The kaleidoscope

by David Brewster

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.

Fig. 9.

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, b O, 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 β b, a α; 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 B O, α O, and b O, and the points of development in the lines A O, a O, and β O, and vice versa, when the motion of the object is from B to A.

If the object moves in an oblique direction m n, 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 B 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 mirrors 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 image b formed at such a distance behind N, that b N = A N; 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 = a M - b N; and adding M N or its equal b M + b N to both sides of the equation, we obtain 2M N = a M - b N + b N + b M; but -b N + b N = 0, and a M + b M = a b; hence 2M N = a b. This result may be obtained otherwise, by considering, that if the mirror M advances one inch towards 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.

Fig. 10.

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 M N = a b, in the same time that the mirror moves through M N, the object being at α when the mirror is at N. Then, since A α = a b and b N = A N, the whole α 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.

Fig. 11.

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 A was stationary, and when M had moved to N, in which case a b = 2 M N, and bʹ the image when A had advanced to α through a space A α = 2 M N, we have b N = A N, and bʹ N = α N, and, therefore, b bʹ = A N - α N = A α = 2 M N, and a b + b bʹ or its equal a bʹ = 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 M 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.

Fig. 12.

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 vision will, of course, be stationary, while the mirrors describe an arch X of 10° for example; but since A O has approached X by 10°, the image of X formed behind A O must have approached X by 20°, and consequently moves with twice the velocity in the same direction as the mirrors. In like manner, since B O has receded 10° from X, the image of X formed by B O must have receded 20° from X, and consequently must have moved with twice the velocity in the same direction as the mirrors. Now, the image of X in the sector b O β is, as it were, an image of the image in B O a reflected from A O. But the image in B O a advances in the same direction as the mirror A O and with twice its velocity, hence the image of it in the sector b O β will be stationary. In like manner it may be shown, that the image in the sector a O α will be stationary. Since α O e is an image of b O r reflected from the mirror B O, and since all images in that sector are stationary, the corresponding images in α O e will move in the same direction α β as the mirrors; and for the same reason the images in the other half-sector β O e will move in the same direction; hence, the image of any object formed in the last sector α O β will move in the same direction, and with the same velocity as the images in the sectors A O b, B O a.

By a similar process of reasoning, the same results will be obtained, whatever be the number of the sectors, and whether the angle A O B be the even or the odd aliquot part of a circle. Hence we may conclude,

1. That during the rotatory motion of the mirrors round O, the objects in the sector seen by direct vision, and all the images of these objects formed by an even number of reflexions are at rest.

2. That all the images of these objects, formed by an odd number of reflexions, move round O in the same direction as the mirrors, and with an angular velocity double that of the mirrors.

3. That when the angle A O B is an even aliquot part of a circle, the number of moving sectors is equal to the number of stationary sectors, a moving sector being placed between two stationary sectors, and vice versa.

4. That when the angle A O B is an odd aliquot part of a circle, the two last sectors adjacent to each other are either both in motion or both stationary, the number of moving sectors being greater by one when the number of sectors is 3, 7, 11, 15, etc., and the number of stationary sectors being greater by one when the number of sectors is 5, 9, 13, 17, etc. And,

5. That as the moving sectors correspond with those in which the images are inverted, and the stationary ones with those in which the images are direct, the number of each may be found from the table given in [page 24].

When one of the mirrors, A O, is stationary, while the other, B O, is moved round, and so as to enlarge the angle A O B, the object X, and the image of it seen in the stationary mirror A O, remain at rest, but all the other images are in motion receding from the object X, and its stationary image; and when B O moves towards A O, so as to diminish the angle A O B, the same effect takes place, only the motion of the images is towards the object X, on one side, and towards its stationary image on the other. These images will obviously move in pairs; for, since the fixed object and its stationary image are at an invariable distance, the existence of a symmetrical arrangement, which we have formerly proved, requires that similar pairs be arranged at equal distances round O, and each of the images of these pairs must be stationary with regard to the other. Now, as the fixed object is placed in the sector A O B, and its stationary image in the sector A O b, it will be found that in the semicircle M b e, containing the fixed mirror, the

