nearer or farther off, or according as atmospheric conditions other than the distance, or thickness of the body of air between the eye and the rain-cloud, allow more or less vivid impressions of the rays of coloured light to affect our eyes. The band of coloured light always appears to us as a curve—apparently an arc of a circle; and, although we sometimes only see a small arc, sometimes a semicircle, under special conditions even a complete or nearly a complete circle, and although the diameter of this circle appears sometimes to be greater than at other times, one peculiarity of the phenomenon was noticed from very early times—viz., that the diameter of the circle always subtended the same angle at the eye, and this was about 82° for the middle of the coloured band (Pl. L., fig. 1). The colours of the primary bow are always in the same order, red being outside and violet inside. This circle of light is always so placed that its centre is in the production of the line drawn from the sun through the observer's eye, or where the shadow of his head would fall on the rain-cloud.

It will assist us to conceive more clearly the circumstances if, as Professor Tyndall happily suggests, we imagine a cone, the axis of which is this line from the sun through the head and eye of the observer to the rain-cloud—the apex at the eye, and the base on the rain-sheet. Such a cone, the surface of which would at its base coincide with the ring or arc of red light on the outer circumference of the rainbow, has the angle at its vertex about 85° (twice 42.½°). A similar cone having a smaller angle at its vertex, of about 81° (twice 40.½°), will at its base coincide with the inner circumference of the rainbow of violet light—the other prismatic colours being ranged between them. The centre of the coloured band would coincide with the base of a cone whose angle at the vertex is 82° (or twice 41°), as before observed.

Now, neglecting for the moment the phenomenon of colour, why is it that this luminous curved band is depicted on the rain-sheet? Why does it not reflect to us the sunlight indifferently from all parts of its surface? It is indeed so reflected to us, or we should not see the rain-cloud and falling rain; but why this brilliant reflection on one special curved band?

Descartes was the first who, in 1637, solved this problem. He drew the section of a globe of water, such as a raindrop, though its centre, which, of course, is a circle, and by laborious arithmetical processes he calculated the courses of 10,000 parallel rays of light falling on one side of this circle, being refracted as they entered it, reflected on the opposite interior surface, and refracted again as they emerged (Pl. L., fig. 2); and he found that, while most of these rays were on emergence scattered in various directions, a very large cluster of rays entering at the points S, S, emerged at the points e, e, parallel

to one another, in the two directions e E, e E; and these directions were inclined at opposite angles of 41° to the sun's rays, or of 82° to one another.

This result is true for all sections of the sphere or drop of water passing through its centre and in the direction of the sun's rays; from which it follows that, viewing the drop in front (Pl. L., fig. 3), the rays entering on the circle s, s, s, s, will be emitted on the circle e, e, e, e, and will issue back at the constant angle of 41° with the original direction of the sun's rays, forming a cone or shell of concentrated parallel rays spreading out from the drop of water which forms the apex of the cone, the angle at the apex being 82°, or twice 41°. We can now see how Tyndall's cone, turned the other way, with its apex at the eye and its base on the rain-sheet, coincides exactly with a particular set of strong pencils of parallel rays proceeding from the raindrops which are momentarily situated at the base of this cone: the divergent rays from these particular raindrops converge exactly in the observer's eye, as they start from the precise spots and are directed at the precise angle to reach the eye. All the rest of the rays, whether strong pencils of parallel rays or weaker scattered rays, merely assist in the general illumination or are lost.

Another eye, however, in another position, will see another rainbow formed in the same way by the divergent pencils of brilliant parallel rays from other raindrops which happen to be in the right position to send their rays so as to converge on that other eye.

The special brilliance of the rainbow as compared to the illumination of the rest of the rain-cloud is due to this maximum of parallel emitted rays, which, in accordance with the laws of refraction and reflection of light, emerge in a cone of brilliant rays from each raindrop that the sun shines on; the circular or quasi-circular form of the brilliant band follows because the angle at which the light emerges is constant in all directions from the impinging parallel rays of the sun.

