An Elementary Treatise on Optics/Chapter 16

by Henry Coddington

CHAP. X.

OPTICAL PHÆNOMENA.

177. The simplest of these appearances are caused by the refraction which the solar light undergoes in passing through the atmosphere.

When a ray of light enters successively several media of different densities it suffers a refraction greater or less at each surface, as represented in Figs. 195 and 196.

It is evident that the greater the number of media and the less their individual thickness, the more continuous is the course of the light, and that if the number be infinite and thickness of each evanescent, this course must be a curve.

Now this is just the case with the atmosphere; for the air being an elastic fluid, is compressed and condensed by its own weight, so that its density increases continually from its extreme bounds to the surface of the earth; and, in consequence of this, all distant objects, whether celestial or terrestrial, except those immediately over-head, are seen by curved rays, and consequently referred by the spectator to places which they do not really occupy.

This is represented on an exaggerated scale in Fig. 197: the actual error or refraction in the apparent height of an object on the Earth varies according to General Roy, from 1/3 to 1/24 of the altitude; Bouguer makes it 1/9, Maskelyne 1/10, Lambert 1/14.

The refractive power of the air varies of course with its density, and that with its temperature, for heat necessarily rarifies it. It is also subject to great irregularities from the aqueous vapours suspended in it.

The Sun has been seen in Nova Zemla, when it was 4 degrees below the horizon, its rays suffering a very extraordinary refraction in their oblique passage through the frozen vapours hanging over the icy sea.

The cliffs of the French coast have been seen from Hastings at the distant of 50 miles, though they are commonly concealed by the convexity of the Globe.^[1]

178. In some instances the refraction of the air is as it were inverted by the great rarification of the lower strata of the atmosphere by heat. When the surface of the sea is very warm, the horizon sometimes appears lower than it naturally should, to the amount of 4 or 5 minutes of a degree.

It will readily occur to the reader, that when the lower part of the atmosphere is very much rarified, so that its density diminishes rapidly to the ground, it is possible for a ray of light tending downwards, to be reflected, as it is in glass, or water, when the angle of incidence exceeds a certain value.

It is also possible, that two rays proceeding from the same point, above the heated air, one horizontally, the other slanting downwards, may meet as in Fig. 202, and thus give to an observer the appearance of two distinct points.

The following account is taken with very little alteration, from that which M. Biot has given in his Astronomie, his Physique, and his Précis Elémentaire.

When a dry and sandy soil is strongly heated by the Sun, the air in contact with it is rarified, so that the density of the atmosphere increases from the surface of the Earth to a certain height, which is generally very small, then continues sensibly the same for a certain space, and decreases slowly, but continually, throughout its remaining extent. Now if the eye of an observer be placed in the stratum of maximum density, and be directed towards an object also situated in that stratum, he will see it in two ways, directly through the uniformly dense air, and indirectly by rays reflected upwards through the lower strata. There will thus be two images seen, one erect by the unrefracted rays, the other inverted by the reflected ones. If the object be detached against the sky, this will also be represented round the inverted image, precisely as if the reflexion took place at the surface of water. See Fig. 203.

Such is the cause of a very curious phænomenon, which the French army observed very frequently in Egypt. The whole of lower Egypt is a vast horizontal plain, broken only by a few eminences, on which the villages are built, to be out of the reach of the inundations of the Nile. In the morning and evening the appearance of the country is such as the real disposition of the objects naturally presents, but in the middle of the day when the ground is heated by an unclouded Sun, the prospect seems bounded on all sides by a general flood; the villages appear like islands in the midst of an immense lake, and under each is seen its image inverted just as it would be given by water. As you approach one of them, the bounds of this apparent inundation retire, the lake which surrounded the village seems to sink away from it, and at length goes to play round some more distant basis. "Thus," says Monge, from whom this description was originally borrowed, "all circumstances concur to complete an illusion, which is sometimes cruel, especially in the desert, where it vainly presents the semblance of water, just at the time when it is most wanted." M. Biot says, that he has observed many phænomena of this kind on the sands at Dunkirk, and he has given the mathematical theory of them in the Memoirs of the Institute for 1809. He has shewn that the successive trajectories which go from the eye of the observer, intersect on their second branches, so as to form a caustic, underneath which, no object is visible. In Fig. 204, LT represents this caustic, and DMS is the limiting trajectory, touching the surface of the Earth.

All objects situated above MS, send but one image to the eye; those in the space SLT send two, one erect, the other inverted; those under the caustic MLT cannot transmit any rays at all to D, and are therefore invisible. In this manner a moving object, a man walking from one, for instance, presents successively the appearances represented in Fig. 205.

The French sailors give the name of Mirage to this phænomenon; the Italian peasants attribute it to the Fata Morgana. According to Dr. Young, it is known in this country by the name of Looming.

Sometimes a distant object appears suspended in the air, without any inverted image. In this case the image does really exist, but it is so extremely thin that it becomes imperceptible.

179. Effects such as we have been describing, may be produced artificially, by introducing some sulphuric acid through a funnel underneath pure water in a glass vessel. The acid being the heavier of the two liquids, raises the water above itself, and remains in part undiluted, but as it has a very strong affinity for water, the fluids combine where they are in contact, and thus there is formed between the pure acid and the pure water, a mixture passing insensibly from the one to the other, and decreasing continuously in refractive power. If now an object be looked at through the acid, it will be seen directly in its natural place, but there will also be an inverted image above it, produced by rays emitted upwards into the mixture, and refracted downwards again to the eye, (see Fig. 206.)

An appearance more exactly like the mirage, may be observed by looking at an object along a heated surface, such as that of a stove, or a hot poker. The air in its vicinity becomes very much rarified, and if the eye be moved about a little, it will be observed, that when a distant object is nearly in a line with the edge of the heated substance, it is seen double.

The Rainbow.

180. This phænomenon is caused by the drops of rain refracting and reflecting the rays of the Sun.

The usual appearance is one bow, consisting of concentric stripes, coloured like the prismatic spectrum, the violet being on the inside: above that is often seen another which is of course wider, but otherwise differs from the former only in having the colours in the inverse order, and rather less distinct. Sometimes a third, and even a fourth bow may be seen, but they are always extremely faint. The manner of this formation is shewn in Fig. 207, where A, B, C, D, &c. represent spheres of water, into which the solar rays entering, are, under certain circumstances refracted and reflected so as to emerge parallel, or nearly so, and thus produce vision in an eye placed at O, with the sensation of some primitive colour, the different homogeneous rays being separated by the refractions.

In the first, or lowest arch there is but one reflexion, in the next two, and so on. The quantity of light lost at each of these reflexions, accounts for the want of distinctness in the upper bows.

181. In the first place, it will be observed that the spectator must turn his back to the Sun, to see a rainbow.

Secondly, that all drops of water similarly situated with respect to the Sun, and to the eye, must produce the same effect.

This similarity of situation, evidently depends on the circumstance of lines drawn from the drop to the Sun, and to the eye, making equal angles.

This includes all the drops that are found on a conical surface of revolution, having for its axis a line OP parallel to the solar rays; since all lines drawn from the vertex along such a surface make equal angles with the axis, or with any line parallel to it.

In order to find in each case the value of the semi-angle of the cone, or the radius of the bow, we have only to determine its equal, the angle which the incident rays make with those emergent rays, which are parallel.

Accordingly, we will find the value of this angle in general, and find what it becomes in the particular case in question.

Let SA, Fig. 208, represent a single ray (belonging to the first bow), which is refracted and reflected to B, C, O.

The incident and emergent rays SA, CO being produced to meet in T, and EA, EC, ET being joined, the latter of which of course passes through B,

Let θ	=	the half angle ATE,
φM	=	the angle of incidence SAM, or EAT,
φ′	=	the angle of refraction EAB,
m	=	the ratio of refraction, sinφ/sinφ′

Then the angles EAB, EBA, EBC, ECB being all equal, we have

ATE=ABE−BAT=ABE−(EAT−EAB),
that is, θ=φ′−(φ−φ′)=2φ′−φ.

Now when two successive emergent rays are parallel, θ remains unchanged, while φ becomes φ+dφ, or in other words, θ is at a maximum or minimum, and dθ/dφ=0;

∴2dφ′/dφ−1=0, or dφ′/dφ=1/2.

But sinφ=m·sinφ′;

∴dφ′/dφ=cosφ/m·cosφ′,
and cosφ/m·cosφ′=1/2, or m·cosφ′=2cosφ;
∴ m²cosφ′²=4cosφ²,
but m²sinφ′²=sinφ²;

---

∴ adding, m2	=	4cosφ2+sinφ2
	=	3cosφ2+1;

∴ cosφ=√m²−1/3.

In order to find θ from this and the equation θ=2φ′−φ, we must put for m the value that it has for any desired sort of homogeneous light refracted between air and water, and by the help of a table of natural sines and cosines, we shall obtain the angles φ′ and θ, and consequently 2θ, which will be the radius of the arc of that particular colour in the primary bow.

The investigation is very similar for the secondary bow, or indeed for any other.

To make it as general as possible, let $p$ represent the number of reflexions within the drop; (Fig. 209.)

θ	=	ATE the half radius, as before,
ATE	=	π−EAT−TEA.

Now EAT=π−TAM=π−φ,

and TEA	=	1/2FEA=1/2{2π−(p+1)AEB}
	=	1/2{2π−(p+1)(π−2φ′)}
	=	(p+1)φ′−(p−1)π/2;

∴ θ	=	π−(π−φ)−(p+1)φ′+(p−1)π/2
	=	φ−(p+1)φ′+(p−1)π/2.
Hence, dθ/dφ	=	1−(p+1)dφ′/dφ; and ∴ dφ′/dφ, or cosφ/m·cosφ′=1/p+1.

We have then mcosφ′	=	(p+1)cosφ;
∴ m2cosφ′2	=	(p+1)2cosφ2,
and m2sinφ′2	=	sinφ2;

---

∴ m2	=	(p+1)2cosφ2+sinφ2
	=	(p2+2p)cosφ2+1;

∴ cosφ=√m²−1/p²+2p.

$\theta$ is of course found as before, 1, 2, 3, . . . . being put for $p,$ according as the question relates to the primary, secondary, or tertiary bow.

In this manner the radius^[2] of the innermost arc^{[errata 1]} of the lower bow, is found to be 40° 17′, that of the outermost 42° 2′. And the extreme values for the second bow, are 50° 57′, and 54° 7′.

182. It is easy to verify these results by observation, for as the center of the bows is in the line joining the center of the Sun and the eye of the spectator, (Fig. 210.) the radius of any arc of which A is the highest point, is equal to the sum of its altitude AOh, and that of the Sun SOH, or hOS. We have therefore only to take with a sextant, or other equivalent instrument, the greatest height of any arc above the horizon, and add that of the Sun, to obtain the radius of the arc.

183. It is sometimes required to determine, from observations on the rainbow, the ratios of refraction, for the different kinds of coloured light, between air and water.

Suppose that we have found the value of θ, or 2φ′−φ for an arc of the primary bow.

Let tan(2φ′−φ)	=	A,
tanφ′	=	t.

We saw that in this case

m·sinφ′	=	sinφ,
and mcosφ′	=	2cosφ;

---

∴ tanφ′=1/2tanφ, or tanφ=2tanφ′.

Then tan(2φ′−φ), or A	=	tan2φ′−tanφ/1+tan2φ′·tanφ
	=	2t/1−t2−2t/1+4t2/1−t2=2t3/1+3t2;

∴ 2t³−3At²+A=0.

From this equation we must find t, and from that by a table φ′ and φ: then dividing sinφ by sinφ′, we shall obtain the particular value of m required.

 

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APPENDIX.

(From Biot's Additions to Fischer's Physique Mechanique.)

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Many optical phenomena relating to the physical properties of light, having of late years acquired some importance, we will here give, not a detailed account of them, which would not suit the plan of this Work, but a sketch which will indicate the principal results.

Coloured Rings.

