The supernumerary colours of the third and fourth bows will be equally imperceptible with the bows themselves; but the portions of light, four times reflected, will cross each other in the point opposite to the sun, where their coincidence will be perfect, and at other neighbouring points will afford an interval nearly proportional to the distance from that point. We shall find that the intervals for different deviations, supposed to be measured in air, are these:

| Deviation | Angle of Reflection | Interval in parts of the Radius | |-----------|---------------------|--------------------------------| | 180° | 24° 49' | .000 | | 185 | 25° 31' | .006 | | 175 | 24° 7' | .096 | | 190 | 26° 14' | .195 | | 170 | 23° 25' | .096 |

Hence, supposing the first bright or greenish ring to appear at the distance of 5° from the observer's head, the radius of the drops must be about .0000225 = .000234, or 4375 of an inch.

It might be questioned whether the light, five times reflected, could retain sufficient force to produce any sensible effects by these interferences; but since it exhibits no appearance of colour between the primary and secondary rainbows, it must necessarily be extremely faint. The interval which it affords, by the comparison of its two portions, agrees sufficiently well with that which is derived from four reflections, to contribute in some measure to the production of an alternation of light and shade; but the separate colours would be rather weakened than strengthened by the mixture: thus, at the deviation of 5°, the interval is found to become .076 instead of .096; and at 10°, .155 instead of .195; and this difference is too considerable to allow us to expect any material increase of brilliancy from the addition of the fifth reflection, however great its intensity might be.

Supposing, now, a cloud to consist of spherules of which the radius is .000234, we may inquire at what distance from the outer edge of the primary rainbow the first additional red of the supernumerary colours ought to be found: the interval being in parts of the radius .0000266 = 116; and we may infer from the table, by taking the successive differences, that this distance will be about 18°; so that the semidiameter of this red ring will be 42—18 = 24°; and the termination of the primitive band of red, supposing it to extend to one fourth of a complete interval only, will be where the difference is .029, or at 7½°; but for the violet the quarter of the interval will be, in parts of the radius, .0000042 = .0183, which answers to a distance from the edge of about 5½°; and this distance, measured from the edge of the violet, which is somewhat less than 2° within that of the red, will extend nearly to the same point as the red space; so that we shall have a circle about 70° in diameter, at the circumference of which all the colours will be united, and which will consequently be white. This magnitude agrees tolerably well with the direct observations of the phenomenon; and if we wish to make the agreement more complete, we have only to suppose the drops a little smaller, and the coloured glories, which they are capable of affording, a little larger. It has already been remarked, that the non-appearance of the ordinary rainbow, in this case, must be referred to the operation of something like diffraction; although it is obvious that its form, under such circumstances, would necessarily be somewhat modified by the diffusion of the colours through a greater space than that which they ordinarily occupy.

SECT. VIII.—Of the Colours of Striated Substances.

It was observed by Boyle that small scratches of any kind on the surfaces of polished substances exhibited, when viewed in the sunshine, a variety of changeable colours; and the observation may easily be repeated with any piece of metal not too highly polished, and placed in a strong but limited light. Dr Young ascertained by experiment that the colours afforded by some regular lines drawn on glass always corresponded to an interval, varying as the sine of the angle of deviation from the position in which an image of the luminous object was exhibited by the regular reflection of the surface; and it is easily shown, that if we suppose two portions of light to be reflected from the opposite edges of the furrow, the difference of their paths must vary in that proportion. Dr Young had conjectured that the colours of the integuments of some of the coleopterous insects might be derived from furrows of this nature; but the conjecture has not been verified by observation. Dr Brewster has, however, very unexpectedly discovered that some similar inequalities are the cause of the colours exhibited by mother of pearl; and he has confirmed the observation by showing that impressions of the surface of this substance taken in black wax, in a hard cement, or in fusible metal, will often exhibit a similar appearance. Where the form of the surface of the mother of pearl is the most regular, it reflects, in an oblique light, a white image of a luminous object, like that which any other polished substance affords; but on one side of this image only, and at some little distance from it, we may observe the first order of recurrent colours, beginning from violet, and occasioned in all probability by the reflections from one side only of an infinite number of parallel striæ, formed by the terminations of a minute lamellated structure, nearly but not perfectly perpendicular to the general surface; one side only of each of the little furrows being situated in such a direction as to reflect an image of the luminous object to the eye, and at such a distance that the whole may constitute a regular series of equal intervals. By transmitted light, this substance generally appears of a red or a green colour, changing more or less according to the obliquity, and apparently belonging to some of the higher orders of recurrent colours.

Dr Young has observed a series of these colours, produced by the parallel lines of some of Coventry's glass micrometers, drawn at the distance of \( \frac{1}{2} \) of an inch from each other, in which the first bright space, or the confine between the green and the red, corresponded to the interval of \( \frac{1}{2} \) of an inch, or .0000232 (Medical Literature, p. 559); and this result agrees very accurately with the general theory, the interval for the yellow, derived from Newton's measurements, being .0000235; but in general these lines exhibit colours much more widely extended, each separate line consisting in reality of two or more scratches at a minute distance from each other.

There is a remarkable peculiarity in the appearance both of these colours, and of those which are exhibited by substances naturally striated, as by mother of pearl, agate, and some other semitransparent stones; they lose the mixed character of periodical colours, and resemble much more the ordinary prismatic spectrum, with intervals completely dark interposed. This circumstance may be satisfactorily deduced from the general law, if we consider that each interference depends not only on two por- tions separated by a simple interval, but also on a number of other neighbouring portions, separated by other intervals which are its multiples; so that unless the difference of the two paths agrees very exactly with the interval appropriate to each ray, the excess or defect being multiplied in the repetitions, the colour will disappear; consequently each of the stripes, which in other cases divide the space in which they appear almost equally between light and darkness, when homogeneous light is employed, becomes here a narrow line; and their succession affords a spectrum exhibiting very little mixture of the neighbouring colours with each other, and nearly resembling that which is afforded by the simple dispersion of the prism; except that, as in all other phenomena of periodical colours, the blue and violet portions are much more contracted than in the common spectrum.

Sect. IX.—Of the Colours of Mirrors, and of thick Plates.

In all the species of periodical colours which have been described, the two portions of light concerned have both been regularly reflected from different surfaces. The methodical division of the subject now leads us to the consideration of the colours exhibited in light separately reflected from the same surface. These may be denominated in general the colours of mirrors; and they will include as a variety those which are called by Newton the colours of thick plates.

The general character of these colours is, that they are observed in light reflected by small particles, or irregularly dissipated by a single surface, first in the passage of the beam of light towards the mirror, and then in its return; the difference of the length of their paths affording, as usual, the interval of retardation. Thus, in Dr Herschel's experiment of scattering a fine powder in a beam of light reflected perpendicularly by a concave mirror, and received on a screen in its return, it may easily be shown that the colours will be precisely such as would be exhibited by light transmitted through a thin plate of air, everywhere half as thick as the plate limited by two spherical surfaces in contact; the centre of the one surface being the particle of powder, and that of the other its image formed by the mirror. For in the direction of the principal ray, which is perpendicular to the mirror, the paths of the light will be of equal length, whether the dissipation takes place before or after the reflection; and in other parts the whole length of the path of the light passing from any local point to its conjugate focus being the same, according to the definition of a conjugate focus in the Huygenian theory, from whatever point of the mirror it may be reflected, the light first dissipated will have advanced, after its reflection, as far as the circumference of a circle, of which the conjugate focus is the centre, at the same instant that the portion coming directly from the powder, after a previous reflection, will reach the circumference of the circle of which the particle of powder is the centre; so that the distance between these two circles must be the difference of the paths of the two portions, and the colours the same as would be exhibited by a plate of air of half the thickness, since such a plate is twice traversed by the retarded light.

A similar appearance of colours had been obtained, by earlier experimenters, from the interposition of a screen of gauze, or of a semitransparent substance, in the path of the beam falling on the mirror. But the colours of thick plates, observed by Newton, are modified by the nature of the transparent substance employed, and by the obliquity of the refracted light. The dissipation here takes place at the anterior surface of a concave mirror of glass, and the reflection at the posterior, which is coated with quicksilver; and if these two portions proceed, each with a slight Chromatic divergence, from a perforation in a screen situated near the centre of curvature of the mirror, they will co-operate perfectly with each other in the circumference of a circle described on the screen, of which the diameter is the distance of the perforation from its image; since all the light passing, in any given section of the mirror, with the same obliquity through the glass as the beam itself passes in the principal section, must be collected into a focal point situated in some part of this circle, and will arrive at this point at the same time, whatever its situation in the section may have been; the obliquity of the incident light being the same in every part of the section, because the point of divergence is at the same distance from the mirror as the centre of curvature. For the other parts of the dissipated light, passing with different obliquities, the interval will be determined by the difference between the lengths of the paths of the two portions of light arriving at the given point, the one by regular refraction, after being first dissipated and then reflected; the other by dissipation, after being first regularly refracted and reflected. And this interval agrees precisely with the law which Newton has deduced from his experiments; but the analogy which he infers from it, between these colours and those of thin plates, is in fact very far from amounting to identity; since, if they belonged to the ordinary colours of thin plates, there is no reason why the series should begin anew from a certain arbitrary thickness, differing in every different experiment, which affords a white of the first order.

Sect. X.—Of the Colours of deflected Light.

We are next to examine the case of light only once reflected, and interfering with a portion of the same beam which has pursued its course without interruption; a case which would scarcely have required a separate consideration, but from the difficulty of including it in a general definition with any others; although it is comprehended in the Newtonian description of the colours of inflected light; but since the light is in this case turned away from the substance near which it passes, it may more properly be termed deflected, especially as the greater number of the appearances mentioned by Newton as depending on inflection, belong more properly to diffraction, and the term inflection might consequently be misunderstood as relating to them.

