passing through its central region as indicated in figure 524. Another kind of spherical aberration is that a pencil of rays from a point, passing obliquely through the lens, becomes astigmatic and converges through two focal lines instead of coming to a single point. These defects cause a lack of clearness and sharpness in the images formed. They are most serious when the diameter of the lens is a large fraction of its focal length. The colors observed at the edges of images formed by lenses are due to the fact that ordinary lenses refract blue light more strongly than red light. This defect, known as chromatic aberration, will be discussed later (§866). FIG. 524.-Spherical aberration. 859. Experimental Illustration.-Form an image of an electric arc on a distant screen by means of the condensing lenses of an ordinary projecting lantern. The image will be blurred and indistinct unless all of the lens is covered except the central portion or a narrow circular zone equidistant from the center, and the lens must be moved nearer to the arc to give a sharp image when the zone is used than when the opening is in the center. Problems. 1. How deep is a tank of water which appears to be 4 ft. deep to a person looking vertically down into it? 2. An incandescent lamp is placed 6 ft. below the surface of a pond. Show why only a fractional part of the light can escape directly from the water. 3. If a beam of light has 50,000 light waves to the inch in air, how many to the inch will there be after it has entered water? 4. Find the velocity of light in water if the critical angle at the surface between water and air is 48° 30'. 5. When the index of refraction of water is 1.33 and that of carbon bisulphide is 1.67, what is the critical angle between water and carbon bisulphide? 6. The object-glass of the Yerkes telescope is a convergent lens 40 in. in diameter and having a focal length of 62 ft. What is the size of the sun's image formed by it? What effect has the size of the lens on the size of the image? 7. An incandescent lamp is 30 cm. from a convergent lens of 10 cm. focal length. Find the position and relative size of the image; is it real or virtual? 8. A candle is placed 1 meter from a divergent lens having a focal length of 1 meter. Where is the image formed and what is its size? Make a construction illustrating the case. 9. A lamp and a screen are 10 ft. apart. Where must a convergent lens of 2 ft. focal length be placed so as to form an image of the lamp on the screen? Show that there are two solutions and find the relative size of the image in each case. 10. A beam of sunlight falls on a divergent lens of focal length 10 inches; 20 inches beyond this lens is placed a convergent lens of 15 inches focal length. Find where a screen should be placed to receive the final image of the sun. II. A convergent lens, focal length 10, is placed 12 in. from a gas flame; then 36 in. beyond the first lens is placed a divergent lens of focal length 16 in. Find the position and size of the final image; is it real or virtual? 12. A certain lens when placed 10 cm. from an object, forms a virtual image 5 times as large as the object. What kind of lens is used and what is its focal length? 13. What must be the focal length of spectacle lenses so that a man who can see distinctly objects 2 meters distant without the glasses can read print at 40 cm. distance with them, and what kind of lenses must be used? Note. The strength of spectacle lenses is expressed in diopters and is the reciprocal of the focal length expressed in meters. 14. A person, who without glasses cannot see distinctly objects more than 12 cm. from the eye, wishes glasses to enable him to see clearly distant objects. What must be the kind used and their focal length and strength in diopters? DISPERSION. 860. Dispersion of Light by a Prism.-When a narrow beam of sunlight passes through a prism, the light is not only bent aside or deviated, it is also dispersed or spread out into a colored band called the spectrum. Sir Isaac Newton placed a second prism (Fig. 525) in the spectrum so that light of only one color might fall on it. This light was refracted on passing through the second prism, but there was no further change in color, showing that the prism itself did not produce the different colors, but simply separated the various kinds of light already present in the beam of sunlight. The separation is effected because the various colored lights are differently refracted by the prism, the red being refracted least and the violet most. White Light RY G B Violet FIG. 525.-Newton's experiment. 861. Cause of Dispersion.-Since the bending of the rays. by a prism depends only on the angle of the prism and the index of refraction of the substance of which it is made, it follows that the index of refraction of the prism must be different for each kind of light in the spectrum, being least for the red which is least refracted and greatest for the violet which is most strongly refracted. Of course the interpretation of this fact is that red light must pass through the substance of the prism with greater velocity than violet light. It will be shown later that the physical difference between one kind of light and another lies in their wave lengths. These vary from one end of the spectrum to the other, the longest waves being at the red end of the spectrum while the shortest are at the violet end. It appears therefore that shorter waves of light are more retarded in passing through glass than longer ones. 862. Dispersive Power. When two prisms of different substances have such angles that each produces the same deviation for yellow light, or light in the middle of the spectrum, the angu lar widths of the two spectra produced will usually not be the same, but are proportional to what are called the dispersive powers of the substances. The dispersive powers of some substances are as follows: Thus for an equal bending of the mean rays, carbon bisulphide will produce a spectrum 3 times as long as that produced by crown glass and 23 times as long as one formed by flint glass. Crown Flint FIG. 526.-Prisms with different dispersive powers. 863. Calculation of Dispersive Power.-We have seen that the index of refraction of a substance depends on the kind of light. The following table gives three indices of refraction for each of four substances. The indices given in the first column are for light near the extreme red end of the spectrum, those in the second are for the yellow sodium light, while those in the third are for light near the violet end of the spectrum. These points in the spectrum correspond to three dark lines in the sun spectrum designated A, D, and H by Fraunhofer ($895). A prism of angle A and index of refraction n causes a minimum deviation D (§848), which is determined by the formula Now, if the angle of the prism is so small that the ratio of the sines is practically equal to the ratio of the arcs themselves, we have So that prisms having a given small angle produce deviations proportional to n−1. The next to the last column in the above table shows the relative deviations of yellow sodium light caused by prisms of the different substances all having the same small angle. Thus it appears that a prism of carbon bisulphide will cause nearly twice as great a deviation as a prism of water of the same angle, if the angles of the prisms are small. In the last column are given the differences between the indices of refraction for red and violet lights. These figures therefore represent the relative angular widths of the spectra produced by prisms of the various substances having the same small angle. The spectrum formed by a thin prism of carbon bisulphide is therefore about 6 times as long as that formed by a similar prism of water. To obtain the relative dispersive powers given in the previous paragraph the figures in the last column must be divided by those in the next to the last. Crown V Y R Flint FIG. 527.-Dispersion without bending the mean ray. 864. Direct-vision Prism.—In consequence of the fact that the dispersive powers of substances differ it is possible to so combine two prisms of different substances as to produce dispersion without deviation of the mean ray, or to produce deviation without dispersion. For example, if a prism of crown glass and one of flint glass are taken whose angles are in the ratio 0.587 to 0.534, respectively, they will each deviate the D line of the spectrum by the same |