 247.
The velocity of propagation is the same as that of the magnetic
force, viz.
Fig. 248.
_n 1+m”
Tom, m2, 2
342. Field at a Distance from the Vibrator.—At a consider-
able distance from the vibrator we may neglect the higher powers of
1/r, so that we have
W = elm cos (mºr – nt) sin” 6,
2
KE= — er sin (mr — nt) sin 6 cos 6,
ežm.” . iº
KR = ar sin (mr – nt) sin” 6,
elmºn,
7"
H=
sin (mr – mt) sin 6.
ART. 343 FIELD NEAR THE WIBRATOR 631
From the second and third equations it follows that
R cos 0+E sin 6–0,
which shows that the direction of the resultant electric force is every-
where perpendicular to the radius vector r; that is, the propagation
takes place in waves of pure transverse vibrations.
The magnitude of the resultant electric force is
2
R. sin 6 – E cos 6–4. sin 6 sin (mr – nt).
Its amplitude is therefore inversely as the distance and directly pro-
portional to the cosine of the angle which the radius r makes with
the equatorial plane.
343. Field near the Vibrator.—In the immediate vicinity of the
vibrator (, very small) we may disregard mr in comparison with
mt, so that the equations of Art. 340 become
e! . e! . º
II = — ; sin 7\t, v=; sin mt sin” 6,
(#4;. 1_ _d’ſ 1
da?' dy?) r" dz2\r/?
we have, by equations (3) of Art. 339,
d / dII d' ſ d'II d /dTI
KP=+#(#) KQ=+#(#) KR=1#(#)
so that the electric forces appear as derived from a potential function
and since
d'II - d. / 1 e! .
\If = -:=+el in u}(i)= -; sin ovt cos 6.
This force distribution will be that due to the action of a very short
rectilinear oscillator of length l on the poles of which the electric
charges are, at the maximum, equal to + 6.
For the magnetic force perpendicular to the meridian planes we
have
I dV elm, º
: cos nt sin 6,
H=##=5
and this represents the magnetic action of an electric current of length
l along the vibrator, and of which the intensity is en cos nt, due to the
oscillation of the charge e, the maximum values of the current being
+ me.
In the axis of 2 we have
- 6=0, dz=dr, dp=7-d6.
Hence
- _1 &W 2elmſ , , y_sin (ºr- *).
E=0, H = 0, - KR Tp dp --a- \cos (mr – mt) 7777°
632 ELECTROMAGNETIC RADIATION CHAP. XXI
The electric force is therefore entirely in the direction of the
Oscillation. It diminishes for small distances as the inverse cube of r
and for large distances as the inverse square.
344. Radiation of Energy.—In order to calculate the rate at
which energy is radiated by the vibrator we have the equations (see
Maxwell's Electricity and Magnetism)
E=;ſſſ(P” + Q?--Rº)dardydz,
and
— P. 2 1 & 2 2.2 -
T=#|ſſe +8°,+ y”)dardydz
where E is the electrostatic energy in the region through which the
integration extends, and T the magnetic energy in the same region.
These equations merely express that work is half the product of the
final stress by the final strain, and in the case of the electric field,
where the electric force is P, the electric displacement is P/4+.
Now if we multiply the equations, 1, 2, 3 of Art. 331 by a, B, y
respectively, and the equations 4, 5, 6 by P, Q, R, we have at once
_c(dB d0 dP_dR d0 dB } *
a(; #)-6(#-#)-(; #) dardyd?
=ſſ(#er-ºf-cº-o-ºp)and.
and this integral taken throughout any volume may be expressed as a
surface integral taken over the enclosing surface in the form
*E +T) =ſſ(BR – y(Q)\+ (YP – c. R)p-i- (a Q — gº)as
where dS is an element of the surface, and A, p, v the direction cosines
of the normal to it. BR – y0, yP – a R, aQ – BP may then be regarded
as the components of energy flow at any point of the surface, and it
follows at once that the direction of this energy flow is perpendicular
to the magnetic force, since the sum of the components multiplied
respectively by a, B, y vanishes; similarly it is seen to be per-
pendicular to the electric force. Considering a ray of light as the
locus of point obstacles, which would prevent light from passing from
one point to another, it is identified with the direction of energy flow,
and its direction is seen to be perpendicular to the electric and
magnetic forces. -
Now in the case of the simple dumb-bell oscillator everything is
ART, 344 RADIATION OF ENERGY 633
symmetrical around the axis of 2, so that in the plane a 2 at a consider-
able distance from the oscillator we have
elmſm . -
a = y = 0, and 3– ––sin (mºr – mt) sin 6,
while -
2
KP = — er sim (mºr – nt) sin 6 cos 6,
…, elm” . e tº
KR = --sin (ma' — nt) sin” 6,
N=sin 6, p = 0, v=cos 6.
