Light

LIGHT

SOUND may be defined as any effect on the sense of hearing, and in the same way Light may be defined as any effect on the sense of sight. This is the purely subjective use of the terms. But both terms are quite as frequently use J in the objective as in the subjective sense. Thus, as Sound may be defined in terms of the motion of the air in the cavity of the external ear, mechanically affecting the tympanum, so Light may be defined by the mechanical effect produced upon the extension of the optic nerve which forms the sensitive surface of the retina.

In treating of Light it will be convenient to use the term in a sort of mixed sense, at least until we come to discuss the different theories which have been devised to account for the propagation of the agent which causes vision. Then we shall have to use the term entirely in the objective sense. On the other hand, in Physiological Optics we are concerned chiefly with the subjective sense of the term.

The present article is intended to give a general sketch of the subject of Optics, so far as it can be treated by the help of elementary mathematics, but with sufficient detail to show the connexion of its various branches, and to enable the reader who desires further information on any point to judge for himself under what heading he will find it in this work. The subject is arranged in the following order :—

Early History of Optics.

Preliminary Statements with regard to Vision, Distinct Vision, the Colour-Sense, and the Duration of Visual Impressions.

Sources of Light.

General Reflexions on the Mechanism of Propagation of Light. Division of the Subject into Geometrical and Physical Optics.

Geometrical Optics.

Rectilinear Propagation of Light in Homogeneous Media, Shadows. Camera Obscura, &c.

Intensity of Illumination as depending on the Distance of the Source and the Obliquity of the Rays. Brightness and Intrinsic Brightness.

Photometry.

Velocity of Light.

Behaviour of Light at the Common Surface of Two Homogeneous Media.

Reflexion. Plane, Spherical, and Cylindrical Mirrors. Real and Virtual Images.

Single Refraction. Composite Nature of White Light. Refractive Index. Dispersion. Prisms; Fraunhofer’s Lines. Irrationality of Dispersion. Achromatism. Lenses. Telescope, Microscope. Pure Spectrum. Refraction by Cylinder. Rainbow.

Refraction in a Non-homogeneous Medium. Hamilton’s Characteristic Function. Mirage.

Absorption, Abnormal Dispersion, Fluorescence, Phosphorescence.

Physical Optics. Undulatory Theory.

Nature and Propagation of Waves. Huygens’s Principle. Explanation of Reflexion and Single Refraction. Disproof of the Corpuscular Theory.

Sketch of the History of the Undulatory Theory. Young’s Discovery of Interference.

Interference Bands. Spectrum formed by Grating. Measure of Wave-Length. Loss of Semiundulatiom Newton’s Rings. Colours of Thin Plates and of Grooved Surfaces.

Relation between Wave-Length and Refractive Index.

Double Refraction. Wave-Surface in Iceland Spar.

Polarization. Transverse Vibrations. Nature of Unpolarized Light.

Plane, Circularly, and Elliptically Polarized Light. Nicol’s Prism. Depolarization by Doubly-Refracting Plate. Fresnel’s Rhomb.

Doppler’s Principle. Measurement of the Relative Velocity of Luminous Source and Spectator.

Under Optics (Geometrical, Physical, and Physiological) further developments will be given; and the connexion between light and radiant heat will be discussed under Radiation. Early History of the Subject. —It is to sight that we are mainly indebted for our knowledge of external things. All our other senses together, except under very special conditions, do not furnish us with a tithe of the information we gain by a single glance. And sight is also that one of our senses which we are able most effectively and extensively to aid by the help uf proper apparatus— not merely (as by spectacles, invented circa 1300) for the cure of natural defects, but (as by the telescope and microscope) for the examination of bodies either too distant or too minute to be studied by the unassisted eye.

It is very remarkable, under these circumstances, to find how slowly men have reached some even of the simplest facts of optics. We can easily understand how constant experience must have forced on them the conviction that light usually moves in straight lines,— i.e., that we see an object in the direction in which it really lies. But how they could have believed for ages that objects are rendered visible by something projected from the eye itself—so that the organ of sight was supposed to be analogous to the tentacula of insects, and sight itself a mere species of touch —is most puzzling. They seem not till about 350 b.c. tc have even raised·the question—If this is how we see, why cannot we see in the dark? or, more simply,—What it darkness? The former of these questions seems to have been first put by Aristotle.

The nature and laws of reflexion were, of course, forced on the ancients by the images seen in still water; and the geometers of the Platonic school were well acquainted with these laws. To Hero of Alexandria we owe the important deduction from them that the course of a reflected ray is the shortest possible.

The general nature of refraction also was known, with some of its special applications, such as, for instance, to burning-glasses and to magnifiers. These were probably either spherical glass shells filled with water (Pliny, 2Z.A r ., xxxvi. 67 [25]; Lact., De Ira Dei, c. 10) or balls of rock crystal (Pliny, xxxvii. 10).

In the first century of our era Cleomedes pointed out how a coin at the bottom of an empty cup, where the eye cannot see it, can be made visible by filling the cup with water; and he showed that, in a similar way, the air may render the sun visible to us while it is still under the horizon. Shortly after this date Ptolemy (the celebrated astronomer) published his great work on Optics. He treats of vision, reflexion, the theory of plane and concave mirrors, and refraction. He measured, with considerable accuracy, the angles of incidence and refraction, for rays passing from air into water and into glass, and from water into glass; it was not, however, till more than fifteen hundred years had passed that the true relation between these angles was discovered. In addition to what has just been mentioned, the ancients’ knowledge of optics was limited to a very superficial acquaintance with some of the properties of rainbows, halos, mirage, Ä’c. But it was fragmentary in the extreme—though it far surpassed in amount as well as in accuracy their knowledge of the other branches of physical science.

It is not easy to understand the ideas of the ancients about colour. That it is a property of a body—just as its density, its hardness, or its smell is a property—was probably held by them. But they also imagined that a body could communicate its colour to light; thus, for instance, the clouds were, bv some of them, sunoosed to communicate their colours to the sunbeams which form a rainbow.

Our next glimpse of real progress dates from the 11th or 12th century, when Alhazen (9.v.) wrote a treatise on optics in Arabic, which for five hundred years or more was a recognized authority on the subject. It was, in many parts, founded on the work of Ptolemy, but with considerable additions and improvements. Alhazen gives an anatomical description of the eye, and points out, fairly enough, how with two eyes we see only one image. But he also points out that we see each object, however small, ∖yya.pencil of diverging rays,—not (as the ancients imagined) by a single ray. Alhazen accounts for twilight, and shows how by it to measure the height of the atmosphere. He also gives the now generally received explanation of the curious fact that the sun and moon appear larger when rising or setting than when they are high in the heavens.

The farther progress of the subject we need not now trace. From the end of 16th century that progress has been extremely rapid. The dates of the more important steps, and the names of their authors, will be given when we treat of these, in their turn, in the course of the article; and we will give them the additional interest of being presented, when this can readily be done, in the author’s own words.

Preliminary Statements. —Before we commence a more rigorous treatment of the subject, it may be well to make a few preliminary statements as to the nature of vision and the conditions for distinct vision. Properly speaking, these belong to Optics (Physiological) (g.v.), but it is impossible to treat intelligibly any part of our subject without presupposing some, generally very slight, knowledge of other parts. And the few preliminary statements we have now to make are in no respect theoretical, while they are so simple that any one may at once test their truth for himself.

Except in the case of a very abnormal eye (extremely short-sighted or long-sighted as the case may be) there is a distance from it—usually somewhere about 10 inches— at which if an object be placed it is seen more distinctly than if placed at any other distance. Almost every one, perhaps without knowing it, habitually places at or about that distance from his eye an object which he wishes to examine carefully. When he.places it at a smaller distance he becomes conscious of the effort required to see it distinctly. He has, in fact, to alter the form of the optical machinery of the eye, by a muscular effort, so that it may become capable of bringing to a focus on the retina rays more divergent than those for which the parts were in their unstrained state adapted. A corresponding effort, but usually much more slight, is commonly felt to be required if the object be at a distance greater than 10 inches.

Hence we arrive at the conclusion that, for the minimum of strain on the eye, rays should fall on it diverging as if they came from a peint about 10 inches distant. But for all ordinary eyes any divergence from double of this Ci.e., divergence as if from a distance of 5 inches) to zero (i.e., parallel rays) is consistent with the possibility of distinct vision. Rays either more divergent than the former limit, or convergent, are unfit to produce distinct vision. Hence every optical instrument, whatever be the reflexions or refractions to which light has been subjected in passing through it, must finally allow the light to escape either in parallel rays or with a divergence within the above specified limits, if it is to be employed by an ordinary eye. The comparatively slight differences which exist among ordinary eyes are easily compensated by the rack-work, or screw adjustment, which is invariably attached to the eye-piece of a good telescope and to the body of a good microscope. Every motion of this rack-work alters the divergence of the rays as they finally escape from the instrument Any eye, however abnormal, if it be capable of producing distinct vision at all, has only to be furnished with suitable spectacles in order that it may behave exactly as does a normal eye. This statement, however, refers only to sharpness of definition, not in any degree to colour. The deficiency which causes colour-blindness cannot be supplied by any conceivable process. A definite part of the ordinary organ of vision is wanting (or inactive) in such cases— while the merely optical parts of the eye are usually in perfect order.

Another fact which must be stated here is that, to produce vision of a body in its natural position, the image on the retina, as seen from the back, must be inverted—not merely as regards up and down, but also as regards right and left. Thus, in the ordinary astronomical telescope, the image on the retina is not inverted, and we therefore see an inverted image.

A third is that our judgment of the relative distances of objects is formed mainly by the use of the two eyes simultaneously. One eye, kept still, can inform us only of relative distance in virtue of the greater or less effort to see distinctly (already spoken of). With both eyes, or with one eye moved from side to side, parallax comes in, and gives us the stereoscopic effect, as it is called. This power of judging distance is, of course, greater as the eyes are set more widely apart. There is, practically, no limit to the effective distance between the eyes when the proper instrumental methods (as with the telestereoscope) are employed.

It is also necessary to premise a few words about colour. The various homogeneous rays of the solar spectrum have each a colour of its own which no refraction can modify. But what about the many colours which do not occur in the spectrum? To such a question as “What is yellow” 1 the answer is, “Each particular kind of yellow may be any one of an infinite number of different combinations of homogeneous rays.” And the same is true, in general, of all other colours. Clerk Maxwell found that a yellow equivalent to that of the spectrum can be obtained by mixing in proper proportions certain homogeneous red and green rays. This single example is sufficient to show that the coloursense is of a very singular nature. This question will be fully treated in Optics (Physiological) ; but for our present purpose it is only necessary to say that we now know (after Wünsch and Young) that the normal eye has only three colour-sensations— a red, a green, and a violet,—and that the apparent colour of any light which falls on it depends merely on the relative intensities of the excitement produced by the light on the three organs of sense corresponding to these sensations. This is true, however, only within certain limits of intensity; for extremely bright light, whatever be its real colour, seems to excite all the three sensations simultaneously, much as white light does; and with very feeble light (as, for instance, that of an ordinary aurora or of a lunar rainbow) we are sometimes scarcely conscious of colours. lu colour-blindness one or more of these organs of sense is wanting, or imperfect. The most common form, Daltonism, depends on the absence of the red sense. Great additions to our knowledge of this subject, if only in confirmation of results already deduced from theory, have been obtained in the last few years by Holmgren; who has experimented on two persons, each of whom was found to have one colour-blind eye, the other being nearly normal. In this way was obtained, what could otherwise have been matter of conjecture only, a description of colour-blind vision in terms of (at least approximately) normal vision.

Finally, the sensation of sight is not limited to the duration of the mechanical action on the eye. It is known that we do not see a sudden flash (an electric spark for instance) until a measurable, though very short, period has elapsed. This depends on the rate at which an excitation is propagated along the optic nerve. But the familiar experiment of whirling a red-hot stick in a dark room shows that the sensation of sight lasts for a short period after the mechanical action which produced it has ceased. This period is probably different for different eyes, and for different amounts of excitement even in the same eye. (If the light be very intense the effect lasts much longer, but completely changes its character.) For our present purpose it may be assumed that the duration is somewhere about ⅜th of a second. Thus, if the end of the red-hot stick describes a circle once in γth of a second, we see the complete circle; if in a longer period, we only see at once such a part of it as was described in 4∙th of a second. Connected with this is the remarkable result obtained experimentally by Swan, that the amount of sensation is, for flashes of short duration, directly proportional, not only to the brightness of the flash, but also to its duration. A flash which lasts for τ 1 σ th of a second produces the full effect on the eye; but an electric spark, as a flash of lightning, which certainly does not endure for more than i 0 ⅛0 000 th of a second, produces at most only τστπππr t ħ °f the effect it would produce if it lasted τ ⅛th of a second. On this short duration of visual impressions depends the action of the thaumatrope, the wheel of life, <fcc, By various kinds of machinery a succession of views of an object in different positions or forms is presented to the eye, each for a brief interval The result is that we fancy we see one and the same object going through a species of continuous motion, or of change of form, which would present it to the eye in these successive positions or forms. Thus, a tadpole may be represented as wriggling about, or as developing continuously into a frog, &c.

Sources of Light. —This subject will be fully treated in other parts of this work under various heads: from the purely scientific point of view under Radiation ; from the more practical side under Lighting (Electric), àc. For our present purpose a very brief summary of the question will suffice; and we do not require to investigate the process by which, in any case, the light is produced.

1. The main source of light is incandescence. (It is usually understood that to be incandescent a body must be at a high temperature.) This may be due to any of a number of causes, such as the following :—

(a) The Potential Energy of Gravitation of Scattered Fragments of Matter.— When these fall together, as in the formation of the sun and stars, heat enough is generated by impact to render the whole vividly incandescent. It is probable that the light of nebulae, and the proper light of comets, is due to this cause. The proximate cause, in all these cases, is the kinetic energy of the fragments before impact. To this class, therefore, can be reduced the light given out when a target is struck by a cannon shot.

(δ) The Kinetic Energy of Cutrent Electricity or of an Electric Discharge.— Here we have lightning, the electric light, and probably also the light of the aurora.

(c) The Potential Energy of Chemical Affinity.— The lime-light, gas-light, candle and lamp-light, fire-light, the magnesium light, &c.; also phosphorus, dead fish (1), <fcc., glowing in the dark.

(<7) Friction, as in the trains of sparks from a grindstone or brake; though here, in general, chemical affinity also has a share.

(e) Sudden great Compression of a Gas, as of air by meteoric stones and falling stars.

2. Another very curious source, not (so far as is known) reducible to incandescence, is the giving out (^usually in an altered form') of light previously absorbed :—fluorescence, phosphorescence, <fcc.

3. A third source is physiological :—fire-flies, glowworms, Medusae, dead fish (?), <fcc., the eye of a cat.

Any not black and not transparent body, exposed to any of these sources of light, becomes in its turn what may for our purpose also be treated as a source.

As will be shown in Radiation, the only bodies which, when incandescent, give every constituent of white light are bodies which are black in the sense of absorbing each and every ray which falls upon them. Such bodies are not necessarily solids—though the best examples we have of them are lamp-black, and (somewhat less perfect) charcoal and gas-coke.

Newton’s speculations on these subjects, taken from the “Queries” at the end of his Optics, give an exceedingly interesting sketch of the state of this subject in his time. We quote a few of the more curious. There is a strange admixture of errors, but a still more strange anticipation of some of the most important of modern discoveries.

“Query 6. Do not Black bodies conceive heat more easily from light than those of other colours do, by reason that the light falling on them is not reflected outwards; but enters the bodies, and is often reflected and refracted within them, until it be stifled and lost?

“Query 8. Do not all Fixed bodies, when heated beyond a certain degree, emit light and shine; and is not this emission performed by the vibrating motions of their Parts. And do not all bodies, which abound with Terrestrial parts, and especially with Sulphureous ones, emit light, as often as those Parts are sufficiently agitated; whether that agitation be made by heat, or by friction, or percussion, or putrefaction, or by any vital motion, or any other cause? . . . .

“Query 9. Is not Fire a body heated so hot, as to emit light copiously? For what else is a red hot iron than fire? And what else is a burning coal than red hot wood?

“Query 10. Is not Flame a vapour, fume or exhalation heated red hot, that is, so hot as to shine? For bodies do not flame without emitting a copious fume, and this fume burns in the flame. The Ignis Fαtτιus is a vapour shining without heat; and is there not the same difference between this vapour and flame, as between rotted wood shining without heat and burning coals of fire? In distilling hot spirits, if the head of the still be taken off, the vapour, which ascends out of the still, will take fire at the flame of a candle, and turn into flame, and the flame will run along the vapour from the candle to the still. Some bodies heated by motion or fermentation, if the heat grow intense, fume copiously; and if the heat be great enough, the fumes will shine, and become flame. Metals in fusion do not flame for Avant of a copious fume, except spelter, which fumes copiously, and thereby flames. All flaming bodies, as oil, tallow, wax, wood, fossil coals, pitch, sulphur, by flaming waste and vanish into burning smoke; which smoke, if the flame be put out, is very thick and visible, and sometimes smells strongly, but in the flame loses its smell by burning; and, according to the nature of the smoke, the flame is of several colours; as that of sulphur, blue; that of copper opened with sublimate, green; that of tallow, yellow; that of camphire, white. Smoke passing through flame cannot but grow red hot; and red hot smoke can have no other appearance than that of flame

“Query 11. Do not Great bodies conserve their heat the longest, their parts heating one another; and may not Great dense and Fixed bodies, when heated beyond a certain degree, emit light so copiously, as by the emission and re-action of its light, and the reflexions and refractions of its rays within its pores, to grow still hotter, till it comes to a certain period of heat, such as is that of the sun? And are not the sun and fixed stars great earths vehemently hot; whose heat is conserved by the greatness of the bodies, and the mutual action and re-action between them, and the light which they emit; and whose parts are kept from fuming away, not only by their Fixity, but also by the vast weight and density of the atmospheres incumbent upon them, and very strongly compressing them, and condensing the vapours and exhalations which arise from them? .... And the same great weight may condense those vapours and exhalations, as soon as they shall at any time begin to ascend from the sun, and make them presently fall back again into him; and by that action increase his heat, much after the manner that in our earth the air increases the heat of a culinary fire. And the same weight may hinder the globe of the sun from being diminished, unless by the emission of light, and a very small quantity of vapours and exhalations.”

Theories of Propagation of Light. —We may begin by assuming that the sensation of light is due to a mechanical action on the retina (see Eye). Now such a mechanical action must have a mechanical cause, and, as far as we can judge with our present knowledge, the latter must consist of impacts on the retina, due to moving matter. This matter may have travelled all the way from the source of light, or it may have been set in motion in the eye by a disturbance (analogous to a wave) which has travelled from the source. What is transferred, or what moves, is a quite independent question. Light must, as far as we can conceive, consist in the motion of particles of some kind from external objects to the eye, or in the propagation of some disturbance or wavemotion in an as yet unknown medium. Though it has been proved, as we will presently show, that some of the consequences of the first supposition are entirely inconsistent with observed facts, the nature of the propagation of the supposed luminous particles is still a very interesting study, and indeed many of the fundamental propositions in optics follow more easily from this hypothesis than from the other. We will therefore not at present dismiss this hypothesis, but will refer freely to it now and then, until its truth is shown to be inconsistent with experiment.

This view, associated with the names of Newton, Laplace, and Biot, is known as the corpuscular theory of light. A formidable objection to it, in limine, will be easily seen to be furnished by the velocity of light. Since every point of every visible body must (on this theory) send such corpuscles to the eye, moving as we shall find at a rate of nearly 200,000 miles per second, their masses must be inconceivably minute in order that their united momentum may not amount to something comparable with that of a cannon shot, a supposition of course utterly destructive of all belief in the hypothesis. But, as we shall see, there are still higher grounds of objection, and such as no mere smallness of mass or size of each corpuscle can explain away.

The rival theory labours under considerable disadvantages, inasmuch as the theory of wave-propagation is very much more obscure and difficult than that of the motion of free particles; but the student, who has mastered the fundamental difficulties of sound (see Acoustics), which presents a fair although not an exact analogy, will find it comparatively easy to obtain a clear conception of the fundamental principles of the explanation offered by the undulatory theory of light.

The difference between these two theories of light may be illustrated by contrasting wind moving at the rate of 1100 feet per second (an inconceivably violent hurricane) and sound, gentle or violent, moving at precisely the same rate—yet how different in its effects!

Division of the Subject. —Optics, or the science of Light, is usually divided into two parts. A simple illustration of the nature of this division will be found in the different conditions of fluid equilibrium according as we do not or do introduce the idea of action between the fluid and the containing vessel (Capillary Action, q.vf. In the first or hypothetical case it is known that the free surface must be horizontal, and that all its separate parts must lie in the same plane; in the second, i.e., the actual, case we find molecular action modifying these results, sometimes indeed to a very large extent, so that no part of the free surface is plane, and no two portions of it are at the same level. So in what is called Geometrical Optics it is assumed from experiment that light moves in straight lines in air, while Physical Optics, or the undulatory theory, agrees with experiment in showing that under certain circumstances a ray of light bends round an obstacle. But as, in obtaining the main facts of fluid equilibrium, capillary forces may be neglected, so, for the explanation of the ordinary phenomena of light, even with accuracy sufficient for the construction of the very finest telescopes and microscopes, it suffices that Geometrical Optics, based on laws nearly verified by experiment, be followed out to its consequences. The residual phenomena then came in to be treated by the undulatory theory. Pouillet divides the subject, in consequence of this distinction, into two parts, viz., (1) that in which we deal with the direction only of the rays, and (2) that in which we deal with the physical properties of the rays themselves.

In this order we will consider the subject, giving the explanations of the approximate experimental laws of Geometrical Optics, as we reach them, in the language of either theory. But before we come to the residual phenomena we shall have found that the corpuscular theory must be rejected, and we will therefore give, without detail, the principles of the undulatory explanation, which will be fully discussed in a special article.

Geometrical Optics. Rectilinear Propagation of Light.

It is approximately true that, in any homogeneous medium, light moves in straight lines.

If an opaque body be placed anywhere in the straight line between the eye and an object, the object is concealed. Through a long straight tube no objects can be seen but those situated in the direction of its axis produced. This is so fundamental a fact, or it is so evident a result of experience, that it is the foundation of every process which involves the direction in space of one object as regards another,—whether it be for the aiming with a rifle, the pointing of a telescope, or for the delicate observations of a geodetic survey. But we must carefully observe the restrictions under which the statement is made. Not merely is it said to be only approximately true, but it is so only in a homogeneous medium. To both of these restrictions we will revert later.

(a) On this is founded the geometrical theory of shadoιοs, —a subject of some importance, especially as regards eclipses. In this application the results may be considered as absolutely true, though, as we shall see in a subsequent page, the statement is liable in certain delicate cases to somewhat startling exceptions. When an opaque body is placed between a screen and a luminous point, it casts a shadow on the screen. (The sun’s image formed by a lens or burning glass of short focus is our best mode of attempting to realize the conception of a luminous point; but a fair approximation may be made by piercing a very small needle-hole in a large plate of thin metal, and placing it close to any bright flame or incandescent body.) The outline of the shadow is, of course, to be found by drawing straight lines from the luminous point so as to touch the opaque body all round. These lines form a cone. The points of contact form a line on the opaque body separating the illuminated from the non-illuminated portion of its surface. Similarly, when these lines are produced to meet the screen, their points of intersection with it form a line which separates the illuminated from the non-illuminated parts of the screen.

This line is called the boundary of the geometrical shadow. A common but beautiful instance of it is seen when a very small gas-jet is burning in a ground-glass shade, near the wall of a room. In this case the cone, above mentioned, is usually a right cone with its axis vertical. Thus the boundary of the geometric shadow is a portion of a circle on the roof, but a portion of an hyperbola on the vertical wall. If the roof be not horizontal, we may obtain in this way any form of conic section. Interesting and useful hints in projection may be obtained by observing the shadows of bodies of various forms cast in this way by rays which virtually diverge from one point: e.g., how to place a plane quadrilateral of given form so that its geometric shadow may be a square; how to place an elliptic disk, with a small hole in it, so that the shadow may be circular with a bright spot at its centre, &c.

When there are more luminous points than one, we have only to draw separately the geometrical shadows due to each of the sources, and then superpose them. A new consideration now comes in. There will be, in general, portions of all the separate geometrical shadows which overlap one another in some particular regions of the screen. In such regions we still have full shadow; but around them there will be other regions, some illuminated by one of the sources alone, some by two, tfcc., until finally we come to the parts of the screen which are illuminated directly by all the sources. There will evidently be still a definite boundary of the parts wholly unilluminated, i.e., the true shadow or umbra, and also a definite boundary of the parts wholly illuminated. The region between these boundaries— i.e., the partially illumined portion—is called the penumbra.

Fig. 1 shows these things very well. It represents the shadow of a circular disk cast by four equal luminous points arranged as the corners of a square,—the disk being large enough to admit of a free overlapping of the separate shadows. The amount of want of illumination in each portion of the penumbra is roughly indicated by the shading. The separate shadows are circular, if the disk is parallel to the screen. If we suppose the number of sources to increase indefinitely, so as finally to give the appearance of a luminous surface as the source of light, it is obvious that the degrees of darkness at different portions of the penumbra will also increase indefinitely; i.e., there will be a gradual increase of brightness in the penumbra from total darkness at the edge next the geometrical shadow to full illumination at the outer edge. It is most instructive to contrast with the above figure that now given (fig. 2), in which the size of the disk is considerably diminished — everything else being unchanged. Here there is no true shadow·—only four equally bright portions of the penumbra, each illuminated by three of the sources.

