LIBRARY OF THE UNIVERSITY OF CALIFORNIA. . 7 1 Accession No. - # Class No. THE ELEMENTS OF PHYSICS VOL. Ill LIGHT AND SOUND THE ELEMENTS OF PHYSICS A COLLEGE TEXT-BOOK BY EDWARD L. NICHOLS AND WILLIAM S. FRANKLIN IN THREE VOLUMES VOL. Ill LIGHT AND SOUND gork THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., LTD. 1897 All rights reserved Engineering JUbiary COPYKIGHT, 1897, BY THE MACMILLAN COMPANY. Norton oti J. S. Cashing & Co. - Berwick 8s Smith Norwoqd Mass. U.S.A. TABLE OF CONTENTS. CHAPTER I. PAGE Light and sound defined ; Measurement of velocities i CHAPTER II. Longitudinal and transverse waves ; Equations of wave motion ; Wave trains, simple and compound ; Fourier's theorem ; The wave front ; Shadows ........... 9 CHAPTER III. Reflection and refraction ; Mirrors and the formation of images ; Re- fraction of plane and spherical waves ; Total reflection ; Spherical aberration . 26 CHAPTER IV. Lenses and lens systems ; Specification of a system ; Principal foci, principal planes, and nodal points 45 CHAPTER V. The correction of lenses and lens systems ; Spherical and chromatic aberration ; Astigmatism ; Distortion ; Curvature of field ; Descrip- tion of lens systems used in practice . . . . . -55 CHAPTER VI. The eye; The photographic camera; The projecting lantern; The microscope ; The telescope ; Measurement of magnifying power . 64 vi CONTENTS. CHAPTER VII. PAGE Newton's experiment ; The spectrum ; The achromatic lens ; The spec- troscope ; Classes of spectra ; The spectrometer ; The spectro- photometer 73 CHAPTER VIII. Interference from similar sources ; Arrangements for producing fringes ; Colors of thin plates ; Newton's rings ; Diffraction past an edge ; Diffraction through a slit ; Zone plates ; The diffraction grating ; The grating spectrometer ; The measurement of wave lengths . 82 CHAPTER IX. Sensations of brightness and color ; Luminosity ; Colors due to homo- geneous and to mixed light ; Color mixing ; Dichroic vision ; Test- ing for color blindness 101 CHAPTER X. Photometry ; The law of inverse squares ; Standards of light ; Bouguer's principle ; Simple photometers ; Distribution of brightness ; The photometry of lights differing in composition 112 CHAPTER XI. Polarization defined ; Behavior of tourmaline ; Polarization by reflection ; Double refraction ; The Nicol prism ; The polariscope ; Rotation of the plane of polarization 125 CHAPTER XII. Radiant heat ; Prevost's principle ; Law of normal radiation ; Selective emission, reflection and transmission ; Selective absorption ; Black bodies ; White bodies ; Surface color ; Methods of measuring radi- ant heat; Fluorescence and Phosphorescence 135 CHAPTER XIII. Vibration of a particle ; Simple and compound vibrations ; Musical tones and noises ; Loudness ; Pitch ; Timbre . . . . 147 CONTENTS. vii CHAPTER XIV. PAGE' Vibrations of air columns ; Organ pipes ; The clarionet ; The cornet ; The vocal organs ; Vibrations of rods and strings ; Kundt's experi- ment ; The compound vibration of strings ; Chladni's figures ; The tuning fork ; Manometric flames . . . . . . .154 CHAPTER XV. Proper and impressed vibrations ; Damping ; Resonance ; Analysis of clang; Vowel sounds ; Reproduction of speech . . . .167 CHAPTER XVI. The ear; Persistence of sound sensations ; Interference of sound sensa- tions ; Combination tones ; The echo ; Influence of diffraction upon the sense of direction ; Changes of pitch due to relative motions . 173 CHAPTER XVII. Pitch intervals; Complete and approximate consonance; Major and minor accords ; Musical scales ; Expression ; Rhythm, melody, har- mony, and modulation 181 HTY THE ELEMENTS OF PHYSICS. VOLUME III. CHAPTER I. LIGHT AND SOUND DEFINED; VELOCITY. 604. Sensory nerves. The sensory nerves of the human body lead from regions near the surface of the body to the central organs of the nervous system. The outer ends of these nerves are exposed in such a way as to be excited or set into commotion by physical disturbances in the region surrounding the body. This commotion is transmitted to the central organs producing commotion there, and we perceive what we call a sensation. The physical disturbance which excites a nerve is called a stimulus. 605. Proper stimuli. End organs ; localization. Those physi- cal disturbances to which a set of sensory nerves are especially sensitive are called the proper stimuli of that set of nerves. A set of sensory nerves is rendered especially sensitive to its proper stimuli by being provided with terminal or end organs which are easily affected by those stimuli, and by being so located as to be very largely protected by the surrounding tissues of the body from other excitation. 606. Optic nerves. Light, the sensation; light, the proper stimuhis. The end organs (rods and cones) of the nerves of 2 ELEMENTS OF PHYSICS. sight are situated in the retina of the eye. They are well pro- tected by the surrounding bones from all physical disturbances, except such as can reach them through the transparent humors of the eye. Severe mechanical shocks and electric disturbances (electric currents) do, however, penetrate to these end organs and excite them. The sensation which is perceived when the optic nerves are excited is called light. That physical disturb- ance which constitutes the proper stimulus of the optic nerves is also called light. 607. Auditory nerves. Sound, the sensation; sound, the proper stimulus, The end organs of the nerves of hearing are situ- ated in the inner ear. They are well protected by the massive bones of the head from all physical disturbances, except such as can travel along the chain of tiny bones which bridges the cavity between the tympanic membrane and the window of the inner ear. Severe mechanical shocks and electric currents do sometimes penetrate to these end organs through the surround- ing bones and excite them. The sensation which is perceived when the auditory nerves are excited is called sound. That class of physical disturbances which constitutes the proper stimulus of the auditory nerves is also called sound. 608. The long-range aspect of the sensations of sight and hearing. We have come by experience to associate more or less remote objects with our sensations of sight and hearing. A physical disturbance reaches our eyes from an object which we see (or our ears from an object which we hear), and this disturb- ance when it reaches us is indicative of such of the characteristics of the object as we can perceive by sight (or hearing}. 609. The corpuscular theory of light. The phenomena of shadows and the obstruction of vision of a distant object by intervening objects show that light travels sensibly in straight lines. In accordance with this fact, it was the accepted theory, until long after the time of Sir Isaac Newton, that light con- LIGHT AND SOUND DEFINED. 3 sisted of particles or corpuscles which were thrown off from luminous bodies at great velocity, traveling in straight lines until reflected (or stopped) by objects upon which they might impinge. This was called the corpuscular theory of light. 610. The wave theory of light and of sound. The most com- prehensive understanding of the phenomena of light and sound is reached if we look upon them as wave-like disturbances which pass out in all directions from luminous and from sonorous bodies respectively. The assumption that light and sound are wave motions is verified by widest experience. No attempt will be made to establish this assumption by any preliminary discussion. The justification of the wave theory, in the reader's mind, will become more and more complete as he has occasion to use it. 611. The luminiferous ether. The conception of light as a wave-like disturbance depends upon the assumed existence of an all-pervading medium, the ether. The fact that light reaches us, from the sun and stars, across the void of interplanetary space, necessitate the assumption of the ether. Many kindred phe- nomena like that of the transparency of vacuum tubes, taken together with the fact that the physical properties of ordinary matter do not enable us to account for the enormous rapidity with which light is transmitted, afford additional support to the assumption. It has been already shown in the second volume of this treatise, that the existence of a similar medium is necessary to the explanation of electro-magnetic phenomena. It is universally considered that the electro-magnetic ether and the luminiferous ether are identical. In the case of sound, which cannot reach us across a vacuum, it is certain that we have to do with a wave-like disturbance of the air or of other material media. In the discussion of wave motion, reflection, refraction, interference, and diffraction, it will be found advantageous to consider light and sound together ; in other portions of the volume they will be treated separately. 4 ELEMENTS OF PHYSICS. 612. The velocity of light and the velocity of sound. It is a familiar fact that sound requires a perceptible time to reach the ear from a sounding body. A Danish astronomer, Roemer (1675), was the first to show that the same is true for light. 613. Methods of measuring the velocity of sound.* The first attempt to measure accurately the velocity of sound was made by a committee of members of the French Academy of Sciences in 1738. The observers were placed at night at the Paris Observatory, and at three stations visible from that point in the surrounding country. Every ten minutes a cannon was fired from one of these stations. At the others observations were made of the time which elapsed between the flash and the sound of the cannon. Since the distance between various sta- tions was known, the velocity of sound could be computed. In 1822 this experiment was repeated at Paris in a slightly modi- fied form ; two stations were selected, and cannon were fired from these alternately at intervals of ten minutes. In this way the influence of the wind was eliminated. The distance between the two stations was 18,622.27 meters. The mean length of time required in one direction was 54.84 seconds, and in the other direction 54.43 seconds. The value of the velocity of sound reduced to a temperature of zero C. of the air was found to be 331.2 meters per second. Similar experiments, made near Amsterdam, gave a velocity of 332.26 meters . sec. As will be shown in Chapter II., the velocity of sound in a given gas varies only with the temperature of the- gas, and is the same for -all pressures (and densities) at the same tem- perature. Experiments upon the velocity of sound at low temperatures, made during a winter spent in the Arctic regions by Lieutenant Greeley, gave the following values : * For a more complete account of the researches described in this article, see Wullner, Experimentalphysik (5th ed., Vol. I, p. 928). LIGHT AND SOUND DEFINED. TEMPERATURES. VELOCITY. - 10.9 326.1 meterS sec. -25-7 3I7-I -37-8 309.7 -45.6 305.6 To establish the fact that the velocity of sound is independent of the density of the air, observations were made by Bravais and Martins between stations at the top of the Faulhorn, a mountain in Switzerland, and upon the shore of Lake Brienz at the foot of that mountain. Shots were exchanged between these sta- tions, the difference in height of which was 2079 meters, and it was found that the velocity of sound was the same as those obtained in experiments near the level of the sea. The average velocity (reduced to o) was found to be 332.37 meters per second. In all these experiments it was assumed that the length of time required to become aware of the flash and of the report following it would be the same ; while the velocity of light is so great, as compared with that of sound, that the time required for the light signal to traverse the measured path is entirely negli- gible. Now the time required for an observer to become cog- nizant of the impression upon his optic or aural nerves is very considerable. Since, in the case of the flash, the observer would be taken by surprise, and the flash would act as a warning so that he would be ready for the reception of the sound which follows it, it was thought that these methods were open to criticism. The French observer, Regnault, therefore performed experiments upon the velocity of sound in which the discharge was automatically recorded by causing a bullet to cut a wire stretched across the muzzle of a pistol, and in which the arrival of the sound wave at the receiving station was likewise recorded by means of an electrical device. The latter arrangement, of which a diagram is given in Fig. ELEMENTS OF PHYSICS. 372, consisted of a very sensitive diaphragm carrying a metal disk. A screw P was turned until its point was almost in contact with the disk. The impulse of the sound wave was sufficient to bring the two into contact, thus completing an electric circuit. The records were made upon a chronograph sheet. The results obtained by this method gave a slightly smaller value A 30. 7 meters ) for the velocity of sound than those obtained in the experiments already described. Taking all the available data together, the most probable value for the velocity of sound at o is found to be 331.76 meters per second. Fig. 372. In addition to the direct methods just described, there are various indirect methods of measuring velocity of sound. To these reference will be made in subsequent chapters. These methods are especially valuable in the determination of the velocity of sound in other gases than air, and in solids and liquids where it is not practicable to employ a path thousands or even hundreds of meters in length. The following table gives the velocity of sound in various substances in terms of the velocity of sound in air : SUBSTANCES. VELOCITY OF SOUND COMPARED WITH THAT IN AlR. Lead 4-257 Gold 6.424 Silver 8.057 Copper 11.167 Platinum 8.467 Iron 15.108 Glass 15.29 Water 4-3 LIGHT AND SOUND DEFINED. 7 614. Methods of measuring the velocity of light.* The ob- servations of Roemer, upon which the first ideas concerning the velocity of light were based, were of an astronomical char- acter. He found that the observed time of the revolution of the satellites of Jupiter varied according to the position of the earth in its orbit. When the earth was so situated as to be moving away from Jupiter, the periodic time of the satellite appeared greater than when the earth was moving toward the latter planet. It was not until the present century that at- tempts were made to measure the velocity of M| light by direct methods. The essential features of Fig. 373. the first of these methods, that employed by Fizeau and later by Cornu, are indicated in Fig. 373- Light from a brilliant source, S, reflected from the face of an unsilvered glass, is sent between the teeth of a cogwheel to a mirror M, situated at a great distance ; in the case of Cornu's measurements at a distance of 23 kilometers. From this mirror the light is reflected back upon its path so that the ray passes through between the same pair of teeth. If, now, light has a finite velocity, and if the distance between the toothed wheel and the mirror is great, it will be found possible to drive the wheel with sufficient rapidity so that in the interval during which the light is traveling from the wheel to the mirror and back again, the opening through which it passed will have been supplanted by the next following tooth of the wheel. An observer at A, looking through the opening in the wheel, would then no longer be able to see the returning ray, because the light passing between each pair of teeth of the wheel to the * For a fuller account of researches upon the velocity of light, see Preston's Theory of Light, Chapter XIX. 8 ELEMENTS OF PHYSICS. mirror would, upon its return, be intercepted by the next fol- lowing tooth. By measuring the velocity of the wheel and the distance between the -mirror and the wheel, the velocity of light may be computed. The result obtained by Cornu, who has made the most trustworthy experiments by this method, was 300,400,000 meters per second. The other method upon which our knowledge of the velocity of light depends is known as Foucault's method. In this method the ray of light is reflected from a rapidly revolving mirror R (Fig. 374), to a distant fixed mirror M\ thence back to the revolving mirror. If the distance between these mirrors be very great and the velocity of the mirror be high, it is found INCIDENT RAY ^ that the path of the returning """r~u^7 N - i ^ ra Y> represented by a dotted line Fi 374 in the diagram, deviates meas- urably from that of the incident ray. A measurement of the angle between these rays, of the distance between R and M, and of the angular velocity of R, makes it possible to compute the velocity of light. The result obtained by Foucault was 298,000,000 meters per second. Foucault's experiments have been repeated by Michelson and by Newcomb under conditions which ensured greater accuracy than did the experiments of the originator of the method. Michelson obtained as the velocity of light, 299,853,000 meters per second. Newcomb, at Washington, found 299,860,000 meters per second. / CHAPTER II. WAVES. 615. Nature of waves. When a portion of an elastic medium is suddenly distorted, and released, a wave of distor- tion passes out in all directions from that portion at a definite velocity. A distant portion of the medium is quiescent until this wave reaches it. It is distorted, and thrown into commo- tion, as the wave passes, after which it again becomes quiet. The movement along a stretched wire, of a wave produced by a blow, and waves upon the surface of water, are familiar examples. 616. Longitudinal and transverse waves. Fluids can trans- mit only those waves in which, as the wave passes, the particles of the medium move to and fro in the direction of progression of the wave. Such waves are called longitudinal waves. Thus waves in the air, sound waves, are longitudinal. On the other hand, solids can transmit longitudinal waves, and also waves in which the particles of the medium move to and fro in a direc- tion at right angles to the direction of progression of the wave. Such waves are called transverse waves. Longitudinal and trans- verse waves have different velocities of progression in the same medium ; and, therefore, if a center of disturbance sends out waves of both kinds, one kind will outstrip the other and become isolated from it. A familiar example of this is afforded by the waves which pass through a long and tightly stretched wire which is struck sharply with a hammer. A person at the other end of the wire will hear a sharp click when the longitu- dinal wave reaches him, and another when the transverse wave 9 I0 ELEMENTS OF PHYSICS. reaches him. The air near the hammer will also be disturbed, and an air wave will reach the person, giving a third click between the other two. Light waves, which belong to the same general class as the electric waves described in Arts. 501, 590, and 603 (Vol. II.), are known to be transverse from the phenomena of polarization. 617. Equations of wave motion. (a) Transverse waves. Let the rectangle in Fig. 375 represent a portion of thickness A# of an elastic medium. At a given instant let it be displaced and distorted, as shown by the dotted rhombus A, by a passing transverse wave. Let Y be the upward displace- ment of the left face, and Y+ AKthe upward displacement of the right face of A. Then the distortion of A is a A y shearing strain, S, equal to - , and, at the limit, the shearing strain at the left face of A is : c dY Fig. 375. S = -. (0 This shearing strain is accompanied by a shearing stress />, such that P=nS, (ii) where n is the slide modulus of the substance. Let 6" + A6* be the shearing strain at the right face of A. Then P + A/* = n(S + A^) is the shearing stress on that face. Let a be the area of the right and left faces of A. Then a A.r is the volume of A, and ap A^r is its mass ; p being the density of the substance. The force pulling downwards on the left face of A is Pa, and the force pulling upwards on the right face is (P + &P)a. The difference a A/* is an unbalanced force which must be equal to the product of the mass of A into its acceleration -. Therefore ap A.r -.= a - A/>; or at 1 at 2 - Substituting in this equation the value of P from (ii), and then the value of S from (i), we have (ti) Longitudinal waves. Let the heavy line rectangle in Fig. 376 represent a portion of an elastic medium which, at a given instant, is dis- WAVES. placed and distorted, as shown by the dotted rectangle A, by a passing longitudinal wave. Let X be the displacement of the left face of A, and AiC Xaxis A Fig. 376. X + AA" the displacement of the right face of A. Then, in a manner pre- cisely similar to the above, it may be shown that d*X = V dP p (v) in which V is the particular elastic modulus which, multiplied by the strain , gives the corresponding stress. For liquids and gases V is the bulk modulus.* Solution of the equation of wave motion. Equation (v) may be written d*X in which (vi) (vii) The solution of equation (vi) is X =f(x + O + F(x - vf), (viii) in which /"and /''signify any functions whatever. F(x vf) is a wave travel- ing in the positive direction along the axis of x, and f(x + vf) is a wave traveling in the other direction. The velocity of progression is v( =-y ) The modulus V is the isentropic bulk modulus. (See Art. 261, Vol. I.) Precisely similar results concerning transverse waves may be obtained from equation (iv), in which case the velocity of progression is (ix) * The modulus V has the following significance for a solid : Consider a strain having but one stretch a. Associated with this strain is a stress of which all three pulls are finite. The pull P, in the direction of a, is taken as the measure of the stress, and we have P Va.. It is this stretch a which is represented above bv ' AJC I2 ELEMENTS OF PHYSICS. The value of n (slide modulus) is zero for liquids and gases, whence it follows that transverse waves are not possible in liquids and gases. In solids, V and n are, in general, different in value, so that longitudinal and transverse waves usually have different velocities in the same medium. In a gas, say air, the isentropic bulk modulus V is equal to kp (by Art. 262, Vol. I.), where k is the ratio of the two specific heats of the gas, and p is the pressure. Therefore, writing kp for V in equation (vii), we have VI = >/T in which v is the velocity of longitudinal waves (sound) in air, k = 1.41, p is the pressure, and p is the density of the air. P The ratio is, by Gay Lussac's Law (Art. 249, Vol. I.), proportional to the absolute temperature. Therefore the velocity of sound in air is propor- tional to the square root of the absolute temperature ; that is : v t :v :: ^273 + / : A/273, /273 -1- / or v t = v \- 27 y-> (319) in which v t is the velocity of sound in air at t C, and z> ( = 331.76 - J is the velocity at o C. 618. Waves from periodic disturbances ; wave trains. A periodic disturbance is one which is repeated, in every detail, in equal intervals of time. The time interval r during which one repetition of the disturbance takes place is called the period of the disturbance, and the number of repetitions per second is called the frequency. A periodic disturbance sends out what is called a train of waves, each one of which is exactly like its forerunner. The distance X between similar parts of the adjacent waves of a train is called the wave length ; it is the distance traveled by the waves during the period r. If the velocity of the waves be v, this distance is in, so that X = VT. (320) 619. Graphic representation of wave trains. Consider a wave train traveling in the direction of the line AB (Fig. 377). At WAVES. each point of AB, erect a perpendicular (as a, b) whose length is proportional to the actual displacement of the medium at that point at a given instant. The curved line ccc, so constructed, represents the wave train graphically; and, if this curve be imagined to move along at the velocity of the wave train, the Fig. 377. actual motion of the various parts of the medium is clearly set forth. If the train is one of transverse waves, then the points of the curve give the actual positions of the particles of the medium which, when the medium is at rest, lie along the line AB. 620. Amplitude ; phase ; energy stream. (a) Amplitude of a wave train. The maximum displacement b (Fig. 377) in a wave train is called the amplitude of the train. (b) Opposition in phase. Two points in a wave train at which the displacements are equal and opposite are said to be opposite in phase. The terms crest and hollow, as applied to water waves, will be used to signify those portions of any wave train where the displacements have the greatest positive, and greatest negative, values respectively. (c) Energy stream in a wave train. It can be shown that, for a given medium, the energy per second streaming across a unit area, which is perpendicular to the direction of progression of a wave train, is proportional to the product of the square of the amplitude divided by the square of the wave length. This energy stream is the physical measure of the intensity of a wave train. (Compare Arts. 598 and 603, Vol. II.) I4 ELEMENTS OF PHYSICS. 621. The principle of superposition. It is a familiar fact that a number of objects remain distinctly visible to a number of observers when the light, in passing from the various objects to the various observers, has to cross the same region at the same time. It is also true that, although a combination of sounds is more or less distracting to the attention, one sound does not sensibly alter the character of another which accompanies it. A number of waves can therefore traverse the same region simultaneously, each one independently of the others. The actual displacement of a particle of the medium at a given instant is the vector sum of the displacements at that instant dtie to the separate waves. Waves on the surface of water are, in the same way, independent of one another. An observer overlooking the sea can trace simultaneously the incoming ocean swell, the smaller waves caused by passing boats, the waves due to local wind, and the waves reflected from the shore. 622. Stationary wave trains. Consider two similar wave trains A and B, Fig. 378 (drawn one above the other to avoid confusion), moving in opposite directions, as indicated by the arrows. By the principle of superposition, the actual displace- ment at each point is equal to the sum of the displacements at that point due to each wave train, and the actual velocity of each particle is equal to the sum of the velocities of that particle due to each wave train. Therefore the medium remains stationary along the lines // ; for the ordinates O, which come up to the line pp as the upper train moves to the right, are at each instant equal and opposite to the ordinates O' which come up to the line pp as the lower train moves to the left. The portions of the medium between the lines // move up and down (to right and left in case of a longitudinal wave), for the ordinates Q and Q r , which come into successive coincidence at q, are of the same sign. The stationary portions of the medium are called nodes, and the intermediate vibrating por- WAVES. j - tions are called vibrating segments. The middle point of a vibrating segment is called an antinode. The resultant of the two wave trains A and B is called a stationary train. ' The result- ant curve (Fig. 378) shows the character of this stationary wave train. This resultant curve is drawn below A and B to avoid Q P ;p Fig. 378. confusion. The straight line is the resultant of the trains A and B when they are in the positions shown, and the arrows represent the velocities at each point. The lines i, 2, (3), (4), 5, 6, and (7) represent the successive stages of the motion as the trains A and B move to right and left. The nodes of a stationary train are evidently the places where the two trains A and B are always opposite in phase, and the antinodes are the places where the two trains A and B are always in the same phase. i6 ELEMENTS OF PHYSICS. The portions of the medium at the antinodes of a stationary wave train have at times considerable velocity, but are never distorted. "The portions at the nodes never move, but are at times much distorted. This, for transverse waves, is shown in Fig. 379. The shaded areas represent portions of the medium. AB represents the state of affairs when the velocities in the vibrat- ing segments are greatest, as indicated by the arrows, and when the displacements are everywhere zero. CD represents the state of affairs one-quarter of a period later, when the velocities are everywhere zero and the displacements are at their greatest. The shaded area at the node is distorted as shown. a Pi P Q PI Fig. 379. Stationary wave trains may result from the superposition of wave trains of any shape, provided only that the advancing train is exactly similar to the receding one turned end for end and upside down. 623. Stationary wave trains by reflection. When a wave train reaches the boundary of a medium, it is reflected. The reflected train is similar to the incident train, and if the incident train strikes the boundary normally, then a stationary train is produced by the superposition of the two. If the boundary were one which separated the medium from a void, then the surface layers of the medium could not be dis- WAVES. i 7 torted, since they would be perfectly free to move with the contiguous portions of the medium. In such a case there would be an antinode of the stationary train at the boundary. If the boundary were a rigid wall, then the medium contiguous to the wall would not be free to move, and there would be a node of the stationary train at the boundary. In all intermediate cases it may be stated that where the boundary separates the medium from some material less dense than itself, an antinode is formed at the boundary, and where the boundary separates the medium from a denser material, a node is formed. The component wave trains of a stationary train are opposite in phase at a node, and in phase at an antinode, so that a reflected wave train is in phase with the incident train at a free boundary, and opposite in phase at a rigid boundary. Reflection of the first kind is called reflection without change of phase, and reflec- tion of the second kind is called reflection with change of phase. A surface separating two media in which the velocities of a wave are different reflects with change of phase in the medium giving the greater velocity, and without change of phase in the other medium. Stationary wave trains may be easily produced by means of a stretched cord or wire, or with a stretched rubber tube. One end of the rubber tube is tied to a rigid support, and the other end is held in the hand. A series of periodic movements of the hand generates a wave train on the tube. This train is reflected from the rigid end with change of phase ; and the tube is broken up into segments with intervening nodes as the reflected train and advancing train come into superposition. The rigid end of the tube is a node. A stationary wave train may likewise be produced in the air within a long glass tube. In this experiment both ends of the tube are open. The lips, applied to one end of the tube as to a bugle, are thrown into periodic motion producing a wave train, which is reflected from the open end without change of phase. A stationary wave train is produced in the tube when the re- i8 ELEMENTS OF PHYSICS. mood a/e fleeted train and advancing train come into superposition. The open end of the tube is the center (nearly) of a vibrating seg- ment. If lycopodium powder is strewn along the inside of the tube, it is swept into the nodal points by the violent to and fro motion of the air, giving a strik- ing indication of the existence of the stationary train. (Com- pare Art. 790.) far 624. Simple and compound wave trains. When the curve f*u which represents a wave train graphically is a curve of sines, - ;< the wave train is said to be sim- ple, otherwise the train is said to be compound. The curves of Fig. 380* represent the wave trains which issue from the mouth when the indicated vowel sounds are produced by a baritone voice. The curve No. i is a simple train. The others are compound. Fig. 380. 625. Fourier's theorem. f A compound wave train is the superposition of a series of simple wave trains, of which the respective wave lengths are i, J, J-, \, -J, etc., of the wave length of the given compound train, and of which the respective am- plitudes are determinate. Example. The heavy line AB (Fig. 381) represents one wave of a wave train, which is compounded of the simple wave train represented by the dotted line AB and another simple wave train CD, of half the wave length. The simple trains of * These curves, by Mr. L. B. Spinney, are copies of photographic tracings. t See Fourier's Series and Sperical Harmonics, W. E. Byerly, pp. 30-38, for a discussion of Fourier's Theorem. WAVES. I9 which the curves in Fig. 380 are built up have been studied by Helmholtz and others. x'B Fig. 381. 626. Wave front. Consider a region, AB (Fig. 382), which is disturbed by a wave coming from a distant center of disturb- ance C. If the medium is isotropic, then all parts of any small plane layer AB of the medium perpendicular to r will be similarly distorted, and the layer will move A as a whole up and down or to Q^ v_ and fro as the wave passes. A Fig. 382. B surface so drawn as to pass through those parts of a wave where the distortion is everywhere the same, or those parts where the displacement is everywhere the same, is called a wave front. The direction of progression of a wave in an isotropic medium is at right angles to its front. When a wave has come from .a large number of centers of disturbance near at hand, it has no definite front. Such a wave may, however, be looked upon as a superposition of elementary waves coming each from a center of disturbance, and these ele- mentary waves have definite fronts. A wave having a definite front may lose this character upon striking an obstacle. 627. Huygens' principle. Let AB (Fig. 383) be the in- stantaneous position of a wave which has come from a disturb- 20 ELEMENTS OF PHYSICS. ance at C. The disturbance produced later at / as the wave passes that point is, of course, to be thought of as having come originally from C ; it may, however, be considered to have come from the disturbance which constitutes x the wave AB. In this latter case each point of the wave AB is to be considered as a center of disturb- ance from which a spherical wave emanates. The waves which thus emanate from each point of a prim- ary wave are called secondary waves or wavelets. The actual disturbance produced at / is the superposition of the effects of all these wavelets. 628. Huygens' construction for wave front. Let AB (Fig. 384) be the front of a wave advancing toward A' B J . Let it be required to find the wave front after a time has elapsed, during which the wave has traveled a distance r. Describe circles (spheres) of radius r from each point of the wave front AB. The envelope A'B' of these circles (spheres) is the required wave front. These spheres are the secondary wavelets described in the previous article. Each of the secondary wavelets ema- nates from an infinitesimal portion of AB, \ , and represents an infinitesimal disturbance. The number of these secondary wavelets which work together (i.e. in the same phase) to produce disturbance in a portion of their enveloping surface A'B' is infinitely greater than the number which work together to produce a disturbance else- where. Therefore, the disturbance along A'B' is infinitely greater than elsewhere. This is equivalent to saying that the disturbance elsewhere is zero. WAVES. 