St., or St Kitts, one of the British West India Islands, 46 miles W.N.W. of Antigua, Lat. 17.20. N., Long. 62.40. W. It is about 20 miles in length from N.W. to S.E., with an average breadth of 4 miles, but narrowing towards its S.E. extremity. It takes its name from Columbus, by whom it was discovered in 1493, at which time it was densely peopled by Caribs. In 1623 a party of English under one Warner first settled on the island, and soon afterwards a party of French arrived under M. D'Esnambeu. The English lived for some time on friendly terms with the natives; but having unwarrantably seized on some of their lands, and being apprehensive that the Caribs would retaliate, they treacherously surprised them during the night, murdered above one hundred, and expelled the rest, reserving the most handsome of the young women. The colony, however, after this inhuman outrage, was far from flourishing, and the two leaders were compelled to return to their respective countries for recruits. Warner returned with about 400, and a plentiful supply of necessaries. The greater part of D'Esnambeu's recruits perished at sea, and the remainder reached the island in a wretched condition. A treaty was now signed, fixing the territory of each party, the upper part, Capis-terre, being allotted to the French, and the lower part, Basse-terre, to the English. The island was afterwards seized by the Spaniards; but these invaders departed in a short time, and the tranquillity of the settlement was restored. In the numerous wars between the The gradual progress of scientific investigation has continued to add, from year to year, a multitude of new discoveries to our knowledge of experimental and physical optics; and no department of this subject has received additions so diversified and so important as those which relate to the phenomena of colours, which have been displayed, with a thousand brilliant and unexpected transformations, under circumstances that in former times could never have been suspected of exhibiting anything resembling them. The successive experiments and calculations of Dr Thomas Young (1801), Dr Wollaston (1802), Mr Malus (1810), Mr Arago, Mr Biot, Dr Brewster, Mr Seebeck, and Mr Fresnel, have all contributed very essentially to the extension and illustration of this interesting branch of science. But, notwithstanding all that has hitherto been done, it appears to be utterly impracticable, in the present state of our knowledge, to obtain a satisfactory explanation of all the phenomena of optics, consi- Sect. I.—Of the Separation of Colours by Refraction.

The separation of white light into different colours, as its component parts, by refraction, though firmly established as an optical fact by Newton, had been in general somewhat negligently examined as to its details, until Dr Wollaston pointed out the inaccuracy of the common subdivision of the colours of the prismatic spectrum into seven different species. There is little reason to doubt that white light consists of an infinite number of rays, varying gradually among each other, without any marked distinctions, and continued, on the one hand, into the dark chemical rays, and, on the other, into the rays of invisible heat; and that all these varieties are separable from each other by refraction, and preserve always a distinct and constant refrangibility. The species of homogeneous light, however, distinguishable from each other by the eye, are only five,—red, yellow, green, blue, and violet; which are uniform in their appearance, and well-defined in their limits, whenever a perfect spectrum is correctly exhibited, whether obtained by interposing a prism between the eye and a small, or rather narrow bright object, or between a lens and the image of such an object formed in its focus; while, in the common method of admitting a beam of the sun's light through a prism, without either employing a lens, or previously limiting the angular extent of the beam, it is obvious that there must be a double source of the mixture of colours; and hence has arisen the Newtonian division of the spectrum into seven parts, which were somewhat fancifully compared, with respect to their extent, to the intervals of the minor diatonic scale in music; although it has been shown by Dr Blair, and still more fully by Dr Brewster, that their proportions are liable to very great variations, according to the nature of the refracting substances employed.