Thus we see at once why the shadows cast by the sun or moon are in general so much less sharp than those cast by the electric light (when it is not surrounded by a semiopaque screen). For, practically, at moderate distances from the electric arc, it appears as a mere luminous point. But, if we place a body at a distance of a foot or two only from the arc, the shadow cast will have as much of penumbra as if the sun had been the source. The breadth of the penumbra when the source and screen are nearly equidistant from the opaque body is equal to the diameter of the luminous source. Simple as is the question from the point of view we have adopted, it may to some persons appear simpler to imagine themselves placed (as spectators) on the screen in different parts of the shadow or penumbra, and to consider what portions of the luminous source they would then be in a position to see.

This is what happens to us when we observe an eclipse of the sun. When the eclipse is total, there is a real geometrical shadow,—very small compared with the penumbra (for the apparent diameters of the sun and moon are nearly equal, but their distances are as 370: 1); when the eclipse is annular, the shadow is all penumbra. In a lunar eclipse, on the other hand, the earth is the shadowcasting body, and the moon is the screen, and we observe things according to our first point of view.

Suppose, next, that the body which casts the shadow is a large one, such as a wall, with a hole in it. If we were to plug the hole, the whole screen would be in geometrical shadow. Hence the illumination of the screen by the light passing through the hole is precisely what would be cut off by a disk which fits the hole Fig. 3, which is the complement of fig. 1, gives therefore the effect of four equal sources of light shining on a wall through a circular hole. And it is evident that, with the change of a word here and there, the previous reasoning may be applied to this case also. The umbra in the former case becomes the fully illuminated portion, and vice versa. The penumbra remains the penumbra, but it is now darkest where before it was brightest, and vice versa. For further information we subjoin the complement (fig. 4) of the second case above,—the same four sources, but the smaller hole. Here we have four equally bright, separate images

—one belonging to each of the sources. Thus we see how, when a small hole is cut in the window-shutter of a dark room, a picture of the sun, and bright clouds about it, is formed on the opposite wall. This picture is obviously inverted, and also perverted, for not only are objects depicted lower the higher they are, but also objects seen to the right are depicted to the left, &c. But it will be seen unperverted (though still inverted) if it be received on a sheet of ground glass and looked at from behind. The smaller the hole (so far at least as Geometrical Optics is concerned) the less confused will the picture be. As the hole is made larger the illuminated portions from different sources gradually overlap; and when the hole bec6mes a window we have no indications of such a picture except from a body (like the sun) much brighter than the other external objects. Here the picture has ceased to be one of the sun, it is now a picture of the window. But if the wall could be placed 100 miles off, the picture would be one of the sun. To prevent this overlapping of images, and yet to admit a good deal of light, is one main object of the lens which usually forms part of the camera obscura.

The formation of pictures of the sun in this way is well seen on a calm sunny day under trees, where the sunlight penetrating through small chinks forms elliptic spots on the ground. During a partial eclipse these pictures have, of course, a crescent form. When detached clouds are drifting rapidly across the sun, we often seethe shadows of the bars of the window on the walls or floor suddenly shifted by an inch or two, and for a moment very much more sharply defined. They are, in fact, shadows cast by a small portion of the sun’s limb, from opposite sides alternately. Another beautiful illustration is easily obtained by cutting with a sharp knife a very small T aperture in a piece of note paper. Place this close to the eye, and an inch or so behind it place another piece of paper with a fine needle-hole in it. The light of the sky passing through the needle-hole forms a bright picture of the T on the retina. The eye perceives this picture, which gives the impression of the T much magnified, but turned upside doινn.

Another curious phenomenon may fitly be referred to in this connexion, viz., the phantoms which are seen when we look at two parallel sets of palisades or railings, one behind the other, or look through two parallel sides of a meat-safe formed of perforated zinc. The appearance presented is that of a magnified set of bars or apertures which appear to move rapidly as we slowly walk past. Their origin is the fact that where the bars appear nearly to coincide the apparent gaps bear the greatest ratio to the dark spaces; he., these parts of the field are the most highly illuminated. The exact determination of the appearances in any given case is a mere problem of convergents to a continued fraction. But the fact that the apparent rapidity of motion of this phantom may exceed in any ratio that of the spectator is of importance,— enabling us to see how velocities, apparently of impossible magnitude, may be accounted for by the mere running along of the condition of visibility among a group of objects no one of which is moving at an extravagant rate.

{b) Another important consequence of this law is that if the medium be transparent the intensity of illumination which a luminous point can produce on a white surface directly exposed to it is inversely as the square of the distance.

The word transparent implies that no light is absorbed or stopped. Whatever, therefore, leaves the source of light must in succession pass through each of a series of spherical surfaces described round the source as centre. The same amount of light falls perpendicularly on all these surfaces in succession. The amount received in a given time by a unit of surface on each is therefore inversely as the number of such units in each. But the surfaces of spheres are as the squares of their radii,—whence the proposition. (We assume here that the velocity of light is constant in the medium, and that the source gives out its light uniformly and not by fits and starts.) When the rays fall otherwise than perpendicularly on the surface, the illumination produced is proportional to the cosine of the obliquity; for the area seen under a given spherical angle increases as the secant of the obliquity, the distance remaining the same.

As a corollary to this we have the further proposition that the apparent brightness of a luminous surface {seen through a transparent homogeneous medium) is the same at all distances.

The word brightness is here taken as a measure of the amount of light falling on the pupil per unit of spherical angle subtended by the luminous surface. The spherical angle subtended by any small surface whose plane is at right angles to the line of sight is inversely as the square of the distance. So also is the light received from it. Hence the brightness is the same at all distances.

The word brightness is often used (even scientifically) in another sense from that just defined. Thus we speak of a bright star, of the question—When is Venus at its brightest! <kc. Strictly, such expressions are not defensible except for sources of light which (like a star) have no apparent surface, so that we cannot tell from what amount of spherical angle their light appears to come. In that case the spherical angle is, for want of knowledge, assumed to be the same for all, and therefore the brightness of each is now estimated in terms of the whole quantity of light we receive from it. It is in this sense only that we use the word when we speak of Venus at its brightest; for if we take the former definition of brightness the solution of this once celebrated problem would be very different from that usually given. As the questiori, however, is an interesting one both in itself and historically, we give an approximate solution of it. The approximation assumes what is certainly not true, that the illuminated portion of Venus always appears uniformly bright, and of the same degree of brightness in all aspects.

Let α be the radius of the earth’s orbit, b that of the orbit of Venus, δ the distance between the planets when Venus is brightest.

Then if θ be the apparent angular distance of the earth from the sun as seen from Venus, the illuminated part of the disk of Venus as seen from the earth is

1 + cos θ 2 of the whole disk. Hence

1 + cos θ

2δ 2 is a maximum,—with the obvious trigonometrical relation

α 2 = δ 2 + b i - 2Z>δ cos θ.

Substituting for cos θ, and putting the differential coefficient =0, we have a quadratic equation of which the only admissible root is the positive one

δ = √8α 2 + δ a -2Z>.

From this the other quantities can be calculated.

But another matter has to be taken into consideration when we apply the above definition of brightness in practice. For the aperture of the pupil is usually very much contracted when we look at a brightly illuminated sky or cloud. Thus there is a rough compensation which, to a certain extent, modifies the effect on the retina.

Founded on the above is Cheseaux’s celebrated argument about the finite dimensions of the stellar universe. For it is easy to see, as below, that if stars be scattered through infinite space, with average closeness and brightness such as is presented by those nearest us, and if stellar space be absolutely transparent, the whole sky should appear of a brightness like that of the sun. Cheseaux and Olbers endeavoured to show that, because the sky is not all over as bright as the sun, there is absorption of light in stellar space. This idea was ingeniously developed by Struve.

Consider a small spherical angle ω. The number of stars included in it whose distances are between r and r + δr from the earth is proportional to ωr 3 δr.

The whole amount of light received from such a portion of the sky must be therefore as

∕*OO ω∕ dr, √0

provided that no star intercepts the light coming from another. This condition is unattainable, so that the conclusion is that the brightness is as great as it can be with the materials employed. Every portion of the background shines as if it were a star.

(c) A third very important fact, connected with our present subject, but not immediately deducible from our principle, is— The brightness of a self-luminous surface does not depend upon its inclination to the line of sight.

Thus a red-hot ball of iron, free from scales of oxide, &c., appears flat in the dark; so, also, the sun, seen through mist, appears as a flat disk. This fact, however, depends ultimately upon the second law of thermodynamics, and its explanation will be fully given under Radiation.

It may be stated, however, in another form, in which its connexion with what precedes is more obvious— The amount of radiation, in any direction, from a luminous surface is proportional to the cosine of the obliquity.

The flow of light (if we may so call it) in straight lines from the luminous point, with constant velocity, leads as we have seen to the expression -ff (where r is the distance from the luminous point) for the quantity of light which passes through unit of surface perpendicular to the ray in unit of time, μ being a quantity indicating the rate at which light is emitted by the source. This represents the illumination of the surface on which it falls. The flow through unit of surface whose normal is inclined at an angle θ to the ray is of course

-4- cos G, r i again representing the illumination. These are precisely the expressions for the gravitation force exerted by a particle of mass μ on a unit of matter at distance r, and for its resolved part in a given direction. Hence we may employ an expression

V = 2— , r which is exactly analagous to the gravitation or electric potential, for the purpose of calculating the effect due to any number of separate sources of light.

And the fundamental proposition in potentials, viz., that, if n be the external normal at any point of a closed surface, the integral ff^ tιken over the whole surface, has the value

- 4vμ 0 , where μ 0 is the sum of the values of μ for each source lying within the surface, follows almost intuitively from the mere consideration of what it means as regards light. For every source external to the closed surface sends in light which goes out again. But the light from an internal source goes wholly out; and the amount per second from each unit source is 4ιr, the total area of the unit sphere surrounding the source.

It is well to observe, however, that the analogy is not quite complete. To make it so, all the sources must lie on the same side of the surface whose illumination we are dealing with. This is due to the fact that, in order that a surface may be illuminated at all, it must be capable of scattering light, i.e., it must be to some extent opaque. Hence the illumination depends mainly upon those sources which are on the same side as that from which it is regarded.

Though this process bears some resemblance to the heat analogy employed by Sir W. Thomson for investigations in statical electricity (Cambridge Mathematical Journal, 1842) and to Clerk Maxwell’s device of an incompressible fluid without mass (Cam. PhU. Trans. , 1856), it is by no means identical with them. Each method deals with a substance, real or imaginary, which flows in conical streams from a source so that the same amount of it passes per second through every section of the cone. But in the present process the velocity is constant and the density variable, while in the others the density is virtually constant and the velocity variable. There is a curious reciprocity in formulae such as we have just given. For instance, it is easily seen that the light received from a uniformly illuminated surface is represented by cos 0

As we have seen that this integral vanishes for a closed surface which has no source inside, its value is the same for all shells of equal uniform brightness whose edges lie on the same cone.

We have said that light moves in straight lines in a homogeneous medium. This rectilinear path follows at once from the corpuscular theory, as well as from the undulatory theory of light: in the first case there is no deflecting cause, so each corpuscle moves in a straight line; in the second, the direction of propagation of a plane wave in an uniform isotropic medium is always perpendicular to its front. Looking along a hot poker or the boiler of a steamboat, we see objects beyond distorted·, i.e., we no longer see each point in its true direction. Here we have a non-homogeneous medium, the air being irregularly expanded in the neighbourhood of the hot body. To this simple cause are due the phenomena of mirage, the fata morgana, the reduplication of images of a distant object seen through an irregularly heated atmosphere, the scintillation or twinkling of stars, and the uselessness of even the best telescopes at certain times, &c. It is interesting to note here that Newton says :—“Long telescopes may cause objects to appear brighter and larger than short ones can do; but they cannot be so formed as to take away that confusion of the rays which arises from the tremors of the atmosphere. The only remedy is a most serene and quiet air, such as may perhaps be found on the tops of the highest mountains, above the grosser clouds.”

Photometry. —The principle above explained suggests many simple methods of comparing the amounts of light given by different sources. If, for instance, a porcelain plate, or even a sheet of paper, of uniform thickness, have one half illuminated directly by one source of light, the other by a different source, and if one or other of these sources be moved to or from the plate till the halves appear equally illuminated, it is obvious that the amounts of light given out by the two sources are directly as the squares of their distances from the screen. This is the principle of Ritchie’s photometer. Rumford suggested the comparison of the intensity of the shadows of the same object thrown side by side on a screen by the two lights to be compared. In this case the shadow due to one source is lit up by the other alone; and here again the amounts of light given out by the sources are as the squares of their distances from the screen when the shadows are equally intense. The shadow-casting object should be near the screen, so as to avoid penumbra as much as possible; yet not too near, so that the shadows may not overlap.

Bunsen has recently suggested the very simple expedient of making a grease-spot on white paper for photometric purposes. When the paper is equally illuminated from both sides, the grease-spot cannot be seen except by very close inspection. In using this photometer, the sources are placed in one line with the grease-spot, which lies between them and can be moved towards one or other. To make the most accurate determinations with this arrangement the adjustment should first be made from the side on which one source lies, then the screen turned round and the adjustment made from the side of the other source,—in both cases, therefore, from the same side of the paper screen. Take the mean of these positions (which are usually very close together), and the amounts of light are as the squares of the distances of the sources from this point. Wheatstone suggested a hollow glass bead, silvered internally, and made to describe very rapidly a closed path, for use as a photometer. When it is placed between two sources, we see two parallel curves of reflected light, one due to each source. Make these, by trial, equally bright; and the amounts of light from the sources are, again, as the squares of the distances. These simple forms of apparatus give results which are fairly accurate, so long at least as the qualities of the light furnished · by the two sources are nearly the same. But, when we endeavour to compare differently coloured lights, the result is by no means so satisfactory. In fact, we cannot well define equality of illumination when the lights are of different qualities. In the undulatory theory, no doubt, we can distinctly define the intensity of any form of radiation. But the definition is a purely dynamical one, and has not necessarily any connexion with what we usually mean by intensity, viz., the amount of effect produced upon the nerves of the retina. Thus the theoretical intensities of a given violet and a given red source may be equal, while one may appear to the eye very much brighter than the other. Think, for instance, of a colour-blind person, who might, under conceivable circumstances, be unable to see the red at all. We are all as it were colour-blind as far as regards radiations whose wave-lengths are longer or shorter than those included in the range of the ordinary solar spectrum.

Other modes of measuring the intensity of light usually depend upon more recondite physical principles,—such as, for instance, the amounts of chemical action of certain kinds which can be produced by an exposure of a given duration to the light from a particular source. But all have the same grand defect as the simpler processes,— they are not adapted to the comparison of sources giving different qualities of light. And those last mentioned are liable to another source of error, viz., the action of radiations which are not called light, only because they are not visible to the eye; for in all other respects they closely resemble light, and are often more active than it is in producing chemical changes.

Velocity of Light. —Light moves with a velocity of nearly 186,000 miles per second. Of this we have four distinct kinds of proof, on each of which depends a method which is capable of giving pretty accurate results.

1. Römer'» Method.— By this the finite velocity of light was discovered in 1676. Suppose, to illustrate this, that at a certain place a cannon is fired precisely at intervals of an hour while the weather is perfectly calm. A person provided with an accurate watch travels about in the surrounding district. When he first hears the cannon let him note the time by his watch, then on account of the non-instantaneous propagation of sound, if at the next discharge he be nearer the gun than before, the report will arrive at his ear before the hour’s interval has elapsed; if he be farther from the gun, the interval between the discharges will appear longer than an hour; and the number of seconds of defect or excess will evidently represent the time employed by sound in travelling over a space equal to the difference of his distances from the gun at the successive observations.

Now the satellites of Jupiter are subject—like our moon, only much more frequently—to eclipse, and the interval between two successive eclipses can easily be observed. The almost sudden extinction of the light is a phenomenon similar to the discharge of the gun; and, if light take time to move from one place to another, we should find the interval between successive eclipses too short when we are approaching Jupiter, too long when we are receding from him. Such was found to be the case by Römer; and he also found that the shortening or lengthening of the interval depended upon the rate at which the earth was approaching to or receding from Jupiter. The inevitable conclusion from these facts is that light is propagated with finite velocity. Römer calculated from them that light takes about 16 m 5 to cross the earth’s orbit. The exact velocity deduced by this method is, after making all corrections, and assuming the most probable value of the solar parallax, about 186,500 miles per second.

2. Bradley's Method.— This depends on the aberration of light, discovered by Bradley in 1728. When in a calm rainy day one stands still he holds his umbrella vertical in order to protect himself. If he walk he requires to hold it forwards, and more inclined the faster he walks. In other words, to a person walking the rain does not appear to come in the same direction as to a person standing still. Now the earth’s velocity in its orbit is a very large quantity, some 18∣ miles per second, or about -∏∏∏∏j4h of that of light Hence the light from a star does not appear to come in the proper direction unless the earth be moving exactly to or from the star, and, as the direction of the earth’s motion is continually changing, so the directions in which different stars are seen are always changing, and thus this phenomenon, called the “aberration of light,” proves at once the finite velocity of light and the earth’s motion round the sun.

As an additional illustration of the phenomenon, suppose a bullet fired through a railway carriage, in a direction perpendicular to the sides of the carriage. If the carriage be standing still, the bullet will make holes in the sides, the line joining which is perpendicular to the length of the carriage; if it be in motion, then the second side of the carriage will have moved through a certain space during the interval occupied by the bullet in passing from side to side, and thus the line joining the holes in the sides (f.e., the line pursued by the bullet relatively to the carriage), will be inclined at an angle greater than a right angle to the direction of the train’s motion.

It is evident that the path apparently described by each star during a year, in conséquence of aberration, will be found by laying off from the star lines which bear the same ratio to the star’s distance as the velocity of the earth does to that of light,—their directions being always the same as that of the earth’s motion at every instant. This is precisely the definition of the Hodograph (<?.v.) of the earth’s orbit. Hence, on account of the finite velocity of light, each star appears to describe in space a circle (not an ellipse) of fixed magnitude in a plane parallel to that of the ecliptic. As seen from the earth, therefore, stars will appear to describe paths which are the projections of these circles on the celestial sphere. These are in general ellipses, but circles for stars at the poles of the ecliptic and straight lines for stars in the ecliptic. This is found to be quite consistent with observation; and the major axes of these ellipses, the diameters of the circles, or the lengths of the lines subtend equally angles of about 41" at the earth. Hence the velocity of light is to the velocity of the earth as 1: tan ⅜ 41"; that is, about 10,000: 1.

Both these methods depend, for their final result, upon a true knowledge of the earth’s distance from the sun. But the most accurate measurements of this quantity are probably to be obtained from the velocity of light itself, this being independently determined by the physical processes next to be described. Thus the earth’s distance from the sun will in future be measured rather by the constant of aberration, or by the acceleration or retardation of the eclipses of Jupiter’s satellites, than by a transit of Venus, by the moon’s motion, or by the parallax of Mars. Thus Römer’s and Bradley’s processes are now applied to the determination of solar parallax.

3. Fizeau's Direct Measurement of the Velocity of Light. —To illustrate the next and by far the most convincing i)opular proof of the finite velocity of light, suppose a persen ooking at himself in a mirror, before which is moving a screen with a number of apertures, the breadth of each aperture being equal to the distance between any two of them. If the screen be at rest with an aperture before the mirror, the light from the observer’s face passes through the aperture and is reflected back, so that he sees himself as if the screen were not present. Suppose the screen to be moving in such a way that, when the light which passed through the aperture returns to the screen after reflexion, the unpierced part of the screen is in its way, it is evident that the observer cannot see himself in the mirror. If the screen pass twice as fast, the light that escaped by one aperture will, after reflexion, return by the next, so that he will see his image as at first. If three times as fast, the second unperforated part of the screen will stop the returning light; so he cannot see his image. To apply this practically a thin metallic disk had a set of teeth cut on its circumference so that the breadth of a tooth was equal to that of the space between two teeth. This disk could be set in very rapid rotation by a train of wheel work, and the rate of turning could easily be determined by Savart’s method (see Acoustics, voL i. p. 108). Light passed between two teeth to a mirror situated at 10 miles’ distance, which sent it back by the same course, so that when the wheel was at rest the reflected light could be seen. On turning the disk with accelerated velocity the light was observed to become more and more feeble up to a certain velocity, at which it was extinguished; turning faster it reappeared, growing brighter and brighter till the velocity was doubled; then it fell off, till it vanished when the velocity was trebled, and so on. It is evident from the first illustration above that the velocity of light in air is to that of the tooth, at the first disappearance of the reflected light, as the distance of the mirror from the disk is to the half breadth of the tooth. It is not to be supposed that the description we have just given embodies all the details of this remarkable experiment. On the contrary, telescopes were used at each station to prevent loss of light as much as possible, and many other precautions were adopted which would be unintelligible without references to later parts of this article. This method and its first results were published in 1849 in the Comptes Rendus. The experiments gave, on their very careful repetition by Cornu in 1874, the value 186,700 miles for the velocity in vacuo (Nature, xi. p. 274).

4. Foucault's Method.— This was described in 1850 to the Academy of Sciences. It depends upon the principle of the rapidly revolving plane mirror introduced by Wheatstone to demonstrate the non instantaneous propagation of an electric discharge. The mirror was made to revolve from 600 to 800 times per second, by means of a siren (see Acoustics) driven by steam. A ray of sunlight fell upon it from a small aperture crossed by a grating of platinum wires. Between the wires and the mirror was placed an achromatic lens—the wires being farther from it than its principal focus, but not twice as far— so that the rays falling on the mirror were slowly convergent. They formed an image of the wires at a distance of about 4 metres from the mirror. In certain positions of the revolving mirror, the rays fell upon a concave mirror of 4 metres radius whose centre of curvature was at the centre of the revolving mirror. They were, therefore, reflected back directly to the revolving mirror, and, passing again through the lens, formed an image of the wire grating which, when the adjustment was perfect, coincided with the grating itself. This coincidence was observed by reflexion from a piece of unsilvered glass, placed obliquely in the track of the rays, the image in which was magnified by an eye-piece. It is obvious that, when the mirror is made to turn, the light which comes back to it after passing to the fixed mirror, finds it in a position slightly different from that in which it left it. That difference is due to the amount of rotation during the time of passage of the light to and fro along an air-space of 4 metres. Accordingly, as soon as the mirror began to rotate with considerable velocity, the coincidence between the wires and their images was destroyed; and the two were separated more and more widely as the velocity of rotation was increased. It was easy to calculate, from the measured dimensions of the apparatus, the amount of deflexion, and the rate of rotation of the mirror, the velocity of light. The rate of rotation was, of course, given by the pitch of the note produced by the siren.

Foucault’s early results with this apparatus showed that the velocity of light which had been deduced from the old methods was too large; and he concludes his first paper by the statement that the determination of the distance of the earth from the sun must now be made by physical instead of astronomical methods. Foucault’s process lias recently been very considerably improved by Mitchelson, who, in 1879, found for the velocity of light in vacuo 186,380 miles per second (Nature, vol. xxi. p. 226).

By interposing a tube filled with water, and having flat glass ends, between the fixed and revolving mirrors, Foucault found that (for the same rate of rotation) the displacement of the image was greater than before in the proportion of the refractive index of water to unity. Thus it was at once evident, by a mode of experimenting exposed to no possible doubt, that light moves faster in air than in water, and, therefore, as will be seen later, that the corpuscular theory of light must be abandoned.

Other methods of determining the velocity of light in air, and for comparing the velocities of light in air and water (on wdιich depends the most definite proof of the erroneousness of the corpuscular theory), and in still and moving water, will be afterwards explained. They give results of very great value, but we cannot introduce then here, as they depend upon somewhat more recondite principles of physical optics.

It is interesting to observe that, as the nearest fixed star is probably about 200,000 times farther from us than the sun is, we now see such a star by light which left it more than three years ago. If, as is now supposed, variable stars (such as Mira Ceti) owe their rapid periodical changes of brightness to eclipses, and if different homogeneous rays travel with different velocities in free space, it is evident that such stars would show a gradual change of colour as they wax, and an opposite change as they wane. Nothing of the kind has as yet been observed, though it has been carefully sought for. Hence we have every reason to conclude that, in free space, all kinds of light have the same velocity. It will be seen later that dispersion has been accounted for by the different velocities of light of different wave-lengths in the same refracting medium,—this being a consequence of the ultimate grained structure of ordinary matter, which is on a scale not incomparably smaller than the average wave-length.

Behaviour of Light at the Common Surface of Two Homogeneous Media.

When a ray of light,'moving in one homogeneous medium, falls on the bounding surface of another homogeneous medium, it is in general divided into several parts, which pursue different courses. These parts are respectively— (a) reflected; (/?) refracted (singly or doubly); (γ) scattered; (δ) absorbed.

In the first two categories the result is two or three rays of light pursuing definite paths according to laws presently to be given. The fraction of the incident light which is reflected is in general greater as the angle of incidence is greater. In one important class of cases the reflexion is total. But at direct incidence the reflected portion is much greater for some bodies, such as mercury, than fpr others, such as water or glass. In bodies which give no scattering, the refracted portion of a ray consists of all the non-reflected portion, and therefore usually diminishes as the angle of incidence increases.

