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, if we hear u sound from a tuning fork, a piano string, a metal bell, etc., it is a simple matter to prove that they are vibrating, and that, when tin- motion ceases, the sound does also. Again, it is a familiar fact that some time elapses between the instants when tin- vibration begins and the sound is heard, and between those when the former stops and the latter ceases; for there is a considerable inter- val of time between the instants when a distant gun is seen to be fired and when the sound of the report is heard, or when a mt steam whistle is seen to blow and when the noise is heard, etc. This proves that \\ a uses the sensation time for its transmission through space. The fact that the product ion of the sensation depends upon the pres- ence of a vmtt' rf'il medium between the vibrating bnd\ and the ear may be proved by suspending the vibrating body in a space from which the air may be more an* 1 more exhausted ; as this is removed, leaving only the ether, the sound becomes less and less intense. Vibrating bodies will produce com- pre»ioiiul waves in a surrounding fluid, provided the fre- quency of vibration is sufficiently great; and the fact that SOUND the sensation of sound is due to these waves may be proved most simply by allowing them to produce vibrations with nodes and loops, and showing that they cause sounds. Methods of doing this will be described in a few pages. We shall discuss first the characteristics of different sounds and the physical cause of these differences, then describe a few typical musical instruments and some acoustic phe- nomena, and finally give the physical explanation of harmony in musical compositions, with a brief description of musical scales. Noise and Musical Notes. — If we analyze our sensations of sounds, we are led at once to recognize two great classes which we call in ordinary language noises and musical notes. The latter have all the characteristics of periodic motion ; they are continuous and uniform in character, and are pleas- ant to the ear. The former are discontinuous, with abrupt changes, and are often extremely disagreeable to -the ear. Thus, the sounds due to a tuning fork, to a piano string, to the column of air in an organ pipe, etc., are musical notes. But the sounds heard when a piece of paper is torn, when a wagon rolls over cobblestones, when a slate pencil is sharpened,
etc., are noises. We can study the nature of the vibrations of a body, as has been explained on page 319, and it is found that a musical note is always due to a periodic vibration ; a noise, to an extremely complex motion, consisting of differ- ent vibrations which differ slightly in frequency and which are rapidly damped. A confused vibration which causes a noise will produce a musical note, if the vibrating body is near a flight of steps ; for, when the pulse reaches the first step, a reflected pulse is produced; and in a similar manner others are produced when it reaches in turn the other steps. Therefore there will be in the air a series of reflected pulses at exactly equal inter- vals apart; and, as they reach the ear, a musical sound is heard. Thus, if one claps one's hands near a staircase, the noise is first heard, but it is followed immediately by a musical note. The same phenomenon occurs if u. noise is produced near a picket fence. ANALYSIS "1 SOUND'395 Simple and Complex Notes; Quality. — If we analyze our sensations of different musical notes. \\c (.ID >eparate them into two classes: one we call "pure" or "simple"; then: "complex." Thus, the note produced by vibrating metal plates like cymbals is complex, while that due to a tuning fork <>r to a stopped organ pipe is pure. In fact, all notes, with the exception of these last, are more or less complex. If we examine the corresponding vibrations, it is found that a pure note is always due to a simple harmonic vibration, and a complex note to a complex vibration. Complex vibrations can be analyzed, in accordance with Fourier's theorem, into simple harmonic components who>e frequencies are in the ratio 1:2:3: etc., or into other com- ponents not so related. In a similar manner, if one listens attentively to a complex sound, various simple pure notes may be distinguished. (This statement that the human ear analy/.es mechanically a complex wa\c into simpler com- ponents and hears the corresponding simple notes separately is known as Ohm's Law for Sound. ) If two complex notes differ, it is found that the corre- sponding complex vibrations differ; but t he converse state- ment that two different complex vibrations produce t\\o different complex notes is true only with one limitation. If the component parts of the two complex vibrations
differ in their frequencies or in their amplitudes, the corresponding notes are different ; but the ditVerenees in phase between the coni\| -nts may be different in the two complex vibrations or may be the same; they have no influence on the note. (This is seen to be in ace. n.l with ohm's law, as stated above.) Two different complex notes are said in "quality." Thus, the complex notes produced by the vibrations of the column of air in an organ pipe, of a violin string, of a piano siring, of the column of air in a horn, etc., all differ in qualit) ; and it is by this property that we recoglii/e the 396 SOUND nature of the source of the sound. This quality depends upon the number and amplitudes of the other component vibrations besides the fundamental present in the complex vibrations ; it does not depend, however, as has been already said, upon the relative phases of these component vibra- tions. Pure notes are never used in music, because -they lack what may be called "character," or individuality. Notes that are useful for musical compositions must be complex to a certain extent, seven or more components often being present. Analysis of Notes. — This process of analyzing a complex sound or a complex vibration into its harmonic components is greatly helped by the use of resonators. Helmholtz in his epoch-making work used those of the form shown in the cut. (One of their advan- tages is that when the inclosed air is set in vibration, the motion is simple harmonic.) He constructed a set of them, and accurately determined the frequencies of the vibrations of their inclosed volumes of air. Then by bringing them in turn near a vibrating body, he could tell by holding one of the ends of the resonator near the ear whether the par- ticular vibration that corresponded to that of the air in the resonator was present in the complex vibration of the body ; for, if it were, the corresponding sound would be intensified. In this manner it is a simple matter to detect the components. In the best and most recent work on analysis of sounds, phonographs are used, and the traces are magnified. FIG. 180. — A Helmholtz resonator. A\AL}'SIS OF SOUND 397 Several other illustrations of resonance are worth mentioning. When a large seashell or a vase is lu-M near the
ear, the roaring that is heard is due to the resonance of the inclosed air produced by certain sounds in tin- room. The sound may be varied by partly closing the opening of the sht'll or vase. The passage leading from the outside of the head into the eardrum forms a small resonator, and its action is often noticed when one is li>t«Miing to an orchestra, by the strong resonance of certain very shrill sounds like the buzzing of insects. Helmholtz performed with his set of resonators the con- verse of the analysis of a complex sound; he produced one by HUM MS of the superposition of simple harmonic vibrations or of pure notes. He arranged in front of each resonator a tuning fork whose frequency of vibration was the same as that of the air in the resonator, and adjusted electro-magnets to these forks in such a manner that he could set them vibrating and maintain them in motion. He could also alter their amplitude. Then by making different forks vibrate he able to produce different complex sounds, and in fact to imitate the sounds characteristic of different instruments. In a complex sound, the component note corresponding to the fundamental vibration is called the "fundamental"; and other notes "overtones." If the component vibrations form a harmonic series, the component notes are also called k* harmonics." Pitch and Loudness. — If we compare two simple notes, we recogni/e the fact that they may differ in two ways, in shrillness and in loudness. Thus, the notes of a piccolo are shriller than those from an nr-^an pipe; and any note of an in may keep the same shrillness and yet may vary in loudness. If we compare the < : ions, we tind that in every case if one note is shrill another, the frequency of its vibration, or rather the number • reaching the ear in a unit of time, is the greater. (This -st observed by G Kurt her, we find that ne note deoreaaei in Umd increases in amplitude, other things remaining unchanged. 398 We cannot measure the shrillness of a note, because we cannot imagine a unit of shrillness nor the idea of shrillness being made up of parts which can be compared. We can, however, give a number to the shrillness of a note by assign- ing it one equal to the frequency of the vibration, if the vibrating body and the
observer are at rest relatively; or, more generally, we assign a number equal to the number of waves reaching the ear in one second. (See Doppler's Prin- ciple, page 345.) This number is called the "pitch" of the sound. Similarly, the pitch of a complex note is denned to be the pitch of its fundamental. The loudness of a sound, either pure or complex, varies as the intensity of the waves producing it. It is thus seen why, when a sounding body approaches the ear, the loudness of the sound heard increases. We can measure.the intensity of the waves, but we have no method of measuring the loudness, for this is a sensation, and not a physical quantity. Audibility of Waves. — In order to produce waves in the air, the frequency of the vibration must exceed a certain limit, as has been explained ; otherwise the air flows, but is not com- pressed. But all waves in the air do not affect our sense of hearing ; for this sense depends upon disturbances being conveyed to the nerve endings from the external air by a mechanism whose parts are set in motion by waves of certain wave lengths, and not by others. Thus, it is found that waves whose wave number is greater than 20,000 per second or less than 30 do not in general produce sounds ; but, of course, these limits are only approximate, and vary greatly with different individuals. In musical compositions as played by orchestras the maximum range of pitch is about from 40 to 4000. The Human Ear. — For a full description of the human ear reference should be made to some treatise on Physiology ; it is necessary to mention only a few details here. The ear consists of three parts : the external ear, which ends at the ear-.1 YALY818 OP SOtTJTD (Inini: the- middle car, which is connected with the throat and mouth I iy a tube, and in which there arc three little hones witli Hex il)lc connect ions, tli us making a mechanism joining the drum to a membrane which closes one opening into the third por- tion of ear; the inner ear, which is entirely inclosed in the hone of the skull, and which consists of several cavities tilled with a liquid. In one of these cavities there is a minute fibres of regularly decreasing length, with which the nerve endings of some of the auditory nerves are connected, and which are thought to
play the part of reso- nators for musical notes. Other branches of the auditory nerve end in another cavity under conditions which have led several scientists to believe that their function was to respond to noises. In any case it is easy to trace a mechanical con- nection between the waves in the air and the nerve endings through the eardrum, the three bones, and the liquid in the inner ear. The student will find in the work of Helmholtz a full discussion of these various steps. CHAPTER XXIII MUSICAL INSTRUMENTS WE shall discuss only two types of musical instruments: stringed and wind instruments. The commonest stringed instruments are the piano, the violin, the violoncello, and the harp; and the commonest wind instruments are the organ, the flute, and the horn. Stringed Instruments. — For present purposes we may regard the vibrations of the strings in any stringed instru- ment as being identical with those of a perfectly flexible cord, although in reality musical strings are far from being per- fectly flexible, and their elasticity plays a part in addition to their tension. Only transverse vibrations are ever used. We saw on page 353 that the frequency of the fundamental 1 \T vibration of a cord was given by the relation N=—-\—'> 2 L * a where L is the length of the cord, T its tension, and d its mass per unit length; and that the parti als had frequencies 2 JV, 3 JV, etc. This formula explains how a string may be "tuned " by altering its tension ; how its frequency may be altered by shortening its length, as is done in violins, etc., by means of the fingers ; and why the different strings of any one instru- ment are made of different densities. When a string is struck at random or is plucked, the vibra- tion is complex ; but those components are absent which would have a node at the point struck or plucked ; thus the quality of the note depends largely upon where this point is, as is shown in the use of violins. The difference between pianos as made by different makers lies to a great extent in 400 MUSICAL 1. \STlirMK\TS 401 the point of the strings which is struck, in the size and hard- - of the fc' hummer," and in the duration of the blow; for these alU influence the quality ofrthe
notes heard. All stringed instruments, with a few exceptions, have the strings stretched between pegs which are fastened to a wooden board or box. Owing to the vibrations of the strings, and the resulting motion of the pegs, this board is made to vibrate; and, since these vibrations are "forced," they imitate more or less closely in character those of the strings. But, of course, there are differences depending upon the thickness, area, stiffness, etc., of the boards. Simi- larly, if there is a box or cavity, the inclosed air may be set vibrating. The vibrations of this "sounding" board or box affect the surrounding air much more than do those of the tine string; and so the sound we hear depends to a great extent upon the former vibrations. We see, therefore, the reasons why the violins of certain makers have such great value, owing to their skill in the construction of the wooden parts. Wind Instruments. — We have given the theory of the vibrations of a enlnmn of air on pages 353-358, and have shown that in the case of a column open at both ends the frequency of the fundamental is N= — -, where V is the velocity of air waves, and L is approximately the length of the column: and that the partial vibrations have frequencies equal to 2 JV, 3 JV, etc.; whereas, a column which is closed 2 /. at one end — a "stopped" pipe — has for the frequency of its fundamental N=. and for those of its partials 8iV, 4 L ate. Wh'-n a column is set vibrating, as a rule both the fundamental and partials are present; but the funda- mental i> in general mm -h more intense than the others, and the partials decrease in intensity as their fr..pinn ies i in- reuse. So, when an organ pipe is blown gently, only t he fundamental AMES'S PHYSICS — S6 402 SOUND • is lu-anl and the note is almost pure; this is specially true of 'a stopped pipe because the vibration whose frequency is "2 N is absent ; and for this reason these pipes are not generally used in orchestras. With but little practice, one may learn to blow a pipe with such pressure as to make the fundamental or a particular partial the most intense. We see from the formula that the frequency does not depend upon the
aking small changes a pipe may be " tuned" accurately. If the pipe is not of uniform cross section, but conical, there are marked differences produced — the position of the nodes and loops is affected ; as is shown by the difference between an organ note and one produced by a horn. The action of "stops" or "pistons" in horns is to vary the length of the vibrating column of air. If an opening is made in the side of the tube containing the column of gas, the vibrations must adjust themselves to this point being a loop ; and if two openings are made, their distance apart determines the length of a vibrating segment and therefore the frequency of the vibration. This explains the effect of making and closing openings in a tube as is done in flutes and similar instruments. Organ Pipes. — The column of air in a pipe or tube is set vibrating in several ways. In the ordinary organ pipe, a section of which is shown in the cut, there is a narrow pas- sage leading to the bellows or wind chest, through which a blast of air is directed against a sharp lip forming the upper edge of a narrow opening at the bottom of the pipe. This blast at the beginning of operations sends a disturbance up the tube, which is reflected at the upper end and returns. J/r>/'-.i/, INSTRUMENTS 403 When it reaches the bottom, its effect is to deflect the blast out of the opening in the tube. When this effect ceases, the' blast returns : and so there is an oscillation of the blast, which has a period determined by that of the stationary vibration of the column of air produced by the two trains of waves, the ••11 direct and the reflected. This bottom of the tube is approximately a loop, because it is open to the air. The vibration of the air would soon cease owing to loss of energy by friction and by the production of waves, 8 it not for the sup- ply furnished periodic- ally by the blast. The MUM method of produc- Fio. 181. —Section of an ordinary organ pipe. ing vibrations is used in flutes and whistles, the lungs or mouth In-ill^ the wind chest. Reed Pipe. —In another form of organ pipe, known as tin- reed pipe, the wind chest is connected directly wit h an elongated l)o\ ; and into this is inserted an inner tube, in which there is a rectangular opening closed by
a strip of brass fastened at one end, called the "reed." When pressure in the wind chest is sufficient, this spring door is pushed open, a 1.1 r passes, and the pressure being ':! 1\. the Nprin^ returns and. o\\ in^ to its inertia. continues to vibrate in its own natural period. In this manner a series of puffs «.i delivered at intervals, 404 SOUND determined by the frequency of the metal spring. There is always attached to the pipe a resonance tube of some kind, the air in which is set in vibration owing to the intermittent puffs. Without this box the sound is most complex, but with it the note becomes fairly simple. The instrument is " tuned " by altering the length of the reed by a clamp. Horns. — In the case of horns, the vibration of the column of air is produced by means of the vibrations of the lips of the player. The column in a horn of fixed length can vibrate in only a limited number of ways ; and the lips must be stretched to exactly the right degree so that, when they are set vibrating by air from the mouth being forced through them, their frequency is one of those to which the horn responds. If one is playing a horn of variable length, like a trombone, and a definite note is being produced, a change in the length changes the frequency of the vibra- tion, and therefore requires a change in the frequency of the lips ; but this change is produced almost auto- matically, owing to the reaction of the column of air itself. The Siren. — There is another acoustic instru- ment which, although not, strictly speaking, a musical one, should be described. It is called the " siren," because its action continues under water as well as above. In principle it is not unlike a reed pipe, inasmuch as it is designed PIG. 182. — Helinholtz's double siren. to deliver a number of M I > 1C A L INSTR UMENTS 4o;> pull's <>f air at iv^ular intervals, only with it this number can be varied at will and can be easily counted. As shown in,the cut, there is a wind chest, which is closed on its upper side by a thick circular plate perforated with a definite number of holes, at regular intervals, around a circle concentric with the plate itself. These passages in
the form of instrument pictured are not perpendicular to the plate, but are inclined slightly, so that the axis of the passage con- Fio. 192 a. -The sidered as a vector has a component parallel to tlu of the circle through the holes at that point. Immediately over tins fixed plate is a movable on*. \\hi< -h can rotate on MI..m«l which is identically like the fixed plate, except that its passages slope the opposite way. Therefore, if the movable plate is in su.-h a position that its openings are over those in the fixed plate, the air rushing out from the wind < -hest will have a momentum against the sloping sides of the passages in the former plate, and will set n m 406 SOUND rotation on its.axis. Each time the openings in the two plates coincide, a puff of air escapes ; and if there are n openings in each plate, there will be n puffs during each rotation. (This would be true, also, if there were n holes in one plate only and but one in the other ; if there are n holes in each, the intensity of the puffs is increased.) So, if the rate of rotation is ra turns per second, the number of puffs in a second is mn ; and this is therefore the pitch of the resulting sound. The speed of rotation may be altered at will by regulating the pressure of the air in the wind chest. The number of openings in the plates is easily observed; and the number of revolutions in any interval of time is determined by using a mechanical counter, such as are seen on steam engines. (A screw thread is cut on the shaft of the rotating plate, so that a worm wheel is turned ; this drives a train of cogwheels, which moves a hand over an indexed dial, like the face of a watch.) In other forms of this instrument, the passages in the plates are not slant- ing, and the movable plate is made to rotate by means of a mechanical or electric motor. The simple siren as just described has been modified in two ways. One is to make in the plates several concentric rows of openings, which contain different numbers, e.g. 8, 10, 12; and thus, if all these are opened at one time, a com- plex sound is heard whose component simple vibrations have frequencies in the ratio 8 : 10 : 12. Again, two sirens con- nected with the
same bellows may be arranged one over the other with their movable plates on the same shaft and facing each other; this enables one to produce two sounds whose pitches have a known ratio, that is, are at a known "inter- val" apart. This instrument was invented by the German physicist Seebeck, and was improved by Cagniard de la Tour and more recently by Helmholtz. It is not, however, as much used now as formerly. MUS1LAL IX^ilil'MENTS 407 Phonograph. — Another acoustic instrument is the phono ;>h, which consists essentially of a hardened wax cylinder ;nst whose surface pre»« •> a sharp point connected with a flexible membrane forming part of a mouthpiece. The cylinder is turned and advanced by clockwork; and, as >ounds are produced near the mouthpiece, the point makes •nding indentations in the wax, which are faithful reproductions of the displacements of the vibrating body. :i. if another point attached to a membrane is made' by mechanical means to pass over these traces on a cylinder at the same rate as that at which the cylinder was turned originally, the membrane will be set in vibration, and its motion will therefore be very nearly the same as that of the one in the mouthpiece that caused the trace. These vibrations will then produce waves in the air which will affect the ear; and so the original sound is reproduced. The Human Voice. — The human voice is due to the vibra- tion of the vocal cords of the larynx and of the various mov- able parts of the mouth. Consonant sounds such as ft, e, etc., are produced by vibration of the lips, the tongue, etc.; while vowel sounds owe their origin directly to the larvnx. This consists of two stretched membranes which have free edges along ;i nearly straight line. Thoe, an beset in vibration •lie air pressure in the Inn^s: and their frequency can be altered by voluntary changes in their tension. Owing to vibrations, which are, however, very "damped." the air in the cavities of the mouth and throat is also set vibrating. When a definite vowel sound, such as <i/t. is produced, no matter what the pitch of its most prominent component, it urn! by analysis that there are present one or more com ponents of definite pitch. These are not partials of the fun- damental vibration
is what may be called " mechanical" refraction. True refraction, similar to that observed in Optics, may, however, be produced. Tyndall made a lens out of a soap bubble filled with nitrous oxide gas, which had all the properties with air waves that an ordinary glass lens has for ether waves. A few photographs of pulses in the air, illustrating these and other properties of wave motion are added. They were obtained by Professor R. W. Wood, using a method devised by Toepler. These pulses are produced by the explosive action of an electric spark. The black knobs seen in the photographs are the centres of the disturbances ; and the wave fronts may be distinguished clearly. The Acoustic Properties of Halls. — Something should be said also in regard to the acoustic properties of halls that are used either for public speaking or for concerts. Great care must be exercised to avoid what is called " reverbera- tion." This, if excessive, is a great objection. It is due, of course, to the echoing and reechoing of the sounds, occa- sioned by the repeated reflection of the waves from the floor, walls, ceiling, seats, auditors, etc. The problem of inves- tigating the exact conditions that determine or prevent 411 reverberation was first undertaken by 1'mfessor Sabine of Harvard I niversity in tin- year 1895. His met hod,,f stndv- ing it was to arrange an organ pipe at one point of a hall so that it could be blown and then stopped at any instant; sta- tioning himself in turn at different positions in the hall, he would note how many seconds he could hear a sound after the organ pipe had ceased acting. It was found that the rU ration was the same practically at all points in the hall and that it was independent of the position of the pipe. For halls of the same volume the reverberation is the same; but, as the size of tin; hall increases, the reverberation in- creases also, other things remaining unchanged, ll is de- creased greatly by putting soft coverings on the floors, walls or seats, by making the walls less rigid, and by the presence of an audience. It was found that for practical purposes the reverberation should not be decreased below 2.3 second >. otherwise the music was not fully appreciated by the audi- ence. Professor Sabine was able to deduce a general formula which can be used to