21 629. Half-period zones. Consider a simple train of plane waves, of very short wave length X, approaching the point O (Fig. 385), as indicated by the arrow, the wave fronts being parallel to AB. This line AB repre- sents a fixed plane perpendicular to the paper. From O as a center, im- agine spheres to be described of which the first has a radius b, equal to OP ; the next has a radius b + -; the next 2 radius # + X: the next a radius + ^, etc. 2 divided into zones. From the figure we have r f2 4- Fig. 385. a radius 0-t-A,; tne next a The plane AB is thus The first of these zones is a circle of radius / ; the second zone is a circular ring of inside radius r' and outside radius r u ; the third zone is a circular ring of inside radius r" and outside radius r'", etc. / ^\2 = f b 4- j ; whence, since X 2 is negligible in comparison with b\ we have similarly, r" =V2^X, (321) etc., etc. The nth zone is called a zone of high order when n is a large number. The intensity of the disturbance reaching O from each zone is slightly less than that coming from the zone of the next higher order, but two adjacent zones of high order produce at O disturbances which are sensibly equal and opposite in phase,* so that their effect is to annul each other. There- * The secondary wavelets reaching O at the same instant from the nth and the ( + i)th zones are on the whole opposite in phase; for, the wavelets coming from the (n -f- i)th zone, having had - farther to travel, must have started half a period earlier than the wavelets from the nth zone. UNIVERSITY 22 ELEMENTS OF PHYSICS. fore the actual disturbance at O comes from the zones of low order j and when X is small these are included in a small portion of the plane AB surrounding the point P. If a small obstacle is placed at P so as to cut off those por- tions of the advancing waves from which most of the disturb- ance at O comes, the disturbance at O will cease. This is the explanation, by means of the wave theory, of the (sensibly) recti- linear propagation of light. Examples. (a) The value of X for yellow light is about 6.io~ 5 centimeters. If b (Fig. 385) is 10 centimeters, the diam- eter of, say the tenth half-period zone is about 0.16 centimeter, which is the order of magnitude of an obstacle required in this case to screen the point O (Fig. 385). (b) The value of X for sound waves from a shrill whistle is, say, 9 centimeters. If b (Fig. 385) is 10,000 centimeters the diameter of the tenth half -period zone is 1880 centimeters, which is the order of magnitude of an obstacle required to screen the point O in this case. 630. The ray of light. Consider a train of waves of wave length X, and of which the fronts are parallel to AB (Fig. 386). When X is small, the disturbance reaching O comes, sensibly, from the immediate neighborhood of P. This disturbance therefore travels along the line PO. > Lines drawn normal to a wave ^ ~~ front of a wave train along which ~~ ~~~ ~~ ~ "7"* "~ __ the disturbance travels are called, l~H11_1.1_1^rZ.1^rr in geometrical optics rays. Rays I~1Z.~I_"IZIin~^r_I drawn from every point of a small * portion of a wave front constitute B Fig. 386. a pencil of rays. Sound rays. The wave length of sound waves is ordinarily from 5 to 500 centimeters. Therefore the disturbance reaching O (Fig. 386) comes, sensibly, from a portion of AB which is so WAVES. 23 large, compared with the dimensions of objects which we en- counter in daily life, that we do not think of sound as being propagated even approximately in straight lines. An obstacle placed between P and O, as has been explained in Art. 629, must needs be very large to screen O. 631. Shadows. When the light from a luminous body is obstructed by an opaque body, as, for example, when the light of a lamp is obscured by the hand, the region beyond the opaque body is more or less darkened. This region is called a shadow. Similarly, the sound of a whistle, for example, is more or less weakened in the region behind a building. Such a region is called a sound shadow. Geometrical shadows. Consider a luminous point O (Fig. 387). If light were propagated in straight lines, the region below the line OB would be GEOMETRICAL SHADOW OBSTACLE Fig. 387. entirely obscured by the ob- LUMINOUS * J POINT stacle. This sharply defined region is called the geometri- cal shadow. The boundary of the actual shadow cast by a luminous point is, however, not sharp, because of the bending of the light into the region of the shadow, as described in the chapter on Diffraction. This indistinctness of boundary of a shadow is very noticeable in sound shadows because of the greater wave length. Umbra and penumbra. Figure 388 represents the shadow of the moon. The region V, from every part of which the sun would be en- tirely invisible to an observer, is called the umbra. The region P, from every part of which the sun would be only partially visible to an observer, is called the penumbra. The Fig. 388. 24 ELEMENTS OF PHYSICS. umbra and penumbra may be easily distinguished in the shadow of the hand by lamplight. In all ordinary cases the bending of light into the region of a shadow is masked by the penumbra, and only becomes notice- able when light is employed, the source of which is very small (a point). 632. Homocentric pencil. When a small portion of a wave front is a sector of a spherical surface, the rays from the por- tion pass through the center of the sphere, and the pencil of rays is said to be homocentric. The center of the sphere is Fig. 389. Fig. 390. called the focal point of the pencil. When the center of the spherical wave is behind the wave, as shown in Fig. 389, the pencil is said to be divergent. When the center of the spheri- cal wave is in front of the wave, as shown in Fig. 390, the pencil is said to be convergent A -r^Tl I I T-> ^ P enc ^ f parallel rays is a homocentric pencil with its center at infinity. 633. Astigmatic pencil. Consider an arc of a circle A A (Fig. 391) with its center at C. Imagine this arc to be rotated about the line DD as Fig. 391. Fig. 392. an aXlS ' The * AA WU1 describe a surface, of which the principal sections are shown in Figs. 391 and 392. It is shown in treatises on geometry that a small portion of any WAVES. 25 surface whatever is either a portion of a sphere (the osculating sphere) or it is of the shape of A A (Figs. 391 and 392.) If a small portion of a wave front is of the shape of AA (Figs. 391 and 392), all the rays drawn from it will pass through the line DD and through the line CC. Such a pencil of rays is called an astigmatic pencil. The central ray of the pencil is called the axis of the pencil. The lines CC and DD are called the/j (326) Substituting the values of h and h' from (i) and (ii) in (326), d drops out also, and we have equation (325). Q.E.D. Remark i. Any wave in passing through a lens has its central portion retarded by the amount (/*, i) (h + h'). Remark 2. The above proof may be adapted to a diverging lens if we think of the lens coming to infinitesimal thickness at the center, and having a thickness h + k' at the edge. Remark 3. The value of //,, and therefore of / also, varies with the wave length of the light (Art. 676 on chromatic aberra- tion). In the following discussion p and/ are assumed to have definite values. 661. Conjugate foci. Two points so situated that light from one, after passing through a lens, is concentrated at, or appears to have come from, the other are called conjugate foci or conju- gate points. Proposition. The distances a and b of a pair of conjugate points from the center of a lens, and the principal focal length p, satisfy the equation 4 8 ELEMENTS OF PHYSICS. Proof. Consider a portion of a wave, which has reached the lens from a distance a, of which the chord is equal to the diameter of the lens d, and of which the versed sine is k ; then a = The action of the lens is to retard the central portion of this wave by the amount (/* i)(/z -h h') [see Art. 660, Remark i], so that the versed sine of the transmitted wave is and its radius of curvature b is (ii) Substituting the values of /, a, and b from equation (326) of Art. 660, and (i) and (ii) in equation (327), we find that equation to be identically satisfied. Q.E.D. 662. Conjugate points out of axis. A pair of conjugate points which do not lie on the axis of a lens, lie on a straight line passing through the center of the lens. Proof. Consider a ray from a point a (Fig. 428) passing through the center of a lens. This ray passes through the two surfaces of the lens at points where these surfaces are parallel. Fig. 428. Therefore the lens acts upon this ray as a thin parallel plate, and the emergent ray is sensibly a continuation of the incident LENSES. 49 ray. (Compare Art. 650.) This emergent ray passes through the point conjugate to the point a. Therefore the proposition is proven. Geometrical construction for the conjugate of a point. Draw a line (ray) from the point a (Fig. 428 or 429) through the center of the lens. Draw a line (ray) from a parallel to the axis of the lens, and from the inter- section of this ray with the lens, which is supposed thin, draw a line through the principal focus F. Then the point b is the conjugate of a. The statements of Arts. 645, 646, and 647 hold for the case of size of object a lens. Ine magnification 01 an imasce, that is, =. - > size of image is equal to the ratio of the distances of object and image from the center of the lens. Fig. 429. 663. Centered system of lenses. Consider a number of trans- parent media, A, B, C, D> E, F (Fig. 456), for example, air and various kinds of glass, separated by spherical surfaces of which the centers lie on a straight line, the axis. Such an arrange- ment is called a centered lens system. A system consisting of two lenses is called a doublet; a system consisting of three lenses is called a triplet. 664. Propositions. (a) A narrow pencil of rays from any luminous point O, in or near the axis * of a lens system is sensibly homocentric after passing through the system, and is concentrated at or appears to have come from a point O', called the conjugate of O. Proof. According to Art. 655, a narrow pencil near the axis will be homo- centric after refraction at the first spherical surface. The resulting homo- centric pencil will be homocentric after refraction at the second spherical surface, and so on. * In the following figures rays are drawn which make considerable angles with the axis for the sake of clearness. OF THK IVERSITY ELEMENTS OF PHYSICS. () A group of luminous points (an object) near the axis in a plane perpendicular to the axis has as its image (with definite magnification, posi- tive or negative) a similar group of points near the axis in another plane perpendicular to the axis. Such planes are called conjugate planes. Proof. By Art. 655 an object has an image formed in accordance with this proposition by the first refracting surface. An image of this image is pro- duced by the second refracting surface, and so on. 'Q.E.D. Corollary i . Any incident ray passing through a point O passes upon emergence through O', the conjugate of O. For, by proposition (), all rays emanating from or passing through O pass through O' upon emergence. An incident ray and its position upon emergence are called conjugate rays. Corollary 2 . Two incident rays intersecting at O, intersect upon emer- gence at O'. Corollary 3. Consider an incident ray passing through the points O and^. This ray upon emergence passes through O' and^'. Remark i . If an emergent ray be reversed, it will retrace its path, as explained in Art. 650. If, therefore, the ray r' is the conjugate of r, then r is the conjugate of r' ; and if the point O' is the conjugate of <9, then O is the conjugate of O'. Remark 2. Any specification which fixes the positions upon emergence of given incident rays is a complete specification of the lens system. 665. Specification of lens systems. The action of a lens system is com- pletely specified when the positions of two pairs of conjugate planes are given, together with the magnification associated with each pair. Proof. Let aa' and bb* (Fig. 430) be two given pairs of conjugate planes. Let the magnification be m, and the magnification be m'. Given an inci- a b' dent ray r' (from the left), cutting the planes a' and V at p' and q', as shown. md md Fig. 430. The transmitted ray r must pass through the points p and q, which are con- jugate to p' and q' respectively. The transmitted ray is thus fixed, and the action of the lens system upon the ray r 1 is completely determined by the given data. Q.E.D. LENSES. 5 1 666. Principal foci of a lens system. Consider an incident ray r' (Fig. 431), from the left, parallel to the axis. The conjugate ray r is as shown. If the distance d changes, the distances md and m'd change in the same ratio, Fig. 431. as shown by the dotted rays R and R', and the point F remains fixed. There- fore all rays from the left, parallel to the axis, pass through F, or seem to have come from F after emergence. This point Fis called the (right) princi- pal focus of the system. Figure 432 shows the construction for the left prin- cipal focus F'. ^V a' b' b a r r> / R s^tf AXIS >- '^' / w 1^ W/= +1^ Fig. 432. 667. Principal planes of a lens system. That pair of conjugate planes for which the magnification is plus one (+ i) are called the principal planes of the system. The following discussion shows that there is always such a pair of planes, and shows their location relative to the given conjugate planes aa' and bb' . Let r (Fig. 433) be an incident ray from the right, and r' its conjugate. Let R', colinear with r, be an incident ray from the left, and R its conjugate. The rays r and R intersect at O, and the rays r' and R', conjugate to r and R. intersect at O'. Therefore O and O 1 are conjugate points, and since they are at the same distance from the axis, it follows that the magnification for the conjugate planes PP is unity, and that P and P are the required principal planes of the system. The distance Fto Pis called the right focal length of the system, and the distance F 1 to P 1 the left focal length of the system. The system represented in Figs. 430, 431, 432, 433, 435, and 436 is a converging 52 ELEMENTS OF PHYSICS. system. The two focal lengths of a lens system have a ratio equal to the ratio of the refractive indices of the media, in which the principal foci are situated. The simplest case is that described in Art. 655. In most lens sys- tems used in practice the two focal lengths are equal. Fig. 433. 668. Example. Figure 434 shows to scale the actual positions of the principal planes /*and P, and of the principal foci Fand F of a symmetrical bi-convex lens not of infinitesimal thickness the glass of which has a re- fractive index of 1.50. The figure also shows the geometrical construction for determining the position of the image of an object. Draw the ray r Fig. 434. parallel to the axis from O to' the principal plane /", thence after emergence through the focus F 1 '. Draw the ray R from F through O to the principal plane P, thence after emergence parallel to the axis. The conjugate of O is at the intersection of r' and R'. It is marked O'. 669. The inverse principal planes of a lens system are conjugate planes for which the magnification is minus one ( i). The following discussion shows that there is always such a pair of conjugate planes, and shows their LENSES. 53 location. Let the principal planes P and P' and the foci F and F' (Fig. 435) be given. This constitutes a complete specification of the system. Let R and R 1 be conjugate rays. Consider the point q', which is on the ray R' and at a distance d below the axis. The conjugate of q' must lie on the ray R at a dis- tance d above the axis. Therefore, the plane Qf, passing through the point g', is one of the inverse principal planes. To determine the other inverse principal Q' P' --,~-*r- r -t *K: F 'R^ Fig. 435. plane, consider the conjugate rays r' and r. The point ^ is the conjugate of g', and the plane Q is the other inverse principal plane. From Fig. 435, it is evident that the distance PQ is 2/, and that the distance P'Q' is 2/' ; where / and /' are the right and left focal lengths of the lens system respectively. 670. The nodal points of a lens system are the two conjugate points in the axis such that any ray passing through them is, upon emergence, parallel to its direction upon incidence. Consider an object in the plane Q (Fig. 435) and its inverted image in the plane Q' . Imagine a ray r (not shown in the figure) passing from the point q to the nodal point N (not shown) and its conjugate * ray r' passing from the nodal point N' to q' '. The image, being of same size as the object and inverted, and r being, by definition of nodal points, parallel to r', it must be that N and N' are at equal distances and in opposite directions from Q and Q' respectively. Similarly, N and N' are at equal distances and in the same direction from P and P' respectively, as shown in Fig. 436. Let x be the distance of N and N' to the left of P and /", and y the distance of N and N' from Q and Q. Then, since Q'P' = 2/' and QP = 2/, we have x-\- y = if and y x = 2f whence xf f\ * It being assumed that N and N' are conjugate points. 54 ELEMENTS OF PHYSICS. It remains to be shown that N and N', the positions of which are determined by (ii), are conjugate points. Draw any parallel lines r and r' (Fig. 436) through jVand N' . Then, since NQ = N'Q' and NP = N' F , we have e = e' and d = d ', so that the points p' and q' are the conjugates of the points p and q ; and r and r' are conjugate rays. The same would be true of another pair of Fig. 436. parallel lines R and R' (not shown) passing through N and N'. R and r intersect at N", and their conjugate rays R' and r' intersect at N', so that N and N' are conjugate points. Corollary. Any object and its image subtend equal angles, as seen from the respective nodal points of a lens system. Remark. When the right and left focal lengths of a system are equal, the nodal points lie in the principal planes P and P ' . CHAPTER V. THE CORRECTION OF LENSES; LENS SYSTEMS. 671. General statement. The elementary theory given in Chapter IV. applies to thin lenses, and to mirrors and centered lens systems of small aperture. It assumes all imaged points to be nearly in the axis and the refractive index of a sample of glass to be the same for all kinds of light. These conditions are not realized in practice, and lenses have therefore certain imperfections. These imperfections may be very largely avoided by using properly designed lens systems instead of simple lenses. The approach to perfection attained in the manufacture of lenses depends upon the fact that the resolving power of the eye is small. Lenses are, in general, used to aid vision, directly or indirectly. In order to secure a satisfactory result, the region within which the light from a point of an object is concen- trated by a lens must be so small as to be, under the conditions in which it is viewed, sensibly a point. Remark. It is a familiar fact that we can see distinct images, more or less distorted, of surrounding objects in almost any polished surface, however uneven, and that objects are not greatly confused when seen through window glass and through smooth glassware. This is due to the fact that in most cases the portion of such a surface which sends light into the eye from a given luminous point is very small, much smaller even than the pupil of the eye. The focal lines of a narrow astigmatic pencil are very short when they are near together, and if these short focal lines are at a considerable distance from the eye, the astigmatic pencil is sensibly homocentric. 672. Definitions. Numerical aperture. The effective free diameter of a lens divided by its principal focal length is called its numerical aperture. 55 56 ELEMENTS OF PHYSICS. Telescopic objectives range up to y 1 ^, photographic objectives up to J or J, and microscopic objectives up to 1.4 numerical aperture. Field angle. The angle at the center of a lens, between lines drawn to the extreme edges of the largest distinct image which the lens will produce, is called the field angle of the lens. A lens so designed as to have a wide field angle is called a wide angle lens. Photographic objectives are made which give excellent definition with a field angle as great as 110. The field angle of telescopic and microscopic objectives is ordinarily very small, seldom exceeding 5. 673. Spherical aberration. When a plane (or spherical) wave passes through a simple lens of wide aperture, those portions of the wave which pass through the outer zones of the lens are focused nearer the lens than those portions are which pass through the central zone. (See Fig. 437.) In other words, the outer zones of a lens are of shorter focal length than the central zone. This fault of lens is called spherical aberration. A lens free from spheri- Fig. 437. cal aberration is said to be aplanatic. The spherical aberration of a simple converging lens, as also its focal length, is considered to be positive. Of a simple diverg- ing lens both are considered to be negative. The greater the refractive index of the glass used, the less the thickness of a lens of given focal length, and the less its spherical aberration. 674. Correction of spherical aberration. A converging lens and a diverging lens of equal and opposite focal lengths, and of equal and opposite spherical aberration, entirely annul each other's action if placed near together. The spherical aberration of a lens of a given focal length depends, however, upon the relative curvature of the two faces of the lens and upon the refractive index of the glass ; so that it is possible to construct LENS SYSTEMS. 57 a converging lens and a diverging lens giving equal and oppo- site spherical aberration, but not having equal and opposite focal lengths. When two such lenses are used together, they form an aplanatic doublet of any required focal length. A lens system can be aplanatic only for a given pair of conjugate points (planes) called the aplanatic points of the system. 675. Abbe's condition for aplanatism. If a lens is to form a distinct image of a small group of points in and near its axis at one of its aplanatic points, the lens must be sensibly aplanatic for the points near the axis as well as for the point in the axis. The condition that must be satisfied in order that this may be true is called, from its discoverer, Abbe's Sine Law. Preliminary statement. Let a and b (Fig. 438) be two conjugate points with respect to a lens system. A spherical wave passing out from a becomes a spherical wave coming in upon b. Let w and w' be the portions of this spherical wave which have traveled along any two rays r and r'. Since iv and w' started out from a at the same instant, and are portions of a spherical surface with its center at , they will reach b at the same instant. It follows, therefore, that the time required for a light wave to travel along any two rays from a point to its conjugate is the same. This time is also the least time in which light can pass from one point to the other. This principle is sometimes called the principle of least time. Consider a very small object aa' and its image bb' (Fig. 439 a}. Let d be the distance aa 1 and md the distance bb', m being the magnification. Con- Fig. 438. Fig. 439 (a). sider the rays R (the axis) and R' from a to b and r and r' from a' to b' . Let be the angle between the incident portions of R' and r' and the axis (these angles are sensibly equal) ; and let 3> be the angle between the emer- gent portions of R' and R and of r' and r (these angles are sensibly equal) . The rays R and r pass through the same portion of the lens, and are of the same length, so that the rays r' and R' must be of the same (optical) length also. The rays r' and R' pass through the same thickness of glass, 58 ELEMENTS OF PHYSICS. the incident portion of R' is dsin(j> longer than the incident portion of r', and the emergent portion of r' is mdsin& longer than the emergent portion of R', as shown in Fig. 439 . Therefore, dsin = sinrf) r; - (328) If the refractive indices of the media in which the object and image are located are /A and p! respectively, then the optical values of Fig. 439 (b). dsin <(> and mdsin $ are and equation (328) becomes and sin $ mdsin (329) The reader will find, by applying equation (329), that the sphere described in Art. 657 satisfies Abbe's Sine Law, and is therefore aplanatic for all points near the axis in a plane passing through the point P (Fig. 423) perpendicular to the axis. If this were not the case, a microscope provided with the objective, shown in Fig. 450, would give sharp definition only in the very central point of the field of view. 676. Chromatic aberration. The value of the refractive index of a given sample of glass varies with the wave length of the incident wave train. Therefore (see Art. 660) the focal length of a lens is different for different wave lengths (colors). Violet light is focused nearer the lens than red light and the intermediate colors between, as shown at r and v (Fig. 440). This variation of focal length of a lens with wave length is called chromatic aberration. It is possible to construct a con- verging lens and a diverging lens, giving equal and oppo- site chromatic aberration (for, r say, red and violet light), but having different focal lengths, so that when used together they will form a doublet of any desired focal length. This doublet will have equal focal lengths for the two colors. Such a lens system is called an achromatic lens. A sketch of the elementary theory of the achromatic lens is given in the chapter on Dispersion. Fig. 440. LENS SYSTEMS. VERSITZ 59 A lens system consisting of two thin lenses of similar glass at a distance apart equal to half the sum of their individual focal lengths is achromatic.* The doublets of Ramsden and Huygens, illustrated in Figs. 448 and 449, are examples. 677. Astigmatism. A pencil of parallel rays (in general any homocentric pencil) becomes an astigmatic pencil when it passes obliquely through a simple lens. Figure 441 shows the actual positions of the focal lines, C and DD, of the pencil RR of parallel rays after passing through a lens of which the principal focal length is OP, and of which the glass has a refractive index equal to f. The astigmatism of a converg- ing lens being considered positive, that of a diverg- ing lens is to be consid- ered negative. The as- tigmatism of a thin lens Fig. 441. of given focal length in- creases with the refractive index of the glass of which the lens is made. It is, therefore, possible to construct a converg- ing lens and a diverging lens, giving (to first order of small quantities) equal and opposite astigmatism, but having different * Proof. It can be shown that the focal length,/ of a doublet is such that 1 I I D in which /i and / 2 are the respective focal lengths of the constituent simple lenses, and D is their distance apart. Let A/i and A/a be the difference in the values of /i and/, respectively, for two colors. Then since the lenses are of similar glass, we have - /i '/a The change A/ in the focal length of the doublet due to the changes A/i and A/ 2 is easily found from (i) . Placing this expression for A/" equal to zero, A/i and A/2 may be eliminated with the help of (ii), giving at once /> = .AA Q.E.D. (330) 6o ELEMENTS OF PHYSICS. focal lengths, so that when used together they may give a doublet of any desired focal length, sensibly free from astig- matism. Such a lens is called an anastigmatic lens. 678. Distortion. The image of an object formed by a lens is, in general, distorted. In case the magnification increases towards the edge of the field, the image of a square network will appear, as Fig. 442 a. Fig. 442. In case the magnification decreases towards the edge of the field, the image of a square network will appear, as Fig. 442 c. In case the magnification is constant, the image will not be distorted. A lens which does not give a distorted image of an object is called an orthoscopic or rectilinear lens. The simplest ortho- scopic combination is the symmetrical doublet first introduced by Steinheil. This consists of a combination of two similar lenses LL (Fig. 443), having a diaphragm with a small opening O between them. It is evident from the symmetry of the combination that any ray rr passing though the center of O is, upon emergence, parallel to its incident direction. Therefore such rays cut the conjugate planes a and b in simi- larly grouped points. This is strictly true if the planes a and b are at equal distances from O (and the magnification is i). Fig. 443. LENS SYSTEMS. fa When the magnification is other than - i, the combination remains sensibly orthoscopic. For photographic purposes it is customary to use in place of the lenses LL two similar achro- matic doublets. Such a combination is shown in Fig. 454. 679. Curvature of field. In order to project upon a screen the most distinct image of an extended flat object, which it is possible to form by means of a simple lens, the screen must be curved, as shown by 55 in Fig. 444. This imperfection of a lens is called curvature of field. The curvature of field of a lens of given focal length varies with the refractive index of the glass, and is of opposite sign for converging lenses and diverging lenses. It is, there- fore, possible to combine a converging lens and a diverging lens of different glass, so as to form a doublet of any desired focal length which will give a fiat field. Flatness of field is very important in photographic objectives, because these are used to project images upon flat glass plates. 680. Simultaneous correction for several imperfections. - There is now a great variety of optical glasses at the disposal of the manufacturing optician, and by the combination of a number of lenses, sometimes as many as ten, each made of chosen glass, lens systems are made which leave but little to be desired.* The apochromatic microscope objective shown in Fig. 451, which consists of ten separate lenses, is an excellent example of a combination designed to annul simultaneously the various errors to which simple lenses are subject. 681. Examples of various lens systems used in practice. (a) Magnifying glasses and eyepieces. Figures 445-447 show standard forms of magnifying glasses. Figure 445 shows the lens known as Brewster ? s * It may be stated that wide field angle is incompatible with large aperture, and that an aplanatic lens of large aperture cannot be orthoscopic. 62 ELEMENTS OF PHYSICS. magnifer; * Fig. 446 the arrangement called Wollaston's doublet. Figure 447 is a modification by Hastings of an earlier form by Brewster. Figures 448 Fig. 445. FRONT Fig. 446. Fig. 447. and 449 are achromatic doublets such as are used for microscope and tele- scope eyepieces. The lenses of Ramsden's doublet (Fig. 448) are placed a little nearer together than is required (by equation 330) to give achromatism. The result is to Fig. 448. Fig. 449. bring the focal points outside of the combination, so that the doublet may be used to view a real image, or an object, just as with an ordinary magnifying glass. The front focal point t of Huygens' doublet (Fig. 449) is between the lenses, so that this doublet cannot be used as an ordinary magnifying glass. The thing viewed with it must be a virtual image (as shown by the arrow in Fig. 449) Telescopes when used for sighting are always provided with Ramsden's doublet eyepiece or some equivalent combination. (b) Microscope objectives. The apla- natic property of the sphere, as described in Art. 657, is utilized in the construction of the microscope objective (homogeneous immersion objective) as follows : A hemis- pherical lens L (Fig. 450) is mounted as shown. The flat face of this lens is sub- merged in oil of the same refractive index as the glass of the lens. Then light from the point P (of an object), after passing through the lens Z,, appears to have come from a point P' . Addi- * A simple lens is practically perfect when used as a magnifying glass of low power. t This point is the front principal focus. See Art. 666. Fig. 450 LENS SYSTEMS. tional lenses are used, as shown by the dotted lines, to correct for the chro- matic aberration of Z, and for concentrating the light from P' at some point beyond, thus forming an image of the object. Figure 451 shows a highly perfected microscope objective, designed by Abbe and constructed by Zeiss. (c} Photographic objectives. Figure 452 shows a simple achromatic lens with diaphragm, such as is used for landscape work. Figure 453 shows a standard ^ J ' J DIAPHRAGM form of wide angle orthoscopic lens for architectural and landscape work. FR Figure 454 shows a standard form of wide aperture orthoscopic lens. Figure 455 shows a very wide aper- ture aplanatic lens by Dalmeyer, used for portrait work. Figure 456 Fig. 451. Fig. 452. shows an anastigmatic photographic lens, designed by P. Rudolph and con- structed by Zeiss and by Bausch and Lomb. Fig. 453. Fig. 454. I Fig. 455. (X) Telescope objectives are usually simple achromatic doublets. Such a doublet may be made both achromatic and aplanatic, as explained in Art. 696. Figure 457 shows the standard form of telescope objective. ZEISS ANASTIGMATIC SYSTEM PHOTOGRAPHIC Fig. 456. Remark. In Figs. 445-457 all the converging lenses are of one or another variety of crown glass, and all the diverging lenses are of one or another variety of flint glass. CHAPTER VI. SIMPLE OPTICAL INSTRUMENTS. 682. Optical instruments defined. The instruments to be described in this chapter are those necessary to vision and those used to aid vision or to supplement it. Certain optical instruments which do not come directly under this definition, such as the spectroscope and the polariscope, will be considered later. 683. The eye. This organ is shown in its essential fea- tures in Fig. 458. The tough membrane of the eyeball is sharply curved and transparent in front, forming the cornea C. Between the cornea and the crystalline lens L is a clear, watery fluid, the aqueous humor. Behind the crys- talline lens, and filling the remainder of the eyeball, is a clear semi-fluid substance, Fig. 458. ^ Q vitreous humor. The front surface of the cornea and the surfaces separating the aqueous humor, crystalline lens, and vitreous humor are sensibly spherical and constitute a lens-system which projects an image of external objects upon a nervous membrane, the retina, at the back of the eye. The retina consists of a vast number of minute end organs of nerve fibers which come together, forming the optic nerve O, and lead to the brain. Two luminous points can be seen as two so long as their images on the retina do not fall upon one and the same end organ. Over a large portion of the retina these end organs are 6 4 SIMPLE OPTICAL INSTRUMENTS. 65 relatively sparse, but in the thin portion /, called from its color the yellow spot, they are very closely packed together. An object can be seen distinctly only when its image falls upon this spot. The line passing through the center of the eye lenses and the center of the yellow spot is called the axis of the eye. Accommodation. For distinct vision, the image on the retina must be sharply defined. The focal length of the eye lenses, to give a sharply defined image of an adjacent object upon the retina, must be different from their focal length to give a sharp image of a distant object. This necessary variation of focal length is provided for by the action of muscles, attached to the edge of the crystalline lens, the contraction of which causes the lens to become thinner at the center. This action is called accommodation. Ordinarily the eye has power of accommoda- tion for objects at any distance greater than about 15 centi- meters from the eye. The distance of most distinct vision, for most individuals, is about 25 centimeters. The imperfections of the eye. Some persons can accom- modate to distant objects only with great effort, or not at all; such persons are said to be nearsighted. Persons who have like difficulty in accommodating to near objects are said to be farsighted. Nearsightedness is relieved by the use of spec- tacles with diverging lenses, farsightedness by the use of spectacles with converging lenses. Nearsightedness (or far- sightedness) may be due to an abnormal elongation (or shorten- ing) of the eyeball in the direction of the axis ; or to an abnormally short focal length (or long focal length) of the eye lenses. Inaccurate centering of the eye lenses, and more or less deviation from true spherical shape of the various refract- ing surfaces, produce astigmatism ; which is sometimes so pronounced as to hinder sharp vision. Such imperfection is corrected by the use of spectacles having cylindrical sur- faces. Apparent size of objects ; visual angle. An object appears 66 ELEMENTS OF PHYSICS. large when its image covers a large portion of the retina. Lines drawn from the extremities of the object through the center* of the eye lenses, as shown in Fig. 459, pass through the ex- OBJECT Fig. 459. tremities of the image. The angle, a, between these lines is called the visual angle of the object and is a measure of its apparent size. 684. The photographic camera consists of a light-tight box, in the front of which a lens is mounted. At the back of this box is a sensitive plate, upon which an image of external objects is projected by means of the lens. The requirements of photo- graphic work have led to the construction of a variety of lens systems, designed to give rapidity of action, wideness of angle, flatness of field, exquisiteness of definition, etc. Special lenses are made for use in portraiture, in architectural work, in land- scape, in instantaneous photography, etc. Some of these forms are described in Chapter V. The color correction of photo- graphic lenses is made with reference to those wave lengths by which the sensitive plates are most strongly affected. 