In the third category the common surface of the two media becomes illuminated, and behaves as if it were itself a source of light, sending rays in all directions. It may be objected to this, that in many cases the rays are scattered while penetrating the second medium. But in such cases the second medium cannot be called homogeneous. This will come up for discussion when we treat of absorption and colours.

In the fourth category the light ceases for an instant to exist as light; but its energy may either become heat in the absorbing body, or it may again be given out by the absorbing body in the form of light, but of a degraded character. This is called fluorescence, or phosphorescence, according as the phenomenon is practically instantaneous or lasts for a measurable time.

In category (a) the light is sent back into the first medium; in (fl) it penetrates into the second; in (γ) it goes, in general, mainly to the first; in (δ) it is shared by both.

It is by scattered light that non-luminous objects are, in general, made visible. Contrast, for instance, the effects when a ray of sunlight in a dark room falls upon a piece of polished silver, and when it falls on a piece of chalk. Unless there be dust or scratches on the silver you cannot see it, because no light is given from it to surrounding bodies except in one definite direction, into which (practically) the whole ray of sunlight is diverted. But the chalk sends light to all surrounding bodies from which any part of its illuminated side can be seen; and there is no special direction in which it sends a much more powerful ray than in others. It is probable that if we could, with sufficient closeness, examine the surface of the chalk, we should find its behaviour to be of the nature of reflexion, but reflexion due to little mirrors inclined in all conceivable aspects, and at all conceivable angles, to the incident light. Thus scattering may be looked upon as ultimately due to reflexion. When the sea is perfectly calm, we see in it one intolerably bright image of the sun only. But when it is continuously covered with slight ripples, the definite image is broken up, and we have a large surface of the water shining by what is virtually scattered light,—though it is really made up of parts each of which is as truly reflected as it was when the surface was flat.

We have spoken above of the behaviour of light at the common surface of two media. Now we do not by this phrase necessarily mean two media different in their chemical composition. We mean merely media optically different. Thus water with steam above it, and in very special cases layers of water or air of different temperatures, give surfaces of separation at which reflexion and refraction may and do take place. But, except in such special cases, we rarely have an abrupt change, such as is necessary for reflexion, between two portions of the same substance in the same molecular state. In general the transition is gradual; and special mathematical methods must be applied for the purpose of tracing the behaviour of the ray, which is now really travelling in a non-homogeneous medium.

Reflexion of Light. —If light be reflected from a plane surface bounding two dissimilar isotropic media, the incident and reflected rays are in one plane with, and are equally inclined (on opposite sides') to, the perpendicular to the reflecting surface at the point of incidence. This is sometimes stated in the form— The angles of incidence and of reflexion are equal to one another, and in one plane. The best experimental proof of the truth of this statement is deduced from the use of a reflecting surface of mercury in observations with the mural circle. The graduation of such an instrument is the most perfect that human skill can accomplish, and no one has ever been able to find by it the slightest exception to the preceding statement.

The principle of Hadley’s quadrant, and of the sextant as now used (an invention of Newton’s), is founded on this fact. If a plane mirror on which a ray falls be turned through any angle about an axis perpendicular to the plane of reflexion, the reflected ray is turned through twice that angle. This is an immediate consequence of the above law. For, if the plane be turned through any angle θ, the perpendicular to it is turned through the same angle. Hence the angle between the incident ray and the perpendicular is increased or diminished by θ, and therefore that between the incident and reflected rays (which is double of this) is altered by 2θ. A plane mirror is now extensively used for the purpose of indicating, by the change of direction of a reflected ray, the motion of a portion ol an instrument to which the mirror is attached. Thus the magnetometers of Gauss, the tuning-forks of Lissajoux, and the electrometers and galvanometers of Sir W. Thomson are all furnished with mirrors. The law of reflexion is also the basis of the goniometer, for the measurement of the angles of crystals and prisms.

It follows from this law that, if a ray pass from one point to another, after any number of reflexions at fixed surfaces, the length of its whole path from one point to the other is the least possible—subject to the condition that it shall meet each of the reflecting surfaces. For the point in a given plane the sum of whose distances from two given points (on the same side of the plane) is the least possible is that to which, if lines be drawn from the points, they are in one plane with the normal (or perpendicular) to the given plane and make equal angles with it. And, as the same is true of each separate reflexion, it is true for the whole course of the ray, since for any one of the reflecting surfaces may be substituted its tangent plane at the point of incidence.

It is to be remarked that there are exceptions to this form of the statement. The true form is that the actual path of a ray, under the given conditions, is less in length than any other path (satisfying the conditions) which is nowhere finitely divergent from it. This may be best seen by another method. Suppose a series of ellipses to be described whose foci are the source of light and an assigned point which is to be reached by the reflected ray. Let this system turn about the line joining the foci; it generates a series of prolate spheroids such that the time of light’s passing from one focus to a point in any one of the surfaces, and thence to the other focus, is the same whatever point be chosen on that particular surface. If we take any point without that surface, for it the corresponding time is obviously greater. Hence to find the path of that one of a system of rays diverging from a given point which, after reflexion at a given surface, shall pass through a given point, we have only to imagine spheroids constructed as before. Of these one at least will touch the given surface. All points on the surface in the neighbourhood of the point of contact (mark this limitation) will in general be outside the spheroid. Hence this point gives the shortest path. But the spheroid and the reflecting surface have the same tangent plane, and therefore the parts of the ray are equally inclined to the surface.

Formation of Images by Reflexion at a Plane Surface.— We may assume here—what is indeed evident from the rectilinear propagation of light—that objects are rendered visible to the eye by rays diverging from them. Hence, if we have a set of reflected or refracted rays diverging from any point, or diverging as if they came from any point, they will convey to the eye the impression of the existence of a luminous source at that point. The eye, in fact, can only tell us what effect is produced upon it, i.e., what sort of mechanical action it is subjected to. Its indications must therefore depend only upon what reaches it, and in no other sense whatever upon the source or the path of light. This point from which rays diverge, or appear to diverge, is called an image. The image of any point in a plane mirror is found by drawing from the point a perpendicular on the mirror and producing it till its length is doubled.

The extremity of the line so drawn is the image of the point; or, in other words, rays proceeding from the point diverge after reflexion as if they came from the image so found. The image in this case is called virtual, to distinguish it from cases, subsequently to be mentioned, where it is real— the distinction being that the rays have actually passed through a real image, while they only appear to come from a virtual one.

To prove this it is only necessary to observe that, if A (fig. 5) be a luminous point or a point from which rays diverge, and CB any section of the mirror by a plane through AB, the perpendicular to it, and if we make Bα = AB, and take any point P, then, joining AP, αP, and producing the latter, the angles APB, αPB, and therefore CPQ are equal; also the plane of the piper contains the perpendicular to the mirror at P. Hence PQ is the reflected ray; or the ray, after reflexion, appears to come from a. Hence a is the image—a virtual one, as before noticed. Also, if R be any point whatever (not P) in the plane of the mirror, we have obviously <ιR = RA. Hence the path AR, RQ is equal to αR, RQ, two sides of the triangle αRQ, of which αQ, which is equal to the actual path (AP, PQ), is the third side.

Fig. G represents the pencils of diverging rays by which two points of the image are rendered visible to an eye placed in front of the mirror. From the requisite modification of this figure it follows that one can see his whole person in a mirror of only half his height and breadth.

Dircks’s ghost, which has played a prominent part in popular entertainments for some years back, is the image, in a large sheet of unsilvered plate glass hung at the front of the stage, of an actor or figure strongly illuminated, and concealed from the audience in a sort of enlarged prompter’s box. Any one can see the phenomenon completely by looking into a plate-glass window on a sunny day, when he sees the passers-by apparently walking inside the house.

The principles already stated suffice fully for the explanation of the curious vistas of images formed by two parallel plane mirrors facing one another at opposite sides of a room. The only additional observation necessary on this subject is that, if the mirrors are silvered on the back, the light at each reflexion has to pass twice through the glass. Thus, if the glass be pinkish or greenish, the various images are more and more coloured as they are due to more numerous reflexions.

These principles also easily explain the Kaleidoscope (g.vi) of Sir D. Brewster, where images are formed by two mirrors inclined to one another. It is easy to see that the series of images of a luminous point produced by such an arrangement after one, two, &c., reflexions must all lie on a circle; also that, if the angle between the mirrors be an aliquot part of four right angles, these images will form a finite number of groups, each consisting of an infinite number of images which have exactly the same position.

The explanation of the law of reflexion which is furnished by the corpuscular theory is excessively simple. We have only to suppose that at the instant of its impact on the reflecting surface the velocity of a corpuscle perpendicular to the surface is reversed, while that parallel to the surface is unchanged. It bounds off, in fact, like a billiard-ball from the cushion. The undulatory theory gives an explanation, which is i in reality, quite as simple, but requires a little more detail for those who are not familiar with the common facts of wave-motion. We therefore reserve it for a time.

Reflexion at a Spherical Surface.— Let APB (fig. 7) be a section of a concave spherical mirror by a plane passing through its centre of curvature O, and through the luminous point U. Then, if any ray from U, as L τ P, meet the surface, it will be reflected in a direction PV, such that UP, PO, and PV are in one plane, and so that PO bisects the angle UPV. (This follows because OP, a radius of the sphere, is normal to the surface at P.) Hence it is rigorously true that, if V be the intersection of PV with UOA,

VO OU

VP = UP.

The full consequences of this exact statement will be developed under Optics (Geometrical). For our present purpose, an approximation will amply suffice. Let us suppose P to be so near to A that no sensible error is introduced by writing A for P in the above formula. This amounts to supposing the mirror’s breadth to be very small in comparison with its radius of curvature. The formula now becomes

VO OU .

AV = AU’

or, what is the same,

AO-AV AU-AO.

AV “AU ’

and V is, to the degree of approximation above stated, independent of the position of the point P. If. we call r the radius AO of the mirror, « = AU the distance of the source, and v = AV, the distance of the point V from the mirror, this becomes

-'→.'irr ι01 l + l.J. . . . ω .

V U U V r

The formula, or the cut, shows at once that this relation between U and V is reciprocal; i.e., all rays from V, falling on the mirror, will be made to converge at U. These points are therefore called conjugate foci.

The simplicity of (a) is remarkable; so, also, is that of its interpretation. For the rays passing from a source to a given object, like the mirror, are less and less divergent as the source is farther off. Hence (a) signifies that the (algebraic) sum of the divergences of the incident and reflected rays is equal to that divergence which the mirror can convert into parallelism.

In fact the rigorous geometrical relation may be written in the obvious form AVP +AUP = 2A0P,—which, when all three angles are small, is simply (a). A similar statement may easily be made in the case of refraction.

Before we proceed to develop the consequences of this simple formula, we may point out that it is applicable to all cases,—to convergent rays falling on a concave mirror, to divergent rays falling on a convex mirror, <tc., <fcc. The reader may easily verify this by trial for himself. But it follows at once from the necessary interpretation of the negative sign in geometry. Thus, if the mirror were convex, O would be to the left of A, as the figure is drawn; and AO, if formerly positive, would now be negative. Thus, for a convex mirror, the formula is

1 + 1=-A.

u v r

If the incident rays be convergent, U is to the left of A, and therefore AU, or u, is negative; and so on.

We must now study the relative positions of U and V, in order to find the size and position of the image for different positions of the object.

Returning to the formula (a) above, we see that the following pairs of values of u and v satisfy it :— [table]

Thus, when the source is at a practically infinite distance (as the sun or a star) the image is formed at a distance from the mirror equal to half its radius of curvature. It is then said to be in the principal focus. As the source comes nearer, the image comes out to meet it, and they coincide at the centre of curvature of the mirror. In fact, a ray leaving the centre of the mirror must meet the surface at right angles, and thus go back by the way it came. When the source comes still nearer, the image goes further off, until, when the source is at the principal focus, the image is at an infinite distance; that is, the rays go off parallel to one another. This is the mode in which a concave reflector is employed for lighthouse purposes. When the source comes still nearer, the image is behind the mirror, i.e., the incident rays are so divergent that part of their divergence remains after reflexion. This remnant of divergence becomes greater and greater as the source is nearer to the mirror, i.e., the (then virtual) image comes closer to the mirror, which finally behaves, for a very near source, almost precisely like a plane mirror.

All of these phenomena can be beautifully seen in a dark room by employing a beam of sunlight, rendered distinctly visible, in the fashion noted by Lucretius, by the motes in the air.

For further explanation pictures are given (figs. 8, 9), showing the course of the pencil of rays when (1) a real and (2) a virtual image is formed by a concave mirror. It will be seen at once that, in the cases figured, the real image is inverted and less than the object, the virtual image erect and larger. In fact the size of a small object is obviously to that of its image in proportion to their distances from O, the centre of curvature of the mirror. Also the image is erect when it lies on the same side of the centre of curvature with the object; inverted, if 0:1 the opposite. In other words, the image is inverted if the rays cross one another’s path, erect if they do not.

When the breadth of the mirror is large compared with its radius, the approximation upon which all these results depend can no longer be made. There is then no definite image even of a luminous point. It becomes spread over what is called a caustic, a section of which is the bright curve familiar to every one who has looked at a cup of milk in sunshine.

Even when the approximation is close enough for ordinary purposes, it is not so for astronomical purposes, and the effect of its inexactness upon the image is known as spherical aberration. For the fine mirrors of reflecting Telescopes {q.v.') the spherical form cannot be employed; the surface of the mirror must be of parabolic section.

Cylindrical Mirrors.— As a simple example of the application of the law of reflexion at curved surfaces, when the rigorous solution is demanded, let us take the case of a vertical right cylinder, the object being a drawing on a horizontal plane. Such mirrors, with the frightfully distorted drawings necessary to give an image of natural proportions, were very common fifty years ago, but are now rarely seen. They are still, however, valuable as illustrations of our subject.

Let the plane of the object cut the axis OB of the cylinder at right angles in 0 (fig. 10), and let A be the position of the eye, and RQA a ray from a point R of the object, reflected at Q. Draw QP perpendicular to the axis. The AQ and QR are in the same plane with QP (the normal to the surface) and make equal angles with it. Hence, when this figure is projected by vertical lines on the plane of the object, it takes the form in fig. 11; and AQ, QR now make equal angles with OQ. Also, if AB be drawn (in fig. 10) perpendicular to OP, the ratio of AQ to QR in fig. 11 is equal to that of BP to PO in fig. 10.

Take QS:QO::QR:QA, and draw ST parallel to OA. Then it is obvious that

SR-ST-jgoA; and also that the angles QSR and QST are equal. Hence the following theorems, which enable us at once to draw a figure on the object plane such that its image shall appear of any assigned form.

1. Any line, such as QR, on the object plane, drawn from a point Q in the section of the cylinder so that the angles OQR and OQA are equal, is seen after reflexion as a generating line of the cylinder.

2. If an epicycloid be described by lines of fixed length OS, SR, turning about 0 with angular velocities 1 and 2, and both coinciding with OA at starting, its image will be a circular section of the cylinder.

Thus, if we imagine as drawn on the cylinder any number of vertical and horizontal sections, forming a network, the object corresponding to them can be traced as a number of intersecting straight lines and epicycloids. Thus we have a well-known means of drawing the required object. A similar process may be applied to other modes of using such mirrors.

When the cylinder has a small diameter, it may be usefully employed to intercept and reflect part of a beam of sunlight entering a dark room. It is easy to see, by a geometrical construction, that the reflected rays will, in this case, form a right cone, whose axis is that of the cylinder; and one of its generating lines will be parallel to the incident ray. Thus the angle of the cone becomes smaller as the inclination of the reflecting cylinder to the ray becomes less. If the ray, at the point of interruption, was at the centre of a spherical dome, after reflexion it will form on the dome a circle, small or great, which passes through its original point of incidence.

In the language of Quaternions (q.v.), let α be the incident ray, β the axis of the cylinder, τ any normal to the cylinder, p the reflected ray. Then the law of reflexion gives

V. τατρ =· 0.

The property of the normal gives

Sj8τ = 0.

Eliminating τ, we have at once

Z.fs≤eY

α a ' ∖8aβJ , the equation of a right cone.

Imitations, more or less perfect, of primary and secondary rainbows can easily be made by this process,—the sunbeam being led through a prism just before it falls on the cylindrical rod. This experiment is a very striking one; but, though capable of giving much information, it is of that dangerous kind which is liable to mislead instead of instructing an audience.

If we look at a great number of thin cylindrical rods, parallel to one another, and illuminated by sunlight, the rays which reach the eye must, by what we have already said, each form a side of some right cone (of definite angle) whose axis is parallel to each of the cylinders. The appearance presented will therefore be that of a luminous circle, passing through the sun. Its angular diameter becomes less as the axes of the cylinders are less inclined to the incident rays.

This phenomenon is beautifully shown by some specimens of crystals, especially of Iceland spar, which are full of minute tubes parallel to one another. In a plate of such a doubly-refracting crystal, however, there are necessarily four images. That which is throughout due to the ordinary ray (this term will be explained later) shows perfectly the phenomenon above described. The light of the luminous circle is white. The other three curves are not circles, and in them the colours are separated. One of them, which is elliptical, is usually very much brighter than either of the remaining two.

Refraction. —If homogeneous light be refracted at a plane surface separating two homogeneous isotropic media, the incident and refracted rays are in one plane u<ith the normal to the surface, and the sines of their inclinations to it are in a constant ratio.

The law of single refraction was put in a form equivalent to this (all but one word) for the first time by Snell in Leyden, some time before 1626. It was first published in 1637 by Descartes, who undoubtedly obtained it from Snell; but he gave it without any mention of its author.

The one word referred to is homogeneous as applied to the incident light. For the fact that white light consists of innumerable different homogeneous constituents, which are separated from one another by refraction, was first established by Newton. We quote his own account of this important discovery from his letter to Oldenburg, dated Feb. , 167½:—

“To perform my late promise to you, I shall without further ceremony acquaint you, that in the year 1666 (at which time I applied myself to the grinding of optick-glasses of other figures than spherical) I procured me a triangular glass-prism, to try therewith the celebrated phaenomena of colours. And in order thereto, having darkened my chamber, and made a small hole in my window-shuts, to let in a convenient quantity of the sun’s light, I placed my prism at its entrance, that it might be thereby refracted to the opposite wall. It was at first a very pleasing divertisement, to view the vivid and intense colours produced thereby; but after a while applying myself to consider them more circumspectly, I became surprised to see them in an oblong form; which, according to the received laws of refractions, I expected should have been circular. They were terminated at the sides with streight lines, but at the ends the decay of light λvas so gradual, that it was difficult to determine justly what was their figure, yet they seemed semicircular.

“Comparing the length of this coloured spectrum with its breadth, I found it about five times greater; a disproportion so extravagant, that it excited me to a more than ordinary curiosity of examining from whence it might proceed. I could scarce think, that the various thickness of the glass, or the termination with shadow or darkness, could have any influence on light to produ<⅛ such an effect: yet I thought it not amiss, first to examine those circumstances, and so tried what would happen by transmitting light through parts of the glass of divers thicknesses, or through holes in the window of divers bignesses, or by setting the prism without, so that the light might pass through it, and be refracted, before it was terminated by the hole: but I found none of those circumstances material. The fashion of the colours was in all these cases the same.

‘ ‘ Then I suspected, whether by any unevenness in the glass, or other contingent irregularity, these colours might be thus dilated. And to try this, I took another prism like the former, and so placed it, that the light passing through them both, might be refracted contrary ways, and so by the latter returned into that course from which the former had diverted it: for by this means I thought the regular effects of the first prism would be destroyed by the second prism, but the irregular ones more augmented, by the multiplicity of refractions. The event was, that the light, which by the first prism was diffused into an oblong form, was by the second reduced. into an orbicular one, with as much regularity as when it did not at all pass through them. So that whatever was the cause of that length, it was not any contingent irregularity.

“I then proceeded to examine more critically, what might be effected by the difference of the incidence of rays coming from divers parts of the sun; and to that end, measured the several lines and angles belonging to the image. Its distance from the hole or prism was 22 foot; its utmost length 13⅛ inches; its breadth 2∣; the diameter of the hole ⅛ of an inch. The angle which the rays, tending towards the middle of the image, made with those lines, in which they would have proceeded without refraction, was 44 deg. 56 min. and the vertical angle of the prism 63 deg. 12 min. Also the refractions on both sides the prism, that is, of the incident and emergent rays, were, as near as I could make them, equal; and consequently about 54 deg. 4 min. And the rays fell perpendicularly upon the wall. Now subducting the diameter of the hole from the length and breadth of the image, there remains 13 inches in the length, and 2§ the breadth, comprehended by those rays which passed through the center of the said hole; and consequently the angle of the hole, which that breadth subtended, was about 31 min. answerable to the sun’s diameter; but the angle which its length subtended, was more than 5 such diameters, namely, 2 deg. 49 min.

“Having made these observations, I first computed from them the refractive power of that glass, and found it measured by the ratio of the sines 20 to 31; and then by that ratio I computed the refractions of two rays flowing from opposite parts of the sun’s discus, so as to differ 31 min. in their obliquity of incidence, and found that the emergent rays should have comprehended an angle of about 31 min. as they did before they were incident.

“But because this computation was founded on the hypothesis of the proportionality of the sines of incidence and refraction, which though by my own experience I could not imagine to be so erroneous, as to make that angle but 31 min. "which in reality was 2 deg. 49 min. yet my curiosity caused me again to take my prism: and having placed it at my window, as before, I observed, that by turning it a little about its axis to and fro, so as to vary its obliquity to the light, more than an angle of 4 or 5 degrees, the colours were not thereby sensibly translated from their place on the wall; and consequently by that variation of incidence, the quan tity of refraction was not sensibly varied. By this experiment, therefore, as well as by the former computation, it was evident, that the difier-ence of the incidence of rays, flowing from divers parts of the sun, could not make them after decussation diverge at a sensibly greater angle, than that at which they before converged; which being at most but about 31 or 32 min. there still remained some other cause to be found out, from whence it could be 2 deg. 49 min.

“Then I began to suspect, whether the rays, after their trajection through the prism, did not move in curve lines, and according to their more or less curvity, tend to divers parts of the wall. And it increased my suspicion, when I remembered that I had often seen a tennis-ball, struck with an oblique racket, describe such a curve line. For, a circular as well as a progressive motion being communicated to it by that stroke, its parts, on that side where the motions conspire, must press and beat the contiguous air more violently than on the other, and there excite a reluctancy and re-action of the air proportiouably greater. And for the same reason, if the rays of light should possibly be globular bodies, and by their oblique passage out of one medium into another acquire a circulating motion; they ought to feel the greater resistance from the ambient aether, on that side where the motions conspire, and thence be continually bowed to the other. But notwithstanding this plausible ground of suspicion, when I came to examine it, I could observe no such curvity in them. And besides (which was enough for my purpose) I observed, that the difference betwixt the length of the image and. the diameter of the hole, through which the light was transmitted, was proportionable to their distance.

“The gradual removal of these suspicions at length led me to the experimentum crucis, which was this. I took two boards, and placed one of them close behind the prism at the window, so that the light might pass through a small hole, made in it for the purpose, and fall on the other board, which I placed at about 12 feet distance, having first made a small hole in it also for some of that incident light to pass through. Then I placed another prism behind this second board, so that the light trajected through both the boards might pass through that also, and be again refracted before it arrived at the wall. This done, I took the first prism in my hand, and turned it to and fro slowly about its axis, so much as to make the several parts of the image, cast on the second board, successively pass through the hole in it, that I might observe to what places on the wall the second prism would refract them. And I saw, by the variation of those places, that the light, tending to that end of the image towards which the refraction of the first prism was made, did in the second prism suffer a refraction considerably greater than the light tending to the other end. And so the true cause of the length of that image was detected to be no other, than that light is not similar or homogeneal, but consists of difform rays, some of which are more refrangible than others ; so that without any difference in their incidence on the same medium, some shall be more refracted than others; and therefore that, according to their particular degrees of refrangibility, they were transmitted through the prism to divers parts of the opposite wall.”

The constant ratio mentioned in the above statement of the law of refraction is called the refractive index. Its numerical value depends upon the nature of the two media, and also upon the quality of the homogeneous light. It is usually greater for orange light than for red, for yellow than for orange, and so on,—so that the violet rays are often called the “more refrangible” rays.

The following experimental facts are additions to the law.

When refraction takes place from a rarer into a denser medium, the angle of refraction is usually less than that of incidence.

If the refractive index for a particular kind of light from a medium A into another B be u, that from B to A is —. In other words, a refracted ray may be sent back by the path by which it came.

If p 1 be the refractive index for a particular ray from A into B, and μ 2 that for the same ray from A into C, that from B into C is 7- .

Mi

These being premised, let us consider a source ⅛f homogeneous light in air shining on a surface of water. Here we may take p as about equal to ∣.

Let MN (fig. 12) be the perpendicular to the water surface at the point where the incident ray AP meets it. In the plane APM make the angle QPN such that sin APM = ⅜ sin QPN, then PQ is the refracted ray. If QP be produced backward to meet the vertical line BA in 2, we may present this statement in the form

PB 4 PB

PA i I⅛ ’ or

PA≡∣Pg r .