predict with great exactness the dura- tion of reverberation in a hall when its dimensions and the materials used in its construction are known. The acoustic property of a hall depends upon other things than reverberation, for its shape may be such as to focus the iraYee at {.articular points, etc. "Sounding boards" which reflect the waves down upon the audience are often used. Another effect whirh must be taken into account is that due to ascending and descending currents of air: for wherever there are changes in the homogeneou of tho air, there are reflections of the waves. CHAPTER XXIV MUSICAL COMPOSITIONS Combinational Notes. — When two instruments are sound- ing at the same instant, there are several interesting phe- nomena besides the production of the two sounds. If the two instruments are setting in vibration directly the same portion of air, as when a double siren is used, or if two wind instruments are blown by the same wind chest, other sounds are heard than the two corresponding to the instruments. Thus, if n^ and n% are the frequencies of the two vibrations, other vibrations of frequencies, n^ -f- n2 and n^ — nv are pro- duced in the air. (The mathematical theory of this was given by Helmholtz.) The corresponding sounds are called "com- binational," or " summational " and " differential " notes. Beats. — If the two vibrations have frequencies which are quite close together, a different phenomenon is observed. Thus, suppose the frequencies are n and n + m, where m is a small number — not necessarily an integer. Then, when the instruments are sounded at the same time, the loudness of the sound heard fluctuates; it rises and falls at regular in- tervals. If m = 4, these intervals are a quarter of a second ; or, in general, this is — th of a second. There are then said m to be 4 "beats" (or, in general, m "beats") per second. This " beating " is due obviously to the fact that as the two trains of waves traverse the same medium before they reach the ear, there will be points at regular intervals apart where the compression of one train will neutralize the expansion of the other, and, at points halfway between these, the com- 412 MUSICAL COMPOSITIONS 413 pressions of the two will coincide. Thus, if the
the same time. This fact has nothing to do with the state of civilization or of musical cultivation ; it is a property of the human ear. It was recognized by the Greeks before the days of Aristotle (probably as long ago as 500 B.C.) that, if two stretched strings of the same size and material and under the same tension, but of lengths in the ratio 1 : 2, were sounded together, the sound heard was agreeable ; and also that, if there were three similar strings whose lengths were in the ratios 4:5:6, the same was true. Mersenne, in 1636, showed that the frequencies of the vibrations of stretched strings varied, other things being equal, inversely as the lengths of the strings. So the problem of explaining the consonance of the sounds produced by the two or three strings as just described became this : why should two notes produced simultaneously by strings making vibrations of frequencies n and 2n, or three notes produced simultane- ously by strings making vibrations of frequencies 4 n, 5 n, 6 n, cause a pleasant sensation? Helmholtz's Explanation of Consonance and Discord. — The answer to these questions was first given by Helmlioltz. He showed that in every case of consonance when two or more notes are produced simultaneously, that is, a " chord " con- sisting of two or more notes, beats were nearly, if not en- tirely, absent ; and that in any other case, when two or more notes were sounded together, the degree of the discord could be predicted from calculations of the number of beats present and from a knowledge of the degree of their disagreeable- ness to the ear. Thus, when a note of pitch n is produced by a vibrating string, notes of pitch 2w, 3 ft, etc., may be heard in the complex sound ; and, when a note of pitch 2 n COJCPO&lTfO 415 is produced, others of pitch I //. b'n, etc., may be heard ; fur- ther, tin- combinational notes all have pitches n, 2n, etc.; so t lie re are no beats, and the two complex notes of pitches n ami 'In are therefore in harmony when produced by wind or stringed instruments. Hut, if the lower note had the pitch n + 5, its partials would have the pitches 2 n + 10, 3 n + 15, etc., and these would beat with the other note
V; those in the octave above this, 4^, 4 0, 4P, 4 #, -, 8N; etc. The notes selected for the octave below the original one have the pitches J JV, JO, J P, J (?, •••, JVi etc. So, in defining a scale, it is necessary to choose only the pitches of the keynote and the notes in its octave. The Diatonic Scale. — The " diatonic " or " natural " scale consists of a series of notes whose pitches may be expressed as follows: If 24 n is the pitch of the keynote, the notes in its oc- tave have the pitches 24 n, 27 n, 30 n, 32 n, 36 n, 40 n, 45 n, 48 n. Thus it is seen that in the interval of an octave there are seven notes, counting only one of the end notes of the octave. If an instrument with strings of fixed lengths, like a piano, is to be constructed to play music written on this scale, some definite keynote must be chosen, and a string must be selected of such a length and size that under suitable tension it will give this note ; then other strings must be selected to pro- duce the other notes of the scale. But suppose a musical composer did not wish to use the same keynote for all his pieces ; suppose, for instance, that he wished to have as the key tone one whose pitch is 20 n. The diatonic scale of this is 20 n, gj x 20 n, fj x 20 n, fj x 20 n, Jf x 20 n,..., 40 n, or, 20 n, 22Jn, 25 n, 26 § n, 30 n, •••, 40 n; and it is seen that, if the instrument described above is pro- vided with strings giving notes in the diatonic scale having 24 n as the keynote, many of the notes required for this composition cannot be produced ; for instance, 25 w, 26£ w, etc. For this reason, and also to make the intervals between two consecutive notes more nearly equal, the diatonic scale was altered by the introduction of five new notes in each octave : between 24 n and 27 n, 27 n and 30 n, 32 n and 36 n, MUSICAL COMPOSIT10 117 36 n and 40//. and 40 H and l~> n : and corresponding st i were introduced
sym- bols have been given the notes in the various octaves of dif- ferent scales. For instance, " Ut3" is sometimes used as the name of the key tone above described. So it is seen that the pitch corresponding to a certain name or symbol in musical notation is not definite. Thus the note called Ut4 had a pitch less than 500 in the early part of the eighteenth cen- tury; in the days of Handel (1750) the note which had this same name and symbol in musical notation was one whose pitch was between 500 and 512 ; and in our days this same symbol is given a note whose frequency varies from 512 to 546, as we have seen above, for Ut4 = 2 Ut3. These facts are expressed by saying that there is a tendency as years go by for the pitch corresponding to a given symbol to rise. The octave of the equally tempered scale starting" from the keynote 264 is made up, then, of the notes whose pitches are as follows: 264, 279.6, 296.3, 314.0, 332.6, 352.4, 373.3, 395.5, 419.0, 443.9, 470.3, 498.3, 528. If this note is used as the keynote of the diatonic or natural scale, the notes in the corresponding octave have the following pitches : 264, 297, 330, 352, 396, 440, 495, 528. It is thus seen how far apart the scales are at certain points. Violins, violoncellos, etc., do not have strings of invariable lengths, because the fingers of the player can alter them at will by " stopping " at any point ; and so one can play with them on the natural scale music written in any key, if the five additional notes are introduced, as previously explained. Musical Notation. — The seven notes in the octave of the diatonic scale are called <?, d, e, f, g,a,b\ and different octaves are assigned different types or marks. Thus, the octave from 132 to 264 is written as above ; the one from 264 to 528, <?', d', etc.; the next one c", d", etc.; the one from 66 to 132, (7, D, etc. ; that from 33 to 66, Cv Dv etc. J/r>/<.\L'Using the tempered scale
experiment* on \\ln.-h Hrlmholtx established his theory of harmony, and also a complete explanation of musical lustrum* Mt- ;iti<l scales. RAY i I vols. London. 1886. i * is a mathematical tren • 1 1 contains • \ i > < 1 1 > experimental knowledge of vibrations, waves, and musical notes. LIGHT CHAPTER XXV GENERAL PHENOMENA OF LIGHT To any one who has the sense of sight, the word " light " conveys a definite meaning, which, however, cannot be put into words ; but to a person who was born blind the word is unintelligible. The attention of all who have this sense of sight is attracted to many phenomena in nature, such as the colors of objects, the action of mirrors, the refraction produced by water, etc. ; and the study of these forms that branch of Physics called "Light." It will be seen, as we go on, that we can subdivide this subject in certain definite ways. Fundamental Facts Light is Due to Ether Waves. — The statement has been made before several times that light is a sensation due to waves in a medium called the ether ; but a brief summary of the facts on which this is based may be given again. Thomas Young showed as early as 1802 that interference phenomena, such as described in Chapter XXI, could be observed with light ; Fresnel soon after performed numerous diffraction ex- periments ; and Wiener and others have obtained evidences of "stationary waves." The experiments of Young and Fresnel may be repeated easily by any student. Thus it is established that light is a wave phenomenon. Again, Fresnel showed conclusively that these waves are transverse, because they admit of polarization (see page 313). The existence of a 420 GE.\Ki;.lL I'l/l-:\n.MENA OF LIGHT 1'Jl medium is proved by the fact of tin- wave motion; and, since we can see objects through spaces which are void of ordinary matter and through glass, water, etc., it is shown that this medium is one which fills all space known to us, even inside ordinary material bodies (see page 19). We often use the expressions " light passes " or travels, etc., meaning that the ether waves which produce light in our eyes pass or travel ; similarly, we speak of "red light," etc., meaning those ether waves which produce in our eyes the
away from a source, and those which are contracting toward a point. It is a familiar fact that an ordinary reading lens, or magnifier, may produce an image of the sun upon some opaque screen — thus acting as a " burn- ing glass " ; this means that the plane waves reaching the lens from any point of the sun are changed 011 passing through the lens into spherical waves which converge to a point on the screen. This process of converging waves is exactly the reverse of that of waves diverging from a point source. Thus, if a spherical wave front is concave when considered from portions in the medium toward which it is advancing, it will -converge to a point ; if it is convex toward that side, it will diverge more and more. Fio. 185. — Dinprram illustrating stellar scintillation. This fact is illustrated in the familiar phenomenon of stellar scintillation. The waves coming from a star are naturally plane ; but if the atmosphere is disturbed by ascending currents of hot air, the wave front is no longer plane, owing to the fact that the velocity of light in cold air is different QKNSRAL PHSNOUBNA o/-' /./<,/// from that in hot. Thus the wave front at any instant may have a "cor- rugated" form, a> indicated in the cut. Therefore the light will be concentrated in certain points Av A* etc., the centres of the concave portions of the wave front ; and, as the heated portions of tin* air change their j these points move; so if the ry«- is at a ]><.iiit.-lj atone instant. th«> n<*xt it may be between two points,.1, ami. lo; etc. So the iiitrn^ity of the light will increase and decrease intermittently. The same phenomenon is observed in the "shadow bands " seen at times of total eclipses of the sun. Homogeneous Light and White Light. — In order to deter- mine tin- wave lengths of these "light waves," it is simply necessary to use the interference method described on page 375. It is found that, if the source of light is white, the interference fringes or bands are all colored, with the excep- tion of the central one, which is white. If, however, the source is colored, the bands are alternately black and colored ; that is, in certain lines disturbances in the
ether are entirely absent. In general, it will be observed that there are apparently two or more sets of fringes superimposed, each set having a definite color. It is possible, however, to secure such a colored source, that the bands are all of one color, separated by the dark fringes. (This is approximately the case with a sodium flame.) Under these conditions the light is said to be "pure," or "homogeneous." When the sourer is white, or when any ordinary colored source is used, we can analyze the complex interference pattern into series of fringes, each series having its own color and its own spacing. It was shown on page 378 that, if the waves have a definite wave length, the fringes are evenly spaced at a distance t which is proportional to the wave length. Therefore, when a source is hom-. it is emitting waves of a •'.«• definite wave Imgih: and, in general, an ordinary source of light is emitting t rains of waves of different wave length-, 'fins, th«- interference apparatus "disperses" t lie complex from the MHirO6 into simpler trains,,-aeh ing a deiimi^ \\.i\r Length, 424 LIGHT Connection between Wave Length and Color. — We can measure the wave length of the waves emitted by any homogeneous source, by using the formula deduced on page 378, viz., distance apart of fringes =—, where I is the wave 6 length, a is the distance from the two sources to the screen, and b is the distance apart of the sources. In this way it is found that pure red light has a wave length greater than that of pure green ; and this in turn is greater than that of pure blue. The longest waves that affect our sense of sight produce the sensation of red and have a length of about 0.000077 cm. (i.e. 770/xft); while the shortest produce the sensation of violet and have a length of about 0.000039 cm. (i.e. 390 pp'). In between these limits there are all possible wave lengths, corresponding to which are all shades of color, ranging from the deepest red, through orange, yellow, green, blue, to the darkest violet. Waves shorter than 390 /i/i can be observed by photography ; and they have been obtained by Schumann as short as 100 /u/x. Waves longer than 770 pp may be studied, as described
on page 292, by various means; and they have been observed as long as 25,000 pp, i.e. 0.025 cm. (Electrical oscillations produce waves in the ether, which are as a rule very long ; but, using minute conductors, waves as short as 0.6 cm. have been obtained. There is thus a gap between 0.6 cm. and 0.025 cm. which has not yet been investigated.) The fundamental facts are, then, that light is due to transverse waves in the ether ; that waves of different wave length produce different colors; that white light is, in general, due to a mixture of waves of all wave lengths which affect our sense of sight ; that the velocity of these waves is less in ether inclosed in matter than in the pure ether. (It will be shown on page 433 that, in the former case, waves of different wave length have different velocities, which is not the case in the pure ether.) mi: i OF LIGHT 425 General Properties of Light as Due to Wave Motion. — \\V described in Chapters XX and XXI certain general properties of wave motion which apply directly to light; viz., reflection, refraction, rectilinear propagation, interfer- ence, and diffraction. Each of these will be discussed more in detail later; but one or two points should be referred to here. (It should be remembered specially that we are now considering extremely short waves, viz., those whose wave length is not far from 500 pp.) Rays, Shadows. — It was shown that, if the waves are short, the disturbance at any point in an isotropic medium due to a train of waves depends directly upon the dis- turbances at previous instants along a straight line drawn backwards perpendicular to the wave front; this line is called a "ray." If the rays are all parallel, we have plane waves, or a " beam " of light. If the waves are spherical, tin- rays are radii ; and, by considering only those rays which are close together, we have a "cone" or a "pencil" of light. (Although this theorem in regard to rays was proved only for a train of waves, it may be shown to hold true for a -pulse" also.) It the waves meet an opaque obstacle, that is. on.- \\ hich does not transmit li^ht, a shadow is cast. If the source of light is a point, the
shadow is that which we have called tin; "geometrical" shadow, if wo neglect diffraction phenomena, as We Shall f,,r the time beiO I f, however, the SOUrce is large, object^, representing the on • tureen rG\y »n opaque,ik\|, \| tl;iMi(i MI. ;m i]lllmm;lted piece of paper, the shadow phenomena are evidently not quite so simple. 426 LiailT Thus, if the large source of light is represented by AB, and the opaque obstacle by CD, the shadow cast by the waves from A on a screen FG- is limited by EG-, and that cast by the waves from B by FH. Therefore the region EH on the screen receives no light ; and the region outside FCr receives light from all points of the source ; but the intermediate regions, EF and GH, receive light from only portions of the source. Therefore, the intensity of light on the screen fades away gradually toward the central region EH, where there is no light. The space back of the screen into which the waves do not penetrate is called the "umbra," while the partly illuminated space surrounding it is called the "penumbra." An illustration is afforded by solar eclipses, where the sun is the source, the moon is the opaque obstacle, and the earth the screen. In the dia- gram, S represents the sun, and M the moon. The umbra and penumbra are indicated by dark spaces. If the earth enters this region, the eclipse is total for all points on its sur- face which are inside the umbra, and par- tial for points outside this- but lying in the penumbra. If the earth just misses the umbra, it may happen that at certain points of its surface a ring of sunlight may be seen around the FIG. 187. — Diagram showing shadow cast in space by the moon, M, owing to the rays from the sun, S, edge of the moon ; this condition is called an "annular" eclipse. Another interesting case of rectilinear propagation is given by the formation of what are called "pin-hole " images. If a small hole is made in an opaque screen, any luminous object — e.g. a building in sunshine — situated on one side of it will produce on a screen on the other side an inverted image of itself, which is comparatively sharply denn
ed. Thus if there is a small opening at O in the screen, and A is a point of an illuminated figure, there will be a cone of light from A passing through the opening. If this meets a screen, it will make at the point /* a bright './•:\/:/,M/. i'in-:.\n.Mi-:\.\ LK.IIT i-JT spot, which will hsivi- tin- >ha\|>«- of tli«- opening. If the opening is small and tin- t\\o.M-m-ns an- close together, the spot at B will be practically a point of light ; and hence, as each point of the illuminated figure pro- duces a point of light on the screen, there will be formed a well-defined image. As tin- images of the various points of the illuminated object overlap, the general appearance of the image is almost independent of the shape of the opening, if it is small. (The round or ellip- tical spotsof light which are v^ m._Pln_holeimg<. seen on the floors near cur- tained windows or under trees are images of the sun formed by minute openings in the curtains or leaves.) This is the principle of pin-hole photographic cameras, of the camera obscura, etc. Opacity; Transparency; Translucency. — A distinction is made between material media which are " transparent " and which are "opaque." If an object when introduced between the eye and a source of light stops all the light, e.g. a board, a piece of tin, etc., we say that it is opaque, meaning that it does not transmit those ether waves which affect our <• of sight. An object may be opaque to some waves and not to others: thus, a piece of red glass is opaque to all visi- ble waves except those which produce the sensation of red. Again, a given material body may be opaque to waves which do not afieet our sense of sight, and may transmit all visible and there may be media with just the reverse prop-. Opacity is due to the fact that the waves which are incident upon the bodv do not pass through it: they are either reflected bn-k from it or are absorbed by it. Thus, a polished metallic surface is opaque largely owing to relleetioii, while a blackened surface is
opaque owing to absorption. If we can look through an object and see sources of light 428 LIU / IT on the other side sharply defined, we say that it is "trans- parent." Here, again, the body may be transparent to some waves and not to others. Waves, then, when incident upon a transparent body, will under ordinary conditions be trans- mitted by the ether in the body, maintaining a definite wave front. This is true only if the transparent body has surfaces which are " smooth," in the sense that there are no inequal- ities comparable in size with the length of the waves. Thus, a window pane of glass, layers of water or alcohol, etc., are transparent. If the surface is rough, the waves suffer irreg- ular reflection. Again, we will see later that in certain cases the waves incident upon the surface of a smooth transparent body are entirely reflected. (This is called total reflection.') It must not be thought, however, that a bod}' which is not transparent is necessarily opaque ; a piece of opal glass or of oiled paper is not transparent, nor is it opaque. One cannot see objects through them, and yet they allow light to pass to a certain extent. Such bodies are said to be " translucent." What happens is this. When waves from any source fall upon a translucent body, they are broken up and scattered by it in such a manner that each point of the body becomes a new and independent source of waves. So when such a body is held between the eye and a source of light, the waves which reach the eye come directly from the points of the body, not from the source ; and what the eye sees, then, is the surface of the body. A transparent body cannot be seen ; it is only owing to dirt on it that we are able to see a surface of water or a window pane. The explanation of translucency will be given in the following pages. Reflection and Refraction. — It has been shown that, if there are two media separated by a bounding surface, waves in one will be reflected at the boundary in general, if the velocity of the waves in the two media is different. The boundary surface must be large in comparison with the length of the waves, otherwise the waves will pass around the "obstacle." 9XNMRAL /•///•: \</.v/-;.\.i or /,/'./// 429 If the surface is large, but
glass, etc., of the third. All these bodies which reflect light diffusively, diffuse it also when they transmit it ; that is, they are translucent. If these foreign particles in a transparent medium are very minute, it may happen that the shorter waves only are affected, while the longer ones will be transmitted through the body. In this case the body would appear bluish when viewed in reflected light. This is the explanation of the blue color of the sky, of fine smoke, of a hazy atmosphere, of blue eyes, etc. In each of these cases there are minute par- ticles existing as foreign bodies in a transparent medium, which diffuse the short waves, but allow the longer ones to pass. Regular Reflection and Refraction. — We gave in Chapter XX the treatment of reflection of plane waves at a plane sur- face, quoting from Huygens. But, as noted at the time, this method is not rigorous. In order to make it so, we must follow the same plan as did Fresnel in discussing rectilinear propagation ; namely, we must divide the wave front up into zones and deduce the effect B/ N \ of the secondary waves, thus combining Huygens's principle of secondary waves and Young's princi- ple of interference. When this is done, we obtain Huygens's solution, and FIG. 189. -Reflection of plane waves by a plane also leam the effect of re- flection upon a ray. Thus, by Huygens's solution of the problem, plane waves, whose wave front at any instant is given in section by AB, inci- dent upon a plane surface MM, are reflected, and form the fXBAL I'llKMtMENA OF LK,11T 431 plane waves whose wa\e front at a later time is given by T)C, where the angles (BAG) and (DC A) are equal. If draw a straight line OjO perpendicular to AB and meeting the reflecting surface at 0, and another OP, per- pendicular to the reflected wave front C/D, it is seen by geometry that the broken line OjOP is the shortest line that can be drawn from P to the wave front AB by way of the surface, and so Ol is the "pole" of P (see page 382) ; and. in drawing the zones around Or which are to be com- pounded in order to deduce
taken for a dis- turbance to pass from 01 to P along this line is less than along any other line. (Huygens proved this.) Therefore 01 is the pole of P on the incident wave front ; and by constructing Fresnel's zones around Ov we are led at once to Huygens's solution. The disturbance at P is due directly to that at 0; and this, in turn, to that at 6^; so the incident ray Of) lias its direction changed at the surface into OP. Drawing a normal NON' to the refracting surface at 0, the angle (01ON^) is the angle of incidence; and the angle (PON'), the angle of refraction. These angles are equal respectively to (BAC) and (DC A). Therefore we can state the law for a ray: the Bine of the angle of incidence = ^ index Jf refraction. the sine of the angle of refraction Calling the angle of incidence i, that of refraction r, and the index of refraction n, this relation is smr QXNSRAL rui-:.\<>Mi,\A OF LIGHT 433 The other law of refraction is evident, viz., the refracted ray is in tin- plane of incidence. Refraction may be studied with ease by allowing a beam •inlight to fall upon the surface of cloudy water in a tank. It will be observed that, if sources of light of different color arc used, the refraction is different; if the waves have the same angle of incidence, the angles of refraction are different. Thus the index of refraction varies with the wave number or color of the light. This is not easily shown in the experiment just described, unless the beam of light is made extremely narrow ; because the differences in the refraction are not great, but by means of a "prism" or a lens this phenomenon is m<>st apparent. If the incident light is white, each of the component trains of waves (see page 423) has its own index of refraction, and so the light is broken up or dispersed, forming a "spectrum." In the case of ordinary transparent media such as glass, water, etc., the waves having the shorter wave lengths are refracted more than those having longer ones; i.e. blue light is refracted more than green, green more than red. This proves that in these media short waV6fl have a less velocity than
