that its existence is easily deducible from the descriptions usually given of the circumstances under which the rainbow is produced.

I will not occupy your time with an enumeration of the various mistakes which seem to exist on this subject, but refer you at once to the Traité de Physique of M. Biot, whose description and analysis, as far as I am able to appreciate them, are the best I have seen. I think, however, with great deference, that even these are, to a certain extent, imperfect and incomplete. M. Biot says

The phenomenon of the rainbow is produced by the coloured spectra which issue from different drops of water after two refractions, separated by one or two intermediate reflexions. But how,' he proceeds, does the superposition of these partial spectra compose the colours of the bow and determine its magnitude? This is what we have to examine.

To do this simply, let us first consider a single incident ray of simple colour-for example, red; then, supposing that it emerges from the drop after a certain number of reflexions and refractions, let us calculate the angle it forms with its primitive direction.

Let S I, Fig. 1. be such a ray entering at I and escaping at I' after a

second refraction, without intermediate reflexion. From the centre of the globe draw CIN, CIN', which will be perpendicular to the surface. Then SIN will be the angle of incidence, which we will call i; and CII' will be the angle of reflexion, which we will call r. Further, in consequence of the symmetry of the figure, the interior incidence at l' will also be r, and the emergence will be i. Prolong the incident and emergent rays till they meet at T, forming the angle I TI', which will be the deviation produced by refraction; we will call this A. Now it will be easy to find its value in functions of the angles i and r; for in the quadrilateral CITI', all the angles are known, except a. In short, the angles at I and I' are both equal to i; further, the triangle CII' being isosceles, the angle I C I' is equal to 180° - 2 r; then, since the sum of the angles of a quadrilateral are equal to four right angles, we have A+2i+1809-2 r = 2, 180°, or A = 180° + 2r - 2 i.

'Let us next consider two refractions separated by one reflexion; the same construction (see Fig. 2.) and reasoning will apply only the angle

Fig. 2.

N

I

R"

2 r); thus we

2 i.

ICI" will be double I C I', that is, equal to 2 (180° have + 2 i + 2 (180° – 2 r) = 2, 180o, and ▲ = 4 r 'Generally, if the ray have n successive incidences in the interior of the globe, the angle IC I" becomes n (180° - 2 r).

The angular deviation will be constant for all rays of the same nature, which penetrate the globe under the same incidence; but the incidence changing, that also will change. To form a clear idea of these variations, let us first consider the case in which the ray suffers but one internal reflexion; after which it escapes from the globule into the air. Then, if we calculate the amount of deviation for several parallel rays, incident at small distances on various parts of the surface, it will be found that the deviation is nothing under a perpendicular incidence, in which the ray passes through the centre of the globule. The deviation gradually increases to a certain limit of incidence, which is about 541° for the red rays, so that a pencil of these rays entering parallel at I (see Fig. 3.) under this incidence, and being once reflected from

I

Fig. 3.

the inner surface, will emerge equally parallel at I", though the general direction of the pencil be deviated 42°. But for more considerable incidences, the deviation diminishes as it had increased; and this diminution

continues as far as the last rays tangent to the globule. Now if these rays are received at such a distance from the globule, that this last may be considered as a point, it is clear that all those which belong to unequal deviations will diverge one from the other, as their distance from the globule increases; so that they will become too feeble to give a perception of the globule to an eye placed in their course; while that eye would be affected even at the same distance by the emergent rays, which correspond to the maximum of deviation, because, being parallel, they are transmitted to any distance without separation.

'Suppose a series of these globules disposed circularly in such manner that the refracted rays which issue from them, and which are supposed to be of the same colour, may thus reach the eye; they will produce the sensation of a luminous line; and several such series placed side by side will produce a coloured band.

The same considerations apply equally to the cases in which the refractions and reflexions are more numerous; there is always for each a limit at which the rays of a small pencil will emerge sensibly parallel, and will be transmitted without becoming enfeebled.

