The conclusion is, therefore, that if homogeneous light fall in parallel lines on the spherical drop, those rays which have been twice refracted at the surface, and once inter nally reflected, will, on emergence, all lie within the cone formed by the revolution of CT about SO, and will be condensed toward the surface of that cone. Hence such an illuminated drop gives off by this particular process a' solid cone of rays, much con densed toward its external boundaries.
So much for each drop. Next, let us inquire what the appearance will be to an eye in any given position. Referring to the next figure, in which the letters are the same as in the former, draw TS' parallel to SO. Then TS' is the direction of the line drawn to the point • 0,1 the heavens diametrically opposite to the sun. So are E,S,' and drawn from any assumed positions, E, and of the spectator's eye.
If the eye be placed in the surface of the cone just described, as at T, it will receive the condensed ray which emerges in the direction CT; if at E, (within the cone), it will receive diffused rays from the drop; if at (outside the cone), it will receive no light at all.
To put this in a simpler form: Draw E,F, and parallel to TC; then we may evidently say that the eye receives a condensed light from any drop whose angular dis tance from the point opposite the sun is CTS', a diffused light if the angular distance be less than this, and none at all if it be greeter. By methods already alluded to, it is found that CTS' is nearly 42° 12' for the index of refraction A.
Hence, if the sun were a luminous point, emitting homogeneous light whose index of refraction in water is a spectator looking through a shower of falling raindrops toward the point immediately opposite to the sun, would see a bright circle of angular diameter 84° 24' surrounding this point, diffused light within that circle, and darkness without it.
The effect of the finite angular diameter of the sun is evidently to widen this circle into a circular luminous band, whose breadth is the sun's apparent diameter, and whose mean radius is 42° 12'.
Next, let its consider the different refrangibilities of the colored constituents of white light. The investigation above hinted at shows that the radius of the luminous circular band is greater, the less the refractive index; the proof, though very simple, would be out of place in this work. Hence the appearance actually observed with sunlight will be formed by the superposition of concentric, overlapping, circular bands, the radii being less and less as we consider the primary colors in the order from red to violet (see SPEC TRUM). That is, we shall have a circular illuminated space, brightest toward the edge,
with a homogeneous red ring as its external boundary, and a gradual mixture of the pris matic colors as we look nearer to the center. This agrees very well with observation, and so do the calculated diameters of the external red (42° 22') and internal violet (40° 35') rings.
But what becomes of the light twice reflected inside the drop, and then refracted out? Let fig. 3 represent again a section of the drop, with sunlight falling on it in lines allel to SO, and let us trace the course of one ray, as SB. The part reflected at B is°to be disposed of as before; it goes moral, to illuminate faehlv the erwise dark background of cloud and vapor. The refracted portion proceeds, as before, to A, where part is reflected internally along AC, and part refracted out. The latter por tion, as we have already seen, cannot possibly reach the eye of a spectator whose back is turned to the sun. Similarly, at C, there is internal reflection along CD, and refraction out of the drop. The refracted part has already been considered, as the cause of the primary rainbow. The reflected part will again at D be sep arated into two; one, reflected inter nally, which proceeds to form the tertiary and higher,orders of bow; drop in the line DT, which goes to form the secondary bow. This we secondary bow, though necessarily fainter will consider with some care, because the than the primary, is usually seen; the tertiary and higher bows, each much fainter than the preceding one, sind3 the beam inside the drop is weakened at each succeeding reflec tion, require no notice, as even the tertiary has never been observed in nature.
As before, we have traced the courses of two other beams, S13, and in their pas :age to form part of the secondary bow. They are respectively S13,A,C,D,T, and and the figure shows us that the final rays D,T, and are each more inclined to SO than DT is. There is, therefore, a particular ray, 613, whose final direc tion, DT, is less inclined to SO than that of any other ray which has suffered two refrac tions and two internal reflections; and, as before, the emergent light is condensed toward this minimum. lf, then, the figure be made to revolve about SO, we see that DT will describe a cone, that inside this cone there is no refracted light, that toward the surface of the cone, part of the light is condensed, and that the rest of it is diffused through exterior space.