If the rays fall nearly perpendicularly on the surface, we may put (approximately) B for P, and we have

Bg=⅜BA.

Hence, an eye placed under water and nearly in the vertical through A, sees a virtual image of A at 2, one-third farther from the surface of the water.

As P is taken farther and farther from A, the angle of incidence becomes more nearly a right angle, and the sine of the angle of refraction becomes more nearly equal to Λ ray cannot go from air into ιvater so as to make, with the perpendicular to the surface, an angle whose sine is greater than ∣ . The true nature of this curious statement is, however, best seen when we suppose the source to be under water, and the light to be refracted into air. If APQ (fig. 13) be the course of a ray, we have as before

AP-⅜P< r .

Hence, if Γ 1 be taken so that

AP 1 -⅜P 1 B, it is clear that q coincides with B, or the ray AP 1 , refracted at P 1 , runs along the surface of the water. If AP. 2 be less than I P 2 B, no point q can be found; so that the ray AP., cannot get out of the water. It is found to be completely reflected in the water. This reflexion unaccompanied by refraction is called total reflexion. The limiting angle of incidence (at P 1 ) which separates the totally reflected rays from those which (at least partially) escape into air is called the critical angle. When an equilateral triangular prism of glass is placed in a ray of sunlight, and made to rotate, we see (besides the spectra formed by refraction) beams of white light reflected alternately from the outside and the inside of each face. The totally reflected ray from the inside is seen to be very much brighter than that reflected from the outside.

To an eye placed nearly in the vertical of A, A appears at A o , where

A 0 B = ≡AB.

Thus a clear stream, when we look vertically into it, appears to be of only ^ths of its real depth. But when we look more and more obliquely, seeing A for instance by the ray QP, the image appears nearer and nearer to the surface; or, if A be at the bottom, the water will appear more and more shallow; and all objects in it will appear to be crowded towards the surface. Thus the part of a stick immersed in water appears bent up towards the surface of the water.

Again, to an eye at A, all objects above the water will be seen within a right cone of which AB is the axis and AP 1 a side. The rest of the water surface, outside the cone just mentioned, shows us objects at the bottom by reflexion in a perfect mirror.

All this is on the supposition that the light is homogeneous. But when white light is emitted by A, the point A o will be nearer the surface for each constituent the greater is the refractive index. Thus a white point at A will appear drawn out into a coloured line whose lower end is red and upper end violet.

It is easily seen from the law of refraction that light, on passing through a plate of homogeneous material with parallel faces, finally emerges in a direction parallel to that at incidence, and therefore white light comes out from it still white. If the plate be water in a glass vessel with parallel sides, a body placed close to one side, while the eye is close to the other, appears to be at fths of its real distance from the eye.

The explanation of the law of refraction, on the corpuscular theory, was given by Newton. It is still of importance, as the earliest instance of the solution of a problem involving molecular forces. Newton shows that, as the molecular forces on a corpuscle balance one another at every point inside either of the media, its velocity must be constant in each, but that in passing through the surface of separation of the two media the square of the velocity perpendicular to the surface undergoes a finite change.

Thus, if V bo the velocity in air, α the angle of incidence, then in glass the velocity parallel to the surface is still υ sin a, but that perpendicula r to t he surface is ^fυ 2 cοsPa + α 2 . Thus the whole velocity is V ν i + α 2 ; and, if a! be the angle of refraction,

Vv 2 + α 3 sin a =ν sin α.

Prisms.— When the surfaces are plane, but not parallel, we have what is called a prism.

The general nature of the action of a prism will be easily understood by the help of the previous illustrations, if we restrict ourselves to the case of a prism of very small angle and to rays passing nearly perpendicular to each of its faces. Thus, the rays falling nearly at right angles to its surface from a point A (fig. 14) will, after the first refraction, appear to diverge from aluminous line RV, red at the end next to A, violet at the other. This line is in the perpendicular AB from A to the first surface of the prism. Draw from R and V perpendiculars RS, VT to the second surface of the prism. Join BS, BT, and draw Ar, Av parallel to them so as to cut RS in r and VT in v. To an eye behind the prism, the bright point A will appear to be drawn out into a coloured line rv, red at the end nearest to A.

If A be a nιrrow bright line of light, parallel to the edge of the prism, it will appear to be drawn out into a rectangle consisting of images of the line ranged parallel to one another, and due to the various homogeneous constituents of white light in order of their refrangibility. If the light do not contain rays of every degree of refrangibility, some of these images will be wanting, and there will be corresponding dark lines or bands crossing this spectrum (as it is called). The amount by which any part of this spectrum is shifted from the true position of the bright slit depends (caeteris paribus) upon the amount of the refraction. It also depends on the angle of the prism. And, for a given angle, the length of the spectrum depends upon the difference between the refractive indices for the red and the violet rays. This is called the dispersion.

If a second prism, of the same glass, and of the same angle, as the first, be placed in a reversed position behind it (as indicated by the dotted lines in the figure), the effect of the two would be simply that of a plate of glass with parallel faces; the emergent rays would each be parallel to its original direction, and there would be no separation of colours. The reversed prism would therefore undo the work of the direct prism. Then we should have no dispersion, but we should also have no refraction. We have, however, as has already been shown, an increase of divergence, 7.e., the image is nearer to the eye than the object. Blair, Brewster, and Amici devised combinations of two pairs of prisms of the same glass, those of each pair having then edges parallel, such that the combination acted as a sort of achromatic telescope of low power.

Newton, from some rough experiments, hastily concluded that the amount of dispersion is in all substances proportional to that of the refraction. If such were the case it is easy to see that prisms of two differently refracting materials and of correspondingly different angles, combined (as above described) so as to annul the dispersion, would likewise annul the refraction. Thus Newton was led to suppose that refraction without dispersion is impossible.

It was found by Hall in 1733, and afterwards (independently) by Dollond, that this idea is incorrect—that, in fact, we have in certain media large refraction with comparatively small dispersion, and vice versa, and thus that the dispersion may be got rid of while a part of the refraction remains. James Gregory had previously conjectured that this might be done by using, as is done in the eye, more media than one. Thus we have for certain specimens of flint and crown glass, whose optical constants were carefully measured by Fraunhofer, the following values of the refractive index for three definite kinds of homogeneous light :— [table]

The rays C and F are in the red and greenish blue respectively, and are given off by incandescent hydrogen. D is the orange-yellow ray of sodium.

When the angle of the prism is very small (the only case we treat here), we may consider Arv as approximately a straight line, whose length is (caeteris panbus) proportional to the angle of the prism. Also the distances Ar, Av, are to one another in the proportion of the refractive indices of red and violet rays, each diminished by unity. Hence, for a prism of flint glass such as was employed by Fraunhofer, the distances from A to its images formed by these three kinds of homogeneous light respectively are very nearly as

630, 635, 648.

When a prism of crown glass is used they are nearly as 527, 530, 536.

The differences between the numbers for C and F are [table]

or as 2: 1. Hence if we make the small angle of the crown-glass prism twice that of the flint, and observe A through the two prisms, with their edges turned in opposite directions, the C and F images will coincide. Both, however, will be displaced from the real direction of A as if a prism had been employed, with its edge turned as that of the crown glass was, and to the same extent as that prism would have displaced them had its refractive index been about 1 ∙21 and the same for the two kinds of light C and F.

In fact, the displacements by the flint prism are as

630, 648, and those by the crown prism (to the opposite side) are as 1054, 1072.

The differences, in favour of the crown prism, are as

424, 424.

This combination of prisms is called achromatic, or colourless, but is not perfectly so. For if we inquire into the displacement of the D image, we find that it is as

635

for the flint prism; but as

1060

in the opposite direction, for the crown prism. Hence its whole displacement is as

425, a little greater than that common to C and F. The reason for this is obvious from Fraunhofer’s numbers given above. The interval from C to D is to that from C to F in a greater ratio in crown than in flint glass,—so that the spectra given by these media are not similar. The rays of higher refrangibility are more separated in proportion to those of lower refrangibility in flint than in crown glass. This is the irrationality of dispersion— which, so far as we yet know, renders absolute achromatism unattainable. Three lenses in combination give a better attempt at achromatism than can be made with two; and some remarkable results were obtained by Blair, 1 with two glass lenses enclosing a lenticular portion of a liquid.

By looking through a prism at a very narrow slit, formed by the window shutters of a darkened room, Wollaston (in 1802) found that the light of the sky (f.e., sunlight) ^ives a spectrum which is not continuous. It is crossed by dark bands, as already hinted. These bands are due to the deficiency of intensity of certain definite kinds of homogeneous light. They were, independently, rediscovered, and their positions measured, by Fraunhofer 2 in 1817 with far more perfect optical apparatus. He also found similar, but not the same, deficiencies in the light from various fixed stars. The origin of these bands will be explained in Radiation, along with the theory of their application in spectrum analysis. In optics they are useful to an extreme degree in enabling us to measure refractive indices with very great precision. Wollaston’s own account of his discovery is as follows :—

“If a beam of day-light be admitted into a dark room by a crevice √ σ th of an inch broad, and received by the eye at the distance of 10 or 12 feet, through a prism of flint-glass, free from veins, held near the eye, the beam is seen to be separated into the four following colours only, red, yellowish-green, blue, and violet, in the proportions represented in lig

“The line A that bounds the red side of the spectrum is somewhat confused, which seems in part owing to want of power in the eye to converge red light. The line B, between red and green, in a certain position of the prism is perfectly distinct; so also are D and E, the two limits of violet. But C, the limit of green and blue, is not so clearly marked as the rest; and there are also on each side of this limit other distinct dark lines f and g, either of which in an imperfect experiment might be mistaken for the boundary of these colours.

“The position of the prism in which the colours are most clearly <livided is when the incident light makes about equal angles with two of its sides. I then found that the spaces AB, BC, CD, DE occupied by them were nearly as the numbers 16, 23, 36, 25.” 3

The mode of formation of a spectrum which was employed by Newton, and which is still used when the spectrum is to be seen by many spectators at a time, differs from that just explained in this, that the light from a source A is allowed to pass through the prism, and to fall on a white screen at a considerable distance from it. In this case the paths of the various rays as they ultimately escape from the prism are found by joining the points r,....ν, with the prism and producing these lines to meet the screen. Unless one surface of the prism be covered by an opaque plate, with a narrow slit in it parallel to the edge of the prism, the spectrum produced iu this way is very impure, i.e., the spaces occupied by the various homogeneous rays overlap one another. To make it really pure an achromatic lens is absolutely requisite. This leads us, naturally, to the consideration of the refraction of light at spherical surfaces.

Refraction at a Spherical Surface.— Following almost exactly the same course as that taken with reflexion above, let O (fig. 15) be the centre of curvature of the spherical refracting surface AB. Let U be the point-source of homogeneous light, and let PV be the prolongation (backwards) of the path pursued, after refraction, by the ray UP.

Then, rigorously, we have sin UPO=∕*sin OPV, where μ is the index of refraction between the two media employed. This may be written (by omitting a common factor) as

OU OV

PU - m PV‘

If, as before, the breadth of the surface be small compared with its radius of curvature, we may approximate

(sufficiently for many important practical purposes) by writing A for P. Thus we have

OU _ OV

AU “m AV'

Retaining the same notation as in the case of reflexion, we get u-r v-r “ μ > u V or

(1).

V u r

Notice that, if we put μ. = - 1, this, becomes the formula for reflexion at a concave mirror.

Lenses.— Suppose now that, after passing a very short Thin distance into the refracting medium, the ray escapes again leuses ∙ into air through another spherical surface whose centre of curvature also lies in the line OA. Let s be the new radius of curvature, w the value of the quantity corresponding to v for the escaping ray. Then, remembering that the refractive index is now —,we have (by the previous formula)

- — - 1

w υ s ’ or = (2).

W V s

Adding (1) and (2) we get rid of v, which indicates the behaviour of the rays in the substance of the lens, and have

÷-⅛→-υ(÷4)∙

This contains the whole (approximate) theory of the behaviour of a very thin lens.

When the source is at an infinite distance, or u= oo, we have

⅛→-<4)

= J suppose.

This quantity f, defined entirely in terms of the refractive index and of the curvatures of the two faces of the lens, is called the principal focal distance. If μ be greater than 1, i.e., as in the case of a glass lens in air,/is positive if

2__J_ r s be so; and it obviously retains the same value, and sign, Reversi-if the lens be turned round. For, in the formula, r and s bi ! it r of change places, and they also change signs: i.e., we must tllinlens ∙

1 Trans. R.S.B., vol. iii. (1791).

2 Gilbert’s Annalen, lvi.

3 “The correspondence of these lines with those of Fraunhofer I have, with some difficulty, ascertained to be as follows :—

A, B, f C, g, D, E, . . . Wollaston’s lines.

B, D, b, F, G, H, . . . Fraunhofer’s lines.

There is no single line in Fraunhofer’s drawing of the spectrum, nor is there any in the real spectrum, coincident with the line C of Wollaston’s, and indeed he himself describes it as not being ‘ so clearly marked as the rest.’ I have found, however, that this line C corresponds to a number of ∏nes half-way between b and F, which, owing to the absorption of the atmosphere, are particularly visible in the light of the sky near the horizon.”—Brewster, Report on Optics, Brit. Association, 1832.

put - s for r and - r for s. This leaves the result unchanged.

All lenses, therefore, whose sections are of any of the forms in fig. 16, whichever way they are turned, render parallel rays which pass through them divergent. Their characteristic is that they are thinnest at the middle. But

2.-2.

r 8 is negative for lenses whose sections are of any of the forms shown in fig. 17. Such lenses, whichever way they are turned, render parallel rays convergent. Their characteristic is that they are thickest at the middle. But these characters are interchanged when p. is less than 1; as, for instance, when the lens is an air-space surrounded by water. The similarity on reversal is not in general true in a second approximation.

The formula for a thin lens now takes the form,

±-J-=Λ

w u f ’ and differs from that for a curved reflecting surface only in the sign of the second term. With the proper allowance for this, then, all that we have said of reflexion at spherical mirrors holds true of refraction through thin lenses with spherical surfaces.

We may now put the whole matter in the excessively simple form which follows :—

Ä thin lens increases or diminishes by a definite quantity the convergence or divergence of all rays which pass through it.

This quantity is the divergence or convergence of rays falling on the lens from or passing from it to its principal focus. Or it is the convergence or divergence which the lens produces in parallel rays. Thus, if the distance of an object from a convex lens is twice the focal length of the lens, the image is formed at the same distance from the lens, and is equal in size to the object.

Figs. 18 and 19 show the production of a real image and of a virtual image by lenses which produce convergence of parallel rays—along with the rays by which these are seen by an eye placed behind the lens. In either case it is obvious that the sizes of object and image are, respectively, as their distances from the centre of the lens.

Fig. 18 shows how a lens produces a real inverted image of a body placed farther from it than its principal focus. This is the case in the camera obscura, in the solar microscope, and in the object glass of a telescope. Fig. 19 shows how a virtual image is formed of a body placed nearer to a lens than its principal focus. This is the case of a single lens used as a microscope. In the former case the divergence of the incident rays is so small that the lens renders them convergent. In the latter the divergence is so great that the lens can only diminish, not destroy it.

In using a hand-magnifier in this way, we so adjust it, by practice, that the enlarged image appears to be formed at the distance from the eye at which vision is most distinct. It is obvious that the amount of magnification must, then, be greater as the focal length of the lens is less.

We can now understand the working of the ordinary astronomical telescope (fig. 20). The object glass furnishes an inverted but real image of a distant body, within our reach. We can, therefore, place the eye-glass (like the single microscope above spoken of) so as to form a virtual magnified image of this real image treated as an object. It is still, of course, inverted. It is easy to see that the angle subtended at the eye by the virtual image seen through the eye-piece is to that subtended by the object at the unaided eye approximately as the focal length of the object lens is to that of the eye lens. These angles are, in fact, those subtended . at the centres of the two lenses by the real image. This ratio is, therefore, called the magnifying power of the telescope.

The compound microscope, in its simplest form, is precisely the same arrangement as the astronomiβal telescope. The only difference is that the object, being at hand, can be placed near to the object-glass (still, however, beyond its principal focus), so that the real image formed is already considerably larger than the object, and is then still further magnified by the eye-glass.

The magnifying power of a single lens, when used as a hand microscope, is to be measured by the ratio of the angle under which the virtual image of an object is seen (at the distance of most distinct vision) to that at which the object itself would be seen (at that same distance); i,e., it is the ratio of 10 inches to the focal length of the lens.

Combinations of Lenses in Contact.— From the formula l ∣ 7-M⅛t) we see that the focal length of a simple lens is less as μ is greater. Thus all that we have just said is true for homogeneous light alone. But if we combine two thin lenses, placing them close together, we may arrive at an approximately achromatic arrangement. For we have, for the first lens

111

w « “f ‘

For a second, close to it, we have

__i i_

X w = f

For the two, considered as one, we have

Now X ' l i f

⅛÷M÷4)+M⅜4)∙

and there is an infinite number of ways in which r , and s can be chosen, when r and s are given, such that the values of the right hand side shall be equal for two values of μ and the corresponding values of μ. Any one of these gives an achromatic combination, of the necessarily imperfect kind described above in considering prisms. But, as we have now the curvatures of four surfaces to deal with, we can adjust these so as not only to make the best attainable approximation to achromatism, but also to reduce the unavoidable spherical aberration to a minimum.

These questions, however, are beyond the scope of this article. We can remark only that the adjustment for two rays, for which the refractive indices are μ and μ + δμ in the first medium, and μ and μ + δμ in the second, requires the one relation

⅛* 1 , V _1 n μ -l ∕ + √-l / -0 ’

which involves only the ratio of the focal lengths of the two lenses—leaving their forms absolutely undetermined. But, if both μ and μ ’ be greater than unity, the signs of ∕and∕ ’ must be different ;—i.e., in an achromatic combination of two lenses one must be convex and the other concave.

The reader must, however, be reminded that we are dealing with a first approximation only, and that spherical aberration does not come in till we reach a second. The details for a proper achromatic combination will be given in Optics (Geometrical).

Before leaving this subject, we must find the behaviour of two thin lenses which are placed at a finite distance from one another. For the first lens we have, as before, w u~* f

If the second lens be placed at a distance a behind the first, the rays which fall on it appear to come from a distance w + a. Hence, for the light emerging from the second lens, we have

_1 L· = 2_

x w+aff

When u is infinite, we have from the last two equations

J--J- + l .

x f+a f ’

It is obvious that a combination of this nature offers the same kind of facilities for the partial cure of dispersion and of spherical aberration as when the lenses are in contact, with one additional disposable constant. Thus we have compound achromatic eye-pieces, which can be corrected for spherical aberration also.

Formation of a Pure Spectrum.— We may now go back to the formation of an image by a prism, and inquire how, by the use of an achromatic lens, we can project a pure spectrum on a screen. We have seen that a thin prism, for rays falling nearly perpendicular to it, forms a virtual and approximately rectilineal image of a luminous point, in which the colours are ranged in order of refrangibility. Suppose the light which passes through the prism to fall on an achromatic lens, placed at a distance greater than its focal length from the virtual image above mentioned. These rays after passing through the lens will proceed to form, at the proper distance, a real linear coloured image of the luminous point, in which (as in the virtual image) the colours do not overlap. Instead of a luminous point, rays diverging from a very narrow slit parallel to the edge of the prism are employed. It is usual to place the lens at double its focal distance from the virtual image, and thus the real image is formed at an equal distance on the other side of it, and is of the same size. It may now, if required, be magnified by means of an achromatic eye-piece. Or, in other words, it may be examined by means of a telescope. In fact a telescope, whose object glass is covered by a thin prism, has been usefully employed during a total eclipse in examining the light of the solar corona. A similar arrangement, made to have an exceptionally large field of view, is employed to find the nature of the spectra of meteorites or falling stars.

Refraction at a Cylindrical Surface.— A very simple, but interesting, case of refraction at a cylindrical surface is furnished by a thermometer tube. It is easily seen that the diameter of the bore appears, to an eye at a distance large as compared with the diameter of the tube, to be greater than it really is, in the proportion of the refractive index of the glass to unity. Thus in flint glass it appears magnified in about the ratio 5: 3. Hence the mercury appears completely to fill the external surface of such a tube, if the bore be only ⅜ths of the external diameter.

But a far more interesting case is that of parallel rays falling on a solid cylinder of glass or water. Its interest consists in the fact that by its aid we can explain the phenomena of the rainbow. We, accordingly, devote special attention to it The problem, without losing any of its applicability to the rainbow, is much simplified by supposing the rays to be incident in a direction perpendicular to the axis of the cylinder; for in this case the whole course of each ray is in a plane perpendicular to the axis. We need not treat here of rays which pass close to the axis of the cylinder. For such the cylinder acts as a lens, and its focal length (to the usual first approximation) can easily be obtained by methods such as those given above.

What we are mainly concerned with is the behaviour of the rays which escape into the air, after one or two reflexions at the inner surface of the cylinder.

Suppose first that we consider a ray once reflected in the interior of the cylinder. Let SP (fig. 21) be one of the set of incident parallel rays, and let its path be SPQP'S ’. This involves refraction at P, reflexion at Q, and again refraction at P'. But it is obvious from the symmetry of the circular section, and from the laws of refraction and reflexion, that this path is symmetrical about the line OQ which joins the axis of the cylinder to the point at which the ray is reflected. Hence SP, S'P' meet OQ in the common point s; and the amount by which the direction of the ray has been turned round by the refractions and the reflexion is twice the supplement of half the angle at s. But the angle POR is double OPQ the angle of refraction, while OPs is equal to the angle of incidence. Hence the half angle at s is the excess of twice the angle of refraction over the angle of incidence.

Turn now to fig. 22, in which we have two concentric circles whose radii are to one another as the refractive index of the cylinder is to unity. If A be any point on a diameter, and tangents Ap and Aq be drawn from it, we see at once that the sines of the angles at A are to one another as the radii of the circles. Hence, if OAp be the angle of incidence, OA# is the corresponding angle of refraction; and it is easy to see that the half angle at s (in fig. 21) will in fig. 22 be represented by the excess of OAy over qAp. Now when OA is large, both of these angles are small, and thus their difference is likewise small. As OA becomes less the difference of the angles becomes greater, but only up to a certain point, for when A is near the outer circle the angle OAp begins to increase much faster than does OAg. Hence there is a single definite position of A for which the difference is a maximum. In the first figure these changes in the angles of incidence and refraction, for the members of a group of parallel rays, correspond to the varying position of P in the circular section of the cylinder. Hence there is one position of P for which the angle at s is a maximum. Now one of the conditions of a maximum or minimum of any quantity is that, near it, the value of the quantity changes very slowly. Thus a number of issuing rays are crowded together near the direction corresponding to this maximum, the others being more widely scattered,—while for all of them the angle at s is smaller. Newton gives us as an illustration of this, the very slow change of length of the day when the sun is near one of the tropics.

To find this Maximum Angle.— If θ be the angle of incidence, φ that of refraction, and μ the refractive index, we have to find the maximum value of

⅛s=2≠-0 (1), with the condition sinβ = μsin≠ (2).

These give at once 2dφ = dθ, and μ cosφ dφ =cosθ dθ. Hence

∕xcos≠=2cosβ (3).

From (2) and (3) we have

3cos 2 β=ju 2 -l; which determines the requisite angle of incidence. The values of the other quantities are easily calculated from this; and we finally have, for the maximum value of the sine of the half angle at s, the expression

⅛(⅛- ’)* <‘>·

This is obviously smaller as μ is greater, at least up to the limit μ = 2.

With the value ⅜ for μ (which is nearly that for yellow rays refracted into water) we have

L sin ⅛s = 9 - 55462, which corresponds very nearly to ⅛s=21° Γ.

Now suppose the diameter of the cylinder to be small compared with the distance of the eye from it. In this case the point s may be considered as being in the axis of the cylinder.

Let SsE 1 (fig. 23) be made equal to the maximum value of s; then an eye placed anywhere in the line sE 1 will receive the rays which are congregated towards the maximum. An eye within the angle SsE 1 (as at E 2 ) will receive some of the straggling rays, while an eye outside that angle (as at E 3 ) will see nothing. Let there now be imagined a great number of parallel cylinders; let E 1 σ be drawn parallel to the incident rays, and make the angle σE 1 s'

equal to σE j s. Then the eye at E 1 will see the concentrated rays (already spoken of) in the directions E 1 s and E 1 s'. From points within sE 1 s' some straggling rays will reach it, from points outside that angle none.

Now suppose cylinders to be placed in great numbers in all directions perpendicular to the incident rays. The eye at E 1 will see a bright circle of light whose centre is in E l σ. Inside that circle there will be feeble illumination; outside it, darkness. This is obviously the case of the rainbow, where we have spherical drops of water instead of the cylinders above spoken of. For each spherical drop is effective only in virtue of a section through its centre, containing the incident ray and the eye; and such sections are the same as those of the cylinders.

Thus far we have been dealing with parallel rays of homogeneous light; and the appearance (to the degree of approximation we have adopted) is that of a bright circle whose centre is diametrically opposite to the source of light, whose radius is (for raindrops) about 42° 2', and whose area is slightly illuminated.