Dr Brewster has remarked, that as, according to the fundamental law of refraction, a prism with a large angle must occasion a dispersion of the several colours somewhat greater than two smaller prisms of the same substance, having together an equal mean refractive power; so also the dispersion of the most refrangible or violet rays amongst themselves will be always somewhat greater in a prism with a larger angle, than in two smaller prisms having an equal mean dispersive power. Hence the green and blue will be less removed from the red towards the violet by the single prism, the refraction of the green remaining in defect, when compared with the mean of the whole. So that, if the two prisms be employed to correct the mean dispersion of the single one, and the extreme Chromatics of the spectrum be brought to a perfect coincidence, the refraction of the green by these prisms being comparatively in excess, the green rays will be found on the side towards which their refraction tends to carry them; and the two extreme portions of red and violet will be left together, forming a crimson, on the side towards which the refraction of the larger prism is directed. It is obvious also that if, instead of the two smaller prisms, a single one of an equal angle, but of twice the dispersive power, were substituted, the joint effect would be nearly the same. Dr Brewster has, however, observed, that in almost all such combinations of different substances, the green is on the side towards which the refraction of the larger prism is directed; so that the original proportion of the space occupied by the different rays in the spectrum must be different for different substances. Dr Brewster has found that the violet is the most dispersed by oil of cassia and by sulphur, and least by sulphuric acid and by water; the distribution afforded by these substances appearing to vary from two parts of red, three green, four blue, and three violet, to about four red, three green, three blue, and two violet; while the yellow is always confined to a narrow line.

The immediate effects of the combinations of the primitive colours on the sense of sight afford an illustration of some of the physiological characters of sensation in general. It is well known that a mixture of red and green light produces a simple sensation, perfectly identical with that which belongs to the minute portion of yellow light originally found in the spectrum; and that a mixture of green and violet makes a perfect blue. The blue colour of the flame of spirit of wine, for example, is derived entirely from a mixture of green and violet rays; while the blue light of the lower part of the flame of a candle is shown by the prism to consist of five different portions, belonging to different parts of the spectrum, nearly resembling those which would be distinguished if we looked through a prism at a small portion of a transparent plate of a certain minute thickness. It is obvious, therefore, that the eye has no immediate power of analysing such light; and if we seek for the simplest arrangement, which would enable it to receive and discriminate the impressions of the different parts of the specimen, we may suppose three distinct sensations only to be excited by the rays of the three principal pure colours falling on any given point of the retina, the red, the green, and the violet; while the rays occupying the intermediate spaces are capable of producing mixed sensations, the yellow those which belong to the red and green, and the blue those which belong to the green and violet; the mixed excitement producing in this case, as well as in that of mixed light, a simple idea only: although it must be observed that no homogeneous light can extend its action so far as to excite at once the sensations of the fibres belonging to the red and the violet; so that every crimson must necessarily be a compound colour. A mixture of red and blue light exhibits an effect which appears unintelligible upon the supposition that a compound light ought to produce a colour intermediate between those of its constituent parts; but this difficulty will vanish, if we assume that the blue of the spectrum contains a greater proportion of violet than of green; so that the green is neutralized into a white by a mixture with the red and part of the violet, and the remaining violet gives its character to the whole, either alone, or with a mixture of red, according to the proportions employed.

When we look through a prism at a luminous object of considerable extent, surrounded by a dark space, the spectra belonging to the several parts of the object are mixed with each other, so as to produce a light perfectly white, except towards the ends of the object, where the extreme parts project beyond each other. At the red end of the spectrum the whole of the red belonging to the extreme point retains its place unaltered, and the green and blue become a greenish yellow, nearly uniform in its appearance, throughout the space which belongs to them, while the place of the violet is scarcely distinguishable from the neighbouring white light; but at the opposite end the violet retains its place and appearance, and the remainder of the length of the spectrum becomes of a green, inclining more or less to blue, and continuing to be very distinctly visible throughout the extent of the simple spectrum, the place of the red included; so that the illuminating power of the red end of the spectrum must be incomparably greater than that of the violet end; as may also be inferred by a direct comparison of the distinctness of objects viewed in these different lights. The portion of light totally reflected at the internal surface of a dense medium, on account of the obliquity of its incidence, is bounded by a fringe or bow resembling the red end of the luminous object viewed through a prism; and the transmitted portion is bounded by the violet and blue fringe; but it requires some caution in observing these colours, to avoid the optical deception which causes the neighbouring space to appear of the complementary colour, especially when the eye is turned towards it immediately after having received the impression of the colours actually exhibited.