do long ones. (In the pure ether all waves, so far as we kno\v. have the same velocity.) This kind of medium is said to have ordinary or " normal " disper- sion. In other media, it may happen that some waves are refracted more than others which have a shorter wave length; they are said to have "anomalous dispersion" (see Chapter \\\,. When waves are passing in the ether inclosed in anv material medium, such as water, the minute particles of this medium an- in motion also to a greater or less extent, owing to the waves; so, if there is a long train of waves, and if \\e consider any one point in space, the effect produced there by the read [on «»f the matter on each •• wave" as it passes it is different from what it would he if the matter were at rest; as, for instance, if a sudden •• pulse " came up to t his point and AMI *'i i iivsics — 28 434 LIGHT passed. Since the velocity of a disturbance through the ether depends upon this reaction of the matter which incloses it, it is evident that the velocity of a train of waves is different from a pulse. Further, the method of Fresnel for consider- ing wave motion presupposes the existence of a train of waves. Thus the laws of refraction apply only to trains of waves. Geometrical Optics. — Other cases of reflection and refrac- tion will be considered in the following chapters : spherical waves upon a plane surface, plane waves upon a spherical surface, and spherical waves upon a spherical surface. There are two modes of procedure possible : one is to study the changes in the wave front produced by reflection or refraction ; the other is to study the changes in the direction of the rays, making use of the theorems just deduced ; for in the case of incidence upon a curved surface, we may con- sider the reflection or refraction of a ray at any point as due to an infinitesimal portion of the tangent plane of the surface at that point. The application of this latter method makes up what is called the science of " Geometrical Optics." We shall use this in these chapters, but shall also outline in cer- tain cases the demonstrations in terms of waves. Real and Virtual Foci. — If spherical waves diverging from a point source are spherical also after
reflection or refraction, we may have either of two conditions : the centre of the reflected or refracted waves may be in the medium in which the waves are advancing, i.e. the waves converge to a "focus"; or the centre of the waves may be in the other medium, i.e. the waves will diverge away from their centre. (Of course, in the former case, the waves, after converging to a point, will diverge again beyond it if no obstacle 'pre- vents.) The centre of the converging waves is called a " real " focus ; it is said to be a " real image " of the source or " object." The centre of the diverging waves is called a "virtual" focus; it is said to be a "virtual image." There GEXKHAL PHENOMENA of LIGHT L86 are cases, however, in which, even though the incident waves are spherical, the reflected <>r retracted waves are not. Homocentric and Astigmatic Pencils. — Similarly, from the standpoint of rays, if we consider any incident pencil of rays proceeding from a point source, it will, after reflection or i-i- fraction, form another pencil with its vertex in the medium into which the rays are advancing, if there is a real focus, or one with its vertex in the other medium, if the focus is vir- tual. But there are cases when, after reflection or refrac- tion, the rays do not form a cone. A pencil of rays which does form a cone is said to be "homocentric"; while one which does not is said to be "astigmatic." In this latter case, as we shall sec, the rays of a homocentric pencil, after reflection or refraction, have as a focus (either real or virtual) not a point, but two short lines perpendicular to each other and a short distance apart; these are called "focal lines." In describing the incidence of a pencil of rays it is simplest to give the direction of its central ray; so by speaking of " normal incidence " of a pencil we mean a case when the central ray of the small pencil is perpendicular to the surface at the point where the pencil meets it; and by "oblique incidence" is meant a case when the central ray of the pen- cil makes an an.^le different from zero with the normal to the surface at the point where this ray meets it. We shall see
shortly that in all eases a pencil which is normal to a surface produces by reflection or refraction a homocentric pencil; and in nearly all cases an oblique pencil produces,in astigmatic one. Properties of a Focus from the Standpoint of Waves. — Since the locus of the points readied by the dis- turbances at any one instant, we may consider the existence of foci from a different standpoint. If \\ a ves di ver^injr from a point BOH 06 '-on verge after reflection or refraction to an- other point, we can draw various rays proceeding out from the former point and all meeting again at the latter. These 436 LIGHT rays have different paths ; but the time taken for the disturb- ances to pass along all of them must be the same. Thus, if Zj is the length of the portion of a ray in one medium in which the velocity of the waves is vv and if?2 is the length of its portion in a second medium in which the velocity is v2, the time taken for the propagation of the disturbance is v\, or /Z, + Sl.0-' Bufc ^ is the index of refraction, n, of the second medium with reference to the first; so this time is (li + nl^)—. The quantity (7X -f- nl2~) is called the " " of the ray. length optical is It dis- tance in the first medium which the waves would advance in the time taken for the actual propagation in the two media. Then we may state that the optical lengths of all rays from the point source to the focus are the same. We shall make use of this principle in discussing lenses. evidently equal the to Another fact in regard to foci should be emphasized. Re- flection and refraction are always produced by pieces of mat- ter of limited size, e.g. looking-glasses, prisms, lenses, etc. ; and so only a portion of the wave front undergoes the change. The effect at any point in the advance of the wave front must then be deduced by following Fresnel's method of combining the principles of Huygens and Young. The point at which the disturbance is greatest is the focus ; but this does not mean that there is no disturbance at other points. The effect at these latter points must in each case be calculated ; for some it is zero, and in no case does
as spherical; and, if Q is the quantity of light emitted by the source in a unit of time, assuming that it radiates uniformly in all directions, the amount falling on a portion of the spherical surface of area A at a distance r is H tne illuminated surface is small and is oblique to the direction of propagation, let its area be B and let the angle made between a perpendicular to it and the direction of propagation be N\ then the projection of this surface on a plane perpendicular to the direction of propagation has the area B cos N. So in the above formula if A is this projected area, A = B cos JV; and the light received in a unit time by the oblique surface whose area is B is — -- — — • The intensity of illumination of this oblique surface is then _Q_ cosjgV 47T r2 4?r r2 This formula offers at once a method for the comparison of the intensities of two sources of light. The general method is to illuminate a portion of a diffusing screen by one source and contiguous portions of the screen by the other, and then to vary the distances of the sources until the two portions of I'UOTOMETHY Jo'.' tin M-iven appear equally bright. When this is the case, the intensity of illumination must be the same for both portions; and, if the angles of inclination of the two sources are the i i Luminosity of Sources of Light. — If the source of light is not a point but an extended surface, like the surface of a white-hot metal or of a diffusing screen, let us consider the radiation of any small portion of the surface, whose area we may call A. If this were a point source, and it emitted a quantity of light Q in a unit of time, the quantity received in that time on a screen of unit area at right angles to the direction of propagation at a distance r, would be, as we have seen, —^—- Therefore the total amount of light actually 4 received per unit area by a screen, parallel to the luminous surface and at a distance r from it, varies inversely as r2, provided the area of the luminous surface and that of the portion of the illuminated screen considered are both small and face each other. That is, calling P the quantity of light received in a unit of time per unit area of the sen-en, P = —, where c is a constant depending upon the properties of the luminous
to our eyes like a luminous disk of uniform brightness. If the area of the oblique surface is A, and if it is inclined at such an angle that a perpendicular to it makes the angle N with a line drawn to the eye, A cos N is the area of the projec- tion of this surface perpendicular to this line ; and a surface of this area placed parallel to the eye appears of the same brightness as the one of area A placed at the angle N. Let L1 be the intrinsic luminosity corresponding to the direction N; i.e. if the area of the surface is unity, the light received Tt per unit area at a distance r in this oblique direction is — • Then the light received by the eye from the oblique surface AU B is — — — ; whereas, if the parallel surface of area A cos N were used, the light received would be A™*N' LB. But PHOTOMETRY 441 •rience proves that these an- practically equal; so L' = L cosiV. This is called " Lamhert's Law." The intrinsic luminosity in any direction of a small luminous surface is, in words, the quantity of light received per unit area by a screen perpendicular to this direction at a unit distance, divided by the area..«• luminous surface. So, if the screen is placed obliquely to this direction, making an angle NI with it ; and if its area is />'. the light received on it in a unit of time from a lu mi nous source of area A, making the angle N with this line referred to, and at a distance r, is AcosN - L- BcosN, A I B cos N cos Nt -I or - H As noted above, this statement is not absolutely true. We may regard L as a quantity which is not a constant factor but varies slightly with N. A method is thus evident for the comparison of the lumi- nosities of different sources of light. Each is surrounded by an opaque screen provided with a rectangular opening, or slit : the sources are so situated that a suitably placed diffusing screen receives light perpendicularly from these two open- ings; one portion of the screen receiving light from one source only, and contiguous portions receiving light from the other source only. The screen is now illuminated by li'^ht coming from the two rectangular openings as sources. Then 1>\ some means the conditions are
to have the angles of inclination the same). In Bunsen's photometer, a screen consisting of white unglazed paper, in the centre of which there is a small round or star-shaped grease spot, is placed between the two sources. Looking at this screen from either side, any portion is illu- minated by the transmitted light from the source on the other side and also by the reflected light from the source on that side. The screen is moved until the grease spot and the other portions appear equally bright when viewed from either PHOTOMETRY 443 side: and then tlu- above relation holds. For, let a be the proportion of light reflected by the unglazed paper, and b that reflected by the greased paper, and assume that there is no absorption. Then, if P1 is the quantity of light per unit area incident upon one side of the screen, and P2 that upon the other, the amount of light reflected per unit area by the unglazed portion on the former side is aPv and that received by transmission from the other side is (1 — a)P2 > similarly, the light reflected by the grease spot is bPr and that received by transmission is (1 — b)P2. Hence, when the two portions are equally bright, aPl + (1 - a)P, = bPl + (1 - &)/>„ or (a - 6) Pl = (a - 6) Py And therefore, since a does not equal 5, Pl = P2. The best photometer in use to-day is one designed by Lummer and Brodhun. For full details of these and other instruments reference should be made to some treatise on Photometry, such as Stine, Photometrical Measurements, or Palaz, Indus- trial Photometry. Naturally, the intensities of two sources of different color cannot be compared directly ; and, in general, if any two sources are to be compared, tin ir luminosities corresponding to each wave length should be investi- gated. This can be done by combining with a photometer a dispersing apparatus such as a prism. The complete apparatus is called a " spectro- photometer," the simplest and most accurate form of which is one devised by Professor Brace of the University of Nebraska. CHAPTER XXVII REFLECTION WHAT is meant by regular reflection, and by a mirror, has already been explained ; and the law of reflection for
a ray has been deduced (see page 431). We will now consider several special illustrations. /N, Plane Waves Incident upon a Plane Mirror. — This is the case already discussed on page 367, and needs no further treatment here. There is one illustration of it, however, which may be described. It is that of a plane mirror which is being rotated when plane waves are incident upon it. Let the trace of the mirror by the paper at any instant be MM, and let PO be any incident ray ; draw ON perpendicular to the mirror at 0; the re- flected ray OR will make with the normal an angle (RON) equal to the angle (PON). Let the mirror now be ro- tated about an axis through 0 perpendicular to the plane of the paper; that is, about an axis parallel to the inter- section of the plane wave At the end of a certain time front with the plane mirror. the mirror will have turned into the position indicated by MlMl ; so, iHXZVi is the position of the normal, the reflected ray will be ORl where the angles (P&ZVj) and (NflR^ are equal. The angle turned through by the reflected ray is 444 Fio. 192. — Rotating mirror: the incident ray iaPO. REFLECTION 445. This equals the difference between the angles (POR) and (POR^j that is, twice the difference between the angles (PON) and (PONJ, or twice the angle (NONJ. But this is the angle of rotation of the mirror ; so the reflected ray turns twice as fast as the mirror. This prin- ciple is made use of in many optical instruments : the sextant, which is used to measure the angle subtended at the eye of the observer by two distant points; the mirror attachment to a galvanometer ; etc. Spherical Waves Incident upon a Plane Mirror. — Let the sheet of paper be perpendicular to the plane of the mirror, and let the trace of the latter be MMr If 0 is the source of the waves, we may consider any two rays OP and OQ. l>v reflection they become PPj and QQ^ where the angles Q, (MPO) and (M^PP^ and (MQO) and (Jfi^Ci) are equal. If PPj is prolonged backwards, it will meet in the point (7 a line drawn from 0 perpendicular
, after reflection at other points of the surface, have directions which pass through 0', provided the surface is only slightly curved, and that only a small portion of the mirror around M is used. For, since the line PC bisects the angle PO-.PO' = C or, putting Jlo = u, MO' = v, MC = r, PO'.PO' = u-r:r-v. If, however, the above conditions as to the curvature of the mirror and the closeness of P to M&TG satisfied, the distance PO nearly equals MO, and PO' nearly equals MO'. That is. Hence for definite values of u and r, that is, for waves from a definite point source 0 falling upon a concave mirror with the radius r, the value of v, which determines the position of the image 0', is independent of P. It is, in the case illustrated in the cut, a real focus of 0. Conversely, if Of is a source of rays, they will after reflection converge to 0. The two points are therefore called "conjugate foci." The equation for v may be put in the form or REFLECTION 449 A simple geometrical method for determining 0' is as follows: Draw OOM through the centre of the mirror (7, draw OP to any point P near M, and OR through C parallel to it; draw a line PF so as to bisect OR at F\ where this line intersects the line OM is the image 0' '. For, as has just been shown, O1 lies on the line OM and on PS, where the angles (£P(7) and (OPC) are equal ; and the intersec- tion of PS with CR may be proved to bisect it. CR is drawn parallel to OP, and F is its point of intersection with PAS'. The angles (RFP) and (FPO) are equal, and (RFP) equals the sum of (FPC) and (FCP) ; hence (FOP) equals the difference between (FPO) and (FPC), i.e. (OP0). But (CPO) and (FPC) are equal (angles of incidence and reflection); therefore (FCP) equals (.FP(7); the triangle (CFP) is isosceles ; and the sides JV and FP are equal. P is supposed to be close to M, and therefore to R ;
* and the virtual is therefore a point bi- secting th«- liii.-r.i/. Thin is called the " principal focus." Thus a normal pencil gives rte to a homocentrio one; and it may be shown, by following the same method as was used for concave mirror.'-, that an oblique pencil produces an astigmatic one by reflection. 454 Plane Waves Incident upon a Parabolic Minor. — This is a mirror whose surface may be imagined described by rotating a parabola around its axis ; it is called a " paraboloid of revo- lution." If the rays are all parallel to the axis, they will after reflection all converge to the focus, F, of the parabola. (To prove this requires a knowledge of the analytical prop- erties of the parabola.) In this case, then, there is no spherical aberration. Conversely, if a point source of light is placed at the focus -F, all the rays which are reflected by the surface proceed out parallel to the axis. This is the reason why such mirrors are used in search lights, the headlights of locomotives, etc. FIG. 206. — Parabolic mirrors. CHAPTER XXVIII REFRACTION Plane Waves Incident upon a Plane Surface. — This is the case already discussed on page 432. The incident plane waves give rise to refracted waves, which are plane and in such a direction that the incident and refracted portions of any ray and the normal to the surface at the point of incidence are all in the same plane, viz., the "plane of incidence"; and, if Ni and JV2 are the angles made with the normal by the two rays, sin is the same for all angles of FIG. 207. - Refraction of a r»y. incidence. As already explained, this ratio, or the index of refraction of the second medium with reference to the Jirxt. as it is called, equals the ratio of the velocities of the waves in the two media. It should be noted that the waves are sup- posed to be homogeneous, i.e. to have a definite wave num- ber. Calling v1 and va these velocities, and writing n^ l for Conversely, the this index of refraction, na> t index of refraction of the first medium with reference to the sn second, n%l equals • Therefore, if v^v^ sin JV^sin
NT and so ^ > N^ ; that is, the refracted ray is bent in closer to the normal than is the incident ray. This is the case illustrated in the cut. On the other hand, if V\|<VP -ZV1<-ZV3; and tin- ivfrartrd ray is In-lit away t'nmi tin- normal. This 456 456 LIGHT may be illustrated by the cut, if the arrows indicating the directions of the rays are considered reversed. Total Reflection. — It is evident in this second class of refraction that, if the angle of incidence, Nv is sufficiently increased, the angle of refraction, Nv may finally equal 90°, i.e. ^. So the refracted ray just grazes the surface. When this occurs, sin JVj = 1 ; and, therefore, according to the for- mula, sin N2 = — — This angle of incidence is called the " critical angle " for the two media and for waves of a defi- nite wave number. If it can FIG. 2Ut>. — i'otal reflection. be measured, 7i2 j, or the corre- sponding index of refraction, may be at once calculated. If the angle of incidence exceeds this critical angle in value, there is no refracted ray, for the sine of an angle cannot exceed unity, and the ray suffers total reflection. The velocity of ether waves in water is less than in air, as is shown by direct experi- ment, or by the fact that a ray in air incident upon a plane surface of water is bent toward the normal. So this phenomenon of total reflection will be observed if rays are incident obliquely upon a surface of water from below. This condition may be secured if one holds a tumbler of water in such a manner that the eye looks up through the glass at the surface of water and turns so as to face an jllu- mmated object. If the direction in which one looks is suffi- ciently oblique to the surface, nothing is seen through it ; for it acts like a plane mirror. A piece of apparatus that is often used to change the direction of a beam of light, called a " totally reflecting prism," consists of a glass "1,0'REFRACTION 457 triangular prism whose cross section is an isosceles right-angle triangle. Light incident normally upon one of the smaller faces
depends upon this. 458 Lid I IT Special Cases. — The refracting matter is generally made into a figure with regular geometrical surfaces. There are three cases of special interest : (1) a " plate," which is a figure bounded in part by two parallel planes ; (2) a " prism," which is a figure bounded in part by two non-parallel planes ; (3) a "spherical lens,'-' which is a figure bounded in part by two spherical surfaces, and which is symmetrical around the straight line joining their centres. We shall discuss briefly the path of a ray in passing through these various figures. 1. Plate. — Let the plate be placed with its parallel faces per- pendicular to the sheet of the paper, and consider a ray incident in this plane. This is illustrated in the cut. If JVj is the angle of incidence upon one plane face and N2 that of refraction, the angle of incidence of the ray upon the FIG. 210. — Refraction of a ray by a plate. other plane face, N2', must by the laws of geometry equal the angle N2 ; and the angle of refraction, or of emergence, out into the original medium, NI, must equal JVj ; for and, since N^ Therefore the emerging ray is parallel to the incident one, but is displaced sidewise an amount depending upon the angle of incidence, the thickness of the plate, and its index of refraction. It follows, then, that plane waves incident upon a plate emerge in the form of plane waves parallel to the incident waves. The case of spherical waves will be considered later. 2. Prism. — The straight line in which the two non-paral- lel surfaces meet (or would meet if prolonged) is called the REFRACTION 459 "edge," and the angle between them is called the "angle" of the prism. Let the prism be placed with its edge perpen- dicular to the plane of the paper, and consider a ray incident in this plum-. This is illustrated in the cut. Call the angle of incidence upon the prism N-^ ; that of refraction, Nz ; that of incidence upon the second face of the prism, N% ; that of refraction out into the original medium, NJ ; that of the angle of the prism, A ; and that between the directions of the entering and the emerging rays, D. Fio. 211. — Refraction of a ray
by a prism. It is evident from the geometry of the figure that the angle b« -tween the two normals drawn to the two surfaces equals the angle of the prism ; and that the following relations are trii.- : Further, JVj and NT and NJ and NJ are connected by the ivt'rart i«»n I'nrinula; and so the value of D, the "de-. " as it is called, can be deduced in terms of A, Nv and n. It \\a\rs nf dilTrnmt wave numbers (or colors) are inci- dent upon a pn>in, it is observed that it deviates these waves 460 LIGHT to different degrees, thus showing that these waves have dif- ferent indices of refraction. If n is large, the ray is refracted more than if n is small ; and therefore the deviation is great, as is evident from the cut apart from the formula ; as a result, if white light enters the prism, it is dispersed into a spectrum of colors. (With glass or water the shorter waves, e.g. the " blue ones," are deviated more than the longer ones, e.g. the "red ones.") (See Chapter XXX.) It may be seen by actual experiment, and it may be proved by methods of the infinitesimal calculus, that as the angle of incidence is varied gradually from normal to grazing inci- dence, i.e. from 0° to 90°, the deviation gradually decreases, reaches a definite minimum value, and then increases ; and, further, that this minimum deviation is obtained when the angle of incidence, Nv equals that of emergence, N^ ; in other words, when the ray is symmetrical on the two sides of the prism. Call this angle of minimum deviation D; then, since N^_ = N^ N2 = NJ, and the two formulae above become It follows that, since we may write n = • A and D may both be measured with accuracy; and so n may be obtained. (Reference for details of the method should be made to some laboratory manual.) If homogeneous plane waves parallel to the edge of a prism are incident upon it, they will therefore emerge in the form of plane waves, but will be deviated through a certain angle. The case of spherical waves will be discussed later. ni: FRACTION 461 3. Spherical Lens
and ° 4 2 Fw.314. — Formation of Image* of a point source 0 by a plane B01 its prolonga- tion backward until •nrfkoe: (1) when n>l ; (2) when n<l.,• •> •, j,-T it meets the normal OA. There are two cases to be con- sidered, depending upon whether the velocity of waves in the first medium is greater or less than that in the second. In either case let n be the index of refraction of the second medium with reference to the first. If the waves in the ° ° 5 5 1 1 ° 5 2 0° REFRACTION 403 latter have a givati'r velocity than in tlie former, n > 1 ; if thi-ir velocity is greater in the former, w<l. These two cases are illustrated by the two cuts. In each the angle of incidence equals (BOA); and that of refraction j&OA. Therefore, nnce the sine of, and the sine of (^01^L)=' the index of 0 B l refraction, n = — —. If B is extremely close to A, that is, OB if the ray OB is one of a normal pencil, we may replace the ratio Qi by -. So n = or O^A = n OA. It fol- lows, then, that Ol is a point at a fixed distance from the plane surface for all the rays of the normal pencil provided the waves are homogeneous, so that n is a constant. If n > 1, Oj is farther from the surface than 0 ; if n< 1, it is nearer the surface. In other words, a normal pencil from a point source 0 gives rise to a pencil of rays by refraction whose centre is Or the virtual image of 0. Conversely, if we imagine the directions of all the rays reversed, a normal pencil of rays in the second medium converging toward a point Oj in the first will actually meet at a point 0. A luminous object in one medium will thus give rise to a virtual image of itself. This image will not be of the same M/.r a^ the object; but their relative dimensions may be easily calculated. The treatment of this case of refraction of spherical waves at a plane surface l.y the method of waves is as follows: Let the source O be in the medium in which the waves hav.
