fyxmll ^nxvmxiia ^itatg THE GIFT OF i:>.SX.^.c/iXxjs^..o^hnS^^ il.3l.3151 ,.■---.-. kiTlilc Cornell University Library arW3843 Elementary text-books of physics. 3 1924 031 362 787 olln.anx The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031362787 ELEMENTARY Text-Book of Physics. PROFESSOR WILLIAM A. ANTHONY, II OF CORNELL UNIVERSITY, PROFESSOR CYRUS F. BRACKETT, OF ^rtB (.ULLEGE OF NEW JERSEY. SEVENTH EDITION, REVISED AND ENLARGED. NEW YORK: JOHN WILEY & SONS, 53 East Tenth Street. i8gi. Copyright, i887t ov John Wiley & Sons. Drummond & Neu, Ferris Bros., Electrotypera, Prin f ej-s, 1 to 7 Hague Street, 326 Pearl Street, New York, New York. PREFACE. The design of the authors in the preparation of this work has teen to present the fundamental principles of Physics, the experimental basis upon which they rest, and, so far as possible, the methods by which they have been established. Illustra- tions of these principles by detailed descriptions of special methods of experimentation and of devices necessary for their applications in the arts have been purposely omitted. The authors believe that such illustrations should be left to the lec- turer, who, in the performance of his duty, will naturally be guided by considerations respecting the wants of his classes and the resources of his cabinet. Pictorial representations of apparatus, which can seldom be employed with advantage unless accompanied with full and exact descriptions, have been discarded, and only such simple diagrams have been introduced into the text as seem suited to aid in the demonstrations. By adhering to this plan greater economy of space has been secured than would other- wise have been possible, and thus the work has been kept within reasonable limits. A few demonstrations have been given which are not usually iv PREFACE. found in elementary text-books, except those which are much more extended in their scope than the present work. This has been done in every case in order that the argument to which the demonstration pertains may be complete and that the stu- dent may be convinced of its validity. In the discussions the method of limits has been recognized wherever it is naturally involved ; the special methods of the calculus, however, have not been employed, since, in most insti- tutions in this country, the study of Physics is commenced be- fore the student is sufficiently familiar with them. The authors desire to acknowledge their obligations to Wm. F. Magie, Assistant Professor of Physics in the College of New Jersey, who has prepared a large portion of the manuscript and has aided in the final revision of all of it, as well as in reading the proof-sheets. W. A. Anthony, C. F. Brackett. September, 1887 CONTENTS. PAGE Introduction, i MECHANICS. Chapter I. Mechanics of Masses, ii II. Mass Attraction 67 III. Molecular Mechanics, 84 IV. Mechanics of Fluids, 120 HEAT. Chapter I. Measurement of Heat 143 II. Transfer of Heat i6i III. Effects of Heat, i68 IV. Thermodynamics, 205 MAGNETISM AND ELECTRICITY. Chapter I. Magnetism, II. Electricity in Equilibrium, III. The Electrical Current, . IV. Chemical Relations of the Current, V. Magnetic Relations of the Current, VI. Thermo-electric Relations of the Current, VII. Luminous Effects of the Current, 223 246 272 282 297 340 348 SOUND. Chapter I. Origin and Transmission of Sound 353 II. Sounds and Music, 365 III. Vibrations of Sounding Bodies 371 IV. Analysis of Sounds and Sound Sensations, . . . 380 V. Effects of the Coexistence of Sounds 385 VI. Velocity of Sound, 3go vi CONTENTS. LIGHT. Chapter I. Propagation of Light, II. Reflection and Refraction, III. Velocity of Light IV. Interference and Diffraction, V. Dispersion, .... VI. Absorption and Emission, . VII. Double Refraction and Polarization, TABLES. Table I. Units of Length 498 II. Acceleration of Gravity, 498 III. Units of Work 498 IV. Densities of Substances at 0°, 499 V. Units of Pressure for ^ = 981, 499 VI. Elasticity 499 VII. Absolute Density op Water 500 VIII. Density of Mercury 500 IX. Coefficients of Linear Expansion, 500 X. Specific Heats — Water at 0° = i, . . . . 501 XI. Melting and Boiling Points, etc 501 XII. Maximum Pressure of Vapor at Various Temperatures, 502 XIII. Critical Temperatures and Pressures in Atmospheres, AT THEIR Critical Temperatures, of Various Gases, 502 XIV. Coefficients of Conductivity for Heat in C. G. S. Units, 502 XV. Energy Produced by Combination of i Gram of Certain Substances with Oxygen 503 XVI. Atomic Weights and Combining Numbers, . . . 503 XVII. Molecular Weights and Densities of Gases, . . 503 XVIII. Electromotive Force of Voltaic Cells 504 XIX. Electro-chemical Equivalents, 504 XX. Electrical Resistance, 504 XXI. Indices of Refraction 505 XXII. Wave Lengths of Light — Rowland's Determinations, 505 XXIII. Rotation of Plane of Polarization by a Quartz Plate, I mm. Thick, Cut Perpendicular to Axis, . . . 506 XXIV. Velocities of Light, and Values of t/, . . . 506 Index 507 INTRODUCTION. I. Divisions of Natural Science. — Everything which can- affect our senses we call matter. Any limited portion of mat- ter, however great or small, is called a body. All bodies, to- gether with their unceasing changes, constitute Nature. Natural Science makes us acquainted with the properties of bodies, and with the changes, or phenomena, which result from their mutual actions. It is therefore conveniently divided into two principal sections, — Natural History and Natural Philosophy. The former describes natural objects, classifies them accord- ing to their resemblances, and, by the aid of Natural Philoso- phy, points out the laws of their production and development. The latter is concerned with the laws which are exhibited in the mutual action of bodies on each other. These mutual actions are of two kinds : those which leave the essential properties of bodies unaltered, and those which effect a complete change of properties, resulting in loss of identity. Changes of the first kind are called physical changes ; those of the second kind are called chemical changes. Nat- ural Philosophy has, therefore, two subdivisions, — Physics and Chemistry. Physics deals with all those phenomena of matter which are not directly related to chemjcal changes. Astronomy is thus a branch of Physics, yet it is usually excluded from works like the present on account of its special character. 2 ELEMENTARY PHYSICS. [2 It is not possible, however, to draw sharp lines of demarca- tion between the various departments of Natural Science, for the successful pursuit of knowledge in any one of them re- quires some acquaintance with the others. 2. Methods. — The ultimate basis of all our knowledge of nature is experience, — experience resulting from the action of bodies on our senses, and the consequent affections of our minds. When a natural phenomenon arrests our attention, we call the result an observation. Simple observations of natural phe- nomena only in rare instances can lead to such complete knowledge as will suffice for a full understanding of them. An observation is the more complete, the more fully we appre- hend the attending circumstances. We are generally not cer- tain that all the circumstances which we note are conditions on which the phenomenon, in a given case, depends. In such cases we modify or suppress one of the circumstances, and ob- serve the effect on the phenomenon. If we find a correspond- ing modification or failure with respect to the phenomenon, we conclude that the circumstance, so modified, is a condition. We may proceed in the same way with each of the remaining circumstances, leaving all unchanged except the single one purposely modified at each trial, and always observing the ef- fect of the modification. We thus determine the conditions on which the phenomenon depends. In other words, we bring experiment to our aid in distinguishing between the real condi- tions on which a phenomenon depends, and the merely acci- dental circumstances which may attend it. But this is not the only use of experiment. By its aid we may frequently modify some of the conditions, known to be conditions, in such ways that the phenomenon is not arrested, but so altered in the rate with which its details pass before us that they may be easily observed. Experiment also often leads to new phenomena, and to a knowledge of activities be 2l INTKODUCTION. fore unobserved. Indeed, by far the greater part of our knowl- edge of natural phenomena has been acquired by means of ex- periment. To be of value,' experiments must be conducted with .system, and so as to trace out the whole course of the phenomenon. Having acquired our facts by observation and experiment, we seek to find out how they are related ; that is, to discover the laws which connect them. The process of reasoning by which we discover such laws is called induction. As we can seldom be sure that we have apprehended all the related facts, it is clear that our inductions must generally be incomplete. Hence it follows that conclusions reached in this way are at best only probable ; yet their probability becomes very great when we can discover no outstanding fact, and especially so when, regarded provisionally as true, they enable us to predict phenomena before unknown. In conducting our experiments, and in reasoning upon them, we are often guided by suppositions suggested by previous experiencr. If the course of our experiment be in accordance with our supposition, there is, so far, a presumption in its favor. So, too, in reference to our reasonings : if all our facts are seen to be consistent with some supposition not unlikely in itself, we say it thereby becomes probable. The term hypothesis is usually employed instead of supposition. Concerning the ultimate modes of exi.stence or action, we know nothing whatever; hence, a law of nature cannot be •demonstrated in the sense that a mathematical truth is demon- strated. Yet so great is the constancy of uniform sequence with which phenomena occur in accordance with the laws which we discover, that we have no doubt respecting their validity. When we would refer a series of ascertained laws to some common agency, we employ the term theory. Thus we find in the "wave theory" of light, based on the hypothesis of a uni- 4 ELEMENTARY PHYSICS. [3 versal ether of extreme elasticity, satisfactory explanations of the laws of reflection, refraction, diffraction, polarization, etc. 3. Measurements. — All the phenomena of nature occur in matter, and are presented to us in time and space. Time and space are fundamental conceptions : they do not admit of definition. Matter is equally indefinable: its distinc- tive characteristic is its persistence in whatever state of rest or motion it may happen to have, and the resistance which it of- fers to any attempt to change that state. This property is called inertia. It must be carefully distinguished from inac- tivity. Another essential property of matter is impenetrability, or th^ property of occupying space to the exclusion of other matter. We are almost constantly obliged, in physical science, to measure the quantities with which we deal. We measure a quantity when we compare it with some standard of the same kind. A simple number expresses the result of the com- parison. If we adopt arbitrary units of length, time, and mass (or quantity of matter), we can express the measure of all other quantities in terms of these so-called fundamental units. A unit of any other quantity, thus expressed, is called a derived unit. It is convenient, in defining the measure of derived units, to speak of the ratio between, or the product of, two dissimilar quantities, such as space and time. This must always be un- derstood to mean the ratio between, or the product of, the numbers expressing those quantities in the fundamental units. The result of taking such a ratio or product of two dissimilar quantities is a number expressing a third quantity in terms of a derived unit. 4. Unit of Length. — The unit of length usually adopted in scientific work is the centimetre. It is the one hundredth part 4] INl^RODUCTION. of the length of a certain piece of platinum, declared to be a standard by legislative act, and preserved in the archives of France. This standard, called the metre, was designed to be equal in length to one ten-millionth of the earth's quadrant. The operation of comparing a length with the standard is often difficult of direct accomplishment. This may arise from the minuteness of the object or distance to be measured, from the distant point at which the measurement is to end being inaccessible, or from the difficulty of accurately dividing our standard into very small fractional parts. In all such cases we have recourse to indirect methods, by which the difficulties are more or less completely obviated. The vernier enables us to estimate small fractions of the unit of length with great convenience and accuracy. It con- sists of an accessory piece, fitted to slide on the principal scale of the instrument to which it is applied. A portion of the ac- cessory piece, equal to n minus one or n plus one divisions of the principal scale, is divided into n divisions. In the former case, the divisions are numbered in the same sense as those of the principal scale ; in the latter, they are numbered in the opposite sense. In either case we can measure a quan- tity accurately to the one «th part of one of the primary divisions of the principal scale. Fig. i will make the construction and use of the ver- nier plain. In Fig. I, let o, I, 2, 3 ... lo be the di- visions on the vernier; let o, i, 2, 3 . . . lO be any set of consecutive divisions on the principal scale. If we suppose the o of the vernier to be in coincidence with the limiting point of the mag- nitude to be measured, it is clear that, from the position shown in the figure, we have 29.7, expressing that magnitude Fig. 6 ELEMENTARY PHYSICS. [4 to the nearest tenth ; and since the sixth division of the ver- nier coincides with a whole division of the principal scale, we have 3% of J^, or y^, of a principal division to be added ; hence the whole value is 29.76. The micrometer screw is also much employed. It consists of a carefully cut screw, accurately fitting in a nut. The head of the screw carries a graduated circle, which can turn past a fixed line. This is frequently the straight edge of a scale with divisions equal in magnitude to the pitch of the screw. These divisions will then show through how many revolutions w^^pD the screw is turned in any given trial ; while the divisions on the graduated circle will show the fractional part of a revolu- tion, and consequently the frac- tional part of the .pitch that must be added. If the screw be turned through n revolutions, as shown by the scale, and through an additional fraction, as shown by the divided circle, it will pass through n times the pitch of the screw, and an ad- ditional fraction of the pitch deter- mined by the ratio of the number of divisions read from o on the di- vided circle to the whole number into which it is divided. The cathctometer is used for measuring differences of level. A graduated scale is cut on an up- right bar, which can turn about a vertical axis. Over this bar sHde 'two accurately fitting pieces, one of which can be clamped to the bar at any point, and serve as the fixed bearing of a micrometer screw. The screw runs in a nut in the second piece, which has Fig. 2. 4] INTRODUCTION. a vernier attached, and carries a horizontal telescope furnished with cross-hairs. The telescope having been made accurately horizontal by means of a delicate level, the cross-hairs are made to cover one of the two points, the difference of level be- tween which is sought, and the reading upon the scale is taken ; the fixed piece is then undamped, and the telescope raised or lowered until the second point is covered by the cross-hairs, and the scale reading is again taken. The difference of scale reading is the difference of level sought. The dividing engine may be used for dividing scales or for Fig. 3. comparing lengths. In its usual form it consists essentially of a long micrometer screw, carrying a table, which slides, with a motion accurately parallel with itself, along fixed guides, resting on a firm support. To this table is fixed an apparatus for making successive cuts upon the object to be graduated. The object to be graduated is fastened to the fixed sup- port. The table is carried along through any required dis- 8 ELEMENTARY PHYSICS. is tance determined by the motion of tlie screw, and the cuts can be thus made at the proper intervals. The same instrument, furnished with microscopes and ac- cessories, may be employed for comparing lengths with a standard. It may then be called a comparator. The spherometer is a special form of the micrometer screw. As its name implies, it is primarily used for measuring the cur- vature of spherical surfaces. It consists of a screw with a large head, divided into a great number of parts, turning in a nut supported on three legs terminating in points, which form the vertices of an equi- lateral triangle. The axis of revolution of the screw is per- pendicular to the plane of the triangle, and passes through its centre. The screw ends in a point which may be brought into the same plane with the points of the legs. This is done by plac- ing the legs on a truly plane sur- face, and turning the screw till its point is just in contact with the sur- face. The sense of touch will en- able one to decide with great nicety when the screw is turned far enough. If, now, we note the reading of the divided scale, and also that of the divided head, and then raise the screw, by turning it backward, so that the given curved surface may exactly coincide with the four points, we can compute the radius of curvature from the difference of the two readings and the known length of the side of the triangle formed by the points of the tripod. 5. Unit of Time. — The unit of time is the mean time second, which is the ^-g-^ of a mean solar day. We employ the clock, regulated by the pendulum or the chronometer balance, to indicate seconds. The clock, while sufificiently ac Fig. 4. 7] INTRODUCTION. 9 curate for ordinary use, must for exact investigations be fre- quently corrected by astronomical observations. Smaller intervals of time than the second are measured by causing some vibrating body, as a tuning-fork, to trace its path along some suitable surface, on which also are recorded the beginning and end of the interval of time to be measured. The number of vibrations traced while the event is occurring determines its duration in known parts of a second. In estimating the duration of certain phenomena giving rise to light, the revolving mirror may be employed. By its use, with proper accessories, intervals as small as forty billionths of a second have been estimated. 6. Unit of Mass. — The unit of mass usually adopted in scientific work is the gram. It is equal to the one thousandth part of a certain piece of platinum, called the kilogram, pre- served as a standard in the archives of France. This standard was intended to be equal in mass tA one cubic decimetre of water at its greatest density. Masses are compared by means of the balance, the con- struction of which will be discussed hereafter. 7- Measurement of Angles. — Angles are usually measured by reference to a divided circle graduated on the system of division upon which the ordinary trigonometrical tables are based. A pointer or an arm turns about the centre of the circle, and the angle between two of its positions is measured in degrees on the arc of the circle. For greater accuracy, the readings may be made by the help of a vernier. To facilitate the measurement of an angle subtended at the centre of the" ■ircle by two distant points, a telescope with cross-hairs is mounted on the movable arm. In theoretical discussions the unit of angle often adopted is the radian, that is, the angle subtended by the arc of a circle equal to its radius. In terms of this unit, a semi-circum- ference equals n = 3.141592. The radian, measured in degrees, is 57" 17' 44.8." lO ELEMENTARY PHYSICS. [8 8. Dimensions of Units. — Any derived unit may be repre- sented by the product of certain powers of the symbols repre- senting the fundamental units of length, mass, and time. Any equation showing what powers of the fundamental units enter into the expression for the derived unit is called its dimensional equation. In a dimensional equation time is represented by T, length by L, and mass by M. To indicate the dimensions of any quantity, the symbol representing that quantity is enclosed in brackets. For example, the unit of area varies as the square of the unit of length ; hence its dimensional equation is [area] = L'. In like manner, the dimensional equation for volume is [vol.] = L\ 9. Systems of Units. — The system of units adopted in this book, and generally employed in scientific work, based upon the centimetre, gram, and second, as fundamental units, is called the centimetre-gram-second system or the C. G. S. system. A system based upon the foot, grain, and second was formerly much used in England. One based upon the milli- metre, milligram, and second is still sometimes used in Ger- many. MECHANICS. CHAPTER I. MECHANICS OF MASSES. 10. The general subject of motion is usually divided, in extended treatises, into two topics, — Kinematics and Dy- navtics. In the first are developed, by purely mathematical methods, the laws of motion considered in the abstract, inde- pendent of any causes producing it, and of any substance in which it inheres ; in the second these mathematical relations are extended and applied, by the aid of a few inductions drawn from universal experience, to the explanation of the motions of bodies, and the discussion of the interactions which are the occasion of those motions. For convenience, the subject of Dynamics is further divided into Statics, which treats of forces as maintaining bodies in equilibrium and at rest, and Kinetics, which treats of forces as setting bodies in motion. In this book it has been found more convenient to make no formal distinction between the mathematical relations of motion and the application of those relations to the study of forces and the motions of bodies. The subject is so extensive that only those fundamental principles and results will be pre- sented which have direct application in subsequent parts of the work. II. Mass and Density. — In many cases it is convenient to speak of the quantity of matter in a body as a whole. It is then called the mass of the body. In case the matter is con- tinuously distributed throughout the body, its mass is often 12 EI^MENTARY PHYSICS. [I2 represented by the help of the quantities of matter in its elementary volumes. The density of any substance is defined as the limit of the ratio of the quantity of matter in any volume within the substance to that volume, when the volume is dimin- ished indefinitely. In case the distribution of matter in the body is uniform, its density may be measured by the quantity of matter in unit volume. Since density is measured by a mass divided by a volume, its dimensions are ML ~ ^ 12. Particle. — A body constituting a part of a material system, and of dimensions such that they may be considered infinitely small in comparison with the distances separating it from all other parts of the system, is called 2. particle. ' 13. Motion. — The change in position of a material particle is called its motion. It is recognized by a change in the config- uration of the system containing the displaced particle ; that is. by a change in the relative positions of the particles making up the system. Any particle in the system may be taken as the fixed point of reference, and the motion of the others may be measured from it. Thus, for example, high-water mark on the shore may be taken as the fixed point in determining the rise and fall of the tides; or, the sun may be assumed to be at rest in computing the orbital motions of the planets. We can have no assurance that the particle which we assume as fixed is not really in motion as a part of some larger system ; indeed, in almost every case we know that it is thus in motion. As it is impossible to conceive of a point in space recognizable as fixed and determined in position, our measurements of motion must always be relative. One important limitation of this statement must be made : by proper experiments it is possible to determine the absolute angular motion of a body rotating about an axis. 14. Path. — The moving particle must always describe a continuous line ov path. In all investigations the path maybe IS] MECHANICS OF MASSES. T3 represented by a diagram or model, or by reference to a set of assumed co-ordinates. 15. Velocity. — The rate of motion of a particle is called its velocity. If the particle move in a straight line, and de- scribe equal spaces in any arbitrary equal times, its velocity is constant. A constant velocity is measured by the ratio of the space traversed by the particle to the time occupied in travers- ing that space. If s^ and s represent the distances of the par- ticle from a fixed point on its path at the instants t„ and t, then its velocity is represented by s — s„ If the path of the particle be curved, or if the spaces described by the particle in equal times be not equal, its velocity is z;«r?'a- ble. The path of a particle moving with a variable velocity may be approximately represented by a succession of very small straight lines, which, if the real path be curved, will differ in direction, along which the particle moves with constant velocities which may differ in amount. The velocity in any one of these straight lines is represented by the formula s — s^ V = 7. As the interval of time t — t„ approaches zero, * '0 each of the spaces s — s^ will become indefinitely small, and in the limit the imaginary path will coincide with the real path. s — s The limit of the expression 7° will represent the velocity of the particle along the tangent to the path at the time t = /„, or, as it is called, the velocity in the path. This limit is usually expressed by -^. The practical unit of velocity is the velocity of a body mov- ing uniformly through one centimetre in one second. The dimensions of velocity are LT~^. 14 ELEMENTARY PHYSICS. ^«d 16. Momentum. — The momentum of a body is a quantity which varies with the mass and with the velocity of the body jointly, and is measured by their product. Thus, for example, a body weighing ten grams, and having a velocity of ten centi- metres, has the same momentum as a body weighing one gram, and having a velocity of one hundred centimetres. The prac- tical unit of momentum is that of a gram of matter moving with the unit velocity. The formula is mv, (2) where m. represents mass. The dimensions of momentum are MLT~\ 17. Acceleration. — When the velocity of a particle varies, its rate of change is called the acceleration of the particle. Acceleration is either positive or negative, according as the velocity increases or diminishes. If the path of the particle be a straight line, and if equal changes in velocity occur in equal times, its acceleration is constant. It is measured by the ratio of the change in velocity to the time during which that change occurs. If v^ and v represent the velocities of the par- ticle at the instants t^ and t, then its acceleration is represented by f= —i; (3)- If the path of the particle be curved, or if the changes in velocity in equal times be not equal, the acceleration is variable. It can be easily shown, by a method similar to that used in the discussion of variable velocity, that the limit of the expression V — v. dv ■ _ r = -y-. will represent the acceleration in the path at the time t =: /„. This acceleration is due to a change of velocity in the path. It is not in all cases the total acceleration of the 17] MECHANICS OF MASSES. I? particle. As will be seen in § 37, a particle moving along a curve has an acceleration which is not due to a change of velocity in the path. The practical unit of acceleration is thatof a particle, the ve- locity of which changes by one unit of velocity in one second. The dimensions of acceleration are L T~^ . The space s — j„ traversed by a particle moving with a con- stant acceleration f, during a time t— t^, is determined by considering that, since the acceleration is constant, the aver- V -\- V. age velocity for the time / — t„ multiplied by ^ — t„, will represent the space traversed ; hence ,_,^ = !i±i^»(^_^„); ■ (4) or, smce - = -, we have, in another form, s-s, = v,{t-Q + ifit-t,y. (4) Multiplying equations (3) and (4), we obtain V' = V,' + 2/{s - s„). (5) When the particle starts from rest, v„ = o; and if we take the starting point as the origin from which to reckon s, and the time of starting as the origin of time, then j„ = o, i^ = O, and equations (3), (4), and (S) become v =ft, s = ^/f, and v' = 2/s. Formula (4) may also be obtained by a geometrical con- struction. At the extremities of a line A£ (Fig. 5), equal in length to i — t^, erect perpendiculars AC and BB, proportional to the i6 ELEMENTARY PHYSICS. [I8 AO!)'-(J initial and final velocities of the moving particle, terval of time Aa so short that the veloc- ity during it may be considered constant, the space described is represented by the „ rectangle Ca, and the space described in*^- ' the whole time t — t„, by a point moving with a velocity increasing by successive equal increments, is represented by a series of rectangles, eb, fc, gd, etc., described on equal bases, ab, be, cd, etc. If ab, ^c ... be diminished indefinitely, the sum of the areas of the rectangles can be made to approach as nearly as we please the area of the quadrilateral ABCD. This area, therefore, represents the space traversed by the point, having the initial velocity v^, and moving with the acceleration f, through the time t — t^. But ABCD is equal to AC {t — t,) -\- {BD -AC){t-i,)-^2; whence s-s, = v„{t-t:)-\-if{t-t:)\ (4) i8 Composition and Resolution of Motions, Velocities, and Accelerations. — If a point a^ move with a constant veloc- ity relative to another point «,, and this point a^ move with a constant velocity relative to a third point a,, then the motion, in any fixed time, of «, relative to ^3 may be readily found. Represent the motion, in a fixed time, oia, relative to a, (Fig. 6) by the line v^, and of «, relative to a, by the line v,. Now, it is plain that the motions v, and v„ whether acting succes- sively or simultaneously, will bring the point a, to B\ and also 7B that if any portions of these motions /"» ^-^^ / Ab and be, occurring in any small por- tion of time, be taken, they will, be- cause the velocities of a, and «, are con- F'G' 6- stant or proportional to v^ and v^, bring the point a, to some point c lying on the line joining A and l8] MECHANICS OF MASSES. '7 B. Therefore the diagonal AB of the parallelogram having the side'; v^ and v, fully represents the motion of a^ relative to a.^. The line AB is called the resultant, of which the two lines v^ and 7/3 are the components. This proposition may now be stated generally. The result- ant of any two simultaneous motions, represented by two lines drawn from the point of reference, is found by completing the parallelogram of which those lines are sides ; the diagonal drawn from the point of reference represents the resultant motion. The resultant of any number of motions may be found by obtaining the resultant of any two of the given components, by means of the parallelogram as before shown, using this re- sultant in combination with another component to obtain a new resultant, and proceeding in this way until all the compo- nents have been used. The same result is reached by laying off the components as the consecutive sides of a polygon, when the line required to complete the polygon is the resultant sought. The components of a given motion in any two given direc- tions may be obtained by drawing lines in the two directions from one extremity of the line representing the motion, taken as origin, and constructing upon those lines the parallelogram of which the line representing the motion is the diagonal. The sides drawn from the origin represent the component motions in direction and amount. A motion may be resolved in thr.ee directions not in the same plane by drawing from the extremity of the line repre- senting the motion, taken as origin, Hnes in the three given direc- tions, and constructing upon those lines the parallelopiped of which the Hne representing the motion is the diagonal. The sides of the parallelopiped drawn from the origin represent the required components. Motions are usually resolved along three rectangular axes by means of the trigonometrical functions. Thus, if a be the i8 ELEMENTARY PHYSICS. [19 line representing the motion, and 6, 0, and ^) the angles which it makes with the three axes, the components along those axes are a cos B, a cos -\- b cos 6 =1 X and a sin -f- (J sin 6* = Y\ then the diago- nal of the rectangle, of which X and y are sides, \s R = {X' + Yy ; or, since the angle between the resultant and the axis of X is known hy Y = X tan ip, it follows that It IS evident that this process may be Fig. 7. R 7- or cos y} sin ip' extended to any number of components in the same plane. It is to be noted that the parallelogram law, though only proved for motions, can be shown by similar methods to be applicable to the resolution and composition of velocities and accelerations. 19. Simple Harmonic Motion. — If a point move in a circle with a constant velocity, the point of intersection of a diameter and a perpendicular drawn from the moving point to this diameter will have a simple harmonic moHon. Its velocity at any instant will be the velocity in the circle resolved at that instant parallel to the diameter. The radius of the circle is the amplitude of the motion. The period is the time between any two successive recurrences of a particular condition of the moving-point. The position of a point executing a simple harmonic motion can be expressed in terms of the interval of time which has elapsed since the point last passed through the 19] MECHANICS OF MASSES. 19 middle of its path in the positive direction. This interval of time, when expressed as a fraction of the period, is the phase. We further define rotation in the positive direction as that rotation in the circle which is contrary to the motion of the hands of a clock, or counter-clockwise. Motion from left to right in the diameter is also considered positive. Displace- ment to the right of the centre is positive, and to the left negative. If a point start from X (Fig. 8), the position of greatest positive elongation, with a simple harmonic motion, its distance s from O or its displacement at the end of the time t, during which the point in the circle has moved through the arc BX, is OC = OB cos 0. Now, OB is equal to OX, the amplitude, 27Ct represented by a, and = —j^, where T is the period ; hence s ^ a cos- 27Tt (6) To find the velocity at the F,G. 8. point C, we must resolve the ve- locity of the point moving in the circle into its components parallel to the axes. The component at the point C along OX 27ta is V sWi. 0; or, smce K= "7^' 2Tca sm 27tt (;) remembering that motion from right to left is considered negative. 20 ELEMENTARY PHYSICS. [19 In order to find the acceleration at the point C directed towards 0, we must find the rate of change of the velocity at C given by Eq. ij). Since, if the point is moving with an accel- eration, the velocity increases with the time, as the time in- creases by a small increment At, the velocity also increases by the increment Av. Eq. (7) then becomes , ^ 2na . 1 27it , 27tAt\ v-\-Av= ^sml^— +-y-j 27ta I 2nt 27tAt , 27rt . 2nAt\ As At approaches zero, cos „ approaches the limit unity, , . 2iiAt , , , , . 2nAt and sm — ~ — can be replaced by its arc — —-; makmg these changes, and transposing. Av \n''a 2nt ^ = T^ ^°^ It ' Av But in the limit where these changes are admissible, —7- ^ ' At becomes -j- ; that is, the acceleration of the point. Hence the acceleration sought is \Tt^ 2nt 7 = — -jTi'a cos -jT ■ (8) This formula shows that the acceleration in a simple har- monic motion is proportional to the displacement. It is of the 19] MECHANICS OF MASSES. 21 opposite sign from the displacement ; that is, acceleration to the right of O is negative, and to the left of O positive. It is often necessary to reckon time from some other posi tion than that of greatest positive elongation. In that case the time required for the moving-point to reach its greatest positive elongation from that position, or the angle described by the corresponding point in the circumference in that time, is called the epoch of the new starting-point. In determining the epoch, it is necessary to consider, not only the position, but the direction of motion, of the moving-point at the instant from which time is reckoned. Thus, if L, corresponding to K in the circumference, be taken as the starting-point, the epoch is the time required to describe the path LX. But if L correspond to the point K' in the circumference, the motion in the diameter is negative, and the epoch is the time required for the moving-point to go from L through O to X' and back toX. The epochs in the two cases, expressed in angle, are, in the first, the angle measured by the arc KX \ and, in the second, the angle measured by the arc K'X'K X. Choosing ^in the circle, or L in the diameter, as the point from which time is to be reckoned, the angle equals angle 2,'7tt KOB — angle KOX, or —j, e, where t is now the time re- quired for the moving-point to describe the arc KB, and e is the epoch or the angle KOX. The formulas then become !2nt s =^ a cosi ^\ T 2Tt . l27tt \ V = — -y asm I ~= el; 4;r' (27tt \ f— — yr « cos \-Y - ej. 22 ELEMENTARY PHYSICS. [19 Returning to our first suppositions, letting X be the point from which epoch and time are reckoned, it is plain that, since I tA (2nt 7t\ BC = a sin = a cosl — — I = « cosl -™ — - I, the projection of B on the diameter OY also has a simple harmonic motion, differing in epoch from that in the diameter ■71 OX by — . It follows immediately that the composition of two simple harmonic motions at right angles to one another, hav- ing the same amplitude and the same period, and differing in epoch by a right angle, will produce a motion in a circle of radius a with a constant velocity. More generally, the co- ordinates of a point moving with two simple harmonic mo- tions at right angles to one another are ;ir = a cos(0 — e) and y ^ b cos 0'. If and 0' are commensurable, that is, if 0' = «0, the curve is re-entrant. Making this supposition. X =■ a cos cos e -)- « sin sin e, and y ■= b cos n(p. * Various values may be assigned to a, to b, and to n. Let a equal' b and 71 equal i ; then X ^ y cos e -|- (a" — y )* sin e ; i9] MECHANICS OF MASSES. 23 from which x' —2xy cos e -|- y cos' e = «' sin" e — y sin" e, or, x' — 2xy cos e -j- J/" = a' sin" e. This becomes, when e = 90°, x^ -\- y^ = a\ the equation for a circle. When e = 0°, it becomes x — y =^ o, the equation for a straight line through the origin, making an angle of 45" with the axis of X. With intermediate values of e, it is the equa- tion for an ellipse. If we make n = J, we obtain, as special cases of the curve, a parabola and a lemniscate, according as e ^ 0° or go°. If a and b are unequal, and « = i, we get, in general, an ellipse. If a line in which a point is describing a simple harmonic motion move uniformly in a direction perpendicular to itself, the moving-point will describe a harmonic curve, called also a sinusoid. It is a diagram of a simple wave. If the ordinates of the curve represent displacements transversely from a fixed line, the curve is the diagram of such waves as those of the ether which constitute light. If the ordinates of the curve represent displacement longitudinally from points of equilibri- um along a fixed line, the curve may be employed to represent the waves which occur in the air when transmitting sound. The length of the wave is the distance between any two iden- tical conditions of points on the line of progress of the wave. The amplitude of the wave is the maximum displacement from its position of equilibrium of any particle along the line of progress. If we assume the origin of co-ordinates such that the epoch of the simple harmonic motion at the axis of ordinates is o, 24 ELEMENTARY PHYSICS. [19 the displacement from the hne of progress of the point describ- ing the simple harmonic motion is represented by s ^^ a cos 2n- 4). The displacement due to any other simple harmonic motion differing from the first only in the epoch is represented by { * s^ = a cos l27r-= We shall now show, in the simplest case, the result of com- pounding two wave motions. The displacement due to both waves is the sum of the dis- placements due to each, hence s -\- s^ = a I cos 27r-:= -{- cos \27r -- — ej = a ( cos 2n-= -\- cos 2n -= cos e -(- sm 2n-=, sm e = a I cos 271 -={i -\- cos 6) -(- sin 27r^ sin e . If for brevity we assume a value A and an angle

