If the object =A= moves in a direction opposite to that of the mirror, and with double its velocity, as is shown in Fig. 11; then, since _b_ would be the image when =A= was stationary, and when =M= had moved to =N=, in which case =_a b_ = 2 M N=, and _bʹ_ the image when =A= had advanced to =α= through a space =A α = 2 M N=, we have =_b_ N = A N=, and =_bʹ_ N = α N=, and, therefore,= _b bʹ_ = A N - α N = A α = 2 M N=, and =_a b_ + _b bʹ_= or its equal =_a bʹ_ = 4 M N=. Hence it follows, _that when the object advances towards the mirror with twice its velocity, the image will move with four times the velocity of the mirror_. If the mirror =M= moves round a centre, the very same results will be obtained from the very same reasoning, only the angular motion of the mirror and the image will then be more conveniently measured by parts of a circle or degrees. [Illustration: FIG. 12.] Now, in Fig. 12, let =X= be a fixed object, and =A O=, =B O=, two mirrors placed at an angle of 60° and moveable round =O= as a centre. When the eye is applied to the end of the mirrors (or at =E=, Fig. 1), the fixed object =X=, Fig. 12, seen by direct 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 =B O= 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 =_b_ O β= is, as it were, an image of the image in =B O _a_= reflected from =A O=. But the image in =B O _a_= advances in the same direction as the mirror =A O= and with twice its velocity, hence the image of it in the sector =_b_ O β= will be stationary. In like manner it may be shown, that the image in the sector =_a_ O α= will be stationary. Since =α O _e_= is an image of =_b_ O _r_= reflected from the mirror =B O=, and since all images in that sector are stationary, the corresponding images in =α O _e_= will move in the same direction =α β= as the mirrors; and for the same reason the images in the other half-sector =β O _e_= will move in the same direction; hence, the image of any object formed in the last sector =α O β= will move in the same direction, and with the same velocity as the images in the sectors =A O _b_=, =B O _a_=. 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.