Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/861

KALEIDOSCOPE 827 dred, and sold in almost every toy-shop. Large cargoes of them were sent abroad ; and it is stated that no fewer than two hundred thousand were sold in London and Paris in the space of three months. Besides being of essential service in the art of the designer, the kaleidoscope consti tutes a very useful piece of philosophical apparatus, as it illustrates, in a very beautiful way, the optical problem of the multiplication of images produced by reflexion when the object is placed between two plane mirrors inclined to each other at a definite angle. The general principle of the instrument will be easily understood from the following description and figures. 1. Let OA, OB (fig. 1) be the sections of two plane mirrors placed perpendicular to the plane of the paper and inclined to each other at a right angle. Let P be a luminous point, or object, placed between them. According to the general law of the reflexion of light from plane mirrors, the image of P formed by the mirror OA will be as far behind OA as P is in front of it ; that is, the image of P is P 1; where PX = P 1 X, the line PPj being perpendicular to OA. Now P! may be regarded as a new object placed before the mirror OB, and hence the image of P x formed by OB will be P 2 where PjYj = P 2 Yj. Similarly the image of P formed bv OB will be P/, where PY-P/Y, and the image of P/ formed by Fig. 1. OA will also be at a point such that P 1 X 1 = P 2 X 1 , that is, the two last formed images will coincide. Hence we have three images placed symmetrically about 0, constituting, with the object P, a symmetrical pattern of four luminous points placed at the corners of a rectangle. 2. Let the mirrors OA and OB (fig. 2) be inclined to each other at any angle a, and let P be the object placed between them. With centre and radius OP describe a circle. Evidently the images formed by successive reflexions from the mirrors will all lie on the circumference of this circle. We shall denote the images formed by a first reflexion at OA, second at B, third at OA, and so on, by the symbols P 2 , P 3 respectively ; P, and the images formed by a first reflexion at OB, second at OA, third at OB, and so on by P/, P 2 , P 3 P 5 respectively. Draw PP X perpendicular to OA, PjP 2 perpendicular to OB, P 2 P 3 perpendicular to AO pro duced, and P 3 P 4 perpen dicular to BO produced. Then P Jf P 2 , P 3 , P 4 are the first set of images formed. Similarly draw the lines PTJ p /p / p p p p i rr l > L 1 A 2 > r 2 r 3 l 3 * 4 ) then P/, P 2 , P 3 , P 4 are the second set of images formed by a first reflexion at OB. Now, when any image falls within the angle vertically opposite to AOB, it is evident that no further reflexion can takcplace, as it is behind both mirrors. Hence the number of images formed depends upon the size of the angle AOB and also upon the position of the point P in relation to the mirrors. "When a symmetrical picture is required, it is essential that the two last formed images, that is, P 4 and P 4 in the figure, should coincide, and we must determine when this will be the case. We shall measure the distances of the several images from P by the arcual distances PP 1} &c. Now it is evident that PiPP/ - 2PA + 2PB = 2AB = 2o. P 2 PP 2 = P P 2 + PP = P B + PjB + PA + P/A = 4AB=4a. P pp _ I 3 i i 3 Da. P n PP )! =2wa. Now, when the last formed imagee coincide, the arcual distance between them must be a whole circumference. Hence if P,, and P,, be the last formed coincident images, we have P n PP n = 2?ia = 2 1 r. Hence a = ; that is, the mirrors must be inclined to each other n at an angle which is an exact submultiple of two right angles, or, which is the same thing, an even submultiple of 360. 3. Next suppose that, instead of a point, we put a line as an object in the angle between the mirrors ; and, first, let us suppose that the mirrors are inclined to each other at an angle which is an odd submultiple of 360 (as one-fifth of 360 in fig. 3). OA, OB are the mirrors, PQ the line placed between them. The image of PQ formed by OA is PQ 1; that formed by OB is QP V The image of PQ 1 formed by OB is PjQ.), and the image of QPj formed by OA is QjP.2. Now it is readily seen that the points P 2 and Q 2 will not, in general, coincide, and, hence, a sym metrical picture of the line cannot in general be formed when the angle is an odd submultiple of 360. If, however, the line OP = OQ, then the points P 2 and Q 2 will coincide, and a sym- Fig. 3. metrical picture of five lines be formed. Secondly, let us suppose that the angle AOB is an even submultiple of 360. By following the course of the images it will be seen that the last-formed images of the line coincide in all positions of PQ, and hence a symmetrical figure can, in all cases, be formed. As the object of the kaleidoscope is to produce symmetrical figures from objects placed in any position between the mirrors, we are necessarily limited to angles which are even submultiples of 360. The simple kaleidoscope consists essentially of two plane mirrors EOA and EOB (fig. 4) inclined to each other at an angle which is an even submultiple of 3GO. A very common angle in practice is 60. The mirrors are usually made of two strips of thin flat glass, the length of each being from 6 to 12 inches, and A the greatest breadth from 1 to 3 inches. The mirrors are first fixed, in any convenient manner, at the Fig. 4. proper angle, and then inserted into a cylindrical tube of brass or paper. At the one end of the tube is a small eye hole opposite the point E, while the other end is closed by what is called the " object box." This consists of a shallow cylindrical box, which fits on to the end of the tube, and contains the objects from whose reflexion the pattern is pro duced. These objects may consist of petals of differently coloured flowers, scraps of differently coloured paper, or, still better, pieces of coloured glass. Very often the objects consist of small glass tubes filled with differently coloured liquids and then hermetically sealed. These produce a very fine effect. The objects are placed in the box between two circles of thin glass which fit into the box, the one of which is transparent and the other obscured by grinding. When in position the transparent glass is close to the end of both mirrors and fills up the sector AOB, while the other, the obscured one, is fixed into the outer end of the object box. The distance between the two glasses is made as small as possible, just room enough being left to allow the objects to fall freely by their own weight into any position between the glasses. Suppose now that the angle AOB is 60, and that the eye is placed at E, a beautiful symmetrical picture of six equal and similar sectors will be seen round the point ; and, by simply turning the tube round, so as to allow the objects to fall into a new position, an endless variety of pictures can be produced. It is important to notice the proper position of the eye. This should be, as nearly as possible, in the plane