In order to understand how this effect is produced, let us take a small sector of white paper of the shape A O B, [Fig. 2], and having laid it on a black ground, let the extremity A O of one of the reflectors be placed upon the edge A O of the sector. It is then obvious that an image A O b of this white sector of paper will be formed behind the mirror A O, and will have the same magnitude and the same situation behind the mirror as the sector A O B had before it. In like manner, if we place the edge B O of the other reflector upon the other side B O of the paper sector, a similar image B O a will be formed behind it. The origin of three of the sectors seen round O is therefore explained: the first, A O B, is the white paper sector seen by direct vision; the second, A O b, is an image of the first formed by one reflexion from the mirror A O; and the third is another image of the first formed by one reflexion from the other mirror B O. But it is well known, that the reflected image of any object, when placed before another mirror, has an image of itself formed behind this mirror, in the very same manner as if it were a new object. Hence it follows, that the image A O b being, as it were, a new object placed before the mirror B O, will have an image a O α of itself formed behind B O; and for the same reason the image B O a will have an image b O β of itself, formed behind the mirror A O, and both these new images will occupy the same position behind the mirrors as the other images did before the mirrors.

A difficulty now presents itself in accounting for the formation of the last or sixth sector, α O β. Mr. Harris, in the xviith Prop. of his Optics, has evaded this difficulty, and given a false demonstration of the proposition. He remarks, that the last sector, α O β is produced “by the reflexion of the rays forming either of the two last images” (namely, b O β and a O α); but this is clearly absurd, for the sector α O β would thus be formed of two images lying above each other, which is impossible. In order to understand the true cause of the formation of the sector α O β, we must recollect that the line O E is the line of junction of the mirrors, and that the eye is placed any where in the plane passing through O E and bisecting A O B. Now, if the mirror, B O, had extended as far as O β, the sector α O β would have been the image of the sector b O β, reflected from B O; and in like manner, if the mirror A O had extended as far as O α, the sector α O β would have been the image of the sector a O α reflected from A O; but as this overlapping or extension of the mirrors is impossible, and as they must necessarily join at the line O E, it follows, that an image α O e, of only half the sector b O β, viz., b O r, can be seen by reflexion from the mirror B O; and that an image β O e, of only half the sector a O α, viz., a O s, can be seen by reflexion from the mirror A O. Hence it is manifest, that the last sector, α O β, is not, as Mr. Harris supposes, a reflexion from either of the two last images, b o β, a o α, but is composed of the images of two half sectors, one of which is formed by the mirror A O, and the other by the mirror B O.

Mr. Harris repeats the same mistake in a more serious form, in his second Scholium, § 240, where he shows that the images are arranged in the circumference of a circle. The two images D, d, says he, coincide and make but one image. Mr. Wood has committed the very same mistake in his second Corollary to Prop. xiv., and his demonstration of that Corollary is decidedly erroneous. This Corollary is stated in the following manner:—“When a (the angle of the mirrors) is a measure of 180° two images coincide,” and it is demonstrated, that since two images of any object X ([Fig. 2]) must be formed, viz., one by each mirror, and since these two images must be formed at 180° from the object X, placed between the mirrors, that is, at the same point x, it follows that the two images must coincide. Now, it will appear from the simplest considerations, that the assumption, as well as the conclusion, is erroneous. The image x is seen by the last reflexion from the mirror B O E, and another image would be seen at x, if the mirror A O E had extended as far as x; but as this is impossible, without covering the part of the mirror B O E, which gives the first image x, there can be only one image seen at x. When the object X is equidistant from A and B, then one-half of the last reflected image x will be formed by the last reflexion from the mirror B O, and the other half by the last reflexion from the mirror A O, and these two half images will join each other, and form a whole image at e, as perfect as any of the rest. In this last case, when the angle A O B is a little different from an even aliquot part of 360°, the eye at E will perceive at e an appearance of two incoincident images; but this arises from the pupil of the eye being partly on one side of E and partly on the other; and, therefore, the apparent duplication of the image is removed by looking through a very small aperture at E. As the preceding remarks are equally true, whatever be the inclination of the mirrors, provided it is an even aliquot part of a circle, it follows,—

1. That when A O B is ¼, ⅙, ⅛, ⅒, ¹/₁₂, etc., of a circle, the number of reflected images of any object X, is 4 - 1, 6 - 1, 8 - 1, 10 - 1, 12 - 1.

2. That when X is nearer one mirror than another, the number of images seen by reflexion from the mirror to which it is nearest will be ⁴/₂, ⁶/₂, ⁸/₂, ¹⁰/₂, ¹²/₂, while the number of images formed by the mirror from which X is most distant will be ⁴/₂ - 1, ⁶/₂ - 1, ⁸/₂ - 1, ¹⁰/₂ - 1; that is, an image more always reaches the eye from the mirror nearest X, than from the mirror farthest from it.

3. That when X is equidistant from A O and B O, the number of images which reaches the eye from each mirror is equal, and is always

4 - 1 , 6 - 1 , 8 - 1 , 10 - 1 , 12 - 1 .
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which are fractional values, showing that the last image is composed of two half images.

When the inclination of the mirrors, or the angle A O B, [Fig. 3], is an odd aliquot part of a circle, such as ⅓, ⅕, ⅐, ⅑, etc., the different sectors which compose the circular image are formed in the very same manner as has been already described; but as the number of reflected sectors must in this case always be even, the line O E, where the mirrors join, will separate the two last reflected sectors, b O e, a O e. Hence it follows,—

Fig. 3.