When a beam of light is received in a dark room, and suffered to fall upon the edges of two extremely sharp knives or razors, meeting each other in a very acute angle, the shadows of the knives, received on a screen at some distance, will be found to be bordered by several fringes of colours; and the angle will be bisected by a dark line. The distances from the shadows at which these fringes appear agree in general with the supposition of their depending on the interference of the light reflected from the edges of the respective knives, with the uninterrupted light of the beam passing between them; but the coincidence of these portions ought to be perfect in the immediate neighbourhood of the point in which the shadows meet, and the two last bright fringes ought to unite there in an angle of light. This, however, does not happen, on account of the modification of the general law (C), which makes it necessary to allow half an interval for the effect of a very oblique reflection; and for the same reason, the space immediately next to the shadow is always dark instead of being light. If the knives are at all blunt, the reflection from one to the other, where they meet, causes the bisecting dark line to disappear; but this source of error may be avoided by causing one of them to advance a little before the plane of the other. Mr Fresnel has repeated these experiments with all possible care, and has ascertained that the points in which the fringes of any one colour are found, at different distances from their origin, belong always to a hyperbola, as they ought to do according to the calculation founded on the general law of interference; a fact which had before been inferred from other measurements, but which had not been so distinctly proved by direct experiments. Newton himself, indeed, was so far from believing that these fringes are rectilinear, as Mr Fresnel supposes, that he expressly mentions their curvature, and infers from it that they are not derived from "the same light" in all their parts; imagining, perhaps, that each fringe was of the nature of a caustic line, formed by reflection or refraction, in which the light is everywhere more condensed than in the collateral spaces, but which is by no means necessarily straight. Mr Fresnel has also shown that all the fringes are found exactly at such distances from the true shadow, as would be inferred from the supposition of the loss of half an interval by reflection; while some of the experiments of Newton appeared to indicate a deviation from this law.

It has been asserted that fringes of the same kind have been observed at the edges of a detached beam of light, reflected into a dark space by a narrow plane and polished surface; and in this case it would be difficult to point out in what manner the supposed oblique reflection could be produced, or how a diffraction of any kind could cause the light to be redoubled back upon itself; but the experiment does not appear to have been hitherto performed with sufficient attention to all possible sources of error.

Sect. XI.—Of the Colours of diffracted Light; including those of Fibres and of Coronae.

The light reflected from each of the knife edges, in experiments like those of Newton, not only produces colours by its interference with the light proceeding uninterruptedly between them, but also with another portion, diverging from the edge of the opposite knife, and spreading into its shadow. This tendency of light to diffuse itself was first described by Grimaldi, under the appropriate name diffraction; but many of the phenomena in which it is concerned having been attributed by Newton to other causes, he appears almost to have overlooked its existence.

The general law of interference is very directly applicable to all phenomena of this kind; the fringes exhibited are broader in the same proportion as the distance between the edges is narrower; and they always depend on the difference of the distance from the edges as the interval of retardation. It is, however, necessary to suppose the same modification to take place in diffraction as in oblique reflection, half an interval being lost in both cases; since the light which deviates the least from a rectilinear direction, and which is derived from the near approach of the two paths to equality, is always white. But it is remarkable, that when the obliquity becomes a very little greater, the diffracted light seems to change its character in this respect; for the colours occupy the same spaces as would have belonged to them if they had begun from a dark centre, one of the portions only having lost a half interval in comparison with the other; and of this circumstance no explanation has yet been attempted.

The diffraction producing these fringes may easily be detected within the eye itself, by holding any object near it in such a position as to intercept nearly all the light of a candle except a narrow line at the edge: this line will then appear to be accompanied by other lines parallel to it, separated from it by a dark space, and becoming wider when the object is brought nearer to the eye. These fringes must be referred to the light diffracted on one side round the object, so as to be spread on the unenlightened part of the retina, and reflected on the other from the margin of the pupil; for if we employ an object narrower than the pupil, so as to observe them on both sides of it, their magnitude will be altered by any change in the aperture of the pupil, occasioned by admitting light to the opposite eye, or otherwise. In such cases as this, where one of the points of divergence is much nearer to the point of interference than the other, the interval increases more rapidly than the distance from the primitive direction; and the first fringes are much broader than those which succeed them; the mode of their formation approaching to that of the fringes seen in deflected light, commonly called the exterior fringes of the shadow; while the interior fringes belong more immediately to the present subject, that of the colours of diffracted light.

When the distance of the points of divergence is more nearly equal, the one being collateral to the other, the breadth of the successive fringes is also more uniform. Such is the appearance of the colours exhibited by a number of equal fibres held between the eye and a distant luminous object; their origin being identical with those of the fringes produced in the shadows of the knives, except that the diffracted rays come from the remoter side of the fibres, and follow the reflected rays instead of preceding them. These colours may easily be observed by looking at a candle through a lock of fine wool, and still more distinctly by substituting for the wool some of the seeds of the lycopodium, strewed on a piece of glass; and they become very large if we employ a few of the particles of the blood, or the dust of the lycoperdon, or puff ball. Dr Young has made this appearance the foundation of a mode of measuring the fineness of wool, which he has recommended for agricultural purposes, though it seems hitherto to have been found much too delicate to be employed by "the hard hands of peasants" with any advantage. The instrument which he has invented for this examination is called the eriometer, and its scale is calculated to express, in semidiameters of a circle, formed round a central aperture in a card or a plate of brass, and marked by minute perforations, the distance at which the lock of wool must be held, in order that the first bright ring of colours, or the limit of the green and the red surrounding it, may coincide with the circle of points; and the actual measure, expressed by a unit of this scale, is found to agree very nearly with the thirty thousandth of an inch. Thus the particles of water which have been found capable of exhibiting a glory 5° from the shadow of the observer, being about \( \frac{1}{3} \) of an inch in diameter, they would correspond to number fourteen of this scale; and the cotangent of the angle subtended by the semidiameter of the bright circle being fourteen, the angle itself will be about 4°; consequently, if we looked at the sun through such a cloud, he would appear to be surrounded by a bright circle of colours, 8° in diameter, green within and red without, and attended by other colours, more or less distinctly marked, according to the degree of uniformity of the magnitude of the drops. These circles are called coronae; their dimensions vary considerably, but they have seldom been observed quite so large as these drops would make them; and more commonly they seem to depend on drops about a thousandth of an inch in diameter, although it is not easy to ascertain the precise parts of the rings from which the measures have been taken by different observers.

In the shadow of a larger substance, formed in a beam of light admitted into a dark room, these colours are still perceptible, beginning from a white line in the middle; but here both the portions on which they depend are diffracted into the shadow, and beyond its limits they are lost in the stronger light that passes on each side of Their appearance is somewhat modified when the shadow is formed by a body terminating in an angle; for the breadth of the fringes being inversely as the breadth of the object which forms them, it is obvious that this breadth must increase towards the point of the shadow, like the distance of the fringes formed in the shadows of Newton's knives; and the fringes seen within the angle must necessarily assume the character of hyperbolas; nor will this form be materially altered when the angle becomes a right one, as in the crested fringes noticed by Grimaldi, although the steps of the calculation for determining their magnitude are in this case a little more complicated.

We find, in an elegant experiment of Mr Biot, on the fringes produced by diffraction, a singular confirmation of the truth of the theory which derives these colours from the difference of the times occupied in the passage of the different portions of light to the point of interference; although this celebrated author does not seem to have been aware of the nature of the inference which may so naturally be drawn from it. He found that the densities of the substances, from the margin of which the diffracted light originated, had no influence whatever on the appearances produced by them; but when they were formed in the light diffracted from substances placed at one end of a long tube, and observed on a piece of glass fixed at the other end, they became contracted, upon filling the tube with water, in the proportion of four to three; as was to be expected from the diminished velocity which must be attributed, according to the modification of the general law (B), to the passage of the light through a denser medium.

Sect. XII.—Of the Colours of Mixed Plates.

The colours of mixed plates depend partly on diffraction, and partly either on reflection or on direct transmission; but their essential character consists in the different nature of the two mediums through which the light passes after its separation.

When a minute quantity of moisture is interposed between two lenses, it readily divides itself into a great number of smaller portions, scarcely distinguishable by the eye; and the light transmitted through the lenses exhibits rings of colours much larger than those which are ordinarily observed, and depending on the interval afforded by the difference of the velocities in the different mediums, according to the inverse proportion of the refractive densities. If they are viewed in a direct and unconfined light, the rings belong to the series commonly seen by transmission, beginning from a light central spot; both portions passing in this case simply through the separate mediums, and arriving at the eye after some slight diffraction only, which affects both of them in an equal degree; but if a distant dark object is situated immediately behind the lenses, and they are illuminated by a light incident a little obliquely, their character is changed, and they resemble the colours commonly seen by reflection, one of the portions of light being necessarily reflected, as in the case of the colours of deflected light; so that, when the dark object is situated behind one half of the glasses only, we observe the halves of two sets of rings, of opposite characters, exhibiting everywhere tints complementary to each other. The diameters of the rings vary according to the refractive density of the liquid employed, diminishing as that density increases, and becoming much larger when two liquids, incapable of mixing with each other, and differing but little in refractive density, as oil and water, are employed instead of air and a single liquid.

The magnitude of the interval may also depend on that of a minute transparent solid substance, immersed in a liquid, instead of being limited by the distance of the two lenses; thus the dust of the lyceperdon, mixed with water, gives it a purplish hue when seen by indirect, and a greenish by direct light; and when salt is added to the water, or oil is substituted for it, the difference of the velocities being lessened, the colours exhibited rise in the series, as if the plate were made thinner.

Mr Arago has very ingeniously applied the principle of the production of these colours to the construction of an instrument for measuring the refractive densities of different elastic fluids, and of air in different states of humidity; the fluids being contained in two contiguous tubes of a given length, through which the two portions of light are made to pass, previously to their re-union, and to the formation of the bands of colours; and it may easily be conceived, that the delicacy of such a test must be great enough for every determination that can be required, either for the correction of astronomical observations, or for the illustration of the optical properties of chemical compounds.

