ON THE EFFECTS PRODUCED BY THE MOTION OF THE OBJECT AND THE MIRRORS.
Hitherto we have considered both the object and the mirrors as
stationary, and we have contemplated only the effects produced by the
union of the different parts of the picture. The variations, however,
which the picture exhibits, have a very singular character, when either
the objects or the mirrors are put in motion. Let us, first, consider
the effects produced by the motion of the object when the mirrors are
at rest.
[Illustration: FIG. 9.]
If the object moves from =X= to =O=, Fig. 9, in the direction of
the radius, all the images will likewise move towards =O=, and the
patterns will have the appearance of being absorbed or extinguished in
the centre. If the motion of the object is from =O= to =X=, the images
will also move outwards in the direction of the radii, and the pattern
will appear to develop itself from the centre =O=, and to be lost or
absorbed at the circumference of the luminous field. The objects that
move parallel to =X O= will have their centre of development, or their
centre of absorption, at the point in the lines =A O=, =B O=, _a_ =O=,
_b_ =O=, etc. where the direction in which the images move cuts these
lines. When the object passes across the field in a circle concentric
with =A B=, and in the direction =A B=, the images in all the sectors
formed by an even number of reflexions will move in the same direction
=A B=, namely, in the direction β _b_, _a_ α; while those that have
been formed by an odd number of reflexions will move in an opposite
direction, namely, in the directions _a_ =B=, =A= _b_. Hence, if the
object moves from =A= to =B=, the points of absorption will be in the
lines =B O=, α =O=, and _b_ =O=, and the points of development in the
lines =A O=, _a_ =O=, and β =O=, and _vice versa_, when the motion of
the object is from =B= to =A=.
If the object moves in an oblique direction _m n_, the images will move
in the directions _m t_, _o n_, _o p_, _q t_, _q p_, and _m_, _o_, _q_,
will be the centres of development, and _n_, _p_, _t_, the centres of
absorption; whereas, if the object moves from _n_ to _m_, these centres
will be interchanged. These results are susceptible of the simplest
demonstration, by supposing the object in one or two successive points
of its path _m n_, and considering that the image must be formed at
points similarly situated behind the mirrors; the line passing through
these points will be the path of the image, and the order in which the
images succeed each other will give the direction of their motion.
Hence, we may conclude in general,
1. That when the path of the object cuts both the mirrors =A O= and =B
O= like _m n_, the centre of absorption will be in the radius passing
through the section of the mirror to which the object moves, and in
every alternate radius; and that the centre of development will be in
the radius passing through the section of the mirror from which the
object moves, and in all the alternate radii: and,
2. That when the path of the object cuts any one of the mirrors and
the circumference of the circular field, the centre of absorption will
be in all the radii which separate the sectors, and the centre of
development in the circumference of the field, if the motion is towards
the mirror, but _vice versa_ if the motion is towards the circumference.
When the objects are at rest, and the Kaleidoscope in motion, a new
series of appearances is presented. Whatever be the direction in which
the Kaleidoscope moves, the object seen by direct vision must always
be stationary, and it is easy to determine the changes which take
place when the Kaleidoscope has a progressive motion over the object.
A very curious effect, however, is observed when the Kaleidoscope has
a rotatory motion round the angular point, or rather round the common
section of the two mirrors. The picture created by the Instrument
seems to be composed of two pictures, one in motion round the centre
of the circular field, and the other at rest. The sectors formed by an
odd number of reflexions are all in motion in the same direction as
the Kaleidoscope, while the sector seen by direct vision, and all the
sectors formed by an even number of reflexions, are at rest. In order
to understand this, let =M=, Fig. 10, be a plane mirror, and =A= an
object whose image is formed at _a_, so that =_a_ M = A M=. Let the
mirror =M= advance to =N=, and the object =A=, which remains fixed,
will have its image _b_ formed at such a distance behind =N=, that =_b_
N = A N=; then it will be found that the space moved through by the
image is double the space moved through by the mirror; that is, =_a b_
= 2 M N=. Since =M N = A M - A N=, and since =A M = _a_ M=, and
=A N = _b_ N=, we have =M N = _a_ M - _b_ N=; and adding =M N= or
its equal =_b_ M + _b_ N= to both sides of the equation, we obtain
=2M N = _a_ M - _b_ N + _b_ N + _b_ M=; but =-_b_ N + _b_N = 0=,
and =_a_ M + _b_ M = _a b_=; hence =2M N = _a b_=. This result may be
obtained otherwise, by considering, that if the mirror =M= advances
_one inch_ towards =A=, one inch is added to the distance of the image
_a_, and one subtracted from the distance of the object; that is, the
difference of these distances is now two inches, or twice the space
moved through by the mirror; but since the new distance of the object
is equal to the distance of the new image, the difference of these
distances, which is the space moved through by the image, must be two
inches, or twice the space described by the mirror.