When two plates of glass whose surfaces are not quite plane, are placed one on the other, the lamina of air naturally adhering to those surfaces, has usually thickness enough to exercise complete action on light, that is, it reflects and refracts all the coloured rays in the same manner as if it were of considerable depth. If, however, one of the glasses be rubbed on the other, and forcibly pressed to it, to exclude a part of the intermediate air, there will soon be perceived a degeee of adhesion, which is generally greater in some parts than in others, either because the surfaces are always a little curved, or because they invariably bend under strong pressure; in this manner there is obtained a lamina of air, thinner than the preceding, and the depth of which increases gradually in all directions from the point in which the surfaces are most closely in contact. If now these glasses be turned so that the eye may receive the light of the clouds, reflected by the lamina of air, there will be perceived a number of concentric coloured rings, which, when the glasses are pressed sufficiently, surround a dark spot, at the point of contact.

These coloured rings may be formed by pressing together transparent plates of any other substance, besides glass; they may be observed, when a glass lens is placed on a plane surface of resin, of metal, of metallic glass, or any other polished body. These rings subsist moreover in the most perfect vacuum that can be produced. Neither is it necessary for their formation, that the interposed lamina be of air, nor that it be contained between two solid substances: a layer of water, of alcohol, of ether, or any other evaporable liquid, spread on a black glass, produces similar colours, when sufficiently attenuated; they may be observed also on soap bubbles, and on blown glass, when thin enough.^[3]

In whatever manner, and under whatever circumstances these rings are formed, the succession of their colours from the central dark spot is invariably the same; the only difference perceptible is in their brightness, which varies with the refracting power of the lamina, and in their form, which depends on the law by which the thickness of the lamina is regulated in different parts. In fact, for any one substance, the colour reflected at any point depends on the thickness of the lamina, and the incidence under which the reflexion takes place.

So far we have supposed the colours of the lamina to be seen only by reflexion; if it be placed between the eye and the light, concentric rings will again be observed similar to the others in form, but not in colour, and fainter, surrounding a bright spot.

This might naturally be expected, for when the incident light is decomposed, so as to give coloured rays in the reflexion, those transmitted must of course be also coloured, and the one set must, in fact, be complementary to the other, that is, both together would produce white.

It follows from all this, that to discover the laws of these phænomena, the best method is to study them in cases where the variation of thickness is regular and known. This is what Newton did; and he conducted his researches with a careful nicety, which could be owing only to the importance which he foresaw would be attached to the consequences of them.

He formed the rings by placing a convex glass of small curvature on a piece of perfectly plane glass; then the thickness of the lamina of air increasing symmetrically in all directions from the point of contact, the rings were perfectly circular round the dark spot formed at that point.

He measured the diameters of these rings, in a particular case, and thence, knowing the curvature of the surface, he was able to calculate the thickness of the lamina at each ring.

Repeating this observation under different angles of incidence, he remarked the variations produced in the rings; he found that they grew wider as the obliquity increased, and by measuring their diameters, he calculated the different thicknesses at which the same colour appeared.

He made similar experiments on thin plates of water, contained between two glasses, and on thin soap bubbles, blown with a pipe. These bubbles being placed on a plane glass, became perfectly hemispherical, and being covered over with a bell-glass, they lasted long enough for him to observe at leisure their brilliant tints. He thus found that the thicknesses, at which the same colours appeared were less than in air, in the ratio of 3 to 4, which is, in fact, that of refraction between those two substances. Other trials with laminæ of glass, led him to generalize this remark, which many other experiments afterwards confirmed. He collected all his results into empiric tables, which express the laws of them in numbers.

These laws were, however, still complicated in consequence of the unequal refrangibilities of the different rays, by which the rings were illuminated. To reduce the phænomenon to its greatest simplicity, Newton formed rings with simple light, by looking, in a dark room, at a white paper, which received in turns all the simple colours of the prismatic spectrum. This paper thus enlightened, and seen by reflexion on the thin laminæ, became like a kind of sky, coloured by that tint alone, which was thrown on it. In this manner the following results were obtained:

(1.) Each kind of simple light produced rings of its own colour, both by reflexion, and by transmission.

(2.) In each case, the rings were separated by dark intervals, which made them much more distinct than in the original experiment, and caused many more to be discerned. They were more and more crowded together as their distance increased from the central spot.

(3.) The dark intervals which separated the bright rings seen by the reflected light, were bright rings themselves by the transmitted rays, and they were separated by dark intervals answering to the former rings. However, those intervals were not exactly black, because the reflexion on a thin lamina of air is far from being perfect, even in the most brilliant part of the reflected rings; and the same thing may be observed of all thin transparent plates of any substance whatever.

(4.) In observing the luminous reflected rings, Newton remarked, that they were not simple geometrical lines, but that each of them occupied a certain space, in which the brightness diminished gradually each way from the middle.

(5.) Measuring the diameters of the reflected rings at their brightest part, he found that for each particular kind of rays, the squares of the diameters followed the arithmetical progression of the numbers 1, 3, 5, 7, &c.; consequently, the thicknesses of the lamina, which are as the squares of their diameters, were in that same progression.

When the glasses were illuminated by the brightest part of the spectrum, which is between the orange and yellow, the diameter of the sixth ring was found to be the same as that of the brightest part of the corresponding ring in the experiment made in full daylight.

(6.) The diameters of the dark rings being likewise measured, he found that their squares, and consequently, the thicknesses of the air below them followed in the progression, 2, 4, 6, 8, &c.

(7.) By other measurements, he discovered that the brightest parts of the transmitted rings answered to the darkest parts of the intervals in reflexion, and vice versâ, the darkest parts here were the brightest in the other case, so that the thicknesses of air which transmitted the bright rings, and those which gave dark intervals, were respectively as 2, 4, 6, 8, &c. and as 1, 3, 5, 7, &c.

(8.) The absolute diameters of corresponding rings of different colours were different, as were also their breadths, both these dimensions being greatest for the extreme red rays, and least for the violet.

(9.) The simple rings of each colour were least when the rays passed perpendicularly through the lamina of air, and increased with the angle of incidence.

These observations explain completely the more complicated phænomenon of the rings formed by the natural light, for this light consisting of different coloured rays mixed together in definite proportions, when a beam of this mixture falls on the thin lamina of air between the glasses, each kind of simple light forms its own rings by itself, according to its own peculiar laws, and as the diameters of these rings are different for the various kinds of light, they are sufficiently separated from each other to be distinguished. However, this separation is by no means so perfect as in observations made with simple rings, because the rings of different colours encroach a little on each other, so as to produce that infinite diversity of tints that the experiment shows. But, though this successive superposition of the simple rings is really the key of the phænomena, one cannot be very sure of the fact without having measured exactly the absolute magnitudes of the diameters and breadths of the rings, formed by the different coloured rays; for when these results are once known, it can only be a simple arithmetical problem to find the species and the quantity of each colour that may be reflected or transmitted at each determinate thickness; and consequently, if the effects of the composition of all these colours be calculated by the rules which Newton has given in his Optics, it will be easy to deduce, with perfect accuracy, the numerical expressions of the tint and intensity of colour which must exist at each point of the compound rings, which may then be compared with experiment. In a word, we have as yet only a suspicion, a probable one no doubt, of the cause of our phænomena; accurate measurements are necessary to convert that probability into certainty.

This is just what Newton did. He measured the diameters of the simple rings of the same order, both at their inner and outer edges, taking successively the various colours of the spectrum, from the extreme violet to the deepest red; afterwards, according to his usual method, he took care to connect these results by a mathematical law, which might represent them with sufficient accuracy. Then comparing the squares of the diameters, he deduced the proportional thickness of the lamina of air at each edge of the observed rings. Similar measurements effected with respect to the different orders of rings, formed by one simple colour, proved to him, that the intervals of thickness, throughout which reflexion took place, were sensibly equal to those which allowed transmission, at least when the light was incident perpendicularly. Thus, designating generally by t the thickness of the air at the beginning of the first lucid ring, for any simple colour, that ring ended at the thickness 3t, and therefore occupied an interval of thickness equal to 2t. Then came the first dark ring, occupying an equal interval 2t; then a second lucid ring from 5t to 7t, and so on.

Combining this law of succession for the different orders, with that of the distribution of the various tints of the same order, one easily conceives that a single absolute thickness, measured at the beginning, the middle, or the end of any ring formed by a simple colour, is sufficient to calculate the value of the first thickness t, relatively to that colour, and thus all thicknesses of the several rings of each colour may be determined.

In this manner Newton, measuring the thickness represented by 2t for the different simple rays, in vacuo, in air, in water, and in common glass, found their values as shown in the following Table, where they are expressed in ten thousandth parts of an inch.

Colours. Values of 2t. In Vacuo. In Air. In Water. In Glass. ——————————— ———— ———— ———— ———— Extreme violet 3,99816 3,99698 2,99773 2,57870 Limit of violet and indigo 4,32436 4,32308 3,24231 2,78908 indigo and blue 4,51475 4,51342 3,38507 2,91188 blue and green 4,84284 4,84142 3,63107 3,12350 green and yellow 5,23886 5,23732 3,92799 3,37891 yellow and orange 5,61963 5,61798 4,21349 3,62450 orange and red 5,86586 5,86414 4,39811 3,78331 Extreme red 6,34628 6,34441 4,75831 4,09317

In this Table the values relating to air were alone immediately obtained by observation; the others were calculated from them by means of the several ratios of refraction, that is, by multiplying them by 3389/3388 for the vacuum, 3/4 for water, and 20/31 for glass. It must be remembered, that these values all suppose the incidence to be perpendicular.

Applying to these results a rule that he had found to determine the nature of the compound colour resulting from any given mixture of simple colours, Newton deduced the following Table, which shows the thickness at which the brightest tints of each ring appear, when seen under the perpendicular incidence. This table is calculated only for air, water, and common glass, but may of course be extended to all other substances, by the method above-mentioned.

The unit is the thousandth part of an inch. By the side of different colours are put the names of certain flowers or metallic substances, just to give more distinct ideas of them.

Colours reflected. Thickness in thousandth
parts of an inch. Names of the Colours, or
substances having them. In air. In water. In glass. 1st Order. Very black 1/2 3/8 10/31 Black 1 3/4 20/31 Beginning of black 2 11/2 12/7 Blue 22/5 14/5 111/20 Whitish sky-blue. White 51/4 37/8 32/5 Tarnished silver. Yellow 71/9 51/3 43/5 Straw colour. Orange 8 6 51/6 Dried orange-peel. Red 9 63/4 54/5 Geranium Sanguineum. 2nd Order. —————— —————— —————— ———————————— Violet 111/6 83/8 71/5 Iodine. Indigo 125/6 95/8 82/11 Indigo. Blue 14 101/2 9 Cobalt blue. Green 151/8 111/3 95/7 Water, aquamarine. Yellow 162/7 121/5 102/5 Lemon. Orange 172/9 13 111/9 Orange. Bright red 181/5 133/4 115/6 Bright May-pink. Scarlet 192/3 143/4 122/3 3rd Order. —————— —————— —————— ———————————— Purple 21 153/4 1311/20 Flax-blossom. Indigo 221/10 164/7 141/4 Indigo. Blue 232/5 1711/20 151/10 Prussian blue. Green 251/5 189/10 161/4 Bright meadow green. Yellow 271/7 201/3 171/2 White wood. Red 29 213/4 185/7 Rose. Bluish red 32 24 202/3 4th Order. —————— —————— —————— ———————————— Bluish green 34 251/2 22 Green 352/7 261/2 223/4 Emerald. Yellowish green 36 27 232/9 Red 401/3 301/4 26 Pale pink. 5th Order. —————— —————— —————— ———————————— Greenish blue 46 341/2 292/3 Sea-green. Red 521/2 393/8 34 Pale pink. 6th Order. —————— —————— —————— ———————————— Greenish blue 583/4 44 38 Light sea-green. Red 65 483/4 42 Paler red. 7th Order. —————— —————— —————— ———————————— Greenish blue 71 531/4 454/5 Very faint. Ruddy white 77 573/4 492/3 Ditto.