685; The magic lantern is an arrangement for projecting the image of a brilliantly illuminated object or picture upon a screen. The light from a brilliant lamp L (Fig. 460) passes through condensing lenses CC, through a transparent picture 55, and through an objective lens O, which throws a greatly * Strictly, lines pass through the extremities of the image, which are drawn through the posterior nodal point of the lens system of the eye, parallel respectively to lines drawn from the extremities of the object to the anterior nodal point. (Com- pare Art. 670.) SIMPLE OPTICAL INSTRUMENTS. 6 7 enlarged image of .S.S upon the distant screen. Symmetrical orthoscopic combinations, such as are designed for photographic work, are much used in magic lantern objectives. Such an objective is shown in Fig. 460. a c c Fig. 460. 686. The simple microscope ; definition of magnifying power. The simple microscope, or magnifying glass, consists of a converging lens (simple or compound) which is held near the eye. The object to be examined is moved up until it is seen sharply. When this is the case, the eye is looking at a virtual image of the object, and this virtual image is at the distance of most distinct vision from the eye. The magnifying power of a microscope is denned as the ratio of the apparent size of an object, as seen with the microscope, divided by its apparent size, as seen with the naked eye at a distance of 25 centimeters. 687. Proposition. The magnifying power of a magnifying glass is =^+'.' (33D in which/ is the principal focal length of the lens in centimeters. The eye is supposed to be accommodated for a distance of 25 centimeters. Proof. Let O (Fig. 461) be the object, and i its virtual image, formed by the magnifying glass. The eye being accom- modated to 25 centimeters, the object will be moved until the 68 ELEMENTS OF PHYSICS. image i is 25 centimeters from the eye. The distance from the lens to the eye may be neglected, so that the distance b of the image from the lens may be taken as 25 centimeters. Since the distance from a lens to a virtual image is considered negative, we have, from equation (327), Whence 25 +/ Now the visual angle a is sensibly the same as the angle at c subtended by O and i t and this angle is as many times as great as the angle that O would subtend at a distance . of '25 centimeters, as 25 is greater than a : that is, m=- (iii) a Substituting the value of a from (ii) in (iii), we have equation (331). Q.E.D. Remark i. When the eye is accommodated for parallel rays, b (Fig. 461) is infinity, and a p. So that m Remark 2. A magnifying power of 125 diameters (focal length of 2 millimeters) is about as great as can be obtained satisfactorily with the simple microscope. For greater magni- fying powers the compound microscope is more satisfactory. 688. The compound microscope consists of a lens A (Fig. 462) which gives an enlarged real image i of an object o t and a SIMPLE OPTICAL INSTRUMENTS. 6 9 magnifying glass B for viewing this image. The lens A is called the objective. It is usually a lens system of the form shown in Fig. 450, or in Fig. Q 451. The lens B is called the eyepiece; it likewise usually consists of a system of lenses as shown in Fig. 448, or Fig. 449. Figure 463 shows, full size, the actual arrangement of the essential optical parts of a compound microscope.* The group of lenses C is a condenser for illuminating the object which is placed upon the stage of the instrument ; O is a homogeneous immersion objective, 2 millimeters focal length, by Bausch and Lomb, and E is a Huy- gens doublet eyepiece. Fig. 463. The magnifying power of a compound 689. Proposition. microscope is in which / is the principal focal length of the eye lens (or system), and a and b are the respective distances of the object and image from the center of the objective lens, as shown in * For more detailed information concerning microscopes see Gage, The Micro- scope, 5th ed., 1896. Andrus and Church, Ithaca, N.Y. 70 ELEMENTS OF PHYSICS. Fig. 462. If the objective is a lens system, then a and b are the distances of object and image from the respective nodal points of the system. Proof. The image is - times as large as the object, and the eyepiece, being a magnifying glass, makes this image appear 2 % + I times as large as it would appear to the naked eye at a distance of 25 centimeters, or-f + I J times as large as the object would appear to the naked eye at that distance. Q.E.D. By the use of high grade objective systems, such as have been described in Chapter V., and of a doublet eyepiece of the form shown in Fig. 449, satisfactory definition and sufficient bright- ness can be obtained with a magnifying power as high as 1 500 diameters. 690. The telescope consists of a lens of long focus O (Fig. 464), which forms an image i of a distant object ; and of a magnifying Fig. 464. glass E for viewing this image. The lens O is called the object glass, and is usually a system of lenses of the kind shown in Fig. 457- The lens E is called the eyepiece, and it is usually a Ramsden or Huygens doublet (Figs. 448 and 449). One of the most important features of an astronomical tele- scope is light-gathering power, which requires an objective lens of large aperture. Until very recently the difficulties of mak- ing large lenses were so great that concave mirrors were used instead in all the largest telescopes. This method of construe- SIMPLE OPTICAL INSTRUMENTS. 7! tion had its culmination in the great reflecting telescope of Lord Rosse (1842), the aperture of which was about 183 centi- meters. The development of the art of glass making is now such that it is possible to construct nearly perfect lenses with an aperture of 100 centimeters. The performance of a large refracting telescope is greatly superior to that of the reflecting telescope, and the latter has become obsolete. 691. The magnifying power of a telescope is defined as the quotient of the visual angle (Fig. 464) of the object as seen with the telescope, divided by the visual angle ft of the object as seen with the naked eye. (Compare Art. 686.) If m be the magnifying power, we have therefore If we consider the object to be, sensibly, at an infinite dis- tance, iO in Fig. 464 is the principal focal length / of the object glass. If the eye is accommodated for parallel rays, the distance iE is the principal focal length of the eyepiece. The angles and ft are then proportional to / and p 1 , so that From (i) and (ii) we have, = (333) The magnifying power of a telescope is therefore equal to the ratio of the focal length of the object glass to the focal length of the eyepiece. 692. The use of the telescope for sighting. The telescope attached to such instruments as transits and levels, and to a great variety of apparatus in physical laboratories, is used for sighting. A cross of very fine wires is fixed in the focal plane of the object glass so as to be seen through the eyepiece (Ramsden's doublet) at the same time with the image of a dis- tant object. When the image of a distant point, as a star, falls ELEMENTS OF PHYSICS. upon the point of intersection of these wires, the axis of the telescope points exactly at the distant point. The axis of the telescope is the line drawn from the intersection of the cross wires to the center of the object glass. 693. The erecting telescope or spyglass. The simple tele- scope shows objects inverted. The spyglass is a telescope modi- Fig. 465. Diag. of spyglass. fied so as to make distant objects appear erect. Figure 465 shows the arrangement of parts in a spyglass. The arrow i represents an inverted image of a distant object formed by the object glass O. A lens vS forms at i 1 an inverted image of i. This image i' is therefore erect, and is viewed by an eye lens E as before. In practice the lenses O and E are lens systems. The lens 5 is usually a symmetrical doublet somewhat similar to Fig. 443. The introduction cf the erecting system 5 lengthens the telescope very materially. 694. The opera glass is a telescope provided with a diverging lens as an eyepiece. The action is shown in Fig. 466, in which i is the position which the (inverted) image formed by the object glass O would take were it not for the di- verging lens E. The action of the lens E is to form at i' an enlarged inverted virtual image of i. In looking in through E, the observer sees the erect image i' of the distant object. This telescope is very short for a given magnifying power. Fig. 466. CHAPTER VII. DISPERSION. 695. Newton's experiment. Homogeneous light; non-homo- geneous light. A beam of parallel rays of white light, such as sunlight or lamplight B (Fig. 467), is changed into a fanlike beam B' by a prism. This fanlike beam falling upon a screen 55 produces an illuminated band R V, called a spectrum, which is red at the end R and passes by insensible gradations through orange, yellow, green, and blue to violet at the end V. The beam of light B is said to be dispersed by the prism. The fanlike beam B' produces white illumination when concentrated by a converging lens upon a small portion of a screen. A photographic plate reveals the existence of invisible rays beyond V, the ultra-violet rays, especially in sunlight; and a thermopile or bolometer shows the existence of rays inside of or below R, the infra-red rays. The portion of the spectrum between R and V is called the visible spectrum. A narrow beam B" passing through a small hole in the screen is deflected by a prism, but not dispersed. The action of the prism upon the beam of white light shows that white light is non-homogeneous, being made up of dissimilar parts. The beam B" , on the other hand, is homogeneous. Homogene- ous light is sometimes "called monochromatic light. 73 Fig. 467. 74 ELEMENTS OF PHYSICS. Since the prism P (Fig. 467) deflects each homogeneous beam B" differently, it is obvious that the glass of which it is made has different refractive indices for the various homogene- ous components of white light. The phenomena of interference (see Chapter VIII.) show that a homogeneous beam of light is a simple wave train of definite wave length, or, more strictly speaking, of definite frequency. A composite beam, such as a beam of sunlight, consists of a group, sometimes infinite in number, of such wave trains. After dispersion, each simple wave train assumes a different path according to its wave length. 696. The achromatic lens. A prism of flint glass which gives a spectrum of the same length (say from red to blue) as a prism of crown glass, produces, on the whole, much less deflection of the beam than does the crown-glass prism. If, therefore, the two prisms are placed together as in Fig. 468, the dispersion of the one will be counterbalanced by that of the other, but the deflection will not be wholly an- nulled. The compound FLINT . ^, f .,, . prism therefore will give deflection without dis- persion. Spectra of the same total length, produced by a flint prism and by a crown prism (Fig. 474), are not of the same length from red to^yellow, yellow to green, etc. ; so that a flint prism cannot be made to wholly neutralize the dispersion of a crown- glass prism. A prism of crown glass which gives the same mean deflection as a prism of flint glass gives a much broader spectrum than the latter, so that two such prisms may be arranged to give a very good spectrum, the middle part of which is not deflected. Such an arrangement of prisms is used in the direct vision spectroscope. The action is rendered more satis- DISPERSION. 75 factory by using two crown prisms. Figure 469 shows the action of such a compound prism. A compound lens, to be achromatic, must satisfy the condi- tion expressed by equation (334), which is derived as follows : /r~i / \ Fig. 469. Let fj/ and p" be the refractive indices of crown glass for red and blue respectively, and ' nj and //,/' the corresponding refrac- tive indices of flint glass. Let C (Fig. 470) be a converging crown-glass lens and F a diverging flint-glass lens. Let * h be the thickness of the crown lens at the center and h' the thickness of the flint lens at the edge, and let d be the diameter of the lenses. Consider the retardation of the central portion, relative to the edge portion, of a plane wave of red light. The retardation by the crown lens is (// i)/z, and the retardation by the flint lens is (/*/ i)h! (see Art. 660); the total retardation is therefore (// i)h (/*/ i)/i'. Simi- larly, the total retardation of the central portion of a plane wave of blue light is (//' i) h (^ i) h 1 . If these two waves are to be focused at the same point, the two retarda- tions must be equal, which gives at once F This is the necessary relation between h and h 1 to give achro- matism. The absolute values given to h and h' depend upon the diameter of the lens and the desired focal length. The * The crown lens is assumed to come to a sharp edge and the flint lens is assumed to be of zero thickness at the center for the sake of simplicity. 76 ELEMENTS OF PHYSICS. curvatures of the various surfaces, in so far as they are not fixed by the absolute values of h and h 1 ', are chosen so as to make the system aplanatic. 697. The spectroscope. In the spectrum obtained by New- ton (described in Art. 695) the beam which falls upon the prism is admitted through a circular hole. Each beam of homogeneous light is as wide, however, as the incident beam ; so that the various homogeneous beams overlap greatly. This difficulty is overcome by means of the spectroscope. In this instrument (Fig. 471) the light to be analyzed is passed through a very narrow slit, S, between two straight metal edges. This slit, -V Fig. 471, which may be regarded as the source of the light, is at the prin- cipal focus of an achromatic lens L, called the collimating lens. The spherical waves from the various points of the slit are plane after passing through L. These plane* waves pass through the prism P, and each homogeneous component of the light (i.e. each simple wave train) appears to have come from a distinct slit, and the lens L' forms images of the slit side by side at R V, one for each homogeneous component. This band of images is called a spectrum ; it is viewed by a magnifying glass or eyepiece E. The images of the slit are called the lines of the spectrum. * Spherical waves are much distorted when refracted at a plane surface, as described in Art. 652; hence the importance of the collimating lens L. DISPERSION. 77 698. Continuous spectra. The light from hot solids and liquids has in it wave trains of every wave length, and when analyzed by the spectroscope the images of the slit form a continuous band, red at one end and violet at the other. Well-known examples of continuous spectra are the spectra of the candle flame, the petroleum flame, the gas flame, etc. The light from such flames is given off by particles of carbon. The spectra of the lime light, of the magnesium flame, of the incandescent lamp, and of the crater of the electric arc are likewise continuous. 699. Bright-line spectra. The light from a hot vapor or gas contains, ordinarily, only wave trains of certain definite wave lengths, and gives, when analyzed by the spectroscope, a group of distinctly separated images of the slit (bright lines) with intervening dark spaces. Such spectra are called bright-line spectra. Every gas or vapor has a characteristic spectrum, that is, a characteristic grouping of images of the slit (bright lines) at R V. The bright-line spectra of some of the metals are of very simple character. When a salt of lithium, for example, is vapor- ized in the flame of a Bunsen burner, a single red line appears in the field of the spectroscope. Sodium is characterized by the presence of two yellow lines. These are so near together that unless the slit is narrow, the dispersion large, and the defi- nition good, they appear as a single line. Thallium has a spec- trum which, as ordinarily observed, possesses a single line of brilliant green. The spectrum of indium in like manner ex- hibits a single blue line. When carefully studied under condi- tions of more intense incandescence, these elements are found to have more complicated spectra. Sodium vapor, in the electric arc, shows some sixteen lines in the visible spectrum ; thallium, twenty-five lines. Even in the infra-red of the spectrum narrow regions of great intensity have been discovered by means of the bolometer. Thus ELEMENTS OF PHYSICS. Snow * found two strong lines due to sodium which are quite invisible to the eye. These are shown in Fig. 472, which 1000 , shows the distribution of energy in the spectrum 950 of that metal. The region between HH and A, in 900 that diagram comprises the entire visible spectrum. 850 - All lying to the right of A constitutes the infra-red. 800 The most complex bright-line spectrum is prob- 750 ably that of iron. Thousands of lines 700 650 due to the vapor of that metal have 600 . already been identified and listed, and 3 550 i- - the catalogue is doubtless incomplete. 500 - z 450 - 700. Dark-line spectra. When an 400 intense beam of light passes through 350 300 a vapor or a gas, those wave trains are 250 absorbed which the gas itself gives off. 200 _ (Kirchhoff and Bunsen.) Such light 150 - in the spectroscope gives missing 100 - images of the slit or dark lines where 50 1 jLU LA \L~S~~ bright lines would W- su^*i_WVW. i f ,,0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 OCCUr in tflC SpCC- trum of the gas or Fig ' 472 - vapor. The most striking example of dark-line spectra is the solar spectrum, which shows a great number of dark lines. These dark lines in the solar spectrum were first carefully studied by Fraunhofer (1819) ; they are called Fraunhofer s lines. The more promi- nent of these lines are designated by the letters A, B, C, D, E, b, F, G, Hp and H 2 , beginning with the red end of the spec- trum. Figure 473 shows the visible portion of the sun's spectrum, as it appears when dispersed with a single prism. The figure is from a drawing by Becquerel. The significance of the Fraunhofer lines was first pointed out by Kirchhoff and Bunsen in 1865, as stated above. They found groups of * Physical Review, Vol. I., p. 95. DISPERSION. 79 dark lines in the sun's spectrum to coincide with the groups of bright lines given by iron vapor, sodium vapor, hydrogen, and other substances, and they inferred the existence of rela- tively cool masses of these vapors in the sun's atmosphere. Kirchhoff and Bunsen devised the following experiment for showing the significance of dark-line spectra : Using the flame of a Bunsen burner, supplied with sodium vapor by the vaporization of common salt, they obtained the ordinary bright-line spectrum of sodium. Then, passing , an intense beam of light from a lime light through the Bunsen flame into the slit, it was found that the absorption of the sodium vapor was such as to leave relatively dark lines in place of the bright lines given by the flame alone. This experiment has become classical under the name of the reversal of sodium lines. 701. The spectrometer. This is a spectroscope which is provided with a divided circle by means of which the lines of the spectrum may be definitely located. The essential parts of a spectrometer are shown in Fig. 471. The lenses L' and E are respectively the objective and eyepiece of a telescope, which turns about a pivot at the center of the circle and is provided with a cross wire in the focal plane RV. Consider a beam of homogeneous light which, upon emer- gence from the prism, is focused by the lens L' . When the telescope is set so that its cross wire coincides with the line of the spectrum, which is due to the homogeneous beam in question, then the axis of the telescope is parallel to the beam, BRA OF THE . ERSITY 8o ELEMENTS OF PHYSICS. and the reading upon the divided circle of a vernier, which moves with the telescope, determines the location of the spec- tral line. Determination of the refractive index. The total angle of deflection of a given homogeneous beam may also be determined by the process just described. When this (minimum) angle of deflection has been measured, and the angle of the prism is known, the refractive index of the prism is easily calculated. i 702. Normal and prismatic spectra. The dispersion by glass prisms is such that the distribution of the images of the slit in the continuous spectrum is by no means uniform. Towards the red there is relative crowding together, while towards the violet end of the spectrum the dispersion increases rapidly. A spec- trum in which the distribution is uniform, i.e. in which equal distances in the spectrum correspond everywhere to equal dif- ferences of wave length, is called a normal spectrum. The relation of the normal spectrum to prismatic spectra produced by prisms of flint glass and of crown glass, is indicated in Fig. 474. NORMAL SCALE 4000 5000 6000 7000 I I I I I I 1 ! I I I Mill 4000 5000 6000 7000 PRISMATIC SCALE (FLINT) Fig. 474. 703. The spectrophotometer. This instrument is a spectro- scope arranged for determining the composition * of light. The measurements consist in comparing the intensity of each part of the spectrum of a given source of light with the intensity at the same part of the spectrum of a source which has been selected as a standard. Figure 475 shows a convenient form of this in- * That is, the relative intensity of the various homogeneous components. DISPERSION. 81 strument. It consists of a direct vision spectroscope (Art. 696) mounted upon a carriage which travels along a track between the two sources, the spectra of which are to be compared. The slit of the instrument, 5 (Fig. 476), is horizontal. Two prisms, /, /, reflect (totally) the light from a standard lamp and from the source of light L, which is to be compared with the standard, into the two ends of the slit. The two spectra are seen side by side in the field of the spectro- scope. The attention is fixed upon a narrow portion of the two spectra (say the red), and the instrument is moved along the line IF until this narrow portion is of equal brightness Fig. 475. STANDARD LIGHT ~T Fig. 476. in the two spectra. From the distances / and /' the relative intensities of the sources, for the given portion of the spec- trum, can then be computed. This process is repeated until the entire spectrum has been explored. (See further the chapter on Photometry.) There are many other devices by means of which the two spectra to be compared in spectrophotometry may be reduced to equal brightness. Sometimes the widths of the two ends of the slit are varied ; sometimes a wedge of neutrally tinted glass is interposed in the path of the brighter beam ; sometimes pairs of Nicol prisms (see the chapter on Polarization) are used. CHAPTER VIII. INTERFERENCE AND DIFFRACTION. 704. Interference from similar sources. Consider two points, O and O f (Fig. 477), which send out continuously simple wave trains of wave length X. The points O and O 1 may be thought of as luminous points, as periodic disturbances on the surface of water, or as two tuning forks in unison sending out sound Fig. 477. waves. Consider a point q whose distance from O is equal to its distance from O', or differs from it by a whole number of wave lengths n\. The wave trains from O and O' are continually alike in phase at such a point, and they work together to pro- duce disturbance there. On the other hand, at a point/, whose distance from O differs from its distance from O 1 by an odd 82 INTERFERENCE AND DIFFRACTION. 83 number of half wave lengths, the wave trains are continually opposite in phase and tend to annul each other. All the points q which satisfy the above condition lie on the dotted hyperbolas * (Fig. 477) of which O and O' are the foci. These dotted hyperbolas intersect the line OO' at equal intervals x (JX). One is a straight line bisecting OO'. All the points/ which satisfy the above condition lie on the full-line hyperbolas midway between the dotted hyperbolas. Along the dotted lines of Fig. 477 the disturbance due to the two sources O and O' is great, and along the full lines the dis- turbance is small or in some cases zero. If the light from O and O' falls on a screen AB, bright bands of illumination will be produced where AB intersects the dotted hyperbolas (hyper- boloids of revolution about OO' as an axis), and dark bands will be left along the intersections of AB with the full-line hyper- boloids. This phenomenon is called interference. The light and dark bands on the screen are called interference fringes ; two such sources as O and O' are called similar sources. 705. Displacement of fringes. If O and O' (Fig. 477) are two tuning forks not exactly in unison ; then as one of them, say O, gains on the other, the hyperboloids will all move downwards (in the figure), and when half a complete vibration has been gained, the / and q hyperbolas will have changed places. Thus at any stationary point strong and weak sound will alternate, producing what are called beats. All the methods for showing interference of light depend upon the production of identically similar sources, and the phenomenon of moving fringes, which are most striking in the case of sound, cannot be observed in that of light. If, however, a slip of thin glass G (Fig. 477) be placed between O' and the screen, the advancing wave train will be retarded in passing through the glass, thus falling behind its * The distances from a point on an hyperbola to the two foci have a constant difference. 8 4 ELEMENTS OF PHYSICS. former place, and the fringes will be displaced along the screen from A towards B. 706. Distribution of fringes over a screen. Consider two similar sources O, O' (Fig. 478). Fig. 478. Let 2 a be the distance between them, and b the distance of a screen AB. Let d and d' be the distances of O and O 1 x from a point p of the screen distant x from the point C. When a'-d = , (i) 2 and ^ is an even number, the ' point / is at the center of a is at the center of a dark bright band. When n is odd, band. Now d' d = O J m, and, since the triangles OO'm and .,,.., , O'm x sensibly similar, we have = -, or OO b (ii) n\ Writing for d' d and 2 a for OO 1 , and solving for x, we have nb\ (335) which expresses the distances x from the center of the screen to the various fringes. For light, X is very small and the ratio - must be very large if the successive fringes are to be far enough apart to be distinguishable. Equation (335) enables the calculation of X when x, a, and b have been observed. The number n may be easily counted, being two for the first bright band, four for the next, and so on. It is found in this way that for the extreme violet light of the spectrum the value of X is INTERFERENCE AND DIFFRACTION. 85 about 39-icr 6 centimeters, and for the extreme red light about 75.icr 6 centimeters. 707. Colored fringes. If the similar sources O, O' (Figs. 477 and 478) give off white light instead of homogeneous light, then a set of fringes will be produced by each simple wave train, and the effect on the screen AB will be the superposition of all the fringes thus produced. The central fringe will be white, for this is a bright fringe for every wave length. Passing out from this center, fringes take on a succession of colors, due to the extinction, one after another, of violet, blue, green, yellow, and red. 708. Arrangements for producing interference fringes. (a) Newton s arrangement. Newton who was among the first to observe interference phenomena, employed two narrow slits, close together, as described in Art. 714. \ \ \ \ \ x/ \ A V \ D \ S Fig. 479. (b) FresneVs mirrors. Light from a very small source S (Fig. 479) falls upon two mirrors M, M', as shown. After re- flection from M and M f , the direction of the rays is such that the light seems to come from the points O and O 1 ', and inter- ference fringes are produced on the screen AB. 86 ELEMENTS OF PHYSICS. (c) Lloyd's mirror. A screen AB (Fig. 480) is illuminated di- rectly by a small source O, also by the light from O which strikes the mirror and is reflected to the screen. The latter appears to have come from O', thus producing interference fringes on the screen. f i MIRROR (d} Sound fringes. Using a shrill Fig 480 whistle giving sound wave trains of about 2\ centimeters wave length, Stevens and Mayer have obtained marked interference effects. They used arrangements similar to Fresnel's mirrors and to Lloyd's mirror. The regions of intense disturbance were indicated by means of a sensitive flame. Such a flame is de- picted in Fig. 481. When undisturbed, the flame is long and narrow (b), and when disturbed, it changes to the shorter form (a) shown in the figure. 709. The colors of thin plates. The interference effects above described seldom or never occur to ordinary observation. Thin plates, on the other hand, present very striking interference phenomena, which are of common occurrence. The colors of soap films, and of films of oil on water, are familiar examples. The action of a thin film in producing interference Fig. 481. i n r n is briefly as follows : Consider a simple train of waves T (Fig. 482) of wave length A,, incident upon a thin transparent plate PP. A portion T 1 of this train is reflected from the surface A with change of phase (see Art. 623). The remainder of the train, passing on through the plate, reaches the second surface B, and is partly reflected (without change of phase). This portion, after passing back INTERFERENCE AND DIFFRACTION. 87 through the plate, emerges (in part) into the air, and travels as the train T 1 parallel to the train T. When the distance 2 a is an odd number of half wave lengths, then, since the reflection at the surface A is with change of phase, the two reflected trains T' and T" are in like phase, and give a resultant train of in- creased intensity. When the dis- tance 2 a is an even number of half wave lengths, the two reflected B trains T and T' are in opposite phase, and tend to annul each other. If the incident light T is non-homogeneous (white light), all those homogeneous com- ponents of the white light are greatly strengthened whose half wave lengths are contained in the distance 2 a an odd number of times, and those homogeneous components are greatly weak- ened whose half wave lengths are contained in the distance 2 a an even number of times. If the plate is very thin, i.e. if a is small, then, of the various visible homogeneous components of white light (X=/5 x icr 6 centimeters for red, to X = 39. io~ 6 centimeters for violet) only one or two will satisfy the above conditions, and the plate will appear brilliantly colored. If the plate is thick, however, then a great number of the components will satisfy the condition. In this case the strengthened components and the weakened components will be distributed more or less evenly throughout the spectrum, and the plate will not show perceptible color. If the reflected light T 1 + T" , in this latter case, is analyzed by the spectroscope, the spectrum will be crossed by numerous dark bands corresponding to the wave lengths which are weak- ened, showing that in case of thick plates the interference takes place, although no color is to be perceived by the eye. The value of a (Fig. 482) depends evidently upon the obliquity of the incident wave train, as well as upon the thickness of the plate. 88 ELEMENTS OF PHYSICS. 710. Newton's rings. The thin film of air between two glass plates laid together presents a fine show of interference colors. Newton made a very minute study of this effect, using the air film between a flat glass plate and the convex surface of a lens laid upon it. In this case the film increases in thickness from the point of contact outwards. With homogeneous light a succession of light and dark rings surrounds this point. With white light these rings are colored.* 711. Diffraction. The spreading of a wave disturbance into the region behind an obstacle is called diffraction. This action has been discussed in Chapter II. to the extent of determining the degree of approximation to which a wave disturbance may be considered to be propagated in straight lines, and the size an object must have in order that it may completely screen off a disturbance from a point behind it. We shall now consider what takes place in the region throughout which a wave dis- turbance does spread sensibly. The diffraction of plane waves past straight edges is the simplest case, and the discussion here given is limited to that case. The diffraction of spherical waves and diffraction past curved edges gives rise to phenomena which are essentially similar to the above. 712. Half-period zones with reference to a line. Let TT (Fig. 483) be a train of plane waves of wave length X approach- ing a screen 55. Let O be a line perpendicular to the paper along which the illumination produced by TT is to be con- sidered, with a view to the determination of the effect of an obstacle in the fixed plane AB, distant b from O. With O as an axis, describe circular cylinders (not shown in the diagram), one of which has a radius b, the next a radius b -f- -, the next a radius b + X, the next a radius b + 3 and so on. These 2 cylinders will cut the plane AB into bands or zones parallel * See Preston, Theory of Light, pp. 114-168. INTERFERENCE AND DIFFRACTION. 8 9 to O. Calling the middle zone No. I, the distance from P to the inner edge of the nth zone is ^/ nb\, as explained in Art. 629. The middle zone is the broadest ; the successive zones grow narrower above and below P and approach the limiting width 1 1 1 1 | s 1 ' 1 1 1 I 1 1 1 ^ 1 > 1 1 1 O B Fig. 483. For the purposejof the following discussion it is sufficient to know that the portion of the disturbance along O which is con- tributed by the nth zone is, on the whole, opposite in phase to that contributed by the n + I th zone, and that this contribution grows less and less as n increases. The illumination along O produced by the unobstructed train of waves is called normal illumination. 713. Diffraction past the straight edge of a large obstacle. (a) For points outside of the geometrical shadow. When the edge of the obstacle (which is parallel to O) is at P, as shown by the dotted line in Fig. 483, half of the middle zone is cut off, together with all the zones below P, and the illumination along O is one-quarter* normal. If the obstacle is moved downwards, the middle zone will gradually be uncovered and the illumination * The amplitude of the disturbance at O is one-half normal, and therefore (Art. 620) the intensity is one-quarter normal. 9 o ELEMENTS OF PHYSICS. along O will increase, reaching a maximum considerably greater than normal at about the time* when that zone is wholly uncovered. As the next zone (No. 2) becomes uncovered, the illumination at O falls off, because the contribution from this zone is nearly opposite in phase to the contribution from the middle zone, and reaches a minimum considerably less than normal at about the time when the second zone is wholly un- covered. As the third zone becomes uncovered, the illumina- tion along O reaches a maximum, and so on. The maxima and minima grow less and less pronounced as the obstacle moves down, and the illumination along O becomes sensibly constant and equal to normal when the obstacle has receded so far as to uncover as many as six or eight zones below P. (b) For points within the geometrical shadow. Let the un- obstructed portion of AB (Fig. 484) be broken up into half- r c i 1 i * i i i P I i . i -> OBSTACLE i i i Fig. 484. period zones beginning at the edge of the obstacle. In this case the band next the edge is more effective in producing disturb- ance at O than the next band, the effect of which at O is opposite in phase, and so on. There is, therefore, a slight resultant disturbance, or illumination, at O produced by all the bands above the edge of the obstacle. This resultant illumination has a quarter of its normal value when the edge is opposite O. * Strictly, when 0.87 of the lower half of the middle zone is uncovered. INTERFERENCE AND DIFFRACTION. As the obstacle moves upward towards A, the illumination at O falls off continuously. The variation of illumination at O, as the obstacle moves, is the same as the variation of illumination along 55 with a given position of the obstacle. The ordinates of the curve in figure 485 represent the intensities of illumination along a screen, as described above under (a) and (). GEOMETRICAL SHADOW. DISTANCE OF OBSTACLE 94 C. M. HORIZONTAL DIMENSIONS ARE AMPLIFIED 10 TIMES. Fig. 485. 