Hence if the integration be extended over the surface of a sphere of
radius r, having its centre at the centre of the oscillator, we may write
dS = 2arr sin 6 . rā6, so that we have
arºſe +T)=ſs& sin 6 — P cos 0) 2-rººsin 6d.6
Tr
4tre27°ºn 30, . o .
****sinº (mr-nl) |* sin” 0d8.
* 0
Consequently we have
2 3
#(E+T)=st *
dt sin”(mr – mt).
Integrating this between the limits T and zero we obtain the expres-
sion for the energy which passes across the whole surface of the sphere
during a complete oscillation; that is, since m = 27/A and n = 27/T,
e2/2mºm,
3K
_*mºnt (21)*
TT3K T 3KX3 °
T
ſ {1 – cos 2(mr – ml)}dt,
0
so that the energy radiated per second is
(21)*.
3KX3T
In one of the oscillators used by Hertz the conductors were two
equal spheres of 15 cms. radius. These were charged in opposite
senses up to a sparking distance of about 1 cm. Taking this to repre-
sent a difference of potential of 120 C. G. S. electrostatic units, so that
one sphere was charged to a potential of + 60, and the other to - 60,
then the charge of each sphere was 15 x 60 = 900 C. G. S. units.
* The energy radiated by a small circular current varying according to the simple
harmonic law was calculated as early as 1883 by G. F. FitzGerald (Trans. Royal
Dublin Society, March 1884).
634 ELECTROMAGNETIC RADIATION CHAP. XXI
Hence the whole stock of energy possessed by the oscillator at the
start amounted to 2 × 4 × 900 x 60 = 54,000 ergs, which is about the
energy acquired by a mass of 1 gramme in falling through a height
of 55 cm. The length l of the oscillator was 100 cm., and the wave
length was about 480 cm., so that the energy lost in the first half-
period oscillation was about 2400 ergs, and after eleven half-period
oscillations about half the total stock of energy was lost. *
APPENI) IX I
The following tables are taken from Watts's Dictionary of Chemistry.
The indices are for the yellow rays, except those of Wollaston, which
are for the extreme red :
INDICES OF REFRACTION OF SOLIDS
Solid. Index Observer.
Lead Chromate 2.5 to 2.97 Brewster.
Diamond 2.47 to 2.75 Br. and Rochon.
Phosphorus e 2-224 Brewster.
Glass of Antimony 2.216 3 y
Sulphur (native) 2-115 3 5
Zircon . º 1.95 Wollaston.
Lead Nitrate. 1.866 Herschel.
Lead Carbonate 1-81 to 2.08 Brewster.
Ruby . 1.779 3 y
Felspar . 1.764 3 5
Tourmaline º 1 - 668 33
Topaz (colourless) . I-610 Biot.
Beryl . º 1.598 Brewster.
Tortoise-shell 1.591 3 x
Emerald 1-585 3 J.
Flint Glass . - 1.57 to 1.58 Br. and Woll.
Rock-crystal (least 1.547 Wollaston.
Rock-salt . º 1.545 Newton.
Apophyllite . 1.543 Brewster.
Colophony 1.543 Wollaston.
Sugar . o 1.535 3 3
Phosphoric Acid 1-534 Brewster.
Copper Sulphate 1-531 to 1.552 33
Canada Balsam 1.532 Young.
Citric Acid l. 527 Brewster.
Crown Glass . 1.525 to 1.534 2 3
Nitre . 1,514 2 x
Plate Glass 1.514 to 1.542 3 3
Spermaceti 1-503 Young.
Crown Glass . º 1-5 Wollaston.
Potassium Sulphate 1.5 Brewster.
Ferrous Sulphate . 1.494 3 3
Tallow ; Wax º 1.492 Young.
