RAINBOW. The ordinary phenomena of the rainbow are usually visible on every occurrence of a "sunny shower," and we need not describe them particularly until we deduce them, one after another, from their cause. The most careless observation shows us that, for the production of a rainbow, we must have a luminous body of moderate angular diameter, and drove of water; for it is never seen except by direct sun or moon light, and never in a cloud unless rain be falling from it. Now, a falling drop of water takes, by its molecular forces, a spherical form. Also, as there is separation of the vari ous colors of which white light is composed, the cause of the phenomenon must involve refraction of light (q.v.), because by reflection (q.v.) these colors are not separated. But, again, the spectator who views the rainbow has his back to the sun, and rays of light merely refracted by a rain-drop could not be thus sent back to the spectator. The phe nomenon must therefore depend upon successive reflections and refractions, and we shall investigate in an elementary manner what appearances we ought to expect as the result of such processes according to the known laWs of optics; merely premising that the funda mental points of the explanation were first given by Newton in the second book of his Optics.
First, then, let sus consider what becomes of parallel rays of light, of one color or refrac tive index (see HEFReeTioN), which are successively refracted and reflected in a single spherical rain-drop.
For our immediate purpose it is sufficient to suppose that the refractive index (see IlEFEAcTrox) of water is I; that is, the incident and refracted rays make with the pendicular to the refracting surface of water, angles whose sines are in the ratio of 4 to 3. Let the circle represent a section of the drop made by any plane passing through its center 0, and the line SO, which joins its center with the sun; the sun being supposed, for the moment, to be a single luminous point, situated at so great a distance that lima drawn to it from different points of the drop are parallel. A ray of light, SB, falling on the drop in the plane of section will be, of course, partly reflected and partly refracted at B. The reflected part does not concern us, as in it all colors would travel together;
and, in fact, the result of xetlection from the external surfaces of the drops is simply to illuminate the background feebly. Join OB, and produce it to Q. Then the refracted ray (see REFRACTION) will have in the drop the direction BA, Where the ratio of the sines of SBQ and OBA is the refractive index of water—i.e., 4 : 3 nearly. Arriving at A the ray will be partly refracted in some such direction as AD, and the rest reflected in the direction AC. Now AD obviously cannot fall on the eye of a spectator whose bark is turned to the sun, and it has, therefore, nothing to do with the rainbow. The internally reflected ray AC, on reaching the surface at C, is partly refracted in the direc tion CT (where BS and CT are symmetrically situated on opposite sides of OA), and partly reflected internally. The latter portion we must consider when we come to the cause of the secondary, or outer rainbow, the former is that which at present concerns us. Let be other incident rays. After a refraction, a reflection, and a second refraction, they emerge in the directions C,T,, respectively. From the figure, which is drawn from calculation, it is obvious that both C,T, and C,T, are less inclined to OS than CT is. Hence for rays, parallel to SO, falling on the drop, and emerging after suffering two refractions and a reflection, the final direction is more and more Inclined to SO, as the point of incidence, B,, is further from P, at least up to some such point as B; after which (for points situated as it diminishes again. By proper inutile. matical methods it is easy to find that the angle SOB is about 59' 24', if the refractive index be 1. Now, by a general property of maxima or minima in optics (see CAusTic), the rays falling on the drop near to B will emerge nearly paralled to CT; while those incident near any other point (as BO will be widely scattered at emergence. And we may evidently extend this reasoning to all other rays by supposing the above figure to rotate about the axis SO.