UC-NRLF
REESE LIBRARY
UNIVERSITY OF CALIFORNIA.
Co
^Accession No . o o 5 9 . Class No .
SCIENTIFIC MEMOIRS
EDITED BY
J. S. AMES, PH.D.
PROFESSOR OF PHYSICS IN JOHNS HOPKINS UNIVERSITY
X.
THE WAVE-THEORY OF LIGHT
THE
WAVE THEORY OF LIGHT
MEMOIRS BY HUYGENS, YOUNG
AND FRESNEL
EDITED BY
HENRY CREW, Pn.D.
\N
PROFKSSOR OK PHYSICS, NORTHWESTERN UNIVERSITY
NEW YORK : CINCINNATI .: CHICAGO
AMERICAN BOOK COMPANY
to *i<&
COPYRIGHT, 1900, HY
AMERICAN BOOK COMPANY
Crew, Light.
W. P. I
PKEFACE
THANKS to the labors of Kirchhoff, Kelvin, Huxley, and
others, there is now a widespread opinion that any physical
phenomenon is " explained" only when some one has devised a
dynamical model which will duplicate the phenomenon. The
completeness of the explanation is to be measured by the com-
pleteness with which the model will duplicate the phenomenon.
Thus, for instance, a refraction model which, like that of
Airy, describes only the path of the refracted ray when the
incident ray is given, does not in any true sense explain how
the refracted ray comes to take one path rather than another.
Such a model illustrates Suell's law, but does not explain the
phenomenon.
If, however, we take a large and shallow tank of water, the
floor of the tank being partly covered with a false bottom, so
as to give two, and only two, different depths of water, we shall
find that the speed of the waves in the deeper portion of the
tank bears to the speed in the shallower portion a constant
ratio ; hence, in passing from one depth to the other, these
waves are refracted according to the sine law.
Such a model may be said to be a " partial explanation" of
refraction in so far as it refers the phenomenon to change of
speed which accompanies change of medium. It represents,
however, only the kinematics of refraction.
If, now, we could go one step further, and make a model in
which the wave -producing forces were duplicated in other
words, if we could make a model of the medium and of the
disturbing forces we should have a fairly complete "expla-
nation" of refraction ; in fact, the dynamics of refraction would
be understood. This would imply not only that we knew the
substance disturbed, but also that we were acquainted with
the laws according to which it is disturbed.
83059
PREFACE
A theory of light may be considered either from a kinemat-
ical or from a dynamical point of view. To assume, on exper-
imental grounds, that a ray of light has a certain speed in one
medium and a different speed in a different medium, and that
it consists in a particular kind of motion, and thence to infer
the laws of refraction, rectilinear propagation, and diffraction,
is to construct a kinematical theory of light. But to assume
a certain structure for the luminous body and for the medium,
and thence to derive the motions and the different speeds
assumed in the kinematical case, is to offer a dynamical ex-
planation of light.
The wave-theory of light is used, nearly always, in the former
and narrower sense to mean the kinematical explanation of
light; it leaves entirely to one side the dynamical questions
hinted at above. It assumes, not without strong experimental
evidence, the existence of waves travelling with different speeds
in different media, and proposes to explain the cardinal phe-
nomena of optics.
To illustrate its limitations, we may cite the instance of the
ordinary and extraordinary ray in crystals. How it happens
that there are two rays is a problem in the dynamics of light ;
but, assuming these two rays, their subsequent behavior, their
inability to interfere, etc., must be accounted for in a general
way, at least by the kinematical theory of light.
It is in this narrow sense that the wave-theory of light is
employed in the memoirs translated in this volume.
The first clear and unmistakable suggestion that light con-
sists in a vibratory motion appears to be due to that brilliant
but unfortunate genius, Robert Hooke (1635-1703), who, in
his Micrograpliia (London, 1665), describes the three charac-
teristic features of the motion which he believes to constitute
light.
Since it has not been deemed advisable to reprint Hooke's
paper in this volume, it may not be out of place here to quote
what few paragraphs are necessary fairly to present his point
of view. This will, perhaps, be accomplished by the follow-
ing selections :
' ' It would be somewhat too long a work for this place Zetet-
ically to examine, and positively to prove, what particular
kind of motion it is that must be the efficient of Light; for
though it be a motion, yet 'tis not every motion that produces
vi
PREFACE
it, since we find there are many bodies very violently mov'd,
which yet afford not such an effect ; and there are other bodies,
which to our senses, seem not mov'd so much, which yet shine.
Thus Water and quick-silver, and most other liquors heated,
shine not; and several hard bodies, as Iron, Silver, Brass, Cop-
per, Wood, &c., though very often struck with a hammer, shine
not presently, though they will all of them grow exceeding
hot; whereas rotten Wood, rotten Fish, Sea Water, Gloworms,
&G. have nothing of tangible heat in them, and yet (where
there is no stronger light to affect the Sensory) they shine some
of them so Vividly, that one may make a shift to read by them.
"It would be too long, I say, here to insert the discursive
progress by which I inquir'd after the proprieties of the mo-
tion of Light, and therefore I shall only add the result.
"And, First, I found it ought to be exceeding quick, such
as those motions of fermentation and putrefaction, whereby,
certainly, the parts are exceeding nimbly and violently mov'd;
and that, because we find those motions are able more mi-
nutely to shatter and divide the body, then the most violent
heats or menstruums we yet know. And that fire is nothing
else but such a dissolution of the Burning body, made by the
most universal menstruum of all sulphureous bodies, namely,
the Air, we shall in an other place of this Tractate endeavour
to make probable. And that, in all extremely hot shining
bodies, there is a very quick motion that causes Light, as well
as a more robust that causes Heat, may be argued from the
celerity wherewith the bodyes are dissolv'd.
"Next, it must be a Vibrative motion. And for this the
newly mentioned Diamond affords us a good argument; since
if the motion of the parts did not return, the Diamond must
after many rubbings decay and be wasted ; but we have no
reason to suspect the latter, especially if we consider the ex-
ceeding difficulty that is found in cutting or wearing away a
Diamond. And a Circular motion of the parts is much more
improbable, since, if that were granted, and they be suppos'd
irregular and Angular parts, I see not how the parts of the
Diamond should hold so firmly together, or remain in the same
sensible dimensions, which yet they do. Next, if they be Glob-
ular, and mov'd only with a turbinated motion, I know not any
cause that can impress that motion upon the pellucid medium,
which yet is done. Thirdly, any other irregular motion of the
vii
PREFACE
parts one amongst another, must necessarily make the body
of a fluid consistence, from which- it is far enough. It must
therefore be a Vibrating motion.
" And Thirdly, That is a very short vibrating motion, I think
the instances drawn from the shining of Diamonds will also
make probable. For a Diamond being the hardest body we yet
know in the World, and consequently the least apt to yield or
bend, must consequently also have its vibrations exceeding
short.
"And these, I think, are the three principal proprieties of
a motion, requisite to produce the effect call'd Light in the
Object." [Micrographia, pp. 54-56.]
The total absence of experimental evidence from the above
statement of the case stands in such marked contrast with the
method of modern physics as initiated by Galileo, that we can-
not for a moment reckon Hooke among the founders of the
wave-theory.
So important, on the contrary, have been the contributions
of Huygens, Newton, Young, and Fresnel, that each has in
turn been considered the founder of the modern science of
optics. What justification there is for each of these views will
be clearer from a brief consideration of optical theory before
and after it had been modified by the work of each of these
four men.
Two questions naturally arise in the consideration of any
theory, viz., (1) What phenomena does it explain? and (2)
How does it explain them ? The answers which have been
given to these two questions at various periods in the develop-
ment of the wave-theory may be outlined as follows:
At the time when Huygens and Newton began their work
on light, the following phenomena were demanding explana-
tion :
1. The existence of rays and shadows, known from the ear-
liest times.
2. The phenomenon of reflection, known from the earliest
times.
3. The phenomenon of refraction, as described by Snell's law.
4. The rainbow and the production of color by the prism.
5. The colors of thin plates Newton's rings.
6. Diffraction bands outside the geometrical shadow, de-
scribed by Grimaldi, 1665.
PREFACE
To these might be added the two following phenomena which
were discovered before the final publication of Newton's Op-
ticks (1704) or Huygens's Traite dela Lumiere (1690).
7. The polarization of light by crystals (Bartholinus, 1670).
8. The finite speed of light (Romer, 1675).
Of these eight cardinal facts, the second, the third, and the
eighth,, were explained by Huygens on the assumption
(a) That a luminous disturbance consists of a wave-motion
in the ether.
(b) That this wave-disturbance travels with a uniform finite
speed through the ether in any homogeneous medium.
(c) That in different media it travels with speeds which are
related inversely as the refractive indices of those media.
But the wave-disturbance as pictured by Huygens was a
single longitudinal pulse, or blow, imparted to an elastic fluid.
Since he did not have in mind either a train of waves or trans-
verse waves, or the idea of " phase," or waves of different
lengths, it is evident that he was unable to explain any of the
remaining five facts.
Turning now to that portion of the work of Newton which
contributed to the wave-theory, we find that the fourth phe-
nomenon prismatic colors was explained by him in 1666, when
he demonstrated that a single ray of white light contains all
the colors of the spectrum, and that color is not produced at
the surface of the prism, as had been hitherto supposed. This
discovery made possible, for the first time, the correct explana-
tion of the rainbow.
In Newton's ingenious, though, as we now know, incorrect
explanation of the fifth phenomenon colors of thin plates we
meet the earliest measurement of the wave-length of light,
viz., the distance traversed by a ray of light during the inter-
val between two successive "fits" of the same kind. We meet
here, also, the first evidence that, in these fits, or, as we now
say, waves, there is a regular periodicity. From this point on
we must consider light as travelling not only in waves, but in
trains of waves.
At the close of the period of Huygens and Newton, we have
then the following facts still demanding explanation :
1. The existence of rays and shadows.
5. The colors of thin plates.
.6. The existence of diffraction fringes.
PREFACE
7. The polarization of light by crystals.
To these must now be added
9. The phenomenon of stellar aberration, discovered by Brad-
ley in 1727.
Considering next the work of Young, we find that he first
suggested the correct explanation for the colors of thin plates,
having shown by experiment that two rays of light can inter-
fere to produce alternately bright and dark bands. From this
experiment and the dark centre in Newton's rings, he con-
cludes that light consists of series of waves which, like other
wave - motions, change phase by 180 on reflection from a
denser medium.
Young, at this period (1802-3), was still laboring under the
impression that light-waves were longitudinal and were propa-
gated in a fluid medium ; fortunately, neither of these assump-
tions affects the validity of his reasoning concerning the colors
of thin plates.
When Fresnel began his optical studies (1814) the following
facts, viz., (1) existence of rays, (6) diffraction fringes, (7)
polarization, and (9) aberration, were still to be accounted for
on the wave-theory. By the union of Huygens's principle with
the principle of interference, Fresnel gave the first satisfac-
tory explanation of the rectilinear propagation of light, and
of the existence of diffraction fringes outside the geometrical
shadow.
FresneFs memoir, in which these discoveries are most sys-
tematically set forth, and which was " crowned" by the French
Academy in 1819, is translated in the following pages. For
the purpose of offering an elementary geometrical explanation
of rays and diffraction bands, Fresnel invented the idea of
dividing the wave-front into a certain series of zones, which in
nearly all text - books are wrongly referred to as " Huygens's
Zones." That this is not only unfair, but also misleading, has
been pointed out by Professor Schuster. Phil. Mag. vol. xxxi.,
p. 77 (1891). The first mention of these Fresnel Zones, as
they should be called, will be found on p. Ill of the present
volume.
It was in order to explain the phenomenon of polarization
that Fresnel introduced the idea of transverse vibrations in the
ether. The boldness of this now universally accepted hypoth-
esis, which was then practically equivalent to supposing the
PREFACE
ether an elastic solid, can be fully appreciated only after one
has carefully studied the views of Fresnel's contemporaries.
The evidence for the transversality of light vibrations rests
.pon the inability of two oppositely polarized rays to interfere.
The memoir of Arago and Fresnel upon this subject is trans-
lated in the present volume.
Of the nine phenomena which we have more or less arbi-
trarily selected as the principal facts of optics, all, save only
the last aberration had received a fairly complete explana-
tion at the close of the labors of Young and Fresnel. This
discovery of Bradley's, which he so easily disposed of on the
corpuscular theory, has received many explanations in terms
of the wave -theory; but none of these can be considered as
thoroughly satisfactory. Young imagines the ether to pass
through ordinary matter "as freely, perhaps, as the wind
passes through a grove of trees." On this view, however, it
is difficult to see how the speed of light in glass, say, should
differ from its speed in a vacuum, or how the aberration con-
stant can remain unchanged when the tube of the telescope
is filled with water, as in Airy's experiment. Proc. Roy. Soc.,
vol. xx., p. 35 (1872).
For it will be remembered that the aberration constant is
vl V radians, where
#=speed of earth in its orbit,
and F^speed of light between the objective and eye-piece of
telescope employed.
Fresnel, accordingly, modified Young's hypothesis by assum-
ing that, in their motion through space, refracting bodies carry
with them only so much ether as is required to increase the
density of free ether from unity to p, where p at any point in
the medium is defined by the following equation:
H being the refractive index at the same point in the body.
This is really equivalent to saying that et the luminiferous
ether is entirely unaffected by the motion of the matter which
it permeates." [Amer. Jour. Sci., vol. cxxxi., p. 386.] And
that this is the fact of nature is exactly the conclusion at which
Fizeau and Michelson and Morley arrive from their experi-
ments upon the effect of motion of the medium upon the speed
of light. LOG dt., p. 377.
xi
PREFACE
When, however, Michelsou and Morley attempt to detect
this relative motion of the earth and the ether as the earth
proceeds in its orbital motion, they do not succeed in certainly
finding that there is any [Phil. Mag., December, 1887]; and they
accordingly conclude that this relative motion is "quite small
enough to refute FresnePs explanation of aberration."
Of the two experimental facts just cited, one apparently
confirms FresnePs view, and makes possible an explanation of
aberration in terms of the wave-theory; while the other leads
us to think that the ether moves with the refracting medium,
in which case the wave-theory appears incompetent to explain
stellar aberration.
It was in the year 1850 that Fizeau and Foucault measured
directly the speed of light in air and in water, and found the
ratio of these speeds numerically equal to the ratio of their
refractive indices. This experiment has sometimes been called
the experimentum crucis of the wave-theory; but with scant
justice we venture to think, inasmuch as no great doctrine in
physics can be said to rest upon any single fact, though mod-
ification may be demanded by a single fact.
We have now followed, in merest outline, the general ex-
planations which Huygens, Newton, Young, and Fresnel have
offered for all, save one, of this group of nine cardinal facts.
It is needless to remind the reader that this enumeration forms
but a small fraction of the phenomena which optical science
has brought to light within the last two centuries, or, indeed,
since the labors of these four men were ended.
No outline of the wave-theory would be complete without
mention of the important addition which was made to it in
.the year 1849 by Sir George Stokes. For he it was who first
completely justified Huygens's principle by showing that if the
primary wave be resolved as proposed by Huygens, no "back
wave" will be produced provided we adopt the proper law
of disturbance for the secondary wave. The discovery of this
law was announced in his memoir on the Dynamical Theory
of Diffraction. [Trans. Oamb. Phil. Soc.,vo\. ix., p. 1; Math.
andPhys. Papers, vol. ii., p. 243.] Mathematically speaking,
this contribution amounts to the introduction of the factor
1 -f- cos into the equation [Eq. 46, loc. cU.~\, which describes
the disturbance in a secondary wave proceeding from an ele-
ment of the primary wave.
xii
P R E F A C K
While, as has been said above, the following memoirs con-
cern themselves only with the kinematics of light-waves and
not at all with the question of what is vibrating, it may not
be out of place to indicate that principally during the last half
of the present century at least four more cardinal facts have
presented themselves and demanded explanation.
10. The speed of light in free space is numerically equal
to the ratio of the electrostatic and electromagnetic units of
quantity.
11. In refracting media, the speed of light varies inversely
as the square root of the product of the electric and magnetic
inductivities.
12-. " Most transparent solid bodies are good insulators, and
all good conductors are very opaque." MAXWELL, Treatise, vol.
ii.,art. 799.
13. The plane of polarization is rotated in a magnetic field.
(Faraday.)
It was to " explain" these additional phenomena that Max-
well proposed, in 1865, to modify the wave-theory of light by
replacing the mechanical shear of the ether by an electric dis-
placement. How thoroughly justified Maxwell was in this
move has been amply proved mathematically by the analogy
of his equations with those of the elastic solid theory, and
experimentally by Hertz (1888).
Within the last decade the wave -theory has shown itself
capable of explaining an entirely new group of phenomena,
viz., the color photography discovered by Lippmann. Wiener
has shown that we have here merely two rays of light the
direct and reflected travelling in opposite directions and inter-
fering to produce stationary light waves.
The flexibility of the wave - theory has still more recently
been exemplified by the beautiful discovery of Zeeman; and
Larmor and Preston have shown that by assuming a particular
kind of electrical displacement, viz., an orbital motion of an
ion, the wave-theory is competent to predict not only the trip-
lets and even the sextet, but also the polarization produced by
placing the source of radiation in a magnetic field. Phil Mag.,
February, 1899.
Striking as the resemblance appears between the kinematics
of wave-motion considered in this volume and the phenomena
of optics, it must never be forgotten that in all probability the
PREFACE
vibrating atom is a structure whose motion is vastly compli-
cated as compared with the few simple motions which the'
experiments of Huygens, Newton, Young, Fresnel, Maxwell,
and Michelson have assigned to it.
H. 0.
EVANSTON, 111., November, 1899.
xiv
GENEBAL CONTENTS
PAGE
Preface v
Treatise on Light. By Christiaan Huygens. (First three chapters).. . 1
Biographical Sketch of Huygens 42
On the Theory of Light and Colors. By Dr. Thomas Young 45
An Account of Some Cases of the Production of Colors not Hitherto
Described. By Dr. Thomas Young 62
Experiments and Calculations relative to Physical Optics. By Dr.
Thomas Young 68
Biographical Sketch of Young 77
Memoir on the Diffraction of Light, crowned by the [French] Acad-
emy of Sciences. By A. J. Fresnel 79
On the Action of Rays of Polarized Light upon Each Other. By
Arago and Fresnel 145
Biographical Sketch of Fresnel 156
Bibliography 161
Index 165
xv
TBEATISE ON LIGHT
CONTAINING
THE EXPLANATION OF REFLECTION AND OF RE-
FRACTION AND ESPECIALLY OF THE RE-
MARKABLE REFRACTION WHICH
OCCURS IN ICELAND SPAR
BY
CHRISTIAAN HUYGENS
(Leyden, 1690)
CONTENTS
rAGE
Preface 3
Table of Contents 7
The Rectilinear Propagation of Rays and some General Considerations
concerning the Nature of Light 9
Explanation of the Laws of Reflection 25
Explanation of the Laws of Refraction 30
TREATISE ON LIGHT
BY
CHRISTIAAN HUYGENS
PKEFACE
THIS treatise was written during my stay in Paris twelve
years ago, and in the year 1678 was presented to the Eoyal
Academy of Sciences, to which the king had. been pleased to call
me. Several of this body who are still living, especially those
who have devoted themselves to the study of mathematics,
will remember having been at the meeting at which I present-
ed the paper; of these I recall only those distinguished gentle-
men Messrs. Cassini, Homer, and De la Hire. Although since
then I have corrected and changed several passages, the copies
which I had made at that time will show that I have added noth-
ing except some conjectures concerning the structure of Iceland
spar and an additional remark concerning refraction in rock-
crystal. I mention these details to show how long I have been
thinking about these matters which I am only just now publish-
ing, and not at all to detract from the merit of those who, with-
out having seen what I have written, may have investigated
similar subjects : as, indeed, happened in the case of two dis-
tinguished mathematicians., Newton and Leibnitz, regarding
the question of the proper figure for a converging lens, one
surface being given.
It may be asked why I have so long delayed the publication
of this work. The reason is that I wrote it rather carelessly in
French, expecting to translate it into Latin, and, in the mean-
time, to give the subject still further attention. Later I
3
PREFACE
thought of publishing this volume together with another on
dioptrics in which I discuss the theory of the telescope and the
phenomena associated with it. But soon the subject was no
longer new and was therefore less interesting. Accordingly
I kept putting off the work from time to time, and now I do
not know when I shall be able to finish it, for my time is large-
ly occupied either by business or by some new investigation.
In view of these facts I have thought wise to publish this
manuscript in its present state rather than to wait longer and
run the risk of its being lost.
One finds in this subject a kind of demonstration which does
not carry with it so high a degree of certainty as that employed
in geometry ; and which differs distinctly from the method
employed by geometers in that they prove their propositions
by well-established and incontrovertible principles, while here
principles are tested by the inferences which are derivable
from them. The nature of the subject permits of no other
treatment. It is possible, however, in this way to establish a
probability which is little short of certainty. This is the case
when the consequences of the assumed principles are in perfect
accord with the observed phenomena, and especially when
these verifications are numerous ; but above all when one
employs the hypothesis to predict new phenomena and finds
his expectations realized.
If in the following treatise all these evidences of probability
are present, as, it seems to me, they are, the correctness of my
conclusions will be confirmed ; and, indeed, it is scarcely pos-
sible that these matters differ very widely from the picture
which I have drawn of them. I venture to hope that those
who enjoy finding out causes and who appreciate the wonders
of light will be interested in these various speculations arid in
the new explanation of that remarkable property upon which
the structure of the human eye depends and upon which are
based those instruments which so powerfully aid the eye. I
trust also there will be some who, -from such beginnings, will
push these investigations far in advance of what I have been
able to do ; for the subject is not one which is easily exhausted.
This will be evident especially from those parts of the subject
which I have indicated as too difficult for solution; and still
more evident from those matters upon which I have not
touched at all, such as the various kinds of luminous bodies
4
PREFACE
and the whole question of color, which no one can yet boast
of having explained.
Finally, there is much more to be learned by investigation
concerning the nature of light than I have yet discovered ; and
I shall be greatly indebted to those who, in the future, shall
furnish what is needed to complete my imperfect knowledge.
THE HAGUE, 8th of January, 1690.
TABLE OF CONTENTS
CHAPTER I
ON THE RECTILINEAR PROPAGATION OF RAYS
PAGE
Light is produced by a certain motion 10
Particles do not pass from the luminous object to the eye 10
Light is propagated radially very much after the manner of Sound ... 11
[As to] whether Light requires time for its propagation 11
An experiment which apparently shows that its transmission is in-
stantaneous 11
An experiment which shows that it requires time 13
Comparison of the Speeds of Light and Sound 15
How the propagation of Light differs from that of Sound 15
They are not each transmitted by the same medium. 15
The propagation of Sound 16
The propagation of Light 17
Details concerning the propagation of Light 19
Why rays travel only in straight lines 22
How rays coming from different directions cross each other without
interference 23
CHAPTER II
ON REFLECTION
Proof that the angles of incidence and reflection are equal to each
other 25
Why the incident and reflected rays lie in one and the same plane
perpendicular to the reflecting surface 27
Equality between the angles of incidence and reflection does not de-
mand that the reflecting surface be perfectly plane 28
CHAPTER III
ON REFRACTION
Bodies may be transparent without any matter passing through them 30
Proof that the ether can penetrate transparent bodies 31
How the ether renders bodies transparent by passing through them. . 32
Bodies, even the most solid ones, have a very porous structure 32
7
TABLE OF CONTENTS
PAGJ?
The speed of light is less in water and in glass than in air 32
A third hypothesis for the explanation of transparency and of the
retardation which light undergoes in bodies 33
Concerning a possible cause of opacity 34
Proof that refraction follows the Law of Sines 34
Why the incident and the refracted rays are each capable of produc-
ing the other 35
Why reflection inside a triangular glass prism suddenly increases
when the light is no longer able to emerge 38
Bodies in which refraction is greatest are also those in which reflec-
tion is strongest 40
Demonstration of a theorem due to Fermat . . 40
CHAPTER IV
ON ATMOSPHERIC REFRACTION
[Not translated.]
CHAPTER V
ON THE PECULIAR REFRACTION OF ICELAND SPAR
[Not translated.]
CHAPTER VI
ON FIGURES OF TRANSPARENT BODIES ADAPTED FOR REFRAC-
TION AND REFLECTION
[Not translated.]
8
CHAPTER I
ON THE RECTILINEAR PROPAGATION OF RAYS
DEMONSTRATIONS in optics, as in every science where geome-
try is applied to matter, are based upon experimental facts; as,
for instance, that light travels in straight lines, that the angles
of incidence and reflection are equal, and that rays of light are
refracted according to the law of sines. For this last fact is
now as widely known and as certainly known as either of the
preceding.
Most writers upon optical subjects have been satisfied to as-
sume these facts. But others, of a more investigating turn of
mind, have tried to find the origin and the cause of these
facts, considering them in themselves interesting natural phe-
nomena. And although they have advanced some ingenious
ideas, these are not such that the more intelligent readers do
not still want further explanation in order to be thoroughly
satisfied.
Accordingly, I here submit some considerations on this sub-
ject with the hope of elucidating, as best I may, this depart-
ment of natural science, which not undeservedly has gained the
reputation of being exceedingly difficult. I feel myself espe-
cially indebted to those who first began to make clear these
deeply obscure matters, and to lead us to hope that they were
capable of simple explanations.
But, on the other hand, I have been astonished to find these
same writers accepting arguments which are far from evi-
dent as if they were conclusive and demonstrative. No one
has yet given even a probable explanation of the fundamental
and remarkable phenomena of light, viz*, why it travels in
straight lines and how rays coining from an infinitude of dif-
ferent directions cross one another without disturbing one an-
other.
I shall attempt, in this volume, to present in accordance with
9
M KM 01 US ON
the principles of modern philosophy, some clearer ;i,nd more
probable reasons, first, for the rectilinear propagation of light,
and, secondly, for its reflection when it meets .other bodies.
Later 1 shall explain the phenomenon of rays which art! said to
undergo refraction in passing through transparent, bodies of
dilTerenl, kinds. Here 1 shall treat, also of refraction efl'i-ets due
to the. varying density of the earth's atmosphere. Afterwards
I shall examine the causes of that peculiar refraction occur-
ring in a certain crystal which comes from Iceland. And last I y,
I shall consider the dill'crenl, shapes required in transparent and
in rellecting bodies to converge! rays upon a single point or to
deflect them in various ways. Hero we shall see with what
ease are determined, by our new theory, not only the ellipses,
hyperbolas, and Other Curves which M. Descartes has so ingen-
iously devised for this purpose, but also the curve which one
surface of a lens must, have when the other surface is given, as
spherical, plane, or of any figure whatever.
We cannot, help believing that light, consists in the motion
of a certain material. lA>r when we consider its production we
find that here on the earth it is generally produced by fire and
llame which, beyond doubt, contain bodies in a state of rapid
motion, since they are able to dissolve and melt numerous
other more solid bodies. And if we consider its effects, we see
that when light is converged, as, for instance, by concave mir-
rors, it is able to produce combustion just as fire does ; i.e., it.
is able to tear bodies apart ; a property that surely indicates
motion, at least in the true philosophy whore one believes all
natural phenomena to be mechanical effects. And, in my opin-
ion, we must admit this, or else give up all hope of ever under-
standing anything in physics.
Since, according to this philosophy, it is considered certain
thai, the sensation of sight is caused only by the impulse of
some form of matter upon tin* nerves at the base of the eye, we
have here still another reason for thinking that light consists
in a motion of the matter situated between us and the lumi-
nous body.
When we consider, further, the very great speed with which
light is propagated in all directions, and the fact that when
rays come from different directions, even those directly op-
posite, they cross without disturbing each other, it must be
evident that we do not see luminous objects by means of matter
10
TIIH \\AYK-TIIKOHY OF LIGHT
translated from the <>l)jcct to us, as a shot or MM arrow travels
through the air. For certainly this would be in contradiction
to the two properties of light which wo have just mentioned,
and especially to the hitter. Light is then propagated in some
other manner, an understanding of which we may obtain from
our knowledge of the manner in which sound travels through
the air.
\\ < know that through the medium of the air, an invisible
and impalpable body, sound is propagated in all directions,
from the point where it is produced, by means of a motion
which is communicated successively from one part of the air
to another ; and since this motion travels with the same speed
in all directions, it must, form spherical surfaces which contin-
ually enlarge until linally they strike our ear. Now there can
be no doubt that, light also comes from the luminous body to
us by means of some motion impressed upon the matter which
lies in the intervening space; for we have already seen that
this cannot occur through the translation of matter from out-
point to the other.
If. in addition, light requires time for its passage a point
we shall presently consider it will then follow that this motion
is impressed upon the matter gradually, and hence is propa-
gated, as that of sound, by surfaces and spherical waves. I
call these 'irdrrti because of their resemblance to those 'which
are formed when one throws a pebble into water and which
represent gradual propagation in circles, although produced by
a different cause and confined <<> ;t plane surface.
As to the ') nest ion of light requiring time for its propaga-
tion, let us consider first whether there is any experimental
evidence to the contrary.
What we can do here on the earth with sources of light placed
at great, distanc.es (alt hough showing that light does not occupy
:i sensible time in passing over these distances) may be objected
to on the ground that these distances are still too srnaM, and
thai, therefore, we can conclude only that the propagation of
light is exceedingly rapid. M. Descartes thought it instanta-
neous, and based his opinion upon much better evidence, fur-
nished by the eclipse of the moon. Nevertheless, as I shall
show, even this evidence is not conclusive. I shall state the
matter in a manner slightly different from his in order that
we may more easily arrive at all the consequences.
11
MEMOIRS ON
Let A be the position of the sun ; BD a part of the orbit or
annual path of the earth ; ABC a straight line intersecting in
C the orbit of the moon, which is represented by the circle CD.
If, now, light re-
quires time say
one hour to trav-
erse the space be-
tween the earth
and the moon, it
follows that when
the earth has
reached the point
fig j B, its shadow, or
the interruption
of light, will not yet have reached the point C, and will not
reach it until one hour later. Counting from the time when
the earth occupies the position B, it will be one hour later that
the moon arrives at the point C and is there obscured ; but this
eclipse or interruption of light will not be visible at the earth
until the end of still another hour. Let us suppose that during
these two hours the earth has moved to the position E. From
this point the moon will appear to be eclipsed at C, a position
which it occupied one hour before, while the sun will be seen
at A. -For I assume with Copernicus that the sun is fixed and,
since light travels in straight lines, must always be seen in its
true position. But it is a matter of universal observation, we
are told, that the eclipsed moon appears in that part of the
ecliptic directly opposite the sun ; while according to our view
its position ought to be behind this by the angle GEC, the
supplement of the angle AEC. But this is contrary to the fact,
for the angle GEC will be quite easily observed, amounting to
about 33. Now according to our computation, which will be
found in the memoir on the causes of the phenomena of Sat-
urn,- the distance, BA, between the earth and the sun is about
12,000 times the diameter of the earth, and consequently 400
times the distance of the moon, which is 30 diameters. The
angle ECB will, therefore, be almost 400 times as great as
BAE, which is 5', viz., the angular distance traversed by the
earth in its orbit during an interval of two hours. Thus the
angle BCE amounts to almost 33, and likewise the angle
CEG, which is 5' greater.
12
f^"
(
THE WAVE-THEORY OF LI'GHT
^^
But it must be noted that in this argument the speed of light
is assumed to be such that the time required for it to pass from
here to the moon is one hour. If, however, we suppose that it
requires only a minute of time, then evidently the angle CEG
will amount to only 33' ; and if it requires only ten seconds of
time, this angle will amount to less than 6'. But so small a
quantity is not easily observed in a lunar eclipse, and conse-
quently it is not allowable to infer the instantaneous propaga-
tion of light.
It is somewhat unusual, we must 'confess, to assume a speed
one hundred thousand times as great as that of sound, which,
according to my observations, travels about 180 toises [1151
feet] in a second, or during a pulse-beat; but this supposition
appears by no means impossible, for it is not a question of carry-
ing a body with such speed, but of a motion passing succes-
sively from one point to another.
I do not therefore, in thinking of these matters, hesitate to
suppose that the propagation of light occupies time, for on this
view all the phenomena can be explained, while on the con-
trary view none of them can be explained. Indeed, it seems to
me, and to many others also, that M. Descartes, whose object
has been to discuss all physical subjects in a clear way, and who
has certainly succeeded better than any one, before him, has
written nothing on light and its properties which is not either
full of difficulty or even inconceivable.
But this idea which I have advanced only as a hypothesis has
recently been almost established as a fact by the ingenious
method of Komer, whose work I propose here to describe, ex-
pecting that he himself will later give a complete confirmation
of this view.
His method, like the one we have just discussed, is astro-
nomical. He proves not only that light requires time for its
propagation, but shows also how much time it requires and that
its speed must be at least six times greater than the estimate
which I have just given.
For this demonstration, he uses the eclipses of the small plan-
ets which revolve about Jupiter, and which very often pass
into its shadow. His reasoning is as follows : Let A denote
the sun; BODE, the annual orbit of the earth; F, Jupiter;
and GN, the orbit of the innermost satellite, for this one, on
account of its short period, is better adapted to this investi-
13
MEMOIRS ON
Cation than is either of the other three. Let G represent the
point of the satellite's entrance into, and H the point of its
emergence from, Jupiter's shadow.
Let us suppose that an emergence of this
satellite has been observed while the earth
occupies the position B, at some time before
the last quarter. If the earth remained in
this position, 42J hours would elapse before
the next emergence would occur. For this
is the time required for the satellite to make
one revolution in its orbit and return to op-
position with the sun. If, for instance, the
earth remained at the point B during 30 rev-
olutions, then, after an interval of 30 times
42-J hours, the satellite would again be ob-
served to emerge. But if meanwhile the
earth has moved to a point 0, more distant
from Jupiter, it is evident that, provided
light requires time for its propagation, the
emergence of the little planet will be record-
ed later at than it would have been at B.
For it will be necessary to add to this in-
terval, 30 times 42 \ hours, the time occupied by ligh't in passing
over a distance MC, the difference of the distances CH and BH.
In like manner, in the other quarter, while the earth travels
from D to E, approaching Jupiter, the eclipses will occur
earlier when the earth is at E than if it had remained at D.
Now by means of a large number of these eclipse observations,
covering a period of ten years, it is shown that these inequali-
ties are very considerable, amounting to as much as ten min-
utes or more; whence it is concluded that, for traversing the
whole diameter of the earth's orbit KL, twice the distance from
here to the sun, light requires about 22 minutes.
The motion of Jupiter in its orbit, while the earth passes
from B to C or from D to E, has been taken into account in
the computation, where it is also shown that these inequalities
cannot be due either to an irregularity in the motion of the
satellite or to its eccentricity.
If we consider the enormous size of this diameter, KL,
which I have found to be about 24 thousand times that of the
earth, we get some idea of the extraordinary speed of light.