(Fig. 13) be the bar, DF snA BG the forces. Their lines of direction will, in general, meet at some point as O. Moving the forces up to O, and applying the parallelogram of forces, we obtain the resultant OJ, which cuts the bar at A. If we resolve both forces separately, parallel to OJ and BD, this re- sultant equals in amount the sum of those components taken paral- lel to OJ. Hence the compon- ents EF and CG, taken parallel to ^^ '^ Fig. 13. DB, annul one another's action, and, being in opposite direc- tions, are equal. Now, by similarity of triangles, OA:AB^BC:CG, and OA:AD = DE: EF; whence, since CG = EF, we obtain AB-BC=AD-DE; 44 ELEMENTARY PHYSICS. [31 Resolving both DE and BC perpendicular to DB, we see that the moments of force about A are equal. Now, if the result- ant OJ be antagonized by an equal and opposite force applied at A, there will be no motion. Hence the tendencies to rota- tion due to the forces are equal, — a result which is in accord with our statement that the moment of force is a measure of the value of the force in producing rotation. The resultant of two forces may be found in general by this method. The case of most importance is the one in which the two forces are parallel. The lines DE and BC in the diagram represent such forces. It is plain, from the dis- cussion, that these forces also will have the force represented by Q/as their resultant, applied at the point A. The resultant of two parallel forces applied at the ends of a rigid bar is then a force equal to their sum apphed at a point such that the two moments of force about it are equal. 31. Couple. — The combination of two forces, equal and oppositely directed, acting on the ends of a rigid bar, is called a couple. By the preceding proposition, the resultant of these forces vanishes, and the action of a couple does not give rise to any motion of translation. The forces, however, conspire to produce rotation about the mid-point of the bar. It follows from the fact that a couple has no resultant, that it cannot be balanced by any single force. 32. Moment of Couple. — The moment of couple is the pro- duct of either of the two forces into the perpendicular distance between them. It follows from what has been already proved, that this measures the value of the couple as respects rota- tion. 33. Centre of Inertia. — If we consider any system of equal material particles, the point of which the distance from any plane whatever,is equal to the average distance of the several particles from that plane, is called the centre of inertia. This point is perfectly definite for any system of particles. It fol- 33] MECHANICS OF MASSES. 45 lows from the definition, that, if any plane pass through the centre of inertia, the sum of the distances of the particles on one side of the plane, from the plane, will be equal to the sum of the distances of the particles on the other side : hence, if the particles are all moving with a common velocity parallel to the plane, the sum of the moments of momentum on the one side is equal to the sum of the moments of momentum on the other side. And, further, if the particles all have a common acceleration, or are each acted on by equal and similarly di- rected forces, the sum of the moments of force on the one side is equal to the sum of the moments of force on the other side. If we combine the forces acting on two of the particles, one on each side of the plane, we obtain a resultant equal to their sum, the distance of which from the plane is determined by the distances of the two particles from the plane. Combining this resultant with the force on another particle, we obtain a second resultant; and, by continuing this process until all the forces have been combined, we obtain a final resultant, equal to the sum of all the forces, lying in the plane, and passing through the centre of inertia. This resultant expresses, in amount, direction, and point of application, the force which, acting on a mass equal to the sum of all the particles, situated at the centre of inertia, would impart the same acceleration to it as the conjoined action of all the separate forces on the separate particles imparts to the system. When the force acting is the force of gravity, the centre of inertia is usually called the centre of gravity. When the forces do not act in parallel lines, the proposi tion just stated does not hold true, except in special cases. Bodies in which it still holds are, for that reason, called centro- baric bodies. The centre of inertia can be readily found in most of the simple geometrical figures. For the sphere, ellipsoid of revolu- tion, or parallelopiped, it evidently coincides with the centre of 46 ELEMENTARY PHYSICS. [34 figure ; since a plane passing through that point in each case cuts the soHd symmetrically. 34. Mechanical Powers. — The preceding definitions and propositions find their most elementary application in the so- called mechanical powers. These are all designed to enable us, by the application of a certain force at one point, to obtain at another point a force, in general not equal to the one applied. Six mechanical powers are usually enumerated, — the lever, pulley, wheel and axle, inclined plane, wedge, and screw. (i) The Lever is any rigid bar, of which the weight may be neglected, resting on a fixed point called 2. fulcrum. From the proposition in § 30, it may be seen, that, if forces be ap- plied to the ends of the lever, there will be equilibrium when the resultant passes through the fulcrum. In that case the moments of force about the fulcrum are equal ; whence, if the forces act in parallel lines, it follows that the force at one end is to the force at the other end in the inverse ratio of the lengthsof their respective lever-arms. If / and /^ represent the lengths of the arms of the lever, and Pand P^ the forces ap- plied to their respective extremities, then Pl^P^l^. The principle of the equality of action and reaction enables us to substitute for the fulcrum a force equal to the resultant of the two forces. We have then a " combination of forces as represented in the diagram (Fig. 14). Plainly any 'p one of these forces may be considered ^•°-'*- as taking the place of the fulcrum, and -ither of the others the power or the weight. The lever is said to be of the first kind if R is fulcrum and P power, of the second kind if P^ is fulcrum and P power, of the third kind if Pis fulcrum and R power. (2) The Pulley is a frictionless wheel, in the groove of which runs a perfectly flexible, inextensible cord. 34] MECHANICS OF MASSES. 47 If the wheel be on a fixed axis, the pulley merely changes the direction of the force applied at one end of the cord. If the wheel be movable and one end of the cord fixed, and a force be applied to the other end parallel to the direction of the first part of the cord, the force acting on the pulley is double the force applied : for the stress on the cord gives rise to a force in each branch of it equal to the applied force ; each of these forces acts on the wheel, and, since the radii of the wheel are equal, the resultant of these two forces is a fbrce equal to their sum applied at the centre of the wheel. From these facts the relation of the applied force to the force ob- tained in any combination of pulleys is evident. (3) The Inclined Plane is any frictionless surface, making an angle with the line of direction of the force applied at a point upon it. Resolving the force P (Fig. 15), making an angle with the normal to the plane, into its coiii- ponents P cos

, sin a = ca^ sin /?. It follows at once that every point on the line OL is at rest. If we consider OL as the axis of rotation, and suppose the angular velocity of every point of the system about this axis to be w, such that oj sin a = w^ sin {a -|- P), this angular velocity will give the. actual velocity of any point. To illustrate by a simple exam- ple, we will show that sin (« + /?). a? sm yS = &, ^^ — -^-^ sm 6 ° sm «■ ' is the velocity at B at unit distance from O. The velocity at B is only due to rotation about OA, and is therefore given by cWj sin (or -(- /J). From our previous equation, , and the angle swept out in any time t is wt sin 0. By such an apparatus has been determined, not only the fact of the earth's, rotation, but even an approximate value of the length of the day. (2) The phenomena presented by the gyroscope also offer an example of the application of the foregoing principles. The construction of the apparatus can best be understood by the help of the diagram (Fig. 18).. The outermost ring rests in a frame, and turns on the points a, « . The inner rests in the outer one, and turns on the pivots b, b^, at right angles to the line of aa^. Within this ring is mounted the wheel G, the axle of which is at right angles to the line bb^, and in a plane passing through ««,. At the point e is fixed a hook, from which weights may be hung. It is evident that if the wheel be mounted on the middle of the axle, the equihbrium of the ap- paratus is neutral in any position, and that a weight hung on the hook e will bring the axle of the wheel vertical, without moving the outer ring. If, however, the wheel be set in ra,pid 54 ELEMENTARY PHYSICS. [35 rotation, with its axle horizontal, and a weight be hung on the hook, the whole system will revolve with a constant angular velocity about the points a, a^, and the axle of the wheel will remain horizontal. The explanation of this phenomenon follows from the principles which we have already discussed. The conditions given are, that a body rotating with an angular velocity in one plane is acted on by a force tending to produce rotation in a perpendicular plane. Let the plane of the paper represent the horizontal plane, f, and the line AB (Fig. 19) represent the direction of the axle at any moment. Lay off on OA a length OP proportional to the angular velocity of the wheel. If ^ °be the point of application of the weight, the weight tends to turn the system about an axis CD at right angles' to AB. Let us suppose, first, that, in the small inter- val of time t, the system acquires an an- gular Velocity about CD proportional to OQ. Compounding the two angular velocities OP and OQ, we obtain the resultant OR. Now, resolving OQ parallel and at right angles to OR, we see that the parallel component is effi- cient in determining the length of OR, the component at right angles, the direction of OR. In the limit, as ^-becomes indefi- nitely small, OQ also becomes indefinitely small, and the re- solved component Ox parallel to OR vanishes in comparison with 0Q\ because from the triangles we have -M,= -^- The OR OQ effect will be a change of direction of the axle AB in the hori- zontal plane, without a change in the angular velocity of the wheel. This change is the equivalent of the introduction of a new angular velocity about an axis perpendicular to the plane of the paper. This new angular velocity, compounded with the angular velocity about OA, gives rise, as before, to a change B Fig. ig. 36] MECHANICS OF MASSES. 55 in the direction of the axis without a change in the angular velocity of the wheel ; and this change in direction is such as to oppose the angular acceleration about CD, introduced by the weight at B. The system will revolve in a horizontal plane about (9 as a centre. Another explanation, leading to the same results, has been given by Poggendorff. As has already been stated, it requires' the application of a force to change the direction of the axis of a rotating body. This force is ex- pended in changing the direction of motion of the component parts of the body. Poggendorf's ex- planation of the movements of the gyroscope is c ■ based on the action of couples formed by these separate forces. Let Fig. 20 represent the rotating wheel of the former diagram, the axle being supposed to be nearly horizontal. If the weight be hung at the point e, it tends to turn the wheel about a horizontal axis CD. The particles moving at A and at B in the plane CD offer no resistance to this change. Those at C moving downwards, and those at D moving upwards, act otherwise. The forces ex- pressed by their momentum in the directions Cp and Dq are re- solved into two each, one of them in the new plane assumed by the wheel, and the other at right angles to it. It will be seen that the latter component acts at C towards the right, and at D towards the left. There is thus set up a couple act- ing to turn the system about the axis AB counter-clockwise, as seen from A. As soon as this rotation begins, the particles moving at A out of the paper, and at B through the paper, are turned out of their original directions, and there arises another couple, of which the component at A is directed towards the left, and at B towards the right. This couple tends to cause the system to rotate about the axis CD counter-clockwise, as seen from C, and thus to oppose the tendency to rotation due to the weight at e. 56 ELEMENTARY PHYSICS. [36 All other points on the wheel except those in the lines AB and CD, are turned out of their paths by both rotations ; and therefore components of the forces due to their motions ap- pear in both couples in the final summation of effects. The result of the existence of these couples is a movement such as has already been described. 36. Moment of Inertia. — The moment of inertia of any body about an axis is defined as the summation of the products of the masses of the particles making up the body into the squares of their respective distances from the axis. This product is the measure of the importance of the body's inertia with respect to rotation, and is proportional to the ki- netic energy of the body having a given angular velocity about the axis ; for, if any particle m, at a distance r from the axis, rotate with an angular velocity w, its velocity is roo and its ki- netic energy is \mw^r\ The whole kinetic energy of the body is, therefore, \oo^'2mr'' ; and since we have assumed •J-ca" to be given, '2,mr^ is proportional to the kinetic energy of the ro- tating body. A distance k such that \k'w^'2m = ^ofSmr' is called the radius of gyration, and is the distance at which a mass equal to that of the whole body must be concen- trated to possess the same moment of inertia as the body possesses. The formula for moment of inertia is r^2mr\ (18) and its dimensions are MU. The moment of inertia of a body with reference to an axis passing through its centre of inertia being known, its moment of inertia with reference to any other axis, parallel to this, is found by adding to the moment of inertia already known, the product of the mass of the body into the square of the distance of its centre of inertia from the new axis of rotation. For if 361 MECHANICS OF MASSES. 57 the centre of inertia of the body of which we know the moment of inertia be C, and if m be any particle of that body, and if O be the new axis to whicli the moment of inertia is to be re- ferred, making the construction as in Fig. 21, we have r" =: fl' + 2rfi + r/. Multiplying by the mass m, performing a similar operation for every particle of the body, and summing the results, we have /, = '2md-\- 2mr'-\- 2'2,mrfi. The term 2'2mrfi on the right van- ishes, for we may write it 2r,'2mb ; and, since C is the centre of inertia, "Zmb is zero (§ 33). Therefore I^ = /+ Mr;. Fig. 21. (19) This equation embodies the proposition which was to be proved. The moment of inertia of the simple geometrical solids may be found by reckoning the moments of inertia for the separate particles of the body, and summing the results. We will show how this may be done in a few simple cases. (i) To find the moment of inertia of a very thin rod AB, of length 2/' and mass 2m', about an axis xx\ passing through the middle point : Suppose the half-length to be divided into a very large m' number n of equal parts. The mass of each will be — . The /' 2/' distance of the first from the axis is -, of the second — , etc. n n 58 ELEMENTARY PHYSICS. [36 Their moments of inertia are w! p m' r m!_ ,r and the moment of inertia of the half-rod is r -j-(i + 4 + 9 • • • + «')• But (I + 4 + 9 . . . + «'), where n is indefinitely large, is — ; hence /' = Fig. I If / equal the whole length of the rod, m the whole mass, and /the entire moment of inertia. mr 12 (20j (2) To find the moment of inertia of a thin plate AB (Fig. 23), of length / and breadth 2b', about an axis perpendicular to it and passing through its centre : 36] MECHANICS OF MASSES. 59 Suppose the half-plate to be divided into n rods, parallel to its length: each rod will have a length / and a breadth -. b' 2b' Their distances from the axis are — , — , etc. Let m be the n n mass of the plate. The moment of inertia of each rod, with respect to an axis passing through its centre of inertia and m I' perpendicular to its length, is ^ X — . The moments of in- ertia of the several rods about the parallel axis xx' are mir b"\ mil" , b'^ and the moment of inertia of the half-plate is m r , m b'\ , , , „ mil'' . b'\ and of the whole plate equals M— ^^. (21) A parallelepiped ot which the axis is xx" may be supposed to be made up of an infinite number of plates, such as AB. Its moment of inertia will be the moment of inertia of one plate multiplied by the number of plates ; or, if M is the mass of the parallelopiped, its moment of inertia is M Y^i'^n (22) 6o ELEMENTARY PHYSICS. [37 The moment of inertia of any body, however irregular in form or density, may be found experimentally by the aid of another body of which the moment of inertia can be computed from its dimensions. We will anticipate the law of the pendu- lum, which has not been proved, for the sake of clearness. The body of which the moment of inertia is desired is set oscillating about an axis under the action of a constant force/. Its time of oscillation is, then, f where / is the moment of inertia. If, now, another body, of which the moment of inertia can be calculated, be joined with the first, the time of oscillation alters to ^,=v^ / where /^ is the moment of inertia of the body added. Com- bining the two equations, we obtain, as the value of the moment of inertia desired, If '= TTZr (23) 37. Central Forces.— If the velocity or direction of motion of a moving body in any way alter, we conceive it to be acted on by some force. In certain cases the direction of this force. 37] MECHANICS OF MASSES. 6 1 and the law of its variation with the position of the body, may be determined by considering tlie path or orbit traversed by the body and tlie circumstances of its motion. We sliall illustrate this by a few propositions, selected on account of their applicability in the establishment of the theory of universal gravitation. The proofs are substantially those given by Newton in the " Principia." Proposition I. — If the radius vector, drawn from a fixed point to a body moving in a curve, describe equal areas in equal times, the force which causes the body to move in the curve is directed towards the fixed point. Let us suppose the whole time divided into equal periods, during any one of which the body is not acted on by the force. It will, in the first period, move over a space represented by a straight line, as AB (Fig. 24). In the second period, it would, if unhindered, move over an equal space BD and in the same line. Let us suppose it, however, deflected by a force acting instantaneously at the point B. It will move in a line BC such that, by hypothesis, triangle OBA = triangle OBC. Now, triangle ODB also = triangle OAB, therefore triangle OCB = triangle ODB, and CD is parallel to OB. Complete the paral- lelogram CDBE ; then it is evident that the motion BC is compounded of the motions BD and BE ; and since forces are proportional to the motions they occasion, the force acting at B is proportional to BE, and is directed along the line BO. If now the periods into which the whole time ^'^ '"^■ is divided become indefinitely small, in the limit the broken line ABC approaches indefinitely near to a curve, and the force which causes the motion in the curve is always directed to the centre O. Proposition II. — If a body move uniformly in a circle, the 6?. ELEMEN TA R Y PH YSICS. [37 force acting upon it varies as its mass and the square of its velocity directly, and as the radius of the circle inversely. If the body m move in a circle (Fig. 25) with a constant angular velocity, and pass over, in any very small time t, the arc ad, which is so small that it may be taken equal to its chord, the motion may be resolved into two components ab and ac, one tangent and the other normal to the Fig. 25. arc. Now /, the acceleration towards O, being constant for that small time, we have s — ac = ^ff. The angle ade is a right angle, and therefore, by similar tri- angles, we have ac ad' ae But ae = 2r\ ad represents the space traversed in the time t, ^ ..... ad and m the limit — - represents v, the velocity in the circle. From the previous reasoning ac represents \ff ; whence 2 2r' ind mf = . r Corollary I. — If two bodies revolve about the same centre, and the squares of their periodic times be in the same ratio as the cubes of the radii of their respective orbits, the forces 37J MECHANICS OF MASSES. 63 acting on them will be inversely as the squares of their radii, and conversely. For, if 2" and 7"^ represent the periodic times of the two bodies moving in circles of radii r and r^, with ve- locities V and V,, then, by hypothesis, 2Ttr 2nr, T -.T, — : = ri : r/: -whence V -.v, = ry. ■.ri. Now whence f:f, = r;:r'. Corollary II. — The relation of Corollary I. holds with refer- ence to bodies describing similar parts of any similar figures having the same centre. In the application of the proof, however, we must substitute for uniform velocity the uniform description of areas ; and instead of radii we must use the distances of the bodies from the centre. The proof is as fol- lows: If D and D, represent the radii of curvature of the paths of the two bodies, R and R^ the distances of the bodies from the centre- of force, then, by hypothesis, letting A represent the area described in one period of time, T: T, ^ ^ -.^ = R^: R} ^ D^: Di from the siftiilarity of figures. 64 ELEMENTARY PHYSICS. ^zr Now hence and A:A, = vR: v,R, ; v:v, = Rj'-.R'- Di : Z», /■■/,= D ■ D, R' : R'. Proposition III. — If a body move in an ellipse, the force acting upon it, directed to the focus of the ellipse, varies in- versely as the square of the radius vector. Suppose the body moving in the ellipse to be at the point P (Fig. 26), and the force to act upon it along the radius vector SP. At the point P draw the tangent PR, and from a point Q on the ellipse draw the line Qv, Fic-=6 cutting SPm x, and complete the parallelogram PRQx. From Q draw (2^ perpendicular to SP. Also draw the diameter (JPand its conjugate DK. The force which acts on the body, causing it to leave the tangent PR and move in the line PQ. acts along SP, and in a time t (supposed very small) causes the body to move in the direction SP over the space Px ; and since, in the small time considered, it may be assumed constant, whence ^_2Px 37] MECHANICS OF MASSES. Again : the area described by the radius vector in the time 5 equal to in unit time, SP . QT t is equal to '- ; and if A represent the area described At = ^-^^. Equating these values of t, we obtain SP\QT' _ 2Px 4A' - /"' whence SA' . Px I / = QT' ■ SP' From Proposition I., the value of A is constant for any Px part of the ellipse. We shall now show that -q^ is also constant. From similar triangles, Px:Pv = PE: PC', or, since by a property of the ellipse PE = AC, Px : Pv = AC : PC. Again, by another property of the ellipse, • Gv .Pv: Qv" = PC" : Ciy. 66 ELEMENTARY PHYSICS. [37 If, now, we consider the time t to become indefinitely small, in the limit, P and Q approach indefinitely near ; whence Qv = Qx and Gv = 2PC. The last proportion then becomes PC.Pv: Qx' = PC : 20^. Again, from similar triangles, Qx : QT ^ PE : PF = AC : PF; and from another property of the ellipse, AC : PF ^ CD : CB; whence Qx:QT= CD:CB. Combining these proportions, Px:Pv = AC -.PC, Pv : Qx' = PC : 2CD\ Qx' :QT'= CD': CB", we obtain, finally, Px: QT'= AC: 2CB'; Px that , since ^c and CB are constant, -^™ is constant. We have now shown that, in the expression for the value of the force on the body at any point in the ellipse, all the factors are constant except -f™- The force, therefore, varies inversely as the square of the radius vector. CHAPTER II. MASS ATTRACTION. 38. Mass Attraction. — The law of mass attraction was the first generahzation of modern science. In its most complete form it may be stated as follows : — Between every two material particles in the universe there is a stress, of the nature of an attraction, which varies directly as the product of the masses of the particles, and inversely as the square of the distance between them. This law is some- times called the law of universal attraction and sometimes the Jaw ol gravitation. Some of the ancient philosophers had a vague belief in the existence of an attraction between the particles of matter. This hypothesis, however, with the knowledge which they possessed, could not be proved. The geocentric theory of the planetary system, which obtained almost universal acceptance, offered none of those simple relations of the planetary motions upon which the law was finally established. It was not until the heliocentric theory, advocated by Copernicus, strengthened by the discoveries of Galileo, and systematized by the labors of Kepler, had been fully accepted, that the discovery of the law became possible. In particular, the three laws of planetary motion published by Kepler in 1609 and 1619 laid the foundation for Newton's demonstrations. The laws are as follows : — I. The planets move in ellipses of which one focus is situ- ated at the sun. II. The radius vector drawn from the sun to the planet sweeps out equal areas in equal times. 68 ELEMENTARY PHYSICS. [38 III. The squares of the periodic times of the planets are proportional to the cubes of their distances from the sun. Kepler could give no physical reason for the existence of such laws. Later in the century, after Iluyghens had discov- ered certain theorems relating to motion in a circle, it was seen that the third law would hold true for bodies moving in concentric circles, and attracted to the common centre by forces varying inversely as the squares of the radii of the cir- cles. Several English philosophers, among them Hooke, Wren, and Halley, based a belief in the existence of an attrac- tion between the sun and the planets upon this theorem. The demonstration was by no means a rigorous one, and was not generally accepted. It was left for Newton to show that not only the third, but all, of Kepler's laws were com- pletely satisfied bj"^ the assumption of the existence of an attraction acting between the sun and the planets, and vary- ing inversely as the square of the distance. His propositions are substantially given in § 37. Newton also showed that the attraction holding the moon in its orbit, which is presumably of the same nature as that existing between the sun and the planets, is of the same nature as that which causes heavy bodies to fall to the earth. This he accomplished by showing that the deviation of the moon from a rectilineal path is such as should occur if the force which at the earth's surface is \^q force of gravity ^^x& to extend outwards to the moon, and vary in intensity inversely as the square of the distance. Two further steps were necessary before the final generali- zation could be reached. One was, to show the relation of the attraction to the masses of the attracting bodies ; the other, to show that this attraction exists between all particles of matter, and not merely, as Huyghens believed, between those particles and the centres of the sun and planets. The first step was taken by Newton. By means of pen- 39] MASS ATTRACTION. 69 dulums having the same length, but with bobs of different materials, he showed that the force acting on a body at the earth's surface is proportional to the mass of the body, since all bodies have the same acceleration. He further brought forward, as the most satisfactory theory which he could form, the general statement that every particle of matter attracts and is attracted by every other particle. The experiments necessary for a complete verification of this last statement were not carried out by Newton. They were performed in 1798 by Cavendish. His apparatus consisted essentially of a bar furnished at both ends with small leaden balls, suspended horizontally by a long fine wire, so that it turned freely in the horizontal plane. Two large leaden balls were mounted on a bar of the same length, which turned about a vertical axis coincident with the axis of rotation of the sus- pended bar. The large balls, therefore, could be set and clamped at any angular distance desired from the small balls. The whole arrangement was enclosed in a room, to prevent all disturbance. The motion of the suspended system was ob- served from without by means of a telescope. Neglecting as unessential the special methods of observation employed, it is sufficient to state that an attraction was observed between the large and small balls, and was found to be in accordance with the law as above stated. 39. Measurement of the Force of Gravity. — When two bodies attract one another, their relative motions are deter, mined by Newton's third law. In the case of the attractior, between the earth and a body near its surface, if we adopt d point on the earth's surface as the fixed point of refere-.Kt; the acceleration of the body alone need be considered. S'nC'* the force acting upon it varies with its mass, and jince its gain in momentum also varies with its mass, it T/yllows that its acceleration will be constant, however its Hi^/j may vary. We may, therefore, obtain a direct measi. > of the 70 ELEMENTARY PHYSICS. [39 earth's attraction, or of the force of gravity, by allowing a body to fall freely, and determining its acceleration. It is found that a body so falling at latitude 40° will describe in one second about 16,08 feet, or 490 centimetres. Its acceleration is there- fore 32.16 in feet and seconds or 980 in centimetres and seconds. We denote this acceleration by the symbol^. The force acting on the body, or the weight of the body, is seen at once to be mg, where m is the mass of the body. On account of the difficulties in the employment of this method, various others are used to obtain the value of ^ indi- rectly. For example, we may allow bodies to slide down a smooth inclined plane, and observe their motion. The force effective in producing motion on the plane is ^ sin is the angle of the plane with the "horizontal ; the space trav- ersed in the time ^ is j = \gf sin 0. By observing s and t, the value of ^ may be obtained. The motion is so much less rapid than that of a freely falling body that tolerably accurate ob- servations can be made. Irregularities due to friction upon the plane and the resistance of the air, however, greatly vitiate any calculations based upon these observations. This method was used by Galileo, who was the first to obtain a measure of the acceleration due to the earth's attraction. The most exact method for determining the value of g is based upon observations of the oscillation of a pendu- lum. A pendulum may be defined as a heavy mass, or bob, suspended from a rigid support, so that it can oscillate about its position of equilibrium. In the simple, or mathematical, pendulum, the bob is assumed to be a material particle, and to be sus- pended by a thread without weight. If the bob be Fig. iy. stationary and acted on by gravity alone, the line of the thread will be the direction of the force. If the bob be withdrawn from the position of equilibrium (Fig. 27), it will be 4'] MASS ATTKACTIOM. 71 acted on by a force at right angles to the thread, in a direction opposite that of the displacement, expressed by — ^ sin (p, where is the angle between the perpendicular and the new position of the thread. The force acting upon the bob at any point in the circle of which the thread is radius, if it be released, and allowed to swing in that circle, varies as the sine of the angle be- tween the perpendicular and the radius drawn to that point. If we make the oscillation so small that the arc may be sub- stituted for its sine without sensible error, the force acting on the bob varies as the displacement of the bob from the point of equilibrium. A body acted on by a force varying as the displacement of the body from a fixed point will have a simple harmonic mo- tion about its position of equilibrium. Hence it follows that the oscillations of the pendulum are symmetrical about the position of equilibrium. The bob will have an amplitude on the one side of the vertical equal to that which it has on the other, and the oscillation, once set up, will continue forever unless modified by outside forces. The importance of the pendulum as a means of determin- ing the value of ^consists in this: that, instead of observing the space traversed by the bob in one second, we may observe the number of oscillations made in any period of time, and de- termine the time of one oscillation ; from this, and the length of the pendulum, we can calculate the value of ^. The errors in the necessary observations and measurements are very slight in comparison with those of any other method. 40. Formula for Simple Pendulum. — The formula con- necting the time of oscillation with the value of ^is obtained as follows : The acceleration of the bob at any point in the arc is, as we have seen, — g sin 0, or — g

. Hence the angular velocity of each particle, and therefore of the pendulum, is expressed by — —^- The kinetic energy of a body, rotating about an axis with an angular velocity oo, has been shown in § 36 to be expressed by '2mr^ — . Substituting in this expression the value obtained for the angular velocity of the pendulum, we A.n'^ , ....... , obtain ^Smr y.^ as the expression for the kinetic energy of 74 ELEMENTARY PHYSICS. [41 the pendulum at the lowest point of its arc. At this point the pendulum possesses no potential energy. Its kinetic energy at this point must therefore be equal to its pot&ntial energy at the highest point of its arc, where it posesses no kinetic energy. If we represent by M the mass of the pendulum, and by R the distance of the centre of gravity from the point of suspension, 7?0 represents the distance traversed by the centre of gravity between the highest and the lowest points of its arc, and \Mg(j> represents the average force acting on the centre of gravity between those points to produce rotation. The poten- tial energy of the pendulum at the highest point of its arc is, therefore, \MRg4?. Hence we have ^■2mr''^=^WRg'; whence ^-V^- '^'> This is the time of oscillation of a simple pendulum of which the length is -,„ . Therefore the moment of inertia of any physical pendulum divided by the product of its mass into the distance of its centre of gravity from the axis of suspension gives the length of the equivalent simple pendulum. An axis paral- lel to the axis of suspension, passing through the point on the line joining the axis of suspension with the centre of gravi- ty of the pendulum and distant -tifh- from the axis of sus- pension, is called the axis of oscillation. A pendulum consisting of a heavy spherical bob suspended by a cylindrical wire was used by Borda in his determinations of the value of ^. The moment of inertia and the centre of 41] MASS ATTRACTION. 75 gravity of the system were easily calculated, and the length of the simple pendulum to which the system was equivalent was thus obtained. 1 2) We may determine the length of the equivalent simple pendulum directly by observation. The method depends upon the principle that, if the axis of oscillation be taken as the axis of suspension, the time of oscillation will not vary. The proof of this principle is as follows : Let rand / — r represent the distances from the centre of gravity to the axis of suspension and of oscillation re- spectively, in the mass of the pendulum, and / its moment of inertia about its centre of gravity. Then, since the moment of inertia about the axis of suspension is /+ mr^, we have T -\- mr' When the pendulum is reversed, we have /"+ m{l - rf /,= m{l — r) From the first equation we have /= mr{l—r), which value substituted in the second gives, after reduction, l^ = l\ that is, the length of the equivalent simple pen- p dulum, and consequently the time of oscillation when the pendulum swings about its axis of suspension, is the same as that when it is reversed, and swings about its former axis of oscillation. A pendulum (Fig. 28) so constructed as to take advantage of this principle was used by Kater in his ^ ^ determination of the value of g\ and this form is known. \_y in consequence, as Kater's pendulum. ELEMENTARY PHYSICS. [42 42. The Balance. — The comparison of masses is of such frequent occurrence in physical investigations that it is im- portant to consider the theory of the balance and the methods of using it. To be of value the balance must be accurate and sensitive ; that is, it must be in the position of equilibrium when the scale-pans contain equal masses, and it must move out of that position on the addition to the mass in one pan of a very small fraction of the original load. These conditions are attained by the application of principles which have already been developed. The balance consists essentially of a regularly formed beam, poised at the middle point of its length upon knife edges which rest on agate planes. From each end of the beam is hung a scale-pan in which the masses to be compared are Fig. 29. placed. Let O (Fig. 29) be the point of suspension of the beam ; A,B, the points of suspension of the scale-pans ; C, the centre of gravity of the beam, the weight of which is W. Represent OA = OB by /, OC by d, and the angle OAB by a. If the weight in the scale-pan at A be P, and that in the one a.t B he P -\-p, where / is a small additional weight, the beam will turn out of its original horizontal position, and as- sume a new one. Let the angle COC„ through which it turns, be designated by /J. Then the moments of force about O are equal ; that is, {P+p)l . cos {a-\-P) = Pl. cos {a- ^)-\^Wd. sin /? ; 42] MASS ATTRACTION. 77 from which we obtain, by expanding and transposing, // cos a The conditions of greatest sensitiveness are readily deduci- ble from this equation. So long as cos a is less than unity, it is evident that tan /?, and therefore /?, increases as the weight 2jPof the load diminishes. As the angle a becomes less, the value of /J also increases, until, when A, O, and B are in the pi same straight line, it depends only on -j^yj, and is independ- ent of the load. In this case tan /? increases as d, the distance from the point of suspension to the centre of gravity of the beam, diminishes, and as the weight of the beam W dimin- ishes. To secure sensitiveness, therefore, the beam must be as long and as light as is consistent with stiffness, the points of suspension of the beam and of the scale-pans rhust be very nearly in the same line, and the distance of the centre of gravity from the point of suspension of the beam must be as small as possible. Great length of beam, and near coincidence of the centre of gravity with the axis, are, however, incon- sistent with rapidity of action. The purpose for which the balance is to be used must determine the extent to which these conditions of sensitiveness shall be carried. Accuracy is secured by making the arms of the beam of equal length, and so that they will perfectly balance, and by attaching scale-pans of equal weight at equal distances from the centre of the beam. In the balances usually employed in physical and chemical investigations, various means of adjustment are provided, by means of which all the required conditions may be secured. The beam is poised on knife edges ; and the adjustment of its centre of gravity is made by changing the position of a nut 78 ELEMENTARY niYSICS. [42 which moves on a screv^r, placed vertically, directly above the point of suspension. Perfect equality in the moments of force due to the two arms of the beam is secured by a similar hori- zontal screw and nut placed at one end of the beam. The beam is a flat rhombus of brass, large portions of which are cut out so as to make it as light as possible. The knife edge on which the beam rests, and those upon which the scale-pans hang, are arranged so that, with a medium load, they are all nearly in the same line. A long pointer attached to the beam moves before a scale, and serves to indicate the deviation of the beam from the position of equilibrium. If the balance be accurately made and perfectly adjusted, and equal weights placed in the scale-pans, the pointer will remain at rest, or will oscillate through distances regularly diminishing on each side of the zero of the scale. If the weight of a body is to be determined, it is placed in one scale-pan, and known weights are placed in the other un- til the balance is in equilibrium or nearly so. The final deter- mination of the exact weight of the body is then made by one of three methods : we may continue to add very small weights until equilibrium is established ; or we may observe the devia- tion of the pointer from the zero of the scale, and, by a table prepared empirically, determine the excess of one weight over the other ; or we may place a known weight at such a point on a graduated bar attached to the beam that equilibrium is established, and find what its value is, in terms of weight placed in the scale-pan, by the relation between the length of the arm of the beam and the distance of the weight from the middle point of the beam. If the balance be not accurately constructed, we can, never- theless, obtain an accurate value of the weight desired. The method employed is known as Borda's method of double weighing. The body to be weighed is placed in one scale-pan, and balanced with fine shot or sand placed in the other. It is 43] MASS ATTRACTION. 79 then replaced by known weights till equilibrium is again estab- lished. It is manifest that the replacing weights represent the weight of the body. If the error of the balance consist in the unequal length of the arms of the beam, the true weight of a body may be ob- tained by weighing it first in one scale-pan and then in the other. The geometrical mean of the two values is the true weight ; for let /, and 4 represent the lengths of the two arms of the balance, P the true weight, and P^ and P^ the values of the weights placed in the pans at the extremities of the arms of lengths /, and /,, which balance it. Then Pl^ = PJ^ and P/^ — PJ.^ ; from which P= \fP^,. 43. Density of the Earth. — One of the most interesting problems connected with the physical aspect of gravitation is the determination of the density of the earth. It has been attacked in several ways, each of which is worthy of consider- ation. The first successful determination of the earth's density was based upon experiments made in 1774 by Maskelyne. He' observed the deflection from the vertical of a plumb-line sus- pended near the mountain Schehallien in Scotland. He then determined the density of the mountain by the specific gravity of specimens of earth and rock from various parts of it, and calculated the ratio of the volume of the mountain to that of the earth. From these data the mean specific gravity of the earth was determined to be about 4.7. The next results were obtained from the experiments of Cavendish, in 1798, with the torsion balance already described. The density, volume, and attraction of the leaden balls being known, the density of the earth could easily be obtained. The value obtained by Cavendish was about 5.5. 