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 {2 and 3} 2} times the images {4 and 5} reflexions, move with 4} velocity of formed {6 and 7} 8} the mirror; by {8 and 9} 16} whereas in the semicircle =M _a e_=, containing the movable mirror,— The {1 and 2} 2 { times the images {3 and 4} reflexions, move with 4 { velocity of formed {5 and 6} 8 { the mirror; by {7 and 8} 16{ a progression which may be continued to any length.