which are fractional values, showing that the last image is composed of two half images. When the inclination of the mirrors, or the angle =A O B=, Fig. 3, is an _odd_ aliquot part of a circle, such as ⅓, ⅕, ⅐, ⅑, etc., the different sectors which compose the circular image are formed in the very same manner as has been already described; but as the number of _reflected sectors_ must in this case always be _even_, the line =O E=, where the mirrors join, will separate the two last reflected sectors, =_b_ O _e_=, =_a_ O _e_=. Hence it follows,— [Illustration: FIG. 3.] 1. That when =A O B= is ⅓, ⅕, ⅐, ⅑, etc., of a circle, the number of reflected images of any object is 3 - 1, 5 - 1, 7 - 1, etc., and,— 2. That the number of images which reach the eye from each mirror is 3 - 1 5 - 1 7 - 1, -----, -----, -----, 2 2 2 which are always even numbers. Hitherto we have supposed the inclination of the mirrors to be _exactly_ either an even or an odd aliquot part of a circle. We shall now proceed to consider the effects which will be produced when this is not the case. If the angle =A O B=, Fig. 2, is made to increase from being an _even_ aliquot part of a circle, such as ⅙th, till it becomes an _odd_ aliquot part, such as ⅐th, the last reflected image =β O α=, composed of the two halves =β O _e_=, =α O _e_=, will gradually increase, in consequence of each of the halves increasing; and when =A O B= becomes ⅐th of the circle, the sector =β O α= will become double of =A O B=, and =α O _e_=, =β O _e_= will become each complete sectors, or equal to =A O B=. If the angle =A O B= is made to vary from ⅙th to ⅕th of a circle, the last sector =β O α= will gradually diminish, in consequence of each of its halves, =β O _e_=, =α O _e_=, diminishing; and just when the angle becomes ⅙th of a circle, the sector =β O α= will have become infinitely small, and the two sectors, =_b_ O β=, =_a_ O α=, will join each other exactly at the line =O _e_=, as in Fig. 3. CHAPTER II. ON THE PRINCIPLES OF THE KALEIDOSCOPE, AND THE FORMATION OF SYMMETRICAL PICTURES BY THE COMBINATION OF DIRECT AND INVERTED IMAGES. The principles which we have laid down in the preceding chapter must not be considered as in any respect the principles of the Kaleidoscope. They are merely a series of preliminary deductions, by means of which the principles of the Instrument may be illustrated, and they go no farther than to explain the formation of an apparent circular aperture by means of successive reflexions.