80 ELEMENTARY PHYSICS. [43 Another method, employed by CarHni in 1824, depends upon the use of the pendulum. The time of the oscillation of a pendulum at the sea-level being known, the pendulum is carried to the top of some .high mountain, and its time of os- cillation again observed. The value of^as deduced from this observation will, of course, be less than that obtained by the observation at the sea-level. It will not, however, be as much less as it would be if the change depended only on the in- creased distance from the centre of the earth. The discrep- ancy is due to the attraction of the mountain, which can, therefore, be calculated, and the calculations completed as in Maskelyne's experiment. The value obtained by Carlini by this method was about 4.8. A fourth method, due to Airy, and employed by him in 1854, consists in observing the time of oscillation of a pendulum at the bottom of a deep mine. By § 29, (i), it appears that the attraction of a spherical shell of earth the thickness of which is the depth of the mine vanishes. The mean density of the earth may, therefore, be determined by the discrepancy between the values of g at the bottom of the mine and at the surface. Still another method, used by Jolly, consists in determining by means of a delicate balance the increase in weight of a small mass of lead when a large leaden block is brought beneath it. Jolly's results were very consistent and give as the earth's density the value 5.69. These methods have yielded results varying from that ob- tained by Airy, who stated the mean specific gravity to be 6.623, to that of Maskelyne, who obtained 4.7. The most elaborate experiments, by Cornu and Bailie, by the method of Cavendish, gave as the value 5.56. This is probably not far from the truth. When the density of the earth is known, we may calculate from it the value of the constant of mass attraction, that is, the attraction between two unit masses at unit distance apart. 44] MASS attraction: 8 1 Representing by D the earth's mean density, by R the earth's -mean radius, and by k the constant of attraction, the mass of the earth is expressed by ^ttR^D. Since by § 29, (2), the at- traction of a sphere is inversely as the square of the distance from its centre, the attraction of the earth on a gram at a point on its surface, or the weight of one gram, is expressed by g = ^n^^k — ^TtRDk. TcR is twice the length of the earth's quadrant, or 2 X 10° centimetres. The value of g at latitude 40° is 980.11, and from the results of Cornu and Bailie we may set D equal to 5.56. With these data we obtain k equal to O.OOOOOCXD66 dynes. 44. Projectiles. — When a body is projected in any direc- tion near the earth's surface, it follows, in .general, a curved path. If the lines of force be considered as radiating from the earth's centre, this path will be, by Proposition III, §37, an ellipse, with one focus at the earth's centre. If the path pursued be so small that the lines may be considered parallel, the centre of force is conceived of as removed to an infinite distance, and the curve becomes a parabola. The fact that ordinary projectiles follow a parabolic path was first shown by Galileo, as a deduction from the principle which he established, — that a constant force produces a uni- form acceleration. The proof is as follows : Suppose the body to be projected from the point O taken as origin, in the direction of the axis OY (Fig. 30), making any angle cf) with OX, a vertical axis, and to move with a velocity 1) ^-■. Owing to the accelerating effect of gravity, it also moves in the vertical direction OX with a velocity v,=gt. At any time t it will have traversed in the direction OY a. space jf = vt, and in the direction OX a space X = iigf. The co-ordinates of the position of the bodj- 6 82 ELEMENTARY PHYSICS. [44 at any time t are, therefore, y ^ vt and x = \gf. The equa- 21)'' X tion connecting x and 7 becomes _;/' = , which is the equa- tion of a parabola referred to the diameter OX and the tan- gent OY. When the body is projected horizontally, the vertex of the parabola is at the origin of the motion. The body be- gins to approach the earth from the start, and reaches it at the same time that it would if allowed to fall freely. One special case of importance in the consideration of the paths of projectiles is that in which the body moves in a circle. It is obvious, that, to bring about this result, the body must be projected horizontally with such an initial velocity that the acceleration due to the earth's attraction shall be precisely equal to the acceleration toward the centre which is necessary in order that the body should move in a circle (Proposition II> § 37)- Hence we must have mv' mM 1i ^ r^'^' where m and M are the masses of the body and the earth re-- spectively, R is the earth's radius, and k the constant of attrac tion. Now V, the velocity of the body, equals 2TtR where T is the time of one complete revolution, and M= \nR'D, where D is the earth's mean density. Substituting these val' 44] A/ASS ATTRACTION. 83 ues, we obtain from which y^2 3^ The result shows that the periodic time of any small body- revolving about a sphere, and infinitely near its surface, is a function of the density only, and does not depend on the radius of the sphere. Upon this principle Maxwell proposed, as an absolute unit of time, the time of revolution of a small satellite revolving in- finitely near the surface of a globe of pure water at its maxi- mum density. . CHAPTER III. MOLECULAR MECHANICS. CONSTITUTION OF MATTER. 45. General Properties of Matter — Besides the proper- ties already deiined in § 3 as characteristic and essential, we find that all bodies possess the properties of compressibility and divisibility. Compressibility. — All bodies change in volume by change of pressure and temperature. If a body of a given volume be subjected to pressure, it will return to its original volume when the pressure is removed, provided the pressure has not been too great. This property of assuming its original volume is called elasticity.. The property of changing volume by the application of heat is sometimes specially called dilatability. Divisibility. — Any body of sensible magnitude may, by mechanical means, be divided, and each of its parts may again be subdivided ; and the process may be continued till the re- sulting particles become so minute that we are no longer able to recognize them, even when assisted by the most perfect ap- pliances of the microscope. If the body be one that can be dissolved, it may be put in solution, and this may be greatly diluted ; and in some cases the body may be detected by the color which it imparts to the diluent, even when constituting so- small a proportion as one one-hundred-millionth part of the solution. 46. Molecules. — We are not, however, at liberty to con- clude that matter is infinitely divisible. The fact, established 481 MOLECULAR MECHANICS. 85 by observation, that bodies are impenetrable, and the one just noted, that they are also compressible, as well as other consid- erations, to be adduced later, lead to the opposite conclusion. To explain the coexistence of these properties, we are com- pelled to assume that bodies are composed of extremely small portions of matter, indivisible without destroying their identity, called molecules, and that these molecules are separated by in- terstitial spaces relatively larger, which are occupied by a highly elastic medium called the ether. > These molecules can be divided only by chemical means. The resulting subdivisions are called atoms. The atom, how- ever, cannot exist in a free state. ' The molecule is the physi- cal unit of matter, while the atom is the chemical unit. 47. Composition of Bodies. — It has Just been said that atoms cannot exist in a free state. They are always combined with others, either of the same kind, forming simple substances, •or of dissimilar kinds, forming compound substances. There are about sixty-seven substances now known which cannot, in the present state of our knowledge, be decomposed, or made to yield anything simpler than themselves. We therefore call them simple substances, elements, or, if we desire to avoid expressing any theory concerning them, radicals. It is not improbable that some of these will yet be divided, perhaps all of them. We can call them elements, then, only provisionally. 48. States of Aggregation. — Bodies exist in three states, — the solid, the liquitl, and the gaseous. In the solid stale the form and volume of the body are both definite. In the liquid state the volume only is definite. In the gaseous state neither form nor volume is definite. Many substances may, under proper conditions, assume either of these three states of aggregation ; and some sub- stances, as, for example, water, may exist in the three states under the same general conditions. 86 ELEMENTARY PHYSICS. [49 It is proper to add, however, that there is no such sharp line of distinction between the three states of matter as our definitions imply. Bodies present all gradations of aggrega- tion between the extreme conditions of solid and gas ; and the same substance, in passing from one state to the other, often presents all these gradations. 49. Structure of Solids. — With the exception of organized bodies, all solids may be divided into two classes. The bodies of one class are characterized by more or less regularity of form, which is called crystalline ; those of the other class, ex- hibiting no such regularity, are called amorphous.. For the formation of crystals a certain amount of freedom of motion of the molecules is necessary. Such freedom of motion is found in the gaseous and liquid states ; and when crystallizable bodies pass slowly from these to the solid state, crystallization usually occurs. It may also occur in some solids spontaneously, or in consequence of agitation of the molecules by mechanical means, such as friction or percussion. Crystallizable bodies are called crystalloids. Some amorphous bodies cannot, under any circumstances, assume the crystalline form. They are called colloids. 50. Crystal Systems. — Crystals are arranged by mineralo- gists in six systems. In the first, or Isometric, system, all the forms are referred to three equal axes at right angles. The system includes the cube, the regular octahedron, and the rhombic dodecahedron'. In the second, or Dimetric, system, all the forms are referred to a system of three rectangular axes, of which only two are equal. In the third, or Hexagonal, system, the forms are referred to four axes, of which three are equal, lie in one plane, and cross each other at angles of 60°. The fourth axis is at right angles to the plane of the other three, and passes through their common intersection. 52] MOLECULAR MECHAiVICS. 8/ The fourth, or Orthorhombic, system is characterized by three rectangular axes of unequal length. In the fifth, or Monoclinic, system, the three axes are un- equal. One of them is at right angles to the plane of the other two. The angles which these two make with each other, as well as the relative lengths of the axes, vary greatly for different substances. In the sixth, or Triclinic, system, the three axes are oblique to each other, and unequal in length. 51. Forces determining the Structure of Bodies. — In view of what precedes, it is necessary to assume the existence of certain forces other than the mass attraction considered in § 38 acting between the molecules of matter. These forces seem to act only within very small or insensible distances, and vary with the character of the molecule. They are hence called molecular forces. In liquids and solids, there must be a force of the nature of attraction, holding the molecules to- gether, and a force equivalent to repulsion, preventing actual contact. The attractive force is called cohesion when it unites molecules of the same kind, and adhesion when it unites mole- cules of different kinds. The repulsive force is probably a manifestation of that motion of the molecules which constitutes heat. In gases this motion is so great as to carry the molecules beyond the limit of their mutual molecular attractions : thus the apparent repulsion prevails, and the gas only ceases ex- panding when this repulsion is balanced by other forces. 52, Structure of the Molecule. — The facts brought to light in the study of crystals compel us to ascribe a structural form to the molecule, determining special points of application for the molecular forces. From this results the arrangement of molecules, which have the requisite freedom of motion, into regular crystalline forms. 88 ELEMENTARY PHYSICS. [53 FRICTION. 53. General Statements. — When the surface of one body is made to move over the surface of another, a resistance to the motion is set up. This resistance is said to be due to fric- tion between the two bodies. It is most marked when the sur- faces of two solids move over one another. It exists, however, also between the surfaces of a solid and of a liquid or a gas, and between the surfaces of contiguous liquids or gases. When the parts of a body move among themselves, there is a similar re- sistance to the motion, which is ascribed to friction among the molecules of the body. This internal friction is called viscosity. 54. Laws of Friction. — Owing to our ignorance of the ar- rangement and behavior of molecules, we cannot form a theory of friction based upon mechanical principles. The laws which have been found are almost entirely experimental, and are only approximately true even in the cases in which they apply. It was found by Coulomb that, when one solid slides over another, the resistance to the motion is proportional to the pressure normal to the surfaces of contact, and is independent of the area of the surfaces and of the velocity with which the moving body slides over the other. It depends upon the na- ture of the bodies, and the character of the surfaces of contact. The ratio of the force required to keep the moving body in uniform motion to the force acting upon it normal to the sur- faces of contact is called the coefficient of friction. It was shown experimentally by Poiseuille that the rate of outflow of a liquid from a vessel through a long straight tube of very small diameter is proportional directly to the difference in pressure in the liquid at the two ends of the tube, to the fourth power of the radius of the tube, and inversely to the length of the tube. The flow of liquid under such conditions can be determined by mathematical analysis, and it is found S6] MOLECULAR MECHANICS. 89 that the results obtained by Poiseuille can only occur if the co- efficient of friction between the liquid and the wall of the tube be very great. In other words, we may think of the liquid par- ticles nearest the wall as adhering to it and forming a tube of molecules of the same sort as those of the liquid. The outflow then depends only upon the coefficient of viscosity of the liquid. From consideration based upon the kinetic theory of gases, Maxwell predicted that the coefficient of viscosity in a gas would be independent of its density. This prediction has been verified by experiment through a wide range of densities. For very low densities. It has been shown that this law no longer holds true. 55. Theory of Friction. — The friction between solids is due largely, if their surfaces be rough, to the interlocking of pro- jecting parts. In order to sHde the bodies over one another, these projections must either be broken off, or the surfaces must separate until they are released. There is also a direct interaction of the molecules which lie in the surfaces of con- tact. This appears in the friction of smooth solids, and is the sole cause of the viscosity of liquids and gases. That this mo- lecular action is of importance in producing the friction of solids is seen in the facts that the friction of solids of the same kind is greater than that of solids of different kinds, and that it requires a greater force to start one body sliding over another than to maintain it in motion after it is once started, CAPILLARITY. 56. Fundamental Facts. — If we immerse one end of a fine glass tube having a very small, or capillary, bore in water, we observe that the water rises in the tube above its general level. We also observe that it rises around the outside of the tube, so that its surface in the immediate vicinity of the tube is curved. If we immerse the same tube in mercury, the surface of the mercury within and just outside the tube, instead 90 ELEMENTARY PHYSICS. [57 of being elevated, is depressed. If we change the tube for one of smaller bore, the water rises higher and the mercury sinks lower within it ; but the rise or depression outside the tube remains the same. If we immerse the same tube in different liquids, we find that the heights to which they ascend vary for the different liquids. If, instead of changing the diameter, we change the thickness of the wall of the tube, no variation occurs in the amount of elevation or depression ; and, finally, the rise or depression in the tube varies for any one liquid with its temperature. 57. Law of Force assumed. — It is found that a force such as is given by the law of mass attraction is not sufficient to produce these phenomena. They can, however, be ex- plained if we assume an additional attraction between the molecules, such as we have already done. The expression, then, of the stress between two molecules m and vt' , at dis- tance r, becomes mm! F= — I — [- mm' f{r). The only law which it is necessary to assign to the function of r in the second term is, that it is very great at insensible distances, diminishes rapidly as r increases, and vanishes while r, though measurable, is still a very small quantity. For adja- cent molecules this molecular attraction is so much greater than the mass attraction, that it is customary, in the discussion m,m' of capillary phenomena, to omit the term — — from the ex- pression for the force. The distance through which this at- traction is appreciable is often called the radius of jnolecular action, and is denoted by the symbol e. It is a very small dis- . tance, but is assumed to be much greater than the distance between adjacent molecules. 59] MOLECULAR MECHANICS. 9I 58. Methods of Development. — The different methods which have been employed to deduce, from this assumed attraction, results which could be submitted to experimental verification, are worthy of notice. They are distinct, though compatible with one another. Young was the first to treat the subject satisfactorily, though others had given partial and im- perfect demonstrations before him. He showed that a liquid can be dealt with as if it were covered at the bounding surface with a stretched membrane, in which is a constant tension tending to contract it. From this basis he proceeded to deduce some of the most important of the experimental laws. Laplace, proceeding directly from the law of the attraction which we have already given, considered the attraction of a mass of liquid on a filament of the liquid terminating at the surface, and obtained an expression for the pressure within the mass at the interior end of the filament. He also was able, not only to account for already observed laws, but to predict, in at least one instance, a subsequently verified result. Some years later. Gauss, dissatisfied with Laplace's assumption, with- out a priori demonstration, of a known experimental fact, treated the subject from the basis of the principle of virtual velocities, which in this case is the equivalent of that of the conservation of energy. He proved, that, if any change be made in the form of a liquid mass, the work done or the energy recovered is proportional to the change of surface, and hence deduced a proof of the fact which Laplace assumed, and also an expression for the pressure within the mass of a liquid identical with his. For purposes of elementary treatment the earliest method is still the best. We shall accordingly employ the idea of surface tension, after having shown that it may be obtained from our first hypothesis. 59. Surface Tension. — Let us consider any liquid bounded by a plane surface, of which the line mn (Fig. 31) is the trace, and let the line m!n' be the trace of a parallel plane at a 92 ELEMENTARY PHYSICS. [S9 distance e from the plane of mn. The liquid is then divided into two parts by the plane of m'n' , — the general mass of the liquid, and a shell of thickness e between the two planes. Then, if we imagine a plane passed through any point within the general mass, it is clear that the attraction of the molecules on opposite sides of that plane will give rise to. a pressure normal to it, which will be constant for every direction of the plane ^ for the number of molecules now acting on the point is the same in all directions. Let, however, the point chosen be P, situated within the shell. With Z' as a centre, and with radius e, describe a sphere. Now, it is evident that the number of mole^ ^- \ n P i \ 7 - V / »' Fig. 31. cules active in producing pressure upon the plane through P, parallel to mn, is less than that of those producing pressure upon the plane through /"normal to mn. The pressure upon the par- allel plane varies as we pass from the mass through the shell, from the value which it has within the mass, to zero, which it has at the plane mu. From this inequality of pressure in the two directions, parallel and normal to the surface, there results a stress or tension of the nature of a contraction in the surface. Provided the radius of curvature of the surface be not very small, this tension will be constant for the surface of each liquid, or, more properly, for the surface of separation between two liquids, or a liquid and a gas. 6o] MOLECULAR MECHANICS. 93 60. Energy and Surface Tension. — We may here show- how the energy of the liquid is related to the surface tension. It is plain, that, if the molecules, which by their mutual attrac- tions give rise to the surface tension, be forced apart by the extrusion from the mass into the shell of a sheet of molecules along a plane normal to the surface, work will be done as the surface is increased. In every system free to move, move- ments will occur until the potential energy becomes a min- imum: hence every free liquid moves so that its bounding surface becomes as small as possible ; that is, it assumes a spherical form. This is exemplified in falling drops of water and in globules of mercury, and can be shown on a large scale by a method soon to be described. If we call the potential energy lost by a diminution in the surface of one unit, the surface energy per unit surface, we can show that it is numeri- cally equal to the surface tension across one unit of length. Suppose a thin film of liquid to be stretched on a frame c D — A Fic. 32. ABCD (Fig. 32), of which the part BCD is solid and fixed, and the part ^ is a light rod, free to slide along C and D. This film tends, as we have said, to diminish its free surface. As it contracts, it draws A towards B. If the length of A be a, and A be drawn towards B over b units, then if E represent the surface energy per unit of surface, the energy lost, or the work done, is expressed by Eab. If we consider the tension acting 94 ELEMENTARY PHYSICS. [6l normal to A, the value of which is T for every unit of length, we have again for the work done during the movement of A, Tab. From these expressions we obtain at once E =^ T; that is, the numerical value of the surface energy per unit of surface is equal to that of the tension in the surface, normal to any line in it, per unit of length of that line. 6i. Equation of Capillarity. — The surface tension intro- duces modifications in the pressure within the liquid mass (§ 85 seq^ depending upon the curvature of the surface. Consider any infinitesimal rect- ^*>\ j j / angle (Fig. 33) on the surface. Let the ^ I 1 I / length of its sides be represented by s and s, \\ \ j I / respectively, and the radii of curvature of '%;, \^l those sides by R and R,. Also let and 0, '\^J I / represent the angles in circular measure sub- \ / / tended by the sides from their respective \ I / centres of curvature. Now, a tension T for \J' every unit of length acts norrhal to s and Fig. 33- tangent to the surface. The total tension across s is then Ts ; and if this tension be resolved parallel and normal to the normal at the point P, the centre of the rect- angle, we obtain for the parallel component Tjsin— , or, (h s since 0, is a very small angle, Ts~ or Ts-^-. The opposite ^ 2/\, side gives a similar component ; the side s^ and the side oppo- site it give each a con^ponent Ts^-^. The total force along the normal at P is then ^-, and W W Wt ^' — W' '^^^"'^^ ^t^'w '^W ~ '^"^'^*' ^^ ^' repre- sent the density of a substance at 0°, A^ — A t{i -\- kf), where k represents the coefficient of voluminal expansion of the sub- stance. 137. Effect of Variation of Temperature upon Specific Heat. — It has already been seen (§ 109) that the specific heat of bodies changes with temperature. With most substances the specific heat increases as the temperature rises. For example, the true specific heat of the diamond Ai 0° is o 0947 At 50° is 0.1435 At 100° is 0.1905 At 200° is 0.2719 138. Effect of Change of Physical State upon Specific Heat. — The specific heat of a substance is not the same when in the solid as when in liquid state. In the solid state of the substance it is generally less than in the liquid. For example : Mean Specific Heat Solid. Liquid. Water 0.504 i.ooo Mercury, 0.0314 o 333 Tin 0.056 0.0637 Lead, 0.0314 0.0402 139. Atomic Heat. — It has been found that the product of the specific heat by the atomic weight of any simple body 176 EIEMENTARY PHYSICS. [140 is a constant quantity. This law is known from its discoverers as the law of Dulong and Petit. This law may be otherwise stated, thus : that to raise the temperature of an atom of any simple substance one degree, an amount of heat is required which is the same for all sub- stances. The experiments of Regnault show that this law may be extended to compound bodies ; that is, for all compounds of similar chemical composition the product of the total chemical equivalent by the specific heat is the same. The following table will illustrate the law of Dulong and Petit. The atomic weights are those given by Clarke. Blements. Iron, Specific Heat of Equal Weights. O.I 14 Atomic Weight. 55-9 Product of Specific Heat into Atomic Weigln. 6.372 Copper, 0.095 63.17 6.00I Mercury, 0.0314 (solid) 199 71 6.128 Silver, . 0.057 107.67 6.137 Gold, . 0.0329 196.15 6.453 Tin, . . 0.056 117.7 6.591 Lead, . 0.0314 206.47 6.483 Zinc, . 0.0955 64.9 6.198 140. Fusion and Solidification. — When ice at a tempera- ture below zero is heated, its temperature rises to zero, and then the ice begins to melt ; and, however high the tempera- ture of the medium that surrounds it may be, its temperature remains constant at zero so long as it remains in the solid state. This temperature is the melting-point of ice, and because of its fixity it is used as one of the standard temperatures in graduating thermometric scales. Other bodies melt at very different but at fixed and definite temperatures. Many sub- stances cannot be melted, as they decompose by heat. Alloys often melt at a lower temperature than any of their constituents. An alloy of one part lead, one part tin, four 142] EFFECTS OF HEAT. 1/7 parts bismuth, melts at 94° ; while the lowest melting-point of its constituents is that of tin, 228°. An alloy of lead, tin, bismuth, and cadmium melts at 62°. If a liquid be placed in a medium the temperature of which is below its melting-point, it will, in general, begin to solidify when its temperature reaches its melting-point, and it will re- main at that temperature until it is all solidified. Under cer- tain conditions, however, the temperature of a liquid may be lowered several degrees below its melting-point without solidi- fication, as will be seen below. 141. Change of Volume with Change of State. — Sub- stances are generally more dense in the solid than in the liquid state, but there are some notable exceptions. Water, on solidi- fying, expands ; so that the density of ice at zero is only 0.9167, while that of water at 4° is i. This expansion exerts consider- able force, as is evidenced by the bursting of vessels and pipes containing water. 142. Change of Melting- and Freezing-Points.— If water be enclosed in a vessel sufficiently strong to, prevent its expan- sion, it cannot freeze except at a lower temperature. The freezing-point of water is, therefore, lowered by pressure. On the other hand, substances which contract on solidifying have their solidification hastened by pressure. The lowering of the melting-point of ice by pressure explains some remarkable phenomena. If pieces of ice be pressed to- gether, even in warm water, they will be firmly united. Frag- ments of ice may be moulded, under heavy pressure, into a solid, transparent mass. This soldering together of masses of ice is called regelation. If a loop of wire be placed over a block of ice and weighted, it will cut its way slowly through the ice, and regelation will occur behind it. After the wire has passed throiigh, the block will be found one solid mass, as before. The explanation of these phenomena is, that the ice is partially melted by the pressure. The liquid thus formed is 12 1/8 ELEMENTARY PHYSICS. [143 colder than the ice ; it finds its way to points of less pressure, and there, because of its low temperature, it congeals, firmly uniting the two masses. Water, when freed from air and kept perfectly quiet, will not form ice at the ordinary freezing-point. Its temperature may be lowered to — 10° or — 12° without solidification. In this condition a slight jar, or the introduction of a small frag- ment of ice, will cause a sudden congelation of part of the liquid, accompanied by a rise in temperature in the whole mass to zero. A similar phenomenon is observed in the case of several solutions, notably sodium sulphate and sodium acetate. If a saturated hot solution of one of these salts be made, and al- lowed to cool in a closed bottle in perfect quiet, it will not crystallize. Upon opening the bottle and admitting air, crys- tallization commences, and spreads rapidly through the mass, accompanied by a considerable rise of temperature. If the amount of salt dissolved in the water be not too great, the so- lution will remain liquid when cooled in the open air, and it may even suffer considerable disturbaiice by foreign bodies without crystallization ; but crystallization begins immediately upon contact with the smallest crystal of the same salt. 143. Heat Equivalent of Fusion. — Some facts that have appeared in the above account of the phenomena of fusion and solidification require further study. It has been seen that, however rapidly the temperature of a solid may be rising, the moment fusion begins the rise of temperature ceases. What- ever the heat to which a solid may be exposed, it cannot be made hotter than its melting-point. When ice is melted by pressure, its temperature is lowered. When a liquid is cooled, its fall of temperature ceases when solidification begins ; and if, as may occur under favorable conditions, a liquid is cooled below its melting-point, its temperature rises at once to the melting-point, when solidification begins. Heat, therefore, dis- 145] EFFECTS OF HEAT. 1/9 appears when a body melts, and is generated when a liquid be- comes solid. It was stated (§ loi) that ice can be melted by friction ; that is, by the expenditure of mechanical energy. Fusion is, therefore, work which requires the expenditure of some form of energy to accomplish it. The heat required to melt unit mass of a substance is the heai equivalent of fusion of that sub- stance. When a substance solidifies, it develops the same amount of heat as was required to melt it. 144. Nature of the Energy stored in the Liquid. — From the facts given above, as well as from the principle of the con- servation of energy, it appears that the energy expended in melting a body is stored in the liquid. It is easy to see what must be the nature of this energy. When a body solidifies, its molecules assume certain positions in obedience to their mu- tual attractions. When it is melted, the molecules are forced into new positions in opposition to the attractive forces. They are, therefore, in positions of advantage with respect to these forces, and possess potential energy. 145. Determination of the Heat Equivalent of Fusion. — The heat equivalent of fusion may be determined by the method of mixtures (§ iii), as follows : a mass of ice, for ex- ample, represented by P, at a temperature t below its melting- point, to insure dryness, is plunged into a mass P' of warm water at the temperature T. Represent by 6 the resulting temperature, when the ice is all melted. If / represent the water equivalent of the calorimeter, (/"+/)( 7"— ^) is the heat given up by the calorimeter and its contents. Let c represent the specific heat of ice, and x the heat equivalent of fusion. The ice absorbs, to raise its temperature to zero, Ptc calories ; to melt it, Px calories ; to warm the water after melt- ing, PB calories. We then have the equation Ptc + PO + P^={P'+P){T- 6), from which x may be found. I So ELEMENTARY PHYSICS. [146 Other calorimetric methods may be employed. The best experiments give, for the heat equivalent of fusion of ice, very nearly eighty calories. GASES AND VAPORS. 14b. The Gaseous State. — A gas may be defined as a highly compressible fluid. A given mass of gas has no definite volume. Its volume varies vs^ith every change in the external pressure to which it is exposed. A vapor is the gaseous state of a substance which at ordinary temperatures exists as a solid or a liquid. 147. Vaporization is the process of formation of vapor. There are two phases of the "process, evaporation, in which vapor is formed at the free surface of the liquid, and ebullition, in which the vapor is formed in bubbles in the mass of the liquid, or at the heated surface with which it is in contact. 148. Nature of the Process of Evaporation. — It has been seen (§ loi) that there are many reasons for believing that the molecules of solids and liquids are in a state of continual motion. It is not supposed that any one molecule maintains continuously the same condition of motion ; but in the inter- action of the molecules the motion of any one may be more or less violent, as it receives motion from its neighbors, or gives up motion to them. It can easily be supposed that, at the ex- posed surface of the substance, the motion of a molecule may at times be so violent as to project it beyond the reach of the molecular attractions. If this occur in the air, or in a space filled with any gas, the molecule maybe turned back, and made to rejoin the molecules in the liquid mass; but many will find their way to such a distance that they will not return. They then constitute a vapor of the substance. As the number of free molecules in the space above the liquid increases, it is plain that there may come a time when as many will rejoin the liquid as escape from it. The space is then saturated with the vapor. 149] EFFECTS OF HEAT. l8l The more violent the motion in the liquid, that is the higher its temperature, the more rapidly the molecules will escape, and the greater must be the number in the space above the liquid before the returning will equal in number the outgoing mole- cules. In other words, the higher the temperature, the more dense the vapor that saturates a given space. If the space above a liquid be a vacuum, the escaping molecules will at first meet with no obstruction, and, as a consequence, the space will be very quickly saturated with the vapor. Experiment verifies all these deductions. Evaporation goes on continually from the free surfaces of many liquids, and even of solids. It increases in rapidity as the temperature increases, and ceases when the vapor has reached a certain density, alwajs the same for the same temperature, but greater for a higher temperature. It goes on very rapidly in a vacuum ; but it is found that the final density of the vapor is no greater, or but little greater, than when some other gas is present. In other words, while a foreign gas impedes the motion of the outgoing molecules, and causes evaporation to go on slowly, it has very little influence upon the number of molecules that must be present in order that those which return may equal in number those which escape. 149. Pressure of Vapors.r— As a liquid evaporates in a closed space, the vapor formed exerts a pressure upon the en- closure and upon the surface of the liquid, which increases so long as the quantity of vapor increases, and reaches a maxi- mum when the space is saturated. This maximum pressure of a vapor increases with the temperature. When evaporation takes place in a space filled by another gas which has no action upon the vapor, the pressure of the vapor is added to that of the gas, and the pressure of the mixture is, therefore, the sum of the pressures of its constituents. The law was announced by Dalton that the quantity of vapor which saturates a given space, and consequently the maximum pressure of that vapor, is the same whether the space be empty or contain a gas. 1 82 ELEMENTARY PHYSICS. [15© Regnault has shown that, for water, ether, and some other substances, the maximum pressure of their vapors is slightly- less when air is present. 150. Ebullition. — As the temperature of a liquid rises, the pressure which its vapor may exert increases, until a point is reached where the vapor is capable of forming, in the mass of the liquid, bubbles which can withstand the superincumbent pressure of the liquid and the atmosphere above it. These bubbles of vapor, escaping from the liquid, give rise to the phenomenon called ebullition, or boiling. Boiling may, there- fore, be defined as the agitation of a liquid by its own vapor. Generally speaking, for a given liquid, ebullition always oc- curs at the same temperature for the same pressure ; and, when tjnce commenced, the temperature of the liquid no longer rises, no matter how intense the source of heat. This fixed temperature is called the boiling-point of the liquid. It differs for different liquids, and for the same liquid under different pressures. That the boiling-point must depend upon the pres- sure is evident from the explanation of the phenomenon of ebullition above given. Substances in solution, if less volatile than the liquid, retard ebullition. While pure water boils at 100°, water saturated with common salt boils at I09°._ The material of ' the contain- ing vessel also influences the boiling point. In a glass vessel the temperature of boiling water is higher than in one of metal. If water be deprived of air by long boiling, and then cooled, its temperature may afterwards be raised considerably above the boiling-point before ebullition commences. Under these con- ditions, the first bubbles of vapor will form with explosive vio- lence. The air dissolved in water separates from it at a high temperature in minute bubbles. Into these the water evapo- rates, and, whenever the elastic force of the vapor is sufficient to overcome the superincumbent pressure, it enlarges them, and causes the commotion that marks the phenomenon of IS2] EFFECTS OF HEAT. 1 83 ebullition. If no such openings in the mass of the fluid exist, the cohesion of the fluid, or its adhesion to the vessel, as well as the pressure, must be overcome by the vapor. This explains the higher temperature at which ebullition commences when the liquid has been deprived of air. 151. Spheroidal State. — If a liquid be introduced into a highly heated capsule, or poured upon a very hot plate, it does not wet the heated surface, but forms a flattened spheroid, which presents no appearance of boiling, and evaporates only very slowly. Boutigny has carefully studied these phenomena, and made known the following facts. The temperature of the spheroid is below the boiling-point of the liquid. The spheroid does not touch the heated plate, but is separated from it by a non-conducting layer of vapor. This accounts for the slowness of the evaporation. To maintain the liquid in this condition the temperature of the capsule must be much above the boil- ing-point* of the liquid ; for water it must be at least 200° C. If the capsule be allowed to cool, the temperature will soon fall below the limit necessary to maintain the spheroidal state, the liquid will moisten the capsule, and there will be a rapid ebul- lition, with disengagement of vapor. If a liquid of very low boiling-point, as liquid nitrous oxide, which boils at — 88°, be poured into a red-hot capsule, it will assume the spheroidal state ; and, since its temperature cannot rise above its boiling- point, water, or even mercury, plunged into it, will be frozen. 152. Production of Vapor in a Limited Space. — When a liquid is heated in a limited space the vapor generated accumu- lates, increasing the pressure, and the temperature rises above the ordinary boiling-point. Cagniard-Latour experimented upon liquids in spaces but little larger than their own volumes. He found that, at a certain temperature, the liquid suddenly disappeared ; that is, it was converted into vapor in a space but little larger than its own volume. It is supposed that above the temperature at which this occurs, which is called the 1 84 ELEMENTARY PHYSICS. [153 critical temperature, the substance cannot exist in the liquid state. 153. Liquefaction. — Only a certain amount of vapor can exist at a given temperature in a given space. If the tempera- ture of a space saturated with vapor be lowered, some of the vapor must condense into the liquid state. It is not necessary that the temperature of the whole space be lowered ; for, when the vapor in the cooled portion is condensed, its pressure is diminished, the vapor from the warmer portion flows in, to be in its turn condensed, and this continues until the whole is brought to the density and pressure due to the cooled portion. Any diminution of the space occupied by a saturated vapor at constant temperature, will cause some of the vapor to be- come liquid, for, if it do not condense, its pressure must in- crease ; but a saturated vapor is already at its maximum pres- sure. If the vapor in a given space be not at its maximum pres- sure, its pressure will increase when its volume is diminished, until the maximum pressure is reached ; when, if the tempera- ture remain constant, further reduction of volume causes condensation into the liquid state, without further increase of pressure or density. This statement is true of several of the gases at ordinary temperatures. Chlorine, sulphur dioxide, ammonia, nitrous oxide, carbon dioxide, and several other gases, become liquid under sufficient pressure. Andrews found that, at a temperature of 30.92°, pressure ceases to liquefy carbon dioxide. This is the critical temperature for that substance. The critical temperatures of oxygen, hydrogen, and the other so-called permanent gases, are so low that it is only by methods capable of yielding an extremely low tempera- ture that they can be liquefied. By the use of such methods any of the gases may be made to assume the liquid state. In the case of hydrogen, however, the low temperature necessary for its liquefaction has only been reached by allowing the gas 1S4] EFFECTS OF HEAT. 185 to expand suddenly from a condition of great condensation, in which it had already been cooled to a very low point. 