CHAPTER VI.
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 |
| 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.
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.
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.