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 PP 1 being perpendicular to OA. Now P 1 may be regarded as a new object placed before the mirror OB, and hence the image of P 1 formed by OB will be P 2 where P 1 Y 1 = P 2 Y 1 . Similarly the image of P formed by OB will be P 1 ', where PY=P 1 'Y, and the image of P 1 ' formed by 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.

S,. 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 0 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 OB, third at OA, and so on, by the symbols P 1 , P 2 , P 3 respectively; and the images formed by a first reflexion at OB, second at OA, third at OB, and so on by P 1 ', P 2 ', P 3 ' respectively. Draw PP 1 perpendicular to OA, P 1 P 2 perpendicular to OB, P 2 P 3 perpendicular to AO produced, and P 3 P 4 perpendicular to BO produced. Then P], P 2 , P 3 , P 4 are the first set of images formed. Similarly draw the lines PP 1 ', P∕P 2 ', P 2 'P 3 ', P 3 , P 4 ', then P 1 ’, P 2 ', P 3 ζ 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 A0B, it is evident that no further reflexion can take*place, 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

P 1 PP 1 , = 2PA + 2PB = 2AB = 2α.

P 2 PP 2 ' = PP 2 + PP 2 ' = PB + P 1 B + PA + P 1 'A

≈PB + PA + AB + P 1 A +AB + P 1 'B

= 4AB = 4α, P 3 PP 3 ' = 6a.

P n PP√=2na,

Now, when the last formed imagée coincide, the arcual distance between them must be a whole circumference. Hence if P n and P√ be the last formed coincident images, we have

P 71 PP√ = 2nα — 2ιr.

Hence a = —; that is, the mirrors must be inclined to each other r H 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 1 . The image of PQ j formed by OB is P 1 Q 2 , and the image of QP 1 formed by OA is Q 1 P 2 . Now it is readily seen that the points P 2 and Q 2 will not, in general, coincide, and, hence, a symmetrical 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 symmetrical 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 360°. 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 the greatest breadth from 1 to 3 inches.

The mirrors are first fixed, in any convenient manner, at the proper angle, and then inserted into a cylindrical tube of brass or paper. At the one end of the tube is a small eyehole 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 produced. 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 O; 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 of both mirrors,—first, because in that position oιdy the direct and reflected sectors are all at the same distance from the eye, and, in consequence, no want of symmetry is introduced by the foreshortening of one sector more than another; and, secondly, because in that position the maximum amount of light is reflected to the eye by the mirrors, and, in consequence, the various sectors appear as nearly as possible equally illuminated. Of course a certain amount of light is necessarily lost at each reflexion, and hence there is always a slight difference between the luminosity of the various sectors. However, this is found not to introduce any serious want of symmetry when the instrument is properly constructed.

A modification of the simple kaleidoscope was introduced by Sir David Brewster, whereby the images of large and distant objects can be introduced into the picture. This is effected by removing the object box and replacing it by a tube carrying at its outer end a double convex lens, represented by LL in fig. 5. By a screw adjustment the lens can be so placed as to focus the distant object exactly in the plane of the sector AOB, and so bring its image into the very best position for producing symmetrical patterns. When this instrument is directed towards a tree in full foliage, or towards an arrangement of flowers in full bloom, a very beautiful effect is produced, which can be varied by gradually moving the instrument. This form was called by Brewster the telescopic kaleidoscope.

Another form is called the polyangular kaleidoscope. (fig. 6). The only essential difference in it is that the mirrors are so arranged that the angle between them can be varied at pleasure. This, being very useful for illustrating the theory of the instrument, is the form usually found in collections of philosophical apparatus.

In all the instruments above described only two mirrors have been employed; but obviously we may have more than two. Suppose we wish to employ three mirrors enclosing a triangular opening, and that we also wish to produce perfectly symmetrical pictures. We are here limited in our choice of angles by the following conditions—first, the sum of the three angles which the mirrors make with each other must be equal to 180°, and, secondly, each angle must be an even submultiplo of 360°. By trial it is easily found that the only angles which satisfy these conditions are 60 δ , 60°, 60°; 90°, 60°, 30°; and 90°, 45°, 45°. Hence with three mirrors we must choose one or other of these three sets. The first is that usually chosen.

Suppose similarly we wish to use four mirrors; then, we must put them either in the form of a square, when all the mirrors are of equal breadth, or in the form of a rectangle, when the opposite mirrors are of equal breadth. It is obvious that in these two cases only will the angle between each pair of mirrors be an even submultiple of 360°.

With more than four mirrors kaleidoscopes cannot be constructed so as to give symmetrical forms, since each of the interior angles of a regular polygon of more than four sides must exceed an even submultiple of 360°.

See Harris’s Optics ; Wood’s Optics ; Parkinson’s Optics ; Brewster’s Treatise οη the Kaleidoscope. The last-mentioned contains an account of the application of the instrument to the art of designing.