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.