Introduce the idea of the different kinds of homogeneous light which make up sunlight, and we find a circular (almost pure) spectrum—the less refrangible rays being on the outside. Next we introduce the consideration of the finite disk of the sun, and we have an infinite series of such arrangements superposed on one another, the centre of each individual of the series being at the point diametrically opposite to the point of the sun’s disk which produced it. This leaves the general aspect of the phenomenon unchanged, but altogether destroys the purity of the spectrum.

If we next consider light which has been twice reflected within the cylinder, we have a figure like the diagram fig. 21; where the lettering is as nearly as possible the same as that in fig. 23. Everything is still symmetrical about the line Os, which obviously cuts at right angles the ray QQ'. .

Reasoning precisely similar to that above given shows that the complement of half the angle at s is now equal to the excess of thmce the angle of refraction (OPQ) over that of incidence (the supplement of OPs), and that this also has a maximum value, i.e., that s itself has a minimum value.

To find it, we have

^-⅛s=3≠-β, with sin θ = μ sin φ.

Differentiating, &c., as before, we find

8 cos 2 0 = m 2 -1, whence, finally,

. , μ 4 + 18μ 2 -27 sm⅜ s

This quantity increases with μ, for its differential coefficient is ⅜(⅜- ι Γ

which is necessarily positive. (It vanishes, no doubt, for ∕* = 3, but then so does θ.)

For μ=⅜ the value of sin ⅜s is

0 ’ 4303 nearly, so that s=50° 58 ’.

Carrying out the same steps of reasoning as before, and applying the result to raindrops, we find a second rainbow concentric with the first, but with a greater radius, viz., about 51° (for yellow light). All the above remarks about the impurity of the spectrum, <fcc., apply to this bow also. In this bow the less refrangible rays are on the inner side, and the straggling rays illuminate feebly the space outside it. Hence the space between the red boundaries of the two bows has no illumination from rays reflected either once or twice within the water drops.

What we have now given is nearly all that geometrical optics can tell us about the rainbow. It seems that the first really important steps in the explanation, viz., (1) that the primary bow is due to rays falling on the outer portions of the drops, which suffer two refractions and one reflexion before reaching the eye, and (2) that the secondary bow is due to rays falling on the inner side, and suffering two refractions and two reflexions, are due to Theodorich, about 1311. His work was not published, and its contents were first announced by Venturi in the present century. These results were, independently, discovered by De Dominis in 1611. Neither of these writers, however, pointed out the concentration of the rays in particular directions. This was done by Descartes in 1637, by the help of Snell’s law. He calculated with great labour the paths of each of 10,000 parallel rays falling on different parts of one side of the drop, and showed that from the 8500th to the 8600th the angle between the extreme issuing rays is measured in minutes of arc,—thus discovering by sheer arithmetic the maximum which, as we have seen above, is so easily found by less laborious methods. Newton’s addition to this theory consisted mainly in applying his discovery of the different refrangibilities of the different homogeneous rays. The explanation was then thought to be complete. For a long time this was held to be one of Newton’s most brilliant discoveries. It is well to notice that he himself speaks of it in its true relation to the work of his predecessors. He merely says :—“But whilst they understood not the true origin of colours, it is necessary to pursue it here a little further?' And he said well; for a full investigation conducted on the principles of the undulatory theory introduces, as was first pointed out by Young, certain important modifications in the above statements. Of these we need mention only one, viz., that in each bow there is more than one maximum of brightness for each homogeneous ray. The spumous bows, as they are called, which often appear like ripples, inside the primary and outside the secondary bow, and which depend upon the fact just mentioned, have no place in even Newton’s theory. About them, in fact, geometrical optics has nothing to say. Young, in 1804, took the first step for their explanation. They were fully investigated, from the undulatory point of view, by Airy, in 1836-38; and his results were completely verified by the measurements of Hallows Miller in 1841. Miller used a fine cylinder of water escaping vertically from a can. This is one of the reasons which induced us to treat the subject as a case of refraction and reflexion in a right cylinder.

The overlapping of the colours in the rainbow, due to the apparent size of the sun’s disk, is occasionally so greatly exaggerated that only faint traces of colour appear. This may happen, for instance, when the sun shines on raindrops in the lower strata of the atmosphere through clouds of icecrystals in the higher strata. By reflexion from the faces of these crystals, the source of light is spread over a much larger spherical angle, and there is no sharp edge to it as in the case of the unclouded disk. The rainbow is then much broader and fainter than usual, and nearly white. The size of the drops also produces modifications which are not indicated by the geometrical theory.

When the moon is the source of light, the rainbow is so faint that it is often difficult to distinguish the colours; but with full moon, and other favourable circumstances, it is easy to assure one’s self that the colours are really present.

The refraction of sunlight, or moonlight, through icecrystals forming cirrhus clouds, gives rise to coloured halos, parhelia, paraselenae, &c. Their approximate explanation depends upon the behaviour of prisms with angles of 60° or 90°, and therefore does not come within the scope of the present article. They must not, however, be confounded with coronae, those rings which encircle the sun or moon when seen through a mist or cloud. Halos have definite radii depending on the definite angles of ice-crystals; the size of a corona depends on the size of the drops of water in a mist or cloud, being smaller as the drops are larger. Thus their diminution in radius shows that the drops are becoming larger, and implies approaching rain. They are due to diffraction, and can only be explained by the help of the undulatory theory.

Refraction in a Nan-Homogeneous Medium.— The principles already explained are sufficient for the purpose of treating this question also. But they require, for their application, the artifice of supposing the medium to be made up of layers, in each of which the refractive power is the same throughout the layer, but finitely differs from one layer to another, and then supposing these layers to become infinitely thin and infinitely numerous. In this case there will of course be only an infinitely small difference in properties between contiguous layers; and the abrupt change of direction which accompanies ordinary refraction is now replaced by a continuous curvature of the path of the ray.

Glimpses of a more general method had been obtained even in the 17th century; and in the 18th these had become a consistent process so far as application to the corpuscular theory is concerned. But it was reserved for Sir W. R. Hamilton (g ’.v.) to discover the existence of what he called the characteristic function, by the help of which all optical problems, whether on the corpuscular or on the undulatory theory, are solved by one common process. Hamilton was in possession of the germs of this grand theory some years before 1824, but it was first communicated to the Royal Irish Academy in that year, and published in imperfect instalments some years later. The following is his own description of it. It is extremely important as showing his views on a very singular part of the more modern history of science.

“Those who have meditated on the beauty and utility, in theoretical mechanics, of the general method of Lagrange, who have felt the power and dignity of that central dynamical theorem which he deduced, in the Mécanique Analytique . . . , must feel that mathematical optics can only then attain a coordinate rank with mathematical mechanics . . . , when it shall possess an appropriate method, and become the unfolding of a central idea. ... It appears that if a general method in deductive optics can be attained at all, it must flow from some law or principle, itself of the highest generality, and among the highest results of induction. . . . [This] must be the principle, or law, called usually the Law of Least Action; suggested by questionable views, but established on the widest induction, and embracing every known combination of media, and every straight, or bent, or curved line, ordinary or extraordinary, along which light (whatever light may be) extends its influence successively in space and time: namely, that this linear path of light, from one point to another, is always found to be such that, if it be compared with the other infinitely various lines by which in thought and in geometry the same two points might be connected, a certain integral or sum, called often Action, and depending by fixed rules on the length, and shape, and position of the path, and on the media which are traversed by it, is less than all the similar integrals for the other neighbouring lines, or, at least, possesses, with respect to them, a certain stationary property. From this Law, then, which may, perhaps, be named the Law of Stationary Action, it seems that we may most fitly and with best hope set out, in the synthetic or deductive process and in the search of a mathematical method.

“Accordingly, from this known law of least or stationary action I deduced (long since) another connected and coextensive principle, which may be called by analogy the Law of Varying Action, and which seems to offer naturally a method such as we are seeking; the one law being as it were the last step in the ascending scale of induction, respecting linear paths of light, while the other law may usefully be made the first in the descending and deductive way.

‘ ‘ The former of these two laws was discovered in the following manner. The elementary principle of straight rays showed that light, under the most simple and usual circumstances, employs the direct, and therefore the shortest, course to pass from one point to another. Again, it was a very early discovery (attributed by Laplace to Ptolemy), that, in the case of a plane mirror, the bent line formed by the incident and reflected rays is shorter than any other bent line having the same extremities, and having its point of bending on the mirror. These facts were thought by some to be instances and results of the simplicity and economy of nature; and Fermat, whose researches on maxima and minima are claimed by the Continental mathematicians as the germ of the differential calculus, sought anxiously to trace some similar economy in the more complex case of refraction. He believed that by a metaphysical or cosmological necessity, arising from the simplicity of the universe, light always takes the course which it can traverse in the shortest time. To reconcile this metaphysical opinion with the law of refraction, discovered experimentally by Snellius, Fermat was led to suppose that the two lengths, or indices, which Snellius had measured on the incident ray prolonged and on the refracted ray, and had observed to have one common projection on a refracting plane, are inversely proportional to the two successive velocities of the light before and after refraction, and therefore that the velocity of light is diminished on entering those denser media in which it is observed to approach the perpendicular; for Fermat believed that the time of propagation of light along a line bent by refraction was represented by the sum of the two products, of the incident portion multiplied by the index of the first medium, and of the refracted portion multiplied by the index of the second medium; because he found, by his mathematical method, that this sum was less, in the case of a plane refractor, than if light went by any other than its actual path from one given point to another, and because he perceived that the supposition of a velocity inversely as the index reconciled his mathematical discovery of the minimum of the foregoing sum with his cosmological principle of least time. Descartes attacked Fermat’s opinions respecting light, but Leibnitz zealously defended them; and Huygens was led, by reasonings of a very different kind, to adopt Fermat’s conclusions of a velocity inversely as the index, and of a minimum time of propagation of light, in passing from one given point to another through an ordinary refracting plane. Newton, however, by his theory of emission and attraction, was led to conclude that the velocity of light was directly, not inversely, as the index, and that it was increased instead of being diminished on entering a denser medium; a result incompatible with the theorem of the shortest time in refraction. This theorem of shortest time was accordingly abandoned by many, and among the rest by Maupertuis, who, however, proposed in its stead, as a new cosmological principle, that celebrated law of least action which has since acquired so high a rank in mathematical physics, by the improvements of Euler and Lagrange.

“Maupertuis gave the name of action to the product of space and velocity, or rather to the sum of all such products for the various elements of any motion,—conceiving that the more space has been traversed and the less time it has been traversed in, the more action may be considered to have been expended; and by combining this idea of action with Newton’s estimate of the velocity of light as increased by a denser medium, and as proportional to the refracting index, and with Fermat’s mathematical theorem of the minimum sum of the products of paths and indices in ordinary refraction at a plane, he concluded that the course chosen by light corresponded always to the least possible action, though not always to the least possible time. He proposed this view as reconciling physical and metaphysical principles which the results of Newton had seemed to put in opposition to each other; and he soon proceeded to extend his law of least action to the phenomena of the shock of bodies. Euler, attached to Maupertuis, and pleased with these novel results, employed his own great mathematical powers to prove that the law of least action extends to all the curves described by points under the influence of central forces; or, to speak more precisely, that if any such curve be compared with any other curve between the same extremities, which differs from it indefinitely little in shape and in position, and may be imagined to be described by a neighbouring point with the same law of velocity, and if we give the name of action to the integral of the product of the velocity and element of a curve, the difference of the two neighbouring values of this action will be indefinitely less than the greatest linear distance (itself indefinitely small) between the two near curves; a theorem which I think may be advantageously expressed by saying that the action is stationary. Lagrange extended this theorem of Euler to the motion of a system of points or bodies which act in any manner on each other; the action being in this case the sum of the masses by the foregoing integrals.

‘ ‘ Laplace has also extended the use of the principle in optics, by applying it to the refraction of crystals, ana has pointed out an analogous principle in mechanics, for all imaginable connexions between force and velocity.”

We give, first, a very brief indication of the nature of Hamilton’s method, as applicable directly to the corpuscular theory. Here the action of a corpuscle is the quantity which possesses the stationary property. Let v be the velocity at any point x, y, z of the medium, ds an element of the path, a, β, y the direction cosines of ds, which are supposed to enter linearly and homogeneously into the expression for v. Then the action V is given by 'V≈fvds.

Hence, for a path nowhere finitely distant from the first, δV =∕(δvds + vdδs}.

But _ dv dv„ dv„

δv = - r δx + - r δy + - r δz dx dy dz dm dv. n dv„

+ -j-δα + -y-δβ + -y-δγ, da dβ dy , the first three terms depending on the translation of the element ds, the others on its change of direction, and all the differential coefficients being partial. The homogeneity of υ gives dv dv dv

"<⅛ + ⅛ + ¼-* , ∙

Also dδx = δdx=δ.ads ≈δa.ds + adδs, with two similar equations in y and z.

By the help of these, and a partial integration of the factors dδx, &c., we have . πτ dv„ , dv„ , ∙dv r , δV= - r δx + ~r^δy + -^-δz da dβ j dy + ∕[ te (⅛*-⅛) + · · · · ]'

where the integrated part is to be taken between proper limits.

If the initial and final points of the path be fixed, δx, &c., vanish in the integrated part, and the stationary condition shows that we must have dv , .dv . -fds - d-r =0, dx da with other two similar conditions, only two of the three being independent because of the necessary relation

α 2 + 0 2 + γ 2 =l.

These may be regarded as the differential equations of the ray, or path of the corpuscle.

But the essence of Hamilton’s method of varying action depends upon a change of the terminal point of the ray, and leads at once to the three equations

δV dv δV dv δV≈dv δx da ’ δy dβ ’ δz dy ’ which follow directly from the general value of δV above, by taking account of the vanishing of the unintegrated part in consequence of the stationary condition. We may now write d for δ everywhere in these expressions.

In any isotropic body, homogeneous or not, it is clear that dV = v{adx + βdy + ydz); and then we have, to determine V, the partial differential equation fdN∖ 2 fdV∖ 2 ∕dV∖ 2 ∖dx) + ∖dy) + ∖dz) V '

The treatment of this equation is precisely the same as that of the corresponding one which will presently be derived from the undulatory view of the question.

We will now illustrate the application of Hamilton’s method to the undulatory theory, in which the time of passage from one point of the path to another is the characteristic which fulfils the stationary condition. For the sake of limitation, we will confine ourselves to its application to single refraction in a non-homogeneous medium. In such a medium the velocity of light, at any point, is the same whatever be the direction of the ray. Hence it depends only upon the coordinates of the point, and upon some characteristic (say the wave-length) of the light considered.

If τ be this time of passage, ds an element of the path, andv the velocity of light in that element, we must have J υ * a quantity fulfilling the stationary condition. This gives ∕ dδs ∕* dsδv

∕ —ä~ ·

V J v i

Now, by what has just been said, if λ be the wave-length, we have an equation expressing the data of the problem, v=∕(λ, a⅞ y, z), where the form of f depends on the arrangement of the parts of the medium. Hence δ ∕^ dxdδx + dydδy + dzdδz ∕~dsidυ^ d υ ^ ∖ τ √ vds ~J v i ∖dx A + dχ x ’ J

Γ 1 fdz ∖-ι sι ∕*1 dv, .

= —( - 5 -δa,∙ + · · I I -δλ∕ — -ds-kc.

L V ∖ds J J J v i dx

The unwritten part consists of an integral which, by the stationary condition, vanishes if the ray be of a definite wave-length and the terminal points through which it passes be given, z.c., if δ.r, δy, δz, be each equal to 0.

The rest of the expression depends on the terminal points of the ray, and on the wave-length, only. It gives the equations

δτ_ 1 dx δτ _ 1 dy δr 1 dz δx V ds ’ δy V ds ’ δz~ v ds ’ and

δτ _ f 1 dv , δλ ~J v i dx

Squaring and adding the first three, we have (SΛS) ’ ÷⅛74∙

It is easily shown, by a process similar to that used for varying action (see Mechanics), that, if we can find a complete integral of this equation, containing therefore two arbitrary constants, in the form τ=F(θ!, y,z,x, a,β), then

- = ¾ - = 33 da ' dβ ω are the equations of two series of surfaces whose intersections give the paths of the rays, 'S and 13 here are also arbitrary constants. (These four constants are necessary, and sufficient, for the purpose of making the two intersecting surfaces pass each through any two given points.)

As an illustration, let us suppose the light to be homogeneous, and the medium to be arranged in concentric spherical shells such that the velocity at a distance r from their centre is expressed by

δ a +r 2 v= , e where b and c are absolute constants. It is easy to see that, on account of the symmetry, the path of every ray is in a plane through the centre of the spheres. We may therefore restrict our work to the plane of x, y passing through that centre.

The equation is then ( d TN + ( d HN ∖dx)^∖dy) (⅛ 2 + r 2 )2 , or, by change to polar coordinates, fdr∖ i , 1 fdτ∖ i c a

UτJ + r 2 U(M (δ 2 + r a ) a '

What we require is a sufficiently general solution. Assume, therefore, dr dθ a, and we have dτ _ ∕ c 2 α 2 dr“V (δ 2 + r a ) 2 r a ’ From these r = ab+J^dr + .

The equation of the path is therefore a-⅛- β -.∕∙— a '

Jr’J c - - 8 r√⅛- 4i *

or

ÈîzT = . ∕⅛-4δ 2 cos(e-¾).

r V α 2

This is the equation of a series of circles, whose one common characteristic is that the rectangle under the segments of any chord which passes through the origin is ⅛ 2 .

Hence every ray in any diametral plane describes a circle; and pairs of conjugate foci are situated on a line through the centre, the rectangle under their distances from the centre being b 2 . The property holds therefore for all rays in the medium. This very singular ideal arrangement was suggested to Clerk Maxwell by the eye of a fish. He has given an investigation of it, by a totally different analysis, in the Cambridge and Dublin Mathematical Journal, vol. ix. As an illustration of those effects of want of homogeneity to which (as already stated) all the complex phenomena of mirage, &c., are due, it may be well to consider this simple case more closely. We will therefore consider how images are seen in such a medium. To get rid of the difficulty which would arise from finite change of density if an eye were supposed to be plunged in the medium, we will suppose it to be cut across by a crevasse whose surface is everywhere nearly at right angles to the rays by which the image is to be seen,—the eye being then placed (in air) close to such a cutting surface.

Let AB (fig. 25) be a small object, 0 the centre of the spherical layers of equal refractive index. Then every ray from A describes a circle which passes through A', where A0A' is a straight line, and

AO.OA'-δ a .

A similar construction gives B' from B.

To an eye placed at E x (in a little crevasse as before explained), and looking towards the object, it will be seen erect,— A being seen in the direction of a tangent to the circle through AE 1 A ’, and similarly for B. Here the rays have not passed through their conjugate focus. But if the eye be now turned away from the object, it (or rather its image) will be seen, A' in the direction opposite to that in which A was seen, B' in the opposite direction to B. The image will now be an inverted one, but it will easily be seen to possess a strange peculiarity. For what is now seen will be the back of the object, the side farthest from the eye. The reader may easily trace for himself the course of the rays which would fall on the eye in any other assigned position. Vision in such cases would usually be of a peculiar character from another point of view, viz., the amount of divergence in the plane of the figure will in general differ from that perpendicular to its plane, and therefore the rays would have different divergence for the height and for the breadth of the image. These would therefore appear at different distances from the spectator. This, however, could be cured by a proper cylindrical lens. It is clear from this example (which has been chosen for its special simplicity) that want of homogeneity in a refracting medium is capable of producing phenomena of the most extraordinary character.

It is difficult to ascertain exactly what is the condition of the atmosphere when multiple images, mirage, &c., are seen; and it is obvious from the remarks and illustrations already given that many very different arrangements will produce sensibly the same result to a spectator in a given position. Comparison of the appearances seen simultaneously by a great number of scattered observers is the only way in which we can expect to obtain definite information on such a point. But the following investigation suggests the general nature of the explanation.

If we suppose the refractive index of the air to depend only upon the vertical height above the earth’s surface, rays will all travel in vertical planes, and Hamilton’s equation (neglecting the curvature of the earth’s surface) takes the very simple form

(S)÷(g)*-∕ω∙

x being measured ho rizontally, and the refractive index being proportional to

This equation gives, as before,

τ =- ax +Jdy∖∕f{y) - aø, and the equation of the path of a ray is

⅛-a-≈-.4 7 Λ=.

da J∖f{y)- a <

Here, on the corpuscular theory, a is the horizontal velocity of the light, and ∖JJ∖y} - α 2 the vertical velocity. If the form of f and the value of a be such that we can have ∕ (∙η') - α 2 = 0, it is clear that at y=n the ray is for a moment horizontal. The form of the equation of the ray shows that it has a vertex at this point, and that it is symmetrical about a vertical axis passing through the vertex. If ξ, η be the coordinates of the vertex for a ray passing through the point 0, b, we have the relation w <⅛

This is the equation of the locus of the vertices of all rays which, starting from a given point, return again to the same level. To find, then, the various rays by which a distant object near the horizon can be seen, all that we have to do is to draw the curve of vertices which passes through the eye of the spectator, and to find the points in which it is intersected by a vertical line situated midway between the object and the eye. Each of these points is the vertex of a ray by which the object can be seen. When the curve of vertices leans forward towards the eye at one of these points, two contiguous rays cross one another, and an inverted image is seen; when it leans from the eye, they do not cross, and the image is erect.

Now, when the curve of vertices is traced, from the above formula, for an arrangement of the air such that the refractive index falls off through a horizontal stratum of air from a greater value below the stratum to a smaller value above it, it is found that the curve of vertices in the· stratum can in general be cut by a vertical line in one point only. But if the refractive index have a nearly stationary value at the upper boundary of the stratum the curve of vertices can be cut twice, or not at all, by a vertical line. When there is no intersection we have only the direct image; but when there are two intersections a distant ship will be seen as usual through the lower uniform air, while there will be seen above it an inverted image, and then a direct image, both due to the stratum. This is a form of mirage very commonly seen at sea. When there is no stationary value of the index at the upper boundary, the upper erect image is not given by the stratum. This arrangement, however, turned upside down, explains the ordinary mirage of the desert—where we see objects directly through the nearly uniform air at some distance above the sand, but also an inverted image (suggesting reflexion from a pool or lake) formed by the refraction in the hot layer of air near the sand.

Absorption, Fluorescence. —We must now take up the third and fourth of the categories under which light incident on the bounding surface of two media may fall —scattering and absorption. We take them together, because in the great majority of bodies, as we have already seen, scattering takes place not merely at the surface but within some distance below the surface, which in general is small, but in some cases considerable. And when the scattering takes place, even in part only, below the surface, the scattered light is usually modified by absorption.

An excellent instance of this scattering from below the surface is afforded by a mass of thin films or small particles of transparent bodies, such as glass, water, or ice. Thus pounded glass, froth or foam, snow, clouds, &c., appear brilliantly white in sunlight, and are, in consequence, opaque when in layers of sufficient thickness. Here the light is obviously scattered by reflexion. What passes through one film, crystal, or particle is, in part, reflected from the next, and so on.

Even when the froth consists of bubbles of a highly coloured liquid, such as porter for instance, it usually shows but slight traces of colour, for the great majority of the scattered rays have passed through very small thicknesses only of the liquid. In the same way, very finely pounded blue or red glass (unless it be exceedingly deeply coloured when in mass) appears nearly white. But when a mass of water is full of air bubbles, as, for instance, is the case in the neighbourhood of a breaker, the light reflected from the surfaces of these bubbles suffers a double absorption by the water before it reaches the eye. This is one of the causes of the exquisite colours of the sea. Near shore, or in shoal water, another cause sometimes comes into play, viz., fine solid particles suspended in the water. When such particles, whether in air or in water, are exceedingly small, they may produce colours due to their minuteness alone, and not to their own colour nor to the absorptive properties of the medium. This, however, is a question of physical optics.

In general, even the most highly coloured opaque or translucent solids, such as painted wood or stained paper, are visible by scattered light whatever portion of the spectrum falls on them. This is very well seen with highly coloured paper-hangings, when illuminated by homogeneous light, such as that of a sodium flame (a Bunsen flame, into which is thrust a platinum wire dipped in strong brine). The red, orange, and yellow parts usually appear very bright under such treatment, the blue parts appearing but slightly illuminated. The colour of all is, of course, that of the incident light. It appears, therefore, that some of the light is scattered from the surface. It is by this, for instance, that the blue parts are feebly visible. But that which is scattered from the portions coloured red, orange, &c., must come mainly from under the surface.

An excellent proof of this is furnished by mixing, in proper proportions, a yellow and a blue powder, or yellow and blue paints. It is commonly imagined that the green colour which is thus produced is a mixture of blue and yellow. Far from it! When a disk divided into alternate sectors, coloured with the same blue and yellow pigments, is made to rotate rapidly in its own plane, it of course produces on the eye the true result of a mixture of these blue and yellow colours. This depends for its exact tint on the pigments employed, and on the angles of the sectors, but is usually a faint pink or a muddy purple,—utterly different from the green produced by mixing the powders or the paints. Helmholtz was the first to point out the true source of the green. It is the one colour which is not freely absorbed either by the yellow or by the blue pigment. For the scattered light by which the mixture is seen comes chiefly from below the surface, and has thus suffered absorption by each of the component powders. The yellow powder removes the greater part of the blue, indigo, and violet rays; the blue, the greater part of the reds, oranges, and yellows. Thus the light which finally escapes is mainly green.

