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].