OPTICS 
one, when the number of fedtors is 3, 7,11, 15, 19, &c. 
and vice verfa. Hence, the number of dire ft pictures of 
the objedt muft always be odd, and the number of in¬ 
verted pictures even, as appears from the following 
Table : 
Inclination of 
the Minors. 
N umber of 
S^dtors. 
N° of Inverted 
Pictures. 
N° of Diredt 
Pictures. 
120° 
3 
2 
I 
7 2 
5 
2 
3 
5 1 3 
7 
4 
3 
4° T 
9 
4 
5 
3 2 TT 
I I 
6 
5 
2 7-ft 
13 
6 
7 
2 4 
15 
8 
7 
2I lV 
17 
8 
9 
1 °1 9 
J 9 
IO 
9 
1 ?7 
21 
IO 
I I 
3. That when the number of fedtors is 3, 7, 11, 15, 
19, &c. the two laft fedtors are inverted ; and when the 
number is 5, 9, 13, 17, 21, &c. the two lalt fedtors are 
■diredt. 
When the inclination of the mirrors is not an aliquot 
part of 366°, the images formed by the lalt reflections do 
not join like every other pair of images, and therefore 
the picture which is created muft: be imperfedt. It has 
already been fhown, that, when the angle of the mirrors 
becomes greater than an even or lei's than an odd aliquot 
part ofa circle, each of the two incomplete fedtors which 
form the laft fedtor becomes greater or lefs than half a fec- 
tor. The image of the objedt comprehended in each of 
the incomplete fedtors muft therefore be greater or lefs 
than the images in half a fedtor; that is, when the laft 
fedlort/Oc, fig. 2, is greater than AOB, the part qv in one 
half muft be the image of more than oz, and vp the image 
of more than t;/ ; and vice verfa when dOc is lefs than AOB. 
Hence it follows, that the fymmetry is imperfedt, from 
the image in the laft fedtor being greater or lefs than the 
other images. But, betides this caufe of imperfedtion in 
the fymmetry, there is another, namely, the difunion of 
the two images qv and vp. The angles Oi/rand Oop are 
obvioufly equal, and alfo the angles Opv, O po; but, fince 
the angle dOt\ or qOp, is by hypothelis greater or lefs 
than pOo, it follows that the angles of the triangle qOp 
are either greater or lefs than two right angles, becaule 
they are greater or lefs than the three angles of the trian¬ 
gle pOo. But, as this is abfurd, the lines qv, vp, cannot 
join fo as to form one ftraight line; and therefore the 
.completion of a perfect figure by means of two mirrors, 
*whofe inclination is not an aliquot part of a circle, is im- 
poflible. When the angle dOc is greater than pOo or 
AOB, the lines qv, vp, will form a re-entering angle to¬ 
wards O ; and, when it is lefs than AOB, the lame lines 
will form a falient angle towards O. 
Hitherto we have confidered both the objedt and the 
mirrors as ftationary, and have contemplated only the 
effedts produced by the union of the different parts of the 
pidture. The variations, however, which the picture 
exhibits, hive a very fingular character, when either the 
objedts or the mirrors are put in motion. Let us, firft, 
confider the effects produced by the motion of the object 
when the mirrors are at reft. 
If the objedt moves from M to O, fig. 2, in the direftion 
of the radius, all the images will likewife move towards 
O, and the patterns will have the appearance of being 
abforbed or extinguiftied in the centre. If the motion of 
the objedt is from O to M, the images will alfo move out¬ 
wards in the direction of the radii, and the pattern will 
appear to develop itfelf from the centre O, and to be 
loft or abforbed at the circumference of the luminous 
field. The objedts that move parallel to MO will 
Iiave their centre of development, or their centre of 
abforption, at the point in the lines AO, BO, aO, 
643 
bO, &c. where the direction in which the images move 
cuts thefe lines. When the objedt pafies acrofs the 
field in a circle concentric with AB, and in the direction 
AB, the images in all the fedtors formed by an even num¬ 
ber of reflections will move in the fame direction AB, 
namely, in the direction cb, ad; while thofe that have 
been formed by an odd number of reflections will move 
in an oppofite direction, namely, in the directions «B, 
A b. Hence, if the objedt moves from A to B, the points 
of abforption will be in the lines BO, cO, and b O, and 
the points of development in the lines AO, aO, and dO; 
and vice verfa, when the motion of the objedt is from 
B to A. 
