LIGHT A TEXTBOOK FOR STUDENTS WHO HAVE HAD ONE YEAR OF PHYSICS By H. M. REESE Professor of Physics in the University of Missouri (Columbia. ffltBsmtri MISSOURI BOOK COMPANY 1921 COPYRIGHT 1921 BY THE MISSOURI BOOK COMPANY. PREFACE. The writing of this book was undertaken because no ex- isting text on the subject quite filled the needs for my own classes. The first draft was mimeographed, and has been used in that form with some success for several years. It is planned for students who have had no training in the calculus, because many of those who take second-year physics at the University of Missouri suffer from that handicap. The first part has purposely been made rather easy, the inten- tion being to lead gradually from less to more difficult matter. A persistent attempt is made to lay stress upon the experimental basis for our theories, and to point out such reasons as exist for and against them; because my own experience has been that many students, though they may learn the facts of a science conscientiously and in a sense thoroughly, fail com- pletely to realize the inductive processes on which the theoreti- cal structure is founded, thus missing one of the chief educa- tional values to be derived from the study of science. It is in line with the same idea that certain matters have been intro- duced, particularly in the last two chapters, whose purpose is to give the reader an idea, incomplete; though it may be, of the present state of optical theory and allied branches of physical science. Thanks are due to my colleague, Professor 0. M. Stewart, for a number of valuable suggestions, and also to Professor Henry Gale, of the University of Chicago, who read the manuscript and suggested changes and additions which I have been glad to make. H. M. R. Columbia, Missouri, October, 1920. 4 65254 CONTENTS CHAPTER I. Article Page 1. Introduction 1 2. Velocity 3 3. Roemer's method 4 4. Bradley's method 5 5. Fizeau's method with the toothed wheel 7 6. The rotating mirror method 9 CHAPTER II. 7. Refraction through a prism 15 8. Newton's conception of color 18 9. Impure colors 19 10. Color due to absorption 20 11. Color due to other causes 22 12. Black and white 23 13. Complementary colors and color mixture 24 14. The eye 26 15. Color vision theories 28 CHAPTER III. 16. The corpuscular theory of light 32 17. The wave theory 33 18. Bending of light into a shadow 34 19. Nature of the ether 36 20. Waves in general. Plane waves . 37 21. Mathematical formula for a wave 39 22. Interference. Fresnel's mirrors 41 23. Interference in white light 48 CHAPTER IV. 24. Reflection and refraction. Huyghens' principle. Index of refraction 50 25. Total reflection. Critical angle . 54 26. Deviation through a prism 59 CHAPTER V. 27. Reflection and refraction of spherical waves at a plane surface 63 vii Vlll LIGHT Article Page 28. Judgment of the distance of an image 67 29. Image of an extended object 69 30. Reflection and refraction at spherical surfaces 69 31. Lenses 77 32. Two lenses in contact 82 33. Chromatic aberration 83 34. Achromatic lenses . 84 35. Image of extended object Undeviated ray 87 36. Magnification 88 37. Micrometer 90 38. Imperfections of mirrors and lenses 91 39. Spherical aberration 91 40. Curvature of field 92 41. Astigmatism 93 42. Lenses for special purposes 94 CHAPTER VI. 43. The telescope 97 44. Magnifying power 98 45. Ramsden eyepiece 100 46. Opera glass 101 47. Prism binocular . . . . 103 48. Reflecting telescopes 103 49. Simple microscope 104 50. Compound microscope 105 51. Projection lanterns 107 CHAPTER VII. 52. Prism spectroscope 110 53. Brightline spectra Ill 54. Spectral series 114 55. Continuous spectra 116 56- Dark-line spectra 117 57. Absorption by solids and liquids 118 58. Continuous spectrum of an absolutely black body 119 59. Planck's theory of "quanta" 120 60. The plane grating 121 61. Why the lines are sharp 126 62. Reflection gratings 129 63. The concave grating 129 64. The ultraviolet region. Fluorescence. Phosphorescence. Photography 130 65. The infrared region 131 66. The bolometer 132 67'. The thermopile 132 68. The Doppler principle. Motion of the stars 133 CONTENTS I CHAPTER VIII. Article Page 69. The approximately rectilinear propagation of light 138 70. Shadow of an edge 142 71. Shadow of a wire 145 72. Diffraction through a rectangular opening 145 73. Resolving-power 148 CHAPTER IX. 74. Young's interference experiment 152 75. The biprism 152 76. Interference in thin uniform films 153 77. Change of phase on reflection 155 78. Non-uniform films 159 79. The Michelson interferometer 160 80. Newton's rings 162 81. Fabry and Perot interferometer 163 82. Interference in white light . . . 163 83. Rainbows 167 84. Motion relative to the ether 171 85. The relativity theory ' 173 CHAPTER X. 86. Simple harmonic motion 175 87. Velocity in S. H. M 178 88. Acceleration in S. H. M 180 89. Energy in S. H. M 182 90. Two parallel S. H. M.'s 184 91. Application to cases of interference 187 92. Two S. H. M.'s at right-angles 190 93. Lissajous figures 193 CHAPTER XI. 94. Inverse square law 195 95. Photometry 196 96. Rumford photometer 196 97. Bunsen photometer 197 98. Lummer-Brodhun photometer 198 99. Light-standards 199 100. Solid angle 199 101. Intrinsic luminosity 200 102. Spectrophotometer 204 X LIGHT CHAPTER XII. Article Page 103. Transverse and longitudinal waves 205 104. Double refraction 206 105. Polarization of the O and E light 209 106. Wave-surface in doubly-refracting crystals 213 107. The lateral displacement of the E-ray 213 108. Special cases of double refraction 215 109. Tourmaline 216 110. Biaxial crystals 217 111. Polarization by reflection . 218 CHAPTER XIII. 112. Methods of polarizing light 223 113. The Nicol prism 224 114. Double-image prisms 225 115. Crossed Nicols and crystal plate 226 116. Elliptic polarization 226 117. Circular polarization ". 228 118. Rotation of the plane of polarization 230 119. Magnetic rotation 233 120. The rings-and-brushes phenomenon 234 121. The nature of elliptic and circular polarization 234 CHAPTER XIV. 122. Plane of polarization and plane of vibration 240 123. Elastic-solid theories 240 124. Electromagnetic theory 241 125. Direction of the vibrations 243 126. Fundamental electromagnetic laws 245 127. Faraday's displacement-currents 245 128. Maxwell's assumption - 247 129- Hertz's experiments . ., 247 130. Propagation of electromagnetic waves 248 131. Velocity of the waves 250 132. Refractive index and dielectric constant . . 252 CHAPTER XV. 133. Dispersion 254 134. Electron theory of matter 255 135. Electromagnetic dispersion formula 255 136. Anomalous dispersion 257 137. Reststrahlen . 259 CONTEXTS XI CHAPTER XVI. Article Page 138. Production of X-rays 262 139. Their properties 263 140. Are X-rays ether waves? 264 141. Crystal reflection of X-rays 266 142. Measurement of wavelengths. Crystal structure 268 143. X-ray spectra 270 144. The K and L series 270 145. Quantum theory applied to X-rays 271 146. Secondary X-rays. Absorption 271 147. Total range of ether waves 272 CHAPTER XVII. 148. Review of the development of light-theory 273 149. Modern lines of investigation 274 150. The Zeeman effect 275 151. Lorentz's theory 276 152. The Stark effect 278 153. The photo-electric effect 278 154. Atom-models 279 155. Bohr's theory of the hydrogen atom 282 Appendix I. Wavelengths for laboratory use 286 Appendix II. The velocity of electromagnetic waves 287 Index . 291 LIGHT. CHAPTER I. 1. Introduction. 2. Velocity. 3. Roemer's method. I. Bradley's method. 5. Fizeau's method with the toothed wheel. 6. The rotating mirror method of Foucault (and Fizeau). 1. Introduction. Simple and familiar observations teach us that the sensation of vision is caused by some agency that emanates from bodies external to us, and enters our eyes. For instance, we cannot see anything in a room with tightly closed blinds where there is no source of artificial illumination, such as a candle, fire, or electric lamp. Incidently, this experience also teaches that we may classify the objects we see into two groups: first, luminous objects, such as a candle, a fire, the sun, or the stars; second, objects such as a book, a tree, the walls of the room, etc., which can be seen only through the agency of a luminous body. From the physicist's point of view, the study of light is the study of this activity, whatever its nature may be, which originates in luminous bodies and causes the sensation of vision when it enters the eye. His interest lies primarily in the way this agency starts into action in a luminous object, how it propels itself through space, how it behaves on striking objects of different kinds, such as glass, crystals, silver, water, etc., and its relations to all other physical phenomena, such as heat, electricity, and magnetism. On the other hand, the physicist proper does not concern himself much with the parts that the eye and the nervous system play, in registering in our consciousness the sensation (vision), whose primary cause is the physical agency that we call light. This question is important and interesting enough, but it belongs primarily to the domains of the physiologist and the psychologist. Let us begin our study by making a summary of such facts as common knowledge gives us about light. In the first place, besides differences in brightness, which may be called a matter of quantity, there are also differences of quality to be considered, as shown in the phenomena of color. 1 2 LIGHT Second, light travels approximately in straight lines, as is shown in the formation of shadows. Nevertheless, we shall see later that light does bend around corners to some slight extent, though not nearly so much as sound does. Third, it differs from sound also in that it travels with- out hindrance through a vacuum. In coming to us from the stars, it travels through millions of miles of the most perfectly empty space obtainable. Fourth, it is either itself a manifestation of energy, or else it carries energy with it, since any object which receives and absorbs it becomes heated. Fifth, when it strikes a surface, more or less of it is gen- erally reflected. (Those exceptional surfaces which reflect no light are said to be black). If the surface is highly polished, the light is reflected at a definite angle, in which case we say the reflection is regular. If the surface is rough, like that of a sheet of paper, the light is scattered in all conceivable direc- tions, and the reflection is said to be diffuse. Sixth, there are many substances, such that when light strikes their surfaces, although part is reflected, part enters the material and passes through it rather freely. Such ma- terials are said to be transparent. Light traverses transparent materials approximately in straight lines, as it does the air or free space, but there is an abrupt bending of the rays at the place where they pass through the surface. This bending is called refraction. Seventh, light travels either instantaneously, or else with enormous velocity. Here again the comparison with sound is very striking. The phenomenon of echoes shows that sound travels with a speed which, though great, cannot be called enormous, and indeed a fairly accurate measurement of this speed could be made by noting with a stop-watch the time required for an echo to be heard from a cliff or large building, whose distance from the observer is known. An exactly analogous experiment with light would be to note with a stop- watch the time that elapses between the flashing of a light and the perception of its reflection in a mirror, whose distance from the observer is known. Such an experiment would fail com- pletely, because no stop-watch could record a short enough VELOCITY 3 time-interval; and even without that objection, no human being has a "reaction-time" constant enough to manipulate a stop-watch with anything like the necessary precision. 2. Velocity. Of the above mentioned seven points of common knowledge about light, the last (in regard to velocity) is of so much interest, and can be so easily discussed without a thorough knowledge of other optical phenomena, that we shall consider it here at some length. It is interesting to note that Galileo actually tried to measure the velocity of light by the method outlined above, except that instead of using a mirror to send back the light (probably none then available were good enough to use over great distances) he stationed two observers with lanterns a great distance apart. Observer number one flashed his lantern, and number two answered by a flash of his own as quickly as possible. Number one then tried to measure the interval of time between his own signal and his perception of the answer- ing signal. Of course no perceptible time-interval was found, and Galileo concluded from this that the velocity of light was too great to measure. Since a velocity is always a distance divided by a time, Galileo's failure shows that in order to measure so great a velocity we may proceed in one of three possible ways. First, we may choose a distance so great that, in spite of the great velocity to be measured, the interval of time will be large enough to measure conveniently by ordinary methods. Second, it might be possible to get a direct comparison between the velocity of light and some known velocity (such as that of the earth in its, orbit) which, although much smaller, is yet far greater than that of anything we can handle in the labora- tory. Third, we may return to the principle of Galileo's method, with a relatively short distance (say a few miles) and correspondingly small time-interval, if we use a mirror to re- turn the light and find some very refined method far measur- ing an exceedingly short time. The last method would have this advantage over the other two, that since the distance concerned is not excessive, it might be possible to measure the velocity, not only in air or in free space, but also in water and other transparent materials. 4 LIGHT It is a matter of historical fact that each of the three possibilities suggested above has been successfully carried out, the first ! by the Danish astronomer Roemer, in 1676, the second by the English astronomer Bradley, in 1728, and the third by two French physicists, Fizeau and Foucault, in 1849 and 1850 respectively. 3. Roemer 's Method. The planet Jupiter, like the earth, revolves about the sun in a nearly circular orbit, but its orbital radius is so much larger that it takes nearly twelve of our years to complete the circuit. It has several satellites, similar to our moon, one of which circles the planet in about 11 hours. Once in every revolution, it enters the shadow of Jupiter and, since it is not a luminous body, but can be seen only by virtue of the sun's light, it then disappears for a short time. The interval of time between two successive eclipses is called the period. We would naturally expect the period to be constant, but it was long known that it seems to vary, accord- ing to the relative positions of Jupiter and the earth. In figure 1, the larger circle represents the orbit of Jupiter, the ^+.~ -^ smaller that of the earth, with the /' sun at the center of both. (Actu- / \ ally each orbit is an ellipse, with "V B^ \ the sun at one focus.) Suppose I J & ( V c? *E, 1 that at a given time the earth is I \ &^..'' i at Ej and Jupiter at J 1? the two ** N ' being in line with the sun. A / \ / little more than six months later, NV V^ ^/ they will again be in line, with the sun between them, the earth at E 2 , Jupiter at J 2 , for Jupiter moves more slowly in its orbit than does the earth. Again, at a still later time, the earth will be at E 3 and Jupiter at J 3 , the former having made something more than a complete circuit, while the latter has travelled only through the arc J^Js- Evidently there are times, as at A, when we are receding from Jupiter, and other times, as at B, when we are approach- ing him. It was noticed that when the earth is receding from Jupiter the period of the satellite seems 1 to be longer, when it is approaching him shorter, than the average. Thus if, BRADLEY 'S METHOD 5 when the earth is in such a position as E x , with Jupiter at J\, a complete schedule of satellite eclipses be made out in ad- vance, on the supposition that they occur with a regular period, it will be found that they appear more and more behind schedule time, till the earth and Jupiter are in the positions E 2 J 2 , and then begin to pick up till they are again actually on schedule time, when the earth is at E 3 and Jupiter at J 3 . Roemer saw that this phenomenon could be explained perfectly by supposing that the eclipses occur at perfectly regular intervals, provided that a finite time is required for the light which brings us the news of an eclipse to travel the very great distances involved. For when we are moving away from Jupiter, as at the position A, each succeeding eclipse is announced to us by light that must travel a somewhat greater distance, and therefore the apparent period would be increased by the time required for the light to travel the additional distance. On the basis of a schedule of eclipses, such as was described in the previous paragraph, it is found that the eclipses observed when the earth and Jupiter are on opposite sides of the sun seem to be about 16.6 minutes behind the schedule. According to Roemer 's views, this would indicate that it takes that much time for light to cross the earth's orbit. Since the mean radius of the orbit is about 92.8 X 10 -_------ -- B Figure 7 far as M,. If the latter had been at rest, the light would have continued to retrace its path as far as the half-silvered mirror mm. Part of it would then penetrate mm and come to a focus at f x , forming there an image that could be seen by the eye at E, aided perhaps by an eye-lens 1. It will be noticed that this returning light would act exactly as if it came from the point I a instead of from M 2 . In fact, I t is the "image by reflection" of the point M 2 , formed by the mirror M t . According to the laws of reflection in a plane mirror, which will be taken up in detail later, the angles A 1 M 1 M 2 and AJM^I! are equal, A l being the point where the plane of the mirror M t cuts the circle. Since the rays striking the lens would appear to come from I,, the image f x would be the ; ' conjugate focus" of I x . According to the law of lenses, the 12 LIGHT . line I 1 f 1 passes through the center of the lens, and f x is at such a distance that + = ! Lfj, LI t F where F is what we call the "focal length" of the lens L. (section 31) In point of fact, however, M instead of being at rest, is rotating in the direction shown by the arrow. Consider the light that starts from it toward M, when M t is in the position indicated. When this light returns to M t the latter will have turned through a small angle a, so that its plane will now intersect the large circle in the point A 2 . Therefore it will reflect the light returned from M 2 as if it came, not from I , but from a new point I 2 , such that A 2 M 1 I 2 = A 2 M 1 M 2 . After reflection then, the lens L will bring it to focus, not at f 1? but at the new point f 2 , such that the straight line I 2 f 2 passes through the center of L. The eye would see an image of the slit at f x if the mirror were at rest, at f 2 if it were rotating. In the latter case, the light would not be really steady, but consist of a series of flashes; but, since the flashes come much more rapidly than 20 per second, it would to all appearance be steady. If the mirror were started from rest and gradually picked up speed, the image would first appear at f t and gradually move away, but would seem perfectly still as long as the speed of the mirror were steady. Evidently, the distance f x f 2 depends upon the velocity of light, together with the distance between the two mirrors M t and M 2 , the speed of rotation of the mirror M 1? and the focal length F; consequently we should be able to get the velocity of light if these other quantities are known. The small dis- tance fjf, is measured with a micrometer (see section 37), the large distance M t M 2 by steel tapes, or by surveyors' methods, and the speed of rotation of the mirror by a special revolution- counter and. stop-watch, or some equivalent mechanical device. The focal length F is supposed to be known, or it can be found by methods to be described later. In order to derive the formula for finding the velocity of light, c, we shall let R = the distance IV^IVLj, d the distance THE ROTATING MIRROR METHOD 13 f t f 2 , t = the time required for light to travel the distance 2R from Mj to M 2 and back, which is also the time required for the mirror to turn through the angle a, and n the number of revolutions per second of the turning mirror. Then the velocity of light is 2R c= (1) L and the angular velocity of the mirror, in radians per second, is 2,rn = - (2) t Since, by the laws of reflection, the angle A 2 M a M 2 = A^IJ.,, and AJNIjM., = A.MJ,, therefore LM^ = 2a. In the actual experiment, the lens L is placed very close to M 1? not more than a few feet away, while the distance I^Ij I 2 M,. = R is quite large, say several hundred meters. Therefore, very nearly, the angle IXI, = IMJ.^ = 2a, and the opposite angle f x Lf 2 is also approximately equal to 2a. With I x and I 2 so far from the lens, the images f 1 and f 2 come practically at the focal distance from the lens, that is f x L = f,L = F. There- fore. in radian measure, (3) Now, from equations (1), (2), and (3) we can eliminate t and a and we get c = 87rnRF/d from which the value of c can be computed, as soon as n, R, F, and d are measured. Professor A. A. Michelson, of the University of Chicago, has made a number of improvements in the details of Fou- cault's method, but has not altered the principles involved. .Measurements of the velocity of light, obtained by ex- perimental methods, vary from. 298,000 to 301,382 kilometers per second. It is usually regarded as sufficiently accurate for all purposes to take the round figure 300,000, or, when ex- pressed in centimeters per second, 30,000,000,000 = 3 X 10 10 . 14 LIGHT Problems. 1. The star Sirius is 5 X 10 13 miles away from us. How many years are required for its light to reach us? 2. Calculate the time required for light to travel four miles and return after being reflected by a mirror. 3. What is the maximum angle by which, owing to aberration, a star may seem to be displaced from its true position ? 4. Derive the formula applying to Fizeau's toothed- wheel method for finding the velocity of light. 5. Referring to figure 1, at what) positions of the earth does the observed period of the satellite eclipses seem longest and shortest respectively? 6. It frequently happensi that the moon passes between the earth and a star (occultation of the star). What would be the effect upon this phenomenon if red light travelled faster than blue, in the space between moon and earth? 7. The "parallax" of a star is the angle which the radius of the earth's orbit. 92.8 X 10 6 miles, subtends as seen from the star. A "parsec" is the distance of a star whose parallax is one second of arc. Find its value in miles, and in "light- years," the distance light travels in a year. CHAPTER II. 7. Refraction through a prism. 8. Newton's conception of color. 9. Impure colors. 10. Color due to absorption. 11. Color due to other causes. 12. Black and white. 13. Complementary colors and color mixture. 14. The eye. 15. Color vision theories. 7. Refraction through a prism. Of the earlier physicists, the one who made greatest progress in the study of light was Sir Isaac Newton. It must be admitted that he was led to believe in certain hypotheses which have since been discarded, but in spite of that fact he accumulated, by experimental methods, a large amount of needed definite information; and his philosophical discussion helped greatly in the development of the theory that later supplanted his own faulty one. Newton was the first to get a clear idea of color, which he | attained through a study of glass prisms. Everyone knows w that a prism of any transparent substance not only bends rays of light, but also makes a beam of white light to show color on the edges. Thus, let W in figure 8 represent a window, through which white light enters a room, passing through the prism and entering the eye placed at E general way the course of the rays. Because we judge the position of an object by the direction of the rays as they enter our eyes, the window appears to be displaced from its true position to some such place as W. But, more than this, the window appears white only in the middle. That edge of it which, as seen through the prism, is nearest to its proper position, is red, the other edge violet. Newton saw that this experiment indicates white light to be a composite of many colors, the color effect at the edges being a result of some property of the prism which causes it to bend, or refract, some of these component colors more than others; for instance, the (15) r \v ,r The Figure 8 arrows indicate in a we judge 16 LIGHT violet more than the red, and other colors to an intermediate degree. According to this hypothesis, the eye would see a red image of the window, as indicated by the rectangle rrrr in figure 9, or as shown in the plan of figure 8 by rr; while, slightly displaced from it, would be seen a violet image (vvw in figure 9, rv in figure 8). Any other color, such as green, would also form an image of the window, displaced less than the violet but more than the red. (Note that, owing to a certain distorting action of the prism, which we shall not here attempt to explain, the vertical edges of the images appear not straight, but curved, as shown by the dotted lines of figure 9). Now, if we bear in mind that there exist in the white light, not three colors, nor only seven, but an infinite number of gradations shading into one another, each of which produces its own image of the window, it is easy to see that all of them will overlap in the middle, so that this part will be white, just like the light as it enters the window. But on passing from the middle toward one edge, we find first the violet missing, then the colors nearest to violet (blue-violet, blue, etc.), until finally, at the extreme edge, only the red is present. On the other hand, passing from the middle toward the other edge, first the red is missing, then the intermediate colors, and at the extreme edge only the violet remains. It is clear that only the extreme colors, red and violet, are seen pure, that is, un- mixed with other colors, because all the intermediate ones over- lap. But it is also evident that the overlapping would be very much reduced if, instead of a wide window, a very narrow slit were used for the admission of the light. Since it is im- possible to use an infinitely narrow slit, there will still be a small amount of overlapping of the images produced by shades of color very close to one another, but none at all in the case of distinctly different colors. This can be understood clearly if the reader will imagine each of the rectangles of figure 9, rrrr, gggg, vvvv, etc., to be made much narrower, without changing the distance between their centers. Any person possessing a prism can try this experiment for himself, by allowing light to stream through the crack in NEWTON'S COLOR EXPERIMENTS 17 a door left slightly ajar, and viewing the crack through the prism held before the eye as in figure 8, with the refracting edge vertical. A band of color will be seen, shading from violet at one edge, through blue, green, yellow, and orange, to red, at the other. The crack in the door acts as a slit, and if this be narrow enough very little overlapping will occur and white will nowhere be seen. Newton's procedure was really somewhat different from the experiment outlined above. He allowed a beam of light direct from the sun to pass through a small hole O in a shut- ter (figure 10) and then through a prism P, which deflected it toward the white screen S. He could have placed his eye at the point E, and by looking into the prism, seen the colored band in the appar- e n t position r'v', since the red light would then have entered his eve as V'' Figure 10 if it came from r', the violet as if it came from v', and the intermediate colors as if they origi- nated at points intermediate between r' and v'. Instead of doing this, he allowed the light to proceed to the white screen, forming a red spot at r, a violet spot at v, etc. Since; the pencil of light coming from the sun through a small hole is rather narrow, including an angle of only about one-half de- gree, there was not much overlapping of the colors, and the whole colored band showed the intermediate, as well as the extreme colors, fairly pure. Newton called this band of color a spectrum, and in technical language it is further defined as a real spectrum because the light actually passes through it, or at least to it, as distinguished from the so-called virtual spectrum, seen in the apparent position rV when one looks into the prism. The light does not actually pass through rV, but merely enters the eye as if it came from there. There is much less overlapping of colors in the virtual than in the real spectrum, -in Newton's experiment, that is, the former is more 18 LIGHT pure. The overlapping in the virtual spectrum can be almost entirely eliminated by making the hole through which the light is admitted very small. The width of the spectrum can be increased by replacing the round hole with a narrow slit, and the overlapping of colors in the real spectrum can be much reduced by the inser- tion, either before or behind the prism, of a lens of suitable focal length, so placed, that each of the colors is brought to a focus on the screen. Under these circumstances, it is possible to regard the real spectrum as made up of an infinite number of images of the slit, side by side, the color of each image being slightly, though imperceptibly, different from that of the image next to it. If the light coming through the slit contains (ivery conceivable gradation of color, as is the case with light coming from an ordinary electric lamp, there will be no gaps in the spectrum. In the case of sunlight there are certain missing shades, and these defects are made evident, if the slit is very narrow and the focussing very good, by certain gaps, or "black lines" across the spectrum, in the positions of those images of the slit which would be supplied by the missing colors if they were only present, (see section 56) 8. Newton's conception of color. Newton's conception of the formation of the spectrum, then, was that the differ- ent colors are already present in the white light, and the prism serves only as a separa- tor. This view was direct- ly opposed to that held by some others, that the prism in some way modi- fies the light so as to change it from white to Fi gur e 11 colored. According to Newton's ideas, if a second prism be introduced behind the first one, but with its refracting edge turned in the opposite direction, as in figure 11, this should reunite the colors into a white beam again. A trial shows that this actually occurs, provided the second prism is of the same kind of glass, and has the same angle. Another test of Newton's theory is this: IMPUBE COLORS 19 If we could pass through a prism only light of a single spectral tint, for instance deep red, or a definite shade of any other color, the prism should simply bend it, and not separate it into more colors. This can be tried by the arrangement shown in figure 12. The white screen, S of figure 10, is re- placed by a screen in which there is a narrow slit, through which any single portion o f the spectrum can be passed to a sec- Figure 12 ond prism. It is found that the second prism does bend the light, but does not spread it out into any more colors; that is, if only red light enters it, only red light leaves it, and that without being spread out to any appreciable extent. For these experimental reasons, we shall accept Newton's view of color as being the correct one.