KALEIDOSCOPE. 



411 



it is necessary to ascertain the effects which are produ. 



ced under these various circumstance*, and to discover 



it' there are any clt terminate positions of tlie re-Hectors, 



the object, and the eye, which enable the reflectors to 



form and carry to the eye i f the oh;i rvir a picture ma- 



mmetric.il. and in which all the reflected 



:ve as nearly as possible the same inten- 



FlATE 



O M.I. 



Fig. 1,*. 



SECT. I. On the Effect* produced, Ixj varying flie rela- 

 tive /'o.ii/io/i of th-- tiro RrJI,-ciort 



Let AB, BO. Fig. 2 be a section of the two reflecting 

 planes shewn in Fig. 1. and let us consider the aperture 

 AOB as an object placed before the mirror AO. By the 

 principles of catoptric*, a similar image AC) A will be 

 formed behind AO ; and for the same reason, a similar 

 image BC) a of the aperture will be formed behind the 

 other mirror BO. But the reflected images AC) h, BO a, 

 may be considered as new objects placed More the re. 

 Hector* BO and AO; and therefore imilar and similarly 

 situated images of these, i. AOa', a()l>, will be 

 formed behind the mirror*. In like manner, these 

 images being cun-iiU-ritl a< new objects, other images of 

 the apertun- A OH will be formed at b'Oa", a'O 6", and 

 a'O / . till a complete circle is formed by their 



combination. If the angle AOB is such ns to make 

 exactly S6f>*, when multiplied by any of the even num- 

 ber* 2, 4, 6. 8, tc. then the OKM MA SB will be com- 

 poaed of an even number of sectors, each of which ie ex- 

 actly equal to A (> H .-t sector o'"OA willcon- 

 aut ot two halve* ./ < < first of which is half 

 of the image of <t*O6'" formed by reflection from 

 while the other is half of the image of A"Oo", formed !>y 

 reflection from BO ; and. in tin. cane, the lat sector is 

 bisected by the common section MN of the two reflect- 



anes. If, on the other hand, the angle AOIf n 

 such as to make exactly S60", when multiplied by any 

 odd number :'. , ach image of the aper- 



ture will be complete by itelf, and the line MN will 

 separate the serie* of image* formed by AC) from the *e- 

 rie* formed by BO. 



. n the angle AOB, after being multiplied by an 

 even or odd number, it either greater or less than 



the last reflected images on Ixrth -id. - <>(' \! S will 

 be incomplete, and the circular field will be inniiMntcl 

 of certain number of complete image* of AOB, and 



...'.. 



of two incomplete images. Hence ^ will always 



r'he number of sectors of which the circular 

 is composed. When the quotient is a whole num- 

 will be exactly equal and complete ; but 

 if it cotuistsof an integer and a fraction, the inti.^er will 

 complete sectors, and half the 



remainder will repr. . tin- magnitude of 



the incomplete sector*. If AOB, for examp c i, IT', then 

 36: J* 



that is, the circular field will be composed 



of 81 complete sectors, and -.' un. .mplete sectors, the an- 

 gular ibignitude of each of which ia equal to one-half 



nied of supposing the angular aperture AOB to 

 be Hit ., .ject. we place any object whatever between 

 AO '"' B' .uei <> " will be formed acco, 



ti we have explain**) ; and th< 



*re created by the combination of these images, will l>e 

 complete or incomplete according aa the anjjft AOB is 

 1 ' or fractional part of the circle. 



There is an exception, however, :o the generality of 

 this result, when the inclination of the reflectors is an Sl '"l'-- 



odd aliquot part of a circle; that is, when ^7;., is an odd 



number. In order to explain this with sufficient per- 

 spicuity, let us take the c;ise where the angle is "2", or 

 ' ; th part of the circle, as shewn in Fig. 3. Let AO, PLATE 

 BO, be the reflecting planes, and m n a line inclint d ri :t -'XLI. 

 to I/if radius itluJi liis-cts >fte angle AOB, so that l ' lg ' 3l 

 Om-^On, then mn' , nm, will be the images formed by 

 the first reflection from AO and BO, and it' m", m'n" the 

 - formed by tlit- second reflection; but by the prin- 

 of catoptrics, Om=Oni'=O m", and On=O/i'= 

 On", consequently since O / is by hypothesis greater 

 than O M, we shall have O m" greater than O n" ; that 

 is, the images m'n", // m" will not coincide. AsO 

 approaches to an equality with O m, O n" approaches 

 to an equality with O m", and when Om=O, we 

 have O "=O in", and at this limit the images are 

 symmetrically arranged. In like manner it may be 

 shewn, that when the rectilineal object mn bisects 

 the angle AOB, the images of it will bisect all the other 

 four sectors, and consequently are symmetrically dispo- 

 sed round the centre of the circle ; anil in general, 

 when the object has such a form or such a position 

 th it its parts are similarly situated with respect to both 

 the mirrors, a symmetrical picture nriy be produced, 

 when the angle formed by the reflectors is an 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 mirrora, we may "conclude, 

 I lnil in order to form a /*r/tc//y si/mmc/ricai picture, 

 from the combination of any olijccis nilh irrcgulir outlines 

 bu titcceuii* reflection* between liv inclined rrflectxrt, 

 the tnclii,ali,in of the re/leclon must be an even aliquot 

 part 0/360. 



SECT. II. On (he Effecti produced hy varying the Po- 

 tilion of tHe Eye. 



If we consider the aperture ABO, Fig. 1, or any ob- 



\ing in a plane passing through ABO, as the 

 which undergoes successive reflections, it is ob- 

 \ious that it may be viewed by placing the eye in any 

 t'tlie 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 repealed images of AOB. and therefore 

 thn field must appear a very eccentric ellipse. As the eye 

 adv.ince< from ( towards K, the ellipse becomes less ec- 

 centric, the >\ iiinieiry oftlie combined images increases, 

 anil when the eye ionic-, to K, the ellipse becomes a 



t circle, and the eye being placed in a line per- 

 pendicular to its centre, observes all the images sym- 

 metrically arranged round tiit centre O. In the pro- 



i of perfectly symmetrical form-, therefore, the 



. e is nei-i --..nly limited to the point I-', 



or r.nher to a point so situated, that the line A'O= AO 



I.) may b. j,,-t teen by the eye; that is, that the 

 line joining the point A' and the eye, may just puss 

 within tlie point E. 



limitation is however necessary, for very diffe- 

 rent reasons. When the eye is placed near , the rays 

 are incident almost perpendicularly u|x>n the mirrors; 

 and it is well known, from the experiments of Bouguer, 

 that the loss of light ny reflection, is in this case extreme- 



t both in the case of mi-taU and poli-hid glass. 



From this cause, the intensity even of the first reflect* 



