CONVEX AND CONCAVE MIRRORS. 1087* which it has been moved is double the angle of the prism, as is obvious from Fig. 767A, in which the directions of the incident and reflected rays are represented by lines marked with arrowheads. The incident rays being parallel, a+ẞ is the angle of the prism, and 2a+23 is the angle between the reflected rays. After measuring the angle of the prism by means of the observing telescope and collimator with slit, the index of refraction of the prism can be determined by illuminating the slit with sodium light or some other monochromatic light, and observing with the telescope the minimum deviation of the image of the slit formed by refraction through the prism. The index of refraction for the particular light employed can then be deduced by formula (4) page 1007. 1066B. Relation between Convex and Concave Mirrors.-The student should notice that every diagram relating to reflection from a concave mirror is equally applicable to a convex mirror. For example Fig. 683, page 988, correctly represents the virtual image AB of a real object ba in front of a convex mirror having C for its centre of curvature and F for its principal focus; and Fig. 677, page 982, shows the effect of interposing a convex mirror in the path of rays which are on their way to form a real image ab. The effect is to produce the virtual image AB. Or we may take AB as representing the real image which the rays were on their way to form, and then ab will be the virtual image which is formed instead. If the real image falls between the principal focus F and the convex mirror, the effect will be that instead of an inverted virtual image, an erect real image will be formed. Thus in Fig. 683, if AB be the image which the rays were on their way to form, the real image ba will be formed instead. Every diagram of an object and its image, as formed by a spherical mirror, has in fact four different interpretations, since the object and image may be interchanged, and the spherical surface may be polished on either side. 1066c. Nodal Points of a lens or system of lenses. When the thickness of a lens is considerable in comparison with the radii of curvature of its faces, the approximate assumption made in the first paragraph of page 1016, "that rays which pass through the centre of a lens undergo no deviation" is no longer admissible. Referring to Fig. 718, page 1015, if we suppose the incident and emergent rays SI and RE to be produced to meet the axis in points N1 and N2, the ultimate positions of N, N, when the rays make very small angles with the axis are called the nodal points of the lens. They are the two images of the centre of the lens formed by refraction out of the lens into air at the two surfaces. Whenever the incident ray passes through the first nodal point, the emergent ray passes through the second nodal point and is parallel to the incident ray. Every system of lenses having a common axis, whether the lenses are in contact with each other or at any distances apart, has two nodal points possessing the above property. An obvious deduction from this property is, that the image subtends the same angle at the second nodal point as the object subtends at the first. The second nodal point of the normal human eye is in the crystalline lens near the back, and the image of a distant object formed on the retina subtends the same angle at this point which the object subtends at the eye. The "line of collimation" of a telescope, which we have defined on page 1058 as "the line joining the cross to the optical centre of the object-glass," would be still more accurately defined as the line joining the cross to the second nodal point of the objectglass. CHAPTER LXXIII. COLOUR. 1067. Colour as a Property of Opaque Bodies.-A body which reflects (by irregular reflection) all the rays of the spectrum in equal proportion, will appear of the same colour as the light which falls upon it; that is to say, in ordinary cases, white or gray. But the majority of bodies reflect some rays in larger proportion than others, and are therefore coloured, their colour being that which arises from the mixture of the rays which they reflect. A body reflecting no light would be perfectly black. Practically, white, gray, and black differ only in brightness. A piece of white paper in shadow appears gray, and in stronger shadow black. 1068. Colour of Transparent Bodies.-A transparent body, seen by transmitted light, is coloured, if it is more transparent to some rays than to others, its colour being that which results from mixing the transmitted rays. No new ingredient is added by transmission, but certain ingredients are more or less completely stopped out. Some transparent substances appear of very different colours according to their thickness. A solution of chloride of chromium, for example, appears green when a thin layer of it is examined, while a greater thickness of it presents the appearance of reddish brown. In such cases, different kinds of rays successively disappear by selective absorption, and the transmitted light, being always the sum of the rays which remain unabsorbed, is accordingly of different composition according to the thickness. When two pieces of coloured glass are placed one behind the other, the light which passes through both has undergone a double process of selective absorption, and therefore consists mainly of those rays