* 9. Impure colors- There is one fact about the spectrum which can hardly fail to strike one who examines it closely, viz., that we fail to find in it certain colors which are more or less common in nature. For instance, there is no purple, and though there are several shades of red, there is none that could be called pink. These two cases are good illustrations of the general fact that most of the colors of nature are not ''pure colors, " in the sense of spectral colors that cannot be further separated by a prism, but are mixtures of two or more of these latter. Purple, for example, is a mixture of red with blue or violet or both. This fact can be shown by placing two small *Certain modern investigations show that the contention of New- ton's opponents, that a prism actually manufactures the different colors from white light instead of merely separating out constituents that are already present, is in a certain sense true. This is embodied in what is called the pulse-theory of white light. But, after all is said, this differs from Newton's theory only in the point of view, and tht latter not only explains all the phenomena in a satisfactory manner, but is much easier to deal with. Therefore we may accept it as true in the pragmatic sense. 20 LIGHT mirrors in the spectrum, one in the deep red part, the other in the blue or violet part, turning them so that each reflects to the same spot on a white screen. This spot, illuminated by both red and, blue or violet light, appears purple. Purple, then is a sensation produced when both red and blue or violet light fall upon the retina of the eye. We can get a pink spot by illuminating a white screen simultaneously with white light and red light. Consequently, pink is produced by a combination of all the spectral colors, with a considerable excess of red. Similarly, pale blue is a combination of all the colors with an excess of blue, pale green a similar combination with excess of green, etc. In technical language, any such combination of one or more definite spec- tral colors with white is said to be unsaturated. Thus, pink is unsaturated red, pale blue is unsaturated blue, etc. That the pigments used ordinarily in painting are very impure colors, can be demonstrated by a very simple experi- ment. Take a strip of paper about % 2 i nc ^ wide, and color it in quarter-inch lengths with the following artists' pigments, each pair of colors being separated by a short length of white : alizarin crimson , alizarin crimson mixed with gam- boge, gamboge alone, gamboge mixed with prussian blue, new blue, and alizarin crimson mixed with new blue. The strip will then appear as a very narrow ribbon showing the follow- ing succession of colors : red, white, orange, white, yellow, white, green, white, blue, white, violet. Now lay it against a dull black background, such as a piece of black felt, illumi- nate it with sunlight, and look at it through a prism whose edges are parallel to the strip. Each of the white portions forms a complete spectrum, with which the spectra of the painted portions can be compared. It will be noted that each of them shows through the prism not only the color which it appears to have when viewed directly, but certain other parts of the spectrum. Not one of them is a pure color, but at best each shows only a strong excess of the color it is meant to have. The most nearly pure of all is the red part, but even it shows quite a little green, with traces of the other colors. A narrow blade of grass, when observed through a prism in such a manner, shows, beside strong green, a great deal of red and yellow, and even some blue and violet. COLOR DUE TO ABSORPTION 21 10. Color due to absorption. Naturally we are led to enquire: why is a blade of grass green, or the petal of a rose red? 'Since grass is not itself luminous, but is seen only be- cause it reflects diffusely the sunlight that falls upon it, its green color, or rather its very mixed color with green predomi- nating, must arise from the fact that some of the chemical substances in the grass have the property of "absorbing," to a greater or less extent, certain colors, or as we often express it, certain parts of the spectrum. For instance, these con- stituents of the grass either do not absorb green at all, or more likely simply absorb it less than they absorb red and yellow, and still less than they absorb blue and violet. But absorption, in the proper sense of the word, can occur only while the light is actually passing through a material, not in the mere act of reflection at the surface. Consequently, it must be that the light penetrates the surface to some appreciable depth, that is, the substance of the grass is to some extent transparent. That this is true, can be readily proved by holding a blade of grass between the eye and a bright source of light. A con- siderable fraction of the incident light passes entirely through the grass to the eye. Indeed, it may truly be said that any material is to some extent transparent, and if made into a thin enough sheet will allow an appreciable amount of light to pass through it. But the farther the light is caused to pass through a material the more of its energy is absorbed. Evidently, what happens in the case of the blade of grass is something like this : Of the white light that strikes the sur- face, part is reflected without penetrating, as would be the case with glass or water; and this part, if it could be seen alone, would be white, like the incident light. The rest of the incident light penetrates the surface; but since the material is not completely transparent, but only what we call translucent, the rays do not pass straight through to the back surface, but are diffused, or scattered, within; the material. In this way part of the light eventually gets back into the air through the front surface, and we call this part "diffusely reflected" light, although it has been within the body of the material. Part also gets out through the back surface, and we call this "transmitted" light. Both the diffusely reflected, and the transmitted, light during its passage through the material, 22 LIGHT suffers losses by absorption in the chemical substances of the leaf. Part of the green is absorbed, more of the yellow, orange, and red, and most of the blue and violet. In this way, both the reflected and the transmitted light become colored, and indeed both parts show about the same color. Most natural objects owe their colors to the same cause. Part of the light penetrates the surface, and part of this emerges again, after suffering absorption. This explanation is satisfactory so far as it goes, but it must not be forgotten that we do not know why, for instance, the leaf-substance absorbs more red than green, while just the reverse is true of the coloring-material known as alizarin-crimson. This question cannot be answered without a far greater knowledge of atomic structure than we now have. 11. Color due to other causes. There are some objects whose color is produced in a different way. The yellow color of gold, as an example, is mostly due to the fact that a certain shade of yellow light seems unable to enter the gold at all. It is completely reflected at the surface. Consequently, a very thin sheet of gold-leaf transmits light which is completely lacking in this color. The transmitted light is dull green, while the reflected light is yellowish. A similar phenomenon occurs in the case of some dyes, which in concentrated form show quite a different color according as they are seen by transmitted or by reflected light. Common red ink is an example; it reflects green very strongly when concentrated, but transmits red. The brilliant colors of rainbows are due, not at all to absorption, but to a separation of the colors something like that which occurs in a prism. The theory of rainbows will be taken up later, (section 83). The blue of the sky is caused by a sort of scattering of the light by the particles of the atmosphere, similar to the scattering produced when a beam of light is sent through milky water. If there were no atmosphere the sky would appear black, and we would receive light only from the sun, moon, and stars directly. In the scattered light from the sky, all colors of the spectrum are represented, with blue in excess. In order to explain this preponderance of blue, we must anticipate to some extent facts that properly come later in BLACK AND WHITE 23 these pages. It will be shown in the next chapter that light consists of waves, the shortest of which are the violet, the longest the red, the intermediate colors having waves of inter- mediate length. Although all these waves are very short, the length of even the shortest" is much greater than any of the dimensions of a molecule. Since the molecules are so small, they are much more efficient in reflecting, or scattering, short waves than longer ones, just as small pieces of wood floating on the surface of water will reflect short ripples, but simply ride on the very long waves. Therefore the molecules in the air reflect scattering in all directions those colors that lie near the violet end of the spectrum to a considerably greater degree than those near the red end, thus giving a bluish color to this scattered light. That it appears blue rather than violet is because the violet is at best very weak. Of course, since a beam of direct light from the sun is thus robbed of a greater percentage of its blue and violet than of its red, that part of it which passes on through, must be abnormally rich in red relatively speaking and therefore / must appear more reddish in color than it was when it emerged from the sun. This is particularly marked when the light has passed through a long distance in air before reaching the eye, as is the case near sunrise or sunset. That the light at sucix times is exceptionally impoverished in blue and violet, is well known to every photographer, for most of the photographic action of light is produced by these colors, and a plate must be exposed several times as long when the sun is low in the sky as at midday. However, the reddish color of the sun when it is near the horizon is familiar to all. 12. Black and white. A black object, strictly speaking, is one which absorbs completely all colors, and reflects none. But an object may appear black simply because the light which illuminates it contains no constituent save those which it absorbs completely. For example, a deep red rose will ap- pear black when placed in the blue or violet part of the spec- trum, because it absorbs these colors completely, and the only color which it can reflect freely, red, is not present. A -ibhite object is one which reflects diffusely, that is ii? all directions, all colors to the same extent. The whiteness of the snow is an interesting case. Snow is really composed of 24 LIGHT numerous little ice crystals, and ice in bulk is not a white body, but ai transparent one. That is, a large chunk of ice allows most of the light which falls upon it to pass, through, and that part which it reflects is reflected, not diffusely, but in a definite direction. But when we have a great mass of very small ice crystals, arranged in an irregular manner, although each crystal surface reflects in a particular direction, the whole mass reflects about as much in one direction as in another. Moreover, the light that passes through the crystals on top of the layer of snow will strike other surfaces below, which again cause reflection, part of the reflected light finding its way out of the mass again. Thus the whole mass reflects irregularly a very large amount of the light, and, since no color is absorbed by the material, this reflected light is white in color. There- fore, the whiteness of snow is due to the great number of re- flecting surfaces, irregularly arranged. The same effect can be produced by crushing a piece of glass with a hammer. The many cracks in the glass cause a multitude of reflecting sur- faces, arranged irregularly, and the mass immediately becomes white. The whiteness of clouds is also due to numberless little reflecting surfaces, the surfaces of millions of small water- drops. Of course, a white object will not appear white unless the light which falls upon it contains all the colors of the spectrum, i. e., is white light. If it be illuminated by red light it will appear red, if by blue light, blue, etc. Thus, the white screens of figures 10 and 11 show whatever color falls upon them, and it is just this property that makes a white screen suitable for such experiments. 13. Complementary colors and color mixture. Any two colors which together produce the sensation of white are called complementary colors. A convenient way of showing these is illustrated in figure 13. White light passes through the slit S to the lens L, which makes the rays parallel. It then passes through the prism P and the second lens L 2 , which' focusses the spectrum in the plane vr. Instead of having a screen at this place, another lens, L 3 , is placed just behind it. The two lenses L 2 and L 3 together form, an image of the face of the prism on a properly placed white screen A. Since light of all colors comes through the whole face of the prism, and all the COLOR 1VIIXTURE 25 light passes through the two lenses, this image will be uniform- ly white. Now, if an opaque obstacle be placed just in front of L,. i. e., just in the plane in which, the spectrum is formed, the obstacle being of sufficient width to cut off a certain spec- tral region, say the green, then only the remaining colors will reach the screen A, and the image of the prism-face will therefore show the color complementary to the color cut out. Figure 13 By this means, we find that the color complementary to the average spectral green is a peculiar shade of red, that com- plementary to spectral blue a sort of golden yellow, etc. Generally, one or both of a pair of complementary colors are impure in the spectral sense. It is found that a combination of the red, the green, and / the-Jalue of the spectrum, in suitable proportions, will produce the sensation of white, without the presence of the other colors, such as violet, orange, and yellow. This can be tried with the arrangement of figure 13 by placing before the lens L 3 a card with holes cut through it so as to let pass only some of these three colors. The best exact location of the holes, and their proper sizes, can be found only by trial. Furthermore, by suitably altering the relative intensities of these three colors, as by stopping down one or two of the holes, any other color, either a pure spectral hue, or such a color as purple, can be closely imitated. The mixing of pigments shows some results which at first sight are very surprising. For instance, since a combination of all the different colors, of proper proportions, produces white, we should naturally expect that when many different paints are mixed, the mixture would tend to become white. 26 LIGHT On the contrary, it tends to become black; and in general, the more different pigments are put into a mixture, the darker it becomes. The reason is that by mixing we combine the absorbing powers rather than the reflecting powers of the constituents. We have seen that the common pigments, alizarin crimson, gamboge, and new blue, have each a distinctive ab- sorption, and in a mixture of the three any part of the spec- trum would be strongly absorbed by one or another, so that little if any of the incident light would escape absorption. An example less extreme than this is seen in the mixture of the crimson and the blue to produce violet. If it were not for the fact that certain parts of the spectrum escape complete absorp- tion in either of these pigments, the mixture would be black instead of violet. It is, as a matter of fact, extremely dark, much darker than the violet seen in the spectrum from direct sunlight, relative to the brightness of the other colors. It is quite plain then, that the mixing of two or mx>re paints produces quite a different result from throwing simul- taneously upon a white screen lights of the corresponding colors; and this fact seriously limits the ability of artists to produce desired effects by mixing paints. The school of painters known as impressionists introduced a new method. Instead of mixing their paints, they lay them on the canvas in little blotches side by side. Thus, where a painter of the older schools would mix crimson and yellow to paint a surface of orange color, the impressionist covers the surface with dots of crimson and dots of yellow close together but arranged in irregular order. Such a painted surface looks very confusing when viewed at close range, but at a greater distance the blotches of red and yellow seem to blend together to produce the effect of a uniform orange, so that the impressionist secures in this way the same effect that we could get in the laboratory by simultaneously illuminating a white surface with red and yellow light. As a result, paintings by impressionists are usually far more brilliant than those of the old masters, though it is true that the latter have a sombre richness which is itself a great charm. 14. The eye. The organ of vision, the eye, is an optical instrument more analogous to the photographic camera than to THE EYE. 27 anything else. It is shown diagrammatically in figure 14. It consists of a shell roughly spherical in shape, of which the front wall, C. the cornea, is transparent. Behind this is the iris I, a screen or diaphragm con- taining a circular hole, the pupil, P, whose diameter contracts in brilliant illumination or expands in dim light, by involuntary muscular action. The lens L is capable of a slight forward and backward mo- tion, like that of a camera lens in focussing, but most of the focus- sing in the eye is accomplished by altering the radii of the lens surfaces. The material of the lens is of course some- what plastic, and in structure it resembles an onion in being built up in layers. The lens forms an image of any object looked at upon the retina R, which is spread over the rear and side surfaces of the shell. The space between the lens and the retina is filled with a jelly-like material called the vitreous humor. The material between lens and cornea is watery, and is called the aqueous humor. By means of the above mentioned muscular distortion of the lens, together with its slight axial movement, any object toward which the eye is directly turned can be sharply f ocussed upon the retina, provided it is not closer than a few inches, in the case of normal eyes. This ad- justment of focus is called accommodation. With the eye re- laxed, very distant objects should be in sharp focus. In view- ing a very small object, however, it is advantageous to bring it close, so that its image upon the retina may be larger; but if it is brought closer than about 10 inches the muscular strain of accommodation becomes unpleasant and is in fact harmful. Consequently, about 10 inches (25 cm.) is the most favorable distance, with people of normal vision, for reading or for careful scrutiny of small objects. This is known as the " dis- tance of distinct vision." A myopic, or short-sighted, eye is one whose focal length in the relaxed condition is abnormally short, so that it cannot focus sharply upon distant objects. Just the reverse is a hypermetropic eye, which requires a certain amount of accom- 28 LIGHT modation even to focus upon an infinitely distant object. Presbyopia, a trouble particularly common among older people, is an impairment of the ability to accommodate, due to a pro- gressive stiffening of the muscles which change the radii of the lens. Astigmatism is caused by lack of axial symmetry in the lens or the cornea or both. It shows itself in an inability to see clearly, with the same accommodation, lines inclined at different degrees with the vertical, though equally distant. The focus upon the retina is sharp only for a limited region near the axis of the lens, but unsharp vision is possible over a very wide angle without moving the eye. The retina is the sensitive part of the eye, which in some manner is stimulated by light falling upon it so that the mind experiences the sensation of vision. It is a network of delicate nerve-fibers which are connected with the brain through the optic nerve O. 15. Color vision theories. The student should understand that the actual mechanism of vision, the connecting link be- tween the light-stimulus upon the retina and the consciousness of light and color, is a thing about which little is known. Even if we knew the physical and chemical processes that go on in a iierve, there would still be a gap or hiatus in our knowledge between that and the actual sensation. Consequently, our notions of light and color perception do not extend very far, and are somewhat uncertain at that. On the face of things, it seems very unlikely that there is a separate type of nerve for every gradation of color. Such an experiment as the pro- duction of the orange sensation by mixing red and yellow light, or the production of any other spectral sensation by a mix- ture in suitable proportions of red, green and blue, suggests that there are only a few distinct color sensations, perhaps three, and that the other sensations are the result of a simul- taneous stimulation of these few. Experiments with color- blind people also support this view. There are in fact two principal theories of color-sensation, the Young-Helmholtz theory and the Hering theory. The former alone is often given in physics texts, but the latter is favored by at least a great many experimental psychologists, and the matter is one of psychology more than of physics. COLOR- VISION THEORIES. 29 According to the Young-Helmholtz theory, the retina con- tains three distinct sets of . nerve-fibers, each giving only a single sensation, no matter what particular part of the spec- trum corresponds to the light that does the stimulating. One set gives a red sensation, the second a green sensation, the third a violet or blue sensation. The three curves of figure 15, which are due to Koenig, show to what extent each of these Dp. R sensations is stimulated by light from different parts of the spectrum. In order to clearly understand these curves, con- sider how the spectrum would appear to a person whose eyes were provided with the red-sensitive nerves, but not with the other two sets. He would be able to see the spectrum through- out its entire extent, with the possible exception of the ex- treme violet end, but all of it would appear of the same color, red. The only differences between different parts would be differences of brightness, as indicated by the varying ordinates of the red curve; In fact, his retina would act in a way quite analogous to the action of a photographic plate, which responds to the influence of light of many different colors, but with a response which is the same in kind for all, differing only in degree. Now consider an eye with all three sets of nerve-fibers, and suppose the retina to be stimulated by light from the yellow-green portion of the spectrum. A comparison of figure 15 shows that this light stimulates all three of the sensations, the green sensation most strongly, the red sensation to a lessi degree, and the violet sensation least of all. The complex of these three sensations acting together is what we are accustomed to call the yellow-green sensation. About one man in thirty 30 LIGHT is "red-color-blind/ 5 which, on the Young-Helmholtz theory, means that his eyes lack the nerve-fibers which give the sen- sation of red. The Hering theory is quite different. Instead of three primary sensations, it postulates certain contrasts, caused by chemical changes, under the influence of light, in three hypo- thetical fluids present in the retina, which we shall designate as A, B, and C. Fluid A undergoes a certain decomposition when any sort of light, irrespective of color, falls upon, it, but recombines, or recovers, in darkness. It reacts upon the nerve fibers differently in its two states, causing a sensation of bright- ness in the one case and of darkness in the other. Fluid B is different. It undergoes a decomposition under the action of light of longer wavelengths, giving a red sensation, and a recombination under the action of light of shorter wavelengths, giving a green sensation, being entirely neutral for light of wavelength corresponding to some part of the yellow. Fluid C acts in a similar way, but the sensation produced by longer wavelengths is yellow, that by shorter wavelengths blue or violet, and the neutral condition would be for wavelengths in the green. Thus we have three contrasts, bright and dark, red and green, yellow and blue. According to this theory, the usual type of color-blindness is due to lack of the B fluid, resulting in an inability to distinguish reds from greens. Color-blindness is not a disease, but a heritable defect, and though a handicap it is not a thing of which one need be particularly ashamed. Many people who have it are not con- scious of it. Recent biological researches have shown the fol- lowing interesting peculiarities about its inheritance. A color-blind man transmits the defect neither to his sons nor to his daughters, but to the sons of his daughters; that is, it passes from the male of the first generation to the male of the third, through the female of the second, but without showing actively in the second generation at all. A woman is never herself color-blind unless she inherits it both from her father and her mother's father. Consequently, cases of color-blind- ness among women are very rare. COLOR-VISION THEORIES 31 Problems. 1. Suppose that a flower whose color is a pure blue is passed slowly through a spectrum, from one end to the other. What would be its appearance in the different parts? Suppose a blade of grass were treated in the same manner? 