714. Diffraction through a slit. Consider the illumination reaching the point O (Fig. 486) through the slit W. When the slit is far above or below P, the illumination at O is sensibly zero. As the slit moves up along AB, the half-wave zones in the slit become broader and the number of them in the slit grows less. When there is an even number of zones in the slit, they neutralize each others' action at O and the illumination is zero. When there is an odd number of zones in the slit, the ^2 ELEMENTS OF PHYSICS. effect of one of them is left outstanding and the illumination at O is a maximum. The intensities of these maxima increase rapidly as the slit approaches P. w Fig. 486. Remark. The light which passes through a slit, not more than half a wave length broad, passes out in all directions from the slit very much as from a row of luminous points. (See Fig. 487.) Two such slits near together act* as similar sources and B Fig. 487. produce interference fringes on a screen SS, as explained in Art. 704. The regions of maximum and minimum disturbance (the / and q regions described in Art. 704) produced by the sound * Provided the light T has come from a very small source. INTERFERENCE AND DIFFRACTION. 93 coming from a shrill whistle through two adjacent narrow slits in a wall have been traced by Mayer and Stevens, and by Rayleigh, who used the sensitive flame as an indicator, as ex- plained in Art. 708. 715. Diffraction past the edges of a narrow strip. Consider an obstacle OO' (Fig. 488). The illumination on the screen 1 1 1 S> 1 i OSSTACL: ft 1 1 1 0' 1 1 1 1 1 3 Fig. 488. outside the geometrical shadow varies according to a law which is nearly the same as that which applies when the obstacle is infinitely broad. Inside the geometrical shadow ab, the light comes from regions very near the edges O and O f , so that O and O 1 act as similar sources and produce interference fringes similar to those described in Art. 704. One of the few cases in which the phenomena of the diffrac- tion of light occur to ordinary observation is that of the shadows of wires and twigs cast upon frosted windows by a distant arc light. 716. Zone plates. A transparent plate having opaque bands painted upon it, so that when placed in the position of AB (Fig. 483) every alternate half-period zone is obscured, is called a zone plate. With such a plate the disturbance from each unobscured zone reaches O in the same phase, and the resul- 94 ELEMENTS OF PHYSICS. tant illumination at O is very intense. The action of a zone plate is shown by Fig. 489 in which AB is a plate on which the middle zone (No. i) and the odd zones above" and below are obscured as shown. Consider the wavelet starting out from the center of the second zone when a given wave of the train TT reaches AB, the wavelet which started out from the center of the fourth zone when the first preceding wave of TT reached AB, the Fig. 489. wavelet which started out from the center of the sixth zone when the second preceding wave of TT reached AB, and so on. These wavelets are tangent to the circle (cylinder) CC with its center at O, and consequently they work together to produce a wave front converging upon O, and the disturbance at O is very intense. 717. The diffraction grating consists of a set of a large num- ber of equidistant slits in an opaque screen. The distance between the slits is called the grating space. Let TT (Fig. 490) INTERFERENCE AND DIFFRACTION. 95 be an incident simple train of plane waves of wave length X, parallel to a diffraction grating AB, of which the grating space is d. Cylindrical wavelets pass out from the slits as the suc- cessive waves of the incident train reach AB. Consider the Fig. 490. nth wavelet from the slit S lt the 2 nth wavelet from the slit S 2 , the 3 nth wavelet from the slit 5 3 , and so on. These wave- lets are tangent to the plane CD, making an angle with AB such that . n n\ sin 6 = r . a (336) These wavelets work together to produce a plane wave CD. The (n i)th wavelet from slit S v the (2n i)th wavelet from slit 5 2 , the (3/2 i)th wavelet from slit S z , etc., work together to produce another plane wave parallel to CD, and at a distance X behind it, and so on. Therefore there is a train of plane waves of wave length X, parallel to CD, passing out from the grating. If the incident light TT is non-homogeneous, as, for example, sunlight, as many distinct wave trains will be pro- duced on either side of the normal to AB as there are simple wave trains in the incident light. If a lens L' be placed in the path of these, each diffracted wave train will be brought to focus at a distinct point O. 9 6 ELEMENTS OF PHYSICS. The points O, O r , O", etc., each illuminated by one homogene- ous component of the incident light, constitute what is called the diffraction spectrum. The two spectra, one on either side of the normal to AB, produced when n = I, are called spectra of the first order ; the two spectra produced when n = 2 are called spectra of the second order, etc. In general, all these spectra are produced simultaneously, and the group of points O, O 1 O", etc., which constitutes one spectrum, often overlaps upon the group which constitutes the spectrum of the next higher order. The complete action of the diffraction grating is very strik- ingly shown by the wire grating sometimes used in front of the object glass of an astronomical telescope. This wire grat- ing consists of a large number of parallel and equidistant wires stretched in a frame. The object glass then produces a central white image of the star. On either side of this are the groups of images constituting the spectra of the first order, beyond these the groups of images constituting the spectra of the second order, and so on. A grating which has a large number of accurately spaced slits produces very pure spectra, that is, each point O (Fig. 490) is illuminated by only one homogeneous component of the incident light. When the slits are not very narrow, compared with the grating space, then some of the spectra of higher orders are weakened. For example, when the width of the slits is -, all spectra of even orders are missing.* 718. Action of a diffraction grat- ing upon obliquely incident plane waves. Consider a train of plane Fi s- 491 - waves TT (Fig. 491), of wave * For a more complete discussion of the diffraction grating, see Preston's Light, pp. 186-201. INTERFERENCE AND DIFFRACTION. 97 length X, approaching a grating AB. Consider a certain wave W of the incident train. Let the wavelets which pass out from the various slits as this wave W reaches the slits be designated as wavelets No. i ; let the wavelets which passed out as the preceding wave passed the slits be designated as wavelets No. 2, etc. The wave W having just reached the slit 5 5 has yet a distance $x to go before reaching the fifth slit below 5 5 , that is, the slit S ; so that the wavelets which pass out from S 5 are $x ahead of the wavelets which pass out from the slit 5. Consider the nth wavelet from the slit S, the 2 nth wavelet from the slit S%, the 3 nth wavelet from the slit 5 3 , etc. These wavelets are tangent to CD (Fig. 491) ; therefore, since the distance SS 5 is 5 d, we have sin = ^ ', and since, y moreover, sin i = , we have a sin0 = ^ + sinf. (337) a 719. Reflection gratings and transmission gratings. Diffrac- tion gratings having small grating space are made by ruling equidistant lines on glass or speculum metal. The former is called a transmission grating; the action of such a grating is discussed in the foregoing articles. The latter is called a reflec- tion grating. When light is reflected from such a grating, it virtually comes through the spaces between the rulings, so that the action of the reflection grating is similar to the action of the transmission grating. 720. The grating spectrometer ; measurement of wave lengths. A reflection grating AB (Fig. 492) is mounted upon an arm R which turns about a pivot at the center of a divided circle by means of which the grating may be turned through any measured angle. The light to be analyzed is admitted through a narrow slit S, which is at the principal focus of a lens L. After passing this lens, the light becomes an incident train of plane waves TT. The various diffracted wave trains H 9 8 ELEMENTS OF PHYSICS. are concentrated at the points O, O", in the focal plane of the lens L' and viewed by the eye lens E. To determine the wave length of a given diffracted wave train, proceed as follows : Turn the arm R until the unscratched portions of the grating surface reflect light from the slit directly back to the slit again, and read the vernier V. Let this reading be r. Turn the grating until light from the cross wire O' is reflected directly back to the cross wire again, and read the vernier. Let this reading be r f . Then having turned the arm R until the light from vS is reflected into the telescope,* turn it until the desired diffracted train CD is concentrated at O' (the first time this occurs will be for the spectrum of the first order, the second time will be for the spectrum of the second order, etc. Having turned to the spectrum of the third order, say, the value of n, equation (337), will be three) and read the vernier a third time. Let this reading be r ff . Then the angle i is equal to rr", and the angle 6 is equal to r" r 1 . The wave length X of the train CD may be calcu- lated from equation (337) when the grating space is known. The following table gives the wave lengths for the principal Fraunhofer lines as measured by Bell : TABLE. Fig. 492. Line. A 7594.06 X I0~ 8 cm. B 6867.46 C 6563.06 Di {5896-15 D 2 15890.18 " E I {5 2 7-5 E 2 (.5269.72 " * The lenses L' and E constitute a telescope. Line. F 4861.49 G (4308.07 ' (.4307.90 HI 4101.85 H 2 3968.62 INTERFERENCE AND DIFFRACTION. 99 The most complete and accurate table of wave lengths is that published by H. A. Rowland in the Astro-Physical Journal (1895-96). Wave length units. It is often convenient to express wave lengths in terms of some small fraction of a centimeter. The micron (symbol /i) is one millionth of a meter or to io~ 4 cm. o o The Angstrom unit (symbol A) is io~ 8 cm. 721. The concave grating. Consider the distances a and b (Fig. 493) of two points 6* and S' from a point / upon a surface AB. As the point p moves along AB, the sum a + b changes continuously. If only those portions of the surface AB are polished (the intervening portions being roughened by scratching) for which p\ where n is an integer and X the wave length of homogeneous light coming from S, then all the wavelets from these polished portions will be in like phase at S', producing intense illumina- tion producing, in fact, an image of 5 at S'. Rowland has realized the condition expressed in equation (i) in his concave grating. AB (Fig. 493) is a speculum metal surface, with its Fig. 493. I Fig. 494. center of curvature at C, and is ruled with lines which are the intersections with AB of equidistant parallel planes. The dotted line is a circle upon Cp as a diameter. The grating AB forms as many distinct images of 5 along the dotted circle near I0 o ELEMENTS OF PHYSICS. S f as there are distinct homogeneous components in the light coming from 5. Any light to be analyzed may be sent through a narrow slit at S, and the group of images at S' may be observed direct or allowed to fall upon a sensitive plate and thus photographed. Figure 494 is a copy of a photograph, taken in this way, of a portion of the solar spectrum in the neighbor- hood of Fraunhofer's lines D l and D 2 . CHAPTER IX. COLOR. 722. Sensations of brightness and color. The special sensa- tions due to light are brightness and color. Brightness depends, for a given homogeneous light stimulus, upon the amplitude of vibration ; that is to say, upon the intensity of the stimulus. The various sensations of color are due to differences of wave length, when the light which produces the sensation is homo- geneous, or to differences in the composition when the light is mixed. 723. Luminosity of homogeneous light. The intensity of the sensation produced by a beam of light is called its luminosity or brightness. The luminosity of the various portions of the visible spectrum differs greatly. 80 GO 40 20 4500 BLUE 5000 5500 6000 6500 Fig. 495. 7000 RED If, for example, we hold a printed page in the different regions of the spectrum and vary the brightness of the source of light, inr IO 2 ELEMENTS OF PHYSICS. from which the spectrum is obtained, until it is just possible to read the text, we find the necessary brightness to be much greater in the red or violet than in the yellow or green. The luminosity of one region of the spectrum is to the luminosity of another region as the necessary brightness of the source in the second case is to the necessary brightness of the source in the first case. The curve given in Fig. 495 was obtained in this way. The ordinates of this curve represent the luminosities of the various portions of the spectrum of gaslight. Remark. The luminosity of the various regions of the spec- trum is not at all proportional to the energy intensity of those regions. The energy intensity of the spectrum of gaslight is a maximum in the infra-red, where the luminosity is of course zero, and falls off rapidly towards the violet. 724. Colors due to homogeneous light. The various wave lengths of homogeneous light produce a variety of color sen- sations. Newton recognized and named seven colors in the spectrum : red, orange, yellow, green, blue, indigo, and violet. About one hundred and fifty steps are made, however, in going through the spectrum from one tint to the next which can barely be distinguished from it. That is to say, there are about one hundred and fifty distinguishable tints in the spec- trum. Various compound names have come into use such as orange-yellow, bluish-green, violet-blue, etc., and these are applied to the regions between those named by Newton. 725. Colors due to mixed light. Definition of white light. Sunlight, or any light approaching it in composition, is called white light. The sensation produced by such light (aside from complications growing out of contrast effects) is called white. Mixed light produces, in general, deeper and deeper color sen- sations as it deviates more and more in composition from white light. Colored bodies occurring in nature owe their color to the fact that they send to the eye light of which the composi- COLOR. 103 tion differs more or less widely from the composition of white light. Color by selective radiation. Hot gases give off light which differs widely from white light in composition. Thus, most of the color effects in fireworks are produced by the use of salts of various metals, such as strontium and copper. These salts are vaporized and the hot vapors give off brilliantly colored light. Color effects by selective reflection and selective transmission. Many substances such as pigments and dyestuffs reflect or transmit in excess certain wave lengths of the light which falls upon them. Such substances appear colored when illuminated by white light. When such substances are illuminated by yel- lowish gaslight, their colors become less Brilliant, and when illu- minated by homogeneous light, as for example sodium light, the differences between their colors disappear entirely. Colors by interference. The light reflected from a thin plate ? a soap film for example, may have one or more of its homo- geneous components extin- guished and others strengthened by interference. Such light dif- fers widely from white light in composition, and produces bril- liant color sensations. The colors produced by diffraction are essentially interference ef- fects. The action of bodies in modi- fying the composition of light by reflection and transmission is discussed at length in Chapter XII. 5000 6000 Fig. 496. 7000 726. , The specification of the composition of light ; exam- ples. Choose any light, say gaslight, as a standard of com- 104 ELEMENTS OF PHYSICS. position. The intensity at each part of the spectrum of this light is to be taken as the unit intensity for that part of the spectrum. The composition of any other light is then easily specified by giving the intensity or brightness of each part of its spectrum in terms of the intensity of the same part of the POTASSIUM CHROMATE ULTRAMARINE BLUE 4000 VIOLET 5000 7000 RED 4000 VIOLET Fig. 497. 5000 Fig. 498. 7000 RED spectrum of the standard light, as determined by means of the spectrophotometer. The curves in Fig. 496 show the composi- tion of daylight, of the light from the glow lamp, and of lime light, each referred to gaslight. The curve in Fig. 497 shows the composition of gaslight after passing through a solution of potassium chromate (referred to gaslight direct). The curve in Fig. 498 shows the composition of gaslight reflected from ultramarine blue (referred to gaslight direct). 727. Color mixing. Dichroic and Trichroic Vision. Consider two (or more) beams of light of different composition. These beams give distinctly different sensations of color. If they are mixed (i.e. allowed to enter the eye and fall upon the same COLOR. I05 portion of the retina), the sensation is still that of a single defi- nite color. The various beams of light are said to blend. Any color may be matched by properly mixing a deep, or saturated, red light, a saturated green light, and a saturated violet light. To this end, the intensity of each colored light must be under control and varied tintil the mixture matches the given color. This is an experimental fact first pointed out by Thomas Young. For some persons (red blind) any color may be matched by properly mixing a saturated green light and a saturated violet light. For other persons (green blind) any color may be matched by properly mixing a saturated red light and a saturated violet light. Persons for whom any color may be matched by mixtures of two saturated colors are said to possess dichroic vision. Normal vision, or the vision of the majority, is called trichroic vision. Dichroic vision is sometimes called color blindness. The color sense of a person having dichroic vision is strikingly different from the color sense of a person having trichroic vision. (See Arts. 729 and 730.) About four per cent of the male population and about four out of every thousand of the female population of the civilized world possess dichroic vision. The color top. The most convenient arrangement for mixing colored light is by means of the color top. This consists of a rotating spindle which carries three colored circular disks (red, green, and blue) slitted in such a way that any desired sector of the face of each disk may be exposed to view. When this triple disk is rotated rapidly, the colors blend and give a single sensa- tion of color, the tint of which may be modified at will by varying the amounts of exposure of the respective disks. 728. The Young-Helmholtz theory of color. The experimen- tal fact stated in the previous article led Thomas Young (1801) to infer the existence of three primary color sensations. Helmholtz attributed each of these primary sensations to a I0 6 ELEMENTS OF PHYSICS. distinct set of nerves in the retina of the eye. The nerves which upon excitation give the primary sensation of red are called the red nerves ; those which upon excitation give the primary sensation of green are called the green nerves ; and the set which upon excitation gives the primary sensation of violet, the violet nerves. Simultaneous excitation of all three sets of nerves gives a blended sensation, the character of which depends upon the degrees of excitation of the respective sets of nerves. The number of distinct color sensations produced by the action in varying proportions of the countless homogeneous rays of which the visible spectrum is composed is very large (according to Titchener,* about 30,000, and according to Rood f a much larger number). A person having dichroic vision has only two primary color sensations and only two sets of color nerves. Persons who do not have the primary sensation of red are said to be red blind, persons who do not have the primary sensation of green are said to be green blind. The Young-Helmholtz theory of color is not the only theory in vogue ; indeed, physiologists are inclined to reject it for lack of microscopical evidence of the existence of the three sets of nerves; and psychologists are inclined to reject it, mainly, because of the difficulty of explaining the great number of distinguishable qualities of color sensation by the varying intensity or quantity of sensation on three sets of nerves. The theory however gives a very clear representation of the experi- mental facts stated in the previous article, and a very satis- factory explanation of contrast effects and of color blindness. Intensity of action of various wave lengths upon the color nerves. According to the Young-Helmholtz theory the message im- parted to the brain by each set of nerves is entirely indepen- dent of the nature of the stimulus. The red nerves, however, * Titchener, An Outline of Psychology, p. 66. t Rood, Modern Chromatics, Chapter IX. COLOR. ID/ are most strongly affected by the wave lengths at the red end of the spectrum, the green nerves by those in the middle of the spectrum, and the violet nerves by the shortest visible waves. It has been found possible to isolate these primary color sen- sations and to determine the intensity of each for each wave length of the spectrum. The curves in Fig. 499 show the HG FE D^CB t S o d S > m o >- o: Fig. 499. result of such measurements made by Koenig. It will be seen that all the rays of the spectrum are capable of producing in some degree the sensation of red, and that the greater portion of them have some effect likewise on the green nerves. Those which affect the violet nerves appreciably lie between the mid- dle of the spectrum and the violet end. Contrast effects. When two colors are viewed in succession the character of the second color is more or less changed. This action is called a contrast effect. The set of nerves most affected by the first color seems to become more or less inactive through fatigue, which results in the preponderance * of the other two primary sensations when the second color is viewed. The fol- lowing example will make this clear. * A color is said to be saturated when one of the primary sensations greatly pre- ponderates over the other two. The mixture of white light with a colored light reduces the saturation by bringing all three sets of nerves into activity. Any process which tends to isolate a single primary sensation increases the saturation. I0 8 ELEMENTS OF PHYSICS. A homogeneous beam of light of a wave length, for example, corresponding to the Fraunhofer line E, will stimulate all three sets of color nerves. The resultant sensation will be composed of a powerful primary sensation of green, a weak primary sensa- tion of red, and a still weaker primary sensation of violet. The resultant sensation of green thus produced is less intense than it would be were the red nerves absent. To make the green more vivid, it is only necessary to reduce the activity of the red nerves. This may be done, for example, by wearing spectacles of ruby glass for several minutes ; during which time the eyes are exposed to bright light. Upon removing these spectacles the green nerves, which have been protected by the opacity of the ruby glass to the rays which most strongly affect them, will remain active, while the red nerves will be greatly fatigued. The result is a greatly increased vivacity and intensity of all sensa- tions of green, with a relative loss of the sensation for red. Extreme fatigue of one set of color nerves produces tempo- rary partial color blindness. Similar effects can be produced by the use of certain drugs, one of which, santonine, renders the violet nerve temporarily inactive. The excessive use of tobacco sometimes produces partial annihilation of the activities of one or more of these color nerves. 729. Peculiarities of dichroic vision. Color-blind persons show marked peculiarities in their classification of colors. These peculiarities are taken advantage of in testing for color blind- ness. They are described in Art. 730. The appearance of the spectrum to a color-blind person is very different from its appearance to a person with normal vision. A color-blind person has two primary color sensations. If these be red and violet, which is the case in green blindness, the spectrum is made up of these two sensations. The violet end appears very much as it would to the normal eye, but of a higher saturation. The red end is also more intense in color, but it merges into white instead of merging into yellow and COLOR. 109 green. The center of the spectrum between the E and F lines appears to such individuals of a neutral shade. To red-blind individuals the spectrum is also made up of two tints ; violet at the violet end and green in the place of red at the red end. The center of the spectrum, to them, is colorless. The red end of the spectrum, which appears most G F E D C B GREEN BLIND WHITE Fig. 500. intense to green-blind persons and to persons with trichroic vision, is invisible to the red blind ; so that the limit of visibility at this end of the spectrum is shifted towards the shorter wave In Fig. 500 the arrangement of colors in the spec- lengths. 40 20 5000 6000 Fig. 501. 7000 RED trum as viewed by normal and by color-blind individuals is indicated. One method of studying color blindness is by means of the curve of luminosity. Figure 501 gives such a curve, from IIO ELEMENTS OF PHYSICS. measurements by Ferry,* together with the curve for a normal eye. The observer was partially red blind. It will be seen, from the figure, that the luminosity of red and yellow light was below the normal, and that in the green the curve joins that for the normal eye. In the blue and violet the luminosity was normal. 730. Testing for color blindness. Since all our systems of signaling, both upon railways and at sea, depend upon the recognition of colored lights, the character of the color vision of employees is a matter of the utmost practical importance. It is fortunately possible to detect dichroic vision with cer- tainty even where the existence of the peculiarity is unsus- pected by the subject himself. The simplest satisfactory method of performing such tests was invented by Holmgren of Upsala, after the occurrence of a terrible railway accident due to the color blindness of an employee. The Holmgren appara- tus consists simply of a collection of colored worsteds, which includes a variety of reds, greens, and purples, together with tints made up of an admixture of these and of a large number of neutral grays of various degrees of brightness. There are also two skeins, called the confusion samples. One of these is a pale apple green : its color corresponds most nearly to the central portion of the spectrum, which in the case of both red- and green-blind persons constitutes the neutral region described in Art. 429. Both red- and green-blind subjects inevitably select, as closely corresponding to this sample, a variety of neutral worsteds. The other confusion sample is a light magenta, which is an admixture, in nearly equal parts, of red and violet, with much white. Red-blind persons when asked to pick out those worsteds which agree most nearly in color with this sample, invariably select violets and blues. The selections made by persons of dichroic vision, when attempting to follow their own judgment * American Journal of Science, Vol. 44 (1892). COLOR. r ! j in matching these confusion samples in the Holmgren test, are very striking to an observer with normal vision. A university student, who was red blind, but who showed great power of discrimination in his choice of colors, as viewed from his own standard, and who was fully aware of the nature of his color sense, made the following selections of the worsteds which seemed to him most nearly related to the two confusion samples. Of the twelve skeins of worsted selected to go with the apple-green sample, only one contained any green pigment. The remainder were very pale yellows and browns, almost with- out coloring matter, and two skeins of very pale rose color. All the worsteds selected on account of their resemblance to the magenta sample were purples in which blue predominated or nearly pure blues. With the scarlet were placed a few other nearly pure reds, together with a much larger number of dark browns and greens. Surprising as these selections seem to one accustomed to the trichroic color system, they are all readily accounted for under the Young-Helmholtz theory of colors. CHAPTER X. PHOTOMETRY. 731. Definitions. A luminous object is one which shines in the dark. The light given off by a luminous body when it strikes surrounding bodies is reflected by them to the eye, and they become visible. These bodies are said to be illuminated. A transparent body is one which, like glass, permits light to pass through it. A body which, like celluloid or horn, appears milky by transmitted light is said to be translucent. An opaque body is one which does not permit light to pass through it. Extremely thin layers of almost all opaque substances, such as the metals, are transparent. 732. The law of inverse squares; brightness. Imagine a spherical surface of radius d described about a luminous body at its center. Light from the luminous body is distributed over this spherical surface. The intensity of illumination of the surface, or of any part of it, is therefore inversely propor- tional to its area, or inversely proportional to the square of its radius. We have, therefore, /=. (338) in which / is the intensity of illumination at a place distant d from the luminous body, and B is the proportionality factor. This quantity B is called the brightness of the luminous body. 733. Standards of brightness, or light standards. In all the photometric operations to be described in this chapter, artificial 112 PHOTOMETRY. light sources such as gas flames, incandescent lamps, etc. are compared with some source of light which has been taken as a standard. The earliest standard of brightness was the candle ; and this standard, although it has been shown to be one of the least constant of artificial sources of light, is still recognized as the official standard in Great Britain, in Germany, and in the United States. The British standard candle is a sperm candle, weighing six to the pound, and burning 120 grains per hour. Many laborious investigations of the behavior of this standard have been made, and it has been shown that it fluctuates through a very great range (more than 20 %). The brightness of the candle depends upon the length and shape of the wick, the height of the flame, and even upon the temperature and humidity of the air of the photometer room. The German standard candle is made of paraffin. It has a uniform diameter of two centimeters. The wick of the German candle is trimmed until the height of the flame is precisely five centimeters, when measurements are taken. Investigations of this standard have shown that even with these precautions it is subject to very considerable fluctuation. I- E35 30 10m. 20m. TIME Fig. 502. The British candle. By means of a bolometer (see Chapter XII.) and a sensitive galvanometer, it is possible to follow the continual fluctuations of a candle flame from moment to moment, and to draw a curve showing graphically the nature of these variations. Ex- ELEMENTS OF PHYSICS. tensive measurements of this kind have been made by Sharp and Turnbull.* Figure 502 gives a portion of such a curve plotted from measurements upon a British standard candle. The interval included in the diagram is half an hour. It will be seen that there are numerous and very sudden diminutions in intensity. Various other flames and light sources have been used as standards. One of the earliest of these was the Carcel lamp, which was invented in France. It is a lamp with a cylindrical Argand burner and central draught. The draught is maintained by clockwork. The fuel used in this lamp is colza oil. Another standard which has been extensively used is the Methven screen. In this apparatus an ordinary argand gas burner is provided with an opaque screen (Fig. 503), through which a rectan- Fig. 503. gular opening is cut of such size as to reduce the flame to two candle power. The opening transmits light only from the central and brightest portions of the gas flame. The variations of such a standard are those due to the varying qualities of the illuminating gas, and to the varying pressures under which it is consumed. 10 m. 30m. TIME Fig. 504. The Methven standard. Figure 504 shows the curve obtained with this lamp. It will be seen that the flame was subject to many large and rapid variations. * Physical Review, Vol. II., p. I. V^ftML^ PHOTOMETRY. In Germany the candle has been largely supplanted as a standard by the Hefner-Alteneck lamp. This is a simple metal lamp, .burning amyl acetate. It is arranged so that the height of the flame can be accurately adjusted and measured. It has been found that this standard is reliable to within two per cent, which is a better result than has been obtained, thus far, with other flames. Figure 505 shows a typical bolometric 45 35 10 m. 20 m. TIME 30 m. Fig. 505. The Hefner lamp. curve for the Hefner lamp. During the half hour included in this test, the brightness of the flame was very nearly constant. The Violle standard. Violle has proposed, as a standard of brightness, the light emitted by a square centimeter of surface of platinum at its melting point. It has been found imprac- ticable to put this standard into general use. The glow lamp. The glow lamp, when supplied with con- stant current, is free from the difficulties which interfere with the accuracy of flame standards. It has been widely adopted in photometric work as a comparison standard, but thus far it has been found impossible to produce glow lamps which do not change in brightness with age. It has not been found practi- cable to so specify the character of the parts of a glow lamp that one could be manufactured which, when provided with a given current, would give the specified candle power. 734. Intensity of illumination; intrinsic brightness. When B (equation 338) is one candle and d is one meter, then / is I ( - - ). This intensity of illumination, namely the illumi- Vmeter 2 / H6 ELEMENTS OF PHYSICS. nation by a standard candle at a distance of one meter, is occa- sionally used as a unit for expressing intensity of illumination. Thus the least intensity of illumination required for comforta- candles (meter)' ble reading is about 4 - - (read four candles-per-meter- square^) The intrinsic brightness of a luminous body is defined as its total brightness, B, divided by the area of its shining sur- face. Thus the intrinsic brightness of a candle flame is about o.i es ; of an Argand gas flame it is about one candle per cm. square centimeter; of a glow lamp it is from 10 to 20 can 2 es > and for the crater of the arc lamp it is as much as 20,000 ^^ ^- cm. 2 735. Bouguer's principle. It was first shown by Bougtier (1726) that the relative brightness of two lamps might be determined by measuring the distances at which those lamps would give equal intensities of illumination upon a screen. This principle is the basis of most practical photometric methods. Consider two lamps. Let B be the brightness of one, and d the distance from a screen at which it gives an illu- mination of intensity /. Then from equation 338 we have Let B 1 be the brightness of the other lamp, and d' the dis- tance from a screen at which it gives the same intensity of illu- mination ; then From these equations we have, upon eliminating /, B' = ^B. (339) 736. Limitations of simple photometry. The photometric comparison of lamps which give light of identically the same PHOTOMETRY. II7 composition is called simple photometry. Lights which differ but slightly in composition, or tint, cannot be compared with any accuracy by the methods of simple photometry, and when the difference in tint is distinctly appreciable, the comparison cannot be made at all. The relative brightness of two lamps of different tint may be estimated with some accuracy by a method devised by Crova, and by a device called the flicker photometer. The complete comparison of lights of different composition can be made only with the help of the spectrophotometer. 737. The shadow photometer was devised by Bouguer, and was used later in a modified form by Lambert, who wrote an extended treatise on photometry * in 1760. It was also used by Rumford, and is often called Rumford 's Photometer. The shadow photom- eter is an arrangement in which the two sources of light to be compared cast overlapping shadows upon a screen, as shown by the overlapping squares, A and B y in Fig. 506. The portion C of the screen which is common to both shadows is illuminated by neither source, while the remaining portions of A and B are illuminated each by a single source. The lamps are moved to such distances from the screen as to bring the contiguous portions of A and B, namely the portions d and e, to equal bright- ness. The distances of the lamps from the screen are measured, and the brightness of one lamp in terms of the other is then given by equation 339. 