14
THE WAVE-THEORY OF LIGHT
Even if we Suppose that KL were only 22 thousand diameters
of the earth, a speed covering this distance in 22 minutes would
be equivalent to the rate of one thousand diameters per minute,
i.e., 16f diameters a second (or a pulse-beat), which makes more
than eleven hundred times one hundred thousand toises [212,222
kilometres], since one terrestrial diameter contains 2865 leagues,
of which there are 5 to the degree, and since, according to the
exact determination made by Mr. Picard in 1669 under orders
from the king, each league contains 2282 toises.
But, as I have said above, sound travels at the rate of only
180 toises [350 metres] per second. Accordingly, the speed of
light is more than 600,000 times as great as that of sound,
which, however, is a very different thing from being instanta-
neous, the difference being exactly that between a finite quantity
and infinity. The idea that luminous disturbances are handed
on from point to point in a gradual manner being thus con-
firmed, it follows, as I have already said, that light is propa-
gated by spherical waves, as is the case with sound.
But if they resemble each other in this, respect, they differ in
several others viz., in the original production of the motion
which causes them, in the medium through which they travel,
and in the manner in which they are transmitted in this
medium.
Sound, we know, is produced by the rapid disturbance of some
body (either as a whole or in part) ; this disturbance setting in
motion the contiguous air. But luminous disturbances must
arise at each point of the luminous object, else all the different
parts of this object would not be visible. This fact will be more
evident in what follows.
In my opinion, this motion of luminous bodies cannot be bet-
ter explained than by supposing that those which are fluid, such
as a flame, and apparently the sun and stars, are composed
of particles that float about in a much more subtle medium,
which sets them in rapid motion, causing them to strike against
the still smaller particles of the surrounding ether. But in the
case of luminous solids, such as red-hot metal or carbon, we
may suppose this motion to be caused by the violent disturb-
ance of the particles of the metal or of the wood, those which
lie on the surface exciting the ether. Thus the motion which
produces light must also be more sudden and more rapid than
that which causes sound, since we do not observe that sonorous
15
MEMOIRS ON
disturbances give rise to light any more than that the motion
of the hand through the air gives rise to sound.
The question next arises as to the nature of the medium in
which is propagated this motion produced by luminous bodies.
I have called it ether ; but it is evidently something different
from the medium through which sound travels. For this lat-
ter is simply the air which we feel and breathe, and which,
when removed from any region, leaves behind the luminiferous
medium. This fact is shown by enclosing a sounding body in a
glass vessel and removing the atmosphere by means of the air-
pump which Mr. Boyle has devised, and with which he has per-
formed so many beautiful experiments. But in trying this it
is well to place the sounding body on cotton or feathers in such
a way that it cannot communicate its vibrations either to the
glass receiver or to the air-pump, a point which has hitherto
been neglected. Then, when all the air has been removed, one
hears no sound from the metal even when it is struck.
From this we infer not only that our atmosphere, which is un-
able to penetrate glass, is the medium through which sound
travels, but also that it is different from that which carries
luminous disturbances ; for when the vessel is exhausted of
air, light traverses it as freely as before.
This last point is demonstrated even more clearly by the
celebrated experiment of Torricelli. That part of the glass
tube which the mercury does not fill contains a high vacuum,
but transmits light the same as when filled with air. This
shows that there is within the tube some form of matter which
is different from air, and which penetrates either glass or mer-
cury, or both, although both the glass and the mercury are im-
pervious to air. And if the same experiment is repeated, ex-
cept that a little water be placed on top of the mercury, it
becomes equally evident that the form of matter in question
passes either through the glass or through the water or through
both.
As to the different modes of transmission of Sound and light,
it is easy to understand what happens in the case of sound
when one recalls that air can be compressed and reduced to
a much smaller volume than it ordinarily occupies, and that
just in proportion as its volume is diminished it tends to re-
gain its original size. This property, taken in conjunction
with its penetrability, which it retains in spite of compression,
16
THE WAVE-THEORY OF LIGHT
appears to show that it is composed of small particles which
float about, in rapid motion, in an ether composed of still finer
particles. Sound, then, is propagated by the effort of these
air particles to escape when at any point in the path of the wave
they are more compressed than at some other point.
But the enormous speed of light, together with its other
properties, hardly allows us to believe that it is propagated in
the same way. Accordingly, I propose to explain the manner
in which I think it must occur. It will be necessary first, how-
ever, to describe that property of hard bodies in virtue of which
they transmit motion from one to another.
If one takes a large number of spheres of equal size, made of
any hard material, and arranges them in contact in a straight
line, he will find that, on allowing a sphere of the same size to
roll against one end of the line, the motion is transmitted in an
instant to the other end of the line* The last sphere in the
row flies off while the intermediate ones are apparently undis-
turbed ; the sphere which originally produced the disturbance
also remains at rest. Here we have a motion which is trans-
mitted with high speed, which varies directly as the hardness
of the spheres.
Nevertheless, it is certain that this motion is not instantane-
ous, but is gradual, requiring time. For if the motion, or, if
you please, the tendency to motion, did not pass successively
from one sphere to another, they would all be affected at the
same instant, and would all move forward together. So far
from this being the case, it is the last one only which leaves the
row, and it acquires the speed of the sphere which gave the blow.
Besides this experiment there are others which show that all
bodies, even those which are considered hardest, such as tem-
pered steel, glass, and agate, are really elastic, and bend to
some extent whether they are made into rods, spheres, or bodies
of any other shape; that is, they yield slightly at the point
where they are struck, and immediately regain their original
figure. For I have found that in allowing a glass or agate
sphere to strike upon a large, thick, flat piece of the same ma-
terial, whose surface has been dulled by the breath, the point
of contact is marked by a circular disk which varies in size
directly as the strength of the blow. This shows that during
the encounter these materials yield and then fly back, a proc-
ess which must require time.
B 17
MEMOIRS ON
Now to apply this kind of motion to the explanation of
light, nothing prevents our imagining the particles of the ether
as endowed with a hardness almost perfect and with an elas-
ticity as great as we please. It is not necessary here to discuss
the cause either of this hardness or of this elasticity, for such a
consideration would lead us too far from the subject. I will,
however, remark in passing that these ether particles, in spite
of their small size, are in turn composed of parts, and that their
elasticity consists in a very rapid motion of a subtle material
which traverses them in all directions and compels them to
assume a structure which offers an easy and open passage to this
fluid. This accords with the theory of M. Descartes, except
that I do not agree with him in assigning to the pores the
form of hollow circular canals. So far from there being any-
thing absurd or impossible in all this, it is quite credible that
nature employs an infinite series of different-sized molecules,
endowed with different velocities, to produce her marvellous
effects.
But although we do not understand the cause of elasticity?
we cannot fail to observe that most bodies possess this prop-
erty : it is not unnatural, therefore, to suppose that it is a char-
acteristic also of the small, invisible' particles of the ether. If,
indeed, one looks for some other mode of accounting for the
gradual propagation of light, he will have difficulty in finding
one better adapted than elasticity to explain the fact of uniform
speed. And this appears to be necessary; for if the motion slowed
up as it became distributed through a larger mass of matter,
and receded farther from the source of light, then its high
speed would be lost at great distances. But we suppose the
elasticity to be a property of the ether so that its particles re-
gain their shape with equal rapidity whether they are struck
with a hard or a gentle blow; and thus the rate at which the
light moves remains the same [at all distances from the source].
Nor is it necessary that the ether particles should be arranged
in straight lines, as was the ease with our row of spheres. The
most irregular configuration, provided the particles are in con-
tact with each other, will not prevent their transmitting the
motion and handing it on to the regions in front. It is to be
noted that we have here a law of motion which governs this
kind of propagation, and which is verified by experiment, viz.,
when a sphere such as A, touching several other smaller ones,
18
THE WAVE-THEORY OF LIGHT
CCC, is struck by another sphere, B, in such a way as to make
an impression upon each of its neighbors, it transfers its mo-
tion to them and remains at rest, as does also the sphere B.
Now, without supposing that ether particles are
spherical (for I do not see that this is neces-
sary), we can nevertheless understand that this
Jaw of impulses plays a part in the propaga-
tion of the motion.
Equality of size would appear to be a more
necessary assumption, since otherwise we should
expect the motion to be reflected on passing
from a smaller to a larger particle, following
the laws of percussion which I published some Fig. 3
years ago. Yet, as will appear later, this equal-
ity is necessary not so much to make the propagation of light
possible as to make it easy and intense. Nor does it appear
improbable that the ether particles were made equal for a pur-
pose so important as the transmission of light. This may be
true, at least, in the vast region lying beyond our atmosphere
and serving only to transmit the light of the sun and the
stars.
I have now shown how we may consider light as propagated,
in time, by spherical waves, arid how it is possible that the
speed of propagation should be as great as that demanded by
experiment and by astronomical observation. It must, how-
ever, be added that although the ether particles are supposed
to be in continual motion (and there is much evidence for this
view), the gradual transmission of the waves
is not thus interfered with. For it does not
consist in a translation of these particles, but
merely in a small vibration, which they are
compelled to transmit to their neighbors in
spite of their proper motion and their change
of relative position.
But we must consider, in greater detail,
the origin of these waves and the manner of
their propagation from one point to another.
And, first, it follows from what has already
been said concerning the production of light
that each point of a luminous body, such as
the sun, a candle, or a piece of burning car-
19
MEMOIRS ON
bon, gives rise to its own waves, and is the centre of these
waves. Thus if A, B, and C represent different points in a
candle flame, concentric circles described about each of these
points will represent the waves to which they give rise. And
the same is true for all the points on the surface and within
the flame. But since the disturbances at the centre of these
waves do not follow each other in regular succession, we need
not imagine the waves to follow one another at equal intervals;
and if, in the figure, these waves are equally spaced, it is rather
to indicate the progress which one and the same wave has made
during equal intervals of time than to represent several waves
having their origin at the same point.*
Nor does this enormous number of waves, crossing one an-
other without confusion and without disturbing one another,
appear unreasonable, for it is well known that one and the
same particle of matter is able to transmit several waves com-
ing from different, and even opposite, directions. And this is
true not only in the case where the displacements follow one
another in succession, but also where they are simultaneous.
This is because the motion is propagated gradually. It is
shown by the row of hard and equal spheres above mentioned.
If we allow two equal spheres, A and D, to strike against the
opposite sides of this row at the same instant, they will be ob-
served to rebound each with the same speed that it had before
collision, while all the other spheres remain at rest, although
the motion has twice traversed the entire row. [This evidently
implies that the spheres A and D have equal speeds justi before
OOO00OO
collision.] If these two oppositely directed motions happen
to meet at the middle sphere, B, or at any other sphere, say 0,
it will yield and spring back from both sides, thus transmitting
both motions at the same instant.
* [From this paragraph it would appear that Huygens had no conception
of trains of light-waves. The experimental evidence for thinking that light-
waves travel in trains seems first to have been furnished by Young. See pp.
60, 61 below. If, however, one prefers to interpret the colored rings of Newton
in terms of the wave-theory, this experimental evidence may be ascribed to
Newton.']
30
THE WAVE-THEORY OF LIGHT
Bnt what is strangest and most astonishing of all is that waves
produced by displacements and particles so minute should spread
to distances so immense, as, for instance, from the sun or from
the stars to the earth. For the intensity of these waves must
diminish in proportion to their distance from the origin until
finally each individual wave is of itself unable to produce the
sensation of light. Our astonishment, however, diminishes
when we consider that in the great distance which separates
us from the luminous body there is an infinitude of waves
which, although coming from different parts of the [luminous]
body, are practically compounded into a single wave which thus
acquires sufficient intensity to affect our senses. Thus the in-
finitely great number of waves which at any one instant leave
a fixed star, as large possibly as our sun, unite to form what
is equivalent to one single wave* of intensity sufficient to affect
the eye. Not only so, but each luminous point may send us
thousands of waves in the shortest imaginable time, on account
of r! e rapidity of the blows with which the particles of the
luminous body strike the ether at these points. The effect of
the waves would thus be rendered still more sensible.
In considering the propagation of waves, we must remember
that each particle of the medium through which the wave
spreads doe not communicate its motion only to that neighbor
which lies in the straight line drawn from the luminous point,
but shares it with all the particles which touch it and resist its
motion. Each particle is thus to be considered as the centre
of a wave. Thus if DCF is a wave whose centre and origin
is the luminous point A, a parti-
cle at B, inside the sphere DCF,
will give rise to its own individual
[secondary] wave, KCL, which will
touch the wave DCF in the point
C, at the same instant in which the
principal wave, originating at A,
reaches the position DCF. And it
is clear that there will be only one D
point of the wave KCL which will
touch the wave DCF, viz., the point Fig. 6
which lies in the straight line from A
drawn through B. In like manner, each of the other particles,
bbbb, etc., lying within the sphere DCF, gives rise to its
21
MEMOIRS ON
own wave. The intensity of each of these waves may, how-
ever, be infinitesimal compared with that of DCF, which is
the resultant of all those parts of the other waves which are at
a maximum distance from the centre A.
We see, moreover, that the wave DCF is determined by the
extreme limit to which the motion has travelled from the point
A within a certain interval of time. For there is no motion
beyond this wave, whatever may have been produced inside by
those parts of the secondary waves which do not touch the
sphere DCF. Let no one think this discussion mere hair-
splitting. For, as the sequel will show, this principle, so far
from being an ultra-refinement, is the chief element in the ex-
planation of all the properties of light, including the phe-
nomena of reflection and refraction. This is exactly the point
which seems to have escaped the attention of those who first
took up the study of light-waves, among whom are Mr. Hooke,
in his Miorographia, and Father Pardies, who had undertaken
to explain reflection and refraction on the wave -theory, as I
know from his having shown me a part of a memoir which he
was unable to finish before his death. But the most important
fundamental idea, which consists in the principle I have just
stated, is wanting in his demonstrations. On other points also
his view is different from mine, as will some day appear in case
his writings have been preserved.
Passing now to the properties of light, we observe first that
each part of the wave is propagated in such a way that its ex-
tremities lie always between the same
straight lines drawn from the lumi-
nous point.
For instance, that part of the wave
BGr, whose centre is the luminous
point A, develops into the arc CE,
limited by the straight lines, ABO
and AGE. For although the sec-
ondary waves produced by the par-
ticles lying within the space CAE
may spread to the region outside,
nevertheless they do not combine at
the same instant to produce one single wave limiting the
motion and lying in the circumference CE which is their
common tangent. This explains the fact that light, pro-
22
Rff.fi
THE WAVE-THEORY OF LIGHT
vided its rays are not reflected or refracted, always travels
in straight lines, so that no body is illuminated by it unless
the straight -line path from the source to this body is unob-
structed.
Let us, for instance, consider the aperture BG- as limited by
the opaque bodies BH, GI ; then, as we have just indicated,
the light- waves will always be limited by the straight lines
AC, AE. The secondary waves which spread into the region
outside of ACE have not sufficient intensity to produce the
sensation of light.
Now, however small we may make the opening BG, the cir-
cumstances which compel the light to travel in straight lines
still remain the same ; for this aperture is always sufficiently
large to contain a great number of these exceedingly minute
ether particles. It is thus evident that each particular part of
any wave can advance only along the straight line drawn to it
from the luminous point. And this justifies us in considering
rays of light as straight lines.
From what has been said concerning the small intensity of
the secondary waves, it would appear not to be necessary that
all the ether particles be equal, although such an equality
would favor the propagation of the motion. The effect of
inequality would be to make a particle, in colliding with a
larger one, use up a part of its momentum in an effort to
recover. The secondary waves thus sent backward towards
the luminous point would be unable to produce the sensation
of light, and would not result in a primary wave similar to
CE.
Another and more remarkable property of light is that
when rays come from different, or even opposite, directions
each produces its effect without disturbance from the other.
Thus several observers are able, all at the same time, to
look at different objects through one single opening ; and
two individuals can look into each other's eyes at the same
instant.
If we now recall our explanation of the action of light and of
waves crossing without destroying or interrupting each other,
these effects which we have just described are readily under-
stood, though they are not so easily explained from Descartes'
point of view, viz., that light consists in a continuous [hydro-
static] pressure which produces only a tendency to motion.
23
MEMOIRS ON THE WAVE-THEORY OF LIGHT
For such a pressure cannot, at the same instant, affect bodies
from two opposite sides unless these bodies have some tendency
to approach each other. It is, therefore, impossible to under-
stand how two persons can look each other in the eye or how
one torch can illuminate another.
24
CHAPTER II
ON REFLECTION
HAVING explained the effects produced by light-waves in a
homogeneous medium, we shall next consider what happens
when they impinge upon other bodies. First of all we shall
see how reflection is explained by these same waves and how
the equality of angles fol-
lows as a consequence.
Let AB represent a plane
polished surface of some
metal, glass, or other sub-
stance, which, for the pres-
ent, we shall consider as
perfectly smooth (concern-
ing irregularities which
are unavoidable we shall
have something to say at
the close of this demon-
stration) ; and let the line
AC, inclined to AB, repre-
sent a part of a light-wave whose centre is so far away that this
part AC may be considered as a straight line. It may be men-
tioned here, once for all, that we shall limit our consideration
to a single plane, viz., the plane of the figure, which passes
through the centre of the spherical wave and cuts the plane
AB at right angles.
The region immediately about C on the wave AC will, after
a certain interval of time, reach the point B in the plane AB,
travelling along the straight line CB, which we may think of
as drawn from the source of light and hence drawn perpen-
dicular to AC. Now in this same interval of time the ^egion
about A on the same wave is unable to transmit its entire
motion beyond the plane AB ; it must, therefore, continue its
25
MEMOIRS ON
motion on this side of the plane to a distance equal to CB,
sending out a secondary spherical wave in the manner described
above. This secondary wave is here represented by the circle
SNR, drawn with its centre at A and with its radius AN equal
to CB.
So, also, if we consider in turn the remaining parts H of the
wave AC, it will be seen that they not only reach the surface
AB along the straight lines HK parallel to CB, but they will
produce, at the centres K, their own spherical waves in the
transparent medium. These secondary waves are here repre-
sented by circles whose radii are equal to KM that is, equal
to the prolongations of HK to the straight line BG which
is drawn parallel to AC. But, as is easily seen, all these cir-
cles have a common tangent in the straight line BN, viz.,
the same line which passes through B and is tangent to
the first circle having A as centre and AN, equal to BC, as
radius.
This line BN (lying between B and the point N, the foot of
the perpendicular let fall from A) is the envelope of all these
circles, and marks the limit of the motion produced by the
reflection of the wave AC. It is here that the motion is more
intense than at any other
point, because, as has been
explained, BN is the new
position which the wave
AC has assumed at the in-
stant when the point C has
reached B. For there is
no other line which, like
BN, is a common tangent
to these circles, unless it
be BG, on the other side
of the plane AB. And BGr
will represent the trans-
mitted wave onlv in case
Fig. 7
the motion occurs in a medium which is homogeneous with
that above the plane. If, however, one wishes to see just how
the wave AC has gradually passed into the wave BN, he has
only to use the same figure and draw the straight lines KO
parallel to BN, and the straight lines KL parallel to AC. It is
thus seen that the wave AC, from being a straight line, passes
26
THE WAVE-THEORY OF LIGH
successively into all the broken lines OKL, and reassumes the
form of a single straight line NB.
It is now evident that the angle of reflection is equal to the
angle of incidence. For the right-angled triangles ABC and BNA
have the side AB in common, and the side OB equal to the side
NA, whence it follows that the angles opposite these sides are
equal, and hence also the angles CBA and NAB. But .CB,
perpendicular to CA, is the direction of the incident ray, while
AN, perpendicular to the wave BN, has the direction of the
reflected ray. These rays are, therefore, equally inclined to the
plane AB.
Against this demonstration it may he urged that while BN
is the common tangent of the circular waves in the plane of
this figure, the fact is that these waves are really spherical and
have an infinitely great number of similar tangents, viz., all
straight lines drawn through the point B and lying in the sur-
face of a cone generated by the revolution of a straight line
BN about BA as axis. It remains to be shown, therefore, that
this objection presents no difficulty ; and, incidentally, we shall
see that the incident and reflected rays each lie in one plane
perpendicular to the reflecting plane.
I remark, then, that the wave AC, so long as it is considered
merely a line, can produce no light. For a ray of light, how-
ever slender, must have a finite thickness in order to be visible.
In order, therefore, to represent a wave whose path is along
this ray, it is necessary to replace the straight line AC by a
plane area, as, for instance, by the circle HC in the following
figure, where the luminous point is supposed to be infinitely
distant. From the preceding proof it is easily seen that each
element of area on the wave HC, having reached the plane AB,
will there give rise to its own secondary wave ; and when C
reaches the point B, these will all have a common tangent
plane, viz.. the circle BN equal to CH. This circle will be cut
through the centre and at right angles by the same plane which
thus cuts the circle CH and the ellipse AB.
It is thus seen that the spherical secondary waves can have
no common tangent plane other than BN. In this plane will
be located more of the reflected motion than in any other, and
it will therefore receive the light transmitted from the wave CH.
I have noted in the preceding explanation that the motion of
the wave incident at A is not transmitted beyond the plane AB,
27
MEMOIRS ON
at least not entirely. And here it is necessary to remark that,
although the motion of. the ether may be partly communicated
to the reflecting body, this cannot in the slightest alter the
speed of the propagation of the waves, which determines the
angle of reflection. For, in any one medium, a slight disturb-
ance produces waves which travel with the same speed as those
Fig. 8
due to a very great disturbance, a consequence of that property
of elastic bodies concerning which we have spoken above, viz.,
the time occupied in recovery is the same whether the com-
pression be large or small. In every case of reflection of light
from the surface of any substance whatever the angles of in-
cidence and reflection are therefore equal, even though the
body be of such a nature as to absorb a part of the motion de-
livered .by the incident wave. And, indeed, experiment shows
that among polished bodies there is no exception to this law
of reflection.
"We must emphasize the fact that in our demonstration there
is no need that the reflecting surface be considered a perfectly
smooth plane, as has been assumed by all those who have at-
tempted to explain the phenomena of reflection. All that is
called for is a degree of smoothness such as would be produced
by the particles of the reflecting medium being placed one near
another. These particles are much larger than those of the
ether, as will be shown later when we come to treat of the
transparency and opacity of bodies. Since, now, the surface
consists thus of particles assembled together, the ether par-
ticles being above and smaller, it is evident that one cannot
demonstrate the equality of the angles of incidence and reflec-
tion from the time-worn analogy with that which happens when
28
THE WAVE-THEORY OF LIGHT
a ball is thrown against a wall. By our method, on the other
hand, the fact is explained without difficulty.
Take particles of mercury, for instance, for they are so
small that we must think of the least visible portion of surface
as containing millions, arranged like the grains in a heap of
sand which one has smoothed out as much as possible; this
surface for our purpose is equal to polished glass. And, though
such a surface may be always rough compared with ether par-
ticles, it is evident that the centres of all the secondary waves
of reflection which we have described above lie practically in
one plane. Accordingly, a single tangent comes as near touch-
ing them all as is necessary for the production of light. And
this is all that is required in our demonstration to explain the
equality of angles without allowing the rest of the motion, re-
flected in various directions, to produce any disturbing elfect.
CHAPTER III
OK REFRACTION
IK the same manner that reflection has been explained by
light-waves reflected at the surface of polished bodies, we pro-
pose now to explain transparency and the phenomena of refrac-
tion by means of waves propagated into and through transpar-
ent bodies, whether solids, such as glass, or liquids, such as
water and oils. But, lest the passage of waves into these
bodies appear an unwarranted assumption, I will first show that
this is possible in more ways than one.
Let us imagine that the ether does penetrate any transparent
body, its particles will still be able to transmit the motion of
the waves just as do those of the ether, supposing them each to
be elastic. And this we can easily believe to be the case with
water and other transparent liquids, since they are composed
of discrete particles. But it may appear more difficult in the
case of glass and other bodies that are transparent and hard,
because their solidity would hardly allow that they should take
up any motion except that of their mass as a whole. This,
however, is not necessary, since this solidity is not what it ap-
pears to us to be, for it is more probable that these bodies are
composed of particles which are placed near one another and
bound together by an external pressure due to some other kind
of matter and by irregularity of their own configurations. For
their looseness of structure is seen in the facility with which
they are penetrated by the medium of magnetic vortices and
those which cause gravitation. One cannot go further than to
say that these bodies have a structure similar to that of a sponge,
or of light bread, because heat will melt them and change the
relative positions of their particles. We infer, then, as has
been indicated above, that they are assemblages of particles
touching one another but not forming a continuous' solid.
This being the case, the motion which these particles receive
30
MEMOIRS ON THE WAVE-THEORY OF LIGHT
in the transmission of light is simply communicated from one
to another, while the particles themselves remain tethered in
their own positions and do not become disarranged among
themselves. It is easily possible for this to occur without in
any way affecting the solidity of the structure as seen by us.
By the external pressure of which I have spoken is not to be
understood that of -'the air, which would be quite insufficient,
but that of another and more subtle medium, whose pressure is
exhibited by an experiment which I chanced upon a long while
ago, namely, that water from which the air has been removed
remains suspended in a glass tube open at the lower end, even
though the air may have been exhausted from a vessel enclosing
this tube.
We may thus explain transparency without assuming that
bodies are penetrated by the luminiferous ether or that they
contain pores through which the ether can pass. The fact,
however, is not only that this medium penetrates ordinary
bodies, but that it does so with the utmost ease, as indeed has
already been shown by the experiment of Torricelli which we
have cited above. When the mercury or the water leaves the
upper part of the glass tube, the ether appears at once to take
its place and transmit light. But following is still another
argument for thinking that bodies, not only those which are
transparent, but others also, are easily penetrable.
When light traverses a hollow glass sphere which is com-
pletely closed, it is evident that the sphere is filled with ether,
just as is the space outside the sphere. And this ether, as we
have shown above, consists of particles lying in close contact
with each other. If, now, it were enclosed in the sphere in such
a way that it could not escape through the pores of the glass,
it would be compelled to partake of any motion which one
might impress upon the sphere ; ^consequently nearly the same
force would be required to impress a given speed upon this
sphere, lying upon a horizontal plane, as if it were filled with
water, or possibly mercury. For the resistance which a body
offers to any velocity one may wish to impart to it varies
directly as the quantity of matter which the body contains and
which is compelled to acquire velocity. But the fact is that
the sphere resists the motion only in proportion to the amount
of glass in it. Whence it follows that the ether within is not
enclosed, but flows through the glass with perfect freedom.
31
MEMOIRS ON
Later we shall show, by this same process, that penetrability
may be inferred for opaque bodies also.
A second and more probable explanation of transparency
is to say that the light-waves continue on in the ether which
always fills the interstices, or pores, of transparent bodies.
For since it passes continuously and with ease, it follows that
these pores are always full. Indeed, it may be shown that these
interstices occupy more space than the particles which make
up the body.
Now if it be true, as we have said, that the force required
to impart a given horizontal velocity to a body is proportional
to the mass of the body, and if this force be also proportional
to the weight of the body, as we know by experiment that it
is, then the mass of any body must be ^also proportional to
its weight. Now we know that water weighs only -fa part as
much as an equal volume of mercury, therefore the substance
of the water occupies only -fa part of the space that encloses
its mass. Indeed, it must occupy even a smaller fraction than
this, "because mercury is not so heavy as gold, and gold is a
substance which is not very dense, since the medium of mag-
netic vortices and that which causes gravitation penetrate it
with the utmost ease.
But it may be objected that if water be a substance of such
small density, and if its particles occupy so small a portion of
its apparent volume, it is very remarkable that it should offer
such stubborn resistance to compression ; for it has not been
condensed by any force hitherto employed, and remains per-
fectly liquid while under pressure.
This is, indeed, no small difficulty. But it may nevertheless
be explained by supposing that the very violent and rapid
motion of the subtle medium which keeps water liquid also
sets in motion the particles of which it is composed, and main-
tains this liquid state in spite of any pressure which has hitherto
been applied.
If, now, the structure of transparent bodies be as loose as we
have indicated, we may easily imagine waves penetrating the
ether which fills the interstices between the particles. Not
only so, but we can easily believe that the speed of these waves
inside the body must be a little less on account of the small
detours necessitated by these same particles. I propose to show
that in this varying velocity of light lies the cause of refraction.
32
THE WAVE-THEORY OF LIGHT
I will first indicate a third and last method by which we may
explain transparency, namely, by supposing that the motion of
the light-waves is transmitted equally well by the ether particles
which fill the interstices of the body, and by the particles which
compose the body, the motion being handed on from one to the
other. A little later we shall see how beautifully this hypothesis
explains the double'refraction of certain transparent substances.
Should one object that the particles of ether are much smaller
than those of the transparent body, since the former pass
through the intervals between the latter, and that consequent-
ly they would be able to communicate only a small portion of
their momentum, we may reply that the particles of the body
are composed of other still smaller particles, and that it is these
secondary particles that take up the momentum from those of
the ether.
Moreover, if the particles- of the transparent body are slight-
ly less elastic than are the ether particles, which we may
reasonably suppose, it would still follow that the speed of
the light waves is less inside the body than outside in the
ether.
We have here, what appears to me, the manner in which
light-waves are probably transmitted by transparent bodies.
But there still remains the consideration of opaque bodies and
the difference between these and transparent bodies, a question
all the more interesting in view of the ease with which ether
penetrates all bodies, a fact to which attention has already been
directed, and which might lead one to think that all bodies
should be transparent. For considering the hollow sphere, by
which I have shown the open structure of glass and the ease
with which ether passes through it, one would naturally infer
the same penetrability as a property of metals and all other
substances. Imagine the sphere to be of silver; it would then
certainly contain luminiferous ether, because this substance,
as well as air, would be present in it when the opening in the
sphere was closed up. But when closed and placed upon a
horizontal plane it would resist motion only in proportion to
the amount of silver in it, showing as above that the enclosed
ether does not acquire the motion of the sphere. Silver, there-
fore, like glass, is easily penetrated by ether. In between the"
particles of silver and of all other opaque bodies this substance
is distributed continuously and abundantly ; and, since it can
c 33
MEMOIRS ON
transmit light, we are led to expect that these bodies should be
as transparent as glass, which, however, is not the fact.
How, then, shall we explain their opacity? Are their con-
stituent particles soft and built up of still smaller particles,
and thus able to change shape when they are struck by ether
particles ? Do they thus damp out the motion and stop the
propagation of the light-waves ? This seems hardly possible ;
for if the particles of a metal were soft, how could polished
silver and mercury reflect light so well ? What seems to me
more probable is that metallic bodies, which are almost the
only ones that are really opaque, have interspersed among their
hard particles some which are soft, the former producing reflec-
tion, the latter destroying transparency ; while, on the other
hand, transparent bodies are made up of only hard and elastic
particles, which, together with the ether, propagate light- waves
in the manner already indicated.
We pass now to the explanation of refraction, assuming, as
above, that light-waves pass through transparent substances
arid in them undergo diminution of speed.
The fundamental phenomenon in refraction is the follow-
ing, viz., when any ray of light, AB, travelling in air, strikes
obliquely upon the polished surface of a transparent body, PGy
it undergoes a sudden change of direction at the point of inci-
dence, B ; and this change occurs in such a way that the angle
CBE, which the ray makes with the normal to the surface, is
less than the angle ABD, which the ray in air made with
the same normal. To determine the numerical value of these
angles, describe about the point B a circle cutting the rays AB,
BC. Then the perpendiculars, AD, CE,
let fall from these points of intersection
upon the normal, DE, viz., the sines of
the angles ABD, CBE, bear to one an-
other a certain ratio which, for any
one transparent body, is constant for
all directions of the incident ray. For
glass this ratio is almost exactly f,
while for water it is very nearly f , thus
varying from one transparent body to
another.
Another property, not unlike the preceding, is that the refrac-
tions of rays entering and of rays emerging from a transpar-
34
THE WAVE-THEORY OF LIGHT
M
Fig. 10
ent body are reciprocal. That is to say, if an incident ray, AB,
be refracted by a transparent body into the ray BO, so also will
a ray, CB, in the interior of the body be refracted, on emer-
gence, into the ray BA.
In order to explain these phenomena on our theory, let the
straight line AB Fig. 10, represent the plane surface bounding a
transparentbodyextendingin
a direction between C and N.
By the use of the word
plane we do not mean to
imply a perfectly smooth sur-
face, but merely such a one
as was employed in treating
of reflection, and for the
same reason. Let the line
AC represent a part of a
light- wave whose source is
so distant that this part may
be treated as a straight line.
The region 0, on the wave
AC, will, after a certain in-
terval of time, arrive at the plane AB, along the straight line
CB, which we must think of #s drawn /from the source of
light, and which will, therefore, intersect AC at right angles.
But during this same interval of time the region about A would
have arrived at G, along the straight line AG, equal and parallel
to CB, and, indeed, the whole of the wave AC would have
reached the position GB, provided the transparent body were
capable of transmitting waves as rapidly as the ether. But
suppose that the rate of transmission is less rapid, say one-third
less. Then the motion from the point A will extend into the
transparent body to a distance which is only two-thirds of CB,
while producing its secondary spherical wave as described
above. This wave is represented by the circle SNR, whose cen-
tre is at A and whose radius is equal to f CB. If we consider,
in like manner, the other points H of the wave AC, it will be
seen that, during the same time which C employs in going to
B, these points will not only have reached the surface AB, along
the straight lines HK, parallel to CB, but they will have start-
ed secondary waves into the transparent body from the points
K as centres. These secondary waves are represented by cir-
35
MEMOIRS ON
cles whose radii are respectively equal to f of the distances
KM that is, f of the prolongations of HK to the straight line
BG. If the two transparent media had the same ability to
transmit light, these radii would equal the whole lengths of
the various lines KM.
But all these circles have a common tangent in the line BN,
viz., the same line which we drew from the point B tangent to
the circle SNR first considered. For it is easy to see that all
the other circles from B up to the point of contact, N, touch,
in the same manner, the line BN, where N is also the foot of
the perpendicular let fall from A upon BN.
We may, therefore, say that BN is made up of small arcs of
these circles, and that it marks the limits which the motion
from the wave AC has reached in the transparent medium, and
the region where this motion is much greater than anywhere
else. And, furthermore, that this line, as already indicated, is
the position assumed by the wave AC at the instant when the
region C has reached the point B. For there is no other line
below the plane AB, which, like BN, is a common tangent to
all these secondary waves.
Accordingly, if one wishes to discover through what in-
termediate steps the wave AC reached the position BN, he has
only to draw, in the same figure, the straight lines KO
parallel to BN, and all the lines KL parallel to ,AC. He will
thus see that the wave CA passes from a straight line into the
successive broken lines LKO, reassuming the form of a straight
line in the position BN. From what has preceded this will be
so evident as to need no further explanation.
If, now, using the same
C figure, we draw EAF nor-
mal to the plane AB at the
point A, and draw DA at
right angles to the wave AC,
the incident ray of light will
then be represented by DA;
and AN, which is drawn per-
pendicular to BN, will be
the refracted ray; for these
rays are merely the straight
lines along which the parts
ffiff- 10 of the waves travel.
36
THE WAVE-THEORY OF LIGHT
From the foregoing it is easy to deduce the principal law of
refraction, viz., that the sine of the angle DAE always bears a
constant ratio of the sine of the angle NAF, whatever may be
the direction of the incident ray, and that the ratio is the
same as that which the speed of the waves in the medium on
the side AE bears to their speed on the side AF.
