Example. A distant luminous sphere subtends a given angle 2 a at the eye of an observer : to find its total illumination of a small plane area A placed at the eye and inclined at a given angle f3 to tho right line joining the eye and the centre of the luminous sphere.
Let B P n represent the small plane; with centre r and radius unity describe a circular arc c A C, of which the measure is 2 a, and which by rotating round its axis r A generates a spherical surface of equal illu minating power with the given sphere. Let the angle n P Taken radium r Q forming an angle A r e= 8, and which, by revolving round I. A, traces the circle n Q D. The plane n r B is taken perpendi :oder to the plane of the diagram. Let w be the inclination of r Q to the given plane. The spherical element at Q is in 8. 5 0 Sip, where ct, is the inclination of the plane A P Q to that of the diagram, and its nominating power is therefore i sin ce sin 0. S 0 50, therefore the total illumination is expressed by ALP sin co sin 0, the limits of ct, being 0 0 ct, and 2 r (where w is the semicircumference to a unit radius), and of 0 being 0 and a. When the intensity is uniform, we get the illumination 1=A iffsin w sin 0. Draw PE perpendicular to the plane it n ; then in the spherical triangle Q A E we have Q A=0, L Q A E= A and QE=,71-0); hence by trigonometry sin w = sin /3 cos 0 + cos $ sin 0 cos cp Hence f sin w sin 0= 2w sin /3 sin 0 cos 0 and now into ct, grating relative to 0, we have 1=Ai. r sin 13 sine a, as the illumination required. In this investigation the whole of the light is supposed to fall on the same side of the plane.
If a small hole be formed in the window-shutters of a darkened chamber, the rays of light passing from opposite parts of any luminous object outside cross each other in entering the orifice, since they necessarily proceed in straight lines, and therefore form on the opposite wall of the chamber a perfectly inverted image of the external object, and if the latter be in motion, the image will also move in the contrary direction. If in be the magnitude of the object, and x its distance from the hole, and a the width of the chamber, then the light being supposed to enter directly, the magnitude of the image, by the known laws of similar figures, will be in . But the total quantity of light which enters the hole (supposed to be of given size) from the object varies as, that is, as the magnitude of the image, a being supposed constant, and therefore the brightness of the image is con stant for all distances of the object. The eye is such a chamber, and therefore a luminous object should appear of equal brightness at all distances, but the absorption of light by the atmosphere causes tho greater dimness of distant atmospheric objects.
If we suppose the quantity of light absorbed by a transparent medium to be a proportional part of the incident light, then denoting by i the intensity of light which corresponds to a space x traversed (or rather, to include the case of divergence, the ratio of the intensity to what it would have been independently of absorption), we shall di have = - k. k being a constant dependent on the particular
nature of the medium, and by integration we find where t is the initial intensity previously to the light entering the medium, and e the base of Napierian logarithms ; therefore the intensity will diminish in a geometrical progression for equal spaces successively traversed.
From these principles we are enabled to calculate the laws which the direct rays of light obey, from their emanation to their incidence. If the body on which the latter takes place be unpolished and opaque, a portion of the light enters into it for a small depth, and is there partially absorbed ; the complementary portion is scattered in all directions; the surface therefore becomes itself, to that extent, a source of light, but the composition of the differently coloured rays [Dunn 810N) may be widely different from that of the incident light : for instance, if the incident light were an equal mixture of red and blue rays, and if the surface favoured the absorption of the latter more than of the former, the scattered or complementary light, then containing more of red blue rays, would proportionally tinge with red the apparent colour of the Solar light is a compound of various homogeneous coloured rays; and by their unequal absorption or transmission bodies acquire these apparent colours ; but the perception of form arises from the variations of light and shade, and the moclifica thins of light on the borders, ridges, and angles of the surfaces; and the painter, when he produces a relief on a plane surface, imitates those modificatious in the colours which he applies. Hence the perception of form is lost when this incident light is excluded, as in a heated square bar of iron in a dark room, which when turned round its axis seems always to be a flat surface, growing wide and narrow alternately as its edges or faces are turned to the eye ; and even when incident light is admitted, a greatness of distance from the eye renders those modifications inappreciable unless under the most favourable circumstances; and thus the heavenly bodies, instead of appearing as round solids, are projected upon a spherical surface, having the eye for the centre. When the body exposed to incident light has even a slight polish, the scattered light will then be most copious in the directions in which the regular reflections take place. Such portions of the surface as are situated, relatively to the eye, properly for regular reflections of the incident light, have therefore a much greater apparent brightness than the parts adjacent, and thus assist in pro ducing the ideas of the position and form of the parts.