596 LIGHT tion at P . But it is obvious from the symmetry of the circular section, and from the laws of refraction and reflexion, that this path is symmetrical about the line OQ which joins the axis of the cylinder to the point at which the ray is reflected. Hence SP, S P meet OQ in the common point s ; and the amount by which the direction of the ray has been turned round by the refractions and the reflexion is twice the supplement of half the angle at s. But the angle POR is double OPQ the angle of refraction, while OPs is equal to the angle of incidence. Hence the half angle at s is the excess of twice the angle of refraction over the angle of incidence. Turn now to fig. 22, in which we have two concentric circles whose radii are to one another as the refractive index of the cylinder is to unity. If A be any point on a diameter, and tangents Ap and A? be drawn from it, we see at once that the sines of the angles at A are to one another as the radii of the Crowding of rays about a position of maxi mum or minimum deviation. circles. Hence, if OAp be the angle of inci dence, OAq is the cor responding angle of re fraction ; and it is easy to see that the half angle at s (in fig. 21) will in fig. 22 be represented by the excess of OA</ over </A/>. Now when OA is large, both of these angles are small, and thus their difference is likewise small. As OA becomes less the difference of the angles becomes greater, but only up to a certain point, for when A is near the outer circle the angle OA/> begins to increase much faster than does OAq. Hence there is a single definite position of A for which the difference is a maximum. In the first figure these changes in the angles of incidence and refraction, for the members of a group of parallel rays, correspond to the varying position of P in the circular section of the cylinder. Hence there is one position of P for which the angle at s is a maximum. Now one of the conditions of a maximum or minimum of any quantity is that, near it, the value of the quantity changes very slowly. Thus a number of issuing rays are crowded together near the direction corresponding to this maximum, the others being more widely scattered, while for all of them the angle at s is smaller. Newton gives us as an illustration of this, the very slow change of length of the day when the sun is near one of the tropics. To find this Maximum Angle. If 6 be the angle of incidence, that of refraction, and /j. the refractive index, we have to find the maximum value of i? = 20-0 (1), with the condition These give at once and Hence From (2) and (3) we have sin 9 = /j. sin <j?> (2). 2d<p = d6, cos(p d<j> = cos QdO. H cos <f> = 2 cos (3). which determines the requisite angle of incidence. The values of the other quantities are easily calculated from this ; and we finally have, for the maximum value of the sine of the half angle at s, the expression This is obviously smaller as /j. is greater, at least up to the limit p-2. With the value for ^ (which is nearly that for yellow rays refracted into water) we have L sin 4s = 9 55462, which corresponds very nearly to Now suppose the diameter of the cylinder to be small compared with the distance of the eye from it. In this case the point s may be considered as being in the axis of the cylinder. Let SsE 1 (fig. 23) be made equal to the maximum value of s ; then an eye placed anywhere in the line sE 1 will receive the rays which are congregated towards the maximum. An eye within the angle SsE 1 (as at E) will receive some of the straggling rays, while an eye outside that angle (as at E 3 ) will see nothing. Let there now be imagined a great number of paral lel cylinders; let EjO- be drawn parallel to the incident rays, and make the angle o-EX equal to crE/. Then the eye at E : will see the concun- trated rays (already spoken of) in the directions E r s- and Ejs . From points within sE A s some straggling rays will reach it, from points outside that angle none. Now suppose cylinders to be placed in great numbers in all directions perpendicular to the incident rays. The eye at E a will see a bright circle of light whose centre is in E^. Inside that circle there will be feeble illumination ; outside it, darkness. This is obviously the case of the rainbow, where we have spherical drops of water instead of the cylinders above spoken of. For each spherical drop is effective only in virtue of a section through its centre, containing the incident ray and the eye ; and such sections are the same as those of the cylinders. Thus far we have been dealing with parallel rays of homogeneous light ; and the appearance (to the degree of approximation we have adopted) is that of a bright circle whose centre is diametrically opposite to the source of light, whose radius is (for raindrops) about 42 2 , and whose area is slightly illuminated. Introduce the idea of the different kinds of homogeneous light which make up sunlight, and we find a circular (almost pure) spectrum the less refrangible rays being on the outside. Next we introduce the consideration of the finite disk of the sun, and we have an infinite series of such arrangements superposed on one another, the centre of each individual of the series being at the point diametrically opposite to the point of the sun s disk which produced it. This leaves the general aspect of the phenomenon un changed, but altogether destroys the purity of the spectrum. If we next consider light which has been twice reflected within the cylinder, we have a figure like the diagram fig. 21 ; where the lettering is as nearly as possible the Primary rainbow. Homo- geneous white liht. Finite disk of sun - Fig. 24. same as that in fig. 23. Everything is still symmetrical about the line Os, which obviously cuts at right angles the ray QQ . _ Reasoning precisely similar to that above given shows that the complement of half the angle at s is now equal to the excess of thrice the angle of refraction (OPQ) over that of incidence (the supplement of OPs), and that this also has a maximum value, i.e., that s itself has a minimum value.
