the picture is destitute of symmetry when the object has not the same position with respect to the two mirrors. [Illustration: FIG. 7.] [Illustration: FIG. 8.] This result may be deduced in a more simple manner, by considering that the symmetrical picture formed by the Kaleidoscope contains half as many pairs of forms as the number of times that the inclination of the mirrors is contained in 360°; and that each pair consists of a direct and an inverted form, so joined as to form a compound form. Now the compound form made up by each pair obviously constitutes a symmetrical picture when multiplied any number of times, whether even or odd; but if we combine so many pair and half a pair, two direct images will come together, the half pair cannot possibly join both with the direct and the inverted image on each side of it, and therefore a symmetrical whole cannot be obtained from such a combination. From these observations we may conclude,— 1. That when the inclination of the mirrors is an _odd_ aliquot part of a circle, the object seen by direct vision through the aperture unites with the images of it formed by repeated reflexions, and forms a complete and symmetrical picture, only in the case when the object is similarly situated with respect to both the mirrors; the two last sectors forming, in every other position of the object, an imperfect junction, in consequence of these being either both direct or both inverted pictures of the object. 2. That the series of parts which compose the symmetrical as well as the unsymmetrical picture, consists of direct and inverted pictures of the object, the number of direct pictures being always equal to half the number of sectors increased by one, when the number of sectors is 5, 9, 13, 17, 21, etc., and the number of inverted pictures being equal to half the number of sectors diminished by one, when the number of sectors is 3, 7, 11, 15, 19, etc., and _vice versa_. Hence, the number of direct pictures of the object must always be odd, and the number of inverted pictures even, as appears from the following table:— Inclination Number Number of Number of of the of Inverted Direct Mirrors. Sectors. Pictures. Pictures. 120° 3 2 1 72 5 2 3 51³/₇ 7 4 3 40 9 4 5 32⁸/₁₁ 11 6 5 27⁹/₁₃ 13 6 7 24 15 8 7 21³/₁₇ 17 8 9 18¹⁸/₁₉ 19 10 9 17⅐ 21 10 11 3. That when the number of sectors is 3, 7, 11, 15, 19, etc., the two last sectors are inverted; and when the number is 5, 9, 13, 17, 21, etc., the two last sectors are direct.