Reduction of the phenomena of the rings to a physical property of light, called Fits of easy reflexion and Transmission.

The phenomena of the rings being reduced to laws extremely exact and well adapted to calculation, Newton concentrated them all in a still simpler expression, making them depend on a physical property, which he attributed to light, and of which he defined all the particulars conformably to their laws.

Considering light as a matter composed of small molecules emitted by luminous bodies with very great velocities, he concluded, that since they were reflected within the lamina of air, at the several thicknesses t, 3t, 5t, 7t, &c., and transmitted at the intermediate thicknesses 1, 2t, 4t, 6t, &c., the molecules must have some peculiar modification, of a periodical nature, such as to incline them alternately to be reflected and refracted after passing through certain spaces. Yet this modification could not be necessary, since the intensity of the reflexion at the second surface varies with the medium contiguous to that surface, so that a given molecule arriving at it, at a given epoch of its period, may be either reflected or transmitted, according to the exterior circumstances which act on it. Newton therefore characterised this property of the luminous molecules as a simple tendency, and designated it appropriately enough by the phrase, Fit of easy reflexion or transmission.

According to this idea of the fits, their duration must evidently be proportional to the thickness t, which regulates, in each substance, the alternations of reflexion and transmission. Thus, in the first table given, we find the measure of it for a vacuum, for air, water, and glass, in the case of perpendicular incidence. In other substances, the duration of the fits must vary as the quantity t, that is, inversely as the refracting power; it will vary also, by parity of reason, with the obliquity of incidence, and the nature of the light: but the laws of these variations are exactly those which regulate the rings themselves; so that, these last being known, it remains only to apply them; this Newton did, and after having defined completely all the characters of the fits, he employed them as a simple property, not only to unite under one point of view the phenomena of the colours produced by thin plates, but also to foresee and to calculate beforehand, both as to their general tenor, and their minutest details, a crowd of analogous phenomena observed to attend reflexion in thick plates, which, in fact, in his experiments, exceeded by as much as twenty or thirty thousand times those on which the calculations had been founded; moreover, applying the same reasoning to the integrant particles of material substances, which all chemical and physical phenomena show to be very minute, and to be separated, even in the most solid bodies, by spaces immense, in comparison of their absolute dimensions, he was able to deduce naturally from the same principles, the theory of the different colours they present to us, a theory which adapts itself with a surprising facility to all the observations to which those colours can be submitted. The number and importance of those applications account sufficiently for the case which Newton took with his experiments on the rings; I am sorry to be obliged to confine myself here to the bare indication of those fine discoveries.

Another explanation of the coloured rings on the hypothesis of undulations. Dr. Young's principle of Interferences.

If light be really a material substance, Newton's fits are a necessary property, because they are only a literal enunciation of the alternations of reflexion and transmission which coloured rings present; but if light be otherwise constituted, these alternations may be accounted for differenly.

Descartes, and after him Huyghens, and a great number of natural philosophers, have supposed that the sensation of light was produced in us by undulations excited in a very elastic medium, and propagated to our eye, which they affect in the same manner as undulations excited in the proper medium of the air, and propagated to the ear, produce in it the sensation of sound. This medium, if it does exist, must fill all the expanse of the heavens, since it is through this expanse that the light of the stars comes to our eyes: it must also be extremely elastic, since the transmission of light takes place with such extraordinary velocity; and at the same time its density must be almost infinitely small, since the most exact discussion of ancient and modern astronomical observations, does not indicate the least trace of resistance in the planetary motions. As to the relations of this medium with earthly bodies, it is plain that it must pervade them all, for all transmit light when sufficiently attenuated; moreover its density must probably differ in them according to the nature of the substances, since unequal refractions appear to prove that the propagation of light takes place in different media with various velocities. But what ought to be the proportions of the densities for these different substances? How is the luminiferous ether brought to, or kept in the proper state for each? How is it inclosed and contained so as to be incapable of spreading out of them? Moreover, how is this medium, so nonresisting, so rare, so intangible, agitated by the molecules of bodies which appear to us luminous? There are so many characters which it would be necessary to know well, or at least to define well, to have an exact idea of the conditions according to which the undulations are formed and propagated; but hitherto they have never been distinctly established.

At any rate, if a body be conceived to have the faculty of exciting an instantaneous agitation in a point of such a medium, supposed at first equally dense in all its extent, this agitation will be propagated in concentric spherical waves, in the same manner as in air, except that the velocity will be much more considerable. Each molecule of the medium will then be agitated in its turn, and afterwards return to a state of rest.

If these agitations are repeated at the same point, there will result, as in air, a series of undulations analogous to those producing sound; and as in these there are observed successive and periodical alternations of condensation and rarefaction, corresponding to the alternations of direction which constitute the vibrations of a sonorous body, in like manner it will be easily conceived that the successive and periodical vibrations of luminous bodies might produce similar effects in luminous undulations: and again, as the succession of sonorous waves, when sufficiently rapid, produces on our ear the sensation of a continuous sound, the quality of which depends on the rapidity of the opposite vibrations, and on the laws of condensation and velocity that the nature of these vibrations excites in each sonorous wave, in like manner, under analogous conditions, the ethereal waves may produce sensations of light in our eyes, and different sensations in consequence of the variety of the conditions. Hence the differences of colours. In this system, the length of the luminous waves correspond to Newton's fits, and their length is, as will be seen hereafter, exactly quadruple; the rapidity of their propagation depends, as in air, on the relation between the elastic force of the fluid and its density.

When a sonorous wave excited in air arrives at the surface of a solid body, its impact produces in the parts of that body a motion, insensible indeed, but nevertheless real, which sends it back. If the body, instead of being solid, is of a gaseous form, the reflexion takes place equally, but there is produced in the gas a sensible undulation depending on the impression that its surface has received.^[4] Luminous undulations ought to produce a similar effect when the medium in which they are excited is terminated by a body in which the density of the ethereal fluid is different; that is to say, there must be produced a reflected wave and one transmitted; which is, in fact, what we call reflexion and refraction. In this system, the intensities of rays of light must be measured by the vis viva of the fluid in motion, that is, by the product of the density of the fluid by the square of the proper velocity of its particles.

To confirm these analogies, already very remarkable, it would be necessary to follow up their consequences further with calculations; but unfortunately this cannot be done rigorously. The subject of undulations thus sent back or transmitted in oblique motions, is beyond the existing powers of analysis. In the case of perpendicular incidence the phænomenon becomes accessible, but then it teaches nothing as to the general direction of the motion communicated, as the propagation must be continued in a straight line, if for no other reason, on account of there being no cause why it should deviate from it; nevertheless in this case theory indicates the proportions of intensity for the incident and reflected waves, which appear, in fact, tolerably conformable to experiments on light, which is, at any rate, a verification as far as it goes.

When the ear receives at once two regular and sustained sounds, it distinguishes, besides those sounds, certain epochs at which undulations of the same nature arrive together or separate. If the periods of these returns are very rapid, a third sound is heard, the tone of which maybe calculated à priori from the epochs of coincidence; but if these happen so seldom as to be heard distinctly, and counted, the effect is a series of beats which succeed each other more or less rapidly. The mixture of two rays, which arrive together at the eye, under proper circumstances, produces an effect of the same kind, which Grimaldi remarked long ago, but of which Dr. Young first showed the numerous applications. The neatest way of exhibiting this phænomenon is the following, which is due to M. Fresnel.

A beam of sun-light, reflected into a fixed direction by a heliostat, is introduced into a darkened room; it is transmitted through a very powerful lens, which collects it almost into a single point at its focus. The rays diverging from thence form a cone of light within which there are placed, at the distance of two or three yards, two metallic mirrors inclined to each other at a very small angle, so that they receive the rays almost under the same angle; the observer places himself at a certain distance, so as to observe the reflexion of the luminous point in both the mirrors. There are thus seen two images separated by an angular interval which depends on the inclination of the two mirrors, their distances from the radiating point, and the place of the observer; but besides these, which is the essential point of the phænomenon, there may be seen, by the help of a strong magnifying lens, between the places of the two images, a series of luminous coloured fringes parallel to each other, and perpendicular to the line joining the images; if the incident light is simple, the fringes are of the colour of that light, and separated by dark intervals. Their direction depends solely on those of the mirrors, and not at all on any influence of the edges of those mirrors, as each of them may be turned round in its own plane without producing the slightest alteration in the phænomenon.

Let us confine our attention, for the sake of greater simplicity, to the case in which the incident light is homogeneous; this case may be easily exhibited in practice by observing the fringes through a coloured glass which will transmit only the rays of a particular tint. In this case if we select any one of the brilliant fringes formed between the two images, we may calculate the directions and paths of the luminous rays which form that fringe coming from each of the mirrors. Now in making this calculation, we find the following results:

(1.) The middle of the space comprised between the two luminous points is occupied by a band of colour formed by rays the lengths of whose paths from the luminous point to the eye are equal.

(2.) The first fringe on each side of this is formed by rays for which the difference of length is constant, and equal for instance to l.

(3.) The second coloured fringe arises from the rays having 2l for the difference of the distances they pass over.

(4.) In general for each fringe this difference is one of the terms of the series 0, l, 2l, 3l, 4l, &c.

(5.) The intermediate dark spaces are formed by rays for which the differences are 1/2l, 3/2l, 5/2l, &c.

(6.) Lastly, the numerical value of l is exactly four times that of the length which Newton assigns to the fit for the particular kind of light considered.

The analogy between these laws and those of the rings is evident. The following is the explanation given of them in the system of undulations: the interval l is precisely equal to the length of a luminous wave, that is, to the distance of those points in the luminiferous ether, which, in the succession of the waves, are at the same moment in similar situations as to their motion. When the paths of two rays which interfere with one another differ exactly by half this quantity at the place where they cross, they bring together contrary motions of which the phases are exactly alike. Moreover the motions produced by these partial undulations take place almost along the same line, as the mutual inclination of the mirrors is supposed to be very small. Consequently the two motions destroy one another, the point of ether at which they meet remains at rest, and the eye receives no sensation of light. The same thing must occur at those points where the differences of the spaces passed over by the rays is 3/2l, 5/2l, or any other such number; whereas at points where the difference is l, 2l, 3l, or any other multiple of l, the undulating motions coincide, and assist each other, so that the appearance of light is produced.

This way of considering of the combination of luminous waves and the alternations of light and darkness which result from it has been called by Dr. Young, the Principle of Interferences.

The phænomenon of the alternations of light and darkness is certain; if, reasoning à priori, it appeared to be possible, only on the hypothesis of undulations, it would reduce the probability of that hypothesis to a certainty, and completely set aside the theory of emission. It does not, however, appear to offer that character of necessary truth which would be so valuable, whichever argument it favoured, because it would be decisive. One may, without violating any rule of logic, conceive equally the principle of interferences in the system of emissions, making the result which it expresses a condition of vision.

In fact, the phænomenon of the fringes does not prove that the rays of light really do affect each other under certain circumstances, it only shows that the eye does, or does not receive the sensation of light, when placed at a point where the rays coincide with those circumstances; it proves also that an unpolished surface placed at such a point, and seen from a distance, appears either bright or dark; now in the former case it is possible that vision may cease when the retina receives simultaneously rays which are at different epochs of their fits; and in the latter, when such rays arrive together at an unpolished surface, and are afterwards dispersed by radiation in all directions, it is clear, that having the same distance to pass over from each of the surfaces to the eye, they will have, on arriving at it, the same relative phases that they had when at the surface; if therefore they were then in opposite states, they will be so likewise in arriving at the retina, and thus there will be no vision produced. I do not pretend that this explanation is the true one, or even that it bears the character of necessity; it is both true and necessary if light be material, for it is but the statement of a phænomenon; but if only it implies no physical contradiction, that is quite sufficient to prevent the phænomenon from which it is derived from being decisive against the system of emission.