air. (n' for water is approximately -; OA \ so 00l is. J A method is thus offered for the measure- ment of w, since both OA and 00l can be measured. If the pencil of rays from 0 is oblique, it forms by refraction an astigmatic pencil. Thus, ^ Fw. 116. — Fnrin.itl..n of focal line* at A\ and A', by the refraction of an oblique pencil at a plane Rurface. if OB and OS are two oblique rays, they form by refraction two rays which if prolonged backward cross at Fv -,\ point off the perpendicular line OA. Therefore, the 217. — Caustic formed by reflection at a plane surface. AMES'S PHTSICS — 30 466 LIGHT whole oblique pencil gives rise to two virtual focal lines: one at Fl perpendicular to the sheet of the paper : the other at JP2 along the line OA. Further, if we consider all the rays from 0 falling upon the surface, they give rise by refraction to a virtual caustic surface with a cusp at Ov the image of 0 for a normal pencil. Special Cases 1. Plate. — We shall consider the plate made of a material in which the waves have a less velocity than in the surround- ing medium, i.e. n>\. Let 0 be the source of the spher- ical waves. The path of any ray is indicated by OB, BB', WC. 0' is the image of 0 in the first surface ; and 0" is the image of 0' in the second one. So, if the pen- cil is a normal one, all the rays leaving 0 will diverge after emerging from the plate as if they came from FIG. 218. — Formation of an image of a point source O by refraction through a plate. 0". Its position may be at once calculated. 2. Prism. — We shall consider the prism made of a material in which the waves have a less velocity than in the surround- ing medium, i.e. n>l. Let 0 be the source of spherical waves. The path of any ray is_indicated by OB, SW, WC. 0' is the image of 0 in the first surface ; and 0" is the image of 0' in the second surface. It should be observed that, if the ray., Fro. 219. — Formation
A is x, the value of u is — x. If n' is the index of refraction of the first medium with reference to the second, n' = - ; and the general formula becomesn't; u n'r & u l=£, or! =!' Conversely, if a pencil of rays in the second medium is converging apparently toward a point C? in the first nn-dium, tlu-y will actually meet at the point 0. Fio. 221. — Formation of an Image of a point source 0 by ™™"™ " • <>«"«* "P»>w leal nurfa, ease when n>l the pencil of rays is oblique, it gives rise to an astig- matic pencil ; and if all the incident rays are considered, the image is a caustic surface. This phenomenon is said to be • In •, as in other similar cases, to spherical aberration. We shall now return to the problem of the refraction pro- duced by a spherical lens. It is evident from what has just been shown that a normal pencil from any point on the axis 470 LIGHT of the lens will give rise to a homocentric pencil emerging from the lens after the two refractions ; the image of the source, though, may be either virtual or real. We shall deduce the formula for a double convex and for a double concave lens, and then show that, by a suitable agreement as to signs, one formula may be used for all lenses. We shall assume at first that the lens is so thin that a ray incident at any point of one surface emerges from the other surface at a point which is at the same distance from the axis as is the former. a. Double Convex Lens. — Consider a section through the axis. Let 0 be the point source. FIG. 222. — Formation of an image of a point source 0 by refraction through a double convex lens. Let PJ be the radius of the first spherical surface of the lens. Let r2 be the radius of the second spherical surface of the lens. Let n be the index of refraction of the lens with reference to the surrounding air. The formula for refraction at the first surface of the lens >i ui ri is, then, — = h n ~~, where u and v* are positive if 0 and its image lie on the right
of the lens, because the centre of the first surface of the lens lies on this side. The refraction REFRACTION 471 at the second surface is from the lens out into the air ; so the formula is — = — h — —, where u2 and v2 are positive if the v2 "2 r2 points to which they refer are to the left of the lens. But, if the lens is thin, v1 = — w2, for the rays incident upon the second surface are those diverging from the virtual image produced at the first ; but a quantity u or v which is positive with reference to one surface is negative with reference to the other, since their centres are on opposite sides of the lens. Therefore we have the two formulae : n _ 1 t n-1 »! tij r, 1 _ n \| 1-n V9 t>! r II. -nee, I + l = -(n- ti, t;, In this formula, MX is the distance 0 lies on the axis to the right of the lens (referring to the cut) ; and v2 is the distance the image produced by the second surface of the lens lies to its left. Therefore, if we agree to call the distance the point source lies to the left of the lens w, and the distance the image lies to the right of the lens v, u = — ur v = — va. So the formula becomes Tin; quantity on the right-hand side of the equation is a con- staiit quantity t<u a -ivcn lens, and it is essentially positive if n>l, as it is in all ordinary cases, e.g. glass, quartz, etc. lenses surrounded by air. We write this quantity — ; and the formula then assumes the final form * 472 LIGHT b. Double Concave Lens. — In this case the formulae for refraction at the two surfaces are, as before, where u^ and v1 are positive if 0 and its image in the first surface lie on the left of the lens ; and Fio. 223. — Formation of an image of a point source 0 by refraction through a double concave lens. where uz and v2 are positive if the points to which they refer are on the right of the lens. Hence, or, introducing the same agreement as to signs as in the pre- vious case, u = uv v = v2 ; and
, if PR is large compared with RQ, we have, on expansion by the binomial theorem and neglecting small terms, 2 PR 2 PS 2 PR* PR-PQ = - Therefore, or. In the formula, as given above, for a lens, then, 2 PS 2 P R'O f f -S 1 CT? Consequently, ± + ^ = („ - 1)(I + I). If the incident ray makes a sufficiently small angle with the axis, and if the curvatures of the two surfaces are small, we may replace OC by OA, i.e. u, and D<y by BO1, i.e. v ; and we have the general formula, as before, rj rzl f REFRACTION 475 We shall now discuss the two types of lenses referred to above : for one/ is positive ; for the other, negative. a. Lenses for which f is positive. — The general formula is A -f _ = _. Let us consider several special cases. u v f If the point source is removed farther and farther from the lens, but kept on the axis, u approaches an infinite value and the waves be- come plane as they reach the lens. When w = oo, it is seen from the for- mula that v =/; and, since / is positive, this means that there is a real imae'e F10-226- — Special case: the source 0 Is at an Infinite distance on at a distance / the axis. from the lens. This point is called a " principal focus " of the lens. We may express this fact in words by saying that plane waves advancing with their normal parallel to the axis are changed by the lens into converging spherical waves whose centre is at a distance / from the lens ; or, rays parallel to the axis are bent by the lens in such a manner as to converge to a point on the axis at a distance/ from the lens. Similarly, if w=/, it is seen from the formula that 0 = 00. In words this states that, if the point source is on the axis of the lens at a distance from it equal to /, the diverging spheri< -al waves are so converged by the lens as to become plane and to advance in a direction parallel to the axis; or, we may say that rays through a point on the axis at a dis- tance / from it are SO detleeted by
the lens as to become parallel tn its axis. Conversely, we can have plane waves incident upon the lens from the other side, which will con- verge to a point at a distance/ from it. There are thus two •I7»i ( LIGHT principal foci ; one on each side of the lens, and at the same distance / from it, if the lens is thin. This distance is called the " focal length." Therefore, if we consider the point source as at an infinite distance from the lens, its image is the principal focus on the other side; and, if the source approaches the lens, the image recedes from it, until, when the source reaches the principal focus, the image is at an infinite distance. When the source is between the principal focus and the lens, i.e. when w</, it is seen from the formula that v < 0, so the image is on the same side of the lens as is the source, and is, therefore, virtual. When the source reaches the lens, i.e. when u = 0, v = 0 also ; so object and image coincide. If u has a negative value, the physical meaning is that rays are converging on the lens apparently toward a point on the other side, which may be called a " virtual " source ; and, in this case, as is seen from the formula, v is positive and less numerically than u ; but it should be noted that v does not exceed / in value so long as u is negative. Therefore, the converging rays are converged still more, and form a real image. It is thus seen that a lens for which / is positive always converges waves which fall upon it; for this reason it is called a "converging lens." The ordinary form of a con- verging lens is double convex ; but any thin spherical lens thicker along the axis than elsewhere is a converging one. We may arrange in tabular form the facts proved above in regard to u and v : U = CO.... V = f °°>w>/.... *>>v>f U =f.... V = 00 />M>0.... «>>-l>>0 0>w>-cc... />y>0 It is a simple matter to find by geometrical methods the position of the image of an object, for we know the effect of UK FRACTION 177 the lens upon three of
the rays from any point source : A ray parallel to the axis is deflected so as to pass through the principal focus on the other side of the lens ; a ray passing through the principal focus on the " incident side " of the lens will emerge on the other side parallel to the axis ; a ray meeting the lens at the point where it is intersected by the FIG. 887. — Formation of images by » converging lens., i.e. the "centre of the lens," keeps its direction unal- tered, because at tliis point the two faces of the lens are par- allel, and it is equivalent to an infinitely thin plate. In showing graphically the formation of images l>y a lens, we shall represent the lens by a straight lino, lot- simplicity. Thus, let Fr F.,. and C!><• the principal foci and the centre <>f the lens, and P any point of the object; its image is atP'. Two cases are xlmwn : if /'is farther from the lens than I he principal fnens. the image is real: if /' is between the lens and the principal focus, the image is virtual. 478 LIGHT If the object is small, close to the axis, and perpendicular to it, as represented by OP, its image O'P' is also perpen- dicular to the axis. So 0' is the image of 0\ and, since these points are on the axis, 00= u and C0' = v. Further, by similar triangles, P 0 : OP1 = u:v. The ratio of the length of the line O'P' to that of OP is called the " linear magnifica- tion " of the lens. It is evident from the geometry of the cut that Therefore, the magnification of the surface of any portion of C O O P O'P'the object perpendicular to the axis is ^-'O C A case of special interest is when the source, P, is placed in a plane perpendicular to the axis at the principal focus, or FIG. 228. — Special case : the point source P is in the focal plane of the converging lens. in the "focal plane," as it is called. It is seen from the geometry of the cut that the two emerging rays which we can draw from known principles are parallel to the line join- ing P to the centre of the lens,
If this angle is small. is the focal length, the length of the line Q(^ <«quals the \| / l.y this nngl.-, for the an (PC/,) an- 1 C<"\',) an- t-.jnal. If these two rays com' from t\\o\| 480 LIGHT on the edge of the distant object, its image will be bounded by the two points Q and Ql ; and the linear dimensions of this image will vary directly as the focal length of the lens. Therefore the area of the image will vary as the square of the focal length. If a lens having a long focal length is used, the size of the image is great; but, if a pho- tograph is to be taken, the time of exposure must be prolonged because FIG. 230. — Formation by a converging lens of an image of an object at an infinite distance. the energy is distributed over a large area. b. Lenses for which f is Negative. — The general formula is - -f- - = -^. If the source is at an infinite distance, u = oo, and therefore v=f; but since f is negative, v is also, and the image is a virtual one on the same side of the lens as the incident waves. This is called a " principal focus " ; and its distance, /, from the lens is called the " focal length." Similarly, there is another F vx Fro. 231. — Special case: the point source is at an Infinite distance on the axis. principal focus on the axis on the opposite side of the lens, and at the same distance from it if the lens is thin. If u=f, i.e. if the rays are converging apparently toward the principal focus on the opposite side of the lens, it is seen from the formula that v = oo, i.e. the emerging rays are all parallel to the axis. We can, moreover, in a similar manner to that used in the previous case, discuss the relation between the positions of the object and image as the point source moves from -f- oo to — oo. It is seen at once that the effect /,'/•;/••/,'.!' T/o.Y 481 of the lens is to make the incident rays or waves diverge; for that reason lenses of this type are called u diverging lenses." All thin lenses which are
thinnest at their centres are diverging. We can also deduce graphically the position of the image of any point source, because we know the effect of the lens upon three ravs : a ray parallel to the axis emerges in such a direction that if pro- longed backward it would ^ F Fio. 282. — Special case: the Incident rays are converging toward the principal focus on the farther side of the lens. meet the axis at the principal focus ; a ray pointed toward the principal focus on the other side of the lens emerges parallel to the axis; a ray through the centre of the lens retains its direction unchanged. A few cases will be drawn, the lens being represented as before by a straight line. F, O'Pia. 988.— Formation of an Image by a diverging lens. A real source P gives rise to a virtual image P' ; and a small object O/' perpendicular to the axis has an ini.r •< 0 /' also perpendicular to the a\ Thelinearmagnification prod viced by the lens, that is the ratio'> /' r r2 —, equals as before - : and the surface magnification is -a. Vr a u* AMES'S PHYSICS — 31 482 LIGHT A case of special interest is when the virtual source is in the focal plane. See Fig. 235. Let this point be P ; draw FIG. 234. — Formation of an image by a diverging lens. The "virtual source" P has a real image P', provided P is between the lens and the focal plane. two rays pointed toward it, one parallel to the axis, the other through the centre of the lens ; it is seen by geometry that the emerging rays are parallel, for the triangles (^(74), ((TAP), (CF2P) are all equal. Conversely, rays which are parallel to each other and inclined slightly to the axis diverge, Fio. 285. — Special case: the virtual point object P lies in the focal plane. after emerging from the lens, as if they proceeded from that point in the focal plane on the incident side where a line through the centre of the lens parallel to the rays meets the plane. This furnishes us with a method for the construction of the emerging portion of any ray. Let PQ be a ray meeting REFli ACTION 4K3 the lens at Q : draw through (7 a line CB parallel to PQ, am
i intersecting the focal plane at Fliu AI draw through A the line AQSi QS is the continuation of the incident ray. For, if there were two parallel rays PQ and BC, they would diverge, on emerging, as if they came from A. F, \S Fio. 286. — Construction for the refraction of any ray PQ by a diverging lens. Spherical Aberration, etc. — It may not be useless to state again the assumptions made in the above treatment of lenses: the lens is supposed to be thin; the object must be small and close to the axis; the pencils must be normal; the lens must have surfaces whose radii are large in comparison with its dimensions. If any of these conditions are violated, the laws cease to hold. Reference should be made to some special treatise such as Lummer, Photographic Optics, for a full discussion of the general subject. We have also assumed ttimughout that the waves were homogeneous, for n has been treated as a constant. Resolving Power of Lenses. — It has already been explained page 436) that when the waves from a point source fall upon a converging lens, the image of this source is the point to which liy far the greater amount of the energy is brought, but that owing to diffraction energy is relieved by other points also. In the case of a lens with a circular edge — such as most lenses have — the diffraction pattern consists of a bright area, whose brightest point is the geometrical image, 484 LIGHT as just explained, and this is surrounded by rings alter- nately dark and bright, if the waves are homogeneous; that is, the light gradually fades, then increases gradually, etc. So, if there are two point sources close together, the line joining which is perpendicular to the axis, there will be two diffraction patterns, which will overlap; and the resultant effect is due to their superposition. It is evident that if the two point sources are so close together that the centres of their diffraction patterns almost coincide, it may be impossi- ble to see these centres as separate bright points; but, if the centre of one pattern coincides with the first dark ring of the other, then the existence of two bright points may be recog- nized. There is thus a limiting value of the nearness of two point sources which can be perceived as such by the use of the lens. This quantity may be
deduced by considering a simple case. Let AOB be the cross section of a converging lens by a plane through the axis, and let 0' be the image of 0, a point on the axis. 0' is the image of 0 because the disturb- ances along the 0' different rays from the latter reach the former in the same time, and so are in Fio.287.-Diagramtoillu8tratethere8olvlngpowerofalen8. the game phage Qf vibration; but, if a point near 0' is considered, the different disturbances from 0 reach there in different phases, because they pass over different "optical" paths; and it may happen that the difference in path is such that owing to the disturb- ances arriving there in opposite phases they annul each other's action. If this is the case, this point is on a dark ring. If there is another point source at P, where OP is perpendicu- lar to the axis, the disturbance produced by its waves at 0' REFRACTION 485 may be calculated in a similar manner. Draw PA and A&, PB and BO\ these are the two lines from P to (7 whose difference in length is the greatest. If this difference amounts to a whole wave length of the waves, it may be assumed that the resultant action at O1 due to the disturbances from P which pass through one half the lens differs in phase by half a wave length from that due to those which pass through the other half of the lens; and therefore 0' will be on the first dark ring of the diffraction pattern due to P. From what was said above, when this is the case, P is as close to 0 as it is possible for it to be and yet to be seen separate from it. This condition may be expressed in an equation: (PB + B&) - (PA + ZO7) = /, where I is the wave length. But 0' is the image of 0\ and, therefore, drawing 03 and OBy we have, since the optical lengths from 0 to O1 are equal, _ (OB + BO') = (OA+AOf). Subtracting this equation from p the previous one, we have ^ ^ (PB-OB) + (OA - PA)=l. These quantities may be ex- *»•««• -Portion of Fig. pressed in a simpler form. Draw- ing the
; and f* or, Therefore, the focal length y2 = FIG. 289.— Formation of an image by a combination of two thin lenses. A special case of this is when the two lenses are placed close together, i.e. when h = 0. Then the focal length equals 0' ; or, calling it /, 1 = 1 + -i- /l-r/2 / /I /2 The reciprocal of the focal length of a lens (or of a combination of lenses) is called its "power." The unit of power adopted by opticians is that of a lens whose focal length is one metre; it is called a " diopter." To find the power of any lens, then, in diopters, its focal length in metres must be measured, and its reciprocal taken. A converging lens is called positive. Thick Lens. — If the lens is of such a thickness that it cannot be assumed to be "thin," in the sense in which this word has been used above, the solution of the problem of re- fraction can be obtained, exactly as in the case of a thin lens, by considering the refraction of a pencil of rays at the two surfaces. But the distance from the first surface to the image produced by it no longer equals its distance REDACTION from the second surface ; that is, v1 does not equal wa numerically (see page 471). If t is the thickness of the lens, M2 = — (Vj + j); and, if this value is substituted in the equations, the iinul formula connecting the positions of the object and image may be deduced. Chromatic Aberration. — Attention has been called re- peatedly to the fact that the index of refraction of a sub- stance is different for trains of waves of different wave number, showing that the velocity of these different waves in the substance is different. (It should be remembered that the wave number of a train of waves does not change as it passes from one medium into another, e.g. air into glass, for the number of "waves" reaching any point in a given interval of time must equal the number that leaves it. But, calling N this wave number, and Fi, L, K, L, the velo- Fio. 240. — Chromatic aborrati-n. cities and wave lengths in the two media, l\ = Nlr V2 = JV72 ; and since Vl is different from
I',. /! is different from lv So, as the waves pass from one medium into another, their wave length changes.) In the case of ordinary dispersion the shorter waves are refracted more than the long ones ; or, in other words, the index of refraction varies in an inverse manner from the wave length. Methods for the study of tin- connection between these two quantities will be described in a later chapter. This fact that n varies with the. wave length is of great importance in dealing with the theory of lenses; for this quantity enters into the two fundamental formulae: and Linear Magnification =-. Therefore, if waves of different wave length are used to illu- minate a Ljiveli nl.j.-ct, or if tllC object itself emits SUci 490 LIGHT not alone will the corresponding images be at different dis- tances from the lens, but they will also be of different sizes. Since light waves of different wave length correspond to different colors, this phenomenon is called " chromatic aberra- tion." For ordinary substances n is, as has just been said, greater for blue light than for green, etc. ; and so, if there are two trains of waves corresponding to these colors, / is less for the blue ones than for the green, and consequently the principal focus for the former is nearer the lens, and the magnification is less. If the object emits white light, there will be a series of colored images of different sizes. This fact is, naturally, most detrimental to the proper use of an optical instrument, for sharp clearly defined images are de- sired. As will be shown in the next paragraph, it is possible by a combination of two lenses of different material to remedy this defect to a certain extent by causing any two definite trains of waves of different wave length to be brought to the same focus, but waves of all wave lengths will not be simi- larly affected. So, when white light is used, if two of its components are thus brought to the same focus, the other components will have different foci — they form what is called a "secondary spectrum." Thus chromatic aberration can be corrected only partially. The choice of the two trains of waves which shall be brought to the same focus is arbi- trary; if the instrument is to be used for visual purposes, two trains are chosen which affect the eye most intensely ; while, if it is to be used for photographic work, the two trains of
waves chosen are those which act most intensely upon a photographic plate. Achromatism. — Since a lens can be considered as made up of prisms, we shall first show how one prism may be so chosen as to neutralize the dispersive action of another prism for two given trains of waves. In speaking of the effect of a prism, it was shown that the deviation produced depended upon the material and angle of the prism, the wave number of the REFRACTION 491 waves, and the angle of incidence. The difference in the de- viations for t\v<» di tie rent trains <>f waves is called the "dispersion" of those waves; and by suitably choosing the material and angle of the prisms and the angles of incidence, it is evidently possible to secure the same dispersion for two dftiiiite trains of waves. When this is done, the dispersion of any two other trains is as a rule different ; for owing to the difference in the material of the prisms there is no connection Fio. 241. — Two different prisms of different materials may produce the same dispersion of two rays lt and /,. 1 "-tween their dispersions in different parts of the spectrum. 'Hi us suppose that, when, white light is incident upon two prisms at angles /and /', as shown in the cut, a ray is dis- persed in such a manner that two of its components, of wave length Jj and lv have the same dispersion D. Let the angle hctween the emerging ray ^ and the normal to the second face of the prism be N and N' in the two prisms. Then if abeam of parallel rays of wave length Zj are ineident at an angle N' upon the second face of the second prism, they will emerge, alon<_r the direction of the incident ray in the cut, i.e. making the angle /' with the normal : and. if a beam of parallel rays of wave length l^ is incident upon the same face at an au^le (N1 + D), they will emerge in the same direction as the former rays. This condition may be secured by inverting the first prism and placing it with its edge parallel to that of the other pi-ism, l.iit M inclined that the angle of incidence upon it of the i. iv /, IV.. m the scc,,nd p] N\ f