2. A story by Ambrose Bierce, entitled "The Damned Thing," has for its subject a supposedly invisible animal. The author argues that such a thing would be possible if the animal's fur reflected only ultraviolet light. What would be the actual appearance of such an animal f 3. Explain the whiteness of soapsuds and other froth. 4. Suppose blue glass were crushed to a powder. What would be the effect upon its color? 5. Explain why tobacco smoke appears blue against a dark, but brown against a bright, background. 6. Why is it that colored cloth can be changed by dyeing to a darker, but not to a brighter color? 7. Birds, animals, and fishes usually have a much lighter color on the lower than on the upper sides of their bodies. Is this fact of any importance in the economy of nature? Explain. CHAPTER III. . 16. The corpuscular theory of light. 17. The wave theory. 18. Bending of light into a shadow. 19. Nature of the ether. 20. Waves in general. Plane waves. 21. Mathematical formula for a wave. 22. Interference. Fresnel's mirrors. 23. Interference in white light. 16. The corpuscular theory of light. We have now learned enough about some of the general properties of light to en- quire with some degree of intelligence as to its nature. The two theories that have had any support may be called the cor- puscular theory and the wave theory. According to the first, light consists of very small weightless material particles; ac- cording to the second, it consists of waves. Either theory strains the imagination greatly. It is hard to think of material corpuscles flying with enormous speed through a solid substance like glass, with so little hindrance as glass seems to offer to the passage of light, though color might well be accounted for by differences in size, in shape, or in some other characteristic among the cor- puscles. It is also extremely difficult to explain how, when these corpuscles strike such a substance as glass or water, some of them should be reflected while others pass into the material, being refracted as they do so. It is true that one might sup- pose that there are two kinds of such particles, a kind that is reflected and a kind that is refracted. But if this were the case, one reflection would completely separate these two kinds, so that if the reflected light struck another such surface all of it would be reflected, none refracted; while if the refracted light struck another surface, all would be refracted, none re- flected. Then if one should observe in a plate glass window the reflected image of his own body, this image would become invisible to him if he held a small piece of glass in front of his eyes. A simple trial shows that this is not true. Further- more, the light that got in through the first surface of the plate glass could not be reflected at all by the second surface. It can easily be proved that this conclusion also is false, for if one stands close to such a window he can distinctly see two images of himself, the brighter image being formed by re- (32) : THE WAVE THEORY 33 ; lection at the first surface, the fainter one at the second. (In reality there is a large number of such images, produced by multiple reflections, but all except the first two are very faint). In order to get around this difficulty, Newton, the chief advo- cate of the corpuscular theory, suggested that, although all the corpuscles are fundamentally alike so far as reflection and refraction are concerned, each one is at times in a state suitable for reflection, at other times in a state suitable for refraction; so that whenever the light strikes a reflecting surface there will be a certain proportion of them ready for reflection, even though some had already been reflected before. But such an hypothesis, though not absolutely absurd, seems clumsy and improbable, and Newton himself was far from being satisfied with it. 17. The wave theory. In the wave theory, there is no difficulty in explaining reflection and refraction. Indeed it is characteristic of all kinds of wave motion that, whenever a wave strikes a surface separating two media in which the velocity of wave propagation is different (such as the surface between air and glass) part of the energy enters the second medium as a refracted wave, while part is sent back into the first as a reflected; wave. Neither is it at all hard to think of waves passing through glass and other transparent media with high velocity and little resistance, for we know that mechanical waves, such as sound waves, do traverse such bodies very easily. Color, also, may be accounted for very simply on the wave theory, by the supposition that differences in color correspond to differences in the length of the waves, just as we know that differences in the pitch of musical notes correspond to differences in the lengths of the sound waves. But it is difficult to understand how waves can pass, as we know that light does, through perfectly empty space, for the use of the names /'wave" implies the existence of some medium in which the waves exist. On account of this difficulty with the wave theory, physicists have been led to assume the existence of a medium filling all space, even a so-called vacuum, to which the name ether has been given. The necessity for this assumption is to this day a very serious load on the shoulders of the wave theory of light, though it becomes less 34 LIGHT objectionable when we find that there are other phenomena ; such as electric and magnetic attractions and repulsions, which also operate through a vacuum, and also seem to indicate the existence of some all-pervading medium. . Newton's chief objection to the wave theory, however, was not the necessity for an ether, but the fact that light apparently travels in straight lines, while other waves, such as sound waves, or waves on the surface of water, will bend freely round an obstacle placed in their path. In this con- nection, it should be noted that the degree to which waves bend round an obstacle, thus departing from the straight-line path, depends to a great extent upon the length of the waves. Long waves do so much more freely than short waves. There- fore, if it can be shown that light really does to some slight extent bend round a corner, this objection will be overcome, and at the same time evidence will be acquired that if light does consist of waves these waves are very short. 18. Bending of light into a shadow. In fact, the bending of light about an obstacle can be shown, by a very simple experiment illustrated in figure 16. represents any opaque obstacle with a straight edge perpendicular to the plane of the paper. The source of light, S, must be of very small dimensions ; otherwise the bending of the light into the shadow, which the ex- periment is designed to show, will be masked by the pe- numbral effect always shown when a shadow is cast by a fairly large source of light. It should be either a very fine hole, or better still a very narrow slit, illuminated by an arc light or other brilliant illuminant. A is a white screen, to receive the shadow. Let a straight line SQ be drawn from the slit through the edge of the obstacle to the screen. Then if light were absolutely rectilinear, all parts of the screen above Q would be fully illuminated, all points below in com- plete shadow. There would be a sharp and abrupt division between the illuminated part and the shadow. BENDING OF LIGHT INTO SHADOW 35 But, as a matter of fact, the light is found to shade off continuously, though rather rapidly, into the shadow; and above, the point Q there are a number of bright and dark bands, parallel to the slit and to the edge of the obstacle. The gradual shading off of the light, merging into the shadow, dis- proves the rectilinear propagation of light, except as an ap- proximation to the truth. As to the bands, it will merely be noted here that they are readily explainable on the wave theory, the actual explanation being deferred to later pages. (See section 70) Probably the crucial reason for the discarding of the corpuscular theory and the definite adoption of the wave theory is the 'following. When a beam of light strikes the surface of glass or water at an angle, it is bent toward the normal, that is, it makes a more acute angle with the perpen- dicular within the glass than in the air. In order to explain this, the advocates of the corpuscular theory were obliged to assume* that the corpuscles are attracted by the glass when they get very close to it, leading to the conclusion that they travel faster in the glass than in the air. On the other hand, the wave theory explains this bending of the rays at the sur- face very easily with the assumption that the waves travel slower in glass or water than in air. Thus the phenomenon of refraction furnishes the occasion for a definite clash between the two theories, one demanding that light travels faster in the refracting medium than in air, the other that it travels slower. During Newton's life, no means of actually measur- ing the velocity of light in such a medium as glass or wa/ter 5 was( known: but after Foucault had devised the rotating mirror method, the velocity in water was measured by filling a long tube, fitted with glass ends, with water, and inserting this in the path of the light. The experiment showed without any possibility of doubt that light travels slower in water than in air. Since this experiment definitely discredits the corpuscular theory, and since the only outstanding objection to the wave theory is the hypothesis of the ether, for whose existence we have additional evidence from electric and magnetic phenom- ena, it is now accepted by physicists that light consists of 36 LIGHT very short waves in a hypothetical medium extending through all space. Even when light passes through a material like glass or water, we regard the ether as the carrier of the waves. "We may think of the molecules of the glass as existing in the ether. Their presence modifies the transmission of the waves, partly by changing the wave-velocity, partly by absorbing part of the wave-energy, which they change into heat-energy, for heat is always produced when light is absorbed. A rough image of the state of affairs may be gotten by considering waves on the surface of water on which are floating many pieces of wood. The water represents the ether, the pieces of wood the atoms or molecules of the material substance. ' 19. Nature of the ether. It is difficult to speculate about the nature of the ether. There aret no reasons for believing that it is atomic or molecular in structure, and it seems to offer absolutely no resistance to the passage of bodies through it. A philosophically minded person might ask the question: Just what do we mean when we say there is an ether! Such a question is worth while because it leads us to take stock of our knowledge; but probably the only answer that can be given i is this; if the wave theory of light is true, (and it is supported by too many facts now for us to doubt it) then the statement that there is an ether means only that empty space has properties other than mere extension, properties that enable disturbances carrying energy to pass through it, the passage requiring finite/ time. Whether we say there is an ether, or that empty space has properties other than those of pure geometry, matters little, but the name "ether" is a convenient one to symbolize these properties, and we shall hereafter use it in this sense. We shall see that it is comparatively easy to devise ex- periments for measuring the wavelength of light, and that the measurements can be carried out to a very high degree of precision. (Chapter Yir and IX) It will be easy to explain the laws of refraction from the fact mentioned above, that the velocity of light is less through such a substance as glass than through air. somewhat less through air than in the free ether. (Chapter IV) "We can explain the formation of a spectrum by a prism, by showing that through glass the velocity of shorter- waves is less than that of longer ones. WAVES IN GENERAL 37 But when all this done, it must not be forgotten that we shall still be much in the dark as to many important, questions. In the first place, are these waves longitudinal, like sound waves, or transverse, like those of a plucked string? Are they purely mechanical waves, or do they consist of rapidly alter- nating electrical disturbances, or are they disturbances of a kind unrelated to any other physical phenomena? What is the nature of the phenomena going on within the atoms of a substance which is emitting light, or in those of a substance which is absorbing it? How and why does the presence of material molecules, in such a substance as glass or water, change the velocity of the waves in the ether surrounding and per- meating them? Some of these questions have been solved satisfactorily, some have been only partially solved, others are still open. It is clear that if we are ever able to answer all of them we shall know a great deal about the inner structure of tHe atom and the molecule; and since the acquisition of such information is one of the principal aims of physical research, the study of light becomes one of the most important branches of science. 20. Waves in general. Plane waves. Before going on with the study of light as a wave-motion, we shall devote some space to the consideration of waves in general. We find cases of waves which advance (1) in only a single dimension, like those that travel along a stretched rope if it is struck or moved in any way. (2) in two dimensions (i. e., spreading over a sur- face) like water waves, and (3) in three dimensions, like sound waves, light waves, or the waves that would spread through a block of jelly if some point in the interior were set vibrating. In the last case, it is clear that the waves would spread out in spheres with the point of origin as 1 center, the direction of advance being along the radii. Under certain circumstances, however, we could have cases where they advance in planes, the direction of advance being perpendicular to these planes. There would be an approximation to this condition, for exam- ple, at a very great distance from the point of origin, for a small section of a sphere of very great radius is nearly plane; but such waves would be feeble because the energy initially given to them would be spread over a great surface. In the case of light, as we shall see in Chapter V, such plane waves 38 LIGHT can be produced by the use of mirrors or lenses, without such a weakening of intensity. The nature of plane waves may be understood by imagining a block of jelly as in figure 17, to one face of which is attached a rigid board that is moved har- monically back and forth in its own plane, in the direction of the Figure 17 line AB. At any instant, each point in a plane parallel to the board will be in the same condition of motion, or, as we say, in the same phase. lln general, any such surface, every point of which is in the same phase, is called a wave-front, whether it is at the begin- ning of a train of waves or not. In figure 17, every plane in the jelly parallel to the board is a wave-front,- and in the case of waves coming from a point-source, every sphere with its center at the source is a wave-front. We may have cases of a single wave, like the sound-wave sent out from an explosion, and also cases where there is a train of waves, such as those sent out by a tuning-fork. The shape of the waves may be simple or complicated. For any medium through which waves of all lengths travel with the same velocity (as is true for the free ether with light-waves) a disturbance of any kind will travel onward without changing its shape, whatever that shape may be. For example, by giv- ing the board of figure 17 a suitable motion, waves of the form shown in figure 18 could be made to travel through the jelly. A mathematical theorem due to Fourier proves that any such periodic form can be made up of a series of simple sine Figure 18 and cosine forms, of different wavelengths. For this reason, we are compelled to make a special study of waves of these simple forms. FORMULA FOR A WAVE 39 Figure 19 21. Mathematical formula for a wave. In figure 19, A represents a sine-curve, whose equation is ir - 2?rX /i\ y = K.sm (1) B a cosine-curve, whose equation is TT y = K.cos a and C a third curve whose equation is y = K.COK (x -. a) a (2) (3) / Evidently, all three are exactly the same in form, and either can be transformed into one of the others by shifting it along the axis of x. In fact, we- can represent either of the three curves by the formula (3) provided a is given a suitable value, different in each case. If a = 0, we have the simple cosine curve. If a a/4 we have T2, 27T f a\ /27TX 7T\ 2?rX y = K.cos (x - = K.COS --- - ) K.sm a 4 / \ a 2 / a the equation of the sine curve. Whatever value a may have, its presence indicates that equation (3) represents a simple cosine curve shifted in the positive direction of x by the dis- tance a. For y has the same value for x in (3) that it has for x a in (2). Equations (1), (2), and (3) do not represent waves, but only stationary curves. For such a curve to become a wave, it must progress steadily to the right (or left). Since the symbol a indicates a shift to the right, (3) can be changed to represent a wave instead of a stationary curve if a is re- 40 LIGHT placed by a term containing the time, which term indicates the movement. If t represents the time and V the velocity of the waves, then in t seconds the curve must be shifted along the x axis a distance Vt. Therefore, putting Vt for a, the wave equation is y K.cos- (x Vt) (4) a a is called the wavelength, for it is evident from figure 19 that it is the distance from crest to crest, or from trough to trough. The quantity K, called the amplitude, gives the maximum value y can have. Equation (4) is an equation of three varia- bles, y, x, t. x and t are called independent variables, y the dependent variable. The equation gives the value of y for any stated distance x from the origin and for' any stated time t. It is often desirable to introduce another constant term, c, within the parenthesis, making the equation read y = K.cos ( x Vt e) (5) a The only difference between (4) and (5) is that the former represents a wave which, at time zero, has the position B of figure 19, while the latter is one which at that instant has the position A. B, C. or any other, depending upon the value of e. This quantity e is called the pilose-constant , the whole quan- tity whose cosine is to be taken, viz., a being known as the phase. A wave advancing to the left would be represented by .equation (4) or (5) with the sign of V changed; thus, y = K.cos 2 -( x + Vt c) a So far we have regarded y as a real mechanical displace- ment, at right angles to the direction in which the waves are propagated. In the case of transverse waves in a string, it is indeed just that. The same thing is true of the mechanical waves in the block of jelly, illustrated in figure 17, and in all such eases the curves of figure 19 give a true picture of the contour of the waves at different times. INTERFERENCE 41 But there are cases of wave-motion ins which this is not the case. For example, if a stiff rod or a stretched metal wire be stroked longitudinally with a rosined piece of leather, longitudinal waves are set up in which the displacement is parallel to the direction of propagation of the waves. In such a case we may still, for convenience, plot y at right angles to x, but the graph so obtained is only a graph, and not a true picture of the wave contour. It may be that a wave does not even consist of mechanical vibrations at all. There are cases, for instance, of temperature waves, in which alternations of temperature, above and below a mean value, are sent through a material. It is quite con- ceivable also that waves should exist which consist of electric disturbances, for instance regions in which there is an electric intensity directed upward, separated by those in which it is directed downward, these alternating regions following one another through space with great rapidity. Since we have no certain knowledge that such electric states in the ether are necessarily accompanied by any real motion, of the ether or of anything else, the quantity y in, such a case would have to represent the intensity of the electric field at the distance represented by x and the time* represented by t. It will be shown later later (Chapter XIV) that such electric waves actually exist, that the waves of wireless telegraphy are un- doubtedly such, and that we have convincing evidence that light waves are also of the same nature, differing from the waves of wireless only in length. But many of the phenomena of waves, such as interfer- ence, diffraction, and some of the phenomena of reflection and refraction, would be the same no matter what the nature of the disturbance might be; and therefore it will be convenient, for the time being, to think of light waves as if they were really waves of mechanical displacement. Whether they are to be thought of as transverse, like those in a plucked string, or longitudinal, like those of sound, need not be considered yet, but evidence will be produced in Chapter XII to help us de- cide between these alternatives. 22. Interference. Fresnel's mirrors. One of the most convincing proofs of the wave theory of light is the phenom- 42 LIGHT enon of interference, in which two separate beams of light annul one another at certain places, producing darkness, and at other places produce a brightness much greater than either alone could cause. The theory of this s, m - phenomenon is as follows : s z m* , Suppose we have two sources of light, Si and S 2 , figure 20, in the form of narrow slits or round holes, through each of which comes light of ex- actly the same wavelength.* At first, we shall also suppose that the two pencils of light are in phase, that is, that when- ever a crest starts from one a crest will also be starting from the other. This means that, if we should write a formula of the type yrzrK.COS ' (X Vt c) a for each pencil, would have the same value in both. It will be easier however to discuss this case without the use of the formulae. The light from each slit falls on the white screen AB, and we shall first investigate what happens at that point, C, which is equally distant from the two slits. Whenever a crest reaches C from S t a crest will also reach it from S 2 , and similarly a trough from S t and a trough from S 2 will reach C at the same instant. Consequently, the amplitude of the vibrations at C will be double that which would exist if light from only one slit reached it, and the screen will be very bright there. There will be other points on the screen for which the same statement *It is customary among physicists to speak sometimes of a small hole or slit through which light is passed as the source of the light, although in fact the real source is a flame, a spark, an arc-lamp, or perhaps the sun. This real source is placed close to the slit or hole, or else an image of it is thrown upon the latter by the use of a lens or mirror. The object of the slit or hole is simply to provide a very narrow opening for the light to come through. If the source proper is itself small enough, the slit may be dispensed with. INTERFERENCE 43 holds true. For instance, if M! is the point just one wave- length nearer to S t than to S 2 , and M\ the point one wave- length nearer to S 2 than to S x , at each of these points .crests will arrive together from the two slits, and also troughs will arrive together, and therefore these too will be points of bright- ness. The same may be said of M 2 , which is two wavelengths nearer to S 1 than to S 2 , and of any point on the screen which is an exact whole number of wavelengths nearer to one of the slits than to the other. On the other hand, consider the points m lt m' lt m,. m' 2 , etc., each of which is so situated that it is either % wavelength, % wavelength, or in general any odd number of half -wave- lengths, nearer to one slit than to the other. Each of these points will receive a crest from one slit at the same time as it receives a trough from the other. In other words, the pencils of light coming from the two sources will at these points be opposite in phase, so that they will annul one another, and the points will be dark. It need hardly be pointed out that there will be points, such as one between C and m^ which will neither be as bright as C nor completely dark as at m 1? since here the two pencils of light meet neither exactly in phase nor exactly opposite in phase. In fact, a moving point, going from C up toward A, would first be in intense illumination, which would fade out to darkness at m 1? then brighten again to a maximum at M,, fade to darkness again at m 2 , etc. We should expect then, according to the wave theory, to find a number of bright regions, separated by dark regions. Tn fact, these should be drawn out into bright and dark streaks, or fringes, as they are called, perpendicular to the pl'ane for which the figure is drawn. For, even if the slits, had no appreciable length in this direction, the loci of bright or dark regions would still be drawn out into lines. Now let us see how all this would be altered if the two, pencils of light were not exactly in phase as they came through the slits. We should still expect to have fringes but they would not occur at quite the same place on the screen. For 1 instance, if the difference in phase were such that a crest would start from Si and a trough from S 2 at the same instant, (and vice versa of course) then the places we have marked as bright 44 LIGHT would be dark, and those we have marked dark would be bright. Herein lies a certain difficulty in subjecting these predictions to experimental proof. The trouble is not so much to keep the two pencils in the same phase, since we don't much care which points are dark and which bright, so long as tihe fringes stay steady long enough for us to see them. But they will not keep steacty unless the two pencils at least keep the same relation to one another in phase, and this they will not do unless they were originally part of the same pencil or beam, that is, unless they originated in the same ultimate source. It must not be supposed that any source of light is perfectly steady. "We might think of it as sending out a regular train of waves perhaps a meter in length, followed by a break and another train, perhaps longer, perhaps shorter, there being no fixed relation between the phase-constant in the first train and that in the second. In other words, the light comes in bunches of waves, rather than in a long uninterrupted series. Now, if changes of this sort are going on in the light coming from each slit, and the breaks are occurring quite independently in the two pencils, it is evident that, although fringes would be present on the screen at any instant, the}^ woiild shift their position with every break in either pencil. If we assume that the aver- age length of an uninterrupted train of waves is one meter, then since light travels 300,000,000 meters per second, there would be at least 300,000,000 shifts per second in the positions of the fringes. Consequently, no fringes could be seen, and the screen would appear uniformly illuminated. Therefore, in order to see such fringes, it is absolutely necessary to get two pencils that have the same origin, so that whenever the phase of one pencil suddenly changes, that of the other will undergo the same change. There are several ways of accomplishing this end, the most satisfactory being one due to the French physicist Fresnel, a diagram of which is shown in figure 21. He used only one slit, S, illuminated by any source of light, but he allowed the beam from S to fall on two mirrors M t and M 2 , very slightly inclined to one another, each of which reflected light to the screen. The light strikes the screen exactly as if it came from the two images S L and S 2 , which may be thought of as replacing the two independent THE FRESNEL MIRRORS 45 slits S t and S 2 of figure 20, with the one important differ- ence that now whatever change occurs in one pencil will also occur at the same instant in the other. Thus they will always have the same relation to one another in phase, and the fringes will be steady and therefore visible. Figure 21 Figure 22 This experiment works very satisfactorily, though the ad- justment is somewhat difficult, since the slit must be very accurately parallel to the line in which the planes of the two mirrors intersect. Figure 22 is a photograph of fringes taken by this method. The light used was the violet of a definite wavelength coming from the "mercury-arc." Referring back to figure 21, let d represent the distance between the two apparent sources S t and S 2 , D the distance from their plane to the plane of the white screen, C the point equally distant from Si and S 2 . Let P be any point on the screen, within the plane of the diagram, and x its distance from C. We shall first calculate what values x may have in order for P to be one of the points of maximum brightness, and from this result find the distance between centers of the bright fringes. If L represents the difference between the distances S X P and S 2 P, P will be a point of maximum or minimum illumina- tion according as L is equal to an even or an odd number of 46 LIGHT half-wavelengths. Therefore our first step will be to express L in terms of measurable quantities d, D, and x. By simple geometry, Therefore, L = S 2 P S,P = \/D 2 + x 2 +^ -f xd Vo 2 + x 2 + I 2 ._xd 4 4 For convenience, we shall represent the quantity D 2 + x 2 + 4 9 which appears in both radicals, by a single term D 2 . Then L = Vl>o 2 + xd VD 2 xd These two simple radicals can be expanded in series form, by using the binomial theorem, or applying the ordinary rules for the extraction of the square-root. The results are n T xd x 2 d 2 , x 3 d s + xd = D., + - _ +*- etc. S n - x2(P x3 d 3 - Pt-- 2D 8D 3 16D 5 " Subtracting the lower from the upper, we get xd x 3 d 3 = i + W 4 In practise, D is about the order of 100 cm., d about half a millimeter, or .05 cm., and x of course has various values, of which the greatest will perhaps be 1 cm. If we substitute these values, we see that in the first place D y will not differ from D itself by more than about 1/20000 cm., so that D may be substituted for D . Furthermore, the value of the first term in the last equation comes out to be about .0005, the next term .0000000000015, and the succeeding terms still more minute. Of course, D, d, and x need not have exactly the values here assumed, but the illustration suffices to show that in any case THE FRESNEL MIRRORS 47 D is so nearly equal to D that the difference is negligible, and that in the final expression for L only the first term is large enough to measure. Consequently, we shall make no error greater than the unavoidable errors of measurement if we adopt as the correct vaiue for L L = xd/D Using the letter \ to represent the wavelength of the light, we must have, in order that P may be a bright point, that L takes one or other of the values 0, \, 2\, 3A, etc. Or, since x DL/d, P is a point of maximum brightness when x has the value 0, D,\/d, 2Dx/d, etc. This shows us that the fringes are, at least approximately, equally spaced, the distance from center to center being Dx/d. This distance can be measured with some degree of accuracy, and also the distance D. The remaining! distance d is more difficult to measure, partly because it is smaller, and partly because it is not the distance between real slits, but between two images. However, it can be measured by indirect means, and then everything necessary is known in order to calculate the wavelength. It is found, as we should expect, that the wavelength as so determined depends upon the kind of light used. If only deep red light enters the slit, the width of the re- sulting red fringes indicates the wavelength to be about .00007 cm. ; while if deep violet is used the fringes are only a little miore than half as far apart as the red, indicating the wave- length of this color to be about .000038 cm. The other colors have wavelengths between these two extremes. But the ex- tremes themselves are not very definite, since it is found that some people can see deeper red or deeper violet than others. This fact makes us suspect the existence of wavelengths longer than the red or shorter than the violet, to which nobody's eyes are sensitive. We shall find later that there are such waves. (Sections 64 to 67) Although this interference experiment gives us a means of measuring the wavelength of light, it is not an accurate method. More complicated interference experiments, to be described later, allow us to measure wavelengths with an accuracy of 1/1000 of 1%. and in a few cases the precision has been car- ried even farther. 