Dr. Young has with equal success applied the principle of interferences to the explanation of the coloured rings, both reflected and transmitted, of thin plates. When such a plate is seen by reflexion, the light coming from the first surface to the eye interferes with that from the second; this interference either does or does not produce the sensation of light, according as the different distances that the rays have to pass over, place them in similar or opposite phases of their undulations; but then, at the point where the thickness is nothing, this difference is nothing, and consequently one would expect to see a bright spot instead of a dark one. To get over this difficulty Dr. Young introduces a new principle, namely, that the reflexion within the plate makes the rays lose an interval 1/2l, exactly equal to half the length of a wave. By means of this modification, the rays reflected from the two surfaces at the point where the thickness is nothing, acquire opposite dispositions, and therefore produce together no sensation of light in the eye; then in the surrounding places, the law of the periods of the undulations gives that of the succession of bright and dark rings: this law, thus modified, agrees with the measurements of the coloured rings observed in the case of perpendicular incidence; but for oblique incidences it is not quite consistent with Newton's statement. Is it possible that the laws which Newton established upon experiments may be inexact, or must we introduce in the case of oblique waves some modification depending on their impact on the surfaces? This point is yet to be decided.

We have hitherto considered only the rings observed by reflected light; the others are formed, according to the undulation system, by the interference of waves transmitted directly, with those which, being reflected at first at the second surface of the thin plate, are again reflected on returning to the first, and are thus sent to the eye at which they arrive without any farther modification. In this case the point where the surfaces touch should give a bright spot, as we find by experience that it does, so that here we have no additional principle to introduce as in reflexion; but this is quite necessary in many other cases.

According to this system, the thicknesses at which the rings are formed indicate the length of the oscillations in any substance. Now for one given mode of vibration of the luminous body, the length of the waves must be equal to the distance that the light passes over whilst the vibration takes place; since therefore the waves are found to be shorter in the more strongly refracting substances, this velocity of transmission must be less in them according to the same law; that is to say, it must be inversely as the ratio of refraction.

By considering the alternations of light and darkness as produced by the superposition of luminous waves of the same or of different nature, we give to the phænomenon a physical character, and it is thus that Dr. Young first announced the principle of interferences; but we may detach it, as he has done, from all extraneous considerations, and present it as an experimental law; it may then be expressed as follows:

1. When two equal portions of light, in exactly similar circumstances, have been separated, and coincide again nearly in the same direction, they either are added together, or destroy one another, according as the difference of the times, occupied in their separate passages, is an even or odd multiple of a certain half interval which is different for the different kinds of light, but constant for each kind.
2. In the application of this law to different media, the velocities of light must be supposed to be inversely proportional to the ratios of refraction for those media, so that the rays move more slowly in the more strongly refracting medium.
3. In reflexions at the surface of a rarer medium, on some metals, and in some other cases, half an interval is lost.
4. Lastly, it may be added that the length of this interval, for a given kind of light, is exactly four times that of the fits attributed by Newton to the same light.

To give an instance of these laws, suppose that when two simple homogeneous rays interfere and form fringes in the experiment with the two mirrors, you interpose across the path of one of these a very thin plate of glass that that ray alone is to pass through. According to the second condition, its motion through the glass must be slower than through the air in proportion as the refracting power is greater. Thus, when after leaving the glass, and continuing its motion, it meets the ray with which it before interfered, its relations with this as to intervals will have been altered; and if the intervals are ever found to be the same, it must be when the ray is so refracted by the glass that the diminution of its velocity be compensated by shortening its path; in this case the fringes will be formed in different places, and their displacement may be calculated from the thickness of the glass and its refracting power; now this is confirmed by experiment with incredible exactness, as M. Arago observes, to whom we are indebted for this ingenious experiment.

By the same rule, if the displacement of the fringes thus produced by a given plate be observed, which may be done with extreme precision, we may evidently find the refracting power of that plate; we may also compare the refractions of various substances by interposing plates of them successively on the directions of the interfering rays. Messrs. Arago and Fresnel tried this method, and found it so exact that they were able to use it to measure differences of refraction that no other method would have given.

Diffraction of light.

When a beam of light is introduced into a dark room, if you place on its direction the edge of some opaque body, and afterwards receive on a white surface placed at a certain distance that portion of the light which is not intercepted, the border of the shadow will be observed to be edged with a bright line; and on increasing the distance, several alternations of coloured fringes are thus seen to be formed. This phænomenon constitutes what is called the Diffraction of light.

To give it all the exactness of which it is capable, it is advisable to use the same disposition as in the experiment with the two mirrors, that is, to take a sunbeam directed by a heliostat and concentrated by a lens almost into a geometrical point: an opaque body is then to be placed in the cone of rays diverging from that point. To fix our ideas, suppose we use an opaque lamina with straight edges, and about a tenth of an inch broad; if then the rays be received on a piece of ground glass placed at a certain distance, and the eye be placed beyond this glass, there will be observed on each side of the shadow of the lamina a numerous series of brilliant fringes parallel to the edges, and separated from each other by dark intervals; the brightness of these fringes diminishes as they recede from the shadow; and the shadow itself is not quite dark, but is formed also of luminous and dark fringes all parallel to the edges of the lamina. Moreover the ground glass is not necessary to exhibit these fringes, for they are formed in the air and may be seen in it, either with the naked eye or by the assistance of a lens placed exactly on their direction. If then a lens be fixed to a firm stand which can be moved horizontally, by means of a screw, along a scale divided into equal parts, its axis may be brought successively opposite each bright and dark fringe; the position of one of these may be determined precisely by referring it to a fine thread stretched in front of the lens, and thus the intervals of the fringes may be measured, on the graduated scale, by the distance through which the lens is moved to set it opposite to each; this advantageous arrangement was devised by M. Fresnel, who made use of it to measure all the particulars of the phænomenon with extreme precision.

Now these particulars, as Dr. Young first announced, may be represented pretty exactly by supposing that the light which falls on the edges of the lamina, spreads over them radiating in all directions from those edges, and interferes both with itself and with the rays transmitted directly.

The first kind of interference forms the interior fringes; the light radiating from one edge interfering with that from the other, these two sets of rays are exactly in the same predicament as the two luminous reflected points in the experiment of the mirrors; thus also the disposition of the interior fringes both bright and dark, and the ratios of their intervals, are exactly similar. If you determine in your mind the series of points in space at which the same kind of interference takes place, at different distances behind the lamina, which gives the succession of the places at which the same fringe appears, you will find that those points are, to all appearance, on a straight line; and their intervals, when measured, are very exactly conformable to what the calculation of the interferences indicates.

As to the exterior fringes, they may be considered as formed by the interference of the light transmitted directly with that radiating from each edge; but we must here, as in the reflected rings, suppose a loss of an interval 1/2l. It thus appears that the points at which each fringe appears at different distances from the lamina are not placed on a straight line, but on an hyperbola of the second order, which experiment confirms completely.

We must not conclude from this that diffracted light does not move in straight lines, for it is not the same ray that forms a fringe of a given order at different distances. That the ray changes, as the distance is altered, may be concluded from this alone, that the fringes may be observed in space either with the naked eye or with a lens; for then it is evident that the rays which form them must converge, and afterwards diverge; otherwise they could not be collected by the lens so as to afford a visible image of their point of concourse.

Very remarkable phænomena of diffraction are again produced, when the cone of light, instead of being intercepted by an opaque lamina, is transmitted between two bodies terminated by straight parallel edges. In this case, the diffracted fringes may, with great appearance of truth, be attributed to the interference of the two portions of light which fall on the opposite edges.

Nevertheless, there are many physical particulars in the phænomenon, which it is difficult to explain on this hypothesis. M. Fresnel has even found that it is not quite consistent with the measurements of the fringes when they are very exact; he has been convinced that the small portion of light which the edges may reflect is not sufficient to produce the observed intensities of the fringes; and that it is necessary to suppose that other rays assist which do not touch the edges. He has thus been induced to consider all the parts of the direct luminous wave as so many distinct centers of undulations, the effects of which must be extended spherically to all the points of space to which they can be propagated; according to which supposition, the particular effect at each point would result from the interferences of all the partial undulations that arrive at it. This consideration, applied to the free propagation of a spherical wave in a homogeneous medium, makes the loss of light proportional to the square of the distance conformably to observation; but when a part of the light is intercepted, it indicates, in the different points of space towards which it is afterwards propagated, alternations of light and darkness, which, in point of disposition and intensity, agree most minutely with those observed in diffracted light.

This introduction of this principle has enabled M. Fresnel to embrace all the cases of diffraction with extraordinary precision; but an exposition of his results, though very interesting, would lead us farther than the plan of this Work would allow.

Double Refraction.

The rays of light, in passing through most crystallized substances, are generally divided into two parcels, one of which, containing what are called the ordinary rays, follows the usual mode of refraction; but the other, consisting of what are termed the extraordinary rays, obeys entirely different laws.

This phænomenon takes place in all transparent crystals, except those which split in planes parallel to the sides of a cube, or a regular octohedron. The separation of the rays is more or less strong, according to the nature of the crystal, and the direction which the light takes in passing through it. Of all known substances, the most powerfully double refracting, is the clear carbonate of lime, commonly called Iceland Spar. As this is a comparatively common substance, and may easily be made the subject of experiment, we take it as a first instance.

The crystals of this variety of carbonate of lime are of a rhomboidal form, as represented in Fig. 211. This rhomboid has six acute angles, and two obtuse; these last are formed by three equal plane angles: in the acute dihedral angles, the inclination of the faces is 74° 55′, and consequently, in the others it is 105° 5′. Malus and Dr. Wollaston have both found these values by the reflexion of light.

If a rhomboid of this description be placed on a printed book, or a paper marked with black lines, every thing seen through it will appear to be double, so that each point under the crystal, must send two images to the eye, and consequently, two pencils of rays. This indicates that each simple pencil must be separated into two in its passage through the rhomboid; and this may be easily shown to be the case, by presenting the crystal to a sunbeam, when it will give two distinct emergent beams. To measure the deviation of these rays, and determine their paths, Malus invented the following simple method: on the paper on which you place the rhomboid, draw, with very black ink, a right angle ABC, (Fig. 212.) of which let the least side BC be, for instance, one-tenth of AC. If this triangle be observed through the rhomboid, it will appear double, wherever the eye be placed; and for each position of the eye there will be found a point T, where the line A′C′, the extraordinary image of AC, will cut the line AB, which I suppose to belong to the ordinary image. Take then on the triangle itself a length AF′ equal to A′F, and the point F′ will be that of which the extraordinary image coincides with the ordinary one of F. The ordinary pencil proceeding from F, and the extraordinary one from F′ are therefore confounded together, on emerging from the crystal, and produce only one single pencil which meets the eye: hence, conversely, a natural pencil proceeding from where the eye is placed to the crystal, would be separated by the refraction into two pencils, one of which would go to F, and the other to F′. This may indeed be easily confirmed by experiment with the heliostat. If then the lines AB, AC be divided each into a thousand parts, for instance, and the divisions be numbered as represented in the Figure, a simple inspection will suffice to determine the points of AB, and AC, of which the images coincide; consequently, if the position of these lines and the triangle be known, relatively to the edges of the base of the crystal, it will be known in any case to what points of the base F and F′ correspond, so that to construct the refracted rays, it will only remain to determine, on the upper surface, the position of their common point of emergence (Fig. 213.). This might be done by marking on that surface the point I, where the images of AB and AC intersect; but as it is useful also to know the direction of the emergent pencil, it is better to make the observation with a graduated circle placed vertically in the plane of emergence IOV. The sights of this circle must be directed to the point I, and, if the precaution has been taken of levelling the plane on which the crystal lies, the same observation will determine at once the angle of emergence IOV, or NIO, measured from the normal, and the position of the point I on the rhomboid. The positions of the points F, F′, are also known à priori, so that the directions FI, F′I may be constructed; whereupon we may remark, that in many cases the extraordinarily refracted pencil F′I does not lie in the plane of emergence NIO.

Such is the process devised by Malus; if we admit it, we may admit also all his observations, and consider them as data to be satisfied, but I will shortly indicate a more simple method, which would allow us to repeat these same measurements with equal facility and accuracy.