[Illustration: FIG. 10.]
Let us now suppose that the object A advances in the same direction as
the mirror, and with twice its velocity, so as to describe a space =A
α = 2 M N = _a b_=, in the same time that the mirror moves through =M
N=, the object being at α when the mirror is at =N=. Then, since =A
α = _a b_= and =_b_ N = A N=, the whole =α N= is equal to the whole
=_a_ N=, that is, =_a_= will still be the place of the image. Hence it
follows, _that if the object advances in the same direction as the
mirror, but with twice its velocity, the image will remain stationary_.
[Illustration: FIG. 11.]
If the object =A= moves in a direction opposite to that of the mirror,
and with double its velocity, as is shown in Fig. 11; then, since _b_
would be the image when =A= was stationary, and when =M= had moved to
=N=, in which case =_a b_ = 2 M N=, and _bʹ_ the image when =A= had
advanced to =α= through a space =A α = 2 M N=, we have =_b_ N = A N=, and
=_bʹ_ N = α N=, and, therefore,= _b bʹ_ = A N - α N = A α = 2 M N=, and
=_a b_ + _b bʹ_= or its equal =_a bʹ_ = 4 M N=. Hence it follows, _that
when the object advances towards the mirror with twice its velocity,
the image will move with four times the velocity of the mirror_.
If the mirror =M= moves round a centre, the very same results will be
obtained from the very same reasoning, only the angular motion of the
mirror and the image will then be more conveniently measured by parts
of a circle or degrees.
[Illustration: FIG. 12.]
Now, in Fig. 12, let =X= be a fixed object, and =A O=, =B O=, two
mirrors placed at an angle of 60° and moveable 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=, Fig. 12, seen by direct vision will, of course,
be stationary, while the mirrors describe an arch =X= of 10° for
example; but since =A O= has approached =X= by 10°, the image of =X=
formed behind =A O= must have approached =X= by 20°, and consequently
moves with twice the velocity in the same direction as the mirrors.
In like manner, since =B O= has receded 10° from =X=, the image of
=X= formed by =B O= must have receded 20° from =X=, and consequently
must have moved with twice the velocity in the same direction as the
mirrors. Now, the image of =X= in the sector =_b_ O β= is, as it were,
an image of the image in =B O _a_= reflected from =A O=. But the image
in =B O _a_= advances in the same direction as the mirror =A O= and
with twice its velocity, hence the image of it in the sector =_b_ O β=
will be stationary. In like manner it may be shown, that the image in
the sector =_a_ O α= will be stationary. Since =α O _e_= is an image of
=_b_ O _r_= reflected from the mirror =B O=, and since all images in
that sector are stationary, the corresponding images in =α O _e_= will
move in the same direction =α β= as the mirrors; and for the same reason
the images in the other half-sector =β O _e_= will move in the same
direction; hence, the image of any object formed in the last sector =α
O β= will move in the same direction, and with the same velocity as the
images in the sectors =A O _b_=, =B O _a_=.
By a similar process of reasoning, the same results will be obtained,
whatever be the number of the sectors, and whether the angle =A O B= be
the even or the odd aliquot part of a circle. Hence we may conclude,
1. That during the rotatory motion of the mirrors round =O=, the
objects in the sector seen by direct vision, and all the images of
these objects formed by an even number of reflexions are at rest.
2. That all the images of these objects, formed by an odd number of
reflexions, move round =O= in the same direction as the mirrors, and
with an angular velocity double that of the mirrors.
3. That when the angle =A O B= is an _even_ aliquot part of a circle,
the number of moving sectors is equal to the number of stationary
sectors, a moving sector being placed between two stationary sectors,
and _vice versa_.
4. That when the angle =A O B= is an _odd_ aliquot part of a circle,
the two last sectors adjacent to each other are either both in motion
or both stationary, the number of moving sectors being greater by one
when the number of sectors is 3, 7, 11, 15, etc., and the number of
stationary sectors being greater by one when the number of sectors is
5, 9, 13, 17, etc. And,
5. That as the moving sectors correspond with those in which the images
are inverted, and the stationary ones with those in which the images
are direct, the number of each may be found from the table given in
page 24.
