1. That when the inclination of the mirror is an _even_ aliquot part of a circle, the object seen by direct vision across the aperture, whether it is simple or compound, is so united with the images of it formed by repeated reflexions, as to form a symmetrical picture. 2. That the symmetrical picture is composed of a series of parts, the number of which is equal to the number of times that the angle =A O B= is contained in 360°. And— 3. That these parts are alternately direct and inverted pictures of the object; a direct picture of it being always placed between two inverted ones, and, _vice versa_, so that the number of direct pictures is equal to the number of inverted ones. When the inclination of the mirrors is an _odd_ aliquot part of 360°, such as ⅕th, as shown in Fig. 3, the picture formed by the combination of the direct object and its reflected images is symmetrical only under particular circumstances. If the object, whether simple or compound, is similarly situated with respect to each of the mirrors, as the straight line 1, 2 of Fig. 6, the compound line 3, 4, the inclined lines 5, 6, the circular object 7, the curved line 8, 9, and the radial line 10, =O=, then the images of all these objects will also be similarly situated with respect to the radial lines that separate the sectors, and will therefore form a whole perfectly symmetrical, whether the number of sectors is odd or even. [Illustration: FIG. 6.] But when the objects are not similarly situated with respect to each of the mirrors, as the compound line 1, 2, Fig. 8, the curved line 3, 4, and the straight line 5, 6, and, in general, as all irregular objects that are presented by accident to the instrument, then the image formed in the last sector =_a_ O _e_=, Fig. 7, by the mirror =B O=, will not join with the image formed in the last sector =_b_ O _e_=, by the mirror =A O=. In order to explain this with sufficient perspicuity, let us take the case where the angle is 72°, or ⅕th part of the circle, as shown in Fig. 7. Let =A O=, =B O=, be the reflecting planes, and _m n_ a line, _inclined to the radius which bisects the angle_ =A O B=, so that =_o m_ > _o n_=; then _m nʹ_, _n mʹ_, will be the images formed by the first reflexion from =A O= and =B O=, and _nʹ mʺ_, _mʹ nʺ_, the images formed by the second reflexion; but by the principles of catoptrics, =o _m_ = o _mʹ_ = O _mʺ_=, and =O _n_ = O _nʹ_ = O _nʺ_=, consequently since =O _m_= is by hypothesis greater than =O _n_=, we shall have =O _mʺ_= greater than =O _nʺ_=; that is, the images _mʹ nʺ_, _nʹ mʺ_, will not coincide. As =O _n_= approaches to an equality with =O _m_=, =O _nʺ_= approaches to an equality with =O _mʺ_=, and when =O _m_ = O _n_=, we have =O _nʺ_ = O _mʺ_=, and at this limit the images are symmetrically arranged, which is the case of the straight line 1, 2 in Fig. 6. By tracing the images of the other lines, as is done in Fig. 8, it will be seen, that in every case