It is therefore of the greatest importance, in the application of the Kaleidoscope to the arts, to have it constructed in such a manner, that patterns composed of any number of pairs of direct and inverted images may be created and drawn. With this view, the instrument may be fitted up in various ways, with paper, cloth, and metallic joints, by means of which the angle can be varied at pleasure; but the most convenient methods are shown in the Figures from Fig. 29 to 35, inclusive, which represent two different kinds of Polyangular Kaleidoscopes, as made by the late Mr. R. B. Bate, Optician, London, who had devoted much time and attention to the perfection of this species of Kaleidoscope. _Bate’s Polyangular Kaleidoscope with Metallic Reflectors._ [Illustration: FIG. 29.] [Illustration: FIG. 30.] The three Figures, viz., 29, 30, and 31, represent the Polyangular Kaleidoscope with metallic reflectors, as made by Mr. Bate. Fig. 29 shows the complete instrument, when mounted upon a stand; Fig. 30 is a section of it in the direction of its length; Fig. 31 is a transverse section of it through the line =S T=, Fig. 30, and Fig. 32 shows the lens of the eye-hole =E=. The tube of this instrument is composed of two cones, =M M=, =N N=, Fig. 30, connected together by a middle piece or ring, =R R=, into which they are both screwed. These two cones enclose two highly polished metallic reflectors, =A O=, =B O=, Fig. 31, only one of them, viz., =B O E=, being seen in Fig. 30. One of these reflectors, =B O E=, is fixed to the ring =R R=, by the intermediate piece =K G L=. The reflector is screwed to this piece by the adjustable screws =K=, =L=; and the piece =K G L= is again fixed to the ring =R R=, by two screws seen above and below =G=, in Fig. 31. Hence the tube, consisting of the cones =M M=, =N N=, and the ring =R R=, are immovably connected with the mirror =B O E=. The surface of the reflector =B O E= is adjusted by the screws at =K= and =L=, till it passes accurately through the axis of the cones and ring as seen in Fig. 31. The other reflector =A O=, is fixed to an outer ring _r r_, by means of an intermediate piece, similar to =K G L=, the arm =F= of which, corresponding to =G=, passes through an annular space or open arch, of more than 90°, cut out of the circumference of the inner ring =R R=. The arm =F= is fixed to the outer ring _r r_ by two screws, seen above and below =F=; and the reflector =A O= is fixed to the bar corresponding to =K L=, Fig. 30, by similar screws, for the purpose of adjusting it. [Illustration: FIG. 31.] [Illustration: FIG. 32.]