When one of the mirrors, =A O=, is stationary, while the other, =B O=,
is moved round, and so as to enlarge the angle =A O B=, the object =X=,
and the image of it seen in the stationary mirror =A O=, remain at
rest, but all the other images are in motion receding from the object
=X=, and its stationary image; and when =B O= moves towards =A O=, so
as to diminish the angle =A O B=, the same effect takes place, only
the motion of the images is towards the object =X=, on one side, and
towards its stationary image on the other. These images will obviously
move in pairs; for, since the fixed object and its stationary image are
at an invariable distance, the existence of a symmetrical arrangement,
which we have formerly proved, requires that similar pairs be arranged
at equal distances round =O=, and each of the images of these pairs
must be stationary with regard to the other. Now, as the fixed object
is placed in the sector =A O B=, and its stationary image in the
sector =A O _b_=, it will be found that in the semicircle =M _b e_=,
containing the fixed mirror, the
1st reflected image and direct object, }
2d 3d reflected image }
4th 5th } are stationary with
6th 7th } respect to each other.
8th 9th }
while in the same semicircle =M _b e_=, the
1st reflected image and 2d reflected image }
3d 4th }
5th 6th } are movable with
7th 8th } respect to each other.
9th 10th }
On the other hand, in the semicircle =M _a e_=, containing the movable
mirror, the phenomena are reversed, the images which were formerly
stationary with respect to each other being now movable, and _vice
versa_.
In considering the velocity with which each pair of images revolves,
it will be readily seen that the pair on each side, and nearest the
fixed pair, will have an angular velocity _double_ that of the mirror
=B O=; the next pair on each side 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 _sixteen_ times as great
as that of the mirror, the velocity of any pair being always double the
velocity of the pair which is adjacent to it on the side of the fixed
pair. The reason of this will be manifest, when we recollect what has
already been demonstrated, 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 =B
O= moves from =A O=, the image in the sector =B O _a_= moves with twice
the velocity of the mirror; but since the image in =_b_ O β= is an
image of the image in =B O _a_= reflected from the fixed mirror =A O=,
it also will move with the same velocity, or twice that of the mirror
=B O=. Again, the image in the sector =_a_ O α=, being a reflexion of
the stationary image in =A O _b_= from the moving mirror, will itself
move with double the velocity of the mirror. But the image in the next
sector =α O β= is a reflexion of the image in =_b_ O β= from the moving
mirror =B O=; and as this latter image has been shown to move in the
direction =_b_ β=, with twice the velocity of the mirror =B O=, while
the mirror =B O= itself moves towards the image, it follows that the
image in =α O β= will move with a velocity four times that of the
mirror. The same reasoning may be extended to any number of sectors,
and it will be found that in the semicircle =M _b e_=, containing the
fixed mirror,
The {2 and 3} 2} times the
images {4 and 5} reflexions, move with 4} velocity of
formed {6 and 7} 8} the mirror;
by {8 and 9} 16}
whereas in the semicircle =M _a e_=, containing the movable mirror,—
The {1 and 2} 2 { times the
images {3 and 4} reflexions, move with 4 { velocity of
formed {5 and 6} 8 { the mirror;
by {7 and 8} 16{
a progression which may be continued to any length.
Before concluding this chapter, it may be proper to mention a very
remarkable effect produced by moving the two plain mirrors along one of
two lines placed at right angles to each other. When the aperture of
the mirrors is crossed by each of the two lines, the figure created by
reflexion consists of two polygons with salient and re-entering angles.
By moving the mirrors along one of the lines, so that it may always
cross the aperture at the same angle, and at the same distance from the
angular point, the polygon formed by this line will remain stationary,
and of the same form and magnitude; but the polygon formed by the
other line, at first emerging from the centre, will gradually increase
till its salient angles touch the re-entering angles of the stationary
polygon; the salient angles becoming more acute, will enclose the
apices of the re-entering angles of the stationary polygon, and at last
the polygon will be destroyed by truncations from its salient angles.
When the lines cross each other at a right angle, the salient angles
of the opening polygon can never touch the salient angles of the
stationary polygon, but always its re-entering angles. If the lines,
however, form a less angle than the complement of the angle formed
by the mirrors, then the salient angles of the opening polygon may
touch the salient angles of the stationary polygon, by placing the
mirrors so as to form re-entering angles in the polygon. When the
lines form an angle between 90° and the complement of the angle formed
by the mirrors, the salient angle of the opening polygon may be made
to touch the salient angles of the stationary one, but in this case
the stationary polygon can have no re-entering angles. The preceding
effects are finely exemplified by the use of a Kaleidoscope with a
draw-tube and lens described in Chapter X., and by employing the
vertical and horizontal bars of a window, which may be set at different
angles, by viewing them in perspective.