To develop the consequences of these results, suppose that an observer placed at O (Fig. 4) views a large cloud composed of spherical

drops of water; draw from the centre of the sun through the eye the line SOC, to designate the direction of the rays, which we will for the present suppose to be exactly parallel, as would be the case if the sun were a

point infinitely distant. This being granted, there will be, from the anterior surface of the drops, a partial reflexion of all the colours which compose the incident light, and which will form a whitish tint, spread over the whole surface of the cloud; but, besides this, there will be seen two concentric arcs, coloured with all the colours of the spectrum. For if through the eye O be drawn a right line OV, forming with O C an angle of 40° 17', and that it be supposed to revolve round O C, describing a conical surface, all the drops which are found in this surface will have the position in which the most refrangible violet rays, after having suffered two refractions and one reflexion, will emerge parallel, and will reach the eye at O, and this will not take place in any other part of the cloud; in virtue, therefore, of these rays alone, the spectator will see upon the cloud a violet bow, of which O C will be the axis, and O the centre. There will, in like manner, be an infinite number of concentric arcs exterior to the last, each formed by one description of simple rays; and as the rays become less refrangible, their arcs will be of larger diameter, so that the largest, composed of the extreme red, will subtend an angle ROC of 42 2'. Thus the total extent of the coloured band will be 42° 2' 40° 17', or 1° 45', the red being without, and the violet

within.

It will be the contrary after two reflexions. If the lines OR, OV', be drawn, making with O C angles of 50° 59′ and 54° 9', and then made to revolve round OC as an axis, the first will intersect all the drops, which, after two refractions and two reflexions of the red rays, will transmit them in parallel lines to the eye; and the second will determine the analogous limit for the extreme violet rays. Between these two arcs there will be others of all the intermediate colours of the prism, and they will form a second coloured band, having a width of 54° 9' 50° 59′, or 3° 10'. This band will have its colours in an order the inverse of the first, that is to say, the red will be inside, and the violet outside; and the distance between the two red arcs will be 8° 57'.

In this account it is assumed that no sensible effect is produced on the eye of the spectator, after one or two internal reflexions, except by those drops which are included within the angles subtended by the coloured bands; although it is said that the whole expanse of the shower will exhibit a degree of whiteness, in consequence of the reflexion of the sun's rays from the anterior surfaces of the drops. These two statements are, to a certain extent, I think, irreconcilable; for if the rays reflected from the internal surfaces would be rendered insensible by their divergence, so also, I conceive, would those reflected from the external surfaces. The dispersion of the former will not account for the supposed differ

VOL. I.

FEB. 1831.

U

ence, because each drop in the shower (with an exception to be presently noticed) would transmit rays of every coloured light, producing by superposition with themselves, and with the rays from other drops, the sensation of white light, differing only in brilliancy from that reflected at the outer surfaces, Mere divergence will not, I think, affect, to the extent supposed, the apparent quantity of light derived from numerous points at a great distance. It is true that a parallel pencil would appear very bright at a distance, which would render a divergent pencil, of equal magnitude, quite insensible. But, in the case under consideration, it is not a single pencil of parallel rays which is compared with another of divergent rays; the eye views a luminous space, part of which is so distant, that a thousand drops might be contained in a line having an inappreciable angular value. If the light from each of these thousand drops proceeded in parallel lines, the eye, although it would receive all the light transmitted by some one drop, would lose all that was reflected by the others. If, on the contrary, the light diverged from the drops, the eye would receive only a very small portion of the light from the one drop, but it would now receive an equal portion of the light reflected from each of the remaining nine hundred and ninety-nine drops; the whole of which proceeding from a space of no sensible magnitude, would produce a general impression of illumination, notwithstanding that the light from any single drop might have been invisible. An instance of the effect produced by numerous simultaneous impressions, each individually imperceptible, is furnished by a room in which silkworms are feeding. A hundred of these animals emit no sound that the ear can detect; but the noise of a very large number in the act of eating has considerable intensity. In a large and crowded theatre no individual is heard to open a play-bill, or turn the leaf of a book; but if any circumstance occasions a large portion of the audience to do either of these nearly at the same instant, a noise is produced like the rushing of a torrent. The correct statement, therefore, I think, would be, that in addition to the reflexion from the anterior surfaces, which is common to all the drops in the shower, every drop, with the exception before alluded to, is rendered visible by