For the accurate study of the absorptive power of a solid or liquid medium, it is necessary to compare the spectrum of white light which has passed through a plate or layer of it with a normal spectrum. This is easily effected by placing the absorbing medium (if a fluid, it must be in a glass trough with parallel sides) in front of the narrow slit through which the light passes, and in such a position that one half of the slit only is thus covered. We have then side by side, under precisely similar circumstances, two spectra to be compared (one altered by absorption, the other not); and very minute differences between them can thus be detected. When the medium produces a general weakening of the whole spectrum, as well as particular local absorptions, the white light passing through the other half of the slit may be weakened to any desired extent by reflexion at the proper incidence from a plate of glass, before it falls on the slit.

To give a satisfactory representation of the phenomena of absorption spectra by the help of a woodcut is not easy. The highest artistic skill could not adequately represent the ordinary solar spectrum by the use of the finest pigments. All optical colour phenomena must be seen, they cannot be reproduced by painting. In such circumstances the simplest method of indicating the locality and amount of the absorption is the best. As we have already seen that we cannot by the eye judge of the relative intensities of lights which differ much in colour, we shall represent the normal spectrum (for our present purpose) as equally bright throughout, and indicate the absorption at different parts by shading of various degrees of depth. A few of the Fraunhofer lines are introduced to indicate (in the absence of colour) the parts of the spectrum which are attacked by the various absorbents. These lines are, of course, in the same absolute positions in all the various spectra; for the spectra are all supposed to be given by the same prism. The line B is in the red, D in the orange, E and F in the green, and G in the indigo. They correspond, as we have already said, to perfectly definite kinds of homogeneous light, and therefore adequately represent the distribution of colours in the spectrum, however much irrationality of dispersion may be shown by the material of the prism.

In fig. 26 α represents the spectrum of light which has passed through diluted blood; β shows the spectrum when the blood has been acted on by a reducing agent; and γ the spectrum when the blood has been altered by acidulation with acetic or tartaric acid.

These figures are taken from an important paper by Stokes {Proceedings of the Royal Society, 1864).

Fig. 27 shows in a rude way the absorption by cobalt glass cut in wedge form, and corrected by an equal prism of clear glass.

The commonly received method of calculating the absorption by layers of gradually increasing thickness is to suppose that, if a layer of unit thickness weakens in any ratio the intensity of any particular homogeneous ray, another unit layer will farther weaken in the same ratio that which reaches it, and so on. Thus the amount which passes through a number of layers diminishes in geometrical progression, while the number of layers increases in arithmetical progression. This is certainly true (neglecting the amount reflected), unless the intensity of the light have an effect on the percentage transmitted. And fig. 27 shows, in a very striking manner, the difference between similar terms of different geometric series as the common ratio becomes less and less. This ratio is not much less than 1 for certain red and blue rays, is smaller for yellow, and is very small for the rest of the red, for orange, and for green. The latter colours are therefore rapidly got rid of with increasing thickness; then the yellow becomes too feeble to be seen; while, even after the blue becomes almost insensible, the specially favoured red rays are still transmitted in sufficient quantity to be observed.

If r be the fraction of any species of homogeneous light which is transmitted by a layer of unit thickness, that transmitted by a layer of thickness x is r x . The following little table will greatly assist the reader in understanding the relative rapidity of extinction of different rays passing through various thicknesses of an absorbing medium. It is a table of double entιy, the first column giving various values of x, and the upper row various values of r, while the value of r z is in the same column as that of r and in the same row as that of x. [table]

Thus a ray which loses 1 per eent. in unit thickness still preserves more than 90 per cent, after passing through ten units. But a ray w·hich loses 10 per cent, in the first unit (and which, therefore, will thus far appear scarcely more weakened than the first) is reduced to 35 per cent, by passage through ten units. After passing through a hundred units the first ray has lost only 63 per cent., the second is practically invisible.

In thin plates cobalt glass is blue, because the particular red which it does not absorb freely forms only a small fraction of the whole transmitted rays; while in thick masses it is nearly red, for then little but this favoured red is transmitted. For a similar reason Condy’s fluid (permanganate of potash) changes its tint in a very singular manner (even when preserved from the action of the air) by gradual dilution with water.

The imperfection of the achromatism of the eye is readily proved by looking through a plate of cobalt glass at a small hole in the window-shutter of a dark room. The hole at first appears red with a blue space round it; but, by an effort of the muscles of the eye, we can see the hole blue, and then there is a red space surrounding it. Rays of so widely different refractive index cannot be seen in focus simultaneously.

Very curious effects are produced when we examine a landscape through such a glass. Foliage of certain kinds scatters scarcely any blue rays, and therefore appears reddish. Bluish-greens, again, which scatter very little red, appear blue. The effects may be exaggerated in a very striking degree by combining the absorptions of two or more media, so as to allow of the free transmission of a few, far detached, portions of the spectrum.

Brewster made the very singular discovery that a solution of oxalate of chromium and potash produces one solitary, narrow, absorption band, almost resembling one of the broader lines in the solar spectrum.

Closely connected with intense local absorption in certain parts of the spectrum is the phenomenon of abnormal dispersion, one of the most singular discoveries of modern times. It seems to have been first observed by Fox Talbot; and he discovered its real nature. But the first published notice of such phenomena is due to Le Roux. Christiansen and others have since greatly extended our knowledge of the subject, and Helmholtz and Ketteler have given theoretical explanations of it. Fox Talbot’s experiment, though the earliest on record, is one of the easiest to perform, and we therefore quote his own account. The experiment was made about 1840, and the following account is from the Proc. Roy. Soc. Edin., 1870-71.

“I prepared some square pieces of window glass, about an inch square. Taking one of these, I placed upon it a drop of a strong solution of some salt of chromium, which, if I remember rightly, was the double oxalate of chromium and potash, but it may have been that substance more or less modified. By placing a second square of glass on the first, the drop was spread out in a thin film, but it was prevented from becoming too thin by four pellets of wax placed at the corners of the square, which likewise served to hold the two pieces of glass together. The glasses were then laid aside for some hours until crystals formed in the liquid. These were necessarily thin, since their thickness was limited by the interval between the glasses. Of course the central part of each crystal, except the smallest ones, was bounded by parallel planes, but the extremities were bevilled at various angles, forming so many little prisms, the smallest or them floating in the liquid. When a distant candle was viewed through these glasses, having the little prisms interposed, a great number of spectra became visible, caused by the inclined edges. Most of these were no doubt very imperfect, but by trying the glass at various points, some very distinct spectra were met with, and these could with some trouble be isolated by covering the glass with a card pierced with a pin-hole. It was then seen that each prism (or oblique edge of crystal) produced two spectra oppositely polarized and widely separated. One of these spectra was normal; there was nothing particular about it. The colours of the other were very anomalous, and, after many experiments, I came to the conclusion that they could only be explained by the supposition that the spectrum, after proceeding for a certain distance, stopped short and returned upon itself.”

Le Roux in 1860 discovered that vapour of iodine, which allows only red and blue rays to pass, refracts the red more than the blue. He, like Talbot, did not at first venture to publish his result, and it appeared only in 1862. Among the many convincing proofs of its accuracy he shows that the dispersion by an iodine-vapour prism can be nearly achromatized by a glass prism which gives refraction in the same direction. He also states that the dispersion in iodine-vapour is-less as the temperature is higher.

Christiansen’s earliest determinations were made in 1870 upon an alcoholic solution of fuchsine (one of the powerful aniline colours). This solution gives a dark absorption band in the green; and it was found that the refractive index rises (as in normal bodies) for rays from the red to the yellow. But all the rest of the transmitted light, consisting of the so-called “more refrangible” rays, is less refracted than the red. Kundt and others shortly afterwards greatly extended these observations.

The explanation of this phenomenon, which has been advanced by Helmholtz, depends upon an assumption as to the nature of the mutual action between the luminiferous ether and the particles of the absorbing medium, coupled with a further assumption connecting the absorption itself with a species of friction among the parts of each absorbing particle.

In 1879 De Klerker made a very curious observation, which shows that the whole subject is still obscure. He employed two hollow prisms of equal angle, turned opposite ways, and filled with alcohol. Through such a combination light passes (as we have seen) w ithout refraction or dispersion. When a few drops of the fuchsine solution were added to the contents of one of the prisms, the yellow, orange, and red rays (in the order named) began to separate themselves from the others. This process could be carried on until the solution was so strong that it transmitted no visible light. All this time the blue and violet rays remained apparently unrefracted—the yellow, orange, and red showing continually increasing refraction. The conclusion from this, on either theory of light, is that the addition of fuchsine to alcohol alters the velocity of propagation of the (so-called) less refrangible rays, but not perceptibly that of the more refrangible.

Fluorescence,— The singular surface appearances presented by “canary” glass, by some specimens of fluor spar, and by certain liquids, such as a solution of sulphate of quinine acidulated with sulphuric acid, had been the source of much speculation long before their true nature was traced by Stokes in 1852. By a series of well-contrived experiments, one or two of which will presently be described, he put it beyond doubt that the cause of these phenomena lies in a change of refrangibility of the light which has been absorbed by the upper layers of the medium, and then given off again. In every case the fluorescent light appears to belong to a less refrangible part of the spectrum than does the incident light which gave rise to it, thus affording an instance of dissipation, or degradation of energy.

The yellowish-green surface-colour of canary glass (coloured with oxide of uranium) is well known, as the substance is, mainly on account of this property, very commonly used for ornaments. If we admit a ray of sunlight (or light from the electric lamp) into a dark room, through a cobalt glass so dark that the feeble violetcoloured light it transmits is scarcely visible, we find that the canary glass shows its yellow-green colour vividly when placed in the track of the ray. Striking as this experiment is, it is not quite conclusive as to the true cause of the appearance. But if we take another piece of glass, slightly tinged of a brownish-yellow (by oxide of gold), we find that it is quite transparent to the brilliant light from the canary glass; if, however, we place it in the track of the violet rays before they fall on the uranium glass, it prevents the production of the phenomenon altogether. That is, rays which cannot pass through the glass coloured with gold are rendered capable of freely passing through it after incidence on the canary glass. That the phenomenon is due to rays which are stopped by the uranium glass itself is proved by the fact that a second piece of the glass, placed in the track of the rays which have passed through the first, does not show the phenomenon. Unless, indeed, the source of light be very bright, the appearance is confined to a mere surfacelayer of the first piece of canary glass. The phenomenon is very well shown by an aqueous infusion of horse-chestnut bark. Some specimens of paraffin oil exhibit it most brilliantly.

To find the rays which are most effective in producing the fluorescence of any substance, we have only to place it in a pure spectrum of sunlight (or, preferably, of the electric light),—prisms and lenses of quartz being used for producing the spectrum, because that material is found to be far less opaque than glass is to the violet and ultraviolet rays. When this is done with uranium glass we find scarcely a trace of effect until the substance reaches the blue rays, and the effect persists through all the higher colours, and even very considerably beyond the bounds of the visible spectrum. Stokes in fact used it as a means of studying the otherwise invisible, but far extending, spectrum of the ultra-violet rays of the electric spark.

The mechanism of the process by which these extraordinary results are produced is still somewhat obscure, and we cannot attempt to explain it here.

The duration of fluorescence is so very short that it is only by specially devised methods that we can make certain that it persists for any measurable time after the exciting light is cut off from the fluorescent body. Becquerel’s ingenious phosphOroscope was invented for the purpose of inquiries of this kind. It consists essentially of a shallow drum, in whose ends two excentric holes, exactly opposite to one another, are cut. Inside it are fixed two equal metal disks, attached perpendicularly to an axis, and divided into the same number of sectors, the alternate sectors of each being cut out. One of these disks is close to one end of the drum, the other to the opposite end, and the sectors are so arranged that, when the disks are made to rotate, the hole in one end is open while that in the other is closed, and vice versa. If the eye be placed near one hole, and a ray of sunlight be admitted by the other, it is obvious that while the sun shines on an object inside the drum the aperture next the eye is closed, and vice versa. If the disks be made to revolve with great velocity by means of a train of toothed wheels, the object will be presented to the eye almost instantly after it has been exposed to sunlight; and these presentations succeed one another so rapidly as to produce a sense of continued vision. By means of this apparatus we can test with considerable accuracy the duration of the phenomenon after the light has been cut off. For such a purpose we require merely to know the number of sectors in the disks and the rate at which they are turned. To guard against deception by the persistence of impressions on the retina, the eye should not be directed fixedly on the object, but should be kept travelling slowly round the position in which it is seen to lie.

Uranium glass shows, with rapid turning, nearly as vivid an effect as when exposed to continuous light, but fades rapidly when the speed of the rotation falls off. A pinkish kind of ruby, exposed to concentrated sunlight in the phosphoroscope, is seen to glow with a bright red like a piece of live coal. With very rapid turning, feeble fluorescence can be detected in a great many substances in which the ordinary methods will not show it. This is due in great measure to the fact that the phosphoroscope entirely does away with the scattered light, which in the ordinary mode of examining these substances overpowers their feeble fluorescence.

What is correctly termed phosphorescence has nothing to do with phosphorus (whose luminosity in the dark is due to slow oxidation), but it is merely a species of fluorescence which lasts for a much longer time after the excitation has ceased than does that just described. Pliny speaks of various gems which shine with a light of their own, and Albertus Magnus knew that the diamond becomes phosphorescent when moderately heated. But the first discovery of phosphorescent substances, such as are now so common, belongs to the early part of the 17th century. During that century the Bologna stone (sulphide of barium) and Homberg’s phosphorus (chloride of calcium) were discovered. Can ton’s phosphorus (sulphide of calcium) dates from 1768. To the substances mentioned may now be added sulphide of strontium. Any of these sulphides, which must be carefully preserved from the air in sealed glass tubes, appears brilliantly luminous when carried from sunlight into a dark room, and for a long time after presents the general aspect of a hot body cooling. The rays which excite their luminosity are (as with the generality of fluorescent bodies) those of higher refrangibilities; but the colours of the phosphorescent light are of the most varied kind, even in specimens of almost precisely the same chemical composition, but prepared at different times. The causes of this strange diversity are as yet quite unguessed at; but the property has been taken advantage of for the production of what are called luminous paints. The behaviour of these substances is one of the most singular phenomena in optics. How they manage to store up so large a supply of energy during a short exposure to bright light, and to dole it out continuously for so long a time and mainly in the form of light, is exceedingly puzzling,— especially as no other physical or chemical change has yet been found to accompany the process. Another curious fact connected with their behaviour was discovered by Becquerel. He found that the less refrangible rays have in some cases the power of arresting the emission of light from these bodies when they have been previously excited by higher rays.

The chemical effects of light will be treated under Photography, so far as they are connected with decomposition. Its effects in causing combination, as of hydrogen and chlorine, have already been treated under Chemistry. Undulatory Theory of Light.

The explanation of the fundamental laws of Geometrical Optics by the wave-theory requires some preliminary remarks. As the subject will be more fully discussed in a special article, we confine ourselves to what is strictly necessary for the immediate purposes of the present article.

(a) The essential characteristic of wave-motion is that a disturbance of some kind is handed on from one portion of a solid or fluid mass to another. In certain cases only, this disturbance is unaltered in amount and in kind as it proceeds.

(δ) So far as light is concerned, the velocity with which each particular species of disturbance passes in any direction through a homogeneous isotropic medium is constant and is the same for all directions. When the medium is not homogeneous, the velocity may vary from point to point. If the medium be not isotropic, the velocity may depend upon the direction of propagation. Examples of each of these peculiarities will be met with presently.

(c) When two or more separate disturbances simultaneously affect the same portion of a medium, the effect may be very complex. But, in the case of light, it has been found that a geometrical (or rather kinematical) superposition or composition agrees, at least to the degree of accuracy of the experiments, with all the observed facts. This would be the case, as a dynamical result, if the distortions due to wave-motion were always, even for the most powerful light, exceedingly small. On this is based the whole doctrine of interference, Young’s grandest contribution to the wave-theory (1801).

(d) The disturbance at any point of a medium, at any instant, is that due to the superposition of all the disturbances which reached it at that instant from the various surrounding parts of the medium. This is (in a somewhat generalized form) what is commonly known as Huygens’s principle, first enunciated in 1678.

(e) The front of a wave is defined at any instant as the continuous locus of all portions of the medium which, at that instant, are equally and similarly distorted. The word continuous is inserted because, in oscillatory wavemotion, such as that of light, a large number of successive waves are exactly equal and similar to one another. Thus we have a semes of wave-fronts following one another, which are not to be considered as parts of one wave-front. The distance between two successive fronts in which the distortions are similar, measured in the direction in which the light is travelling, is called the wave-lempth. ( f) The colour of homogeneous light depends entirely on the period of a wave, i.e., on the time of passage from one wave-front to the next. This is obviously the same thing as the time of a complete vibration of any one particle of the medium—whatever be the velocity of light in the medium, or the consequent wave-length.

These being premised, let us take the propagation of homogeneous light from a luminous point in a homogeneous isotropic medium. Here we have simply a succession of concentric spherical wave-fronts, their radii differing by one or more whole wave-lengths. The disturbance in any portion of one of these fronts is propagated radially. But we may consider it from a different point of view, as hinted in (<7) above. Simple as this particular case is, the reader will probably find that it will greatly assist him in understanding the more complex ones which follow.

Every disturbed portion of the medium may be looked upon as a centre of disturbance from which a new set of spherical waves is constantly spreading. Take then, as common radius, the space described by a disturbance in any very short interval; and, with centres at every point of any one wave-front, describe a series of spheres. The ultimate intersections of these spheres will lie on a surface which is the envelop of them all. In the case considered, it is obviously a sphere whose radius exceeds that of the wave-front from which we started by the common radius of the set of spheres. This is shown in a central section in fig. 28 below, which suffices to prove that we arrive l>y this mode of construction at the result which we know in this simple case to be the correct one. It will be seen that the centres of the construction-spheres lie on a certain part of one wave-front, while their ultimate intersections lie on the corresponding part of the future wave-front. This holds for spheres of all radii, and for continually increasing radii shows that a plane wave moves perpendicularly to its front. This is so important a part of Huygens’s work that we give it in his own words lfTraite de la Lumière, 1690, pp. 18-20):—

“Pour venir aux proprietez de la lumière; remarquons premièrement que chaque partie d’onde doit s’étendre en sorte, que les extremitez soient tousjours comprises entre les mesmes lignes droites tireés du point lumineux. Ainsi la partie de l’onde BG, ayant le point lumineux A pour centre, s’étendra en l’arc CE, terminé par les droites ABC, AGE. Car bien que les ondes particulières, produites par les particules que comprend l’espace CAE, se repandent aussi hors de cet espace, toutesfois elles ne concourent point en mesme instant, à composer ensemble une onde qui termine le mouvement, que précisément dans la circonférence CE, qui est leur tangente commune.

“Et d’icy l’on voit la raison pourquoy la lumiere, à moins que ses rayons ne soient réfléchis ou rompus, ne se répand que par des lignes droites, en sorte qu’elle n’éclaire aucun objet que quand le chemin depuis sa source jusqu’à cet objet est ouvert suivant de telles lignes. Car si, par exemple, il y avoit une ouverture BG, borneé par des corps opaques BH, GI; l’onde de lumiere qui sort du point A sera tousjours termineé par les droites AC, AE, comme il vient d’estre de-monstré: les parties des ondes particulières, qui s’étendent hors de l’espace ACE, estant trop foibles pour y produire de la lumiere.

“Or quelque petite que nous fassions l’ouverture BG, la raison est tousjours la mesme pour y faire passer la lumiere entre des lignes droites; parce que cette ouverture est tousjours assez grande pour contenir un grand nombre de particules de la matière etherée, qui sont d’une petitesse inconcevable; de sorte qu’il paroit que chaque petite partie d’onde s’avance nécessairement suivant la ligne droite qui vient du point luisant. Et c’est ainsi que l’on peut prendre des rayons de lumiere comme si c’estoient des lignes droites.

“Il paroit au reste, par ce qui à esté remarqué touchant la foiblesse des ondes particulières, qu’il n’est pas necessaire que toutes les particules de,l’Ether soient égales entre elles, quoique l’égalité soit plus propre à la propagation du mouvement. Car il est vray que 1 inégalité fera qu’une particule, en poussant une autre plus grande, fasse effort pour reculer avec une partie de son mouvement, mais il ne s engendrera de cela que quelques ondes particulières en arriere vers le point lumineux, incapables de faire de la lumiere: & non pas d’onde composeé de plusieurs, comme estoit CE.

“Une autre, et des plus merveilleuses proprietez de la lumiere est que, quand il en vient de divers costez, ou mesme d’opposez, elles font leur effet l’une à travers l’autre sans aucun empêchement. D’on vient aussi que par une mesme ouverture plusieurs spectateurs peuvent voir tout à la fois des objets differens, et que deux personnes se voyent en mesme instant les yeux l’un de l’autre. Or suivant ce qui a esté expliqué de l’action de la lumiere, et comment ses ondes ne se détruisent point, ny ne s’interrompent les unes les autres quand elles se croisent, ces effets que je viens de dire sont aisez à concevoir. Qui ne le sont nullement à mon avis selon l’opinion de Des-Cartes, qui fait consister la lumiere dans une pression continuelle, qui ne fait que tendre au mouvement. Car cette pression ne pouvant agir tout a la fois des deux costez opposez, contre des corps qui n’ont aucune inclination à s’approcher; il est impossible de comprendre ce que je viens de dire de deux personnes qui se voyent les yeux mutuellement, ni comment deux flambeaux se puissent éclairer l’un l’autre.”

We will now, for the purposes of this elementary article, assume that something similar holds in all cases, and will not trouble ourselves with the fact that our construction, if fully carried out, would indicate a retrograding wave as well as a progressive one. The obvious fact that a solitary wave can be propagated in water, or along a stretched string, may assist the reader in taking the bold step which we have proposed to him. And we will also assume that this mode of representation leads to correct results even when we do not choose a wave-front as the locus of the centres of disturbance,—that in fact we may choose for our purpose any surface through which the rays pass, provided always that the radii of the spheres are so chosen that the length of each ray from some definite wave-front to the centre of the sphere, together with the radius of that sphere, always corresponds to a path described in a given time.

We are now prepared to explain the reflexion of light, and we need do so for a plane reflecting surface alone, because the length of a wave, as we shall soon see, is an almost vanishing quantity in comparison with the radius of curvature of any artificial mirror, be it even the smallest visible drop of mercury.

Let a plane wave-front be approaching a plane mirror, and at any instant let fig. 29 represent a section by a plane perpendicular to each, cutting the wave-front in AB and the mirror in AC. From what has been already said, the motion of every part of AB is perpendicular to that line, and in the plane of the figure. During the time that the disturbance at B takes to reach C, the disturbance which had reached A will have (in part, for there is usually a refracted part also) spread back into the medium in the form of a spherical wave whose radius, AD, is equal to BC. Its section is of course a circle. That from any other point P will have reached Q, and then (in part) diverged into a spherical wave whose centre is Q and radius QT( = QT') = BC - PQ. Obviously all the circles which can be thus drawn ultimately intersect in the straight line CD. This is a section of the reflected wave-front. A plane wave, therefore, remains a plane wave after reflexion, each part of it obviously moves in the plane of incidence, and the similarity of the triangles ABC and CDA proves the equality of the angles of incidence and reflexion, for the ray is everywhere perpendicular to the wave-front. It is to be particularly noted that this is independent of the velocity of the light, so that all rays are reflected alike. In this, as in the preceding and the immediately following instances, the diagram has been taken (with but slight change) from Huygens.

This being true of any plane wave-front, large or small in area, is necessarily also true of any wave-front of finite curvature. Thus, if a set of rays be drawn perpendicular to any wave-front, they will after reflexion be perpendicular to a new wave-front; and the lengths of all the rays, from wave-front to wave-front, will be equal.

This is merely another way of stating that if a set of rays can be cut at right angles by a surface (of finite curvature) they will always be capable of being cut at right angles by such a surface, even after any number of reflexions at surfaces of finite curvature, provided they move in a homogeneous isotropic medium.

This proposition will be seen to be capable of extension to refraction, provided always that both media are homogeneous and isotropic. For a plane wave, falling on a plane refracting surface, our construction (fig. 30) is as follows: —

Let AB be, as before, a plane wave-front in the first medium, and AC the plane surface of the second medium. As before, let BC be perpendicular to AB. Also let CD' be drawn parallel toBA. With centre A and radius AD equal to the space described in the second medium while BC is described in the first, let a sphere be described. The disturbance at A will have diverged in this sphere, while that at B has just reached C. The disturbance at any other point, as P, will have passed to Q, and then have diverged into a sphere of radius QT such that

QT: QT':: AD; BC.

Obviously all spheres so drawn ultimately intersect along CD, which is therefore the front of the refracted wave. The angles of incidence and refraction, being the inclinations of the incident and refracted rays to the normal, are the inclinations BAG and DCA of the incident and refracted wave-fronts to the refracting surface. Their sines are evidently in the ratio of BC to AD, i.e., they are directly as the velocities of propagation in the two media.

Hence the law of refraction also follows from this hypothesis. But there will now be separation of the various homogeneous rays, because the ratio of their velocities in the two media is not generally constant.