Magnesium Sulphate . 1-488 Brewster.
Iceland Spar (greatest) . 1.654 Malus.
Obsidian º º 1. 488 Brewster.
Gum 1.476 Newton.
Borax 1.475 Brewster.
Alum 1.457 Wollaston.
Fluorspar 1. 436 Brewster.
Ice e 1.310 Wollaston,
Tabasheer 1.1115 Brewster.
APPENDIX II
INDICES OF REFRACTION OF LIQUIDs
Liquid. Index. Observer.
Sulphide of Carbon . 1,678 Brewster.
Oil of Cassia e 1.631 Young.
Bitter Almond Oil . 1.603 Brewster.
Nut: Oil . º 1-500 3 3
Linseed Oil . 1.485 Wollaston.
Oil of Naphtha 1-475 Young.
Rape Oil . 1.475 5 3 1'.
Olive Oil. º º 1.470 Brewster.
Oil of Turpentine . . 1,470 Wollaston.
Oil of Almonds 1-469 5 y
Oil of Lavender tº & 1.457 Brewster.
Sulphuric Acid (sp. gr. 1-7) 1.429 Newton.
Nitric Acid (sp. gr. 1-48) g 1-410 Young, Woll.
Solution of Potash (sp. gr. 1:41) 1-405 Fraunhofer.
Hydrochloric Acid (concentrated) 1-410 Biot.
Sea Salt (saturated). g * 1.375 5 y
Alcohol (rectified) & 1.372 Herschel.
Ether . e 1.358 Wollaston.
Alum (saturated) 1.356 Herschel.
Human Blood . 1-354 Young.
White of Egg . g 1.351 Euler (jun.).
Vinegar (distilled) . 1.372 Herschel.
Saliva gº tº 1-339 Young.
Water 1.336 Woll., Br.
APPENDIX III
STANDARD WAVE LENGTHS IN DRY AIR AT 15° C. AND 760 MM. PRESSURE
Benoit, Fabry, and Perot—Cadmium (red), 6438°4696 Ångström units.
Perot and Fabry–Cadmium (green), 5085-8240 Ångström units.
For STANDARD IRON LINES :
See (1) Buisson and Fabry, C.R. cxliii. p. 165 (1906).
(2) Eversheim, Astrophys. J. xxxi. p. 76 (1910).
[These two tables differ at most by six units in the seventh signifi-
*
cant figure.]
INDEX
The Ireferences are to Pages
Abbe, 137, 328
Aberration, 20
chromatic, 140
of lenses, 134
of light, 12, 555, 558, 559
spherical, 134
Abney, 42
Absent spectra, 267
Absorption bands, 501
coefficients, 504
colour produced by, 506
double, 344
lines, 510
selective, 501
Achromatic interference bands, 185
lines, 430
Achromatism, 201
Actinic rays, 44
Airy, 192, 311, 326, 452, 478, 488, 562,
573, 580, 581
Alhazen, 5, 6
Ames, 273
Amplitude, 39
Analyser, 489
Jellett, 490
Laurent, 491
Poynting, 494
Ångström, 275, 511
Aplanatic points, 134
surface, 93, 111
Arago, 18, 186, 214, 346, 349, 393, 409,
454, 457, 540
Aragonite, 382
Aristophanes, 2
Aristotle, 2, 4, 24
Arndtsen, 621
Astigmatism, 120
Atmosphere, height of, 6
Axes, crystal, 341
Axes, optic, 341, 372, 437
of external conical refraction, 384
of internal conical refraction, 379
of single ray velocity, 383, 387
of single wave velocity, 379
Babinet's principle, 254
Bacon, Roger, 7
Bartholinus, 350
Becquerel, 516, 521, 526
Bell, 274, 276
Biaxal crystals, 369–388
isochromatic surface in, 428
wave surface in, 374
Billet, 187
Biot, 360, 405, 584
formula, 523, 621
laws, 458, 499
polariscope, 338
Biplates, 173
Bi-prism, 170
fringes, 173
Bi-quartz, 492
Boscovitch, 21
Boussinesq., 524
Brace, 569
Bradley, 12, 555, 558, 561
Brewster, 193, 214, 224, 228, 405, 407,
459, 464, 518
Brewster's bands, 224
law, 345
Brilliancy, intrinsic, definition of, 47
Broch, 494
Brühl, 162
Burning glass, 1
Calorescence, 43, 517
Camera obscura, 8
Carbon bisulphide, 554, 622