Sect. XIII.—Of the Laws of the Polarisation of Light.

The colours first observed by Mr Arago in doubly refracting crystals, and since more particularly analysed by Mr Biot, afford by far the most striking and interesting examples of the colours of mixed plates. In order to understand the laws of these phenomena, it is necessary to be previously acquainted with the affections of polarised light, which were first accurately investigated by Malus, and with the theory of extraordinary refraction, derived by Huygens, with equal elegance and precision, from his peculiar hypothesis respecting the nature of the transmission of light.

1. Mr Malus discovered, that at a certain angle of incidence, the light partially reflected by a transparent substance receives a peculiar modification, with respect to the plane of reflection, which is called polarisation in that plane.

2. Dr Brewster observed, that the angle of complete polarisation is such, that the mean direction of the transmitted light is perpendicular to that of the reflected portion; the tangent of the angle of incidence being equal to the index of the refractive density of the medium.

3. A ray of polarised light is again subdivided, in the usual proportion, by a second refraction in the plane of polarisation; but when it is refracted in a plane perpendicular to the plane of polarisation, by a surface properly inclined, there is no partial reflection; and in intermediate positions, the intensity of the reflection is nearly as the square of the cosine of the angular distance of the two planes.

4. A portion of the transmitted light is polarised in a direction perpendicular to that of the plane of refraction; so that none of this portion is reflected by a second surface parallel to the first; and when there are several parallel surfaces in succession, the whole of the transmitted light becomes at last so polarised, that none of it is partially reflected.

5. The same transverse polarisation will happen in a greater number of transmissions, when the angle differs from that of complete polarisation; and in the same manner a second partial reflection, by a surface parallel to the first, will produce a more complete polarisation, when the first is imperfect.

6. A perfect polarisation in any new plane, by a partial reflection at the appropriate angle, completely supersedes the former polarisation; but a reflection or refraction void of any polarising effect, which may be called a neutral reflection or refraction, changes the direction of the plane of polarisation, according to Mr Biot's experiments, into that of the image of the former plane, supposed to be formed by the action of the given surface.

7. The light ordinarily refracted by a doubling crystal in the plane of the principal section of the crystal, passing through its axis, is polarised in that direction; the light extraordinary refracted, in the transverse direction.

8. Light previously polarised is transmitted by the ordinary refraction when its plane of polarisation coincides with the principal section, and by the extraordinary when it is perpendicular to it. In intermediate directions, the quantity of light transmitted by each refraction is, according to Malus, as the square of the cosine and sine of the angle formed by the planes passing through the paths of the ray, and a line parallel to the axis in each crystal, supposing the species of refraction to be exchanged.

9. The rays of light ordinarily transmitted by doubling crystals appear in general to retain their previous polarisation, like rays transmitted through simple substances; but the extraordinary refraction polarises them, according to Biot, like a neutral reflection at a surface coinciding with the principal section; the new plane of polarisation taking the place of the image of the former.

10. Reflections at metallic surfaces are generally neutral with respect to polarisation; but in oblique planes they seem, according to some experiments of Malus, to mix or depolarise the light subjected to them.

Sect. XIV.—Of the Laws of Extraordinary Refraction.

The extraordinary refraction of regular doubling crystals may be correctly determined in all circumstances, by means of the Huygenian supposition of an undulation diverging in the form of a spheroid from every point of the medium, the velocity in any given direction being always proportional to the corresponding diameter, so that the successive spheroidal surfaces remain always similar to each other. The relations of the angles of incidence and refraction may be calculated by finding the point in which any of the spheroids, supposed to represent the forms of the elementary undulations, at a given instant, is touched by a plane passing through that point of the surface at which the original beam of light would have arrived, at the same instant, through the external medium; it may also be deduced, somewhat more simply, from the determination of the velocity with which an expanding spheroidal undulation must extend itself on any given surface; a velocity which immediately gives us the direction of the ray in the surrounding medium; and the relation thus obtained will also obviously hold good with respect to a ray returning in the opposite direction. (Quarterly Review, No. xxii.)

In common refractions, if we compare the space described by an undulation or any given surface with the radius, the velocities appropriate to the different mediums will be represented by the sines of the respective angles; but the velocity with which a spheroidal undulation advances on any surface is evidently determined by the increment, or the fluxion, of the perpendicular to the circumference of the section of the spheroid formed by that surface; and calling this perpendicular \( y \), the velocity may be considered as proportional to its increment \( y' \); but the velocity of the surrounding medium is to that with which the axis \( x \) increases, as \( r \) to 1, \( r \) being the index of the ordinary refractive density of the crystal, compared with that of the surrounding medium, since the velocity in the direction of the axis is the same as that which belongs to the ordinary refractive density; consequently, the increment of the path of the undulation in the surrounding medium will be expressed by \( rx' \), and \( s \), the sine of refraction or incidence without the crystal, will be to

\[ \frac{dx}{dy} = \text{the evanescent increments of any quantities being always in the ratio of their fluxions: and the plane of refraction or incidence, without the crystal, will always be perpendicular to the tangent of the section formed by the refracting surface. The determination of the relation of the angles is therefore reduced to the calculation of the value of } y \text{ and of its fluxion.} \]

Supposing, then, the ratio of the greatest and least refractive densities of the crystal or of the equatorial diameter of the spheroid \( 2AB \) to the axis \( 2AC \) to be that of \( n \) to 1, \( n \) being greater than unity, and the tangent of the angle \( ADE \), formed by the axis with the refracting surface \( DE \), being called \( p \); the magnitude of the semidiameter \( AF \), parallel to the surface, may be found by comparing the secants of the angles \( FAG, HAG \), subtended at the centre by the corresponding ordinates of the ellipse and the inscribed circle; for their tangents \( FG, HG \), being represented by \( P \) and \( \frac{P}{n} \), the secants will be

\[ \sqrt{(1 + p^2)} \text{ and } \sqrt{\left(1 + \frac{PP}{nn}\right)} \]

and the semidiameter of the circle \( AH \) being \( x \), that of the ellipse, \( AF \), will be \( n \sqrt{\frac{1 + PP}{nn + pp}} x \). But the tangent of the angle \( GIF \), made by the tangent of the ellipse with the axis, is to that of the angle made by the corresponding tangent of the circle, \( GIH \) or \( GHA \), that is, \( \frac{n}{P} \) as \( n \) to 1; consequently \( \frac{nn}{P} \) will be the tangent of the angle made with the axis by the elliptic tangent \( IF \) or by the conjugate diameter \( AK \); and if we substitute \( \frac{nn}{P} \) for \( p \), we shall find the length of this diameter \( AK = \sqrt{\frac{n^4 + p^2}{nn + pp}} x \), which is to that of the former \( AF \) in the ratio of \( n \sqrt{\frac{n^4 + p^2}{1 + pp}} \) to 1. Hence, for the lesser semiaxis of the section formed by the given surface \( EL \), calling \( AL \) the distance of its centre from that of the spheroid \( z \), we have the mean proportional between the segments of the diameter \( \sqrt{\left(\frac{AK + AL}{AK - AL}\right)} \cdot \left[\frac{AK - AL}{AK + AL}\right] = \sqrt{(AK^2 - AL^2)} = \sqrt{\left(\frac{n^4 + p^2}{nn + pp}\right)} \), which must be reduced in the ratio of the conjugate diameters \( AK \) and \( AF \), so that it becomes \[ n \sqrt{\left( \frac{1 + pp}{nn + pp} x^2 - \frac{1 + pp}{n^4 + p^2} z^2 \right)} = EL. \]

But from the known similarity of the parallel sections of a spheroid, the axes will be to each other as the semidiameter \( AF = n \sqrt{\frac{1 + pp}{nn + pp}} x \) is to \( nx \) the equatorial semi-diameter, a ratio which may be called that of 1 to \( m \), \( m \) being \( \sqrt{\frac{nn + pp}{1 + pp}} \); so that the lesser axis \( EL \) being

\[ = n \sqrt{\left( \frac{xx}{nn + pp} - \frac{1 + pp}{n^4 + p^2} z^2 \right)}, \]

the greater \( LP \) will be

\[ = n \sqrt{\left( x^2 - \frac{1 + pp}{n^4 + p^2} m^2 z^2 \right)}. \]

Now, if \( q \) be the cotangent of the angle MNE, formed by the plane of the ray's motion in the external medium, with the lesser axis of the section, or the tangent of the angle ELO formed by the conjugate semidiameter LO with the same axis, this semidiameter may be found by substituting \( q \) for \( p \), \( m \) for \( n \), and the value of the semiaxis of the section for \( x \), in the expression for \( AF \), the semidiameter parallel to the refracting surface, and it becomes

\[ m \sqrt{\frac{1 + qq}{mm + qq}} EL = n \sqrt{\frac{1 + qq}{mm + qq} \left( \frac{xx}{nn + pp} - \frac{1 + pp}{n^4 + p^2} z^2 \right)} = LO. \]

Hence, since all parallelograms described about an ellipse are equal, dividing the product of the semiaxes \( EL \cdot LP \) by this semidiameter, we shall have the required perpendicular \( y = MQ = \frac{EL \cdot LP}{LO} = \frac{LP}{m} \sqrt{\frac{mm + qq}{1 + qq}} = \frac{n}{m} \sqrt{\frac{mm + qq}{1 + qq}} \sqrt{\left( x^2 - \frac{1 + pp}{n^4 + p^2} m^2 z^2 \right)}. \)