Sect. II.—Of the Colours of Halos and Parhelia.

The immediate effect of the different refrangibility of light in the production of colours is sometimes spontaneously exhibited, in the atmospherical phenomena of halos and parhelia, or paraseleenes, attending the sun or moon; the edge nearest to the luminary being generally reddish, and the remoter parts more or less green and blue, although without any well-marked separation of the different tints. These appearances had been long ago referred by Mariotte to the refraction of the prismatic crystals of snow floating in the atmosphere, and descending through it in all possible positions, but more especially in a vertical or horizontal direction, on account of the effect of gravity, combined with that of the resistance of the air; and sometimes perhaps, from their connection with other crystals, making angles of 60° with either of these positions. This theory, however simple and satisfactory, had been very unaccountably neglected for more than a century, and even superseded by the awkward and unsupported conjectures of Huygens respecting the existence of spherical or cylindrical particles of hail, including opaque nodules, related to them in a certain constant ratio; or by the equally inadmissible calculation of Newton, which assigns a partial maximum to the density of the light simply refracted through a spherical drop of water when the deviation is about 25°; and it is only a few years since that the doctrine of Mariotte was revived and extended by Dr Young, and approved by Mr Cavendish and Mr Arago.

In some of the highest northern latitudes these appearances of halos and parhelia are almost constant; and in warmer countries they are confined to the light clouds which occupy the higher and colder regions of the atmosphere. The halos are broad circles, with their interior margin tolerably well defined, and about the distance of 22° and 46° from the sun or moon, but less distinctly terminated externally. Now the angle of 22° exactly corresponds to the deviation produced by a prism of ice, with a refracting angle of 60°, when it becomes a minimum from the equality of the angles of incidence and emergence; and in other positions of the prism the deviation increases very slowly, till it becomes a few degrees greater. Hence the breadth of the circles of each colour being considerable, the colours must fall principally on each other, and become very indistinctly separated. The external circle may be referred to the effect of two such refractions in succession. Mr Cavendish seems to have thought the angle somewhat too great to be derived from this source; and he suggested that it might depend on a single refraction by the rectangular terminations of the crystals; but it does not appear that such terminations are very commonly observable; and it may easily be shown that the greatest intensity of the light of a halo formed by two refractions must be at more than twice the distance of the edge of the inner halo, derived from one only.

These halos are commonly accompanied by a white horizontal circle passing through the sun, derived from the reflection of the vertical faces of the crystals, which are scattered equally throughout all possible azimuths. There are also generally coloured parhelia on each side, depending on the refraction of these vertical prisms; they are commonly a little without the halos, because the deviation of the light passing obliquely through these crystals is somewhat greater than that of the light transmitted by the crystals which have their axes perpendicular to the plane of incidence and refraction. For a similar reason, the light passing through the crystals horizontally in various azimuths is variously modified, so as to produce the appearance of inverted arches, touching the halos at their highest points, and sometimes expanding in the form of a pair of wings, with a point of contrary flexure on each side.

The antihelia seem to be referrible to two refractions and an intermediate reflection within the same crystal, causing a deviation of about $120 + 22 = 142°$; and sometimes with two intermediate reflections, producing an angle of $60 + 22 = 82°$ only. It is not, however, very easy to assign a reason for the appearance of an antihelion exactly opposite to the sun, which is said to have been sometimes seen in the horizontal circle; but it has been delineated with the accompaniment of an oblique cross, and of other unusual appearances, which must have been derived from extraordinary forms of the compound crystals of snow existing at the time of the observation in the atmosphere.

Sect. III.—Of the Colours of the Rainbow.