637
638 THEORY OF LIGHT
Carvallo, 526
Catoptrics, 1
Cauchy formula, 161, 523
Caustic, 120
by reflection, 12ſ, 123
by refraction, 121
Change of tint by absorption, 505
de Chaulmes, 237
Chlorophyll, 518
Christiansen, 514
Circle of least confusion, 122
Cleomedes, 5
Collimeter, 146
Colour, 42
of interference bands, 166
Newton's experiments, 129
at polarising angle, 345
of skylight, 300
surface, 515
and velocity, 104, 106
Colours, by superposition, 415
of thin plates, 196
of thin crystalline plates, 409
Compensator, Babinet's, 441
Soleil, 495
strain, 447
Complementary screens, 256
Conductor, isotropic, 610
Conrady, 331
Convection current, 614
Corbino, 622
Cornu, 534, 539, 550, 552
Cornu's spiral, 297
Coronas, 257
Corpuscular theory, development of,
11, 15
Creosote, 622
Critical angle, 99, 357
thickness, 505
Crookes, 16
Crystals, positive, 352, 357
negative, 351, 357
Curvature, impressed, 108
of image, 136
measurement of, 86
Cusp of caustic, 123
Cuthbertson, 615
Dale, 153
law of Gladstone and, 155
Desains, 393 p
Descartes, 9, 24, 95, 151, 573
Deslandres, 512
Deviation, by prism, 104
minimum, 131, 264, 574
Dichromatism, 505
Dielectric, crystalline, 607
isotropic, 599
Diffraction, 10, 242
at circular aperture, 250
grating, 261, 311
in optical instruments, 324
patterns, 279
at rectangular aperture, 249
at straight edge, 245
Diffusion rings, 240
Dioptrics, 1
Discharge, oscillating, 588
Dispersion, 104, 147, 201, 265, 575
anomalous, 513, 524
anomalous rotatory, 621
of electromagnetic waves, 612
formulae, 521, 524
of planes of polarisation, 461
Dispersive power, 140
de Dominis, 8, 572
Doppler effect, 558
Doppler principle, 512
Double refraction, 32, 340
Fresnel's theory of, 366
of quartz, 497
in uniaxal crystals, 350
Dragoumis, 595
Drift, produced by moving matter,
561
Fizeau's experiment on, 564
law of, 562
Drude, 339, 620, 622
Dufet, 157
Dulong, 160
Eagle mounting, 271
Earnshaw, 397 -
Ehert, 512
Echelon (spectroscope), 316
Ehrenfest, 556
Elastic solid theory, 31
Elasticity, of ether, 31, 390
ellipsoid of 369
Electromagnetic equations, 599,
theory of light, 34.
theory of radiation, 588
Electromagnetic waves, interference of,
592 -
reflection of, 592, 604
refraction of, 604
INDEX s 639
Electromagnetic waves, radiation of, 596
transmission through dielectrics,
613
transmission through metals, 619
Electron, 586
Elements, half period, 66
Empedocles, 2
Energy, conservation of, 13
distribution of, 49
Energy, in medium, 45
modes of propagation, 14
transmission of, 13
Energy relation, 100 .
Fresnel's, 101
MacCullagh's, 101
Epicurus, 5
Eriometer, 257
Etalon, 162
Ether, 15, 29, 32, 390
Ether density, 563
Euclid, 5
Examples, 50, 57, 92, 97, 100, 114, 123,
154, 166, 189, 194, 203, 209, 240,
264, 291, 298, 333, 489, 500
Eyepiece, Huygens's, 142
Kellner's, 142
Ramsden’s, 142
Faraday, 459, 463
Feddersen, 590
Fermat, law, 102
Field of view, 140
FitzGerald, 474, 568, 595, 596, 614
FitzGerald-Lorentz hypothesis,
57.1
Fizeau, 181, 529, 554, 556, 564
Fluorescence, 43, 517
Focal distance, definition of, 108
length, definition of 88.
