Let AB, BO, Fig. 2, be a section of the two reflecting planes shown in Fig. 1. and let us consider the aperture AOB as an object placed before the minor AO. By the principles of catoptrics, a similar image AO b will be form ed behind AO ; and for the same reason, a similar image BO a of the aperture will be formed behind the other mir ror BO. But the reflected images AO b, BO a, may he considered as new objects placed before the reflectors BO and AO ; and therefore similar and similarly situated ima ges of these, viz. b 0 a', a 0 b',, will be formed behind the mirrors. In like manner, these images being consideredas new objects, other images of the aperture AOB will be formed at b'Oa", a'0 b", and a"O b"', b"O a", till a complete circle is formed by their combination. If the angle AOB is such as to make exactly 360°, when multiplied by any of the even numbers 2, 4, 6, 8, &c. then the circle MANE will be composed of an even number of sectors, each of which is exactly equal to AOB; hut the last sector. a"'Ob" will consist of two halves a'"ON, b"'ON, the first which is half of the image of anOr formed by reflection from AO, while the other is half of the image of Maw, formed by teflection from BO ; and, in this case, the last sector is bi sected by the common section MN of the two reflecting planes. If, on the other hand, the angle AOB is such as to make exactly 560°, when multiplied by any odd number 3, 5,7, 9, &c. each image of the aperture will be complete by itself, and the line MN will separate the series of images formed by AO from the series formed by BO.
When the angle AOB, after being multiplied by an even or odd number, is either greater or less than :360°, then the last reflected images on both sides of MN will be incom plete, and the circular field will be of a certain number of complete images of AOB, and of two incomplete 360° images. Hence will always represent the number of sectors of which the circular field is composed. When the quotient is a whole number, the images will be exactly equal and complete ; but if it consists of an integer and a fraction, the integer will represent the number of complete ectors, and half the remainder will represent in degrees the magnitude of the incomplete sectors. If AOB, for ex 60" ample is 17°, then that is, the circular field will be composed of 21 complete sectors, and 2 incomplete sectors, the angular magnitude of each of which is equal to one-half of 3°, or 1,1°.
If, instead of supposing the angular aperture AOB to be the object, we place any object whatever between AO and 110, the images of it will be formed according to the prin ciples which we have explained ; and the picture created by the combination of these images will be complete or in complete, according as the angle.AOB is an integral or frac tional part of the circle.
There is an exception, however, to the generality of this result, when the inclination Of the reflectors is an odd ali 30 quot part of a circle ; that is, when A6 11 — is an odd number.
0 In order to explain this with sufficient perspicuity, let us take the case where the angle is 72°, or 1th part of the cir cle, as shewn in Fig. 3. Let AO, BO, be the reflecting planes, and m n a line inclined to the radius which bisects the angle AOB, so that 0 m 7 On, then inn', nm', will be the images formed by the first reflection from AO and BO, and n' in", n" the images formed by the second reflection; but by the principles of catoptrics, 0 m=0 m'=O 7n", and 0 n=0 ni=0 n", consequently since 0 ni is by hypothesis greater than 0 n, we shall have 0 m" greater than 0 n"; that is, the images tn' n", nz" will not coincide. As 0 n approaches to an equality with 0 m, 0 n"- approaches to an equality with 0 m", and when 0 m=0 n, we have 0 n"=0 and at this limit the images are sym metrically arranged. In like manner it may be shewn, that when the rectilineal object ni n bisectsthe angle AOB, the images of it will bisect all the other four sectors, and conse quently are symmetrically disposed round the centre of the circle ; and in general, when the object has such a form or such a position that its parts are similarly situated with re spect to both the mirrors, a symmetrical picture may be produced, when the angle formed by the reflectors is ar odd aliquot part of a circle ; and therefore, since all irregu lar objects are composed of lines not similarly situated with respect to both the mirrors, ive may conclude, That in order to form a perfectly symmetrical picture, from the combina tion of any objects with irregular outlines by successive re flections between two inclined reflectors, the inclination cy the reflectors must be an even aliquot part of 360°.
If we consider the aperture ABO, Fig. 1, or any object lying in a plane passing through ABO, as the object which undergoes successive reflections, it is obvious that it may be viewed by placing the eye in any point of the quadrant comprehended between OC and OE. When the eye is placed at a small distance from C, it will be a little above the plane of the circular field formed by repeated images of AOB, and therefore this field must appear a very ceceteric ellipse. As the eye advances from C towards E, the ellipse becomes less eccentric, the symmetry of the combined images increases, and when the eye comes to E, the ellipse becomes a perfect circle, and the eye being. placed in a line perpendicular to its centre, observes all the images symme trically arranged round the centre G. In the production of perfectly symmetrical forms, therefore, the position of the eye is necessarily limited to the point E, or rather to a poirit so situated, that the line A'O=AO (Fig. 1.) may be just seen by the eye ; that is, that the line joining the point A' and the eye may just pass within the point E.