The secondary or outer bow often observed is produced in a similar manner, except that the rays which produce it have been twice reflected in the raindrop, as shown in Pl. L., figs. 4 and 4a. This double reflection diminishes the force of the light, for each reflection is only partial, a portion of the impinging rays passing out, and consequently the secondary bow is always fainter than the primary bow; but the secondary cones of light act in the same way as the primary, the angle of divergence on each side of the sun's rays being in this case 52°, or the angle at the vertex of the cone 104°. Hence, this bow must be about 11° outside the primary bow.

The colours of the rainbow are due to the varying refrangibility of the coloured rays, which, when combined, give us the

sensation of white light. The laws of the refraction of the different-coloured rays as they pass from one transparent medium into another of different density are known, and the angles at which the extreme visible rays, violet and red, are emitted in the cases of the primary and the secondary bow are noted in Pl. L., fig. 5, from which it will be observed that in the case of the primary bow the less refrangible red rays will reach the eye from raindrops on the outside of the band of light, the violet rays from raindrops on the inside; while in the case of the secondary bow, owing to the second reflection the reverse will be the case, and the outer part of the band of light is violet, the inner edge red.

Two other facts connected with the ordinary rainbow must be noted before we pass to the special cases which are to be discussed: First, that owing to the apparent size of the sun a rainbow is formed by the light coming from each point in the sun's disc: this results in the formation of a number of rainbows superimposed upon one another, and producing the blending of the colours in the spectrum of the rainbow which we always observe. From this also follows a degree of uncertainty in the exact measurement of the angles subtended at the eye by the circles forming the edges of the bows, and of the bands of colour in them. Different observers have obtained slightly varying results. The angles I have quoted can therefore only be taken as near approximations. Second, that, though there is a principal maximum of emergent parallel rays produced, as before stated, both by a single and by a double reflection inside the drop, there are also secondary fainter maxima which produce the spurious rainbows sometimes seen inside the primary and outside the secondary bow, and described in Mr. Harding's paper.

We may now pass on to the case of a rainbow produced by the reflected light of the sun.

An observer in a boat on a calm sheet of water may see on a rain-cloud in front of him not only a primary and a secondary rainbow having the same centre as usual, but also a third bow having its centre at a higher elevation than that of the others. He will ascertain the source of this third rainbow by turning round to face the sun, when he will notice that by reflection a virtual image of the sun is seen at the same depression below the horizon as the elevation of the sun itself above the horizon. It is the light from this reflection which produces the third rainbow, the centre of which is in the line inclining upwards from the reflected sun through the observer's eye. If the sun be close to the horizon the sun and its reflection will be so close together that the centres of the primary rainbow and of that formed by the reflection of the sun will nearly coincide, and the third bow may be difficult to distinguish, or it

may only be observed in a slight increase in the width of the band of coloured light, and some blending of the colours at the top of the arc; but if the sun be at some considerable elevation the rainbow formed by the light from the reflected sun will have its centre high up in the heavens, while that of the primary and secondary bows will be below the horizon. (See Pl. LI., fig. 6.)

