ON THE INTENSITY OF THE LIGHT IN DIFFERENT PARTS OF
THE FIELD, AND ON THE EFFECTS PRODUCED BY VARYING
THE LENGTH AND BREADTH OF THE REFLECTORS.
When we look through a Kaleidoscope in which the mirrors are placed
at an angle of 18° or 22½°, the eye will perceive a very obvious
difference in the intensity of the light in different parts of the
field. If the inclination of the mirrors be about 30°, and the eye
properly placed near the angular point, the intensity of the light is
tolerably uniform; and a person who is unaccustomed to the comparison
of different lights, will find it extremely difficult to distinguish
the direct sector from the reflected ones. This difficulty will be
still greater if the mirrors are made of finely polished steel, or
of the best speculum metal, and the observer will not hesitate in
believing that he is looking through a tube whose diameter is equal to
that of the circular field. This approximation to uniformity in the
intensity of the light in all the sectors, which arises wholly from the
determination of the proper position of the eye, is one of the most
curious and unexpected properties of the Kaleidoscope, and is one which
could not have been anticipated from any theoretical views, or from
any experimental results obtained from the ancient mode of combining
plain mirrors. It is that property, too, which gives it all its value;
for, if the eye observed the direct sector with its included objects
distinguished from all the rest by superior brilliancy, not only would
the illusion vanish, but the picture itself would cease to afford
pleasure, from the want of symmetry in the light of the field.
[Illustration: FIG. 18.]
Before we proceed to investigate the effects produced by a variation in
the length of the reflecting planes, it will be necessary to consider
the variation of the intensity of the light in different parts of the
reflected sectors. In the direct sector =A O B=, Fig. 2, the intensity
of the light is uniform in every part of its surface; but this is far
from being the case in the images formed by reflexion. In Fig. 17, take
any two points _m_, _o_, and draw the lines _m n_, _o p_, perpendicular
to =A O=, and meeting =β O= in _n_ and _p_. Let =O E=, Fig. 18, be a
section of the reflector =A O= seen edgewise, and let =O _p_=, =O _n_=,
be taken equal to the lines _m n_, _o p_, or the height of the points
_n_, _p_, above the plane of the reflector =A O=. Make =O R= to =R E=
as =O _p_= is to =E _e_= the constant height of the eye above the
reflecting plane, and =O _r_= to =_r_ E= as =O _n_= to =E _e_=, and the
points =R, _r_=, will be the points of incidence of the rays issuing
from _p_ and _n_; for in this case =O R _p_ = E R _e_=, and =O _r n_ =
E _r e_=. Hence it is obvious, that =E R _e_= is less than =E _r e_=,
and that the rays issuing from _p_, by falling more obliquely upon
the reflecting surface, will be more copiously reflected. It follows,
therefore, that the intensity of the light in the reflected sector =A
O β= is not uniform, the lines of equal brightness, or the _isophotal_
lines, as they may be called, being parallel to the reflecting surface
=A O=, and in every sector parallel to the radius, between the given
sector and the reflecting surface by which the sector is formed.
As it is easy from the preceding construction to determine the angles
at which the light from any points _m_, _n_, is reflected, when the
length =O E= of the reflectors, and the position of the eye at =E= is
given, we may calculate the intensity of the light in any point of
the circular field by means of the following table, which shows the
number of rays reflected at various angles of incidence, the number
of incident rays being supposed to be 1000. Part of this table was
computed by Bouguer for plate glass not quicksilvered, by means of a
formula deduced from his experiments. By the aid of the same formula I
have extended the table considerably.
_Table showing the quantity of light reflected at various angles of
incidence from plate glass._
+---------------+---------------+
| Complement of | Rays Reflected|
| the Angles | out |
| of Incidence. | of 1000. |
+---------------+---------------+
| 2½° | 584 |
| 5 | 543 |
| 7 | 474 |
| 10 | 412 |
| 12½ | 356 |
| 15 | 299 |
| 20 | 222 |
| 21 | 210 |
| 25 | 157 |
| 26 | 149 |
| 30 | 112 |
| 31 | 105 |
| 34 | 85 |
| 35 | 79 |
| 36 | 74 |
| 37 | 69 |
| 38 | 65 |
| 39 | 61 |
| 40 | 57 |
| 46 | 40 |
| 50 | 34 |
| 55 | 29 |
| 60 | 27 |
| 70 | 25 |
| 80 | 25 |
| 90 | 25 |
+---------------+---------------+
In order to explain the method of using the table, let us suppose that
the angle of incidence, or =O R _p_=, Fig. 19, is 85°: then the number
of rays in the corresponding point π of the reflected sector =A O _b_=
(Fig. 17) will be 543. By letting fall perpendiculars from the points
μ, π, upon the mirror =B O=, and taking =O _p_=, =O _n_=, Fig. 19,
equal to these perpendiculars, we may ascertain the angles at which the
light from the points μ, π, suffer a second reflexion from the mirror
=B O=. Let the angle for the point π be 10°, then the number of rays
out of 1000 reflected at this angle, according to the table, is 412;
but as the number of rays emanating from π, and incident upon =B O=, is
not 1000, but only 543, we must say as 1000: 412 = 543: 224, the number
of rays reflected from =B O=, or the intensity of the light in a point
in the line =O _bʹ_= corresponding to π.