Now, in order to find the fluxion of this quantity, increasing as the spheroid increases, while the place of the centre of radiation remains unaltered, we must make \( z \) constant while \( x \) varies, and we shall have

\[ dy = \frac{n}{m} \sqrt{\frac{mm + qq}{1 + qq}} \frac{dx}{x} \cdot \sqrt{\left( x^2 - \frac{1 + pp}{n^4 + p^2} m^2 z^2 \right)}, \]

which must be equal to \( \frac{dx}{s} \); consequently,

\[ \sqrt{\left( x^2 - \frac{1 + pp}{n^4 + p^2} m^2 z^2 \right)} = \frac{ns}{mr} \sqrt{\frac{mm + qq}{1 + qq}} x; \]

and the lesser semiaxis of the section \( EL \), which was found

\[ = \frac{n}{m} \sqrt{\left( x^2 - \frac{1 + pp}{n^4 + p^2} m^2 z^2 \right)}, \]

becomes

\[ \frac{mmr}{mm} \sqrt{\frac{mm + qq}{1 + qq}} x, \]

whence the semidiameter \( LM \) at the point of incidence, which may be called \( w \), and which is analogous to the conjugate diameter \( AK \) in the former section, will be

\[ \frac{m^4 + q^2}{mm + qq} \cdot \frac{mmr}{mm} \sqrt{\frac{mm + qq}{1 + qq}} x = \frac{mmr}{mm} \sqrt{\frac{m^4 + q^2}{1 + qq}}. \]

Hence it is obvious that this semidiameter, in any one plane of incidence, will be in a constant proportion to the sines, as Huygens himself demonstrated; so that, supposing \( x \) to be constant, and \( z \) to vary, the semidiameter \( w \) may be considered as an ordinate in the elliptic section passing through the point of incidence \( M \) and the diameter \( AK \) conjugate to the refracting surface, which is also the path of a ray falling perpendicularly on that surface from without; and the tangent of the angle \( ELM \) formed by this semidiameter with the lesser axis of the given section, will be \( \frac{mm}{q} \), which determines the intersection of this oblique plane with the refracting surface.

But in order to find the angle made with the refracting surface in a plane perpendicular to it, we must compute \( LR \), the distance of the centre of the refracting section from the point nearest to the centre of the spheroid; and the tangents of the inclinations of the diameters to the axis being \( p \) and \( \frac{mm}{P} \), that of their mutual inclination will be \( \frac{mm + pp}{P(1-mm)} \).

Since \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \), and the sine of the same angle being expressed by \( \frac{\tan a + \tan b}{\sec a \sec b} \), it becomes here \( \frac{mm + pp}{\sqrt{(1 + p^2)} \sqrt{(n^4 + p^2)}} = \sin FAK = \sin ALR, \)

which we may call \( r \), and the cosine \( \frac{1 - \tan a \tan b}{\sec a \sec b} \)

\[ = \frac{p(1-mm)}{\sqrt{(1 + p^2)} \sqrt{(n^4 + p^2)}} = t; \]

and the required distance \( LR \) will be \( tr \), and the distance of the centre of the spheroid from the refracting surface \( AR = rz \). But \( MS \), the perpendicular falling from the point of incidence on the lesser axis of the section formed by the surface, being called \( u \), the tangent of the angle MLS subtended by it at the centre being \( \frac{mm}{q} \) and its sine consequently \( \frac{mm}{q} \),

we have \( u = \frac{mmr}{\sqrt{(m^4 + q^2)}} = \frac{mm}{r \sqrt{(1 + q)}} \);

and the distance of this perpendicular from the centre, \( LS = v = \frac{qq}{\sqrt{(m^4 + q^2)}} \); or if we call the sine of ordinary refraction \( \frac{s}{r} = g \), and the sine of the inclination of the plane of the ray's motion to the lesser axis \( \frac{1}{\sqrt{(1 + qq)}} = h \), and its cosine \( \frac{q}{\sqrt{(1 + qq)}} = k \), we have \( u = n^2 h r \), and \( v = \frac{mm}{mm} \cdot \frac{qq}{\sqrt{(m^4 + q^2)}} \).

Hence the cotangent of the angle \( ERM \), formed by the line nearest to the ray in the section with the lesser axis, will be \( \frac{v + tz}{u} \) if the value of \( s \) be considered as positive, when the ray is inclined on the refracting surface towards the axis of the crystal; for in this case the sign of \( t \) being negative, \( tz \) or \( LR \) will be subtracted from \( v \) or \( LS \); and the reverse when \( s \) is negative. We have also for the hypotenuse \( RM \), or the distance of the point of incidence from the point nearest to the centre of the spheroid,

\[ \sqrt{(u^2 + [v + tz]^2)}; \]

consequently the tangent of \( RAM \), the angle of incidence or refraction within the crystal, will be \( \sqrt{(u^2 + [v + tz]^2)} \).

Now, since it has been shown that \( \sqrt{(x^2 - \frac{1 + pp}{n^4 + p^2} m^2 z^2)} = \frac{ns}{mr} \sqrt{\frac{mm + qq}{1 + qq}} x \), we have \( z^2 = \frac{n^4 + p^2}{mm(1 + pp)} \left( 1 - \frac{mmr}{mm} \cdot \frac{mm + qq}{1 + qq} \right)^2 \), and the cotangent of the inclination of the plane of refraction \( ERM \), or

\[ \frac{v + tz}{u} = \frac{q}{mm} + \frac{tz}{u}, \]

becomes

\[ \frac{q}{mm} + \frac{p(1-mm)}{m(1+pp)} \sqrt{\left( 1 - \frac{mm}{mm} \left[ m^4k^2 + h^2 \right] \right)} \cdot \frac{1}{mmk}; \]

and since \( r^2z^2 = (m^4 - m^2 \left[ m^4k^2 + h^2 \right]) \), the tangent of the angle of incidence or refraction within the crystal, which is \( \sqrt{\left( \frac{mm}{rrzz} + \frac{ve}{rrzz} + \frac{2t}{rr} \cdot \frac{v}{rr} + \frac{tt}{rr} \right)} \) will be represented by \( \sqrt{\left( \frac{mm}{m^4(m^2 - m^2 \left[ m^4k^2 + h^2 \right])} \right)} \). The value of the perpendicular to the surface, AR or rZ, is also of importance, as immediately indicating, by its proportion to the axis x, the velocity of the undulation in the direction of the depth, which is therefore represented by \( \sqrt{(m^2 - n^2)} \left( m^2 k^2 + h^2 \right) \).

These expressions become somewhat simpler in many cases of common occurrence. Thus, when the axis is parallel to the surface, \( p = 0, m = n, \) and \( t = 0, \) consequently the tangent of refraction is \( \frac{t}{n} \sqrt{1 - \left( \frac{n^2 k^2 + h^2}{m^2 k^2 + h^2} \right)} \), and the perpendicular velocity \( \sqrt{(1 - \left[ \frac{n^2 k^2 + h^2}{m^2 k^2 + h^2} \right])} \).

When the axis is perpendicular to the surface, \( p \) is infinite, \( m = 1, \) and \( t \) is again \( 0; \) and the tangent of the angle of refraction is \( \frac{mng}{\sqrt{(1 - \left[ \frac{n^2 g^2}{m^2 g^2} \right])}}, \) the perpendicular velocity being \( \sqrt{(1 - \left[ \frac{n^2 g^2}{m^2 g^2} \right])}. \)

The retardation, produced by the passage of light through such a plate, being equal to the time occupied within the plate, diminished by a time proportional to the product of the tangent of the angle of refraction and the sine of the angle of incidence (see Sect. V.), it will be expressed, in the case of a plate parallel to the axis, by

\[ \frac{r}{n \sqrt{(1 - \left[ \frac{n^2 k^2 + h^2}{m^2 k^2 + h^2} \right])}} = \frac{s}{s' \sqrt{1 - \left( \frac{n^2 k^2 + h^2}{m^2 k^2 + h^2} \right)}}; \]

and when the axis is perpendicular to the plate, by

\[ \frac{r}{\sqrt{(1 - \left[ \frac{n^2 g^2}{m^2 g^2} \right])}} = \frac{smg}{\sqrt{(1 - \left[ \frac{n^2 g^2}{m^2 g^2} \right])}} = \frac{r}{\sqrt{(1 - \left[ \frac{n^2 g^2}{m^2 g^2} \right])}} = \frac{r}{\sqrt{(1 - \left[ \frac{n^2 g^2}{m^2 g^2} \right])}}. \]

The effect of any small change in the form of the spheroid, on the retardation, may be found from the fluxions of these quantities, supposing \( n \) to vary; which, when properly reduced, making \( n = 1, \) will be

\[ -\frac{1}{\sqrt{(1 - \left[ \frac{n^2 g^2}{m^2 g^2} \right])}} \text{ds}, \text{ and } -\frac{\text{ds}}{\sqrt{(1 - \left[ \frac{n^2 g^2}{m^2 g^2} \right])}} \text{dn} \text{ respectively.} \]

The values of \( r \) and \( n, \) for the principal substances, exhibiting the extraordinary refraction, which have been examined, are these:

| Substance | \( r \) | \( n \) | |--------------------|------------|------------| | Iceland crystal | 1.657 | 1.1140 | | Arragonite | 1.693 | 1.1030 | | Ice | 1.310 | 0.9989 | | Quartz | 1.558 | 0.9944 | | Sulphate of lime | 1.525 | 0.99432 | | Sulphate of barita | 1.635 | 0.99295 |

In mica, according to Mr Biot, and in arragonite, according to Dr Brewster, there are two axes of crystallization; and the refraction of such substances may probably be represented, by supposing all the circular sections of a spheroid to become ellipses, so that the undulation may assume the shape of an almond.

