of

possible brilliancy: making the angle S B M equal to 42° 2′, in the case of red light. If the line MO is now drawn parallel to S B, it may be regarded as a ray the sun passing through the eye of the spectator; and since all the rays of the sun are parallel to each other at the earth, the angle B M O will also be equal to 42° 2′.

420. The observer then being placed with his back to the sun, and his eye at M, will receive the impression of red light from the drop PLQ D, in the direction BM; and not only from this drop, but also from every other drop, whose angular distance from the line M Ŏ is, at that moment, the same.

that

It is therefore evident, if we suppose the line M B to turn about M O, like the legs of a pair of compasses, all the points at which red light is seen lie in the circumference of a circle whose centre is O; and that around this centre an arch of red light will appear in

the heavens.

421. The breadth of this arch will be equal to the apparent diameter of the sun, or about 32'; for what has been said in regard to rays proceeding from any one point in the sun, viz., that some of them will reach the eye under the angle of greatest brilliancy, is equally true of those which emanate from every point of his disk. 422. The explanation of the origin of the red arch is equally applicable to the rest of the colored arches. The latter will be found, however, below the former; for, since their angles of greatest brilliancy are each less than that of the red, they must consist of portions of smaller, concentric circles.

Thus, the violet arch can only be seen from drops below and within B, when the light that meets the eye coming in the direction B2 M, makes the angle B2 M O equal to 40° 17'. Between the violet and red arches the other colored bows will be seen arranged in the order of the spectrum; the whole forming, by their union, the primary bow.

Explain the manner in which the red arch is caused. What is its breadth?
Apply the same mode of reasoning to the other colored arches.
How is the primary bow formed?

• 423. SECONDARY Bow. The secondary bow is formed when the sun's rays, entering the bottom of the drop, suffer two reflections from the interior surface, and emerging at the top, reach the eye of the spectator after two refractions.

The course of the ray is seen in figure 22., where S E A is the incident ray, B and C the two points of reflection, and DEH the emergent ray, supposed to meet the eye of the observer at H.

424. So much light is lost by these successive changes in direction, that only at certain inclinations a sufficient quantity reaches

Fig. 22.

the eye, from each of the prismatic colors, to produce the secondary bow. Its tints, after all, are faint compared with those of the primary.

425. The violet light can only be seen when the angle of deviation SE H is 54° 9', and the red when it is 50° 59. Suppose, as in the case of the primary bow, that H L is the direction of a ray from the sun passing through the eye of the observer, and making with HE the angle L H E equal to the angle of deviation. If then, the line H E revolves about HL, the spectator, with his back to the sun and his eye at H, will behold in the heavens, between the limits of 54° 9′ and 50° 59' a prismatic bow consisting of similar portions of seven concentric circles; the violet arch assuming the highest position and the red the lowest. The other colors occupy intermediate places; the greater their refrangibility the greater their elevation.

Under what circumstances does the secondary bow occur?

Trace the course of the ray in figure 22.

What is said of the brilliancy of the secondary bow?

What must be the size of the angle of deviation that the violet light can be seen? What the size that the red ray may be visible?

How is the secondary bow formed?

426. The subject is further illustrated by the following figure, where the four parallel lines represent rays of the sun falling upon four drops of water, and OP the direction of another ray imagined to pass through the eye of the spectator, RO and VO are the red and violet rays of the primary bow; R' O and V O the red and violet rays of the secondary; and the positions of the red and violet arches of the two bows are indicated by the dotted lines. The other colored arches are found between the red and violet, following the order of colors in the prismatic spectrum. P is the centre of the rainbow.

427. In the explanation just given, we have reasoned as if the rain-drops were stationary, which of course is not the case; but this supposition leads to no error, inasmuch as the air is filled with rain-drops during the prevalence of a shower, and before one set of drops, by sinking too low, ceases to present to the eye the colors of the bow, another set has descended, taken their place, and is performing their office. While the observer is stationary the rainbow is fixed in position, but the drops that give rise to its glowing tints are continually changing.

428. BREADTH OF THE BOWS. The angular distance from the middle of the red arch to the middle of the violet in the inner bow, is the difference between 42°

2′ and 40° 17′ or 1° 45'; a quantity nearly equal to three and a half times the apparent breadth of the sun. This space is occupied by the remaining five colored arches, and, as each is 32′ in width, (Art. 421,) they necessarily overlap one another, and cause, by their mutual blending, an indistinctness in the boundary of the several hues. The two half-breadths of the red and violet arches added to 1° 45′ give the whole width of the bow, which is equal to 20 17, or about four and a half times the apparent diameter of the sun.

429. The breadth of the exterior bow, from the middle of the red to that of the violet, is found in like manner to be 3o 10'-the difference between 54° 9′ and 50° To this quantity 32' must be added to obtain the entire breadth.

59'.

The interval between the bows, computing from the red of the primary to that of the secondary, is 8° 57'. All these results, deduced theoretically, precisely agree with those obtained by actual measurement.

430. POSITION AND SIZE OF THE RAINBOW. Since the centre of the rainbow is in the direction of the line imagined to be drawn from the sun through the eye of the spectator, its position will evidently vary with that of the spectator, and its size with the altitude of the sun. If this luminary is 42° 2' above the horizon, the top of the inner bow will be just visible; but if upon the horizon, the bow will be a semicircle, having an elevation of 42° 2. If the observer, in the latter case, were upon the summit of a mountain, the arch would be somewhat greater than a semicircle; since the line of direction. from the sun through his eye, would strike the sky opposite, at a point above the horizon.

Should a person happen to be upon a mountain, when the sun is high in the heavens, and a shower at the same time occur in the vale below, he will sometimes perceive a rainbow forming a complete circle.

State what is said in regard to the breadth of the bows.
What in respect to the position and size of the rainbow.
When are entire circles beheld?

Such are said by Ulloa to be frequently seen on the mountains of Peru above Quito.

The foaming waters of cataracts are often spanned by richly tinted bows, caused by the rising spray. They are regularly seen at the falls of Schaffhausen, on the Rhine, and at the cataract of Niagara. At Terni, in Italy, where the river Velino rushes over a precipice 200 feet high, a bow of rare beauty is beheld. It appears, to a spectator below, arching the falls with its glowing tints, while two other bows are reflected on the right and left.

431. RAINBOWS IN THE NORTH. Rainbows are sometimes seen at mid-day. On the 13th of Dec. 1847, at one o'clock, P. M., Prof. Olmstead beheld at Yale College an entire bow in the north. During the same week, the writer observed at Hartford a similar bow at nearly the same hour of the day. Such a phenomenon can never arise, in the case of the primary bow, unless the sun's altitude at the time is considerably less than 42°, which only happens in the winter.

times observed. On

the 6th of August, D

* 1698, Dr. Halley beheld, while walking

on the walls of Chester, by the river Dee, a rainbow of the form represented in figure 24., where A B C is the primary bow, DE F the secondary, and AHGC the extraordinary bow, cutting the secondary at H and G. Its colors were arranged like those of the primary.

Give the instances of rainbows over cataracts.

When can rainbows appear in the north?

Explain from figure 24. the extraordinary bow seen by Halley.