For if we consider AB as the radius of a circle, the sine
of the angle BAG is BC, and the sine of the angle ABN _is
AN. But the angles BAG and DAE are equal; for each is
the complement of CAE. And the angle ABN is equal to
NAF, since each is the complement of BAN". Hence the sine
of the angle DAE is to the sine NAF as BC is to AN. But
the ratio of BO to AN is the same as that of the speeds of
light in the media on the side towards AE and the side tow-
ards AF, respectively ; hence, also, the sine of the angle DAE
bears to the sine of the angle NAF the same ratio as these two
speeds of light.
In order to follow the refracted ray when the light-waves en-
ter a body which transmits them .more rapidly than the body
from which they emerge (say in the ratio of 3 to 2), it is necessary
only to repeat the same construction and the same demonstra-
tion which we have just been
using, substituting, however,
f in place of f . And we find,
by the same logical process,
employing this other figure,
that when the region of
the wave AO reaches the
point B of the surface AB,
the whole wave AC will have
advanced to the position BN,
such that the ratio of BC, per-
pendicular to AC, is to AN,
perpendicular to BN, as 2 is
to 3. The same ratio will
also hold between the sine of the angle EAD and the sine of
the angle FAN.
The reciprocal relations between the refractions of a ray
on entering and on emerging from one and the same me-
dium is thus made evident. If the ray NA is incident upon
the exterior surface AB, and is refracted into AD, then
37
Fig. 11
MEMOIRS ON
the ray DA on emerging from the medium will be refracted
into AN".
We are now able to explain a remarkable phenomenon which
occurs in this refraction. When the incident ray DA exceeds
a certain inclination it loses its ability to pass into the other
medium. Because if the angle DAQ or OBA is such that, in
the triangle AOB, OB is equal to or greater than f of AB, then
AN, being equal to or greater than AB, can no longer form one
side of the triangle ANB. ' Therefore the wave BN does not
exist, and consequently there can be no line AN drawn at right
angles to it. And thus the incident ray DA cannot penetrate
the surface AB.
When the wave-speeds are in the ratio of % to 3, as in the
case of glass and air, which we have considered, the angle
DAQ must exceed 48 11' if the ray DA is to emerge. And
when the ratio of speeds is that of 3 to 4, as is almost exactly
the case in water and air, this angle DAQ must be greater than
41 24'. And this agrees perfectly with experiment.
But one may here ask why no light penetrates the surface,
since the encounter of the wave AC against the surface AB
must produce some motion in the medium on the other side.
The answer is simple, if we recall what has already been said.
For although an infinite number of secondary waves may be
started into the medium on the other side of AB, these waves
at no time have a common tangent* line, either straight or
curved. There is thus no line which marks the limit to which
the wave AC has been transmitted beyond the plane, AB, nor is
there any line in which the motion has been sufficiently con-
centrated to produce light.
In the following manner one may easily recognize the fact
that, when OB is greater than f AB, the waves beyond the
plane AB have no common tangent. About the centres K
describe circles having radii respectively equal to f LB. These
circles will enclose one another and will each pass beyond the
point B.
It is to be noted that just as soon as the angle DAQ becomes
too small to allow the refracted ray DA to pass into, the other
medium, the internal reflection which occurs at the surface AB
increases rapidly in brilliancy, as may be easily shown by means
of a triangular prism. In terms of our theory, we may thus
explain this phenomenon : While the angle DAQ is still large
38
THE WAVE-THEORY OF LIGHT
enough for the ray DA to be transmitted, it is evident that the
light from the wave-front AC will be concentrated into a much
shorter line when it reaches the position BN. It will be seen
also that the wave-front BN grows shorter in proportion as the
angle CBA or DAQ becomes smaller, until finally, when the
limit indicated above is reached, BN is reduced to a point.
That is to say, when the region about C, on the wave AC,
reaches B, the wave BN", which is the wave AC after trans-
mission, is entirely compressed into this same point B ; and, in
like manner, when the region about H has reached the point K
the part AH is completely reduced to this same point K. It
follows, therefore, that in proportion as the direction of propa-
gation of the wave AC happens to coincide with the surface AB,
so will be the quantity of motion along this surface.
Now this motion must necessarily spread into the transparent
body and also greatly reinforce the secondary waves which pro-
duce internal reflection at the face AB, according to the laws
of this reflection explained above.
And since a small diminution in the angle of incidence is
sufficient to reduce the wave -front BN from a fairly large
quantity to zero (for if this angle in the case of glass be
49 11', the angle BAN amounts to as much as 11 21'; but if
this same angle DAQ be diminished by one degree only, the
angle BAN becomes zero and the wave-front BN is reduced to
a point), it follows that the internal reflection occurs suddenly,
passing from comparative darkness to brilliancy at the instant
when the angle of incidence assumes a value which no longer
permits refraction.
Now as to ordinary external reflection, i. e., reflection which
occurs when the angle of incidence DAQ is still large enough
to allow the refracted ray to pass through the face AB, this
reflection must be from the particles which bound the trans-
parent body on the outside, apparently from particles of air
and from others which are mixed with, but are larger than, the
ether particles.
On the other hand, external reflection from bodies is pro-
duced by the particles which compose these bodies, and which
are larger than those of the ether, since the ether flows through
the interstices of the body.
It must be confessed that we here find difficulty in explain-
ing the experimental fact that internal reflection occurs even
39
MEMOIRS ON
where the particles of air can cut no figure, as, for instance, in
vessels and tubes from which the air has been exhausted.
Experiment shows further that these two reflections are of
almost equal intensity, and that in various transparent bodies
this intensity increases directly as the refractive index. Thus
we see that reflection from glass is stronger than that from
water, while in turn that from diamond is stronger than that
from glass.
I shall conclude this theory of refraction by demonstrating a
remarkable proposition depending upon it, namely, that when
a ray of light passes from one point to another, the two points
lying in different media, refraction at the bounding surface
occurs in such a way as to make the time required the least
possible ; and exactly the same thing occurs in reflection at a
plane surface. M. Fermat discovered this property of refrac-
tion, believing with us and in opposition to M. Descartes that
light travels more slowly through glass than through air.
But, besides this, he assumed what we have just proved from
the fact that the velocities in the two media are different, viz.,
that the ratio of the sines is a constant ; or, what amounts to
the same thing, he assumed, besides the different velocities,
that the time employed was a minimum ; and from this he
derived the constancy of the sine ratio.
His* demonstration, which may be found in his works and in
the correspondence of M. Descartes, is very long. It is for this
reason that I here offer a simpler and easier one.
Let KF represent a plane surface ; imagine the point A in
the medium through which the light travels more rapidly, say
air ; the point C lies in anoth-
er, say water, in which the speed
of light is less. Let us sup-
pose that a ray passes from A,
through B, to 0, suffering re-
fraction at B, according to the
law above demonstrated ; or,
what is the same thing, having
drawn PBQ perpendicular to
the surface, the sine of the
angle ABP is to the sine of the
angle CBQ in the same ratio as
the speed of light in the medium
40
THE WAVE-THEORY OF LIGHT
containing A is to the speed in the medium containing C. It
remains to show that the time required for the light to traverse
AB and BC taken together is the least possible. Let us assume
that the light takes some other path, say AF, FC, where F, the
point at which refraction occurs, is more distant than B from A.
Draw AO perpendicular to AB, and FO parallel to BA ; BH
perpendicular to FO, and FG perpendicular to BC. Since, now,
the angle HBF is equal to PBA, and the angle BFG is equal to
QBC, it follows that the sine of the angle HBF will bear to the
sine of BFG the same ratio as the speed of light in the medium
A bears to the speed in the medium 0. But if we consider BF
the radius of a circle, then sines are represented by the lines
HF, BG. Accordingly, the lines HF, BG are in the ratio of
these speeds. If, therefore, we imagine OF to be the incident
ray, the time of passage from H to F will be the same as the
time of passage from B to G in the medium C.
But the time from A to B is equal to the time from to H.
Hence the time from to F is the same as the time from A to
G, via B. Again, the time along FC is greater than the time
along GC ; and hence the time along the route OFC is greater
than that along the path ABC. But AF is greater than OF;
hence, a fortiorij the time along AFC is greater than that
along ABC.
Let us now assume that the ray passes from A to C by the
route AK, KC, the point of refraction, K, being nearer to A
than is B. Draw CN perpendicular to BC ; KN" parallel to BC ;
BM perpendicular to KN ; and KL perpendicular to BA.
Here BL and KM represent the sines of the angles BKL and
KBM that is, the angles PBA and QBC ; and hence they are
in the same ratio as the speeds of light in the media A and C
respectively. The time, therefore, from L to B is equal to the
time from K to M ; and, since the time from B to C is equal to
the time from M to N, the time by the path LBC is the same
as the time via KMN. But the time from A to K is greater
than the time from A to L, and, therefore, the time by the
route AKN is greater than the route ABC.
Not only so, but since KC is greater than KN, the time by
the path AKC will be so much the greater than by the path
ABC. Hence follows that which was to be proved, namely,
that the time along the path ABC is the least possible.
41
MEMOIRS ON
BIOGRAPHICAL SKETCH
WHILE there are no sharp lines in nature, there is a very
true sense in which the year 1642, marking the death of
Galileo and the birth of Newton, serves as a line of demar-
cation between the foundation and the superstructure of mod-
ern physics.
Galileo, by his careful study of gravitation, by his clear grasp
of force as determining acceleration, by his careful search after
causes and their respective effects, by his profound faith in
experiment, had more than cleared the ground for the build-
ers of modern physics. The rapid rise of this structure at
the hands of Newton and his brilliant contemporaries, Boyle,
Leibnitz, Bonier, Du Fay, Bradley, and Hooke, marks a dis-
tinctly modern era compared with that of Galileo.
The work of Christiaan Huygens, the "Dutch Archimedes,"
occupies, as regards both time and character, a position inter-
mediate between these two periods. He was born at The
Hague in 1629, and died there in 1695. A splendid ancestry,
three years of university training at Leyden and Breda, much
travel, and a rare group of associates, combined to give him
an education which left little to be desired. Most of his life
was spent in Holland, but for the fifteen years following 1666
he lived and worked in Paris, where he was the guest of Louis
XIV. and the then recently founded French Academy of Sci-
ences. This was for him a happy period of great activity, and
it was only in anticipation of the revocation of the Edict of
Nantes, in 1685, that he returned to his own country, where
in private retirement and study he spent most of his remain-
ing years.
His intellectual achievements fall into three not very dis-
tinct departments of science namely, mathematics, physics,
and physical astronomy. In mathematics, his chief accom-
plishments refer to
(a) The quadrature of conies.
(#) The theory of probabilities.
(c) A discussion of the evolutes and involutes of curves and
the introduction of the idea of the envelope of a moving
straight line.
42
THE WAVE-THEORY OF LIGHT
In physics he gave
(a) A general solution of the problem of the Compound Pen-
dulum, and in the demonstration enunciated the very
general principle that in any mechanical system acting
under gravity the centre of gravity can never rise to a
point higher than that from which it fell a principle
which we now recognize as a special case of the law
that the potential energy of any mechanical system tends
to a minimum.
(b) The invention of the pendulum clock and its application
to the measurement of gravity at various points on the
earth's surface.
(c) An accurate description of the behavior of bodies in
collision.
(d) The laws governing centrifugal forces.
(e) The undulatory theory of light arid its application to
the explanation of reflection, ordinary refraction, and
double refraction.
Among his contributions to physical astronomy are
(a) The construction of the first powerful telescope of the
refracting kind.
(b) The discovery of the rings of Saturn and its sixth
satellite.
(c) Improvements in the methods of grinding lenses and the
addition of a tube to the object-glass and another to
the eye-piece of the aerial telescope.
All his mechanical inventions are characterized by practica-
bility, and all his intellectual work by clearness and elegance.
Those who wish a more detailed account of his activity will
find it in the superb edition of his works* recently published
by the Societe Hollandaise des Sciences, while that delightful
sketch of his life and work given 'by Dr. Bosschaf should be
read by every one.
* (Euvres Computes de Christiaan Huygens (La Haye : Martin us Nijhoff,
1888 to 19).
f Bosscha : Christiaan Huygens, Rede am 200sten Ged^chtnistage seines
Lebensende. Ubersetzt von Engelmann. (Eugelmann : Leipzig, 1895),
pp. 77.
43
>*-' ov
LIBRA
Oc
ON THE THEORY OF LIGHT AND
COLOES
From the Philosophical Transactions for 1802, p. 12.
AN ACCOUNT OF SOME CASES OF THE
PRODUCTION OF COLORS NOT
HITHERTO DESCRIBED
From the Philosophical Transactions for 1802. p. 387.
EXPERIMENTS AND CALCULATIONS
RELATIVE TO PHYSICAL OPTICS
From the Philosophical Transactions for 1804.
.BY
THOMAS YOUNG.
These three papers are reprinted in Young's Miscellaneous Works, vol. i.,
and also in his Lectures on Natural Philosophy and Mechanical Arts.
45
CONTENTS
PAGE
General Statement of Wave - Theory, including the Principle of Inter-
ference 47
Diffraction in che Shadow of a Narrow Obstacle , 62
Observations on the Interference Bands in the Shadow of a Narrow
Obstacle. . . 68
46
ON THE THEORY OF LIGHT AND
COLORS*
A BAKERIAN LECTURE
Read November 12, 1801.
ALTHOUGH the invention of plausible hypotheses, indepen-
dent of any connection with experimental observations, can be of
very little use in the promotion of natural knowledge, yet the
discovery of simple and uniform principles, by .which a great
number of apparently heterogeneous phenomena are reduced
to coherent and universal laws, must ever be allowed to be of
considerable importance towards the improvement of the hu-
man intellect.
The object of the present dissertation is not so much to pro-
pose any opinions which are absolutely new, as to refer some
theories, which have been already advanced, to their original
inventors, to support them by additional evidence, and to apply
them to a great number of diversified facts, which have hither-
to been buried in obscurity. Nor is it absolutely necessary in
this instance to produce a single new experiment ; for of ex-
periments there is already an ample store, which are so much
the more unexceptionable as they must have been conducted
without the least partiality for the system by which they will
be explained ; yet some facts, hitherto unobserved, will be
brought forward, in order to show the perfect agreement of
that system with the multifarious phenomena of nature.
The optical observations of Newton are yet unrivalled ; and,
excepting some casual inaccuracies, they only rise in our esti-
mation as we compare them with later attempts to improve on
* From the Philosophical Transactions for 1802, p. 12
47
MEMOIRS ON
them. A further consideration of the colors of thin plates, as
they are described in the second book of Newton's Optics, has
converted that prepossession which I before entertained for the
undulatory system of light into a very strong conviction of its
truth and sufficiency, a conviction which has been since most
strikingly confirmed by an analysis of the colors of striated
substances. The phenomena of thin plates are indeed so sin-
gular that their general complexion is not without great diffi-
culty reconcilable to any theory, however complicated, that
has hitherto been applied to them ; and some of the principal
circumstances have never been explained by the most gratui-
tous assumptions; but it will appear that the minutest particu-
lars of these phenomena are not only perfectly consistent with
the theory which will now be detailed, but that they are all the
necessary consequences of that theory, without any auxiliary
suppositions ; and this by inferences so simple that they be-
come particular corollaries, which scarcely require a distinct
enumeration.
A more extensive examination of Newton's various writings
has shown me that he was in reality the first that suggested
such a theory as I shall endeavor to maintain ; that his own
opinions varied less from this theory than is now almost univer-
sally supposed ; and that a variety of arguments have been ad-
vanced, as if to confute him, which may be found nearly in a
similar form in his own works ; and this by no less a math-
ematician than Leonard Euler, whose system of light, as
far as it is worthy of notice, either was, or might have been,
wholly borrowed from Newton, Hooke, Hnygens, and Male-
branche.
Those who are attached, as they may be with the greatest
justice, to every doctrine which is stamped with the Newtonian
approbation, will probably be disposed to bestow on these con-
siderations so much the more of their attention, as they appear
to coincide more nearly with Newton's own opinions. For
this reason, after having briefly stated each particular po-
sition of my theory, I shall collect, from Newton's various
writings, such passages as seem to be the most favorable to its
admission; and although I shall quote some papers which may
be thought to have been partly retracted at the publication of
the Optics, yet I shall borrow nothing from them that can be
supposed to militate against his rnaturer judgment.
48
THE WAVE-THEORY OF LIGHT
HYPOTHESIS I
A luminiferous ether pervades the universe, rare and elastic
in a high degree.
PASSAGES FROM KEWTOtf
"The hypothesis certainly has a much greater affinity with
his own/' that is, Dr. Hooke's, " hypothesis than he seems
to be aware of; the vibrations of the ether being as useful and
necessary in this as in his." Phil. Trans., vol. vii., p. 5087.
Abr., vol. i., p. 145. Nov., 1672.
" To proceed to the hypothesis : first, it is to be supposed
therein that there is an ethereal medium, much of the same
constitution with air, but far rarer, subtler, and more strongly
elastic. It is not to be supposed that this medium is one uni-
form matter, but compounded, partly of the main phlegmatic
body of ether, partly of other various ethereal spirits, much
after the manner that air is compounded of the phlegmatic
body of air, intermixed with various vapors and exhalations :
for the electric and magnetic effluvia and gravitating princi-
ple seem to argue such variety." BIRCH, Hist, of R. 8., vol.
iii., p. 249, Dec., 1675.
" Is not the heat (of the warm room) conveyed through
the vacuum by the vibrations of a much subtler 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
reflection and easy transmission ? And do not the vibrations
of this medium in hot bodies contribute to the intenseness and
duration of their heat ? And do not hot bodies communicate
their heat to contiguous cold ones by the vibrations of this me-
dium propagated from them into the cold ones ? And is not this
medium exceedingly more rare and subtle than the air, and ex-
ceedingly more elastic and active ? And doth it not readily per-
vade all bodies ? And is it not, by its elastic force, expanded
through all the heavens ? May not planets and comets, and
all the gross bodies, perform their motions in this ethereal me-
dium ? And may not its resistance be so small as to be incon-
siderable ? For instance, if this ether (for so I will call it)
should be supposed 700,000 times more elastic than our air,
and above 700,000 times more rare, its resistance would be
D 49
MEMOIRS ON
about 600,000,000 times less than that of water. And so small
a resistance would scarce make any sensible alteration in the
motions of the planets in ten thousand years. If any one
would ask how a medium can be so rare, let him tell me how
an electric body can by friction emit an exhalation so rare and
subtle, and yet so potent? And how the effluvia of a mag-
net can pass through a plate of glass without resistance, and
yet turn a magnetic needle beyond the glass ?" Optics,
Qu. 18, 22.
HYPOTHESIS II
Undulations are excited in this ether 'whenever a body becomes
luminous.
Scholium. I use the word undulation in preference to vi-
bration, because vibration is generally understood as implying
a motion which is continued alternately backward and for-
ward by a combination of the momentum of the body with an
accelerating force, and which is naturally more or less perma-
nent; but an undulation is supposed to consist in vibratory
motion transmitted successively through different parts of a
medium without any tendency in each particle to continue its
motion, except in consequence of the transmission of succeeding
undulations from a distinct vibrating body; as in the air the
vibrations of a chord produce the undulations constituting
sound.
PASSAGES FROM NEWT02ST
" Were I to assume an hypothesis, it should be this, if pro-
pounded more generally, so as not to determine what light is
further than that it is something or other capable of e-xciting
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 invented." BIRCH, Hist, of R. S., vol. iii., p.
249. Dec., 1675.
" In the second place, it is to be supposed that the ether is a
vibrating medium like air, only the vibrations far more swift
and minute ; those of air, made by a man's ordinary voice,
succeeding one another at more than half a foot or a foot dis-
tance, but those of ether at a less distance than the hundred-
thousandth part of an inch. And as in air the vibrations are
50
THE WAVE-THEORY OF LI
some larger than others, but yet all equally swift (for in a ring
of bells the sound of every tone is heard at two or three miles
distance in the same order that the bells are struck), so, I sup-
pose, the ethereal vibrations differ in bigness, but not in swift-
ness. Kow, these vibrations, besides their use in reflection and
refraction, may be supposed the chief means by which the parts
of fermenting or putrefying substances, fluid liquors, or melted,
burning, or other hot bodies, continue in motion." BIRCH,
Hist, of R. S., vol. iii., p. 251, Dec., 1675.
"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 refracting or
reflecting medium ? 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 succes-
sively, do not they put them into the fits of easy reflection and
easy transmission described above ?" Optics, Qu. 17.
"Light is in fits of easy reflection and easy transmission
before its incidence on transparent bodies. And probably it is
put into such fits at its first emission from luminous bodies, and
continues in them during all its progress/' Optics, Book ii.,
part iii., prop. 13.
HYPOTHESIS III
The sensation of different colors depends on the different
frequency of vibrations excited by light in the retina.
PASSAGES FROM NEWTON
"The objector's hypothesis, as to the fundamental part of it,
is not against me. That fundamental supposition is, that the
parts of bodies, when briskly agitated, do excite vibrations in
the ether, which are propagated every way from those bodies in
straight lines, and cause a sensation of light by beating and
dashing against the bottom of the eye, something after the
manner that vibrations in the air cause a sensation of sound
by beating against the organs of hearing. Now, the most free
and natural application of this hypothesis to the solution of
phenomena I take to be this that the agitated parts of bodies,
according to their several sizes, figures, and motions, do excite
vibrations in the ether of various depths or bignesses, which,
51
MEMOIRS ON
being promiscuously propagated through that medium to our
eyes, effect in us a sensation of light of a white color; but if
by any means those of unequal bignesses be separated from one
another, the largest beget a sensation of a red color; the least,
or shortest, of a deep violet, and the intermediate ones of inter-
mediate colors ; much after the manner that bodies, according
to their several sizes, shapes, and motions, excite vibrations in
the air of various bignesses, which, according to those bignesses,
make several tones in sound : that the largest vibrations are
best able to overcome the resistance of a refracting superficies,
and so break through it with least refraction ; whence the vi-
brations of several bignesses that is, the rays of several colors,
which are blended together in light must be parted from one
another by refraction, and so cause the phenomena of prisms
and other refracting substances ; and that it depends on the
thickness of a thin transparent plate or bubble whether a vi-
bration shall be reflected at its further superficies or transmit-
ted ; so that, according to the number of vibrations interced-
ing the two superficies, they may be reflected or transmitted for
many successive thicknesses. And since the vibrations which
make blue and violet are supposed shorter than those which
make red and yellow, they must be reflected at a less thickness
of the plate, which is sufficient to explicate all the ordinary
phenomena of those plates or bubbles, and also of all natural
bodies, whose parts are like so many fragments of such plates.
These seem to be the most plain, genuine, and necessary con-
ditions of this hypothesis; and they agree so justly with my
theory that if the animadversor think fit to apply them, he
need not, on that account, apprehend a divorce from it ; but
yet, how he will defend it from other difficulties I know not."-
Phil Trans., vol. vii., p. 5088. Abr., vol. i., p. 145. Nov., 1672.
"To explain colors, I suppose that as bodies of various
sizes, densities, or sensations do by percussion or other action
excite sounds of various tones, and consequently vibrations in
the air of different bigness, so the rays of light, by impinging
on the stiff refracting superficies, excite vibrations in the
ether of various bigness, the biggest, strongest, or most po-
tent rays, the largest vibrations ; and others shorter, according
to their bigness, strength, or power : and therefore the ends of
the capillamenta of the optic nerve, which pave or face the ret-
ina, being such refracting superficies, when the rays impinge
52
THE WAVE-THEORY OF LIGHT
upon them, they must there excite these vibrations, which vi-
brations (like those of sound in a trunk or trumpet) will run
along the aqueous pores or crystalline pith of the capillamenta,
through the optic nerves, into the sensorium ; and there, I
suppose, affect the sense with various colors, according to their
bigness and mixture ; the biggest with the strongest colors,
reds and yellows ; the least with the weakest blues and violets ;
the middle with green, and a confusion of all with white
much after the manner that, in the sense of hearing, nature
makes use of aerial vibrations of several bignesses to generate
sounds of divers tones, for the analogy of nature is to be ob-
served." BIRCH, Hist, of R. 8., vol. iii., p. 262, Dec., 1675.
' ' Considering the lastingness of the motions excited in the bot-
tom of the eye by light, are they not of a vibrating nature ? Do
not the most refrangible rays excite the shortest vibrations, the
least refrangible the largest ? May not the harmony and dis-
cord of colors arise from the proportions of the vibrations
propagated through the fibres of the optic nerve into the brain,
as the harmony and discord of sounds arise from the propor-
tions of the vibrations of the air ?" Optics, Qu. 16, 13, 14.
[Scholium omitted.]
HYPOTHESIS IV
All material bodies have an attraction for the ethereal medium,
by means of which it is accumulated within their substance,
and for a small distance around them, in a state of greater
density but not of greater elasticity.
It has been shown that the three former hypotheses, which
may be called essential, are literally parts of the more compli-
cated Newtonian system. This fourth hypothesis differs per-
haps, in some degree from any that have been proposed by
former authors, and is diametrically opposite to that of New-
ton ; but both being in themselves equally probable, the op-
position is merely accidental, and it is only to be inquired
which is the best capable of explaining the phenomena. Other
suppositions might perhaps be substituted for this, and there-
fore I do not consider it as fundamental, yet it appears to be
the simplest and best of any that have occurred to me.
53
OK THli
UNIVERSIT
MEMOIRS ON
PROPOSITION I
All impulses are propagated in a homogeneous elastic medium
with an equable velocity.
Every experiment relative to sound coincides with the ob-
servation already quoted from Newton, that all undulations
are propagated through the air with equal velocity ; and this is
further confirmed by calculations. (LAGRANGE, Misc. Taur.,
vol. i., p. 91. Also, -much more concisely, in my syllabus of
a course of lectures on Natural and Experimental Philosophy,
about to be published. Art. 289.) If the impulse be so
great as materially to disturb the density of the medium, it
will be no longer homogeneous; but, as far as concerns our
senses, the quantity of motion may be considered as infinitely
small. It is surprising that Euler, although aware of the mat-
ter of fact, should still have maintained that the more frequent
undulations are more rapidly propagated. (Tlieor. mus. and
Conject. phys.} It is possible that the actual velocity of the
particles of the luminiferous ether may bear a much less pro-
portion to the velocity of the undulations than in sound, for
light maybe excited by the motion of a body moving at the rate
of only one mile in the time that light moves a hundred millions.
Scholium 1. It has been demonstrated that in different me-
diums the velocity varies in the snbduplicate ratio of the force
directly and of the density inversely. (Misc. Taur., vol. L,
p. 91. Young's Syllabus. Art. 294.)
Scholium 2. It is obvious, from the phenomena of elastic
bodies and of sounds, that the undulations may cross each
other without interruption ; but there is no necessity that the
various colors of white light should intermix their undula-
tions, for, supposing the vibrations of the retina to continue
but a five-hundredth of a second after their excitement, a mill-
ion undulations of each of a million colors may arrive in dis-
tinct succession within this interval of time, and produce the
same sensible effect as if all the colors arrived precisely at the
same instant.
PROPOSITION II
An undulation conceived to originate from the vibration of a
single particle must expand through a homogeneous medium
54
THE WAVE-THEORY OF LIGHT
in a spherical form, but with different quantities of motion
in different parts.
For, since every impulse, considered as positive or negative,
is propagated with a constant velocity, each part of the undula-
tion must in equal times have passed through equal distances
from the vibrating-point. And, supposing the vibrating particle,
in the course of its motion, to proceed forward to a small dis-
tance in a given direction, the principal strength of the undula-
tion will naturally be straight before it ; behind it the motion
will be equal in a contrary direction ; and at right angles to
the line of vibration the undulation will be evanescent.
Now, in order that such an undulation may continue its
progress to any considerable distance, there must be in each
part of it a tendency to preserve its own motion in a right line
from the centre ; for if the excess of force at any part were
communicated to the neighboring particles, there can be no
reason why it should not very soon be equalized throughout,
or, in other words, become wholly extinct, since the motions in
contrary directions would naturally destroy each other. The
origin of sound from the vibration of a chord is evidently of
this nature ; on the contrary, in a circular wave of water every
part is at the same instant either elevated or depressed. It may
be difficult to show mathematically the mode in which this in-
equality of force is preserved, but the inference from the mat-
ter of fact appears to be unavoidable ; and while the science of
hydrodynamics is so imperfect that we cannot even solve the
simple problem of the time required to empty a vessel by ;i
given aperture, it cannot be expected that we should be able
to account perfectly for so complicated a series of phenomena
as those of elastic fluids. The theory of Huygens, indeed, ex-
plains the circumstance in a manner tolerably satisfactory. He
supposes every particle of the medium to propagate a distinct
undulation in all directions, and that the general effect is only
perceptible where a portion of each undulation conspires in
direction at the same instant ; and it is easy to show that sucli
a general undulation would in all cases proceed rectilinearly,
with proportionate force ; but, upon this supposition, it seems
to follow that a greater quantity of force must be lost by the
divergence of the partial undulations than appears to be con-
sistent with the propagation of the effect to any considerable
55
MEMOIRS ON
distance ; yet it is obvious that some such limitation of the
motion must naturally be expected to take place, for if the in-
tensity of the motion of any particular part, instead of co^itinu-
ing to be propagated straight forward, were supposed to affect
the intensity of a neighboring part of the undulation, an im-
pulse must then have travelled from an internal to an external
circle in an oblique direction, in the same time as in the direc-
tion of the radius, and consequently with a greater velocity,
against the first proposition. In the case of water the velocity
is by no means so rigidly limited as in that of an elastic medium.
Yet it is not necessary to suppose, nor is it indeed probable, that
there is absolutely not the least lateral communication of the
force of the undulation, but that, in highly elastic mediums,
this communication is almost insensible. In the air, if a chord
be perfectly insulated so as to propagate exactly such vibra-
tions as have been described, they will, in fact, be much less
forcible than if the chord be placed in the neighborhood of a
sounding-board, and probably in some measure because of this
lateral communication of motions of an opposite tendency.
And the different intensity of different parts of the same cir-
cular undulation may be observed by holding a common tun-
ing-fork at arm's-length, while sounding, and turning it, from
a plane directed to the ear, into a position perpendicular to
that plane.
PROPOSITION III
A portion of a spherical undulation, admitted through an
aperture into a quiescent medium, toill proceed to be further
propagated rectilinearly in concentric superficies, terminated
laterally by weak and irregular portions of newly diverging
undulations.
At the instant of admission the circumference of each of
the undulations may be supposed to generate a partial undula-
tion, filling up the nascent angle between the radii and the sur-
face terminating the medium ; but no sensible addition will be
made to its strength by a divergence of motion from any other
parts of the undulation, for want of a coincidence in time, as
has already been explained with respect to the various force of
a spherical undulation. If, indeed, the aperture bear but a small
proportion to the breadth of an undulation, the newly gener-
56
THE WAVE-THEORY OF LIGHT
ated undulation may nearly absorb the whole force of the por-
tion admitted ; and this is the case considered by Newton in
the Principia. But no experiment can be made under these
circumstanced with light, on account of the minuteness of its
undulations and the interference of inflection; and yet some
faint radiations do actually diverge beyond any probable lim-
its of inflection, rendering the margin of the aperture distinctly
visible in all directions. These are attributed by Newton to
some unknown cause, distinct from inflection (Optics, Book
iii., obs. 5)/and they fully answer the description of this
proposition.
Let the concentric lines in Fig. 13 represent the con-
temporaneous situation of similar parts of a number of succes-
sive undulations diverging from the point A; they will also
represent the successive situations of each individual undula-
tion: let the force of each undulation be represented by the
breadth of the line, and let the cone of light ABO be admitted
through the aperture BO; then the principal undulations will
proceed in a rectilinear direction towards GrH, and the faint
radiations on each side will diverge from B
and as centres, without receiving any ad-
ditional force from any intermediate point
D of the undulation, on account of the in-
equality of the lines DE and DF. But if
we allow some little lateral divergence from
the extremities of the undulations, it must
diminish their force, without adding materi-
ally to that of the dissipated light ; and their -
termination, instead of the right line BG,
will assume the form OH, since the loss of
force must be more considerable near to
than at greater distances. This line corre-
sponds with the boundary of the shadow
in Newton's first observation, Fig. 13 ; and
it is much more probable that such a dissi-
pation of light was the cause of the increase
of the shadow in that observation than that
it was owing to the action of the inflecting
atmosphere, which must have extended a
thirtieth of an inch each way in order to pro-
duce it ; especially when it is considered that
57
Fig. 13
MEMOIRS ON
the shadow was not diminished by surrounding the hair with
a denser medium than air, which must in all probability have
weakened and contracted its inflecting atmosphere. In other
circumstances the lateral divergence might appear to increase,
instead of diminishing, the breadth of the beam.
As the subject of this proposition has always been esteemed
the most difficult part of the undulatory system, it will be
proper to examine here the objections which Newton has
grounded upon it.
''To me the fundamental supposition itself seems impossi-
ble namely, that the waves or vibrations of any fluid can, like
the rays of light, be propagated in straight lines, without a
continual and very extravagant spreading and bending every
way into the quiescent medium, where they are terminated by
it. I mistake if there be not both experiment and demonstra-
tion to the contrary." Phil. Trans., vol. vii., p. 5089. Abr.,
vol. L, p. 146. Nov. 1672.
" Motus omnis per flu id urn. propagatus divergit a recto
tramite in spatia immota.",
"Quoniam medium ibi," in the middle of an undulation ad-
mitted, " densius est, quam in spatiis hinc inde, dilatabit sese
tarn versus spatia utrinque sita, quam versus pulsum rariora
intervalla ; eoque pacto pulsns eadem fere celeritate sese in
medii partes quiescentes hinc inde relaxare debent ; ideoqne
spatium totum occupabunt Hoc experimur in sonis." Prin-
cip., lib. ii., prop. 42.
"Are not all hypotheses erroneous in which light is sup-
posed to consist in pression or motion propagated through a
fluid medium ? If it consisted in pression or motion, propa-
gated either in an instant, or in time, it would bend into the
shadow. For pression or motion cannot be propagated in a
fluid in right lines beyond an obstacle which stops part of the
motion, but will bend and spread every way into the quiescent
medium which lies beyond the obstacle. The waves on the
surface of stagnating water passing by the sides of a broad ob-
stacle which stops part of them, bend afterwards, and dilate
themselves gradually into the quiet water behind the obstacle.
The waves, pulses, or vibrations of the air, wherein sounds
consist, bend manifestly, though not so much as the waves of
water. For a bell or a cannon may be heard beyond a hill
which intercepts the sight of the sounding body ; and sounds
58
THE WAVE-THEORY OF LIGHT
are propagated as readily through crooked pipes as straight
ones. Bat light is never known to follow crooked passages
nor to bend into the shadow. For the fixed stars, by the inter-
position of any of the planets, cease to be seen. And so do
the parts of the sun by the interposition of the moon, Mer-
cury, or Venus. The rays which pass very near to the edges
of any body are bent a little by the action of the body ; but
this bending is not towards but from the shadow, and is per-
formed only in the passage of the ray by the body, and at a
very small distance from it. So soon as the ray is past the
body it goes right on." Optics, Qu. 28.