154. Pressure and Density of Non-saturated Gases and Vapors. — If a gas or vapor in the non-saturated condition be maintained at constant temperature, it follows very nearly Boyle's law (§§ "JJ and 98). If its temperature be below its critical temperature, the product of volume by pressure dimin- ishes, and near the point of saturation the departure from the law may be considerable. At this point there is a sudden diminution of volume, and the gas assumes the liquid state. The less the pressure and density of the gas, the more nearly it obeys Boyle's law. It has been stated already (§ 99) that gases expand as the temperature rises. The law of this expansion, called, after its discoverer, Gay-Lussac s law, is that, for each increment of temperature of one degree, every gas expands by the same constant fraction of its volume at zero. This is equivalent to saying that a gas has a constant coefficient of expansion, which is the same for all gases. Let Vo , Vt , represent the volumes at zero and t respectively, and a the coefficient of expansion. Then, the pressure remain- ing constant, we have V,= VXi + oa). (51) If d„, dt, represent the densities at the same two tempera- tures, we have, since densities are inversely as volumes, Later investigations, especially those of Regnault, show that this simple law, like the law of Boyle, is not rigorously true, though it is very nearly so for all gases and vapors which are 1 86 ELEMENTARY PHYSICS. [155 not too near their points of saturation. The common co- efficient of expansion is a = 0.003666 = ^^ very nearly. From the law of Boyle we have, for a given mass of gas, if the temperature remain constant, Vpp = Vp'p' = volume at pressure unity, where Vp, Vf, represent the volumes at pressure / and/' re- spectively. From the law of Gay-Lussac we have, if the pressure remain constant, ° 1^ at I H- at' ^^^' If the temperature and pressure both vary, we have I _j_ «^ - 1 4- «r ' '^54; that is, if the volume of a given mass of gas be multiplied by the corresponding pressure and divided by the factor of ex- pansion, the quotient is constant. 155. Pressure and Density of Saturated Gases and Vapors. — It has been seen that, for each gas or vapor at a temperature below the critical temperature, there is a maximum pressure which it can exert at that temperature. To each temperature there corresponds a maximum pressure, which is higher as the temperature is higher. A gas or vapor in contact with its liquid in a closed space will exert its maximum pressure. The relation between the temperature and the correspond- ing maximum pressure of a vapor is a very important one, and has been the subject of many investigations. The vapor of water has' been especially studied, the most extensive and accurate experiments being those of Regnault. iSS] EFFECTS OF KEAT. 187 Two distinct methods were employed, one for temperatures below 50°, and the other for higher temperatures. The first consisted in observing the difference in height of two barom- eters placed side by side, the vacuum chamber of one con- taining a little water. The temperature was carried from zero to about 50°. Both barometers were surrounded by the same medium, and in every way under the same conditions, except that water and its vapor were present in one and not in the other. The difference between the heights of the two gave the pressure of the vapor at the temperature of the experiment. The second method was founded on the principle that the vapor of a boiling liquid exerts a pressure equal to that of the atmosphere above it. The experiment consisted in boiling water in a closed space in which the air could be rarefied or con- densed to a known pressure, and noting the temperature of the boiling liquid and that of the vapor above it. To prevent the accumulation of the vapor and the consequent change of pres- sure, a condenser communicated with the boiler, consisting of a tube surrounded by a larger tube, forming an annular space, through which a stream of cold water was kept flowing. By this means the vapor was condensed as fast as formed, and the water from its condensation flowed back into the boiler. By rarefy.ng or compressing the air in the closed space, an artificial atmos- phere of any desired pressure could be obtained, and maintained constant as long as was necessary for making the observations. The temperature was determined by means of four thermometers placed in the boiler, two of them in the liquid m » so 40 50 60 70 80 90 luu hjG 51. l88 ELEMENTARY PHYSICS. [156 and two in the vapor. The bulbs of the thermometers were placed in metal tubes, to protect them from the pressure, which otherwise would compress the bulb, and cause the thermome- ter to register too high a temperature. The results of Regnault's observations may be repre- sented graphically, as in Fig. 51, where pressures are measured in the vertical, and temperatures in the horizontal, direction. It is seen that the pressure varies very rapidly with the tem- perature. 156. Kinetic Theory of Gases. — ^According to the kinetic theory of gases, a perfect gas consists of an assemblage of free, perfectly elastic molecules in constant motion. Each mole- cule moves in a straight line with a constant velocity, until it encounters some other molecule, or the side of the vessel. The impacts of the molecules upon the sides of the vessel are so numerous that their effect is that of a continuous constant force or pressure. The entire independence of the molecules is assumed from the fact that, when gases or vapors are mixed, the pressure of one is added to that of the others ; that is, the pressure of the mixture is the sum of the pressures of the separate gases. It follows from this, that no energy is required to separate the molecules ; in other words, no internal work need be done to expand a* gas. This was demonstrated experimentally by Joule, who showed that when a gas expands without perform- ing external work, it is not cooled. The action between two molecules, or between a molecule and a solid wall, must be of such a nature that no energy is lost ; that is, the sum of the kinetic energies of all the mole- cules must remain constant. Whatever be the nature of this action, it is evident that when a molecule strikes a solid stationary wall, it must be reflected back with a velocity equal to that before impact. If the velocity be resolved into two components, one parallel to the wall and the other normal to 156] EFFECTS OF HEAT. 1 89 it, the parallel component remains unchanged, while the nor- mal component is changed from -f- v, its value before impact, to — I', its value after impact. The change of velocity is there- fore 2v ; and if represent the duration of impact, the mean . . . '2.1) 2V acceleration is -^ , and the mean force of impact p = m-yr , where m represents the mass of the molecule. Since the effect of the impacts is a continuous pressure, the total pressure P exerted upon unit area is equal to this mean force of impact of one molecule multiplied by the num- ber of molecules meeting unit area in the time 0. To find this latter factor, we suppose the molecules confined between two parallel walls at a distance s from each other. Any molecule may be supposed to suffer reflection from one wall, pass across to the other, be reflected back to the first, and so on. What- ever may be the effect of the mutual collisions of the mole- cules, the number of impacts upon the surface considered will be the same as though each one preserved its rectilinear mo- tion unchanged, except when reflected from the solid walls. The time required for a molecule moving with a velocity v to 2S pass across the space between the two walls and back is — ; and the number of impacts upon the first surface in unit time is — . 2S Represent by n' the number of molecules in a rectangular prism, with bases of unit area in the walls. These molecules must be considered as moving in all directions and with various velocities. But the velocity of any molecule may be resolved in the direction of three rectangular axes, one normal to the surface and the other two parallel to it ; and, since the number of molecules in any finite volume of gas is practically infinite, the effect upon the wall due to their real motions will be the same as would result from a motion of one third the total 190 ELEMENTARY PHYSICS. [156 number of molecules in each of the three directions with the mean velocity. Hence the number of molecules moving, in a manner similar to that of the single molecule already consid- ered, normal to the walls is \n' . The number of impacts upon I n'v unit area of the first surface in unit time is ; and in time d 3 '2S . I n'vd ^^ , , r. • . IS . Hence the total pressure F on unit area is „ 2V I «'Z'^ I ,«' P — m-a X = - mir- . e I 2s 3 s n But — is the number of molecules in unit volume. Repre- senting this by n, we have P = ^nmi/'. (55) That is, the pressure upon unit area is equal to one third the number of molecules in unit volume at that pressure multiphed by twice the kinetic energy of each molecule. Suppose, now, the volume of the gas be changed from unity to V, without change of temperature. The number of mole- cules in unit volume is now v?, and the pressure P, = - -^mv", whence P^V= ^nmv'. This is a constant quantity, since n and m are constant for the same mass of gas, and v is constant if there be no change of temperature. But PV equal to a con- stant is Boyle's law. From the law of Gay-Lussac we have, if P represent the pressure at t°, and P„ the pressure at zero, P=PXl + at). 1S6] EFFECTS OF HEAT. IQI We have a = -^^ very nearly ; hence i'=^.(. + ^). (56) It / = - 2;3", that is, at 273° below zero the pressure vanishes. Since P = \nmv'', it follows that, at this temperature, z/ = o, or the molecules are at rest. This temperature is therefore called the absolute zero. In studying the expansion of gases, it is very convenient to use a scale of temperatures the zero-point of which is at the absolute zero. Temperatures reckoned upon this scale are called absolute temperatures. Let T represent a temperature upon the absolute scale: then T = t -\- 273, and Eq. (56) be- T comes P— P^ . Substituting the value of P from (55), we 273 have whence T ^ °273' T = ^n^^-mv\ (57) That is, the absolute temperature of a gas is proportional to the kinetic energy of the molecules. It has been already stated (§ 100), that, when a gas is com- pressed, a certain amount of heat, is generated. Suppose a cylinder with a tightly-fitting piston. So long as the piston is 192 ELEMENTARY PHYSICS. [157 at rest, each molecule that strikes it is reflected with a velocity equal to that before impact : but if the piston be forced into the cylinder, each molecule, as it is reflected, has its velocity increased ; and, as was shown above, this is equivalent to a rise in temperature. It can be shown that the increase of kinetic energy in this case is precisely equal to the work done in forc- ing the piston into th.e cylinder against the pressure of the gas. On the other hand, if the piston be pushed backward by the force of the impact of the molecules, there will be a loss of velocity by reflection from the moving surface, kinetic energy equal in amount to the work done upon the piston disappears, and the temperature falls. The phenomena exhibited by the radiometer afford a strong experimental confirmation of the kinetic theory of gases. These phenomena were discovered by Crookes. In the form first given to it by him, the instrument consists of a delicate torsion balance suspended in a vessel from which the air is very corripletely exhausted. On one end of the arm of the torsion balance is fixed a light vane, one face of which is blackened. When a beam of light falls on the vane, it moves as if a press- ure were applied to its blackened surface. The explanation of this movement is, that the molecules of air remaining in the vessel are more heated when they come in contact with the blackened face of the vane than when they come in contact with the other face, and are hence thrown off with a greater velocity, and react more strongly upon the blackened face of the vane. At ordinary pressures the free paths of the mole- cules are very small, their collisions very frequent, and any in- equality in the pressures is so speedily reduced, that no effect upon the vane is apparent. At the high exhaustions at which the movement of the vane becomes evident, the collisions are less frequent; and hence an immediate equalization of pressure does not occur. The vane therefore moves in consequence of the greater reaction upon its blackened surface. 158] EFFECTS OF HEAT. 193 157. Mean Velocity of Molecules.— Equation (55) enables us to determine the mean velocity of the molecules of a gas of which the density and pressure are known, since nm is the mass of the gas in unit volume. Solving the equation with reference to v, and substituting the known values of the constants for hydrogen, namely, P^ 1013373 dynes per square centimetre, and nm, or density, = 0.00008954 grams per cubic centimetre, we have 184260 cen- timetres per second, or a little more than one mile per second, as the mean velocity of a molecule of hydrogen. 158. Elasticity of Gases. — It has been shown (§ "]•]) that the elasticity of a gas, obeying Boyle's law, is numerically equal to the pressure. This is the elasticity for constant temperature. But, as was seen (§ 156), when a gas is compressed it is heated ; and heating a gas increases its pressure. Under ordi- nary conditions, therefore, the ratio of a small increase of pres- sure to the corresponding decrease of unit volume is greater than when the temperature is constant. It is important to consider the case when all the heat generated by the compres- sion is retained by the gas. The elasticity is then a maximum, and is called the elasticity when no heat is allowed to enter or escape. Let mn (Fig. 52) be a curve representing the relation be- tween volume and pressure for con- stant temperature, of which the ab- scissas represent volumes and the ordinates pressures. Such a curve is called an isothermal line. It is plain that to each temperature must correspond its own isothermal line. If, now, we suppose the gas to be compressed, and no heat to escape, it is plain that if the volume dimin-o ish from OC to OG, the pressure will become greater than GD\ suppose it to be GM. If a number ij 194 ELEMENTARY PHYSICS. [159 of such points as M be found, and a line be drawn through them, it will represent the relation between volume and pres- sure when no heat enters or escapes. It is called an adiabafic line. It evidently makes a greater angle with the horizontal than the isothermal. 159. Specific Heats of Gases. — In § 1 56 it is seen that the temperature of a gas is proportional to the kinetic energy of its molecules. To warm a gas without change of volume is, therefore, only to add to this kinetic energy. If, however, the gas be allowed to expand when heated, the molecules lose energy by impact upon the receding surface ; and this, together with the kinetic energy due to the rise in temperature, must be supplied from the source of heat.' It has been seen that the loss of energy resulting from impact upon a receding surface is equal to the work done by the gas in expanding. The amount of heat necessary to raise the temperature of unit mass of a gas one degree, while the volume remains un- changed, is called the specific heat of the gas at constant volume. The amount of heat necessary to raise the temperature of unit mass of a gas one degree when expansion takes place without change of pressure, is called the specific heat of the gas at con- stant pressure. From what has been said above, it is evident that the differ- ence between these two quantities of heat is the equivalent of the work done by the expanding gas. The determination of the relation of thest two quantities is a very important problem. The specific heat of a gas at constant pressure may be found by passing a current of warmed gas through a tube coiled in a calorimeter. This is the method of mixtures (§ 1 1 1). There are great difficulties in the way of an accurate determination, because of the small density of the gas, and the time required to pass enough of it through the calorimeter to obtain a reason- able rise of temperature. The various sources of error produce 159] EFFECTS OF HEAT. 195 effects which are sometimes as great as, or even greater than, the quantity to be measured. It is beyond the scope of this work to describe in detail the means by which the effects of the disturbing causes have been determined or eHminated. The specific heat of a gas at constant volume is generally determined from the ratio between it and the specific heat at constant pressure. The first determination of this ratio was accomplished by Clement and Desormes. The theory of the experiment may be understood from the following considerations : Let a unit mass of gas at any temperature / and volume Vt be confined in a cylinder by a closely fitting piston of area A. Suppose its temperature to be raised one degree, by com- munication of heat from some external source, while its volume remains unchanged. It absorbs heat, which we will suppose measured in mechanical units, and will represent by Cj, the specific heat at constant volume. Now let the gas expand, at the constant temperature t -\- i, -until it returns to its original pressure. During this expansion the piston will be forced out through a distance d, and an additional quantity of heat will be absorbed from the source. Represent by P the mean pressure on unit area of the piston exerted by the gas during this operation. Then the work done during expansion, which is the equivalent of the heat absorbed, is PAd. .(4 aT represents the increase in volume of the gas during this process. The same increase in volume would have occurred had the gas been allowed to expand at constant pressure, while its temperature was rising. But, for a rise in temperature of one degree, the increase in volume of any mass of gas is aV^, where V^ repre- sents the volume at zero. Hence we have Ad^^ aV„, and the work done during the expansion is PAd = Pa V^. The heat absorbed, therefore, in raising the temperature of the gas one degree at constant pressure is Cp ^ Cz, -\- Pa. F„. Cp represents the specific heat of the gas at constant pressure, measured in 196 ELEMENTARY PHYSICS. [lS9 mechanical units. The ratio of the two specific heats is ^=i + ^i'«f'.. (58) If, in the case considered above, the gas had expanded without receiving any heat, the work PaV^ would have been done at the expense of its own internal energy, and the temperature would have fallen. The performance of this work is equivalent to abstracting the quantity of heat, PaV„ which would lower the temperature -^ -PaV^ degrees, since the ab- straction of a quantity C-^ of heat would lower the temperature one degree. Represent this change of temperature by 6. Re- membering that the supposed change of volume was or F,, which OiVt equals """377"^. and that the original volume was Vt, it is seen a. that the change of — , — - in unit volume would cause a fall in *= \ A^ at temperature of ^degrees. Substituting B ior -p^-paV„ in Eq. (58), we have -^ = i -\- 6. It is the object of the experiment to find 0. The method of Clement and Desormes is as follows t A large flask is furnished with a stopcock having a large opening, and a very sensitive manometer which shows the difference between the pressure in the flask and the pressure of the air. The air in the flask is first rarefied, and left to assume the temperature of the surrounding atmosphere. Sup- pose its pressure now to he If — k, H representing the height of the barometer, and k the difference between the pressure in the flask and the pressure of the atmosphere, as shown by the manometer. The large stopcock is then suddenly opened for a very short time only ; the air rushes in, re-establishes th& 159] EFFECTS OF HEAT. 197 atmospheric pressure, compresses the air originally in the flask, and raises its temperature. ,The volume of the air becomes I — 0, where its original volume is taken as unity and repre- sents its reduction ; and, if there were no change of tempera- H —h ture, the pressure would be -r. If the temperature m crease B' degrees, and become t -\- 0', the pressure will be r^r-0 X —T^Ta— - ^' (59) the atmospheric pressure. The flask is now left until the air within it returns to the temperature of the atmosphere t, when the manometer shows a fall of pressure h' , and we have H — k ^ = H-h'. (60) From these two equations we have ^ _ ^- ^' B' - (1+^)1' ^- H-'h" - a{H-k')' Suppose, now, the change of volume had been -j- — -, then the change of temperature would have been 6 ; and, since change of volume is proportional to change of temperature, we have 0:— f~ = ^':^; hence a 6'- ^ = — 0— : 198 ELEMENTARY PHYSICS. \\tii or, substituting the values of and B' , we have H-h' ^ h-h'~ h Now we have shown that ^=1 + ^; hence 160, The Two Specific Heats of a Gas have the Same Ratio as the Two Elasticities. — Suppose a gas, of which the mass is unity and volume V, to rise in temperature at constant pressure from the temperature t to the temperature {t -\- At), At representing a very small increment of tempera- ture. The heat consumed will be C^At, and the increase of volume aV„At. Now, if the volume had remained constant, the amount of heat required to cause the rise of temperature At would have been C^At. Hence if the gas be not allowed to expand, the amount of heat, CpAt, will cause a rise of tempera- ture -J At ; and the same rise of temperature will occur if the gas, after first being allowed to expand, be compressed to its initial volume. Such a compression would be attended by an increase of pressure, which we will call Ap. The ratio between this and the corresponding change of volume is Ap l6o] EFFECTS OF HEAT. 199 where £/, is the elasticity under the condition that no heat enters or escapes. If, now, tlie heat produced by compression be allowed to escape, there will remain the quantity C-j^t, and the increment of pressure will be reduced to Sp = Ap-~. This is the increase of pressure that will occur if the gas be compressed by the amount aV^^t without change of temperature; hence ^■^. = £. (63) aVM where Et is the elasticity for constant temperature. Dividing (62) by (63), we have Eh_ aV,At Ap _^_ C; Et~ Sp ~ dp- C^~ C' that is, the two elasticities have the same ratio as the two specific heats of a gas. It may be shown that the velocity of sound in any medium is equal to the square root of the quotient of the elasticity divided by the density of the medium ; that is, velocity =■ \ — (64) In the progress of a sound-wave, the air is alternately com- 200 ELEMENTARY PHYSICS. [l6i pressed and rarefied, the compressions and rarefactions occur- ring in such rapid succession that there is no time for any transfer of heat. If Eq. (64) be applied to air, the E becomes Eh, or the elasticity under the condition that no heat enters or escapes. Since we know the density of the air and the velocity of sound, Eh can be computed. In § jj it is shown that Et is numerically equal to the pressure ; hence we have the values of the two elasticities of air, and, as seen above, their ratio is the ratio of the two specific heats of air. 161. Examples of Energy absorbed by Vaporization. — When a liquid boils, its temperature remains constant, however intense the source of heat. This shows that the heat applied to it is expended in producing the change of state. Heat is absorbed during evaporation. By promoting evaporation, in- tense cold may be produced. In a vacuum, water may be frozen by its own evaporation. If a liquid be heated to a,, temperature above its ordinary boiling-point under pressure, relief of the pressure is followed by a very rapid evolution of vapor and a rapid cooling of the liquid. Liquid nitrous oxide at a temperature of zero is still far above its boiling-point, and its vapor exerts a pressure of about thirty atmospheres. If the liquid be drawn off into an open vessel, it at first boils with extreme violence, but is soon cooled to its boiling-point for the atmospheric pressure, about — 88°, and then boils away slowly, while its temperature remains at that low point. 162. Heat Equivalent of Vaporization. — It is plain, from what has preceded (§ 148), that the formation of vapor is work requiring the expenditure of energy for its accomplishment. Each molecule that is shot off into space obtains the motion which projected it beyond the reach of the molecular attrac- tion, at the expense of the energy of the molecules that remain behind. A quantity of heat disappears when a liquid evapo- rates ; and experiment demonstrates, that to evaporate a kilo- gram of a liquid at a given temperature always requires the l64] EFFECTS OF HEAT. 201 same amount of heat. This is the heat equivalent of vaporiza- tion. When a vapor condenses into the Hquid state, the same amount of heat is generated as disappears when the liquid assumes the state of vapor. The heat equivalent of vaporiza- tion is determined by passing the vapor at a known tempera- ture into a calorimeter, there condensing it into the liquid state, and noting the rise of temperature in the calorimeter. This, it will be seen, is essentially the method of mixtures. Many experimenters have given attention to this determina- tion ; but here, again, the best experiments are those of Reg- nault. He determined what he called the total Iteat of steam at various pressures. By this was meant the heat required to raise the temperature of a kilogram of water from zero to the temperature of saturated vapor at the pressure chosen, and then convert it wholly into steam. The result of his experi- ments give, for the heat equivalent of vaporization of water at ioo°, 537 calories. "That is, he found that by condensing a kilogram of steam at ioo° into water, and then cooling the water to zero, 637 calories were obtained. But almost exactly 100 calories are derived from the water cooling from 100° to zero ; hence 537 calories is the heat equivalent of vaporization at 100°. 163. Dissociation. — It has already been noted (§ 99), that, at high temperatures, compounds are separated into their ele- ments. To effect this separation, the powerful forces of chem- ical affinity must be overcome, and a considerable amount of energy must be consumed. 164. Heat Equivalent of Dissociation and Chemical Union. — From the principle of the conservation of energy, it may be assumed that the energy required for dissociation is the same as that developed by the reunion of the elements. The heat equivalent of chemical union is not easy to deter- mine because the process is usually complicated by changes of physical state. We may cause the union of carbon and 202 ELEMENTARY PHYSICS. [165 oxygen in a calorimeter, and, bringing the products of com- bustion to the temperature of the elements before the union, measure the heat given to the instrument ; but the carbon has changed its state from a solid to a gas, and some of the chem- ical energy must have been consumed in that process. The heat measured is the available heat. The best determinations of the available heat of chemical union have been made by Andrews, Favre and Silbermann, and Berthelot. HYGROMETRY. 165. Object of Hygrometry. — Hygrometry has for its ob- ject the determination of the state of the air with regard to moisture. The amount of vapor in a given volume of air may be de- termined directly by passing a known volume of air through tubes containing some substance which will absorb the mois- ture, and finding the increase in weight of the tubes and their contents. The quantity of vapor contained in a cubic metre of air is called its absolute humidity. Methods of determining this quantity indirectly are given below. 166. Pressure of the Vapor. — It has been seen (§ 149), that, when two or more gases occupy the same space, each exerts its own pressure independently of the others. The pres- sure of the atmosphere is, therefore, the pressure of the dry air, with that of the vapor of water added. If we can determine this latter pressure it is easy to compute the quantity of mois- ture in the air. It has also been seen that the pressure exerted by the vapor in the air is at a certain temperature its maximum pres- sure. Now, if any small portion of the space be cooled till its temperature is below that at which the pressure exerted is the maximum pressure, a portion of the vapor will condense into liquid. If, then, we determine the temperature at which con- densation begins, the maximum pressure of the vapor for this 167] EFFECTS OF HEAT. 203 temperature, which may be found from tables, is the real pres. sure of the vapor in the air. The mass of vapor in a cubic metre of air may then be computed as follows : A cubic metre of dry air has a mass of 1293.2 grams at zero and at 760 milli- metres pressure. At the pressure / of the vapor, and tem- perature t of the air at the time of the experiment, the same space would contain P I 1293-2 X -^ X 760 \-\- at grams of air ; and, since the density of vapor of water referred to air is 0.623, a cubic metre would contain 1293.2 X ^ X ^-^ X 0.623 (65) grams of vapor. 167. Dew Point. — The temperature at which the vapor of the air begins to condense is called the dew point. It is deter- mined by means of instruments called dew-point hygrometers, which are instruments so constructed that a small surface exposed to the air may be cooled until moisture deposits upon it, when its temperature is accurately determined. The AUuard hygrometer consists of a metal box about one and a half centimetres square and four centimetres deep. Two tubes pass through the top of the box — one terminating just inside and the other extending to the bottom. One side of the box is gilded and polished, and is so placed that the gilded surface lies on the same plane with, and in close proximity to, a gilded metal plate. The box is partly filled with ether, and the short tube is connected with an aspirator. Air is thus drawn through the longer tube, and, bubbling up through the ether, causes rapid evaporation, which soon cools the box, and causes a deposit of dew upon the gilded surface. The presence of the gilded plate helps very much in recognizing the beginning of the deposit of dew, by the contrast between it and the dew- 204 ELEMENTARY PHYSICS. [i68 covered surface of the box. A thermometer plunged in the ether gives its temperature, and another outside gives the tem- perature of the air. The temperature of the ether is the dew point. From it the pressure of the vapor in the air is deter- mined as described in the last section, and this pressure sub- stituted for/ in Eq. 65 gives the absolute humidity. 168. Relative Humidity. — The amount of moisture that the air may contain depends upon its temperature. The damp- ness or dryness of the air does not depend upon the absolute amount of moisture it contains, but upon the ratio of this to the amount it might contain if saturated. The relative hiiinid- ity is the ratio of the amount of moisture in the air to that which would be required tt) saturate it at the existing temper- ature. Since non-saturated vapors follow Boyle's law very closely, this ratio will be very nearly the ratio of the actual pressure to the possible pressure for the temperature. Both these pressures may be taken from the tables. One corre- sponds to the dew point, and the other to the temperature of the ail CHAPTER IV. THERMODYNAMICS. 169. First Law of Thermodynamics. — The first law of thermodynamics may be thus stated : When heat is trans- formed into work, or work into heat, the quantity of work is equivalent to the quantity of heat. The experiments of Joule and Rowland establishing this law, and determining the me- chanical equivalent, have already been described (§ 114). 170. Second Law of Thermodynamics. — When heat is converted into work by any heat-engine under the conditions that exist on the earth's surface, only a comparatively small proportion of the heat drawn from the source can be so trans- formed. The remainder is given up to a refrigerator, which in some form must be an adjunct of every heat-engine, and still exists as heat. It will be shown that the heat which is con- verted into work bears to that which must be drawn from the source of heat a certain simple ratio depending upon the tem- peratures of the source and refrigerator. The second law of thermodynamics asserts this relation. The ratio between the heat converted into work and that drawn from the source is called the efficiency of the engine. To convert heat into mechanical work, it is necessary that the heat should act through some substance called the working substance ; as for instance, steam in the steam-engine or air in the hot-air engine. In studying the transformation of heat into work, it is an essential condition that the working sub- stance must, after passing through a cycle of operations, return to the same condition as at the beginning ; for if the substance be not in the same condition at the end as at the beginning, internal work may have been done, or internal energy expend- 200 ELEMENTARY PHYSICS. [170 ed, which will increase or diminish the work apparently de- veloped from the heat. To develop the second law of thermodynamics, we make use of a conception due to Carnot, of an engine completely re- versible in all its mechanical and physical operations. In the discussion of the reversible engine we employ a principle, first enunciated by Clausius. Clausius' principle is, that heat cannot pass of itself from a cold to a hot body. In many cases this principle agrees with common experience, and in other cases results in accordance with it have been obtained by experiment. It is so fundamental that it is often called the second law of thermodynamics. Suppose a heat-engine in operation, running forward. It will receive from a source a certain quantity of heat H, transfer to a refrigerator a certain quantity of heat h, and perform a cer- tain amount W oi mechanical work. If it be perfectly reversible, it will, by the performance upon it of the amount of work W, take from the refrigerator the quantity of heat A, and restore to the source the amount H. Such an engine will convert into work, under given conditions, as large as possible a proportion of the heat taken from the source. For, let there be two engines, A and B, of which B is reversible, working between the same source and refrigerator. If possible let A perform more work than B, while taking from the source the same amount of heat. If W be the work it performs, and w the work B performs, B will, from its reversibility, by the performance upon it of the work w, less than W, restore to the source the amount of heat, H, which it takes away when running forward. Let A be em- ployed to run B backward : A will take from the source a quantity of heat, H, and perform work, W. B will restore the heat H to the source by the performance upon it of work, w. The system will then continue running, developing the work W— w, while the source loses no heat. It must be, then, that A gives up to the refrigerator less heat than B takes 170] THERM OD YNAMICS. 207 away ; and the refrigerator must be growing colder. For the purposes of this discussion, we may assume that all surround- ing bodies, except the refrigerator, are at the same tempera- ture as the source; hence the work W—w, performed by the system of two engines, must be performed by means of heat taken from a body colder than all surrounding bodies. Now this is contrary to the principle of Clausius. The hypothesis with which we started must, therefore, be false ; and we must admit that no engine, no contrivance for converting heat into work, can under similar conditions, and while taking the same heat from the source, perform more work than a rever- sible engine. It follows that all reversible engines, whatever the working substance, have the same efficiency. This is a most important conclusion. In view of it, we may, in study- ing the conversion of heat into work, choose for the working substance the one v/hich presents the greatest advantage for the study. Since of all substances the properties of gases are best known, we will assume a perfect gas as the working sub- stance. The cycle of four operations which we will study is perfectly reversible. It is known as Carnofs Cycle. Suppose the gas to be enclosed in a cylinder having a tightly-fitting piston. Suppose the cycle to begin by a depres- sion of the piston, compressing the gas, without loss or gain of heat, until the temperature rises from (9 to / ; where t repre- sents the temperature of the source, and d that of the refriger- ator. In Fig. 53, let Oa represent the volume, and Aa the pressure at the be- ginning. If the gas be compressed un- til its volume becomes Ob, its pressure will be bB. AB representing the pres- sures and corresponding volumes dur- ing the operation, is an adiabatic line. This is the first operation. For the b aK c d Fig. 53. second operation, let the piston rise, and the volume increase 208 ELEMENTARY PHYSICS. [l70 from b to c -aX the constant temperature of the source. The pressure will fall from bB to cC. BC is the isothermal line for the temperature t. During this operation, a quantity of heat represented by H must be taken from the source, to maintain the constant temperature t. For the third operation let the piston still ascend, and the volume increase from Oc to C?a? with- out loss or gain of heat until the temperature falls from t to B, the temperature at which the cycle began. CD is an adiabatic line. For the fourth operation, let the piston be depressed to the starting point, and the gas maintained at the constant temperature 6 of the refrigerator. The volume becomes Oa and the pressure aA, as at the beginning. DA is the isother- mal line for the temperature 6. Now let us consider the work done in each operation. While the piston is being depressed through the volume represented by ab, work must be performed upon it equal to ab X the mean pressure exerted upon the piston. This mean pressure lies between Aa and Bb, and the product of this by ab is evi- dently the area ABba. In the same way it is shown that when the gas expands from b to c\\. performs work represented by the area BCcb; and again, in the third operation, it performs work represented by CDdc. In the fourth operation, when the gas is compressed, work must be done upon it represented by the area ADda. During the cycle, therefore, work is done by the gas represented by the area BCDdb, and work is done upon the gas represented by the area BADdb. The difference rep- resented by the area ABCD is the work done by the engine during the cycle. Since the gas is in all respects in the same condition at the end as at the beginning of the cycle, no work can have been developed from it ; and the work which the en- gine has done must have been derived from the heat communi- cated to the gas during the second operation. Now it has been shown that when a gas expands no inter- nal work is done in separating the molecules, and when it ex- 170] THERMODYNAMICS. 209 pands at constant temperature no change occurs in the in- ternal kinetic energy; the heat which is imparted to the gas during the second operation is, therefore, the equivalent to the work done by the gas upon the piston, and may be represented by the area BCcb. It will be seen, also, that the heat given up to the refrigerator during the fourth operation is represented by the area ADda, and that heat, the equivalent of the work performed by the engine, represented by the aiea ABCD, has disappeared. Of the heat withdrawn from the source, then, area ABCD . , . , _, . only the fraction Hy^'^~ is converted mto work. 1 his ■' area BLco fraction is the efificiency of the engine. Now let the operation of the cycle be reversed. Starting with the volume Oa the gas expands at the temperature d, ab- sorbs a quantity of heat represented by h, the same as it gave up when compressed, and performs work represented \yy ADda; next, it is compressed, without loss of heat, until its tempera- ture rises to t, and work represented by DCcd is done upon it; next, it is still further compressed at the temperature t, until its volume becomes Ob, and its pressure Bb. During this operation it gives up the heat H which it absorbed during the direct action, and work represented by CBbc is done upon it. Lastly, it expands to the starting-point, and falls to its initial temperature. It will be seen that each operation is the reverse in all respects of the corresponding operation of the direct action, and that during the cycle work represented by the area ABCD must be performed upon the engine while the quantity of heat h is taken from the refrigerator, and the quantity of heat H is transferred to the source. Such an engine is therefore a reversible engine ; and it converts into work as large a pro- portion of the heat derived from the source as is possible under the circumstances. An inspection of the figure shows that, since the line BC remains the same so long as the amount of heat H and the temperature t of the source remain constant, 14 2IO ELEMENTARY PHYSICS. [170 the only way to increase the proportion of work derived from a given amount of heat H is to increase the difference of temperature between the source and the refrigerator; that is, to increase the area ABCD, the line AD must be taken lower down. The proportion of heat which can be converted into work depends, therefore, upon the difference of temperature between source and refrigerator. To determine the nature of this dependence, suppose the range of temperature so small that the sides of the figure ABCD may be considered straight and parallel. Produce AD to e, and draw gh representing the mean pressure for the second operation. Now ABCD = eBCf z= Be X be ^ gi X be. Also BCeb = gh X be. Then we have H-h area ABCD gi X be gi H ~ area BCeb ghXbc' ~ gh But gh is the pressure corresponding to volume Oh and tem- perature t, hi is the pressure corresponding to the same volume and temperature B. These pressures are proportional to the absolute temperatures (§ 156); that is, if t and d are tempera- tures on the absolute scale,. and sn H —h t—d hence (67) In another form the result contained in Eq. (66) may be written he . H=7- (68) ih gh 8 ~1' gi gh H-h t-9 ~ H ~ t • area ABCD t — d ney ■■ ~ area BCeb ~ t 170] THERMODYNAMICS. 