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CHAPTER III.

ON THE EFFECTS PRODUCED BY THE MOTION OF THE OBJECT AND THE MIRRORS.

HITHERTO We have considered both the object and the mirrors as stationary, and we have contemplated only the effects produced by the union of the different parts of the picture. The variations, however, which the picture exhibits, have a very singular character, when either the objects or the mirrors are put in motion. Let us, first, consider the effects produced by the motion of the object when the mirrors are at rest.

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If the object moves from x to o, Fig. 9, in the direction of the radius, all the images will likewise move towards o,

and the patterns will have the appearance of being absorbed or extinguished in the centre. If the motion of the object is from o to x, the images will also move outwards in the direction of the radii, and the pattern will appear to develop itself from the centre o, and to be lost or absorbed at the circumference of the luminous field. The objects that move parallel to x o will have their centre of development, or their centre of absorption, at the point in the lines A O, B o, a o, bo, etc. where the direction in which the images move cuts these lines. When the object passes across the field in a circle concentric with A B, and in the direction A B, the images in all the sectors formed by an even number of reflexions will move in the same direction A B, namely, in the direction B b, a a; while those that have been formed by an odd number of reflexions will move in an opposite direction, namely, in the directions a B, A b. Hence, if the object moves from A to B, the points of absorption will be in the lines в O, a 0, and b o, and the points of development in the lines A o, a o, and 6 o, and vice versa, when the motion of the object is from B to A.

If the object moves in an oblique direction mn, the images will move in the directions m t, o n, o p, q t, q p, and m, o, q, will be the centres of development, and n, p, t, the centres of absorption; whereas, if the object moves from n to m, these centres will be interchanged. These results are susceptible of the simplest demonstration, by supposing the object in one or two successive points of its path m n, and considering that the image must be formed at points similarly situated behind the mirrors; the line passing through these points will be the path of the image, and the order in which the images succeed each other will

give the direction of their motion. Hence, we may conclude in general,

1. That when the path of the object cuts both the mirrors A o and в o like m n, the centre of absorption will be in the radius passing through the section of the mirror to which the object moves, and in every alternate radius ; and that the centre of development will be in the radius passing through the section of the mirror from which the object moves, and in all the alternate radii: and,

2. That when the path of the object cuts any one of the inirrors and the circumference of the circular field, the centre of absorption will be in all the radii which separate the sectors, and the centre of development in the circumference of the field, if the motion is towards the mirror, but vice versa if the motion is towards the circumference.

When the objects are at rest, and the Kaleidoscope in motion, a new series of appearances is presented. Whatever be the direction in which the Kaleidoscope moves, the object seen by direct vision must always be stationary, and it is easy to determine the changes which take place when the Kaleidoscope has a progressive motion over the object. A very curious effect, however, is observed when the Kaleidoscope has a rotatory motion round the angular point, or rather round the common section of the two mirrors. The picture created by the Instrument seems to be composed of two pictures, one in motion round the centre of the circular field, and the other at rest. The sectors formed by an odd number of reflexions are all in motion in the same direction as the Kaleidoscope, while the sector seen by direct vision, and all the sectors formed by an even number of reflexions, are at rest. In order to understand this, let M,

Fig. 10, be a plane mirror, and a an object whose image is formed at a, so that a M = A M. Let the mirror M advance to N, and the object A, which remains fixed, will have its

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image b formed at such a distance behind N, that b N AN; then it will be found that the space moved through by the image is double the space moved through by the mirror; that is, a b 2 M N. Since M N = A M-A N, and since A M = a M, and A N = b N, we have M N = α M —b N; and adding M N or its equal bм + 6 N to both sides of the equation, we obtain 2 M Na M- bN+bN + 6 M ; but-b NbN 0, and a м+ b м = a b; hence 2 M N = a b. This result may be obtained otherwise, by considering, that if the mirror м advances one inch to A, one inch is added to the distance of the image a, and one subtracted from the distance of the object; that is, the difference of these distances is now two inches, or twice the space moved through by the mirror; but since the new distance of the object is equal to the distance of the new image, the difference of these distances, which is the space moved through by the image, must be two inches, or twice the space described by the mirror.

Let us now suppose that the object A advances in the same direction as the mirror, and with twice its velocity, so as to describe a space A α = 2 MN ab, in the same time

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that the mirror moves through MN, the object being at

a when the mirror is at N. Then, since A α = a b and b N

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AN, the whole a N is equal to the whole a N, that is, a will still be the place of the image. Hence it follows, that if the object advances in the same direction as the mirror, but with twice its velocity, the image will remain stationary.

If the object A moves in a direction opposite to that of the mirror, and with double its velocity, as is shown in Fig. 11; then, since b would be the image when ▲ was stationary, and when м had moved to N, in which case a b = 2 MN, and b' the image when A had advanced to

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a through a space A a = 2 M N, we have b NA N, and b' N = a N, and, therefore, bb' = A N—α N = A α = 2 M N, and a b+b b' or its equal a l' = 4 M N. Hence it follows, that when the object advances towards the mirror with twice its velocity, the image will move with four times the velocity of the mirror.

If the mirror м moves round a centre, the very same results will be obtained from the very same reasoning, only the angular motion of the mirror and the image will then be more conveniently measured by parts of a circle or degrees.

Now, in Fig. 12, let x be a fixed object, and A O, B O, two mirrors placed at an angle of 60° and moveable round o as a centre. When the eye is applied to the end of the mirrors (or at E, Fig. 1), the fixed object x, Fig. 12, seen by direct

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