Now the proposition quoted from the Principia does not di-
rectly contradict this proposition ; for it does not assert that
such a motion must diverge equally in all directions ; neither
can it with truth be maintained that the parts of an elastic
medium communicating any motion must propagate that mo-
tion equally in all directions. All that can be inferred by rea-
soning is that the marginal parts of the undulation must be
somewhat weakened and that there must be a faint divergence
in every direction ; but whether either of these effects might
be of sufficient magnitude to be sensible could not have been
inferred from argument, if the affirmative had not been ren-
dered probable by experiment.
As to the analogy with other fluids, the most natural infer-
ence from it is this : "The waves of the air, wherein sounds
consist, bend manifestly, though not so much as the waves of
water "; water being an inelastic and air a moderately elastic
medium ; but ether being most highly elastic, its waves bend
very far less than those of the air, and therefore almost imper-
ceptibly. Sounds are propagated through crooked passages,
because their sides are capable of reflecting sound, just as light
would be propagated through a bent tube, if perfectly polished
within.
The light of a star is by far too weak to produce, by its faint
divergence, any visible illumination of the margin of a planet
eclipsing it ; and the interception of the sun's light by the
moon is as foreign to the question as the statement of inflec-
tion is inaccurate.
To the argument adduced by Huygens in favor of the rec-
tilinear propagation of undulations Newton has made no reply;
perhaps because of his own misconception of the nature of the
59
MEMOIRS ON
motions of elastic mediums, as dependent on a peculiar law of
vibration, which has been corrected by later mathematicians.
On the whole, it is presumed that this proposition may be
safely admitted as perfectly consistent with analogy and with
experiment.
PROPOSITION IV
When an undulation arrives at a surface which is the limit
of mediums of different densities, a partial reflection takes
place proportionate in force to the difference of the densities.
This may be illustrated, if not demonstrated, by the analogy
of elastic bodies of different sizes. " If a smaller elastic body
strikes against a larger one, it is well known that the smaller is
reflected more or less powerfully, according to the difference of
their magnitudes : thus, there is always a reflection when the
rays of light pass from a rarer to a denser stratum of ether ;
and frequently an echo when a sound strikes against a cloud.
A greater body striking a smaller one propels it, without losing
all its motion : thus, the particles of a denser stratum of ether
do not impart the whole of their motion to a rarer, but, in their
effort to proceed, they are recalled by the attraction of the
refracting substance with equal force ; and thus a reflection is
always secondarily produced when the rays of light pass from
a denser to a rarer stratum." But it is not absolutely necessary
to suppose an attraction in the latter case, since the effort
to proceed would be propagated backward without it, and the
undulation would be reversed, a rarefaction returning in place
of a condensation ; and this will perhaps be found most con-
sistent with the phenomena.
[Propositions F., VI., and VII. omitted.]
PROPOSITION VIII
When two undulations, from different origins, coincide
either perfectly or very nearly in direction, their joint effect
is a combination of the motions belonging to each.
Since every particle of the medium is affected by each undu-
lation, wherever the directions coincide, the undulations can
proceed no otherwise than by uniting their motions, so that
the joint motion may be the sum or difference of the separate
60
THE WAVE-THEORY OF LIGHT
motions, accordingly as similar or dissimilar parts of the undu-
lations are coincident.
I have, on a former occasion,, insisted at large on the appli-
cation of this principle to harmonics ; and it will appear to be
of still more extensive utility in explaining the phenomena of
colors. The undulations which are now to be compared are
those of equal frequency. When the two series coincide ex-
actly in point of time, it is obvious that the united velocity of
the particular motions must be greatest, and, in effect at least,
double the separate velocities ; and also that it must be smallest,
and, if the undulations are of equal strength, totally destroyed
when the time of the greatest direct motion belonging to one
undulation coincides with that of the greatest retrograde motion
of the other. In intermediate states the joint undulation will
be of intermediate strength ; but by what laws this intermediate
strength must vary cannot be determined without further data.
It is well known that a similar cause produces in sound that
effect which is called a beat ; two series of undulations of nearly
equal magnitude co-operating and destroying each other alter-
nately, as they coincide more or less perfectly in the times of
performing their respective motions.
[Proposition IX. and five corollaries to Proposition VIII.
are here omitted.]
61
AN ACCOUNT OF SOME CASES OF THE
PRODUCTION OF COLORS NOT
HITHERTO DESCRIBED*
READ JULY 1, 1802
WHATEVER opinion maybe entertained of the theory o
and colors which I have lately had the honor of submitting
to the Royal Society, it must at any rate be allowed that it
has given birth to the discovery of a simple and general law
capable of explaining a number of the phenomena of col-
ored light, which, without this law, would remain insulated
and unintelligible. The law is, that " wherever two portions
of the same light arrive at the eye by different routes, either
exactly or very nearly in the same direction, the light becomes
most intense when the difference of the routes is any multiple
of a certain length, and least intense in the intermediate state
of the interfering portions ; and this length is different for
light of different colors."
I have already shown in detail the sufficiency of this law
for explaining all the phenomena described in the second
and third books of Newton's Optics, as well as some others
not mentioned by Newton. But it is still more satisfactory
to observe its conformity to other facts, which constitute new
and distinct classes of phenomena, and which could scarcely
have agreed so well with any anterior law, if that law had
been erroneous or imaginary : these are the colors of fibres
and the colors of mixed plates.
As I was observing the appearance of the fine parallel lines'
of light which are seen upon the margin of an object held near
*From the Philosophical Transactions for 1802, p. 387.
ERSITY
MEMOIRS OK THE WAVE-THEORY OF LIGHT
the eye, so as to intercept the greater part of the light of a
distant luminous object, and which are produced by the fringes
caused by the inflection of light already known, I observed
that they were sometimes accompanied by colored fringes,
much broader and more distinct ; and I soon found that
these broader fringes were occasioned by the accidental inter-
position of a hair. . In order to make them more distinct, I
employed a horse-hair, but they were then no longer visible.
\\ r ith a fibre of wool, on the contrary, they became very large
ard conspicuous; and, with a single silk -worm's thread,
their magnitude was so much increased that two or three of
them seemed to occupy the whole field of view. They ap-
peared to extend on each side of the candle, in the same order
as the colors of thin plates seen by transmitted light. It oc-
curred to me that their cause must be sought in the interfer-
ence of two portions of light, one reflected from the fibre, the
other bending round its opposite side, and at last coinciding
nearly in direction with the former portion ; that, accordingly,
as both portions deviated more from a rectilinear direction, the
difference of the length of their paths would become gradual-
ly greater and greater, and would consequently produce the
appearances of color usual in such cases ; that supposing
them to be inflected at right angles, the difference would
amount nearly to the diameter of the fibre, and that this dif-
ference must consequently be smaller as the fibre became
smaller ; and, the number of fringes in a right angle becoming
smaller, that their angular distances would consequently be-
come greater, and the whole appearance would be dilated. It
was easy to calculate that for the light least inflected the
difference of the paths would be to the diameter of the fibre
very nearly as the deviation of the ray at any point from the
rectilinear direction to its distance from the fibre.
I therefore made a rectangular hole in a card, and bent its
ends so as to support a hair parallel to the sides of the hole ;
then, upon applying the eye near the hole, the hair, of course,
appeared dilated by indistinct vision into a surface, of which
the breadth was determined by the distance of the hair and
the magnitude of the hole, independently of the temporary
aperture of the pupil. When the hair approached so near to
the direction of the margin of a candle that the inflected light
was sufficiently copious to produce a sensible effect, the fringes
63
M E M O I R S ON
began to appear ; and it was easy to estimate the proportion
of their breadth to the apparent breadth of the hair across the
image of which they extended. I found that six of the bright-
est red fringes, nearly at equal distances, occupied the whole
of that image. The breadth of the aperture was T {Hb-> and its
distance from the hair -f$ of an inch ; the diameter of the hair
was less than -g-J-g- of an inch ; as nearly as I could ascertain
it was ^. Hence, we have y-j-J^ for the deviation of the
first red fringe at the distance ^ ; and as ^ : T^-Q : : -g-g-g-
rro O-OTT* or OTST f r tne difference of the routes of the rea
light where it was most intense. The measure deduced from
Newton's experiments is 36 | 00 . I thought this coincidence,
with only an error of one-ninth of so minute a quantity, suffi-
ciently perfect to warrant completely the explanation of the
phenomenon, and even to render a repetition of the experi-
ment unnecessary ; for there are several circumstances whicft
make it difficult to calculate much more precisely what ought
to be the result of the measurement.
When- a number of fibres of the same kind for instance, a
uniform lock of wool are held near to the eye, we see an ap-
pearance of halos surrounding a distant candle ; but their
brilliancy, and even their existence, depends on the uniformity
of the dimensions of the fibres ; and they are larger as the
fibres are smaller. It is obvious that they are the immediate
consequences of the coincidence of a number of fringes of the
same size, which, as the fibres are arranged in all imaginable
directions, must necessarily surround the luminous object at
equal distances on all sides, and constitute circular fringes.
There can be little doubt that the colored atmospherical
halos are of the same kind ; their appearance must depend on
the existence of a number of particles of water of equal dimen-
sions, and in a proper position with respect to the luminary
and to the eye. As there is no natural limit to the magnitude
of the spherules of water, we may expect these halos to vary
without limit in their diameters, and accordingly Mr. Jordan
has observed that their dimensions are exceedingly various,
and has remarked that they frequently change during the time
of observation.
I first noticed the colors of mixed plates in looking at a
candle through two pieces of plate-glass with a little moisture
between them. I observed an appearance of fringes resembling
64:
THE WAVE-THEORY OF LIGHT
the common colors of thin plates ; and, upon looking for the
fringes by reflection, I found that these new fringes were
always in the same direction as the other fringes, but many
times larger. By examining the glasses with a magnifier, I
perceived that wherever these fringes were visible the moist-
ure was intermixed with portions of air, producing an appear-
ance similar to dew. I then supposed that the origin of the
colors was the same as that of the colors of halos ; but, on
a more minute examination, I found that the magnitude of the
portions of air and water was by no means uniform, and that
the explanation was, therefore, inadmissible. It was, however,
easy to find two portions of light sufficient for the production
of these fringes ; for the light transmitted through the water,
moving in it with a velocity different from that of the light
passing through the interstices filled only with air, the two
portions would interfere with each other and produce effects
of color according to the general law. The ratio of the
velocities in water and in air is that of 3 to 4 ; the fringes
ought, therefore, to appear where the thickness is six times as
great as that which corresponds to the same color in the com-
mon case of thin plates ; and, upon making the experiment
with a plane glass and a lens slightly convex, I found the sixtli
dark circle actually of the same diameter as the first in the
new fringes. The colors are also very easily produced when
butter or tallow is substituted for water; and the rings then
become smaller, on account of the greater refractive density
of the oils ; but when water is added, so as to fill up the in-
terstices of the oil, the rings are very much enlarged ; for here
the difference only of the velocities in water and in oil is to
be considered, and this is much smaller than the difference
between air and water. All these circumstances are sufficient
to satisfy us with respect to the truth of the explanation ; and
it is still more confirmed by the effect of inclining the plates
to the direction of the light; for then, instead of dilating, like
the colors of thin plates, these rings contract: and this is the
obvious consequence of an increase of the length of the paths
of light, which now traverse both mediums obliquely ; and the
effect is everywhere the same as that of a thicker plate.
It must, however, be observed that the colors are not pro-
duced in the whole light that is transmitted through the
mediums : a small portion only of each pencil, passing through
E 65
MEMOIRS ON
the water contiguous to the edges of the particle, is sufficient-
ly coincident with the light transmitted by the neighboring
portions of air to produce the necessary interference ; and it
is easy to show that, on account of the natural concavity of the
surface of each portion of the fluid adhering to the two pieces
of glass, a considerable portion of the light which is beginning
to pass through the water will be dissipated laterally by re-
flection at its entrance, and that much of the light passing
through the air will be scattered by refraction at the second
surface. For these reasons the fringes are seen when the
plates are not directly interposed between the eye and the
luminous object; and on account of the absence of foreign
light, even more distinctly than when they are in the same
right line with that object. And if we remove the plates to a
considerable distance out of this line, the rings are still visible
and become larger than before ; for here the actual route of
the light passing through the air is longer than that of the
light passing more obliquely through the water, and the differ-
ence in the times of passage is lessened. It is ?< however, im-
possible to be quite confident with respect to the causes of
these minute variations, without some means of ascertaining
accurately the forms of the dissipating surfaces.
In applying the general law of interference to these colors,
as well as to those of thin plates already known, I must con-
fess that it is impossible to avoid another supposition, which is
a part of the undulatory theory that is, that the velocity of
light is the greater the rarer the medium ; and that there is
also a condition annexed to the explanation of the colors of
thin plates which involves another part of the same theory
that is, that where one of the portions of light has been re-
flected at the surface of a rarer medium, it must be supposed
to be retarded one-half of the appropriate interval for in-
stance, in the central black spot of a soap-bubble, where the
actual lengths of the paths very nearly coincide, but the effect
is the same as if one of the portions had been so retarded as to
destroy the other. From considering the nature of this cir-
cumstance,- I ventured to predict that if the two reflections
were of the same kind, made at the surfaces of a thin plate of
a density intermediate between the densities of the mediums
containing it, the effect would be reversed, and the central
spot, instead of black, would become white ; and I have now
66
THE WAVE-THEORY OF LIGHT
the pleasure of stating that I have fully verified this predic-
tion by interposing a drop of oil of sassafras between a prism
of fiint-glass and a lens of crown-glass ; the central spot seen
by reflected light was white and surrounded by a dark ring.
It was, however, necessary to use some force in order to pro-
duce a contact sufficiently intimate ; and the white spot dif-
fered, even at last, in the same degree from perfect whiteness
as the black spot usually does from perfect blackness.
[Three pages of speculation concerning dispersion are here
omitted.]
67
EXPEEIMENTS AND CALCULATIONS REL-
ATIVE TO PHYSICAL OPTICS*
A BAKEKIAN LECTUEE
Read Noveniber 24, 1803
I. EXPERIMENTAL DEMONSTRATION OF THE GENERAL LAW OF
THE INTERFERENCE OF LIGHT.
IN making some experiments on the fringes of colors ac-
companying shadows, I have found so simple and so demon-
strative a proof of the general law of the interference of two
portions of light, which I have already endeavored to estab-
lish, that I think it right to lay before the Royal Society a
short statement of the facts which appear to me so decisive.
The proposition on which I mean to insist at present is simply
this that fringes of colors are produced by the interference
of two portions of light ; and I think it will not be defied by
the most prejudiced that the assertion is proved by the ex-
periments I am about to relate, which may be repeated with
great ease whenever the sun shines, and without any other ap-
paratus than is at hand to every one.
Experiment 1. I made a small hole in a window-shutter, and
covered it with a piece of thick paper, which I perforated with
a fine needle. For greater convenience of observation I placed
a small looking-glass without the window-shutter, in such a
position as to reflect the sun's light in a direction nearly hor-
izontal upon the opposite wall, and to cause the cone of di-
verging light to pass over a table on which were several little
screens of card-paper. I brought into the sunbeam a slip of
*From the Philosophical Transactions for 1804.
MEMOIRS ON THE WAVE-THEORY OF LIGHT
card about one-thirtieth of an inch in breadth, and observed
its shadow, either on the wall or on other cards held at differ-
ent distances. Besides the fringes of color on each side of
the shadow, the shadow itself was divided by similar parallel
fringes of smaller dimensions, differing in number according
to the distance at which the shadow was observed, but leaving
the middle of the-' shadow always white. Now these fringes
were the joint effects of the portions of light passing on each
side of the slip of card, and inflected, or rather diffracted, into
the shadow ; for a little screen being placed a few inches
from the card so as to receive either edge of the shadow on
its margin, all the fringes which had before been observed in
the shadow on the wall immediately disappeared, although the
light inflected on the other side was allowed to retain its course,
and although this light must have undergone any modification
that the proximity of the other edge of the slip of card might
have been capable of occasioning. When the interposed screen
was more remote from the narrow card, it was necessary to
plunge it more deeply into the shadow, in order to extinguish
the parallel lines ; for here the light diffracted from the edge
of the object had entered farther into the shadow in its way
towards the fringes. Nor was it for want of a sufficient in-
tensity of light that one of the two portions was incapable of
producing the fringes alone ; for when they were both unin-
terrupted, the lines appeared, even if the intensity was reduced
to one- tenth or one-twentieth.
Experiment 2. The crested fringes described by the ingenious
and accurate Grimaldi afford an elegant variation of the pre-
ceding experiment and an interesting example of a calcula-
tion grounded on it. When a shadow is formed by an object
which has a rectangular termination besides the usual external
fringes there are two or three alternations of colors, beginning
from the line which bisects the angle, disposed on each side of
it in curves, which are convex towards the bisecting line, and
which converge in some degree towards it as they become
more remote from the angular point. These fringes are also
the joint effect of the light which is inflected directly towards
the shadow from each of the two outlines of the object ; for if
a screen be placed within a few inches of the object, so as to
receive only one of the edges of the shadow, the whole of the
fringes disappear ; if, on the contrary, the rectangular point
MEM OIKS ON
of the screen be opposed to the point of the shadow so as
barely to receive the angle of the shadow on its extremity, the
fringes will remain undisturbed.
II. COMPARISON OF MEASURES DEDUCED FROM VARIOUS EX-
PERIMENTS.
If we now proceed to examine the dimensions of the fringes
under different circumstances, we may calculate the differences
of the lengths of the paths described by the portions of light
which have thus been proved to be concerned in producing
those fringes ; and we shall find that where the lengths are
equal the light always remains white ; but that where either
the brightest light or the light of any given color disappears
and reappears a first, a second, or a third time, the differences
of the lengths of the paths of the two portions are in arithmet-
ical progression, as nearly as we, can expect experiments of this
kind to agree with each other. I shall compare, in this point
of view, the measures deduced from several experiments of
Newton and from some of my own.
In the eighth and ninth observations of the third book of
Newton's Optics some experiments are related which, together
with the third observation, will furnish us with the data neces-
sary for the calculation. Two knives were placed, with their
edges meeting at a very acute angle, in a beam of the sun's
light, admitted through a small aperture, and the point of con-
course of the two first dark lines bordering the shadows of the re-
spective knives was observed at various distances. The results
of six observations are expressed in the first three lines of the
first table. On the supposition that the dark line is produced
by the first interference of the light reflected from the edges of
the knives, with the light passing in a straight line between
them, we may assign, by calculating the difference of the two
paths, the interval for the first disappearance of the brightest
light, as it is expressed in the fourth line. The second table
contains the results of a similar calculation from Newton's ob-
servations on the shadow of a hair ; and the third, from some
experiments of my own of the same nature ; the second bright
line being supposed to correspond to a double interval, the sec-
ond dark line to a triple interval, and the succeeding lines to
depend on a continuation of the progression. The unit of all
,the tables is an inch.
70
THE WAVE-THEORY OF LIGHT
TABLE I. Observation 9. N.
Distance of the knives from the aperture 101
Distance of the
paper from
the knives 1 ^ &t 32 96 131
Distance b e -
tween the
edges of the
knives o p-
posite to the
point of
concourse 012 .020 .034 .057 .081 .087
Interval of dis-
appearance 0000122 .0000155 .0000182 .0000167 .0000166 .0000166
TABLE II. Observation 3. N.
Breadth of the hair ^
Distance of the hair from the aperture 144
Distances of the scale from the aperture 150 252
(Breadths of the shadow ^ )
Breadth between ihe second pair of bright lines -g? T 4 T
Interval of disappearance, or half the difference of the
paths 0000151 .0000173
Breadth between the third pair of bright lines . .. ^ T 3
Interval of disappearance, one-fourth of the difference.. .0000130 .0000143
TABLE III. Experiment 3.
Breadth of the object 434
Distance of the object from the aperture 125
Distance of the wall from the aperture 250
Distance of the second pair of dark lines from each other 1.167
Interval of disappearance, one-third of the difference 0000149
Experiment 4.
Breadth of the wire 083
Distance of the wire from the aperture 32
Distance of the wall from the aperture 250
(Breadth of the shadow, by three
measurements 815. .826, or .827 ; mean, .823)
Distance of the first pair of dark lines 1.165, 1.170, or 1.160; mean, 1.165
Interval of disappearance .......... 0000194
Distance of the second pair of dark
lines 1.402. 1.395, or 1.400; mean. 1.399
Interval of disappearance 0000137
Distance of the third pair of dark
lines. , 1.594, 1.580, or 1.585 ; mean, 1.586
Interval of disappearance 0000128
71
MEMOIRS ON
It appears, from five of the six observations of the first
table, in which the distance of the shadow was varied from
about 3 inches to 11 feet, and the breadth of the fringes was
increased in the ratio of 7 to 1, that the difference of the
routes constituting the interval of disappearance varied but
one-eleven tli at most ; and that in three out of the five it
agreed with the mean, either exactly or within y^- part.
Hence we are warranted in inferring that the interval appro-
priate to the extinction of the brightest light is either accu-
rately or very nearly constant.
But it may be inferred from a comparison of all the other
observations that when the obliquity of the reflection is very
great some circumstance takes place which causes the inter-
val thus calculated to be somewhat greater ; thus, in the elev-
enth line of the third table it comes out one-sixth greater than
the mean of the five already mentioned. On the other hand,
the mean of two of Newton's experiments and one of mine is
a result about one-fourth less than the former. With respect
to the nature of this circumstance I cannot at present form a
decided opinion ; but I conjecture that it is a deviation of
some of the light concerned, from the rectilinear direction as-
signed to it, arising either from its natural diffraction, by which
the magnitude of the shadow is also enlarged, or from some
other unknown cause. If we imagined the shadow of the
wire and the fringes nearest it to be so contracted that the
motion of the light bounding the shadow might be rectilinear,
we should thus make a sufficient compensation for this devia-
tion ; but it is difficult to point out what precise track of the
light would cause it to require this correction.
The mean of the three experiments which appear to have been
least affected by this unknown deviation gives .0000127 for the
interval appropriate to the disappearance of the brightest
light ; and. it may be inferred that if they had been wholly ex-
empted from its effects the measure would have been some-
what smaller. Now the analogous interval, deduced from the
experiments of Newton on this plate, is .0000112, which is
about one-eighth less than the former result ; and this appears
to be a coincidence fully sufficient to authorize us to attribute
these two classes of phenomena to the same cause. It is very
easily shown, with respect to the colors of thin plates, that
each kind of light disappears and reappears where the differ-
72
THE WAVE-THEORY OF LIGHT
ences of the routes of two of its portions are in arithmetical
progression*} and we have seen that the same law may be in
general inferred from the phenomena of diffracted light, even
independently of the analogy.
The distribution of the colors is also so similar in both cases
as to point immediately to a similarity in the causes. In the
thirteenth observation of the second part of the first book
Newton relates that the interval of the glasses where the rings
appeared in red light was to the interval where they appeared in
violet light as 14 to 9 ; and, in the eleventh observation of the
third book, that the distances between the fringes, under the
same circumstances, were the twenty-second and the twenty-sev-
enth of an inch. Hence, deducting the breadth of the hair and
taking the squares, in order to find the relation of the difference
of the routes, we have the proportion of 14 to 9, which scarcely
differs from the proportion observed in the colors of the thin
plate.
We may readily determine from this general principle the
form of the crested fringes of Grimaldi, already described } for
it will appear that, under the circumstances of the experiment
related, the points in which the differences of the lengths of
the paths described by the two portions of light are equal to
a constant quantity, and in which, therefore, the same kinds
of light ought to appear or disappear, are always found in
equilateral hyperbolas, of which the axes coincide with the
outlines of the shadow, and the asymptotes nearly with the
diagonal line. Such, therefore, must be the direction of the
fringes ; and this conclusion agrees perfectly with the observa-
tion. But it must be remarked that the parts near the out-
lines of the shadow are so much shaded off as to render the
character of the curve somewhat less decidedly marked where
it approaches to its axis. These fringes have a slight resem-
blance to the hyperbolic fringes observed by Newton ; but the
analogy is only distant.
[///. Application to the Supernumerary Rainbows, omitted.]
IV. ARGUMENTATIVE INFERENCE RESPECTING THE NATURE
OF LIGHT.
The experiment of Grimaldi on the crested fringes within
the shadow, together with several others of his observations
73
MEMOIRS ON
equally important, has been left unnoticed by Newton. Those
who are attached to the Newtonian theory of light, or to the
hypothesis of modern opticians founded on views still less
enlarged, would do well to endeavor to imagine anything like
an explanation of these experiments derived from their own
doctrines ; and if they fail in the attempt, to refrain at least
from idle declamation against a system which is founded on
the accuracy of its application to all these facts, and to a thou-
sand others of a similar nature.
From the experiments and calculation which have been pre-
mised, we may be allowed to infer that homogeneous light
at certain equal distances in the direction of its motion is pos-
sessed of opposite qualities capable of neutralizing or destroy-
ing each other, and of extinguishing the light where they
happen to be united; that these qualities succeed each other
alternately in successive concentric superficies, at distances
which are constant for the same light passing through the
same medium. From the agreement of the measures, and from
the similarity of the phenomena, we may conclude that these
intervals are the same as are concerned in the production of the
colors of thin plates ; but these are shown, by the experi-
ments of Newton, to be the smaller the denser the medium;
and since it may be presumed that their number must neces-
sarily remain unaltered in a given quantity of light, it follows,
of course, that light moves more slowly in a denser than in a
rarer medium'; and this being granted, it must be allowed that
refraction is not the effect of an attractive force directed to a
denser medium. The advocates for the projectile hypothesis
of light must consider which link in this chain of reasoning
they may judge to be the most feeble, for hitherto I have
advanced in this paper no general hypothesis whatever. But
since we know that sound diverges in concentric superficies,
and that musical sounds consist of opposite qualities, capable
of neutralizing each other, and succeeding at certain equal
intervals, which are different according to the difference of
the note, we are fully authorized to conclude that there must
be some strong resemblance between the nature of sound and
that of light.
I have not, in the course of these investigations, found any
reason to suppose the presence of such an inflecting medium
in the neighborhood of dense substances as I was formerly
74
THE WAVE-T11EOKY OF LIGHT
inclined to attribute to them ; and, upon considering the phe-
nomena of the aberration of the stars, I am disposed to believe
that the luminiferous ether pervades the substance of all ma-
terial bodies, with little or no resistance, as freely, perhaps, as
the wind passes through a grove of trees.
The observations on the effects of diffraction and inter-
ference may, perhaps, sometimes be applied to a practical pur-
pose in making us cautious in our conclusions respecting the
appearances of minute bodies viewed in a microscope. The
shadow of a fibre, however opaque, placed in a pencil of light
admitted through a small aperture, is always somewhat less dark
in the middle of its breadth than in the parts on each side. A
similar effect may also take place, in some degree, with respect
to the image on the retina, and impress the sense with an idea
of a transparency which has no real existence ; and if a small
portion of light be really transmitted through the substance,
this may again be destroyed by its interference with the dif-
fracted light, and produce an appearance of partial opacity,
instead of uniform semi-transparency. Thus a central dark spot
and a light spot, surrounded by a darker circle, may respec-
tively be produced in the images of a semi-transparent and
an opaque corpuscle, and impress us with an idea of a com-
plication of structure which does not exist. In order to detect
the fallacy, we make two or three fibres cross each other, and
view a number of globules contiguous to each other; or we
may obtain a still more effectual remedy by changing the mag-
nifying power; and then, if the appearance remain constant in
kind and in degree, we may be assured that it truly represents
the nature of the substance to be examined. It is natural to
inquire whether or not the figures of the globules of blood
delineated by Mr. Hewson in the Phil. Trans., vol. Ixiii., for
1773, might not in some measure have been influenced by a
deception of this kind ; but, as far as I have hitherto been able
to examine the globules with a lens of one-fiftieth of an inch
focus, I have found them nearly such as Mr. Hewson has de-
scribed them.
[ V. Remarks on the Colors of Natural Bodies, omitted.]
VI. EXPERIMENT ON THE DARK RAYS OF RITTER
Experiment 6. The existence of solar rays accompanying
light, more refrangible than the violet rays and cognizable by
75
MEMOIRS ON
their chemical effects, was first ascertained by Mr. Bitter ; but
Dr. Wollaston made the same experiments a very short time
afterwards without having been informed of what had been
done on the Continent. These rays appear to extend beyond
the violet rays of the prismatic spectrum, through a space
nearly equal to that which is occupied by the violet. In order
to complete the comparison of their properties with those of
visible light, I was desirous of examining the effect of their re-
flection from a thin plate of air, capable of producing the well-
known rings of colors. For this purpose I formed an image
of the rings, by means of the solar microscope, with the appa-
ratus which I have described in the Journals of the Royal
Institution, and I threw this image on paper dipped in a solu-
tion of nitrate of silver, placed at the distance of about nine
inches from the microscope. In the course of an hour portions
of three dark rings were very distinctly visible, much smaller
than the brightest rings of the colored image, and coinciding
very nearly in their dimensions with the rings of violet light
that appeared upon the interposition of violet glass. I thought
the dark rings were a little smaller than the violet rings, but
the difference was not sufficiently great to be accurately ascer-
tained ; it might be as much as fa or fa of the diameters, but
not greater. It is the less surprising that the difference should
be so small, as the dimensions of the colored rings do not by
any means vary at the violet end of the spectrum so rapidly as
at the red end. For performing this experiment with very
great accuracy a heliostat would be necessary, since the motion
of the sun causes a slight change in the place of the image ;
and leather impregnated with the muriate of silver would
indicate the effect with greater delicacy. The experiment,
however, in its present state, is sufficient to complete the anal-
ogy of the invisible with the visible rays, and to show that they
are equally liable to the general law which is the principal sub-
ject of this paper. If we had thermometers sufficiently delicate,
it is probable that we might acquire, by similar means, infor-
mation still more interesting with respect to the rays of invis-
ible heat discovered by Dr. Herschel ; but at present there is
great reason to doubt of the practicability of such an experi-
ment.
76
THE WAVE-THEORY OF LIGHT
BIOGRAPHICAL SKETCH
THOMAS YOUNG was born at Milverton. England, in 1773,
and died at London in 1829. His education, in respect to the
amount of ground it co.vered, is quite as remarkable as his later
scientific work. As a lad he showed marked proficiency in
linguistic studies, acquired great mechanical skill, distin-
guished himself in drawing, music, and athletics. As a young
man he pursued his university studies at London, Edinburgh,
Gb'ttingeu, and Cambridge.
The following programme of his daily work at Gottingen in
the autumn of 1795 characterizes at once the lad, the youth,
and the mature man:
" At 8, I attend Spittler's course on the History of the Principal States
of Europe, exclusive of Germany.
" At 9, Arnemann on Materia Medica.
"At 10, Richter on Acute Diseases.
" At 11, twice a week, private lessons from Blessman, the academical
dancing-master.
"At 12, I dine at Ruhlander's table d'hote.
"At 1, twice a week, lessons on the clavichord from Forkel; and twice
a week at home, from Fiorillo on Drawing.
" At 2, Lichtenberg on Physics.
"At 3, I ride in the academical manege, under the instruction of Ayrer,
four times a week.
" At 4, Stromeyer on Diseases.
*' At 5, Blumenbach on Natural History.
"At 6, twice Blessman with other pupils, and twice Forkel."
He was born of a well-to-do Quaker family; he inherited
ample money ; he had all that travel, leisure, and good society
could do for a man. Only in one particular does his education
appear to have been defective viz., in the absence of any
training in advanced dynamics or in higher mathematical
analysis.
In 1800 he completed his medical studies at Cambridge, and
settled as a practising physician in London. In the year fol-
lowing he was appointed to the professorship of natural philos-
ophy in the then newly founded Royal Institution, a position
from which he resigned at the end of two years in order to
devote himself more completely to the practice of medicine.
It was during his occupancy of this chair that he published the
77
MEMOIRS ON THE WAVE-THEORY OF LIGHT
three papers reprinted in this volume, the first of which is pos-
sibly the most important of his contributions to physics. It
was during this period also that he wrote his Lectures on Natural
Philosophy, which must always be reckoned as a potent factor
in the spread of sound physical science in the nineteenth cen-
tury, while its bibliography of more than four hundred quarto
pages is to-day valuable as well as classic.
But nothing short of a catalogue of his papers can give one
an adequate idea of the varied activity of this man during the
remaining quarter-century of his life. His contributions cover
fields as diverse as the physiology of the human eye, hydro-
dynamics, music, paleography, atmospheric refraction, theory
of tides, tables of mortality, theory of structures. His expla-
nation of color-vision as due to the presence of three sets of
nerve fibres in the retina, which, when excited, give respectively
sensations of red, green, and violet, has been adopted and modi-
fied by Helmholtz, and is to-day perhaps the most widely
accepted of the various theories on this subject.
After all, it rnnst^ be confessed, even by his most ardent
admirers, that Young's style is, in general, far from clear.
Whether this is in any way connected with his lack of mathe-
matical training, or whether it is due to the fact that his own
clear intuitions bridged most of the gaps in his written work,
it is difficult to say ; but in any event many of his papers are
obscure, and few of them are read. The reader who desires a
full biography will find it in Dr. Peacock's Life of Young
(London, 1855). This biographer also edited his Miscellaneous
Works, 3 vols. (London, 1855). All his papers, however, which
are of especial interest to the student of physics are contained
in the lectures on Natural Philosophy (London, 1807).
78
MEMOIR ON THE DIFFRACTION OF LIGHT
o
" CROWNED " BY THE FRENCH ACADEMY OF SCIENCES IN 1819
BY
A. FRE8NEL
Natura simplex et fecunda.
-*
MEMOIR ON THE ACTION OF RAYS OF
POLARIZED LIGHT UPON EACH OTHER
4.
V
BY
MESSRS. ARAGO AND FRESNEL
(Annales de Ghimie et cto ffysique. t. x., p. 288, 1819.)
79
CONTENTS
PAGE
On tlie Insufficiency of the Corpuscular Theory and of Young's Views
concerning Interference 81
PJienomena of Diffraction explained by the Combination of Huygentfs
Principle with that of Interference 108
On the Interference of Polarized Light ; . 145
80
FRESNEL'S PRIZE MEMOIR ON THE DIP-
FRACTION OF LIGHT
[The Introduction, covering fourteen pages and describing in
the most general, way the defects of the emission-theory and some
of the merits of the wave-theory, is omitted.]
SECTION I
11. It might appear that on the emission -theory nothing
would be simpler than the phenomena of shadows, especially
when the source of light is merely a point ; but, on the con-
trary, nothing is more complicated. If we suppose the surface
of the body producing the shadow to be endowed with a re-
pulsive property capable \>f changing the direction of rays of
light passing very near it, we should then expect only to see
the shadows increase in size and, towards their edges, to blend
a little with the illuminated area; while, as a matter of fact,
they are bordered with three very distinct colored fringes
when one employs white light, and with a still greater number
of bright and- dark bands when one uses light which is practi-
cally homogeneous. These fringes we shall call exterior, and
those which are observed in the midst of very narrow shadows
we shall call interior fringes.