**Sect. XV.—Of the Colours of doubly refracting Substances.**

In the case of doubly refracting substances, the first difficulty is, not to explain why the colours of double lights are sometimes produced, but why they are not more universally observable; since it might naturally be expected, as a consequence of the general law of interference, that two portions of the same beam, passing through a moderately thin plate of such a substance, in paths differing but little from each other, and coinciding again in direction, should, in all common cases, exhibit colours nearly similar to those of ordinary thin plates. It would, however, be difficult to conjecture whether they ought to resemble the colours seen by transmission or by reflection; and the fact is, that both these series of colours are at once produced by the substances in question; but they are so mixed that, without a particular arrangement, they always neutralise each other; and their formation appears to be also limited to certain peculiar conditions of polarisation, consistent with Mr Arago's observation on the non-interference of two portions of light polarised in transverse directions. Several of the cases, indeed, in which they are exhibited remain still involved in some degree of obscurity; but it is easy to analyse the most important of the phenomena, and to reduce them, with great precision, to the general laws of periodical colours.

Mr Malus has demonstrated, by satisfactory experiments, that a beam of light, admitted into a doubly refracting crystal, is as much divided by partial reflection at the second surface as by transmission at the first; the directions and the relative intensities of the two portions being precisely the same as those of the two portions of a ray similarly polarised, and returning to the second surface from without in an equal angle; so that, after a further transmission at the first surface, all the portions become again parallel. When the ray is in the direction of the principal section, there is no separation, each of the pencils proceeding undivided, as they would do if they passed through a second crystal parallel to the first; and the separation becomes most complete when the plane of incidence makes an angle of about 45° with the principal section; each of the portions \( o \) and \( e, \) into which the ray is divided upon its admission, affording then two reflections, \( oO \) and \( oE, \) \( eO \) and \( eE, \) of nearly equal intensity. The times occupied by the portions \( oO, \) \( eE, \) will differ most from each other, while \( oB \) and \( eO \) will describe their paths in equal times of intermediate length; but of these, \( eO \) only will commonly interfere with \( oO, \) which has a similar polarisation in the plane of incidence, and \( eE \) with \( eE, \) both being polarised in a transverse direction; so that we have two series of colours, depending on an equal interval, except so far as they are distinguished by the inversion of one of the portions belonging to the extraordinary reflection, which renders the series of colours exhibited by them similar to that of the colours of common thin plates seen by reflection, while the ordinary reflection exhibits colours analogous to those of thin plates seen by transmission.

Mr Biot's usual mode of exhibiting these colours is to place a thin plate of sulphate of lime, or of any other crystal, on a black substance, to allow it to reflect the white light of the clouds at an angle of incidence of about 55°, and to receive this light on a black glass, at an equal angle of incidence, in a plane transverse to the former, so that the plate may be viewed by reflection in the black glass. In this arrangement, the light reflected from the upper surface of the plate, being polarised in the first plane of reflection, is not reflected by the black glass, and consequently is incapable of rendering the colours less easily perceptible by admixture with them; the beams \( oO \) and \( eO, \) returning by the ordinary reflection, are also similarly polarised, and will be transmitted or absorbed by the glass; but the beams \( oE \) and \( eE, \) being polarised in a transverse direction, will be partially reflected by it, and will exhibit a very brilliant colour, depending on their mutual interference. If, on the contrary, the black glass be turned round the ray, so that the second plane of incidence may coincide with the first, the ordinary rays only will be partially reflected by it, and the complementary colour will be exhibited by the union of the portions \( oO \), \( eO \); but this colour will be less distinct, on account of its mixture with the white light reflected by the first surface.

Appearances of a similar nature may also be observed in the transmitted light, each of the refractions exhibiting the colour complementary to that which it affords by reflection, as happens in the ordinary colours of thin plates; and we must seek for the portions of light which afford them in the successive partial reflections at the two surfaces of the plate, as in the case of the ordinary colours; the light simply transmitted by the separate refractions not exhibiting the ordinary effects of interference, for want of a similarity of polarisation. The obliquity of the incident light produces similar effects on both series.

Under some circumstances of the reflection of rays near the perpendicular, Mr Biot observes that the plate assumes the colour which is usually exhibited by a plate of twice the thickness viewed a little more obliquely; and in such cases it is probable that the polarisation of the beams \( oO \) and \( eE \) has been so modified as to afford a partial interference; and if this is not the true explanation, it will not be difficult to suppose the interval to be doubled in some other manner by a repeated reflection.

The effect of a plate of a double thickness is also produced by two equal and parallel plates, through which the light passes in succession, provided that their axes of crystallization be parallel, and that they be of such a thickness as to exhibit in conjunction a colour more easily observable than those which they afford separately; a condition which is more generally applicable to the case in which the axes are transverse to each other, and one of the thicknesses is to be subtracted from the other; since in this situation the two portions of light must always interchange their refractions, and that which has moved the more slowly in its passage through one of the plates, will move the more rapidly in the other. This result is very accurately confirmed by experiment, and certainly affords a very striking illustration of the truth of the law of interference.

When we wish to examine the effects of the different obliques of the incident light, it is most convenient to employ a beam previously polarised, which renders the separation of the different portions by a subsequent reflection or refraction more easily practicable; and for these purposes we may either make use of plates of black glass, placed in proper situations, or polarising piles, consisting of a number of oblique thin plates, which produce the effect on the light transmitted through them, with less diminution of its intensity than would take place in a single partial reflection. In some cases, also, the light may be analysed, by causing it to pass through a piece of Iceland crystal; or through a thin plate of agate, which Dr Brewster has found to transmit only such light as is polarised in a particular plane.

The measurements of the thickness of plates of doubly refracting substances agree in general very accurately with the various tints exhibited by them in various situations with respect to the axis, and with various obliques of the incident light, according to the theory of periodical colours; and the agreement is always sufficiently perfect to convince us of the dependence of the phenomena on the law of interference, even if it should happen to require some unknown modification in particular cases. In the first place, when the incidence is perpendicular, the thickness of the plates is precisely such as would be inferred from the theory, at least as nearly as the theory is founded on observations sufficiently accurate, although this thickness is often many hundred times as great as that of the thin plates with which it is to be compared; thus the greatest disproportion of the ordinary and extraordinary refraction of rock crystal, according to Malus's experiments, is that of 159 to 160; so that the difference of the times occupied by light in passing through this substance is to the interval, in virtue of which a similar plate exhibits the common colours, as 1 to 320, and to the interval in a plate of crown glass as 1 to 318; while the experiments of Mr Biot make the observed proportion that of 1 to 360; the difference being no greater than would arise from an error of less than a thousandth part of the whole, in the determination of one of the refractive densities.

The effect of the obliquity of the incident light, on the colours exhibited by plates of rock crystal, agrees also perfectly with the theory. The difference of the times required for the ordinary and extraordinary refractions, which is always comparatively small, will vary as the fluxion of the retardation when the obliquity varies; and the sine of ordinary refraction being \( \theta \), the interval will be expressed by \( -r \frac{1-h\theta}{\sqrt{(1-\theta^2)}} \) da when the axis is parallel to the surface of the plate, and by \( -r \frac{\theta}{\sqrt{(1-\theta^2)}} \) da when it is perpendicular. Taking, for example, an experiment of Mr Biot, on a plate in which the axis was nearly perpendicular, the mean angle of refraction being 21° 38' 3", the tint was a reddish white of the seventh order, answering to the reflection from a plate of glass '0000496 of an inch thick, in the experiments of Newton, while the colour exhibited, in a perpendicular light, by a plate of the same crystal, in which the axis was parallel to the surface, would have been expressed by the thickness '000332. In these two cases, the values of the fluxion become \( -rda \) and \( -14633rd \); and reducing the interval '000332 in this proportion, we find '0000486 for the thickness of a plate of glass which ought to exhibit the tint corresponding to the oblique incidence; the difference from the experiment being only one millionth of an inch, which would scarcely make a sensible alteration in the colour observed. When the thickness of such a plate is more considerable, or when the eccentricity of the extraordinary refraction is greater, the colours differ, with the incidence, in different parts of the plate; and they are generally disposed in rings concentric with the axis. These rings have been particularly described by Dr Brewster, as observed in the topaz; they are always interrupted by a dark cross, occasioned by the want of light properly polarised to afford them, in the two transverse directions.

Mr Biot has made a great number of experiments on the colours of the plates of sulphate of lime, in the form denominated Muscovy tale. They exhibit a general agreement with the results of the calculation, particularly with respect to the constancy of the tint, in all moderate obliques, when the inclination of the axis to the plane of incidence is 45°; but in other cases the agreement is somewhat less perfect, and the difference is too great to be attributed altogether to accident. The most probable reason for this irregularity, under circumstances so nearly similar to those which accord with the theory in the case of rock crystal, is the want of a perfect identity of the two refractions, in the direction of the supposed axis; or, in the language employed by Mr Biot with respect to mica, the existence of a double axis of extraordinary refraction; and it is the more credible that such a slight irregularity may have existed in the sulphate of lime without having been observed, as Dr Brewster has detected a similar property in the arragonite, though both Malus and Biot had examined this substance very carefully without being aware of it. The calculation of the extraordinary re- fraction, in such a case, would afford but little additional difficulty; if its characters were well determined: the form in which the undulations must be supposed to diverge might properly be termed an amygdaloid, and the velocities with which the sections formed by the given surface would extend themselves might be deduced from the properties of the ellipse, nearly in the same manner as they have been determined for the spheroid. The difference of the results of the calculation from the spheroid, and of Mr Biot's experiments, or rather of the empirical formula derived from them, may be seen in the subjoined table; the first part of which, deduced from the theory, is applicable to all substances affording a regular extraordinary refraction, when the axis is either perpendicular or parallel to the surface of the plate. The first column of decimals shows the equivalent variation of thickness where the axis is perpendicular to the plate, being equal to \( \frac{2\pi}{\sqrt{(1-\varepsilon^2)}} \) the product of the sine and tangent of refraction; the second represents the variation for an ordinary thin plate, being proportionable to the cosine \( \sqrt{(1-\varepsilon^2)} \); and the subsequent columns are found by adding to the numbers of the second column those of the first, multiplied by \( k^2 \), the square of the sine of the inclination of the plane of incidence to the axis, since

\[ \frac{1-\varepsilon^2}{\sqrt{(1-\varepsilon^2)}} = \sqrt{(1-\varepsilon^2)} + \frac{k^2}{\sqrt{(1-\varepsilon^2)}}. \]

There are also some circumstances in the experiments of Mr Biot on plates of rock crystal cut perpendicularly to the axis, which cannot be sufficiently explained on any hypothesis, without some further investigation. These plates seem to transmit the beam of light subjected to the experiment, without materially altering its polarisation, and then to produce different colours, according to the situation of the substance subsequently employed for analysing the light; so that Mr Biot supposes the rays of light to be turned more or less by the crystal, round an axis situated in the direction of their motion; and he has observed some similar effects in oil of turpentine, and in some other fluids. But it is highly probable that all these phenomena will ultimately be referred to some simpler operation of the general law of interference.

