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A, B, C, D, &c. represent spheres of water, into which the solar rays entering, are, under certain circumstances refracted and reflected so as to emerge parallel, or nearly so, and thus produce vision in an eye placed at O, with the sensation of some primitive colour, the different homogeneous rays being separated by the refractions.
In the first, or lowest arch there is but one reflexion, in the next two, and so on. The quantity of light lost at each of these reflexions, accounts for the want of distinctness in the upper bows.
181. In the first place, it will be observed that the spectator must turn his back to the Sun, to see a rainbow.
Secondly, that all drops of water similarly situated with respect to the Sun, and to the eye, must produce the same effect.
This similarity of situation, evidently depends on the circumstance of lines drawn from the drop to the Sun, and to the eye, making equal angles.
This includes all the drops that are found on a conical surface of revolution, having for its axis a line OP parallel to the solar rays; since all lines drawn from the vertex along such a surface make equal angles with the axis, or with any line parallel to it.
In order to find in each case the value of the semi-angle of the cone, or the radius of the bow, we have only to determine its equal, the angle which the incident rays make with those emergent rays, which are parallel.
Accordingly, we will find the value of this angle in general, and find what it becomes in the particular case in question.
Let SA, Fig. 208, represent a single ray (belonging to the first bow), which is refracted and reflected to B, C, O.
The incident and emergent rays SA, CO being produced to meet in T, and EA, EC, ET being joined, the latter of which of course passes through B,
| Let θ | = | the half angle ATE, |
| φM | = | the angle of incidence SAM, or EAT, |
| φ′ | = | the angle of refraction EAB, |
| m | = | the ratio of refraction, sinφsinφ′ |