If one adopts the Newtonian theory, he is tempted at first to
explain the exterior fringes as produced by a force which is al-
ternately attractive and repulsive, and which has its source in
the surface of the body producing the shadows. I shall now
consider the consequences of this theory and show that its re-
sults are not justified by experiment; but, first of all, I must
explain the experimental method which I have employed.
12. We know that the effect of a magnifying-glass placed in
front of the eye is to reproduce accurately upon the retina any
object or image which is located at its conjugate focus ; at least
F 81
MEMOIRS ON
this is the case whenever all the rays which go to make up the
image are incident upon the surface of the glass. In place,
then, of projecting the fringes upon a white card or a ground
glass, one may observe them directly with a magnifying-
glass, and he then sees them as they are at its focus. One
has then only to look towards the luminous point and place
the glass between his eye and the opaque body in such a way
that the point where the refracted rays cross each other falls
in the middle of the pupil ; this position is recognized by the
fact that the entire surface of the magnifying-glass appears to
be filled with light. This method is much preferable to the
other two in that it enables us to study conveniently phenom-
ena of diffraction in a weak light, and has, at the same time,
the further advantage of allowing us to follow the exterior
fringes right up to their source. Using a lens of 2 mm. focus
and light which is practically homogeneous, I have been able
to follow these fringes very close to their origin and yet ob-
serve the dark band of the fifth order. The interval which sep-
arates this band from the edge of the shadow I have measured
on the micrometer and find it to be less than 0.015 mm., while
the first three fringes are comprised within a space not exceed-
0.01 mm. ; by using a lens of shorter focus, one would doubt-
less still further diminish this distance. We may thus regard
the dark and bright bands as beginning at the very edge of the
opaque body, so long as we do not push the accuracy of our
measures beyond the hundredth part of a millimeter an ac-
curacy which proves to be sufficient, and which it is difficult to
exceed except when the fringes are somewhat larger, as is the
case with those most frequently observed.
13. This point established, suppose that we measure the ex-
terior fringes at any given distance from the [opaque] screen
and then allow the luminous point to approach ; the fringes
are observed to grow much larger. Meanwhile the angle which
the incident ray passing through the origin of the fringes makes
with the tangent drawn from the luminous point to the edge of
the screen will be almost zero. And since these fringes take
their rise at a distance less than 0.01 mm. from the edge, the
variation of this angle would not be able to sensibly affect the
size of the fringes. To explain this enlargement, we must there-
fore assume that the repulsive force increases in proportion as
the opaque body approaches the luminous point. But this is
82
THE WAVE-THEORY OF LIGHT
impossible, for the intensity of this force can evidently depend
only upon the distance at which the light corpuscle passes the
opaque body, upon the size and form of the surface of this
body, upon its density, mass, or nature ; and by hypothesis
these all remain constant.
But even if we suppose the origin of the dark and bright
bands to lie at a greater distance from the edge of the screen,
a supposition which would explain the fact that they grow
larger in proportion as one approaches the luminous point, it
is still impossible to make the results of experiment agree with
the formula deduced from the [Newtonian] hypothesis which
we are here discussing.
14. The following table gives the distance between the dark-
est point in the dark band of the fourth order and the edge of
the geometrical shadow* for different distances of the opaque
body from the luminous point. These measures have been
taken with a micrometer eye-piece which carries in its focal
plane a silk fibre, the whoje being moved by a micrometer
screw. By the aid of a head divided into one hundred parts,
passing an index, fixed with reference to the screw, one is
able to read the displacement of the silk thread to within
about 0.01 mm.
Distance from edge of
No. of
Observation
Distance of luminous point
from opaque screen
Distance of opaque body
from micrometer
geometrical shadow to
the middle of the dark
band of the fourth order
m.
m.
mm.
1
0.100
0.7985
5.96
2
0.510
1.005
3.84
3
1.011
0.996
3.12
4
2.008
0.999
2.71
5
3.018
1.003
2.56
6
4.507
1.018
2.49
7
6.007
0.999
2.40
These experiments were made with practically homogeneous
red light, which was obtained by means of a colored glass trans-
* I define geometrical shadow as the space included between the straight
lines drawn through the luminous point and tangent to the edges of the
screen ; this is the shadow which the light would project if it were not
diffracted.
83
MEMOIRS ON
mitting only the red rays and a small portion of the orange rays.
One might obtain more homogeneous light by use of a prism,
but he would not be so certain as to its identity in the various
observations a condition which it is very necessary to sat-
isfy.
15. Let us represent by a and b the respective distances of
the opaque body from the luminous point and from the mi-
crometer ; let d be the distance from the edge of the body to
the origin of the dark band of the fourth order, and r the tan-
gent of the small angle of inflection resulting from the action
of the repulsive forces. We then have the following expression*
for the distance between the edge of the geometrical shadow
and the darkest point in the dark band :
Now since r and d remain constant whatever be the distances
of the luminous point from the opaque body and from the mi-
crometer respectively, two observations suffice to determine
their value. Combining the first and the last observations, we
find fl? = 0.5019 mm. and r = 1.8164. We are thus compelled to
suppose that at its origin the dark band of the fourth order is
distant one -half a millimeter from the edge of the opaque
body. If, now, we substitute these values in the formula and
apply it to the intermediate observations, we obtain the follow-
ing values, several of which evidently differ widely from the
results of experiment.
* [In diagram S is luminous point, O is edge of opaque body, A is edge of
geometrical shadow, O' is origin of dark band of fourth (or any) order. Hence
Ar> . d(a-\-b)
AB is -- - ; and
a
the centre of the dark
band is a distance
br farther, where
A.C = br. The unit
which Fresnel here
employs for r is evidently one hundred times smaller than that in which we
ordinarily express natural tangents.']
84
THE WAVE-THEORY OF
No. of
Observation
Distance of lu-
minous point
from opaque
body
Distance of opnque
body from mi-
crometer
Distance between the
edge of geometrical
shadow and darkest
point of fourth baud
Differences
Observed
Computed
from formula
d(a+b)
br+ JT~
1
2
3
4
5
6
7
m.
0.1000
0.510
1.011
2.008
3.018
4.507
6.007
m.
0.7985
1.005
0.996
0.999
1.003
1.018
0.999
m.
596
3.84
3.12
2.71
2.56
249
2.40
mm-
3.32
281
2.57
2.49
246
mm.
-0.52
-0.31
-0.14
-0.07
-0.03
16. In attributing the production of fringes to the alternate
expansion and contraction of rays which pass very near the
opaque body, we are led to still another inference which is
contradicted by experiment viz., that the centres of the dark
and bright bands ought to lie along straight lines which would
be the axes of the expanded or condensed pencils of rays.
But experiment shows that in the case of exterior fringes their
trajectories are hyperbolas, of which the curvature is quite
sensible whenever the body which produces the shadow is suf-
ficiently distant from the luminous point.
The screen being placed at a distance of 3.018 m. from the
luminous point, I measured in succession the deviation of the
darkest point of the dark band of the third order, first at
0.0017 m. from the screen, then at 1.003 m., and lastly at
3.995 m. from the screen ; and I found for its distance from
the edge of the geometrical shadow first 0.08 mm., secondly 2.20
mm., and thirdly 5.83 mm. If, now, we join the two extreme
points by a straight line, we find for the ordinate correspond-
ing to the intermediate point 1.52 mm. in place of 2.20 mm.,
the difference being 0.68 mm. that is to say, about one and
one-half times the interval between the middle of the third
and the middle of the second bands. For this interval at a
distance of 1.003 m. from the opaque body was only 0.42 mm.,
from which it is evident that the difference of 0.68 mm. cannot
be attributed to an error resulting from lack of definition
in the fringes observed. Nor is one able to explain this dis-
crepancy by supposing an error in the observation made at
3.995 m. from the opaque body. From the fact that the
85
MEMOIRS ON
fringes are larger, the measures should be less accurate ; but in
repeating them several times I find variations which at most
amount to three or four hundredths of a millimeter. Even
supposing that there were an error of one-half a millimeter in
this measure, it would produce only a difference of 0.13 mm.
at a distance of 1.003 m. ; so that experiment shows conclu-
sively that the exterior fringes lie on curved lines with their
convex side outwards.
The following table gives these trajectories, referred to their
chords, for different series of observations, in each of which
the distance of the opaque body from the luminous point re-
mains constant. In the fourth series I suppose first that the
chord joins the two extreme readings, and next I suppose it to
be drawn from the edge of the opaque body itself where the
deviation of the fringes from their origin is, as we have already
seen, very small. In the other series the chord joins the edge
of the opaque body and the point most distant from it.
Lance from lu-
nous point to
aque screen,
the value of a
gl!s
<*" -o 5 o
Ji*
llli
3=.2
Ordinates of Trajectories of dark bands referred
to their chords
s aoS
g&ss
1st order
2d order
3d order
4th order
5th order
IST SERIES
f
m.
mm.
mm.
mm.
mm.
mm.
m.
0.510
J 0.110
1 0.501
019
0.14
0.29
0.21
035
0.25
0.40
0.30
0.44
0.34
[ 1.005
2o SERIES
f
m.
mm.
mm.
mm.
mm.
mm.
m.
1 0.116
0.23
0.35
0.42
0.49
055
1.011
1 0.502
0.27
0.40
0.51
0.57
0.63
0.996
0.21
0.30
0.38
0.42
0.49
[ 2.010
3D SERIES
f o
m.
mm.
mm.
mm.
mm.
mm.
m.
2.008
J 0.118
1 0.999
0.26
0.34
0.38
0.48
0.47
0.60
0.54
0.68
0.60
0.76
[ 2.998
THE WAVE-THEORY OF LIGHT
4-TH SERIES referred to the chord joining the extreme readings
m. r"
0.0017
mm. mm.
linn.
0.253
0.30 0.45
056
m.
3.018
<
0.500
1.003
0.38 0.53
0.38 0.56
0.65
0.68
z
1.998
0.31 0.45
0.54
3.002
0.17 0.23
0.28
1 3.995
4TH SERIES referred to the chord drawn from the edge of the opaque body
(
m.
mm.
mm.
mm.
00017
0.04
0.06
0.08
mm.
mm.
m
0.253
0.34
0.50
0.63 0.73
0.83
3018
-
0.500
0.41
0.58
72 0.85
0.94
1.003
041
0.60
0.74 0.87
0.97
1.998
0.32
0.48
57 67
075
3.002
0.18
0.25
30 38
0.39
13995
STH SERIES
r
m.
mm.
mm.
mm.
mm.
mm.
4.507
J 0.131
1.018
0.27
032
0.40
0.48
0.50
0.59
0.58
071
0.66
081
^2.506
GTH SERIES
f
m.
m.
mm
mm.
mm.
mm.
mm.
6007
1 0.117
023
033
0.42
0.49
0.53
LO. 999
It is thus evident that the hypothesis of contraction and
expansion produced by the action of the body upon rays of
light is insufficient to explain the phenomena of diffraction.
Introducing the principle of interference, however, we are able
to predict not only the variation in size of the exterior fringes
when the screen is made to approach or recede from the lumi-
nous point, but also the curved path of the bright and dark
bands. The law of interference, or the mutual influence of
rays of light, is an immediate consequence of the wave-theory;
not only so, but it is proved or confirmed by so many different
experiments that it is really one of the best-established prin-
ciples of optics.
17. Grimaldi was the first to observe the effect which rays
87
MEMOIRS ON
of light produce upon one another. Recently the distinguish-
ed Dr. Thomas Young has shown by a simple and ingenious
experiment that the interior fringes are produced by the meet-
ing of rays inflected at each side of the opaque body. This
he proved by using a screen to intercept one of the two pen-
cils of light; and in this way lie was able to make the interior
fringes completely vanish, whatever might be the form, mass,
or nature of the screen, and whether he intercepted the lu-
minous pencil before or after its passage into the shadow.
18. Brighter and sharper fringes may be produced by cut-
ting two parallel slits close together in a piece of cardboard
or a sheet of metal, and placing the screen thus prepared in
front of the luminous point. We may then observe, by use
of a magnifying -glass between the opaque body and the eye,
that the shadow is filled with a large number of very sharp-
colored fringes so long as the light shines through both open-
ings at the same time, but these disappear whenever the light
is cut off from one of the slits.
19. If we allow two pencils of light, each coming from the
same source and regularly reflected by two metallic mirrors,
to meet under a very small angle, we obtain similar fringes,
the colors of which are even purer and more brilliant than
before. To obtain these bands, it is necessary to be very care-
ful that in the region where the two mirrors come into con-
tact, or at least throughout a portion of their line of contact,,
the surface of the one is not shifted sensibly past that of the
other. This is necessary in order that the difference of path
traversed by two reflected rays meeting in the area common
to the two luminous* fields may be very small. I may re-
mark in passing that the theory of interference alone will
* In the case of white light, or even in light as homogeneous as possible,
the number of fringes which one can see is always limited, because even
when the light has reached a degree of simplicity as great as possible
without too far diminishing its intensity, it is still composed of rays which
are heterogeneous; and since the bright and dark bands thus produced do
not all have the same size, they encroach the one upon the other in pro-
portion as their order increases, and finally they completely destroy each
other ; and this is why one does not see any fringes when the difference
of paths becomes slightly sensible. Concerning the details of this ex-
periment and its explanation on the principle of interference, see the arti-
cle upon Light in the French translation of Thomson's Chemistry, already
cited.
THE WAVE-THEORY OF LIGHT
explain this experiment, and that the experiment calls for
manipulation so delicate and effort so continued that it is
almost impossible that one should strike upon it by accident.
If we raise one of the mirrors or intercept the light which it
reflects either before or after reflection, the fringes disappear
as in the preceding case. This furnishes still further evidence
that the fringes are produced, not by the action of the edges
of the mirrors, but by the meeting of two pencils of light.
For these fringes are always at right angles to the line which
joins the two images of the luminous point, whatever be its
inclination with respect to these edges, at least throughout the
extent of the area which is common to the two regularly re-
flected pencils.*
20. Since the fringes which one sees in the interior of
the shadow of a very narrow body and those which one ob-
tains by the use of two mirrors result evidently from the mut-
ual influence of rays of light, analogy would indicate that the
same thing ought to be true for the exterior fringes of the
shadows of bodies illuminated by a point source. The first
explanation which occurs to one is that these fringes are pro-
duced by the interference of direct rays with those which are
reflected at the edge of the opaque body, while the interior
fringes result from the combined action of rays inflected into
the shadow from the two sides of the opaque body, these in-
flected rays having their origin either at the surface or at
points indefinitely near it. This appears to be the opinion of
Mr. Young, and it was at first my o\vn opinion ; but a closer
examination of the phenomena convinced me of its falsity.
Nevertheless, I propose to follow it to its logical conclusion
and to state the formula which I have derived in order to facil-
itate comparison of this theory with that which I offer as a
substitute.
Let R, Fig. 14, be the radiant point, AA' the opaque body, and
FT' either a white screen upon which the shadow of this body
falls or the focal plane of a magnifying-glass with which the
* When the fringes extend outside, all their exterior portions resulting
from the meeting of rays regularly reflected by one of the mirrors and rays
inflected near the edge of the other should have different directions. If
one observes this phenomenon carefully, he will see that the form and
position of the fringes are in each case in accord with the theory of inter-
ference.
MEMOIRS ON
fringes are observed. RT and RT' are rays tangent at the
edge of the opaque body, T and T' being the limits of the ge-
ometrical shadow. Let us indicate by
a the distance RB from the luminous
point to the opaque body, by b the dis-
tance BO of the body from the white
screen, and by c its diameter, AA',
which we shall consider very small com-
pared with the distances a and b. This
assumption is made in order that we
may measure the size of the fringes
either in a plane perpendicular to RT
or perpendicular to the line RC,
which passes through the middle of the
shadow.
With these conventions we shall con-
sider, first, the exterior fringes. Let F be any point on the re-
ceiving screen outside the shadow. The difference of path
traversed by the direct ray, and by the ray reflected at the edge
of the opaque body, and meeting the direct ray at this point,
is RA + AF RF. Let us represent FT by x, and express in
series the values of RF, AR, and AF. Then, if we neglect all
terms involving any power of x or of c higher than the second,
since they are very small compared with distances a and b, the
terms which contain c will disappear and we shall have for the
difference of path traversed
*=-<
whence follows
21. If we call X the length of a light-wave, that is to say,
the distance between two points in the ether where vibrations
of the same kind are occurring at the same time and in the
same sense, then A/2 will be the distance between two ether
particles whose velocities of vibration are at any one instant
equal but oppositely directed. Thus two trains of waves sepa-
rated by an interval equal to X are in perfect accord as to their
vibrations; but when the distance between corresponding points
is A/2, then their vibrations are directly opposed. Accordingly
90
THE WAVE-THEORY OF LIGHT
the above formula gives for the value of x, corresponding to
the centre of the dark band of the first order, the following
value : \J * '- ; while observation shows that, as a matter
v ct
of fact, this is the brightest part of the first fringe. On the
same theory, the edge of the geometrical shadow, where the
difference of path vanishes, ought to be brighter than the rest
of the fringe, while, as a matter of fact, this is precisely the
darkest region outside the geometrical shadow. In general,
the position of the dark and bright bands deduced from this
formula is almost exactly the inverse of that determined by
experiment. This is the first difficulty presented by this
theory. To avoid it, we must suppose that the rays reflected
at the edge of the screen suffer the loss of half a wave-length;
adding A/2 to the difference of path, d, the general expression
becomes
y ~ a
Replacing d in this formula by A/2, 3 A/2, 5 A/2, 7 A/2, etc., we
have for the values of x corresponding to dark bands of the
first, second, third, fourth, etc., orders:
/2\b(a + d) /\b(a + b) /Q\b(a+b) .
v - ^r v <r~ V~ ^T-> V~ ^--> etc -
These formulae appear to agree fairly well with the observations;
however, closer measurements show that the ratios between the
sizes of the fringes derived from these expressions are not ex-
actly correct, as we shall see later.
22. I pass now to the consideration of interior fringes pro-
duced in the shadow by the meeting of two pencils of light
inflected at A and A'. Let M, Fig. 14, be any point located in
the interior of the shadow ; the intensity of the light at this
point depends upon the amount of disagreement between the
vibrations of the rays AM and A'M, which meet at this point,
or upon the difference of path A'M AM. I shall denote by x
the distance MC of the point M from the middle of the shadow,
and by d the difference of paths, and hence
Expanding the radicals and neglecting the higher powers of
x, since this quantity is very small compared with b, we have
dcxjb,
91
MEMOIRS ON
or x = bd/c.
If in place of d in this expression we substitute successively X/2,
3X/2, 5X/2, 7X/2, etc., we obtain the values of x correspond-
ing to dark bands of the first, second, third, fourth, etc.,
order, namely,
b\ 3X 5b\ 7#X ,
7\~> ~T ' ~T > ~z\ > etc.,
2c 2c 2c 2c
and consequently for the distance between the middle points of
two consecutive dark bands, b\/c.
The general expression for n such intervals is, therefore,
rib\lc.
23. So long as the extreme fringes are sufficiently distant
from the edges of the shadow, this formula agrees fairly well
with experiment ; but when they approach very near or pass
beyond the edges, one detects a slight difference between
their actual position and that deduced from the formula. In
general, the calculated values are always a little larger than
the observed. The reason for this I shall show when we come
to the true theory of diffraction. It also follows from this
formula that the size of the interior fringes ought to be entire-
ly independent of the distance, a, of the luminous point from
the opaque body; this prediction, however, is not completely
verified by experiment, especially when the fringes completely
fill the shadow; their position then varies distinctly with the
distance a.
24. According to the formula
which we have just derived for the exterior fringes, their
position depends upon a as well as upon b. Experiment shows
that, in fact, their size increases or diminishes according as the
opaque body approaches or recedes from the luminous point,
and that the ratios between the different sizes of one and the
same fringe deduced from the formula are precisely those given
by observation. But the most remarkable inference from this
formula is that, when a remains constant, the distance of any
dark or bright band from the edge of the geometrical shadow
is not directly proportional to b as in the case of interior
fringes, but varies in such a way that this band traces out, not
a straight line, but a hyperbola of sensible curvature. This is
92
THE WAVE-THEORY OF LIGHT
also confirmed by experiment, as may be seen from the observa-
tions given above.
Considering the striking agreement of these formulae with
experiment, it is natural to suppose that they are accurate ex-
pressions of fact, and therefore natural to attribute any small
differences between calculated and observed values to the errors
which are unavoidable in such delicate measurements.*
But a closer examination of the hypotheses from which they
are derived, and of the inferences derivable from them, shows
that they do not agree with the facts of nature.
25. If the fringes at the edge of a shadow are really due to
the meeting of the direct rays with those reflected at the edge
of the screen, their intensity ought to depend upon the area
and the curvature of its surface, and the fringes produced by
the back of a razor, for instance, ought to be much more vis-
* It might appear at first sight that one would be able to adapt this
theory to the ideas of Newton by introducing the principle of interference,
as I have indicated above ; but besides the complication of fundamental
hypotheses and the small probability of any of them, this principle, it ap-
pears to me, would lead to consequences which contradict the emission-
theory.
M. Arago has remarked that the interposition of a thin transparent plate
at the edge of an opaque body sufficiently narrow to produce interior
fringes in its shadow displaces these fringes and shifts them towards the
side [see paper by Arago (Ann. Chim. et Phys.,\., p. 199, 1816)] where is
placed the transparent plate. This being so, it follows from the principle
of interference that the rays which have traversed the plate have been re-
tarded in their path, because the same fringes in each case must correspond
to equal intervals between the times of arrival of rays. This inference at
once confirms the wave-theory and manifestly contradicts the emission-
theory, in which one is compelled to assume that light travels more rapidly
in dense than in rare media.
This objection can be avoided only by substituting for difference of
path difference of "fit"; but we lose all that was gained by the principle
of interference in thus replacing a sharp idea by a hazy one, a satisfactory
explanation by one which does not aid our understanding of the phenome-
na ; for one can readily see how two light particles striking the retina at the
same point may produce sensations more or less intense, according as the
interval of time which separates two consecutive impacts is sufficient to
produce unison or dissonance between the vibrations at the optic nerve;
while it is by no means so easy to see how this effect could be produced
by a difference of " fit " between two light particles, or how by simultaneous
impact on the optic nerve they would produce no effect at all when they
were in opposite " fits," even though their mechanical impacts were in per-
fect unison.
MEMOIRS ON
ible than those produced by the edge; but, using a magnifying-
glass at a distance of some centimeters, one detects practi-
cally no difference in intensity in these two cases. This test
is more easily made by using a steel plate one edge of which
is round throughout a part of its length and sharp through-
out the remainder of its length, these two edges lying in
the same straight line. One is thus easily convinced that
the fringes have the same intensity throughout their entire
length.
26. We know that under large angles of incidence dull sur-
faces reflect light almost as well as polished mirrors. This is
easily explained either on the emission-theory or on the wave-
theory. But although one can understand how difference of
polish cuts a small figure when the angle of incidence is large,
it is not easy to see how the intensity of the reflected light can
be independent of the curvature of the reflecting surface ;
indeed, it is clear that as the radius of curvature diminishes
the reflected rays will diverge more and more, whatever- be
their angle of incidence.
27. Not only so, but I have convinced myself by another
simple experiment of the incorrectness of the hypothesis which
I had first adopted, and which I am now opposing. I cut a
sheet of copper into the shape represented in Fig. 15, and
placed it in a dark room about four meters in front of a
luminous point, and examined its shadow with a magnifying-
glass. What I observed, on slowly re-
ceding, was as follows: When the large
fringes produced by each of the very
narrow openings CEE'C' and DFF'D'
had spread out into the geometrical
shadow of CDFE, which received prac-
tically only white light from each sep-
arate slit, the interior fringes produced
by the meeting of these two pencils of
light showed colors much sharper and
Fig i 5.
purer than the interior fringes of the shadow of ABDO, and
were, at the same time, much brighter. On receding still
farther, I noticed that the light diminished throughout the
whole of the shadow of ABFE, but much more rapidly back
of EFDC than in the upper part of the shadow, so that there
was one particular instant when the intensity of the light ap-
94
THE WAVE-THEORY OF LIGHT
peared to be the same above and below, after which the fringes
remained less intense in the lower* part, although their colors
were always much purer.
If, now, the only inflected light were that which grazed the
edges of the opaque bodies, the fringes of the upper part ought
to be sharper and ought to show purer colors than those of
the lower part ; for* the first are produced by the meeting of
two systems of waves which have their centres upon the edges
AC and BD, while the others are formed by the meeting of
four systems of waves having their origin at the edges C'E',
CE, DF, DF'; and this would necessarily diminish the differ-
ence of intensity between the dark and bright bands, in the
case of homogeneous light, or the purity of the colors, in the
case of white light, because the fringes produced by the rays
reflected and inflected at C'E' and DF would not exactly coin-
cide with those produced by the meeting of rays coming from
CE and D'F'. Now experiment shows, as I have just said, that
exactly the reverse of this is true. One might explain on this
same hypothesis how it happens that the shadow of ECDF is
much brighter than that of ABDC arising from the double
source of light presented by the two edges of each slit ; but from
this it would follow that the lower part ought always to be
brighter, and we have just seen that this is not the fact.
28. From the experiments which I have just described it is
evident that we cannot attribute the phenomena of diffraction
solely to rays which graze the edge of the body ; but we must,
on the contrary, admit that there is an infinitude of other rays
sensibly distant from the body and yet deviated from their
original direction, so as to meet and form these fringes.
29. The spreading out of a pencil of light in passing through
a very narrow opening shows in an even better manner that the
inflection of light occurs at a sensible distance from the edges
* In order that this difference of intensity between the two parts of the
shadow shall be as marked as possible, it is necessary that the slits CE aud
DF be very narrow as compared with the distance which separates them,
and that the sheet of copper should be as far away as possible from the
luminous point.
[In repeating this experiment, it will be found very convenient to use, in-
stead of sheet copper, an unfieed photographic plate : lantern slide is best.
The two slits can be cut either with a pocket-knife or, better still, by means of a
dividing engine.]
95
MEMOIRS ON
of the diaphragm. It was in the consideration of this phenom-
enon that I discovered the error into which I had previously
fallen. When one brings the edges of two opaque screens very
close together in front of a luminous point in a dark room, he
observes that the region illuminated by the aperture greatly in-
creases. Such screens were Newton's two knife-edges. I shall
suppose, as in his- experiment, that the edges of the aperture
are thin and perfectly sharp ; not that this has any effect upon
the phenomena, but simply for making clearer the conclusion
which is to be drawn. The small number of rays which graze
these sharp edges, being spread out over a rather large area,
could produce only an insensible amount of illumination, or, at
most, an exceedingly feeble light, and in the midst of it one
ought to see a bright band traced out by the pencil of direct
rays. This, however, is not the fact ; for white light of almost
uniform intensity fills a space much larger than the projection
of the aperture,* and gradually grows weaker, shading into the
dark bands of the first order. It was doubtless in order to ac-
count for the large amount of light inflected that Newton sup-
posed the action of the body upon rays of light to extend to
sensible distances, but this hypothesis will not bear careful
scrutiny.
30. If the expansion of a pencil of light which passes through
a narrow opening were brought about by attractive and repul-
sive forces having their origin at the edges of the aperture, the
intensity of these forces, and consequently their effect upon
the light, would necessarily vary with the nature, the mass, and
the surface of the edges of the screen. All forces produced by
a body acting at a sensible distance and taking their rise in any
considerable extent of its mass or its surface would depend
upon the relative positions and upon the number of particles
contained within this sphere of activity, or, what is the same
thing, upon the shape of the surface. If, then, the phenomena
in question are due to the action of such forces, one would ex-
pect that, on placing a sharp body opposite a round body, the
rays of light would be inflected more to the one side than the
* The illuminated space increases so rapidly in comparison with the width
. of the conical projection of the aperture as the receiving screen recedes from
the aperture, and likewise when the aperture itself is further withdrawn
from the luminous point, that by making these two distances sufficiently
great one can obtain the same effect with an opening of any size.
96
THE WAVE-THEORY OF LIGHT
other ; but, as I have shown by a very simple experiment, this
is not the fact. I passed a pencil of rays between two steel
plates whose vertical edges were brought very close together
and were carefully straightened throughout their entire length.
A part of each edge was sharp, the rest of it round, and these
edges were arranged so that the round portion of one plate cor-
responded to the sharp one of the other. Thus, if a sharp edge
were located on the right in the upper part of the opening, an-
other was located on the left in the lower part, so that if there
had been any difference in the action of the two edges upon the
rays, I should have noticed it in the relative positions of the
upper and lower parts of the bright interval at the middle, and
especially in the fringes in that neighborhood, as they would be
interrupted at the point of passage from the sharp to the round
edge ;. but, on observing them closely, I noted that they were
perfectly straight throughout their entire length, even at the
bright interval in the middle, exactly in the same way as when
two edges of the same kind are opposed one to the other. The
experiment may be varied by using plates made of two different
substances, but the result* obtained will certainly remain the
same.
31. All the experiments which I have tried so far have
shown that the nature of the body interposed has in other
respects no more influence upon the inflection of light than is
exerted by the mass or the shape of the two edges. I shall
cite only one experiment, in which I have taken every precau-
tion necessary to determine the correctness of this principle,
which, indeed, is already well established by the preceding
experiment.
I covered an unsilvered mirror with a layer of India ink
spread over a thin layer of paper, forming together a thickness
of one-tenth of a millimeter. With a sharp point I traced two
parallel lines, and then carefully removed from between these
two lines the paper and the India ink which adhered to the
surface of the glass. This aperture, as measured by the microm-
* Messrs. Berthollet and Mains found a long while ago that the nature
of the body had no effect upon the diffraction of light. For screens they
employed plates composed of different substances, with edges made up,
for instance, of very dense metal and a piece of ivory ; but they had no
means of observation so convenient and accurate iis mine, and consequently
one might suspect that some small difference might have escaped them.
G 97
MEMOIRS ON
eter, was 1.17 mm. I then placed opposite each other two
copper cylinders, each having a diameter of 14.5 mm., and by
means of a graduated wedge I made the interval between these
cylinders also 1.1? mm. The cylinders, placed alongside the
blackened glass, were at a distance of 4.015 m. from the lumi-
nous point, and at a distance of 1.663 m. from the micrometer.
I then measured the size of the fringes produced by these two
openings, and found that they were absolutely the same. The
following are the results of the two observations made with
white light :
The distance between the darkest point ~\ mm.
of the two dark bands of the first I First reading, 1.49
order at the point of separation of [Second reading, 1.49
the brownish red from the violet . J
The interval between the two fringes 1 Firgt reading> 3.33
of the second order at the point of Y Second reading) 3>2 2
separation of red and green . '
It is hardly possible that two sets of circumstances should
differ more than these as regards the mass and the nature of
the edges of the aperture. In the one case there is a single
layer of India ink producing the fringes, for the glass to which
it adheres completely fills the aperture ; in the other case we
have two massive cylinders of copper, 14.5 mm. in diameter,
giving us an aperture whose edges have very considerable
masses and areas, but we observe 110 difference in the expan-
sion of the pencil of light.
32. It is therefore certain that the phenomena of diffraction
do not at all depend upon the nature, the mass, or the shape of
the body which intercepts the light,* but only upon the size of
the intercepting body or upon the size of the aperture through
which it passes. We must, therefore, reject any hypothesis
which assigns these phenomena to attractive and repulsive
* This is so, at least provided one does not consider the shadow too close
up to the edge of the screen, or provided the surface grazed by the rays of
light has not too large an area compared with this distance ; for in this
case it may happen that the reflected rays sensibly affect the phenomenon
as, for instance, occurs when the surface grazed by the rays is a plane
mirror of one or two decimeters in size and when one observes the fringes
at a short distance. Besides, there would then be successive diffractions
over an area too considerable for one to neglect.
THE WAVE-THEORY OF LIGHT
forces whose action extends to a distance from the body as
great as that at which rays are inflected. We are equally
unable to admit that diffraction is caused by a shallow atmos-
phere which has the same thickness as the sphere of activity
of these forces, and whose refractive index differs from that of
the neighboring medium ; for this second hypothesis, like the
first, would lead us.'to think that the inflection of light ought
to vary with the form and the nature of the edge of the screen,
and ought not to be the same, for instance, at the edge and at
the back of a razor. Now, on the emission-theory it is impos-
sible to explain in any other manner the expansion of a beam of
light passing through a narrow opening, and this expansion is
a well-established fact.* Consequently, the phenomena of dif-
fraction cannot be explained on the emission-theory.
SECTION II
33. In the first section of this memoir I have shown that
the corpuscular theory, and even the principle of interference
when applied only to direct rays and to rays reflected or in-
flected at the very edge of the opaque screen, is incompetent
to explain the phenomena of diffraction. I now propose to
show that we may find a satisfactory explanation and a general
theory in terms of waves, without recourse to any auxiliary
hypothesis, by basing everything upon the principle of Huygens
and upon that of interference, both of which are inferences
from the fundamental hypothesis.
Admitting that light consists in vibrations of the ether sim-
ilar to sound-waves, we can easily account for the inflection
of rays of light at sensible distances from the diffracting
body. For when any small portion of an elastic fluid under-
* The rise of a liquid in a capillary tube occurs between two surfaces
separated by a finite distance, although the attraction which these sur-
faces exert upon the liquid extends only to an infinitely small distance.
The reason of this is, that the molecules of the liquid, attracted by the
surface of the tube, also in their turn attract other molecules of the liquid
situated within their sphere of action, and so on, step by step ; but in the
emission-theory an analogous explanation is not admissible, for the funda-
mental hypothesis is that the luminous particles never exert any sensible
effect upon the path of neighboring particles. No interdependence of
motion is here admissible, for such an assumption would be the assumption
of a fluid medium.
99 s^0^*
S V* OJ- THB -Y
I UNIVERSITY
\f
MEMOIRS ON
goes condensation, for instance, it tends to expand in all direc-
tions ; and if throughout the entire wave the particles are dis-
placed only along the normal, the result would be that all
points of the wave lying upon the same spherical surface would
simultaneously suffer the same condensation or expansion, thus
leaving the transverse pressures in equilibrium ; but when a
portion of the wave-front is intercepted or retarded in its path
by interposing an opaque or transparent screen, it is easily seen
that this transverse equilibrium is destroyed and that various
points of the wave may now send out rays along new direc-
tions.
To follow by analytical mechanics all the various changes
which a wave-front undergoes from the instant at which a part
of it is intercepted by a screen would be an exceedingly diffi-
cult task, and we do not propose to derive the laws of diffrac-
tion in this manner, nor do we propose to inquire what hap-
pens in the immediate neighborhood of the opaque body, where
the laws are doubtless very complicated and where the form
of the edge of the screen must have a perceptible effect upon
the position and the intensity of the fringes. We propose
rather to compute the relative intensities at different points
of the wave-front only after it has gone a large number of wave-
lengths beyond the screen. Thus the positions at which we
study the waves are always to be regarded as separated from
the screen by a distance which is very considerable compared
with the length of a light-wave.
34. We shall not take up the problem of vibrations in an
elastic fluid from the point of view which the mathematicians
have ordinarily employed that is, considering only a single
disturbance. Single vibrations are never met with in nature.