Among all the positions that may be given to the crystal, resting always on the same face, there is one which deserves particularly to be remarked, because the extraordinary refraction takes place, like the ordinary, in the plane of emergence. To find this position, it is necessary to conceive a vertical plane to pass through the side BC of the triangle, to place the eye in this plane, and slowly turn the crystal round on its base, till the two images of BC coincide; then, as the ordinary image is always in the plane of emergence, the extraordinary must in that case be in it likewise. The particular plane for which this takes place is called the principal section of the rhomboid. If the crystal used in the experiment be of the primitive form, for the carbonate of lime, the bases of the rhomboid will be perfect rhombs, and the principal section will be that containing the shorter diagonals of the upper and lower faces. This section of the rhomboid will be a parallelogram ABA′B′, (Fig. 214.) in which AB, A′B′ are the diagonals just mentioned, and AB′, A′B edges of the rhomboid. The line AA′ is called the axis of the crystal; it is equally inclined to all the faces, forming with them angles of 45° 23′ 25″. It is to this line that all the phænomena of double refraction are referred.

Let us examine at first the manner of this refraction in the principal section. All its general phænomena are exhibited in Fig. 215, in which SI represents an incident ray, IO the ordinary refracted ray, IE the extraordinary: IN is the normal. When the incidence is perpendicular, the ordinary ray is confounded with the normal, and passes through the crystal without deviation; but the extraordinary is refracted at the point of incidence, and is more or less deflected towards the lesser solid angle B′. A similar effect is observed in every other case, as shown in the Figure, the extraordinary ray lying always on the same side of the ordinary.

The inference to be drawn from this is, that there exists in the crystal some peculiar force which abstracts from the incident pencil a part of its molecules, and repels them towards B′. But what is this force? We shall soon see that it emanates, or seems to emanate from the axis of the crystal; that is, that if through each point of incidence there be drawn a line IA′ parallel to that axis, and representing its position in the first strata in which the pencil is divided, all the phænomena take place just as if there emanated from that line a repulsive force, which acted only on a certain number of luminous particles, and tended to drive them from its direction. This force always throws the rays towards B′, because they are always found on that side of the axis, under whatever angle of incidence they may have entered.

Let us follow up this idea, which does not appear repugnant to the few observations that have been made, and to verify it by a direct experiment, let us divide the crystal by two planes perpendicular to its axis, (Fig. 216.) so as to form two new faces abc, a′b′c′, parallel to each other. Now if we direct a ray SI perpendicularly to those faces, it will penetrate them in a direction parallel to the primitive axis of the crystal. Supposing then that the repulsive force emanates from that axis, it will be nothing in this case, and the incident rays will not be separated. This is, in fact, what takes place: there is in this case but one image.

It is even found, in making the experiment, that the image remains single when the second face of the plate is inclined to the axis, provided the first be perpendicular to it, and to the incident rays. This would happen, for instance, if only the first solid angle A of the primitive rhomboid were taken oft. The incident ray SI would continue its progress parallel to the axis, as before, and on emerging from the second surface, it would be refracted in one single direction, according to the law of ordinary refraction. Hence, we may conclude, conversely, that an incident ray R′I′, which passed out of air into such a prism under the proper angle of incidence, would be refracted in one single ray parallel to the axis, and emerge at I in the same manner. This again is confirmed by experience. If, after having cut a rhomboid in the manner described, the eye be applied to the face, which is perpendicular to the axis, so as to receive only the rays which arrive in that direction, all the images of external objects will be single; they only undergo at their edges the diffusion which belongs to the general phænomenon of the decomposition of light by the unequal refractions.

But if the repulsive force, which produces the extraordinary refraction, really emanates from the axis, as the phænomena seem to indicate, it cannot disappear, except when the incident ray is parallel to the axis. The section, then, which we have described, is the only one in which a crystal prism can give a single image: this again is confirmed by experience, and we might avail ourselves of this character, to find the position of the axis in any piece of Iceland spar, not in the primitive form.

To return to our plate with parallel faces, cut perpendicularly to the axis. We have seen that the rays are not separated when they are incident perpendicularly; but when they enter obliquely, they ought to be separated, since they then form a certain angle with the axis, from which the repulsive force emanates. This is really what takes place; and moreover, for equal angles of incidence, the extraordinary refraction is the same on all sides of the axis, which shows that the repulsive force acts from the axis equally in all directions.

Many other crystallized substances, very different from the Iceland spar, exhibit like it a certain single line or axis, round which their double refraction is exerted symmetrically, being insensible for rays parallel to that axis, and increasing with their inclination to it, so as to be strongest for those which are at right angles to the axis. Crystals thus constituted are called crystals with one axis. For instance, quartz, commonly called rock crystal, has an axis parallel to the edges of the hexahedral prism, under the form of which it is generally found. But there is between its double refraction and that of the spar, this capital difference, observed by M. Biot, that in the spar the deviation of extraordinary rays from the axis, is greater than that of the ordinary, whereas in quartz crystals it is less. All crystals with one axis, that he has examined, have been found to possess one or other of these modes of action, which has occasioned their distinction, by him, into crystals of attractive and repulsive double refraction; these denominations, which express at once the phænomena, are useful in innumerable cases, to indicate how the extraordinary ray is disposed with respect to the other, since it is only necessary afterwards to know the direction of the axis at the point where the refraction and separation of the rays take place. The progressive and increasing separation of the rays, as their direction deviates more and more from the axis in each of these classes of crystals, may also be conveniently expressed by saying, that the phænomena take place as if there emanated from the axis a force attractive in the one class, and repulsive in the other; which does not, however, imply a belief that such forces do actually exist, or are immediately exerted.

There are, however, other crystals in great number, in which the double refraction disappears in two distinct directions, forming an angle more or less considerable, so that rays are singly refracted along those two lines, but are separated more and more widely as their incident direction deviates from them, crystals of this kind have been called crystals with two axes. In those which have hitherto been examined, it has been found that one of the refractions is always of the ordinary kind, as if the substance was not crystallized, whilst the other follows a law analogous to that of the crystals with one axis, but more complex, which will be afterwards explained. There are here, as in the simpler case, two classes distinguished by attractive and repulsive double refraction. No crystals have as yet been discovered, possessing more than two directions of single refraction, except indeed those in which it is single in all directions, which is the case with those of which the primitive form is either a cube, or a regular octohedron.^[5]

The general circumstances which characterise the phænomenon of double refraction, being thus recognised, its effects must be exactly measured in each class of crystals, in order to try and discover the laws of it. In order to this, there is no better plan to be pursued, than to cut them into plates, or prisms in different directions, relatively to the axes, to observe the extraordinary refractions, under different incidences, and endeavour to comprise them in one general law. This Huyghens has done for Iceland spar. The empiric law inferred by him, has been since verified by Dr. Wollaston, and subsequently by Malus, by means of direct experiments, which have confirmed the exactness of it. M. Biot has made similar experiments with other crystals of both classes, by means of a very simple apparatus, which affords very exact measurements of the deviations of the rays, even in cases where the double refraction is very weak. As observations of this kind are indispensable, as the foundations of all theory, it will be as well to give here a detailed description of the apparatus.

It consists principally of two ivory rulers $AX,\ AZ,$ (Fig. 217.) divided into equal parts, and fixed at a right angle. The former, $AX$, is placed on a table; the other becomes vertical. A little pillar $Hh,$ of which the top and bottom are parallel planes, is moveable along $AX,$ and may therefore be placed at any required distance from $AZ.$

This disposition is sufficient, when the extraordinary refraction to be observed takes place in the same plane as the ordinary, which we have seen to be the case under particular circumstances. As this is the simplest case, and is all that is necessary to understand the method, I will explain it first.

If the substance to be observed, had a very strong refracting power, it would be sufficient to form a plate of it with parallel surfaces, upon which experiments might be made in the manner about to be described; but this case being of rare occurrence, we will suppose, in general, that the crystal is cut into a prismatic form, to make its refraction more sensible; it is even advisable to give the prism a very large refracting angle, a right angle, for instance, which has the particular advantage of simplifying calculations. As, however, the rays of light cannot pass through both sides of such a prism, of any ordinary solid substance placed in air, being reflected at the second surface, there must be fixed to this surface, represented by $CD$ in Fig. 218, another prism, or parallelepiped of glass $CFED,$ of which the refracting angle $D,$ is nearly equal to the angle $C$ of the crystal prism, so that the faces $CB,\ DE$ of the crystal and glass, may be nearly parallel. The two prisms are to be joined together, by heating them, and melting between them a few grains of very pure gum-mastic, which on being pressed, will spread into a very thin transparent layer. This, when cooled, will be quite sufficient to make the prisms cohere together very strongly, and to let the rays pass from one into the other.

The double prism is to be placed on the pillar $Hh,$ as in the Figure, and the observer is to look through it at the vertical scale $AZ.$ This scale will appear double, the ordinary and extraordinary image being, in the simple case here considered, in the same vertical line. Now whatever be the law of the two refractions, the corresponding lines of the two scales seen, are never equally separated in all places, so that if in one part the separation amounts to half a degree of the scale, a little further on it will be a whole degree, in another place a degree and a half, two degrees, and so on. If, for instance. No. 451, of the extraordinary division, which we will represent by 451_e, coincides with No. 450, of the ordinary (450₀), so that here the separation of the images is of one degree, it will perhaps be found that 502_e falls on 500₀. This shows that the extraordinary rays coming from 502, enter the eye together with the ordinary from 500, and since the glass prism can produce no effect beyond simple refraction on these rays, it is certain that the rays from 500₀ and 502_e, must coincide at their emergence from the crystal. This condition furnishes a very accurate method to verify the law followed by the extraordinary rays in the crystal. In fact, the directions of incidence of the two pencils may be determined, since one of them $EI,$ proceeds from the point $E$ of the scale, of which the place is known from the graduation, and arrives at the point of incidence $I,$ the position of which is also determined by the known height of the pillar $Hh,$ and its position on the horizontal scale. There are similar data for the other $Oi,$ which undergoes only the ordinary refraction, whether its point of incidence be supposed the same as that for $EI,$ or whether the small distance of those points be estimated by calculation, taking into account the thickness of the crystal prism, as will be hereafter mentioned.

Now if the ordinary refracted pencil $OI$ be followed through the crystal, which may be done by the common law of refraction, it may be traced to its emergence from the second surface $CD$. Thus it will only remain to calculate the position of the extraordinary pencil, which should enter the crystal by the same surface, accompanying the exterior ray $I''\ I';$ and following back this pencil through the prism, to the first surface by an assumed law, for the extraordinary refraction, it will be seen whether it coincides, as it ought, with the incident pencil $EI.$ It is not irrelevant to remark that this condition, and indeed every part of the observation, is quite independent of the greater or less refracting power of the glass prism $CDEF,$ which serves merely to receive the rays refracted into the crystal, and make their emergence possible.

In the above instance, I have supposed the crystal to be cut so that the extraordinary refraction took place in the vertical plane, like the ordinary: that is the simplest case; but when there is a lateral deviation, I place perpendicularly to the vertical division, a divided ruler $RR,$ (Fig. 219,) which is fixed at the point from which the refracted rays proceed. Then there are observed certain lateral coincidences on the scale of $RR,$ on each of the vertical rods, if the direction of the point or line of incidence be marked on the first surface of the crystal, by a small line drawn on it, or by means of a little strip of paper stuck to it, to limit the incidence of the rays of which the common incidence is observed.

Similar means are used to fix the heights of the points of incidence on the crystal, when the coincidences are observed on the vertical scale, but then the edge of the strip of paper must be put horizontal.

One may even observe the coincidences on the horizontal scale $AX,$ on which the pillar stands. Then the places of incidence on the crystal must be limited as before.

One of the data of the calculation must be the ordinary refracting power of the crystal. This may be measured by observing on what line of the horizontal, or vertical scale another line falls, which is observed by ordinary refraction through the double prism, or through a crystal prism of a smaller angle, without a glass one. One may even see whether the ordinary refraction follows, in all cases, the law of the proportionality of the sines.

