vision will, of course, be stationary, while the mirrors describe an arch x of 10° for example; but since a o has approached x by 10°, the image of x formed behind a o must have

approached x by 20°, and consequently moves with twice the velocity in the same direction as the mirrors. In like manner, since B o has receded 10° from x, the image of x formed by Bo must have receded 20° from x, and consequently must have moved with twice the velocity in the same direction as the mirrors. Now, the image of x in the sector boẞ is, as it were, an image of the image in BO a reflected from A O. But the image in BO a advances in the same direction as the mirror A o and with twice its velocity, hence the image of it in the sector boẞ will be stationary. In like manner it may be shown, that the image in the sector a oa will be stationary. Since a oe is an image of bor reflected from the mirror BO, and since all images in that sector are stationary, the corresponding images in a oe will move in the same direction aẞ as the mirrors; and for the same reason the images in the other half-sector Boe will move in the same direction; hence, the image of any object

formed in the last sector a o 8 will move in the same direction, and with the same velocity as the images in the sectors дов, воа.

By a similar process of reasoning, the same results will be obtained, whatever be the number of the sectors, and whether the angle A O B be the even or the odd aliquot part of a circle. Hence we may conclude,

1. That during the rotatory motion of the mirrors round o, the objects in the sector seen by direct vision, and all the images of these objects formed by an even number of reflexions are at rest.

2. That all the images of these objects, formed by an odd number of reflexions, move round in the same direction as the mirrors, and with an angular velocity double that of the mirrors.

3. That when the angle A O B is an even aliquot part of a circle, the number of moving sectors is equal to the number of stationary sectors, a moving sector being placed between two stationary sectors, and vice versa.

4. That when the angle A O B is an odd aliquot part of a circle, the two last sectors adjacent to each other are either both in motion or both stationary, the number of moving sectors being greater by one when the number of sectors is 3, 7, 11, 15, etc., and the number of stationary sectors being greater by one when the number of sectors is · 5, 9, 13, 17, etc. And,

5. That as the moving sectors correspond with those in which the images are inverted, and the stationary ones with those in which the images are direct, the number of each may be found from the table given in page 24.

When one of the mirrors, a o, is stationary, while the other, B O, is moved round, and so as to enlarge the angle A O B, the object x, and the image of it seen in the stationary mirror a 0, remain at rest, but all the other images are in motion receding from the object x, and its stationary image; and when Bo moves towards a o, so as to diminish the angle A O B, the same effect takes place, only the motion of the images is towards the object x, on one side, and towards its stationary image on the other. These images will obviously move in pairs; for, since the fixed object and its stationary image are at an invariable distance, the existence of a symmetrical arrangement, which we have formerly proved, requires that similar pairs be arranged at equal distances round o, and each of the images of these pairs must be stationary with regard to the other. Now, as the fixed object is placed in the sector A OB, and its stationary image in the sector a ob, it will be found that in the semicircle мb e, containing the fixed mirror, the

1st reflected image and direct object,

On the other hand, in the semicircle м a e, containing the movable mirror, the phenomena are reversed, the

C

images which were formerly stationary with respect to each other being now movable, and vice versa.

In considering the velocity with which each pair of images revolves, it will be readily seen that the pair on each side, and nearest the fixed pair, will have an angular velocity double that of the mirror B 0; the next pair on each side will have a velocity four times as great as that of the mirror; the next pair will have a velocity eight times as great, and the next pair a velocity sixteen times as great as that of the mirror, the velocity of any pair being always double the velocity of the pair which is adjacent to it on the side of the fixed pair. The reason of this will be manifest, when we recollect what has already been demonstrated, that the velocity of the image is always double that of the mirror, when the mirror alone moves towards the object, and quadruple that of the mirror when both are in motion, and when the object approaches the mirror with twice the velocity. When B O moves from A o, the image in the sector Bo a moves with twice the velocity of the mirror; but since the image in bo ẞ is an image of the image in B O a reflected from the fixed mirror a o, it also will move with the same velocity, or twice that of the mirror B O. Again, the image in the sector a o a, being a reflection of the stationary image in A o b from the moving mirror, will itself move with double the velocity of the mirror. But the image in the next sector a o B is a reflection of the image in bo 6 from the moving mirror B 0; and as this latter image has been shown to move in the direction bẞ, with twice the velocity of the mirror B O, while the mirror B o itself moves towards the image, it follows that the image in a o ẞ will move with a velocity

four times that of the mirror.

The same reasoning may be

extended to any number of sectors, and it will be found that in the semicircle м b e, containing the fixed mirror,

whereas in the semicircle м a e, containing the movable

a progression which may be continued to any length.

Before concluding this chapter, it may be proper to mention a very remarkable effect produced by moving the two plain mirrors along one of two lines placed at right angles to each other. When the aperture of the mirrors is crossed by each of the two lines, the figure created by reflexion consists of two polygons with salient and re-entering angles. By moving the mirrors along one of the lines, so that it may always cross the aperture at the same angle, and at the same distance from the angular point, the polygon formed by this line will remain stationary, and of the same form and magnitude; but the polygon formed by the other line, at first emerging from the centre, will gradually increase till its salient angles touch the re-entering angles of the stationary polygon; the salient angles becoming more acute, will enclose the apices of the re-entering angles of the stationary polygon, and at last the polygon will be destroyed by truncations from its salient angles.