Dr Seebeck and Dr Brewster have discovered appearances of colours, like those of doubly refracting substances, in a number of bodies which can scarcely be supposed to possess any crystalline structure. They are particularly conspicuous in large cubes of glass which have been somewhat suddenly cooled, so that their internal structure has been rendered unequal with regard to tension. The outside of a round mass thus suddenly cooled, being too large for the parts within it, must necessarily be held by them in a state of compression with respect to the direction of the circumference, while they are extended in their turn by its resistance; although in the direction of the diameter the whole will generally be in a state of tension; so that the refractive density may naturally be expected to be somewhat different in different directions, which constitutes the essential character of oblique refraction; and when the proportions of the external parts to the internal are modified by the existence of angles, or other deviations from a spherical form, the arrangement of the tensions must be altered accordingly; and there is no doubt that all the apparently capricious variations of the rings and bands of colours which are observed, might, by a careful and minute examination, be reduced to the natural consequences of these inequalities of density, so far at least as the laws of the extraordinary refraction alone are concerned, although the separation of the light into two portions might still remain unexplained. Effects of the same kind are produced by the temporary operation of partial changes of temperature, producing partial compression and extension of the internal structure of the substance; and even a mechanical force, if sufficiently powerful, when applied externally in a single direction, has been shown, by the same observers, to produce a double refraction; although the difference of the densities thus induced is much too minute to be perceived in any other way than by means of these colours, which are in general so much the more easily seen, as the cause which excites them is the feebler.

Dr Brewster has also shown that the total reflection of light within a denser medium, and the brilliant reflection at the surfaces of some of the metals, are capable of exhibiting some of the appearances of colour; as if the light concerned were divided into two portions, the one partially reflected in the first instance, the other beginning to be refracted, and caused to return by the continued operation of the same power. In the case of silver and gold, it has already been observed that there appear to be two kinds of reflection, occasioning opposite polarities; and these may possibly be concerned in the production of this phenomenon. The original interval appears to be extremely minute, but it is capable of being increased by a repetition of similar reflections, as well as by obliquity of incidence. Mr Biot has also found that such surfaces, combined with plates of doubly refracting substances, either increase or diminish the equivalent thickness, according to the direction of the polarisation which they occasion. In these, and in a variety of similar investigations, a rich harvest is opened, to be reaped by the enlightened labours of future observers; and the more difficulty we find in fully explaining the facts, upon the general principles hitherto established, the more reason there is to hope for an extension of the bounds of our knowledge of the optical properties of matter, and of all the laws of nature connected with them, when the examination of these apparent anomalies shall have been still more diligently pursued.

Sect. XVI.—Of the Nature of Light and Colours.

Notwithstanding the acknowledged impossibility of fully explaining all the phenomena of light and colours by any imaginable hypothesis respecting their nature, it is yet practicable to illustrate them very essentially, by a comparison with the known effects of certain mechanical causes, which are observed to act in circumstances somewhat analogous; and, as far as a theory will enable us to connect with each other a variety of facts, it is perfectly justifiable to employ it hypothetically, as a temporary expedient for assisting the memory and the judgment, until all doubts are removed respecting its actual foundation in truth and nature. Whether, therefore, light may consist merely in the projection of detached particles with a certain velocity, as some of the most celebrated philosophers of modern times assert; or whether in the undulations of a certain ethereal medium, as Hooke and Huygens maintained; or whether, as Sir Isaac Newton believed, both of these causes are concerned in the phenomena; without positively admitting or rejecting any opinion as demonstrably true or false, it is our duty to inquire what assistance can be given to our conception and recollection, by the adoption of any comparison which may be pointedly applicable even to some insulated facts only. It has, however, been thought desirable to separate this investigation as much as possible from the relation of the facts, in order to avoid confounding the results of observation with the deductions from mere hypothesis; an error which has been committed by some of the latest and most meritorious authors in this department. It may be objected to some of the preceding sections, that this forbearance has not been exercised with respect to the general law of interference and its modifications; but it would have been impossible to give any correct statement of the facts in question, without determining whether the appearances depend upon one or both of the portions of light supposed to be concerned.

Art. I. (Sect. I.) The separation of colours is explained, in the hypothesis of emission, by the supposition of an elective attraction, different in intensity for the different rays of the spectrum; but for this difference no anterior cause is assigned. Any original difference of velocity is contrary to direct experiment; and even the alterations of relative velocity, which must inevitably be occasioned by a variety of astronomical causes, have not been detected by the most accurate observations, instituted for the express purpose of discovering them; so that it has been suggested that there may possibly be a multitude of rays of the same colour, moving with various velocities, and only affecting the sense when they have the velocity appropriate to that colour in the eye. The name of elective attraction is indeed little more than a mode of expressing the fact, without referring it to any simpler mechanical cause; and in chemical elective attractions the substances concerned are under very different circumstances with respect to contact, and with respect to the probable influence of the form and bulk of their integral particles; at the same time it seems impossible to show any absurdity in the supposition of the existence of such an elective attraction with regard to the different kinds of light. On the other hand, if we consider colours as depending on a succession of equal undulations, of different magnitudes as the colours are different, we may discover an analogy, somewhat more approaching to a mechanical explanation, in the motions of waves on the surface of a liquid; the largest waves moving with the greatest rapidity, although the approximate calculation, derived from the most approved theory, leads us to the same expressions for the velocity as are applicable to the transmission of an impulse through an elastic fluid. The fact is, that a larger wave moves more rapidly than a smaller, because the pressure is not precisely limited to a perpendicular direction, as the simplest calculation supposes, but operates also more or less in an oblique direction, principally within a certain angular limit; so that the utmost depth at which any difference of pressure can affect the liquid as a motive force, is that at which this angle may be imagined to comprehend virtually the exact breadth of a wave; and since the velocity depends on the depth of the fluid affected at once by the pressure, the breadth becomes in this manner an element of the determination. Thus also the larger undulations, constituting red light, are found to move more rapidly than those of the violet, which are supposed to be smaller; and there are many ways in which the difference may be supposed to be occasioned, although not depending exactly on the same cause as in the case of the waves on the surface of Chromatics. It is well known that sounds of all kinds move with an equal velocity through the air, and all colours arrive through the supposed elastic ether in the same time from the remotest planets; but a refractive medium, however transparent, is not to be considered as perfectly homogeneous; in many instances, two mediums, of different qualities, seem to pervade every part of a crystal, which is completely uniform in its appearance; and it seems to be necessary, in every case, to suppose the particles of material bodies scattered at considerable distances through a medium which passes freely through their interstices; so that we may conceive the undulations of light to be transmitted partly through the particles themselves, and partly through the intervening spaces, the two portions meeting continually after a certain very minute difference in the length of their paths; we may then suppose the portion transmitted through the interstices to be weakened by the irregularity of its passage, which will affect the smaller undulations more than the larger; and when these portions are combined with the portions more slowly transmitted through the particles themselves, these last will bear a greater proportion to the former in the violet than in the red light, and will have more influence on the ultimate velocity, which will therefore be smaller for the violet than for the red. This explanation may perhaps be far from the best that the hypothesis in question might afford; but it will serve as an illustration of a possible mode, in which the phenomenon may be referred to the established laws of mechanics, without the continual introduction of new principles and properties.

Art. 2. (Sect. IV. A.) Most of the ordinary phenomena of optics are capable of a sufficiently satisfactory explanation, on either of the hypotheses respecting the nature of light and colours; but the laws of interference, which have been shown to be so extensively applicable to the diversified appearances of periodical colours, point very directly to the theory of undulation; so directly, indeed, that their establishment has been considered by many persons on the Continent as almost paramount to the establishment of that theory. It might not, however, be absolutely impossible to invent some suppositions respecting the effects of light, which might partially reconcile these laws to the theory of emission. Thus, if we suppose, with Newton, the projected corpusescles of light to excite sensation by means of the vibrations of the fibres of the retina and of the nerves, we may imagine that such vibrations must be most easily produced by a series of particles following each other at equal distances, each having its appropriate distance in any given medium; it will then be demonstrable that any second series of similar particles interfering with them, in such a manner as to bisect their intervals, will destroy their effect in exciting a vibratory motion, each succeeding particle meeting the fibre at the instant of its return from the excursion occasioned by the stroke of the preceding, and thus annihilating the motive effect of that stroke. But the illustration ends here; for it seems impossible to adapt it to the greater number of the alternations which occur, during the passage of a ray, through a given space, in a denser medium; since it is an indispensable condition of the projectile theory, that the velocity of light should be increased upon its entrance into a medium of greater refractive density. The Newtonian theory, of fits of easy reflection and easy transmission, is still more limited in its application, since it attributes to one portion of light those effects which have been strictly demonstrated to depend on the presence of two.