..r. as is seen from geometry, the angle of incidence <»t the ray £s is, 492 LIGHT under these conditions, (N -f D). Therefore a ray incident upon the second prism at the angle I1 is dispersed by it ; and two of its components, ^ and Z2, fall upon the first prism, are refracted by it, and emerge parallel to each other, making an angle I with the normal to the last face of the prism. Therefore a beam of parallel rays of white light incident upon the second prism at an angle I' will be dispersed by the two prisms ; but all the rays of wave length l^ and £2 will emerge parallel to each other in the direction defined by /. In practice it is found _ja that it is possible to choose the prisms so that, when combined as described above, FIG. 242. — One prism may neutralize the disper- their adjacent faC6S may be sion of two rays produced by another prism. n i i • j_i parallel ; but owing to the differences in the material and angles of the prisms the directions of the incident and emerging parallel rays are not necessarily the same. Therefore this double prism still deviates the two rays ^ and Z2, but does not disperse them. Such a prism is called an " achromatic " one ; or, either one of its component prisms is said to be "achromatized." (It is evident also that two prisms may be compounded which will not deviate one particular train of waves, but will deviate the others, thus producing dispersion.) It follows at once that, by combining two lenses of different materials, one diverging, the other converging, a double lens may be secured which will deviate two rays of definite wave length, but not disperse them. Such a lens is shown in the cut. The converging lens is as a rule made of "crown" glass ; the diverging one, of " flint " glass. It is called an achromatic lens for the two definite trains of waves. FIG. 248. — Achromatic lens. FLINT QLA88 CROWN_GUSS REFRACTION 41 »o If the component lenses are thin and have focal lengths /! and /2, the focal length of the combination is given by — = — \| Therefore, we can express the facts in regard / /i /2 to the achromatic combination by saying that
if one were to look through the other with the eye accommodated for an infinitely distant object, as is the case when one is using a telescope. Therefore it is better to make the distance of the lenses apart slightly less, although by so doing the chromatic aberra- tion is not exactly corrected. Thus, if k = $/j, the distance of the prin- cipal focus from either end is Fio. 244. — A Ramsden eyepiece: the focal lengths of the two lenses are equal. In this case, any object placed at this distance from one of the lenses will be seen on looking through the other lens at apparently an infinite distance. In other words, this eye- piece has the focal properties of an ordinary converging lens of this focal length, but it is approximately achromatic. 2. Huygens eyepiece. — In this there are two plano-convex lenses, placed as shown in the cut, with focal lengths /, and /2, where /: — %f.,. In order to secure achromatism their distance apart should satis- fy the equation h = The distance of the principal focus from the first lens is given by /l(2/2~/2) _ 3/2 /2 2/,-/,-/« 2 - 2 This means that, if a ray is incident upon the first lens at such an angle as to be pointed toward a point between the lenses at a dis- tance £ from the first lens, it will emerge from the second lens parallel Fio. 245. — A Huygens eyepiece : the focal length of the first lens is three times that of the second. to the axis. Therefore, since the object must be virtual in order to have an image at infinity, this eyepiece cannot be used, like a converging lens, to magnify ordinary objects. CHAPTER XXIX OPTICAL INSTRUMENTS OPTICAL instruments, so called, are pieces of apparatus so designed as to make use of light waves in order to form images of luminous objects. In general they consist of com- binations of mirrors, lenses, and prisms; but some instruments do not have such parts. The number of these instruments is great, but in this chapter only the simplest and most gen- erally important ones will be described. Optical instruments may be divided roughly into three <lasses : " pin-hole," reflecting, and refracting. In
the first, no mirror or lens is used ; in the second, some form of mirror is the essential feature ; in the last, some prism or lens, or a combination of prisms or lenses. " Pin-hole " Instruments These have been described in Chapter XXV, and nothing further need be said here. They all depend upon the "recti- ir propagation" of light. The pin-hole camera and the •• e.imera obscura" are the only instruments of this class that of importance. Reflecting Instruments The action..I plane, spherical, and parabolic mirrors \\ as explained in Chapter \\YII: and their use as looking glasses, in searcl} lights, etc., was described. The great advantage of all forms of reflecting iiM rnmcnis over refract- ing om-s is that they are necessarily five from chromatic 4'.':, 4'.t»i LIGHT aberration. Spherical aberration, however, can be avoids 1 only by using normal pencils of light and mirrors with small curvatures. One important application of a concave spherical mirror was first made by Newton, namely, to form a "telescope." When we look at a distant object, it subtends at the eye a comparatively small angle. Thus, if A and B are two points on opposite edges of the object, and if 0 marks the position of the eye, the linear angle subtended by the object is (AOB). We estimate the distance of an object from us by the angle which it subtends at our eyes ; the nearer it is, the larger is this angle. FIG. 246. -Diagram representing the effect of a The purpose of a telescope telescopejn increasing the angle between two rays jg to brinff a distant ob- AO&nlJBO.. ', ject apparently nearer, by changing the direction of a ray AO to Af 0, and a ray B 0 to B' 0 ; so that the angle subtended by the rays from A and B is now (A' OB'), an angle greater than (AOB). The "power of the telescope" is defined to be the ratio of the angles (A1 0&) and (A OB). In all telescopes either a concave mirror or a converging lens converges the rays so as to form a real image ; and this is viewed by a lens, or a combination of lenses, called the "eyepiece." The simplest
form of eyepiece is a converg- ing lens, but a Ramsden or Huygens eyepiece is ordinarily used. In doing this the eyepiece is so placed that the image to be viewed comes at its principal focus (or just inside it), and it forms a virtual magnified image at apparently an infinite distance. Reflecting Telescope. — In all forms of reflecting telescopes a concave spherical mirror is used to receive the rays from OPTICAL INSTRUMENTS 497 the distant object. The mirror is mounted in some frame- work or tube, so that it may be turned to point in any direction. In the cut, let C be the centre of this spher- ical surface, and I\C and PC be lines drawn.from two opposite points ill the edge Of the F'°- 247- — Dia?rnin representing a reflectinp telescope. P remote object. All rays parallel to Pj(7are brought to a focus at a point Fv on PjC', halfway between C and the mirror; and all rays parallel to PC will be brought to a focus at a point F, on PC, halfway between C and the mirror (see page 449). Therefore, there will be a real image of the object in the plane FFr and I\ are two points at an infinite distance. In order to see the magnified image of this image produced by the eyepiece, some plan must be adopted which will ren- der it unnecessary for the observer to stand in front of the mirror and so obstruct the view. Several have been tried : 1, the mirror is tipped slightly so that its centre lies outside the telescope tube, and the principal focus for parallel rays lies near the edge of this tul>e, and thus the image may be viewed — this arrangement was used by Herschel ; 2, a small plane mirror (or a totally reflecting prism ) is placed close to the principal focus, but between it and the mirror, and is so inclined as to turn the rays off sidewise through an opening in the side of the telescope tube; thus the image is formed outside or near the edge of the telescope — this was Newton V plan; 8, a small convj mirror is used in place of the plan, mirror, and is so turned as to reflect the rays back through a small opening at the centre of the concave mirror and form an image at iinur this arrangement is due to Casse- grain. 4. or. finally
. therefnre, the great ad \, intake of having a teles< whose concave minor has a long focal length. 500 LIGHT The limit to the resolving power of a telescope is deter- mined, as we have seen on page 487, by the diameter of the mirror ; the larger it is, so much the greater is this power. A large mirror also gathers more light, and therefore enables fainter objects to be seen. Newton constructed his first reflecting telescope in 1668. It had a diameter of one inch, and magnified thirty or forty times. He later made a larger instrument. Previous to this, such instruments had been designed ; and the invention is attributed to Niccolo Zucchi (1586-1670) of Rome. Refracting Instruments All instruments containing lenses are subject to a certain amount of chromatic aberration, although this may be mini- mized by using only achromatic lenses and combinations. There is also always some spherical aberration ; but this may be avoided largely by using only normal pencils of light, and exposing only the central portions of the lenses. The number of the defects which are possible with a lens is so great that some special treatise on the subject should be consulted. Only a few of the simplest refracting instru- ments will be described here. Photographic Camera. — In this instrument a real image of an object, more or less remote, is formed on a plane photo- graphic plate. This image is produced by a converging lens or system of lenses. If the camera is used for taking photo- graphs of landscapes, the lens is ordinarily a single achroma- tic one, and a diaphragm with a circular opening of variable size is placed in front of it. If it is used for photographing buildings, special pains must be taken to avoid spherical aberration and the consequent distortion; two achromatic lenses symmetrical with reference to a plane halfway between them are used, the diaphragm being in this plane - this is called a " rectilinear " or " orthoscopic " lens. For portrait work, lenses of large diameter must be used in order to secure as much light as possible. OPTICAL IN STRUM E* TS 501 When photographs of distant objects are taken, the images are, in general, small ; but by combining a diverging lens with the converging system, a larger image may be secured. Sueh a compound lens system is called a "
teleobjective." Projection Lantern. — This instrument consists of a lamp, or a strong source of light, which by means of lenses illumi- nates a drawing or photograph on glass, or some object which is transparent in parts, and of a converging system of lenses which throws a real image of this illuminated object upon a suitable screen. The lens system between the light and the object — called the "condenser" — consists, in gen- \ \ Fie. SCO. — Projection lantern ; (.-1) arc light; (C) condenser; (5) slide to be Illuminated; (/•') focusing lens. eral, of two plano-convex lenses with their curved surfaces in contact. Its function is to deflect down upon the object to be " projected " as much light as possible, so as to render it strongly luminous. Then by means of the "focusing lens " a real image of it is formed upon the screen. The focusing and condensing lenses must be achromatic, and the former must be corrected for spherical aberration also. Astronomical Telescope. — This instrument consists of a C()nver«rin^ achromatic lens, called the "object gla—." \\liieh forms a real image of the distant object in its focal plane, and this is viewed by an eyepiece. (In the cut a Ramsden eye- piece is represented.) The eyepiece is so placed that the image formed by the object glass comes at or just inside its 502 LIGHT principal focus, so the object is seen apparently at infinity. The power of this telescope is, therefore, like that of the FIG. 251. — Astronomical telescope: O is source at an infinite distance; O' is image formed by object glass ; 0" is virtual image formed at an infinite distance by the eyepiece. reflecting instrument (see page 499), equal to the ratio of the focal length of the object glass to that of the eyepiece. The resolving power of a telescope is, as we have seen, determined by the size of the object glass, as is also the quantity of light received. (See page 500.) The astronomical telescope forms, as just explained, a virtual image of the distant object which is inverted; that is, if the object is a tree, the image will have the tree pointed down instead of up. This is a disadvantage if the instrument is to be used for purposes that are not purely astronomical
. Dutch Telescope (or Galileo's telescope). — This telescope is free from the disadvantage of the astronomical instrument FIG. 252. —Dutch telescope. to which reference has just been made. It consists of a converging lens, forming the object glass, and a diverging <U>TK AL INSTRUMENTS 503 I- us so placed that the principal focus on the side next the observer coincides with the principal focus of the object glass. Thus, parallel rays from a point of the distant object are con- verged by the object glass toward a point behind the diverg- ing lens, which is in the focal plane of both lenses ; these rays are diverged again by the second lens and emerge parallel to a line drawn from the centre of the second lens to the point in the focal plane toward which the rays were converging. The image of the distant object is therefore a virtual one at an inlinite distance; but it is erect; that is, the image formed of a distant tree represents the tree in an upright position. The first telescope was probably constructed by Hans Lippershey of Middleburg in the Netherlands in 1608 ; and Galileo, upon hearing of the invention, but without knowing any details of the construction, made an exactly similar instrument in 1009. Kepler was the first to suggest the use of a convex lens for the eyepiece. Galileo immediately used his telescope for observing the heavenly bodies, and made many most im- portant discoveries. Previous to the construction of the reflecting telescope by Newton and the invention of the achromatic 1ms by Pollond, the only means of minimi/ing the color effects produced by telescopes was to use lenses of great focal length, which were clumsy and ott* «.1 many disadvantages. 1 1 _cens presented to the Royal Society of London a lens whose focal length was 123 ft. Microscope. — This instrument is one designed to -magnify" a small ohjert, that is, to increase the apparent distance apart <>t any t \vo of its point* wh'u-h are close together. a. Simple microscope. — As was said on page 494, in speak- iii'_r of eyepieces, a single converging lens or.t K'amsden eye- piece can be used as a microscope, the object being placed inside the principal focus. There is thus formed an virtual image of the object (see page 477). The dis- tance at
which this image is formed is arbitrary: but it is generally chosen as about 2"> om. (or 1" in. > fn>ni the eye, because this U t: < <• at which j.copl,- with normal eyes hold an object in «>rder to see it most distinctly. 504 LIGHT b. Compound microscope. — The magnification secured by a single lens is not great ; and it is in general combined with another lens system, as shown in Fig. 253. The latter forms a real magnified image of the object, which is viewed and again magnified by the eyepiece. (In this instrument a Huygens eyepiece may be used, as is illustrated in the cut.) The lens system which is nearest the object is called the FIG. 258. — Compound microscope: OP is the object; 0"P", the virtual image formed by the eyepiece. "objective." It consists of several lenses so chosen as to give an image as free as possible from chromatic and spher- ical aberration ; and it is so constructed as to give as much light as possible to the image and at the same time to have a large value of n sin ^V (see page 487), so that the resolving power is large. The magnifying power can be deduced by simple geometrical methods. The use of converging lenses as simple magnifiers was known to the ancients; but the compound microscope was probably invented by Zach- arias Joannides of Middleburg in the Netherlands and his father, some years before 1610. It was invented independently by others also, among OPT1< AL INSTRUMENTS 505 whom was Galileo. All the early microscopes had a concave lens for the eyepiece ; and Franciscus Fontana of Naples was the first to suggest the sul»titution of a convex lens. Spectrometer. — A spectrometer is an instrument primarily <lfsigned to measure the angle of deviation produced in the direction of a beam of light by reflection or refraction. It consists essentially of four parts. There is a substan- tial metal base to which is rigidly connected an upright metal cylinder, whose axis is called the "axis of the instrument." To this is attached a metal plate whose edge is divided into degrees, minutes, etc. Two bent metal arms are also attached to this cylin- der by means of collars, so tliat they
can turn around it as an axle ; they carry metal tubes which lie in a plane perpendicular to the axis of the instrument and are thus movable in this plane. One of these tubes is a simple form of astronomical telescope; the other carries at one end an achromatic converging lens whusf focal length is that of the tube, and at the other end it is.-IMN.-.I by a metal cap in which there is a fine slit with straight parallel edges, which can be opened or closed — this is called a 4i enllimator." If the slit is illuminated, it serves as a s<>mvc <>f diverging waves which will emerge from the lens of the collimator in the f,.rm of plane waves. 506 LIGHT The telescope and collimator are turned until their axes intersect at the axis of the instrument ; and the object which is to produce the deviation of the light — a plane mirror or prism — is placed on a platform in the middle of the metal plate referred to above, having the normals to its reflecting or refracting faces in the plane of the axes of the collimator and telescope. If now by means of the former a beam of parallel rays falls upon the mirror or prism, they will be deviated, and their new direction may be found by turning the arm carrying the telescope until there is formed in the centre of the field of view an image of the illuminated slit. In order to determine this condition FIG. 255. -cross hairs exactly, it is customary to insert in the focal plane of the object glass a metal ring across which are stretched two fine silk fibres or spider lines, which are called the "cross hairs." These are made to cross exactly at the centre of the tube. Since they are at the focal plane of the object glass, they will be seen through the eyepiece at the same apparent distance as the object which sends the parallel rays into the object glass. The positions of the collimator and telescope may be noted by means of the divided scale on the edge of the plate, if suitable pointers or verniers are attached to them. used in telescope. * With this instrument the laws of reflection and refraction may be verified ; the angle of a prism and the angle of minimum deviation produced by it for any train of waves may be measured; etc. Therefore, the index of refraction of any substance which can be made in a prism may be deter-. A + D sin
— \|- mined ; for n = ^ (See Ames and Bliss, Manual of sin- Experiments in Physics, pages 459-475.) OPTICAL M->T/,T.I//-;.\ :,«»; If the laws of reflection are to be studied, ordinary white light may be used to illuminate the slit; but, if the phe- nomena of refraction are to be observed, special precautions must In- taken which will be described in the next chapter. Effect of Diaphragms. — One fact that must be taken into account in the description of all reflecting or refracting instruments is that the wave front of the waves is always limited by certain apertures or diaphragms. Thus, in the case of a telescope, the only portion of the wave front that enters the instrument has the size of the object glass. Again, if there are diaphragms in the tubes, they may limit the cone of rays which proceed from any point of the object and enter the instrument. Further, the only rays that are of practical use in the case of an instrument that is used visually are those which have such a direction as they leave the eyepiece as to enter the pupil of the eye. The effect of the presence of these various circular openings is felt in many ways. The brightness of the image, the resolving power of the instrument, the contrast of the background, the spherical aberration, etc., all depend upon their size and position. For details in regard to the matter, reference should In- made to some advanced text-book, such as Drude, Optics, or Luiuiuer, Photographic Optics. CHAPTER XXX DISPERSION IN speaking of chromatic aberration, achromatic lenses, etc., it was noted that the index of refraction of a given substance varies with the wave length of the light which suffers refraction, and that, in general, the shorter the wave length, the greater is the index of refraction. In this chapter the method of determining the connection between these two quantities will be discussed. Pure Spectrum. — Before any relation between index of refraction and wave length can be established, it is necessary to devise a method for securing homogeneous light of a FIG. 256. — Prism spectroscope. definite wave length. The method ordinarily adopted is to make use of the dispersive action of a prism. If waves of a definite index of refraction, emitted by a small source,
fall upon one face of a prism, they suffer refraction and emerge 608 509 on the other side, diverging apparently from a virtual image of the source. They do not, therefore, of themselves come to a focus ; but, if a converging lens is introduced, the rays may be focused upon a screen. This real image is due to the virtual iina^e formed of the small source by the prism ; and it should be noted that the latter image is an astigmatic one, unless the pencil of incident rays meets the prism at the angle corresponding to minimum deviation. (See page 467.) The intensity of the effect can be increased if the point source is replaced by a series of such sources forming a line parallel to the edge of the prism. In general, a fine slit is made in an opaque solid, and the source of light, in the form of a flame, etc., is placed behind it. The image on the screen is in this case a narrow rectangle, practically a line, parallel to the slit. If the source is emitting several trains of waves of different wave length, there will then be formed as many images of tin- slit as there are separate trains of waves. The light is said to be "analyzed"; or a "spectrum" of the lii?ht is said to be formed. If the slit is very fine and the adjustment for minimum deviation as exact as possible the spectrum is said to be "pure," because in this case each train of waves affects only a very narrow rectangle on the screen, and so there is only a small amount of overlapping of the images. (Since t}i«- angle of minimum deviation is different for waves of different wave lengths, it is best to make the adjustment of the prism for the mean wave length of the light which is Further, since the focusing lens — even if aohrom — has a different focus for different waves, the screen must be curved so as to obviate the error, or one must be satis- fied with a slightly i 114)11 re spectrum.) In this manner the nature of the light — or, speaking in a more general manner, of the ether waves — emitted by any source can ivestigated; and we shall discuss this matter in a later chapter. 510 LIGHT Resolving Power. — Owing to the fact that the waves which leave the slit have passed through several "apertures" on their way to the screen (or to the eye), and that their wave
front has been limited by this means to the shape of these apertures, the final image is not a line. These apertures are the edges of the lenses, of the prism, and of any diaphragms there may be in the tubes. There is, therefore, a diffraction pattern produced, exactly as described for a lens on page 484. The image of the slit, provided the light is homogeneous, is a broadened line accompanied on each side by a series of alter- nately dark and light fringes, whose intensity is much less than that of the central line. The distance apart of these fringes varies with the FIG. 257. — Diagram to illustrate the resolving power of a prism. f,.,, width of the aper- ture. So, if the source of light is emitting two trains of waves of wave length I and I + AZ, there will be two overlapping diffraction patterns ; and the two trains of waves will produce two distinct images if the central line of one pattern is so far displaced by dispersion as to coincide with the first dark fringe of the other. If #j is the length of the shortest path of any of the rays through the prism, and £2 is the length of the longest one, t2 — t± is practically equal to the length of the base of the prism ; call its value b. If n and n + Aw are the indices of refraction of the two trains of waves, of wave length I and I + AZ, which are just resolved by the prism, it may be shown that An = Y • In other words, if the prism has a thick base 0 An is small, the image formed of the slit is extremely nar- row, and the resulting spectrum is pure ; but, if the prism is thin, the image formed is broad. D/> /•/-:/,•> ION 'A 1 Spectroscope. — The essential parts, then, of an instrument to In- used in order to form a pure speeti urn are a slit, a pi ism (or other means for securing dispersion), and a converging lens. This is called a "spectroscope." In general, the arrange- ment is slightly different, a collimator and telescope heing used as described for a spectrometer on page 505. When the slit is illuminated, plane waves emerge from the lens; these enter the prism and are dispersed by it ; and, finally