Focal lines for lens, 135
primary, 122
secondary, 122
* Focal power, 88, 108
Foci, conjugate, 90, 110, 111
dispersion of, 130
primary, 121
secondary, 121
Focus, principal,
100
of a lens, 113
Forbes, 533, 537, 552, 554
Foucault, 181, 540, 552, 553
Foucault's prism, 361
569,
definition of, 88,
Fraunhofer, 153, 257, 261, 275
Fraunhofer fringes, 258, 26.1, 303
lines, 510
Frequency, 42
Fresnel, 24, 76, 101, 167, 175, 186, 244,
349, 389, 405, 563
Fresnel's, bi-prism, 170
energy equation, 390
fringes, 284
hypothesis, 368
integrals, 293
mirrors, 167
rhomb, 402
Fringes, circular, 188
Fresnel's, 284
Haidinger's, 197, 205, 208, 210
Herschel's interference, 217
limit to number, 179
Galileo, 8, 24, 528
Gernez, 461
Gibbs, Willard, 557
Gilbert, 296
Gladstone, 153
law of, 155
Goldhammer, 620
Gouy, 76, 182
Gratings, choice of 271
calibration of, 276
curved, 268
difficulties of construction, 273
diffraction, 261
Eagle mounting, 271
Nobert, 275
reflection, 267
Rowland's, 269
Green, 34
theorem of, 77
Grimaldi, 9, 24, 177, 243
Grimaldi's experiment, 25
Gumlich, 621
Hagen, 620
Haidinger's brushes, 452
fringes, 197, 205, 208, 210
Hale, 512
Halos, 257
Hamilton, 381
Havelock, 556
Helmert, 552
Helmholtz, 156, 453, 526
equation of, 137
Herschel, 217, 221, 459, 518, 520
640 THEORY OF LIGHT
Hertz, 44, 588, 591
Hertz, vibrator experiments, 591, 592
Hertzian waves, 44, 76
Hilger, 318
Hoek, 562
Hooke, 12, 24, 198.
Hull, 16, 17
Hurion, 514
Huygens, 20, 24, 47, 77, 98, 338, 344, 350
Huygens's, construction, 353, 379, 399
principle, 77, 244
Iceland spar, 341
Image, definition of 86
Induced currents, 585
Intensity of illumination, 45
variation of 46
Interference, 25, 47, 64
bands, 27
conditions for, 177, 414
fringes. See Fringes
of polarised light, 409
Interferometer, Fabry and Perot, 208,
233 -
Lummer-Gehrcke, 319
Michelson and Morley, 229
Isochromatic lines, 219, 484, 485
Jamin, 156, 226, 340, 405, 406, 441, 444,
453
Janssen, 511
Jones, 624
Jupiter's moons, eclipse of, 11, 561
Kayser, 555
Kelvin, 526, 588
Kennedy, 569
Kerr, 447
effects, 466-472
experiments of, 466
Kettler, 156
formula, 524, 527
Kirchhoff, 79
Kohlrausch, 153
Kundt, experiments, 470
law of, 515, 516
Landolt, 621
Langley, 270, 523
Larmor, 7.9, 569
Lea, Alice, 598
Least time, principle of 85, 102
Lebedev, 16
Lens, definition of, 111 -----.
focal power of 113 -
general formula for, 113
kinds of, 113
refraction through, 111
split, 187
Leyden jar, 591
Light, bending of 71.
common, 398, 450.
emission theory, 3, 15, 19, 20, 27,
97, 156. -
incident, 81
pressure of 16
propagation of, 62, 71, 76
reflection of, 18
refraction of, 19
scattering of by very small particles,
300 - .*
white, 200, 263
velocity of, determination by
Bradley, 555 ; by Cornu, 534;
by Fizeau, 529; by Foucault,
540; by Galileo, 528 ; by
Michelson, 545; by Michelson and
Pearson, 551; by Newcomb, 547;
by Perrotin, 552; by Young and
Forbes, 537; by Römer, 554
Light, velocity of, in different media, 552
table of values, 551, 552
Lippershey, 8
Lloyd, 385
Lloyd's experiment, 381
mirror, 174, 178, 185
Lockyer, 511
Lodge's experiment, 570, 595 .
Lorentz, 156, 161, 163, 566, 568
Lorenz-Lorentz formula, 161
constant, 615
Lucretius, 5
Macaluso, 622 .