211 This proportion has been derived upon the supposition that the range of temperature was very small : but it is equally true for any range; for, let there be a series of engines of small range, of which the second has for a source the refrigerator of the first, the third has for a source the refrigerator of the second, and so on. The first takes from the source the heat H, and gives to the refrigerator the heat h, working between the temperatures t and 6. The second takes the heat h from the refrigerator of the first, and gives to its own refrigerator the heat h^, working between the temperatures and 0^. The operation of the others is similar ; then, from Eq. 68, we have h__6_ H~ t' k~ e' K_e^ K ~ e: k„ On f^n — I "« — I multiplying, we obtain or and h K K K 6 e^ e^ e„ H^ h ^^. "" '" h„^,'~ t -^ e ^ ^, ^' • • e„.^ h„ d„ H~ i H-K t-e„ H Hence it appears that, in a perfect heat-engine, the heat con- 212 ELEMENTARY PHYSICS. [lyr verted into work is to the heat received as the difference of temperature between the source and the refrigerator is to the absolute temperature of the source. This ratio can become unity only when d — o°, or when the refrigerator is at the ab- solute zero of temperature. Since the difference of tempera- tures between which it is practicable to work is always small compared to the absolute temperature of the source, a perfect heat-engine can convert into work only a small fraction of the heat it receives. The formulas developed in this section embody what we have called the second law of thermodynamics. 171. Absolute Scale of Temperatures. — An absolute scale of temperatures, formed upon the assumed properties of a perfect gas, has already been described (§ 156). No such sub- FlG. 54. stance as a perfect gas exists; but, since (§ 170) any two temperatures on the absolute scale are to each other as the heat taken from the source is to the heat transferred to the refrigerator by a reversible engine, any substance of which we know the properties with sufficient exactness to draw its iso- thermal and adiabatic lines, may be used as a thermometric 1 7 1 ] THERMOD YNA MICS. 2 1 3 substance, and, by means of it, an absolute scale of tempera- tures may be constructed. For example, in Fig. 54 let BB' be an isothermal line for some substance, corresponding to the temperature t of boiling water at a standard pressure. Let yS^' be the isothermal line for the temperature t^ of melting ice, and let bb' be an isothermal line for an intermediate tempera- ture. Let BP, B'ft', be adiabatic lines, such that, if the sub- stance expand at constant temperature t from the condition B to the condition B' , the equivalent in heat of one mechani- cal unit of energy will be absorbed. Now, the figure BB' fi' 13 represents Carnot's cycle, and the heat given to the refrigerator at the temperature /„, measured in mechanical units, is less than the heat taken from the source at the temperature t, by the energy represented by the area BB'^'fi ; or, the heat given to the refrigerator is equal to i — area Bfi': hence t„ _i — are a B^^ J~ i '' and t — t^ _ area Bfi' Now ,iit — fi, = icxf, as in the Centigrade scale, we have 100 area Bfi' and 214 ELEMENTARY PHYSICS. [lyz If 6 be the temperature corresponding to the isothermal line dd', we have, as above, I — area BV t~ \ :' whence lOO 6 = t{\ — area Bb') = ,3-57(1 — area BbX (70) ^ ' area Bp^ ' ^ ' If, now, it be proposed to use the substance as a thermometric substance by noting its expansion at constant pressure, take Om to represent that pressure, and draw the horizontal line mnop ; mn is the volume of the substance at temperature ^„, mo the volume at temperature B, and mp the volume at tem- perature t. This method of constructing an absolute scale of tempera- ture was proposed by Thomson. 172. The Steam-Engine.— The steam-engine in its usual form consists essentially of a piston, moving in a closed cylin- der, which is provided with passages and valves by which steam can be admitted and allowed to escape. A boiler heated by a suitable furnace supplies the steam. The valves of the cylin- der are opened and closed^ automatically, admitting and dis- chargirig the steam at the proper times to impart to the piston a reciprocating motion, which may be converted into a circular motion by means of suitable mechanism. There are two classes of steam-engines, condensing and non-condensing. In condensing engines the steam, after doing its work in the cyhnder, escapes into a condenser, kept cold by a circulation of cold water. Here the steam is condensed into water ; and this water, with air or other contents of the con- denser, is removed by an "air-pump." In non-condensing engines the steam escapes into the open air. In this case the 173] THEKMOD YNA MICS. 2 1 5 temperature of the refrigerator must be considered at least as high as that of saturated steam at the atmospheric pressure, or about ioo°, and the temperature of the source must be taken as that of saturated steam at the boiler pressure. Applying the expression for the efficiency (§ 170), t-e e = — 7—. it will be seen, that, for any boiler pressure which it is safe to employ in practice, it is not possible, even with a perfect en- gine, to convert into work more than about fifteen per cent of the heat used. In the condensing engine the temperature of the refrigera- tor may be taken as that of saturated steam at the pressure which exists in the condenser, which is usually about 30° or 40° : hence t — ^ is a much larger quantity for condensing than for non-condensing engines. The gain of efficiency is not, however, so great as would appear from the formula, because of the energy that must be expended to maintain the vacuum in the condenser. 173. Hot-air and Gas Engines.^— Hot-air engines consist essentially of two cylinders of different capacities, with some arrangement for heating air in, or on its way to, the larger cylinder. In one form of the engine, an air-tight furnace forms the passage between the two cylinders, of which the smaller may be considered as a supply-pump for taking air from out- side, and forcing it through the furnace into the larger cylinder, where, in consequence of its expansion by the heat, it is enabled to perform work. On the return stroke, this air is expelled into the external air, still hot, but at a lower temperature than it would have been had it not expanded and performed work. This case is exactly analogous to that of the steam-engine, in which water is forced by a piston working in a small cylinder. 2l6 ELEMENTARY PHYSICS. 1 173 into a boiler, is there converted into steam, and then, acting upon a much larger piston, performs work, and is rejected. In another form of the engine, known as the " ready motor," the air is forced into the large cylinder through a passage kept sup- plied with crude petroleum. The air becomes saturated with the vapor, forming a combustible mixture, which is burned in the cylinder itself. The Stirling hot-air engine and the Rider " compression engine" are interesting as realizing an approach to Carnot's cycle. These engines, like those ' described above, consist of two cylinders of different capacities, in which work air-tight pistons ; but, unlike those, there are no valves communicating with the external atmosphere. Air is not taken in and rejected; but the same mass of air is alternately heated and cooled, alter- nately expands and contracts, moving the piston, and per- forming work at the expense of a portion of the heat imparted to it. It is of interest to study a little more in detail the cycle of operations in these two forms of engines. The larger of the two cylinders is kept constantly at a high temperature by means of a furnace, while the smaller is kept cold by the circu- lation of water. The cylinders communicate freely with each other. The pistons are connected to cranks set on an axis, so as to make an angle of nearly ninety degrees with each other. Thus both pistons are moving for a short time in the same direction twice during the revolution of the axis. At the in- stant that the small piston reaches the top of its stroke, the large piston will be near the bottom of the cylinder, and de- scending. The small piston now descends, as well as the large one, the air in both cylinders is compressed, and there is but little transfer from one to the other. There is, therefore, comparatively little heat given up. The large piston, reaching its lowest point, begins to ascend, while the descent of the 174] THERMODYNAMICS. 217 smaller continues: The air is rapidly transferred to the larger heated cylinder, and expands while taking heat from the highly heated surface. After the small piston has reached its lowest point, there is a short time during which both the pistons are rising and the air expanding, with but little transfer from one cylinder to the other, and with a relatively small absorption of heat. When the descent of the large piston begins, the small one still rising, the air is rapidly transferred to the smaller cylinder : its volume is diminished, and its heat is given up to the cold surface with which it is brought in contact. The completion of this operation brings the air back to the condi- tion from which it started. It will be seen that there are here four operations, which, while not presenting the simplicity of the four operations of Carnot, — since the first and third are not performed without transfer of heat, and the second and fourth not without change of temperature, — still furnish an example of work done by heat through a series of changes in the working substance, which brings it back, at the end of each revolution, to the same condition as at the beginning. Gas-engines derive their power from the force developed by the combustion, within the cylinder, of a mixture of illuminat- ing gas and air. As compared with steam-engines, hot-air and gas engines use the working substance at a much higher temperature. ^ — ^ is, therefore, greater, and the theoretical efficiency higher. There are, however, practical difficulties connected with the lubrication of the sliding surfaces at such high temperatures that have so far prevented the use of large engines of this' class. 174. Sources of Terrestrial Energy. — Water flowing from a higher to a lower level furnishes energy for driving machin- ery. The energy theoretically available in a given time is the weight of the water that flows during that time multiplied by the height of the fall. If this energy be not utilized, it devel- 2l8 ELEMENTARY PHYSICS. [174 ops heat by friction of the water or of the material that may be transported by it. But water-power is only possible so long as the supply of water continues. The supply of water is de- pendent upon the rains; the rains depend upon evaporation; and evaporation is maintained by solar heat. The energy of water-power is, therefore, transformed solar energy. A moving mass of air possesses energy equal to the mass multiplied by half the square of the velocity. This energy is available for propelling ships, for turning windmills, and for other work. Winds are due to a disturbance of atmospheric equilibrium by solar heat ; and the energy of wind-power, like that of water-power, is, therefore, derived from solar energy. The ocean currents also possess energy due to their motioo, and this motion is, like that of the winds, derived from solar energy. By far the larger part of the energy employed by man for his purposes is derived from the combustion of wood and coal. This energy exists as the potential energy of chemical separation of oxygen from carbon and hydrogen. Now, we know that vegetable matter is formed by the action of the solar rays through the mechanism of the leaf, and that coal is the carbon of plants that grew and decayed in a past geological age. The energy of wood and coal is, therefore, the transformed energy of solar radiations. It is well known that, in the animal tissues, a chemical action takes place similar to that involved in combustion. The • oxygen taken into the lungs and absorbed by the blood com- bines by processes with which we are not here concerned with the constituents of the food. Among the products of this combination are carbon dioxide and water, as in the combus- tion of the same substances elsewhere. Lavoisier assumed that such chemical combinations were the source of animal heat, and was the first to attempt a measurement of it. He com- 174] THERMODYNAMICS. 2I9 pared the heat developed with that due to the formation of the carbonic dioxide exhaled in a given time. Despretz and Dulong made similar experiments with more perfect apparatus, and found that the heat produced by the animal was about one tenth greater than would have been produced by the formation by combustion of the carbonic acid and water ex- haled. These and similar experiments, although not taking into account all the chemical actions taking place in the body, leave no doubt that animal heat is due to atomic and molecular changes within the body. The work performed by muscular action is also the trans- formed energy of food. Rumford, in 1798, saw this clearly; and he showed, in a paper of that date, that the amount of work done by a horse is much greater than would be obtained by using its food as fuel for a steam-engine. Mayer, in 1845, held that an animal is a heat-engine, that every motion of the animal is a transformation into work of the heat developed in the tissues. Hirn, in 1858, executed a series of interesting experiments bearing upon this subject. In a closed box was placed a sort of treadmill, which a man could cause to revolve by stepping from step to step. He thus performed work which could be measured by suitable apparatus outside the box. The tread- wheel could also be made to revolve backward by a motor placed outside, when the man descended from step to step, and work was performed upon him. Three distinct experiments were performed ; and the amount of oxygen consumed by respiration, and the heat developed, were determined. In the first experiment the man remained in repose ; in the second he performed work by causing the wheel to revolve ; in the third the wheel was made to revolve backward, and work was performed upon him. In the second experiment, the 220 ELEMENTARY PHYSICS. [174 amount of heat developed for a gram of oxygen consumed was much less, and in the third case much greater, than in the first ; that is, in the first case, the heat developed was due to a chemi- cal action, indicated by the absorption of oxygen ; in the second, a portion of the chemical action went to perform the work, and hence a less amount of heat was developed ; while in the third case the motor, causing the treadwheel to revolve, performed work, which produced heat in addition to that due to the chemical action. It has been thought that muscular energy is due to the waste of the muscles themselves : but experiments show that the waste of nitrogenized material is far too small in amount to account for the energy developed by the animal : and we must, therefore, conclude that the principal source of muscular energy is the oxidation of the nori-nitrogenized material of the blood by the oxygen absorbed in respiration. An animal is, then, a machine for converting the potential energy of food into mechanical work : but he is not, as Mayer supposed, a heat-engine ; for he performs far more work than could be performed by a perfect heat-engine, working between the same limits of temperature, and using the food as fuel. The food of animals is of vegetable origin, and owes its energy to the solar rays. Animal heat and energy is, therefore, the transformed energy of the sun. The tides are mainly caused by the attraction of the moon upon the waters of the earth. If the earth did not revolve upon its axis, or, rather, if it always presented one face to the moon, the elevated waters would remain stationary upon its surface, and furnish no source of energy. But as the earth revolves, the crest of the tidal wave moves apparently in the opposite direction, meets the shores of the continents, and forces the water up the bays and rivers, where energy is wasted in friction upon the shores or may be made use of for turning mill-wheels. It is evident that all the energy derived from the 176] THERMODYNAMICS. 221 tides comes from the rotation of the earth upon its axis ; and a part of the energy of the earth's rotation is, therefore, being- dissipated in the heat of friction they cause. The internal heat of the earth and a few other forms of energy, such as that of native sulphur, iron, etc., are of little consequence as sources of useful energy. They may be con- sidered as the remnants of the original energy of the earth. 175. Energy of the Sun. — It has been seen that the sun's rays are the source of all the forms of energy practically avail- able, except that of the tides. It has been estimated that the heat received by the earth from the sun each year would melt a layer of ice over the entire globe a hundred feet in thickness. This represents energy equal to one horse-power for each fifty square feet of surface, and the heat which reaches the earth is only one twenty-two-hundred-millionth of the heat that leaves the sun. Notwithstanding this enormous expenditure of en- ergy, Helmholtz and Thomson have shown that the nebular hypothesis, which supposes the solar system to have originally existed as a chaotic mass of widely separated gravitating par- ticles, presents to us an adequate source for all the energy of the system. As the particles of the system rush together by their mutual attractions, heat is generated by their collision ; and after they have collected into large masses, the conden- sation of these masses continues to r^rnerate heat. 176. Dissipation of Energy. — It has been seen that only a fraction of the energy of heat is available for transformation into other forms of energy, and that such transformation is possible only when a difference of temperature exists. Every conversion of other forms of energy into heat puts it in a form from which it can be only partially recovered. Every transfer of heat from one body to another, or from one part to another of the same body, tends to equalize temperatures, and to diminish the proportion of energy available for transformation. Such transfers of heat are continually taking place ; and, so 222 ELEMENTARY PHYSICS. [176 far as our present knowledge goes, there is a tendency toward an equality, of temperature, or, in other words, a uniform mo- lecular motion, throughout the universe. If this condition of things were • reached, although the total amount of energy existing in the universe would remain unchanged, the possibil- ity of transformation would be at an end, and all activity and change would cease. This is the doctrine of the dissipation of energy to which our limited knowledge of the .operations of nature leads us; but it must be remembered that our knowl- edge is very limited, and that there may be in nature the means of restoring the differences upon which all activity de- pends. ..0*0- ^» I UNIVERSITY MAGNETISM AND ELECTRICITY, CHAPTER I. MAGNETISM. 177. Fundamental Facts. — Masses of iron ore are some- times found which possess the property of attracting pieces of iron and a few other substances. Such masses are called natu- ral magnets or lodestones. A bar of steel may be so treated as to acquire similar properties. It is then called a magnet. Such a magnetized steel bar may be used as fundamental in the investigation of the properties of magnetism. If pieces of iron or steel be brought near a steel magnet, they are attracted by it, and unless removed by an outside force they remain permanently in contact with it. While in contact with the magnet, the pieces of iron or steel also ex- hibit magnetic properties. The iron almost wholly loses these properties when removed from the magnet. The steel retains them and itself becomes a magnet. The reason for this differ- ence is not known. It is usually said to be due to a coercive force in the steel. The attractive power of the original magnet for other iron or steel remains unimpaired by the formation of new magnets. A body which is thus magnetized or which has its mag- netic condition disturbed is said to be affected by magnetic induction. 224 ELEMENTARY PHYSICS. [178 In an ordinary bar magnet there are two small regions, near the ends of the bar, at which the attractive powers of the mag- net are most strongly manifested. These regions are called i\\& poles of the magnet. The line joining two points in these regions, the location of which will hereafter be more closely defined, is called the magnetic axis. An imaginary plane drawn normal to the axis at its middle point is called the equatorial plane. If the magnet be balanced so as to turn freely in a horizon- tal plane, the axis assumes a direction which is approximately north and south. The pole toward the north is usually called the north or positive pole ; that toward the south, the south or negative pole. If two magnets be brought near together, it is found that their like poles repel and unlike poles attract one another. If the two poles of a magnet be successively placed at the same distance from a pole of another magnet, it is found that the forces exerted are equal in amount and oppositely directed. The direction assumed by a freely suspended magnet shows that the earth acts as a magnet, and that its north magnetic pole is situated in the southern hemisphere. If a bar magnet be broken, it is found that two new poles are formed, one on each side of the fracture, so that the two portions are each perfect magnets. This process of making new magnets by subdivision of the original one may be, so far as known, continued until the magnet is divided into its least parts, each of which will be a perfect magnet. This last experiment enables us at once to adopt the view that the properties of a magnet are due to the resultant action of its constituent magnetic molecules. 178. Law of Magnetic Force. — By the help of the torsion balance, the pnnciple of which is described in §§ 82, 188, and by using very long, thin, and uniformly magnetized bars, in which the poles can be considered as situated at the extrcmi- 178] MAGNETISM. 225 ties, Coulomb showed that the repulsion between two similar poles, and the attraction between two dissimilar poles, is in- versely as the square of the distance between them. Coulomb also demonstrated the same law by another method. He suspended a short magnet so that it could oscil- late about its centre in the horizontal plane. He first ob- served the time of its oscillation when it was oscillating in the earth's magnetic field. He then placed a long magnet verti- cally, so that one of its poles was in the horizontal plane of the suspended magnet, and in the magnetic meridian passing through its centre, and observed the times of oscillation when the pole of the vertical magnet was at two different distances from the suspended magnet. If we represent by / the moment of inertia of the suspended magnet, by M its magnetic moment, by H the horizontal intensity of the earth's magnetism, by //, and h^ the force in the region occupied by the suspended mag- net due to the vertical magnet in its two positions, it may be shown as in § 183 that the times of oscillation of the suspended magnet should be respectively ^^ = ;r V jj^ ^h = -^ Y ,j^ , ^^j^^ =v; ' t^^^ Tt y . From such equations, by elimination the ratio of k^ and /i, was obtained, and was found to be in accordance with the law of magnetic force already given. All theories of magnetism assume that the force between two magnet poles is proportional to the product of the strengths of the poles. The law of magnetic force is then the same as that upon which the discussion of potential (§§ 28, 29) was based. The theorems there discussed are in general applicable in the study of magnetism, although modifications in the details of their application occur, arising from the fact that the field of force about a magnet is due to the combined action of two dissimilar and equal poles. If m and m^ represent the strengths of two magnet poles, r IS 226 ELEMENTARY PHYSICS. [179 the distance between them, and k a factor depending on the units in which the strength of the pole is measured, the formula expressing the force between the poles is k — 5-'- 179. Definitions of Magnetic Quantities. — The law of magnetic force enables us to define a unit magnet pole, based upon the fundamental mechanical units. If two perfectly similar magnets, infinitely thin, uniformly and longitudinally magnetized, be so placed that their positive poles are unit distance apart, and if these poles repel one an- other with unit force, the magnet poles are said to be of unit strength. Hence, in the expression for the force between two poles, k becomes unity, and the dimensions of -^ are those of a force. That is, from which the dimensions of a magnet pole are This definition of a unit magnet pole is the foundation of the magnetic system of units. The strength of a magnet pole is then equal to the force which it will exert on a unit pole at unit distance. The product of the strength of the positive pole of a uni- formly and longitudinally magnetized magnet into the distance between its poles is called its magnetic moment. The quotient of the magnetic moment of such a magnet by its volume, or the magnetic moment of unit of volume, is called the intensity of magnetization. The dimensions of magnetic moment and of intensity of l8o] MAGNETISM. 22/ magnetization follow from these definitions. They are respec tively [w/] = 7l/*Z5r-' and K] = J/iZ-i^-^ 180. Distribution of Magnetism in a Magnet. — If we con- ceive of a single row of magnetic molecules with their unlike poles in contact, we can easily see that all the poles, except those at the ends, neutralize one another's action, and that such a row will have a free north pole at one end and a free south pole at the other. If a magnet be thought of as made up of a combination of such rows of different lengths, the ac- tion of their free poles may be seen to be the same as that of an imaginary distribution of equal quantities of north and south magnetism, on the surface and throughout the volume of the magnet. If the magnet be uniformly magnetized, the volume distribution becomes zero. The surf ace distribution of magnet- ism will sometimes be used to express the magnetization of a magnet. In that case what has hitherto been called the mag- netic intensity becomes the magnetic density. It is defined as the ratio of the quantity of magnetism on an element of sur- face to the area of that element. To illustrate this statement, we will consider an infinitely thin and uniformly magnetized bar, of which the length and cross-section are represented by I and s respectively. Its magnetic intensity is -^- or — . If, now, for the pole m we substitute a continuous surface distri- tn bution over the end of the bar, then - is also the density of that distribution. The dimensions of magnetic density follow from this defini- tion. They are E-] = 228 ELEMENTARY PHYSICS. [l8r Coulomb showed, by a method of oscillations similar to that described in § 178, that the magnetic force at different points aloiig a straight bar magnet gradually increases from the mid- dle of the bar, where it is imperceptible, to the extremities. This would not be the case if the bar magnet were made up of equal straight rows of magnetic molecules in contact, placed side by side. With such an arrangement there would be no force at any point along the bar, but it would all appear at the two ends. The mutual interaction of the molecules of contig- uous rows make such an arrangement, however, impossible. In the earth's magnetic field, in which the lines of magnetic force may be considered parallel, a couple will be set up on any magnet, so magnetized as to have only two poles, due to th,i action of equal quantities of north and south magnetism distributed in the magnet. The points at which the forces mak- ing up this couple are applied are the poles of the magnet, and the line joining them is the magnetic axis. These definitions are more precise than those which could be given at the outset. 181. Action of One Magnet on Another. — The investiga- tion of the mechanical action of one magnet on another is im- portant in the construction of apparatus for the measurement of magnetism. (i) To determine the potential of a short bar magnet at a p point distant from it, let NS (Fig. 55) represent the magnet of length 2/, the poles of which are of strength m, and g Q ^ let the point P be at a distance r from p,G. 55. the centre of the magnet, taken as ori- gin. Let the x axis coincide with the axis of the magnet. The potential at P is then V = '^((y + (;r-//)* - (/-f(;r + /)»)*) = ^[{r- + P- 2xiy ~ {r'' ■+ r -\- 2xl)ij- I8l] MAGNETISM. 229 This expression expanded gives _2mlx 3W;tr S;«/V '^ ~ ~7= Z5 I 15 . (71) if we assume r so large that we may neglect terms of higher order in /. The first term is the most important, and if r be very great compared with /, the other terms may be neglected. The ratio — is the cosine of the angle PON or B. If we rep- resent the magnetic moment 2ml, as is generally done, by M, the potential at any very distant point becomes M — , cos e. (72) Since cos 6 is zero for all points in a plane through the ori- gin at right angles to the magnetic axis, that plane is an equi- potential surface of zero potential. It is the plane defined as the equatorial plane. The lines of force evidently originate at the poles and pass perpendicularly through this surface. This system of lines of force can be easily illustrated by scattering fine iron filings on a sheet of paper held over a bar magnet. They will arrange themselves approximately along the lines of force. At a point on the line of the axis where r = jr, the poten- tial becomes ^^ M , Ml' ^=-= + ^+--- (73)_ (2) In one method of application of the instrument called the magnetometer it is necessary to know the expression for the moment of couple set up by the action of a magnet at right 230 ELEMENTARY PHYSICS. [i8r angles to another, the centre of which is in the prolongation of the axis of the first magnet. Let the centre of the first magnet be the origin, and its axis the x axis. Represent the strength of its poles by m, and the strength of the poles of the second magnet by ;%„ the lengths of the two magnets by 2/ and 2y respectively. To determine the moment of couple due to the action of the first magnet on the second, we must first find the component along the x axis of the force due to the first magnet on a pole m^, at a point distant y from the x axis. ^ The force due to the pole of the first magnet at N (Fig. 56) on a pole »«, is Fig. 56. f^{x-iY The cosine of the angle made by this force with the x axis is X — I 11,, — _ /,a\^ - Hence the component of this force along the X axis is mntiix — /) (/+(;.-/)■)?• Hence the component along the x axis of the whole force on the pole m^, due to the first magnet, is X — I X -\- 1 mm' ''{• (y + (^-/y)i {/ + (^x + i)yj- When this expression is expanded in increasing negative pow- ers of X, neglecting all terms containing higher powers of x than the fifth, we obtain J 1 , 2/" 3y\ i82] MAGNETISM. 23 1 An equal and oppositely directed component acts upon the other pole — ;«^ of the second magnet. Hence the moment of couple due to the action of the first magnet upon the sec- ond is 8^,Vj'(ii + '^-^). (74) If y be such that jy = 2^, or if the ratio of the lengths of the two magnets used be i : "/i-S, the second and third terms vanish, and the expression for the moment of couple depends only on the first term of the series. In practice it is not pos- sible to completely neglect the other terms, on account of the uncertainty as to the position of the poles in the figure of a magnet, but by making the lengths of the two magnets as i to i/1.5, the numerator of the term having x" in the denominator is made very small, and is eliminated by the methpd of obser- vation employed, as will be explained in the discussion of the magnetometer. 182. The Magnetic Shell. — A magnetic shell may be de- fined as an infinitely thin sheet of magnetizable matter, mag- netized transversely ; so that any line in the shell normal to its surfaces may be looked on as an infinitesimally short and thin magnet. These imaginary magnets have their like poles con- tiguous. The product of the intensity of magnetization at any point in the shell into the thickness of the shell at that point is called the strength of the shell at that point, and is de- Jioted by the symbol/. The dimensions of the strength of a magnetic shell follow at once from this definition. We have [/] equal to the dimen- sions of intensity of magnetization multiplied by a length. Therefore {j'\ = M^'D'T-^. We obtain first the potential of such a shell of infinitesi- 232 ELEMENTARY PHYSICS. fi82 mal area. Let the origin (Fig. 57) be taken half-way between ^^P the two faces of the shell, and let the -^ ' shell stand perpendicular to the x axis. Let a represent the area of the shell, supposed infinitesimal, 2/ the "li? X thickness of the shell, and d the mag- '^"^ "• netic intensity. The volume of this infinitesimal magnet is 2al, and from the definition of mag- netic intensity 2ald is its magnetic moment. The potential at the point P is then given by Eq. 72, since / is so small that all but the first term in the series of Eq. 71 may be neglected. We have M 2ald V= — rcos = — 5- cos 0. r r Now a cos is the pjojection of the area of the shell upon a plane through the origin normal to the radius vector r, and, . . ^ . , a cos 6 . , smce a is mfinitesmal, 5 — is the solid angle 00 bounded by the lines drawn from P to the boundary of the area a. The potential then becomes V — 2ldoo = Jo), since 2ld is what has been called the strength of the shell. The same proof may be extended to any number of con- tiguous areas making up a finite magnetic shell. The potential due to such a shell is then 2ja}. If the shell be of uniform strength, the potential due to it becomes j2oo and is got by summing the elementary solid angles. This sum is the solid angle £1, bounded by the lines drawn from the point of which the potential is required to the boundary of the shell. The potential due to a magnetic shell of uniform strength is there- fore jfl. (75) It does not depend on the form of the shell, but only on the angle i83j MAGNETtSM. 233 subtended by its contour. At a point very near the positive face of a flat shell, so near that the solid angle subtended by the shell equals 2;r, the potential is 2nj ; at a point in the plane of the shell outside its boundary where the angle subtended is zero, the potential is zero ; and near the other or negative face of the shell it is — 2nj. The whole work done, then, in moving a unit magnet pole from a point very near one face to a point very near the other face is £,tij. This result is of im- poitance in connection with electrical currents. 183. Magnetic Measurements. — It was shown by Gilbert in a work published in 1600, that the earth can be considered as a magnet, having its positive pole toward the south and its negative toward the north. The determination of the mag- netic relations of the earth are of importance in navigation and geodesy. The principal magnetic elements are the de- clination, the dip, and the horizontal intensity. The declination is the angle between the magnetic meridian, or the direction assumed by the axis of a magnetic needle suspended to move freely in a horizontal plane, and the geo- graphical meridian. The dip is the angle made with the horizontal by the axis of a magnetic needle suspended so as to turn freely in a verti- cal plane containing the magnetic meridian. The horizontal intensity is the strength of the earth's mag- netic field resolved along the horizontal line in the plane of the magnetic meridian. A magnet pole of strength m in a field in which the horizontal intensity is represented by H is urged along this horizontal line with a force equal to mH. From this equation the dimensions of the horizontal intensity, and so also of the strength of a magnetic field in any case, are L ;« -I 234 ELEMENTARY PHYSICS. [183 The horizontal intensity can be measured relatively to some assumed magnet as standard, by allowing the magnet to oscil- late freely in the horizontal plane about its centre, and noting the time of oscillation. The relation between the magnetic moment M of the magnet and the horizontal intensity H is calculated by a formula analogous to that employed in the computation of g from observations with the pendulum. If the magnet be slightly displaced from its position of equilib- rium, so as to make small oscillations about its point of sus- pension, it can be shown as in § 39 that it is describing a simple harmonic motion, and as in § 41 (i) that the kinetic energy of the magnet when its axis coincides, during an oscillation, with ' the magnetic meridian is a-* -T-a • The potential energy at the extremity of its arc is due to the magnetic force mH acting on the poles. The component of this force which is efficient in moving the magnet is mH sin « or mHa, if a be always very small. Since a varies between O and 0, the average force efficient in turning the needle is \mH. The poles upon which this force acts move from the position of maximum kinetic energy to the position of no kinetic energy, through a distance /0, if / represent the half length of the magnet. The potential energy of the couple formed by the two poles of the magnet is then mHl^, and this is equal to the kinetic energy at the point of equilibrium ; that is, \I^^ = mHl\ Hence if we write 2ml = M, the magnetic moment of the mag- i«3] MAGNETISM. "21$ net, we obtain MH = -~r ; or if we take the time of oscillation as / = — , we have 2 MH=^. (76) The moment of inertia / may be either computed directly from the magnet itself, if it be of symmetrical form, or it may be determined experimentally by the method of § 36, Eq. 23, Avhich applies in this case. The horizontal intensity is then determined relative to the magnetic moment of the assumed standard magnet. This measure may be used to give an absolute measure of H by combining with it another observation which gives an independent relation between M^ and H. In one arrangement of the apparatus two magnets are used : one, the deflected mag- net, so suspended as to turn freely in the horizontal plane ; and the other, the deflecting magnet, the one of moment M used in the last operation, carried upon a bar which can be turned about a vertical axis passing through the point of suspension of the deflected magnet. The centre of the deflected magnet is in the prolongation of the axis of the deflecting magnet, and, when the apparatus is used, the carrier bar is turned until the two magnets are at right angles to one another. The equilib- rium established is due to two couples acting on the deflected magnet, one arising from the action of the earth's magnetism, and the other from that of the deflecting magnet. This latter has been already discussed in § 181. The couple acting on the deflected magnet is expressed by 4MM^j'{—i-\ — ^j, where P represents the small numerator of the correction term. This correction can be made very small in practice by giving to the 236 ELEMENTARY PHYSICS. [183 magnets, as already explained, lengths in the ratio of i to 1/1.5. The opposing equal couple is 2miHy sin 0, where represents the angle of deflection from the magnetic meridian. We have then ^Mmj\^ + ^j ~ '^^J^ ®^" ^' °'' ^^ + ^ = 2 M^^"^ ^• Since P is always a very small quantity, this equation may be written H ix" sin 0(1 - ^). (;;) P is determined by measuring the angles