The general nature of the primary rainbow was cursorily explained by De Dominis; but Descartes first applied the true law of refraction, which had recently before been discovered, to the determination of the angular magnitude both of this and of the secondary rainbow; although no sufficient reason could be assigned for the appearance of colours in either of them, until Newton ascertained the different refrangibilities of the different kinds of rays; but as soon as this discovery was established, the method of fluxions at once enabled him to determine precisely the limit at which the broad expanse of light belonging to each colour must necessarily terminate in an edge of greater brilliancy; the bright edges of the different colours projecting gradually beyond each other, so as to form a spectrum somewhat mixed, but still approaching to the common appearance of a spectrum obtained by the refraction of a prism; and in fact the angular distances of the exterior termination of the primary rainbow and of the interior of the secondary, from the sun, are found to agree very accurately with the calculation of the extreme deviations of the red rays reflected once and twice respectively within the spherical drops of rain; although the whole breadth of the coloured appearances is liable to variations dependent on the magnitude of the drops, and belonging to the phenomena of supernumerary rainbows, to be described hereafter.

The light reflected from very small portions of water appears to be incapable of producing a regular rainbow. Thus we scarcely ever see a rainbow in a cloud, unless it has united its drops, so that they begin to descend in the form of rain. Dr Smith has observed this circumstance, and has attributed it to a tendency of the bright edge of the expanse of light to lose its intensity, by being gradually dissipated into the neighbouring dark space; a tendency which he would probably have been much at a loss to explain from any of the received doctrines of optics, but which bears some analogy to the effects more commonly observed in beams of light admitted into dark spaces, and sometimes designated by the term diffraction.

Sect. IV.—Of Periodical Colours in general.

By far the greater part of the phenomena of colours, except their separation by simple refraction, are referrible to the description of periodical or recurrent colours; being characterized by an alternation which is generally repeated, where the observation is sufficiently extensive, several times in succession, while the circumstances on which they depend are varied uniformly and by slow degrees. The number of these alternations, when light perfectly homogeneous is employed, appears to be continued without any discoverable limit, although it is always smaller, for any given change of circumstances, when the least refrangible or red light is employed, than when the observation is made on the most refrangible or violet; so that mixed or white light always produces a combination of alternations arranged according to a series of different intervals, which are at first more or less distinct, but by degrees are so mixed with each other as again to be lost in the general effect of white light. In all these cases the appearances may be reduced to calculation by means of the general law of the interference of two portions of light, with its appropriate modifications and corrections.

A. The law is, that when two equal portions of light, in circumstances exactly similar, have been separated and coincide again in nearly the same direction, they will either cooperate or destroy each other, according as the difference of the times occupied in their separate paths is an even or an odd multiple of a certain half interval, which is different for the different colours, but constant for the same kind of light.

B. In the application of this law to different mediums, the velocity must be supposed to be inversely as the refractive density.

C. In reflections at the surface of a rarer medium, and of some metals, in all very oblique reflections, in diffraction, and in some extraordinary refractions, a half interval appears to be lost.

D. It is said that, according to some late observations of Mr Arago, two portions of light, polarised in transverse directions, do not interfere with each other.

E. The principal intervals in air arc, for the Extreme Red.............. -0000286 = 37 1/40 Yellow....................... -0000235 = 43 3/40 Green....................... -0000211 = 47 3/40 Blue......................... -0000189 = 53 3/40 Extreme Violet............. -0000167 = 59 3/40 Mean, or White............ -0000225 = 34 1/40 inch.

Sect. V.—Of the Colours of Thin Plates.

The colours exhibited by very thin plates of transparent or semitransparent substances have been well known to optical philosophers, from the time that they were first noticed by Boyle, and more particularly examined by Hooke and Newton. They may be readily observed by pressing together any two clean pieces of common plate-glass, which have always sufficient convexities and concavities to exhibit them, touching each other in some points, and leaving elsewhere a thin plate of air between them; or, still more conveniently, by selecting from the plano-convex lenses, kept by the opticians, such as have their flatter sides very slightly convex, and are consequently calculated to throw the spaces of equal thickness, and the colours dependent on them, into the form of rings. The colours are most distinct when they are formed in the light reflected from the two surfaces in contact, especially when care is taken to exclude the foreign light reflected by the surfaces not concerned in their production; and in this case they begin from a central dark spot, immediately surrounded by a bright ring, and then by rings more distinctly coloured, while the colours, exhibited in light transmitted through the glasses, begin from a bright spot in the centre, surrounded by a dark ring, being always exactly complementary to the colours seen by reflection; to which they are also, as Mr Arago has demonstrated, either exactly or very nearly equal in intensity, although they have generally been supposed to be much less vivid, on account of the diminution of their effect on the eye by their mixture with the whole of the beam of light which affords them. But if we employ for the observation two flattish pieces of glass, held in such a position as to transmit the light received from one part, and to reflect an image of another part, of an object equally illuminated throughout its extent, the two series of colours will destroy each other, and the whole appearance of rings will vanish. When, on the contrary, the illumination of the object varies materially, the rings will re-appear in one or other of their forms, according to the different intensities of the lights received from its different parts; so that, as Mr Arago has ingeniously suggested, this test might be employed to answer the purpose of a photometer, for ascertaining the equality of the lights of two distant objects.