48 LIGHT 23. Interference in white light. Let us consider what would happen if white light, instead of light of only one color, were admitted through the slit in figure 21. Evidently every wavelength would produce its own set of fringes, the spacing being different for each wavelength, and there would be much overlapping of fringes of different color. Only one point, the point C. would be bright for all colors, since it is equally dis- tant from the two slit-images. The central fringe would there- fore be white. But the first red fringe on either side of the center would be slightly farther away than the first violet fringe. Consequently, the totality of each of these two fringes would be composed of fringes due to all the different wave- lengths, for no two of which would the maxima come in ex- actly the same place, violet being on the side nearest to C, red on the other side. One might regard each of these fringes as a very short and impure spectrum. It would be white near its middle, with a violet inner edge and a- red outer. This effect would be more pronounced for the second fringe on each side, still more for the third, etc., as if each succeeding fringe were a longer and longer spectrum. At the distance of two or three fringes away from 0, they begin to seriously overlap ; and at the distance of six or eight, the overlapping becomes so complex that all color-effect is lost, the fringes are no longer visible, and the screen becomes uniformly white. When white light is used, therefore, only a small number of fringes are ever visible, whereas with light of a single wavelength, spoken of as monochromatic light, a great number may be seen, pro- vided the two beams overlap over a sufficiently extended region. A further discussion of interference in white light will be found in section 82. Problems. 1. If, in Foucault's rotating mirror experiment, figure 7, a cylinder of some material, in which the velocity of light is less than in air, is inserted between the mirrors M t and M 2 , what would be the effect upon the distance f^? What would be the effect if the material were inserted between M x and L? 2. Plot to scale, on the same diagram, the curves repre- sented by formulae (2) and (3), letting K = 1, a = 4, THE FRESNEL MIRRORS 49 a = 1.5. (It will suffice to plot only the peaks and troughs, and the points where the curves cross the x-axis) Show by scaling that the curve (3) lies to the right of (2) by the amount predicted in the text. 3. Plot the curve for equation (4), giving any desired value to the time. 4. If the two slit-images of the Fresnel Mirror experiment are l/10mm. apart, how far away must the screen be to have fringes 3mm. apart, if the wavelength is .00005cm.? 5. Suppose the light coming from the two sources of figure 20 had not the same wavelength. What would be the result? CHAPTER IV. 24. Reflection and refraction. Huyghens' principle. Index of re- fraction. 25. Total reflection. Critical angle. 26. Deviation through a prism. 24. Reflection and refraction. Huyghens' principle. In- dex of refraction. The laws of reflection and refraction, so far as concerns only the direction of the reflected and re- fracted waves and not their intensity, are easy to derive by an application of geometry to the wave theory. Figure i 23 Let MN, in figure 23, represent the surface of a sheet of water, or the plane and polished surface of glass or some other reflecting and refracting medium. We shall suppose the water, glass, or other material to fill the space below MN, the medium above being the free ether. Plane waves are advancing through the ether, in the direction indicated by the arrow P, toward the surface. The lines ab, ajbj, etc., represent successive posi- tions of an advancing wavefront, as it approaches MN. Our problem is to determine the position of the reflected wave and the refracted wave to which this incident wave gives rise. (50) HUYGHENS' PRINCIPLE 51 In order to do this we shall make use of a principle enunciated by the Dutch physicist Huyghens, which applies to all types of waves. It may be stated as follows: A wave-front propagates itself by virtue of the fact that each point in the medium, as the wave-front reaches it, becomes itself a center of disturbance from which a spherical wave is sent out; and the further-advanced position of the original wavefront is nothing more nor less than the envelope of all the secondary wavelets sent out from the totality of points taken as centers. "When the wavefront ab reaches the position a^b.,, the point a 2 therefore becomes the center of such a spherical wavelet, not only in the medium above MN, but also in that below. But the wavelets in the two media will not advance equally fast, because the velocity of light is less in the lower medium than in the free ether. Let us suppose the velocity in the lower medium to be 1/n of that in the upper. Then, while the incident wavefront is travelling onward from the position a 2 b., till it reaches' the reflecting surface at b 3x the secondary wavelet from a 2 will have acquired a radius equal to b 2 b 3 in the upper medium, but! a radius of only 1/ri of b 2 b 3 in the lower. Therefore an arc is drawn in the upper medium with radius b 2 b 3 , and one in the lower with radius b 2 b 3 /n, both hav- ing a-, as center. What has occurred at a 2 will occur at every point on MN as the advancing incident wavefront reaches it, except that, if we want to construct the reflected and refracted wavefronts for the time when the incident wave is at b 3 , we must take the radii of the secondary wavelets shorter and shorter for centers nearer and nearer to b 3 . Thus, for the point c as center, we take the radius equal to mb,, in the upper medium, mb,/n in the lower. For, when the incident wave- front has reached c it .has also reached m, and still has tine distance mb : , to travel. For d as center, the proper radii are ob c and ob 3 /n, and so on. There should be an infinite number of such secondary wavelets, of which only a few are drawn in the figure. A plane passing through b 3 and tangent to all the secondary wavelets in the upper medium gives the wavefront of the reflected light, and another through b 3 tangent to all those in the lower medium gives that of the refracted light. Each of these advances perpendicular to its own plane, as 52 LIGHT shown by the arrows Q and R respectively. The student may ask what becomes of those parts of the secondary wavelets which do not lie on the common tangent plane. It can be shown that they mutually annul one another by interference, if we take account not only of the crests of waves but also of the troughs. The angle b 2 a 2 b 3 , which is a dihedral angle between the reflecting surface and the plane of the incident wavefront, is called the angle of incidence. The dihedral angle between the the reflecting surface and the reflected wavefront, a 2 b 3 a 3 , is the angle of reflection, and that between the reflecting surface and the refracted wavefront, a 2 b 3 a 4 , is the angle of refraction. a.a, is drawn perpendicular to the reflected wavefront, a 2 a 4 perpendicular to the refracted wavefront. Since a 2 a 3 is equal to b 2 bo,, the triangles a 2 a 3 b 3 and b 2 b 3 a 2 are equal, and the angles of incidence and reflection are equal. The triangles b 2 b 3 a 2 and a^bg are not equal, but are both right-angled triangles, the angles at b 2 and a 4 being each equal to 90. Therefore, repre- senting 1 the angle of incidence by i and the angle of refraction by r, we have b 2 b- sin. i = -v^ a, a, sin. r = ^~ Sjb, Therefore, sin. i _ b a b, ; _ ^ sin. r " a 2 a 4 By the method of constructing the figure, a 2 a 4 , bears the same relation to b 2 b 3 that the velocity of light in the lower medium bears to that in the upper. That is, a 3 a (i = b,b 3 /n, or b 2 b 3 /a 2 a t = n. Therefore, sin. i = n sm. r Since the velocity of light through a non-crystalline material such as glass or water is the same no matter what the direction of the rays may be, and depends only upon the nature of the material and the wavelength of the light, the above equation INDEX OF * REFRACTION 53 indicates that the ratio of the sines of the angles of incidence and refraction is canst ant, that is, it has the same value for all different angles of incidence. This statement, known as Snell's law, was first proved by direct measurements of differ- ent sets of angles of incidence and refraction. It. holds good / for all isotropic (non-crystalline) materials, but not as we shall see later for all crystals. The quantity n, which was originally defined simply as the ratio of the sines of i and r, is called the index of refraction of the lower medium; in tlier figure, that is of the medium into which the light is refracted. The index differs slightly for different colors or wave- / lengths. It is for this reason that a prism not only bends or refracts a beam of light, but also separates it into a spectrum. Since violet is bent more than red, evidently the index is greater for the shorter waves than for the longer, at least in ordinary media such as glass. We have assumed, in discussing figure 23, that the upper medium is the free ether, but the conclusion would be exactly the same if it were any other isotropic medium, except that the corresponding index of refraction would have a different value. If the first medium were air, the change in the index would be extremely slight, since light travels almost as fast through air as through a vacuum. But if it were water or some other transparent solid or liquid the change would be great. In such a case, we say that the ratio of the sines or, what comes to the same thing, the ratio of the light velocities is the index of the second medium with respect to the first. Let Ui be the index of the first medium (with respect to the ether), n 2 that of the second, v x the velocity of light in the first medium, v 2 that in the second, and v the velocity in the ether. Then * sin, i __ _y_ 1 ___ v^ _ VQ . v,_ _ ^ _^_ v^ _ n^ sin. r v 2 v 2 v v 2 x V(i v 2 ' v, "~ U L therefore, if n 12 be used to indicate the index of refraction of the second medium with respect to the first, (light passing from the first to the second), n, 2 = n 2 /n 1 54 LIGHT Tabulated values of the refractive indices of various solids and liquids are to be found in such collections of physical data as the Smithsonian Physical Tables, Recueil de Constantes Physique of the French Physical Society, the Physikalische- Chemische Tabellen of Landolt and Btrnstein, and Kaye and Laby's Physical Tables. So far as glass is concerned, there is an almost endless variety of different glasses, having all sorts of variations of refractive index, dispersion, and absorption. For rough calculations, we may take 1.53 as the approxi- mate index of refraction of crown glass for yellow light, and 1.63 as that of flint glass. The index of water for yellow light is very nearly 1.33, that of diamond 2.42. It is customary to speak of substances having very high refractive indices as being "optically dense." This is only a technical expression, for refractive index has nothing to do with real density, or specific gravity, beyond the fact that in general the heavier kinds of glass, have the higher indices. Thus, we say that diamond is an extremely dense medium, and that flint glass is denser than crown. 25. Total reflection. Critical angle. So far, we have taken up principally cases where the light passes from the less dense to the more dense medium, as from air to water, but obviously cases of the reverse type are almost as common. Whenever we see anything that lies below the surface of water, for instance, the light must pass out of water into air in order to reach our eyes. In such a case we may still write sin. i - n sin. r where i means the true angle of incidence (on the water side of the boundary), r the true angle of refraction (on the air side), and n is the index of air with respect to water, just the reciprocal of the index of water as found in tables, that is, .75 instead of 1.33. Of course, beside the light that is refracted out into the air, there is always in addition a reflected beam going back into the water, for which the angle of reflection is equal to the angle of incidence, exactly as it would be if the light had been incident on the air side. Sometimes with light incident on the denser side of a dividing surface, it will hap- pen when the angle of incidence is large enough, that there TOTAL REFLECTION 55 is no refracted light at all, all instead of part of the incident beam being reflected back into the first medium. A considera- tion of the formula of refraction shows that this must be so. Solving for sin. r, we get sin. i sin. r = Since, for such cases as we are now considering, n is less than unity, this equation shows that sin. r is greater than sin. i, and therefore r is greater than i, both being acute angles. It is possible, then for r to be equal to 90 and sin. r equal to 1, while i is still considerably less than 90. If i becomes any greater, sin. r as calculated from the above equation becomes greater than 1 ; and this means, since the sine of a real angle* cannot exceed 1, that there is no refracted wavefront. Obvious- ly, the largest value that i can have for refraction still to occur, is that value which makes sin. r = 1. Such a value for the angle of incidence is called the critical angle. If we let 7 represent the critical angle, we can find its value from the equation for refraction, by substituting 7 for -i, and 1 for sin. r. This gives sin. 7 = n As an example, let us calculate the critical angle for crown glass, in contact with air. We have taken 1.53 as the index from air to the glass, which gives 1/1.53, or .654 as the index from the glass to air. Therefore sin. 7 = .654 7 = 4049' So much for the mathematical side of the question. The physical interpretation can be gotten by considering figure 23 again, with the modification that now the velocity of light in the lower medium is greater than that in the upper. Suppose, for instance, that the velocities in the two media, and the angle of incidence, have such values that while light travels the dis- tance bob s in the upper medium it will travel a distance in the lower medium greater than a 2 b 3 . Under these circumstances the radius a 2 a. t of the secondary wavelet from a 2 will be so great that the point b 3 will lie ivithin the sphere of the wave- 56 LIGHT let and it will be impossible to draw a plane through b 3 tangent to this sphere. When the angle of incidence has exactly the critical value, a 2 a 4 the radius of the secondary wavelet from a 2 , is just equal to a 2 b 3 . One of the most effective ways of showing total reflection is that illus- trated in ^figure 24. ABC represents a right-angled prism of crown glass. The eye is held at some such point as E, close to one of the shorter prism-faces. SS' is any rather bright- ly illuminated surface, such as the sky, the whitewashed wall of a room, or a sheet of paper. The eye sees, reflected in the hypothenuse BC, an image of the bright area SS', the upper part of which is almost as bright as SS' itself, the lower part much fainter. There is a fairly sharp boundary between the bright upper part, seen by total reflection, and the fainter lower part, seen by ordinary partial reflection. Following is the explanation. Any point of SS', such as a, sends out rays of light in all directions, but only a small bundle of these, comprising a cone, will reach such a position, after reflection at the face BC and refraction at the faces AB and AC, that they can enter the pupil of the eye and contribute to vision. For simplicity's sake, a single ray aa'a"E is drawn to represent this slender cone. Similar rays are drawn from a few other points on SS'. In every case, some light is lost by reflection at the two surfaces AB and AC, but this is not indicated on the drawing in order that the diagram may not become too intricate. There will be some point on SS', such as b, which is so situated that the cone of light from it which enters the eye will strike the hypothenuse with an angle of incidence, b'b"K, which is exactly equal to the critical angle. The light from any point to the left of b, such as c or d, will strike BC at an angle less than the critical angle, so that the greater part of the) light in the small cone will be refracted through BC into the air, below and to the right of the prism, leaving only a small fraction to be reflected into the eye. Therefore such TOTAL REFLECTION 57 points as c and d will appear faint in the reflected image. On the other hand, the light from a, or any other point to the right of b, will be reflected from BC at an angle greater than the critical angle, none will be refracted through the hypothe- nuse, and all the light in the cone, except for the small amount reflected by the other two faces of the prism, will enter the eye. Therefore, points to the right of b will appear very bright in the reflected image of SS', as bright as if they had been reflected by a silvered mirror, or even brighter, since a silver mirror does not reflect by any means all the light that falls upon it. Naturally, the critical angle for any material depends not only upon the nature of the material itself, but also upon that of the material in contact with it. For example, if the prism were submerged in water instead of being in air, many of the rays which are totally reflected against air would be refracted through into the water. The principle of the totally reflecting prism is utilized in a number of optical instruments, a few of which we shall con- sider later. But one of the most interesting cases of total reflection is seen in the case of a cut diamond. The index of / diamond with respect to air being so large, its critical angle is correspondingly small, and a diamond owes its brilliance partly to this fact and partly to the additional fact that its index for different colors differs large- ly, so that for certain angles of in- cidence the shorter waves are totally reflected while the longer ones are not. Let figure 25 represent a cross- section of a cut diamond. Not only the ray aaa, but the very oblique one bbbb may undergo total reflection at the two surfaces XZ and YZ. More- over, if the b ray happens to strike one of these surfaces at the critical angle for green light, for in- stance, the waves of shorter length will strike it at an angle greater than the critical, the waves of greater length at an angle less. For, since the index is greater for short than for long waves, the critical angle is greater for long than for short. Consequently, red rays, will escape total reflection and be par- 58 LIGHT tially refracted out of the diamond for angles which make re- fraction impossible for violet rays. Industrial applications of the total reflection principle have been made in the manufacture of so-called "shades" flight- distributors) for incandescent electric lighting, sidewalk-lights for illuminating basements, etc. The appearance, to a fish, of things outside the water, is largely affected by refraction, "and also illustrates total re- flection. Figure 26 represents a pond, the eye of the fish being at E. XEY is the section of a cone, whose axis is verti- cal and whose half- angle K is equal to the critical angle of water, 4845'. Any object outside the water, as at a, would be seen within the Figure 26 con e, for obviously light passing from air to water at an angle of incidence less than 90, as it must be, will be refracted with an angle of refraction which is less than what would be the criti- cal angle for light incident on the under side of the surface. The object a would appear in some such position as a'. It is clear then that the whole array of objects outside the water would appear to the fish very much distorted from their true relative positions, being crowded within a relatively small cone of view. Professor R. "W. Wood, of Johns Hopkins University, has taken some curious photographs which illustrate what he calls "fish-eye views" reproductions of which can be found in his book "Physical Optics." Whatever the fish sees by looking at the surface outside the cone XEY would be totally reflected images of objects in the water. For instance, it would see the objects C and D, not only directly, but also by reflection in the surface. In other words, the whole top surface, outside the circle whose diameter is XY, would appear as a perfect mirror. Inside this circle, the fish would see, not only objects that are outside the water, as already stated, but also faint reflections of objects within DEVIATION THROUGH A PRISM 59 the water between M and N. It need hardly be pointed out that as the fish swims about the cone XEY moves with it, the diameter XY becoming smaller as the fish approaches the sur- face, larger as it sinks toward the bottom. A man with his head under water would see things the same way as a fish but for one thing. Our eyes are adapted for seeing in the air, and the index of refraction of the cornea (the forward portion of the eye)' with respect to air is such that the lens within the eye can bring* to focus upon the retina objects anywhere from about eight inches to an infinite dis- tance away. The substitution of water for air, as the medium in contact with the cornea, alters the refraction so that it is impossible, at least without extreme eye-strain, to focus upon the retina. Therefore human vision with the eyes in contact with water is very much blurred and indistinct. 26. Deviation through a prism. We shall now consider the deviation of light by a prism (figure 27) ; and since, in the actual use of prisms for the production of spectra, the light waves are practically always first made plane by the use of a lens, we shall take only the case of plane waves. At each surface of the prism, not only refraction occurs, but also reflection; but in this discussion we shall ignore the reflected light. For the sake of symmetry, we call i x the angle of incidence at the first surface, r t the corre- sponding angle of refraction, r 2 the angle of incidence at the sec- ond surface and i, the correspond- ing angle of refraction; so that i^ and i, are angles in air, r t and r., angles in glass. Then n = sin. i sin. i 2 sm. sin. r 2 where n is the index of glass with respect to air. The drawing shows a series of wavefronts sup- posed to be just one wavelength apart (for instance the lines of the crests) both in the air and in the glass, though of course the actual length of the v/ t Figure 27 60 LIGHT light waves, as compared to the dimensions of a practicable prism, is enormously exaggerated in the figure. *It will be i noticed that the wavelength is shorter in glass than in air) Indeed this must be so, for the following reasons : The period, or time of one vibration, must be the same in glass as in air, for only so many waves can in a given time leave thle surface toward the glass side as come up to it on the air side. Also, since a train of waves advances the distance of one wavelength during the time of one vibration, wavelength velocity ^ period Therefore, since the period is the same in the two media, wavelength in air velocity in air wavelength in glass velocity in glass Consequently, whenever light of any wavelength A passes from air into another medium whose index of refraction with re- \ spect to air is n, the wavelength within this medium is reduced to the value A/n. The angle D, between the wavefronts of the light before entering the prism and those after leaving it, or what comes to the same thing the angle between the rays before and after passage through the prism, is called the angle of deviation. The refracting angle of the prism itself we call A. It can be easily proved by simple geometry from the figure that A . r, + r 2 = 1,4-12 A From these two equations, together with the two gotten by applying the law of refraction to each surface of the prism, we can, if A, i it and n are given, solve for r 1? i 2 , r 2 , and D. A and n are necessarily constant for a given prism and a given wavelength of light, but by turning the prism about an axis perpendicular to the plane of the figure i : can be made to take any value from to 90. Changing the value of i t in such a manner will naturally cause changes in the value of D. If we plot the values of i t as abscissae, and the corresponding values of D as or din at es, the curve will be found to have the form MINIMUM DEVIATION 61 shown in figure 28, which shows that for a certain value of i x the value of D is less than for any other value of i x . Both ex- periment and theory by the D application of the differential calculus show that this mini- mum value of D occurs when / ij = i 2 and r t = r 2 , that is when the light passes through the prism symmetrically. In such a case, Figure 28 n = This equation has a great deal of importance in practical optical work. For, by the use of a spectrometer, both the re- fracting angle of a prism and the angle of minimum devia- tion, D. can be measured with great accuracy. Therefore, by applying this equation we can get very accurate determinations of the index of refraction for any piece of glass that can be obtained in the form of a prism. It is the most convenient method for finding not merely the average index for white light, but the separate indices for different wavelengths. Problems. 