738. The Bunsen photometer. This instrument, which has been more widely used in practical photometry than any other, depends upon Bouguer's principle and is peculiar in the method employed in judging the equality of illumination produced upon the screen by the two lamps to be compared. The screen is * Photometria sive de mensura et gradibus luminis, colorum et umbrae. Aug. Vind. 1760. U8 ELEMENTS OF PHYSICS. made of unsized paper thick enough to be opaque. All but a central spot is made translucent by soaking with oil or paraf- fin. If this screen be observed by reflected light, the unoiled portion will appear bright and the oiled portion will appear dark. A considerable proportion of the light striking the oiled surface penetrates the same and is transmitted, and when the screen is viewed by transmitted light it presents the precisely opposite appearance ; the central spot is dark upon a bright background. The appearance of the Bunsen screen under these conditions is illustrated in Fig. 507. When the two sides of such a screen are equally illu- minated, the central spot appears of the same brightness as the surround- ing oiled surface. The effect of the oil is to vary slightly the reflecting power of the surface ; so that complete identity of appearance is never secured : the spot of light is, therefore, not absolutely lost to view even when the illumina- tion from the two sides is the same. Under such conditions, however, the two faces of the paper are identical in appear- ance. In practice the screen of the Bunsen photometer is placed between two oblique mirrors, within a box or carriage which slides or rolls upon a track, at the ends of which are situated the light sources to be compared. The track is called the photometer bar. It is provided with a scale, usually of 1000 equal parts. The carriage is moved along the bar until equal illumination upon the two sides of the screen has been secured, when the distances d^ and d^ between the disk and the sources of light are read upon the scale. Equation 339 then gives the brightness of one lamp in terms of the other. The object of the two mirrors between which the screen is mounted, as shown in Fig. 508, is to enable the observer to see the images of both sides of the screen simultaneously. The observation consists in so placing the carriage that the two images shall be identical in appearance. PHOTOMETRY. 119 The accurate use of the Bunsen photometer supposes identity in the color of the two light sources to be compared. As in the case of the shadow photometer, and of all instruments based upon the principle of the equality of two fields of view, differences in the color of the light reflected from the two surfaces of the screen, even when very slight, vitiate the judgment as to their relative brightness. Fig. 508. 739. The Lummer-Brodhun photometer. This instrument is a modification of the Bunsen photometer, which has been found to possess one great advantage. In the attempt to observe the two images of the Bunsen disk simultaneously all experimenters with that instrument acquire the habit of observing with the two eyes independently. They view the right-hand image with the right eye, and the left-hand image with the left eye. Owing to differences in the sensitiveness of the two eyes, the images appear to be equally bright when they are not so in reality. The result is a false setting of the pho- s, tometer, which is persistent and of nearly constant value with each individual. It is found that a very large proportion of observers set the disk to the left of its true position ; as though the right eye were the more sensitive organ. A few observers have the opposite tendency. Observations are made with one eye in the Lummer- Brodhun photometer, and this personal error is avoided. In this instrument light from the two sources S 1 S 2 , Fig. 509. falls upon the two faces of an opaque screen AB, which is whitened with magnesium oxide. 120 ELEMENTS OF PHYSICS. FROM S 2 FROM S 2 LIGHT FROM S, LIGHT FROM 8 2 The observer at the small telescope sees portions of the two sides of this opaque screen side by side in the same field of view. The cube of glass CD is made of two right-angled prisms, the diagonal face of one of which is partly cut away, as shown in Fig. 510. These prisms are then cemented together with Canada balsam so that light from the mirror M 2 passes through the cemented portion unobstructed, while it is totally reflected else- where. The light from the mirror FROM 8l j^ is tota iiy reflected from the " Sz free portions of the diagonal plane Fig - 510< and passes through the cemented portion unobstructed. The cemented portion is commonly circular, and the appearance of the field of view of the tele- scope is shown in Fig. 511. The whole arrangement is moved along the photometer bar, as in the case of Bunsen's photometer, until the field of view of the telescope is uni- formly illuminated. The distances of Fig - 511> the screen AB from the sources S l and S% are then observed, and the brightness of one source in terms of the brightness of the other is given by equation 339. The Lummer-Brodhun arrangement is slightly more sensi- tive than that of Bunsen. 740. Distribution of brightness. Light from a single point travels outward in spherical waves, and the intensity of illumi- nation at a given distance from the point is in all directions the same. In the case of the light from the various flames used in artificial illumination, from the filaments of glow lamps, and from the carbons of the arc light, the distribution is not uniform. The study of the distribution of brightness is made by turn- PHOTOMETRY. 121 ing the flame about into various positions and measuring its brightness. The distribution of light in a given plane is then represented by means of a polar diagram in which the radius vector gives the brightness at all angles. If the flat flame of a bat's-wing gas burner be thus measured in a horizontal plane, it will be found that the distribution is uniform, and that the curve is a circle. The fact that such a flame is as bright when viewed edgewise as when its entire breadth is exposed, shows that the light comes from a comparatively small number of isolated particles of glowing carbon, which are so few and far between that they do not, to any considerable extent, screen each other. If the flame from a richer fuel such as petroleum be tested in like manner, it will be found that the flame is not so fully transparent. Figure 512 shows the curve of horizontal distribution in the case of the flame of an ordinary petroleum lamp. The falling off of brightness in the plane of the flame shows that there is a considerable screening action. The length of the radius vector of the curve in any direction repre- sents the candle power of the lamp in that direction. The distribution of bright- ness in the case of the arc lamp is very far from uniform. In commercial direct-current arc lamps the current is always sent through the lamp in such a direction that the crater is formed at the end of the upper carbon. The vertical distribu- , . , . . .. . Fig. 513. Fig. 514. tion of brightness in this case is shown by the curve in Fig. 513. The vertical distribution of brightness in case of the alternating current arc is shown by the curve in Fig. 514. The carbons in an alternating cur- rent arc are equally heated. Fig. 512. 122 ELEMENTS OF PHYSICS. The distribution of brightness of an arc lamp varies in a rapid and irregular manner on account of the shifting of the arc. Figures 513 and 514 show the average distribution of light ; the momentary distribution may be very different. 741. The photometry of lights differing in composition. The rigorous method for comparing lights of different compo- sitions is by means of the spectrophotometer, which has been described in Art. 703. In addition to the method, there explained, for determining the brightness of each part of the spectrum of the light which is being studied, in terms of the brightness of the same part of the spectrum of a standard light, the following methods are frequently employed. (a) Vierordt's slit. This is the earliest device used in spectro- photometry. The two ends of the slit of the spectrometer are arranged to be opened or closed independently of each other by means of micrometer screws. The instrument is so arranged that the beams of light from the sources to be compared pass through these halves of the slit. The spectra are brought to equality at a given region by opening that half of the slit through which the weaker beam enters. The relative bright- ness of the sources for the given region of the spectrum is known from the relative widths of the slit halves. This is an exceedingly simple and convenient method, but its usefulness is limited to cases where the ratio of intensities is not large. Such a slit cannot be made very wide without affecting the purity of the colors of the spectrum by overlapping of the images. As soon as this change in the quality of the spec- trum begins to be noticeable, the limit of usefulness of the apparatus has been reached. (b) The method of the Nicol prisms. In this, which is the method most frequently employed, a pair of Nicol prisms are mounted, as in the polariscope (Art. 751), and are placed in the path of the brighter of the two beams of light. This beam enters one end of the slit of the spectrophotometer. One of PHOTOMETRY. I23 the Nicol prisms is turned until the spectra are of equal inten- sity at a given region. The angle between the principal planes of the prisms is then observed, and the relative brightness of the sources for the given spectral region is calculated as ex- plained in Art. 750. 742. Approximate methods. The following approximate methods do not give exact values. They consist in the employ- ment of some simple device for overcoming more or less completely the difficulties arising from the difference in the color of the lights to be compared. (a) Crovas method. When incandescent carbon, which is the glowing material in nearly all artificial illuminants, rises in temperature, all the wave lengths of the spectrum increase in brightness simultaneously. The rate of increase is small- est in the red, and becomes continually greater as the wave length diminishes. The result is a gradual change in the composition of the light which the glowing carbon emits. Since the rate of increase of intensity varies continuously from red to violet, there must, in every case, be some par- ticular wave length the change of which is the same as the change in brightness of the light taken as a whole. Having determined the portion of the spectrum for which this is true, we may confine our attention entirely to that. From the rela- tive brightness of that particular region we thus obtain the relative brightness of the sources to be compared. This method was suggested by Crova, who found the proper wave length for such measurements to be in the yellow (5800 Angstrom units). Crova's method consists in the use of a spectrophotometer, by means of which the spectra of the two sources are brought to equal brightness in the yellow. It is applicable only in cases where the law of radiation of the material in the two sources of light is the same, and where the difference in temperature is not very great. In more extreme cases it affords only a rough approximation to the true photometric ratio. 124 ELEMENTS OF PHYSICS. (b) The flicker photometer is an arrangement by means of which the two sides of a photometer screen are brought into the same field of view in rapid succession. If the frequency of interchange is very rapid, the illumination will appear uni- form whatever the relative brightness of the two sides of the screen may be; but if the frequency is only moderately rapid, the flickering will cease only when the light shining upon the two faces of the screen brings them to the same luminosity. This fact was discovered by Rood. The device employed by Whitman,* who has based a successful photometric method upon it, is as follows : T T Fig. 515. Fig. 516. A white cardboard disk, A, Figs. 515 and 516, is mounted upon the axis //, and a stationary piece of the same cardboard is placed at B. When the disk A is set in rapid rotation, the cardboard B and the wing of the disk are seen in rapid suc- cession by the eye placed at e. The wing of the disk is il- luminated by the source S v and the stationary cardboard is illuminated by the source 5 2 . The distances of the sources are adjusted until the "sense of flickering" disappears. * F. P. Whitman, Physical Review, Vol. III., p. 241. CHAPTER XI. POLARIZATION AND DOUBLE REFRACTION. 743. The side aspect of transverse waves. Consider a stretched cord, for example a violin string. Such a string set vibrating longitudinally by rubbing it lengthwise presents iden- tically the same appearance from whatever side it is viewed. If the string passes loosely through a slit in a card, the longi- tudinal waves (vibrations) are affected, if at all, in a similar man- ner whatever the direction of the slit. The transverse waves, on the other hand, such as are produced by the action of a violin bow drawn across the string, pass the card freely if the slit is parallel to the direction in which the particles vibrate, but they are stopped by the card if the slit is perpendicular to the vibra- tions. The transverse vibrations of the string may be such that each point of the string describes a circle, the plane of which is perpendicular to the string, or they may change rapidly and irregularly from one direction to another. In either case the action of the slit, whatever its direction, is the same, so far as the intensity of the transmitted waves is concerned. Trans- verse waves, whatever the character of the vibration, which pass through a slit in one card, would be entirely stopped by a second card, the slit in which is held at right angles to the slit in the first. Transverse waves in which the vibration takes place continu- ously in the same direction, that is in the same plane, are said to \$, plane polarized or svcw^ky polarized. When the vibrations are irregular, but those in a certain direction predominate, the I2 5 126 ELEMENTS OF PHYSICS. waves are said to \>e partially polarized. When the particles of the medium describe circular or elliptical paths, the waves are said to be respectively circularly or elliptically polarized. 744. The optical behavior of tourmaline crystals ; polarized light. A plate of this mineral cut parallel to the axis of the crystal transmits only a portion of the light which falls upon it. The light which passes through such a plate passes freely through a similar plate when the axes of the plates are parallel, and is shut off entirely when the axes of the plates are at right angles. The beam of light transmitted by the first plate is polarized, as is evident from the side properties which it exhibits with respect to the second plate. When the second plate is slowly turned about an axis parallel to the beam of light, the intensity of the transmitted beam changes slowly from maximum in- tensity when the axes of the plates are parallel to zero intensity when the axes are crossed. This is shown by the shading in Fig. 517. The fact that light can be polarized shows that light waves are transverse waves. 745. Polarization of light by reflection. Light which is re- flected from a polished non-metallic surface is polarized. In general such a reflected beam of light is only partially polar- ized (i.e. it cannot be completely shut off by a plate of tourmaline). The de- gree of polarization varies with the angle of incidence. At normal incidence the reflected beam is not at all polarized. As the incidence becomes more oblique, the degree of polarization increases, POLARIZE BEAM reaches a maximum when the reflected and refracted beams are at right angles Flg< 518- (Brewster), and then decreases. The angle of incidence for which the degree of polarization of the POLARIZATION AND DOUBLE REFRACTION. 127 reflected beam is a maximum is called the polarizing angle. Its tangent is equal to the refractive index of the reflecting substance. For glass the polarizing angle is about 57 ; for pure water it is 53 11'. Substances of which the refractive index is about 1.46 give complete polarization by reflection at the polarizing angle. Figure 518 shows a glass plate ar- ranged to give a beam of completely polarized light by reflec- tion. 746. Plane of polarization. That plane passing through the reflected beam which is perpendicular to the reflecting surface is called the plane of polarization of the reflected beam. The vibrations of plane polarized light must be either parallel to or perpendicular to this plane of polarization thus conventionally defined. According to the theory of Fresnel, the vibrations of the medium are perpendicular to this plane. According to the electro-magnetic theory of light, the electric force is perpendicu- lar to and the magnetic force is parallel to the plane of polariza- tion. The character of plane polarized electro-magnetic waves is described in Art. 603 (Vol. II.). 747. Reflection of polarized light from a polished surface. Consider a beam of polarized light incident at the polarizing Fig. 519. angle upon a glass plate. If the glass plate is turned about the incident beam as an axis, keeping the angle of incidence 128 ELEMENTS OF PHYSICS. il constant, the amount of light which is reflected varies from a maximum, when the plane of polarization of the incident beam is perpendicular to the surface of the glass, to zero when the glass plate is turned one quarter of a revolution from this position. Figure 519 shows the two positions of a pair of glass plates, A and B, for which the polarized beam reflected from the one is most completely reflected by the other. If either plate is turned one quarter of a revolution about the ver- tical line AB, then no portion of the polarized beam is reflected from the plate B. 748. Double refraction. Many crystalline substances have the property of dividing a beam of homogeneous light into two beams by refraction. This phenomenon is called double refraction. All allo- tropic substances are doubly refracting. The crystalline mineral, Iceland spar (calcium carbonate), separates the two refracted beams widely, and therefore shows the effect very distinctly. Consider first a plate of glass, AB (Fig. 520). A beam of light from e, falling upon the glass, reaches the point /, and if / is a luminous point, it will be seen by an eye held at e, as though it were at q. If the plate is turned about the line / as an axis, while the point / is stationary, then the /' point q will remain stationary. A beam of light R (Fig. 521), falling upon a plate AB of Ice- land spar, is broken up into two rays, and a point / sends out two rays o and x, parallel to O and X, respectively, which enter an eye at e, so that the point / is seen as two points q and g'. If the plate AB is rotated about the line / as an axis, one of the images Fig. 520. POLARIZATION AND DOUBLE REFRACTION. 129 q remains stationary, just as if AB were a plate of glass, and the other q' moves round it. The ray o (or O) in the crystal corresponding to the stationary image q is called the ordinary ray, inasmuch as it is refracted in the ordinary way (as in glass) ; and the ray x (or X) in the crystal corresponding to the moving image q' is called the extraordinary ray, inasmuch as it is not refracted in the ordinary way. Some crystals (biaxial crystals) divide a beam of common light into two beams, neither of which follows the ordinary law of refraction. The rays r and r' (Fig. 521) are completely polarized, and their planes of polarization are at right angles. This may be shown by holding a tourmaline plate (or Nicol prism) before the eye at e. As the tourmaline plate is turned, one and then the other of the images q and q' becomes invisible. 749. Huygens' theory of double refraction. The phenomena of double refraction in Iceland spar were fully analyzed by Huygens. He assumed two secondary wavelets to pass out from each point of the surface of a plate of Iceland spar as an incident wave reached that point ; one of these wavelets being a sphere and the other an ellipsoid of revolution. The en- C [ ( P ] \ D velope of the spherical wavelets deter- mines the ordinary refracted wave, as explained in Art. 635, and the envelope of the ellipsoidal wavelets determines Fig. 522. the extraordinary refracted wave. Let/ (Fig. 522) be a center of disturbance in Iceland spar; for example, a point of a wave from which secondary wavelets pass out. One wavelet is a sphere. It travels out from p at a velocity - - as great as the velocity of light in air. The 1.658 other wavelet is an oblate spheroid which touches the sphere at A and B, and of which the major diameter CD is ^ times 1.486 130 ELEMENTS OF PHYSICS. as great as the diameter of the sphere. The axis AB of the spheroid is parallel to the axis of symmetry of the crystal. The axis of symmetry of a crystal of Iceland spar is called its optic axis. Any plane which includes the optic axis is called a principal plane. The vibrations (electric force) in the spheroidal wavelet are everywhere in the principal planes. The vibrations in the spherical wavelet are everywhere perpendicular to the principal planes. Figure 523 shows Huygens' construction for the ordinary and extraordinary waves in Iceland spar. Let ww be the position which an incident wave would reach at a given instant in a homogeneous medium. Con- sider the wavelets which ema- nated from the point / as the wave passed that point. Let the distance from / to ww be R, then the radius of the spher- r> Fig. 523. ical wavelet from / is - , r> and the major semi-diameter of the spheroidal wavelet is 1.486 The axis of the spheroidal wavelet, that is, the optic axis of the crystal, is supposed to be given. The line o represents the ordinary wave, and r the ordinary ray. The line x represents the extraordinary wave, and r' the extraordinary ray. It is to be noticed that / is not perpendicu- lar to x. In fact, the extraordinary wave does not progress in a direction at "right angles to its front. When the ordinary and extraordinary rays coincide with the optic axis, there is no double refraction. Crystals like Iceland spar, which have only one direction in which there is no double refraction, are said to be uniaxial crystals. Many crystals have two such directions or two optic axes. Such crystals are said to be biaxial. POLARIZATION AND DOUBLE REFRACTION. 750. The Nicol prism. A beam of completely polarized light may be easily obtained by reflection from a glass plate, as explained in Art. 745. A much more convenient arrange- ment, however, for obtaining a beam of completely polarized light is the Nicol prism, which A constructed as follows : A / \__ -\ -^ ^ is crystal of Iceland spar is reduced to the form shown in Fig. 524; Fig. 525. B Fig. 524. the faces of this rhomb being cleavage planes of the crys- tal. This rhomb is divided along AB perpendicular to the plane of the paper.* The faces along AB are polished, and the pieces cemented together with a thin layer of Canada balsam, the re- fractive index of which is be- tween 1.658 and 1.486 (the ordi- nary and extraordinary indices of Iceland spar). An incident ray r (Fig. 525) is broken into two rays by the spar, and the ordinary ray o is totally reflected from the layer of balsam as shown. The extraordinary ray passes on through and is com- pletely polarized. 751. The action of a Nicol pnsm on a beam of polarized light. Let ABCD (Fig. 526) be the A- front face of a Nicol prism, and let the line a represent the amplitude and direction of vibra- tion of an incident beam of polarized light. This beam is broken up by the prism into ordinary and extraordinary rays, of which the vibrations are parallel to the diagonals AB and CD respectively. The amplitudes of these vibrations are represented by the lines o and e. The ampli- * The plane of the paper is a principal plane of the crystal, as drawn, and the vibrations of the ordinary ray and of the extraordinary ray are in the plane of the paper, and perpendicular thereto respectively. Compare Art. 748. o Fig. 526. 132 ELEMENTS OF PHYSICS. tude of the extraordinary ray, which passes through the prism, is a cos cf>. Its intensity is to the intensity of the incident beam as the square of its amplitude is to the square of the ampli- tude of the incident beam. That is, T \ I = # 2 cos 2 c/> : a 2 , or r=/cos 2 <, (340) in which / is the intensity of the incident beam, and T is the intensity of the transmitted beam. T evidently varies from a maximum when cos$ = I, to zero, when cos = o. 752. The polariscope consists of a Nicol prism P (Fig. 527) called the polarizer, which produces a beam of polarized light ; and another Nicol prism A, called the analyzer, through which Fig. 527. this polarized beam passes, in whole or in part, to the eye. Both prisms are arranged to turn about the axis of the instru- ment. The object which is to be examined is placed in the polarized beam between the prisms. Sometimes it is desired to examine an object in a convergent beam of polarized light as at cc. To this end a lens L is provided which gives a con- vergent beam. A second lens L' renders the beam again par- allel before it reaches the analyzer. Two glass plates placed as shown in Fig. 519, and arranged to turn about the polarized beam, as an axis, also constitute a polariscope. Two tourmaline plates used as polarizer and ana- lyzer are likewise sometimes employed ; they form a very con- venient and inexpensive polariscope. 753. Appearance in the polariscope of a thin plate of a doubly refracting crystal. The polarized beam from the polarizing Nicol is resolved by the crystalline plate into two beams (see Art. 751). These two beams (ordinary and extraordinary) pass through the plate at different velocities, so that one of them is POLARIZATION AND DOUBLE REFRACTION. 133 retarded with respect to the other. A component of each of these beams passes through the analyzer. Upon emergence these transmitted components are polarized in the same plane, and one of them being retarded relatively to the other, they interfere.* Some wave lengths are thus strengthened while others are weakened, and brilliant color effects are produced. This action is shown beautifully by thin plates of mica and by glass plates which have been rendered doubly refracting by elastic strain. 754. Rotation of plane of polarization ; the saccharimeter. Many substances have the property of turning the plane of polarization of a transmitted ray about the ray as an axis. The amount of turning is proportional to the distance which the waves travel through the substance, and varies with the wave length of the light, and with the nature of the substance. Crystals of potassium chlorate and of quartz, turpentine, and solutions of sugar, are examples. In the case of sugar solutions the rotation of the plane of polarization, for light of a given wave length, is proportional to the distance traveled by the light in the solution and to the strength of the solution ; so that a = k Im. (34 1 ) In this equation a is the angle (in degrees) of rotation pro- duced when polarized light passes through / cm. of a solution containing m grams per ex. of cane sugar. The proportionality constant k, for sodium light, at the ordinary room temperatures has the value - When k is known, and a and / have been 1-504 observed, the strength of the syrup may be computed by means of equation (341). A polariscope provided with a tube, to con- tain the syrup, and arranged for the measurement of a is called a saccharimeter. * Two beams of which the planes of polarization are at right angles do not inter- fere, inasmuch as the displacements at a point due to the respective beams are at right angles to each other. 134 ELEMENTS OF PHYSICS. 755. Electro-magnetic rotation of the plane of polarization. When polarized light is passed parallel to the lines of force through a transparent medium in a magnetic field, a rotation of the plane of polarization is produced. For example, carbon bisulphide in a tube surrounded by a coil of wire carrying cur- rent, rotates the plane of polarization of light which is sent through the tube. This phenomenon has been utilized by Crehore and Squier, in their photo-chronograph,* which is used in measuring the velocity of projectiles. It consists of a tube of carbon bisul- phide C surrounded by a coil of wire. This tube is mounted between two Nicol prisms (Fig. 528). These are crossed, and so Fig. 528. long as no current flows through the coil, no light can pass. The momentary currents which form the chronographic signals, and which it is desired to record, turn the plane of polarization within the tube, and flashes of light pass through the second Nicol prism JV Z . These flashes are recorded photographically upon a rotating disk Z>.f * Journal of the United States Artillery, Vol. VI., No. 3. t For a fuller treatment of polarization and double refraction the student is referred to Preston, Theory of Light; Groth, Physikalische Krystallographie, and to the original memoirs of Fresnel. CHAPTER XII. RADIATION. 756. Radiant heat. The various homogeneous components of the radiation from the sun, or from any hot body or luminous body, have this common property ; namely, that they generate heat in a body which absorbs them. Such radiation is there- fore called radiant heat.* The energy per second streaming across unit area at right angles to the ray is taken as the physical measure of its intensity. The radiation from a hot body, such as the sun, is composed of numerous homogeneous components which differ from one another only in wave length. The range, as to wave length, however, seems to be almost infinite. Rubens and Nichols f (E. F.) have isolated and identi- fied homogeneous components of the radiation from hot zirconia of ^ mm. wave length. These waves were isolated by repeated reflection from fluorite (compare footnote to Art. 767), and the wave length was determined by the use of a coarse diffraction grating made of wires. The wide gap in wave length between the radiation due to electrical disturbances and the longest waves previously recog- nized in the radiation from hot bodies is thus bridged, and homogeneous radiation of every wave length from Hertz waves, several meters or even kilometers long, down to ultra-viplet rays of less than 2O-io~~ 6 cm. in wave length, is definitely known*. * Sometimes called radiant energy. Radiation, in systems in equilibrium, how- ever, conforms to both laws of thermodynamics, and may more properly be called radiant heat. t Physical Review, Vol. IV., 1897. 135 OF THK IVERSITT 136 ELEMENTS OF PHYSICS. Becquerel* has described rays which are emitted by the ura- nium salts, and which appear to be of much shorter wave length than any of the ultra-violet rays hitherto observed. 757. Luminous effects and chemical effects of radiant heat. - Homogeneous radiation, of which the wave length lies between 39-icr 6 cm. and 75-io~ 6 cm., affects the optic nerves and pro- duces sensations of light. These limits are not sharply defined, but vary greatly with different persons, with the intensity of the radiation, and with the degree of fatigue of the optic nerves. The chemical effect of radiation is exemplified in the reduc- tion of carbon dioxide in the growth of plants, in the bleaching action of bright sunlight, in the action of light upon photo- graphic sensitive plates, and so on. The intensity of this chemi- cal action varies greatly with the wave length. The particular wave length for which the chemical action is greatest (for a given energy intensity of the radiation) varies with the sub- stance upon which the action is exerted. For the salts of silver used in photography the green, blue, and violet rays are most active, while the extreme red rays are almost wholly inactive. 758. Prevost's principle of exchanges. A number of bodies in an inclosed region exchange heat by radiation until they reach uniform temperature. The whole system is then in ther- mal equilibrium. Radiation persists in a region after the region has settled to thermal equilibrium. It is almost necessary to suppose that the molecular commotion in the various bodies continues to produce waves even after all the bodies in a region have reached uniform temperature. Each body then gives off as much radiant heat as it receives. If the temperature of one body is low, it radiates less than it receives, and its tempera- ture rises ; if its temperature is high, it radiates more than it receives, and its temperature falls. These facts were first pointed out by Prevost. They comprise what is known as the principle of exchanges. * Comptes Rendus, 122 (1896). RADIATION. 137 759. The law of normal radiation. The radiation coming from a body, at a given temperature, is of three distinct parts : (i) rays emitted by the body ; (2) rays reflected from its sur- face ; (3) rays transmitted by the body from radiating surfaces behind it. These, taken together, constitute what is called the total radiation. A system or region which neither gains nor loses heat is called a closed system or region. The law of radia- tion for such a system may be stated as follows : In a closed region at a given uniform temperature, the total radiation from any body, whatever its nature, is of definite com- position. In other words, each of the various homogeneous com- ponents of the total radiation is of definite intensity.* Proof. Experience shows that a number of bodies come to a state of thermal equilibrium when left to themselves in a closed region ; and that this state is not disturbed in any way when a foreign body at the same temperature is introduced into the region. Remark. The radiation in a closed region at a given uniform temperature is called the normal radiation for that temperature. The term normal applies both to composition and to inten- sity. Both the radiation impinging upon a body and that sent out from it in such a closed region, are normal. 760. Proposition. Emission and absorption are equal. Con- sider a body in a region in thermal equilibrium. The total radiation falling upon the body and the total radiation from it are normal. A portion of the latter is reflected. Let this por- tion be removed, and an equal amount be subtracted from the incident radiation. Another portion of the total radiation from the body is transmitted ; imagine this to be removed and a cor- responding amount to be subtracted from the incident radiation. * In 1866 Kirchhoff published a theorem, known as KirchhoflPs law, viz. : for a given temperature, the relation between emissive power and absorbing power is for all bodies the same. The import of this theorem is in accordance with the statement given above. 138 ELEMENTS OF PHYSICS. The remaining portion of the incident radiation is absorbed by the body ; the remaining portion of the radiation from the body is emitted, and these two portions are equal. 761. Selective emission, reflection and transmission. In the case of many substances, the total radiation for certain wave lengths consists almost entirely of emitted light, while for other wave lengths the emission is very slight, and the total radiation is made up chiefly of transmitted or reflected light. Gases fur- nish the most striking example. Such substances are said to exhibit selective emission. When the total radiation for certain wave lengths contains a greater proportion of transmitted (or reflected) light, than is the case for other wave lengths, the substance is said to exhibit selective transmission (or reflection}. 762. Selective absorption. It is frequently convenient to express the behavior of a body by reference to its absorbing power for radiation. Since emission and absorption are always equal, wave length for wave length, all bodies which show selec- tive emission show selective absorption also. 763. Ideal cases of selective action. There are four ideal cases, the consideration of which will help towards the under- standing of the behavior of various substances which approxi- mate to them. i (a) Bodies which do not reflect perceptibly. Such bodies do not transmit (that is, they do absorb) those wave lengths which they emit in excess. In this case the transmitted and emitted radiations are complementary. (b) Bodies which are opaque (i.e., which do not transmit). Such bodies emit radiations which are complementary to the radiations which they reflect. (c) Bodies which do not radiate perceptibly. Such bodies reflect best those wave lengths which they do not transmit. RADIATION. 