In Sir David Brewster's treatise on optics published in 1853 he mentions such a third arc as having been observed by Dr. Halley, in 1698, from the walls of Chester, the River Dee, which was unruffled by wind, forming the necessary reflector. Dr. Halley supposed that the third bow “was only that part of the circle of the primary bow that would have been under the castle, bent upwards by reflection from the river.” He was evidently wrong. By no operation of the laws of optics could such a bending-up of an image which had no existence occur. It was, indeed, the result of reflection on the mirror of the unruffled Dee; but the reflection was that of the sun, and this reflected sun acted as a second sun, and produced a second rainbow. In this case the sun must have been about 5.½° above the horizon when the rainbow was formed (Pl. L., fig. 7), as the top of the arch of the rainbow coincided with the top of the arch of the original secondary bow, and, the colours being in reverse order, this portion of the two bows was white, the two spectra counteracting on each other. Other examples of rainbows formed by the reflection of the sun in calm water are recorded, and the mode in which such appearances are produced is sufficiently simple; but the question as to whether the sun's rays refracted and reflected in the raindrops may be so reflected again from the surface of a calm sheet of water as to produce in the observer's eye an image of a rainbow in the water, and, if so, what would be the form and colouring of that image, is much more difficult of solution. Professor Tyndall puts the question thus: “Whether a rainbow which spans a tranquil sheet of water is ever seen reflected in the water,” and his reply is, “The rays effective in the rainbow are emitted only in the direction fixed by the angle of 41°. Those rays, therefore, which are scattered from the drops upon the water do not carry along with them the necessary condition of parallelism, and hence, though the cloud on which the bow is painted may be reflected from the water, we can have no reflection of the bow itself.” Of the bow itself, you observe; but he does not say of any other bow that a reflection is impossible. Of the bow itself it is evidently impossible; for, think only of the concentrated parallel rays which produce in our eyes the impression of the highest point of the arch: they come directly from the raindrop there to our eyes, and therefore cannot touch the water and

be reflected to our eyes. The particular raindrops at that part of the falling sheet of raindrops send their effective concentrated parallel rays direct to our eyes, and produce the brilliant image apparently at that spot, and, although scattered rays from that raindrop may reach the water at the right angle to be reflected to our eyes, they will be weak, scattered rays, and will not produce any effect beyond that of general illumination of the water. And the same is true of the particular raindrops forming the image in our eyes of any part and all parts of the rainbow we see. But the question arises, Are there not other raindrops at other parts of the rain-sheet the effective parallel rays from which, falling on the surface of the water, and reflected therefrom, will reach our eyes and produce there the brilliant sensation of a rainbow reflected in the water?

Professor Tait, in his treatise on light, puts the question and the answer in this way: “Can a rainbow be seen by reflection in still water?” “To this, of course, the answer is that a spectator sees, by reflection in still water, the rainbow he would have seen had the water been removed, and had his eye been at the position formerly occupied by its image in the water.” He adds, “But a reflected rainbow differs from a rainbow seen directly, in the fact that, as the light forming the latter is partially polarised, the intensity of the former is modified differently at different points in the act of reflection.” His reply therefore is that a reflected rainbow can be seen, though, as before observed, it will not be a reflection of the particular rainbow perceived by direct vision. And, although his reply is not quite easy to understand, owing to its extreme brevity, it gives the key to the solution of the question. The diagram (Pl. LI., fig. 8) will explain. By the words “its image” he means the image of the observer's eye as it would be reflected in still water vertically beneath it, where it would appear to be as deep below the surface as his eye was above it. The geometrical construction shows that a line from this point at an angle of 41° with the direction of the sun's rays cuts the water-line at a point where a ray, proceeding from a drop in the rain-sheet at an angle of 41° with the direction of the sun's rays, would be reflected to the observer's eye, the angles of incidence and reflection at the surface of the water being equal. The ray reflected to the eye would give an image of the highest point of a rainbow which would appear to be at the same distance below the surface of the water at the rain-sheet as the drop which emitted the ray was above it. Knowing that the centre of the circle must be at the point where the parallel sun's ray, passing through the reflected image of the eye, cuts the vertical rain-plane (produced below the surface of the water), we have no difficulty in describing the arc of the rain-

bow that would be seen from the position of the reflected eye, and in drawing its reflection which might be seen by the actual observer.