Disturbances occur in groups, as is seen in the pendulum and
in sounding bodies. We shall assume that vibrations of lumi-
nous particles occur in the same manner that is, one after
another and series after series. This hypothesis follows not
only from analogy, but as an inference from the nature of the
forces which hold the particles of a body in equilibrium. To
understand how a single luminous particle may perform a large
series of oscillations all of which are nearly equal, we have only
to imagine that its density is much greater than that of the
fluid in which it vibrates and, indeed, this is only what has
already been inferred from the uniformity of the motions of
100
THE WAVE-THEORY OF LIGHT
the planets through this same fluid which fills planetary space.
It is not improbable also that the optic nerve yields the sensa-
tion of sight only after having received a considerable number
of successive stimuli.
However extended one may consider systems of wave-fronts
to be, it is clear that they have limits, and that in considering
interference we cannot predicate of their extreme portions
that which is true for the region in which they are superposed.
Thus, for instance, two systems of equal wave-length and of
equal intensity, differing in path by half a wave, interfere de-
structively only at those points in the ether where they meet,
and the two extreme half wave-lengths escape interference.
Nevertheless, we shall assume that the various systems of
waves undergo the same change throughout their entire ex-
tent, the error introduced by this assumption being inap-
preciable ; or, what amounts to the same thing, we shall
assume in our discussion of interference that these series of
light-waves represent general vibrations of the ether, and are
undefined as to their limits.
THE PROBLEM OF INTERFERENCE
35. Given the intensities and relative positions of any number
of trains of light-waves of the same length* and travelling in the
same direction, to determine the intensity of the vibrations pro-
duced by the meeting of these different trains of loaves, that is,
the oscillatory velocity of the ether particles. \
* We shall not here consider light-waves of different lengths which, in
general, come from different sources and which cannot, therefore, give
rise to simultaneous disturbances and cannot by their interaction produce
any phenomena which are uniform ; and even if they were uniform, the
rise and fall of intensity produced by the interference of two different
kinds of waves, after the manner of beats in sound, would be far too
rapid to be detected, and would produce only a sensation of constant in-
tensity. ^
f It was Mr. Thomas Young who first introduced the principle of inter-
ference into optics, where he showed much ingenuity in applying it to
special cases; but in the problems which he has thus solved he has con-
sidered, I think, only the limiting cases, where the difference in phase be-
tween the two trains of waves is either a maximum or a minimum, and has
not computed the intensity of the light for any intermediate cases or for
any number whatever of trains of waves, as I here propose to do.
101
MEMOIRS ON
Employing the general principle of the superposition of
small motions, the total velocity impressed upon any particle
of a fluid is equal to the sum of the velocities impressed by
each train of waves acting by itself. When these waves do not
coincide, these different velocities depend not only upon the
intensity of each wave, but also upon its phase at the instant
under consideration. We must, therefore, know the law ac-
cording to which the velocity of vibration varies in any one
wave, and for this purpose we must trace the wave back to the
origin whence it derives all its characteristics.
36. It is natural to suppose that the particles whose vibra-
tions produce light perform their oscillations like those of
sounding bodies that is, according to the laws which hold for
the pendulum ; or, what is the same thing, to suppose that the
acceleration tending to make a particle return to its position
of equilibrium is directly proportional . to the displacement.
Let us denote this displacement by x. A suitable function of
this displacement can then be represented by the expression
Ax + Bx* + Cx s -\-etc., since this will vanish when #=0. If, now,
we suppose the excursion of the particle to be very small when
compared with the radius of the sphere throughout which
the forces of attraction and repulsion act, we can neglect in
comparison with Ax all other terms of the series and con-
sider the acceleration as practically proportional to the dis-
tance x. This hypothesis, to which we are led by analogy,
and which is the simplest that one can make concerning the
vibrations of light particles, ought to lead to accurate results,
since the laws of optics remain the same for all intensities of
light.
Let us represent by v the velocity of vibration of a light par-
ticle at the end of a time t. We shall then have dvAxdt ;
but v=dxldt, or dt = dx/v. Substituting in the first equation,
we have vdvAxdx. Integrating, we have v*=C Ax 2 ; and
hence
A
Substituting this value of x in the first equation, we have
dv
THE WAVE-THEORY OF LIGHT
which, on integration, gives
VA Vc'
If we measure time from the instant at which the velocity
is zero, the constant C' becomes zero, and we have
1 . , v
- or v=
If we employ as unit of time the interval occupied by the par-
ticle in one complete vibration, we have v= VU sin (M).
Thus, in isochronous vibrations, the velocities for equal values
of t are always proportional to the constant V 0, which, there-
fore, measures the intensity of the vibration.
37. Let us now consider the wave produced in the ether by
the vibrations of this particle. The energy of motion in the
ether at any point on the wave depends upon the velocity of
the point-source at the instant when it started a disturbance
which has just reached this point. The velocity of the ether
particles at any point in space after an interval of time t is
proportional to that of the point-source at the instant txl\, x
being the distance of this point from the source of motion and
\ the length of a light-wave. Let us denote by u the velocity
of the ether particles. We then have
uci sin
We know that the intensity a of vibration* [oscillatory ve-
locity'} in a fluid is in inverse ratio to the distance of the wave
from the centre of disturbance; but, considering how minute
these waves are when compared with the distance which sep-
arates them from the luminous point, we may neglect the va-
riation of a and consider it as constant throughout the extent
of one or even of several waves.
38. By the aid of this expression one can compute the in-
tensity of vibration produced by the meeting of any number
of pencils of light whenever he knows the intensity of the
different trains of waves and their respective positions.
Let us first determine the velocity of a luminous particle in
a vibration which results from the interference of two trains
* [See last sentence of section 57 below. ]
103
MEMOIRS ON
of waves displaced, one with respect to the other, by a quar-
ter of a wave-length [i.e., differing in phase ~by 90], and hav-
ing intensities which we shall denote by a and a'. We shall
count time, t, from the moment at which the vibrations of the
first train begin. Let u and u' be the velocities which the first
and second trains of waves would impress upon a light particle
whose distance from the source of motion is x. We then have
ua sin 2-* It j and u' = a' sin \2-n- ( 4 ) ,
or
=- -cos
Hence, the resultant velocity 7 will be
a sin ^TT It -- I a' cos 2?r 1 1 -- j.
Putting a = A cos i and a' = A sin i, this expression may always
be placed in the following form :
A cos i sin 2-n- It j \A sin i cos 2* If J ,
or
A sin
in \2TT ( ^ ) i ' .
Thus the wave produced by the meeting of two others will be
of the same nature, but will have a different position [phase]
and a different intensity. From the equations A cos i=a and
A sin i=a', we have for the value of A (that is, for the in-
tensity of the resultant wave) -v/ rt 2 _j_#' 2 ; but this is exactly the
value of the resultant of two mutually rectangular forces, a
and a'.
From the same equations it is easily seen also that the new
wave exactly corresponds in angular position [phase] to ,the
resultant of the two mutually rectangular forces a and a'; for
the equation
U=A sin
shows that the linear displacement of this wave with respect
i\
to the first is ; but i is also the angle which the force a
104
THE WAVE-THEORY OF LIGHT
makes with the resultant A, because A cos i a. Thus we
have complete analogy between the resultant of two mutually
rectangular forces and the resultant of two trains of waves dif
fering in phase by a quarter of a wave-length.
39. The solution of this particular case for waves differing
by a quarter of a wave-length suffices to solve all other cases.
In fact, whatever be the number of the trains of waves, and
whatever be the intervals which separate them, we can always
substitute for each of them its components referred to two ref-
erence points which are common to each train of waves and which
are distant from each other by a quarter of a wave-length;
then adding or subtracting, according to sign, the intensities of
the components referred to the same point, we may reduce the
whole motion to that of two trains of waves separated by the
distance of a quarter of a wave-length; and the square root
of the sum of the squares of their intensities will be the inten-
sity of their resultant; but this is exactly the method employ-
ed in statics to ftrid the resultant of any number of forces ;
here the wave-length corresponds to one circumference in the
statical problem, and the interval of a quarter of a wave-
length between the trains of waves to an angular displacement
of 90 between the components.
40. It very often happens in optics that the intensities of
light or the particular tint which one wishes to compute is
produced by the meeting of only two trains of waves, as in the
case of [Newton's] colored rings and the ordinary phenomena of
color presented by crystalline plates. It is, therefore, well to
know the general expression for the resultant of two trains of
waves differing in phase by any amount whatever. The result
is easily predicted from the general method which I have
explained, but I think it will be wise to emphasize somewhat
the theory of vibrations, and to show directly that the wave
resulting from two others, separated by any interval whatever,
corresponds exactly in intensity and position to the resultant
of two forces whose intensities are equal to those of the two
pencils of light, making an angle with each other which bears
to one complete circumference the same ratio that the in-
terval between the two trains of waves bears to one wave-
length.
Let x be the distance from the origin of the first train of
waves to the light particle under consideration, and t the
105
MEMOIRS ON
instant for which we wish to compute its velocity. The speed
impressed by the first train of waves will be
a sin [fc (<-)],
where a represents the intensity of this ray of light.
Let us call a' the intensity of the second pencil, and let us
denote by c the distance between corresponding points on the
two trains of waves; the [oscillatory] velocity due to the second
train will then be
a' sin %TT ft - j ,
and hence the total velocity impressed upon the particle will be
a sin 2nl t \ \-\-a' sin 2*1 1 ) ,
or
an expression to which may always be given the following
form :
A cos i sin 2?r u - J A sin i cos %* u T J \,
or
arfn[fc-(<-|)-<],
where
#4-^' cos ( 2?r- ) = J cos i f
A V A/
and
a' sin f 2r | = ^4 sin i.
Squaring and adding, we have
A*=a?-i-a
Hence,
A = \/ ^ 2 + ^' 3 4- ###' cos ( 2?r J .
But this is precisely t*he value of the resultant of two forces,
a and a', inclined to each other at an angle %*.
A
106
THE WAVE-THEORY OF LIGHT
41. From this general expression it is seen that the resultant
intensity of the light vibrations is equal to the sum of intensi-
ties of the two constituent pencils when they are in perfect
agreement and to their difference when they are in exactly
opposite phases, and, lastly, to the square root of the sum of
their squares when their phase difference is a quarter of a
wave-length, as we' have already shown.
It thus follows that the phase of the wave corresponds ex-
actly to the angular position of the resultant of two forces,
a and a'. The distance from the first wave to the second is c,
to the resultant wave ^-, and from the resultant wave to the
/*7T
second is c - ; accordingly, the corresponding angles are
(* C '
2TT., i, and 2ir. i. Let us multiply the equation
A A
+ ' COS ( 2TT \=:A COS i
by sin t, and the following equation
a' sin ( %TT J =A sin i
by cos i. Subtracting one from the other, we have
a sin i=a' sin t 2?r i V
which, together with
a' sin ( 27T j = A sin i,
gives the following proportion:
I 2?r i\: sin i : sin 2?r- : : a : a' : A.
42. The general expression, A sin gyff -Y_t|, for the
velocity of the particles in a wave produced by the meeting of
two others shows that this wave has the same length as its
components and that the velocities at corresponding points are
proportional, so that the resultant wave is always of the same
nature as its components and differs only in intensity that is
to say, in the constant by which we must multiply the velocities
in either of the components in order to obtain the correspond-
107
sn
MEMOIRS ON
ing velocities in the resultant. In combining this resultant
with still another new wave., one again arrives at an expression
of the same form a remarkable property of a function of this
kind. Thus in the resultant of any number of trains of waves
of the same length the light particles are always urged by veloc-
ities proportional to those of the components at points located
at the same distance from the end of each wave. [This is seen
~by multiplying each of the last three terms in the preceding pro-
portion by sin tat. For then,
a sin wt : a 1 sin wi : A sin ut ::a : a' :A'
: : constant ratio. ]
APPLICATIONS OF HUYGENS'S PRINCIPLE TO THE PHENOMENA
OF DIFFRACTION
43. Having determined the resultant of any number of trains
of light-waves, I shall now show how by the aid of these inter-
ference formulae and by the principle of Huygens alone it is
possible to explain, and even to compute, all the phenomena
of diffraction. This principle, which I consider as a rigorous
deduction from the basal hypothesis, maybe expressed thus:
The vibrations at each point in the wave-front may be considered
as the sum of the elementary motions which at any one instant
are sent to that point from all parts of this same wave in any
one of its previous* positions, each of these parts acting inde-
pendently the one of the other. It follows from the principle
of the superposition of small motions that the vibrations pro-
duced at any point in an elastic fluid by several disturbances
are equal to the resultant of all the disturbances reaching this
point at the same instant from different centres of vibration,
whatever be their number, their respective positions, their
nature, or the epoch of the different disturbances. This gen-
eral principle must apply to all particular cases. I shall sup-
pose that all of these disturbances, infinite in number, are of
the same kind, that they take place simultaneously, that they
*I am here discussing only an infinite train of waves, or the most gen-
eral vibration of a fluid. It is only in this sense that one can speak of two
light, waves annulling one another when they are half a wave-length apart.
The formulae of interference just given do not apply to the case of a sin-
gle wave, not 'o mention the fact that such waves do not occur in nature.
108
THE WAVE-THEORY OF LIGHT
are contiguous and occur in the single plane or on a single
spherical surface. I shall make still another hypothesis with
reference to the nature of these disturbances, viz., I shall sup-
pose that the velocities impressed upon the particles are all
directed in the same sense, perpendicular to the surface of the
sphere,* and, besides, that they are proportional to the compres-
sion, and in such a/ way that the particles have no retrograde
motion. I have thus reconstructed a primary wave out of par-
tial [secondary] disturbances. We may, therefore, say that the
vibrations at each point in the wave-front can be looked upon
as the resultant of all the secondary displacements which reach
it at the same instant from all parts of this same wave in some
previous position, each of these parts acting independently one
of the other.
44. If the intensity of the primary wave is uniform, it fol-
lows from theoretical as well as from all other considerations
that this uniformity will be maintained throughout its path,
provided only that no part of the wave is intercepted or re-
tarded with respect to its neighboring parts, because the re-
sultant of the secondary displacements mentioned above will
be the same at every point. But if a portion of the wave be
stopped by the interposition of an opaque body, then the in-
tensity of each point varies with its distance from the edge of
the shadow, and these variations will be especially marked near
the edge of the geometrical shadow.
Let be the luminous point, AG the screen, AME a wave
which has just reached A and is partly intercepted by the
opaque body. Imagine it to be divided into an infinite num-
ber of small arcs Am', m'm, wM, Mw, nn', n'n", etc. In order
to determine the intensity at any point P in any of the later po-
sitions of the wave BPD, it is necessary to find the resultant of
*It is possible for light- waves to occur in which the direction of the ab-
solute velocity impressed upon the particles is not perpendicular to the
wave surface. In studying the laws of interference of polarized light, I
have become convinced since the writing of this memoir that light vibra-
tions are at right angles to the rays or parallel to the wave surface. The
arguments and computations contained in this memoir harmonize quite
as well with this new hypothesis as with the preceding, because they are
quite independent of the actual direction of the vibrations and pre-sup-
pose only that the direction of these vibrations is the same for all rays
belonging to any system of waves producing fringes.
109
MEMOIRS ON
Fig. 16
all the secondary waves which each of these
portions of the primitive wave would send to
the point P, provided they were acting inde-
pendently one of the other.
Since the impulse communicated to every'
part of the primitive wave was directed along
the normal, the motion which each [part of the
wave] tends to impress upon the ether ought
to be more intense in this direction than in
any other ; and the rays which would emanate
from it, if acting alone, would be less and less
intense as they deviated more and more from
this direction.
45. The investigation of the law according to which their
intensity varies about each centre of disturbance is doubtless a
very difficult matter ;* but, fortunately, we have no need of
knowing it, for it is easily seen that the effects produced by
these rays are mutually destructive when their directions are
sensibly inclined towards the normal. Consequently, the rays
which produce any appreciable Qifect upon the quantity of
light received at any point P may be regarded as of equal in-
tensity, f
Let us now consider the rays EP, FP, and IP, which are sen-
* [This is the problem solved by Stokes; Math, and Phys. Papers, vol. ii., p.
243.]
f When the centre of disturbance has been compressed, the force of ex-
pansion tends to thrust the particles in all directions ; and if they have no
backward motion, the reason is simply that their initial velocities forward
destroy those which expansion tends to impress upon them towards the
rear ; but it does not follow that the disturbance can be transmitted only
along the direction of the initial velocities, for the force of expansion in a
perpendicular direction, for instance, combines with a primitive impulse
without having its effect diminished. It is clear that the intensity of the
wave thus produced must vary greatly at different points of its circumfer-
ence, not only on account of the initial impulse, but also because the com-
pressions do not obey the same law around the centre of disturbance ; but
the variations of intensity in the resultant wave must follow the law of
continuity, and may, therefore, be considered as vanishing throughout a
small angle, especially along the normal to the primitive wave. For the
initial velocities of the particles in any direction whatever are proportion-
al to the cosine of the angle which this direction makes with that of the
normal, so that these components vary much less rapidly than the angle
so long as the angle is small.
110
THE WAVE-THEORY OF LIGHT
sibly inclined and which meet at P, a point whose distance from
the wave EA I shall suppose to include a large number of wave-
lengths. Take the two arcs EF and FI of such a length that the
differences EP FP and FP IP shall be equal to a half wave-
length. Since these rays are quite oblique, and since a half
wave-length is very small compared with their length, these
two arcs will be very nearly equal, and the rays which they send
to the point P will be practically parallel ; and since corre-
sponding rays on the two arcs differ by half a wave-length, the
two are mutually destructive.
We may then suppose that all the rays which various parts
of the primary wave AE send to the point P are of equal in-
tensity, since the only rays for which this assumption is not
accurate produce no sensible effect upon the quantity of light
which it receives. In the same manner, for the sake of simpli-
fying the calculation of the resultant of all the elementary
waves, we may consider their vibrations as taking place in the
same direction, since the angles which these rays make with
each other are very small ; so that the problem reduces itself
to the one which we have already solved namely, to find the
resultant of any number of parallel trains of light-waves of the
same length, the intensities and relative positions being given.
The intensities are here proportional to the lengths of the il-
luminating arcs, and the relative positions of the wave trains
are given by the differences of path traversed.
46. Properly speaking, we have considered up to this point
only the section of the wave made by a plane perpendicular to
the edge of the screen projected at A. We shall now consider
it in its entirety, and shall think of it as divided by equidistant
meridians perpendicular to the plane of the figure into infinitely
thin spindles. We shall then be able to employ the same proc-
ess of reasoning which we have just used for a section of the
wave, and thus show that the rays which are quite oblique are
mutually destructive.
In the case we are now considering these spindles are indef-
initely extended in a direction parallel to the edge of the screen,
for the wave is intercepted only on one side. Accordingly the
intensity of the resultant of all the vibrations which they send
to the point P would be the same for each of them ; for, owing
to the extremely small difference of path, the rays which em-
anate from these spindles must be considered as of equal in-
Ill
MEMOIRS ON
tensity, at least throughout that region of the primitive wave
which produces a sensible effect upon the light sent to P.
Further, it is evident that each elementary resultant will differ
in phase by the same quantity with respect to the ray coming
from that point of the spindle nearest P, that is to say, from
the point at which the spindle cuts the plane of the figure.
The intervals between these elementary resultants will then be
equal to the difference of path traversed by the rays AP, m'P,
mP, etc., all lying in the plane of the figure; and their inten-
sities will be proportional .to the arcs Aw', m'm, mM., etc. In
order now to obtain the intensity of the total resultant, we
have to -make the same calculation which we have already
made, considering only the section of the wave by a plane per-
pendicular to the edge of the screen.*
47. Before deriving the analytical expression for this result-
ant I propose to draw from the principle of Huygens some of the
inferences which follow from simple geometrical considerations.
Let AG represent an opaque body suffi-
ciently narrow for one to distinguish fringes
in its shadow at the distance AB. Let be
the luminous point and BD be either the fo-
/ i i ca l pl^ne of the magnifying-glass with which
one observes these fringes or a white card
upon which the fringes are projected.
Let us now imagine the original wave di-
vided into small arcs Am, mm', m'm", etc.,
Gn, nri, n'n", ri'ri", etc. in such a way
that the rays drawn from the point P in the
Fi shadow to two consecutive points of division
will differ by half a wave-length. All of the
secondary waves sent to the point P by the elements of each
of these arcs will completely interfere with those which emanate
* So lone: as the edge of the screen is rectilinear we can determine the
position of the dark and bright bands and their relative intensities by con-
sidering only the section of the wave made by a plane which is perpendic-
ular to the edge of the screen. But when the edge of the screen is curved
or composed of straight edges inclined at an angle it is then necessary to
integrate along two directions at right ansrles to each other, or to integrate
around the point under consideration. In some particular cases this latter
method is simpler, as, for instance, when we have to calculate the intensity
of the light in the centre of the shadow produced by a screen or in the
projection of a circular aperture.
112
THE WAVE-THEORY OF LIGHT
from the corresponding parts of the two arcs immediately ad-
joining it "^ so that, if all these arcs were equal, the rays which
they would send to the point P would be mutually destructive,
with the exception of the extreme arc mA. Half of the in-
tensity of this arc would be left, for half the light sent by
the arc mm' (with which mA is in complete discordance)
would be destroyed, by half of the preceding arc m"m'. As
soon as the rays meeting at P are considerably inclined with
respect to the normal, these arcs are practically equal. The
resultant wave, therefore, corresponds in phase almost ex-
actly to the middle of mA, the only arc which produces
any sensible effect. It is thus seen that it differs in phase
by one-quarter of a wave-length from the element at the edge
A of the opaque screen. Since the same thing takes place in
the other part of the incident wave Qn, the interference be-
tween these two vibrations occurring at the point P is deter-
mined by the difference of length between the two rays sP and
tP, which take their rise at the middle of the arcs Am and Gn,
or, what amounts to the same thing, by the difference between
the two rays AP and GP coming from the very edge of the
opaque body. It thus happens that when the interior fringes
under consideration are rather distant from the edges of the
geometrical shadow, we are able to apply practically without
error the formula based upon the hypothesis that the inflected
waves have their origin at the very edges of the opaque body;
but in proportion as the point P approaches B the arc Am be-
comes greater in comparison with the arc mm', the arc mm'
with respect to the arc m'm", etc. ; and likewise in the arc mA
the elements in the immediate vicinity of the point A become
sensibly greater than the elements which are situated near the
point m, and which correspond to equal differences of path. It
happens, therefore, that the effective* ray, sP, will not be the
mean between the outside rays, mP and AP, but will more
nearly approach the length of the latter. On the other side
of the opaque body we have slightly different circumstances.
The difference between the ray GP and the effective ray tP ap-
proximates more and more nearly a quarter of a wave-length
* I have given this name to the distance of the resultant wave from the
original wave because the positions of the dark and bright bands are the
same as they would be if these effective rays alone produced them.
H 113
MEMOIRS ON
as the point P moves farther and farther away from D, so that
the difference of path traversed varies more rapidly between
the effective rays sP and tP than between the rays AP and GP ;
consequently, the fringes in the neighborhood of the point B
ought to be a little farther from the centre of the shadow than
would be indicated by the formula based upon the first hy-
pothesis.
48. Having considered the case of fringes produced by a
narrow body, I pass to the consideration of those which are
caused by a small aperture.
Let AG be the aperture through which
the light passes. I shall at first suppose
that it is sufficiently narrow for the dark
bands of the first order to fall inside the
geometrical shadow of the screen, and
at the same time to be fairly distant
from the edges B and D. Let P be the
darkest point in one of these two bands;
it is then easily seen that this must cor-
respond to a difference of one whole wave-
length between the two extreme rays AP
and GP. Let us now imagine another
ray, PI, drawn in such a way that its
length shall be a mean between the other
two. Then, on account of its marked in-
clination to the arc AIG, the point I will
fall almost exactly in the middle. We now have the arc di-
vided into two parts, whose corresponding elements are almost
exactly equal, and send to the point P vibrations in exactly
opposite phases, so that these must annul each other.
By the same reasoning it is easily seen that the darkest
points in the other dark bands also correspond to differences
of an even number of half wave-lengths between ihe two rays
which come from the edges of the aperture ; and, in like man-
ner, the brightest points of the bright bands correspond to
differences of an uneven number of half wave-lengths that is
to say, their positions are exactly reversed as compared with
those which are deduced from the interference of the limiting
rays on the hypothesis that these alone are concerned in the
production of fringes. This is true with the exception of the
point at the middle, which, on either hypothesis, must be
114
THE WAVE-THEORY OF
bright. The inferences deduced from the theory
fringes result from the superposition of all of the disturbances
from all parts of the arc AGr are verified by experiments,
which at the same time disprove the theory which looks upon
these bands as produced only by rays inflected and reflected at
the edges of the diaphragm. These are precisely the phenom-
ena which first led me to recognize the insufficiency of this
hypothesis, and suggested the fundamental principle of the
theory which I have just explained namely, the principle of
Huygens combined with the principle of interference.
49. In the case which we have just considered, where, by
virtue of a very small aperture, the dark bands of the first or-
der fall at some distance from the edges of the geometrical
shadow, it follows from theory, as well as from experiment,
that the distance comprised between the darkest points is al-
most exactly double that of the other intervals between the
middle points of two consecutive dark bands, and this is all
the more nearly true in proportion as the aperture becomes
smaller or more distant from the luminous point and from the
focus of the magnifying-glass with which one observes the
fringes ; for, by sufficiently increasing these distances one may
produce the same effects with an aperture of any size what-
ever.
But when these distances are not very great, and when their
aperture is too large for the rays producing the fringes to be
very much inclined to the wave-front, AG-, it follows that
corresponding elements of the arcs into which we have sup-
posed a wave to be divided can no longer be considered as each
equal to the other, for they are sensibly larger on the side next
the band un'der consideration. Under these conditions we
can rigorously deduce the positions of maximum and mini-
mum intensity only by computing the resultant of all the
small secondary waves which are sent out by the incident
wave.
50. But there is one very remarkable case where a knowl-
edge of this integral is not needed for the determination of
the law of the fringes by an aperture of very considerable
size. This is the case where a lens is placed in front of the
diaphragm, and brings the refracted rays to focus upon the
plane in which the fringes are observed. The problem is now
greatly simplified by the fact that the centre of curvature of the
115
MEMOIRS ON
emergent wave now lies in this plane instead of at the lumi-
nous point.
Let be the projection of the middle
point of the aperture upon this plane.
From the point as centre, and with a
radius equal to AO, let us now describe
the arc AI'Gr, which will now represent
the incident wave as modified by the inter-
position of the lens. If, now, from the
point P as centre, and with a radius AP,
we describe the arc AEF, those portions
of the luminous rays meeting at the point
P which are comprised between the arc
AI'G- and the arc AEF will be the differ-
ences of path traversed by the secondary
waves ; and, since these two arcs have
equal curvatures and are convex towards
the same side, it follows that equal differ-
ences of path will correspond to equal intervals upon the wave-
front AI'G. Let us suppose this wave divided in such a manner
that any two consecutive rays drawn through the points of di-
vision shall differ by one-half a wave-length. If, then, the point
P be located in such a way that the total number of these arcs is
even, it will no longer receive any light. For these arcs, taken
two and two, are mutually destructive, since the vibrations due
to corresponding elements are at the same time of equal in-
tensity and opposite phase. The light reaching any point
P will be a maximum when the total number of arcs is un-
even. The brightest points of the bright bands, therefore, cor-
respond to a difference of an uneven number of half wave-
lengths between the two rays coming from the edges of the
diaphragm, and the darkest points on the dark bands to a dif-
ference of an even number of half wave-lengths. Consequently,
all the dark bands will be equally spaced among themselves,
with the exception of the first two, where the interval is ex-
actly double that which separates the others. This result,
which had already been suggested by theory, I found to be
thoroughly confirmed by experiment. I shall cite only one
experiment of this kind made in homogeneous red light. In
order to bring the centre of the incident wave to the plane of
the micrometer wire, I used, instead of an ordinary lens, a
116
THE WAVE-THEORY OF LIGHT
glass cylinder, which, in order to get the full length of the
fringes, I placed with its generating line parallel to the edges
of the aperture in the diaphragm.
mm.
Size of the aperture 2.00
m.
Distance from the luminous point to the diaphragm, or a 2 507
Distance from the diaphragm to the micrometer, or b 1.140
mm.
Interval between the middle points of the two dark bands of the first
order 0.72
Interval between the band of the first order and the third 0.73
Interval between the band of the third order and the fifth 0.72
It will be observed that the first interval is double that of
the others.
I have observed that the same law holds, even at distances
which are not very great, for apertures which are much wider,
a centimeter or even a centimeter and a half ; but if we further
increase the aperture of the diaphragm, the fringes become
confused, however much care be taken to place the microme-
ter in the focus of the cylindrical lens; which goes to show
that the rays refracted by this glass vibrate in unison [in the
same phase} only within rather narrow limits, just as happens
with ordinary lenses.
51. When the aperture of the diaphragm thus backed with a
cylindrical lens is not too great, the dark and bright bands
produced are as sharp as the fringes which result from the
union of rays reflected from two mirrors. But, in the latter
case, the intensity of the light is the same for all fringes, or, at
least, whatever differences there are appear to arise merely from
the fact that the light employed is not perfectly homogeneous ;
and if it happen that the bright bands diminish in brilliancy,
the dark bands become less dark, so that the sum of the light
in one entire fringe remains practically constant. But in the
other phenomenon, as one recedes from the centre he observes
a rapid diminution of the light, which is easily accounted for
by the theory we have just explained. For, indeed, all the
rays which leave the wave-front AI'Gr and meet at the centre
of the bright band of the first order have traversed equal
paths ; so that all the small secondary waves which they bring
to this point coincide \in phase] and strengthen each other.
117
MEMOIRS ON
But this is not the case with the other bright bands. The
brightest band of the second order, for instance, corresponds
to a division of the wave AI'G- into three arcs, the extreme
rays of which differ by one-half a wave-length ; the effects
produced by two of these arcs annul each other. Consequent-
ly, this band receives light from only one-third of the incident
wave-front, while even the effect produced by this third is
somewhat diminished by the fact that there is a difference of
one-half a wave-length between the rays from its edges. A
similar process of reasoning shows that the middle of the
bright band of the third order is illuminated by only one-fifth
of the wave-front AI'G, the light of this one-fifth being still
further diminished by opposition of phase in its extreme
rays.
[Here are omitted six pages, including a geometrical discussion
of the general relations between size of aperture (or obstacle), dis-
tance of screen, distance of luminous point, etc.]
56. I have just explained the general relations between the
size of any particular fringe and the respective distances of the
obstacle from the luminous point and from the micrometer.
As we have seen, these laws may be derived from theory quite
independently of any knowledge of the integral which at each
point represents the resultant of all the secondary waves ; but
in order to find the absolute size of these fringes, it is essential
that we compute this resultant, for the positions of maxima and
minima of intensity can be determined only by a comparison
of the different values of this resultant, or at least by knowing
the function which represents it.
In order to do this, we propose to apply to the principle of
Huygens the method which we have already explained for com-
puting the resultant of any number of trains of waves when
their intensities and relative positions [phases'] are given.
APPLICATION OF THEORY OF INTERFERENCE TO HUYGENS'S
PRINCIPLE
57. Let the waves from any luminous point be partly inter-
cepted by an opaque body AG. To begin with, we shall sup-
pose that this screen is so large that no light comes around the
edge G, so that we need consider only that part of the wave
which lies to the left of the point A. Let DB represent the
118
THE WAVE-THEORY OF LIGHT
plane upon which are received the shadow and its fringes. The
problem then is to find the intensity .of the light at any point
P in this plane.
If from C as centre and with a radius CA we describe the
circle AMI, it will represent the light- wave at the instant it
is partly intercepted by the opaque body.
It is from this position of the wave that I
have computed the resultant of the sec-
ondary waves sent to the point P.. If we
consider the wave in an earlier position,
say A'MT, i.t then becomes necessary to
calculate the effect of the obstacle on each
of the secondary waves arising from the
arc A'MT ; and if we consider the wave
in a later position, say A"M"I", it becomes
necessary to first determine the intensities
of its various points, for they are no longer
equal, having been changed by the inter-
position of the screen. In this case the
computation is vastly more complicated,
possibly quite impracticable. If, however,
we consider the wave at the instant it
Fig. 19
reaches A, the process is simple ; for then all parts of the wave
have the same intensity. Not only so, but none of the second-
ary waves are now affected by the opaque screen. However
numerous the subdivisions into which we may consider these
elementary waves divided, it is evident that the number will
be the same for each, since they are transmitted freely in all
directions. And, therefore, we need only consider the axes of
these pencils of split rays?, e., the straight lines drawn from
the various points on the wave AMI to the point P. The dif-
ferences of length in these direct rays are the differences of
path traversed by the elementary or partial resultants meeting
at P.*
In order to compute the total effect, I refer these partial re-
sultants to the wave emitted by the point M on the straight line
CP, and to another wave displaced a quarter of a wave-length
with reference to the preceding. This is the process already
employed (p. 101) in the general solution of the interference
* [Afoot-note is here omitted.']
119
MEMOIRS ON
problem. We shall consider only a section of the wave made
by a plane perpendicular to the edge of the screen, and shall
indicate by dz an element, nn', of the primary wave, and by z its
distance from the point M. These, as I have shown, suffice to
determine the position and the relative intensities of the bright
and dark bands. The distance nS included between the wave
AMI and the tangential arc, EMF, described about the point P
as centre is | = ^ where. a and b are, as before, the distances
ab
CA and AB. If we denote the wave-length by X, we have for
the component in question, referred to the wave leaving the
point M, the following expression
dz cos ( TT - ) ;
\ abX J '
while for the other component,* referred to a wave displaced a
quarter of a wave-length from the first, we have
If, now, we take the sum of all similar components of all the
other elements, we shall have
r dz cos , + and r dz siri
J ab\ I J
ab\
Hence the intensity of the vibration at P resulting from all
these small disturbances is
The intensity of the sensation, being proportional to the
square of the speeds of the particles, is
This is what I have called the intensity of the light in order to
conform to ordinary usage, while reserving the expression in-
tensity of vibration to designate the speed of an ether particle
during its oscillation.
* [ TJiese expressions for amplitude follow directly from sec. 40, when in the
ke a.
120
general expression for velocity we make ao, a'=dz, and c=^- .]
THE WAVE-THEORY OF LIGHT
58. In the case we are now considering, where the body, AG,
is so large that we can neglect any light coming around the
edge G, the integration extends from A to infinity on the side
towards I. This integral naturally divides into two parts, one
extending from A to M, the other from M to infinity. This
latter integral remains constant, while the former varies with
the position of the point P. This variation, indeed, is the de-
termining factor in the size and relative intensity of the bright
and dark bands.
The integrals
r, / z*(a+b)\ -,/,.
Jdz cos (T-^) and / dz sin
may be evaluated in finite terms when the limits of z are taken
at zero and infinity ; but between any other limits their values
can be expressed only in terms of a series or by means of par-
tial integration.