139 That is to say, the reflected radiations and transmitted radia- tions are complementary. (d) Opaque bodies which do not reflect (black bodies). Such bodies emit normal radiation. These peculiarities persist even when the incident radiation is not the normal radiation corresponding to the temperature of the body, although in such a case the two portions into which the radiations are divided are not to be thought of as comple- mentary in the complete sense in which this term is used above. 764. Existing cases of selective action. (i) Gases do not re- flect perceptibly, and when hot they radiate those wave lengths in excess which they absorb excessively when cool (Arts. 699 and 700). Ruby glass, which is red by transmitted light, and which shows no marked selective action by reflection, is green when it is heated to incandescence. (2) Metallic copper is sensibly opaque. It is red by reflected light, but when incandescent, it is green. (3) Fuchsine, of the emitting power of which nothing defi- nite is known, transmits red and violet light and reflects green light almost completely. The approximately complementary character of the transmitted and reflected radiation shows indeed that the emitting power of fuchsine is relatively small. In dilute solution its behavior is somewhat different. Gold is yellow by reflected light, and gold leaf is beautifully green by transmitted light. It follows that an extremely thin sheet of gold would not exhibit selective emission when hot. A thick sheet (opaque) appears greenish when incandescent, but the effect is not so marked as in the case of copper, because the color of the metal by reflected light is not so ruddy. Remark. An opaque body which reflects all wave lengths well radiates very little, and an opaque body which reflects very little radiates well. Thus dead black steam pipes radiate much better than brightly polished ones. Water cools much more slowly in a polished metal vessel than in a blackened one. I4 ELEMENTS OF PHYSICS. 765. Black bodies. An opaque body which reflects very little of the radiation which falls upon it is said to be black. Such bodies, when heated to a given temperature, emit very nearly the normal radiation for that temperature. Amorphous carbon, graphitic carbon, and the black oxide of iron are exam- ples of nearly black bodies. Perfectly black substances do not occur. A perfectly black body would be an opaque body which reflected no portion of the rays falling upon it. A small hole in the opaque wall of a large closed chamber is perfectly black, for only an infinitesimal portion of the rays which enter find their way out again. If such a chamber be maintained at a uniform temperature, the radiation emitted from the hole will be the same as the radiation within, which is normal. The appearance of a large, uniformly heated tile kiln, as seen through a hole in the wall, is very striking. Even though the interior be partially free from tiles, nothing can be seen but a flood of soft yellow light. The radiation reflected, transmitted (if any), and emitted by a tile, or by a piece of the air for that matter, is normal, so that identical radiations reach the eye from every portion of the interior. If the peep hole is large enough to cool the adjacent tiles, they become faintly visible ; or if a beam of sunlight (it must be light from something hotter than the kiln) is reflected into the opening, the tiles become visible, as if they were in a dark chamber. 766. White bodies. A body which reflects (approximately) the same proportion of the various homogeneous components of the radiation which falls upon it is called a white body. The light reflected by a white body is of the same composition as the incident light. Thus white paper is red in red light, green in green light, etc. A perfectly white body would be one reflect- ing the whole of the incident radiation, as opposed to a perfectly black body which absorbs the whole. No substance is perfectly white. There is no substance, even, which reflects exactly the RADIATION. I4I same proportion of each homogeneous component *"of the inci- dent radiation. Most white powders for example, sugar, magnesium oxide, etc. reflect the longer wave lengths in excess, and have a yellowish tint. This is counteracted by the admixture of a small amount of a powder, e.g. ultramarine blue, which reflects the shorter wave lengths in excess. Substances, like glass and ice, which transmit (and reflect) every part of the visible spectrum equally well (approximately), are dazzling white when powdered. This is exemplified in the whiteness of snow. Nearly the whole of the light falling upon a sheet of snow is reflected (diffusely), because of the repeated reflections by the successive particles as they are reached by the light as it penetrates deeper and deeper into the snow. 767. Surface color.* A substance which shows marked selective reflection is said to have surface color. All such substances appear different in color by reflected and by trans- mitted light. Thus gold is yellow by reflected light, and gold- leaf is beautifully green by transmitted light. The aniline dyes, especially when concentrated, are different in color by reflected and by transmitted light. Red wine is of a beautiful bronze color by reflected light. 768. Absorption color. Most colored substances, such as colored glass, colored solutions, etc., show color, perceptibly, by transmitted light only. Many colored substances which * The electro-magnetic theory of light shows : (i) That a perfect electrical con- ductor would totally reflect all radiations falling upon it; (2) That a substance of which the structural elements have a proper period of undamped electrical vibration would totally reflect wave trains of that particular period; and (3) That in pro- portion as these proper vibrations are damped the substance would reflect less and absorb more of the trains having its proper period. An example of the first case is furnished by metals, which, especially for the longer wave lengths, give almost complete reflection. Such reflection is called metallic reflection. Many substances such as fluorite and fuchsine reflect certain wave lengths almost completely. Repeated reflection from fluorite, for example, isolates the wave train which is most completely reflected. Reflection of this kind, for want of a better name, is called anomalous reflection (surface color). It might well be called reso- nant reflection. 142 ELEMENTS OF PHYSICS. appear colored by reflected light, such as the pigments used in painting, really owe their color to selective transmission or absorption. The light falling upon their grains is partially transmitted, becomes colored, and is reflected by numerous foreign particles and by breaks in the continuity of the grains. This is shown by the fact that a mixture of two pigments reflects only those wave lengths which are reflected by both pigments unmixed, just as a pair of colored glasses transmits only those wave lengths which can pass through both. If the pigments were colored mainly by true selective reflection, each isolated grain of each pigment would reflect independently, and all the wave lengths reflected by each pigment would be found in the light reflected by the mixture. 769. Methods of measuring radiant heat. The study of the composition of radiation is a matter of great difficulty. In the experimental determination of the distribution of energy in the spectrum of a glowing body, the selective transmission of the prisms and lenses of the spectrometer introduces uncertainties which cannot be wholly eliminated ; and if a concave diffraction grating is used, it is impossible to know the details of the grating with sufficient accuracy to be able to calculate the intensity of a homogeneous component of the radiation from the observed intensity of the corresponding part of the spectrum. The instruments used in such work are the thermopile, the bolometer, and the radiometer. The thermopile has been described in Vol. II. (Art. 558). The bolometer, invented by Langley * in 1880, consists of a Wheatstone bridge, one of the arms of which is made of a thin strip of blackened metal. This is exposed to the rays, the intensity of which is to be indicated. The radiation causes a rise in temperature and a consequent increase in the resistance of the metal strip. This affects the balance of the bridge, and a sensitive galvanometer in the bridge circuit shows a deflection. * See Transactions of the National Academy of Sciences, Vol. V. RADIATION. 143 jf Y The radiometer used in the measurement of radiation differs essentially from the radiometer of Crookes. It was invented by Ernest Nichols* in 1896. It consists of two similar thin vanes of blackened mica (aa) attached to a horizontal arm, and suspended in a high vacuum by means of a fine quartz fiber (Fig. 529). The radia- tion to be indicated falls upon one of these vanes and warms it slightly. This causes the few remaining molecules of air to rebound with increased velocity from the blackened face. The reaction pushes the vane backwards, and turns the arm about the fiber as an axis. The deflec- tion is observed by means of a telescope and scale. The sensitiveness of this in- strument is such that the rays from a candle at a distance of 450 meters (nearly one-third of a mile) produce a noticeable deflection. 770. The comparison of various sources of light by means of the bolometer. Langley's measurements of the spectra of various sources of radiation afford good illustrations of the use of the bolometer. aHa 1 - 1 Fig. 529. 600- 400 200- V R l.O/i 2.0// 3.0/i Fig. 530. The ordinates of the curves in Fig. 530 show the relative intensities of the different homogeneous components in sun- * See Physical Review, Vol. IV., p. 297. I44 ELEMENTS OF PHYSICS. light, gaslight, and in the light of the electric arc lamp. The deep notches in the curve of sunlight are absorption bands, due to relatively cool vapors around the sun and in the earth's atmosphere. The dotted line shows approximately what the radiation from the sun would be were it not for this selective absorption. 771. Coefficients of absorption. Let / be the intensity of a homogeneous beam of light as it enters a plate of indefinite thickness. Let i be the intensity of the beam after having penetrated to a distance x into the plate ; and let Az be the further decrease in intensity of the beam as it penetrates to an addi- tional distance kx into the plate. The portion Az of the beam which is absprbed is, by experiment, sensibly proportional to i and to A^r, so that A*' ki . Ar, or In this equation k is the proportionality factor. The negative sign is chosen inasmuch as A/ is a decrement of intensity. By integration, equation (i) gives Loge i = kx + a constant, or in which C is an undetermined constant. When x=o, thenz' = /. These values substituted in (ii) give C /, so that equation (ii) becomes i = I*'**, (342) in which / is the intensity of a beam of homogeneous light as it enters an absorbing substance, i is its intensity when it has penetrated to a depth x, e is the Naperian base, and k is a constant called the coefficient of absorption, of the substance for the particular kind of homogeneous light. For substances which exhibit selective absorption, the value of k depends upon the wave length of the homogeneous beam. The law of decrease of intensity of a beam, as expressed by equation (342), is as follows : Of the beam which remains unabsorbed the same fractional part is absorbed in each successive layer of the substance. 772. Helmholtz's theory of dispersion. Consider a stretched rubber heavier at one end than at the other, AB (Fig. 531), along which trans- verse waves may be sent by moving the end A back and forth. Let c, c, c, etc., be equal weights suspended from the tube by similar spiral springs, so that each of these weights has the same proper period of vibration T. Wave trains of all wave lengths will move along AD at the same velocity v, and ignoring the action of the weights, along DB, the heavier end, at a RADIATION. velocity - v. The action of the weights is to cause /u, to vary with the peri- odic time T of the wave train. Helmholtz's theory of dispersion is represented mechanically by this arrangement. Helmholtz showed * that the velocity of o C C C ETC. Fig. 531. a wave train along DB is increased (/x decreased) by the action of the weights, so long as r is less than T, and decreased (/x, increased) so long as T is greater than T, and that the effect of the weights becomes greater as T approaches T. When T = T, an incident wave train along AD is almost totally reflected from A, especially if the motion of the weights is frictionless ; and, in any case, the action along DB when T = T is not of the nature of a wave train at all. The ordinates of the curve (Fig. 532) represent the values of u f velocit y DB\ \ velocity in AD I for different values of the period r of the incident wave train. The abscissa of the ordinate T rep- resents the periodic time of the suspended weights. Examples. The Helmholtz theory shows that the refractive index of a substance is very greatly increased below an absorption band (r>Z > ), and greatly decreased above an ab- sorption band (T < Z 1 ), and that wave trains for which r = T are almost wholly reflected. This, in fact, is found to be the case for those sub- stances which exhibit powerful selective action and show surface color. This wide variation of the refractive index in the neighborhood of an absorption band has been called anomalous dispersion, or, better, resonant dispersion. T Fig. 532, 773. Example of resonant dispersion. The ordinates of the dotted curve in Fig. 533 show the refractive index of a solution of fuchsine for rays of various wave lengths. The band AB shows the actual appearance of a narrow spectrum CD when viewed through a prism containing a solution of fuchsine held in such a way as to deflect the spectrum CD laterally. Cl ID Fig. 533. * See Berliner Berichte, 1874, p. 667. 2'(i), p. 674. L Also Winkelmann, Handbuch der Physik, I4 6 ELEMENTS OF PHYSICS. 774. Phosphorescence and fluorescence. Some substances, while undergoing slow chemical action, emit radiation of the shorter wave lengths greatly in excess of the normal amount cor- responding to the temperature of the substance. Thus phos- phorus oxidizes slowly when it is exposed to the air at a low temperature, and emits a pale white light. This action is some- times called phosphorescence. The term is, however, more generally applied to the phenomenon described below. Many substances for example, calcium sulphide glow for a time in the dark, after being exposed to intense radiation. Such substances are said to be phosphorescent. The light given off by a phosphorescent substance is generally of greater wave length than the light which is needed to bring the substance to phosphorescence. A very remarkable exception is the case of the salts of uranium. Becquerel (See Art. 756) has found that these salts send out rays of extremely short wave lengths for a very long time after being exposed to sunlight. These rays (Becquerel rays) are of much shorter wave length than any hitherto recognized in the ultra-violet of the solar spectrum. Some substances as sulphate of quinine, kerosene, and uranium glass emit light of medium wave length when exposed to radiation of very short wave length. Such sub- stances are said to be fluorescent or luminescent. The Rontgen rays produce strong luminescence in some substances. (See Vol. II., Art. 498.) CHAPTER XIII. LOUDNESS, PITCH, AND TIMBRE. 775. The vibration of a particle. Simple and compound vibrations. When a particle moves to and fro along a straight line, performing simple harmonic * motion, its vibrations are said to be simple. When the motion of a particle is periodic, but not simply harmonic, its vibrations are said to be compound. A B Fig. 534 a. Graphical representation of simple and compound vibrations. Consider a point / (Fig. 534 a) vibrating up and down along the line AB. Imagine the paper to move with uniform velocity to the right, then the point p will trace a line cc. If the vibrations of / are simple, this curve cc will be a curve of sines. If the vibrations of / are compound, the curve cc will be a periodic curve, i.e. each section of it exactly similar to every other section, but not a curve of sines. The curve in Fig. 534 b, which shows the vibratory motion of a point of a violin string, is such a compound curve. The number of (complete) vibrations per second of a particle is called the frequency of the vibrations. * See Vol. I., Art. 59. I4 8 ELEMENTS OF PHYSICS. The time of one complete vibration is called the period of the vibrations. (See Art. 59, Vol. I., for definitions of ampli- tttde, phase, and phase difference^ Fig. 534 b. Remark. The conception of simple and compound vibra- tions of a particle may be applied to the vibrations of a body which moves up and down or to and fro sensibly as a whole. For example, the end of the prong of a tuning fork performs simple vibrations ; a portion of the sounding board of a piano performs compound vibrations. 776. Fourier's theorem applied to the vibrations of a particle. Any periodic vibration of frequency n is equivalent to a number of superposed simple vibrations of which the respective frequencies are n, 2/2, 3/2, 4/2, and so on, and of which the respective amplitudes are determinate. It is for this reason that such vibrations are called compound vibrations. The state- ment here given of Fourier's theorem is identical, from the mathematical point of view, with the statement of Art. 624, referring to simple and compound wave trains. 777. Musical tones defined. When a simple or compound wave train from a vibrating body falls upon the ear, the result is a sensation of sound. It is necessary to the production of such a sensation, that the frequency of some of the components of the wave train lie within certain limits called the limits of audibility. (See further, Art. 780.) OP THE { - -SRSITY LOUDNESS, PITCH, AND TIMBRE. When a wave train is simple, or when, although compound, it contains some component which is so much stronger than the others as to hold the attention of the hearer, the result is a musical tone. A simple tone is one produced by a simple wave train. A compound tone is one produced by a compound wave train. 778. Noises. Sound sensations not falling under the defini- tion of musical tones, or not consisting of some simple arrange- ment or combination of musical tones such that the hearer can distinguish the various parts and recognize their relations to one another, may be classified as noises. The distinction between musical tones and noises is not a definite one. Rattling noises are due to irregular successions of sharp clicks. Hissing and roaring noises are due to complex and rapidly varying combinations of tones. In the case of hissing noises the tones, which may be few in number, are of very high pitch, and in the case of roaring noises the tones are of lower pitch. All manner of combinations of rattling, roaring, and hissing noises occur, from those combinations of musical tones which begin to be so complicated that a hearer cannot distin- guish the various parts and recognize their relations to one another, to the extremely complex sounds from waterfalls, trains, and busy streets. The prevalence in a roar of a dominant tone of medium pitch is apt to engage the attention and leave the accompanying noise to a great extent unnoticed. Tones of higher pitch than those used in music (hissing) engage the attention, if they are at all prominent, and give a noisy charac- ter to the sound. Musical tones are generally accompanied by characteristic noises. Thus, the whispering noises of the breath, and the sounds of the consonants used in articulation, accompany the musical tones of the singer ; and the faint noises produced by the fingers, keys, and pedals always accompany piano music. Many noises, on the other hand, are accompanied by character- ISO ELEMENTS OF PHYSICS. istic musical tones. A light blow upon a floor, or upon a piece of furniture, produces a faint musical tone, of short duration, which is often prominent enough to be easily distinguishable. 779. Loudness. That quality of sound which depends upon the intensity of the sensation is called loudness. The loudness of a given tone, in so far as it is not affected by fatigue of the organs of the ear, and by attention, depends upon the energy of the vibrations which produce it. Tones of medium or high frequency are very much louder, for the same energy of vibra- tion, than tones of low frequency. 780. Pitch. That quality of a musical tone which depends upon the frequency of vibrations of the wave train is called pitch. The pitch of a tone is high or low according as the frequency is great or small. "^Simple vibrations of lower frequency than about 34 (com- pleteV vibrations per second are not heard as a musical tone, and wh^n the frequency exceeds 35,000 or 40,000 per second the tone becomes inaudible. These limits vary greatly for different persons. 781. The measurement of pitch. Direct and indirect methods. The direct measurement of pitch consists in counting the number of vibrations per second, in the production of a given musical tone. The instrument commonly used for this purpose is the siren. The siren consists of a circular metal disk (Fig. 535) mounted upon a shaft upon which a screw thread is cut. This screw thread engages a gear wheel which actuates a device for counting the revolutions of the disk. The disk has one or more rows of holes. These rows are circular, with their Fig. 535. centers at the axis of the shaft. This disk rotates very near the wall of a chamber containing air under pressure, and the holes in the disk come before apertures in LOUDNESS, PITCH, AND TIMBRE. AIR-CHAMBER / this wall in rapid succession. (See Fig. 536.) The puffs of air thus produced blend into a musical note, the pitch of which is known from the observed speed of the disk and the number of holes in the row. The disk is sometimes driven by an electric motor and sometimes by the ac- tion of the issuing air. In the latter case, the holes in the disk are inclined like the vanes of a windmill. Fi s- 536. Another direct method consists in recording upon the cylin- der of a rapidly moving chronograph the vibrations of the sounding body to the oscillation of which the tone to be meas- ured is due. Sometimes the record is made by means, of a stylus attached to the vibrating body and tracing a curve upon a smoked surface. Sometimes a small mirror is attached to the vibrating body and a beam of light is thrown from it upon a sensitized surface. When the wave train is simple, the tracing is sinuous. Figure 537 is a facsimile of such a tracing. The pitch is determined by counting the number of undulations recorded in a second of time. Fig. 537. The indirect measurement of pitch consists simply in compar- ing the pitch of one sounding body with- that of another, which is taken as a standard. The standard is commonly a tuning fork. (See Art. 795.) Where the tones to be compared are nearly of the same pitch, the method of beats is frequently employed. This method depends upon the fact, which is more fully discussed in Art. 806, that when two wave trains, nearly alike in frequency, reach the ear simultaneously they are alter- nately in the same phase (and reinforce each other) and in opposite phase (and annul each other). The result is a series of pulsations which grow slower as the two wave trains approach 152 ELEMENTS OF PHYSICS. in frequency and disappear altogether when complete unison is attained. Another method of comparing vibrations is by means of Lissajous figiires. It may be used whenever the two frequen- cies are in a simple ratio such as i : i ; 1:2; 2:3, etc. To the two vibrating bodies mirrors are attached. The adjustment is such that the planes of the two vibrations are at right angles. If a beam of light be reflected from one mirror to the other and thence to a screen or into the eyepiece of a telescope, the spot of light thus formed moves in a closed curve. The pattern thus produced is called a Lissajous figure. It is that which naturally arises from the combination of the two rectilinear motions imparted to the beam by the individual vibrations. Figure 538 shows the Lissajous figures correspond- ing to the ratios i : i ; 1:2; 1:3 and 2 : 3. Each figure is shown in four phases. I : I 2: 3 Fig. 538. Both of the indirect methods described above are used in tuning instruments. The vibration microscope of von Helm- holtz, an apparatus for the accurate adjustment of tuning forks, is based upon the method of Lissajous' figures. 782. Timbre. The sound sensation produced by a compound vibration, i.e. a compound tone, may be regarded as composed of a series of simple tones corresponding to the various simple LOUDNESS, PITCH, AND TIMBRE. ^3 vibrations which enter into the composition of the compound vibration. Such a sound is usually perceived as a whole, even by a practiced ear, and may be called, following the German, a clang. A very little practice enables one to distinguish the various simple tones which enter into the composition of a clang, such as a note from a violin, a piano, or an organ. It is much more difficult to distinguish the tones which enter into the composi- tion of a vowel sound or of the note produced by a singer. The reason may be, perhaps, that the habit of perceiving the sound of the human voice as a whole is more nearly fixed. The resonator (Art. 801) makes it easy to distinguish the compo- sition tones of almost any clang. The composition tones of a clang are called harmonic overtones, and their vibration fre- quencies are as the successive whole numbers I, 2, 3, etc. This is in accordance with Fourier's theorem, and is fully confirmed by experiment. The various overtones in a clang are designated by the numbers which express the relation of their vibration fre- quency to that of the fundamental tone. Thus we speak of the fundamental, the first, the second, third, etc., tones or overtones of a clang. That quality of a clang which depends upon the relative intensities of the various overtones is called timbre. The tones of various musical instruments, for example, owe their peculiar quality largely to differences in timbre. CHAPTER XIV. FREE SONOROUS VIBRATIONS. 783. Vibrations of air columns. Let AB (Fig. 539) be an indefinitely long tube filled with air, and open at the end B. i i A K rj- 1 ya 1 ! VB Fig. 539. Consider a simple train of sound waves of wave length X advan- cing in the tube from A towards B. When this train reaches B, it is almost entirely reflected, without change of phase (Art. 623). The reflected train superposed upon the advancing train produces a stationary train (Art. 623) of which the nodes are distant - from each other, the first node being at a distance of - from the open end, which is an antinode, as shown by the dotted wave lines. The air in each vibrating segment of this stationary wave surges back and forth in the direction of the tube ; the air on the two sides of a node surges towards the node simultaneously, and then away from it simultaneously, so that the air at the node is alternately compressed and expanded, but does not move. The time r required for one back and forward move- ment, that is, for one vibration of the inclosed air, is the period of the wave train. From equation (320), Art. 618, we have \ = TV, (i) FREE SONOROUS VIBRATIONS. i$$ in which v is the velocity of progression of sound waves in the tube. This velocity is sensibly the same as the velocity in the open air. The frequency /of the vibration of the inclosed air is equal to -; whence, using the value of r from equation (i), we have /-J do Air column open at both ends. After the stationary wave train (or vibration) is once established in the tube A (Fig. 539), the tube may be (ideally) cut across at any antinode without altering * the subsequent action in the detached portion of the tube ; except that the vibrations will soon die away as the energy of the vibrations is dissipated. This dissipation is partly be- cause of friction against the walls of the tube, and partly because of the incomplete reflections from the open ends of the tube. The tube being cut across at an antinode, the length / of the detached portion will be , n\ - ,... / = , (m) in which n is any whole number. Therefore, substituting the value of X from (iii) in (ii), we have /=J7 (343) in which f is the frequency of vibration of an air column of length / open at both ends, v is the velocity of sound in air, and n is any whole number. When n = i, the frequency of vibration of the air column is least, and the tone produced is called the fundamental tone of the column. When n is 2, 3, etc., the tone produced is called the second, third, etc., harmonic, or overtone. The char- * The wave train reflected from B will be again reflected without change of phase from the cut, and this second reflected train takes the place of the original advancing train. ELEMENTS OF PHYSICS. acter of the vibration when = i, when n = 2, and when n 3 is shown by the dotted wave lines in Fig. 540. Air column closed at one end. When the stationary wave train is once established in the tube AB (Fig. 539), a rigid dia- Fig. 540. phragm may be (ideally) placed across the tube at any node without altering the subsequent behavior ; except that the vibrations will soon die away. In this case, the length / of the detached portion of the tube will be . n\ r . / = - , (iv) 4 in which n is any odd number. Substituting the value of X from (iv) in (ii), we have /=T7 (344) 4/ in which f is the frequency of vibration of an air column of length /, closed at one end. Since n in this case is necessarily an odd number, an air column open at one end has only odd harmonics. The character of the vibrations when n = I, when ;/ = 3, and when n = 5 is shown by the dotted wave lines in Fig. 541. 784. Simple and compound vibrations of an air column. An air column vibrating so as to give its fundamental or one of its harmonics alone is said to vibrate simply. An air column gen- FREE SONOROUS VIBRATIONS. 157 erally vibrates so as to sound its fundamental and its various harmonics or overtones simultaneously, thus producing a clang. In such a case the vibration of the air column is said to be com- W7////A Fig. 541. pound. The relative loudness of the various overtones depends upon the shape of the column and upon the manner in which the vibrations are excited. This is exemplified in the strikingly different clangs of organ pipes, whistles, horns, and clarionets. 785. Organ pipes. The vibrating elements of the pipe organ are columns of air called organ pipes. These are very similar to the bark whistles known to children. A metal or wooden tube AB (Fig. 542) incloses the vibrat- ing air column, and the vibrations are excited by an air jet which, issuing from a narrow slit, blows across the opening against the sharp edge s. This air jet makes a slight noise, which starts the air column vibrating feebly. These vibrations react upon the air jet and cause it to play from one side to the other of s. This reinforces the vibrations, so that they quickly become energetic. When the end A is closed, the air column vibrates in accord- ance with equation (344), and only the odd over- tones are present. When the end A is open, the air column vibrates in accordance with equation (343), and both even and Fig. 54-2. 158 ELEMENTS OF PHYSICS. odd overtones are present. With broad pipes the overtones are very weak, and the tone approaches a pure tone in char- acter. With narrow pipes the overtones up to the fifth or sixth are audible. When an organ pipe is blown strongly, the over- tones become more prominent. If blown very strongly, the second or third overtone may become so prominent as to com- pletely dominate the tone. The reed pipe is an air column into which a stream of air flows through an opening in and out of which a spring or reed vibrates, so as to convert the air stream into a series of puffs of the proper rhythm to excite the vibrations. The clarionet, the cornet, and the vocal organs of man are types of the reed pipe. 786. The clarionet. The tones of the clarionet are produced by the vibrations of an air column, the length of which may be altered at will by uncovering openings in the side of the tube. The end of the tube which is placed in the mouth is covered by a light reed or tongue. When the player blows into the instru- ment, this reed, acting like a valve, suddenly closes the opening, and a wave passes down the tube, is reflected from the open end of the tube, and, returning, strikes the reed and causes it to open again. The impulse is then repeated. The movement of the reed is not, however, entirely controlled by the surging of the air in the tube of the instrument. The lips of the player are pressed against the reed in such a way that its tendency is to open and close in the proper rhythm, independently of the surging of the air. The abrupt motion of the reed of the clarionet, as it opens and closes the mouth end of the instrument, is very far removed from simple vibration. The result is, that the various (odd) har- monics, as well as the fundamental tone of the air column, are sounded quite loudly, and the instrument gives a characteristic clang, which is strikingly different from the sound produced by the flute or the organ pipe. FREE SONOROUS VIBRATIONS. 59 787. The cornet. In the case of the cornet the lips of the performer take the place of the reed of the clarionet, and the length of the vibrating column of air is altered at will by means of valves or keys which include or exclude auxiliary lengths of tube. The keys provide for six distinct lengths of air column and by sounding the various harmonics of these six lengths at will, the player produces any desired note. The bugle is similar to the cornet ; except that it has a tube of fixed length, the harmonics of which are sounded at the will of the player. The harmonics ordinarily used are the 2d, 3d, 4th, 5th, and 6th. 788. The vocal organs. These consist of the vocal cords, two muscular membranes which are stretched across the top of the windpipe, and the motith cavity. The vocal cords are tuned to any desired pitch by muscular effort and are set vibrating by air forced from the lungs. The smooth tones produced in singing arise from vibrations in which the vocal cords do not strike against each other. In speech the cords strike against each other as they vibrate and produce sounds which contain a great many simple component tones. The vowel sounds are produced by so shaping the mouth cavity as to strengthen (by resonance) certain of these component tones at will. The consonant sounds, so common in speech, are characteristic noises with which, in articulation, the vowel sounds are begun and ended. 789. Longitudinal vibrations of rods and strings. The lon- gitudinal vibrations of a rod free at both ends are similar to those of an air column open at both ends. The frequency of vibration is given by equation (343), in which v is the velocity of longitudinal waves (sound waves) along the rod. The dotted wave lines in Fig. 540 show the character of the longitudinal vibration of a rod with one, two, and three nodes, respectively. A string stretched between rigid supports vibrates longitudi- ELEMENTS OF PHYSICS. rially in a manner similar to the vibrations of an air column closed at both ends. 790. Kundt's experiment. A rod (Fig. 543) is supported, say at its center, and set vibrating longitudinally by rubbing it with a rosined cloth. One end of this vibrating rod extends loosely into a tube of air AB. A train of waves passes out from the end R of the rod as it vibrates back and forth, and this wave train upon reflection from the closed end B forms a stationary Fig. 543. train. When the rod is adjusted until its end is near a node, this stationary train acquires great intensity. Lycopodium or other light powder, which is strewn inside the tube AB, is swept out of the vibrating segments by the surging motion of the air and is heaped up at the nodes. The distance between nodes is then easily measured. This distance is half the distance traveled by the sound waves in the air of the tube during one vibration of the rod. If the rod is giving its fundamental tone, for which n = I, the length of the rod is half the distance traveled by a sound wave in the rod during one vibration. Therefore the ratio of the length of the rod, divided by the distance between nodes in AB, is equal to the velocity of sound in the material of the rod divided by its velocity in air. Know- ing the velocity of sound in air, its velocity in the rod is thus easily determined. The tube AB may then be filled with any gas and the distance between the nodes again measured; whence the velocity of sound in the gas may be found. Some of the velocities given in Art. 613 were determined in this way. 791. The transverse vibration of strings. Preliminary state- ment. If a stretched string be struck or distorted and released, the wave produced will travel at a velocity FREE SONOROUS VIBRATIONS. !6i in which T is the tension of the string in dynes and m is its mass per unit length. Let AB (Fig. 544) be an indefinitely long stretched string fixed to a rigid support at B. Consider a simple transverse A Fig. 544. wave train of wave length \ advancing from A towards B. Upon reflection at B (with change of phase) a stationary wave train will be formed, of which the nodes are distant from each 2 other. Once this stationary train is established, a rigid support may be placed under the string at any node, giving a vibrating string of which the length is / ^ /-\ / = , (11) in which n is any whole number. The time r of one complete vibration of the string is equal to the period of the wave train ; so that \ = TV, and the frequency of the vibrations is /=- We have, therefore, Substituting the value of X from (ii), and the value of v from (i) in (iii), we have in which f is the frequency of vibration of a string of length 7, stretched with tension T, and weighing m grams per centi- meter ; and n is any whole number. When n is unity, the whole string is one vibrating segment. The tone given in this case is the fundamental of the string. When n equals 2, 3, or 4, etc., the string has 2, 3, or 4, etc., vibrating segments. The tone given by the string when n is UNIVERSITY 1 62 ELEMENTS OF PHYSICS. greater than unity is called, as in the case of vibrating air col- umns, a harmonic of the string, or an overtone. 