If any thin plate affording colours be inclined to the direction of the light passing through it, the appearance of the colours will be changed either precisely or very nearly in the same manner as if the thickness were reduced in the ratio of the radius to the cosine of the inclination within the plate; at least, if this proportion is not perfectly accurate, the deviations from it, in the experiments of Newton, are manifestly within the limits of the unavoidable errors of observation.

We are indebted to Mr Arago for the important fact, that the colours observed in transmitted light are distinguished by a polarisation opposite or transverse to that which is appropriate to transmitted light in general, and possessing the ordinary character of the polarisation produced by partial reflection. It is in light thus reflected that we must seek for one of the two portions which are to be combined according to the laws of interference, in the case of the colours seen by transmission, and for both in the case of reflection. The light transmitted simply through the plate will be followed by a portion which has been reflected back from the second surface to the first, and forwards again from the first to the second; and the difference of the times occupied in these different paths will obviously be proportional to the thickness of the plate, and also, according to the modification (B) of the law, to its refractive density; so that the number of alterations of any given colour between the central spots of the rings and any given point will be as the thickness of the plate at that point; and the numbers for different colours will be inversely as the magnitudes of the appropriate inter- vals; the plate appearing light when illuminated by a ho- mogeneous colour, only where the thickness corresponds to any exact multiple of the interval, and dark at the in- termediate points; and this proportion is found to agree perfectly with experiment. The two reflections within the plate being always of the same kind, will either not require any correction on account of their nature (C), or will altogether add a whole interval to the length of the path; an alteration which makes no change in the appear- ances.

When the incidence is oblique, the actual length of the two passages of the reflected ray across the plate AB, BC, is as twice the secant of the angle of refraction ABD, and its advance upon the surface AC, as twice the tangent; and this advance, reduced to the direction of the transmi- tted ray AE without the plate, must be subtracted from the retardation within the plate; the reduction being in the proportion of the radius to the sine of the angle of in- cidence ACE, for which, if we substitute that of the radi- us to the sine of the angle of refraction ADF or CDG, we

shall have the deduction required to be made from the length of the path within the plate, since the velocities vary directly as these sines; and by this deduction the se- cants AB, BC, will be reduced to the cosines BF, BG: so that the true retardation will always be proportional to the cosine of refraction.

The same demonstration is applicable to the difference of the paths of the two portions of light reflected once only, from the upper and lower surfaces of the plates respec- tively, supposing A, the point of emergence of the trans- mitted ray, to become the point of incidence of a new re- flected ray HA. Hence it might be expected that all the phenomena of colours should be the same as in the case of transmitted light; and this really appears to happen when the observation is made on a plate of air contained between a transparent substance and a polished surface of gold or silver; or on a plate of a refractive density intermediate between the densities of the neighbouring substances, as in the instance of a thin coat of smoke or of an oxide, adher- ing to any polished metallic surface, which is at first of a yellowish white, and, as it becomes thicker, changes to a yellow and an orange colour; but in more common cases there is a loss of half an interval in one of the two reflec- tions only, so that the thicknesses affording a perfect coin- cidence for any species of colour, are always intermediate between the thicknesses affording the same colour by trans- mission; and hence the tints of the two series of rings are always complementary to each other; the series seen by reflection always beginning from a dark central spot, when they are exhibited by any detached transparent substance, as a soap bubble, a thin film of glass or of talc, or by a plate of air contained between two plates of glass, or be- tween a plate of glass and a piece of polished steel.