, and D^, connected with the plates A^ and A^, and the pins Ci and C^, insulated from the plates, but joined to the prime conductors F^ and F^. Suppose the prime conductors to be in contact and the car- riers so placed that B^ is between D^ and C^, and suppose the plate A.^ to be at a slightly higher potential than the rest of the machine. The carrier B^ is then charged by induction. When the carriers are turned in the direction of the arrows, and the carrier B^ makes contact with the pin C„ it losed a part of its positive charge and the prime conductors become positively charged. At the same time the carrier B^ becomes positively charged. As the carrier B^ passes over the upper part of the plate A^, the lower part of the plate A^ is charged positively by induction. This positive charge is neutralized by the nega- tive charge of the carrier B^, when contact is made at Z>,. The plate A^ is then negatively charged. The carrier B^ at its con- tact at D^ shares its positive charge with the plate A^. The carriers then return to the positions from which they started, and the difference of potential between the plates^, and A^ is greater than it was at first. When, after sufficient repetition of this process, the difference of potential has become suffi- ciently great, the prime conductors may be separated, and the transfer of electricity between the points F^ and F^ then takes place through the air. Obviously the number of carriers may be increased, with a corresponding increase in the rapidity of action of the machine. This improvement is usually effect- ed by attaching disks of tin-foil at equal distances from each other on one face of a glass wheel, so that, as the wheel re- volves, they pass the contact points in succession. Another induction machine, invented by Holtz, differs in plan from the one just described in that the metallic carriers 199] ELECTRICITY IN EQUILIBRIUM. 27I are replaced by a revolving glass plate, and the two metallic inductor plates, by a fixed glass plate. In the fixed plate are cut two openings, diametrically opposite. Near these open- ings, and placed symmetrically with respect to them, are fixed upon the back of the plate two paper sectors or armatures, terminating in points which project into the openings. In front of the revolving plate and opposite the ends of the arma- tures nearest the openings are the combs of two prime con- ductors. Opposite the other ends of the armatures, and also in front of the revolving wheel, are two other combs joined to- gether by a cross-bar. In order to set this machine in operation, one of the paper armatures must be charged from some outside source. The surface of the revolving plate performs the functions of the carriers in the induction machine already explained. The armatures take the place of the inductors, and the points in which they terminate serve the same purpose as the contact points in connection with the inductors. The explanation of the action of this machine is, in general, similar to that already given. The effect of the combs joined by the cross-bar is equivalent to joining to ground that portion of the outside face of the revolving plate which is passing under them. CHAPTER III. THE ELECTRICAL CURRENT. 200. Fundamental Effects of the Electrical Current. — In 1791 Galvani of Bologna published an account of some experiments made two years before, which opened a new de- partment of electrical science. He showed that, if the lumbar nerves of, a freshly skinned frog be touched by a strip of metal and the muscles of the hind leg by a strip of another metal, the leg is violently agitated when the two pieces of metal are brought in contact. Similar phenomena had been previously observed, when sparks were passing from the conductor of an electrical machine in the vicinity of the frog preparation. He ascribed the facts observed to a hypothetical animal electricity or vital principle, and discussed them from the physiological standpoint; and thus, although he and his im- mediate associates pursued his theory with great acuteness, they did not effect any marked advance along the true direc- tion. Volta at Pavia followed up Galvani's discovery in a most masterly way. He showed that, if two different metals, or, in general, two heterogeneous substances, be brought in contact, there immediately arises a difference of electrical po- tential between them. He divided all bodies into two classes. Those of ^\\& first class, comprising all simple bodies and many others, are so related to one another that, if a closed circuit be formed of them or any of them, the sum of all the differences of potential taken around the circuit in one direction is equal to zero. If a body of the second class be substituted for one of 200] THE ELECTRICAL CURRENT. 273 the first class, this statement is no longer true. There exists then in the circuit a preponderating difference of potential in one direction. Volta described in 1800 his famous voltaic battery. He placed in a vessel, containing a s6lution of salt in water, plates of copper and zinc separated from one another. When wires joined to the copper and zinc were tested, they were found to be at different potentials, and they could be used to produce the effects observed by Galvani. The effects were heightened, and especially the difference of potential be- tween the two terminal wires was increased, when several such cups were used, the copper of one being joined to the zinc of the next so as to form a series. This arrangement was called by Volta the galvanic battery, but is now generally known as the voltaic battery. Volta observed that, if the terminals of his battery were joined, the connecting wire became heated. Soon after Volta sent an account of the invention of his battery to the Royal Society, Nicholson and Carlisle observed that, when the terminals of the battery were joined by a column of acidulated water, the water was decomposed into its constituents, hydrogen and oxygen. In 1820 Oersted made the discovery of the relation be- tween electricity and magnetism. He showed that a magnet brought near a wire joining the terminals of a battery is de- flected, and tends to stand at right angles to the wire. His discovery was at once followed up by Ampfere, who showed that, if the wire joining the terminals be so bent on itself as to form an almost closed circuit, and if the rest of the circuit be so disposed as to have no appreciable influence, the magnetic potential at any point outside the wire will be similar to that due to a magnetic shell. In 1834 Peltier showed that, if the terminals of the battery be joined by wires of two different metals, there is a produc- tion or an absorption of heat at the point of contact of the 18 274 ELEMENTARY PHYSICS. [201 wires, depending upon which of the wires is joined to the ter- minal the potential of which is positive with respect to the other. This fact is referred to as the Peltier effect. 201. Electromotive Force. — In 1833 Faraday showed con- clusively that if a Leyden jar be discharged through a circuit, it will produce the same thermal, chemical, and magnetic effects as those just described as produced by the voltaic battery. We know that, in the discharge of a jar, a charge of elec- tricity is transferred from a point at a higher potential to one at a lower. It is reasonable, therefore, to suppose the phe- nomena under consideration to be also due, in some way, to the transfer of electricity from a higher to a lower potential. Since, these phenomena continue without interruption while the circuit is joined up, it is necessary to assume that the vol- taic battery maintains a permanent difference of piotential. This power of maintaining a difference of potential is ascribed to an electromotive force existing in the circuit. In an actual circuit containing a voltaic battery, if two points on the circuit outside the battery be tested by an elec- trometer, a difference of potential between them will be found. If the circuit be broken between the two points considered, the difference of potential between them becomes greater. This maximum difference of potential is the sum of finite differences of potential supposed to be due to molecular inter- actions at the surfaces of contact of different substances in the circuit, and is the measure of the electromotive force. An electromotive force may exist in a circuit in which there are no differences of potential. These cases will be considered later. It is sufficient for the present to consider two points between which a difference of potential is maintained, and which are connected by conductors of any kind whatever. The dimensions of electromotive force in the electrostatic system are those of difference of potential, or [.£] = M^L^ T- '. 202] THE ELECTRICAL CURRENT. 2JC, 202. Electrostatic Unit of Current. — Let us denote the potentials at the two points i and 2 in the circuit by V^ and V„ and let V, be greater than F, ; then if, in the time i, a quan- tity of electricity equal to Q passes through a conductor join- ing those points from potential V^ to potential V,, the amount of work done by it is Q{V, — V^. If the conductor be a single homogeneous metal or some analogous substance, and no motion of the conductor or of any external magnetic body take place, the whole work done is expended in heating the conductor. If we suppose the transfer to be such that equal quantities of heat are developed in equal times, we may represent the heat produced in the time t by Ht, if H represent the heat developed in one unit of time. If all the quantities considered are expressed in terms of the same fundamental units, we have Q{V,-V:) = Ht, or H=^-{V,-K). The transfer of electricity in the circuit is called the electrical current, and the rate of transfer — = / is called the current strength, or often simply the current. The current, as here de- fined, is independent of the nature' of the conductor, and is the same for all parts of the circuit. This fact was experimentally proved by Faraday. Employing this quantity /, we have the fundamental equation H = I{V^-V^. (85) If heat and difference of potential be measured in absolute units, this equation enables us to determine the absolute unit /}f current. The system of .units here used is the electrostatic system. The dimensions of current strength in the electro- 276 ELEMENTARY PHYSICS. [203 static system are obtained from the equation above. We have [/] = y- = M^L^ Z-^ the dimensions of current. 203. Ohm's Law. — In § 187 it was remarked that a body- is distinguished as a good or a poor conductor by the rate at which it will equalize the potentials of two electrified conduc- tors, if it be used to connect them. Manifestly this property of the substances forming a circuit, of conducting electricity rapidly or otherwise, will influence the strength of the current in the circuit. It was shown on theoretical considerations, in 1827, by Ohm of Berlin, that in a homogeneous conductor which is kept constant, the current varies directly with the difference of potential between the terminals. If R represent a factor, constant for each conductor, Ohm's law is expressed in its sim- plest form by IR=V,- K. (86) The quantity R is called the resistance oi the conductor. If the difference of potential be maintained constant, and the conduc- tor be altered in any way that does not introduce an internal electromotive force, the current will vary with the changes in the conductor, and there will be a different, value of i? with each change in the conductor. The quantity R is therefore a function of the nature and materials of the conductor, and does not depend on the current or the difference of potential between the ends of the conductor. Since it is the ratio of the current to the difference of potential, and since we know the:je quantities in electrostatic units, we can measure R in electro- static units. From the dimensions of /and {V^ — V^ we may obtain the dimensions of R. They are in electrostatic units [R] = l^^]=L-': 203] THE ELECTRICAL CURRENT. 277 M' To generalize Ohm's law for the whole circuit, let us con- sider a special circuit which may serve as a type. It shall consist of a voltaic cell contain- ing acidulated water, in which are immersed a zinc and a platinum plate, joined together by a platinum wire outside the liquid (Fig. 62), Consider a point in the liquid just outside the zinc : if the potential of a point near it, just inside the zinc, be Vz, then the potential at the point considered \% Vz-\- Z/L, if Z/L represent the sudden change in potential across the surface of separation. The potential at a point in the liquid just outside the platinum is V L' and by the elementary form of Ohm's law already con- sidered we have Vz + Z/L - Vl Fig. &. / = Rl In the same way the current in the platinum and platinum wire is expressed by / = and in the zinc by / = Vl^L/P- Vp Rp Vp+P/Z- Vz Rz Now these currents are all equal, for there is no accumulation of electricity anywhere in the circuit. Hence Vz + Z/L - Vl ^ Vl+L/P-Vp ' ~ Rl Rp Vp + P/Z - Vz Rz 278 ELEMENTARY PHYSICS. [204 or Z/L + L/P + P/Z 1 = Rl + Rp + R-z But the numerator is the sum of all the differences of potential in the circuit taken in one direction, or the measure of the electromotive force, and the denominator is the total resistance of .the circuit. It may then be stated more generally as Ohm's law that in any circuit the current equals the electromotive force divided by the resistance, or / = f. (8;) 204. Specific Conductivity and Specific Resistance. — If two points be kept at a constant difference of potential, and joined by a homogeneous conductor of uniform cross-section, it is found that the current in the conductor is directly propor- tional to its cross-section and inversely as its length. The cur- rent also depends upon the nature of the conductor. If con- ductors of similar dimensions, but of different materials, are used, the current in each is proportional to a quantity called the specific conductivity of the material. The numerical value of the current set up in a conducting cube, with edges of unit length, by unit difference of potential between two opposite faces, is the measure of the conductivity of the material of the cube. The reciprocal of this number is the specific resistance of the material. If p represent the specific resistance of the conducting material, 5 the cross-section and / the length of a portion of the conductor of uniform cross-section between two poirxts at potentials V^ and V^, Ohm's law for this special case car be presented in the formula 206] THE ELECTRICAL CURRENT. 279 The specific resistance is not perfectly constant for any one material, but varies with the temperature. In metals the spe- cific resistance increases with rise in temperature; in liquids and in carbon it diminishes with rise in temperature. Upon this fact of change of resistance with temperature is based a very delicate instrument, called by Langley, its inventor, the bolometer, for the measurement of the intensity of radiant energy. 205. Joule's Law. — If we modify the equation H = /(F, — V^ by the help of Ohm's law, we obtain H=rR. (89) The heat developed in a homogeneous portion of any cir- cuit is equal to the square of the current in the circuit multi- plied by the resistance of that portion. This relation was first experimentally proved by Joule in 1841, and is known after his name as Joule's law. It holds true for any homogeneous circuit or for all parts of a circuit which are homogeneous. The heat which is sometimes evolved by chemical action, or by the Peltier effect, occurs at non-homogeneous portions of the circuit. 206. Counter Electromotive Force in the Circuit.— In many cases the work done by the current does not' appear wholly as heat developed in accordance with Joule's law. Besides the production of heat throughout the circuit, work may be done during the passage of the current, in the decom- position of chemical compounds, in producing movements of magnetic bodies or other circuits in which currents are passing, or in heating junctions of dissimilar substances. Before discussing these cases separately we will connect them all by a general law, which will at the same time present the various methods by which currents can be maintained. They differ from the simple case in which the work done ap- 28o ELEMENTARY PHYSICS. [206 pears wholly as heat throughout the circuit, in that the work done appears partly as energy available to generate currents in the circuit. To show this we will use the method given by Helmholtz and by Thomson. The total energy expended in the circuit in the time /, which is such that, during it, the current is constant, is lEt. It appears partly as heat, which equals PRt by Joule's law, and partly as other work, which in every case is proportional to /, and can be set equal to lA, where A is a factor which varies with the particular work done. Then we have lEt = PRt -\- lA, whence '=-r^- fee) It is evident from the equation that E is an electromo- tive force, and that the original electromotive force of the cir- cuit has been modified by the fact of work having been done by the current. In other words, the performance of the work lA in the time t by the circuit has set up a counter electromo- tive force — . The separated constituents of the chemical com- pound, the moved magnet, the heated junction, are all sources of electromotive force which oppose that of the original circuit. If then, in a circuit containing no impressed electromotive force, or in which E = o, there be brought an arrangement of uncombined chemical substances which are capable of com- bination, or if in its presence a magnet or closed current be moved, or if a junction of two dissimilar parts of the circuit be heated, there will be set up an electromotive force -, and a A current / = — . Any of these methods may then be used as 206] THE ELECTRICAL CURRENT. 28 1 the means of generating a current. The first gives the ordi- nary battery currents of Volta, the second the induced cur- rents discovered by Faraday, and the third the thermo-electric currents of Seebeck. CHAPTER IV. CHEMICAL RELATIONS OF THE CURRENT. 207. Electrolysis. — It has been already mentioned that, in certain cases, the existence of an electrical current in a circuit is accompanied by the decomposition into their constituents of chemical compounds forming part of the circuit. This process, called electrolysis, must now be considered more fully. It is one of those treated generally in § 206, in which work other than heating the circuit is done by the current. That work is done by the decomposition of a body the constituents of which, if left to themselves, tend to recombine, is evident from the fact that, if they be allowed to recombine, the combina- tion is always attended with the evolution of heat or the ap- pearance of some other form of energy. The amount of heat developed, or the energy gained, is, of course, the measure of the energy lost by combination or necessary to decomposi- tion. A free motion of the molecules of a body, associated with close contiguity, seems to be necessary in order that it may be decomposed by the current. Only liquids, and solids in solu- tion or fused, have been electrolysed. Bodies which can be decomposed were called by Faraday, to whom the nomencla- ture of this subject is due, electrolytes. The current is usually introduced into the electrolyte by solid terminals called elec- trodes. The one at the higher potential is called the positive electrode, or anode; the other, the negative electrode, or cathode. The two constituents into which the electrolyte is decom- posed are called ions. One of them appears at the anode and 207] CHEMICAL RELATIONS OF THE CURRENT. 283 is called the anion, the other at the cathode and is called the cation. For the sake of clearness we will describe some typical cases of electrolysis. The original observation of the evolution of gas when the current was passed through a drop of water, made by Nicholson and Carlisle, was soon modified by Carlisle in a way which is still generally in use. Two platinum elec- trodes are immersed in water slightly acidulated with sulphuric acid, and tubes are arranged above them so that the gases evolved can be collected separately. When the current is pass- ing, bubbles of gas appear on the electrodes. When they are collected and examined, the gas which appears at the anode is found to be oxygen, and that which appears at the cathode to be hydrogen. The quantities evolved are in the proportion to form water. This appears to be a simple decomposition of water into its constituents, but it is probable that the acid in the water is first decomposed, and that the constituents of water are evolved by a secondary chemical reaction. An experiment performed by Davy, by which he dis- covered the elements potassium and sodium, is a good example of simple electrolysis. He fused caustic potash in a platinum dish, which was made the anode, and immersed in the fused mass a platinum wire as cathode. Oxygen was then evolved at the anode, and the metal potassium was de- posited on the cathode. This is the type of a large series of decompositions. If, in a solution of zinc sulphate, a plate of copper be made the anode and a plate of zinc the cathode, there will be zinc deposited on the cathode and copper taken from the anode, so that, after the process has continued for a time, the solution will contain a quantity of cupric sulphate. This is a case simi- lar to the electrolysis of acidulated water, in which the simple decomposition of the electrolyte is modified by secondary chemical reaction. 284 ELEMENTARY PHYSICS. [208 If two copper electrodes be immersed in a solution of cu- pric sulphate, copper will be removed from the anode and de- posited on the cathode, without any important change occur- ring in the character or concentration of the electrolyte. This is an example of the special case in which the secondary reac- tions in the electrolyte exactly balance the work done by the current in decomposition, so that on the whole no chemical work is done. 208. Faraday's Laws. — The researches of Faraday in elec- trolysis developed two laws, which are of great importance in the theory of chemistry as well as in electricity. (i) The amount of an electrolyte decomposed is directly pro- portional to the quantity of electricity which passes through it ; or, the rate at which a body is electrolysed is proportional to the current strength. (2) If the same current be passed through different electro- lytes, the quantity of each ion evolved is proportional to its chemical equivalent. If we define an electro-chemical equivalent as the quantity of any ion which is evolved by unit current in unit time, then the two laws may be summed up by saying : , The number of electro-chemical equivalents evolved in a given time by the passage of any current through any electro- lyte is equal to the number of units of electricity which pass through the electrolyte in the given time. The electro-chemical equivalents of different ions are pro- portional to their chemical equivalents. Thus, if zinc sulphate, cupric sulphate, and argentic chloride be electrolysed by the same current, zinc is deposited on the cathode in the first case, copper in the second, and silver in the third. The amounts by weight deposited are in proportion to the chemical equiva- lents, 32.6 parts of zinc, 31.7 parts of copper, and 108 parts of silver. 209. The Voltameter. ^These laws were used by Faraday 2io] CHEMICAL RELATIONS OF THE CURRENT. 285 to establish a method of measuring current by reference to an arbitrary standard. The method employs a vessel containing an electrolyte in which suitable electrodes are immersed, so arranged that the products of electrolysis, if gaseous, can be collected and measured or, if solid, can be weighed. This ar- rangement is called a voltameter. If the current strength be desired, the current must be kept constant in the voltameter by suitable variation of the resistance in the circuit during the time in which electrolysis is going on. Two forms of voltameter are in frequent use. In the first form there is, on the whole, no chemical work done in the electrolytic process. The system consisting of two copper electrodes and cupric sulphate as the electrolyte is an example of such a voltameter. The weight of the copper de- posited on the cathode measures the current. The second form depends for its indications on the evolu- tion of gas, the volume of which is measured. The water vol- tameter is a type, and is the form especially used. The gases evolved are either collected together, or the hydrogen alone is collected. The latter is p/eferable, because oxygen is more easily absorbed by water than hydrogen and an error is thus introduced when the oxygen is measured. 210. Measure of the Counter Electromotive Force of Decomposition. — In the general formula developed in § 206, the quantity lA represents the energy expended in the circuit which does not appear as heat developed in accordance with Joule's law. In the present case it is the energy expended during electrolysis in decomposing chemical compounds and in doing mechanical work. In many cases the mechanical work done is not appreciable ; but when a liquid like water is decom- posed into its constituent gases, work is done by the expan- sion of the gases from their volume as water to their volume as gases. Let e represent the electro-chemical equivalent of one of the ions, and^ the heat evolved by the combination of a 286 ELEMENTARY PHYSICS. [sil unit mass of this ion with an equivalent mass of the other ion, in which is included the heat equivalent of the mechanical work done if the state of aggregation change. Then / will represent the number of electro-chemical equivalents evolved, and led -will rep/esent the energy expended, which appears as chemical sepa- ration and mechanical work. This is equal to lA ; whence A — eO. All these quantities are measured in absolute units. The quantity ed represents the energy required to separate the quantity e of the ion considered from the equivalent quantity of the other ion, and to bring both constituents to their normal condition. If the electrolytic process go on uniformly for a time /, so that equal quantities of the ion considered are evolved in equal A ed ^^ A times, we have — = — JN ow, — represents the counter elec- tromotive force set up in the circuit by electrolysis. Hence the electromotive force set up in the electrolytic process may be measured in terms of heat units ; or, since these heat units are measures of chemical affinity, the same relation gives a measure of chemical affinity in terms of electromotive force. It often is the case that the two ions which appear at the electrodes are not capable of direct recombination, as has been tacitly assumed in ihe definition of d. A series of chemical exchanges is always possible, however, which will restore the ions as constituents of the electrolyte, and the total heat evolved for a unit mass of one ion during the process is the quantity 0. The theory here presented is abundantly verified by the ex- periments of Joule. Favre and Silbermann, Wright and others. 211. Positive and Negative Ions. — Experiment shows that certain of the bodies which act as ions usually appear at the cathode, and certain others at the anode. The former are called electro-positive elements: the latter, electro-negative ele- ments. Faraday divided all the ions into these two classes, and thought that every compound capable of electrolysis was 2l2l CHEMICAL RELATIONS OF THE CURRENT. 287 made up of one electro-positive and one electro-negative ion. But the distinction is not absolute. Some ions are electro- positive in one combination and electro-negative in another. Berzelius made an attempt to arrange the ions in a series, such that any one ion should be electro-positive to all those above it and electro-negative to all those below it. It is questionable whether a rigorous arrangement of the ions is at the present time possible. 212. Theory of Electrolysis. — When any attempt is made to explain the behavior of the ions in the process of electroly- sis, grave difificulties are met with at once. The foundation of all the present theories is found in the theory published by Grotthus in 1805. He considers the constituent ions of a molecule as oppositely electrified to an equal amount. When the current passes, owing to the electrical attractions of the electrodes, the molecules arrange themselves in lines with their similar ends in one direction, and then break up. The electro- negative ion of one molecule moves toward the positive elec- trode and meets the electro-positive ion of the neighboring molecule, with which it momentarily unites. At the ends of the line an electro-negative ion with its charge is freed at the anode, and an electro-positive ion with its charge at the cathode. This process is repeated indefinitely so long as the current passes. Faraday modified tRis view, in that he ascribed the arrange- ment ox polarization of the molecules, and their disruption, to the stress in the medium which was the cardinal point in his electrical theories. Otherwise he held closely to Grotthus' theory. He showed that the state of polarization existed in the electrolyte by means of fine silk threads immersed in it. These arranged themselves along the lines of electrical stress. Other phenomena, however, show that Grotthus' hypothesis can only be treated as a rough mechanical illustration of the main facts. 288 ELEMENTARY PHYSICS. [212 Joule showed that during electrolysis there is a development of heat at the electrodes, in certain cases, which is not accounted for by the elementary theory above given. It must depend upon a more complicated process of electrolysis than the one we have described. The results of researches on the so-called wandering of the ions are also at variance with Grotthus' theory. If the electro- lysis of a copper salt, in a cell with a copper anode at the bot- tom, be examined, it will be found that the solution becomes more concentrated about the anode and more dilute about the cathode. These changes can be detected by the color of the parts of the solution, and substantiated by chemical analysis. If this result be explained by Grotthus' theory, the explanation furnishes at the same time a numerical relation between the ions which have wandered to their respective regions in the electrolyte which is not in accord with experiment. Another peculiar phenomenon, known as electrical endos- mose, may also be mentioned in this connection. It is found that, if the electrolyte be divided into two portions by a porous diaphragm, there is a transfer of the electrolyte toward the cathode, so that it stands at a higher level on the side of the diaphragm nearer the cathode than on the other. This fact was discovered by Reuss in 1807, and has been investigated by Wiedemann and Quincke. They found that the amount of the electrolyte transferred is proportional to the current strength, and independent of the extent of surface or the thick- ness of the diaphragm. Quincke has also demonstrated a flow of the electrolyte toward the cathode in a narrow tube, without the intervention of a diaphragm. Those electrolytes which are the poorest conductors show the phenomenon the best. In a very few cases the motion is towards the anode. The material of which the tube is composed influences the direction of flow. It has also been shown that solid particles move in the electro- lyte, usually towards the anode. 212] CHEMICAL RELATIONS OF THE CURRENT. 289 To explain these phenomena, Quincke has brought forward a theory of electrolysis which is widely different from Grotthus' simple hypothesis, but is too complicated for presentation here. It is an objection against Grotthus' theory, and indeed against Thomson's method given in § 210 of connecting chemi- cal affinity and electromotive force, that, on those theories, it would require an electromotive force in the circuit greater than — , the counter electromotive force in the electrolytic cell, to set up a current, and that the current would begin sud- denly, with a finite value, after this electromotive force was reached. On the contrary, experiments show that the smallest electromotive force will set up a current in an electrolyte and even maintain one constantly, though the current strength may be extremely small. This is explained by Clausius by the help of the theory of the constitution of liquids which is now generally adopted. He conceives the molecules of the electrolyte to be moving about with different velocities. He thinks that occasionally the at- traction between two opposite ions of two neighboring mole- cules may become greater than that between the constituents of the molecules. In that case the molecules are broken up, the two attracting ions combine to form a new molecule, and two opposite ions are set free. These may at once combine to form another new molecule, or they may wander through the mass until they meet with other ions, with which they can unite to again form molecules. He thinks that the electro- motive force in the circuit, while not great enough to effect a decomposition of the electrolyte, may yet be sufficient to deter- mine the direction of motion of these unpaired ions, so that they move, on the whole, towards their respective electrodes. Every theory of electrolysis assumes that the transfer of elec- tricity is, in some way, connected with the transfer of the ions; hence on Clausius' theory there will be a current and an evolu- 19 290 ELEMENTARY PHYSICS. [213 tion of the ions with any electromotive force in the circuit, however low. This current would at once cease if the ions were to collect on the electrodes, and set up a permanent counter electromotive force ; but the same reasoning as has just been used will show that the liberated ions, if not formed in such quantities as to collect and pass out of the liquid as in true electrolysis, will wander back into the liquid again. On this theory the number of free ions of either kind ought to be greater near the electrode to which they tend to move. While Clausius' theory fully accounts for the behavior of the ions, it does not explain their relations to the electrical current. No satisfactory theory of the relations of electricity to the molecules of matter has as yet been given. 213. Voltaic Cells. — From the discussion given in § 206 it is obvious that, if an arrangement be made, in a circuit, of sub- stances capable of uniting chemically and such as would result from electrolysis, there will result an electromotive force in such a sense as to oppose the current which would effect the electrolysis. If, thenj the electrodes of an electrolytic cell in which this electromotive force exists be joined by a wire, a current will be set up through the wire in the opposite , direc- tion to the one which would continue the electrolysis, and the ions at the electrodes will recombine to form the electrolyte. There is thus formed an independent source of current, the voltaic cell. The electrode in connection with the electro-nega- tive ion is called the positive pole, and that in connection with the electro-positive ion the negative pole. Thus, if after the electrolysis of water in a voltameter, in which the gases are collected separately in tubes over platinum electrodes, the electrodes be joined by a wire, a current will be set up in it, and the gases will gradually, and at last totally, disappear, and the current will cease. The current which de- composes the water is conventionally said to flow through the liquid from the anode to the cathode, from the electrode above 213] CHEMICAL RELATIONS OF THE CURRENT. 29 1 which oxygen is collected to the electrode above which hydro- gen is collected. The current existing during the recombina- tion of the gases flows through the liquid from the hydrogen electrode to the oxygen electrode, or outside the liquid from the positive to the negative pole. Such an arrangement as is here described was devised by Grove, and is called the Grove s gas battery. A combination known as Smee's cell consists of a plate of zinc and one of platinum, immersed in dilute sulphuric acid. It is such a cell as would be formed by the complete electrolysis of a solution of zinc sulphate, if the zinc plate were made the cathode. When the zinc and platinum plates are joined by a wire, a current is set up from the platinum to the zinc outside the liquid, and the zinc combines with the acid to form zinc sulphate. The hydrogen thus liberated appears at the platinum plate, where, since the oxygen which was the electro-negative ion of the hypothetical electrolysis by which the cell was formed does not exist there ready to combine with it, it col- lects in bubbles and passes up through the liquid. The pres- ence of this hydrogen at once lowers the current from the cell, for it sets up a counter electromotive force, and also dimin- ishes the surface of the platinum plate in contact with the liquid, and thus increases the resistance of the cell. It may be partially removed by mechanical movements of the plate or by roughening its surface. The counter electromotive force is called the electromotive force of polarization. It occurs soon after the circuit is joined up in all cells in which only a single Uquid is used, and very much diminishes the currents which are at first produced. Advantage is taken of secondary chemical reactions to avoid -this electromotive force of polarization. The best example, and a cell which is of great practical value for its cheapness, durability, and constancy, is the DanielVs cell. Two liquids are used, solutions of cupric sulphate and zinc sulphate. They 292 ELEMENTARY PHYSICS. [213 are best separated from one another by a porous porcelain diaphragm. A plate of copper is immersed in the cupric sul- phate, and a plate of zinc in the zinc sulphate. The copper is the positive pole, the zinc the negative pole. When the circuit IS made and the current passes, zinc is dissolved, the quantity of zinc sulphate increases and'that of the cupric sulphate de- creases, and copper is deposited on the copper plate. To pre- vent the destruction of the cell by the consumption of the cupric sulphate, crystals of the salt are placed in the solution. The electromotive force of this cell is evidently due to the loss of energy in the substitution of zinc for copper in the solution of cupric sulphate. It may be calculated by the for- mula of §210. The experiments of Kohlrausch give for zinc in C. G. S. units, e = 0.00341 1, where the system of units em- ployed is the electromagnetic (§ 218). Favre and Silbermann give for 6, in the chemical process here involved, 714 gram- degrees or lesser calories. The mechanical equivalent of one gram-degree is 41,595,000. Hence we obtain for the electro- motive force of a Daniell's cell in C. G. S. electromagnetic units the value 1.013-10". The value as found by direct ex- periment is about I.I ■ 10° in C. G. S. electromagnetic units. There are many other forms of cell, which are all valuable for certain purposes. One of the best known is the Grove s cell. It has for positive pole a platinum plate, immersed in strong nitric acid, and for negative pole a zinc plate, immersed in dilute sulphuric acid. The two liquids are separated by a porous porcelain diaphragm. When the current passes, the zinc is dissolved. The hydrogen freed is oxidized by the nitric acid, which is gradually broken up into other compounds. The electromotive force of the Grove's cell is very high, being about 1.88- \ — 0) and sb sin (^ — {n — 0)). The lever arms on and OS are equal, and it is assumed that, since the poles are at equal distances from the current, the forces na and sb are equal; therefore sin (^ — 0) must equal sin (ip — {n — 0)), and this is true only when =— . The lines of magnetic force about an infinite straight current are therefore circles, and the equipotential surfaces determined by these lines are planes passing through the current. From the times of oscillation observed, it was proved that the force exerted is proportional directly to the strength of the magnet pole and to the strength of the current, and in- versely to the distance between the pole and the current. Biot hence deduced a law for the action of each element of length of the current upon a magnet pole, which is expressed in the formula mi sin ads , , /= --. • (91) In this formula m represents the strength of the magnet pole, i the current strength measured in electromagnetic units, ds the element of the current, r the distance "between that ele- ment and the magnet pole, and a the angle between r and ds. It is easy to show that the force exerted by a long straight current, observed by Biot to be inversely as the distance from 2l6] MAGNETIC RELATIONS OF THE CURRENT. 299 the current, is consistent with this law. For simplicity we will consider an infinitely long straight current. Let the magnet pole m be at the point P (Fig. 64). Let QR be the current element ds, and PO the perpendicular distance between the pole and the current. Then Biot's law gives for the force ex- . miQR PO erted by the element QR the expression -pn?" ' ~5B- I" the limit, as QR becomes indefinitely small, the triangles QRS and POR become similar. Hence QS equals — 5^5^ — , and the ex- miQS PR pression for the force becomes „„, -. If about P, with PO as radius, we draw the arc OU, the elementary arc a in the limit QS.PO equals -, and the projection b of the arc a on the line PR aPO ,, , , , . miQS , PC/' equals -^p- Usmg these values, the expression „„; be- i7ttb comes "H^- There will be a similar expression for the force due to any other element. The total force due to the whole mi current will be equal to the constant factor -„-—, multiplied by 300 ELEMENTARY PHYSICS. [217 the sum of all the projections corresponding to b. This sum, for the infinite current, is manifestly 2PU =■ 2PO. Hence the . 2mi total force is -p^ ; or, it is inversely as the distance PO be- tween the pole and the current. 217. Equivalence of a Closed Circuit and a Magnetic Shell. — The law of the force between a pole and a current, which has been stated, leads to the conclusion that a very small closed plane circuit, carrying a current, will act upon a magnet pole at a distance from it in the same way as a magnetic shell, of which the edge coincides with the contour of the circuit, and the strength equals the strength of the current. To show this we will use a rectangular circuit with indefinitely small sides. We will place the origin (Fig. 65) at the centre of the rectan- gle, and draw the x axis perpendicular to the plane of the rect- angle, and the y and z axes parallel with its sides. For con- venience, we will call the length of the sides parallel to the y axis 2^, and of those parallel to the z axis 2s'. We assume that a current of strength i traverses the bound- ary of the rectangle in a direction related to the positive di- rection of the X axis, as the motions of rotation and propul- sion are related in a right-handed screw. If the magnet pole be at the point {xyz), the ^orce on it due 217] MAGNETIC RELATIONS OF THE CURRENT. 3°! to one side, 2s, is, as stated in Eq. (90), proportional to the length 2s, is inversely as the square of the distance and is proportional to the sine of the angle between the line join- ing (xyz) and the element 2s and the direction of that element. (■*■" + (■^ — ■y')')* This sine is expressed by -7-^-. — ^r"? T^i^- The total force . , mi2s(x' -\-(z- sjy ^, . , due to the element is then -f-r-y — r-p"? t^^- This force is at right angles to the plane passing through the direction of the element 2s and the perpendicular from {xyz) on the direc- tion of that element. We shall investigate in turn the compo- nents along the three axes. That along the x axis is found by z — s' multiplying the total force by j^T ^^^yy The expression for the component along the x axis then becomes 2mis{z — j') (^+/ + (^-O0^" We will expand {z — s'y in the denominator, reject the term J'^ remembering that the sides are indefinitely small, and write for brevity x^ +7' -}-£•" — r'. We then have this component 2mis(z — s') „. ., . 1. 11 r 4.U expressed by ■ , ., _ 7^. Similar expressions hold for the components due to each of the other sides, with the difTerencc that those due to opposite sides must have different signs. We call those positive which are directed along the positive direction of x. We will write the four components, and opposite them their expansions in ascending powers of s or /, rejecting all terms containing the second or higher powers of s. 302 ELEMENTARY PHYSICS. [217 2mis(z — j') . , ,, , , , , ~ (f - 2Zs')i = ~ ^"^"^^ ~ ^ )('■ " ' + y^""') ' , 2mis{z -\- s') , .,.,., , V + (r" + zT/)! = + ^''''■'(^ + •')(''"' - s^^'' - ; 2mis'{y—s) ... . , , . - (y' - ay J)!" ^ ~ ^'^^ ^^-'^ ~ ^)^~' + S^C'-') ; , 2mis'(y-\-s) ,■,,,■., If we write out the sum of these expressions, rejecting all terms of the dimensions of /, we obtain as the component along the x axis of the force due to the whole circuit the ex- pression — ^miss'[ 1 ^1. The term in parenthesis 3/ + 32-' — 2r' r' — ix' „, can be written t = — — 5 — - The factor ass is equal to a, the area of the rectangle. The force along the x axis is then finally - mia\^ - ^J. (92) For the component along the y axis we have to consider only the forces due to the sides 2/, for the other sides have no tendency to move the pole parallel with themselves. The com- ponents of these forces along j/, that one being called positive X which is in the positive direction of y, are + 2mis'-r-^ -r ^ \r — 2ysf and — 2mis'-r-r-, ^- The sum of these components is . ,?,xy . ^xy 4mtss — ;- = mza—r. (gza) 217] MAGNETIC RELATIONS OF THE CURRENT. 