| 1st reflected image and direct object, |  | are stationary with respect to each other. | |
| 2d | 3d reflected image | ||
| 4th | 5th | ||
| 6th | 7th | ||
| 8th | 9th | ||
while in the same semicircle M b e, the
| 1st reflected image and | 2d reflected image | | are movable with respect to each other. |
| 3d | 4d | ||
| 5th | 6th | ||
| 7th | 8th | ||
| 9th | 10th |
On the other hand, in the semicircle M a e, containing the movable mirror, the phenomena are reversed, the 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 O; 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 B O a moves with twice the velocity of the mirror; but since the image in b O β 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 α, being a reflexion 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 α O β is a reflexion of the image in b O β from the moving mirror B O; and as this latter image has been shown to move in the direction b β, 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 α O β 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 M b e, containing the fixed mirror,
| The images formed by |  | 2 and 3 | | reflexions, move with | 2 | | times the velocity of the mirror; |
| 4 and 5 | 4 | ||||||
| 6 and 7 | 8 | ||||||
| 8 and 9 | 16 |
whereas in the semicircle M a e, containing the movable mirror,—
| The images formed by | | 1 and 2 | | reflexions, move with | 2 | | times the velocity of the mirror; |
| 3 and 4 | 4 | ||||||
| 5 and 6 | 8 | ||||||
| 7 and 8 | 16 |
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.