and therefore, if sufficiently dense, they may form coloured arches within the primary bow, but they must be fainter than the colours of the bow, because but few rays are thus irregularly reflected in comparison with such as are regularly reflected. These arches will be variously mixed with the colours of the bow, and some reach below it; and thereby alter their colours a little.

THE REFLEXION OF LIGHT,

BY PLANE SURFACES.

THE Surfaces may be either plane, concave, or convex, which reflect the rays of light. The whole doctrine of reflexion depends upon this fundamental principle, which we have already demonstrated, viz. That the angle of incidence is equal to the angle of reflexion; and when any point of an object is seen by reflected light, it is always seen in the direction of the reflected ray, and at its intersection with the perpendicular drawn to the reflecting surface from the object.

In a plane mirror, rays diverging from any point of an object are reflected back diverging, as if they had proceeded from an imaginary radiant point in the perpendicular, as far behind the mirror as the object is before it. Let EBC* be a plane mirror, AED, HB, IC, perpendicular to its surface, A, any point of an object, from which rays diverge and fall upon the mirror at B and C; these rays will be reflected back diverging from the point D, in the directions BF and CG, making ED=AE. Because ABH=FBH=BDE, and the angles at E, are right, and EB is common to both triangles AEB, DEB, therefore AE=ED. Hence DBF is the reflected ray. In the same manner DCG will be the reflected ray, when incident on the

*See Plate 11, fig. 3.

surface at C from the same point A. And as both the rays BF and CG are reflected from the same point D, this point will be the focus of the rays, which diverge from A: and it is in the perpendicular to the surface of the speculum, as far behind it as A is be fore it.

Hence, as this is true of every point of the object, it must necessarily follow, that the image will be similar to, and of equal magnitude with the object. It follows also, from the same proposition, that the dis-. tance of the image from the eye will be equal to the sum of the incident and reflected rays.

If the object be parallel to the mirror, it is not necessary that the mirror should be more than half of the length and breadth of the object, to represent the whole of the image, to an eye at the place of the object. Yet when the mirror is so short, that we cannot see the whole of the object, we may nevertheless see it completely, either by bringing the eye nearer to the mirror, or by removing the object farther from it. Yet this change of position will be of no advantage to assist a person in seeing the whole of his own figure in a mirror, which is shorter than one half of his own length; because as the eye approaches to the mirror, the image approaches to it at the same time, whereas it should continue in the same place, to give this advantage. So that at all distances he will see an equal portion of himself.

The whole of the object AC* may be seen by an eye at either extremity of it, in a mirror EF, which is half its length. But if the eye approach the mirror to I, while the object continues stationary, it may be seen, although it extended to G.

In a plane mirror, that, which is on the right hand

* See Plate 11, fig. 4.

of the object appears on the left hand of the image, and vice versa, because the face of the image is turned the contrary way from that of the object. While a person moves his right hand, before the glass, his image appears to move its left; and all the letters of a book held before it, will, for the same reason, be reversed, but not inverted, as the image is always erect, behind such a speculum, or in a positive focus.

But if a plane mirror be held parallel to the horizon, any object that is perpendicular to it, will have its image inverted. This is seen in the inverted images of trees and buildings on the bank of a river, because the roots being nearer to the reflecting surface, must have their images below it nearer to the surface, than those of the tops. In like manner, if any person hold the speculum above his head parallel to the horizon, he will see his image inverted above him. But if he incline it in an angle of 45° to the horizon, his image will be parallel to the horizon, and the images of all objects, that are parallel to the horizon, will appear erect. Hence the image of any object, that is stationary, will appear, by the motion of the speculum, to move over double the space through which the speculum moves, whether the motion be angularly or in a right line.

If an object be placed between two plane mirrors, inclined to each other in any angle, a number of ima ges will be formed by them, which will be placed in the circumference of a circle, whose center is the point of concurrence of the planes, and whose radius is the distance of the object from the said point. One set of these images, formed first by one of the specula, will exhibit one side of the object, and the other set will exhibit the other side of it. The first image

* See Plate 12, fig. 1. O the object, N, P, the two first images.

formed by one mirror will become the object, from whence a second image will be formed by the other mirror, and as far behind it as the other image was before it. This second image will also become the object, from whence a third image will be formed by the first mirror, and so on, till this set of images, which began from the first mirror shall end. A second set of images, beginning from the second mirror, and continued alternately between them, will be formed at the same time.

The number of images will be determinate; for whenever the place of the last image of either set falls between the planes of both the mirrors, produced beyond their concurrence, no more images can be formed from it considered as an object; because it is then behind the reflecting surfaces of both the mirrors, and no rays from it can fall upon them...

That these images will be all formed in the circumference of a circle may be proved geometrically from this consideration, that each image is as far behind the mirror by which it is formed, as the object is before it, and in a line drawn perpendicular to the plane of the mirror from the object to the image. Now as this line is bisected at right angles by the plane of the mirror, the plane must pass through the center of a circle, whose circumference passes through the places of the object and image.

The angular distanceof the two first images, formed by the two mirrors, is equal to twice the inclination of the mirror. Because each image being as far behind the mirror that formed it, as the object is before it, the sum of the distances of the object from the mirrors, or the inclination of the mirrors, can be but half the distances of the images.

The angular distance of the next two images, form

ed by two reflexions each, exceeds the angular distance of the first two, by twice the angle of inclination. And each succeeding pair of images, formed by an equal number of reflexions, will exceed the angular distance of the prceding pair by the same quantity, viz. twice the angle of inclination.

If KL, KM,* be two mirrors, and O, an object between them, there will be one set of images formed at A, B, C, D, E, and another set at a, b, c, d, e. Now the angular distance of the two first images, A, a, viz. AKa, is equal to 2 OKL+2 OKM=2 LKM= twice the inclination of the mirrors. And the angular distance BKb, of the two succeeding images, exceeds the angular distance AKa, of the first two images, by the angles BKa+AKb=AKO+aKO=2LKM.

Hence as the distance between each succeeding pair of images increases by double the angle of inclination of the planes, there must be as many images formed by the two mirrors, as there are angles in the circle equal to the angle of inclination. Thus, if the angle of inclination were 10°, 20°, 30°, 40°, or 45°, the number of images would be 36, 18, 12, 9, or 8 respectively. And if the double angle of inclination were commensurate to a whole circle, the two last images would coincide and be formed in the same place, whereby one of them would disappear.

As all the images are formed in the circumference of a circle, if the radius of the circle were increased ad infinitum, or the angle of inclination between the mirrors were diminished ad infinitum, so that the planes would become parallel, the number of images would become infinite, and all be placed in a right line.

The angular distance between the object and either of the second pair of images is double to the inclina

* See Plate 12, fig. 2.