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.

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
the Angles of
Incidence.		 Rays Reflected
out of 1000.
   2½°584
5543
7474
10412
  12½356
15299
20222
21210
25157
26149
30112
31105
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.

Fig. 19.