But when the objects are not similarly situated with respect to each of the mirrors, as the compound line 1, 2, [Fig. 8], the curved line 3, 4, and the straight line 5, 6, and, in general, as all irregular objects that are presented by accident to the instrument, then the image formed in the last sector a O e, [Fig. 7], by the mirror B O, will not join with the image formed in the last sector b O e, by the mirror A O. In order to explain this with sufficient perspicuity, let us take the case where the angle is 72°, or ⅕th part of the circle, as shown in [Fig. 7]. Let A O, B O, be the reflecting planes, and m n a line, inclined to the radius which bisects the angle A O B, so that o m > o n; then m nʹ, n mʹ, will be the images formed by the first reflexion from A O and B O, and nʹ mʺ, mʹ nʺ, the images formed by the second reflexion; but by the principles of catoptrics, O m = O mʹ = O mʺ, and O n = O nʹ = O nʺ, consequently since O m is by hypothesis greater than O n, we shall have O mʺ greater than O nʺ; that is, the images mʹ nʺ, nʹ mʺ, will not coincide. As O n approaches to an equality with O m, O nʺ approaches to an equality with O mʺ, and when O m = O n, we have O nʺ = O mʺ, and at this limit the images are symmetrically arranged, which is the case of the straight line 1, 2 in [Fig. 6]. By tracing the images of the other lines, as is done in [Fig. 8], it will be seen, that in every case the picture is destitute of symmetry when the object has not the same position with respect to the two mirrors.

Fig. 7.

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
of the
Mirrors.		 Number of
Sectors.		 Number of
Inverted
Pictures.		 Number of
Direct
Pictures.
120°321
72523
51³/₇743
40945
32⁸/₁₁11 65
27⁹/₁₃13 67
2415 87
21³/₁₇17 89
18¹⁸/₁₉19 10 9
17⅐21 10 11