MacCullagh, 101, 377, 391, 402, 610
MacCullagh's theory, 497
| Magic lantern, 7
Magnification, transverse, 110
Malus, 154, 338, 355, 405, 409
law of, 346
Mascart, 265, 275
Maxwell, 16, 34, 586
Meslin, 187
Meteors, 4
Michelson, 181, 332, 545, 551, 552, 554,
565 -
INDEX 641
Michelson and Morley, experiments, 566
Micrometer, double image, 363
Miller, 569, 581
Minimum deviation, measurement of,
149
Mirror, aplanatic, 93
concave, 89
convex, 90
metallic, 1
Mixed plates, 238
Molecular index, 157
Monoclinic system, 438
Moore, 520
Morley, 181, 565, 569
Neumann, 402
Neutral lines, 430
Newcomb, 547, 552
Newton, 9, 20, 21, 22, 104, 129, 145,
152, 222, 240, 243, 257, 259, 572
Newton's rings, 210, 216, 233
Nichols, 16, 527, 616
Nicol's prism, 360
varieties of, 362
Nobert, 275
Obliquity, law of, 65, 295
Optical bench, 176
Optical testing of surfaces, 201
Optically active media, 620
Optics, history of, 1
geometrical, definition of, 1
physiological, definition of, 1
physical, definition of, 1
Orthorhombic system, 437
Oscillator, Hertz, 234, 624
Oxley, 447
Paalzow, 591
Parallax, 146
Paris, E. T., 459
Paschen, 527
Pearson, 551, 552, 598
Peirce, 275
Periodic time, 39
Phase, definition of, 36, 42
Phillips, 162, 615
Phosphorescence, 520
Phosphoroscope, 521
Pitch, 42
Planck, 156, 184
Plane, principal, 344
primary, 122
Plane, secondary, 122
Plane wave, refraction of, 107
reflection of 87
Plates, pile of, 345
quarter-wave, 415, 446
thick, 415
Plato, 2,
Pockels, 157
Poggendorff, 25
Polarisation, 32
angle of, 339
circular, 448
by double refraction, 342
electric, 587
elliptical, 406, 449
partial, 339, 450
plane, 447
plane of, 340
plane of, change by reflection, 394
plane of rotation of, by liquids and
vapours, 457, 458, 463
plane of, magnetic rotation of, 622
by reflection, 335
of refracted light, 346
of skylight, 454
Polarised light, 56
circularly, 396 ; convergent, 435,
487 ; parallel, 432
detection of, 440
elliptically, 396, 436, 444
interference of 348
partially, 348 ; convergent, 420 ;
divergent, 420
reflection of, 389 ; by metals, 405
refraction of, 389
study of, 440
Polariscope, de Senarmont, 497
Savart's, 440
Polarising angle, 405
Pole, 71
Poles, definition of, 66
Porta, 8
Porter, 459
Posejpal, 162
Potter, 573, 579
Potter's experiment, 191
Powell, 191, 397
Powell’s bands, 193
Poynting, 17, 494
Praseodymium, chloride, 622
Prism, definition of, 103
refraction through, 103
Rochon, 362
2
T
642 THEORY OF LIGHT
Prism, Wollaston, 364
Provostaye, 393
Ptolemy, 5
Pupil, entrance and exit, 140
Pythagoras, 2
Quarter-wave plate, 415, 446
Quetelet, 238
Quincke, 156
Quinine sulphate, 518
Radiant heat, 43
Radiation, electromagnetic, 585
from moving molecules, 512
transformation of, 517
Radiometer, 16, 17
Rainbow, 8, 572
complementary, 579
intersecting, 578
lunar, 578
primary, 572
reflected, 519
secondary, 572
spurious, 579
supernumerary, 573, 579
Ramsden, 140
circle, 140
Ray, definition of, 92
extraordinary, 342
ordinary, 342
velocity, 358
Rayleigh, 183, 303, 323, 328, 331, 339,
554, 557, 569, 613
Reflection, multiple, 204
of spherical wave, 85, 89
total, 99, 152
of wave of any form, 91
Refraction, astronomical, 5
external conical, 384
internal conical, 381
laws of 95, 96 -
Refractive index, 96, 515, 635, 636
determination of, 144, 186
effect of mechanical stress on,