If the objedt moves in an oblique direction mil, the 
image will move in the directions mu, on, op, qt, qp; 
and m, 0, q, will be the centres of development, and 
11, p, t, the centres of abforption ; whereas, if the 
objedt moves from n to to, thefe centres will be inter¬ 
changed. Thefe refults are fufceptible of the limpleft: 
demonftration, by fuppofing the object in one or two 
fuccefilve points of its path mn, and confldering that the 
image muft be formed at points ffmilarly lituated behind 
the mirrors ; the line palling through thefe points will be 
the path of the image, and the order in which the images 
fucceed each other will give the direction of their motion. 
Hence we may conclude in general, 1. That, when the 
path of the objedt cuts both the mirrors AO and BO like 
tom, the centre of abforption will be in the radius palling 
through the feftion of the mirror to which the objedt 
moves, and in every alternate radius; and’that the cen¬ 
tre of development will be in the radius palling through 
the fedtion of the mirror from which the objedt moves, 
and in all the alternate radii. 2. That, when the path of 
the objedt cuts any one of the mirrors and the circumfe¬ 
rence of the circular field, the centre of abforption will 
be in all the radii which feparate the fedtors, and the cen¬ 
tre of development in the circumference of the field, if 
the motion is towards the mirror, but vice verfa if the mo¬ 
tion is towards the circumference. 
When the objedts are at reft, and the kaleidofcope in 
motion, a new feries of appearances is prefented. What¬ 
ever be the direction in which the kaleidofcope moves, 
the object feen by diredt vifion mult always be flationary, 
and it is ea'fy to determine the changes which take place 
when the kaleidofcope has a progreflive motion over the 
objedt. A very curious elfeft, however, is o'ol'erved when 
the kaleidofcope has a rotatory motion round the angular 
point, or rather round the common fedtion of the two 
mirrors. The picture created by the inltrument leems to 
be compofed of two pictures, one in motion round tire 
centre of the circular field, and the other at reft. The 
fectors formed by an odd number of reflections are alt in 
motion in the fame direction as the kaleidofcope, while 
the fedtor feen by diredt vifion, and all the fedtors formed 
by an even number of reflections, are at reft. In order 
to underftand this, let M, fig. 8, be a plane mirror, and 
A an objedt whofe image is formed at a, fo that«M=AM; 
let the mirror M advauceto N, and the objedt A, which 
remains fixed, will have its image b formed at fudh a dis¬ 
tance behind N, that 6N=AN; then it will be found, 
that the (pace moved through by the image is double the 
lpace moved through by the mirror; that is, ab— 2MN 
Since MN=AM—AN, and fince AM=aM, and AN= 
bN, we have MN=«M— b N ; and, adding MN or its equal 
bM+bN to both tides of the equation, we obtain zMN= 
nM—iN-j-iN-ftiM; but —6N+iN=2o, and a\I-\-bMz=ab ; 
hence zMN—ab. This refult may be obtained without 
the aid of mathematics, by conlidering that, if the mir¬ 
ror M advances one inch to A, one inch is added to the 
diitance of the image a, and one fubtradled from the dis¬ 
tance of the objedt; that is, the difference of thefe dif- 
tances is now two inches, or twice the fpace moved 
through by the mirror; but, fince the new diitance of 
the objedt is equal to the diitance of the new image, tire 
difference of thefe diftances, which is the fpace moved 
through 