Besides, it is clear from the investigation above that, in the refracting medium, the rays are still perpendicular to the wave-front. Thus the proposition lately given may now be extended in the following form :—

If a series of rays travelling in homogeneous isotropic media be at any place normal to a wave-front, they will possess the same property after any number of reflexions and refractions. And it is clear from the investigations already given that the time employed by light in passing from one of these wave-fronts to another is the same for every ray of the series.

We now see how crucial a test of theory is furnished by the simple refraction of light. On the corpuscular theory the velocity of light in water is to its velocity in air as 4: 3 nearly; on the undulatory theory these velocities are as 3: 4, since, as we have seen, the refractive index of water is about ⅜. But Foucault’s experimental method showed at once that the velocity is less in water than in air. This finally disposed of the corpuscular theory. Though it had been conclusively disproved long before, by certain interference experiments whose nature will presently be described, the argument from these was somewhat indirect and not well suited to convince the large non-mathematical class among optical students and experimenters. The true author of the undulatory theory is undoubtedly Huygens. Grimaldi, Hooke, and others had expressed more or less obscure notions on the subject, but Huygens in 1678 first gave it in a definite form, based to a great extent upon measurements of his own. It was read to the French Academy, but not published till 1690, when it appeared with the title Traité de la Lumière. Huygens gives the explanation of the double refraction of Iceland spar, which had been described by Bartholinus in 1670. Unfortunately the remarkable step taken by Newton in explaining the law of refraction on the corpuscular theory—the earliest solution of a problem connected with molecular forces—had for some time been before the scientific world. The authority of Newton was paramount in such matters, and the work of Huygens produced no effect at the time. Even the genius of Young, who at the commencement of the present century recalled attention to this all-but-forgotten theory, and enriched it by the addition of the principle of interference, as well as by many important applications, failed to secure its recognition. It was not till 1815 and subsequent years that, in the hands of Fresnel, the undulatory theory finally triumphed, and, even then, the battle was won against determined resistance on the part of the upholders of the corpuscular theory. Witness what Laplace said, in 1817, in the following excerpt from a letter to Young :—

“J’ai reçu la lettre que vous m’avez fait l’honneur de> m’écrire, et dans laquelle vous cherchez à établir que, suivant le système des ondulations de la lumière, les sinus d’incidence et de réfraction sont en rapport constant, lorsqu’elle passe d’un milieu dans un autre. Quelque ingénieux que soit ce raisonnement, je ne puis le regarder que comme un aperçu, et non comme une démonstration géométrique. Je persiste à croire que le problème de la propagation des ondes, lorsqu’elles traversent différens milieux, n’a jamais été résolu, et qu’il surpasse peut-être les forces actuelles de l’analyse. Descartes expliquoit ce rapport constant, au moyen de deux suppositions; l’une, que la vitesse des rayons lumineux parallèlement a la surface du milieu refringent ne changeoit point par la réfraction; l’autre, que sa vitesse entière dans ce milieu étoit la même, sous toutes les incidences; mais comme il nerattachoit aucune de ces suppositions aux lois de la mécanique, son explication a été vivement combattue et rejettée par les plus grand nombre des physiciens jusqu’ à ce que Newton ait fait voir que ces suppositions résultoient de l’action du milieu refringent sur la lumière; alors on a eu une explication mathématique du phénomène dans le système de l’émission de la lumière: système qui donne encore l’explication la plus simple du phénomène de l’aberration, que n’explique point le système des ondes lumineuses. Ainsi les suppositions de Descartes, comme plusieurs aperçus de Kepler sur le système du monde, ont été vérifiées par l’analyse: mais le mérite de la découverte d’une vérité appartient tout entier à celui qui la démontre. Je conviens que de nouveaux phénomènes de la lumière sont jusqu’ à présent très difficiles à expliquer; mais en les étudiant avec un grand soin, pour découvrir les lois dont ils dépendent, ou parviendra peut-être un jour à reconnaître dans les molecules lumineuses des propriétés nouvelles qui donneront une explication mathématique de ces phénomènes. Remonter des phénomènes aux lois et des lois aux forces, est, comme vous le savez, la vraie marche des sciences naturelles.”

Poggendorff remarks that there is no other instance, in the whole history of modern physics, in which the truth was so long kept down by authority. Poggendorff further remarks that of the six chief phenomena of light known in Huygens’s time he fully explained three—reflexion, refraction, and the double refraction of Iceland spar—at least so far as concerns the direction of the reflected or refracted rays. Phenomena such as diffraction, and the colours of thin plates, required the principle of interference for their explanation, which was first given by Young; and dispersion (not yet quite satisfactorily disposed of) was first accounted for in comparatively recent times by Cauchy. Huygens himself was the discoverer of polarization, but he could not account for it. Even Young also, because like Huygens he supposed the undulations to be in the direction of the ray, failed to account for it; and it was not explained till Fresnel reintroduced with the most brilliant success a guess of Hooke’s (of date 1672), that the vibrations of light in an isotropic medium are perpendicular to the direction of the ray.

Taking the undulatory theory as the only one left possible by the experiments of Foucault, we will now consider the explanation it offers of various phenomena. It will be remembered that we have as yet made no assumption whatever as to the precise nature of a wave ; and it will be found that a large class of important phenomena can be explained by it without our making any such assumption, but that other classes of phenomena compel us to adopt certain limitations of the very general hypothesis with which we started. As long as we deal with the first class of phenomena, we may take for granted those properties which are common to all ordinary forms of wave-motion, such as those in water or air. In ordinary water-waves the motion of a particle is partly to and fro in the direction in which a wave is travelling, partly up and down and therefore perpendicular to that direction. This is obvious to every one who watches a floating cork. In sound-waves, whether in air or in water, the displacement of each particle of the medium is wholly in the direction in which the wave is travelling. Directly connected with this there is another distinction between these classes of waves. In ordinary water-waves the water-elements change only their form as the wave passes; in sound-waves there is change of volume also. A third distinction, also directly connected with the first, is that sound-waves in water travel at a much greater rate than the swiftest, ⅛.e,, the longest, of surface waves. But, in either case, when two similar and equal series of waves arrive at a common point they interfere, as it is called, with one another, so that the actual disturbance of the medium at any instant is the resultant of the disturbances which it would have suffered at that instant from the two series separately. Thus if crests, and therefore troughs, arrive simultaneously from the two series, the result is a doubled amount of disturbance. If, on the contrary, a crest of the first series arrive along with a trough of the second, the next trough of the first series will arrive along with the next crest of the second, and so on. One series is then said to be half a wave-length behind the other. In this case, the portion of the medium considered will remain undisturbed. Thus, at the port of Batsha in Tong-king, the ocean tide-wave arrives by two different channels, one part being nearly six hours, or half a wave-length, behind the other. As a result, there is scarcely any noticeable tide at Batsha itself, though at places not very far from it the rise and fall are considerable. This was known to Newton, and is noticed by him in the Principia, iii. 24. See also Phil. Trans., vol. xiv. p. 677, for the observed facts and Halley’s comments. Thus also (see Acoustics) two sounds of the same wavelength and of equal intensity produce silence if they reach the external ear with an interval of half a wave-length, or any odd multiple of half a wave-length.

It is not remarkable that Young’s Bakerian Lecture (1801), in which the principle of interference is for the first time described and applied, should consist in great part of extracts from the Principia. For there are many passages in Newton’s works which might have been written by an upholder of the wave-theory. Unaccountably, however, Newton in the context almost always brings in a reference to the “rays of light” as something different from the vibrations of the ether, yet capable of being acted on by them so as to be put into “fits of easy reflexion or of easy transmission.” These allusions are the most obscure parts of all Newton’s scientific writings; and it is very difficult to form a precise conception of what he meant to express in them.

The following passage, extracted from Young’s temperate reply (Works, vol. i. p. 202) to the violent but ignorant assault on him by Lord Brougham in the Edinburgh Review, is chosen as showing his own estimate of his own work and of its relation to what was already known :—

“It was in May 1801 that I discovered, by reflecting on the beautiful experiments of Newton, a law which appears to me to account for a greater variety· of interesting phenomena than any other optical principle that has yet been made known. I shall endeavour to explain this law by a comparison.

“Suppose a number of equal waves of water to move upon the surface of a stagnant lake, with a certain constant velocity, and to enter a narrow channel leading out of the lake. Suppose then another similar cause to have excited another equal series of waves, which arrive at the same channel, -with the same velocity, and at the same time with the first Neither series of waves will destroy the other, but their effects will be combined: if they enter the channel in such a manner that the elevations of one series coincide with those of the other, they must together produce a series of greater joint elevations; but if the elevations of one series are so situated as to correspond to the depressions of the other, they must exactly fill up those depressions, and the surface of the water must remain smooth; at least I can discover no alternative, either from theory or from experiment.

“Now I maintain that similar effects take place whenever two portions of light are thus mixed; and this I call the general law of the interference of light. I have shown that this law agrees, most accurately, with the measures recorded in Newton’s Optics, relative to the colours of transparent substances, observed under circumstances which had never before been subjected to calculation, and with a great diversity of other experiments never before explained. This, I assert, is a most powerful argument in favour of the theory which I had before revived: there was nothing that could have led to it in any author with whom I am acquainted, except some imperfect hints in those inexhaustible but neglected mines of nascent inventions, the works of the great Dr Robert Hooke, which had never occurred to me at the time that I discovered the law; and except the Newtonian explanation of the combinations of tides in the port of Batsha.”

Young’s first application of the principle of interference was made to the colours of striated surfaces, the next to the colours of thin plates. These, however, are not so easily intelligible as the application to an experiment devised by Fresnel several years later. We therefore commence with Fresnel’s experiment, which gives the most simple arrangement yet contrived, but it must be understood that the explanation is really due to Young. BCD (fig. 31) is an isosceles prism of glass, with the angle at C very little less than two right angles. A luminous point is placed at O, in the plane through the obtuse edge of the prism and perpendicular to its base. If homogeneous light be used, the light which passes through the prism will consist of two parts, diverging as if from points O 1 and O 2 symmetrically situated on opposite sides of the line CO. Suppose a sheet of paper to be placed at A with its plane perpendicular to the line OCA, and let us consider what illumination will be produced at different parts of this paper. As O 1 and O 2 are images of O, crests of waves must be supposed to start from them simultaneously. Hence they will arrive simultaneously at A, which is equidistant from them, and there they will reinforce one another. Thus there will be a bright band on the paper parallel to the edges of the prism. If P 1 be chosen so that the difference between P 1 O 2 and P 1 O 1 is half a wavelength (i.e., half the distance between two successive crests), the two streams of light will constantly meet in such relative conditions as to destroy one another. Hence there will be a line of darkness on the paper, through P 1 , parallel to the edges of the prism. At P 2 , where O 2 P 2 exceeds O 1 P 2 by a whole wave-length, we have another bright band; and at P 3 , where O 2 P 3 exceeds O 1 P 3 by a wave-length and a half, another dark band; and so on. Hence, as everything is symmetrical about the bright band through A, the screen will be illuminated by a series of bright and dark bands, gradually shading into one another. If the paper screen be moved parallel to itself to or from the prism, the locus of all the successive positions of any one band will (by the nature of the curve) obviously be an hyperbola whose foci are O 1 and O 2 ∙ Thus the interval between any two bands will increase in a more rapid ratio than does the distance of the screen from the source of light. But the intensity of the bright bands diminishes rapidly as the screen moves farther off; so that, in order to measure their distance from A, it is better to substitute the eye (furnished with a convex lens) for the screen. If we thus measure the distance AP 1 between A and the nearest bright band, measure also AO, and calculate (from the known material and form of the prism, and the distance CO) the distance O 1 O 2 , it is obvious that we can deduce from them the lengths of O 1 P 2 and O 2 P 2 ∙ Their difference is the length of a wave of the homogeneous light experimented with. Though this is not the method actually employed for the purpose (as it admits of little precision), it has been thus fully explained here because it shows in a very simple way the possibility of measuring a wave-length.

The difference between O 1 P 1 and O 2 P 1 becomes greater as AP 1 is greater. Thus it is clear that the bands are more widely separated the longer the wave-length of the homogeneous light employed. Hence when we use white light, and thus have systems of bands of every visible wave-length superposed, the band A will be red at its edges, the next bright bands will be blue at their inner edges and red at their outer edges. But, after a few bands are passed, the bright bands due to one kind of light will gradually fill up the dark bands due to another; so that, while we may count hundreds of successive bright and dark bars when homogeneous light is used, with white light the bars become gradually less and less defined as they are farther from A, and finally merge into an almost uniform white illumination of the screen.

In this example, and in all others of a similar character which will be introduced into this elementary article, the solution is only approximate. The utmost resources of mathematics are in most cases required for the purpose of complete solution.

We are now in a position to prove that light moves slower in glass than in air, by the process which was merely indicated while we were discussing the velocity of light. For, if we could slightly lengthen the paths of the rays which come from O 1 , leaving those from O 2 unaltered, the system of bands would obviously be shifted in the direction from A to P in the figure. This happens if a very thin film of glass be interposed in the path of the rays which appear to come from O 1 . The best mode of making the experiment is to put a piece of very uniform plate glass, cut into two parts, between the prism and the screen, so that rays from O 1 pass through one part and those from O 2 through the other. So long as these pieces are parallel, no shifting takes place. But if one be slightly turned, so as to give the rays a longer path through it, the system of bands is at once displaced to the side at which it is situated.

Also, we can now see how it is possible to discover whether light has its velocity affected by that of the medium in which it is travelling. We know that sound travels faster with the wind, and slower against it, than it does in still air. We may, therefore, suppose a disposition of the interference apparatus such that the two rays which interfere have each passed through a long tube full of water. A rapid current may be established, in either direction, in one or other of the tubes, or in opposite directions in the two, and the shifting of the interferencebands will at once indicate the nature of the effect. We cannot describe the details of the process. The result, however, is analogous to that of wind on sound, but of course very much smaller; and it seems that the actual change of the velocity of light is less than the velocity of the current. See Ether.

Let us next consider the effect of a grating, a series of fine parallel wires placed at small equal intervals, or a piece of glass or of speculum metal on which a series of equidistant parallel lines have been ruled by a diamond point. We take only the case in which plane waves of homogeneous light are incident in a direction perpendicular to the plane of the grating, and when the bars and openings of the grating are all equal in breadth.

Consider the effect on an eye or screen at a considerable distance, in the direction BE (fig. 32). If there were no grating, practically no light would reach tlae eye from the aperture AD unless ABE were very nearly a right angle.

This is, of course, the statement of Huygens already quoted. But Young’s principle enables us to say why this is the case. Let us divide AD into a series of equal parts by lines perpendicular to BE, and distant from one another by half a wave-length of the homogeneous light employed. The portions coming to the eye from any two adjacent parts AB, BC will be practically of the same intensity, and will exactly neutralize one another’s effects on the eye. For if we take points a and b similarly situated with regard to A and B respectively, the distances of a and b from the eye differ by half a wave-length, and rays from a neutralize those from δ. This is true wherever a be taken between A and B. Hence, under the conditions assumed, no light reaches the eye.

Now suppose the alternate parts AB, CD, &c., to be opaque. Similar reasoning will show that the remaining rays conspire to strengthen one another. Thus, when homogeneous light from a distant point falls perpendicularly on a grating in which the breadth of the bars is equal to that of the interstices, it will be seen brightly in a direction inclined at an angle θ (ABE) to the plane of the grating,—the angle θ being such that

AC cos θ= wave-length.

Similar reasoning shows that the light is reinforced whenever θ is such that

AC cos θ is an integral multiple of the wave-length. The appearance presented when a long narrow slit is the luminous object, and the bars of the grating are placed parallel to it, is therefore (with homogeneous light) a central image with others equidistant from it on each side—their angular distances from it being the values of the angle corresponding to the sines

Δ , ?λ j 3λ i &c α a a

Here λ is the wave-length, and a is the sum of the breadths of a bar and an interstice. It is found in practice, and it is also deducible from the complete theory, that the ratio of the breadths of the bar and interstice has but little effect on the result, unless it be either very large or very small. Hence if λ be expressed as a fraction of an inch, and n be the number of lines per inch in the grating, the angular deviations of the bright bands have the sines ηK, 2πλ, 3nλ, &c.

The mean wave-length of visible rays in air is about g 0 ⅜ τ∏yth of an inch. Thus a grating with 5000 equidistant lines per inch will give with such light an angular deviation of about 6° for the first bright diffraction line.

If we notice that the sine of the deviation is proportional to the wave-length, it will be obvious that when white light is used the result will be a series of spectra on each side of the central white image, their more refrangible ends being turned towards that image. When the grating is a very regular one, and the appearances are examined by means of a telescope adjusted for parallel rays, the spectra formed in this way show the Fraunhofer lines with as great perfection as do the best prisms. And they have one special advantage, which prisms do not possess. The relative angular separation of the various colours depends solely on their wave-lengths, and thus the spectra formed by different gratings are practically similar to one another. There is, in fact, almost no irrationality in this kind of dispersion. In glass prisms and especially in those of flint glass, the more refrangible part of the spectrum is much dilated, while the less refrangible part is compressed.

The counting of the number of lines per inch in a grating is not difficult, nor is the accurate measurement of the angle of deviation of any particular Fraunhofer line. Hence, by the help of the very simple formula given above, the wave-lengths of light corresponding to the various Fraunhofer lines have been determined with very great accuracy from the diffraction spectra of gratings. The following are, according to Angström, a few of the chief values, λ is expressed in ten-millionths of a millimetre. [table]

For the sake of a discussion to be entered on later, we have appended the refractive index from air into water for each of these rays, as given by Fraunhofer himself.

If now we suppose AB, CD, &c., to be transparent, while BC, <fcc., become opaque, it is obvious that the new grating will be the complement of the old one, and will give precisely the same appearances at points outside the course of the direct beam. For when there is no grating there is practically no illumination at such points. This statement of course is equally true of any grating, whatever be the ratio of the breadths of the bars to those of the interstices.

Another very curious result of the theory of interference, fully verified by experiment, is furnished by the fact that the central spot of the shadow of a small circular disk, cast by rays diverging from a distant point in its axis, is as brightly illuminated as if the disk had not been interposed.

The final example of interference which we can give here is noteworthy on account of a peculiarity which it presents. Let us consider the case of homogeneous light reflected by a thin plate or film of a transparent material.

Let AB (fig. 33) be the direction of the incident ray, BdE the direction in which part of it is reflected to an eye E at a considerable distance: and let DE be the direction in which another part escapes after refraction into the plate at B and partial reflexion at the second surface of the plate at C. Then if D<7 be drawn perpendicular to BE, the re tardation of the wave in DE as compared with that in BE will be (2μ,BCB√)∕λ wave-lengths, where p is the refractive index into the plate.

If α' be the angle of refraction, and t the thickness of the plate, it is easily seen that

BC cos α' = t, and BD = 2BC sin α' = 2i tan α'.

Hence 2gBC - Bd= 2μt cos α'.

Hence whenever, for a given thickness of plate, α' is such that 2μt COS α'

is an integral multiple of λ, the two rays should reinforce one another at E. The same will happen for a given angle of incidence when the thickness of the plate is such that 2μt cos α'

is an integral multiple of λ. When, on either account 2 pt cos a is an odd integral multiple of λ∕2; the rays at E will weaken (perhaps destroy) one another.

Hence, in homogeneous light, a thin plate, turned about, alternately reflects and does not reflect to an eye in a given position. And a fixed plate of non-uniform thickness reflects light from some parts and not from others. When white light is used there will in general be colours seen which vary with the angle of incidence, and also with the thickness. If the plate is infinitely thin it would appear that there should be infinitely slight retardation only, and the plate should thus be bright in homogeneous light (and of course white in white light) at all incidences.

In general this is not the case. Thus when a soap bubble, or a vertical soap-film, is screened from currents of air, and allowed to drain, the uppermost (i.e., the thinnest) part becomes perfectly black. It can, in fact, be seen only by the feeble light scattered by little drops of oil or particles of soap or dust on its surface. Here, again, Young’s sagacity supplied the germ at least of the explanation. It is given in the following extract from his Theory of Light and Colours, the Bakerian Lecture for 1801 already referred to :—

“Proposition IV. — ΤVhen an undulation arrives at a Surface which is the Limit of Mediums of different Densities, a partial Reflexion takes place, proportionate in Force to the Difference of the Densities.

“This may be illustrated, if not demonstrated, by the analogy ol 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 reflexion 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 reflexion 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 backwards without it, and the undulation would be reversed, a rarefaction returning in place of a condensation; and this will perhaps be found most consistent with the phenomena.”

This idea, of a rarefaction returning by reflexion when a condensation is incident, is equivalent to a loss or gain of half a wave-length when light in a denser body is reflected at the surface of a rarer body. Whether, then, the plate be denser or rarer than the medium surrounding it, one or other of the two interfering rays loses half an undulation more than the other in the mere act of reflexion. This completely removes the difficulty. But Young went farther, and pointed out that if a thin plate be interposed between two media, one rarer, the other denser than the plate, this half wave-length effect should disappear. He verified this conjecture by direct experiment, founded on a modification of a process due to Newton.

Newton had, long before, devised and carefully employed an excessively ingenious (because extremely simple and effective) method of studying the colours of thin plates. It consisted merely in laying a lens of long focus on a flat plate of glass. The film of air or other fluid between the spherical surface and its tangent plane has a thickness which is directly proportional to the square of the distance from the point of contact When such an arrangement is looked at in homogeneous light, the lens having been pressed into contact with the flat plate, there is seen a central black spot, surrounded by successive bright and dark rings, whose number appears to be practically unlimited. The radii of the successive bright rings were found by Newton to be as the square roots of the odd numbers 1, 3, 5, &c. Hence the thicknesses of the film of air are directly as these numbers. When rays of higher refrangibility are used the rings diminish in diameter. Hence when white light is employed we have a superposition of coloured rings of all sizes, but it is no longer possible to trace more than four or five alternations of bright and dark rings—the colours being then more and more compound. This series of coloured rings is named after Newton, and the successive colours, gradually more and more composite, form Newton’s scale of colours. Thus we read, in books more than thirty years old, of a red or blue of the third order, meaning those colours as seen in the third bright ring round the central dark spot.

Many of the most vivid colours of natural and artificial bodies are due to one or other of the forms of interference we have roughly explained. Thus Barton’s buttons (once employed for ornament as they produce an effect very similar to that of diamonds) were simply polished metal plates stamped by a die of hardened steel, on whose surface a pattern had been engraved consisting of small areas ruled in different directions with close equidistant parallel grooves. That the colours of a pearl and of mother-of-pearl are due to a similar surface corrugation was proved by Brewster, who took impressions from such substances in black wax, and found that it was thus rendered capable of giving the same play of colours. The scales from the wings of butterflies owe their bright colours to a delicate ribbed structure. On the other hand, the thin transparent wings of the house-fly, earwig, <fcc., owe their colours to their thinness. The same is true of the temper colour of steel, Nobili’s rings, &c. Very beautiful examples of thin plates scaled off from decayed glass (found in Roman excavations) have been figured, with their play of colours, by Brewster. 1

Here we can only say a word or two about the probable relation between the wave-length of homogeneous light and its refractive index for any isotropic medium. The existence of dispersion was attributed by Cauchy to the fact that even the most homogeneous media, such as water, have grained or heterogeneous structure of dimensions not incomparably smaller than the average length of a wave of light. This grained structure has been recently proved to exist, by several perfectly independent processes arising from totally unconnected branches of physics; and its dimensions have been assigned, at least in a roughly approximate manner. See Atom, and Constitution of Bodies.

It appears from the theory of disturbances in such a medium that the velocity of a ray depends upon its wavelength in a manner which is expressed by a series of even inverse powers of that wave-length. Hence we have a relation such as

M-+f.+⅛÷

in which, from our present ignorance of the precise connexion between matter and ether, we must be content to find the multipliers of the various terms by direct measurement. If we neglect all but the first two terms, we may determine α and β from the known wave-lengths of two of Fraunhofer’s lines, and their refractive indices for a particular medium. We can then test the accuracy of the formula by its agreement with the corresponding numbers in the same medium for others of the fixed lines. Thus, taking the data for water given above, we have, from the numbers for the two hydrogen-lines C and F, the values

α = 1*3243,

0-0*00000000319.

Calculating from these, and the wave-length of H, we have for its refractive index D3447, instead of D3442 as determined by Fraunhofer. So far as we may trust this theory, which certainly accords fairly with the experimental data for substances of moderate dispersive powers, though by no means well with those for substances of high dispersive power such as oil of cassia, the value of the quantity α is the refractive index for the longest possible waves; i.e., it is that of the inferior limit of the spectrum.