It is necesary to make the edge of the crystal prism as sharp as possible, in order that the corrections made for its thickness be inconsiderable. In fact the best way of making the observation, when it can be done, is to let the rays pass actually through the edge, for then the two refracted pencils have but an infinitely small space to pass through, before they emerge together. For a similar reason, the pillar should, in the experiments, not be placed very near the vertical scale, on which the coincidences are observed, because the corrections for thickness, which are nearly insensible at moderate distances, might become more considerable.

Besides these precautions, the faces of the prisms should be ground very smooth, and plane, and their inclinations should be accurately determined, by the reflecting geniometer. Moreover, it is necessary that the direction in which the prism is cut, relatively to the axis, or axes of the crystal, should be accurately known; in order to which these axes should be previously determined, either by immediate observation of the directions in which the reflexion is single, or by inferences drawn from the experiments themselves, or by other processes that will be hereafter detailed. By following these rules, the observer will be, I believe, perfectly satisfied, as to the nicety and accuracy of the mode of experiment. These advantages are derived from the multiplicity of the coincidences, seen on the doubly-refracted scale. The alternate superpositions and separations of the lines of division produce, if I may so express myself, the effect of verniers, and enable one to judge with extreme precision, of the point where the coincidence is most perfect.

Suppose then, that by this, or some analogous process, we have determined for some given crystal, the deviation of the rays in different directions round the axis, it remains to find out the general law, which regulates the phænomenon in all cases. This Huyghens has done, as has been before mentioned, for crystals with one single axis, by means of a remarkable law that he connected with the system of undulations: but this same law has since been deduced by M. Laplace, from the principle of material attraction.

If light is to be considered as a material substance, the refraction of its rays must be produced by attractive forces, exerted by the particles of other bodies on the luminous molecules, forces which can be sensible only at very minute distances, and which are therefore quite analogous to those which are exerted in chemical affinities. It follows, that when particles of light are at a sensible distance from a refracting body, the effect they experience from it is quite inappreciable, so that their natural rectilinear direction is not altered; they begin to deviate from this direction only at the moment when they are in the immediate vicinity of the refracting surface, and the action takes place only for an infinitely short period of time; for as soon as the particles have penetrated within the surface to a distance ever so small, the forces exerted on them by the molecules of the medium become sensibly equal in all directions, so that the path of the light becomes again a straight line, though different from the preceding. It is therefore clear that the curved portion of the path being infinitely small, it must appear to consist, on the whole, of two straight lines forming an angle, which, in fact, is quite conformable to experience. But for the very reason that the curve is not perceptible, it is useless to seek, from experiment, any notions of its form that might lead to a knowledge of the laws which produce it, as observations on the orbits of the planets have led to a knowledge of the laws of gravitation. We must therefore have recourse to some other characters derived from experiment.

Isaac Newton has succeeded in the case of ordinary refraction, by considering each luminous molecule passing through a refracting surface, as acted on before, during, and after its passage, by attractive forces sensible only at very small distances, and emanating from all parts of the refracting medium. This definition specifies nothing as to the law of the attracting forces; it allows us only to calculate their resultant for any distance, and to suppose that they become evanescent when the distance is of sensible magnitude. Now these data are sufficient to calculate, not indeed the velocity of the molecules in their curvilinear motion, nor the nature of that motion, but only the relations of the final velocities and directions, which ensue, either in the medium or out of it, when the distance of the luminous molecules from the refracting surface is become so considerable that the trajectory is sensibly rectilinear, which will comprehend all distances that we can observe.

For extraordinary refraction, we have not the advantage of being able to define the origin of the molecular force, nor the manner in which it emanates individually from each particle of the crystal; for what we have said about accounting for the phænomena by the supposition of attractive and repulsive forces emanating from the axes is only the indication of a complicated result, and not the expression of a molecular action. What is known then, in this case, or at least what may be supposed, when the idea of the materiality of light is adopted, is that the forces, whatever they may be, which act on the rays of light, in these as in other circumstances, are attractive or repulsive, or both, and emanate from the axes of the crystal. Now in all cases when a material particle is subjected to the action of such forces, its motion is subjected to a general mechanical condition called the principle of least action. Applying this principle here, and joining the particular condition that the forces are sensible only at insensible distances, M. Laplace has deduced two equations which determine completely and generally the direction of the refracted ray for each given direction of incidence, when you know the law of the final velocity of the luminous molecules in the interior of the medium, at a sensible distance from the refracting surface.

In the case of ordinary refraction the final velocity is constant; for the deviation of the ordinary ray is the same in a given substance in whatever direction the experiment be made, provided the angle of incidence and the nature of the ambient medium be unchanged. Accordingly if the interior velocity is supposed to be constant, the equations deduced from the principle of the least action, show that the refraction takes place in the same plain as the incidence, and that the ratio of the sines is invariable, as it appears to be from all observations hitherto made.

Reasoning by analogy, it appeared natural to suppose that the extraordinary refraction was produced by a velocity varying according to the inclination of a ray to the axes of the crystal. Now taking at first crystals with one axis, we have seen that the extraordinary refraction takes place symmetrically all round the axis, that it disappears when a ray lies along the axis, and is at its maximum when they are at right angles. We must then, in the case of these crystals, limit ourselves to the laws of velocity that satisfy these conditions. M. Laplace has tried the following:

V²=v²+Ksinθ²,

where $v$ represents the ordinary velocity, $V$ the extraordinary, $\theta$ the angle between the extraordinary ray and the axis, and $K$ is a coefficient which is constant for any one given crystal. Introducing this law of the velocity in the equations of the principle of least action, he obtained immediately Huygens's law. This law had been completely verified only for Iceland spar, but M. Biot has found it true for quartz and beril; only the coefficient $K$ is positive in crystals of attractive double refraction, and negative in the others. Its absolute value is different in different substances, and it is even found to vary in specimens of the same mineralogical species; but with these modifications it is probable that the law applies equally to all crystals with one axis.

As to those having two axes, it is clear that the extraordinary velocity $V$ must depend on the two angles $\theta$ and $\theta '$ made by the refracted ray with the two axes. Analogy leads us to try whether the square of the velocity $V$ cannot be expressed here also by a function of the second degree, but more general, that is, depending on both the angles; now in such crystals the refractions become equal when the ray coincides with one or the other axis. This proves that the ordinary velocity must then be equal to the ordinary. This condition limits the generality of the function, and reduces it to the following form:

$V^{2}=v^{2}+K.\sin .\theta .\sin \theta ',$

that is, there must remain only the product of the two sines. Introducing this formula into the equations of the principle of least action, the path and motion of the rays is found for all cases, and it remains only to try whether it is conformable to experiment. M. Biot has done this for the white topaz which has two axes of double refraction, and the formula agreed perfectly with observation. One may, besides, judging by other phænomena that will be hereafter indicated, be convinced that the same law applies to other crystals with two axes on which experiments have not yet been made; and it is highly probable that it is universally applicable.

It may be remarked that the general law comprises Huyghens's as a particular case, for crystals with only one axis, considering these as having two axes which coincide, for then $\theta$ and $\theta '$ become equal, and the equation for $V$ contains the square of $\sin \theta .$

It will be seen farther on that the same analogy extends also to another species of action that crystallized substances exert on light, which will be explained in the following article.

Polarization of Light.

The polarization of light is a property discovered by Malus, which consists in certain affections that the rays of light assume on being reflected by polished surfaces, or refracted by these same surfaces, or transmitted through substances possessing double refraction.

Though it would be impossible here to give a complete exposition of the details of these phænomena, we will at least describe some of the experiments by which they may be exhibited.

The first and principal of these consists in giving to light a modification, such that the rays composing a pencil will all escape reflexion when they fall on a reflecting surface under certain circumstances.

As an instance, suppose a beam of sun-light $SI$ (Fig. 220.) falls on the first surface $LL$ of a plate of glass, smooth but not silvered, making with the surface an angle of 35° 25′: it will be reflected in the direction $II'$, making the angle of reflexion equal to that of incidence. Let it then be received on another plate of glass, smooth but unsilvered, like the former; generally speaking it will be again reflected with a partial loss. But the reflexion will cease altogether if the second glass be placed like the first, at an angle of 35° 25′ to the line $II'$, provided also it be so turned that the second reflexion take place in a plane $I'L$ perpendicular to that of the first, $SIL$.

In order to make this disposition of the glasses more clearly intelligible, we may imagine that $II'$ is a vertical line, that $IS$ lies north and south, and $I'L$ east and west.

Before we enter upon the inferences to be drawn from this remarkable experiment, I will make a few observations on the manner of performing it conveniently and accurately.

Many kinds of apparatus may be devised to attain this end. That which M. Biot usually employs, is represented in Fig. 221. It is very simple, and is sufficient for all experiment on polarization. It consists of a tube TT′, to the ends of which are fixed two collars which turn with sufficient friction to keep them fast in any position. Each of them bears a circular division which marks degrees. From two opposite points of their circumference proceed two brass stems TV, T′V′, parallel to the axis of the tube, and between them is suspended a brass ring AA, which can turn about an axis XX perpendicular to the common direction of the stems. The motion of the ring is likewise measured by a circular graduation, and it may be confined in any position by screws. When a plate of glass is to be exposed to the light, it must be fixed on the surface of the ring; then it may be placed in any situation whatever with respect to the rays of light which pass through the tube; for the collar, turning circularly round the tube, brings the reflecting plane into all possible directions, preserving a constant inclination to the axis, and this inclination may be varied by means of the proper motion of the ring round its axis XX. The graduated circle which regulates this motion should mark zero when the plane of the ring is perpendicular to the axis of the tube, and the divisions on the two collars should have their zeros on the same straight line parallel to the axis. In constructing the apparatus one should take care that these conditions are fulfilled; but it is of no great consequence that they be so exactly, as any error may be compensated by repeating each observation on both sides of the axis, and taking the mean of the numbers of degrees found in the two opposite positions.

If it be desired, for instance, to repeat Malus's experiment described above, a plate of glass must be placed on each ring, and they must be disposed so as to be inclined to the axis at angles of 35° 25′. Then the graduated circle of one of the collars must be brought to mark zero, and the other 90°, that the places of reflexion may be perpendicular to each other. The tube must then be secured, and a candle placed at some distance in such a position that its rays may be reflected by the glass along the axis TT′. This will happen when on looking through the tube the reflexion of the candle is seen in the first glass. Every thing being thus arranged, the reflected rays will meet the second glass at the same angle of 35° 25′; then according to the different positions given to the collar T′T′ which carries this glass, the light proceeding from the second reflexion will be more or less intense, and there will be two particular positions in which there will be no rays reflected at all, of those at least which are regularly reflected by the first glass. Care must be taken to put a dark object behind the glass L′L′ on the side opposite to the reflected light, in order to intercept the extraneous rays which might be sent on this side from exterior objects, and which, passing through the glass, and arriving at the eye, would mix with the reflected rays that are the subject of the observation. The same precaution should be taken for the glass LL; and indeed as this is never used except to reflect light at its first surface, the back of it may be blackened once for all with Indian ink, or smoked over a lamp; it would not do to silver it for a reason that will be given hereafter.

For the light of the candle mentioned above may be substituted that coming from the atmosphere, which may be received into the tube when reflected by the first glass LL; but in this case to preserve to the rays the precise inclination required for the phænomenon, the field of the tube should be limited by some diaphragms with very small apertures placed inside it. The first glass should be blackened or smoked as before mentioned to intercept any rays that might come by refraction from objects situated under it. In this manner, on looking through the tube, when the glass LL is turned towards the sky a small brilliant white speck will be seen, on which all the experiments may be made. The perfect whiteness of this spot is a great advantage; it is an indispensable qualification in many cases, where different tints are to be observed and compared: it is impossible to succeed as well with the flame of a candle or any other inflamed substance, as none of these flames are perfectly white. Lastly, the brightness of the incident light must be modified, so that the portion irregularly reflected by the two glasses may not be sensible; for this portion, being after such reflexion in the state of radiant light, cannot be polarized in one single direction: the other part, which is regularly reflected, alone undergoes polarization, and therefore alone escapes reflexion at the second glass.

