3. _On combinations of three reflectors at angles of 60°._ [Illustration: FIG. 46.] When three reflectors are combined at angles of 60°, as shown at =A O B=, Fig. 46, they form an equilateral triangle, and therefore all the images will also be equilateral triangles. The figure =C D E F G H=, which is a truncated equilateral triangle, is obviously composed of three hexagonal patterns, of which the sectors, or rather triangles, are arranged round the three centres =A=, =O=, =B=; the triangle =A O B= being common to all the three. The three triangles, adjacent to the sides of =A O B=, are formed by one reflexion from each mirror. The three which spring from the vertices =A=, =O=, =B=, of the triangle, consist of two halves, each of which is formed by three reflexions, the last reflexion of the one half being made from one of the nearest mirrors, and that of the other half from the other nearest mirror. If we consider the formation of a more extended figure, =I L M P N K=, which is also a truncated equilateral triangle, with its truncations corresponding to the sides of the former figure, we shall find that it has been completed by an addition, to each side of the former, of three equilateral triangles, two of which are formed by three reflexions, and the third, consisting of two halves, formed by four reflexions. This figure consists of three entirely separate hexagons, =I C A O H K=, =L D A B E M=, and =B O G N P F=, all of which are formed of reflected images;—of one triangle =A O B= seen by direct vision,—and of three triangles =A C D=, =B E F=, =O G H=, consisting of half sectors. In constructing this Kaleidoscope, which, like the two former, has the equally luminous images symmetrically arranged round the aperture =A O B=, it is unnecessary to shape all the reflectors with accuracy. When two of them, both of which have a greater breadth than is wanted, are placed together, with the edge of the one resting upon the face of the other, the third reflector, which must be ground with great accuracy to the desired shape of the tapering equilateral prism, may then be placed so that each of its edges rests upon the faces of the other two. When this instrument is nicely executed with metallic plates, and when all the junctions are perfect, the effects which it produces are uncommonly splendid. 4. _On combinations of three reflectors at angles of 90°, 45°, and 45°._ [Illustration: FIG. 47.]