1. Calculate the angle of refraction when li^ht strikes crown glass with an angle of incidence of 60. 2. Find the index of refraction of crown glass with respect to water, for yellow light. 3. Light within a piece of crown glass strikes the surface at an angle of incidence of 40. At what angle does it emerge? 62 LIGHT 4. Calculate the critical angle for diamond, yellow light. 5. Show that if light strikes a pile of parallel -sided plates with different refractive indices, all in contact with one another, it enters each plate with the same angle of refraction as if the others were absent, and finally emerges parallel to its original direction. 6. A star seems displaced from its proper position owing to refraction in the earth's atmosphere. Show that, despite the changes in the density of the air at different levels, we can calculate the refraction by considering that all the air has the same index as that at the earth's surface. (See problem 5). 7. Find the approximate length of wave, in water, of the extreme red and the extreme violet light, assuming the index for both to be 1.33. 8. Calculate the angle of minimum deviation for a 60 prism, the light having index 1.68. 9. What must be the properties of a body which, in air, would be invisible under any illumination? Would such a body be visible if immersed in water f 10. Certain aquatic bodies are "nearly invisible in water. What are their properties'? 11. Show that any colorless and transparent object would be invisible if surrounded completely by uniformly illuminated walls. 12. Prove that, in figure 27, A = r t + r, and D = i t + i 2 A. 13. A real diamond will continue to glitter when immersed in water, while an imitation will not. Explain this. 14. Plot four points on a curve like figure 28, for a prism of 60, having an index 1.54. CHAPTER V. 27. Reflection and refraction of spherical waves at a plane surface. 28. Judgment of the distance of an image. 29. Image of an extended object. 30. Reflection and refraction at spherical surfaces. 31. Lenses. 32. Two lenses in contact. 33. Chromatic aberration. 34. Achro- matic lenses. 35. Image of an extended object. Undeviated ray. 36. Magnification. 37. Micrometer. 38. Imperfections in mirrors and lenses. 39. Spherical aberration. 40. Curvature of field. 41. Astig- matism. 42. Lenses for special purposes. 27. Reflection and refraction of spherical waves at a plane surface. It is shown in the preceding chapter that when plane waves strike a plane surface, both the refracted and the re- flected wavefronts are plane. In this chapter we shall show that when the incident wavefronts are spherical (diverging^ from a point) and the reflecting surface plane, the reflected wavefronts are also spherical but the refracted wavefronts are not, except as an approximation to the truth. We shall also show that when both the incident wavefronts and the reflecting surface are spherical, neither the reflected nor the refracted wavefronts are truly spherical, except in the one special case that the center of the incident waves coincides with that of the reflecting surface. In figure 29, let CBD represent the plane boundary between two media, the in- dex of refraction of the low- er with respect to the upper being n. A is the center of a system of spherical wave- fronts, advancing from A toward the surface CBD. B is the foot of the perpendic- ular from A upon this sur- face, that is, the first point on CBD reached by each Figure 29 wavefront. If the reflecting surface had not been in its place, a wavefront, after reaching B would continue to travel with its original velocity and in a short time would reach some such (63) 64 LIGHT position as CxD. With the surface in its place, B becomes the center for a secondary wavelet in the upper medium whose radius, at the instant when the incident wavefront reaches C and D, will have acquired the length Bx' = Bx. In the mean- time, other points such as M and N will also have been reached by the incident wave and become centers of secondary wave- lets, whose radii in the upper medium will be respectively MO' = MO and NP' = NP. The reflected wavefront will therefore be CO'x'P'D, the envelope of all such secondary wavelets. From the manner of its formation, it is obviously exactly symmetrical with the hypothetical incident wavefront COxPD, and therefore is truly spherical, with center at A'. A and A' are equally distant from the reflecting surface, and the line AA' is perpendicular to the latter. An eye placed any- where in the upper medium would receive reflected light which would appear to come from A', though really from A, and we . therefore say that A 7 is the reflected image, or image by Te-\ flection, of A. The refracted wavefront is formed in a simitar manner, except that in the lower medium the radii of the secondary wavelets are shorter than in the upper if n is greater than unity, longer if n is less than unity. The radius By of the secondary wavelet from B as center is not equal to Bx, but to Bx/n. The radius of the secondary wavelet from M is Mu =: MO/n, that from N'is Nv = NP/n, etc. The envelope of all these secondary wavelets in the lower iBedium, CuyvD, turns out to be, not a sphere, but a surface of higher order. There- fore, we cannot say that there is a refracted image in the same strict sense in which we speak of a reflected image. It is true * that an eye placed in the lower medium would receive light that appeared to come from some point in the upper medium other than the true source A, but this apparent "image" would have a different position for every change in the location of the eye. However, it is always possible to describe a sphere which approximates more or less closely to the refracted wavefront. Suppose, for example, that a circular arc be drawn through the three points C, y, and D, and a spherical segment be formed by rotating this arc about the axis AA'. This surface would SPHERICAL WAVES AT A PLANE SURFACE 65 coincide exactly with the refracted wavefront at the three given points, and would also pass very close to other points such as u and v. The nearer the three points C, y, and D are together, the closer would sphere and wavefront coincide for all the region between C and D, and if the three points are distant from one another by only an infinitesimal amount we may say that in the immediate neighborhood of these points the coincidence is exact. If A" represents the center of this sphere, then to an eye located in the lower medium anywhere along the line AA' or AA' produced, or in the close neighbor- hood of this line, as at the point E, the light would appear to come from A" instead of from A, and we therefore define A" as the image by refraction of A. But, if the eye be located at some distance from the line AA', as at E', the light appears to come from a different point, such as Q. In order to find the position of the image A", it is best to consider the refracted wavefront just as it breaks through the surface CBD, that is, we imagine C and D to be very close together and By and Bx to be infinitesimally small. Then the radius of the sphere through C, y, and D will be the radius of the refracted wavefront just as it breaks through the surface, and equal to the distance of the image A" from the surface. To find A" is merely a matter of plane geometry. CD is a chord common to the two arcs CxD (hypothetical inci- dent wavefront) and CyD (refracted wavefront). The line drawn from the middle point of a chord, perpendicular to the latter, till it meets the arc, is called the sagitta of the arc. Thus, Bx is the sagitta of the arc of the incident wave- front, By that of the refracted wave- front. We must first find what relation the sagitta bears to the radius. This can be best done by a consideration of figure 30, where the complete circle is shown. The triangles KBD and DBy are similar, therefore KB _BD BD "BY If we let s represent the sagitta, R the radius of the circle, and 66 LIGHT a the half -chord, KB = 2R s, BD = , and By = s. There- fore 2R s We are supposing that C, y and D are very close together, so that both s and a are quantities of infinitesimal magnitude, but R remains a quantity of finite size, therefore in the limit 2R s becomes equal to 2R, and JL !_ a 2 " 2R or, E = Equation (1) shows that although a and s both become in- finitesimally small, the ratio o 2 /s remains a finite quantity, equal to the diameter of the circle. Now let us apply equation (1) to both the hypothetical incident wavefront and the refracted wavefront. For the former, s=Bx, R=BA. For the latter, s=By, R=BA", the required distance. Therefore, since a is the same for both, viz., BD, If we divide the last equation by the one above it, we get B A" _ Bx BA ~~ By But, by the method of constructing the refracted wavefront, Bx/By .- n. Therefore, BA" ^-r- =n JUDGMENT OF DISTANCE 67 or in general, if d x represent the distance of any point source of light from a plane refracting surface, d 2 the distance of the corresponding refracted image, and n the index of refraction of the second medium with respect to the first, then. As an application of formula (2) suppose a certain object to be 2 feet above the surface of a pond. A person above the surface would of course see a reflected image of it, apparently 2 feet beneath the surface, but a fish in the water would see a refracted image. If the fish is directly beneath the object, we can apply equation 2, putting d = 2, n = %, or 1.33. This gives d 2 = 2 X ^r-~ 2.67 feet o Tha-t is, the object would appear to the fish to be 2 ft. 8 in. above the surface. If the source of light lies within the denser medium, the refracted light travels faster than the incident, the refracted wavefront of figure 29 will be bulged out more than the inci- dent instead of being flattened, and n has a value less tihan unity. Thus, suppose we look straight down to the bottom of a pool 2 feet deep. Then, in equation (2), d t = 2, n = % .. and d 2 = 2Xj =1-5 feet 4 The pool appears to be only % its actual depth. The fact that the wavefronts are not spherical is shown clearly by the observation that, if we look very obliquely to the bottom, the depth appears to have much less than % its actual value. This explains a curious phenomenon which anyone standing in a pool a few feet deep with perfectly level bottom can hardly fail to notice. The bottom always appears to be bowl-shaped, with the greatest depth just underfoot, and that depth of course just about % the true depth of the whole pool. 28. Judgment of the distance of an image. The student may wonder why the curvature of the wavefront has anything to do with our judgment of the distance of an object perceived, \ 68 LIGHT for one is apt to think that only a single ray ever enters the eye. This is incorrect, for the pupil of the eye has a finite size and therefore always receives a finite area of the wavefront. Ac- cording as the curvature of this section of wavefront is greater or less, we must exert more or less muscular strain upon the lens in the eye in order) to focus) the light upon the retina, and the degree of this strain enables us to judge distance to some extent. Still more important is 1 the fact that we ordinarily see with both e^yes at once, thus taking in at the same time two separate sections of wavefront. A person with only one eye is far less accurate in estimating distance than a normal per- son. For instance, a one-eyed man usually has much greater difficult} 7 - in hitting a nail with a hammer for this reason. Naturally enough, estimation of distance becomes much more difficult, even for normal two-eyed vision, when the dis- tance becomes great. It is comparatively easy to tell whether an object is five feet away or ten feet, for the difference in curvature of a five foot and a ten foot sphere is comparatively great. But it is not easy to tell whether a distant object is, nearer to a mile or to two miles from us. For a small section of a sphere of either one or two miles radius is nearly flat, and there is no perceptible difference in the focussing and align- ment of the eyes for such great distances. Our estimation of great distances is a result of subconscious consideration of such details as size, speed of motion (if the object seen happens to- be in motion), distinctness of vision, etc., and at best it is very uncertain and subject to queer illusions. Soldiers are given a long and systematic training in judging distance. If our eyes were set three feet apart instead of a few inches, judgment of distance would become much easier. No doubt small animals with their eyes close together are less adept in this respect than human beings. Indeed there is some reason fori thinking that many animals and birds are much less keen than men in observing stationary objects, although a moving objects almost instantly arrests their attention. Most birds have the eyes set in the sides of the head, and therefore seie an object with only ones eye at a time. This fact must seriously hinder them in the estimation of distance; and it is no doubt to counteract this deficiency that birds have the IMAGE OF AN EXTENDED OBJECT 69 habit, particularly when alarmed, of darting the head rapidly forward and backward, so as to get two or more points of view of any object that excites their suspicion. The parallax of the observed object gives some ground for a judgment of its remote- ness. Thus the mechanism by which a bird estimates distance is perhaps an automatic and unconscious application of the same principle used by a surveyor in finding the width of a river, or by an astronomer in finding the distance of the nearer fixed stars. 29. Image of an extended object. We have so far con- sidered only cases where a single point acted as a source of light, and have found the positions of the reflected image and the refracted image of this point source. Actually, we always have to deal with objects more or less extended; but in order to find the images of such an object we have only to apply the principles al- ready learned to each point of it. In figure 31, AB represents any object. The point A has an image by reflection, _ A', and an image by refraction, A", found by these principles, and similar- ly B has an image by reflection, B', and an image by refraction B", and so on. Figure 31 So far as the image by reflection goes, it is exactly symmetri- cal to the object, with respect to the reflecting plane. 30. Reflection and refraction at spherical surfaces. We shall now consider reflection and refraction at spherical sur- faces, a subject which is of great importance because of the use of curved mirrors and lenses in optical instruments. Figures 32 to 35 show four different cases of the reflection and refraction of spherical wavefronts at a spherical surface, the source of the incident waves being in each case on the con- cave side. All these figures are drawn on the supposition that the medium on the convex side (second medium), is denser than that on the concave side (first medium), and both media are supposed to extend indefinitely. The reflecting and re- fracting surface is indicated by a heavy line, the incident wave- fronts by normal unbroken lines, and the reflected and refracted wavefronts by dotted lines. The two last are not truly spheri- cal, but are near enough to be considered so as long as the Incidence is nowhere very oblique. Therefore, the center of 70 LIGHT the incident wavefronts is represented by 0, the center of the reflected wavefronts (except in figure 35) by I, the center of the refracted wavefronts by I', and the center of the surface itself by C. Arrows show the direction of advance of the waves. Figure 33 t'o ,-v^r Figure 34 Figure 35 In figure "B3, the source is farther from the mirror than is the center C. The reflected waves converge to the point I, betAveen C and the mirror, but nearer to the former, and diverge again after* passing through I. The refracted waves diverge more than the incident, appearing therefore to come from a point I', between C and 0. In figure &2i -the source is between C and the mirror,' nearer to the former. The point I, to which the reflected waves converge, and from which' they later diverge, lies outside of C. The refracted waves diverge less than the incident, appearing to come from I' between and C. ^ In figure 34 is nearer to the surface than to C. Here the reflected waves do not converge at all, but diverge at once, seeming to come from the point I within the second medium. SPHERICAL SURFACES 71 The center of the refracted waves is again between and C. In figure 35 is just halfway between C and the surface. The reflected waves are plane, and may be said either to converge to an infinitely distant point on the left, or to diverge from an infinitely distant point on the right. The latter state- ment is preferable, since an eye placed in the path of the re- flected light would see an image of infinitely distant on the right of the figure. In fact these waves would strike the eye exactly as waves coming from a very distant star. The points I in figures 32 and 33 are said to be real images of the corresponding' sources 0, because the light is actually converged to these points; while in the other two figures I is only a virtual image, the reflected light not actually being brought to.' focus at these points, but simply diverging as if it had come from them. In each of the figures, the refracted image, I' is virtual. Figure 36 Figure 37 ?. I Af'\ \U1 Figure 38 Figure 39 Figures 36 to 39 differ from the four preceding figures in that the reflecting and refracting surface is convex, instead of concave, to the incident light and the rarer medium. In each 72 LIGHT of these cases, the reflected rays diverge from the surface at once, and I is therefore a virtual image. As shown in figure 38, there is a certain position for the source 0, depending upon the radius of the surface and the index of refraction, for which the refracted waves are plane. If lies any nearer to the surface, as in figure 39, they diverge, and V is virtual. If O is farther from the surface, they converge, and I' is real, as in figures 36 and 37. In figure 36 the source is infinitely distant, and the incident waves are plane.* From what precedes, it is obvious that there is a great diversity of typical cases for reflection and refraction at spherical surfaces. The surface may be concave or convex to the incident light: the source may lie in the medium of less or of greater optical density, and may be at any distance from the surface. To develop and remember for each case a special formula giving! the positions of I and V in terms of that of 0, would be unduly laborious. Fortunately, we can derive a pair of very general formulae, which are applicable to any case that may arise, provided we make consistent and rational con- ventions in regard to the algebraic sign of the distances in- volved. We shall suppose all distances to be measured from the reflecting surface as a base, distances to one side being regarded as positive, those to the other side negative; and it will be most convenient to take the side from which the light comes as the positive side. We let r stand for the radius of the re- flecting and refracting surface, u for that of the incident wavefronts, v for that of the reflected wavefronts, and v' for that of the refracted wavefronts. In figures 32 and 33 all these quantities are positive. In figure 34 v has become negative, while it is 00 in figure 35. v' is negative in figures 36 and 37, -+-_ oo in figure 38, and positive in all the other cases shown, r is always positive for a concave mirror, nega- tive for a convex one. u is always and necessarily positive, unless the incident waves are rendered convergent before striking the surface, by means of another mirror or a lens. *In figures 32 to 35 the index of refraction has been taken as 1.5, but in figures 36 to 39 it has been taken as 1.67 to avoid making some of these figures inconveniently long. SPHERICAL SURFACES 73 It is easily seen that v is positive when the reflected image is real, negative when it is virtual. On the other hand, v' is negative when the refracted image is real, positive when it is virtual. It is easiest to derive the two formulae by considering a case where all the quantities, r, u, v, and v' are positive, as in figure 32. Figure 40 represents the case of figure 32 some- J?_-_-L-9- Figure 40 what exaggerated to make the diagram clearer. I, C, If and have the same significance as in the preceding figures. XPY is the reflecting and refracting surface (center at C), AdB an incident wavefront as it would be if it advanced into the second medium without being retarded (center at 0), AfB the actual refracted wavefront (center at I'), and AeB the reflected wavefront (center at I). In practice, mirrors are seldom used in which the diameter XY of the mirror-faee, is more than % of the radius of curvature CP = r. In this figure XY is made about equal to r in order that the different arcs having a common chord AB may be more clearly seen as separate. The formula for the reflected wave is based upon the fol- lowing physical fact: While the incident light would, but for retardation in the second medium, travel from P to d, the re- flected light travels back from P to e. Therefore the distances Pd and Pe are equal. Putting this statement into the form of an equation, Pe = Pd (3) 74 LIGHT But we can write Pe=:PK eK Pd=dK PK Therefore PK eK = dK PK dK + eK = 2PK But dK, eK, and PK are respectively the sagittas of the in- cident wavefront, the reflected wavefront, and the reflecting surface. We can therefore apply to each of them the general geometrical formula (1), getting dK = a 2 /2u eK a 2 /2v PK = a 2 /2r Making these substitutions, we get _o^ _a?_ _ 2a^ 2u " h 2v " 2r or, ' This is the general formula for a mirror. The symbol f, equal to r/2, is called the focal length of the mirror. Its physical meaning can be shown by supposing that u = oo , that is, that we are dealing with an object infinitely distant. Then 1/u = 0, 1/v = 1/f , and v f . Then f is the distance from the mirror to that point (called the principal focus) where parallel rays, or plane waves are brought to a focus in the reflected light. Conversely, if u = f , 1/v = 0, and v = oo ; that is, if the source of light is at the principal focus, the reflected waves are plane. If neither u nor v is infinite, the points O and I are called conjugate foci. If the source is at 0, the reflected image is at I, and conversely if the source is at I the image will be at 0, for equation (4) shows that the relation between these two points is reciprocal, u and v appearing in it in ex- actly the same way. The formula for the refracted wave is found in a similar way. The physical fact upon which it is based is this : While SPHERICAL SURFACES 75 the incident light would, but for the retardation in the second medium, travel from P to d, the actual refracted wave travels only from P to f, where Pd and Pf are to each other as the velocities of light in the two media. That is, Pd FF= n or Pd = n X Pf But Pd=:dK PK Pf ^:fK PK (5) therefore dK PK=rn(fK PK) nXfK dK= (n 1)PK But fK, dK, and PK are the sagittas respectively of the re- fracted wavefront, the incident wavefront, and the refracting surface. Applying to each of these the geometrical formula (1), we get f K = a 2 /2v' dK = a 2 /2u 'PK = a 2 /2r With these substitutions, nXa 2 _** _ (n l)a 2 2v' " 2u ~ 2r or This formula, for the refracted light, is necessarily more com- plicated than that for the reflected, because it involves the index of refraction, which does not affect reflection, and there- fore does not appear in (4). Formula (6) is of less common use than (4). but there are certain problems where it becomes necessary, for example, the following. 76 LIGHT A spherical globe, one meter in diameter, and made of thin glass, is filled with water. A small fish is located 40cm. from the glass wall at a certain side of the globe, (a) Where would be the image which the fish would see of himself in the farthest part of the surface! (b) Where would the fish appear, to a person outside the globe on the side farthest from the fish? Question (a) is easily answered, for the fish would see his own reflected image, and we only need to apply equation (4) as if the glass wall were non-existent, since its thinness pre- vents it from affecting the problem to any extent. Then we must put r = 50, u = 60, giving v = 42.9cm. Therefore the fish would see, reflected from the farthest part of the boundary of the globe, an image of himself 42.9cm. from the boundary, 17.1 cm. from himself. (If we had tried to find the location of the image of the fish formed by reflection in the nearest part of the wall, it would have come out to be behind the fish and therefore not discernable by him as an image). To answer question (b), we must apply equation (6), for we have to do with refracted light. Since the light passes from water to air, the appropriate index of refraction is not %, but the reciprocal of this, %. Therefore 31 1 4v' 60- "4X50 Giving v' = 64.3. Therefore the person outside the globe would see the fish apparently 4.3 cm farther away than it really is. The fact that v' comes out positive shows that the refracted image is on the same side of the bounding surface as the fish itself. The reflected image seen by the fish is real, the re- fracted image seen by the observer outside is virtual. Mirrors for experimental work in optics are usually either flat or concave, though convex mirrors are occasionally used. They are made by taking a disc of homogeneous and thoroughly annealed glass, and reducing one surface to the required radius of curvature by careful grinding and polishing. This surface is then covered with a deposit of silver by chemical deposition from a solution of silver nitrate, and the silver film is thorough- ly dried and then lightly polished with chamois and rouge. The silver of course prevents any appreciable refracted light so that the major part of the incident light is turned into the LtfNSES 77 reflected beam. Flat mirrors such as were used in the experi- ments of Fizeau and Foucault for finding the velocity of light, whose function is to reflect only part of the light and transmit the rest, must of course be ground and polished flat on both, sides. One face is then covered with a very thin film of silver, the best result being obtained when the film reflects approxi- mately half the incident light, transmitting the rest, except for some unavoidable absorption. Such a mirror is said to be half -silvered. When a silvered mirror becomes tarnished and dull, it is a comparatively easy matter to dissolve off the old silver with nitric acid and put a new silver coating upon it. 31. Lenses. A lens is a disc of transparent refractive ma- terial, such as glass, bounded by two surfaces, one or both of which is curved, usually spherical. Each surface produces some reflection, which not only weakens the transmitted beam but also causes annoyance in other ways. The light reflected from the lens-surfaces is therefore a hindrance, but is abso- lutely unavoidable. In the following discussion of lenses we shall ignore the reflected light. Figure 41 When spherical wavefronts, with center at (figure 41) strike a lens, they are refracted at the first surface, and again at the second surface, finally emerging approximately spherical, so that they either converge to a point I on the side opposite to 0, as in the figure, or diverge from a point on the same side as 0. Our task is to derive a formula by means of which, knowing the distance of from the lens, the radii of curvature of the \ two lens-surfaces, and the index of refraction, we can calculate the distance of I. This might of course be done by applying equation (6) once for. each surface, taking due ac- count of the fact that the appropriate index to be used at the second surface is the reciprocal of that for the first. However, we shall develop our lens formula by a different method, chiefly because by so doing we can introduce a con- 78 LIGHT vention as to algebraic sign which will prove more convenient for our purpose than the one used in equations (4) and (6). We assume that the lens is so thin that its greatest thickness may be neglected in comparison with the distance from source to image. Call the distance from the lens to the center of the incident wavefronts u, that from the lens to the center of the refracted wavefronts v. u is considered positive when the center of the incident wavefronts lies on the