It will be evident that the lower and nearer to the reflecting surface the eye of the observer, the less will be the distance between the eye and its reflected image, and, consequently, the locus of the invisible rainbow to be seen by reflection will more nearly correspond with that seen by direct vision, and the reflection would seem very like a reflection of the visible rainbow were it not for the polarisation of the rays emitted from the raindrops after reflection and refraction. This polarisation has been found to correspond with the radii of the arch. Thus the light coming from the summit of a semicircular arch would be polarised at right angles to the light coming from the feet of the arch, and the direction of polarisation would change gradually between these points. The polarised rays from the top of the arch would, I think, be more strongly reflected than those from the sides, and I should therefore expect to see the lower part of the inverted reflection more distinctly than that of the sides.

The question still remains, Has any one ever seen such a reflection? I can trace no record of such an experience, and I attribute it to the fact that almost always the surface of the water is strongly illuminated by the sun when a rainbow is formed over still water, and that, consequently, the refracted and reflected rays which would form this reflected image are overpowered by the strong illumination of the water-surface. To see distinctly the reflection of a bright object in a mirror, the mirror itself must be in comparative shade. This condition might be produced by the shadow of a ridge of land, however, and, although the meteorological conditions at Wellington are rarely favourable to the exhibition of a rainbow reflection in calm water, yet, were a shower to fall over the harbour towards sunset without wind, the shadow thrown by the hills on the west of the harbour might enable us to observe the evidently very rare but possible phenomenon of an inverted rainbow seen by reflection on the water. From a yacht becalmed in mid-harbour, or from Somes Island, the bow, or bows, formed by the sun reflected in the calm water might be observed.

In considering the subject of rainbows geometrically, I was at first led to the conclusion that they must often appear not as arcs of circles but as portions of ellipses. I have not seen any allusion to this in books, nor had I hitherto thought of looking for or expecting any such distortion of the circular arch, but I think there can be no question that wherever the raindrops emitting the pencils of parallel rays which reach our eyes are not equidistant from the eye they cannot lie in

the arc of a circle; and this must be the case whenever the rain-sheet is inclined towards or away from us by wind, or when one side of it is farther from us than the other, or, finally, when the sun is comparatively high above the horizon, and the rain-sheet is vertical. In all these cases the cone having its vertex at our eyes, and its axis in the line from the sun, through our head and eyes, to the rain-sheet, must be intersected by the rain-sheet in a plane to which the axis is not perpendicular, and the intersection must be the conic section known as an ellipse.

In Plate LI., fig. 9, the geometric elevations of the raindrops forming two rainbows are shown as they would appear if occurring when the sun's rays were nearly horizontal, soon after sunrise or sunset: b a, c, a semicircle on a vertical rain-sheet, and b a c, an elliptical curve with the major axis vertical, occurring on a rain-sheet inclined by the wind 15° from the vertical.

In Plate LI., fig. 10 are shown the geometric elevations of the rain-drops forming two rainbows—one, a, o b, a semicircle, as before; the other, a o b, a semi-ellipse with the major axis horizontal, formed on a rain-sheet oblique in plan to the sun's rays, the ellipse being thus thrown altogether towards the side where the rain-sheet was most distant.

In Plate LI., fig. 11, are shown similarly the geometric elevations of the drops forming two rainbows as seen from the top of a mountain 8,000ft. high, the rain-sheet being vertical and about 8,000ft., or, say, a mile and a half, distant. In this case, if the sun were near setting, and the sun's rays consequently nearly horizontal, the rainbow would appear as the geometric elevation of the raindrops forming it, a complete circle, excepting the part obscured by the shadow of the mountain—were the observer in a balloon the circle would be complete. And it is to be noted that great altitude is not required in order that a circular or nearly circular rainbow may be observed⁴⁵⁷—the conditions are that the elevation shall be about the same as the distance of the rain-cloud from the observer.

In this same figure (fig. 11) is shown the elliptical geometric elevation of the raindrops forming the rainbow which would be observed under the same conditions except that the sun is higher up in the heavens, the axis of the cone having thus become oblique instead of perpendicular to the plane of the rain-sheet. The rainbow would, however, appear as the lower and less complete circle.