In the undulatory theory, the analogy between the laws of interference, and the phenomena of the tides, and the effects of the combination of musical sounds, is direct and striking. The existence of an undulation of an elastic medium depends on the recurrence of opposite motions, alternately direct and retrograde, at certain equal distances, in the same manner as a series of waves consists in a number of alternate elevations and depressions, and the succession of the tides in a number of periods of high and low water. The spring and neap tides, derived from the combination of the simple solar and lunar tides, afford a magnificent example of the interference of two immense waves with each other; the spring tide being the joint result of the combination, when they coincide in time and place, and the neap when they succeed each other at the distance of half an interval, so as to leave the effect of their difference only sensible. The tides of the port of Batsha, described and explained by Halley and Newton, exhibit a different modification of the same opposition of undulations; the ordinary periods of high and low water being altogether superseded, on account of the different lengths of the two channels by which the tides arrive affording exactly the half interval which causes the disappearance of the alternation. It may also be very easily observed, by merely throwing two equal stones into a piece of stagnant water, that the circles of waves which they occasion obliterate each other, and leave the surface of the water smooth, in certain lines of a hyperbolic form, while, in other neighbouring parts, the surface exhibits the agitation belonging to both series united. The beating of two musical sounds, nearly in unison with each other, appears also to be an effect exactly resembling the succession of spring and neap tides, which may be considered as the beatings of two undulations related to each other in frequency as 29 to 30; and the combination of these sounds is still more identical with that which this theory attributes to light, since the elementary motions of the particles of the luminiferous medium are supposed to be principally confined to the line of direction of the undulation, while the most sensible effects of the waves depend immediately on their ascent and descent, in a direction perpendicular to that of their progressive motion.

Art. 3. (Sect. IV. B.) The diminution of the velocity of light upon its entrance into a denser medium, in the direct proportion of the refractive density, is one of the fundamental principles of the undulatory theory, and is perfectly inadmissible on the supposition of projected corpuscles. But it must be remembered that the demonstration of the actual existence of this proportion is somewhat indirect, being only derived from the necessity of admitting it in the application of the laws of interference to the observed phenomena; and we have no means of obtaining an immediate measure of the velocity of light in different mediums.

Art. 4. (Sect. IV. C.) The loss of the half interval may be explained, in particular cases, without difficulty, although, in other instances, the circumstances are too complicated to allow us to appreciate their effects. In the direct transmission of a ray of light through a plate of a transparent substance, we may compare the denser medium to a series of elastic balls, larger and heavier than another series in contact with them on each side. Now, it is well known that a series of elastic balls transmits any motion from one end to the other, while each ball remains at rest, after having communicated the motion to the next in order, so that the last only flies off, from having none beyond it to impel; and if the balls, instead of being only possessed of repulsive forces, were connected by elastic ligaments of equal powers, a motion in a contrary direction would be transmitted with equal ease; the last ball, being retained by the ligament, instead of flying off, would draw the last but one in the same direction, itself remaining at rest after this negative impulse; and the motion would be communicated backwards in the same manner throughout the series to the first ball; and then, for want of further resistance, this ball would not remain at rest after receiving the negative impulse, but would be drawn forwards by it, so as to strike the second, precisely in the same manner as at the beginning of the experiment; and this second positive impulse would proceed through the whole series like the first. Such is the nature of the longitudinal vibrations of elastic rods, first observed by Chladni; the cohesion of the substance supplying the place of the supposed elastic ligaments; and in the case of an elastic fluid, the pressure of the surrounding parts performs the same office, a negative impulse being always propagated through it with the same facility as a positive one. If, instead of a single series of balls, we now consider the effect of two series, the second consisting of larger balls than the first, the last ball of the smaller series will not remain at rest after striking the first of the larger, but will be reflected, so as to strike the last ball but one in a retrograde direction; and this retrograde impulse will be continued to the first ball, constituting a positive impulse with respect to the new direction in which it is propagated. But if the first series of balls be larger than the second, the last of the larger balls will not be deprived of all its motion by striking the first of the smaller, but will continue to move more slowly in its first direction; and the elastic ligaments will then be called into action, so as to carry back step by step to the first ball this remaining impulse, which will become negative with respect to the new direction of its transmission. And the same must happen in the case of two elastic mediums in contact, supposing them to be of equal elasticity, but of different densities; the direction of the elementary motions either coinciding with that of the general impulse, or being opposite to it in both mediums at once, when the reflection is produced by the arrival of the undulation at the surface of a denser medium, and being reversed when at the surface of a rarer; and it is obvious that such an inversion of any regular undulation is paramount to its retardation or advancement, to the extent of half of the interval which constitutes its whole breadth; every affection of such an undulation being precisely inverted at the distance of half the breadth of a complete alternation; and these effects will not materially differ, whether the impulse be supposed to arrive perpendicularly at the surface, or in an oblique direction.

Art. 5. (Sect. IV. D.; Sect. XIII.) The experiments of Mr Arago, which show that light does not interfere with light polarised in a transverse direction, lead us immediately to the consideration of the general phenomena of polarisation, which cannot be said to have been by any means explained on any hypothesis respecting the nature of light. It is certainly easier to conceive a detached particle, however minute, distinguished by its different sides, and having a particular axis turned in a particular direction, than to imagine how an undulation, resembling the motion of the air, which constitutes sound, can have any different properties with respect to the different planes which diverge from its path. But here the advantage of the projectile theory ends; for every attempt to reduce the phenomena of polarisation to mechanical laws, by the analogy of magnetism, has completely failed of enabling us to calculate the results of the actions of the forces supposed to be concerned, in any correct manner; to say nothing of the extreme complication of the properties which it would be necessary to attribute to the simplest and minutest substances, in order to justify the original hypothesis of a polarity existing in all the particles of light, and a directive attraction, that is, a combination of attraction and repulsion, in every reflecting or refracting sub- In the undulatory theory we may discover some distant analogies sufficient to give us a conception of the possibility of reconciling the facts with the theory, and perhaps even of reducing those facts to some general laws derived from it; although it will be necessary, in this intricate part of the inquiry, to proceed analytically rather than synthetically, and to rest satisfied for the present, without bringing the analysis to a termination by any means explanatory of all the phenomena. Some of the supporters of this theory may perhaps be of opinion that its deficiencies are too strongly displayed by this attempt; but it is for them to find a more complete solution of the difficulties, if any such can be discovered.

In the case of a wave moving on the surface of a liquid, considering the motion of the particles at some little distance below the surface as concerned in the propagation of an undulation in a horizontal direction, we may observe that there is actually a lateral motion, throughout the liquid, in a plane of which the direction is determined by that of gravitation; but this happens because the liquid is more at liberty to extend itself on this side than on any other, the force of gravitation tending to bring it back with a pressure of which the operation is analogous to that of elasticity; and we cannot find a parallel for this force in the motions of an elastic medium. It is indeed very easy to deduce a motion, transverse to the general direction, from the combination of two undulations proceeding from two neighbouring points, and interfering with each other, when the difference of their paths amounts to half an interval; for the result of this combination will be a regular though a very minute vibration in a transverse direction, which will continue to take place throughout the line of the propagation of the joint motions, although certainly not with any force that would naturally be supposed capable of producing any perceptible effects. There must even be a difference in the motions of the particles in every simply diverging undulation, in different parts of the spherical surface to which they extend; for, supposing it to originate from a vibration in a given plane, the velocity of the motion constituting the undulation will be greatest in the direction of that plane, and will disappear in a direction perpendicular to it, or rather will there become transverse to the direction of the diverging radii; and in all other parts there must be a very minute tendency to a transverse motion, on account of the difference of the velocities of the collateral direct motions, and of the compressions and dilatations which they occasion. When, also, a limited undulation is admitted into a quiescent medium, it loses some of its force by diffraction on each side, where it is unsupported by the progress of the collateral parts; and if an undulation were admitted by a number of minute parallel linear apertures or slits, or reflected from an infinite number of small wires, parallel to each other, it would still retain the impression of the incipient tendency to diffraction in all its parts, producing a modification of the motion, in a direction transverse to that of the slits or wires.

It is true that all these motions and modifications of motion would be minute beyond the power of imagination, even when compared with other motions, themselves extending to a space far too minute to be immediately perceived by the senses; and this consideration may perhaps lessen the probability of the theory as a physical explanation of the facts; but it would not destroy its utility as a mathematical representation of them, provided that such a representation could be rendered general, and reducible to calculation; and, even in a physical sense, if the alternative were unavoidable, it is easier to imagine the powers of perceiving minute changes to be all but infinite, than to admit the portentous complication of machinery, which must be heaped up, in order to afford a solution of the difficulties which beset the application of the doctrine of simple projection to all the phenomena of polarisation and colours. It is not however possible at present to complete such a mathematical theory, even on imaginary grounds; although a few further analogies between polarisation and transverse motion force themselves on our observation.

In the theory of emission, the resemblance of the phenomena of polarisation to the selection of a certain number of particles, having their axes turned in a particular direction, supposing these axes, like those of the celestial bodies, to remain always parallel, will carry us to a certain extent, in estimating the quantity of light contained in each of the two pencils into which a beam is divided and subdivided; but it would soon appear that, after a few modifications, this parallelism could no longer be supposed to be preserved: we should also find it impossible to assign the nature and extent of any forces which might be capable of changing the former directions of the axes, and fixing them permanently in new ones. The distinction of a fixed, a movable, and a partial polarisation, which has been imagined by Mr Biot, must vanish altogether, upon considering that all the effects which he attributes to the partial polarisation are observable in experiments like those of Mr Knox, in which there is confessedly no polarisation at all.

