188 EAINBOW into a fixed number of different colored rays, refracted or bent at the same time in different but definite degrees, so that they must appear, under given circumstances, separated just so much, and always in the same successive or- der. This result will follow, then, whether sunlight is dispersed by prisms or by transpar- ent spheres, as water drops. The mathemati- cal theory, which belongs to Descartes, may be found in the higher text books of optics, and is illustrated by the accompanying diagram taken from DeschanePs " Natural Philosophy." If a ray of light pass through the centre of a sphere or drop, its course is in an axis of the sphere or drop; it is not refracted. A ray parallel with this, and very near it, is refracted within the drop, toward this axis, but very slightly. Other rays, further and further from the axis, are refracted more and more toward it, but yet so as to fall, by lessening degrees, further from it on the inner or second sur- face of the drop ; until, as Descartes proved, a ray, S b or S a, entering the upper side of the drop, when this is above the eye, and at a point for which its angle of incidence is 60, will strike on the inner surface as far as any ray can do from the axis; the rays incident at greater angles than this, up to 90, deviating again toward the axis. Of course, near this limit, the deviation is very slight for rays com- ing on either side, so that much more. light within the drop will be accumulated just at this point of the second surface than at any other ; and though part of it emerges here, a sufficient quantity is reflected, and that in rays which preserve a parallel course (ft O or a O), after leaving the drop in the direction toward the spectator, to form a compact, parallel beam, bright enough to affect the eye at a great dis- tance. The apparent radii of the arcs con- stituting the rainbow are constant, or nearly so ; they are expressed by the angles between the axis O Z and the lines O a, O ft, &c., and are as follows: in the primary bow, for the violet 40 17', for the red 42 2' ; in the sec- ondary bow, for the red 50 57', for the violet 54 7'. A tertiary bow, formed by rays that have been thrice reflected within the rain- drops, is possible at a distance of about 43 50' from the sun ; but this is very rarely visible, owing to its faintness and other causes. From the above explanation, the following conse- quences are obvious: that the ordinary rain- bows must be on the side of the observer op- posite the sun ; that their centres must be di- rectly opposite the sun ; that they must move with the motion of the sun, declining in the morning, and rising if seen at evening; that when the sun and the observer are in the same horizontal plane, as at sunset, the bows will be semicircles, and their altitudes then about 42 and 54; that they can never ap- proach nearer than this to the zenith, unless the observer be on an elevated position, so that the sun can shine from below the horizontal plane in which he is ; that at the tops of high mountains they may be seen as complete cir- cles; and that, to one at the ordinary level, in the low and middle latitudes, they are nev- er seen between about 9 o'clock in the morn- ing and 3 o'clock in the afternoon ; while in higher latitudes, where the sun is always very low in the sky, they may occur even at mid- day. If the rain is near, the bows may some- times be seen prolonged upon the landscape. The small water drops constituting spray may afford a rainbow ; hence it is seen in the mist arising near cataracts, and, because near, is then small, and may appear as a complete circle. A partial bow may be observed at times in drops of dew or rain upon herbage or grass. The formation of the supernumerary bows was explained by Young (1804), as due to interfer- ence of sets of rays emerging at angles very nearly those of the proper colors of the bows. Biot, and afterward Brewster, have shown that in all rainbows the light is polarized in the radial planes passing through the axis O Z, and hence polarized by refraction and re- flection. The lunar rainbow is usually sin- gle, the primary bow only, and is often white; when colored, it is but faintly so. When the drops of rain are exceedingly fine, as in the case of clouds and fog, the rainbow prop- er is replaced by bows formed by the reflec- tion and interference of light from these fine particles. The laws of these fog bows are deducible from the same principles that hav served to explain the rainbow. The phe- nomena themselves are exceedingly brilliant; they were observed by Sykes in 1829 (see "Philosophical Transactions," 1835), but far more perfectly by the aeronauts of the past few years; beautiful examples are recorded in Glaisher's " Travels in the Air " (London, 1870). The floating ice spiculse or crystals that compose those higher clouds called cirri