, as they leave the prism they enter the telescope which is focused for plane waves ; and a series of colored images of the slit is seen 1)\ the observer looking in the eyepiece. Measurement of Dispersion. —If the eyepiece is removed, and the images formed by the object glass are focused sharply on an opaque screen, and, if a fine slit, parallel to that in the collimator, is made in this screen, it will be illuminated by iiract i( -ally homogeneous waves, and will serve as a source K h waves. This entire instrument is called a "mono- chromatic illuminator." By varying the position of this second slit (or by turning the prism) the wave length of the waves transmitted through it is altered..Methods will be discussed later which enable one to measure tin- wave length of any train of waves; and, granting that the wave lengths of the waves emitted by the source are known, we have thus a method for measuring the refractive index of any substance for waves of definite wave lengths. The substance to be studied is made in the form of a prism, and is placed on a spectrometer table. The entire instru- ment is turned until the slit of its eollimator eoinrides with the slit of the inonnehromatie illuminator. The angle of the prism and the angle of minimum deviation t<u different waves are m- md I mm a knowledge of these quantities the indi -inn may be calculated. Fraunhofer Lines. — It will be sho\\ n later that, when HUH- light, liy a spectroscope, it is found to OO1 trains of waves of varying lengths, forming \\hal a; 512 LIGHT first sight to be a "continuous" spectrum; that is, one in which waves of all wave lengths within certain limits are present. But, if examined with an instrument of fair resolv- ing power, the solar spectrum is seen to be distinguished by the absence of a great many trains of waves, as is shown by the presence of black lines across the bright colored spectrum. (The fact that this absence of the waves is manifested by the presence of lines is due, of course, to the use of a slit illumi- nated with sunlight as the source of the light in the spectro- scope.) These lines are called " Fraunhofer lines," because they were first carefully studied
by him. He measured — by methods to be discussed later — the wave lengths of the waves which correspond to these lines ; and to the most prominent of them he gave names, in the form of letters. (Thus the A line is at the limit of vision in the red end of the spectrum ; B and 0 are also in the red ; Dl and D2 are in the yellow, etc. ; K is at the limit of vision in the violet end of the spectrum.) It is customary, therefore, in studying the dispersion of any prism to illuminate the slit of the spectrom- eter with sunlight, and then to note the angle of minimum deviation that corresponds to the waves in the immediate neighborhood of the various strong Fraunhofer lines. Thus, nE means the index of refraction for waves whose wave length is that of the E line, etc. Dispersive Power. — The differences in the refractive index of various substances for different waves is shown in the fol- lowing table, in which the columns give the values for the A, B, C, etc., lines : A B C D E F 0 Water, 16° C... Carbon bisulphide, 10° Crown glass Flint glass. Rock salt, 17°.. 1.330 l.«16 1.528 1.678 1.537 1.630 1.681 1.539 1.332 1.626 1.531 1.583 1.640 1.334 1.635 1.534 1.587 1.644 1.338 1.661 1.540 1.597 1.653 1.537 1.502 1.549 1.546 1.606 1.661 // 1.344 1.708 1.561 1.614 1.568 DISPERSION 513 It is evident from this table that not alone is the index of refraction different for different substances, but that also the dispersion of any two rays, e.g. (nD — wc), may be the same for two substances and the dispersion of two other rays may differ ; thus for crown glass and rock salt nB — nA = 0.002, but for the former nn — n?= 0.011, while for the lat- ter it equals 0.015. Further, what is called the "dispersive power " is different. This is defined to be the ratio n»~nA. (In practice, the n
greater. So, if a prism is made of this substance, it will deviate the long waves near the ab- sorption band more than the short ones ; and the resulting S8iS3Sg§§2gg§8S!Sg 2.3'5>p'^>« oooSoo pc-t-V-t- WAVE LENGTHS IN fJ.fJL FIG. 2GO. — Dispersion curve of cyanine. spectrum will appear, in general terms, as if it were divided in two parts by the absorption band and as if these two halves were shifted toward and across each other. Whenever there is an absorption band, there is anomalous dispersion, and many substances show these phenomena in the visible spectrum ; such are the aniline dyes, the vapors of sodium and other metals, thin layers or films of metals, etc. It will be seen later that all these substances have other opti- cal phenomena which are closely connected with this fact. Rainbows. — An interesting illustration of dispersion is furnished by the phenomenon in nature called the " rainbow." After a rain shower, DISPERSION 517 if tl.r sun is not far from the horizon, and the rain clouds have passed in such ;i.liivrtion that the observer is between them and the sun, a series of colored arcs or circles may be seen on looking away from the sun. These colored arcs are in the following order : violet on the inside shad- ing off to red on the outside, then a dark space, and another arc colored red on the inside and violet on the outer edge. They are all portions of circles whose centres coincide at a point lying in the prolongation of the line joining the sun with the eye of the observer. There are often also u supernumerary " bows inside the primary one. G — - FIG. 261. — Rainbow : refraction oi a ray by a drop of water, single reflection. A complete explanation of these phenomena requires a consideration of the size of the drops, the nature of sun -light, the size of the sun, etc.; this was first given by Sir George Airy. An elementary, but imperfect, theory was given by Descartes ; an outline of which is as follows. Con- sider a raindrop as a sphere of water, and draw the paths of the rays inci- dent upon it from the sun. Certain rays will enter
the drop, suffer reflection once and be refracted out, as shown in the cut, in which 0 is the centre of a drop, OS is the direction of the sun, and AB, BC, CD, DE represent the portions of a ray. The deviation of the ray is shown in the cut by the angle (GFE), where F is the intersection of the prolongations of the incident and emerging portions of the ray, AB and DE. By means of higher mathematics it may be shown that if there in a homo- geneous beam of rays, all parallel to OS, falling upon the drop, the angles of de- viation all exceed a definite value. In..—Rainbow: JVI» angle of mini- mum deviation. N other words, there is a minimum value of the deviation, and this fixes a certain direction with reference to the line OS. If tins minimum angle is N, draw the line AB making such an angle with SA. Then of all the 518 LIGHT rays parallel to OS falling upon the drop, none are so deviated as to • •iin-rge outside the angle (SAB)] some emerge in the directions ABr A Bp etc. This minimum angle is different for different wave lengths ; for violet it is about 140°, for red about 138°, etc. Therefore, if the observer looks up at the rain cloud in such a direction that his line of sight makes an angle of 180° -140°, or 40° (or less), with a line joining his eye to the sun, he will receive violet light from the rain- l / FIG. 268. — Kainbow. drops. If he looks up at an angle greater than 40°, he receives no violet light at all. Therefore, the observer will see a violet arc, each point of which subtends at his eye an angle of 40° with the line drawn from the sun. Similarly, with the other colors, there is a red arc corresponding to the angle 180°— 138°, or 42°, which is sharply denned on its outer edge, and to see which one must look higher up in the sky than was necessary in order to see the vio- let arc. But some of the rays from the sun suffer two reflections in the rain- drop, as shown in the cut. As before, we may show that there are FIG. 264. — Rainbow: refraction of a ray by a drop of water, double reflection.
angles of minimum de- viation for the different colors, which give rise to a red bow at the angle 51° and a violet one at 54°, these bows being sharply defined on their lower edges. CHAPTER XXXI INTERFERENCE OF LIGHT Interference Fringes. — The general phenomena of interfer- ence of waves were described in Chapter XXI, page 374, and the special ones dealing with light were discussed in Chap- ter XXV, page 420. It was shown that the simplest mode of illustrating interference was to place two identical sources of light close together and to allow them to illuminate a screen or to enter the eye directly. If the two sources are parallel slits, the interference pattern is a series of parallel colored fringes ; if homogeneous light is used, these are alternately bright and dark, and at regular intervals apart proportional to the wave length ; if white light is used, the fringes are a superposition of different sets, each of whirl i is due to a different color, thus proving that white light is equivalent to a superposition of waves of different wave length. Unless the two sources are identical, there is no permanent phase relation between the two sets of waves emitted l.y them, and so these cannot interfere. In demonstrating the interference fringes which are pro- duced by the two identical sources, a converging lens is always used, the arrangement being as shown in Fig. 265. Ol and 02 are the two sources ; L is the lens : M \^ a sen-en placed at a distance fmm the lens equal t<> its focal length for parallel rays. The two sources are emitting niys in all directions; let OlAl and O^A^ be two parallel rays. After n-f motion by tin- 1ms, tln-\ will meet at the point B in the screen, \vhn. < I: a line drawn through the centre of tin- lens parallel to the incident rays. (See page IT1.'. > If 619 520 LIGHT the difference in length of the optical paths of these rays is half a wave length (or an odd number of half wave lengths), there will be complete inter- ference at B. This difference in path is found exactly, as on page 377, by drawing a line from 02 perpendicular to 0-^Ar For, if B were a source of waves, two of its rays would be BAfli and BA202i and the position of the wave front
linn. This slit, if illuminated, will emit waves, which will suffer refraction and deviation by the two hal\«-s of the prism. If the slit is at 0, as sho\\ n in the cut, one half will form a virtual image at 0,, 522 LIGHT the other at 02. So the waves as they emerge from the biprism will come apparently from the two sources Ol and 02 ; and there are then two identical trains of waves, which will inter- fere, and may be focused on a screen by a lens, as described above. Therefore, if the distance 0^0^ is known (and it may easily be determined by experiment), the wave length of the light may be measured. Fresnel's Mirrors. — These are two plane mirrors which are carefully adjusted until their faces are slightly inclined to each other, but are in actual contact along a line. FIG. 267. — Fresnel's mirrors: two virtual images 0j and 02 of the source 0 WV are produced. A slit 0 is placed parallel to this line ; and therefore two virtual images of it, 01 and 02, are formed by the two mirrors. Let B be the line of contact of the two mirrors. Then a pencil of rays P^OB falling upon the first mirror will be reflected into the pencil Q^ OlBl ; and the pencil B OP2 falling upon the second mirror will be reflected into B202Q2. Therefore there will be a region included in the angle (^B^B,^) which is traversed by two trains of waves coming from identical sources. Lloyd's Mirror. - In this arrangement a slit is placed paral- lel to a plane mirror, at some distance from it, but only a slight distance above its plane. There will be a virtual image formed by the reflected rays ; and so any Pie. 268.— Lloyd's mirror: a virtual image 0, of the source 0, is formed by the mirror. INTERFI-:I;I-:\<-E OF LIGUT 523 point above the mirror will receive waves directly from the slit, and also by reflection, apparently coming from the vir- tual linage of the slit. There will therefore be interference. (These two sources of waves are only approximately iden- tical, for one is the inverted image of the other.) Colors of Thin Plates. — The first interference phenomenon which was recognized as such and so explained is the pro- duction
of the brilliant color effects by such thin films of transparent matter as soap bubbles, films of oil on water, layers of air between two pieces of nearly parallel glass, etc. These colors are due to the interference of the trains of waves which suffer reflection directly at one surface of the lil in with the waves which are refracted out from the film after one or more internal reflections. If we consider any point on the surface of a film which is receiving homoge- neous light from any point source, one ray from the latter is reflected at the point, and other rays emerge there which have entered the film at other points and have suffered reflection at the surfaces of the film. It is evident that these rays have had paths of different lengths ; and that also the ray directly reflected has suffered reflection when incident upon the surface of the film from the surround \\\g medium, while the emerging rays have suffered reflection when inci- dent upon the surface of the Him fmm its interior. Owing to this last cause there is a difference in phase introduced, in addition to that caused by the difference in path, because the reflection in the one case is from a "fast" to a "slow*' medium, and is the opposite in the other. (See page 834.) This additional difference in phase is equivalent, as was shown, to half a period of the vibration. If the total effect of all the rays at the point on the surface of the film is null owing to interference, this point will appear dark : while, if th. ravs do not destroy each other's action, the point will be bright. It is evident also that a film of such a thickness as to cause intc for waves of a definite wave length will 524 LIGHT not, in general, cause interference for other waves; so, if white light is used, a point where there is complete inter- ference for a definite train of waves will appear colored, owing to the fact that the other waves are not cut off ; and the effect is as if one color were completely removed from the constituent colors of white light. The trains of waves which interfere at the top surface of the film are not destroyed, for energy cannot be annihilated ; they are trans- mitted through the film and emerge on the lower side. Thus, when white light is incident upon a transparent film, some is reflected at the upper surface, the rest enters the film ; of this a certain amount is reflected once or more times, and is finally refracted
out through the upper surface, while the rest is either directly, or after two or more reflections, refracted out through the lower surface. By far the greater amount of the light is transmitted, owing to the poor reflect- ing power of the film. There will then be colors visible if one looks at either of the two surfaces of the film ; but those seen by looking back at the second surface are much weaker than those at the other, owing to the presence of so much white light. If the film does not absorb the waves, the com- bined effects on its two sides are exactly equivalent to the incident waves; that is, they are "complementary." We shall now consider in detail the case of a thin film with parallel faces, and we shall suppose that the film has a greater index of refraction than the surrounding medium, e.g. a film of water in air. Let the rays come from a homogeneous point source at such a distance compared with the area of the film that they may be regarded as parallel. Let 00 in the cut be an incident ray ; it undergoes reflection at O and gives rise to a ray OD in general. Other rays emerge by refraction at C\ one of these is due to the incident ray PA. This is refracted into the ray AB, then reflected into BO, and finally refracted out. We can calculate the difference in optical path of these rays from the source to O. Draw UtTEBFEREN( /•; LIGHT 525 AE perpendicular to the incident rays; then the phase at A.tiid K is the same because they are on the same wave front; draw FC perpendic- ular to AB, it repre- sents the refracted wave front; and there- fore the phase at C and F is the same. The difference in op- tical path of the two rays meeting at C is, then, n(FB + TC), where n is the index of refraction of the film with reference to the surrounding medium. Calling the thickness of the film h, and the angle of refraction into the film r, it is seen from geometry that FB + BC= 2 h cos r. Therefore the difference in optical path is 2 nh cos r ; and FIG. 269.— Colors of thin plates. (The tram are omitted.) >ittd rays'_ the corresponding time lag is, where v is the velocity There is also the of the waves in the outer medium. v
of r in the formula is so small that cos r = 1 ; and the condition for complete interference is that 2 nh = ml, which is the same for all these face. INTERFERENCE OF LIGHT 527 pairs of rays. If the thickness of the film at the point C satisfies this condition, and if the eye is focused on the film, C will be a dark point. So, in general, when an extended source of light is used and the incidence is practically normal, the dark or light bands are to be seen by looking at the surface of the film. Another illustration of these formulae is given when the convex surface of a plano-convex glass lens is pressed closely against a glass plate. The film of air between the two pieces of glass may be regarded as made up of a great number of concentric rings, each ring having the same thickness at ^ — -~ — -"" all points. The film is then like that of a FIG. 271.— Apparatus for New. wedge one of whose surfaces is curved; and ton's rings. there is, of course, symmetry around the centre, or point of contact. Therefore if homogeneous light is incident nor- mally upon this film, there will be a central dark spot surrounded by alternately bright and dark rings. The dark rings are given by h = 0, — -, —, etc., where h is the thickness of the film. These are known L' // 'J n as " Newton's rings." h can be expressed in terms of the radius of the spherical surface of the lens and of the radius of the dark ring ; and a method is thus offered for the measurement of the wave length of light; or, if this is known, for the measurement of the radius of the spherical surface of the lens. If in any of the above experiments white light is used, there are no dark bands or rings — all are colored ; and the details can in each case be deduced from the general prin- ciples given on page 524. Interference over Long Paths.— In the cases so far treated, the inter- l rays have been considered to differ in path only by a small amount, 1'iit there are many interesting and important phenomena in which HUH condition is not fulfilled. Tims let 0 be a homogeneous point source of liLjht : /' \te a plate of tran-par.-ut material; C be a converging lens
; It be a screen placed in tin- />rincipal focus of the lens. Draw from C, the centre of the lens, a line CE at random. Then, all rays falling upon the lens parallel to this line \\ill be brought to a focus at E. The point source 0 is emitting rays in all directions; one of them, OP,, is parallel to / '/•;. ThN ray aft.-r incidence upon the plate gives rise after successive reflections and refractions to a series of rays Q,/?P Q9K.f etc., 528 all parallel to each other and to the original ray. Therefore they all uuite at E. The difference in path of two consecutive rays is 2 h cosr; and, if this equals an odd number of half wave lengths, there is complete interference at E, and also, by symmetry, for all points in a circle drawn around D with ED as a radius. If this differ- ence equals a whole number of wave lengths, the ring is bright. Thus, corresponding to the rays from 0 in all directions there will be a series of circular rings, alter- nately dark and light, around D. Similarly, if instead of having a point source, an extended one is used, each point will give rise to identically the same series of cir- cular rings around D ; and so the effect is more intense. PIG. 272. — Interference over long paths. If the source emits two trains of waves of different wave length, there will be two sets of concen- tric rings around Z); at certain points a ring of one set may coin- cide with one of the other, and at others it may fall between two rings of the other set. A connection may be established between the radius of any one ring, say the tenth, the focal length of the lens, the thickness of the transparent plate, and the wave length of the light ; and it is not difficult to see how a method can be devised for measuring the relation in wave length of the two trains of waves emitted by the source. The transparent plate is in general a layer of air included between two plane parallel glass plates. If one of these is kept fixed and the other is moved, h may be varied at will. Professor Michelson of Chicago has obtained interference fringes in this manner, using radiations from mer- cury vapor, when h was so great that the difference in path between two interfering rays amounted to 540,000
a slit, or opening be- tween two scratches, serves as a source of waves, and sends out rays in all directions. Let 630 PIG. 274,— Transmission diffraction grating. DIFFRACTION 531 a converging lens be placed with its axis perpendicular to the plane of the grating, and let a screen be placed in its focal plane. In the cut, draw an arbitrary line OP from the centre of tin.- leii> to the screen; all the rays from the various open- ings of the grating which are parallel to this line will be brought to a focus at P. The difference in path of any two surh rays from the corresponding edges of two consecutive openings, or from two corresponding points in two consecu- tive openings, may be deduced at once. Call the distance from the edge of one opening to the corresponding edge of the next, i.e. the "grating space," a; and the angle (P&A), jy. Then, the difference in path referred to is a sin N-, and, it this is a whole number of wave lengths, parallel rays from corresponding points in all the openings will coincide in phase at P\ and it will be a bright point. (There is an exception to this, when rays from some of the points in any one opening interfere with those from other points in the same opening ; but this case need not be discussed here.) There will there- fore be a line of light through P parallel to the openings in the grating. The condition, then, that P should be bright is : where m = 0, 1, 2, 3, etc., and I is the wave length. Conse- quently, there is a series of bright lines determined by a sin jv"0 = 0, a sin N{ = /, a sin Nt = 2 /, a sin N3 = 3 /, etc. ; that is, by riii JV9 = Ob «nJ\r, = 8inN9 =? sin JV8 = if, etc. The light along any line through a point P defined by these relations is bright ; thru for neighboring portions of the screen as one takes points farther ami farther away from P the light fades gradually away and vanishes. It rises to a maximum in another line detined l»y the next value of N, etc. Then- Si.1 maximum!'«»r tin- dnvetion
sin ^ = 0, or ^V0=0; that is along a line through the point A in the cut where 532 LIGHT the axis of the lens meets the screen; this is called the u central image " ; the next maximum, at the angle Nv is called the "first spectrum," etc. It is evident that there are maxima also on the other side of A, corresponding to negative values of N. The number m is said to give the "order" of the spectrum. If white light is used, instead of homogeneous light, each constituent train of waves has its own series of spectra : a central image, and spectra of different orders on its two sides. The spectra of the different colors overlap ; and the spectrum of any one order is not pure unless the individual FIG. 275. — Photograph of spectra produced by a grating, showing the different orders. spectra formed for any one train of waves are extremely narrow. We can easily determine how wide any one spec- trum " line " is by calculating the position of the point next it on either side where the intensity is zero. This condition involves complete interference at that point of all the rays reaching it from the grating. Let P be a point where there is a maximum, and let P1 be the nearest minimum on the side toward A. Draw the line OPV and call the angle (PjO^.), iVj ; then all rays parallel to OP1 are brought to a focus at.Pj. Let us suppose that there is an even number of openings in the grating ; if there is an odd number, we DIFFRACTION 583 may consider the last one by itself, and its effect in com- parison with that of the others may be neglected. If the number of openings is 2n, the condition that P should be the position of the mth spectrum may be expressed by saying that the difference in path of the two rays reach- ing it from corresponding points in the first opening and the middle one, equals //////. For this difference in path is na sin N* and it has been shown thai the con- dition for a maximum is that P F> A FIG. 276. — Diagram Illustrating the resolving power of a grating. a sin N= ml. If P1 is to be the nearest minimum, the difference in path between two rays from these same points must differ from this value for P by half a wave length;
for, if this is true, the rays from the first and //th. the second and the (n+ l)th, etc., will interfere completely. The con- < N dition for a minimum at Pl is, then. //'/ sin NI = nml — -• illaily, for a minimum point on the other side of P, the vain.- \\onhl be nml + ^- ) The condition for a maxi- mum at P is, Ilrlier, :Jnn wi Mil.V - mill. / The entire number of openings in the grating is 2n, and each has the grating space a ; so 2 na is the width of the grating; and the formula shows that in order for N to be nearly e.pial to \}. that is. toi />f to be very close to P, the 534 LIGHT width of the grating must be large. Under these conditions the spectrum "lines" are narrow. The distance from one spectrum line produced by a homo- geneous train of waves to the next one is determined by giv- ing m two consecutive values in the formula a sin N= ml. Thus, the fourth spectrum is at an angle -ZV4 whose sine satisfies the equation 4 / sin N. = — ; a and the corresponding formula for the fifth spectrum is Thus, sin N6 — sin N4 = -. This relation is general ; and it is seen, therefore, that in order to have consecutive spectra far apart the grating space a must be small. If white light is used, or waves from some complex source, the central image will receive light of all wave lengths ; and, in addition, a series of spectra will be produced on both sides of this. In order to have these spectra long, i.e. the disper- sion great, the grating space must be small ; and to have the spectra pure, i.e. the "lines" narrow, the grating must be wide. This condition for " purity " may be expressed differently. If there are two trains of waves of wave lengths I and I + AZ, which differ only slightly, their spectral images will, as a rule, overlap; but, if A£ is so large that the maximum of the waves of length / + AZ coincides with the minimum of the other waves, the two images or " lines " may be seen distinct from each other. The condition for
