(Ml 
OPTIC S. 
through by the image, nuift be two inches, or twice the 
fpace defcribed by the mirror. 
Let us now fuppofe that the objeCt A advances in the 
fame direction as the mirror, and with twice its velocity, 
fo as to defcribe a fpace Ac=zzMN=ab, in the fame tirne 
that the mirror moves through MN, the object being at 
c when the mirror is at N. Then, iince Ac=ab and 6N 
tea AN, the whole <?N is equal to the whole «N, that is, a 
will ilill be the place of the image. Hence it follows, 
that, if the object advances in the fame direction ns the mir¬ 
ror, but with twice its velociti/, the image will remain Jla- 
tionar//. 
If the objeCt A moves in a direction oppofite to that of 
the mirror, and with double its velocity, as is fhown in 
fig. 9. then, iince b would be the image when A was fla¬ 
tionary, and when M had moved to N, in which cafe ab 
=aMN, and // the image when A had advanced to c 
through a fpace AcmiMN, we have 6N=AN, and b'N 
=fN, and therefore, bb’—AN —eN==.Acz=:zMN, and ab 
f-bb 1 or its equal Hence it follows, that, when 
the object advances towards the mirror with twice its velociti/, 
the image will move with four times the velociti/ of the mirror. 
If the mirror M moves round a centre, the very fame 
refuits will be obtained from the very fame reafoning, 
only the angular motion of the mirror and the image will 
then be more conveniently meafured by degrees. 
Now, in fig. 2., let X be a fixed objeft, and AO, BO, 
two mirrors placed at an angle of 6o° and movable 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 feen by 
direct vifion will of courfe be llationary, while the mir¬ 
rors defcribe an arch AM of io° for example ; but, fince 
AO has approached X by i-o°, the image of X formed be¬ 
hind AO rnult have approached X by 20 0 , .and conf'e- 
quently. moves with twice the velocity in the fame direc¬ 
tion as the mirrors. In like manner, fince BO has re¬ 
ceded io° from X, the image of X formed by BO muff 
have receded 30° from X, and confequently inufc have 
moved with twice the velocity in the fame direction as 
the mirrors. Now, the image of X in the feCtor bOd is, 
as it were, an image of the image in B 0 « reflected from 
AO. But the image in BOa advances in the fame direc¬ 
tion as the mirror AO, and with twice its velocity ; hence 
the image of it in the feCtor bOd will be llationary. In 
like manner it may be fhown, that the image in the feCtor 
eOcwill be llationary. Since cOs is an image of bOr re¬ 
flected from the mirror BO, and fince all images in that 
feCtor are ftationary, the correfponding images in cOe will 
move in the fame direction cd as the mirrors ; and, for the 
fame reafon, the images in the other half-feltor dOs will 
move in the fame direction ; hence, the image of any ob¬ 
ject formed in the la ft feCtor cOd will move in the fame 
direction, and with the fame velocity, as the images in the 
feCtors AOi, BO«. 
By a fimiiar procefs of reafoning, the fame refuits will 
be obtained, whatever be the number of the feCtors, and 
whether the angle AOB be the even or the odd aliquot 
part of a circle. Hence we may conclude, 1. That, dur¬ 
ing the rotatory motion of the mirrors round O, the ob¬ 
jects in the f’ettor feen by direct vifion, and all the images 
of thefe objects' formed by an even number of reflections, 
are at reft, 3, That all the images of thefe objeCts, formed 
by an odd number of reflections, move round O in the 
fame direction as the mirrors, and with an angular velo¬ 
city double that of the mirrors. 3. That, when the angle 
AOB is an even aliquot part of a circle, the number of 
moving feCtors is equal to the number of llationary fee- 
tors, a moving feCtor being placed between two.llationary 
feCtors, and vice verfa. 4., That, when the angle AOB 
is an odd aliquot part of a circle, the two laft feCtors ad¬ 
jacent to each other are either both in motion or both lta- 
tionary, the number of moving feCtors being greater by 
one when the number, of feCtors is 3, 7, 11, 15, &c. and 
the number of ftationary lectors being greater by one 
when the number of feCtors is 5, 9, 13, 17, &c. And, 5. 