303 Similarly the total component along the z axis is trna-p^. (92^) Now to compare these forces with those due to a magnetic shell of the indefinitely small area a and strength/, we use the result of the discussion in § 182, that the potential of such a shell at any external point \sjw. In that discussion the con- vention was made that the positive fac? of the shell was turned toward the positive direction of the x axis. We then have go, the solid angle subtended by the shell as seen from the point P, equal to a cos 6 ax x = — , = «7 The potential at the point P is then 17- ■ ^ To find the forces along the three axes we must find the rate of change of this potential with respect to space. To do this for the X axis, let x increase by a small increment ^x ; then the potential will take a small increment /I V. We will have and as ^;ir becomes indefinitely small, ' (^ + 2xAx)i 304 ELEMENTARY PHYSICS. [217 Expanding this expression, rejecting all terms containing the second or higher powers of Ax, we obtain x^ . ^rr . (X %X'AX , Ax\ From this we have further AV . (I 3^\ Jx -■^'^\r'~ r'r Ax In the limit, as Ax becomes indefinitely small, this is the rate of change of potential along the x axis at the point {xyz). The force along the x axis on a unit magnet pole at the point {xyz) is this rate of change of potential taken with the opposite sign. Hence the force, on the magnet pole m at that point is — mja\—i ^j. Similarly the forces along the ^ and , . , . 3,xy , . XX2 z axes can be found to be respectively mja — i- and mja — r- If these expressions be compared with the expressions for the components of force arising from the action of the rect- angular current, they will be seen to be completely identical, provided that the unit of current be so selected that the factors / andy are equal. If the current in the circuit be reversed, the components of force due to it remain the same in amount but are opposite in direction. The direction of current in the circuit which will render its action completely identical with that of the mag- netic shell may be readily stated. Let us draw a line through the magnetic shell, tangent to the lines of force, from the negative to the positive face, and call its direction the positive direction of the lines of force. Then the current in the equiv- 217] MAGNETIC RELATIONS OF THE CURRENT. 305 alent circuit is such that its direction is related to the positive" direction of the Hnes of force as the motions of rotation and propulsion are related in a right-handed screw. It may now be shown that a finite circuit of any form carrying a current i is equivalent to a magnetic shell of uni- form strength /, the edge of which coincides with the circuit. For a finite circuit may be conceived to be made up of an assemblage of elementary circuits of the kind considered, lying contiguous to one another in the surface bounded by the contour of the circuit. Everywhere the currents of one of these elementary circuits is neutralized by the equal and in- finitely near currents in the opposite direction of the contigu- ous circuits, except at the boundary, where all the elementary currents are in the same direction and are equivalent to the current in the circuit. This reasoning will be plain at once from Fig. 66. The forces due to such a current will then be equal to the forces due to a magnetic shell made up of elements which corre- spond to the elementary circuits. The systems of lines of force due to Fig. ee. the shell and the equivalent circuit will be precisely similar in form and distribution. They will differ, however, in this, that the line of force joining two contiguous points on opposite faces of the shell will be interrupted by the shell, while in the case of the circuit it passes through the circuit as a continuous line enclosing the current. If a unit positive magnet pole were placed at a point on the positive face of a magnetic shell, it would move along a line of force to a point infinitely near the one from which it started, but on the opposite or negative face of the shell, and during the movement it would do an amount of work expressed by i^nj. This same amount of work would be done upon it if it were brought back by any path to the point from which it started, so that the total work done in the 306 ELEMENTARY PHYSICS. [217 closed path is zero. If, on the other hand, the pole were mov- ing undfr the influence of the circuit equivalent to the mag- netic shell, it would move, as in the case of the shell, along the line cif force from the positive to the negative face of the circuit, and in so doing would do work equal to /i^ni. But from the fact that the line of force on which it is moving is continuous, and that the force in the field is everywhere finite, it would pass over the infinitJSBimal distance between the point on the negative face and the one on the positive face, from which it started, without doing any finite work. The system would then have returned to its original condition, and work equal to \ni would have been done. This is expressed by saying that the potential of a closed current is multiply-valued. The work done during any movement depends not only on the position of the initial and final points in the path, as in the case of the ordinary single-valued gravitational, electrical, and magnetic potentials, but also on the path traversed by the moving magnet pole. Every time the path encloses the cur- rent, work equal to i^ni is done. The work done in moving by a path which does not enclose the current, from a point where the solid angle subtended by the circuit is w^ to one where it is <», is, as in the case of the magnetic shell, equal to iioo^ — 00). If the path further enclose the current n times, the work done is /ifTtni, so that the total work done, or the total difference of potential between the two points, is V,-V= i{oo, - CO + 4;r«), (93) where n may have any value from o to infinity. The fact that the potential of a current is multiply-valued is well illustrated by any one of a series of experiments due to Faraday. If we imagine a wire frame forming three sides of a rectangle to be mounted on a support so as to turn freely about one of its sides as a vertical axis, while the free end, of the 2l8] MAGNETIC RELATIONS OF THE CURRENT. 3O7 opposite side dips in mercury contained in a circular ^ough of which the axis of rotation passes through the centre, and if we suppose a current to be sent through the axis and the frame, passing out through the mercury ; then if a magnet be placed vertically with its centre on the level of the trough, and with either pole confronting the frame, the frame will rotate con- tinuously about the axis. Other arrangements are made by which more complicated rotations of circuits can be effected. If the circuit be fixed and the magnet movable, similar arrangements will give rise to motions of the magnet or to rotations about its own axis. 218. Electromagnetic Unit of Current. — The relation which has been discussed between a circuit and the equivalent magnetic shell affords a means of defining a unit of current dif- ferent from that before defined in the electrostatic system. That current is defined as the unit current, which will set up the same magnetic field as that due to a magnetic shell of which the edge coincides with the circuit, and the strength is unity. This definition is equivalent to the following one, which is sometimes given. If the force between a unit magnet pole and a current flowing in a plane circuit of unit length, every part of which is at unit distance from the pole, be the unit force, then the current is the unit current. The equivalence of the two definitions may be shown as follows: Consider z. circular plane magnetic shell of strength/ set up normal to the x axis with its centre at the origin. We will determine the force at the pointy on the x axis, along that axis, due to the shell. Designate the radius of the circular shell by R, and the angle contained between the x axis and a line drawn from the point p to any point on the boundary of the shell by 0. Then 2jr^'(i — cos (j>) rep- resents the area of the spherical calotte, of which the centre is the point /, and the edge the boundary of the shell. Hence 27r(i — cos 0) is the solid angle subtended at the point p by the shell. We may express cos in other terms by (^»^!^8)i - and the potential V at the point / by K = iitj [x - ^-^-±-^\ The rate of change of potential along the x axis measures the force required. 308 ELEMENTARY PHYSICS. [218 Let X change by a small increment Ax. Then V will take an increment A V, and we have If we expand this expression, we have in the limit, as Ax becomes in- definitely small, since we may neglect higher powers of Ax than the first, = 2^y (i -{x + Ax) ^-^-jL_ _ j^^^^^ _ .1 x Ax x^Ax \ ~ ^^-^ V (x^ + R')i ix' + i?')i "^ W+^if Hence we obtain AV _ .1 x^ I \__„„- -g' Ax ^ ^ \{x' + ^2)3 (;,! _). Si)ij ^ (j;!! + ips) j- The force at the point / is therefore equal to zitj . When the (x -\- R^ point/ is at the origin, x equals o, and the force is expressed by —^. If we adopt the first definition of unit current, and sety = i, the force on a magnet zicim pole m due to a circuit equivalent to the shell is „ . If we adopt the second definition of unit current, and use Biot's formula for the action of a cuftent on a magnet pole^ the force due to a circular current, made up of current elements of length s, upon a pole at its centre is -^-ni- The sum of all the elements of the circle is 2nR. Hence the force 2nim on this definition is also R ■ The unit based upon these definitions is called the electro- magnetic unit of current. It is fundamental in the construction of the electromagnetic system of units, in just the same way as the unit of quantity is fundamental in the electrostatic system. 219] MAGNETIC RELATIONS OF THE CURRENT. 3O9 In practice another unit of current, called the ampere, is used. It is equal to lo""' C. G. S. electromagnetic units. The dimen- sions of the electromagnetic unit of current are those of the strength of a magnetic shell, or [?] = M^DT^'^. 219. Lines of Magnetic Force. — It is convenient, in much of the discussion of the action of currents, to use the notion of lines of force, and to measure the strength of field, as explained in § 21, by the number of hnes of force. For example, we may conceive the field about a magnet pole to be filled with conical tubes of force, of an angular aperture which is very small, and equal for all the cones, but otherwise entirely arbitrary. It is commonly assumed that each one of these cones represents a line of force. Then the solid angle sub- tended by any magnetic shell in the field, which is measured by the number of the cones contained in that solid angle, can be replaced by the number of lines of force which the bound- ary of the shell encloses. If the magnet pole be free to move, it will move from a point of higher to a point of lower potential ; that is, it will move in general to a point as near as possible to the negative face of the shell. If we make the convention that a line of force passes through a shell in the positive direction when it passes from the negative to the positive face, we may describe this mo- tion as one of which the result is, that as many lines of force as possible pass through the shell in the positive direction. If the magnet pole be fixed, and the shell free to move, it follows, from the equality of action and reaction, that the shell will set itself so that as many lines of force as possible will pass through it in the positive direction. When the shell is not perfectly free to move, and in certain other special cases, it is sometimes convenient to use an equivalent statement, that the shell will move so that as few lines of force as possible pass through it in the negative direction. These last conclusions are independent of the particular 310 ELEMENTARY PHYSICS. [220 character of the magnetic field in which, the shell is situated. It may then be stated generally, as a law governing the motions of magnetic shells or their equivalent electrical circuits in a magnetic field, that they tend to move so that as many lines of force as possible will pass through them in the positive direction. From the discussion in §217 it may be seen that the positive direction of a line of force due to a current is re- lated to the direction of the current in the circuit as the direc- tions of propulsion and of rotation in a right-handed screw. To one looking at the negative face of a magnetic shell, the current in the equivalent circuit will travel with the hands of a watch. If a part only of the closed circuit be free to move, it may be considered by itself as a magnetic shell, and it will move in accordance with the same law. We can therefore use this law to investigate the movements of circuits or parts of circuits due to the magnetic field in which they are placed. 220. Mutual Action of Two Currents. — In general, two plane circuits, if they be free to move, will so place themselves that the lines of force from the positive face of one will pass through the other in the positive direction, or through its negative face. The currents in the two circuits will then have the same direction. If they be placed so that unlike faces are opposed, they will move towardsone another; if so that similar faces are opposed, they will move away from one another. Since in the first case the currents are in the same direction, and in the second in opposite directions, the law may be stated in another form : that circuits carrying currents in the same direction attract one another ; in opposite directions, repel one another. Parts of the circuits, if movable, follow the same law. For example, consider a circuit in the form of a wire square, free to turn about a vertical line passing through the centres of two opposite sides. If now a vertical wire, forming part of another 22i] MAGNETIC RELATIONS OF THE CURRENT. 31I circuit, be brought near one of the vertical sides of the square, that side will move towards the vertical wire, or away from it, according as the currents in the two wires are in the same or in opposite directions. It is clear that the maximum number of lines of force due to the fixed circuit pass through the mov- able circuit in the positive direction, when the two parallel portions carrying currents in the same direction are as near one another as possible ; and that as few lines of force as pos- sible pass through the movable circuit in the negative direction, when the two parallel portions carrying currents in opposite directions are as far from one another as possible. 221. Ampere's Law for the Mutual Action of Currents. — The laws of the action between electrical currents were first investigated by Ampere from a different point of view. From a series of ingenious experiments he deduced a law which ex- presses the action of a current element on any other current element. The action of any circuit on any other can be ob- tained from this law by summing the effects of all the elements. The complete deduction of the law from the experimental facts is too complicated to be given, but the experiments themselves are of great interest. Ampere's method consisted in submitting a movable circuit or part of a circuit carrying a current to the action of a fixed circuit, and so disposing the parts of the fixed circuit that the forces arising from different fn^ parts exactly annulled one another, so that the - movable circuit did not move when the current in the fixed circuit was made or broken. In the first two of his experiments the movable circuit consisted of a wire frame of the form shown in Fig. 67. The current passes into the frame by the points a and b, upon which the frame is supported. It is evident that the two halves of the frame tend to face in opposite directions in the earth's mag- I t t 1 Fig. 67. 312 ELEMENTARY PHYSICS. [221 netic field, so that there is no tendency of the frame as a whole to face in any one direction rather than any other. If a long straight wire be placed near to one of the extreme vertical sides of the frame and a current be sent through it, that side will move towards the wire if the currents in it and in the wire be in the same direction, and will move away from the wire if the currents be in opposite directions. If now this wire be doubled on itself, so that near the frame there are two equal currents occupying practically the same position, but in opposite directions, then no motion of the frame can be observed when a current is set up in the wire. This is Ampere's first case of equilibrium. It shows that the forces due to two currents, identical in strength and in posi- tion, but opposite in direction, are equal and opposite. If the portion of the wire which is doubled back be not left straight, but bent into any sinuosities, provided these be small compared with the distance between the wire and the frame, still no motion of the frame occurs when a current is set up in the wire. This is Ampere's second case of equilibrium. It shows that the action of the elements of the curved conductor is the same as that of their projections on the straight conduc- tor. To obtain the third case of equilibrium, a wire, bent in the arc of a circle, is arranged so that it may turn freely about a vertical axis passing through the centre of the circle of which the wire forms an arc, and normal to the plane of that circle. The wire is then free to move only in the circumference of that circle, or in the direction of its own length. Two vessels filled with mercury, so that the mercury stands above the level of their sides, are brought under the wire arc, and raised until conducting contact is made between the wire and the mercury in both vessels. A current is then passed through the mova- ble wire through the mercury. Then if any closed circuit whatever, or any magnet, be brought near the wire, it is found 221] MAGNETIC RELATIONS OF THE CURRENT. 313 that the wire remains stationary. The deduction from this observation is that no closed circuit tends to displace an ele- ment of current in the direction of its length. In the fourth experiment three circuits are used, which we may call respectively A,B, and C. They are alike in form, and the dimensions of B are mean proportionals to the correspond- ing dimensions of A and C. B is suspended so as to be free to move, and A and C are placed on opposite sides of B, so that the ratio of their distances from B is the same as the ratio of the dimensions of A to those of B. If then the same current be sent through A and C, and any current whatever through B, it is found that B does not move. The opposing forces due to the actions of A and C upon B are in equilib- rium. From this fourth case of equilibrium is deduced the law that the force between two current elements is inversely as the square of the distance between them. Ampfere made the assumption that the action between two current elements is in the line joining them. From the four cases of equilibrium he then deduced an expression for the attraction between two current elements. It is ii' ds ds -I2 cos e — 3 cos cos 0'). (94) r' In this formula ds and ds' represent the elements of the two circuits, i and i' the strength of current in those circuits meas- ured in electromagnetic units, r the distance between the cur- rent elements, e the angles made by the two elements with one another, 6 and 6' the angles made by ds and ds' with r or r produced, the direction of the two elements being taken in the sense of their respective currents. A remarkable result of this equation is that two current elements of the same circuit in the same straight line repel 314 ELEMENTARY PHYSICS. [222 one another. The angle e becomes = o, and ^ = ^' = o ; 1 r . , , . . it' ds ds' therefore the force given by the equation is — j — . Since this is negative it expresses a repulsion. 222. Solenoids and Electromagnets. — Ampfere also showed that the action between two small plane circuits is the same as that between two small magnetic shells, and that a cir- cuit, or system of circuits, may be constructed which is the complete equivalent of any magnet. A long bar magnet may be looked on as made up of a great number of equal and simi- lar magnetic shells arranged perpendicular to the axis of the magnet, with their similar faces all in one direction. In order to produce the equivalent of this arrangement with the circuit, a long insulated wire is wound into a close spiral, straight and of uniform cross-section. The end of the wire is passed back through the spiral. When the current passes, the action of each turn of the spiral may be resolved into two parts, that due to the projection of the spiral on the plane normal to the axis, and that due to its projection on the axis. This latter component, for every turn, is neutralized by the current in the returning wire, and the action of the spiral is reduced to that of a number of similar plane circuits perpendicular to its axis. Such an arrangement is called a solenoid. The poles of a sole- noid of very small cross-section are situated at its ends, and it is equivalent to a bar magnet uniformly magnetized. If a bar of soft iron be introduced into the magnetic field within a solenoid it will become magnetized by induction. This combination is called an electromagnet. Since the strength of the magnetic field varies with the strength of the current in the solenoid, and with the number of layers of wire wrapped around the iron core, the magnetization of any bar of iron whatever may be raised to its maximum by in- creasing the current or the number of turns of wire. 224] MAGXETIC RELATIONS OF THE CURRENT. 315 223. Ampere's Theory of Magnetism. — "Ampere based upon these facts a famous theory of magnetism which bears his name. He assumed that around every molecule of iron there circulates an electrical current, and that to such molecular currents are due all magnetic phenomena. He made no hy- pothesis with regard to the origin or the permanency of these currents. The theory agrees with Weber's hypothesis that magnetization consists in an arrangement of magnetic mole- cules. If we further adopt Thomson's explanation of the dia- magnetic phenomena (§ 184), we may extend Ampere's theory to all matter, and assume that an electrical current circulates about every molecule. In order to account for the different magnetic susceptibilities of different bodies, it must also be assumed that these molecular currents are of different intensi- ties in different kinds of matter. Ampere's theory, however, admits another explanation of diamagnetism, which was given by Weber. He assumes that all diamagnetic molecules are capable of carrying molecular currents, but that those currents, under ordinary conditions, do not exist in them. When, however, a diamagnetic body is moved up to a magnet, an induced current due to the motion (§ 226) is set up in each molecule, and in such a direction that the molecules become elementary magnets, with their poles so directed towards the magnet in the field that there is repulsion between them. If this theory be true, it ought to be possible, as suggested by Maxwell, to lessen the intensity of magnetiza- tion of a body magnetized by induction, by increasing the strength of the field beyond a certain limit. 224. The Hall Effect. — Hitherto it has been assumed that when currents interact, it is their conductors alone which are affected, and that the currents in the conductors are not in any way altered. Hall has, however, discovered a fact which seems to show that currents may be displaced in their conduc- tors. If the two poles of a voltaic battery be joined to two op- 3l6 ELEMENTARY PHYSICS. [225 posite arms of a cross of gold foil mounted on a glass plate, and if a galvanometer be joined to the other two arms at such points that no current flows through it, then if a magnet pole be brought opposite the face of the cross a permanent current will be indicated by the galvanometer. The same effect ap- pears in the case of other metals. Thfe direction of the per- manent current and its amount difTer under the same circum- stances for different metals. The coefficient which represents the amount of the Hall effect in any metal is called the rota- tional coefficient of that metal. Since the rotational coefficients of such metals as have been tested agree in sign and in relative magnitude with their thermo-electric powers (§ 235), it is argued by Bidwell, Etting- hausen, and others that the Hall effect is due to thermo-electric action. 225. Measurement of Current. — Instruments which are used to detect the presence of a current, or to measure its strength by means of the deflection of a magnetic needle, are commonly called galvanometers. The simplest form of the galvanometer is the old instru- ment called the Schweigger' s multiplier. It consists of a flat spool upon which an insulated wire is wound a number of times. The plane of the coils is vertical, and usually also co- incides with the plane of the magnetic meridian. A magnetic needle is suspended in the interior of the spool. When a cur- rent is passed through the wire, the needle is deflected from the magnetic meridian. Usually, in order to make the indica- tions of the apparatus more sensitive, a combination of two needles is used. They are joined rigidly together, so that when suspended the lower one hangs in the interior of the spool, and the other in the same plane directly above the spool. These needles are magnetized so that the positive end of one is above the negative end of the other. If they are of nearly equal strength, such a combination will have very little 225] MAGNETIC RELATIONS OF THE CURRENT. 317 directive tendency in the earth's magnetic field. It is there- fore called an astatic system. When a current passes in the wire, however, the lines of force due to the current form closed curves passing through the coil, and both needles tend to turn in the same direction. Since the earth's field offers almost no resistance to this tendency, an astatic system will indicate the presence of very feeble currents. The apparatus here described is no longer used to measure currents, but only to detect their presence and direction. The sine galvanometer consists of a circular coil of insulated wire, set in the vertical plane, in the centre of which is a sup- port for a magnetic needle. The needle can turn in the hori- zontal plane. When a current is sent through the coil, the magnet is deflected. The coil is then turned about the ver- tical axis, until the magnet lies in the plane of the coils. When this is the case, the equilibrium of the needle is due to the equality of the couples set up by the cur- rent in the coils and by the horizontal com- ponent of the earth's magnetism. The couple due to the horizontal component (Fig. 68) is Hmlsin f)i ^"° (r" + (^+/sin0)y If another precisely similai^coil be set at the same distance d from the point of suspension of the needle, on the opposite side of it, and if the current be sent through it in the same direction, two other forces equal to those just stated will act upon the needle, tending to turn it in the same direction. There will thus arise two couples with moments equal to A Ttmir'l cos , /^nmir'l cos and (f +{d-l sin 0/)3 " " {r' + (^ + / sin (p)y 320 ELEMENTARY PHYSICS. [225 both tending to turn the magnet in the same direction. The factors 7-5—; — 7^—, — 7—: — TTKi are equal to (r" -\r(d ± I sm 0)')^ ^ (r' + ^') - ' T |(r^ + ^0 " ^ {2dl^m ± /» sin= 0) + V-(^' + d'y^^d'P sin'' 0, if we neglect all terms containing higher powers of / than the second. In this expression the upper or the lower signs must be used throughout. When we add the two moments of couple, we obtain for the total moment of couple acting on the needle the expression, after reduction, 8;fmVVcos0 / 3 (r' - 4^ ') . \ {f 4- d'f \ 2 (r' + dy ^'" '' This moment of couple is equal to that due to the horizon- tal intensity of the earth's magnetism, or 2inHl sin 0. Setting these expressions equal, we obtain for i, if we neglect powers of / higher than the second, The best form of the tangent galvanometer is so constructed r '■*■■ that d =^ ' . In this case the second term in the parenthesis 5« Hr disappears, and we have ?' = — . tan 0. The current is pro- portional to the tangent of the angle of deflection. If the galvanometer coils contain a number of turns equal in each 71 coil to -, the proportion of the breadth to the depth of the 226] MAGNETIC RELATIONS OF THE CURRENT. 321 coils may be so determined that the current is given by the equation • 5* ^^ ^ 2 = -> . tan 0. (c7) i6 nn ^^" In this equation R is the mean radius of the coil. All the quantities in this expression for i, except H, are either num- bers or lengths, and H can be measured in absolute units. The tangent galvanometer can therefore be used to measure current in absolute units. Weber's electro-dynamometer is an instrument with fixed coils like those of the tangent galvanometer, but with a small suspended coil substituted for the magnet. The small coil is usually suspended by the two fine wires through which the current is introduced into it, and the moment of torsion of this so-called hifilar suspension enters into the expression for the current strength. The same current is sent through the fixed and the movable coils, and a measurement of its strength can be obtained in absolute units, as with the tangent galvanome- ter. By a proper series of experiments, this measurement is made independent of the horizontal intensity of the earth's magnetism. When the current is reversed in the instrument, the couple tending to turn the suspended coil does not change. If the effects of terrestrial magnetism can be avoided, the electro-dynamometer can therefore be used to measure rapidly alternating currents. 226. Induced Currents. — It was shown in § 206 that the movement of a magnet in the neighborhood of a closed circuit will give rise, in general, to an electromotive force in the cir- cuit, and that the current due to this electromotive force will be in the direction opposite to that current which, by its action upon the magnet, would assist the actual motion of the mag- net. This current is called an induced current. From the 322 ELEMENTARY PHYSICS. l^2.t equivaknce between a magnetic shell and an electrical cur- rent, it is plain that a similar induced current will be produced in a closed circuit by the movement near it of an electrical current or any part of one. Since the, joining up or breaking the circuit carrying a current is equivalent to bringing up that same current from an infinite distance, or removing it to an infinite distance, it is further evident that similar induced currents will be produced in a closed circuit when a circuit is made or broken in its presence. The demonstration of the production of induced currents in § 206 depends upon the assumption that the path of the magnet pole is such that work is done upon it by the current assumed to exist in the circuit. The potential of the magnet pole relative to the current is changed. The change in potential from one point to another in the magnetic field due to a closed current is (Eq. 93) equal to i{po^ — 00 -\- ^Ttn), and the work done on a magnet pole m, in moving it from one point to another, is mi{oo^ — go -\- j^nn). In the demonstration of § 206 we may substitute m{Go^—Go-\-^7tti) ior A, and, provided the change in potential be uniform, we m(oo. — GO -\- 47rn) , obtam at once the expression for the elec- tromotive force due to the movement of the magnet pole. If the change in potential be not uniform, we may conceive the time in which it occurs to be divided into indefinitely small intervals, during any one of which, t, it may be considered uni- „, ,,..,, . m(G0, — ao-\- ATTfi) form. Then the limit of the expression -^- -, as i becomes indefinitely small, is the electromotive force during that interval. The current strength due to this electromotive force is m{Go^ — Go-\- 47rn) ''= Rt 226] MAGNETIC RELATIONS OF THE CURRENT. 323 If the induced current be steady, the total quantity of electricity flowing in the circuit is expressed by _ mipOi — a? -|- i^nri) t,t - - . The total quantity of electricity flowing in the circuit de- pends, therefore, only upon the initial and final positions of the magnet pole, and the number of times it passes through the circuit, and not upon its rate of motion. The electro- motive force due to the movement of the magnet, and conse- quently the current strength, depends, on the other hand, upon the rate at which the potential changes with respect to time. A more general statement, which will include all cases of the production of induced currents, may be derived by the use of the method of discussion given in § 2ig. The change in potential of a closed circuit, carrying a current in a magnetic field, may be measured by the change in the number of lines of force which pass through it in the positive direction. Any movement which changes the number of lines of force will set up in the circuit an electromotive force, and an induced current in a sense opposite to that current which would by its action assist the movement. As in the elementary case which has just been discussed, the total quantity of electricity passing in the circuit depends only upon the total change in the num- ber of lines of force passing through the circuit in the positive direction, but the electromotive force and current strength depend on the rate of change in the number of lines of force. It is often convenient, especially when considering the movement of part of a circuit in a magnetic field, to speak of the change in the number of lines of force enclosed by the circuit as the number of lines of force cut by the moving part of the circuit. The direction of the induced current in the 324 ELEMENTARY PHYSICS. \p.2iy moving part of the circuit, if it be supposed to move normal to the lines of force, is related to the direction of motion and to the positive direction of the lines of force cut, in such a way that the three directions may be represented by the posi- tive directions of the three co-ordinate axes of x, y, and s, when the x axis represents the direction of motion, the y axis the lines of magnetic force, and the z axis the direction of the induced current. The positive directions of the three axes are such that, if we rotate the positive x axis through a right angle about the z axis, clockwise as seen by one looking along the positive direction of the z axis, it will coincide with the posi- tive y axis. The fac); that induced currents are produced in a closed circuit by a variation in the number of lines of magnetic force included in it was first shown experimentally by Faraday in 1 83 1. He placed one wire coil, in circuit with a voltaic battery, inside another which was joined with a sensitive galvanometer. The first he called the primary, the second the secondary, cir- cuit. When the battery circuit was made or broken, deflections of the galvanometer were observed. These were in such a direction as to indicate a current in the secondary' coil, when the primary circuit was made, in the opposite direction to that in the primary, and when the primary circuit was broken, in the same direction as that in the primary. When the positive pole of a bar magnet was thrust into or withdrawn from the secondary coil, the galvanometer was deflected. The currents indicated were related to the direction of motion of the posi- tive magnet pole, as the directions of rotation and propulsion in a left-handed screw. The direction of the induced currents in these experiments is easily seen to be in accordance with the law above stated, that the induced currents are always in the opposite direction to those currents which would, by their action, assist the motion. This law of induced currents in its general form was first 227] MAGNETIC RELATIONS OF THE CURRENT. 325 announced by Lenz in 1834, soon after Faraday's discovery of the production of induced currents. It is known as Lend s law. The case in which an induced current in the secondary cir- cuit is set up by making the primary circuit is, as has been said, an extreme case of the movement of the primary circuit from an infinite distance into the presence of the secondary. The ex- periments of Faraday and others show that the total quantity of electricity induced when the primary circuit is made is exactly equal and opposite to that induced when the primary circuit is broken. They also show that the electromotive force induced in the secondary circuit is independent of the materials consti- tuting either circuit, and is proportional to the current strength in the primary circuit. These results are consistent with the formula already deduced for the induced current. 227. Self-induction. — When a current is set up in any cir- cuit, the different parts of the circuit act on one another in the relation of primary and secondary circuits. In a long straight wire, for example, the current which is set up through any small area in the cross-section of the wire tends to develop an op- posing electromotive force through every other area in the same cross-section. The true current will thus be temporarily weak- ened, and will require a certain time to attain its full strength. On the other hand, when the circuit is broken, the induced electromotive force is in the same direction as the electromo- tive force of the circuit. Since the time occupied by the change of the true current from its full value to zero, when the circuit is broken, is very small, the induced electromotive force is very great. The current formed at breaking is called the extra cur- rent, and gives rise to a spark at the point where the circuit is broken. The extra current may be heightened by anything which will increase the change in the number of lines of force, as by winding the wire in a coil and by inserting in the coil a piece of soft iron. This action of a circuit on itself is called J^//"- induction. 526 ELEMENTARY PHYSICS. [228 228. Electromagnetic Unit of Electromotive Force.— If the circuit considered in § 226 move from a point where its po- tential relative to the magnet pole is moo^ to one_where it is moa, provided that the magnetic pole do not pass through the cir- cuit, and that the movement be so carried out that the induced current is constant, the electromotive force of the induced cur- m (00 — go) ^^ , , , . . . rent is — — ^— ^t • If the movement take place m unit time, and if m (ca, — w) also equal unity, the electromotive force in the circuit is defined to be unit electromotive force. The expression m {00^ — ao) is equivalent to the change in the number of lines of force passing through the circuit in the positive direction. More generally, then, if a circuit or part of a circuit so move in a magnetic field that, in unit time, the number of lines of force passing through the circuit in the posi- tive direction increase or diminish by unity, at a uniform rate, the electromotive force induced is unit electromotive force. The simplest way in which these conditions can be presented is as follows : Suppose two parallel straight conductors at unit distance apart, joined at one end by a fixed cross-piece. Sup- pose the circuit to be completed by a straight cross-piece of unit length which can slide freely on the two long conductors. Sup- pose this system placed in a magnetic field of unit intensity, so that the lines of force are everywhere perpendicular to the plane of the conductors. Then, if we suppose the sliding piece to be moved with unit velocity perpendicular to itself along the parallel conductors, the electromotive force set up in the circuit will be the unit electromotive force. The unit of electromotive force thus defined is the electro- magnetic unit. In practice another unit is used, called the volt. It contains 10^ C. G. S. electromagnetic units. To obtain the dimensions of electromotive force in the elec- tromagnetic system we need first the dimensions of number of lines of force. From the convention adopted by which lines of 229] MAGNETIC RELATIONS OF THE CURRENT. 327 force are used to measure the strength of a magnetic field we have -=, = [//"] ; whence [«] = M^ L^ T~ '. Since the electromo- tive force is measured by the rate of change of the number of lines of force we have [£•]=-==: M^L^ T~'^. The definition of electromotive force is consistent,' as it must be, with the equation ie = rate of work, or work divided by time. This equation is the same as that discussed in § 202, and holds whichever system of units is adopted. In the deter- mination of the unit of electromotive force the arrangement given above is, of course, impracticable. In those experiments which have been made, the induced electromotive force which was due to the rotation of a circular coil in a magnetic field was determined by calculation. 229. Apparatus employing Induced Currents. — The pro- duction of induced currents by the relative movements of con- ductors and magnets is taken advantage of in the construction of pieces of apparatus which are of great importance not only for laboratory use but in the arts. The telephonic receiver consists essentially of a bar magnet around one end of which is carried a coil of fine insulated wire. In front of this coil is placed a thin plate of soft iron. When the coils of two such instruments are joined in circuit by conducting wires, any disturbance of the iron diaphragm in front of one coil will change the magnetic field near it, and a current will be set up in the circuit. The strength of the mag- net in the other instrument will be altered by this current, and the diaphragm in front of it will move. When the diaphragm of the first instrument, or transmitter, is set in motion by sound- waves due to the voice, the induced currents, and the conse- quent movementsof the diaphragm of the second instrument, or receiver, are such that the words spoken into the one can be recognized by a listener at the other. 