There is a peculiarity in the surface of silver and gold, and perhaps of some other metals, that, besides the regu-

lar reflection at an angle equal to that of incidence, a con- siderable quantity of light is dispersed irregularly; and this light, as Mr Arago has observed, is polarised in a di- rection transverse to that of the usual polarisation by re- flection; there is also in the irregular reflection no loss of a half interval; so that it exhibits, with a piece of glass, a series of rings resembling those which are produced by polished steel, except that their dimensions are not varied exactly in the same proportion by the obliquity of the in- cidence, because the light which forms them is not re- quired to pass towards the metal in an angle exactly equal to that which it makes upon its return after reflection; and there will probably be considerable irregularities in the interval of retardation, according to the mode of per- forming the experiment; although in general the irregu- lar dispersion or diffraction from the glass is too weak to afford colours easily observable, when the position of the plate differs considerably from that in which the light is regularly reflected. If a portion of polarised light is in- capable of interfering with another portion polarised in a transverse direction, these rings ought to disappear when the angle of incidence on the plate of glass is about 55°, since in this case the light reflected by it is completely polarised in the plane of incidence; and this disappear- ance seems actually to have been observed in some of Mr Arago's experiments, though in others, where the metal- lic surface was less highly polished, the polarisation of the dispersed light may have been less complete, and the rings may still have been visible at this angle. (Mémoires d'Arceuil, vol. iii. p. 354, 359.)

Sect. VI.—Of the Colours of Double Plates.

When light is transmitted in succession through two plates, differing but little in thickness, they exhibit an ap- pearance of colour similar to that which would be produc- ed by a single plate equal in thickness to their difference; and this appearance is wholly independent of the distance of the plates from each other. It was first noticed by Mr Nicholson in the glasses employed for the sights of sex- tants, and is attributed by Dr Young to "the rays twice reflected in the second only." In some circumstances, how- ever, the light returning from the second glass to the first, and again reflected by it, may co-operate in the effect, the interval of retardation being the same in both cases. Mr Knott has more lately described some very striking appear- ances of colours, obtained in this way, by the combination of two pairs of lenses, each exhibiting their appropriate rings when viewed separately, and affording together a third series of rings of larger dimensions when the two former are unequal in magnitude, and of straight bands when they are equal. It is in fact easily demonstrable, that in order that the thicknesses of the plates of air, con- tained between two unequal pairs of lenses, may be equal, the distances from the centres of contact must be in a con- stant proportion; and it is well known that all the points from which the lines drawn to two given points are in a constant proportion, will be found in the circumference of a circle, the diameter of which is a third proportional to the difference and sum of the segments of the given dis- tance of the points; so that the colours, depending on this difference, instead of beginning as usual from a white cen- tral spot, will begin from a white ring, and will be arrange- d in concentric rings on each side of it, precisely in the same order as when they form concentric rings round an actual point of contact; and when the curvatures of the two pairs of lenses are equal, the diameter of the circle becoming infinite, it will obviously be converted into a right line.

Dr Brewster has observed a series of similar phenomena produced by two plates, of equal thickness, but forming a small angle with each other, so as to be differently inclined to the light passing through them. The effect of the inclination being to reduce the virtual thickness of the plate in the ratio of the cosine, and the difference of the cosines of equi-different arcs being simply as the sine of their half sum, it is evident that the colours must correspond to a thickness which varies nearly as the sine of the angle of incidence, considered with regard to a plane bisecting the angle formed by the plates; and this result agrees correctly with Dr Brewster's experiments.

Sect. VII.—Of the Colours of Supernumerary Rainbows and Glories.