which are abundantly transmitted by both glasses; or to speak broadly, the colour which we see in looking through the combination is not the sum of the colours of the two glasses, but their common part. Accordingly, if we combine a piece of ordinary red glass, transmitting light which consists almost entirely of red rays, with a piece of ordinary green glass, which transmits hardly any red, the combination will be almost black. The light transmitted through two glasses of different colour and of the same depth of tint, is always less than would be transmitted by a double thickness of either; and the colour of the transmitted light is in most cases a colour which occupies in the spectrum an intermediate place between the two given colours. Thus, if the two glasses are yellow and blue, the transmitted light will, in most cases, be green, since most natural yellows and blues when analysed by a prism show a large quantity of green in their composition. Similar effects are obtained by mixing coloured liquids. 1069. Colours of Mixed Powders.-"In a coloured powder, each particle is to be regarded as a small transparent body which colours light by selective absorption. It is true that powdered pigments when taken in bulk are extremely opaque. Nevertheless, whenever we have the opportunity of seeing these substances in compact and homogeneous pieces before they have been reduced to powder, we find them transparent, at least when in thin slices. Cinnabar, chromate of lead, verdigris, and cobalt glass are examples in point. "When light falls on a powder thus composed of transparent particles, a small part is reflected at the upper surface; the rest penetrates, and undergoes partial reflection at some of the surfaces of separation between the particles. A single plate of uncoloured glass reflects of normally incident light; two plates, and a large number nearly the whole. In the powder of such glass, we must accordingly conclude that only about of normally incident light is reflected from the first surface, and that all the rest of the light which gives the powder its whiteness comes from deeper layers. It must be the same with the light reflected from blue glass; and in coloured powders generally only a very small part of the light which they reflect comes from the first surface; it nearly all comes from beneath. The light reflected from the first surface is white, except when the reflection is metallic. That which comes from below is coloured, and so much the more deeply the further it has penetrated. This is the reason why coarse powder of a given material is more deeply coloured than fine, for the quantity of light returned at each successive reflection depends only on the number of reflections and not on the MIXTURE OF COLOURS. 1089 thickness of the particles. If these are large, the light must penetrate so much the deeper in order to undergo a given number of reflections, and will therefore be the more deeply coloured. "The reflection at the surfaces of the particles is weakened if we interpose between them, in the place of air, a fluid whose index of refraction more nearly approaches their own. Thus powders and pigments are usually rendered darker by wetting them with water, and still more with the more highly refracting liquid, oil. "If the colours of powders depended only on light reflected from their first surfaces, the light reflected from a mixed powder would be the sum of the lights reflected from the surfaces of both. But most of the light, in fact, comes from deeper layers, and having had to traverse particles of both powders, must consist of those rays which are able to traverse both. The resultant colour therefore, as in the case of superposed glass plates, depends not on addition but rather on subtraction. Hence it is that a mixture of two pigments is usually much more sombre than the pigments themselves, if these are very unlike in the average refrangibility of the light which they reflect. Vermilion and ultramarine, for example, give a black-gray (showing scarcely a trace of purple, which would be the colour obtained by a true mixture of lights), each of these pigments being in fact nearly opaque to the light of the other.” 1 1070. Mixtures of Colours.-By the colour resulting from the mixture of two lights, we mean the colour which is seen when they both fall on the same part of the retina. Propositions regarding mixtures of colours are merely subjective. The only objective differences of colour are differences of refrangibility, or if traced to their source, differences of wave-frequency. All the colours in a pure spectrum are objectively simple, each having its own definite period of vibration by which it is distinguished from all others. But whereas, in acoustics, the quality of a sound as it affects the ear varies with every change in its composition, in colour, on the other hand, very different compositions may produce precisely the same visual impression. Every colour that we see in nature can be exactly imitated by an infinite variety of different combinations of elementary rays. To take, for example, the case of white. Ordinary white light consists of all the colours of the spectrum combined; but any one of the elementary colours, from the extreme red to a certain point in yellowish green, can be combined with another elementary colour 1 Translated from Helmholtz's Physiological Optics, § 20. |