Schaw. — On Bainboius. 457 



the arc of a circle ; and this must be the case whenever the 

 rain-sheet is inclined towards or away from us by wind, or 

 when one side of it is farther from us than the other, or, finally, 

 when the sun is comparatively high above the horizon, and 

 the rain-sheet is vertical. In all these cases the cone having 

 its vertex at our eyes, and its axis in the line from the sun, 

 through our head and eyes, to the rain-sheet, must be inter- 

 sected by the rain-sheet in a plane to which the axis is not 

 perpendicular, and the intersection must be the conic section 

 known as an ellipse. 



In Plate LI., fig. 9, the geometric elevations of the rain- 

 drops forming two rainbows are shown as they would appear 

 if occurring when the sun's rays were nearly horizontal, soon 

 after sunrise or sunset : b a l c, a semicircle on a vertical rain- 

 sheet, and b a c, an elliptical curve with the major axis ver- 

 tical, occurring on a rain-sheet iuclined by the wind 15° from 

 the vertical. 



In Plate LI., fig. 10 are shown the geometric elevations of 

 the rain-drops f owning two rainbows — one, a t ob t , a semicircle, 

 as before ; the other, a o b, a semi-ellipse with the major axis 

 horizontal, formed on a rain-sheet oblique in plan to the sun's 

 rays, the ellipse being thus thrown altogether towards the 

 side where the rain-sheet was most distant. 



In Plate LI., fig. 11, are shown similarly the geometric ele- 

 vations of the drops forming two rainbows as seen from the top 

 of a mountain 8,000ft. high, the rain-sheet being vertical and 

 about 8,000ft., or, say, a mile and a half, distant. In this case, 

 if the sun were near setting, and the sun's rays consequently 

 nearly horizontal, the rainbow would appear as the geometric 

 elevation of the raindrops forming it, a complete circle, ex- 

 cepting the part obscured by the shadow of the mountain — 

 were the observer in a balloon the circle would be complete. 

 And it is to be noted that great altitude is not required 

 in order that a circular or nearly circular rainbow may 

 be observed* — the conditions are that the elevation shall 

 be about the same as the distance of the rain-cloud from the 

 observer. 



In this same figure (fig. 11) is shown the elliptical geometric 

 elevation of the raindrops forming the rainbow which would 

 be observed under the same conditions except that the sun is 

 higher up in the heavens, the axis of the cone having thus 

 Income oblique instead of perpendicular to the plane of the 

 rain-sheet. The rainbow would, however, appear as the lower 

 and less complete circle. 



* The curvature of the earth has but little influence on the result : 

 at 10,000ft. altitude only about 2° of depression would be given to the 

 sun when setting or rising. 