side from which the light comes, that is, when the incident light is diverging, as is practically always the case. Otherwise, u is 1 negative. On the other hand, we consider v as positive when the center of the refracted wavefronts is on the side opposite to that from which the light comes, that is, when the light leaves the lens in a converging beam. The radius of curvature of the first surface of the lens will be considered positive when that sur- face is convex to the incident light ; that of the second surface is positive when it is concave to the incident light. By this convention, all four of these quantities will be positive m the most common case, viz., when a double-convex lens forms a real image. In figure 42 LAL'B is a somewhat exaggerated diagram of a lens. is the center of. the incident waves, or source, I is the center of the emergent waves, or image. Let r t be the radius of curvature of the first surface, LAL', r a that of the second, LBL'. Figure 42 In developing the! formula, we shall use the following principle : In order that I shall be the image of 0, there must be the same number of wavelengths in every path from to I. In particular, there are as many in the distance OL -f- LI as in the straight path OABI. Of course this can be true only be- cause in part of the shorter path, viz., in the distance AB, the wavelength is shorter than in the air. We have already seen LENSES 79 that, if A is the wavelength in air, A/n will be that in glass whose index of refraction is n. Therefore the total number of wavelengths in the straight path is (OA + BI)/A + n X AB/A = (OA + BI + n X AB)/A The number in the path OL + LI is (OL + LI)/A Equating these two expressions, OL + LI = OA + BI + n X AB or OL OA + LI BI = n X AB Now draw the arcs LxL', from as center, and LyL', from I as center. Then OL = Ox, and LI = Iy. Therefore, Ox OA 4- yl BI = n X AB Ax + B y = n X AB But Ax is the sum of the sagittaVof the incident wavefront and the first lens-surface; By is the sum of the sagittas of the emergent wavefront and the second lens-surface: and AB is the sum of the sagittas of the two lens-surfaces, all with the same chord LL'. Therefore, we may substitute the appropriate values obtained from equation (1), and get 2u or or This is the approximate formula for a lens. It is ad- mittedly not accurate, and indeed* no perfectly accurate formula can be found. For the wavefront emerging from a lens is not accurately spherical. Consequently it has no true center and 1.1 11 /I , 1\ - -I 1- 4 n ( I u T L ' v ' r 2 \r l T 2 I 80 LIGHT there is no perfect focus. It is possible, by giving the lens- surfaces a special non-spherical form, to make the emergent wavefronts really spherical; but this can be done only for a certain fixed distance of from the lens, and the emerging waves are no longer accurately spherical if the source is moved closer to the lens or farther away. There is therefore no ad- vantage to be gained by using the mathematically correct form, except in the case of telescopic and microscopic objec- tives, which are always used under the same conditions. With spherical lens-faces, formula (7) is accurate enough for all ordinary purposes, provided' the lens is thin and its diameter is not more than 1/20 the distance u or v. For photographic lenses and microscopic objectives, which are thick and have relatively large diameters, it becomes very inaccurate. Since the right-hand member of (7) contains only terms which are constant for a given lens (r t , r 2 , and n) it is con- venient to replace it by a single symbol, 1/f, where f is known as the focal length of the lens. The formula then becomes -+-=* (8) u v f which is identical with one of the forms of the equation (4). The meaning of the focal length f is also the same in the case of mirror and lens. That is, f is the distance from the lens to the principal focus, which is the point to which incident plane waves would be brought to focus by the lens, or the point such that if the center of the incident wavefronts were located there the emergent wavefronts would be plane. In fact, the only differences between (4) and (8) are first the difference in convention as to sign, already explained, second, the fact that the focal length of a mirror is simply half the radius, while that of a lens is a function of two radii and an index of re- fraction, having the value f= ( *'** . . (9) If we solve equation (8) for v, we get uf v = 1 T=f LENSES 81 As already noted, u is practically always positive, so that if f is also positive, v will be + if u>f, infinite if u = f , and - if u < f . That is, the emergent wavefronts will be convergent if the source lies beyond the principal focus, plane if it is at the principal focus and divergent if it lies between the prin- cipal focus and the lens itself. A negative value for f itself means that plane waves fall- ing upon it from the left would not be converged to a point on the right, but diverged as if they came from a point on the same side as the incident light. Equation (9) shows that in order for f to be negative either r l and r, must both be nega- tive, or the larger one must be positive and the smaller nega- tive, since in all practical cases n is greater than 1. This is the same as saying that f is negative if the lens is thinner in the middle than at the edges, positive if thicker at the middle than at thp edges. In the latter case we say that the lens is converging or convex, in the former case diverging or concave, In figure 43 are shown three different types of converging, and three of diverging lens. In order from left to right, they are named planoconvex, double convex, concavoconvex (or menis- cus), convexoconcave, double concave, and planoconcave. Figure 43 From the elementary theory of lenses that we have given here, it is immaterial which face is turned toward the incident light, for equation (9) shows that r x and r 2 can be interchanged without affecting the value of f , and such an interchange would be the only effect of turning the lens around. That is, for example, the meniscus type has the same focal length no mat- ter whether the convex or the concave face be turned toward the incident light. But a more thorough study of lenses shows that there usually is a choice, depending upon the circumstances under which the lens is to be used. In some cases, it is best to use a meniscus or planoconvex lens, with the faces turned in a certain way, while in others a symmetrical double convex 82 LIGHT will function better, etc. The complete theory of lenses is a long- and difficult study in itself, and cannot be taken up in this book. Figure 44 The quantity u can never be negative so long as the lens receives the light directly from the source. But figure 44 shows a case where, for the second lens, L 2 , u is negative. (Here the wavefronts are not drawn, but the course of the light is suffi- ciently well indicated by the limiting rays of the beam.) is a point-source, the light from which would be brought by lens L t to a focus at I it Lens L, therefore receives convergent light whose center is at I x , and in order to find the position of the final image L we must substitute for u in equation (8) the numerical value of the distance L.,!^ with a negative sign in front of it, and then solve as usual for v. 32. Two lenses in contact. We can now prove that when two thin lenses are placed very close together they act approxi- mately as a single lens, the reciprocal of whose focal length is equal to the sum of the reciprocals of the focal lengths of the < r . Figure 45 two given lenses. See figure 45. Let f t be the focal length of L,, f 2 that of L,. Aplying equation (8) to each lens, we get - u, v, f, i+.L i U 2 V 2 ~~ . Adding these two equations, we get u, f, CHROMATIC ABERRATION 83 Since the center of the emergent wavefronts for the first lens, I,, is also the center of the incident wavefronts for the second, and since, L x and L 2 being very close together, they are almost the same distance from I 1? v, and u 2 are numerically practically equal, but opposite in algebraic sign. Therefore l/u 2 and l/v t cancel one another, and we have left 14- - 1 1 *i v 2 f, ' h f 2 Uj for the first lens is simply u for the combination, and v 2 for the second is v for the combination. Therefore, if we replace 1/f ! -f- 1/f 2 by the single constant 1/f, we get for the combina- tion the ordinary equation for a single lens 1/u 4- 1/v = 1/f where K+C y < -&; The reciprocal of the focal length of a lens is sometimes spoken of as its dioptric strength, and practical opticians adopt a lens of one meter focal length as the unit, calling it a lens of one diopter. A lens of two diopters would then be one of focal length 50 cm., a lens of % diopter one of 400 cm focal length, etc. The above demonstration tlien shows that when two thin lenses are placed very close together their dioptric strengths are added. This relation holds good even if one of the lenses is diverging, provided we take the sum of the reciprocals of the focal lengths in the algebraic sense, the focal length of the diverging lens being negative. We shall find the principle very useful in discussing "achromatic," or color-free, lenses. 33. Chromatic aberration. We have already seen that the index of refraction of a substance is different for different wavelengths, or colors; and since the focal length depends upon the index it is obvious that a lens, unlike a mirror, focusses different colors at different points. This is a serious defect in simple lenses, and it would be impossible to have very effective lenses for telescopes, microscopes, or cameras, if it were not. possible to avoid it in some degree. Figure 46 is a diagram, plotted to scale, which shows the variation in focal length with 84 LIGHT the wavelength of the light, for two different lenses, one of crown glass (dotted line) and one of flint glass (full line). i Each lens has a focal length of 100 inches for light of wavelength .0000589 cm. (yellow), and the ordinates show the differences between this and the focal length for any other wavelength plot- ted as abscissa. It is seen Figure 46 that the focal length for the blue differs by more than an inch from that for the yellow in the flint lens, by something less than this in the crown. This defect is known as chromatic aberration. The figure shows that it is greater for flint than for crown lenses. Not only does flint have a greater index than crown, but its relative dispersion, that is, the percent change in index for a given change in wavelength, is also greater. This fact enables us, by combining a crown converging with a flint diverging lens, to produce a combination known as an achromatic lens, in which, though the focal length still varies for different wave- lengths, the variation is relatively small. The plan adopted is to figure the two lenses so that the .focal length of the com- bination is the same for two chosen wavelengths, say one in the brighter red and one in the greenish blue. It will then be slightly less for wavelengths intermediate between these two, somewhat greater for the deep red and the blue and violet. 34. Achromatic lenses. In order to explain the production of achromatic lenses by a concrete example, we shall calculate in detail the radii of curvature for an achromatic of 100 inches focal length. We first choose, from a catalogue of optical glasses, two known respectively ias "S.40, medium phosphate crown," and "0.335, dense silicate flint." The refractive in- dices of each of these glasses is given for five different locations in the spectrum, known as the points A' (wavelength = .00007677cm., deep red), C (wavelength = .00006563cm., bright red), D (wavelength = .00005893cm., orange-yellow), F (wavelength = .00004862cm., blue-green), and G' (wavelength = .00004341cm., deep blue). The table of indices follows: ACHROMATIC LENSES 85 S.40 (crown) 0.335 (flint) A' 1.55354 1.62621 C 1.55678 1.63197 D 1.5590 1.6372 F 1.56415 1.65028 G' 1.56953 1.66152 We are to find what must be the radii of curvature of the crown glass converging and the flint glass diverging lens, in order that the combination shall have a focal length of 100 inches for the C and also for the F light. To simplify the problem, we shall assume that the second surface of the flint lens is flat, and that its first surface fits exactly over the second surface of the crown, so that the combina- tion will appear like figure 47, which is a very common type of achromatic. Then the radii of the four surfaces, beginning with the left hand, will be indicated by a, b, b, and o> . 47 Let F c ' and F c " represent respectively the focal lengths ot the crown lens and the flint lens, for the C light. Then, in order that the combination shall have a focal length of 100 inches for C light, we must have, by equation (10), 1 1 1 160~F C ' + F C " From equation (9), or from (7) and (8), we have that for any lens therefore, substituting the appropriate values of n, r l and r,, for both lenses, we get .jL- = _. 63197 jj Therefore 1 /1 1 \ 1 (11) 86 LIGHT Of course we get an exactly analogous equation from the fact that the focal length of the combination is also 100 inches for the F light, viz., the equation 4-56415(1 + J.)-. 86028 J- (12) In the two equations (11) and (12), we may regard I/a + 1/b on the one hand, and 1/b alone on the other, as two unknown quantities, and solve for their numerical values. The result is - 4- -i = .037269 4 = .016998 an b a = 49.33 inches. b= 58.84 inches. Therefore, if the crown lens be ground with convex sur- faces of radius 49.33 in. and 58.84 in. ! respectively, and the flint lens with one surface concave of radius 58.84 in. and the other surface plane, then the combination will have exactly the same focal length, viz., 100 in., for the bright red and the green- blue light. In order to find the focal length of the combina- tion for other colors, we can make the calculations very simply by using the already found values for the radii, and the appro- priate values for the' refractive indices. Thus, if F a , F^, and F g represent the focal lengths for the A' light, the D light, and the G' light respectively, *- = .55354 (- + -r ) -62621 ~= .55354 X -037269 .69621 X. 016996 F a =100.13 in. -i- = -55900 (- + 4 W .63720 4= -55900 X .037269 .63720 F d \ a b/ b X .016996 F d = 99.94 in. ^ = .56953 (- + 4) .66152 -= .56953 X .037269 .66152 .r g \ a b/ D X .016996 F g = 100.17 in. These results are plotted in figure 48, to the same scale used in figure 46. Since the lenses for which the latter figure is drawn are supposed to be ma,de from the identical glasses which we have used in our calculation, a comparison of the two figures shows very clearly the superiority of an achromatic IMAGE OF AN EXTENDED OBJECT 87 lens over a single-piece lens made from either of the glasses composing the achromatic. Simple lenses of crown glass are practically never used, ex- cept as spectacle-lenses, as condensers for lantern or mi- croscopes, and in some few other cases where good defini- tion is not required. Lenses are never made from flint glass alone. It would be less suitable than crown, not only Figure 48 because of its greater relative dispersion, but also because flint glasses are generally softer and more easily scratched than crown. Whenever, as in the example calculated above, the crown and flint components of an achromatic lens have one radius of curvature in common, so that they fit to one another, they are cemented together with Canada Balsam. This procedure prevents part of the loss of light that would otherwise occur by reflection at the two surfaces. 35. Image of extended object. Undeviated ray. Up to this point in our discussion of mirrors and lenses, we have always supposed the source to be located somewhere on the axis of the mirror or lens. But when we consider the image of an extended object we must enquire what happens when the source lies off the axis, for evidently not all points of an object of any size can lie on the axis. In this book we shall not at- tempt to give a mathematical treatment of this problem, on account of its difficulty, but merely state the result yielded by such an investigation, as follows: Let and I inl figure 49 Figure 49 be the positions respectively of a source on the axis of the lens and its image, as found by formula (8). Also let 0' be a point off the axis, but lying in a plane perpendicular to the axis through 0. Then, provided 00' is small compared to the dis- 88 LIGHT tance from the lens, the image of 0', which we shall call I', is found to lie very nearly in a plane perpendicular to the axis through I. These two planes, both perpendicular to the axis, and so situated that a point in one finds its image in the other, are called confocal planes. In order to locate, in the plane through I, that particular point which is the image of 0', we reason as follows: Among all the rays which diverge from 0' there will be one which, striking the first surface of the lens near the point where the axis penetrates it, will be deflected into the glass in such a way that it strikes the second surface at the same angle at which it left the first. For this ray, the lens acts merely as a flat plate of glass with parallel sides, and the ray on emerging will take a direction parallel to that which it had on entering. The desired point I' will be the point where this ray strikes the confocal plane through I. The ray in question may be called the undeviated ray, for although it suffers a slight lateral displacement in traversing the lens, its direction is not changed. The thinner the lens, the smaller will be the lateral displacement of the undeviated ray, and the closer will its point of entrance into the lens and its point of exit coincide with the geometrical center of the lens. There- fore, for thin lenses, we find the image of such a point as 0' by drawing a line from 0' through the center of the lens, and another through I perpendicular to the axis, their intersection giving the location of the image of to an accuracy sufficient for most practical purposes. 36. Magnification. Incidentally, this construction enables us to find the size of the image of an extended object such as the arrow 00' of figure 49. For, since OO' and II' subtend equal angles from the center of the lens, they must be propor- tional to the distance from the lens. That is, n/ v n-n 00' "a It is also evident that if object and image lie on opposite sides of the lens, as in figure 49, the image is inverted, while if they lie on the same side it is erect. MAGNIFICATION Figure 49 is drawn for a converging lens arranged to pro- duce a real image, but the facts stated above hold good whether the lens be converging or diverging, the image real or virtual. Similar conclusions hold for the reflected images from a mirror. Here also we have confocal planes, such that a point in one has its image in the other, but there is of course no such thing as an undeviated ray, since a change of direction is always present in reflection. However, if we draw from 0' in figure 50 the ray OT, to the point where the line 01 meets Figure 50 the mirror, the ray O'P will be reflected in the direction PI', where, by the laws of reflection, the angles O'PO and I'PI are equal, and the intersection of this line with the plane confocal to the plane of 0' will give the image I'. It follows at once that equation (13) holds for mirrors as well as lenses. But in the case of a mirror, the image is inverted if it lies on the same side as the object, erect if on the opposite side. An important application of the principles just stated is illustrated in figure 51 .' Suppose there are two stars, prac- tically at an infinite distance, in the direction from the lens indicated by the letters C and D, the arrows indicating the Figure 51 direction in which the light is propagated from them, in plane waves. If either star lay on the prolongation of the axis BM, its light would be focussed at the principal focus F. Other- wise, if the angles CBM and DBM are small, not more than a few degrees, the image; of each star will lie in what is called the principal focal plane of the lens, a plane through F per- 90 LIGHT pendicular to the axis. For plane waves, every line perpen- dicular to the wavefronts is a ray. Therefore, if we draw through the center of the) lens a line perpendicular to each set of incident wavefronts, these will be the undeviated rays from the two stars ; and the points where these lines meet the prin- cipal focal plane, c and d, will be the images of the two stars. The angle cBd, subtended by the images from the center of the lens, is then equal to the angle subtended by the stars them- selves from the center of the lens or indeed from any terres- trial point since the distance of the stars is so great. This angle is given in radian measure, to a sufficiently close approxi- mation, as the quotient of the distance cd divided by the focal length. This principle is used in practical astronomy for measur- ing the angular distance between double stars. There are two methods of measuring the distance cd between the images. One is to place a photographic plate directly in the focal plane, expose it to the light from the stars, and then develop it by the usual photographic processes, which leave a little blackened dot where each star-image falls. The distance between these dots is accurately measured on a dividing-engine. 37. Micrometer. The other method is to use a micrometer, the essential part of which is a small metal frame arranged so that it can slide in a plane perpendicular to the axis' of the lens. A fine spider-thread is stretched across this frame, so that it lies in the focal plane, perpendicular to the direction in which the frame slides. The whole micrometer is turned about the axis of the lens, till the direction in which the frame slides is parallel to the line joining the two star-images, and the frame is then moved by a fine-pitched screw to which it is attached, so that the spider- thread, commonly called the cross-hair, lies first on one image, then on the other. The pitch of the screw is known, so that the number of its revolutions necessary to move the cross-hair from one image to the other gives the distance. In order to make the cross-hair and star- images clearly visible, a short-focus lens, or combination of Itmses, called the eyepiece, is placed just behind the focal plane. The eye sees magnified images of the cross-hair and the two original star-images. DEFECTS OF MIRRORS AND LENSES 91 A similar method is used in certain surveying instruments, although the conditions are somewhat different. The objects observed are not infinitely distant, and the images are con- sequently not formed exactly in the principal focal plane. 38. Imperfections of mirrors and lenses. The student will no doubt have drawn for himself the conclusion that, quite apart from chromatic aberration in lenses, both lenses and mirrors are far from perfect optical instruments, since our formula? are only approximations. Such a conclusion is un- doubtedly correct, and it will be worth while to enumerate the more common faults. . We shall speak principally of lenses, but what is said applies- also to mirrors, for all the faults found in lenses, except those due to chromatic aberration and absorption, are also shared by mirrors, in many cases to a greater degree. In the first place, owing to the fact that light is a wave motion, and therefore does not travel absolutely in straight lines, no optical instrument, whether it be made up of lenses, mirrors, or other elements, and no matter how perfect the workmanship may be. can produce from a point source an image which is a real mathematical point. For instance, al- though a star, on account of its great distance, may be regarded as a point source, its image as produced by the most perfect telescope is not a mathematical point, but a very small disc surrounded by a series of faint rings. If the lens is well made and of large diameter, the diameter of the disc and the sur- rounding rings is so small that the latter can be seen only by highly magnifying them, and they become a source of trouble only in the most exacting work with telescope or microscope. The nature of this fault will be considered later under the head of diffraction, sections 72 and 73. 39. Spherical aberration. Another fault is known as spherical aberration. Quite apart from the just-mentioned difficulty of diffraction, and from chromatic aberration, the rays coming through the edges of a lens are not brought to the same focus as those coming through near the center. This follows from the fact mentioned above, that when a spherical wavefront is refracted at a spherical surface, it emerges not truly spherical. Figure 52 illustrates this defect in an exag- gerated manner. The rays are drawn, but not the wavefronts. 92 LIGHT is a point source, from which all the rays originate. When they emerge from the lens, they do not converge to a single point. Since the central part of the emergent wavefront, say the part to which rays between those marked 5 and 7 belong, Figure 52 is very nearly spherical, these rays will all intersect nearly at a single point f. Rays 4 and 5, however, cross before reaching f; at such a point as e, and the corresponding rays 7 and 8 at e'. Rays 3 and 4 will cross still nearer the lens, as at c, 3 and 2 at b, 2 and 1 at a, etc. Consequently, instead of having a single point f as the image of 0, we may say that the image is the line abcefe'c'b'a', or rather, the surface formed by re- volving this line about the axis of the lens. This surface is roughly conical, with a point or "cusp" at f, and by far the greater part of the light is concentrated at this point, which we commonly regard as the proper image. Nevertheless, much light fails to pass through f, and if a screen were placed at that point we should see a sort of halo surrounding the bright center, caused by light which came to focus before reaching the screen. The curve abcefe'c'b'a' is called a caustic, and the corresponding surface a caustic surface. A familiar example of a caustic is the socalled "cow's hoof" seen on the ^urf ace of a glass of milk. It is formed by reflection from the inner sur- face/ of the rim of the glass, which acts as a concave cylindrical mirror, reflecting light from a nearby window, or any other conveniently placed source of light. 40. Curvature of field. Still another defect is curvature of the field. Referring to figure 49, it was stated in the text that if 0' lies in the plane of 0, its image I' is very nearly in the plane of I, provided that the distance 00' is small com- pared to the distances of and 0' from the lens; and the two planes perpendicular to the axis were called confocal plane. More accurately, the surface which is confocal to the ASTIGMATISM 93 plane through O is not a plane but a slightly curved surface, concave toward the lens and nearly plane in the neighborhood of the axis. As a special case, suppose O ' and 0' are so far away that the waves reaching the lens from them are prac- tically plane. We may then regard 0, 0', and all other suf- ficiently distant objects as being in a plane perpendicular to the axis of the lens. Under these circumstances, if a screen be placed perpendicular to the axis at the principal focus, those of the distant objects which subtend only a small angle with the axis will be sharply in focus on the screen, but the others will be blurred, and the screen must be moved closer to the lens to bring them into sharp focus. 41. Astigmatism. This is a fault which shows itself par- ticularly for pencils of light which strike a lens or mirror diagonally. Under such circumstances, the image of a true point tends to become a pair of short lines, perpendicular to one another, but not intersecting. Figure 53 is intended to Figure 53 make this plain. The ellipse L represents a lens seen in per- spective. For convenience in explanation, suppose an opaque piece of paper pasted over the^ face of the lens, with a square hole, so that the beam that comes through is limited to what passes through this square aperture. Only the rays coming from the four corners are shown. Rays A and B intersect at the point R, but A and C intersect at P, nearer the lens. Similarly, C and D intersect at S, B and D at Q. Rays B and C do not intersect at all, neither do A and D. There will be a horizontal focal line PQ, for the intersection ' of rays in a vertical plane, and a vertical focal line SR, for the inter- 94 LIGHT . section of rays in a horizontal plane. The two focal lines can be shown very nicely by holding a converging lens, or better still a concave mirror, so that it receives somewhat obliquely the light from the sun, and moving a white card back and forth till the two focal lines are found. 