The latter method seems to me more convenient, and I have,
therefore, employed it in the computation of the following
table, where the limits of integration are taken so close together
that we can neglect the square of half the arc included between
them.*
* Let * and i+t be the narrow limits between which it is proposed to in-
tegrate dv cos q^ and dv sin qv' 2 . Neglecting the square of -, we then
&
timl the following approximate values for these integrals:
These are the formulae which I have used in the computation of the table.
When the limits are sufficiently narrow for us to neglect < 2 instead of
5 j j the following still simpler formnlae may be employed :
/ dv cos 9^=2^ sin qi (i+2t)-sin qi*
i+t
/** i r
I dv sin qv*=p-T-\ cos otf (i
/ 2iq\
/t+t L
121
MEMOIRS ON
This arc here amounts to ^ of a quadrant, since this fur-
nishes results of an accuracy greater than is attainable in the
observations. In place of the integrals mentioned above, I have
substituted fdv sin qv" 1 and fdv cos qv*, where q stands for
quadrant or -.
/v
To pass from one of these forms to the other is a simple
matter.
[Following is a derivation of these formulae, which Verdet found in one of
Fresnel's journals.
Let the limits of integration he denoted by a and a -f 2p.
Put v=a+p-\-u. Then du=dv ; and when v=a, u=p; but when
v=a+2p, u=+p.
Substituting for v,
I dv cos qv*= I du cos q\ u' 1 + 2(a +p)u + (a +_^) 2 1.
If the limiting values of u are +p and p, we may take p so smalL say
Y 1 ^, that we may neglect its square, u 1 . We then have
C a i* +p \ 1
I dv cos qv*= I du cos q\ Zu(a+p)+(a+p)*
Ja+* P J -p
2q(a+p)
122
THE WAVE-THEORY OF LIGHT
TABLE OF THE NUMERICAL VALUES OF THE INTEGRALS
fdv cos qv* and J dv sin qv*.*
Limits
of
Integrals
J'dv cos qv*
J'dv sin gt> 3
Limits
of
Integrals
fdv cos 5 a
J'dv sin qv*
From v=W
From v=W
to fl=(K10
0.0999
0.0006
to 0=2*. 90
0.5627
0.4098
to =0.20
0.1999
0.0042
to 3.00
0.6061
0.4959
0.30
0.2993
0.0140
3.10
0.5621
0.5815
0.40
0.3974
0.0332
3.20
0.4668
0.5931
0.50
0.4923
0.0644
3.30
0.4061
0.5191
0.60
0.5811
0.1101
3.40
0.4388
0.4294
0.70
0.6597
0.1716
3.50
0.5328
0.4149
0.80
0.7230
0.2487
3.60
0.5883
0.4919
0.90
0.7651
0.3391
3.70
0.5424
0.5746
1.00
0.7803
0.4376
3.80
0.4485
0.5654
1.10
0.7643
0.5359
3.90
0.4226
0.4750
1.20
0.7161
0.6229
4.00
0.4986
0.4202
1.30
0.6393
0.6859
4.10
0:5739
0.4754
1.40
0.5439
0.7132
4.20
0.5420
0.5628
1.50
0.4461
0.6973
4.30
0.4497
0.5537
1.60
0.3662
0.6388
4.40
0.4385
0.4620
1.70
0.3245
0.5492
4.50
0.5261
0.4339
1.80
0.3342
0.4509
4.60
0.5674
0.5158
1.90
0.3949
0.3732
4.70
0.4917
0.5668
2.00
0.4886
0.3432
4.80
0.4340
0.4965
2.10
0.5819
0.3739
.4.90
0.5003
0.4347
2.20
0.6367
0.4553
5.00
0.5638
0.4987
2.30
0.6271
0.5528
5.10
0.5000
0.5620
2.40
0.5556
0.6194
5.20
0.4390
0.4966
2.50
0.4581
0.6190
5.30
0.5078
0.4401
2.60
0.3895
0.5499
5.40
0.5573
0.5136
2.70
0.3929
0.4528
5.50
0.4785
0.5533
2.80
0.4678
0.3913
* From the text, and also from the first column of this table, one would
be led to think that the second and third columns in the table give the
values of the
values of v :
integrals / a
v o
dv cos V 2 and
/
dv sin -
for the following
etc.
That this is not the case, however, may be shown by using the approxima-
tion formulae of Fresnel to compute any pair of consecutive values of
123
MEMOIRS ON
Either of the integrals i dv cos qv* and jdv sin qv* taken
from zero to infinity have the value -3-. We may thus by the
aid of the above table find the intensity of light corresponding
to any given position of the point P, or, what is the same
thing, corresponding to any definite value of v, where v is one
limit of integration and infinity the other. We have only to
take from the table the values of J civ cos qv* and / dv sin qv*,
using the value of v as an argument, then add to each , and
finally take the sum of their squares.
59. Simple inspection of this table shows a periodic change
in the intensity of light as one leaves the geometrical shadow.
To obtain the values of v corresponding to maxima and minima,
i. e., the brightest and darkest points in the respective bright
and dark bands, I take from the table the numbers which most
nearly correspond to them and then compute the correspond-
ing intensities. Finally, by means of these data and a simple
formula of approximation, I determine with sufficient accu-
racy the values of v. which give maxima and minima.
Let us represent by * the approximate value of v taken di-
rectly from the table, by / and Y the corresponding values of
_[_ jdv cos qv* and i+jdv sin qv*, and by t the small arc by
which v must be increased in order to give the maximum or
minimum of light. Neglecting the square of t, we find that
the following formula gives the value of t which yields a
maximum or a minimum.
f /-a, o'Al- %qil sin qi* _
n i A. -
[A foot-note containing the derivation of this expression is
here omitted.}
If in this formula we substitute the numbers taken from the
table, we obtain the following results :
either integral. The successive values of v employed in the first column
are
6=0.1,
0=0.2,
v=0.3,
etc.
The same remark applies to the following tables. [E. Verdet.]
124
THE WAVE-THEOKY OF LIGHT
TABLE OF MAXIMA AND MINIMA FOR EXTERIOR FRINGES
AND OF THE CORRESPONDING INTENSITIES
Values of V
Intensities
of Light
Maximum of 1st order
1 2172
2 7413
Minimum of 1st order
1 8726
1 5570
Maximum of 2d order. . . .
2 3449
2 3990
Minimum of 3d order
2 7392
1 6867
Maximum of 3d order
3 0820
2 3022
Minimum of 3d order
3 3913
1 7440
Maximum of 4th order
3 6742
2 2523
Minimum of 4th order
3 9372
1 7783
Maximum of 5th order
4 1832
2 2206
Minimum of 5th order
44160
1 8014
Maximum of 6th order
4 6369
2 1985
Minimum of 6th order
4 8479
1 8185
Maximum of 7th order
5 0500
2 1818
Minimum of 7th order . . .
5 2442
1 8317
It is to be observed that here none of the minima become
zero, as in the case of Newton's rings, or in fringes produced
by the meeting of two beams of light of equal intensities ; here
the difference between maxima and minima diminishes as one
goes farther away from the edge of the opaque screen.
This explains why the fringes which border shadows are not
so bright or so numerous as the colored rings, or as the bands
produced by the reflection of a luminous point in two slightly
inclined mirrors.
60. To employ the above table in computing the size of the
exterior fringes, we must first recall the substitution of the in-
tegrals / dv cos qv* and / dv sin qv* for the integrals * in
question,
cos -
_ .
(fesin
whence
and
[In what follows Fresnel replaces = by q. the initial letter of "quadrant."]
125
MEMOIRS ON
Therefore
fdz cos fa ^+*>) .__ /~^2 fdv cos
J \ ab\ I V 2(^fS)J
and
Also
c "
Now the factor ^ rr- is constant ; whence we infer that
4{a+o)
these two quantities,
and
/ r , A 2 / r ,
sin
I I dv cos gt^J 4- ( ^
will each reach their maximum or minimum values at the same
time. Let us now denote by n the value of v which yields a
maximum or minimum value for these integrals ; the corre-
sponding value of z will then be
. , ##X
Z^r
The size of the fringe, x, [that is, its distance from the edge of
the opaque screen] then follows from the proportion
a : z ::
whence
or, substituting for z,
This radical, it may be remarked, is exactly the distance
between the edge of the geometrical shadow and that point
126
THE WAVE-THEORY OF LIGHT
which corresponds to a difference* of a quarter of a wave-length
between the direct ray and the ray coming via the edge of the
opaque screen. But this is precisely what might have been
predicted, inasmuch as the corresponding value of v [viz., the
quadrant] has been taken as unity in the table of numerical
values of jdv cos qv 2 and / dv sin qv 2 .
If in the formula
V 2a~
we substitute for n the value corresponding to a minimum of
the first order i. e., to the darkest part of the first dark band,
we have
z=1.873
61. If, however, we assume that the fringes are produced
by the meeting of the direct rays with those reflected at the
edge of the opaque screen, and if we suppose further that the
reflected rays lose half a wave-length, we have [section 20] for
the same band
or x=
Accordingly, these two quantities are in the ratio of 2 to 1.873.
The second is measurably smaller than the first, differing as
they do by nearly a fifteenth; so that by accurate observations
on homogeneous light of well-determined wave-length one
might distinguish between these two theories by means of ex-
periment.
62. The method which I at first thought best adapted to
the determination of the wave-length was to measure the size
of the fringes produced by two mirrors slightly inclined to one
another, and also the distance between the two images of the
luminous point; but the slightest curvature in the mirrors
diminishes the accuracy, and so I preferred to use the bands
* [The general expression for this difference of path, d, has been given
above section 20.
ax*
~2b(a+b)'
Put d=, and we haw the value of x in question.]
127
MEMOIRS ON
produced by a narrow slit combined with the cylindrical lens
of which I have already spoken. We have already found that
the distance between any two consecutive dark bands, either to
the right or the left of the aperture, is , where X is the wave-
c
length, c the width of the aperture, and b its distance from the
micrometer. The distance between the two dark bands of
the first order is just twice this amount. With these data, it
is an easy matter to determine X from measurements on the
fringes.
The following table gives the results of five observations of
this kind, together with the wave-lengths computed from them.
In order to describe all the conditions of the experiment, I
include in the table the various values of #, the distance from
the luminous point to the screen, even though this quantity is
not employed in the calculation. These measures have been
made with practically homogeneous red light, obtained by use
of the same colored glass which, for the purpose of getting
results that are comparable, I have used in all my observa-
tions. Each measure recorded in the table is the mean of four
observations.
Number of
Distance from
luminous point
to diaphragm
Distance from
diaphragm to
micrometer
Size of
aperture
bands in-
c
eluded in each
Mean of
micrometer
measures
Wave lengths
computed from
these measurer
measure
a
b
m.
m.
mm.
mm.
mm.
2.507
1.140
2.00
6
2.185
0.000639
2.010
1.302
4.00
10
2.075
0.000637
2.010
1.302
3.00
8
2.222
0.000640
1.304
2.046
3.00
8
3.466
0.000635
1.304
2.046
2.00
6
3.922
0.000639
Sum of these results^ 0.003190
Fifth of the sum or mean 0.000638
The agreement of these results with each other is very satis-
factory, differing, as they do, among themselves by less than one
per cent. Accordingly I have adopted the value 0.000.638mm.,
and have employed it in all my comparisons of theory and ex-
periment.
[Four pages devoted to verifications of the results in this table
are here omitted.]
128
THE WAVE-THEORY OF LIGHT
65. Having thus, by means of simple and well-known meth-
ods, verified the wave-length determination made with the
single slit and cylindrical lens, I have used this same value to
compute the exterior fringes by use of the formula
S = IH> + *)* X ,
2a
substituting for w those values of v which, according to the
table, give maxima and minima.
The table on page 130 summarizes the results of calculation
and observation. In my experiments I have measured the
positions of the minima only, because I considered this a suf-
ficient test of the theory, and because my eye can determine the
darkest point of a dark band with greater accuracy than it can
set upon the brightest point of a bright baud.
[Only every fifth observation in the table is reproduced.]
More striking agreement between theory and experiment
could scarcely be expected. W^hen these small differences are
compared with the quantities measured, and when the great
variations in the quantities a and b are noted, one can no longer
doubt that the integrals which led to these Jesuits accurately
describe the law governing the phenomena. The probability
in favor of the new theory is still further increased by the fact
that the wave-length here employed has been deduced from
different and simpler phenomena.
[Four pages, devoted to a description of some experimental
precautions and to a computation of exterior fringes on Young's
hypothesis of reflection from the edge of the opaque body, are
here omitted.}
68. We have just seen that both the formation and the posi-
tion of the exterior fringes can be explained in a satisfactory
manner by considering them as produced by the meeting of an
infinitely great number of secondary waves which originate on
that part of the primary wave which is not intercepted by the
opaque screen. From this view it follows that the light which
is inflected into the shadow ought not to produce any bright or
dark band, but ought to diminish gradually in intensity, pro-
vided the screen is sufficiently large to allow no light to go
around the other side; and this is true, even though this
inflected light, like that which gives rise to the exterior fringes,
is the resultant of an infinitude of secondary waves. This will
i 129
MEMOIRS ON
TABLE COMPARING THE RESULTS OF EXPERIMENT WITH
THOSE OF THEORY
EXTERIOR FRINGES IN RED LIGHT OP WAVE-LENGTH 0.000638 MM.
Number
of obser-
vation
Distance
from lumi-
nous point
to 0]),'lc|llf
screen
a
Distance from
op;. quo body
to microme-
ter
/)
Order of
dark
band
Distil noe from darkest
point in each band to
edge of geometrical
shadow
Difference
Observed
Computed
in.
m.
mm.
mm.
ri
2.84
283
-1
3
414
4 14
1
0.1000
07985
is
5.14
5.13
-1
14
5.96
5.96
15
6.68
6.68
fl
1.05
1 05
N
1.54
1 54
5
0.510
0.501
i 3
1.90
1.91
+1
4
2.21
222
+1
15
2.49
2.49
fl
2.59
259
2
3.79
3.79
10
1.011
2.010
^3
4.68
4.69
+1
14
5.45
5.45
15
6.10
6.11
+1
fl
0.54
0.55
+1
2
0.80
0.81
+1
15
3.018
0.253
I 3
1.00
1.00
I 4
1.16
1.16
15
1.31
1.31
fi
3.19
3.22
+ 3
2
4.70
4.71
+ 1
20
3.018
3995
\z
5.83
5.84
+ 1
4
6.73
6.78
+ 5
U
7.58
7.60
+ 2
fl
1.13
1.14
+ 1
2
1.67
.1.67
25
6.007
0.999
<S
2.06
2.07
+ 1
I 4
2.40
2.49
16
2.69
2.69
be easily seen by looking at the table below, which gives the
intensity of light in the shadow for rays inflected at various
angles. These intensities have been computed by means of the
table giving the numerical values of the integrals
/ dv cos qv 2 and / dv sin qv 2 ,
130
THE WAVE-THEORY OF LIGHT
by taking the sums of the squares of the corresponding numbers
and subtracting ^. In spite of the inaccuracy introduced by
the method of partial integration employed in the first table,
it is seen that the intensity of light diminishes rapidly as v
increases, presenting none of the maxima and minima observed
outside of the shadow.
INTENSITIES OF LIGHT DIFFRACTED INTO THE SHADOW
UNDER DIFFERENT ANGLES
Values of v
Corresponding
intensities
Values of v
Corresponding
intensities
q
0.10
04095
q
2.90
0.0121
020
03359
3.00
0.0113
030
0.2765
3.10
0.0105
0.40
0.2284
3.20
0.0098
0.50
0.1898
3.30
00092
060
0.1586
3.40
0.0087
0.70
0.1334
3.50
0.0083
0.80
0.1129
3.60
0.0079
0.90
0.0962
3.70
0.0074
1.00
0.0825
3.80
00069
1.10
00711
3.90
0.0066
1 20
0.0618
4.00
0.0064
1 30
0.0540
4.10
0.0061
1.40
0.0474
4.20
0.0057
1.50
0.0418
430
0.0054
1.60
0.0372
4.40
0.0052
1.70
00332
4.50
0.0051
1.80
0.0299
4.60
00048
1 90
0.0271
4.70
0.0045
2.00
0.0247
4.80
00044
210
0.0226
4.90
0.0043
2.20
0.0207
5.00
0.0041
2.30
0.0189
5 10
00038
2.40
0.0173
520
00037
2.50
00159
5.30
0.0036
260
0.0147
5.40
0.0035
2.70
0.0137
5.50
0.0033
2.80
0.0129
As usual, a and b represent the distances of the screen from
the luminous point and from the plane in which the shadow lies,
while x is the distance from the edge of the geometrical shadow
to the point in this plane under consideration, so that we have
131
MEMOIRS ON
and therefore
69. By the aid of these formulae we can find the value of the
distance x or the angle x/b of the inflected ray corresponding
to the various values of #; and vice versa, if x or the slant x/b
be given, we can find v, and thus determine the intensity of the
inflected light. One striking inference from this formula,
\ a ~\ ' , is that the values of x are not directly pro-
6(1
portional to those of b, but are related to them as the ordinates
of a hyperbola are to its abscissas. It thus follows that points
of equal intensity along the edge of the geometrical shadow do
not lie upon a straight line as we vary b, but upon a hyperbola
of appreciable curvature, like the corresponding loci in exterior
fringes.
70. I have not yet succeeded in verifying by direct experi-
ment the ratios of intensity in the inflected light as predicted
by the theory of interference applied to the principle of Huy-
gens. A measurement of this kind is very difficult [foot-note
omitted], and I hardly think that one would be able to reach
the same accuracy as in the determination of the darkest and
brightest points in fringes. The results already obtained for
fringes, however, appear to me as verifications indirect, it
must be confessed of these very ratios of intensity; for
whenever the positions of maxima and minima have been de-
duced from the general expression for the intensity of light
and have been found to coincide accurately with experiment,
it becomes more and more probable that this integral cor-
rectly represents all the variations of intensity in the inflected
light.
71. In the case of exterior fringes one may, as we have seen,
use the table of maxima and minima to compute the positions
of the darkest and brightest points in the dark and bright bands
for all values of a and b. This, however, is not the case with
regard to the interior fringes in the shadow of a narrow body
or in the case of a narrow aperture. The limits of the integral
vary all the while, and it is therefore impossible to give general
results applicable to every case, so that one is obliged to deter-
mine the maxima and minima for each particular case, using
132
THE WAVE-THEORY OF LIGHT
the table, which gives the numerical values of j dv cos qv 2 and
/ dv sin qv 2 . I propose to give the results of all the computa-
tions of this kind which, up to the present, I have made for the
purpose of testing the theory. They are very long, and I have
not been able to finish as many as I had desired, but this lack in
quantity is, perhaps, compensated by the variety of the cases
which I have studied, for in trying the theory on the observa-
tions, I have, by preference, selected cases in which the disposi-
tion of the fringes is somewhat unusual.
72. And, first, I propose to consider the case of a narrow
aperture which presents at once the
case of exterior and interior fringes.
Let C be a luminous point, AG
a narrow aperture whose edges A
and G are straight and parallel ;
let BD be the central projection of
this aperture upon the plane in
which the fringes are observed,
and P be any point in this plane
at which the intensity is to be de-
termined. For this purpose we
must integrate
// g*
dz cos ( 2q
and
''(ab]\
ab\ I
I dz sin l*2q
ab\ 1
between the limits A and G, afterwards taking the sum of the
squares of these integrals. This will give us the intensity of
light at the point P, but we must not forget that the origin
from which z is measured lies upon the direct line CP, and
that therefore the two limits A and G correspond to z=MG
and z= AM. The next step is to compute the corresponding
values of v from the formula
/2
v=z y-
or
133
MEMOIRS ON
where x is the distance from the point P to the edge of the
geometrical shadow. From the table of integrals,
/ dv cos qv 2 and / dv sin qv 2 ,
we then find the values most nearly corresponding to those of v.
Let us call t the difference between the value for which the
integral is desired and the. number i [for which it is computed]
in the table. The proper integral can then be found by means
of the formulas of approximation.
/i+t / i -|
dv cos qv 2 = I dv cos qv* + (sin qi (i'-\-*Zt) sin qi 2 )
Jo 2iq
/ i+t r l i
dv sin qv 2 =.J dv sin qv 2 -f- T-T- (cos qi (i + '2t)-{-cos qi 2 ).
Having made this computation for the two values of v which
represent the edges A and G- of the aperture, if the point is in-
side we add these integrals; if, however, it is on the outside,
we subtract them ; and, lastly, take the sum of the squares of
the two numbers thus found. In like manner, one finds the
intensity of the light for any other point whose position is
given, and in comparing these various results the positions of
maxima and minima may be found.
[Half a page concerning the method of interpolation is here
omitted.]
73. In order to apply this method of computation to the ob-
servations, I first determined the tabulary value of c, that is to
say, the size of the aperture, by means of the formula
f , =
/2(a
V
so that I thus obtained the tabular interval between the limits.
By a few easy trials I find between what numbers of the table
the maxima and minima lie ; afterwards I determine their posi-
tion more accurately by the process which I have just de-
scribed. Having thus obtained the values of v corresponding to
maxima or minima, I subtract them from the half of the tabulary
value of c, in order to refer them to the middle of the aperture.
And, last of all, the for inn hi
2a
gives me the distance of these same maxima or minima from
the middle of the projection of aperture, which is the point of
reference used in my observations.
134
THE WAVE-THEORY OF LIGHT
COMPARISON OP THEORY AND EXPERIMENT
REGARDING THE POSITIONS OF MAXIMA AND MINIMA IN THE FRINGES
PRODUCED BY A NARROW APERTURE
Number of bright
or dark bands
counted from mid-
dle
Approximate
value of*
counted from
edge of aper-
ture
Corresponding
intensity
Value of* cor-
responding to
itftxitH't or
minim* i
Distance of maxima
or minima from
projection of centre
of aperture
Difference
Computed
Observed
FIRST OBSERVATION
m. m. mm.
a=2.QW ; 6=0.617; c=0.50 ; tabulary value of e=1.288
mm.
mm.
mm.
( +0.812
0.03495 )
1. Minimum
4 4-0.912
0.01645 L
+0.913
0.79
0.77
+0.02
( +1.012
0.03406 ) .
( +2.412
0.00238 )
2. Minimum
] +2.512
0.00235 }
+2.463
1.58
1.58
0.00
( +2.612
0.00541 )
THIRD OBSERVATION
m. m. mm.
a=2.010; 6=0.401; c = 1.00; tabulary value of c=3.062
mm.
mm.
mm.
( -1.262
2.2575 )
1. Minimum
- -1.162
2.2153 [
-1.181
0.14
0.16
-0.02
( -1.100
2.2577 )
( -0.300
0.7135 )
2. Minimum
-0.262
0.6925 [
-0.215
0.51
0.48
+0.03
( -0.162
0.6950 )
( +0.400
0.1501 )
3. Minimum
< +0.438
0.1477V
+0.431
0.77
0.76
+0.01
( +0.500
0.1604 )
( +0.938
0.0799 )
4. Minimum
1 +1.038
0.0417V
+1.084
1.02
1.01
+ .01
( +1.138
0.0432 J
( +1.800
0.0170 )
5 Minimum
\ +1.738
0.0128 [
+ 1.736
1.28
1.28
( +1.700
0141 )
FIFTH OBSERVATION
m. m. mm.
=2.010; 6=0.492; c = 1.50; tabulary value of c= 4. 224
mm.
mm.
mm.
( -1.300
2.7239 )
1. Maximum
- -1.200
3.0466 -
i -1.168
042
0.43
-0.01
/ -1.100 2.9780) A
[The second, fourth, and sixm observations are omitted.]
135
CA, ,,-,
MEMOIRS ON .
Evidently theory and observations agree in general quite
well, although in the second and fourth observations the dis-
agreement is quite marked and rather more than one would
expect from the size of the fringes ; for in the second observa-
tion the individual measures differ at most by 0.04 mm., and
the fourth observation, which I have already described, agrees
perfectly, as has been seen, with another experiment in which
the same fringes appear. This disagreement, therefore, can
only be explained by assuming that the theory is wrong or that
constant errors have entered the observations through optical
illusion.
74. Our theory rests upon a hypothesis which is at once so
simple and so inherently probable, and which besides has been
so strikingly verified by many varied experiments, that one can
scarcely doubt the truth of the fundamental principle. It is
quite possible that this anomaly is only apparent, and that the
eye does not correctly estimate the position of the minima in
question. We must remember that they are not very sharp,
and that they are always bounded on each side by two bright
bands of very different intensities. Now, in order to deter-
mine the position of the minimum, my eye must include a part
of each of these two bands, so that that part of the dark band
on the side next the brightest appears to me darker still on
account of its environment ; thus attracting, as it' were, the
apparent minimum to its side, and, indeed, all the discrepancies
lie in this direction. That the eye includes a sufficiently large
portion of the fringes for correctly estimating the position of
maxima and minima is evident from the fact that in repeating
the fourth observation, using a diaphragm of small aperture in
the focus of the micrometer eye-piece, nothing was left but a
band which was uniformly dark and in which the minimum
was no longer distinguishable. If I have succeeded in getting
the correct positions of the minima in the exterior fringes
even in regions of poor definition, it is owing to the fact that
the bright bands between which these are included differ very
slightly in intensity ; and if, in the case of the narrow aperture
and cylindrical lens, experiment and theory happen to agree in
spite of great differences of intensity between two adjoining
bright bands, this is because the dark band, especially in the
first and second orders, is almost perfectly black. In general,
whenever the maximum or minimum is very sharp, I find.ox-
136
THE WAVE-THEORY OF LIGHT
perirnent and calculation in thorough agreement. In the fifth
observation, for instance, I measure the distance from the centre
to the maximum of the first order because this bright band is
very well defined, and I am therefore able to determine its most
brilliant point with great precision. The difference between
the computed and observed values is indeed only 0.01 mm.
75. But our theory does more than merely give us the posi-
tions of maxima and minima, for it enables us to predict the
general appearance of the phenomena, so that, without experi-
mental determination, we can foretell the variations of intensity
in the light ; thus, for instance, in the fifth observation the part
of the shadow corresponding to the middle of the aperture was
filled by a large dark band of a tint that was practically uni-
form up to 0.26 mm. on each side of the centre, after which
the intensity of the light increased rapidly so as to form the
bright band of the first order which I have just mentioned.
Now in computing the intensity of the light within these
limits, we find that in fact its intensity varies scarcely at all,
but that in passing from these limits to the bright band it in-
creases very rapidly. In the following table are given the re-
sults of computation for different points of the dark band and
the two bright bands which include it. The position of each
point is denoted by the corresponding value of v, measured al-
ways from one of the edges of the aperture.
Number of
observation
Value ofv
Intensity
1
1.100
2.9780
2
1.200
30466
3
1.300
2.7239
4
1.400
2.2843
Limit of the colored )
region as determined >
5
1.524
1.9671
by observation. )
6
1.824
1.9100
7
2.112
1.9802
The distribution of intensities on the other side of the centre is
the same.
If the distances of these various points from a common ori-
gin be plotted as abscissas and the corresponding intensities as
ordinates, we shall obtain the curve MCM', which gives us in
fat t~a picture of the phenomenon just as one finds it in the
137
MEMOIRS ON
M
J 1 l__l I 1 L
12345 6 7
*iu. 2 1
Fiy.
experiment. I should like to have made similar drawings for
all the other observations in order to facilitate the comparison
of theory with experiment, but the length of the computation
and the time at my disposal did not permit.
[Five pages, in which the case of a narrow opaque obstacle is
discussed, are here omitted.]
79. I have now applied the principle of Huygens to the
three general classes of phenomena in which diffraction occurs,
namely, first, to the fringes produced by a screen whose edges
are straight and infinitely long, and which is so large that the
light passes practically only one edge of the screen ; secondly,
to the fringes produced by a system of two similar screens
brought very near together ; thirdly, to those fringes which
accompany the shadow of a very narrow screen.*
Comparing observations with the predictions of the theory,
I have shown that it suffices to explain the most diverse phe-
nomena, and that the general expression for the intensity of
light derived from it gives us a faithful picture of the phenom-
ena, even when they are most bizarre and apparently irregular,
Besides the three general classes, one might devise a large
number of others by combining these among themselves. The
theory would doubtless apply here with the same success and
the same ease. The computation would be more tedious in
proportion as the variety of limits assigned to the integrals be-
came greater and greater ; the experiments would also demand
more complicated apparatus.
80. In the first section of this memoir I have described a
phenomenon which results from a combination of two of the
principal cases of diffraction, namely, the fringes produced by
* I do riot here include those fringes which are produced by the biprism,
or two mirrors slightly inclined to each other, for, properly speaking,
these are not diffraction effects, since they are not produced by rays which
are diffracted or inflected, but by two pencils which are regularly re
fleeted or refracted.
138
THE WAVE-THEORY OF LIGHT
light in passing through two apertures, each very narrow and
each near to the other. Having prepared a sheet of copper in
the form drawn in Fig. 15, I noted that
when the large fringes produced by each
of the slits CEC'E' and DFD'F', expand-
ing as I moved away from the screen, had G_
filled the shadow 6f CDFE so that it con- "
tained only the bright band of the first
order, the interference bands resulting
from the twc pencils of light became
much sharper and brighter than the in-
terior fringes of the part ABCD. The
Fig. 15
lower part, CEDF, which was at first brighter than the other,
became darker the farther I went away from the screen, but
its fringes continued to show colors which in white light were
purer and bands which in homogenous light were sharper.
With the simple apparatus which I employed one could not ob-
tain exact measures, and I have not therefore carried out the
computations for this experiment ; accordingly I limit myself
to the explanation of these phenomena by means of some gen-
eral considerations.
Let L be the luminous point, and IK the horizontal projec-
tion of the part AEBF of the screen represented in Fig 15. P is
any point in the interior of the shadow lying upon the straight
line LO. From the point L as centre, and
with a radius equal to LI, describe the arc
IMM', representing the incident wave. Now
with a point P as centre, arid with a radius
equal to IP, describe the arc Imm'. The vari-
ous distances between these two arcs give us
the differences of path traversed by the sec-
ondary waves meeting at P. We shall first
consider the upper part of the screen that is,
the case where the wave IMM' is not inter-
cepted on the other side of the point I. Let
us now imagine this wave divided into a
large number of small arcs, IM, MM', etc.,
in such a manner that the straight lines
drawn to P from any two consecutive points
of division differ by half a wave-length ; and,
for sake of simplicity, let us suppose that the
139
MEMOIRS ON
point P lies well within the edge of the shadow, or, what is the
same thing, let us imagine the ray IP sufficiently inclined to
the incident ray to make these arcs practically equal. Then
each of these arcs, excepting the one at the end IM, will lie be-
tween two others, which will combine to annul its effect at the
point P. In the case of the arc IM, which lies at the extreme
edge of the wave, we have, however, an exception ; for this arc
loses only one-half its intensity by interference with the vibra-
tions of the neighboring arc, MM'. If, therefore, we intercept
this arc [MM'] and all the rest of the incident wave, the light
which is received by the point P will actually be increased ;*
this is precisely the effect which, at a certain distance, is pro-
duced by the part of the screen G'C'E" (Fig. 15). But in
proportion as the point P (Fig. 22) recedes from the opaque-
screen, the arc \mrti approaches the wave IMM' ; and in the
case where the luminous point L is at an infinite distance,
these two approach indefinitely near to each other. The di-
visions M, M', etc., being determined by the separation of
these two arcs, keep spreading apart from the point I in pro-
portion as the arcs approach each other. It follows, therefore,
that the part MI of the incident wave will grow larger and
larger, and the rays from this part passing the edge (Fig. 15)
retain at least half their intensity in the region behind the
upper part of the screen. But in the lower part of the screen
the aperture CEC'E' does not increase in size, so that if the
luminous' point is far enough away the effective arc IM (Fig. 22)
will finally become so large compared with this aperture that
[most of the rays from MI are intercepted by GC'E', and hence]
the point will receive less light in the lower part of the shadow
than in the upper.
Let us now pass to the consideration of fringes produced by
the meeting of rays coming from both edges of the screen,
AEBF (Fig. 15). Behind the upper part, ABOD, the inflected
light diminishes rapidly in intensity .as one recedes from the
edge of the geometrical shadow, and therefore all the fringes
except those which are very near the middle are produced bj
two rays of very unequal intensities ; consequently the dark
* The light at P would be increased still more if the screen were perfo-
rated in such a way as to permit all the arcs of even order to pass through
and at the same time intercept all the arcs of odd order.
140
THE WAVE-THEORY OF LIGHT
bands are not very sharp when one uses homogeneous light,
and the colors are mixed with gray when one uses white light.
Behind the lower part, CEDF, the two pencils of light coming
from the slits CEO'E ' and DFD'F ' have a practically uniform
intensity throughout a considerable portion of the bright band
from each of these apertures ; and if these apertures are so nar-
row compared with the distance between them that the region
of uniform intensity in the inflected light includes all the
fringes produced by the two pencils, then in those points where
the vibrations are in complete discordance the light-waves will
almost completely destroy one another ; accordingly the dark
bands will be very much sharper than in the upper part of the
shadow when homogeneous light is employed, and the colors
will be very much purer when white light is used. When one
looks at these points close up to the screen before the larger
fringes which are produced by each slit have spread out into
the shadow AEBF, the phenomenon becomes very complicated
and changes rapidly with the distance of the rnagnifying-glass,
especially when the distance between the two slits is not very
great when compared with their size. It would be interesting
to determine by computation the positions of the maxima and
minima of the bright and dark bands, and to compare these re-
sults with those of observation. I have no doubt that the the-
ory would thus acquire fresh confirmation.
81. Hitherto we have considered all waves as coming from a
single centre, but, in actual experiment, luminous points are
always made up of a very large number of centres of vibration,
and it is to each one of these by itself that the preceding dis-
cussion applies. So long as these are not very widely sepa-
rated from each other, the fringes which they produce practi-
cally coincide, but the dark bands from one overlap the bright
bands of the other in proportion as we increase the dimensions
of the luminous point, until finally they completely annul each
other. In the case of the exterior fringes this effect is more
and more appreciable as one gets farther and farther away
from the screen, because it increases directly as the distance,
while the size of the bright and dark bands increases less rapid-
ly. And this is why a luminous source sufficiently small to
produce fringes which, in the near neighborhood of an opaque
body, are very sharp will, at a considerable distance from this
body, give only ill-defined fringes.
141
MEMOIRS ON
82. It is not necessary that the interposed body should be
opaque in order to produce the phenomena of diffraction at its
edges ; all that is required is that a part of the wave should be
retarded with respect to its neighboring parts, but this is ex-
actly what a transparent body does when its refractive index
differs appreciably from that of the medium surrounding it ; it
thus gives rise to fringes which border both the inside and the
outside of their shadow. They are exactly like the exterior
fringes of opaque bodies when the difference of path between
the rays which have traversed the transparent screen and the
outside rays contains a considerable number of wave-lengths, be-
cause their mutual influence [interference] is no longer appre-
ciable and we have simply the addition of two uniform illu-
minations. But this is not the case when the transparent screen
is very even or when its refractive index differs very slightly
from that of the surrounding medium, for now the fringes are
altered in a very marked way by the mutual influence of those
rays which traverse the transparent plate and those which pass
its edge. It is from similar reasons that the striae in layers of
mica resulting from slight differences of thicknesses give rise,
in white light, to colored fringes in the very peculiar manner
described by M. Arago.