If we assume as a mathematical postulate, in the undulatory theory, without attempting to demonstrate its physical foundation, that a transverse motion may be propagated in a direct line, we may derive from this assumption a tolerable illustration of the subdivision of polarised light by reflection in an oblique plane. Supposing polarisation to depend on a transverse motion in the given plane, when a ray completely polarised is subjected to simple reflection in a different plane, which is destitute of any polarising action, and may therefore be called a neutral reflection, the polar motion may be conceived to be reflected, as any other motion would be reflected, at a perfectly smooth surface, the new plane of the motion being always the image of the former plane; and the effect of refraction will be nearly of a similar nature. But when the surface exhibits a new polarising influence, and the beams of light are divided by it into two portions, the intensity of each may be calculated, by supposing the polar motion to be resolved instead of being reflected, the simple velocities of the two portions being as the cosines of the angles formed by the new planes of motion with the old, and the energies, which are the true measure of the intensity, as the squares of the sines. We are thus insensibly led to confound the intensity of the supposed polar motion with that of the reflected light itself; since it was observed by Malus, that the relative intensity of the two portions into which light is divided under such circumstances, is indicated by the proportion of the squares of the cosine and sine of the inclination of the planes of polarisation. The imaginary transverse motion might also necessarily be alternate, partly from the nature of a continuous medium, and partly from the observed fact, that there is no distinction between the polarisations, produced by causes precisely opposed to each other, in the same plane.

Why light should or should not be reflected at certain surfaces, when it has been previously polarised, cannot, even with the greatest latitude of hypothesis, be very satisfactorily explained; but it is remarkable that the transmission is never wholly destroyed, or even weakened in any considerable proportion. We might, indeed, assign a reason for the occurrence of a partial reflection or a total transmission in the constitution of the surface concerned, since every abrupt change of density must necessarily produce a partial reflection, while a gradual transition by insensible steps must transmit each impulse with undimi- nished energy, and without any reflection of finite intensity, as in the well-known case of a collision supposed to be performed with the interposition of an infinite number of balls of all possible intermediate magnitudes. If, therefore, we could find any modification of light, which could cause it to be transmitted from one medium to another in a more or less abrupt manner, we should thus be able to discover a cause of a variation of the intensity of the partial reflection; and this seems to be the nearest approach that we can at present make to an explanation of the phenomenon, according to the undulatory theory.

Art. 6. (Sect. V.) The equal intensity of the colours of thin plates, seen by reflection and by transmission, is a fact which would not have been expected from the immediate application of the law of interference, and which seems, therefore, at first sight to militate against its general adoption. But this is only one of the many modifications of the law, which are the immediate consequences of its connection with the undulatory theory; and it may be demonstrated, from the analogy of a series of elastic bodies, that no material difference in the intensity of the two kinds of colours ought to be expected in such circumstances. The intensity of a ray of light must always be considered as proportional to the energy or impetus of the elementary motions of the particles concerned, which varies as the square of the velocity, and not simply as the velocity itself; for if the velocity were made the measure of intensity, there would be an actual gain of joint intensity whenever a ray is divided by partial reflection; since it follows from the laws of the motion of the centre of inertia, that when a smaller body strikes a larger, not the sum, but the difference of the separate momenta, will remain unchanged by the collision, while the sum of the energies remains constant in all circumstances; the square of a negative quantity being equal to that of the same quantity taken positively. Thus, supposing an elastic ball, 1, to strike another, of which the mass is r, with the velocity 1, the velocity of the transmitted impulse will be

\[ \frac{2}{r+1} \]

and that of the reflected,

\[ \frac{2}{r+1} - 1 = \frac{r-1}{r+1}, \]

the sum of the momenta in the opposite directions being

\[ \frac{3r-1}{r+1}. \]

instead of 1, the original momentum; but the energies expressed by the products of the masses into the squares of the velocities will be

\[ \frac{4r}{(r+1)^2}, \]

and

\[ \frac{(r-1)^2}{(r+1)^2} \]

respectively; and the sum of these is

\[ \frac{(r+1)}{(r+1)^2} = 1. \]

Now, when an impulse arrives at the last of a series of larger particles, and is reflected in an inverted form, if we substitute \( \frac{1}{r} \) for \( r \), the energies will be in the proportion of

\[ \frac{4}{r} \text{ and } \left( \frac{1}{r} - 1 \right)^2, \]

or of \( 4r \) and \( (1-r)^2 \), which is the same as the former; so that, according to this analogy, the subdivision of the light at the second surface of a plate must be in the same proportion as at the first. We may call this proportion that of \( m^2 \) to \( n^2 \); \( m^2 + n^2 \) being equal to 1; we have then \( n^2 \) for the energy of the first partial reflection, \( m^2n^2 \) for the second, and \( m^2n^4 \) for the third; for the first transmission, into the substance, \( m^2 \); for the second, out of it, \( m^4 \); for the third, after an intermediate reflection, \( m^2n^4 \); and for the fourth, after two reflections, \( m^2n^8 \); and the elementary velocities in either medium, compared among themselves, will be as the square roots of the respective energies. But it may be proved that, in all collisions of two moving bodies, each of the motions produces its effect on the velocities after impulse, independently of the other;

so that the changes introduced, in consequence of the motion of one of the bodies concerned, are the same as it would have occasioned if the other had been at rest; and, consequently, if two undulations interfere in any manner, the joint velocities of the particles must always be expressed by the addition or subtraction of the separate velocities belonging to the respective undulations. When, therefore, the beam first partially reflected, of which the elementary velocity is expressed by \( n \), interferes with the beam transmitted back, after reflection at the second surface, with the velocity \( m^2n \), the joint velocity, in the case of the perfect agreement of the motions, will be \( n + m^2n \); and in case of their disagreement, \( n - m^2n \); the energies being \( (n + m^2n)^2 \) and \( (n - m^2n)^2 \); the difference, which is the true measure of the effect of the interference, being \( 4m^2n^2 \), that is, four times the product of the respective velocities. But when the light simply transmitted at the second surface, with the velocity \( m^2 \), interferes with the light transmitted after two reflections, with the velocity \( m^2n^2 \), the quadruple product becomes \( 4m^2n^4 \), only differing from the former in the ratio of \( m^2 \) to 1, which is that of the intensity of the light transmitted by the single surface to the intensity of the incident light, the difference being much too slight to be directly perceived by the eye; so that this result may be considered as agreeing perfectly with Mr Arago's observation.

We may also obtain, from the analogy with the effects of collision, an illustration of the intensity of the partial reflection in different circumstances; although it is not easy to say what ought to be the precise value of \( r \) in the comparison. If we imagined the two mediums to differ only in density, while their elasticity remained equal, which is the simplest supposition, the density must be conceived to vary as the square of the velocity appropriate to the medium; but the value of \( r \) thus determined makes the partial reflection in general much too intense, and it becomes necessary to suppose it weakened by the intervention of a stratum of intermediate density, such as there is every reason to attribute to the surfaces of material substances in general, from the considerations stated in the article Conson. However this may be, we shall in general approach sufficiently near to a representation of the phenomenon, by taking the mass \( r \) in the simple proportion of the refractive density: thus, in the case of water, making \( r = \frac{4}{3} \), we have for the energy of the first partial reflection

\[ \frac{(r-1)^2}{(r+1)^2} = \frac{1}{49} = 0.0204, \]

while the result of Bouguer's experiments is 0.018; and the agreement is as accurate as could have been expected, even if the whole calculation had not been an imaginary structure. In the case of glass, the difference is somewhat greater; and it is natural to expect a greater loss of light from a want of perfect polish in the surface; for, taking \( r = \frac{3}{2} \), we have \( n^2 = 0.40 \), and Bouguer found the reflection only 0.025. The surface of mercury reflected nearly 60°; whence \( r \) should be about 8. Whether the index of the refractive density can be so great as this, we have no precise mode of determining; but there seems to be something in the nature of metallic reflection not wholly dependent on the density. Thus it may be observed, that potassium has a very brilliant appearance, though its specific gravity is very low; at the same time, its great combustibility might give it a much higher rank among refractive substances than could otherwise have been expected from its actual density.

Art. 7. (Sect. XIII.) Although the ingenuity of man chromium has not yet been able to devise any thing like a satisfactory reason for the reflection of a polarised ray in one case, and its transmission in another; yet several attempts have been made, with various success, to reconcile the different hypotheses of light with the other phenomena of oblique refraction. The illustrious M. Laplace has undertaken to reduce the laws of this refraction, according to the projectile system, from the general doctrines of motion; and he has sufficiently demonstrated that the path followed by the light is always such as to agree with the principle of the least action, supposing the law of the velocities previously established; or, in other words, that the sum of the products of the spaces described, into the respective velocities, is always the least possible. To this demonstration it has been objected, that notwithstanding the complication of its steps, it is in fact nothing more than the simple translation of the fundamental law of Huygens into another language; for it is assumed in this theory, upon obvious and intelligible grounds, that the path of light must always be such that the time may be equal with respect to two neighbouring collateral parts of the undulation, which is the well-known condition of a minimum of the whole time employed; and the time being always expressed by the space divided by the velocity, if we suppose the proportions of the velocities to be inverted, as in the two theories respecting light, the expression of the time in the one will be identical with that of the action in the other; consequently the conditions of the propagation of light, in the Huygenian doctrine, must always imply the observation of the law of least action in the opposite hypothesis; and this general proposition M. Laplace has taken great pains to prove with respect to a particular instance, in which the Huygenian calculation had been found, notwithstanding Newton's doubts, to agree perfectly with the phenomena. It has also been observed, that the law of the least action is wholly inadmissible as a fundamental principle of motion; that it is completely unphilosophical to multiply unnecessarily the number of postulates or elementary laws; and that although in many cases the principle may be capable of being established as a derivative proposition, yet, in order to demonstrate it, we must assume that the velocity must be the same in all directions, between the same parallel or concentric surfaces, or at any rate must limit our reasoning by conditions incompatible with the nature of the actions, to be considered as the foundation of the laws of extraordinary refraction.

In this single point, the undulatory theory has every possible advantage over its rival. For the difference of the velocities in different directions, no force has been assigned as a cause, in the projectile system, at all more general than the individual directions of the rays with respect to the axis. But upon the hypothesis of undulations, it has been demonstrated, without any gratuitous supposition, that every lamellar or fibrous substance must transmit every diverging impulse in the form of a spheroidal surface, supposing only the elasticity to act more powerfully in one direction than another, as it naturally must do in such circumstances. (Quarterly Review, No.)