Within the common primary rainbow, and without the secondary, we sometimes observe a partial repetition of colours, more or less distinctly marked, and extending occasionally to several alternations; the repetitions occupying somewhat narrower spaces, as they are more remote from the ordinary bows. These appearances seem to have been first described by Mariotte; they have been since noticed by Langwith, Daval, and Dicquemare; and the term supernumerary rainbows has been very properly applied to them. The coloured circles called glories may generally be seen surrounding the shadows of our heads when we have an opportunity of standing on a high hill, and observing them in a cloud below us; they are also sometimes accompanied by a large white circle, which, in an observation of Ulloa, was 67° in diameter; and such a circle may frequently be distinguished when the sun shines on a mass of vapour rising from a warm bath, of nearly the same dimensions, or sometimes a little smaller. The whole of these phenomena may be explained, from the interference of some of the portions of light regularly reflected within the minute drops of water, with other portions, incident at a different angle, but, after an equal number of reflections, coinciding ultimately with them in direction; supposing only the clouds in question to afford a number of these drops, varying but little from each other in diameter. We find, by the well-known mode of calculating the greatest deviation, that each order of reflections exhibits a zone from 30° to 10° in breadth, through which a double light is diffused by each drop; and, besides this, when there have been more than three reflections, the portions belonging to the opposite sides cross each other in one or more points, and surround the drop, or rather the observer, if we consider the effect of the refraction of a multitude of drops situated in all directions. Supposing the index of refraction for the extreme rays 1:336, and its logarithm '1258000, the results will be these:

| After | Extreme Deviation | Final Deviation | |-------|-------------------|-----------------| | 1 | 41° 40' | 13° 52' | | 2 | 51° | 69° 12' | | 3 | 40° 39' | 27° 44' | | 4 | 45° 2' | 55° 20' | | 5 | 50° | 41° 36' | | 6 | 34° 14' | 41° 28' |

We may obtain a more distinct idea of these duplicatures if we represent them in a diagram, showing the angular extent of the diffusion of light derived from each order of reflections, and distinguishing by different kinds of lines the portions belonging to the opposite halves of the drops; and it will be obvious, from the inspection of this figure, that the appearances in question have only been observed within some of the duplicatures of the orders to which they belong, between the angles of extreme and of final deviation. The tertiary and quaternary bows (III. IV.) are evidently too near the luminary to be visible; the quinary (V.) ought to be seen in the space between the primary and secondary; but it is probably much too faint to be visible under any circumstances. The duplicature belonging to the primary rainbow exhibits two portions, for which we may calculate the interval of retardation in parts of the radius of the drop, supposing the velocity to be that which is appropriate to the air, by taking twice the difference of the cosines of incidence on the drop, and multiplying twice that of the cosine of refraction by the index 1:336; the difference of these differences giving the interval for the two portions, of which the direction has been found to coincide by a previous calculation.

| Distance from the Edge | Angle of Reflection | Difference of the Paths | |------------------------|---------------------|-------------------------| | 0° | 40° 2' | -0000 | | 1 | 42° 59' | -0014 | | 2 | 36° 23' | -0040 | | 3 | 44° 2' | -0074 | | 4 | 44° 45' | -0113 | | 5 | 45° 46' | -0160 | | 6 | 46° 9' | -0210 | | 8 | 46° 45' | -0327 | | 10 | 47° 12' | -0461 | | 12 | 47° 32' | -0612 |

Hence it may be inferred, that, supposing the extreme red to reappear at the distance of 2° from the primitive external termination of the rainbow, the radius of the drop must be \(-\frac{0000266}{004} = -00665\), or \(1\frac{1}{10}\) of an inch; the fourth alternation of the red being at the distance of 5°, where the interval is -016. The magnitude of the interval, at an equal distance from the edge, varies but little with the refractive density: thus, for violet light, the index of refraction being probably about 1:346, and its logarithm '1290000, the greatest deviation will be found 40° 14'; and for a deviation 2° less, the angles of refraction must be 43° 30' and 33° 37', and the interval will be little different from -00100.

The supernumerary bands of the secondary bow, formed by the same drops, will be a little broader than these, since it appears, from a similar calculation, that the rays inter-