' So much for one drop; let us now, as before, consider what will be seen by an eye in any position with regard to this particular drop. In fig. 4, the letters denote the same gs as to fig. 3. Hence 11 uyt,. be placed at T, it will receive the maximum of light, in a direction making an angle DTS'• with the point in the heavens oppo site to the sun. If at 13,, it will receive some of the diffused light from a drop whose angular distance from the point opposite the sun is greater than bTS'; and if at E, it will receive no light at all, the drop's angular distance from the point opposite the sun being less titan DTS'. Hence the appearance presented by a shower of drops is, for homogeneous light coming in parallel lines, a bright circle, whose angular radius is DTS'; diffused light outsidr that circle, and no light within it. 'When the light comes from a source of finite angular diameter, as the sun, the only effect is, as in the primary bow, to widen the bright circular baud. When we consider the various compo nents of white light, calculation shows us that MR' is least for red smilernstest for violet. Ilence we have a series of conceit tric colored bands superposed, their diam eters increasing from the red to the violet. Hence the secondary rainbow has its inner edge red, and its outer violet; the intermediate space being an exceedingly mixed, or impure spectrum (q.v.). The results of geometrical optics show us that the •angular diameter of the red is 100° 48', and of the violet 100° 44'; so that the breadth of the bow is 3° 30' nearly.
In nature, these rough results are pretty closely verified; but a more profound inves tigation into the circumstances of the problem shows us some modifications. In the first
place, we find that for each kind of homogeneous light the actual maximum of brightness is in a rather less angular diameter than that given by the more elementary investigation for the primary bow, and rather greater for the secondary. Secondly. and still with hoinogeneons light. there is a succession of feebler and feebler concentric circles of maximum brightness—inside the principal maximum in the primary bow, and out side it in the secondary. 'These give rise to what is always seen in a fine rainbow, the so called spurious or supernumerary bows, lying close inside the violet of the primary bow, and outside that of the secondigy. These are fainter and more impure as they pro ceed from the principal and finally merge into the diffused white light inside the primary bow, and outside the secondary.
The angular dimensions of these bows. principal and spurious, were calculated from theory by Airy, and carrfully measured by Miller ih the artificial bow formed by passing light through a very fine column of water descending through a small aperture, and the accordance was perfect.
The lunar rainbow, which is a comparatively rare, but very beautiful phenomenon, differs from the solar simply in ths source and intensity.of the light by which it is pro duced; and, as in all cases of feeble light, the distinction of the colors is very difficult. In fact, except under the most favorable circumstances, the lunar rainbow rarely shows colors at all, giving a pale ghostly gleam of apparently white or yellow light.