Page:Encyclopædia Britannica, Ninth Edition, v. 14.djvu/617

 LIGHT 597 To find it, we have -*-*-* with sin Q = p. sin &amp;lt;p. Differentiating, &c., as before, we find 8 cos 2 = ju-- 1, whence, finally, This quantity increases with ^, for its differential coefficient is which is necessarily positive. (It vanishes, no doubt, for /t = 3, but then so does 8.) For n = | the value of sin s is 0-4303 nearly, so that s = 5Q 58. econd- Carrying out the same steps of reasoning as before, and ry rain- applying the result to raindrops, we find a second rainbow concentric with the first, but with a greater radius, viz., about 5T (for yellow light). All the above remarks about the impurity of the spectrum, &c., apply to this bow also. In this bow the less refrangible rays are on the inner side, and the straggling rays illuminate feebly the space outside it. Hence the space between the red boundaries of the two bows has no illumination from rays reflected either once or twice within the water drops. What we have now given is nearly all that geometrical optics can tell us about the rainbow. It seems that the first really important steps in the explanation, viz., (1) that the primary bow is due to rays falling on the outer portions of the drops, which suffer two refractions and one reflexion before reaching the eye, and (2) that the secondary bow is due to rays falling on the inner side, and suffering two refractions and two reflexions, are due to Theodorich, about 1311. His work was not published, and its contents were first announced by Yenturi l in the present century. These results were, independently, discovered by De Do- minis ~ in 1611. Neither of these writers, however, pointed out the concentration of the rays in particular directions. This was done by Descartes in 1637, by the help of Snell s law. He calculated with great labour the paths of each of 10,000 parallel rays falling on different parts of one side of the drop, and showed that from the 8500th to the 8600th the angle between the extreme issuing rays is measured in minutes of arc, thus discovering by sheer arithmetic the maximum which, as we have seen above, is so easily found by less laborious methods. Newton s addition to this theory consisted mainly in applying his discovery of the different refrangibilities of the different homogeneous rays. The explanation was then thought to be complete. For a long time this was held to be one of Newton s most brilliant discoveries. It is well to notice that he himself speaks of it in its true relation to the work of his predecessors. He merely says : &quot; But whilst they understood not the true origin of colours, it is necessary to pursue it here a little further.&quot; And he said well ; for a full investigation con ducted on the principles of the undulatory theory intro duces, as was first pointed out by Young, certain important modifications in the above statements. Of these we need mention only one, viz., that in each bow there is more than one maximum of brightness for each homogeneous ray. Spurious The spurious bows, as they are called, which often appear ws. like ripples, inside the primary and outside the secondary bow, and which depend upon the fact just mentioned, have 1 Commentari sopra la storia e le feorie dell Ottica, Bologna, 1814. 2 Newton, in his Optics, says the work of De Dominis was written twenty years before it was published. no place in even Newton s theory. About them, in fact, geometrical optics has nothing to say. Young, in 1804, took the first step for their explanation. They were fully Complete investigated, from the undulatory point of view, by Airy, theory in 1836-38; and his results were completely verified by? U1 the measurements of Hallows Miller in 1841. 3 Miller used a fine cylinder of water escaping vertically from a can. This is one of the reasons which induced us to treat the subject as a case of refraction and reflexion in a right cylinder. The overlapping of the colours in the rainbow, due to the White apparent size of the sun s disk, is occasionally so greatly bows, exaggerated that only faint traces of colour appear. This may happen, for instance, when the sun shines on raindrops in the lower strata of the atmosphere through clouds of ice- crystals in the higher strata. By reflexion from the faces of these crystals, the source of light is spread over a much larger spherical angle, and there is no sharp edge to it as in the case of the unclouded disk. The rainbow is then much broader and fainter than usual, and nearly white. The size of the drops also produces modifications which are not indicated by the geometrical theory. &quot;When the moon is the source of light, the rainbow is so Lunar faint that it is often difficult to distinguish the colours ; rain - but with full moon, and other favourable circumstances, it is easy to assure one s self that the colours are really present. The refraction of sunlight, or moonlight, through ice- Halos. crystals forming cirrhus clouds, gives rise to coloured halos, parhelia, paraselense, &c. Their approximate explanation depends upon the behaviour of prisms with angles of 60 or 90, and therefore does not come within the scope of the present article. They must not, however, be confounded with coronse, those rings which encircle the sun or moon Corona?, when seen through a mist or cloud. Halos have definite radii depending on the definite angles of ice-crystals ; the size of a corona depends on the size of the drops of water in a mist or cloud, being smaller as the drops are larger. Thus their diminution in radius shows that the drops are becoming larger, and implies approaching rain. They are due to diffraction, and can only be explained by the help of the undulatory theory. Refraction in a Non- Homogeneous Medium. The prin- Non- ciples already explained are sufficient for the purpose of llom - treating this question also. But they require, for their ge f? u application, the artifice of supposing the medium to be made up of layers, in each of which the refractive power is the same throughout the layer, but finitely differs from one layer to another, and then supposing these layers to become infinitely thin and infinitely numerous. In this case there will of course be only an infinitely small difference in properties between contiguous layers ; and the abrupt change of direction which accompanies ordinary refraction is now replaced by a continuous curvature of the path of the ray. Glimpses of a more general method had been obtained even in Hamil- the 17th century ; and in the 18th these had become a consistent ton s process so far as application to the corpuscular theory is concerned, general But it was reserved for Sir W. R, HAMILTON (q.v.) to discover the investi- existenee of what he called the characteristic function, by the help gations. of which all optical problems, whether on the corpuscular or on the undulatory theory, are solved by one common process. Hamilton was in possession of the germs of this grand theory some years be fore 1824, but it was first communicated to the Royal Irish Academy in that year, and published in imperfect instalments some years later. The following is his own description of it. It is extremely important as showing his views on a very singular part of the more modern history of science. &quot; Thosa who have meditated on the beauty and utility, in theo retical mechanics, of the general method of Lagrange, who have felt the power and dignity of that central dynamical theorem which he deduced, in the Mecanique Analytique. . ., must feel that mathe- 3 Airy s paper is in vol. vi. of the Cambridge Phil. Trans., Miller s in vol. vii.