1. _On combinations of four mirrors forming a square._ [Illustration: FIG. 45.] The first of these Kaleidoscopes is represented in Fig. 45, where =A B=, =B C=, =C D=, =D A=, are the four equal and similar reflectors placed accurately at right angles to each other. If we consider the effect only of the two reflectors =A B=, =B C=, and regard =A D=, =D C= as only the limits of the aperture, it is obvious, from the principles explained in Chap. II., that we shall have a regular figure =D _m k h_ D=, composed of four squares, one of which =D B= is seen by direct vision; other two =A _l_=, =C _i_= formed by one reflexion from each mirror; and the fourth =B _k_= composed of two half squares, each half being formed by a second reflexion from each mirror. In like manner, if we suppose =A D=, =D C= to act alone, they will form a square pattern =B _b d f_=, composed like the last; and the same result will be obtained by supposing =B A=, =A D=, and =B C=, =C D= to act alone. The combination of these effects will produce a square _a d g k_, composed of nine squares, four of which, formed by second reflexions, are placed at the angles; other four formed by first reflexions in the middle; and one, seen by direct vision, in the centre. Hence, it follows, that the light of the different squares is symmetrical as well as the patterns, a property which does not belong to all polycentral instruments. The pattern, however, does not terminate with the square _a d g k_, but extends indefinitely on all sides till the squares become invisible, from the extinction of the light by repeated reflexions. In order to discover the law according to which the squares succeed each other, we shall examine in what manner a still larger square =E F G H= is completed round the central square, seen by direct vision. By considering every square in the large square _a d g k_ as an object placed before the four reflectors, and recollecting that the reflected images must be similarly situated behind the reflectors, we shall find that the larger square =E F G H= is completed by images that have suffered two, three, and four reflexions, as marked in the figure, and that all these are symmetrically arranged with regard to the central square. The squares which are crossed with a dotted diagonal line, are those composed of two halves, each half being formed by a different reflector. When a Kaleidoscope is formed out of the preceding combination, the aperture, or the breadth of the plates next the eye, should not exceed one-sixth of an inch. The effect is very pleasing when the reflectors are accurately joined and nicely adjusted, and when distant objects are introduced by means of a lens. 2. _On combinations of four mirrors forming a rectangle._ When the reflectors are of different breadths, so as to form a rectangle, the very same effects are produced as in the preceding combination, with this difference only, that the images are all rectangular, in place of being square.