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.
2. That all the images of these objects, formed by an odd number of reflexions, move round O 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 O, remain at rest, but all the other images are in motion receding from the object X, and its stationary image; and when B O 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 O B, and its stationary image in the sector A O b, it will be found that in the semicircle M b e, containing the fixed mirror, the