, and if 0 is the centre of curvature of this surface, draw a circle with 0(7 as a diameter which is tangent to the grating surface at C. Then, if the point source S is at any point of this circle, the spectral images will be formed at points P on this same circle. If the source is emitting white light, there will be a central bright image formed by ordinary reflection, and on each side of this there will be series of spectra, all on this circle. Those spectra formed in the immediate neighborhood of 0, the centre of curvature, are normal (see page 536), because CO is perpendicular to the grating, and so N in the formula is small. There is a simple kinematic method of maintaining this normal con- dition for a grating, and yet varying the waves which are brought to a focus at 0. If O is joined_to 5 by a straight line, the triangle (OSC) is a right-angled one^having OCas a hypotenuse. Therefore, if two rigid beams, SB and SA, are set up at right angles to each other, and are g furnished with tracks along which small car- riages may run, and if a beam of fixed length equal to CO is pivoted at each end to such car- riages, then the points at the ends of this movable beam and the one at the intersection of the two fixed beams are always on the circumference of a circle whose diameter equals OC however the cross beam is moved, its ends always being on the two fixed ones. In practice, then, the slit or source of light is put at 5, the intersection of the fixed beams ; the concave grating is placed at one of the ends, f ', of the movable beam and so turned that its centre of curvature comes at O, the other end of this beam ; the observations of the spectra are FIG. 279. — Rowland's arrangement of the concave grating. Dirn;A< TION 539 nia'l'\ cither visually or by photographic means, at O. For am definite po>ition of the crossbeam, certain waves in overlapping spectra an- in locus at O; but, as the beam is moved, these change, owing to the change in the angle of incidence upon the tt; rating, (SCO). (WhendiffrmctioD takes place through two rows of rectangular
axis are called "uniaxal" ; the others, which have two axes, are called " biaxal." All crystals which belong to the cubical system, so called, are single refracting; those that belong to the pyramidal or second system are doubly refracting and uniaxal ; the other crystals are doubly refracting and biaxal. Any ordinary iso- tropic transparent substance, such as glass, becomes doubly refracting if it is strained in one direction by pressure, by unequal annealing, etc. Uniaxal Substances. — In the case of uniaxal doubly re- fracting bodies, it is found that one of the rays obeys both of the ordinary laws of refraction, while the other in general obeys neither of them. The former is called the " ordinary ray " ; the latter, the "extraordinary." In such a substance, then, a centre of disturbance gives rise to a spherical wave front, which accounts for the ordinary ray, and also to another wave front which advances with a different velocity, and which cannot be a sphere ; otherwise the extraordinary ray would obey the laws of ordinary refraction, but would have a different index of refraction from the ordinary ray. If a plate of Iceland spar is held between the eye and a bright object, two images of it are seen ; and if the plate is turned DOUBLE REFRACTION 543 around an axis perpendicular to its faces, one image will Ive ;ir<>mi<l the other. Huygens suggested that this ml wave front was the surface of an ellipsoid; and by his work and that of later investigators, this idea has been confirmed. Since the velocity of both disturbances is the same along the op- tic axis, these two surfaces, which make up at any instant the wave surface produced by a point source, the sphere and the ellipsoid, must be tangent to eacli other at the extremities of a diameter having the direction of the axis; and, since the phenomena are symmetrical around this axis, the ellip- soid must be one of revolution around this diameter. Tin- ellipsoid may lie inside the sphere or outside. Plane sec- tions of the two types of wave surface through the axis are ;i uniaxal doubly re- fracting substance. shown in the cut. It is a simple matter of geometry
incident upon such a compound plate, it will, on entering it, be broken up into two beams, ordinary and extraordinary, which will have different directions ; the former will suffer total reflection at the section of balsam, the latter will be transmitted and will emerge at the end of the plate. The sides of the plate are generally painted black ; so that the ordinary rays are all absorbed. This piece of apparatus was invented by William Nicol of Edinburgh, and is called a "Nicol's prism," or a "nicol." Biaxal Crystals. — In the case of biaxal crystals, neither of the rays, as a rule, obeys the ordinary laws of refraction. The form of the wave surface which was proposed by Fresnel is the true one so far as is known ; but it has not been veri- fied in all particulars. Its properties will be found discussed in any advanced book, such as Preston, Theory of Light, or Drude, Optics. CHAPTER XXXIV POLARIZATION Huygens's Experiment. — In his investigation of the doubly refracting properties of Iceland spar, Huygens no- ticed a remarkable fact concerning the two rays transmitted by a plate made of it. They both appeared to be like ordi- nary beams of light; they could be reflected and refracted; they affected the sense of sight; etc.; yet they were differ- ent in one respect, as was shown by an ingenious experiment. The simplest form of this is a slight modification of Huy- If8 original one. Two identical prisms are cut out of a piece of Iceland spar, so that the optic axis in each makes the same angle with the normal t<> the surface. These are made "achro- matic" (see page 545), and are mounted in tubes so that each can be turned around a line perpendicular to its faces. (This line may be called the "axis of figure.") If light from a small source falls upon one of these prisms, two pen- cils emerge : one, the ordinary : the other, the extraordinary. It now the second prism is placed parallel to the first, these two pencils emerging from the latin- \\ill fall upon the for- ii!••!•; and each will give rise to two pencils, one ordinary, th« other extraordinary; thus, four pencils in all will emerge, two ordinary and two extraordinary. This
is true n. ral ; l.ut llnvgens observed that as the tube contain- ing the second \|>rism Was turned around its axis of figure, there were f t mns, 90° apart, dm inir a complete revo- lution «.f the tube, in which only two pencils emerged ; and in one of tlieqp positions the two emerging pencils coincide 647 548 LIGHT in direction. For an intermediate position between any two of these there are, as said, four pencils. They all appear equally bright for a position halfway between any two con- secutive ones of these four positions ; but, as the tube is turned, two of these grow feeble and vanish, while the other two grow more intense ; then, as the tube is turned farther, these two grow feebler and finally vanish, while two others appear and grow more intense. A rotation of 90° is required to turn from one of these positions into the other. It is thus evident that the two pencils which emerge from the first prism are not like the incident light. Further, they are not like each other. For, calling the two pencils emerging from the first prism 0 and E ; and the two pencils produced in the second prism by the former pencil, 00 and OE; and those produced by the latter, EO and EE, we may state the above facts as follows: in general, 00, OE, EO, and EE are present ; as the second prism is turned, a position is reached for which only 00 and EE appear; as the rotation is con- tinued for 90°, these disappear, and OE and EO only are present; etc. Thus for one position of the second prism, one of the two pencils incident upon it gives rise to an ordi- nary pencil, while the other produces an extraordinary one ; and after a rotation of 90° this condition is reversed. Huy- gens noticed that when the principal sections of the two prisms were parallel, 00 and EE were transmitted; but when these planes were perpendicular to each other, OE and EO were transmitted. The explanation of all these phenomena is simple if we consider the two rays 0 and E as plane polarized with their planes of vibration at right angles to each other. (See page 313.) The waves are then to be thought of as transverse, and the vibrations of any one beam are all in parallel straight lines
A, 00 = A cos N, OE = A sin N. Similarly, if there is a vibration along CB of ampli- tude A, we have E = A, EO = A sin N, EE=A cos JV. We must consider the light received from ordinary sources, such as the sun, flames, etc., as being made up of vibrations in all directions in the wave front, because it is not polarized in any way, and when these vibrations are analyzed by the first prism into two sets, along CA and CB, their intensities are equal; that is, their amplitudes are the same. Therefore the four pencils of light transmitted by the second prism have amplitudes given as above : 00 = A cos N; OE = AsmN; E0 = A sin N; EE = A cos N. For N= 0, 00 = A, OE = 0, EO = 0,EE = A. As N increases, 00 decreases, OE and EO increase, EE decreases. For N = 90°, 00 = Q,OE = A, EO = A, EE = 0. In this manner the phenomena observed by Huygens are all explained; but the hypothesis on which it is based, viz., that ether waves are transverse, was not advanced until the early part of the nineteenth century, when it was proposed independently by Young and Fresnel. The credit of explain- ing the various phenomena of polarization and of defending this hypothesis must be given the latter. Huygens recog- nized that the only way possible to account for his observa- tions was to assume a two-sided character for the ether waves ; but the only waves known to him were the longitudinal air waves which produce sounds, and the idea of transverse Waves does not seem to have occurred to him. POLARIZATION 551 Phase Differences. — These emerging rays have different phases, partly because 0 and E are not necessarily in the same phase at any instant, and also because they take dif- ferent times to traverse the plates, owing to their different velocities. Thus, if the incident waves are homogeneous, and if Vl is the velocity of the ordinary waves and V2 that of the extraordinary ones, and if h is the thickness of a plate, the difference in phase introduced by the plate, expressed in V>i V terms of time, between the two emerging rays is h ( Y The period of
; while, if the principal section is at right angles to the plane of incidence, only the extraordinary rays are transmitted. For positions of the plate between these two, both rays are transmitted, but with different intensities except for the position halfway between. The direction of the vibration of the rays reflected from the plane mirror must by symmetry be either in the plane of incidence or at right angles to it, i.e. parallel to the plane of the mirror; and so the fact just described proves the statement made above in regard to the connection between the principal section of the Iceland spar plate and the possible directions of vibrations. (For many reasons it is believed that the vibrations in the rays polarized by reflection from a glass plate are parallel to its plane. This is in accord with the statement that the vibrations of the extraordinary rays are in the principal sec- tion. Therefore the vibrations transmitted by a Nicol's prism are in the principal section.) If the light is not reflected at the polarizing angle, only a portion of the reflected light is plane polarized ; the rest is POLARIZATION 563 like the incident light, made up of rays whose vibrations are in all directions in tin- wave front. At the polarizing angle more of the transmitted light is plane polarized than for any other angle ; but, as said above, these vibrations are at right angles to those of the reflected light, as may be shown by viewing it through an Iceland spar plate. Brewster's Law. — It is found by experiment that light reflected from a plane mirror of any transparent isotropic substance which shows ordinary normal dispersion, such as all kinds of glass, water, etc., may be plane polarized for certain definite polarizing angles. (This is not true of re- flection from metallic mirrors or from substances showing anom- alous dispersion.) A connection between the polarizing angle of any substance and its index of refraction was established by.vster. He found that at the polarizing angle the reflected and retracted rays were perpen- dicular to each other. If, in the Fio.m-Bwwst«r'8Lawiiir<»Tdto cut, MM is the plane surface of msparent substance, whose index of refraction with ref< T- ence to the surrounding nu-d in m is n, and if BA AD, and AC are the incident, tin- refracted, and the reflected rays respec- tively, experiments sliow, u
just said, that if the angle (BA /') is the polari/mg angle, {CAD) is a right angle. Therefore the angle of refraction {DAP') equals (CAM) ; so, calling tin- angle of incidence N, (DAP') = -N. The index of refraction n satisfies the emiat inn n s= 8in \ _'; and hence, ut the polarizing angle, n = - -—= tan N\ or, the angle of cos A polari/.atinn -nUunrr ii such that its tangent equals the index of refraction. This is known as " Brews ter's jy \r\(DAP') 554 LIGHT Law." (More careful experiments have shown that there is no angle of incidence for which all the reflected light is plane polarized ; in reality there is always a small amount not so polarized, but at the polarizing angle this is small.). The value of the polarizing angle of pure water is 53° 11' ; of crown glass, about 57° ; etc. Plane of Polarization. — The light which is plane polarized by reflection from a plane transparent surface is by definition said to be " polarized in the plane of incidence " ; or its "plane of polarization" is said to be that of the plane of incidence. (This definition is entirely independent of any conception of the directions of the vibrations of the rays in this reflected beam; but, accepting the statements made above in regard to these directions, it is seen that in a plane polarized beam the direction of vibration is in the wave front and at right angles to the plane of polarization.) Pile of Plates. — A means is obviously offered of securing plane polarized waves by reflecting ordinary sunlight or light from a flame, etc., from a glass or water surface at the polarizing angle. In general, only a small quantity of light is reflected owing to the poor reflecting power of glass or water; but the effect can be increased greatly in the case of glass by using several thin plates placed one on top of the other, thus forming a "pile of plates." When light falls upon such a pile at the polarizing angle, the reflected light is plane polarized, but part of the refracted light is not ; this falls at the polarizing angle upon the surface where the top plate meets the next one, is partially reflected, and is refracted out so as to coincide in direction with the beam reflected from the plate,
56 LIGHT with the observed position of the plane of polarization, it is proved that the vibrations are at right angles to it. (See page 554.) Interference of Plane Polarized Waves. — Fresnel and Arago performed by means of two piles of plates a most ingenious experiment to determine whether the vibrations in the waves transmitted by them were exactly in planes at right angles to the direction of propagation of the light. If such is the case, by using independently the two piles with their planes of incidence at right angles to each other, two beams of light may be secured in which the vibrations are in the wave fronts but are perpendicular to each other. Two such trains of waves as this cannot " interfere " ; because, in order to have one train interfere with another, the vibrations of both must be in the same straight line. The experiment re- ferred to consisted in modifying Young's original interference one by introducing a pile of plates in front of each of the two slits. We shall quote from their own description, follow- ing Crew's translation in his Memoirs on the Wave Theory of Light : " It has been known for a long time that if one cuts two very narrow slits close together in a thin screen and illuminates them by a single luminous point, there will be produced be- hind the screen a series of bright bands resulting from the meeting of the rays passing through the right-hand slit with those passing through the left. In order to polarize at right angles the rays passing through these two apertures,... we selected fifteen plates as clear as possible and superposed them. This pile was next cut in two by use of a sharp tool. So that now we had two piles of plates of almost exactly the same thickness, at least in those parts bordering on the line of bisection ; and this would be true even if the component plates had been perceptibly wedge shaped. The light trans- mitted by these plates was almost completely polarized when the angle of incidence was about thirty degrees. And it was I'ULAHI/.ATION 557 : ly at this angle of incidence that the plates were inclined when they were placed in front of the slits in the copper screen. •• When the two planes of incidence were parallel, i.e. when the plates were inclined in the same direction; — up and down. for instance, — one could very distinctly see the interference hands produced by the two polarized pencils. In fact, they behave exactly as two rays of ordinary
will bo extin- guished. (They suffer total reflection at the surface win i. the two halves of the last nicol are cemented together.) Other cases, in which the two nicols are not crossed, m ax- be found discussed in advanced text-books. So, if white light is used, certain waves will be absent in the transmitted light, ami it will he colored. Which par- ti< Milar waves are absent depends upon the thickness of the double refracting plate, as is evident from the formula. 560 LIGHT The first nicol, which polarizes the incident light, is called the "polarizer"; the second one is called the " analyser." If there is no such plate between the crossed nicols, no light at all is transmitted ; but, if a plate is introduced, cer- tain waves appear, as just explained. The phenomenon is called " depolarization " ; and the experiment serves as an extremely delicate test of the double refraction of a substance. It at first sight appears as if, in the above experiment, the first nicol might be removed so that the light would fall directly upon the double refracting plate and then upon the second nicol, and there might still be interference ; for the plate would break up the light into two beams and introduce a dif- ference of phase between them before they were combined again by the second nicol. But the relations between the ampli- tudes and the phases in this case are not definite, because the light incident upon the plate is not polarized, but consists of vibrations in all directions ; and so there is no permanent interference. This fact is the fundamental one established by the experiments of Fresnel and Arago. Circular and Elliptical Polarization. — The fact that a plate of a doubly refracting substance breaks up an incident beam of plane polarized light into two such beams, polarized at right angles to each other and with a differ- ence in phase between them which varies directly with the thickness of the plate, renders it possible to secure a circular or an elliptical vibration. A beam of light whose vibrations are A of this character is said to be "circu- FIG. 292. — Formation of cir- J J F larly" or "elliptically" polarized. cularly and elliptically polarized Thus let OA and OB DG the tions of possible vibrations in the plate, and let OP be the direction of the principal section
of the nicol through which the light is incident upon the plate. POL A UIZA T1ON If the amplitude of this plane polarized light is OP, those of the two beams transmitted by the plate are OP1 and OP2. Whatever the difference of phase between them introduced by the plate, these two vibrations will combine to form an elliptical one. (See page 324.) If this differ- ence in phase is equivalent to a quarter of a period, or to any odd number of quarter periods, the vibration is an ellipse whose axes coincide in direction with the lines OA and OB. If, in addition to this condition for the thickness of the plate being satisfied, the incident vibration OP bisects the angle between OA and OB, the amplitudes of the two trans- mitted beams will be the same ; and they will combine to form a circular vibration.'Such a plate is called a "quarter wave plate" ; and obviously plates of different thicknesses must be used for waves of different wave length or color. Fresnel's Rhomb. — If plane polarized light is totally re- flected from the surface of a transparent substance such as glass or water, it becomes, in general, elliptically polarized ; for, if the light is plane polarized in such a manner that the direction of the vibration is neither parallel to the surface nor in the plane of incidence, it is resolved by reflection into two plane polarized beams, one with its vibrations parallel to the surface, the other with its vibration in the plane of incidence. Their amplitudes are different, unless the direction of vibration in the incident beam bisects the an^le between two lines in the wave front, one parallel to the surface, the other in the plane of incidence. A change of phase is introduced by the reflection, which is not the same for the two beams; and the difference for the two depends upon the angle of incidence and the material at whose surface the reflection takes place. Fresnel m.ide a rhomb of such a particular kind of glass AMES'S PHYSICS — 86 562 LIGHT and with such angles that when light was incident perpen- dicularly upon one of its end faces, it would suffer total reflection twice and emerge perpendicular to the opposite face with a difference of phase equivalent to a quarter of a period between the two plane polarized beams. By this means it is possible to obtain light circularly polarized, or elliptically polarized with one axis in the plane of incidence and the other at
right angles to it. Detection of circularly or elliptically Polarized Light. - If circularly polarized light is incident upon a Fresnel's rhomb or upon a quarter wave plate, it will emerge plane polarized, for the effect of these pieces of apparatus is to in- troduce a difference of phase of a quartet* of a period between the two component plane polarized waves into which the incident waves are resolved. The existence of this plane polarized light may be detected by a nicol. If elliptically polarized light is passed through a Fresnel's rhomb or a quarter wave plate, it will, in general, emerge elliptically polarized ; but, if the plate is turned in its own plane, or the rhomb is turned around an axis perpendicular to the planes of its end faces, there will be four positions in one complete revolution for which this light will be plane polarized. This may be detected by a nicol. Other methods are described in advanced text-books. Ordinary Light. — The light which we receive from ordi- nary sources, such as diffused sunlight, etc., is not polarized in any manner ; yet it can be transformed into plane, circu- larly, or elliptically polarized light by methods which have been discussed. When ordinary light is passed through a doubly refracting substance, both the transmitted plane polar- ized beams are of equal intensity, and there is no permanent phase relation between them. This shows that we must consider ordinary light as due to transverse waves in which the vibration at any instant may be rectilinear or circular, etc.. but in which the vibration is continually changing its POLARIZATION 563 form. We may regard it, then, as due to an ever changing mixture of transverse rectilinear vibrations. Rotation of the Plane of Polarization. — There are certain substances which have a most remarkable property in regard to plane polarized light. If one of them is made in the form of a plate and a beam of homogeneous light plane polarized in a } (articular direction is transmitted through it, the emerg- ing light is plane polarized, but its plane of polarization has been rotated through a certain angle. Such substances are said to be "optically active." Thus, if the incident light is produced by the use of a nicol, and a second nicol is " crossed " with it, no light passes before the " active " substance is intro- duced between them ; but after this is
done, the second nicol must be turned on its axis of figure through a definite angle before the light is again extinguished. This angle varies directly with the thickness of the substance, and is different for waves of different wave lengths, being much _ri' ater for the short waves than for the long ones. This last phenomenon is called " rotatory dispersion." There are two classes of these substances; one is made up of bodies which are naturally active, while the other contains bodies which are active only when they are under the influ- ence of a magnetic force. The former phenomenon was discovered by Hint ; the latter, by Faraday. A kinematic explanation of this rotation was given by Fresnel; but it is not necessary to state it here. It may be found in any advanced treatise. a. Naturally active bodies. — Examples of these bodies are quart/, when cut at ri^ht angles to its optic axis, an aqueniis solution of certain tartaric acids, of many of the irs, etc. In all these cases, if the plane polar i/ed li^ht is made to pass through a plate and then by means of a m is reflected back,i.;,nn, the plane of polarization is rotated in one direction and then in the opposite, so it emerges the second time polari/ed exactly as it was on incidence; 564 LIGHT it is as if one screwed a screw into a board and then unscrewed it. In certain bodies the plane of polarization is rotated in a right-handed direction, while in others it is turned in the opposite sense. Thus, if the light is emerging in a direction_perpendicularly up from the X paper, and if AB is the direction of the prin- cipal section of the second nicol, in the experi- ment described above, when it is so placed as to extinguish the light before the active FIG 294 - Rota- substance is introduced ; and, if after this tion of the plane of takes place, the nicol must be turned in the direction shown by the arrow in order to extinguish the light again, the rotation is said to be "right- handed." If, on the other hand, the rotation of the nicol must be in the opposite direction, it is called "left-handed." There are two varieties of quartz, left-handed and right- handed ; two varieties of active tartaric acids, etc. It was discovered by Paste