Whatever be the nature of the apparatus employed, the process will always be the same, and the same phænomena of reflexion will be observed on the second glass. To exhibit them in a methodical manner, which will allow us easily to take them all in at one view, we will suppose, as above, that $STL$ the plane of incidence of the light on the first glass coincides with that of the meridian, and that the reflected ray $II'$ is vertical. Then if the collar $T'T'$ which bears the second glass be turned round, this glass will also turn all round the reflected ray, making always the same angle with it, and the second reflexion will be directed successively to all the different points of the horizon: this being premised, the phænomena that will be observed are as follows:

When the second or lower glass is placed so that the second reflexion takes place in the plane of the meridian like the first, the intensity of the light finally reflected is at its maximum. As this glass is turned round it reflects less and less of the light thrown on it.

Finally, when the lower glass faces the east or west point, the light passes altogether through it without being reflected at either surface.

If the collar be turned still farther round, the same phænomena recur in an inverse order, that is, the intensity of the light reflected increases by the same degrees as it before diminished, and attains the same maximum state when directed towards the meridian, and so on through the whole circle.

It appears then, that during a whole revolution of the glass the intensity of the reflected light has two maxima answering to the azimuths 0 and 180°, and two minima answering to 90° and 270°. Moreover, the variations are quite similar on different sides of these positions. These conditions will be completely satisfied by supposing, as Malus does, that the intensity varies as the square of the cosine of the angle between the first and second planes of reflexion.

The results of this interesting observation being thus collected into one point of view, we may draw this general consequence from them, that a ray reflected by the first surface is not reflected by the second, (under a particular incidence) when it presents its east or west side to the surface, but that in all other positions it is more or less reflected. Now if light be a matter emitted, a ray of light can be nothing else but the rapid succession of a series of molecules, and the sides of it are only the different sides of these molecules. We must therefore necessarily conclude that these have faces endowed with different physical properties, and that in the present case the first reflexion turns towards the same point of space, faces, if not similar, at least endowed with similar properties. This arrangement of the molecules Malus denominated the Polarization of light, assimilating the operation of the first glass to that of a magnet which turns the poles of a number of needles all in the same direction.

Hitherto we have supposed that the incident and reflected rays made angles of 35° 25′ with the glasses: it is indeed only under that angle that the phænomenon takes place completely. If while the first glass remains fixed, the inclination of the second to the ray be ever so little altered, it will be found that the second reflexion will not be entirely destroyed in any position, though it will still be at a minimum in the east and west plane. If again the inclination of the ray to the second glass, being preserved, that on the first be changed, it will be seen that the ray will never pass entirely through the second glass, but the partial reflexions which take place at its surfaces are at a minimum in the above-mentioned position.

Similar phænomena may be produced by means of most transparent substances besides glass. The two planes of reflexion must always be at right angles, but the angle of incidence varies with the substance. According as the refracting power of this is greater or less than that of the ambient medium, the angle of polarization, measured from the surface, is greater or less than half a right angle. We have seen that for glass this angle is 35° 25′: for sulphate of barytes it is only 32°, and for diamond only 23°. If glass plates be placed in essential oil of turpentine which has a refracting power almost exactly equal to that of glass, the angle of polarization will be found to differ very little indeed from 45°. The reflexion at the second surface is supposed to take place on the ambient medium which bounds the glass. In general, according to an ingenious remark of Dr. Brewster's, the angle of polarization is characterised by the reflected ray being perpendicular to the refracted. The angles calculated on this hypothesis agree singularly well with experiment, and also confirm the rule given above for the different magnitudes of them, as will easily appear from Figs. 222, 223, and 224^{[errata 2]}, in which the refracting power is supposed to be respectively greater than unity, equal to that number, and less than it.

This law applies equally well to substances which, like the diamond and sulphur, never produce more than an incomplete polarization, for the quantity of light reflected is invariably a minimum for the angle so determined.

If the mode of observation which we have applied to smooth glass plates be universally employed, it may serve to show that polarization when complete is always a modification exactly of the same kind, for all substances: for when a beam of light has been once polarized, it will equally pass through all substances, with the exception mentioned above, provided each be presented to it under its proper angle, and whatever be the nature of the first or second substance employed, the variation of intensity in the light after the second reflexion is always subject to the same laws.

To represent these circumstances geometrically, let us consider a ray $II'$ (Fig 225^{[errata 3]}.) polarized by reflexion on a glass plate $LL$, and through any one of the molecules composing it, let there be drawn three rectangular axes $cz,cx,cy$, the first coinciding with the ray, the second in the plane of reflexion $SIC$, the third perpendicular to both the others. Then when the ray $II'$ meets a second glass $L'L'$ placed so as to produce no reflexion, the reflecting forces which emanate perpendicularly from the glass, must be perpendicular to the axis $cx$; moreover they must act equally on molecules lying towards $cx$, and towards $cx'$, for if the glass be turned a little from the position of no reflexion, the effects are found to be symmetrical on all sides of that position. The action, therefore, of these reflecting forces, in this position, cannot make the axis $xcx'$ turn either to the right or left, any more than the force of gravity can turn a horizontal lever with equal arms. They cannot bring the axis into their own plane, in which we see it was in the first reflexion, by which the polarization took place on the glass $LL$. This proves that it is on that axis that the properties of the luminous molecules depend. We will for that reason call it the axis of polarization, and suppose its direction similarly and invariably determined for each molecule. Farther, for the sake of conciseness, we will call $cz$ the axis of translation; but we do not suppose this invariable in each molecule, and we will consider it only as relative to its actual direction, in order to leave each molecule at liberty to turn round its axis of polarization. According to these definitions all the results that we have hitherto obtained may be enounced very simply and clearly in the following manner:

When a ray of light is reflected by a polished surface, under the angle which produces complete polarization, the axis of polarization of every reflected molecule is situated in the plane of reflexion, and perpendicular to the actual axis of translation of that molecule.

If the incident molecules are turned so that this condition cannot possibly be fulfilled, they will not be reflected, at least under the angle of complete polarization. That happens when the axis of polarization of an incident molecule is perpendicular to the plane of incidence, the angle of incidence being properly determined à priori.

Generally speaking, when a polished surface receives a polarized ray under the angle at which it would itself produce polarization, if it be made to turn round the ray without changing that angle, the quantity of light reflected in different positions varies as the square of the cosine of the angle between the plane of incidence, and the axis of polarization.

When a ray of light has undergone polarization in a certain direction, by the process above described, it carries that property with it, and preserves it without sensible alteration, when made to pass perpendicularly through even considerable thicknesses of air, water, and, in general, any substance that exerts only single refraction, but double refracting media alter, in general, the polarization of a ray, and in a manner, to all appearance, sudden, communicating to it a new polarization of the same nature in a different direction. It is only when crystals are held in certain directions, that the ray can escape this disturbing influence. Let us endeavour to compare more closely these two kinds of action.

That of single-axed crystals has been studied by Malus, who has comprised its effects in the following law. When a pencil of light naturally emanating from a luminous body, passes through a single-axed crystal, and is divided into two pencils having different directions, each of these pencils is polarized in one single direction; the ordinary one in the plane passing through its direction and a line parallel to the axis of the crystal, the extraordinary one perpendicularly to a plane similarly situated with respect to its direction. Either of these rays, when received on a plate of glass after its emergence, shows all the characters of polarization that we have described.

This law subsists equally, when the ray has been polarized by reflexion before its passage through the crystal. The two refracted pencils are always polarized, as if they had been composed of direct rays, but their relative intensities differ according to the direction of the primitive polarization given to them; this direction must therefore have predisposed the particles to undergo in preference one or other of the refractions.

These two laws were discovered by Malus. The analogy remarked above, between the single and double-axed crystals indicates sufficiently how it is to be extended to the latter; to find the direction of polarization for the ordinary pencil, draw a plane through its direction, and through each of the axes of the crystal. If either of these axes existed alone, the ordinary pencil would be polarized in the plane belonging to it. Now it is really found polarized in a plane intermediate to those two, and the extraordinary pencil perpendicularly to the analogous plane drawn through its direction between the two planes containing the axes. If the angle between these be equal to nothing, the crystal is single-axed, and the direction of polarization is conformable to Malus's indications: this law has been directly verified on the two pencils refracted by the topaz; as for other crystals in which it has not been possible to verify it directly, we may, by the consideration of some other phænomena that will shortly be mentioned, judge that it applies to them also.

These laws of polarization are applicable in all cases where the two pencils transmitted by a crystal are observed separately, but when they are received simultaneously, and in nearly the same direction, that of their apparent polarization is found to be modified, and at the same time their coincidence produces certain colours, which M. Arago first observed, and of which M. Biot determined the experimental laws. The most simple arrangement to exhibit these colours, is to place a thin lamina of some crystallized substance, in the direction of a white ray, previously polarized by reflexion, and to analyze the transmitted light by means of a double-refracting prism. The light is thus separated into two portions, of which the colours are complementary to each other, and identical with those of the rings between two glasses. One of these portions appears to have preserved its primitive polarization, whilst the other exhibits a new polarization, of which the direction depends on that given to the axes of the crystal by turning the lamina round in its own plane.

Following gradually in this manner the direction of the polarization given to a molecule of light, transmitted through different thicknesses of a crystalline medium, it will be found to undergo periodical alternations, which, if light be a matter emitted, indicate an oscillatory motion of the axes of the molecules accompanying their progressive motion. M. Biot has designated this fact by the name of moveable polarization, which is merely the expression of results observed.

If the system of undulations be adopted, the colours of the two images may be attributed to the interference of the two pencils into which the incident polarized light separates, in passing through the lamina. This is what Dr. Young does, and it is remarkable that calculations founded on this principle gave him the nature of the tints, and the periods after which they recur, precisely as M. Biot had determined them by experiment. As to the alternations of polarization, they become, in the undulation system, a compound result produced by the mutual influence of the interfering rays, and it is easy to deduce from observation the conditions to which the mixture of the waves must be subjected to produce the new direction of apparent polarization. M. Fresnel has done this, and the indications of his formulæ have been found conformable in all respects to the laws deduced by M. Biot from observation.

These interferences of the rays may be produced without the assistance of crystalline laminæ; we may equally employ thick plates, provided the rays pass through them at very small inclinations to their crystalline axes. If the experiment be made with a conical pencil of light, large enough to give the various rays composing its inclinations sensibly different to the axes, so that they experience double refractions sensibly unequal, these rays, analyzed after they emerge, offer different colours united in the same system of polarization; and the union of these colours forms round the axes coloured zones, the configuration of which indicates the system of polarizing action exerted by the substance under consideration. This kind of experiment is therefore very proper to exhibit the axes and to indicate the mode of polarization with which any given substance affects the rays.

Upon the whole, the interferences of polarized rays offer very remarkable properties, many of which have been discovered and analyzed by Messrs. Arago and Fresnel with great ingenuity and considerable success, but as the limits of this Work do not allow of a full exposition of them, I will only cite one, which is, that rays polarized at right angles do not affect each other when they are made to interfere, whereas they preserve that power when they are polarized in the same direction. It is not only crystalline bodies that modify polarization impressed on the rays of light: Messrs. Malus and Biot found by different experiments made about the same time, that if a ray be refracted successively by several glass plates placed parallel to each other, it will at length be polarized in one single direction perpendicular to the plane of refraction. Malus, by a very ingenious analysis of this phænomenon, has moreover shown that it is progressive, the first glass polarizing a small portion of the incident light, the second a part of that which had escaped the action of the first, and so on. M. Arago, measuring the successive intensities by a method of his own invention has shown that they are exactly equal to the quantity of light polarized in contrary directions at each reflexion. A phænomenon analogous to this is produced naturally in prisms of tourmaline, which appear to be composed of a multitude of smaller prisms, united together, but without any immediate contact. All light passing through one of these prisms perpendicularly is found to be polarized in a direction perpendicular to the edges, so that if two such prisms be placed at right angles, on looking through them a dark spot is seen where they cross. This property of the tourmaline affords a very convenient method to impress on a pencil of rays a polarization in any required direction, or to discover such polarization when it exists.