792. Simple and compound vibrations of strings. A string vibrating so as to give one of its harmonics, or its fundamental, is said to vibrate simply. A string may (and generally does) vibrate so as to give simultaneously its fundamental and various harmonics. In such a case its vibration is said to be compound. The relative intensities of the fundamental and the various har- monics of a vibrating string depend upon the manner of excit- ing the vibrations. Thus the plucked string of the guitar, the struck string of the piano, and the bowed string of the violin give very different clangs, and these clangs change very percep- tibly with the point at which the string is plucked or struck or bowed. A guitar string plucked near the center gives only Fig. 545. odd harmonics and those not strongly. If plucked about one- seventh of the length of the string from one end, the overtones up to the sixth are prominent, and the clang is correspondingly rich. The use of the bow enables the skilled performer upon FREE SONOROUS VIBRATIONS. 163 stringed instruments to give a great variety of complex vibra- tions to the strings. Figure 545 shows a few curves illustrating the character of vibration of bowed strings. They are selected from the numerous traces published by Krigar-Menzel and Raps.* The curves were obtained by photographing upon a moving plate the motions of an isolated point (or element) of the string. 793. The use of sounding boards. A vibrating string pro- duces a comparatively feeble sound when it is stretched over rigid supports upon a metal bed, because of the very small dis- turbance of the air produced by the string directly. Stringed instruments are therefore provided with thin wooden boards or shells to which the bridges which support the strings are fixed. The vibrations are transmitted through these supports to this sounding board, and thence to the air. 794. Transverse vibrations of stiff rods and plates. The velocity of propagation of a transverse wave (a bend) along a stiff rod or plate varies with the wave length. The frequencies of the various simple modes of vibra- -f- very weak. tion of a rod or of a plate are not so ~Q jf 9 strong tremulo. simply related, as are those of vibrat- :fi) fo wea k. ing air columns and strings. The ^ -j- very strong. compound vibration of a stiff rod or L of an elastic plate gives therefore a -^- ff- ^ e weak clang of which the overtones are . . Fig. 546. unharmonic. The discordant sound produced by striking a steel rod and the sound produced by cymbals and gongs are examples. The overtones of the tun- ing fork and of the bell are likewise unharmonic. Bell found- ers have, however, learned to model bells in such a way as to make the more prominent tones harmonic. A bell of the most approved model gives a rich mellow tone. Figure 546 shows the more prominent tones of a Russian bell in the library of Cornell University. * Wiedemann's Annalen, Vol. 44, p. 623. 164 ELEMENTS OF PHYSICS. Chladni' s figures. The nodal lines on a vibrating plate may be shown by fixing the plate horizontally in a clamp, strewing sand upon it, and causing it to vibrate in a simple mode by means of a violin bow. The sand collects along the nodal lines. These sand figures, which were discovered by Chladni, may be obtained in a great variety of forms, corresponding to the vari- ous simple modes of vibra- tion of the plate. To this end the vibrations of the plate must be controlled by holding the fingers against it while using the bow. Figure 547 shows some of the figures depicted by Chladni in his treatise on Acoustics. (Leipzig, 1787.) 795. The tuning fork is a stiff rod bent into the form shown in Fig. 548. The character of its fundamental mode of vibration is shown by the lines B. The two nodes n, n are near together, and the intervening segment, together with the metal post P, moves up and down through a small am- plitude and causes the sounding board upon which the fork is mounted to vibrate in unison with the fork. The second tone of a fork, which is some three or four oc- taves above its fundamental, dies out quickly after the fork is thrown into vibration, leaving the fundamental alone. Fig. 547. / \ Fig. 548. FREE SONOROUS VIBRATIONS. I6 5 The tuning fork, therefore, gives a pure tone. Carefully tuned forks are much used by musicians as standards of pitch. (See Art. 781.) 796. Vibrating diaphragms. The sound produced by a vibrating membrane owes its peculiar quality largely to the characteristic quickness with which the vibrations die out. The sound of the drum, for example, is so brief that there is scarcely time for a distinct sensation of pitch to be produced. The over- tones of a vibrating membrane or diaphragm are unharmonic. A diaphragm, being light and exposing a broad surface to the air, vibrates very perceptibly with the air when any sound strikes it. It is this property of a light diaphragm which renders it useful in the telephone and the phonograph. 797. Manometric flames. An apparatus has been devised by Koenig which shows, with some distinctness, the character of the vibrations produced by a sound. It depends upon the action of the waves upon a gas flame. A thin diaphragm covers ACETYLENE Fig. 549. a hole in the side of a chamber through which gas passes to a small flame. The pressure of the gas fluctuates with the move- ments of the diaphragm and causes the height of the flame to vary accordingly. When the flame is viewed in a rotating mir- !66 ELEMENTS OF PHYSICS. ror, or is photographed upon a moving plate, it presents a saw- toothed appearance which varies with the character of the sound falling upon the diaphragm. Photographs of the manometric flame are shown in the accompanying plate. These photo- graphs were taken from a brilliant acetylene flame burning in oxygen. The movement of the photographic plate was such as to make it necessary to read -the figure from right to left. The apparatus* used in taking these photographs, which is shown in Fig. 549, is that devised by Merritt in 1893. The lens forms an image of the flame upon the sensitive plate AB, which moves rapidly in a direction perpendicular to the paper. * Physical Review, Vol. I., p. 166. CHAPTER XV. IMPRESSED VIBRATIONS AND RESONANCE. 798. Proper vibrations ; impressed vibrations. The vibra- tions which a body performs when struck or disturbed in any way, and left to itself, are called its proper vibrations. When a simple train of sound waves of any wave length strikes a body, the body is made to vibrate in unison, or in the same rhythm, with the impinging waves. Such vibrations are called impressed vibrations. A compound wave train causes a body to perform simultaneously the simple vibrations corre- sponding to the various simple wave trains which enter into the composition of the compound train. The quickness with which a body assumes a steady state of impressed vibration under the action of sound waves depends upon the mass of the body and upon the extent of the surface which it exposes to the action of the waves. The violence of the impressed vibrations depends upon the intensity of the impinging waves; upon the extent to which the vibrations are damped; and upon the relation between the fre- quency of the impinging waves and the frequency of the proper vibrations of the body. 799. Damping. Vibrations are said to be damped when they die out quickly. This is generally due, in part, to the dissipa- tion of energy in the body in the form of heat, as it is repeat- edly distorted, and in part to the giving up of energy to the surrounding air. Thus the vibrations of light bodies, which expose considerable surface to the air (diaphragms, etc.), and 167 T 68 ELEMENTS OF PHYSICS. of bodies which are imperfectly elastic, are greatly damped. On the other hand, the vibrations of a heavy, elastic body, such as a tuning fork, are but slightly damped. A heavy tuning fork performs several thousands of perceptible vibrations when struck. The column of air in an organ pipe performs several hundreds of perceptible vibrations after the exciting cause ceases. A drum head performs only a very few per- ceptible vibrations when struck. 800. Resonance. Very perceptible vibrations are impressed upon diaphragms and stretched membranes by sound wave trains of any frequency. The vibrations impressed lipon a heavy body, of which the damping is slight, are much more violent when the impressed frequency approaches the proper fre- quency of the body. Thus the sound of a tuning fork (removed from its sounding board) is perceptibly enforced when it is held near the open end of a tube of any length. If the length of the air column is adjusted, for example, by pouring water into the tube, the sound becomes louder as the proper frequency of the air column approaches that of the fork; and it reaches a very distinct maximum when the impressed vibrations become proper to the air column. With bodies which exhibit less and less damping the maximum violence of impressed vibrations at proper frequency becomes more and more pronounced; and at the same time the impressed vibrations of improper frequency become more and more nearly imperceptible. Thus, a massive tuning fork, mounted upon its sounding board, is thrown into quite violent vibration by a tone of its proper frequency sus- tained for four or five seconds. This pronounced maximum violence of impressed vibration at proper frequency is called resonance, and the vibrating body or air column is called a resonator. When impressed vibrations are proper to a body, the action of the impulse due to each successive wave is to add to the exist- ing motion, and the vibrations increase in violence until the IMPRESSED VIBRATIONS AND RESONANCE. iftg energy given to the vibrating body by the impinging waves is all dissipated by damping. It is for this reason that the impressed vibrations become quite violent when the damping is small. On the other hand, when improper vibrations are impressed upon a body, the periodic forces, with which the waves act upon the body, must take the place more or less of the elastic forces, which ordinarily cause a body to vibrate. 801. Analysis of clangs by means of resonators. Any over- tone of a clang may be easily detected or brought to notice by intermittently strengthening it by means of a resonator tuned to unison with it. A convenient resonator for this purpose is made by drawing the end of a glass tube to such size as will fit tightly into the ear. The other end of the tube is left open. The inclosed air will strengthen very perceptibly any tone in unison with it. The action is more striking if the tube is repeatedly removed from the ear and replaced. For tones of low pitch such a tubular resonator would be unwieldy. In this case it is more convenient to use a hollow globular vessel of glass or metal, with a small neck on one side to project into the ear and an opening on the other side several centimeters in diameter. In order to analyze a clang, one after another of a series of such resonators is applied to the ear. Figure 550 shows an ad- justable resonator designed by Koenig. It consists of two cylindrical brass F . 550 tubes fitted- to one another. The outer tube is contracted to a narrow opening which enters the outer ear of the observer. The other tube is partially closed by a cap, which contains a circular opening (1-3 centimeters in diameter). 802. Vowel sounds. The various vowel sounds are charac- terized each by one or two tones of definite pitch. In producing a given vowel, the mouth cavity is shaped so as to strengthen by I/O ELEMENTS OF PHYSICS. resonance the tones which characterize the vowel. The charac- teristic tones of some of the vowels, as determined by Helm- holtz, are as follows : VOWEL. TONE. VIBRATION FREQUENCY.* u as in rude / 173 6 as in no c" 517 a as in paw g" 775 a as in part d\>'" 1096 a as y&pay f and tip" 346 and 1843 e as in pet c"" 2068 e as in see f and d"" 173 and 2322 Figure 551 gives these characteristic tones of the vowels in terms of the ordinary musical notation. In ordinary speech the rough sound from the vocal cords easily excites the proper resonance in the mouth cavity for the / uw li 1 u 6 a a f a e e Fig. 551. production of any required vowel. The smooth tone of a singer, however, may not contain the characteristic tones of a vowel, so that these cannot be strengthened by resonance. In this case those overtones, of the note which is sung, which are nearest the characteristic tones of a vowel, for which the mouth cavity is set, are strengthened, and in this way the vowel sound is (incompletely) characterized. It is a well-known fact that spoken words are much more easily understood than words * Complete vibrations. IMPRESSED VIBRATIONS AND RESONANCE. I/I which are sung. The difference in distinctness is due largely to the imperfect character of vowels when sung. Overtones of a given pitch are more widely separated in a note of high pitch than in a note of low pitch, so that the mouth cavity, shaped to give the characteristic tone of a vowel, is less likely to produce the desired effect with high notes than with low. The words of a soprano singer are, in fact, less distinct than the words of a bass singer of similar schooling. The proper tones of the mouth cavity for the production of the vowels o, a, and a may be easily heard by thumping against the cheek when the mouth is prepared to sound those vowels. Helmholtz determined the characteristic tones of the vowels by finding which of a series of tuning forks placed before the mouth in succession would produce resonance in the mouth cavity shaped to produce a given vowel. He also determined by the help of resonators, as described in the previous article, the actual tones present in the various vowel sounds ; and by combining the sounds of suitable tuning forks he was able to imitate successfully these various vowels. 803. The artificial reproduction of speech. The phonograph. Vowel sounds, and indeed all the sounds used in speech, can be reproduced by means of any device which is capable of giving out the necessary combination of simple tones with the proper relative intensities. Such an instrument is the phonograph (and its modification the gramophone). It consists essentially of a thin diaphragm, to which is fastened a light tool which scratches a minute groove in a rotating smooth cylinder made of a hard wax-like compound, or of soap. A sound striking the diaphragm impresses vibrations upon it, and causes the attached tool to cut a groove of varying depth. A record of the sound is thus made upon the cylinder. The cylinder is driven with a very nearly uniform motion of rotation by means of an electric motor. In some cases clockwork is employed. 1/2 ELEMENTS OF PHYSICS. To reproduce the sound a round-ended tool, which is attached to the diaphragm, is adjusted to follow this groove and the cyl- inder is set rotating at its former speed. The varying depth of the groove causes the diaphragm to vibrate and the sound is reproduced. The phonograph may be regarded as a develop- ment of an earlier instrument, the phonautograph, a mechanism by means of which a curve, showing the character of the vibrations producing a sound or produced by the sound, is traced. The movements of a diaphragm are transmitted to a tracing point which marks a line upon smoked paper. The paper is carried on a rotating cylinder. CHAPTER XVI. THE EAR AND HEARING. 804. The human ear.* The fact that the various overtones of a clang may be distinctly heard, shows that the conception of simple and compound vibrations as developed in Arts. 773 and 774 is not merely a mathematical fiction, but that it has real physical significance. This significance is briefly this ; namely, that a body, of which the proper vibration frequency is in unison with any simple tone of a clang, is set into violent vibration by the clang. This property has been discussed in the articles on Resonance (Chapter XV.). It was first pointed out by Helmholtz that our perception of the various simple tones in a clang must depend upon the existence of a series of organs (the end organs of the auditory nerves) in the ear, each of which has a proper vibration frequency and is sensitive (by resonance) to simple tones nearly in unison with it. This action may be illustrated by means of the piano, as follows : A musical sound of characteristic timbre, for example, a vowel sound, is sung loudly against the sounding board of a piano, of which the damper is raised so as to leave the strings free. Those strings which are in unison with the various simple tones of the clang are set into vibration and we hear a continuation, by the piano, of the vowel sound after the singing ceases. Imagine each string of a piano to be connected * The anatomy of the ear is too complex to be described here with any fullness. See Helmholtz, Die Lehre von den Tonempfindungen, pp. 209-250. 173 174 ELEMENTS OF PHYSICS. to a nerve fiber, and we have an apparatus which would perceive sounds as they are actually perceived by the ear. The process of perception is as follows : Sound waves enter the ear and strike against the tympanic membrane. The vibra- tions of this membrane, of which the area is about 70 square millimeters, are reduced in amplitude and concentrated* upon another diaphragm, of about 5 square millimeters in area, by the action of a chain of three tiny bones. This second dia- phragm covers a small window (the oval window] of a bony cavity, called the labyrinth, which is filled with a watery fluid. Another opening of the labyrinth, the round window, is cov- ered with a diaphragm which is entirely free. The vibrations of the diaphragm in the oval window cause the watery fluid of the labyrinth to surge back and forth between the two windows and through the chambers of the labyrinth. One of these cham- bers, the cochlea, is very long, and is coiled upon itself like a snail shell. The end organs of the auditory nerves are located in membraneous structures, which are in part suspended in the watery fluid of the labyrinth, and in part constitute an elastic diaphragm, the basilar membrane, which divides the cavity of the cochlea longitudinally. The various shreds of this basilar membrane seem to be the resonating elements of the ear. Remark. Those shreds of the basilar membrane which are in unison with a given simple tone are most strongly excited by the tone ; and the intensity with which an adjacent shred is excited falls off as its proper vibration frequency differs more and more from the vibration frequency of the tone. The pro- duction of audible beats by the interference of two tones depends upon the simultaneous action of the two tones upon the same shreds of the basilar membrane. When the two tones work together to produce commotion upon the shreds which they both affect, the sound is a maximum, and when they are opposite in phase they produce but little commotion, and the sound is a minimum. * See Helmholtz, Loc. cit. THE EAR AND HEARING. 175 805. Persistence of the sensation of sound. When the stimu- lation of a nerve ceases, the accompanying sensation continues for a length of time, which depends upon the intensity of the stimulation, and which varies greatly with different nerves. Sensations of light persist much longer than sensations of sound. A periodic light becomes sensibly continuous when the flashes follow one another at the rate of forty per second. A periodic sound for example, the sound of a tuning fork of high pitch which is shut off from the ear intermittently has been found by Mayer to become smooth when the fluctua- tions reach a frequency of one hundred and thirty-five per second. The sensation produced by an intermittent tone is very rough and unpleasant, an effect which is called discord, or dissonance. The discordant effect increases with the fre- quency of fluctuation, reaches a maximum, and finally disap- pears, when the sensation becomes smooth. 806. Interference. The shortness of the interval during which a tone sensation continues after the stimulation ceases is very intimately connected with the phenomenon of inter- ference of two tones. The general character of this phenome- non is described in Chapter VIII. Consider two simple tones of vibration frequencies / and /', which are very nearly equal. At a certain instant the wave trains which constitute these tones will be in like phase as they enter the ear. The disturbance produced and the correspond- ing sensation will then be a maximum. When the tone of higher pitch has gained half a vibration (or half a wave length) over the other, the wave trains will be opposite in phase as they enter the ear, and the sensation will be a minimum. When the higher tone has gained a whole vibration, the waves will again enter the ear in like phase, and the sensation will again be a maximum. These successive maximum sensations are called beats. The number of beats which occur in one second is ff. ELEMENTS OF PHYSICS. As the pitch of one of the tones increases, the difference ff increases; the beats become more frequent; the inter- mittent sound sensation becomes more disagreeable or dis- cordant, and soon reaches a point of maximum discord, after which the discord decreases again. When the beats become sufficiently frequent, the sensation becomes smooth because of the persistence of the sound sensations. The number of beats per second which produces maximum discord, and the number per second which leaves the resulting sensation smooth, vary with the absolute pitch of the interfering tones in a manner shown approximately in the following table. FREQUENCY OF FLUCTUATIONS. VIBRATION FREQUENCY. WHEN TONE BECOMES SMOOTH. WHEN DISCORD is A MAXIMUM. 6 4 16 6. 4 128 26 10-4 2 5 6 47 18.8 384 60 24.0 5 I2 78 31.2 640 90 36.0 7 68 I0 9 43- 6 1024 135 54.0 807. Combination tones. The principle of superposition (Art. 621), viz., that two tones may exist simultaneously with- out affecting each other, is true only when the forces producing a distortion are strictly proportional to the distortion. In this case an added distortion produces exactly the same additional forces no matter what the initial distortion may be. No actual material satisfies this condition except for very small distortions, a fact which is ordinarily expressed by saying that all materials are imperfectly elastic. Two weak (primary) tones do not sen- sibly affect each other ; but as they grow louder certain other tones, produced by their mutual action, are found to accompany THE EAR AND HEARING. 177 them. These accompanying tones are called combination tones. Combination tones are most pronounced in those cases in which the primary tones cause the same substance or the same portion of air to vibrate violently. The imperfectly elastic chain of bones in the ear, for example, is conducive to the formation of com- bination tones. Difference tones. The most prominent combination tone is that of which the frequency is equal to the difference of the frequencies of the two primary tones. This is called the differ- ence tone of the first order. This difference tone forms differ- ence tones with each of the primary tones, in like manner. These are called difference tones of the second order; and so on. Summation tones. Less prominent than the difference tones is the combination tone of which the frequency is equal to the sum of the frequencies of the two primary tones. This is called a summation tone of the first order. Summation tones of the first order form summation tones with each primary tone. These are called summation tones of the second order. Difference tones of the first order are very noticeable when two tuning forks are sounded simultaneously. Helmholtz has shown that difference tones of higher order and summation tones are audible in the sound of a two-voiced siren, and in the sound produced by two notes of a reed organ sounding simultaneously. It often requires the help of a resonator, however, to bring them into notice. 808. Miscellaneous phenomena depending upon the reflection, refraction, and diffraction of sound. The reflection, refraction, and diffraction of sound waves have been discussed in the earlier chapters of this volume. Certain phenomena relating to hearing which result from those processes, however, are still to be considered. These are briefly described in the following articles. 809. Echo. This well-known phenomenon is produced by the reflection of sound. The echo from the side of a large I 7 8 ELEMENTS OF PHYSICS. building is very distinct, and the smooth face of a cliff, or a well-defined forest front, may produce an echo sufficiently dis- tinct to repeat words. If the reflecting surface is sufficiently distant, an entire sentence may be repeated by the echo. An echo grows less distinct the more irregular the reflecting surface, and it becomes an indistinct roar when the reflecting surface is very irregular. With multiple reflections, as in the case of the two walls of a tunnel or of a canon, a sharp loud sound, such as the report of a gun, is prolonged in a manner resembling thunder. Reflection often produces the effect of an apparent change of direction of a sound, when from any cause the direct waves are masked or diverted from any cause, so that the hearer perceives only the reflected wave trains. 810. The influence of the refraction of sound upon hearing at a distance. Phenomena due to regular refraction, as sound waves pass from one medium to another, in which the velocity is dif- ferent, do not occur to ordinary observation. The following phenomena, however, which are due essentially to refraction, are of common occurrence. The velocity of the wind is usually less near the ground than higher up, and the upper portion of a sound wave W (Fig. 552), WIND GROUND Fig. 552. proceeding against the wind, is retarded. The direction of pro- gression of the wave is thus thrown upwards and the sound tends to leave the region near the ground. When the wave THE EAR AND HEARING. 179 travels with the wind, the tendency is to concentrate the sound near the ground. It is a familiar fact that it is much more difficult to make one's self heard against than with the wind. Sound travels faster in hot than in cold air. When the air near the ground is warmer than it is higher up, the upper por- tion of a sound wave is retarded and the sound tends to leave the ground. When the air near the ground is relatively cool, the tendency is for the sound to be concentrated near the ground. The greater distinctness of distant sounds by night than by day is no doubt due largely to this cause. 811. The influence of diffraction upon the sense of direction of a sound. Our sense of the direction of a sound seems to depend, in part, upon diffraction. We have this sense only with com- plex sounds, not with a sound which consists of a single simple wave train. The approaching complex waves reach one ear without much obstruction, while the other ear is more or less shaded from them by the head. This shading action is greater the shorter the wave length, so that different sensations are produced in the two ears. Differences of sensation brought about in this manner are significant of direction, and have come to be perceived as such without entering consciousness in any other way. 812. Changes of pitch due to the motion of the sounding body or of the hearer. Since pitch depends upon the frequency with which successive waves fall upon the ear, it is obvious that motion in the direction of the wave (either with or against the wave) will respectively lower or raise the pitch. The velocity of sound is so small that the motion of a railway train is by no means inappreciable in comparison. Thus the whistle of an approaching engine is distinctly raised and that of a receding train is distinctly lowered by the motion of the sounding body. The effect is especially noticeable to an observer upon one train who listens to the whistle of an engine passing in an opposite direction. OF THB 'CTNIVERSITY !8o ELEMENTS OF PHYSICS. 813. Sounds of all wave lengths have the same velocity. We have very direct evidence of this fact in the case of both speech and music. If different components of a spoken sound or of a concerted piece traveled at different velocities, we should have sensations varying 'with the distance from the source. Both speech and music indeed would become unrecognizable at considerable distances because the various components to which the timbre is due would fail to reach the ear simultaneously.* In point of fact, the only effect of distance, aside from general loss of intensity, is to diminish the loudness of sounds varying in pitch by somewhat unequal amount. Some waves seem to carry further than others. * This remark, although true of light and sound, fails with water waves. Let A (Fig. 553) represent a water wave produced by a quick movement of an oar. When Fig- 553. this wave has traveled a short distance, it will be seen to have assumed the form shown by the wave line B. The various simple wave components of A are separated because of their different velocities. The movement of a chip produced by the wave B would be very different in character from any motion which could possibly pro- duce the wave A. This phenomenon is shown very beautifully by the ripple produced by dipping an oar edgewise into still water. CHAPTER XVII. MUSICAL INTERVALS AND SCALES. 814. Pitch intervals. It is shown in the next following arti- cle [815] that the consonant relation of two tones depends mainly upon the ratio of their vibration frequencies. Two pitch intervals are therefore said to be equal when they are expressed by the same frequency ratio. Consider a number of tones of which the vibration frequencies are n, an, a 2 n, a 8 n, etc. The frequency ratio of two successive tones in this series is a. Let their pitch interval be /. The frequency ratio of the first and third tones is # 2 , and their pitch interval will be 2/ ; the frequency ratio of the first and fourth tones is a?, and their pitch interval will be 3/, etc. Let / be the logarithm of a. Then 2 / is the logarithm of a 2 , 3 / is the logarithm of a 3 , etc. Therefore we have for these various pitch intervals the following relation : Frequency ratios a a? a s a* etc. Logarithms of frequency ratios / 2 / 3 / 4 / etc. . Pitch intervals p 2p *$p 4p etc. It will be seen that the pitch interval between two tones is pro- portional to the logarithm of their frequency ratio. Pitch intervals are ordinarily expressed in terms of frequency ratios. When the relative magnitude of a number of pitch intervals is the object of consideration, it is, however, con- venient to express the intervals in terms of the logarithms of their frequency ratios. As an example, see the discussion of the tempered musical scale. (Art. 824.) 181 l$2 ELEMENTS OF PHYSICS. 815. Complete and approximate consonance of compound tones.* Preliminary statement. Tones ordinarily used in music are compound. The fundamental tone usually predominates, in a compound tone, and the overtones 2, 3, 4, 5, 6, and 8 usually occur, decreasing in loudness in the order given. The follow- ing discussion of consonance is limited to the influence of these six overtones. When two compound tones A and B are in unison, their respective overtones are in unison also ; the combined sound of the two tones is entirely free from roughness due to beats, and the two tones are completely consonant. When the two tones are not in unison, then, even if their difference in pitch is so great that the fundamental tones of A and B do not produce audible beats, some of the overtones of A will generally be near enough to some of the overtones of B to produce marked roughness in the combined sound of A and B\ that is, to pro- duce distinct dissonance. If the pitch of the tone B is slowly raised or lowered, starting from unison with A, this dissonance passes through a very marked minimum value every time one (or more) of the overtones of A come into unison with one (or more) of the overtones of B. The two tones A and B are approximately consonant when their dissonance thus reaches a minimum value. The following example will make this clear. Let the tone B be a very little higher than A. Then the fundamentals and each pair of the overtones produce beats, and the dissonance is great. As the pitch of B is raised, this dissonance falls off as each pair of jarring tones becomes more widely separated in pitch. This falling off in the dissonance continues until the fifth overtone of B comes to be adjacent to the sixth overtone of A. The jarring action of this pair of tones then causes a dis- tinct rise in the dissonance, followed by a rapid fall as the pair come into unison. As the tone B continues to rise in pitch, there is a rapid rise in the dissonance, and so on. When the * The consonance of simple tones is discussed in Art. 805. MUSICAL INTERVALS AND SCALES. 183 C5 All of B ELEMENTS OF PHYSICS. fifth overtone of B is in unison with the sixth overtone of A, the frequency ratio of B : A is equal to 6:5. The ordinates of the curve in Fig. 554 show the values of the dissonance of two violin tones A and B, in so far as the overtones 2, 3, 4, 5, 6, and 8 are concerned. The fractions z? below the line BAB show the values of the frequency ratio for the various minima of dissonance. This curve is adapted from a more complex one by Helmholtz. The numerical evalua- tion * of dissonance is an approximation. Remark. Combination tones have some action in the pro- duction of dissonance when two tones are sounded together. The dissonance due to combination tones passes through mini- mum values for the same pitch intervals as does the dissonance due to overtones. 816. Consonant intervals. A pitch interval between two tones, for which the tones are approximately consonant, is called a consonant interval. Figure 554 shows the various consonant intervals. The following table exhibits the various consonant intervals in the order of the completeness of their consonance, together with their names. TABLE OF CONSONANT INTERVALS. i : i Unison 3 : 5 Major Sixth i : 2 Octave 4 : 5 Major Third 2 : 3 Fifth 5 : 6 Minor Third 3 : 4 Fourth 5 : 8 Minor Sixth The consonance of the octave is complete. That of the fifth is very nearly so. The bounding of consonant intervals. Those overtones and combination tones which determine a consonant interval by their coincidence, and which produce the greater part of the dissonance when the interval is slightly out of tune, are said to bound the interval. * See Helmholtz, Tonempfindungen, Beilage XV. MUSICAL INTERVALS AND SCALES. 185 The great increase in dissonance, due to a slight error of tuning of a consonant interval, is the basis for a remarkably acute sense which we have of the accuracy of these intervals. This acute sense of pitch of consonant tones has a great deal to do with the effectiveness of consonant intervals in music ; for there can be no refinement of musical expression without an acute sense to seize upon it, and it is the ultimate depend- ence of this acute sense upon the presence of prominent over- tones which explains the peculiar musical value of such tones as those of the violin and of the human voice. 