LIGHT 583
Substituting for cos 6, and putting the differential coefficient =0, we have a quadratic equation of which the only admissible root is the positive one
From this the other quantities can be calculated.
Effect of contraction of pupil.
But another matter has to be taken into consideration when we apply the above definition of brightness in practice. For the aperture of the pupil is usually very much contracted when we look at a brightly illuminated sky or cloud. Thus there is a rough compensation which, to a certain extent, modifies the effect on the retina.
Argument for finite number of stars.
Founded on the above is Cheseaux's celebrated argument about the finite dimensions of the stellar universe. For it is easy to see, as below, that if stars be scattered through infinite space, with average closeness and brightness such as is presented by those nearest us, and if stellar space be absolutely transparent, the whole sky should appear of a brightness like that of the sun. Cheseaux and Olbers endeavoured to show that, because the sky is not all over as bright as the sun, there is absorption of light in stellar space. This idea was ingeniously developed by Struve.
Consider a small spherical angle . The number of stars in
cluded in it whose distances are between r and r + 8r from the earth
is proportional to
The whole amount of light received from such a portion of the sky
must be therefore as
provided that no star intercepts the light coming from another.
This condition is unattainable, so that the conclusion is that the
brightness is as great as it can be with the materials employed.
Every portion of the background shines as if it were a star.
Brightness at different obliquities.
(c) A third very important fact, connected with our present subject, but not immediately deducible from our principle, is – The brightness of a self-luminous surface does not depend upon its inclination to the line of sight.
Thus a red-hot ball of iron, free from scales of oxide, &c., appears flat in the dark; so, also, the sun, seen through mist, appears as a flat disk. This fact, however, depends ultimately upon the second law of thermodynamics, and its explanation will be fully given under RADIATION.
It may be stated, however, in another form, in which its connexion with what precedes is more obvious – The amount of radiation, in any direction, from a luminous surface is proportional to the cosine of the obliquity.
General principles of the theory of illumination.
The flow of light (if we may so call it) in straight lines from the luminous point, with constant velocity, leads as we have seen to the expression -^ (where r is the distance from the luminous point) for the quantity of light which passes through unit of surface perpendicular to the ray in unit of time, /j. being a quantity indi cating the rate at which light is emitted by the source. This represents the illumination of the surface on which it falls. The flow through unit of surface whose normal is inclined at an angle 6 to the ray is of course
again representing the illumination. These are precisely the
expressions for the gravitation force exerted by a particle of mass
fj. on a unit of matter at distance r, and for its resolved part in a
given direction. Hence we may employ an expression
which is exactly analagous to the gravitation or electric potential, for the purpose of calculating the effect due to any number of separate sources of light.
And the fundamental proposition in potentials, viz., that, if n be the external normal at any point of a closed surface, the integral
taken over the whole surface, has the value
-4*7i ,
where /i is the sum of the values of p for each source lying within the surface, follows almost intuitively from the mere consideration of what it means as regards light. For every source external to the closed surface sends in light which goes out again. But the light from an internal source goes wholly out; and the amount pet second from each unit source is 4ir, the total area of the unit sphere surrounding the source.
It is well to observe, however, that the analogy is not quite com plete. To make it so, all the sources must lie on the same side of the surface whose illumination we are dealing with. This is due to the fact that, in order that a surface may be illuminated at all, it must be capable of scattering light, i.e., it must be to some extent opaque. Hence the illumination depends mainly upon those sources which are on the same side as that from which it is regarded.[1]
Though this process bears some resemblance to the heat analogy employed by Sir W. Thomson for investigations in statical electricity (Cambridge Mathematical Journal, 1842) and to Clerk Maxwell's device of an incompressible fluid without mass (Cam. Phil. Trans., 1856), it is by no means identical with them. Each method deals with a substance, real or imaginary, which flows in conical streams from a source so that the same amount of it passes per second through every section of the cone. But in the present process the velocity is constant and the density variable, while in the others the density is virtually constant and the velocity variable. There is a curious reciprocity in formulæ such as we have just given. For instance, it is easily seen that the light received from a uniformly illuminated surface is represented by
S cos 6
As we have seen that this integral vanishes for a closed surface which has no source inside, its value is the same for all shells of equal uniform brightness whose edges lie on the same cone.
Theoretical explanations of rectilinear propagation. Non-homogeneous medium.
We have said that light moves in straight lines in a homogeneous medium. This rectilinear path follows at once from the corpuscular theory, as well as from the undulatory theory of light: in the first case there is no deflecting cause, so each corpuscle moves in a straight line; in the second, the direction of propagation of a plane wave in an uniform isotropic medium is always perpendicular to its front. Looking along a hot poker or the boiler of a steamboat, we see objects beyond distorted; i.e., we no longer see each point in its true direction. Here we have a non-homogeneous medium, the air being irregu- larly expanded in the neighbourhood of the hot body. To this simple cause are due the phenomena of mirage, the fata morgana, the reduplication of images of a distant object seen through an irregularly heated atmosphere, the scintillation or twinkling of stars, and the uselessness of even the best telescopes at certain times, &c. It is interest ing to note here that Newton[2] says: – "Long telescopes may cause objects to appear brighter and larger than short ones can do; but they cannot be so formed as to take away that confusion of the rays which arises from the tremors of the atmosphere. The only remedy is a most serene and quiet air, such as may perhaps be found on the tops of the highest mountains, above the grosser clouds."
Photometers. Ritchie's. Rumford's.
Photometry. – The principle above explained suggests many simple methods of comparing the amounts of light given by different sources. If, for instance, a porcelain plate, or even a sheet of piper, of uniform thickness, have one half illuminated directly by one source of light, the other by a different source, and if one or other of these sources be moved to or from the plate till the halves appear equally illuminated, it is obvious that the amounts of light given out by the two sources are directly as the squares of their distances from the screen. This is the principle of Ritchie's photometer. Rumford suggested the com- parison of the intensity of the shadows of the same object thrown side by side on a screen by the two lights to be compared. In this case the shadow due to one source is
1 From the formula of which the proof has been indicated Green's theorem and its consequences follow immediately. But we need not give these here.
2 Optics, end of part i.