157
extraordinary, 353
formula for, 151
of gases, 159
of mixtures, 158
of solutions, 158
values, 161
variation with density, 155
Refractive index, variation with tem-
ºperature, 157, 158 -
Refractivity of gases, 162
Refractometer, 181, 226
Refrangibility, 129
Relative motion, of earth and ether, 566
of matter and ether, 557
Resolving power, 215
of microscope, 328
of prism, 322
Reststrahlem, 617
Reynolds, Osborne, 61
Rochon's prism, 362
von Rohr, 140
Römer, 11, 554
Rotation of plane of polarisation, 475
Rotatory power, 458
Jmolecular, 460 -
relation to crystalline form, 462
values, 460
Le Roux, 514
Rowland, 269
Rubens, 527, 616, 620
Runge, 277, 555
Saccharimeter, 460
Jellett, 490
Laurent, 491
Soleil, 495
Schmauss, 622
Schuster, 146, 183, 557, 618, 624
Schwerd, 307
Sellmeier, 156
formula, 525
Shadows, 62
Simple harmonic motion, 39
Sine condition, 137
Snell, 9
law, 95
Spectra, anomalous, 513
irrationality of, 265
'normal, 265
overlapping of, 263
photographs, 271
purity of, 266
secondary, 141
Spectral lines, width of, 512
Spectrometer, description of, 146
Spectroscope, 146
echelon, 316
Spectroscopic analysis, 181
Sphere, refracting, 574
Stellar diameter, measurement of, 332
INDEX
643
Stokes, 20, 34, 65, 189, 194, 237, 382,
518, 519
law, 65
theory of fluorescence, 518
Stop, definition of, 139
Strutt, 301
Superposition, principle of, 47
Surface colour, 508
Surface of equal phase, 38
spherical, 87
Sylvine, 616
Talbot, 218, 220, 514
Talbot's bands, 192, 310, 326
Telescopes, introduction of, 8
Terquem, 152, 153
Thalén, 511
Tint of passage, 461, 496
Tourmaline, 336, 344
Trannin, 152, 153
Transmission, coefficients, 504
oblique, through quartz, 478
Triclinic system, 439
Trouton, 76, 595
Trowbridge, 527
Turmeric paper, 520
Tyndall, 454, 517.
Ultramicroscopy, 331
Uniaxal crystals, 377, 421
Huygens's construction, 352
isochromatic surface in, 424
wave surface in, 350
Uranium oxide, 518
Vector, line integral of 598
Velocity, group, 59, 554
normal surface, 373
ray, 358
Verdet, 239, 258, 460, 466, 557
Verdet's constant, 466
Vibrations, composition of, 48, 54
curve, 52 -
equivalent, principle of, 391
graphic representation of, 51
of permanent type, 5
polygon, 52 -
rectangular, 51
spiral, 72
Vinci, Leonardo da, 24
Visibility, 232
Vitellio (or Vitello), 7
d
Voigt, 79
Wolke, 339
Wave, envelopes, 63
equation of surface, 374, 376
evolute of, 120
front, 37, 38
of any form, 74
plane, 37, 69, 602, 610
Wave motion, 35
detection of, 43 ; equation of, 40;
transverse, 35
Wave theory, 15, 24, 25, 97, 243
special forms of, 33
velocity test, 97
velocity, 358, 554
Waves, aerial, capillary water, 60
deep water, 60
electric, 587
flexural, 60
polarisation of, 594
reflected, 81
refracted, 81
secondary, 62
Wave length, 37, 166
dependence of velocity on, 521
determination of, 212
difference of 54
measurement of, 274
standard, 636
Wheatstone, 540
Wheeler, 339
Whewell, 237
Whittaker, 612
Wiedemann, 458, 591
Wiener, 607
Wien's law, 183
Windows, 140
Wollaston, 152, 154, 510
Wollaston's prism, 364
Wood, 239, 509, 520, 526, 622
Young, Thomas, 20, 24, 25, 26, 29, 47,
165, 177, 214, 238, 243, 249, 257,
258, 533, 537, 552, 554, 573, 579
Young's experiment, 26, 165
Zeeman, 566, 622
effect, 472
Zone, Huygens's, 66
plates, 252
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