817. The variation of the character of the consonant inter- vals with timbre. A consonant interval is the more striking in character in proportion as it is more sharply bounded. Therefore the character of a given consonant interval varies with the timbre of the tones used; that is, with the relative loudness of the various overtones. This is exemplified by clarionet tones, which have only odd overtones. All of the consonant intervals are indeed, in this case, bounded by combination tones, but the major sixth (3 : 5) is more striking in character than the fourth (3 14), and perhaps even as sharply defined as the fifth (2:3); while the minor third (5 :6) and the major third (4 : 5) are ill defined. A major third (4 : 5), formed by a violin tone and a clarionet tone, is very much more striking when the clarionet tone is the lower, so that its fifth overtone coincides with the fourth overtone of the violin, than it is when the violin tone is the lower ; and a minor third (5 : 6) is much more striking when the violin tone is the lower. In the case of pure tones, such as the tones of tuning forks and of wide organ pipes closed at one end, combination tones, only, serve to bound the consonant intervals. With such tones, the octave (i : 2) is pretty sharply defined, the fifth (2:3) less sharply, while the remaining intervals are scarcely bounded at all. Helmholtz, indeed, has found that the sound of two tuning 1 S6 ELEMENTS OF PHYSICS. forks is smooth, or consonant, whatever the pitch interval, provided only that this interval is not so near to the fifth (2 : 3) or to the octave (1:2) as to bring out the dissonances which bound these two intervals. 818. The major and minor accords. Three tones which form a consonant combination are called an accord, or chord. Thus three tones, of which the vibration frequencies are as 4:5:6, form an accord. Any tone of an accord may be replaced by its octave, or may be accompanied by its octave, without greatly altering the character of the accord. This is evident when we consider that no new overtones are introduced into the sound by the octave. The major accord and its modifications. The three tones of which the vibration frequencies are as 4:5:6 constitute what is called the major accord. By replacing the first tone (4) by its octave (8), we obtain a modification of this accord, and by re- placing the third tone (6) by its lower octave (3), we obtain another modification. The three forms of the major accord are, therefore, 3 4 5 456 5 6 8 The minor accord and its modifications. The three tones of which the vibration frequencies are as 10 : 12 : 15 constitute what is called the minor accord. The interval between the first two tones is a minor third (5 : 6), between the last two tones is a major third (4 : 5), and between the first and last the interval is a fifth (2:3); so that the minor accord contains the same con- sonant intervals as the' major accord (4:5:6). The modifica- tions of the minor accord are 10 12 15 12 15 20 15 20 24 MUSICAL INTERVALS AND SCALES. I8 7 The primary forms of the major and minor accords, viz., 4:5:6 and 10: 12 : 15, are those in which the three tones are separated by the smallest pitch intervals. Difference in character of major and minor accords. The major and minor accords contain the same consonant intervals, and the coincident overtones are identical in the two cases. The combination tones, however, are very different. The fol- lowing schedule shows the combination tones of the first and second orders. The major accord. Primary tones 456 First difference tones I 2 Second difference tones 2345 The minor accord. Primary tones 10 12 15 First difference tones 235 Second difference tones 7 8 9 10 12 13 This schedule shows that the difference tones of the major accord are exact duplications, or duplications in the lower octaves, of the primary tones. That is, no foreign tones are introduced into the major accord by the combination tones. On the other hand, some of the difference tones of the second order, viz., 7, 8, 9, and 13, which occur in the minor accord^ are dissonant, and give to this accord a character very different from that of the major accord. 819. Musical scales. The successive tones in a melody (see Art. 822), and the simultaneous tones in harmony (see Art. 823), are chosen with reference to their consonance. The major scale. Consider a given tone c\ The tones which can be used with c' with musical effect are those desig- nated by e'b,-e', f ', g', a'\>, and a' in Fig. 554. Ignoring the tones e'b and a'b, which have low degrees of consonance with c f , !88 ELEMENTS OF PHYSICS. we have the following series of musical tones, each of which is consonant with c 1 : Tones c' e' f g' a! c" } Vibration frequencies i f f f f 2 J Remark. The tone c 1 , with reference to which a series of tones is selected, is called the tonic of the series. The tone g' , having next to the octave the most complete consonance with c' , is called the dominant ; and the tone f', which is next in order of consonance, is called the subdominant of the series. For purposes of harmony, it is desirable to be able to build major accords (4:5:6) upon the tonic, upon the dominant, and upon the subdominant of a series of musical tones. Two of these major accords may be built up with the tones in series L, namely, c', e' y g f (4:5:6) and /', a', c" (4:5:6). To build a major accord upon g 1 , two additional tones, say b 1 and d' } ', are required such that g' : b' : d n = 4 : 5 : 6. Therefore the vibra- tion frequencies of b 1 and d" are J^- and |- respectively. Taking a tone d } ', an octave below d" , we have the series : Tones c' d' e' f g' a' b' c" \ Vibration frequencies i f f f f f - 1 /- 2 J This is the ordinary musical scale, called the major scale. The minor scale. Choosing, with the help of Fig. 554, the tones below c', which are most nearly consonant with c' t we have the series : Tones c fo f g d& c' Vibration frequencies \ fill I or, in order that this series may be more easily compared with L, we may choose all of these tones an octave higher, whence we obtain the following series of musical tones : Tones c' e>\> f g' a'b c" Vibration frequencies i III! 2 Remark. This series III. is more melodious when sounded in the order of descending pitch than when sounded in the reverse order, for the reason that the tones of the series are MUSICAL INTERVALS AND SCALES. 189 more nearly consonant with c" than with c', and whichever of these tones is sounded first is made correspondingly promi- nent. The series I. (and also II.) is more melodious when sounded in the order of ascending pitch. The series III. includes the two minor accords (10, 12, 15) c f e ^> g '> an d f y & r b, c" . To build a minor accord upon g 1 , two additional tones, say tf\> and d u ', are required, such that g' : b'\>\ d" 10 : 12 : 15, so that the vibration frequency of b'\> is f and the vibration frequency of d n is |-. Taking a tone d l r , an octave below d", we have the series : Tones c' d 1 e'\> f g' afy b'\> c" } Jy Vibration frequencies iffff I 1 2 J This series of tones is called the descending minor scale. For purposes of melody, this scale is changed to the following for ascending movements : Tones c' d' e'\> f g' a! V c" j y Vibration frequencies i f f f f f - 1 /- 2 J This is called the ascending minor scale. Remark. The tones which are consonant with the tonic c 1 are called related tones of the first order. The tones d' and b' of the major scale and d' and b'b of the minor scale which are consonant with g' are called related tones of the second order (that is, related to the tonic c'). The scales II. and IV. are better suited to the require- ments of harmony than is scale V. The scale II. is suited to harmony in which major accords predominate. It is for this reason called the major scale. The scale IV. is suited to harmony in which minor accords predominate. It is for this reason called the minor scale. The following schedules exhibit all of the major and minor accords which can be formed of the tones of the major and minor scales. The major scale. major accord major accord major accord cf~ e' ^' V ~d" minor accord minor accord o ELEMENTS OF PHYSICS. The minor scale. major accord major accord / aft c' e'\> g 1 b'\) d 1 minor accord minor accord minor accord The tonic accord is shown in each case by the bold-faced type. In the major scale the tonic accord, the dominant accord, and the subdominant accord are major accords. In the minor scale these accords are minor accords. The naming of consonant intervals. The intervals between the tonic and the third and sixth tones of the major scale are called the major third and major sixth respectively. The inter- vals between the tonic and the third and sixth tones of the minor scale (IV) are called the minor third and minor sixth respectively. The intervals between the tonic and the fourth and fifth tones of either scale are called the fourth and fifth respectively. 820. Musical expression. Music is said to be expressive when it appeals in a distinct manner to the emotions. The primary forms of musical expression are rhythm, melody, har- mony, and modulation. The artistic use of these elements is largely traditional and conventional ; still the development of musical method has been largely determined by physical facts or laws. The following articles give brief statements of the essential nature of rhythm, melody, harmony, and modulation. 821. Rhythm. The rapidity of succession of the tones in music and the manner in which successive tones are set off in groups is called rhythm. The rhythmic grouping of tones enables a listener to appre- hend them more clearly, and therefore permits the effective use of more extended phrases in melody and modulation than would otherwise be possible. Rhythm is also effective in giving expression to vigor or languor according to the rapidity of the movement ; and changes of rhythm serve to mark the MUSICAL INTERVALS AND SCALES. 191 progress of long compositions, and at the same time to give variety. 822. Melody. A sequence of tones is called a melody. The shades of expression produced by different sequences of tones depend mainly upon the coincidence of overtones of successive notes. When two successive notes have several coincident overtones, the progressive effect of the melody, or, as it is technically called, the movement comes to a momentary stop of a more or less decided character. Thus a repetition of the same tone is a full stop in the movement, and the progressive effect becomes more and more distinct as the consonant rela- tion of successive notes becomes less and less marked. When successive tones are dissonant, the movement is abrupt, and is expressive of sudden emotions, such as surprise. The interval if between b' and c" of the major scale is an exception to this last statement. The note b' seems to serve as a catch note, or a threshold, to be passed in reaching c" . 823. Harmony. The simultaneous use of a number of tones in music is called harmony. The shades of musical expression produced by different combinations of tones depend mainly upon the degree of consonance of the combination. Harmony built upon major accords is strikingly different in expression from that built upon minor accords. The first gives an expres- sion of contentment and cheerfulness, while the latter is expres- sive of discontent and sadness. 824. Modulation. Two accords are said to be related when they have a common tone or two common tones. A sequence of accords, each of which is related to the one preceding it, is called a modulation. The possibility of modulation in the major and minor scales are exhibited by the following schedule, taken from Art. 819. Major scale. major accord major accord major accord / a c' e' g' b' d" minor accord minor accord ICJ 2 ELEMENTS OF PHYSICS. Minor scale. major accord major accord c e'fr minor accord minor accord minor accord More extended modulations than those exhibited in this schedule require the use of tones related, in the third order, to the tonic c'. Let us consider, for example, an extension in both directions of the modulation of the major scale. We have: iv x f a c' e' g 1 b' d" y z The brackets represent major accords. The tones y and z are not related to c', but they are related to g' exactly as b' and d" respectively are related to c'. Thus the extension of this modulation upwards leads to a set of tones having a new tonic, namely, g* . In like manner the tones w and x are related to / exactly as f and a respectively are related to c', so that an extension of this modulation downwards leads to a set of tones having a new tonic, namely,/ Such an extended modulation in one direction or the other is called a change of key. In popular music, for example dance music, the modulation is confined to the tonic, dominant, and, subdominant accords, which succeed each other periodically. The greater musicians often use very extended and very complex modulations, making use of major and minor accords in one and the same phrase, with occasional dissonances for the sake of contrast. 825. The tempered scale. A great number of distinct tones is required for extended modulation, and it would be impracti- cable for a player to use a piano or an organ having a separate key (and string or organ pipe) for each tone. This difficulty is overcome, at the expense of accuracy of tuning of the various consonant intervals, by the use of what is called the tempered scale. This scale consists of twelve tones (thirteen, counting both end tones) in each octave, the pitch intervals between successive tones being equal. The octave is thus divided into MUSICAL INTERVALS AND SCALES. 193 twelve equal pitch intervals. The logarithm of the frequency ratio of each of these intervals is therefore one-twelfth of the logarithm of 2, which is the frequency ratio of the octave. (Compare Art. 814.) The following table shows the logarithms of the frequency ratios of each tone of the major scale to the tonic, and also the logarithms of the frequency ratios of each tone of the tempered scale to the tonic. Tonic. Number of tone in tempered scale I 2 3 4 5 6 7 Log of ratio of tone of tempered scale to the tonic .0251 .0502 0753 .1003 .1254 .1505 Nearest tone in major scale c' d' e' / Log of ratio of tone of major scale to tonic .0511 .0979 .1250 Difference of logs .0009 + .0024 + .0004 Octave. Number of tone in tempered scale 8 9 10 ii 12 13 Log of ratio of tone of tempered scale to the tonic .1756 .2007 .2258 .2509 .2759 .3010 Nearest tone in major scale f a' V c" Log of ratio of tone of major scale to tonic .1761 .2219 .2730 .3010 Difference of logs - .0005 -f .0039 + .0029 o 194 ELEMENTS OF PHYSICS. This table shows that the first, third, fifth, sixth, eighth, tenth, twelfth, and thirteenth tones of the tempered scale are very nearly in unison with the successive tones of the major scale. These tones may, in fact, be used for the tones of the major scale ; and since the intervals of the tempered scale are all equal, it is clear that any tone of the tempered scale may be chosen as a tonic, and that the third, fifth, sixth, eighth, tenth, twelfth, and thirteenth tones, counting from the chosen one, con- stitute a major scale. All such major scales are equally well in tune. Indefinite modulations may be carried out on this scale inasmuch as any tone reached in a modulation has a group of tones related to it as a tonic. The minor scale may be made up also from the tempered scale and indefinite modulations may be performed. OF THB DIVERSITY ^ -1 I VER! V ',- , INDEX. NUMBERS RELATE TO PAGE. ABBE'S LAW, 57. Aberration, chromatic, 58. spherical, 43, 56. Absorption, coefficients of, 144. selective, 138. Accommodation of the eye, 65. Accords, major and minor, 186. Achromatic lenses, 58, 74. Air columns, vibration of, 1 54. Amplitude of vibration, 148. of a wave train, 13. Anastigmatic lens systems, 63. Angle, the polarizing, 127. visual, 65. of incidence defined, 37. of a lens, 56. of refraction defined, 37. Antinodes, defined, 15. Aperture, defined, 55. Aplanatic points, 57. surface, defined, 43. Aqueous humor, 64. Astigmatic pencils, 24. Astigmatism of lenses, 59. Audibility, limits of, 150. Auditory nerves, 2. Angstrom units, 99. BASILAR MEMBRANE, THE, 174. Beats, nature of, 175. the method of, 150. Becquerel on radiation from uranium, 136. Becquerel rays, the, 146. Bell, clang of a Russian, 163. Bells, overtones of, 163. Black bodies, properties of, 140. Bouguer's principle, 116. Bravais and Martins, velocity of sound, 5. Brewster's magnifier, 61. Bright line spectra, 77. Brightness and color, 101. distribution of, 1 20. intrinsic, 115. standards of, 112. Bunsen's photometer, 117. CAMERA, THE PHOTOGRAPHIC, 66. Candles, British and German, 113. Carcel lamp, the, 1 14. Caustic, defined, 30. Change of phase by reflection, 17. Chemical effects of radiation, 136. Chladni's figures, 164. Chords, major and minor, 1 86. Chromatic aberration, 58. Circular polarization, 126. Clang, analysis of, 169. defined, 153. of a Russian bell, 163. Clarionet, the, 158. Cochlea, the, 174. Coefficients of absorption, 144. Collinator, 76. Color and brightness, 101. Color blindness, 105. testing for, 1 10. Color by absorption, 141. by interference, 103. by selective radiation, 103. by selective reflection and transmis- sion, 103. mixing, 104. '95 196 ELEMENTS OF PHYSICS. Color sensations, the primary, 105. the Young-Helmholtz theory of, 105. top, 105. Colors due to homogeneous light, 102. due to mixed light, 102. of thin plates, 86. saturated, 107. surface, 141. Combination tones, 176. Composition of light, how measured, 103. Concave and convex mirrors, 31. Concave gratings, 99. Conjugate foci, of a lens, 47. of a mirror, 32. out of axis, 34. of a lens system, 50. planes, 34. points out of axis, 48. Conjugates, geometrical construction for, 49. Consonance of tones, 159, 182. Consonant intervals, 184. Continuous spectra, 77. Contrast effects, 107. Convergent lenses, 45. Cornea, the, 64. Cornet, the, 159. Cornu's measurement of the velocity of light, 7. Corpuscular theory of light, 2. Correction for spherical aberration, 56. Corrections, simultaneous, of lenses, 61. Crehore and Squier, on the photochrono- graph, 134. Crova's method in photometry, 123. Crystals, axial and biaxial, 130. Curvature of field, 61. DAMPING OF VIBRATIONS, 167. Dark line spectra, 78. Daylight and gaslight compared, 103. Diaphragms, motion of, 165. Dichroic vision, 104. peculiarities of, 108. Difference tones, 177. Diffraction, defined, 88. gratings, 94. Diffraction past an edge, 89. past the edges of a narrow strip, 93. through a slit, 91. of sound and the sense of direction, 179. Direct vision spectroscopes, 74. Direction of sounds, the sense of, 179. Discord and beats, 176. Dispersion, the Helmholtz theory of, 144. resonant, 145. Displacement of fringes, 83. Distortion of lenses, 60. Distribution of brightness, 1 20. Divergent lenses, 45. Double refraction, 128. Doublet, defined, 49. Huygens', 62. Ramsden's, 62. Wollaston's, 62. Doublets, aplanatic, 57. symmetrical, 60. EAR, THE, 173. Echo, the, 177. Elliptical mirrors, 31. polarization, 126. Emission, selective, 138. Emission and absorption, equality of, 137- Energy, distribution of, in spectrum, 144. radiant, 135. stream in a wave train, 13. End organs, I. Ether, the luminiferous, 3. Exchanges, the principle of, 136. Expression, musical, 190. Extraordinary ray, the, 129. Eye, the, 64. Eyepieces used in practice, 61. FARSIGHTEDNESS, 65. Field, angle, of lenses, 56. curvature and flatness of, 61. Fizeau and Cornu on the velocity of light, 7. Flatness of field, 61. Flicker photometer, the, 124. INDEX. 197 Fluorescence and phosphorescence, 146. Focal length of a mirror, 33. Foci, virtual and real, 33. of a lens, 47. of a mirror, 32. Foucault, velocity of light, 8. Fourier's theorem, 18. applied to vibration, 148. Fraunhofer lines, 78. Frequency of a wave motion, 12. of vibration, defined, 147. Fringes, displacement of, 83. Fundamental tone, of a pipe, 155. of a string, 160. GASLIGHT AND DAYLIGHT COMPARED, 103. German candles, 113. Grating, the concave, 99. the diffraction, 94. Greeley, velocity of sound at low temper- atures, 4. HALF-PERIOD ZONES, 21, 88. Harmonic overtones, 153. Harmony and modulation, 191. Hastings' magnifier, 62. Hearing at a distance, influence of wind and temperature on, 178. Heat, radiant, 135. Hefner lamp, the, 115. Helmholtz's theory of dispersion, 144. Holmgren test for color blindness, 1 10. Homocentric pencils, 24. Homogeneous light, 73. color due to, 102. luminosity of, IOI. Huygens' construction for wave front, 20. doublet, 62. principle, 19. principle applied to reflection and refraction, 26. theory of double refraction, 129. ILLUMINATION, INTENSITY OF, 115. Image in a plane mirror, 29. Images, distortion of, 60. magnification of, 35. in spherical mirrors, 34. Incidence, angle of, 37. Index of refraction, 37. Infra-red rays, 73. Interference, color by, 103. from similar sources, 82. of sound, 175. fringes, defined, 83. Lloyd's mirror for, 86. Newton's arrangement for, 85. Intervals of pitch, 181. Inverse squares, law of, 112. KIRCHHOFF'S LAW, 137. Kirchhoff and Bunsen on reversal of lines, 79- Koenig on color curves, 107. Krigar-Menzel and Raps' experiment, 163. Kundt's experiment, 160. LABYRINTH OF THE EAR, 174. Langley's bolometer, 142. Lantern, the magic, 66. Law of aplanatism (Abbe), 57. of Kirchhoff, 137. of normal radiation, 137. of Snell, 37. of inverse squares, 112. Lens, foci of a, 47. of the eye, 64. Lens systems, anastigmatic, 63. defined, 49. foci of, 51. inverse principal planes of, 52. nodal points of, 53. principal planes of, 51. specification of, 50. Lenses, achromatic, 38, 74. distortion of, 60. diverging and converging, 45. field angle of, 56. numerical aperture of, 55. orthoscopic, 60. rectilinear, 60. 198 ELEMENTS OF PHYSICS. Light, corpuscular theory of, 2. homogeneous or monochromatic, 73. the sensation of, 2. standards, 112. velocity of, 7. wave theory of, 3. white, defined, IO2. Limits of audibility, 1 50. Lissajous' figures, 152. Lloyd's mirrors, 86. Longitudinal transverse waves, 9. Loudness of sound, 150. Luminescence, 146. Luminiferous ether, 3. Luminosity, curve of, for the red-blind eye, 109. Luminosity of homogeneous light, 101. Lummer-Brodhum photometer, 119. MAGIC LANTERN, 66. Magnification of images, 35. Magnifying glasses, 61, 67. Magnifying power, defined, 67. of microscopes, 68. Major and minor accords, 186. Major scale, the, 189. Manometric flames, 165. Measurement of radiant heat, 142. Melody and harmony, 191. Membrane, the basilar, 174. Methven screen, the, 114. Michelson's measurement of the velocity of light, 8. Micron, the, 99. Microscope objectives, 62. Microscopes, compound, 68. magnifying power of, 68. simple, 67. Minimum deviation, 38. Minor accords, 186. Minor scale, the, 190. Mirror, foci of a, 32. Mirrors, concave and convex, 31. elliptical, 31. images in convex, 34. parabolic, 31. plane and spherical, 29. Modulation, 191. Monochromatic light, 73. Musical expression, 190. scales, 187. tones defined, 148. NEARSIGHTEDNESS, 65. Nerves, the auditory, 2. the optic, i. the sensory, I. Newcomb's measurement of the velocity of light, 8. Newton, seven spectrum colors of, 102. Newton's rings, 88. Nichols radiometer, the, 143. spectrophotometer, the, 80. Nicol prism, the, 131. Nicol prisms in photometry, 122. Nodal points of a lens system, 53. Nodes, defined, 14. Noises, defined, 149. Normal and prismatic spectra, 80. Normal radiation, the law of, 137. OBJECTIVES FOR MICROSCOPES, 62. for telescopes, 63. photographic, 63. Opaque bodies, defined, 112. Opera glasses, 72. Optic nerves, I, 64. Ordinary and extraordinary rays, 129. Organ pipes, 157. Orthoscopic lenses, 60. Overtones, defined, 153. PARABOLIC MIRRORS, 31. Pencils, homocentric and astigmatic, 24. Penumbra, the, 23. Period of a wave motion, 12. of a vibration, 148. Phase, change of, by reflection, 17. of vibration, 148. of a wave train, 13. Phonautograph, the, 172. Phonograph, the, 171. Phosphorescence and fluorescence, 146. Photographic camera, 66. Photographic objectives, 63. INDEX. 199 Photometer, the Bunsen, 117. the flicker, 124. the Lummer-Brodhun, 119. the Rumford, 117. the shadow, 1 1 7. Photometry, simple, 116. of light differing in composition, 122. Whitman's method, 124. Pipe, the organ, 157. Pitch, defined, 150. influence of relative motion on, 179. intervals, 181. -measurement of, 150. Plane of polarization, 127. of polarization, the rotation of, 133. waves, refraction of, 36. Planes, conjugate, 34. Polariscope, the, 132. Polarization, defined, 125. circular and elliptical, 126. by double refraction, 131. by reflection, 126. by tourmaline, 126. Polarizing angle, the, 127. Potassium chromate, color of, 104. Prevost's principle of exchanges, 136. Primary color sensations, 105. Principal foci of a lens system, 51. Principal focus of a lens, 45. of a mirror, 32. Principal planes of a lens system, 51. Principle of exchanges, the, 136. Prism, defined, 38. the Nicol, 131. Proper stimuli, i. RADIANT ENERGY, 135. Radiant heat, 135. luminous and chemical effects of, 136. measurement of, 142. Radiation in a closed system, 137. color by, 103. the law of normal, 137. Radiometer, the, 143. Ramsden's doublet, 62. Ray, the ordinary, 128. Rays, defined, 22. Real and virtual foci, 33. Rectilinear lenses, 60. Reed pipes, 158. Reflection, application of Huygens' prin- ciple to, 26. color by, 103. from a plane surface, 27. polarization by, 126. regular and diffuse, 26. selective, 138. total, 41. with and without change of phase, 17- Refraction, defined, 26, angle of, 37. application of Huygens' principle to, 26. double, 128. at spherical surfaces, 42. of a plane wave, 36. of spherical waves, 38. Refractive index, 37. Regnault's measurement of the velocity of sound, 5. Resonance and resonators, 168. Resonant dispersion, 145. Retina, the, 64. Reversal of sodium lines, 79. Rhythm, 190. Rods and strings, longitudinal vibrations of, 159- Roemer's measurement of the velocity of light, 7. Rotation of the plane of polarization, 133. Rubens and Nichols, on radiation of long waves, 135. Rumford's photometer, 116. Russian bell, clang of, 163. SACCHARIMETER, THE, 133. Scale, the tempered, 192. Scales, major and minor, 189. musical, 187. Segments, defined, 15. Selective absorption, 138. emission, 138. reflection, 138. 200 ELEMENTS OF PHYSICS. Selective transmission, 138. Sensations, defined, I. of light, 2. of sound, 2. Sensory nerves, I. Shadow, the geometrical, 23. photometers, 116. Shadows, defined, 23. Sharp and Turnbull on light standards, 114. Siren, the, 150. Snell's law, 37. Snow, on lines in the infra-red, 78. Solar spectrum, the, 78. Sound rays, 22. fringes, 86. interference of, 175. loudness of, 150. the sensation of, 2. persistence of, 175. Sound shadows, 23. Sound, velocity in air, 4. velocity at low temperatures, 4. velocity at high and low levels, 5. velocity in metals, 5. velocity in water and glass, 6. wave theory of, 3. Sounding boards, use of, 163. Specification of a lens system, 51. Speech, reproduction of, 171. Spectra, bright line, 77. continuous, 77. dark line, 78. normal and prismatic, 80. Spectrometer, the grating, 97. Spectrometers, 79. Spectroscopes, described, 76. Spectro-photometers, 80. Spectro-photometry, Crova's method in, 123. method of the Vierordt slit, 122. Spectroscope, the direct vision, 74. Spectrum, the, 73. appearance of, to color-blind observ- ers, 109. energy, curves of the, 143. the solar, 78. Spherical aberration, 43, 56. mirrors, 29. waves, refraction of, 38. Spinney ; curves of wave motions, 18. Spy glasses, 72. Standards of brightness, 112. Stationary wave trains, 14. Stevens and Mayer on sound fringes, 86. Stimulus, defined, I. Strings and rods, longitudinal vibrations of, 159. Strings, simple and compound, vibrations of, 162. transverse vibration of, 160. Summation tones, 177. Sun, the spectrum of, 78. Superposition, the principle of, 14. Surface color, 141. Systems of lenses, 49. TELESCOPES, 70. magnifying power of, 71. Tempered scale, the, 192. Thin plates, colors of, 86. Timbre, defined, 152. Tones, combination, 176. consonance of, 182. musical, 148. summation and difference, 177. Total reflection, 41. Tourmaline, optical behavior of> 126. Transmission, color by, 103. selective, 138. Transparent and translucent bodies de- fined, 112. Transverse and longitudinal waves, 9. vibrations of strings, 160. Trichroic vision, 104. Triplet, the, defined, 49. Tuning forks, the motion of, 164. overtones of, 163. Turnbull and Sharp on light standards, 114. Tympanic membrane, the, 174. ULTRAMARINE BLUE, COLOR OF, 104. Ultra-violet rays, 73. INDEX. 201 Umbra and penumbra, 23. Unit, the Angstrom, 99. VELOCITY OF LARGE AND SMALL WATER WAVES, 1 80. of light, 7. of sound, 4. of sound (independent of wave length), 1 80. of sound in water and glass, 6. of sound in metals, 6. Vibrating segments, defined, 15. Vibration, amplitude and phase of, 148. of air columns, 154. of a particle, 147. period of, 148. Vibrations, damping of, 167. proper and impressed, 167. simple and compound, 147, 148. of rods and strings (longitudinal), 159. of rods and plates (transverse), 183. Vibrations, of strings (simple and com- pound), 162. Vierordt's slit, 122. Violle standard of light, the, 115. Virtual and real foci, 33. Vision, dichroic and trichroic, 104. peculiarities of dichroic, 108. Visual angle, 65. Vitreous humor, the, 64. Vocal organs, the, 159. Vowel sounds analyzed, 169. how produced, 159. WATER WAVES, VELOCITIES OF, 180. Wave front, defined, 19. Huygens' construction of, 2O. Wave length, defined, 12. measurement of, 97. Wave motion, equations of, 10. Wave theory of light, 3. Wave trains, 12. simple and compound, 18. stationary, 14. Waves, nature of, 9. White bodies, properties of, 140. light, defined, 102. Whitman's flicker photometer, 124. Wollaston's doublet, 62. Word, manometric flame-image of, 166. YELLOW SPOT, THE, 65. Young-Helmholtz theory of color, the, 105. ZONE PLATES, 93. Zones, half-period, 21, 88. THE ELEMENTS OF PHYSICS. BY EDWARD L. NICHOLS, B.S., Ph.D., Professor of Physics m Cornell University, AND WILLIAM S. FRANKLIN, M.S., Professor of Physics and Electrical Engineering at the Iowa Agricultural College, Ames, la. WITH NUMEROUS ILLUSTRATIONS. (Vol. I. ;: j II. ( III. Mechanics and Heat. PART I. In Three Volumes : -J II. Electricity and Magnetism. Sound and Light. Price $1.50, net, per volume. It has been written with a view to providing a text-book which shall correspond with the increasing strength of the mathematical teaching in our university classes. In most of the existing text-books it appears to have been assumed that the student possesses so scanty a mathematical knowledge that he cannot understand the natural language of physics, i.e., the language of the calculus. Some authors, on the other hand, have assumed a degree of mathematical training such that their work is unreadable .for nearh/ all under- jjraduates. The present writers having had occasion to teach large classes, the members of which were acquainted with the elementary principles of the calculus, have sorely felt the need of a text-book adapted to their students. The present work is an attempt on their part to supply this want. It is believed that in very many institutions a similar condition of affairs exists, and that there is a demand for a work of a grade intermediate between that of the existing elementary texts and the advanced manuals of physics. No attempt has been made in this work to produce a complete manual or compendium of experimental physics. The book is planned to be used in connection with illustrated lectures, in the course of which the phenomena are demonstrated and described. The authors have accordingly confined themselves to a statement of principles, leaving the lecturer to bring to notice the phenomena based upon them. In stating these principles, free use has been made of the calculus, but no demand has been made upon the student beyond that supplied by the ordinary elementary college courses on this subject. Certain parts of physics contain real and unavoidable difficulties. These have not been slurred over, nor have those portions of the subject which contain them been omitted. It has been thought more serviceable to the student and to the teacher who may have occa- sion to use the book to face such difficulties frankly, reducing the statements involving them to the simplest form which is compatible with accuracy. In a word, the Elements of Physics is a book which has been written for use in such institutions as give their undergraduates a reasonably good mathematical training. It is intended for teachers who desire to treat their subject as an exact science, and who are prepared to supplement the brief subject-matter of the text by demonstration, illustration, and discussion drawn from the fund of their own knowledge. THE MACMILLAN COMPANY. NEW YORK: CHICAGO: 66 FIFTH AVENUE. ROOM 23, AUDITORIUM. A Laboratory flanual OF Physics and Applied Electricity. ARRANGED AND EDITED BY EDWARD L. NICHOLS, Professor of Physics in Cornell University. IN TWO VOLUMES. VoL L JUNIOR COURSE IN GENERAL PHYSICS. BY ERNEST MERRITT and FREDERICK J. ROGERS. Cloth. $3.00. VoL IL SENIOR COURSES AND OUTLINE OF ADVANCED WORK. BY GEORGE S. MOLER, FREDERICK BEDELL, HOMER J. HOTCHKISS, CHARLES P. MATTHEWS, and THE EDITOR. Cloth, pp. 444. $3.25. The first volume, intended for beginners, affords explicit directions adapted to a modern laboratory, together with demonstrations and elementary statements of principles. It is assumed that the student possesses some knowledge of analytical geometry and of the cal- culus. In the second volume more is left to the individual effort and to the maturer intel- ligence of the practicant. A large proportion of the students for whom primarily this Manual is intended, are pre- paring to become engineers, and especial attention has been devoted to the needs of that class of readers. In Vol. II., especially, a considerable amount of work in applied elec- tricity, in photometry, and in heat has been introduced. COMMENTS. " The work as a whole cannot be too highly commended. Its brief outlines of the various experiments are very satisfactory, its descriptions of apparatus are excellent; its numerous suggestions are calculated to develop the thinking and reasoning powers of the student. The diagrams are carefully prepared, and its frequent citations of original sources of information are of the greatest value." Street Railway Journal. " The work is clearly and concisely written, the fact that it is edited by Professor Nichols being a sufficient guarantee of merit." Electrical Engineering. " It will be a great aid to students. The notes of experiments and problems reveal much original work, and the book will be sure to commend itself to instructors." San Francisco Chronicle. THE MACMILLAN COMPANY, NEW YORK: CHICAGO: 66 FIFTH AVENUE. ROOM 23, AUDITORIUM. A LABORATORY MANUAL OF EXPERIMENTAL PHYSICS, BY W. J. LOUDON and J. C. McLE*NNAN, Demonstrators in Physics, University of Toronto, Cloth. 8vo. pp. 302. $1.90 net. FROM THE AUTHORS PREFACE. At the present day, when students are required to gain knowledge of natural phe- nomena by performing experiments for themselves in laboratories, every teacher rinds that as his classes increase in number, some difficulty is experienced in providing, during a limited time, ample instruction in the matter of details and methods. During the past few years we ourselves have had such difficulties with large classes; and that is our reason for the appearance of the present work, which is the natural outcome of our experience. We know that it will be of service to our own students, and hope that it will be appreciated by those engaged in teaching Experimental Physics elsewhere. The book contains a series of elementary experiments specially adapted for stu- dents who have had but little acquaintance with higher mathematical methods : these are arranged, as far as possible, in order of difficulty. There is also an advanced course of experimental work in Acoustics, Heat, and Electricity and Magnetism, which is intended for those who have taken the elementary course. The experiments in Acoustics are simple, and of such a nature that the most of them can be performed by beginners in the study of Physics; those in Heat, although not requiring more than an ordinary acquaintance with Arithmetic, are more tedious and apt to test the patience of the experimenter; while the course in Electricity and Magnetism has been arranged to illustrate the fundamental laws of the mathematical theory, and involves a good working knowledge of the Calculus. THE MACMILLAN COMPANY, 66 FIFTH AVENUE, NEW YORK. OF THK TERSITT UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed below. Afty YC 32681