When the inclination of the mirrors is not an aliquot part of 360°, the images formed by the last reflexions do not join like every other pair of images, and therefore the picture which is created must be imperfect. It has already been shown at the end of Chap. I. that when the angle of the mirrors becomes greater than an even or less than an odd aliquot part of a circle, each of the two incomplete sectors which form the last sector becomes greater or less than half a sector. The image of the object comprehended in each of the incomplete sectors must therefore be greater or less than the images in half a sector; that is, when the last sector =β O α=, Fig. 2, is greater than =A O B=, the part _q v_ in one half must be the image of more than _o z_, and _v p_ the image of more than _t y_, and _vice versa_, when =β O α= is less than =A O B=. Hence it follows that the symmetry is imperfect from the image in the last sector being greater or less than the other images. But besides this cause of imperfection in the symmetry, there is another, namely, the disunion of the two images _q v_ and _v p_. The angles =O _q v_= and =O _o p_= are obviously equal, and also the angles =O _p v_=, =O _p o_=; but since the angle =β O α=, or= _q_ O _p_=, is by hypothesis greater or less than =_p_ O _o_=, it follows that the angles of the triangle =_q_ O _p_= are either greater or less than two right angles, because they are greater or less than the three angles of the triangle =_p_ O _o_=. But as this is absurd, the lines _q v_, _v p_, cannot join so as to form one straight line, and therefore the completion of a perfect figure by means of two mirrors, whose inclination is not an aliquot part of a circle, is impossible. When the angle =β O α= is greater than =_p_ O _o_=, or =A O B=, the lines _q v_, _v p_, will form a re-entering angle towards =O=, and when it is less than =A O B=, the same lines will form a salient angle towards =O=. CHAPTER III. ON THE EFFECTS PRODUCED BY THE MOTION OF THE OBJECT AND THE MIRRORS. Hitherto we have considered both the object and the mirrors as stationary, and we have contemplated only the effects produced by the union of the different parts of the picture. The variations, however, which the picture exhibits, have a very singular character, when either the objects or the mirrors are put in motion. Let us, first, consider the effects produced by the motion of the object when the mirrors are at rest. [Illustration: FIG. 9.]