Double Refraction. —We now come to phenomena which cannot be even roughly explained by processes based on the vague analogies of sound and water waves which have hitherto sufficed for our elementary treatment of the subject

These phenomena were first observed in Iceland spar. They were described in a general way by Bartholinus, who showed that one of the two rays into which a single incident ray is divided by this substance follows the ordinary law of refraction. Huygens, who studied the subject only eight years later, verified the greater part of the results of Bartholinus, and added many new ones. From his point of view it was of course obvious that the ordinary ray is propagated by spherical waves, i.e., its velocity is the same in all directions inside the crystal. To explain the extraordinary ray, he assumed that it was propagated in waves of the form of an ellipsoid of revolution, the simplest assumption he could make. To test its accuracy he first noticed that a rhombohedral crystal of Iceland spar behaves in precisely the same way whichever pair of parallel faces light passes through. Hence he acutely concluded that the axes of the ellipsoids of revolution (if such were the form of the waves for the extraordinary ray) must be symmetrically situated with regard to each of these planes. The only such lines in a rhombohedron are parallel to that which joins those corners which are formed by the meeting of three equal plane angles. In the case of Iceland spar these equal angles are obtuse. Huygens then verified, by experiments well contrived, though carried out by a very rough mode of measurement, the general agreement of his hypothesis with the fact; and he further tested it by comparing its indications as to the position of the two images for any position of the crystal with the results of direct observation. There can be no question that the whole investigation was, for the age in which it was made, of an exceedingly high order. But it must not be left unsaid that far more accurate measurements than those of Huygens were necessary before it could be asserted that the form of the extraordinary wave is an ellipsoid of revolution, and not merely a surface closely resembling such an ellipsoid. These improved measurements were made 1802 by Wollaston, and they have recently been repeated with far more perfect optical means by Stokes, Mascart, and Glazebrook. The result has been the complete verification of Huygens’s conjecture. The generating ellipse of the extraordinary waves is found to have its minor axis, which is that of revolution, equal to the diameter of the corresponding sphere for the ordinary ray. Its major axis is to the minor nearly in the ratio 1-654:1∙483.

We are now in a position to trace the paths of the two rays into which a ray falling in any direction on a surface of the crystal is divided by refraction.

Let fig. 34 represent a plane wave front AB (in air) falling on the surface AC of a piece of Iceland spar cut in any way. The figure is a section perpendicular to the surface, and parallel to the incident ray. The wave-front AB cuts the surface of the spar in a line (not shown) at right angles to the plane of the paper. Draw from A the axis Aα (not necessarily in the plane of the paper) and the sphere and ellipsoid of revolution which have Aα for a common axis. Then, if C be taken such that BC is to Aα as the velocity of light in air is to that of the ordinary ray in the crystal, the wave-front of the ordinary ray is found by drawing a tangent plane to the sphere, passing through C and perpendicular to the plane of the paper. This touches the sphere in a point o (in the plane of the paper) and AoO is the ordinary ray. To find the direction of the extraordinary ray, a plane perpendicular to the paper, and passing through C, must be drawn so as to touch the ellipsoid. Let e be the point of contact, which will in general not be in the plane of the paper unless Aα is in or perpendicular to that plane; then AeE is the extraordinary ray.

Thus, in general, the extraordinary ray is not in the plane of incidence. Also the ratio of the sines of the angles of incidence and refraction is generally different for different directions of incidence, in the case of the extraordinary ray.

In an elementary article we cannot attempt more fully to study these phenomena; so we merely state that all the observed appearances, so far as the directions of the refracted rays are concerned, are explained by supposing the wave-surface in the crystal to be made up of the sphere and the ellipsoid of revolution above described. Thus, when both eyes are used, the two images of a plane object seen through a crystal of Iceland spar appear in general to be situated at different distances above the plane. One of them maintains its apparent position as the crystal is made to rotate about a perpendicular to the two faces employed; the other’s position varies as the crystal is turned.

But we have now to inquire why the incident ray is divided into two, and why one of them follows the ordinary law of refraction. Here another experimental result of Huygens comes to our assistance. We paraphrase the author’s description :—

“I will, before concluding, mention another remarkable phenomenon which I discovered after the above was written. For, although I have not yet been able to find the cause of it, I do not wish on that account to refrain from pointing it out, in order that others may have an opportunity of seeking to explain it. It appears that it will be necessary to make hypotheses additional to those already given,—though these will lose none of their probability, confirmed as they have been by so many tests. The phenomenon is that, taking two fragments of the crystal (Iceland spar) and laying them on one another, or even holding them apart, if all the faces of the one be parallel to those of the other, a ray of light divided into two by the first fragment will not be farther subdivided by the second. The ordinary ray from the first will be refracted ordinarily by the second, the extraordinary ray extraordinarily. And the same thing happens not only in this arrangement but in all others in which the principal sections of the two fragments are in the same plane, -whether the surfaces turned towards one another be parallel or not. It is, in fact, marvellous that these rays, falling on the second fragment, do not divide like the ray incident on the first. One would say that the ordinary ray from the first fragment had lost what is necessary for the production of extraordinary refraction, and the extraordinary ray that which is necessary for ordinary refraction; but there is something else which upsets this view. For when one places the fragments so that their principal sections are at right angles, whether the opposed surfaces be parallel or not, the ordinary ray from the first suffers only extraordinary refraction by the second, and vice versa.

“But in all the infinite number of positions other than those named, both rays from the first fragment are divided into two by the second. Thus the single incident ray is divided into four, sometimes equally sometimes unequally bright, according to the varying relative position of the crystals. But all together do not seem to have more light than has the single incident ray.

“When we consider that, the two rays given by the first crystal remaining the same, it depends upon the position of the second crystal whether they shall be divided into two or not, while the incident ray is always divided, it appears that we must conclude that the waves of light which have traversed the first crystal have acquired a form or disposition which in some positions enables them to excite the two kinds of matter which give rise to the two kinds of refraction, in other positions to excite only one of them. But I have not yet been able to find any satisfactory explanation of this.”

So far Huygens. His statements are perfectly in accordance with fact; and they were reproduced by Newton in very nearly the same form. Newton adds :—“The unusual refraction is, therefore, performed by an original property of the rays. And it remains to be enquired, whether the rays have not more original properties than are yet discovered. Have not the rays of light several sides, endued with several original properties ?”

It is very curious to notice how near each of these great men came to the true explanation, and yet how long time elapsed before that explanation was found. The date of Huygens’s work is 1690, that of Newton’s 1704. It was not till 1810 that farther information on the subject was obtained. Then one brilliant observation opened the way for a host of discoveries in a new and immense field of optics.

In the last-mentioned year Malus, while engaged on the theory of double refraction, casually examined through a doubly refracting prism of quartz the sunlight reflected from the windows of the Luxembourg palace. He was surprised to find that the two rays alternately disappeared as the prism was rotated through successive right angles, —in other words, that the reflected light had acquired properties exactly corresponding to those of the rays transmitted through Iceland spar. Even Malus was so imbued with the corpuscular theory of light that he named this phenomenon polarization, holding it as inexplicable on the wave theory, and as requiring a species of polarity (akin to the magnetic) in the light-corpuscles—a close reproduction of one of Newton’s guesses.

But after a short time Hooke’s old guess was independently reproduced, and in the hands of Young and others, but most especially of Fresnel, the consequences of the assumption, that the vibrations of the luminiferous medium take place perpendicularly to the direction of the ray, were the almost complete explanation of the cause of double refraction, and the discovery (often the prediction) of a long series of the most gorgeous phenomena known to science.

The real difficulty in the way of this conception probably lay in the fact that most of the familiar forms of wavemotion—such as sound-waves in air or in water, and ordinary water waves—are not of this character. In sound-waves the vibrations are wholly in the direction of the ray, while in surface-waves in water they are partly parallel to and partly perpendicular to the direction in which the wave is travelling. That a body may transmit waves in which the vibration is perpendicular to the direction of a ray, it must have the properties of an elastic solid rather than of a fluid of any kind. And our experience of the almost entire absence of resistance to the planetary motion seems, at first sight at least, altogether incompatible with the idea that the planets move in a jelly-like solid, filling all space through which light can be propagated.

Without going into difficult dynamical details, we may obtain a notion of the nature of the motion now to be considered, by observing the propagation of a wave when a long stretched wire or string is struck or plucked near one end. Here the line of motion of each part of the wire is almost exactly perpendicular to the direction of the wire, i.e. to the line along which the wave travels. (When the string is extensible there may be another wave, due to extension; but this, which is analogous to sound, has its vibrations along the string, and it usually travels at a very different rate from the other, so that the two are not in any way associated).

Now it is clear that waves of this wholly transverse character can have, in Newton’s language, sides. And it is also clear that they cannot interfere so as mutually to destroy one another unless their corresponding sides are parallel to one another; nor can they interfere at all if their sides are perpendicular to one another. Hence a very severe test of the theory will be furnished by examining various cases of interference of polarized light, which ought to present in general marked differences from those of ordinary light. It was by experiments of this kind that Fresnel and Arago first firmly established the bases of the theory of polarization. The important fact discovered by Malus was soon generalized into the following statement :—

Light reflected from the surface of substances so different as water, glass, polished wood, &c., at a certain definite angle, which depends on the nature of the substance, is found to possess all the properties of one of the rays transmitted through Iceland spar. If the plane of reflexion is parallel to the axis of the spar, the properties of the reflected light are those of the ordinary ray; if perpendicular to it, those of the extraordinary ray.

It was reserved for Brewster to discover, as the resul of an extraordinary series of experimental measurements, the very simple law which follows :—

The tangent of the polarizing angle is equal to the refractive index of the reflecting substance.

This may be put in another form, in which its connexion with theory is a little more evident :—

When the reflected ray is completely polarized, it is perpendicular to the refracted ray.

Bearing in mind Huygens’s observations on light which has passed through two crystals of Iceland spar, we can now see that a ray of light polarized by reflexion is in general divided into two by a crystal of Iceland spar. But there is only one ray when the principal plane of the crystal is parallel to the plane of reflexion, and none when these planes are perpendicular to one another.

We may now much simplify matters by suppressing the Iceland spar, and using two reflecting plates of glass, so placed that a ray meets each of them in succession at the polarizing angle. It is then found that when the planes of reflexion are parallel the ray is reflected (almost without loss) from the second plate, but when they are perpendicular to one another there is complete extinction. In intermediate positions the intensity was found by Arago to be as the square of the cosine of the inclination of these planes.

This very simple experiment, which any one may easily make for himself, by putting two pieces of glass at the proper angle in the ends of two wooden tubes which fit into one another, enables us to form a general notion of the modification which is called polarization. The “sides” of the reflected ray are obviously in, and perpendicular to, the plane of incidence; for a ray can be reflected over and over again if the successive planes of incidence are parallel, but is stopped at once if one of them be perpendicular to the others.

Here, however, two new difficulties come in at once :— (1) Are the vibrations of the reflected ray in, or perpendicular to, the plane of reflexion? (2) As ordinary sun or lamp light, reflected at the proper angle from a polarizing surface, shows no variation of intensity when the azimuth of the plane of reflexion is changed, what can be then the direction of its vibrations? These questions have not yet been answered in a thoroughly satisfactory manner Many important phenomena are explained in terms quite independent of the proper answer to (1); and, in others which do depend on the answer, the theoretical differences between the results of the two hypotheses are so small as to have hitherto remained undetected. In an important test, suggested by Stokes, the experimental results have been at variance in a way not yet explained. It is quite possible that, as is required by Clerk Maxwell’s electromagnetic theory of light (see Ether), there may be simultaneous displacements, but of different characters, in each of these planes, and then the question would be reduced to—Which of these displacements is the luminous one? But on this theory, both are probably essential to vision.

As to the second question, it may be said— first, that, so far as the test of double refraction can inform us, a polarized ray whose plane of polarization is made to rotate rapidly produces precisely the same effects as a ray of ordinary light; and, secondly, that, so great is the number of vibrations even of red light in one second, it would be impossible to make the plane of polarization rotate fast enough to affect the circumstances of any of the phenomena of interference, even when they take place between two portions of the same ray, one of which is retarded thousands of wave-lengths more than the other. But, thirdly, the fact that, when homogeneous light is used, Newton’s rings have been counted up to the 7000th shows that, whatever be the actual nature of the vibrations of unpolarized light, they must for at least 7000 waves in succession be almost precisely similar to one another. Then, for other 7000 waves or so, we may have a totally different type of vibration. But, fourthly, in the course of γth of a second, at the very utmost, the vibrations must have been almost uniformly distributed over all directions perpendicular to the ray. Again, however, fifthly, another quite different view may be suggested. All common light has its origin from a practically infinite number of sources, consisting of the vibrating particles of the luminous body. The contributions from each of these sources (so far as one definite wave-length is concerned) may be and probably are at any one point as different in direction of vibration as they certainly must be in phase. From this point of view, which we cannot develop here, the uniformity of optical phenomena becomes quite analogous to the statistical species of uniformity which is now found to account for the behaviour of the practically infinite group of particles forming a cubic inch of gas. The reader need only think of the fact that, so numerous are those particles, it is practically (though not theoretically) impossible that even a cubic millimetre of air should, even for τπ ⅞ π θth of a second, contain oxygen particles alone.

When light is reflected at an incidence either less or greater than the polarizing angle, it behaves as if part of it only were polarized and the rest ordinary light; and it is said to be partially polarized. Tested by a crystal of Iceland spar, it gives two images in all positions of the crystal; but their brightness is unequal except in the special positions where they would be of equal brightness were the ray wholly polarized.

From the fourth of the remarks made above regarding common light, and the facts of double refraction, it follows at once that, when light is to any extent polarized by reflexion, there must be an exactly equal amount of polarized light in the refracted ray, and its plane of polarization must be perpendicular to that of refraction. This was established by experiment soon after Malus’s discovery. But as the reflected ray from glass, water, &c., is in general much weaker than the refracted ray, the percentage of polarized light is generally much greater in the former. It was found, however, by experiment that refraction at a second glass plate parallel to the first increases the proportion of polarized to common light in the transmitted ray, and thus that light may be almost completely polarized by transmission, at the proper angle, through a number of parallel plates. The experimental data of this subject were very carefully obtained by Brewster. He has found, for instance, how the angle of incidence for the most complete polarization varies with the number of plates. The plane of polarization of such a bundle is perpendicular to the plane of refraction.

This, however useful on many occasions, is at best a rough arrangement for producing polarized light. By far the most perfect polarizer for a broad beam of light is a crystal of Iceland spar, sufficiently thick to allow of the complete separation of the two rays. But such specimens are rare and costly, so that the polarizer in practical use is now what is called Nicol's prism, invented in 1828 {Jameson's Journal, p. 83). By cutting a rhomb of Iceland spar in two, and cementing the pieces together with Canada balsam (after carefully polishing the cut faces),

Nicol produced an arrangement in which one only of the two rays is transmitted, the other being totally reflected at the surface of the balsam. The reason is simply that the refractive index of Canada balsam is intermediate to those of the ordinary and extraordinary rays in the spar. The ordinary ray, falling very obliquely on a medium of a smaller refractive index, is totally reflected; the extraordinary ray, falling on a medium of greater, but very little greater, refractive power, is almost wholly transmitted. The only defect of the Nicol’s prism is that, to secure the total reflexion, its length must be considerably greater than its breadth; and thus it necessarily limits the divergency of the beam it allows to pass.

Certain doubly refracting crystals exert considerable absorption on one of the two rays they produce, and can therefore, when in plates of sufficient thickness, be employed as polarizers. This is the case with some specimens of tourmaline when cut into plates parallel to the axis of the crystal. It is also found in the flat crystals of several artificial salts, such as, for instance, iodo-sulphate of quinine.

Let us now suppose that by one or other of these pieces of apparatus, say a Nicol’s prism, light has been polarized. If we examine this ray by means of a second Nicol, placed in a similar position to the first, it passes practically unaltered. As the second Nicol is made to rotate, more and more of the light is stopped, till the rotation amounts to a right angle. Two well-constructed Nicols, placed in this position, are practically opaque to the strongest sunlight. During the next quadrant of rotation the transmitted ray gradually increases in brightness, until at 180° of rotation it passes practically unaltered. Precisely the same phenomena occur in the same order during the next half of a complete rotation. The reader will observe that this is merely Huygens’s original statement, limited to one of the four rays which are produced by passing common light successively through two crystals of Iceland spar.

Whatever be the true mechanism of polarized light, there can be no doubt that its vibrations are symmetrical with respect to the ray, and also with respect to the plane of polarization. Hence we may, for many important purposes, symbolize them by simple harmonic vibrations taking place either id or perpendicular to the plane of polarization. But, if they be supposed to take place simultaneously in these two planes, their quality or nature must be essentially different in the two, else the symmetry above referred to would be violated. Hence it will be sufficient for the present to assume that they take place perpendicular to the plane of polarization. The nature of the resulting effects, so far as the eye is concerned, will not be different for the different hypotheses. Also, as no instance has yet been observed, even with the most intense beams of light, in which the joint effects observed are not those due to simple superposition, we may assume that the elastic force of the luminiferous medium, called into play by a displacement, is directly proportional to the displacement, and therefore that the vibrations for each wave-length follow the simple harmonic law, that of the cycloidal pendulum.

The subject of the composition of simple harmonic motions of equal period falls to be discussed as an important branch of kinematics (see Mechanics). We will therefore here assume the following results,—referring to the above-quoted article for their proof :—

1. Two simple harmonic motions of the same period, in lines perpendicular to one another, give, in general, elliptic motion, which may be in the positive or negative direction of rotation.

2. The ellipse becomes a straight line, and the resultant motion therefore simple harmonic, when the phases of the components are the same, or differ by an integral multiple of τr.

3. It becomes a circle when the amplitudes of the components are equal, and their phases differ by an odd multiple of ⅜ τr. The motion takes place in one direction (say right-handedly) in the circle when this multiplier is 1, 5, 9, 13, <fcc., and in the opposite (left-handed) when it is 3, 7, 11, 15, &c. . .

Now, suppose a plane polarized ray to fall on a plate of a doubly-refracting crystal (a thin plate of mica or selenite, for instance). Within the plate it will in general be divided into two, which are polarized in planes at right angles to one another. The directions of vibration in these rays are determined by the physical properties of the material. Let them be represented by the lines Oz, Oy in fig. 35. Then, if OA represents the semiamplitude of vibration in the incident ray, it may be looked on by (2) above as the resultant of two simple harmonic motions of the same period, whose semiamplitudes are OM and ON, and which are in the same phase. Each of these will pass through the plate of crystal unchanged. But one will, in general, travel faster than the other; for the essential cause of double refraction is the difference of velocities of the two rays. The portions of the two rays which simultaneously escape from the crystal, and which travel together outside it, will therefore differ in phase. Hence, to find the nature of the transmitted light, we must recombine the vibrations in OM, ON, taking account of this difference of phase. By (1) above the result will be in general elliptic motion. The ellipse will necessarily be one of the infinite number which can be inscribed in the rectangle AA'BB', whose construction is obvious. We have then, in general, what is called elliptically polarized light. This degenerates (by (2) above) into plane polarized 1 light, whose vibrations are along OA or OA' according as the difference of phase is 0, 2π∙, 4τr, <fcc., or τr, 3τr, 5τr, <fcc. And it will become circularly polarized light if OM = ON (i.e., if AO∙r = ∣τr) and the difference of phase be an odd multiple of ⅜τr. By (3) above this will be right or left handed, according to the value of the odd multiplier.

This conclusion from the assumption above made is fully borne out by experiment. When a plate of mica, of such a thickness as to retard one of the two rays a quarter of a wave-length more than the other, is interposed between two Nicols, we observe the following phenomena :—

If the Nicols were originally placed so as to extinguish the light, the introduction of the mica plate in general partially restores it. Now, let the mica plate be made to rotate in its own plane. The light vanishes for successive positions, differing by a quadrant of rotation, ie. whenever the directions of vibration in the crystal coincide with the principal planes of the Nicols. In each of these positions the light from tlm first Nicol passes unchanged through the mica, and is therefore entirely stopped by the second Nicol. Half-way between these positions the light transmitted through the system is at its brightest; and in these cases it is not altered in brightness by rotating the second Nicol. It is then circularly polarized, and in whatever direction the second Nicol is placed the component of the circular motion which is ready to pass through it is of the same amplitude. Here, then, is a case in which a Nicol (the second) cannot enable us to distinguish between common light and light very seriously modified.

In what precedes, we have assumed that homogeneous light was used. In general, a doubly-refracting plate produces a difference of phase in its two rays which will depend on their wave-length; and thus when white light is used we have a display of colour, sometimes extremely gorgeous, and we may distinguish light thus circularly polarized from common light by slight changes of colour and intensity as the second Nicol is turned.

Hitherto we have spoken of the polarizing angle for light reflected in air from bodies such as glass, water, &c., which have a higher refractive index than air, and we have seen that an equal amount of light is polarized in the refracted beam. But what if there be no refracted beam? This is the case of total reflexion inside the denser body. Fresnel discovered that in this case the two kinds of polarized light (in planes at right angles to one another) co-exist in the totally reflected ray, but that they differ in phase, and therefore in general recombine into elliptically polarized light. Guided by peculiar theoretical considerations, he was led to construct a piece of glass (Fresnel's rhomb), inside which light is twice totally reflected at a certain angle with the result that, if it be originally polarized in a plane inclined at 45° to the plane of reflexion, the emergent light is circularly polarized.

Reflexion from the surface of metals, and of very highly refractive substances such as diamond, generally gives at all incidences elliptically polarized light. Attempts have been made to determine from such effects the refractive indices of metals and other opaque substances. These are all based upon theory, and cannot as yet command much confidence. With certain doubly-refracting substances the light reflected at a definite angle is differently polarized, and sometimes even differently coloured, for different azimuths of the plane of incidence.

When a thin plate of doubly-refracting crystal, which gives a bright colour when placed between two Nicols, is slightly inclined to the ray, the colour changes as the difference of phase of the two refracted rays is increased. If, now, we take a plate of Iceland spar cut perpendicularly to the axis, no colour will be produced by parallel rays passing through it perpendicularly, because both rays have a common velocity parallel to the axis; but, if divergent light be used, there is a gorgeous display of circular coloured rings surrounding the axis, which depends upon the increasing retardation of the ordinary ray behind the extraordinary as their inclination to the axis increases. When the principal planes of the Nicols are at right angles, this system of rings is intersected by two black diameters, in these planes respectively. When the second Nicol is turned through a right angle, we have exactly the complement of the former appearance, i.e., a figure such that, if superposed on the former, it would give an uniform field of white light.

It is to be noticed that none of these phenomena can be observed without the use of the second Nicol. This arises from the fact that, where the vibrations in any direction interfere so as to destroy one another, those in the direction perpendicular to the former interfere so as to strengthen one another. The second Nicol enables us to select one of these portions, and examine it independently of the other.

The only double refraction we have considered particularly is that of Iceland spar, where everything is symmetrical about the axis of the crystal. Such crystals, and they include as a rule all those of the second and third systems in Crystallography (q.v.), are called uniaxαl. Crystals of the first system are not doubly refractive. But it was one of the most valuable of Brewster’s discoveries that the great majority of non-isotropic substances are doubly refracting, and in general are biaxal, i.e., have two equally important axes inclined to each other at angles of all values from 0° to 90°. The form of the wave-surface in such bodies was, at least very approximately, assigned by Fresnel. This forms one of the most brilliant of his many grand discoveries; and it led to Hamilton’s prediction of the existence of the two species of conical refraction, which was experimentally verified by Lloyd.

Fresnel also made the striking discovery that glass and other simply refracting bodies are rendered doubly refracting when in a state of strain. To this Brewster added the observation that the requisite strain might be produced by unequal heating instead of by mechanical stress, and also that unannealed glass is usually doubly refractive. Clerk Maxwell in 1873 (Proc. Roy. Soc.) showed that shearing stress in viscous liquids, such as Canada balsam, renders them temporarily doubly-refractive. This subject has been elaborately investigated by Kundt {Pogg. Ann., 1879).

The details of these subjects, with those of the polarization of light reflected from small particles, the rotatory polarization produced by quartz, sugar, transparent bodies uιrder the influence of magnetism, <fcc., must be deferred to Optics (Physical).

There is, however, one elementary point which must not be omitted here, as it is intimately connected with the wave-theory,—that is, the alteration which light undergoes in consequence of the relative motion of the source and spectator in the line of vision.

When a steamer is moving in a direction perpendicular to the crests of the waves, she will encounter more of them in a given time if her course is towards them than if she were at rest, while, if she be moving in the same direction as the waves, fewer of them will overtake her in a given time than if she were at rest. The same thing is true of soundwaves. When an express train passes a level crossing at full speed, the pitch of the steam whistle is higher during the approach to and lower during the recess from the listener at the gate than it would be if the engine were at rest. The successive sound-pulses are emitted at the same intervals as before, but from points successively nearer to or farther from the listener. Hence more or fewer reach his ear in a given time. The principle is precisely the same as that of Romer’s observation of the frequency of eclipse of Jupiter’s satellites, which we have already given: the number of light-waves which reach the eye per second is increased if the -source is approaching, and diminished if it be receding. We are now dealing with a phenomenon which occurs some 600,000,000,000,000 times per second instead of once every forty-two hours. Now, increased wave-frequency, with unaltered velocity of light, certainly implies shorter wave-length and most probably greater refrangibility, and vice versa. There is, undoubtedly, a weak link in this reasoning, due to our ignorance of the true nature of the luminiferous medium and of the species of vibration on which light depends. If we knew something definite about the nature of the ether, and the mechanism of its vibrations, this weakness would be at least in part got rid of. Observation has not yet settled the question of the relative motion of bodies, the ether they contain, and the ether in free space.

This principle has been applied with success by Huggins and others to find the rate at which we are approaching to or receding from different fixed stars, and the rate of motion in solar cyclones; and it may even be applied, as was ingeniously suggested by Fox Talbot lβ. A. Report, 1871), to determine (from the relative velocities of the components of a double star in the line of sight, measured by its aid) the distance of the star itself from our system.

The reader is advised to consult, in connexion with the whole of the second part of this article, the previous article Ethek. (p. g. t.)