3 
That, as the moving feCtors correfpond with thofe in which 
the images are inverted, and the ftationary ones with 
thofe in which the images are direCt, the number of each 
may be found from the Table given in p. 64.3. 
When one of the mirrors, AO, fig. 2, is ftationary^ 
while the other, BO, is moved round, and fo as to en¬ 
large the angle AOB, the object X, and the image of it 
feen in the ftationary mirror AO, remain at reft, but all 
the other images are in motion, receding from the objedt 
X, and its ftationary image ; and, when BO moves towards 
AO, fo as to diminifh the angle AOB, the fame elfeCt 
takes place, only the motion of the images is towards the 
objeCt X on one fide, and towards its ftationary image 011 
the other. Thefe images will obvioufly move in pairs; 
for, fince the fixed objeCt and its ftationary image are at 
an invariable distance, the exiftence of asymmetrical ar¬ 
rangement, as formerly proved, requires that fimiiar pairs 
be arranged at equal distances round O, and each of the 
images of thefe pairs mult be ftationary withregard to the 
other. Now, as the fixed objeCt is placed in the feCtor 
AO b, and its ftationary image in the feCtor AO b, it will 
be found that, in the femicircie M be, containing the fixed 
mirror, the 
ift reflected image and direCt objeCt, 
2d and 3d reflected image, § are ftationary 
4th - - 5th - > with refpeCt to 
6th - 7th 4 each other. 
8th - - 9th - - J 
while, in the fame femicircie M be, the 
1 ft reflected image and 2d reflected image > 
3d 4th e/ are movable 
5th - 6th - - - > with refpect 
7th ... 8th - - i to each other. 
9th ... 10th - - - J 
On the other hand, in the femicircie Mae, containing the 
movable mirror, the phenomena are reverfed, the images- 
which were formerly llationary with refpeCt to each other 
being now movable, and vice verfa. 
In conlidering the velocity with which each pair of 
images revolves, it will be readily feen, that the pair on 
each fide, and nearell, the fixed pair, will have an angular 
velocity double that of the mirror BO; the next pair on 
each fide will have a velocity four times as great as that of 
the mirror; the next pair will have a velocity eight times 
as great, and the next pair a velocity fixleen times as great 
as that of the mirror, the velocity of any pair being always 
double the velocity or the pair which is adjacent to it on 
the fide of the fixed pair. The reafon of this will be rna- 
nifeft, when we recolleCt what has already been demon-, 
ftrated, that the velocity of the image is always double 
that of the mirror when the mirror alone moves towards 
the objeCt, and quadruple that of the mirror when both 
‘are in motion, and when the objeCt approaches the mirror 
with twice the velocity. When BO moves from AO, the 
image in the feCtor BOa moves with twice the velocity of 
the mirror; but, fince the image in bOc is an image or the 
image in BOa reflected from the fixed mirror AO, it alfo 
will move with the fame velocity, or twice that of the 
mirror BO. Again, the image in the feCtor «Oc, being a 
reflection of the ftationary image in AO b from the moving 
mirror, will itfelf move with double the velocity of the 
mirror. But the image in the next feCtor cOd is a reflec¬ 
tion of the image in bOd from the moving mirror BO; 
and, as this latter image has been fhown to move in the 
direction bd with twice the velocity of the mirror BO, 
while the mirror BO itfelf moves towards the image, it- 
follows that the image in cOd will move with a velocity 
four times that of the mirror. The fame reafoning may 
be extended to any number of feCtors, and it will be 
found, that in the femicircie M be, containing the fixed 
mirror, 
The /■ 2 and 
images) 4 and 
formed! 6 and 
by \2 and 
reflections, move with/ f 
. 16 
. times the 
' velocity 
> of the.: 
1 mirror; 
whereas 