[Illustration: FIG. 19.]
The preceding method of calculation is applicable only with strictness
to the two sectors =A O _b_=, =B O _a_=, formed by one reflexion, for
the intensity of the light in the other sectors which are formed by
more than one reflexion, must be affected by the polarization which
the light experiences after successive reflexions; for light which has
acquired this property is reflected according to laws different from
those which regulate the reflexion of direct light.
When the mirrors are metallic, the quantity of reflected light is also
affected by its polarization, but it is regulated by more complicated
laws.
In Kaleidoscopes made of plates of glass, the last reflected image
=β O ω=, Fig. 2, is more polarized than any of the rest, and is
polarized in a plane perpendicular to =X E=, or in the same manner as
if it had been reflected at the polarizing angle from a vertical plane
parallel to =X E=.
[Illustration: FIG. 20.]
Let us now consider what will take place by a variation in the length
of the reflecting planes, the angular extent of the field of view
remaining always the same. If =A O E=, =A O Eʹ=, Fig. 20, be two
reflecting plates of the same breadth =A O=, but of different lengths,
it is manifest that the light which forms the direct sector must be
incident nearer the perpendicular, or reflected at less obliquities in
the short plate than in the long one, and, therefore, that a similarly
situated point in the circular field of the shorter instrument, will
have less intensity of light than a similarly situated point in a
larger instrument. But in this case, the field of view in the short
instrument is proportionally enlarged, so that the comparison between
the two is incorrect. When the long and the short instrument have equal
apparent apertures, which will be the case when the plates are =A O E=,
=Aʹ O Eʹ=, then similarly situated points of the two fields will have
exactly the same intensity of light.
This will be better understood from Fig. 19, where =O E= may represent
the long reflector and =Oʹ E= the short one. Then, if these two have
exactly the same aperture, or a circular field of the same angular
magnitude, the rays of light which flow from two given points, _p_,
_n_, of the long instrument, will be reflected at a certain angle
from the points =R=, _r_; but as the points _pʹ_, _nʹ_, are the
corresponding points in the field of the shorter instrument, the rays
which issue from them will be reflected at the same angles from the
points =R=, _r_, the eye being in both cases placed at the same point
_e_. Hence it is obvious, that the quantity of reflected light will
in both cases be the same, and, therefore, that there is no peculiar
advantage to be derived, in so far as the light of the field is
concerned, by increasing the length of the reflectors, unless we raise
the eye above _e_, till every part of the pupil receives the reflected
rays.
There is, however, one advantage, and a very important one, to be
derived from an increase of length in the mirrors, namely a diminution
of the deviation from symmetry which arises from the small height of
the eye above the plane of the mirrors, and of the small distance
of the objects from the extremity of the mirrors. As the height of
the eye must always be a certain quantity, =E _e_=, Fig. 17, above
the angular point =E=, whatever be the length of the reflectors, it
is obvious, that when the length of the reflectors is =_e_ O=, the
deviation from symmetry will be only =P _oʹ_=, whereas when the length
of the reflectors is reduced to =_eʹ_ O=, the height of the eye =_eʹ_
Eʹ= being still equal to =_e_ E=, the aberration will be increased
to =P _o_=. This advantage is certainly of considerable consequence;
but in practice the difficulty of constructing a perfect instrument,
increases with the length of the reflectors. When the plates are long,
it is more difficult to get the surface perfectly flat; the risk of a
bending in the plates is also increased, which creates the additional
difficulty of forming a good junction, on which the excellence of the
instrument so much depends. By augmenting the length of the reflectors,
the quantity of dust which collects between them is also increased,
and it is then very difficult to remove this dust, without taking the
instrument to pieces. From these causes it is advisable to limit the
greatest length of the reflectors to seven or eight inches.