83. As to fringes of the kind which we have called interior,
they are not to be obtained with a narrow transparent body,
because the direct light which traverses it is so much brighter
than the inflected rays as to mask the^effects of interference;
and, besides, the bright and dark bands which this transparent
body tends to produce, when considered as a narrow aperture,
do not coincide with those which it tends to produce when
considered as a small obstacle.
84. The phenomena of diffraction, once explained for the
case of homogeneous light, are easily predicted for the case of
white light. These fringes come from the superposition of all
the bright and dark bands of the various sizes produced by the
different kinds of waves which go to make up white light, so
that when we have once computed the intensity of each of the
principal kinds of rays at the point under consideration, using
the proper wave-length, according to the theory which I have
just explained, we can find the resultant tint by substituting
these values in Newton's empirical formula for determining the
result obtained by mixing any set of colored rays.
142
THE WAVE-THEORY OF LIGHT
85. Polished surfaces illuminated by a point-source present
a set of diffraction phenomena exactly like those which we ob-
serve in direct light. The field of light reflected by a mirror
is bordered with fringes similar to those which surround the
shadows of bodies. If the surfaces be very narrow or so black-
ened that only a single bright line remains, or indeed if one
inclines the mirror in such a way as to diminish greatly the
size of the field [foot-note here omitted], the phenomenon of a
pencil of light dilated by passing through a very narrow aperture
will be reproduced. If a mirror be blackened throughout its
entire extent, with the exception of two bright lines, it gives
rise to a set of fringes identical with those produced by two
parallel slits in an opaque screen. If, instead of blackening the
large part of the reflecting surface, one, on the contrary, mere-
ly traces a single fine black line, it will produce fringes similar
to those observed in the shadow of a narrow screen. In short,
the phenomena are absolutely the same as if the mirror were
transparent and the rays came from the image of the luminous
point. The explanation is very simple ; for we know that the
image (which lies upon the perpendicular drawn from the lu-
minous point to the mirror, and which is situated at a distance
from the surface of the mirror equal to the distance of the lu-
minous point from the mirror), has this remarkable property,
namely, that its distance from any point on the surface of the
mirror is equal to the distance of the same point from the lu-
minous centre. When, therefore, we consider the rays as orig-
inating in the image of a luminous point, we do not alter the
difference of path traversed by the elementary waves which
produce the fringes, and consequently there is no change in
the size, or in the relative intensities, of the bright and dark
bands.
I may here remark that the position [pliase] of the resultant
of the secondary waves at any point, depending as it does merely
upon differences of path, ought, in the case of reflection, to be
the same as if the rays were emitted by the image just men-
tioned. Consequently, in the case of a polished surface of
large area, all the partial resultants will be situated at the
same distance from this point, thus making it the centre of
the reflected wave.
86. It is by means of these secondary waves that Huygens
has explained in such a simple manner the laws of reflection
143
MEMOIRS ON THE WAVE-THEORY OF LIGHT
and refraction, showing that they are phenomena of the same
kind as the propagation of light in a homogeneous medium;
but his explanation leaves much to be desired. He has not
proved that there will be only one system of waves resulting
from this multitude of systems of secondary waves, for he has
not used the principle of interference. He assumes that the
light is appreciable only in those points where the secondary
waves coincide [in phase] exactly ; while the complete absence
of any luminous disturbance can occur only when the second-
ary disturbances are in [direct] opposition. It was this, doubt-
less, that led him to think that light was not inflected to any
appreciable extent into shadows, and which prevented him
from discovering the phenomena of diffraction, the laws of
which his theory could have given him without recourse to ex-
periment.
This theory, when combined with the principle of interfer-
ence, gives us not only the path of the ray in the particular
case where reflection occurs at a polished surface of indefinite
extent, but also in those cases where the surface is very nar-
row or even discontinuous ; it shows us how diminution in
size of the surface produces the dilation of the reflected ray,
and how a system of very narrow mirrors placed side by side
and very close together can produce colored images, owing to
the mutual influence of pencils of light thus dilated. This is
the phenomenon of ruled surfaces. With the same ease it ex-
plains the images and colored rings produced by a thin fabric
or even an irregular combination of very fine threads or small
particles, provided they are almost equal in size, when placed
between the eye of the observer and the luminous point.
I think it hardly necessary to emphasize these phenomena,
since they are merely combinations of those described above,
and since I have attempted to give, for all of them, a general
theory.
144
ON THE ACTION OF RAYS OF POLARIZED
LIGHT UPON EACH OTHER*
BY
ARAGO ASTD FRESNBL
1. Before describing the experiments which form the sub-
ject of this memoir it will perhaps be well to recall the ex-
quisite results obtained b^Dr. Thomas Young, who, with rare
sagacity and characteristic skill, has already studied the effects
which rays of light exert upon each other.
First. Two rays of homogeneous light coming from the same
source and reaching a certain point in space by paths which
are different and slightly unequal in length, either strengthen
one another or annul one another, and produce upon the re-
ceiving screen a bright or a dark point according as the differ-
ence of path has one value or another.
Second. Two rays always intensify each other at any point
tor which their paths are equal ; if their intensities are added
foi another point where the difference of path is equal to a
quantity d, their intensities will be added also for all differ-
ences of path included in the series 2d, 3d, etc. The interme-
diate values 0-f J^, d+\d, 2d+%d, etc., represent the points in
which the rays annul each other.
Third. The quantity d does not have the same value for all
homogeneous rays. In air its value for the extreme red rays of
the spectrum is TTrffoo- mm., while for violet rays it is only
TF ii_ mm. For other colors the corresponding values are in-
termediate between these which we have just given.
The periodicity of color which is seen in Newton's rings, in
* [Annale* de Chimie et de Physique, t. x., p. 288 (1819).]
K 145
MEMOIRS ON
halos, etc., seems to depend upon the influence exerted upon
one another by rays whose paths at first diverge, and later are
so inclined as to again meet ; but in order to bring these vari-
ous phenomena into harmony with the laws just stated, we are
forced to adm.it that difference of path alone is not sufficient
to determine the mutual action of two rays at their point of
meeting except when they are both travelling in the same
medium ; and it must be recognized also that differences of re-
fractive index, or thickness in the transparent bodies traversed
by the respective rays, produce the same effect as difference
of path. In this journal, vol. i., p. 199, there is described a
direct experiment due to M. Arago, which shows the same
thing, and proves also that a transparent body diminishes the
speed of light traversing it in the ratio of the sine of the angle
of incidence to the sine of the angle of refraction ; so that in
all the phenomena of interference * two different media produce
similar effects when their thicknesses are in inverse ratio to
their refractive indices. These considerations at once suggest
a new method for measuring slight differences of refrangi-
bility.
2. While we were trying to determine what accuracy was
attainable by this method, one of us (M. Arago) thought that
it would be interesting to find out whether the actions which
ordinary rays exert one upon another were in any way modi-
fied when two previously polarized pencils of light were made
to interfere. We know that if a narrow body be illuminated
by light coming from a point-source, its shadow is bordered
on the outside by a series of fringes produced by the interfer-
ence of the direct light with the rays inflected near the opaque
body. It is known also that a part of this same light, passing
into the geometrical shadow from the two opposite sides of the
body, there gives rise to fringes of the same kind.
Now the fact was easily recognized that these two systems
of fringes are absolutely the same, whether the incident light
has received no modification whatever, or whether it has been
polarized previous to incidence. Rays which are polarized, in
one plane, therefore mutually affect one another in the same
manner as rays of ordinary light.
* Tliis is the name which Mr. Youug has given to the phenomena pro-
duced by the meeting of two or more rays of light.
146
THE WAVE-THEORY OF LIGHT
3. It was still to be determined whether two rays originally
polarized at right angles would not produce phenomena of the
same kind when they met inside the geometrical shadow of an
opaque body. For this purpose we placed in front of the
point -source* sometimes a rhomb of calc-spar, sometimes an
achromatic prism of rock crystal, and thus obtained two lumi-
nous points. In each case we had a divergent pencil, and'these
two pencils were polarized at right angles. Behind the two
radiant points and midway of the space between them was
placed a cylinder of metal. In this manner a part of the po-
larized light from the first pencil reached the interior of the
shadow via the right-hand side of the cylinder; while a part
of the light from the second pencil, polarized in a plane at right
angles to the first, entered the shadow from the left-hand side
of the cylinder. Some of these rays met along the line joining
the centre of the cylinder and the middle point of the straight
line drawn from one luminous point to the other. Here these
rays had traversed equal paths, and one might expect them to
produce fringes. On the contrary, not the slightest trace of
fringes could be seen, even with a magnifying-glass. In fact,
the rays here cross without either affecting the other. The
only fringes which make their appearance in this experiment
arise from the interference of rays which come from only one
of the radiant points and enter the shadow from each side of
the cylinder. Those which we tried to produce by the inter-
ference of rays polarized at right angles to each other would have
occupied a position intermediate between those just mentioned.
Since the images which we employed were not very widely
separated, the thicknesses of crystal traversed by the ordinary
and the extraordinary rays must have been very nearly equal.
Nevertheless, similar experiments have already shown us, only
too frequently, how sensitive the phenomena of interference
are to the slightest difference of speed in the rays, to the
length of path, and to the refractive index of the medium. No
argument was needed, therefore, to convince us of the neces-
sity of repeating these experiments under conditions which
would eliminate these various sources of inaccuracy. This has
been attempted by each of us.
* For all the experiments described in this paper our source of light was
the focus of a small magnifying-glass.
147
MEMOIRS ON
4. M. Fresnel at once devised two distinctly different meth-
ods. The principle of interference shows us that pencils of
light from two luminous points, originally from a single point-
source, produce bright and dark bands at points of intersection
even though no opaque body be interposed. (See Annales de
Chimie et de Physique, t. i., p. 332.)
To solve the problem it is then only necessary to determine
whether, when two images are produced by placing a rhomb of
calc-spar in front of a luminous point, they will behave in this
same way ; but since, from the theory of double refraction,
we know that the extraordinary ray traverses carbonate of
lime more rapidly than the ordinary ray, it becomes necessary
to compensate this extra speed before the two rays are allowed
to intersect. In order to accomplish this a method was em-
ployed which has been described by M. Arago in this journal,
vol i., p. 199. M. Fresnel placed in the path of the extraor-
dinary pencil alone a plate of glass whose thickness had been
determined by computation in such a way that, under perpen-
dicular incidence, this pencil lost nearly all the ground which,
in the crystal, it had gained over the ordinary ray. By slightly
inclining the plate the compensation could be made exact. In
spite of these precautions, the two rays, polarized at right
angles, gave not the slightest trace of interference bands.
In another experiment, M. Fresnel compensated for the dif-
ference of speed in the two rays by allowing them each to fall
upon a small unsilvered mirror whose thickness had been so
computed that the extraordinary ray, when reflected at the sec-
ond face, lost by twice traversing the glass more than it had
gained in traversing the crystal ; a gradual inclination of the
plate brought about complete compensation.
Under no angle of incidence, however, would the ordinary
rays, reflected at the first surface, interfere with the rays re-
flected from the second surface to produce bands.
5. In order to avoid the theoretical consideration introduced
into the preceding experiment, and to maintain the original
intensity of the light, M. Fresnel adopted the following meth-
od : A rhombohedron of calc-spar was sawed through the mid-
dle, and the two parts were placed one in front of the other
with their principal sections at right angles. In this position,
the ordinary ray from the first crystal was refracted as an ex-
traordinary ray in the second ; while, conversely, the extraor-
148
THE WAVE-THEORY OF LIGHT
dinary ray in the first crystal suffered ordinary refraction in
the second. On viewing a luminous point through this com-
bination, one sees only a double image. Each pencil has ex-
perienced in succession the two kinds of refraction. The sum
of the paths of each pencil through the two crystals ought,
therefore, to be equal, since by hypothesis the crystals have
the same thickness ; so that everything is compensated, both
as regards speed and length of path. Nevertheless, two systems
of rays polarized at right angles never gave rise to any interfer-
ence fringes. Lest the two parts of the rhombohedron did not
have quite the same thickness, we took pains in each test to
vary slightly and slowly the angle of incidence at the face of
the second crystal.
6. The method devised by M. Arago for solving this same
problem was independent of double refraction. It has been
known for a long time that if one cuts two very narrow slits
close together in a thin screen arid illuminates them by a
single luminous point, there will be produced behind the
screen a series of bright bands resulting from the meeting of
the rays passing through the right-hand slit with those passing
through the left. In order to polarize at right angles the rays
passing through these two apertures, M. Arago at first thought
of using a thin piece of agate, sawed through the middle and
placed one piece in front of each slit, in such a way that the
edges formerly meeting along the line of the cut are now at
right angles to each other. This arrangement ought certainly
to produce the effect expected. But not having at hand a
suitable piece of agate, M. Arago proposed to supply its place
by two piles of plates, of proper thickness, built up from sheets
of mica.
For this purpose we selected fifteen plates as clear as possi-
ble and superposed them. This pile was next cut in two by
use of a sharp tool. So that now we had two piles of plates of
almost exactly the same thickness, at least in those parts bor-
dering on the line of bisection ; and this would be true even if
the component plates had been perceptibly wedge-shaped.
The light transmitted by these plates was almost completely
polarized when the angle of incidence was about thirty degrees.
And it was exactly 'at this angle that the plates were inclined
when they were placed in front of the slits in the copper
screen.
149
MEMOIRS ON
When the two planes of incidence were parallel, i. e., when
the plates were inclined in -the same direction, up and down,
for instance, one could very distinctly see the interference
bands produced by the two polarized pencils. In fact, they be-
have exactly as two rays of ordinary light. But if one of the
piles be rotated about the incident ray until the two planes of
incidence are at right angles to each other, the first pile, say,
remaining inclined up and down while the second is inclined
from right to left, then the two emergent pencils will be polar-
ized at right angles to each other and will not, on meeting,
produce any interference bands.
The pains we took to make these two piles of equal thick-
ness would indicate that we also took care in placing them be-
fore the slits to have the light traverse those parts which were
originally contiguous. But all difficulties of this kind are
really solved by the fact that the two rays when polarized in the
same plane interfere like ordinary light. Moreover, we could
not produce interference by slowly and gradually changing the
inclination of one of the plates so long as the planes of inci-
dence were at right angles.
7. The same day that we tried the combination of these two
piles we also tried an experiment suggested by M. Fresnel^ an
experiment which, it must be confessed, is less direct than the
preceding, but which is also more easily performed and which
demonstrates equally well the impossibility of producing fringes
by bringing together rays polarized at right angles to each
other.
In front of a sheet of copper in which are cut two slits we
placed, for instance, a thin plate of selenite. Since this is a
doubly refracting crystal, there will be two pencils of light
polarized at right angles passing through each slit. Now if
rays polarized in one plane can affect rays polarized in a plane
at right angles, we should expect with this arrangement to see
three distinct systems of fringes. The ordinary rays from the
right-hand slit would combine with the ordinary rays from the
left-hand slit to form a first system symmetrical with respect
to the line bisecting the space between the two slits. The
bands formed by the two extraordinary pencils would fall in
the same position as the preceding, increasing their intensity,
but remaining indistinguishable from them. As to those which
would result from the action of the ordinary rays from the
150
THE WAVE-THEORY OF LIGHT
right upon the extraordinary from the left, and conversely, it
is clear that they would form a system to the right and to the
left of the central band. The distance of either of these sys-
tems from the centre would increase with the thickness of the
plate, for, as we have seen, difference of speed is quite as ef-
fective as difference of path in shifting the position of fringes.
Now, since the fringes in the centre are the only ones visible,
even though the plate of selenite be so thin as not to shift the
other two systems very much, we must conclude that rays of
light polarized at right angles do not affect one another.
8. In order to verify this conclusion, suppose that we cut
the selenite plate in two, and that we place one half in front
of the first slit and the other half in front of the other slit;
and instead of placing their axes parallel as in the case of a
single plate, let us put them at right angles to each other. In
this way the ordinary ray coming through the right-hand slit
will be polarized in the same plane as the extraordinary ray
from the left-hand slit, and vice versa. These rays will then
form fringes ; but their speeds in the crystal will not be equal,
and they will not lie symmetrically about the middle of the
space between the two slits. Central fringes will be produced
only by ordinary or extraordinary rays from the one slit meeting
rays from the other slit which are polarized in the same plane.
But when the two parts of crystal are arranged as we have
here supposed them, those rays which are polarized at right
angles to each other ought not to affect one another. One
would, therefore, see simply the first two systems of fringes
separated by an interval of white or of some uniform shade.
[An unimportant foot-note is here omitted.]
If, without changing the experiment in any other respect,
we simply set the two plates of selenite so that their axes make
an angle of 45 instead of 90, we should at once see three
systems of fringes ; for now, since their planes of polarization
are no longer at right angles, each pencil from the right will in-
terfere with the two pencils from the left, and vice versa. It
should be observed also that the middle system is the most in-
tense, resulting, as it does, from the exact superposition of in-
terference bands of polarized pencils of the same kind.
9. Let us return to the combination of the two piles and
imagine that the planes of incidence are mutually perpendicular,
so that the two pencils are polarized at risjht angles to each
151
MEMOIRS ON
other. Between the copper screen and the eye place a doubly
refracting crystal in such a way that its principal section
makes an angle of 45 with the planes of incidence. In ac-
cordance with the well-known laws of double refraction, the
rays which are transmitted by the piles will afterwards, in pass-
ing through the crystal, each be divided into two others.
These two will be of equal intensity, will be polarized in planes
which are mutually perpendicular, and one of these planes will
coincide with the principal section of the crystal. One might
therefore expect to see, in this experiment, one series of
fringes due to the meeting of the ordinary pencil from the
right with the ordinary pencil from the left, and a second se-
ries similar to the preceding, but arising from the interference
of the two extraordinary pencils. Such, however, is not the
case ; for these four pencils meet and produce only a uniform
illumination, showing not the slightest interference.*
This experiment shows that two rays originally polarized at
right angles to each other may subsequently be brought into
the same plane of polarization without again acquiring the
power of interference.
10. In order to produce interference between two rays po-
larized at right angles and afterwards reduced to the same
plane it is necessary that they should originally have been po-
larized in one and the same plane. This is shown by the fol-
lowing experiment, which was devised by M. Fresnel.
A plate of selenite, backed with a sheet of copper in which
two apertures have been made, is illuminated by a pencil of
polarized light coming from a point-source and striking the
selenite plate at perpendicular incidence. The axis of the
plate makes an angle of 45 with the original plane of polariza-
tion. As in all similar experiments the shadow of the copper
screen is observed with a magnifying-glass ; but in this case
* If the plate interposed between the copper screen and the eye were so
thin as to only slightly separate the two images, one might explain the ab-
sence of interference as follows : viz., suppose the two systems of bands
are superposed in such a fashion that the bright bands of one system
coincide with the dark bands of the other system, and vice versa. But the
insufficiency of this explanation is shown by placing a rhombohedron of
Iceland spar between the eye and the preceding crystal. In certain posi-
tions this Iceland spar separates the two systems of bands, because they
are polarized at right angles. But even under these circumstances one
sees no trace of bands.
152
THE WAVE-THEORY OF LIGHT
a rhombohedron of Iceland spar, in which [the separation of
images due to] double refraction is perceptible, is placed in
front of the focus.
The principal section of the Iceland spar makes an angle of
45 with that of the plate of selenite. Accordingly we find in
each image three systems of fringes, one falling exactly in the
middle of the shaoow, the others being situated on the right
and left respectively.
Let us now consider one of these two images, say the or-
dinary, and see what gives rise to these three systems of
bands.
The pencils which pass through the two slits are polarized
in the same plane, but on emergence from the plate of selenite
they are divided into two pencils polarized at right angles.
Since double refraction in this plate is inappreciable, the or-
dinary and extraordinary pencils each follow practically the
same route, though with different speeds.
Each of these double pencils, say the one from the right-hand
slit, will be divided, in passing through the Iceland spar, into four
pencils, two ordinary and two extraordinary ; but, as a matter of
fact, one will see only two, since components in the same plane
will coincide. It is also evident, from the well-known laws of
double refraction and from the relative positions of the selenite
and the Iceland spar, that at emergence from this latter crys-
tal the ordinary pencil will be composed partly of the ray
which "was ordinary in the selenite and partty of the ray which
was extraordinary ; while the other two components of these
same rays go to form the extraordinary image which we are
not now considering. The pencil which emerges from the left-
hand slit behaves in the same way. We see, in fact, that the
ordinary pencil coming either from the right or left hand slit
will, after traversing the two crystals in this new instrument,
be composed partly of light which has followed the ordinary
path in each crystal and partly of light which started out as an
extraordinary ray.
Eays coming from the two slits and following the ordinary
path through each of the two crystals will have traversed
routes of the same length and with the same speed. On meet-
ing, they ought, therefore, to give rise to central bands. The
same is true of rays which have pursued the extraordinary
path both in the selenite and in the Iceland spar. The bands
153
MEMOIRS ON
in the middle of the shadow result, therefore, from the super-
position of these two different systems.
Now- as to that portion of light from the right-hand slit
which has traversed the selenite as an extraordinary ray, for
instance, but passed the Iceland spar as an ordinary ray, it is
evident that it will have traversed a path which in length is
equal to that of the left-hand pencil which made the whole
trip as an ordinary ray. But since in the selenite the speeds
are different, those points where they meet to form fringes will
not lie symmetrically between the two slits, but will be shifted
to the right, i. e., to the side opposite the ray which for a
while travelled as an extraordinary ray, but now travels more
slowly. Finally, as a last combination, we have interference
between that component of the right-hand pencil which trav-
ersed both crystals as an ordinary ray and that component of
the left-hand pencil which in the selenite was an extraordinary
ray and in the Iceland spar an ordinary ray. This interfer-
ence gives rise to a system of bands situated on the left of the
centre.
We have now explained the paths of the rays which meet to
form the three systems of fringes in the experiment under discus-
sion. And it may be remarked that the right and left systems
were produced by the interference of rays which were previous-
ly polarized at right angles in the selenite and afterwards re-
duced to the same plane in the Iceland spar. Two rays polar-
ized at right angles and later reduced to the same plane-of po-
larization can, then, meet and produce interference bands ;
but for this purpose it is an essential condition that the rays
should ORIGINALLY have been polarized in the same plane.
So far we have not considered the interaction of the two
pencils which suffered extraordinary refraction in the Iceland
spar. These pencils also furnish three systems of bands, but
they are separated from the others. If we allow all the condi-
tions of the experiment to remain the same, except that we
substitute for the Iceland spar a plate of selenite or quartz
which does not give two distinct images, the six systems, in-
stead of being reduced to three by superposition, will result in
one central system. This remarkable fact shows, first, that
the fringes resulting from the interference of the ordinary
rays are complementary to those produced by the interference
of the extraordinary rays ; and, secondly, that these two sys-
154
THE WAVE-THEORY OF LIGHT
terns are so located that a bright band in the one system corre-
sponds to a dark band in the other system. Were these two
conditions not satisfied, one would not find uniform and con-
tinuous illumination on each side of the central fringes. We
meet here, then, the same difference of half a wave-length that
is found in the phenomena of colored rings.
From the experiments just described we may, therefore, infer
the following facts :
(1.) Two rays of light polarized at right angles do not pro-
duce any effect upon each other under the same circumstances
in which two rays of ordinary light produce destructive inter-
ference.
(2.) Rays of light polarized in the same plane interfere like
rays of ordinary light ; so that in these two kinds of light the
phenomena of interference are absolutely identical.
(3.) Two rays which were originally polarized at right an-
gles may be brought to the same plane of polarization -without
thereby acquiring the ability to interfere.
(4.) Two rays of light polarized at right angles and after-
wards brought into the same plane of polarization interfere
like ordinary light provided they were originally polarized in
the same plane.
(5.) In the phenomena of interference produced by rays
which have experienced double refraction the position of the
interference bands is determined not only by difference of path
and difference of speed, but in some cases, as above indicated,
it is necessary to take into account also a difference of one-
half a wave-length.
All these laws are, as we have seen, based directly upon ex-
perimental evidence. In starting from the phenomena of
crystalline plates, they can be derived more simply ; but then
we have to assume that the colors of the plates when illumi-
nated by polarized light are produced by the interference of
several systems of waves. The evidence which we have just
presented has the advantage of establishing the same laws
quite independently of hypothesis.
155
MEMOIRS ON
BIOGRAPHICAL SKETCH
AUGUSTIN JEAN FRESNEL was born in Normandy in 1788,
and died near Paris in 1827.
As a child he was quite the reverse of precocious ; but at
the age of sixteen he was ready to enter the Ecole Poly tech-
nique at Paris, where he received sound mathematical training
and attracted to himself the attention of Legendre. His edu-
cation was completed at the Ecole des Fonts et Ohausees where
he received an engineer's training. Several years were next
spent in professional work in various parts of France.
In 1816, through the influence of Arago, he received an ap-
pointment in Paris, where he remained during the rest of his
life. When we recall that his first studies in optics date from
1814, his accomplishments during the eleven years of his Paris
residence must ever fill us with wonder. New ideas were not
only rapidly acquired, but were also rapidly perfected. They
were at once submitted to the test of experiment and as quick-
ly received elegant mathematical description.
The wave-theory of light had lacked neither merit nor able
support ; Grimaldi, Hooke, Efuygens, and Young had been its
advocates; but it was only in the hands of Fresnel that the
problem and its solution received such clear and simple state-
ment as to command acceptance. The work of Fresnel lies ex-
clusively in the domain of optics, each of his investigations fall-
ing into one of two distinct groups, viz., the kinematics of light
and the dynamics of light.
His earlier papers deal with questions of diffraction, inter-
ference, and polarization, in which the chief factors of the dis-
cussion are displacements, velocities, and squares of velocities
the quantities of kinematics.
His later papers, however, refer more to the medium through
which luminous energy is transferred; they deal with the forces
of elasticity here brought into play, and seek to determine the
speed of light as a function of the mechanical properties of the
matter through which the light travels ; they deal, in short,
with the dynamics of light.
But the particular achievements with which the name of
Fresnel must always be associated are
156
THE WAVE-THEORY OF LIGHT
(1.) The introduction of the idea of transverse vibrations.
(2.) The combination of the principle of Huygens with that
of interference.
An excellent and appreciative sketch of Fresnel will be
found in Arago's Notices BiograpMques, vol. i. It is here that
he paraphrases Newton's remark concerning Cotes by saying
" que nous savons quelque chose quoique Fresnel ait peu vecu."
Between the years 1866 and 1870 the French government
published the works of Fresnel in three worthy quarto vol-
umes, ably edited by Senarmont, Verdet, and the author's
brother, Leonor Fresnel.
157
OOOOCfocTo O^r-TT-T^i-^i-^ "'-
BIBLIOGRAPHY
HISTORICAL
'Hooke. Micrographia. London, 1665.
Posthumous Works of R. Hooke. London, 1705.
Priestley. History and Present State of Discoveries concerning Vision^
Light, and Colours. 2 vols. London, 1772.
JWilde. Geschichte der Optik. 2 vols. Berlin, 1838. This history
covers only the period from Aristotle to Euler.
JVerdet Lecons d'Optique Physique. 2 vols. Paris, 1869. The
second chapter fifty pages is devoted entirely to
the history of the wave-theory.
Introduction aux ceuvres d'Augustin Fi'esnel.
See CEuvres (Completes de Fresnel, t, i., pp. 1-99.
cLloyd. Report of the Progress and Present State of Physical Op-
tics. Brit. Assoc. Rep. 1834.
cArago. (Euvres Completes. Paris, 1854. The first volume con-
tains valuable biographies of Fresnel and Young.
""Peacock. Life of Thomas Young. London, 1855.
Bosscha. Christian Huygens. Rede am SOOstcn. Gedachtnistage
seines Lebensendes. Leipsig, 1895.
DIFFRACTION AND INTERFERENCE
--Grimaldi. Physico-Mathesis de lumine, coloribus, et iride. Bononiae,
1665. Those to whom Grimaldi's work is not acces-
sible will find an excellent resume of his observa-
tions in Priestley's History.
^Newton. Opticks. London, 1704. Newton's description of dif-
fraction phenomena (Bk. III.) and of the behavior of
the prism (Bk. II.) should be read by every student
of optics.
Fraunhofer. Neue Modification des Lichtes, etc. 1821 . Translated by
Ames in Harper's Scientific Memoirs.
Schwerd. Beugungserscheinungen. Mannheim, 1835.
Stokes. Dynamical Theory of Diffraction. Trans. Camb. Phil.
8oc., 9, 1 (1849). R-printed in his Math, and Phys.
Papers, vol. ii., p. 243.
Loramel. Abh. der IT. Cl. der Kon. Bayer. Akad. der Wiss., vol. xv.
160
MEMOIRS ON THE WAVE-THEORY OF LIGHT
Cornu. Journal de Physique, 3, p. 1, 1874. Interpretation of
Fresnel's Integrals in terms of " Cornu's Spiral."
Rayleigh. Treatise on Sound, Second Edition, vol. i. On Group
Velocity of Waves.
Phil. Mag., 27, 460 (1889). On Interference phenomena
with a source of white light.
Schuster. Phil. Mag., 31, 77 (1891). Elementary treatment of prob-
lems in diffraction.
Phil. Mag., 37, 509 (1894). Interference phenomena with
a source of white light.
Gouy. Jour, de PJiysique, p. 354 (1886). On Interference phe-
nomena with a source of white light.
Michelson. Amer. Jour. Sci., 39, Feb., 1890.
Phil. Mag., Mch.. 1891.
Phil Mag., April, 1891.
Phil. Mag., Sept., 1892.
Comptes rendus, 17th April, 1893.
Astronomy and Astrophysics, 12, 556(1893). Compari-
son of meter with wave-length of cadmium light.
Light -waves and their application to Metrology,
Nature, 16, Nov., 1893.
Rowland. Gratings in Theory and Practice ; Astronomy and Astro-
physics, 12, 129 (1893).
ABERRATION
v Young. Lectures on Natural Philosophy, vol. i., p. 462.
-Fresnel. Ann. de Chimie et de Physique, 9, 57 (1818).
Stokes. Phil. Mag., 27, 9 (1845) ; 28, 76 (1846) ; 29, 6 (1846).
Fizeau. Ann. de Chimie, [3] 57 (1859).
Hoek. Arch. Neerlandaises, 3, 180 (1868).
Airy. Proc. Roy. Soc., 2O, 35 (1872) ; 21, 121 (1873). Meas-
urement of the aberration constant by means of a
telescope whose tube is filled with water.
Michelson. Amer. Jour. Sci., 122, 120 (1881).
Michelson and Morley. Amer. Jour. Sci.., 131, 377 (1886).
Phil. Mag.. 24, 449 (1887).
Rayleigh. Nature, 45, 499 (1892). A splendid presentation of facts
and theories up to 1887.
Glazebrook. Report on Optical Theories. Brit. Assoc. Rep. (1895).
Lodge, O. Phil. Trans., 184, 727(1893).
Pellat. Jour, de Physique, [3] 4, 21 (1895). Discusses case of
telescope filled with water.
Larmor Ether and Matter. Cambridge, 1900.
STATIONARY LIGHT- WAVES
Zenker. Lehrbuch der Photochromie. Berlin, 1868.
Rayleigh. Phil. Mag.. 24, 158, note (1887).
L 161
MEMOIRS ON THE WAVE-THEORY OF LIGHT
Lippmanu. Comptes rendus, 112, 274 (1891); 114, 961 (1892);
115, 575(1892).
Nature, 46, 12.
Becquerel. Comptes rendus, 112, 275 (1891).
Wiener. Wied. Ann., 4O, 203 (1890). Shows that photographic
effect of light-waves is due to vibration of electric,
not magnetic, forces.
Wied. Ann., 55, 225 (1895).
SOME IMPORTANT TREATISES ON THE WAVE-THEORY
Knockenhauer. Die Undulationstheorie des Lichtes. Berlin, 1839.
Airy. Undulatory Theory of Optics. London, 1866.
Lord Kelvin. Molecular Dynamics. (Baltimore Lectures.) Baltimore
1884.
Ketteler. Theoretische Optik. Braunschweig, 1885.
Rayleigh. Art. Wave Theory of Light. Ency. Brit , 1888.
Mascart. Traite d'Optique. Paris, 1889.
Preston. The Theory of Light. London, 1890.
Kirchhoff. Optik. Leipzig, 1891.
Basset. A Treatise on Physical Optics. Cambridge, 1892
Poincare. La Lumiere. 2 vols. Paris, 1892.
Winklemann. Handbuch der Physik. Breslau, 1893.
Ilelmholtz. Electromagnetische Theorie des Lichtes. Hamburg, 1897.
Gray and Matthews.
Bessel's Functions.
fraction.
162
London, 1895. Chapter on Dif-
INDEX
Aberration (see Preface).
Arago, 148, 149, 157.
Bosscha, 43.
Cassini, 3.
Color, Newton's Explanation of, 52.
Crested Fringes of Grimaldi, 69.
D
De la Hire, 3.
Descartes' Idea of the Ether, 11, 18,
23.
Diffraction in the Shadow of a Nar-
row Obstacle, 63, 82.
Diffraction Past an Edge, 130.
Diffraction Through a Narrow Aper
ture, 114, 135.
Diffraction Through Parallel Slits,
88.
Diffraction, Young's Idea of, 56
E
Effective Ray, 113
Emission Theory of Newton, Pres-
nel's Objections to, 99.
Ether, Newton's Idea of, 49.
Euler, 48.
Fermat's Principle, 40.
Fresnel, Biographical Sketch of, 156.
Fresnel's Integrals, 123.
Fresnel's Zones. 111.
Fringes, Interior and Exterior, 81.
G
Glazebrook, 161.
Griiualdi, 69-
H
Herschel, 76.
Hooke, 22 (see Preface).
Huygvns, Biography of, 42.
Huygens's Principle, 108, 118.
I
Integrals of Fresnel, 123.
Intensity of Light, 120.
Intensity of Vibration, 120.
Interference, 146.
Interference of Polarized Light,
145.
Interference, Young's Explanation
of, 60, 68 ; Fresnel's Explanation
of, 101.
L
Larmor, 162.
Light, Intensity of, 120.
M
Michelson. 162.
Micrographia of Hooke (see Preface).
Newton's Optics, 48.
Pardies. 22.
Phase, 104, 143.
Pioard. 15.
Polarized Light, Interference of, 145.
163
INDEX
Rayleigh, 161.
Reflection of Light, 25.
Refraction of Light, 30.
R5mer, 3, 13.
Rowland, 161.
Schuster, 162
Secondary Waves, 21, 143.
Simple Harmonic Motion, 102.
Speed of Light, 13.
Stokes, 110.
Trains of Waves, 20.
Transparency, 31.
Transverse Vibrations, 156.
Verdet, 121, 160.
Vibration, Intensity of, 120.
W
Wave-Length of Light as Determined
by Fresuel, 128 ; as Determined by
Young, 71.
\Vollaston. 76.
Young's Idea of Diffraction, 56
i 2'ones of Fresnel, Hi.
164
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