1st reflected image and direct object,			are stationary with respect to each other.
2d	3d reflected image
4th	5th
6th	7th
8th	9th

while in the same semicircle M b e, the

1st reflected image and	2d reflected image		are movable with respect to each other.
3d	4d
5th	6th
7th	8th
9th	10th

On the other hand, in the semicircle M a e, containing the movable mirror, the phenomena are reversed, the images which were formerly stationary with respect to each other being now movable, and vice versa.

In considering the velocity with which each pair of images revolves, it will be readily seen that the pair on each side, and nearest the fixed pair, will have an angular velocity double that of the mirror B O; the next pair on each side will have a velocity four times as great as that of the mirror; the next pair will have a velocity eight times as great, and the next pair a velocity sixteen times as great as that of the mirror, the velocity of any pair being always double the velocity of the pair which is adjacent to it on the side of the fixed pair. The reason of this will be manifest, when we recollect what has already been demonstrated, that the velocity of the image is always double that of the mirror, when the mirror alone moves towards the object, and quadruple that of the mirror when both are in motion, and when the object approaches the mirror with twice the velocity. When B O moves from A O, the image in the sector B O a moves with twice the velocity of the mirror; but since the image in b O β is an image of the image in B O a reflected from the fixed mirror A O, it also will move with the same velocity, or twice that of the mirror B O. Again, the image in the sector a O α, being a reflexion of the stationary image in A O b from the moving mirror, will itself move with double the velocity of the mirror. But the image in the next sector α O β is a reflexion of the image in b O β from the moving mirror B O; and as this latter image has been shown to move in the direction b β, with twice the velocity of the mirror B O, while the mirror B O itself moves towards the image, it follows that the image in α O β will move with a velocity four times that of the mirror. The same reasoning may be extended to any number of sectors, and it will be found that in the semicircle M b e, containing the fixed mirror,

The images formed by		2 and 3		reflexions, move with	2		times the velocity of the mirror;
		4 and 5			4
		6 and 7			8
		8 and 9			16

whereas in the semicircle M a e, containing the movable mirror,—

The images formed by		1 and 2		reflexions, move with	2		times the velocity of the mirror;
		3 and 4			4
		5 and 6			8
		7 and 8			16

a progression which may be continued to any length.

Before concluding this chapter, it may be proper to mention a very remarkable effect produced by moving the two plain mirrors along one of two lines placed at right angles to each other. When the aperture of the mirrors is crossed by each of the two lines, the figure created by reflexion consists of two polygons with salient and re-entering angles. By moving the mirrors along one of the lines, so that it may always cross the aperture at the same angle, and at the same distance from the angular point, the polygon formed by this line will remain stationary, and of the same form and magnitude; but the polygon formed by the other line, at first emerging from the centre, will gradually increase till its salient angles touch the re-entering angles of the stationary polygon; the salient angles becoming more acute, will enclose the apices of the re-entering angles of the stationary polygon, and at last the polygon will be destroyed by truncations from its salient angles.

When the lines cross each other at a right angle, the salient angles of the opening polygon can never touch the salient angles of the stationary polygon, but always its re-entering angles. If the lines, however, form a less angle than the complement of the angle formed by the mirrors, then the salient angles of the opening polygon may touch the salient angles of the stationary polygon, by placing the mirrors so as to form re-entering angles in the polygon. When the lines form an angle between 90° and the complement of the angle formed by the mirrors, the salient angle of the opening polygon may be made to touch the salient angles of the stationary one, but in this case the stationary polygon can have no re-entering angles. The preceding effects are finely exemplified by the use of a Kaleidoscope with a draw-tube and lens described in [Chapter X]., and by employing the vertical and horizontal bars of a window, which may be set at different angles, by viewing them in perspective.