Moreover, M. Biot has discovered that certain solid bodies, and even certain fluids, possess the faculty of changing progressively polarization previously impressed on rays passing through them; and by an analysis of the phænomena produced by those substances he has shown that the same faculty resides in their smallest molecules, so that they preserve it in all states solid, liquid, and acriform, and even in all combinations into which they may happen to enter. M. Fresnel has found certain analogies between these phænomena and those of double refraction, which seem to connect the two together most intimately through the intermediation of total reflexion.

Since reflexion and refraction, even of the ordinary kind, modify the polarization of light, we may expect to find this effect produced when rays of light are made to pass through media of regularly varying density. It is accordingly found that all transparent bodies which are sufficiently elastic to admit of different positions of their particles round a given state of equilibrium, as glass, crystals, animal jellies, horn, &c. produce phænomena of polarization when they are compressed or expanded, or made unequally dense by being considerably heated and then cooled suddenly and unequally. These phænomena, discovered originally by M. Seebeck, have been since studied and considerably extended by Dr. Brewster, who has moreover remarked, that successive reflexions of light on metallic plates produced phænomena of colours in which both M. Biot and he have recognized all the characters of alternate polarization.

Knowing, by what has preceded, the experimental laws, according to which light is decomposed in crystals endued with double refraction, we may consider these effects as proofs proper to characterise the mode of intimate aggregation of the particles of such bodies, and to give some insight into the nature of their crystalline structure. Light becomes thus, as it were, a delicate sounding instrument with which we probe the substance of matter, and which, insinuating itself between their minutest parts, permits us to study their arrangement at which Mineralogists previously guessed only by inspection of their external forms. M. Biot has shown the use of this method, applying it to a numerous class of minerals designated by the general name of Mica, and he thinks he has decisive reasons to believe that several substances of natures so extremely different as to their composition and structure have been improperly comprised under that name. He has also made use of the phænomena of alternate polarization, to construct an instrument which he calls a colorigrade, which, producing in all cases the same series of colours in exactly the same order, merely by the nature of its construction, affords a mode of designation just as convenient for comparison as that furnished by the thermometer for temperatures.

Many other experiments have been made, and are daily making; many other properties have been discovered in polarized light; but the limits of this Work do not allow us to give any detailed account of them, so that we have been obliged to confine ourselves to the results, which are, perhaps not the most important part of the subject, but the easiest to explain; our aim in this rapid sketch being rather to stimulate than satisfy the desire of knowledge on this branch of science which presents so vast a field for research both in theory and experiment, and which, though so lately discovered, has already furnished some useful applications to Physics and Mineralogy.

TABLE

Of the Refractive and Dispersive Powers of different Substances, with their Densities compared with that of Water which is taken as the Unit.

The substances marked (*) are combustible.

The refraction is supposed to take place between the given substance and a vacuum.

Substance. Ratio of
refraction. Dispersive
power. Density.  ————————————— —————— —————— —————— Chromate of lead (strongest) 2.974 0.4 5.8 Realgar 2.549 0.267 3.4 Chromate of lead (weakest) 2.503 0.262 5.8 *Diamond 2.45 0.038 3.521 *Sulphur (native) 2.115 2.003 Carbonate of lead (strongest) 2.084 0.091 6.071 —— weakest 1.813 0.091 4.000 Garnet 1.815 0.033 3.213 Axinite 1.735 0.030 Calcareous Spar (strongest) 1.665 0.04 2.715 —— weakest 1.519 2.715 *Oil of Cassia 1.641 0.139 Flint glass 1.616 0.048 3.329 —— another kind 1.590 Rock crystal 1.562 0.026 2.653 Rock salt 1.557 0.053 2.130 Canada basalm 1.549 0.045 Crown glass 1.544 0.036 2.642 Selenite 1.536 0.037 2.322 Plate glass 1.527 0.032 2.488 Gum arabic 1.512 0.036 1.452 *Oil of almonds 1.483 0.917 *Oil of turpentine 1.475 0.042 0.869 Borax 1.475 0.030 1.718 Sulphuric acid 1.440 0.031 1.850 Fluor spar 1.436 0.022 3.168 Nitric acid 1.406 0.045 1.217 Muriatic acid 1.374 0.043 1.194 *Alcohol 1.374 0.029 0.825 White of egg 1.361 0.037 1.090 Salt water 1.343 1.026 Water 1.336 0.035 1.000 Ice 1.307 0.930 ——— Air 1.00029 0.0013 Oxygen 1.00028 0.0014 *Hydrogen 1.00014 0.0001 Nitrogen 1.00029 0.0012 Carbonic acid gas 1.00045 0.0018

 

MISCELLANEOUS QUESTIONS.

1. A luminous point is placed at the distance of 5 feet from a concave mirror of one foot radius; to what point will the rays be reflected?

2. In the above instance, determine the initial velocity of the focus, supposing the luminous point to advance towards the mirror at the ratio of 2 feet per second.

3. Supposing two rays inclined to each other at an angle of one degree to fall nearly perpendicularly on a convex mirror of which the radius is 5 feet, placed at 31/2 feet from the point of intersection of the rays; what will be their mutual inclination after reflexion?

4. A light is placed behind a screen, at the distance of two inches from a concave reflector of 9 inches principal focal length; whence do the rays appear to proceed?

5. What is the extreme aberration when rays diverging from a point fall perpendicularly on a convex mirror of 3 inches diameter, and 10 inches focal length, at the distance of 30 feet?

6. Supposing the focal length and aperture to be the same, which gives the greater aberration, a concave mirror or a convex one?

7. A concave mirror being formed by the revolution of an ellipse (whose axes are 3 and 2 feet), about the axis major, to what point will those rays be reflected which diverging from the centre of the figure, fall on the circle whose diameter is the latus rectum?

8. A small object is placed between two plane mirrors inclined to each other, so that a perpendicular drawn from the object to their intersection makes an angle of 3° with one, and of 12° with the other: what is the number of images formed?

9. A straight line is placed at the distance of 3 inches from a concave mirror of 9 inches radius: find the dimensions of the image.

10. Define the image of a portion of a parabola placed before a concave mirror, and having the center of the mirror for its focus.

11. There are three transparent plates A, B, C, bounded by plane surfaces. The ratio of refraction between A and B placed in contact is 4/3, and between A and C 7/8; what is it between B and C?

12. A ray of light falls at an angle of 45° on the plane surface of a refracting medium; the ratio of refraction is 3/2: what is the deviation?

13. What must be the refracting power of a transparent sphere in order that it may just collect parallel rays to a point within itself?

14. At what distance from a luminous point should a convex lens of two feet focal length be placed, in order that the focus of refracted rays may be at the same distance on the other side of it?

15. What equiradial lens is equivalent to a meniscus, the radii of which are 6 and 10 inches?

16. A double convex lens whose thickness is 3 inches and radii 30, and 20, is placed in air: what is its focal length?

17. What is the focal length of a lens composed of water contained between two meniscus-shaped watch-glasses, the radii of the surfaces being 5 and 7 inches, and the thickness supposed inconsiderable.

18. Supposing a diamond sphere to be just inclosed in a cube of glass, what would be the focus of rays incident perpendicularly at one of the points where the surfaces touch?

19. The caustic by refraction of a plane surface, for rays diverging from a point, is the evolute of an ellipse or an hyperbola, according as the passage is into a denser or a rarer medium.

20. What is the form of the caustic produced by a parabolic conoid of glass, the incident rays being all parallel to the axis.

21. A small rectilinear object is placed before a double convex lens, of inconsiderable thickness, inclined to the axis at an angle of 30°, and the distance of its intersection with the axis from the lens, is four times the focal length; shew that the image is an arc of an ellipse, and find the axis major of it.

22. If an object be placed in the principal focus of a convex lens, the visual angle is the same, whatever be the place of the eye on the axis.

23. In a convex lens with surfaces of equal curvature, the spherical aberration exceeds the chromatic, if the semi-aperture be greater than 1/9 of the radius.

24. If the dispersive powers of two prisms be inversely as their refracting angles, they will form an achromatic compound prism, when placed against each other in opposite directions.

25. It is required to achromatize a double concave lens of rock crystal, by means of a meniscus of iceland spar, which is just to fit into it. What must be the radius of the inner surface of the meniscus, supposing those of the concave lens to be each 5 inches, the refracting powers of the substances being 1.547, and 1.657?

26. How many times will the surface of a minute object be magnified by a globule of spirit of wine 1/40th of an inch in diameter, supposing the least distance of correct vision to be 5 inches?

27. What must be the limit of the angular distance of two stars, that they may be both seen at once through an Astronomical telescope 4 feet long which magnifies 47 times?

28. Compare the fields of view of an Astronomical and a Galilean telescope, supposing the object glasses to be each 4 inches in diameter, and of 3 feet focal length, and the eye glasses of 1/4 inch aperture, and 1 inch focal length.

29. How much must the Galilean telescope, mentioned in the last question, be lengthened, to be used as a microscope, supposing that the object to be examined would be placed at 3 ft. 2 in. from the object glass?

30. What is the focal length of the eye-glass of Sir W. Herschel's great telescope? See page 131, Note.

31. Whereabouts should a plane reflector be placed in that telescope, according to Newton's construction, so that the eye glass, of 1 inch focal length, may be set in the side of it?

32. What must be the ratio of refraction for yellow light, between air and water, supposing the radius of the arc of that colour in the secondary rainbow to be 52° 10′.

33. A plane mirror two feet in height is placed against a vertical wall, so that its lower edge is 4 ft. 2 in. from the ground. What part of his figure can a man 6 feet high see in it, when standing upright on the ground, supposing the vertical distance of the eyes from the crown of the head to be 1/16 of the whole height?

34. If parallel rays be incident on a sphere of a given refracting power, find that ray of which, when produced, the part included within the sphere, is to the analogous part of the refracted ray in a given ratio.

35. Given the distance of the points of incidence of two parallel rays on a transparent sphere, one of which passes through the center; required the distance between the points of emergence.

36. Given the apparent perpendicular depth of a fish under water; find the direction in which an arrow should be shot to hit it.

37. Let the surface of a plane reflector be always perpendicular to a line which revolves about one of its extremities, and cuts two other lines given in position; it is required to determine the focus of the reflector, so that an object moving in the intersection of the revolving line, with one of the given lines, the image shall move in its intersection with the other.

38. If a ray of light refracted into a sphere emerge from it after any given number of reflexions; determine the distance of the incident ray from the axis, when the arc of the circle intercepted between the axis and the point of emergence, is a minimum.

 

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1. In the Annals of Philosophy for Nov. 1818, there is an account of some barometrical measurements on the Jura mountains, in which it is stated, that a signal in form of a pyramid, and a large poplar tree which stood by it, presented in the different parts of the day, the extraordinary varieties of appearance represented in Figs. 198, 199, 200, 201.
2. These arcs are considered as parts of small circles of the celestial sphere, and the radius is the distance of each from its pole.
3. Of the same nature are the coloured stripes often seen in cracked ice, in transparent calcareous spar, selenite, and other substances.
4. This phænomenon may be observed in the sounds produced by organ-pipes when filled with successive strata of gases of unequal densities, for instance, with atmospheric air and hydrogen. The sounds which should be produced under such circumstances have been calculatad by Mr. Poisson, and his results agree perfectly with experiment.
5. This important remark of the connexion between the primitive form of a crystal, and its single or double refraction, is due to Dufay, who was likewise the discoverer of the distinction between the vitreous and resinous electricities.

Errata

1. Original: innermost, or was amended to innermost arc: detail
2. Original: 220, 221, and 222 was amended to 222, 223, and 224: detail
3. Original: 223 was amended to 225: detail