ur that all optically active sub- stances were made up entirely or in part of certain crystals which had a "hemihedral" form ; of which there are for any substance two possible states. These two are symmetrical with reference to a plane, like the two hands of any individ- ual. Thus right-handed quartz has imbedded in it minute hemihedral crystals of one form ; while left-handed quartz has hemihedral crystals of the symmetrical form, etc. Crys- tals of tartaric acid are hemihedral; and when dissolved in water the molecules retain their asymmetric character. These facts are the basis of what is called "stereochemistry," which is a branch of chemistry dealing with conceptions of the atomic arrangement in certain organic molecules. (See Richardson, Foundations of Stereochemistry, New York.) b. Magnetically active substances. — When any transpar- ent substance, such as glass, is placed in an intense magnetic field, it acquires the power of rotating the plane of polariza- POLARIZATION 565 tion ; but this rotation is different from that just described, because, if the rotated light is reflected back on its path, the plane is rotated still farther, in the same direction as before ; it is not turned back into its previous position. When the subject of electricity is discussed, it will be shown that if an electric current is passed through a wire wound in a helix, there is a strong magnetic field inside it, and that the direction of this field is reversed if that of the current is reversed. It is found by experiment that, if a piece of transparent matter is introduced in this helix, the direction of the rotation of the plane of polarization is that of the electric current in the helix. Metallic Reflection. — When plane polarized light falls upon a polished metal surface, it is reflected according to the ordinary laws; but the light is elliptically polarized, unless the incidence is normal. This is owing to the fact that the incident light is broken up into two plane polarized beams which have a difference in phase. If the metal surface is magnetic, that is, if it is made of iron, steel, nickel, etc., the character of the reflected light — the shape of the resulting ellipse — depends upon whether the metal is in its natural condition or is magnetized; if the latter is the case, the effect of the reflection also varies with the direction and the intensity of the magnetization. This II
MWII as the " Kerr effect," and will l>e found fully described in any advanced text-book. CHAPTER XXXV VELOCITY OF LIGHT THE first experiments to determine whether, like sound, light traveled with a measurable velocity were performed by Galileo. They consisted in having one observer flash a light which was seen by a second one at a considerable dis- tance, who then flashed another light as quickly as possible, and this was seen and noted by the first observer. In any case there would necessarily be an interval between the instant when the first observer flashed his light and when he saw that flashed by the second one, owing to the time required to perform the manipulation, but, if time were re- quired for the passage of the light across the space between the two observers, this interval would vary directly with the distance between the observers. No such effect was observed. The interval of time between the events referred to was ap- parently independent of the distance apart of the observers, and was conditioned only by their quickness of motion and perception. It was therefore concluded that light traveled with an infinite velocity. Method of Roemer. — This opinion was maintained by every one until the year 1676, when Roemer, a Danish as- tronomer, then living in Paris, made certain observations on the eclipses of one of the satellites of Jupiter by the planet, which he interpreted as proving that the velocity of light was finite, but very great. As a satellite revolves around its planet — e.g. the moon around the earth — its motion is periodic, or may be assumed to be so to a very high degree of accuracy. So the interval of time which elapses between 566 7MLOCITT oi' LK.nr two consecutive instants of disappearance of a satellite of Jupiter behind the edge of the planet, when viewed from a point in space fixe<l with reference to the planet, must be a constant quantity. Roemer observed that this interval of time, as noted here on the earth, was not, however, the same when the earth in its motion around the sun was moving away from Jupiter, as when it was approaching it; it was longer in the former case than in the latter. Further, if this period of revolution of the satellite was noted when the earth was nearest Jupiter, and if a calculation was made, based on this, of the instant at which at the end of half a year, when the earth was farthest from Jupiter, an
eelipse should take place, there was found to be a differ- ence of 996 sec. between this calculated instant and the observed one. Roemer saw that these facts could all be explained if the assumption were made, that it takes time for the propagation of light across space. As the earth is receding from Jupiter, light has to travel a greater distance at the instant of the second eclipse than at that of the first : and so the apparent period,,f the satellite is greater than it would l>e if the earth were at rest with reference to J uj liter. Just the reverse is true when the earth is approach- ing the planet. So, when the earth is farthest from Jupiter, light has to travel an additional dUtanee equal to the diameter lie earth's orbit ; and the interval of 996 SCC. observed, as stated above, between the calculated instant of an eclipse and the icc,,rd.-d one is the time required for light to pass ii stance; and so the velocity of light may be rmined. More recent observations have given 1002 sec. as the interval of tin)..'oenier; and the diameter of the earth*! orl.it may be taken as 2998 x 106 Km., as will be shown immediately; so the velocity of Iiurht in space given l.\ this method is this quantity divided by 1003, or 2.984 x 1010 cm. per second. 568 LIGHT The angle subtended at the sun by the radius of the earth, N in the diagram, is called the " solar parallax " ; and it is a quantity whose value may be determined by astronomical observations with a high degree of accuracy. The accepted value of this constant at the present time is 8.79 sec. of arc. The radius of the earth is known, its value being 6378 Km. So the distance from the earth to the sun may be calculated. Referring to the cut, AE = 6.378 x 108 cm., But 27r in angular measure = 360 degrees of arc FIG. 295. — Solar parallax : S is the sun ; E is the earth. and N= 8.79 sec. of arc. = 360 x 60 x 60 sec. ; so 1 sec. = -, and 8.79 sec. =- 360x60x60" 360x60x60 27TX8.79 Therefore, since N is so
small, we may write AE = ES x N, or t™ AE 6.378 x 108 x 360 x 60 x 60 27TX8.79 = 1.4966 x 1018cm. The diameter of the earth's orbit is, then, twice this, or 2.993 x 1018 cm. Method of Bradley. — Another method by which the veloc- ity of light could be determined was discovered by Bradley, the great English astronomer, in 1727. He had observed that if a fixed star was observed through a telescope, this instrument had to be pointed in slightly different directions at different times of the year ; so that, if the image of the star were kept on the cross hairs of the telescope during the whole year, the instrument had to be kept in continual motion in such a manner that its end described a small curve. The explana- tion given by Bradley was extremely simple. Consider a long tube closed at its two ends by caps in which there are two openings directly opposite each other. A particle entering at one open- ing, with a motion parallel to the axis Fio. 296. — Diagram repre- senting stellar aberration. 1 C B of the tube, will escape through the opening at the other end, if the tube is at rest. But, if the tube is moving at right VELOCITY OF LIGHT 569 angles to its length, the opening in the opposite end from tin- one at which the particle enters must be displaced in a direction opposite to the motion of the tube if the particle is to escape. The line of motion of the particle, with refer- ence to the tube, makes an angle with the axis of the tube whose tangent equals the ratio of the velocity of the tube to that of the particle; for, referring to the cut, if Vl is the velocity of the tube and V% that of the particle, and if t is the time taken for the particle to pass through the tube, 03= Vj and AB= F3*; so tan (BAG) = =. Simi- larly, if we consider light from a fixed star entering a tele- scope, if the instrument is at rest, the light will emerge directly from its farther end ; but, if the telescope is mov- ing at right angles to the direction from which the light is coming, it must be inclined forward in order to see the star, and the reason for this, according to Bradley, is because the
"path of the light" is along the line AC. The angle through which the telescope has to be turned when it is pointed approximately perpendicular to the path of the earth in its orbit, is called the "constant of aberration." (Its value according to recent astronomical observations is slightly less than 20.5 sec. of arc.) If this angle is determined by observations on the fixed stars, the velocity of liijht may be calculated, assuming that in the above formula Vl is the velocity of the earth in its orbit and \\ that of liurht, because the velocity of tin- earth in its orbit is known. The value thus deduced is 2.982 x 1010 cm. per second. This explanation of stellar aberration is insufficient to account I'm- all the facts. In the above formula T, w«>ul«l be the velocity of light inside the telescope; therefore, if the telescope tube is filled with water, V^ is diminished in the ratio of the index of refraction of water to air, and so the aberration angle should l»e increased. This experiment was actually performed by Sir George Airy; 570 LIGHT and no change in the angle was observed. When we con- sider light as due to waves in the ether, and if we assume that the ether as a medium does not move as the earth travels through it in its orbital motion, the same formula, as given above, may be deduced for the aberration angle, only in it V% is the velocity of light in the pure ether, not when it is inside matter. Method of Fizeau. — The first method for measuring the velocity of light directly here on the earth, without making use of any astronomical data, was devised and applied by Fizeau in 1849. The principle is extremely simple. A source of light is placed in such a manner that it shines FIG. 297. — Fizeau's toothed wheel apparatus for measuring the velocity of light. through the space between two teeth of a cogwheel, which may be driven at a high speed. At some distance on the other side of this wheel is a mirror which reflects the light it receives back toward the source of light. If then the cogwheel is rotating so rapidly that the light which passes between any two teeth travels to the mirror and is reflected back at such an interval of time that the wheel has turned through a distance which brings a tooth to where an opening was, no light passes back through the
2 7) in the time - — — ; and the velocity of light '_' nN is, then, 2 D divided by - J—, or * M fwAw 572 LIGHT In one of Fizeau's original experiments, the distance from La to L3 was 8.633 Kin. or 8.633 x 108 cm. ; the toothed wheel had 720 teeth ; and it was found that the first obscuration of the reflected light occurred when the wheel was making 12.6 revolutions per second. Therefore, in the above formula n = 12.6, N = 720, D = 8.633 x 1010, and so the velocity of light is determined to be 3.13 x 1010 cm. per second. The great experimental difficulty is to maintain a uniform speed of the wheel and to measure it accurately. This method has been used in more recent years by Cornu and Perrotin. In the work of the latter, the distance apart of the lenses L2 and L3 was about 12 Km. The results of these experiments is to give 2.99820 x 1010 cm. per second as the velocity of light in air. The object of using a concave mirror at S with its centre of curvature at the centre of Z-3, instead of a plane mirror, is apparent if it is remem- bered that the source of light at P is not a point, but is extended. So the waves from a point near P are converged at /, and are made plane by the lens L2 ; but their line of propagation is inclined slightly to that of the waves from P, and they are converged by the lens Ls upon a point of the mirror S, a short distance away from its middle point. If the mirror were plane, these waves would then be reflected off one side, and so would not return through the lens L3 ; but, since the mirror is concave, with its centre of curvature at the centre of L3, these incident waves are reflected back through the lens and finally reach the wheel. Therefore the reflected light is brighter than it would otherwise be. Method of Foucault. — Another method was suggested by Arago, but was first put in practical use by Foucault in 1850, and is always called by his name. (Fizeau also made some valuable suggestions in regard to it.) It consists of making use of a rotating plane mirror in the following manner : Referring
to Fig. 298, there is a source of light at P ; a plate of glass at p ; a revolving mirror at m, whose axis of rotation is perpendicular to the plane of the paper ; a converging lens.L, which focuses upon a concave mirror S the light from P reflected at m ; the centre of curvature of this mirror S is at m. For a suitable position of the revolving mirror the light from the source P will be focused at the middle point of the concave mirror, and will be reflected back on its path until it reaches the glass plate, when it will be in part reflected and VELOCITY OF LIGHT 573 will form an image at a point P'. The revolving mirror is, of necessity, small, almost linear ; and so, in order to col- lect more light from the source P, the I;P" mirror S is concave. (Another method, adopted later, was to place the lens L at a distance from m equal to its focal length ; and in this case the mirror S may be plane.) If, then, a reflected image Of P FIG. 298. — Foucault's revolving mirror apparatus for i is formed at P' when uring the veloclty of "ht- the revolving mirror is at rest, a displaced image P" will be formed when the mirror is turning rapidly, because in the time taken for the waves to pass from m to S and back again, the mirror m will have turned a short distance, and so a reflected ray coming from S will have an angle of inci- dence upon m different from that which it would have if the minor had not moved. This ray, after reflection from m, \vill have a different patli from the incident ray. If the mirror is making ti revolutions per second, its angu- lar velocity is 2 Trn, and that of the reflected ray is 4 Trn. (See page 445.) If the distance from m to S is D, if that from m to P is r, and if v is the velocity of light, - - is the time 2 D required for the light to pass from m to A^and back again; in this tim«. the mirror will have tnrne<l through an angle 27) o /> - • 2 Trn, and the reflected ray through - -- 4 Trn ; therefore v the image of P will be displaced by a distance r times this. So, calling PP".
:>74 LIGHT Foucault increased the effective distance D by having the light reflected several times back and forth between five mirrors before it was finally returned to the revolving mirror ; but in no case did he obtain a very large displacement P'P". Michelson, however, by changing the arrangement of the apparatus was able to increase D to 600 m.; and even when the mirror was turning at the moderate speed of 200 revolutions per second, he obtained a displacement of 13 cm. This method was improved still more by Newcomb, who FIG. 298 a. — Michelson's modification of Foucault's apparatus. operated over a distance of 3721 m. The final result ob- tained for the velocity of light in air by this method was 2.999778 x 1010 cm. per second. The mean of all the best values for the velocity of light in the ether is 2.999880 x 1010 cm. per second, with a probable error of about 20 Km. It should be noted that the values of this velocity obtained directly by the methods of Fizeau and Foucault are for the velocity in air ; and, since the index of refraction of air is 1.00029, the velocity in the pure ether is greater in this ratio. The figure given above for the final value is corrected in this manner so as to apply to the pure ether. Velocity of Waves of Different Periods. — In the pure ether of interstellar space all ether waves, of whatever period, travel with the same velocity so far as is known, as is shown by the fact that the color of any one of Jupiter's satellites is VELOCITY OF LIGHT 575 the same to our eyes when we observe it as it goes into eclipse and as it emerges. If the short waves traveled faster than the longer ones, the satellite would appear red as it disappeared and blue as it reappeared ; and the converse would be true if the long waves traveled more rapidly. In ordinary transparent matter, however, not alone do all waves travel more slowly than in the pure ether, but the waves of different periods have different velocities. It is this which explains refraction and dispersion. Foucault showed by direct experiment that the velocity of light was less in water than in air, by placing a long tube of water in his apparatus immediately in front of the concave mirror. (Actually he used two concave mirrors, one for the light passing in air, the other for light passing
in water, and thus olitaim-d two displaced images.) Michelson showed the same for water and for carbon bisulphide; and he also proved directly that in these substances red light travels more rapidly than blue. CHAPTER XXXVI RADIATION AND ABSORPTION SPECTRA Discovery by Newton of Nature of White Light. In the year 1672 Newton made the interesting discovery that when sunlight was admitted into a darkened room through a small opening and was allowed to traverse a glass prism, the trans- mitted light was no longer white, but consisted of beams of different colors, each color having a different refrangibility and therefore a direction differing slightly from that of its neighbors. He recognized as distinct colors violet, indigo, blue, green, yellow, orange, and red; but all other intermedi- ate shades were present also. He performed further the reverse experiment of combining these colors by means of a second prism, and produced white light again. He also showed that it was impossible by means of a second prism to break up any of these spectrum colors into parts. These observations prove that white light is due to a combination of simple elementary causes; and we know from Young's experiments that these are trains of waves of definite wave lengths, each train being characteristic of a definite color if it is perceived by the eye. These experiments of Newton form the basis of our expla- nation of the color of natural objects and of the science of spectrum analysis. Various bodies in the universe are emit- ting light (or, more generally, all bodies are emitting ether waves) ; all bodies reflect light (or ether waves) to a greater or less extent ; so, if we look at any object, the light (or ether waves) which we receive is due to various causes. We can analyze this radiation into its component parts by means 676 RADIATION AND ABSORPTION SPECTRA 577 of suitable dispersive apparatus, and can then detect these separate trains of waves by proper means. We shall con- sider in this chapter (1) methods of producing ether waves, especially those which appeal to our sense of sight ; (2) dif- ferent forms of dispersive apparatus and different modes of recognizing trains of waves of different wave length ; (3) the results of the examination of the radiations from different sources. Sources and Cause of Radiation Radiation owing to Temperature. — All substances in the universe are, so far as kumvn to us, emitting ether waves, owing
to the vibrations of certain parts inside their mole- cules. If the body when placed in a darkened room can be seen by the eye, it is said to give off light. The ordinary method of making a body luminous is to raise its tempera- ture ; thus a body may be exposed to a hot flame, such as one from a Bun sen burner, or it may be placed in the poles of an electric arc light. (See page 665.) At such high tem- peratures, many bodies are vaporized, and their vapors are then at this temperature. The laws of radiation due to this cause have been discussed in Chapter XIV. Electro-luminescence. — Again, if an electric spark is made to pass between two metal points, they are vaporized ami the vapors are luminous ; not, however, owing entirely to the temperature being raised. (The same statement is true of the luminosity of the vapors in the electric arc ; it is only in part due to the temperature of the vapors.) Similarly, if a gas or vapor is inclosed in a hollow vessel, such as a glass bulb, and an electric discharge1 through it is produced 1>\ any means, it becomes luminous. These cases of luminosity are said to be due to "electro-luminescence." Chemical Luminescence. — In certain chemical reactions light is emitted ; for instance, when a piece of decayed wood slowly oxidizes, or when phosphorus is oxidized. These are illustrations of "chemical luminescence." AMES'S PHYSICS — 578 LIGHT Fluorescence and Phosphorescence. — There are many bodies which emit waves as a result of their having absorbed other ether waves, quite apart from any radiation due to tempera- ture alone. Some bodies emit these waves only while they are absorbing the other waves; while others continue to emit them even after the absorption ceases. All bodies of this kind are called " fluorescent," and the phenomenon itself is called " fluorescence " ; while the second division of these bodies, as just described, are called "phosphorescent," and the phenomenon is called "phosphorescence." If a beam of light is passed into a fluorescent substance, certain trains of waves are absorbed and others are transmitted ; the energy of these absorbed waves is not spent in producing heat effects, but in emitting other ether waves, which proceed out in all directions. So this fluorescent light may best be seen by looking
at the substance from one side. This phenomenon was first observed by Herschel and Brewster, but was first thoroughly investigated by the late Sir George Stokes. He showed that in all cases observed by him, the fluorescent light was of a wave length longer than that of the waves whose absorption caused the fluor- escence. This relation is not, however, true in all cases. Some common illustrations of fluorescence are the colors seen in certain forms of fluor spar (whence the name of the phe- nomenon); the color of canary glass — which is ordinary glass containing traces of certain salts of uranium ; the color of a decoction of the bark of chestnut trees ; the color of the surface layers of kerosene oil ; etc. Phosphorescence is exhibited by the sulphides of barium, calcium, strontium, etc., and by a great many ordinary sub- stances to a certain extent. Sometimes the light is emitted for only a minute fraction of a second ; but in other cases it continues for hours. Conclusion. — In many cases it is impossible to say exactly what is the cause of the luminosity ; and in nearly all there RADIATION AXD ABSORPTION -1'ECTRA 579 are several phenomena involved. We can, however, divide all cases of radiation into two classes : in one, the substance that is radiating does not change so long as its temperature is maintained constant; in the other, the substance does change even if its temperature is kept unchanged. In the tirst class of bodies, the radiation is a purely temperature effect; and to them Balfour Stewart's or KirchhofFs law — as it is more often called (see page 301) — and the other laws of radiation may be applied. This is not true of the bodies of the second class, in which molecular changes are going on. Spectroscopes Different Forms. — In order to study the radiation of any l><»dy, some method of dispersing it into a pure spectrum, an-1 some instrument which is sensitive to the various radia- tions, must be used. As we have seen, there are three ways in which dispersion may be secured: by the use of a prism. a grating, or some interference apparatus. Further, a slit (or small source of light) and a converging lens must be used. Thus we have prism, grating, and interference spectroscopes. The conditions as to the purity of the
there are no gaps in them. In the latter only certain isolated trains of waves are emitted, thus forming separate "lines." Investigations show that all solids and liquids — with possibly a few exceptions — emit continuous spectra ; while all gases and vapors emit discontinuous ones. (This is obviously what one would expect to be the case from the kinetic theory of different forms of matter.) Gaseous Spectra. — The spectrum of a gas depends, natu- rally, upon the manner in which it is rendered luminous. So we have "flame spectra," "arc spectra," "spark spectra." " fluorescent spectra," etc. If, however, a gas is made lumi- nous in any definite manner, the waves it emits are definite and characteristic of the gas. Thus, different gases may be identified by their spectra ; and in many cases the discovery of new lines in the spectrum of a gas that was supposed to be pure has led t<> the identification of new elements. Absorption Spectra. — If the radiation from a solid or liquid falls upon any body, certain waves are absorbed ; and so only a portion of the incident waves are transmitted. The spectrum of this transmitted radiation is called the -orption spectrum " of the body which produces the al KOI -j.t ion. This absorption takes place in many ways, as has l>cen already stated. In all, the absorption is due, in the main, to the resonance of the minute parts of the mole- cule or of the molecules themselves; and in the greater number of bodies the energy absorbed is distributed among the molecules of the body, and is manifest by heat effects. 582 LIGHT This is called "body absorption." In other substances the energy absorbed in the interior is spent in emitting other waves of longer wave length, thus producing fluorescence. In certain bodies, the absorption takes place in a very thin surface layer ; but the larger portion of the energy incident upon the surface is reflected directly. This is the case with the metals and a few other bodies, and is therefore called " metallic absorption." The law of Kirchhoff in regard to the equality of radiating and absorbing powers may be applied to a substance which exhibits body absorption only. Thus, if a substance absorbs certain trains of waves of definite wave lengths, it has the power of emitting them if rendered luminous by means of temperature alone (and, also, often if other means are used), and the intens

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