42. Lenses for special purposes. In spite of this long list of faults, lenses function in a very satisfactory manner for most purposes. Consider for instance the lenses used as objectives in telescopes. (Telescopes are considered in detail in the following chapter.) Chromatic aberration is hardly perceptible if the lens is properly constructed of flint and crown glass in the manner already described, for the focal length is very nearly the same for all wavelengths except the blue and violet; and since these colors in most light-sources have feeble luminosity, their being somewhat out of focus hardly affects the sharpness of the images. Spherical aber- ration is negligible because the area of the lens is made small enough so that only that part of the emergent wavefront is used which is nearly spherical. As a rule, the diameter of a telescope objective is somewhere between 1/30 and 1/16 of the focal length, making the angular diameter of the cone of light which it transmits relatively small. Finally, the curvature of the field and astigmatism produce a negligible effect, on ac- count of the smallness of the field. In astronomical telescopes, it is seldom necessary to use a field of as much as one degree. Consequently, only the flatter portion of the field is used, and no pencil of light that is visible in the eyepiece traverses the lens with enough obliquity to cause appreciable astigmatism. -The design of a telescope objective is therefore relatively sim- ple, and its excellence depends mainly on the quality of the workmanship and the homogeneity of the glass. This latter condition is by no means easy to fulfill in such large discs of glass as were used in making the objective of the Lick tele- scope (36 inches in diameter) or that of the Yerkes (40 inches). Camera lenses are used under more exacting circumstances. In order that the lens may be "fast," that is, give sufficient illumination with very short exposure-time, it must have a diameter as great as 1/5 or 1/6 the focal length, giving great opportunity for spherical aberration. Moreover, the extent of field used is large, since the dimensions of the photographic LENSES FOR SPECIAL PURPOSES 95 plate are nearly as great as the focal length. Consequently, troubles due to curvature of the field and astigmatism are likely to appear. The manufacturers of photographic lenses have, however, achieved remarkable success in combating these difficulties, so that a first-class lens shows them to only a limited extent. Even with the best lenses, however, if they are used with full aperture, the corners of the picture are slightly out of focus due to curvature of field, and show a slight drawing out of points into lines, which is the result of astigmatism. A good photographic lens is made in two parts, separated by an air-space, and each part is composed of several pieces of glass. It is by altering the composition of these separate pieces and the curvature of their surfaces that the designers have suc- ceeded in reducing largely, but not entirely, the inherent lens- defects. A photographic lens cannot properly be regarded as a thin lens. Microscopic objectives also, if of high power, are objects of elaborate design, consisting of many pieces of glass. The thickness of such a so-called lens (really it is a combination of a number of lenses) is much greater than the equivalent focal length of the combination. Problems. 1. If a plate- glass window, index 1.58, appears to one looking into it to be 8 mm. thick, what is the actual thickness? 2. What must be the radius of curvature of a symmetrical converging lens of crown glass, to have a focal length 1 meter f 3. An object is 4 ft. from a white screen. Find two posi- tions in which a lens of 8 inch focus can be placed, to form an image of the object on the screen. 4. An object is 8 inches from a screen. Where should a concave mirror of 2 foot radius be placed to form an image of the object on the screen. 5. Show that problem 3 cannot be solved if the focal length of the lens is more than 12 inches. 6. A lens of 3 foot focus forms images of two stars in its principal focal plane, and a micrometer is used to find tho distance between the images. It takes 12.85 turns of the screw to move the cross-hair from one image to the other, and the 96 LIGHT screw has 50 threads to the inch. Find the angle between the stars. 7. Show how an achromatic diverging lens can be made, aad write the equations from which the curvature of its sur- faces can be found, using the data for the glasses given in paragraph 38. 8. In what position must the eye be placed, to see an image formed by a lens ox* mirror, if a screen is not used? Why is it usually easier to find a virtual than a real image? 9. If a camera lens has a focal length of 8 inches, find the proper position for focus on an object 5 ft. away, and the length of this image if the object is 2 ft. long. 10. Show that if a camera of focus 6 inches is focussed for infinitely distant objects, any object more than 40 ft. away will be less than 5/64 inch out of focus. 11. Explain why " depth of focus" in a camera is impos- sible to obtain without sacrificing "speed." CHAPTER VI. 43. The telescope. 44. Magnifying power. 45. Ramsden eyepiece. 46. Opera glass. 47. Prism binocular. 48. Reflecting telescopes. 49. Simple microscope. 50. Compound microscope. 51. Projection lan- terns. 43. The telescope. The essential part of a telescope is two lenses, a long-focus, large diameter achromatic, turned toward the object in view and therefore known as the objective, and a smaller lens (or, as we shall see later, more commonly a pair of lenses) called the eyepiece. Figure 54 shows a sim- ple diagram of a telescope. The object viewed is supposed to be an arrow, very far away, but so large that in spite of distance it covers an angle of a degree or so. If this conception seems too artificial, we may think of the point of the arrow as repre- senting one star, the butt another. Wavefronts are not indi- cated, but lines are drawn to show the course, through the in- strument, of the cone of light from each end of the arrow. Dotted lines show the undeviated rays for each lens. A real inverted image of the object is formed in the focal plane of the objective, from which the waves continue on, diverging from this image exactly as if it were a material object, except that the light is limited to a comparatively small cone. This light falls upon the eyepiece, which forms with it a second image, really an image of an image. Since the rays that form any point of the first image are limited to the cone that comes through the objective, it may well happen that the undeviated ray drawn from this point through the center of the eyepiece lies outside the cone and therefore does not exist as a real ray. But the position of the second image must cer- (97) 98 LIGHT tainly be independent of the diameter of the objective, and therefore we are at liberty in such a case to find that position by drawing fictitious undeviated rays just as if they really did exist. The figure is drawn for such a case. The position of the second image depends of course upon the location of the eyepiece which is mounted so that the observer can slide it at will through a short distance toward or away from the objective. Most observers place it so that the first image lies a little within its principal focus. Then the second image, the one which the eye sees, is virtual, still inverted, and on the same side as the first image, but farther away. This is shown by AB in the figure. If the principal focus x were placed just at the real image ab, as is sometimes done by persons of far- sighted or normal vision, AB would be thrown back to infinity, like the original object, but would still subtend a much greater angle than the latter. 44. Magnifying power. We take as a measure of the mag- nifying power of the telescope the ratio of the angle subtended by the image AB to that subtended by the original object, and in calculating its numerical value we assume, for the sake of definiteness, that the principal focus of the eyepiece coincides exactly with that of the objective, that is, that x lies exactly on ab, putting AB at an infinite distance. With both the original object and the final image so far away, it does not matter what point is chosen as the apex of the angles sub- tended. That subtended by the object is aOb, which, in radian units, has the value ab/F, F being the focal length of the objective. That subtended by the image AB is ApB, whose value is ab/f, f being the focal length of the eyepiece. There- fore the magnifying power is ab f F 1 z L i IT ab f F Therefore, for high magnifying power, we should use a long- focus objective and a short-focus eyepiece. Usually, a large telescope is provided with several eyepieces of different focal length, so that the magnifying power can be changed at will. For some purposes, high magnification is desirable, for others MAGNIFYING POWER OF TELESCOPES 99 not. The higher the magnification, the smaller the visible field: that is the smaller the area that can be seen at once. For instance, with high magnification only a very small part of the surface of the sun or moon can be seen. There are other practical limitations to the magnifying power that can be used with advantage. As we have previous- ly stated, the real imag'e produced by the objective is not a true picture of the object, but is slightly hazy at the edges, and is surrounded by faint diffraction rings. Any increase in mag- nification beyond the point where these rings or bands become visible is useless, for it merely magnifies the bands and the haziness along with the rest of the image and contributes noth- ing to distinctness of vision. Astronomers also find that cer- tain conditions of the atmosphere, caused no doubt by irregu- larities in density, produce a haziness or fuzziness of image known as "bad seeing," which is worse than the diffraction difficulty. In fact, with a large telescope, the appearance of the diffraction bands oh high magnification indicates that the "seeing is good,' 7 for poor seeing conditions cause them to be blurred out of recognition. With only moderately good seeing, an astronomer will use moderate magnification, and with very bad seeing he will abstain from observing at all. Figure 54 shows that at a certain place cd the two cones of light from the head and the butt of the arrow cross. In fact, there is a little circle at this place through which passes every cone of light that traverses the telescope, and it Is not hard to show that this circle is nothing more nor less than the image of the objective lens formed by the eyepiece. For best vision, the eye should be held so that its pupil coincides with this small circle, which, by the way, is called the exit -pupil of the telescope. For the figure shows clearly that if the eye is held much closer to the eyepiece, or much farther from it, only cones of light from the middle part of the arrow (cones not drawn in the figure) would enter the pupil of the eye, unless the latter were very large. To see the whole image AB at once would then be impossible, though one could see different parts of it at a time by moving the eye up or down, so as to receive the light from those parts. That part of the image that can be seen in any one position of the eye is called the field of view, 100 LIGHT and it is greatest when the pupil of the eye coincides with the exit-pupil. If the eye is in thisi most favorable position, the field of view is still limited by the diameter and the position of the eyepiece. Reference to the figure shows that if the arrow were much larger than it is there made, the cones of light from the ends of the image would partly or wholly miss the eye- piece. If they missed the latter entirely, these ends would be completely invisible in the telescope, while if only part of the cone fell on it, the image would be faint at the ends. The simple eyepiece shown in the figure is not suitable for produc- ing a large and uniformly illuminated field of view. Jt is desirable that the eyepiece should come very close to the real image ab; but in order to have this occur with a single-lens eyepiece, the focal length of the latter would have to be inordi- nately short; and if, in addition, the diameter were made great, spherical aberration and other lens defects would become too pronounced. 45. Rainsden eyepiece. For the reasons outlined above, much ingenuity has been applied in devising eyepieces which consist of combinations of lenses instead of single lenses, so as to give a larger field for the same magnifying power. The best known of these is the Ramsden eyepiece, which functions very satisfactorily, and is used on most telescopes. It is shown in figure 55. The objective of the telescope has been omitted, from the drawing, but the real image ab and cones of light forming its ends are shown just as they are in figure 54. The eyepiece consists of two identi- cal planoconvex lenses, mounted rigidly in a metal tube, and separated by a distance equal to % the focal length of either. It can be shown that such a pair- is equivalent, so far as magnification is concerned, to a single lens whose focal length is % that of either component. The front lens, called the field-lens, is placed very close to the real image ab, and therefore forms from it a virtual image a'b', slightly larger, and slightly farther away. In fact, a'b', comes just at the principal focus of the rear lens of the combination (called the eye-lens) or just within it. Accordingly, this latter forms the final virtual image, AB of figure 54, either at infinity OPERA GLASS 101 or at whatever distance from the eye is most suitable for the observer, adjustment being secured by sliding the whole eye- piece toward or away from the real image ab. Since ab lies close to the field lens, the latter receives and transmits light from an area of the image practically equal to the area of the lens itself, thus giving a field whose diameter is approximately equal to the diameter of the field-lens. The pencils from all parts of the image cross the axis just behind the eye-lens. This is the most convenient place for the exit-pupil, for the eye can then be placed close up to the end of the eyepiece. It is found further that this combination of two planoconvex lenses ia almost free from spherical aberration, and the chromatic aber- ration is of such a nature asi to be hardly perceptible. If a micrometer is used with the telescope, it is placed so that the crosshairs move exactly in the plane ab in which the real image lies. The proper method of adjusting the telescope, known as focussing, is as follows : First, the eyepiece is pushed in or out until the crosshairs are visible, clearly and without eyestrain. Then, by means of a suitable slide in the telescope tube, the eyepiece and crosshairs together are pushed in or out till the image of the object viewed is also seen clearly, and there is no parallax between the image and the crosshairs. One defect of the kind of telescope we have been describ- ing is that the image is inverted. This is not a disadvantage in astronomical telescopes, but for terrestrial telescopes it is inconvenient. In such instruments the image is usually rein- verted by making the tube of the instrument very long, and inserting between the eyepiece and the real image formed by the objective another lens or pair of lenses of rather short focal length, whose function is to receive the light from the real image (ab of figure 54) and form therefrom another real image which is reinverted and therefore right side up. The eye- piece then forms from this image a magnified virtual image which the eye sees. The great length of tube necessary in this form of instrument makes it inconvenient except for small spy- glasses. 46. Opera glass. In the old-fashioned opera glass, shown in diagram in figure 56, the erection of the image is provided for by using a diverging lens for the eyepiece. This form of telescope is quite short; for, in order that the eyepiece may 102 LIGHT magnify the image, it is necessary that it should intercept the light between the objective and the latter 's principal focus. Thus the real image 1 ab is not actually formed at all, for it would come behind the eyepiece instead of before it. Con- Figure 56 verging waves strike the eyepiece, with their centers on ab. Therefore, in the formula or its equivalent uf -u f a must be taken as negative. Since f is also negative, because the lens is diverging, the numerator of the fraction in the last equation is positive, and the sign of v depends upon the sign of u f , where both u and f are essentially negative. In order to have a virtual image AB, v must be negative, therefore u must be greater in absolute amount than f ; that is, the prin- cipal focus of the eyepiece must lie between the latter and the place where the image ab would come if the lens were removed For example, let f = - 5cm., u = - 5.2cm. Then v = 130 cm. The observed image AB would then lie 130 cm. away on the side from which the light comes, and would be virtual. Also, it would be inverted as regards ab, erect as regards the original object. The erectness of image, and its shortness make this type of telescope convenient, but unfor- tunately its field is quite small. The diagram shows that there is no real exit-pupil, as there is in the ordinary form of tele- scope; that is there is no place where the eye can be placed so that it will receive every cone of light that does not miss PRISM BINOCULAR. REFLECTING TELESCOPE 103 the eyepiece. The eye must be moved about in order to see the whole image of any object viewed,; unless the object be relatively small. For this reason, such instruments are made only with small magnifying powers, say two or three diameters. 47. Prism binocular. The modern prism-binocular is a great improvement over the form of telescope described above. It is essentially a telescope of the form described (or rather, a pair of such telescopes, one for each eye), in which the length is very much reduced by four reflections in totally re- flecting prisms. Incidentally, the series of reflections rein- verts the image, so that it is possible to use an eyepiece like the Ramsden, with, its large field. An additional advantage lies in the fact that the arrangement of the two telescopes brings the objectives farther apart than the two eyes. This, being equivalent to a wider spacing of the eyes themselves, greatly increases the stereoscopic effect, or parallax, and brings the field into strong and pleasing relief. 48. Reflecting telescopes. A telescope composed of ob- jective lens and eyepiece is known, among astronomers as a refractor. A reflector is a telescope in which a concave mirror is substituted for the objective lens. At the present time the use of reflectors is confined almost exclusively to photographic work, for which purpose they possess several decided advan- tages. In the first place, they are free from chromatic aberra- tion. Secondly, they have no absorption, and this is very im- portant in photography, for the transmission of light through glass causes much of the photographically active ultraviolet light to be lost} by absorption. Finally, by making the con- cave reflecting surface parabolic instead of spherical, the prin- cipal focus is rendered absolutely free from spherical aberra- tion, and a small region in its neighborhood almost so, so that most beautiful definition is secured in photographing objects of such small angular dimensions as a star-cluster or a small nebula. The concave mirror is usually placed at one end of a long tube or frame work, the other end of which is open and pointed toward the celestial body to be photographed. Between the mirror and its principal focus, is placed a small plane mirror, set at an angle of 45 with the axis of the instrument. This reflects the light coming from the concave mirror to the side 104 LIGHT of the tube, where the photographic plate is exposed in the reflected position of the focal plane. A Ramsden eyepiece may also be placed in position to receive the light, but this has no function during the photographic process. It may be used however in pointing the instrument to the desired object. There are, however, other forms of reflecting telescopes. 49. Simple microscope. The word microscope usually means an instrument used for magnifying small objects close at hand which, like a telescope, has two optical parts, objective and eyepiece: but in stricter language such an instrument is called a compound microscope, while the name simple micro- scope is applied to a single ' lens used as a magnifier of low power. A simple microscope is held as close to the eye as con- venient, and the object to be examined is placed somewhat within the principal focus, so that the eye sees a magnified virtual image of it at the distance which is most suitable for distinct vision. For the normal eye this is about 25cm., though it differs with different individuals. Figure 57 shows the ar- rangement. L is the magnifying lens, F its principal focus, Figure 57 AB the object, A'B' the image seen by the eye. We take as a measure of the magnifying power the ratio of the angle which the image subtends at the eye to that which the object itself would subtend if the lens were removed and the object put back where it could be seen most distinctly, that is, where the image is in the figure. Since magnifying powers need only be known roughly, and since the eye is placed so close to the lens, it will suffice to consider the center of the lens, instead of the pupil COMPOUND MICROSCOPE 105 of the eye, as the apex of the angles. The angle subtended by the image is A'B'/v = AB/u The angle which, the object would subtend if placed at the distance v is AB/v Therefore the magnifying power is AB/u -f- AB/v = From the law of lenses I II u"~ v~~ --!=!- u f -=!+!- u f Therefore the magnifying power is 1 + v/f or 1 + 25/f if f is expressed in centimeters. As an example, the magnifying power of a lens of 5cm. focal length is 6. A Bamsden eyepiece used as a single lens makes a very good simple microscope. 50. Compound microscope. It is not practicable to get very high magnifying power with a single lens, for that would require such a short focal length that the lens defects such as chromatic and spherical aberration, curvature of field, etc., would be very prominent. Therefore, wherever high magnify- ing power is necessary, it is provided by a compound micro- scope. Figure 58 shows the arrangement of parts. The "ob- ject" is usually in the form of a slide, a very thin slice of the material to be examined enclosed between two thin glass plates. The slide is represented by the short arrow A in the diagram. Slides being more or less transparent, they are examined by transmitted light". A mirror M and a condenser C concentrate upon the slide, from below, a beam of light from a window or other broad illuminated area. The set of lenses forming the condenser are of low grade, for their function is merely to provide illumination, not to form a clear image of anything. 306 LIGHT The mirror and condensers are accessories rather than parts of the microscope proper. The latter consists of an objective and an eyepiece FE. The former is shown in the figure as a Figure 58 single hemispherical lens, but it is in fact compounded of several distinct units in order to secure chromatic and other corrections. The eyepiece shoAvn. in the figure is what is called the Huyghens type, which has certain advantages over the Ramsden eyepiece described in connection with telescopes, though it has also the disadvantage that it cannot be used in connection with crosshairs or micrometer. If it is necessary, as sometimes happens, to put a micrometer on a microscope, the Huyghens eyepiece must be exchanged for one of the Ramsden type. The objective would, but for the eyepiece, form a magni- fied real inverted image of the slide at I 15 but the field-lens F intercepts the converging light directed toward this image, and converges it still more, forming the real image I 2 , slightly PROJECTION LANTERNS 10? smaller and slightly lower. Usually a diaphragm is placed in the plane of L so as to limit the visible field to a small circle over which the illumination is uniform. The light diverging from I, then passes through the eye-lens E, which forms a mag- nified virtual image at I 3 , the image which the eye sees. For very high powers, an "oil-immersion" objective is used. The objective comes very close to the slide, and the space between is filled with a drop of oil having an index of refraction nearly the same as that of glass, so that one might regard the object as being imbedded in the objective. Under these circumstances, the resolving power of the microscope is somewhat increased, and the brightness is also increased because less light is lost by reflection from the bottom surface of the objective and the top surface of the slide. Certain particles too small to be seen with the bright back- ground illumination commonly used in a microscope, such as the particles in certain colloidal solutions, can be seen as bright points against a dark background if the illumination comes from the sides instead of from below. This is the principle of the so-called ' ' ultramicroscope. ' ' In the figure, the fainter lines represent the full beam of light from the point of the arrow through the objective and eyepiece. The same rays are shown below, from before they impinge upon the mirror till they strike the slide. Similar rays from the butt of the arrow, or from any other part of it, couid be drawn, but they are omitted for the sake of simplicity. Figure 59 51. Projection lanterns. Figure 59 shows the ordinary projection lantern, for throwing upon a screen an enlarged image of a lantern-slide. The slide S is so placed with refer- ence to the projecting lens L that the screen comes at the con- jugate focus, and the focal length of L must be chosen with due regard to the distance of the screen and the desired magni- fication. The rest of the apparatus is for obtaining the neces- sary illumination of the slide. Since the latter is more or less trasparent, the illumination is supplied by transmitted light. 108 LIGHT p is the positive, n the negative carbon of an arc-lamp, and C a condenser consisting of two plano-convex rough lenses. The slide should lie as close to the condenser as convenient, and the arc should be so placed that its light is focussed by the condenser through the slide upon the center of the projection lens, forming an image of the arc there. The lens L should be achromatic, in order to avoid chromatic aberration at the screen; and it should consist of two units with a diaphragm between, otherwise the picture on the screen may show some distortion. Except for these two points L need not be a high- grade lens. The pencil of light coming from any point on the slide is so narrow, as shown in the figure, that there is little opportunity for spherical aberration, astigmatism, or curva- ture of field to show any bad effect. The focussing is done by moving the lens L. If the illumi- nation of the image is not uniform, this is an indication that the arc is not in the proper place, or that the negative carbon is shutting off some of the light from the positive carbon. Lately it has become more common to substitute a high-power filament lamp instead of the arc. This makes the lantern much easier to operate, and the illumination, though slightly weaker, is strong enough in most cases. The opaque projec- tion lantern, one form of which is shown in figure 60, is used for projecting images of postcards, pic- tures or printed matter in books, etc. It differs from the slide-lantern in two es- 6u sential points. First, it is necessary to introduce a reflection (mirror m in the figure) be- tween picture and screen, to prevent the image from being either upside down or right side to left. Second, the original picture must be illuminated from the front, since transmitted light is out of the question, and this makes it difficult to get the illumina- tion strong enough. The projection lens L must be of excep- tionally large diameter for its focal length, and it must be well corrected for all the defects of lenses, since it receives a full beam of light. The arc is made to give an exceptionally large PROJECTION LANTERNS 100 amount of light by using large carbons and a very heavy cur- rent, and the condenser system is made to cover a very large angle from the arc. Usually a glass cell containing water is interposed, to cut out much of the infrared light, which would unduly heat the picture. For use in small rooms, where the picture on the screen need not be very large, a high-power tungsten filament lamp may replace the arc, but in such a case it is necessary to use a special screen made of filled can- vas covered with aluminum paint, which reflects more strongly than a simple white screen. Problems. 