As the full elucidation of this subject involves a variety of optical and mathematical investigations, I shall do little more than explain the general principle on which the prominent phenomena of the rainbow may be accounted for, and some of the facts and results which theory and observation have deduced.
We have just now alluded to an experiment with a glass globe:—If, then, we take either a solid glass globe, or a hollow globe filled with water, and suspend it so high in the solar rays above the eye, that the spectator, with his back to the sun, can see the globe red;—if it be lowered slowly, he will see it orange, then yellow, then green, then blue, then indigo, and then violet; so that the drop at different heights, shall present to the eye the seven primitive colours in succession. In this case, the globe, from its form, will act in some measure like a prism, and the ray will be separated into its component parts. The following figure will more particularly illustrate this point. Suppose A (fig. 35.) to represent a drop of rain—which may be considered as a globe of glass in miniature, and will produce the same effect on the rays of light—and let Sd represent a ray from the sun falling upon the upper part of the drop at D. At the point of entering the drop, it will suffer a refraction, and instead of going forward to C, it will be bent to N. From N a part of the light will be reflected to Q—some part of it will, of course, pass through the drop. By the obliquity with which it falls on the side of the drop at Q, that part becomes a kind of prism, and separates the ray into its primitive colours. It is found by computation that, after a ray has suffered two refractions and one reflection, as here represented, the least refrangible part of it, namely the red ray, will make an angle with the incident solar ray of 42° 2´, as Sfq; and the violet, or greatest refrangible ray will make with the solar ray, an angle of 40° 17´, as Scq; and thus all the particles of water within the difference of those two angles, namely 1° 45´—(supposing the ray to proceed merely from the centre of the sun)—will exhibit severally the colours of the prism, and constitute the interior bow of the cloud. This holds good at whatever height the sun may chance to be in a shower of rain. If he be at a high altitude, the rainbow will be low; if he be at a low elevation, the rainbow must be high; and if a shower happen in a vale, when the spectator is on a mountain, he will sometimes see the bow in the form of a complete circle below him. We have at present described the phenomena only of a single drop; but it is to be considered that in a shower of rain there are drops at all heights and at all distances; and therefore the eye situated at G will see all the different colours. All those drops that are in a certain position with respect to the spectator will reflect the red rays, all those in the next station the orange, those in the next the green, and so on with regard to all the other colours.
figure 35.
It appears, then, that the first or primary bow is formed by two refractions and one reflection; but there is frequently a second bow, on the outside of the other, which is considerably fainter. This is produced by drops of rain above the drop we have supposed at A. If B (fig. 35.) represent one of these drops, the ray to be sent to the eye enters the drop near the bottom, and suffers two refractions and two reflections, by which means the colours become reversed, that is, the violet is lowest in the exterior bow, and the red is lowest in the interior one, and the other colours are reversed accordingly. The ray T is refracted at R: a part of it is reflected from S to T, and at T it suffers another reflection from T to U. At the points S and T part of the ray passes through the drop on account of its transparency, towards W and X, and therefore we say that part only of the ray is reflected. By these losses and reflections the exterior bow becomes faint and ill-defined in comparison of the interior or primary bow. In this case the upper part of the secondary bow will not be seen when the sun is above 54° 10´ above the horizon; and the lower part of the bow will not be seen when the sun is 60° 58´ above the horizon.
figure 36.
For the further illustrations of this subject, we may introduce the following section of a bow, (fig. 36.) and, in order to prevent confusion in attempting to represent all the different colours—let us suppose only three drops of rain, and three different colours, as shown in the figure. The spectator O being in the centre of the two bows, here represented,—the planes of which must be considered as perpendicular to his view—the drops A,B, and C produce part of the interior bow by two refractions and one reflection as stated above, and the drops D,E,F will produce the exterior bow by two refractions and two reflections, the sun’s rays being represented by 3,3. It is evident that the angle COP is less than the angle BOP, and that the angle AOP is the greatest of the three. The largest angle, then, is formed by the red rays, the middle one consists of the green, and the smallest the purple or violet. All the drops of rain, therefore, that happen to be in a certain position with respect to the spectator’s eye, will reflect the red rays, and form a band or semicircle of red, and so of the other colours from drops in other positions. If the spectator alters his station, he will see a bow, but not the same as before; and if there be many spectators, they will each see a different bow, though it appears to be the same.
The rainbow assumes a semicircular appearance, because it is only at certain angles that the refracted rays are visible to our eyes, as is evident from the experiment of the glass globe formerly alluded to, which will refract the rays only in a certain position. We have already stated that the red rays make an angle of 42° 2´, and the violet an angle of 40° 17´. Now, if a line be drawn horizontally from the spectator’s eye, it is evident that angles formed with this line, of a certain dimension, in every direction, will produce a circle, as will appear by attaching a cord of a given length to a certain point, round which it may turn as round its axis; and, in every point will describe an angle with the horizontal line of a certain and determinate extent.
Sometimes it happens that three or more bows are visible, though with different degrees of distinctness. I have more than once observed this phenomenon, particularly in Edinburgh, in the month of August, 1825, when three rainbows were distinctly seen in the same quarter of the sky; and, if I recollect right, a fragment of a fourth made its appearance. This happens when the rays suffer a third or fourth reflection; but, on account of the light lost by so many reflections, such bows are, for the most part, altogether imperceptible.