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.
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.
Fig. 9.
If the object moves from X to O, [Fig. 9], in the direction of the radius, all the images will likewise move towards O, and the patterns will have the appearance of being absorbed or extinguished in the centre. If the motion of the object is from O to X, the images will also move outwards in the direction of the radii, and the pattern will appear to develop itself from the centre O, and to be lost or absorbed at the circumference of the luminous field. The objects that move parallel to X O will have their centre of development, or their centre of absorption, at the point in the lines A O, B O, a O, b O, etc. where the direction in which the images move cuts these lines. When the object passes across the field in a circle concentric with A B, and in the direction A B, the images in all the sectors formed by an even number of reflexions will move in the same direction A B, namely, in the direction β b, a α; while those that have been formed by an odd number of reflexions will move in an opposite direction, namely, in the directions a B, A b. Hence, if the object moves from A to B, the points of absorption will be in the lines B O, α O, and b O, and the points of development in the lines A O, a O, and β O, and vice versa, when the motion of the object is from B to A.
If the object moves in an oblique direction m n, the images will move in the directions m t, o n, o p, q t, q p, and m, o, q, will be the centres of development, and n, p, t, the centres of absorption; whereas, if the object moves from n to m, these centres will be interchanged. These results are susceptible of the simplest demonstration, by supposing the object in one or two successive points of its path m n, and considering that the image must be formed at points similarly situated behind the mirrors; the line passing through these points will be the path of the image, and the order in which the images succeed each other will give the direction of their motion. Hence, we may conclude in general,