1. The objective of a telescope has a focal length of 30 ft. "What is the magnifying power, when an eyepiece of focal length y 2 inch is used? 2. Explain why it is that dirt, or even a large opaque obstacle, on the surface of the objective of a telescope, is never visible to a person looking through; the eyepiece, the only apparent effect being a general dimming of the image. 3. Prove that the " exit-pupil " is the image of the objec- tive as formed by the eyepiece. 4. A projection lantern is being planned for use in a certain room. The screen is to be 30 ft. from the slide, and it is desired that the image of the slide on the screen shall measure 58.5 X 72 inches. (A slide is 3.25 X 4 inches). What must be the focal length of the projection lensf 5. Explain completely why strong illumination is so much harder to obtain with the opaque projection lantern than with the ordinary lantern for slides. CHAPTER VII. 52. Prism spectroscope. 53. Bright-line spectra. 54. Spectral series. 55. Continuous spectra. 56. Dark-line spectra. 57. Absorp- tion by solids and liquids. 58. Continuous spectrum of an absolutely black body. 59. Planck's theory of "quanta." 60. The plane grating. 61. Why the lines are sharp. 62. Reflection gratings. 63. The con- cave grating. 64. The ultraviolet region. Fluorescence. Phosphor- escence. Photography. 65. The infrared region. 66. The bolometer. 67. The thermopile. 68. The Doppler principle. Motion of the stars. 52. Prism spectroscope. We have already considered the spectroscope to some extent, in the chapter on color, but we are now in a better position to understand its principles. The essential parts of a prism spectroscope, shown in figure 61, are the collimator C, the prism P (or a train of prisms), and the telescope T. The colli- mator is a tube, at one end of which is an achromatic lens, at the other end a fine slit. The latter is carefully adjusted at the principal focus of the former, so that light which enters the slit and passes through the lens emerges in accurately plane waves. The beam then passes through the prism and is dispersed, that is, the different wavelengths are deviated to different amounts. It is best to have the prism turned so that the region of the spectrum to be examined traverses it at minimum deviation. It should be borne in mind that when the light leaves the prism ; although the different wavelengths take different direc- tions, yet all the rays of -any one wavelength remain parallel to one another till they strike the objective of the telescope. Therefore the latter converges the waves of each particular length to a definite place in the principal focal plane. Con- sequently, there will be in this plane an image of the slit for each particular wavelength that enters the slit, and the whole (110) BRIGHT-MNE SPECTRA 111 array of these images constitute the spectrum of the light in question. The eyepiece of the telescope forms an enlarged vir- tual image of the spectrum, just as it would form a virtual image of any very distant object which the objective focussed in its focal plane. A spectrometer is a spectroscope provided with a large divided circle, so that the angular position of the prism, of the telescope, or of both, can be accurately measured. There is also a crosshair at the principal focus of the telescope. The spectrometer is used for making accurate measurements of the refractive indices of prisms, and for other angular measure- ments. A spectrograph is a spectroscope so arranged that the spectrum can be photographed instead of viewed directly. Any spectroscope can be converted into a spectrograph by removing the eyepiece from the telescope and putting a photographic plate in the focal plane of the objective; but it is better to remove the whole telescope, replacing it by what is really a long-focus camera, a light- tight box with a specially corrected photographic lens at one end and a holder for photographic plates at the other. An ordinary hand camera, focussed for distant objects, might be used instead, but in most cameras the focal length is rather short, and this causes the spectrum also to be short, since its length is proportional to the focal length of the projecting lens, other things being equal. 53. Bright-line spectra. The character of the spectrum seen in a spectroscope varies greatly with the chemical nature and physical condition of the body emitting the light that passes in through the slit. The flame of a Bunsen burner, except for the small bluish inner cone, is practically invisible, and if such a flame is placed before the slit, nothing is seen on looking into the spectroscope, as we should expect. But if a piece of asbestos soaked in a solution of common salt (sodium chloride, NaCl) or of any other compound of sodium, is put into the edge of the flame, the latter immediately becomes yel- low in color. If this yellow light enters the slit, the spectrum shows two fine yellow lines, that is two yellow images of the slit. The light has a slightly different wavelength in these 112 LIGHT images, .00005890cm. in one and .00005896cm. in the other.* We interpret the appearance of these lines as follows: atoms of sodium pass into the flame and, under the conditions exist- ing there, start into vibration with two different periods, thus starting in the ether waves of the two different wavelengths given. The fact that these particular lines appear in the spectrum, whatever compound of sodium be used, proves that it is the sodium, not the chlorine, that is responsible for them. Chlorine produces no color in a flame, though it can be excited to radia- tion by an electric spark. The appearance, in the spectrum of any source of light, of two bright lines of wavelength .00005890 cm. and .00005896 cm. is therefore a sure and delicate test for the presence of sodium. The merest traces of sodium, far too slight for detection by chemical methods, produce the distinc- tive coloration in a flame. One cannot, however, assume that a yellow color alone indicates sodium, for yellow includes a considerable range of wavelengths, and some other elements have in their spectra yellow lines, of wavelength different from those attributable to sodium. For instance, if the light from a Cooper-Hewitt electric lamp (mercury arc) be examined with, a spectroscope, it shows a number of lines, two of which have wavelengths .00005769cm. and .00005790cm., which brings them in the yellow region, but a slightly different part of the yellow from the sodium lines. In the Bunsen flame, sodium never shows anything but the two above-mentioned lines and a very faint green one, but there are circumstances in which it gives in addition a number of other lines ; as when metallic sodium or a salt of sodium is put into the crater of a carbon arc-light. *The two lines are so nearly the same in wavelength that when a single small prism is used in the spectroscope they appear as a single line. Spectroscopes of higher power show them as separate and dis- tinct, and the most powerful even show that the wavelength is not absolutely definite in either line. Each line has a small but perceptible ividth, showing that for each the wavelength varies between certain narrow limits. This statement is also true of all other spectrum lines, and it is impossible to obtain a bean) of light all of which has exactly the same wavelength. BRIGHT-LINE SPECTRA 113 In order to study the spectra oi the different elements, various means must be employed to get them into a condition where they emit their characteristic radiations. There are only a few elements which give their spectra in a flame, like sodium. An effective method in the case of metallic elements which do not rapidly oxidize or suffer other chemical change in air, is to pass an electric spark between points made from the metal in question, the spark being operated by an induction coil or transformer in parallel with a condenser. Another method is to take two carbon rods, bore a hole in one of them, fill it with the metal, and connect both rods to the terminals of a direct-current supply of not too low voltage with some resistance in series. When the ends of the rods are touched together and separated about a quarter of an inch, the intense heating at the point of contact vaporizes the carbon and forms a bridge of glowing vapor called the arc, across which the cur- rent continues to flow. Some of the element packed into the hole in the rod also vaporizes and contributes its vapor to the formation of the arc. If the light from the arc itself (the bridge of vapor, not the glowing ends of the rods) is passed through the slit of the spectroscope, a large number of lines appear, some of which are due. to the element in question, some to gaseous compounds of carbon, and some to such impurities as are always present in the rods. The spectrum of a gas is usually obtained by the use of a so-called vacuum-tube. This is a glass tube with a restricted middle portion, into opposite ends of which are sealed metallic terminals. The tube is evacuated of air, and enough of the gas is put in to exert a pressure of a few millimeters of mer- cury, after which the tube is hermetically sealed. The electric discharge of an induction coil is sent through the tube, from one terminal to the other, causing the gas inside to become luminous and emit its characteristic wavelengths. Every known element has more than one line in its spec- trum. No two elements have identical spectra, and so far as is known no two elements show the same line in common, with, the possible exception of hydrogen and helium. Besides the elements, certain compounds also emit spectra composed of lines. Thus, the blue inner cone of a Bunsen burner shows the spectrum of carbon monoxide, and there are several 114 LIGHT groups of lines in the spectrum from the carbon arc which are believed to originate in cyanogen gas, produced by the action of atmospheric nitrogen on the carbon poles. The lines in the spectra of compounds are arranged in groups of more or less regular order, technically known as bands. 54 Spectral series. The fact that a single element emits light of several wavelengths shows that the atom is capable of vibrating in several different frequencies, just as a stretched string or column of air can execute the vibrations that produce sound waves in several definite frequencies. A string can vibrate not only with its fundamental frequency, which we may call N, giving a sound of wavelength L, but also with a frequency twice as high, 2N (the octave), giving a wavelength L/2, a frequency 3N, giving wavelength L/3, and so on indefinitely. Therefore we might represent the whole series of sound wave- lengths emitted by the string with the single formula L A = n where L is a certain constant for the string, and n may have any integral value from 1 to oo. When n = 1, A is the wave- length of the fundamental, when n = 2, A is the wavelength of the first overtone, etc. A most natural question is the following : Are not also the different wavelengths of light, given out by such an ele- ment as sodium or hydrogen, related to one another in some simple numerical way, so that a single formula will represent all of them if different integral values are given to one of the symbols? This cannot be answered by an unqualified yes or no, but we can say that in some of the elements (particularly the metallic ones of small atomic weight) some of the lines can be represented in this way, though not by so simple a formula as applies to the acoustical vibrations of a string. The simplest case is that of hydrogen. Figure 62 is a photograph of the visible, and part of the ultraviolet, regions in the spectrum of this gas. A careful examination shows that the lines may be classified into two groups- First, there is a large number of lines without any apparent regularity of arrangement what- ever. Second, there are several lines, marked a, /?, y, etc., on SPECTRAL SERIES 115 the photograph, which show regularity in two respects. The one of greatest wavelength, a, is the strongest line in the whole spectrum, and each succeeding line of the group is weaker than the one before it- Moreover, when the wavelengths Figure 62 of these lines are measured, it is found that in passing through the series, from a through (3, y, 8, etc., they come closer and closer in wavelength, like a mathematical series approaching a limit. Balmer showed that the wavelengths of the whole group, including lines too far in the ultraviolet to show in this photo- graph, can be represented with a considerable degree of accura- cy by the single formula A. =. 00003646 Q2 ^ 4 where n may take any integral value from 3 on. If n = 3, we get the wavelength of the a line, if n = 4. we get /?, and so on*. This whole group of lines, represented by a single formula, constitutes a spectral series. Our knowledge of series has been greatly increased by the work of Rydberg, Kayser and Runge, and many other investigators. The hydro- gen spectrum shows another series in the far ultraviolet region, and still another in the infrared. Other elements also show series in their spectra, although a slightly more complicated formula is necessary to represent them. A comprehensive ac- count of series in spectra is found in French in the " Rapports Presentes au Congres International de Physique," 1900. A very good resume in English is contained in pages 559 to 621 of Baly's ' ' Spectroscopy, " second edition. In this chapter we shall not take up the attempts to explain the peculiar form of *The measured wavelengths of the first four of these lines, in centimeters, are .00006563, .00004862, 00004341, .00004102. 116 LIGHT the series relation, since the subject conies more appropriately after the introduction of the electromagnetic theory of light. 55. Continuous spectra. Spectral lines tend to become widened when the density of the radiating gas or vapor is in- creased. The hydrogen lines produced when a spark is passed between platinum points in the gas at atmospheric pressure are much broader and hazier than when it is in the rarefied con- dition of the vacuum-tube. If a large amount of sodium is present in the carbon arc, the two strong lines in the yellow, instead of being fine and sharp, become very broad, and can easily be made to run together and extend some distance be- yond their original positions on both sides, causing them to have the appearance of a single broad yellow band with hazy edges. This of course means that the wavelengths emitted are no longer confined even approximately to two definite numeri- cal values, but extend over a relatively wide range. Several causes contribute to this effect of increased density, one of which is probably the fact that any given atom is hindered, by the very close proximity of other atoms, in its natural free vibrations. At any rate, when an exceedingly dense gas, or a solid or liquid body, becomes luminous, the widening of the characteristic lines is so extreme that all possible wavelengths are emitted within the range of the visible spectrum and be- yond, and all appearance of definite lines is lost. The spec- trum is then said to be continuous, since it extends throughout a very wide range of wavelengths without a break in- continuity anywhere. In contradistinction from this, the kind of spectrum that we have found to be given out by rare gases, metallic vapors, etc., is called a bright-line spectrum. As an example, the hot carbon pole of an arc light gives a continuous spectrum, but the bridge of vapor between the two poles gives a bright- line spectrum. The following application of the principles of the spectro- scope to astronomical problems is interesting and instructive. A nebula is a celestial object which appears in the telescope as a cloud of gas, but the possibility exists that it may really be a swarm of stars, so close together and so far from us that the telescope is incapable of resolving them into discrete bodies. The spectroscope, however, shows that some nebulae have a con- DARK-LINE SPECTRA 117 tinuous spectrum, others a bright-line spectrum, so that with the aid of this instrument it is easy to pick out those that are gaseous. 56. Dark-line spectra. The sun and most of the stars show still a third type of spectrum, which may be said to be an exact reversal of the bright-line type. While the latter is an assemblage of scattered bright lines, in colors appropriate to the spectral region in which they fall, against a blank that is a black background, the former is an assemblage of fine black lines against an otherwise continuous colored background. It is therefore called a dark-line spectrum. It may be described as a continuous spectrum with certain definite wavelengths . miss- ing. The following experiment explains the cause of these dark lines in the solar spectrum. An arc lamp, a Bunsen burner, a converging lens, and a spectroscope are set up in line, so that the lens forms an image of the bright carbon pole on the slit of the spectroscope, and the light passes through the flame of the burner before reaching the slit. Some of the light passes into the collimator, and of course produces a continuous spec- trum. A little common salt is inserted in the flame, and im- mediately the sodium lines appear, not bright, however, as they would be in the absence of the light from the arc, but as apparently dark lines against the bright continuous spectrum of the arc. The sodium vapor absorbs from the light passing through it those particular wavelengths which, it is capable of emitting, and -absorbs more than it emits, thus making the lines appear black against the brighter arc spectrum, though in reality they are not absolutely black. Undoubtedly, the dark lines of the sun's spectrum are produced by absorption in the same way. The main bulk of the sun is believed to be an exceedingly dense gaseous mixture, as viscous as a liquid, and like a hot liquid it gives a strictly continuous spectrum. But surrounding this dense luminous portion is an envelope of cooler and rarer vapor containing many chemical elements. These absorb from the light passing' through them just those wavelengths which they can emit. When the light reaches the earth it is therefore deprived of these particular wavelengths, and the spectroscope shows black lines at the corresponding 118 LIGHT positions in the spectrum. By comparing- the positions of the black lines with the positions of the bright lines emitted by various terrestrial elements, it has been possible to identify on the sun most of the elements known to us on the earth. The moon and the planets, since they send us only light which they receive from the sun, have the same spectrum. Most of the fixed stars have spectra of the same character as the sun's, and in some cases only a careful examination can distinguish them from the latter. Thus the spectroscope proves that the fixed stars are bodies of the same general physical condition as our sun, in spite of great differences in size, mass, and temperature; and it also shows that the chemical elements present in the earth are distributed throughout the universe and probably make up the major part of its material, a fact which could hardly have been proved by any other means. 57. Absorption by solids and liquids. Solids and liquids also produce absorption of light, as we have already seen in the chapter on color. If a colored liquid, such as a solution of copper sulphate, potassium permanganate, or chlorophyll, is placed in front of the slit of a spectroscope, and the light from a source that would of itself alone give a continuous spectrum is passed through it, certain parts of the spectrum are absorbed in whole or in part. The black, or darkened, regions which then appear in the spectrum are not fine and sharp, like those figure 63 produced by sodium vapor in a Bunsen flame, but broad and hazy. They are called absorption bands. Figure 63 (A) is a photograph of the absorption spectrum of an alcoholic solution of chlorophyll, the green coloring matter of plants. Figure 63 (B) is a photograph of the complete spectrum, with the chloro- phyll absent, to serve as a comparison. BLACK-BODY SPECTRUM 119 58. Continuous spectrum of an absolutely black body. The strictly continuous spectra given out by hot solids and liquids differ from one another only as regards the distribu- tion of the energy in different wavelengths. Thus, the spec- trum from the pole of a carbon arc is not only brighter through- out than that from the filament of a tungsten incandescent lamp, it also has a greater proportio.n of its energy in; the shorter wavelengths than that from the same body when cool- er; but differences in material and surface condition have also great effect. The only kind of continuous spectrum that can be theoretically studied is that from what is called an absolute - Figure 64 ly black body, which is defined as one which absorbs all light, of whatsoever wavelength, that falls upon it, reflecting 1 none. Strictly speaking no objects are absolutely black, though such materials as lampblack and platinum-black nearly fulfill the definition. But it can be shown by theory that the inside of an enclosure, the walls of which are kept at a uniform and constant temperature, acts exactly like a theoretical absolutely black body; and it is possible to make use of this fact for experimental purposes by making a small hole in the wall of such an enclosure, through which light from the interior can pass out and enter the slit of a spectroscope. Figure 64 is a series of graphs, drawn for different temperatures of the radiating enclosure, of the results of measurements of such black-body continuous spectra. For each point on a curve, the abscissa represents the wavelength, the ordinate the correspond- ing energy. It will be noticed that for higher temperatures 120 LIGHT the energy is greater throughout, but particularly in the short- er wavelengths. 59. Planck's theory of "quanta". The relation between the energy of black body radiation and the wavelength, for a given temperature, as represented by the curves of figure 64, has been the subject of many exhaustive theoretical investiga- tions. It is extremely difficult to give a complete theoretical explanation of the exact form of these curves. Indeed, the man who made most progress toward this end, Max Planck, came to the conclusion that an explanation is impossible unless we make a very remarkable hypothesis in regard to the be havior of a radiating atom, which amounts to this that al- though an atom can absorb energy steadily and continuously, it can radiate only if and when it has acquired by absorption a certain definite quantum of energy, or an integral number of times that quantum; and when it does radiate it radiates away all the energy it contains. Just as by adopting the atomic theory of matter we abandon the ancient notion that matter is continuous, so Planck's hypothesis would lead to the con- clusion that energy also, so far as the radiation of it is con- cerned, is composed of discrete amounts. For an atom can radiate one quantum, or two, or three, etc., but not one and a fraction. This hypothesis, which has been named the "quantum theory," is so very different from our previous notions about energy that, in spite of Planck's success in deriving a formula for black body radiation which fits the experimental curves, it is doubtful if it could obtain much support were it not that a number of other phenomena, including such diverse things as the variation of specific heats with temperature, X-rays, and the explanation of spectral series, are made much more under- standable by means of the same hypothesis. Planck offers no explanation of why an atom should radiate in such a manner, and the whole question of the quantum theory is one of the puzzles of modern physics. The size of the quantum is not the same for all wave- lengths, but is directly proportional to the frequency, or in- versely to wavelength. That is, for any frequency v the small- est unit of energy radiated is PLANE GRATING 121 The multiplier h is an absolute constant, whose numerical value, in the c. g. s. system of units is h = 6.55 X 10- 27 It is commonly known as "Planck's constant." 60. The plane grating. So far, we have learned no way of measuring wavelengths except by simple interference experi- ments, such as that of Fresnel with the two mirrors, as described in section 22. That such a method is not capable of much, ac- curacy can be seen from the following considerations. Refer- ring again to figure 20, section 22, it will be recalled that we found that certain points C, M^, M/, M 2 , M/, etc., are very bright, and the points midway between dark. The determination of wavelength is made by measuring the interval between two successive bright spots, and also the distance between the two sources S T and S 2 , and their distance from the plane of the screen. Not only is the distance SjS,, difficult to measure, but also the distance between the bands, CMi,, M^IVL, M 2 M 3 , etc., cannot be measured accurately, because the bright points are not sharply defined but shade off gradually into darkness. The difficulty might be illustrated graphically by plotting abscissas to the right of the line AB in figure 20, the length of each abscissa representing the intensity of the illumination at the corresponding point on the screen. The graph that would be obtained would be like fig- ure 5 (X). Evidently the loca- tion of the places of maximum brilliancy is subject to consid- erable error, which would be much lessened if, instead of these broad maxima, we had sharp and clearcut bright lines separated by broad dark spaces, as indicated in figure 65 (Y). x Y Two other decided advantage? Figure 6S would also accrue : first, the maxima would be much brighter, since the light would be confined to a very narrow instead of a broad band, second, if more than one wavelength were pres- ent, the maxima due to the different colors would be much less likely to overlap. 122 LIGHT The desired result, making the maxima narrow, sharp and bright, can be secured by getting interference from more than two points at once. For instance, if Fresnel's mirror experi- ment could be arranged so that there were three regularly spaced apparent sources of light, instead of only the two, S : and S 2 , the maxima would be sharper and brighter, if there were four, they would be still sharper, and so on; but one of the best devices for the purpose is what is called a grating. In its theoretically simplest form, a grating is an opaque plate containing a large number of slits, parallel and spaced olose together at equal intervals. Light from a narrow source, like a distant slit, or a star, falls upon it and passes through the many narrow slits, producing interference bands on the other 1 side. Let AB, figure 66, represent a section of such a Figure 66 grating. The slits, shown in section at a, b, c, etc., are sup- posed to run perpendicular to the plane of the paper. We shall suppose that monochromatic plane waves are falling perpendic- ularly upon it, the advancing wavefronts being indicated at C. The slits, or transparent portions of the grating, may be regarded as centers froni which new wavelets start out. As these get farther from the grating, where their curvature is less, they 1 tend to combine into several different sets of plane wavefronts, moving in different directions. For instance, the 12th wave out from a, together with the 12th from each of the other openings, tends to form a plane wave, parallel to the PLANE CRATING 123 incident waves C. There will ber a continuous train of such waves, a few of which are indicated farther out at X. These, after passing through the lens, will, be brought to the prin- cipal focus P, and will form a bright spot there, exactly as if the grating were removed and the original wavefronts C fell directly upon the lens. An entirely different set of wavefronts will be formed by a combination of the 12th wavelet out from a, the llth from b, the 10th from c, and so on, the resulting wavefront being inclined at a certain angle to the original wavefronts. A few of the wavefronts formed in this manner are shown at Y. After traversing the lens, they will be brought to focus at a point 1^ in the principal focal plane. This point will of course be the intersection of the plane with the un- deviated ray for this set of waves, which is a line drawn through the optical center perpendicular to the wavefronts Y. Still a third set of wavefronts will be formed by a combination of the -12th wavelet from a, the 10th from b, the 8th from! c, etc. These, a few of which are drawn at Z, are still- more inclined to the original wavefronts, and are brought to focus farther out in the focal plane, at such a point as I,. It is not difficult to prove that if X represents the angle between the Y and the X wavefronts (which is the same as the angle IjOP) and 6 2 the angle between the Z and X wavefronts (the angle Tv>OP), A the wavelength of the light, and A, each bright point for A' will be focussed farther from P than the corresponding point for A, and con- versely if A'