328 ELEMENTARY PHYSICS. [229 Other transmitters are generally used, in which the dia- phragm presses upon a small button of carbon. A current is passed from a battery through the diaphragm, the carbon but- ton, and the rest of the circuit, including the receiver. When the diaphragm moves, it presses upon the carbon button and alters the resistance of the circuit at the point of contact. This change in resistance gives rise to a change in the current, and the diaphragm of the receiver is moved. The telephone serves in the laboratory as a most delicate means of detecting a change of current in a circuit. The various forms of magneto-electrical and dynamo-elec- trical machines are too numerous and too complicated for de- scription. In all of them an arrangement of conductors, usually called the armature, is moved in a powerful magnetic field, and a suitable arrangement is made by which the currents thus in- duced may be led off and utilized in an outside circuit. The magnetic field is sometimes established by permanent magnets, and the machine is called a magneto-machine. In most cases, however, the circuit containing the armature also contains the coils of the electromagnets to which the magnetic field is due. When the armature rotates, a current starts in it, at first due to the residual magnetism of some part of the machine : this cur- rent passes through the field magnets and increases the strength of the magnetic field. This in turn reacts upon the armature, and the current rapidly increases until it attains a maximum due to the fact that the magnetic field does not increase pro- portionally to the current which produces it. Such a machine is called a dynamo-machine. The induction coil, or Ruhmkorff's coil, consists of two cir- cuits wound on two concentric cylindrical spools. The inner or primary circuit is made up of a comparatively few layers of large wire, and the outer, or secondary, of a great number of turns of fine wire. Within the primary circuit is a bundle of iron wires, which, by its magnetic action, increases the electro- 230] MAGNETIC RELATIONS OF THE CURRENT. 329 motive force of the induced current in the secondary coil. Some device is employed by which the primary circuit can be made or broken mechanically. The electromotive force of the induced current is proportional to the number of windings in the sec- ondary coil, and as this is very great the electromotive force of the induced current greatly exceeds that of the primary current. The electromotive force of the induced current set up when the primary circuit is broken is further heightened by a device pro- posed by Fizeau. To two points in the primary circuit, one on either side of the point where the circuit is broken, are joined the two surfaces of a condenser. When the circuit is broken, the extra current, if the condenser be not introduced, forms a long spark across the gap and so prolongs the fall of the primary cur- rent to zero. The electromotive force of the induced current is therefore not so great as it would be if the fall of the primary current could be made more rapid. When the condenser is in- troduced, the extra current is partly spent in charging the con- denser, the difference of potential between the two sides of the gap is not so great, the length of the spark and consequently the time taken by the primary current to become zero is lessened, and the electromotive force of the induced current is proportionally increased. 230. Resistance. — As in the discussion of § 203, we may here define the ratio of the electromotive force to the current in any circuit as the resistance in that circuit. The electromag- netic unit of resistance is the resistance of that circuit in which unit electromotive force gives rise to unit current, when both these quantities are measured in electromagnetic units. In the example given in § 228, if we insert a galvanometer in that part of the circuit occupied by the fixed cross-piece, and assume that the resistance of every -part of the circuit ex- cept the sliding piece is zero, the resistance of the sliding piece will be unity when, moving with unit velocity, it gives rise to unit current in the galvanometer. If it move with 330 ELEMENTARY PHYSICS. [230 any other velocity v, and still produce unit current in the gal- vanometer, its resistance will be numerically equal to the veloc- ity V. For the electromotive force produced by a movement with that velocity is v, and the ratio of that electromotive force to unit current is v, which is the resistance by definition. A unit of resistance, intended to be the C.G.S. electromagnetic unit, was determined by a committee of the British Association by the following method ; A circular coil of wire, in the centre of which was suspended a small magnetic needle, was mounted so as to rotate with constant velocity about a vertical diameter. From the dimensions and velocity of rotation of the coil and the intensity of the earth's magnetic field, the induced electro- motive force in the coil was calculated. The current in the same coil was determined by the deflection of the small magnet. The ratio of these two quantities gave the resistance of the coil. In practice another unit of resistance is used, called the ohm. It would be the resistance of a sliding piece in the arrangement before described which would give rise to the C. G. S. unit of cur- rent if it were to move with a velocity of one billion centi- metres in a second. The true ohm thus contains 10° C. G. S. elec- tromagnetic units. The dimensions of resistance in the elec- tromagnetic system are [r] = - | = Z. 7" \ The dimensions of resistance are therefore those of a velocity, as might be in* ferred from the measure of resistance in terms of velocity in the example given above. The standard of resistance, usually called the B. A. unit, de- termined by the committee of the British Association, has a resistance somewhat less than the true ohm as it is here defined. In practical work resistances are used which have been compared with this standard. The Electrical Congress of 1884 defined the legal ohm to be " the resistance of a column of mercury of one square millimetre section and of 106 centimetres of length 231] MAGNETIC RELATIONS OF THE CURRENT. 331 at the temperature of freezing." The legal ohm contains i.oi 12 B. A. units. Boxes containing coils of wire of definite resistance, so arranged that by different combinations of them any desired resistance may be introduced into a circuit, are called resistance boxes or rheostats. , 231. Kirchhoff's Laws. — In circuits which are made up of several parts, forming what may be called a network of con- ductors, there exist relations among the electromotive forces, currents, and resistances in the different branches, which have been stated by Kirchhoff in a way which admits of easy appli- cation. Several conventions are made with regard to the positive and negative directions of currents. In considering the currents meeting at any point, those currents are taken as positive which come up to the point, and those as negative which move away from it. In travelling around any closed portion of the net- work, those currents are taken as positive which are in the di- rection of motion, and those as negative which are opposite to the direction of motion. Further, those electromotive forces are positive which tend to set up a positive current in their respec- tive branches. With those conventions Kirchhoff ' s laws may be stated as follows : 1. The algebraic sum of all the currents meeting at any point of junction of two or more branches is equal to zero. This first law is evident, because, after the current has become steady, there is no accumulation of electricity at the junctions. 2. The sum, taken around any number of branches forming a closed circuit, of the products of the currents in those branches into their respective resistances is equal to the sum of the elec- tromotive forces in those branches. This law can easily be seen to be only a modified statement of Ohm's law, which was given in § 203. These laws may be best illustrated by their application in a form of apparatus known as Wheatstone's bridge. The circuit 332 ELEMENTARY PHYSICS. [231 of the Wheatstone's bridge is made up of six branches. An end of any branch meets two, and only two, ends of other branches, as shown in Fig. 71. In the .branch 6 is a voltaic ceil with an electromotive force E. In the branch 5 is a galvan- ometer which will indicate the presence of a current in that ^"^' ''■ branch. In the other branches are conductors, the resistances of which may be called respec tively r„ r„ r,, r,. From Kirchhoff' s first law the sum of the currents meeting at the point C is i^ -\- i^ -\- i^ = o, and of those meeting at the point D is i^ -\- i^ -\- 4 = o. By the second law, the sum of the products ir in the circuit ADC is «,r, + i^r, -\- i^r^ = o, and in the circuit DBC is i^r^ -\- i^r, -\- i^r^ = o, since there are no electromotive forces in those circuits. If we so arrange the resistances of the branches i, 2, 3, 4 that the galvanometer shows no deflection, then the current i^, is zero, and these equa- tions give the relations, i^ = — i^, i, = — i„ i^r^ = — t^r^, i^r, = — z^r,. From these four equations follows at once a relation between the resistances, expressed in the equation r^r, = r,r,. (98) If, therefore, we know the value of r, and know the ratio of r, to r„ we may obtain the value of r^. This method of comparing resistances by means of the Wheatstone's bridge is of great importance in practice. By the use of a form of apparatus known as the British Association bridge the method can be carried to a high degree of accuracy, la this form of the bridge, the portion marked ^C5 (Fig. 71) is a straight cylindrical wire, along which the end of the branch CD 23i] MAGNETIC RELATIONS OF THE CURRENT. 333 is moved until a point C is found, such that the galvanometer shows no deflection. The two portions of the wire between C and A, and C and B, are then the two conductors of which the resistances are r^ and r„ and these resistances are proportional to the lengths of those portions (§ 204). The ratio of r, to r^ is therefore the ratio of the lengths of wire on either side of C, and only the resistance of r^ need be known in order to obtain that of r,. It is often convenient in determining the relations of current and resistance in a network of conductors to use Ohm's law (§203), directly, and consider the difference of potential between the two points on a conductor as equal to the product ir. When a part of a circuit is made up of several portions which all meet at two points A and B, the relation between the whole resistance and that of the separate parts may be obtained easily in this way. For convenience in illustration we will sup- pose the divided circuit (Fig. 72) made up of only three portions, i, 2, 3, meeting at the points A and B, and that no electromotive force exists in those portions. Then the difference of potential between A and B is V^— Vb= i,r^ = i^r, = i,r^. We have also by Kirchhoff's first law — 2, = /, + /j + ^3. By the combination of these equations we obtain -^;=(f^.-f^.)C-+p+9. (99) The current in the divided circuit equals the difference of potential between A and B multiplied by the sum of the recip- rocals of the resistances of the separate portions. If we set this sum equal to-, and call r the resistance of the divided circuit, 334 ELEMENTARY PHYSICS. [231 we may say that the reciprocal of the resistance of a divided circuit is equal to the sum of the reciprocals of the resistances of the separate portions of the circuit. When there are only two portions into which the circuit is divided, one of them is usually called a shunt, and the circuit a shunt circuit. An arrangement devised by Clark, called the Clark' s poten- tiometer, used to compare the electromotive forces of voltaic cells, depends for its action on the principles here discussed. It consists of a spiral of evenly drawn wire coiled about a rubber cylinder, with arrangements by which contact can be made with it at both ends and at any point along it. Let us call the cells to be compared cell i and cell 2, and let the electromotive force of cell I be the greater. To the two ends of the spiral are joined the terminals of a circuit which we will call A, containing a coxi- stant voltaic battery, of which the electromotive force is greater than that of either cell i or cell 2, and a set of resistances which can be varied. To the same points are joined the terminals of a circuit which we will call B, containing cell i, and a sensitive galvanometer. The positive poles of the constant battery and of cell I are joined to the same end of the spiral. The resist- ance is then modified in circuit A until the galvanometer in circuit B shows no deflection. The difference of potential between the ends of the spiral is, therefore, equal and in the opposite direction to the electromotive force of cell i. The positive pole of cell 2 is now joined to the end of the spiral to which the positive poles of the other circuits are joined, and with the free end of a circuit C, containing cell 2 and a sensitive galvanometer, contact is made at different points on the spiral until the point is found at which, when contact is made, the galvanometer in C shows no deflection. The difference of poten- tial between that point and the end of the spiral joined to the positive poles is equal and opposite to the electromotive force of cell 2. The electromotive forces of the two cells are then proportional to the lengths of the wire between the points of 33l] MAGNETIC RELATIONS OF THE CURRENT. 335 contact of their terminals ; that is, the electromotive force of cell I is to that of cell 2 as the length of the wire spiral is to that portion of its length between the two terminals of cell 2. For, since the wire is uniform, its resistance is proportional to its length, and if we represent the potential of the common point of contact of the positive poles by V, the potentials of the points of contact of the two negative poles by V^ and F^, the current in the spiral by i, and the resistances of the lengths of wire considered by r, and r,, we have y — — • The rules for joining up sets of voltaic cells in circuits so as to accomplish any desired purpose may be discussed by the same method. Let us suppose that there are n cells, each with an electromotive force e and an internal resistance r, and that the external resistance of the circuit is s. If in be a factor of n, and if we join up the cells with the external resistance so as to form a divided circuit of m parallel branches, each containing — cells, we shall have for the electromotive force in such a m ne nT circuit — , and for the resistance of the circuit s -\ j- The m ni IffHS current in the circuit is therefore i = —„ — ; Two cases ms -\- nr may arise which are common in practice. The resistance s of the external circuit may be so great that, in comparison with n^s, nr may be neglected. In that case i is a maximum when m= I, that is, when the cells are arranged tandem, or in series, with their unlike poles connected. On the other hand, if in's be very small as compared with nr, it may be neglected, and i becomes a maximum when m—n, that is, when the cells are 336 ELEMENTARY PHYSICS. [233 arranged abreast, or in multiple arc, with their like poles in con- tact. 232. Ratio between the Electrostatic and Electromag- netic Units. — When the dimensions of any electrical quantity derived from its electrostatic definition are compared with its dimensions derived from its electromagnetic definition, the ratio between them is always of the dimensions of some power of a velocity. The ratio between the electrostatic and electro- magnetic unit of any electrical quantity is, therefore, of the dimensions of some power of a velocity. If, therefore, this ratio be obtained for any set of units, the number expressing it will also express some power of a velocity. This velocity is an absolute quantity or constant of nature. Whatever changes are made in the units of length and time, the number express- ing this velocity in the new units will also express the ratio of the two sets of electrical units. This ratio, which is called v, can be measured in several ways. The first method, used by Weber and Kohlrausch, depends upon the comparison of a quantity of electricity measured in the two systems. From the dimensions of current in the elec- tromagnetic system we have the dimensions of quantity \_q\ = [«' Z] = 31^ L^. The dimensions of quantity in the electro- static system are [(2] =.^*i^^ 7"'- The ratio of these dimen- sions is — = LT'\ or, the number of electrostatic units of quantity in one electromagnetic unit is the velocity v. In Weber and Kohlrausch's method the charge of a Leyden jar was measured in electrostatic units by a determination of its capacity and the difference of potential between its coatings. The current produced by its discharge through a galvanometer was used to measure the same quantity in electromagnetic measure. Thomson determined w by a comparison of an electromotive 232] MAGNETIC RELATIONS OF THE CURRENT. 337 force measured in the two systems. He sent a current through a coil of very high known resistance, and measured it by an electro-dynamometer. The electromagnetic difference of po- tential between the two ends of the resistance coil was then equal to the product of the current by the resistance. The electrostatic difference of potential between the same two points was measured by an absolute electrometer. From the dimen- sional formulas we have [4]= M\L^T2_ _ ^. The number of electromagnetic units of electromotive force in one electrostatic unit is v. The ratio of the numbers express- ing the electromagnetic and the electrostatic measures of the electromotive force in Thomson's experiment is therefore the quantity v. This experiment was carried out by Maxwell in a different form, in which the electrostatic repulsion of two simi- larly charged disks was balanced by an electromagnetic attrac- tion between currents passing through flat coils on the back of the two disks. Other methods, depending on comparisons of currents, of resistances, and other electrical quantities, have been employed. The methods described are historically interesting as being the first ones used. The values of v obtained by them differed rather widely from one another. Recent determinations, how- ever, give more consistent results. It is found that v, considered as a velocity, is about 3-io'° centimetres in a second. This velocity agrees very closely with the velocity of light. The physical significance of this quantity v may be under- stood from an experiment of Rowland. The principle of the experiment is as follows. If we consider an indefinitely ex- tended plane surface on which the surface density of electrifica- 338 ELEMENTARY PHYSICS. 232] (T tion is cr, measured in electrostatic units, or - measured in elec- V tromagnetic units, since the ratio of the electrostatic to the electromagnetic unit of quantity is v ; and conceive it to move in its own plane with a velocity x ; the charge moving with it may be considered as the equivalent of a current in that sur- face, the strength of which, measured by the quantity of elec- tricity which crosses a line of unit length, perpendicular to the (TX direction of movement, in unit time, is — The force due to V such a current on a magnet may be calculated. Conversely, if the force on the magnet be observed, and the surface density o- and the velocity x be also measured, the value of v may be calculated. The probability of such an action as the one here described was stated by Maxwell. The experiment by which Rowland verified Maxwell's view consisted in rotating a disk cut into numerous sectors, each of which was electrified, under an astatic magnetic needle. Dur- ing the rotation of the disk, a deflection of the needle was ob- served, in the same sense as that in which it would have moved if a current had been flowing about the disk in the direction of its rotation. From the measured values of the deflecting force, of the surface density of electrification on the disk, and the velocity of rotation, Rowland calculated a value of v which lies between those given by Weber and Maxwell. It may be seen that, if the velocity x of the moving surface which we at first considered be equal to v, the equivalent cur- rent strength in the surface will be a. If we imagine another such surface near the one already considered, the repulsion be- tween them due to their opposite charges is 2n(f for every unit of surface (§ 198). It can be shown, by a method too extended to be given here, that the attraction between two currents in the same surfaces, of which the strengths in the surface are both <7, is also expressed by 2;r(r'' for every unit of surface. Hence 232] MAGNETIC RELATIONS OF THE CURRENT. 339 if the surfaces, so charged that the surface density of their elec- trification is o", can move with a velocity in their own planes equal to v, the repulsion of the charges will exactly counter- balance the attraction of the currents due to their movement. CHAPTER VI, THERMO-ELECTRIC RELATIONS OF THE CURRENT. 333. Thermo-electric Currents. — The heating or cooling of a junction of two dissimilar metals by the passage of a cuiTent, referred to in § 200 as the Peltier effect, is the reverse of a phenomenon discovered in 1822-23 by Seebeck. He found that, when the junction of two dissimilar metals was heated, a current was sent through any circuit of which they formed a part. It has since been shown that the same phe- nomenon appears if the junction of two liquids, or of a liquid and a metal, be heated. This fact, as has been already shown in § 206, follows as a result of the Peltier phenomenon. If we designate by P the heat developed at the junction by the passage of unit current for unit time, we may substitute it for the expression — in the general equation of § 206, and obtain E — P I = — — — . The counter electromotive force set up at the R heated junction is the coefficient P, and is the measure of the true electromotive force of contact (§ 214). The contact elec- tromotive force of Volta does not agree in magnitude and not always in sign with this electromotive force. From this fact it is evident that the contact electromotive force of Volta is at least partially due to the air or other medium in which the bodies which are tested are placed. If the electromotive force E and the current / be reversed E-\-P in the circuit, the junction is cooled and we obtain / = — — . 2Zli\ THERMO-ELECTRIC RELATIONS OF THE CURRENT. 34 1 The electromotive force at the junction, therefore, tends to increase the electromotive force of the circuit. Since this is opposite to the electromotive force of the circuit in the case in which the junction is heated, the direction of the electro- motive force at the junction is the same as that found in the other case. If, then, there be no electromotive force E in the P circuit, we have / = h" in case a unit of heat is communi- K cated to the junction and absorbed by it in unit time, and P / = — - in case a similar quantity of heat is removed from the R junction by cooling. If two strips of dissimilar metals, for example antimony and bismuth, be placed side by side, and united at one end of the pair, being everywhere else insulated from one another, the combination is called a thermo-electric element. If several such elements be joined in series, so that their alternate junctions lie near together and in one plane, as indicated in Fig. 73, such an arrangement is called a thermo- pile. When one face of the pile is heated, the electromotive force of the pile is the sum of the elec- fig. 73. tromotive forces of the several elements. Such an instrument was used by Melloni, in connection with a delicate galvanom- eter, in his researches on radiant heat. When a thermo-electric element is constructed of any two metals, that metal is said to be thermo-electrically positive to the other from which the current flows across the heated junction. 234, Thermo-electric Series. — It was found by the experi- ments of Seebeck himself, and those of others, that the metals may be arranged in a series such that any metal in it is thermo- 342 ELEMENTARY PHYSICS. [235 electrically positive to those which follow it, and thermo-elec- trically negative to those which precede it. If a circuit be formed of any two metals in this series, and one of the junctions be kept at the temperature zero, while the other is heated to a fixed temperature, there will be set up an electromotive force which can be measured. If now the circuit be broken at either junction, and the gap filled by the intro- duction of any other metals of the series, then, provided that the junction which has not been disturbed be kept at the tem- perature which it previously had, and that the other junctions in the circuit be all raised to the temperature of the junction which was broken, there will be the same electromotive force in the circuit as existed before the introduction of the other metals of the series. It is manifest, then, that in a circuit made up of any metals whatever, at one temperature, no electromo- tive force can be set up by changing the temperature of the circuit as a whole. • Thomson showed that it is not necessary for the production of thermal currents that the circuit should contain two metals ; but that want of homogeneity arising from any strain of one part of an otherwise homogeneous circuit will also admit of the production pi such currents. It has also been shown that when a portion of an iron wire is magnetized, and is heated near one of the poles produced, a thermal current will be set up. Gumming discovered in 1823 that, if the temperature of one junction of a circuit of two metals be gradually raised, the cur- rent produced will increase to a maximum, then decrease until it becomes zero, after which it is reversed and flows in the opposite direction. The experiments of Avenarius, Tait, and Le Roux show that, for almost all metals, the temperature of the hot junction at which the maximum current occurs is the mean between the temperatures of the two junctions at which the current is reversed. 235. Thermo-electric Diagram. — The facts hitherto dis- covered in relation to thermo-electricity may be collected in a 235] THERMO-ELECTRIC RELA TIONS OF THE CURRENT. 343 general formula or exhibited by means of a thermo-electric dia- gram. Let us consider a circuit of two metals, copper and lead, in which both junctions are at first at the same temperature. We may assume that there is an equal electromotive force of contact at both junctions acting from lead to copper. If one of the junctions be gradually heated, a current will be set up, passing from lead to copper across the hot junction. The heating has disturbed the equilibrium of electromotive forces, and has in- creased the electromotive force across the hot junction from lead to copper. The rate at which this electromotive force, changes with change in the temperature is called the thermo- electric power of the two metals. That is, if E represent the electromotive force, / the temperature, and 6 the thermo- E — E electric power, we have — ^ '- = ^„ in the Hmit where t^ and /„ are indefinitely near one another. Hence if we lay off on the axis of abscissas (Fig. 74) an infinitesimal length /, — t„ and erect as ordinate the corresponding thermo-electric power B^, the area of the rectangle formed by the two lines will represent the electromotive force £, — E„ due to the change in tempera- ture. If, beginning at the point /,, we lay off the similar infini- tesimal length t^ — t^, and erect as ordinate the thermo-electric power 6^, we shall obtain another rectangle representing the electromotive force ^^ — E^. So for any temperature changes the total area of the figure bounded by the axis of tem- peratures, by the ordinates representing the fhermo-elec- tric powers at the temper- atures /„ and /,, and by the curve A A' passing through the summits of the rectangles F-'^- t- so obtained, will represent the electromotive force due to the heating of the junction from ?„ to Ix- 344 ELEMENTARY PHYSICS. [235 It was found by Tait and Le Roux that the thermo-electric power, referred to lead as a standard, of all metals but iron and nickel, is proportional to the rise in temperature. The curve A A' is therefore for those metals a straight line. For iron and nickel the curve is not straight. For another metal in comparison with lead, the line BB' , cor- responding to the line A A' for copper, may have a different direction. From what has been said about the possibility of arranging the metals in a thermo-electric series, it is evident that the thermo-electric power between copper and the other metal is the difference of their thermo-electric powers referred to lead, and that the electromotive force at the junction of the two metals, due to a rise of temperature from /?„ to t„ is repre- sented by the area of the figure contained by the two terminal ordinates and the two lines A A' and BB' . The thermo-elec- tric power is reckoned positive when the current sets from lead to copper across the hot junction. In the diagram the ther- mo-electric power A'B' is positive, and the electromotive force indicated by the area is from copper to the other metal across the hot junction. At the point where the lines A A' and BB' intersect, the thermo-electric power for the two metals vanishes. The temperature at which this occurs is called the neutral temperature and is designated by t„. When the temperature t:c lies on the other side of the neutral temperature from t„ the thermo-electric power becomes negative, and the electromotive force due to the rise in tempera- ature from ^„ to tx is negative. In Fig- 75 it is at once seen that A'B' is negative for t„ and that the area NA'B' is also negative- The electromotive force due to a rise of temperature from t^ increases until the temperature of the hot junction is t„, when it is a maximum, and then de- 235] THERMO-ELECTRIC RELATIONS OF THE CURRENT. 345 creases. When the area NA'B' becomes equal to the area ANB, the total electromotive force is zero; when NA'B' is greater than ANB, the electromotive force becomes negative, and the current is reversed. In case A A' and BB' are straight lines it is plain that the temperature t^,, at which this reversal occurs, will be such that the neutral temperature t^ is a mean between /„ and ^^. The same facts can be represented by a general formula. Thomson first pointed out that the fact of thermo-electrical in- version necessitates the view that the thermo-electric power at a junction is a function of the temperature of that junction. Avenarius embodied this idea in a formula, which his own re- searches, and those of Tait, show to be closely in agreement with experiment. Let us call the hot junction i and the cool junction 2, and set the electromotive force at each junction as a quadratic function of the absolute temperatures. We have E, = A-\-bt,-\- ct^ and E., = A^bt,-\- ct^, where A, b, and c are constants. The difference E^ — E^, or the electromotive force in the circuit, is E,-E, = b{i,-t:)+c{t,^~t:) ^{K-t:){b+c{t,+t:j) This equation may be put in the form used by Tait, if we write b = at„ and c = '-. We then have 2 E,-E, = a{t, - Q {t„ - KA + 1.)) (loo) The electromotive force in the circuit can become zero when either of these terms equals zero. It has been already stated that when t, = t,, or when both junctions are at the same temperature, there is no electromotive force in the circuit. 346 ELEMENTARY PHYSICS. [235 When ^(^, + O equals t„, or when the mean of the tempera- tures of the hot and cold junctions equals a certain temperature, constant for each pair of metals, there will be also no electro- motive force in the circuit. This temperature t^ is that which has already been called the neutral temperature. The formula also assigns the value to that temperature t^ at which, for fixed values of t„ and t„ the electromotive force in the circuit is a maximum. If we represent the difference between t^ and t^ by X, then t^-= tn ±. X. Using this value in the formula, we ob- tain £, — £, = - ((/„ — t^ — x'). This is manifestly a max- imum when X ^o. The electromotive force in a circuit is then^ according to the formula, a maximum when the temperature of one junction is the neutral temperature. The formula also shows that the thermo-electric power is a zero when /, = t„. We may set E^ — A -\- at„t^ — - ^/. Now if t^ take any small increment At^ , E^ has a corresponding in- crement /i^,. Hence we have E, + AE, = A-j- at J, — -- t," -f at„ At, — at. At, , if we neglect the term containing At,^. From this equation AE, we obtain . = at^ — at,, which in the limit, as At, becomes indefinitely small, is the thermo-electric power at the tempera- ture t,. It is positive for values of t, below t„ ; is zero for t, = t„, and negative for higher values of t,. That is, if we as- sume t, = t^ lower than t„ , and then gradually raise the tem- perature t, , the thermo-electric power at the heated junction is at first positive, but continually decreases in numerical value, until at t, = t„ it becomes zero. At that temperature, then, the metals are thermo-electrically neutral to one another, and a 236] THERMO-ELECTRIC RELA TIONS OF THE CURRENT. 347 small change in the temperature does not change the ' electro- motive force at the junction. 236. The Thomson Effect. — Thomson has shown that, in certain metals, there must be a reversible thermal effect when the current passes between two unequally heated parts of the same metal. Let us suppose a circuit of copper and iron, of which one junction is at the neutral temperature, and the other below the neutral temperature. The current then sets from copper to iron across the hot junction. In the hot junction there is no thermal effect produced, because the metals are at the neutral temperature. Across the cold junction the current is flowing from iron to copper, and hence is evolving heat. The current in the circuit can be made to do work, and since no other energy is imparted to the circuit this work must be done at the expense of the heat in the circuit. Since heat is not absorbed at either junction, it must be absorbed in the unequally heated parts of the circuit between the junctions. . To show this, Thomson used a conductor the ends of which were kept at constant temperatures in two coolers, while the central portion was heated. When a current was passed through this conductor, thermometers, placed in contact with exposed portions of the conductor between the heater and the coolers, indicated a rise of temperature different according as the cur- rent was passing from hot to cold or from cold to hot. The heat seems therefore to be carried along by the current, and the process has accordingly been called the electrical convection of heat. In copper the heat moves with the current, in iron against it. In another form of statement, it may be said that, in unequally heated copper, a current from hot to cold heats the metal, and from cold to hot cools it, while in iron the reverse thermal effects occur. The experiments of Le Roux show that the process of electrical convection of heat cannot be detected in lead. For this reason, lead is used as the standard metal in constructing the thermo-electric diagram. CHAPTER VII. LUMINOUS EFECTS OF THE CURRENT. 237. The Electric Arc. — If the terminals of an electric circuit in which is an electromotive force of forty or more volts be formed of carbon rods, a brilliant and permanent luminous arc will appear between the ends of the rods if they be touched together and then withdrawn a short distance from each other. The temperature of the arc is so high that the most refractory substances melt or are dissipated when placed in it. The carbon forming the positive terminal is hotter than the other. Both the carbons are gradually oxidized, the loss of the positive terminal being about twice as great as that of the nega- tive. The arc is, however, not due to combustion, since it can be formed in a vacuum. The current passing in the arc is, in ordinary cases, not greater than ten amperes, while the measurements of the resist- ance of the arc show that it is altogether too small to account for this current when the original electromotive force is taken into account. This fact has been explained by Edlund and others on the hypothesis that there is a counter electromotive force set up in the arc, which diminishes the effective electro- motive force of the circuit. The measurements of Lang show that this counter electromotive force in an arc formed between carbon points is about thirty-six volts, and in one formed be- tween metal points about twenty-three volts. 238. The Spark, Brush, and Glow Discharges. — When a conductor is charged to a high potential and brought near an- other conductor which is joined to ground, a spark or a series 238] LUMINOUS EFFECTS OF THE CURRENT. 349 of sparks will pass from one to the other. This phenomenon and others associated with it are most readily studied by the use of an electrical machine or an induction coil, between the electrodes of which a great difference of potential can be easily produced. If the spark be examined with the spectroscope, its spectrum is found to be characterized by lines which are due to the metals composing the electrodes, and to the medium between them. The passage of the spark through air or any dielectric is attended with a sharp report, and if the dielectric be solid, it is perforated or ruptured. If the electrodes be separated by a considerable distance, the path of the spark is usually a zigzag one. It is probable that this is due to irregularities in the dielectric, due to the presence of dust particles. With proper adjustment of the electrodes, the discharge may sometimes be made to take the form of a long brush spring- ing from the positive electrode, with a single trunk which branches and becomes invisible before reaching the negative electrode. Accompanying this is usually a number of small and irregular brushes starting from the negative electrode. Another form of discharge consists of a pale luminous glom covering part of the surface of one or both electrodes. If a small conducting body be interposed between the electrodes when the glow is established, a portion of the glow will be cut off, marking out a region on the electrode which is the projec- tion of the intervening conductor by the lines of electrical force. This phenomenon is called the electrical shadow. The difference of potential required to set up a spark be- tween two slightly convex metallic surfaces, separated by a stratum of air 0.125 centimetres thick, has been shown by Thomson to be about 5500 volts. The difference of potential which produces the sparks between the electrodes of an elec- trical machine, which are sometimes fifty or sixty centimetres long, must therefore be very great. The quantity of electricity 350 ELEMENTARY PHYSICS. [239 which passes during the discharge is, however, exceedingly small, on account of the great resistance of the medium through which the discharge takes place. Faraday showed that many of the phenomena of the dis- charge depend to some extent upon the medium in which it occurs. The differences in color and in the facility with which various forms of the discharge were set up in the gases upon which he experimented were especially noticeable. It was proved by Franklin that the lightning flash is an electrical discharge between a cloud and the earth or another cloud at a different electrical potential. The differences of potential to which such discharges are due must be enormous, and the heat developed by the discharge shows that the quantity of electricity which passes in it is not inconsiderable. Slowly moving fire-balls are sometimes seen, which last for a considerable time and disappear with a loud report and with all the attendant phenomena of a lightning discharge. It is not improbable that they are glow discharges which appear just before the difference of potential between the cloud and the earth becomes sufficiently great to give rise to a lightning flash. 239. The Electrical Discharge in Rarefied Gases. — If the air between the electrodes of an electrical machine be heated, it is found that the discharge takes place with greater facility and that the spark which can be obtained is longer than before. Similar phenomena appear if the air about the electrodes be rarefied by means of an air-pump. After the rarefaction has reached a certain point the discharge ceases to pass as a spark and becomes continuous. The arrangement in which this dis- charge is studied consists of a glass tube into which are sealed two platinum or, preferabJy, aluminium wires to serve as electrodes, and from which the air is removed to any required degree of exhaustion by an air-pump. Such an arrangement is usually called a vacuum-tube. 239] LUMINOUS EFFECTS OF THE CURRENT. 35 1 As the exhaustion proceeds there appears about the negative electrode ' in the tube a bright glow, separated from the electrode by a small non-luminous region. The body of the tube is filled with a faint rosy light, which in many cases breaks up into a succession of bright and dark layers transverse to the direction of the discharge. The discharge in this case is called the stratified discharge. A vacuum-tube in which the exhaus- tion is such that the phenomena are those here described is often called a Geissler tube. As the exhaustion is raised still higher, the rosy light in the tube fades out, the non-luminous . space around the negative electrode becomes very much greater, and the phenomena in the tube become exceedingly interesting. They were discovered and have been carefully studied by Crookes, and the vacuum-tubes in which they appear are hence called Crookes tubes. They may be most conveniently de- scribed by assuming that there is a special discharge from the negative electrode, which we will usually call the discharge. This view receives some support from the fact that the relations of current and resistance in the tube are such as to indicate a counter electromotive force at the negative electrode. The region occupied by the discharge from the negative electrode may be recognized by a faint blue light, which was not visible in the former condition of the tube. At every point on the wall of the tube to which this discharge extends occurs a brilliant phosphorescent glow, the color of which depends on the nature of the glass. The discharge seems to be indepen- dent of the position of the positive electrode, and to take place in nearly straight lines, which start normally from the negative electrode. If two negative electrodes be fixed in the tube, the discharge from one seems to be deflected by the other, and two discharges which meet at right angles seem to deflect one another. If the discharge from a flat electrode be made to fall upon a 352 ELEMENTARY PHYSICS. [239 body which can be moved, such as a glass film, or the vane of a light wheel, mechanical motions will be set up. If the negative electrode be made in the form of a spherical cup, and a strip of 'platinum foil be placed at its centre, the foil will become heated to redness when the discharge is set up. Two discharges in the same direction repel one another as if they were similarly electrified, and a magnet, brought near the outside of the tube, will deflect a discharge as if it were an electrical current. The explanation of these phenomena is probably that given by Crookes, and adopted by Spottiswoode and Moulton. It is assumed that they are due to the presence of the molecules of gas left in the tube after the exhaustion has been brought to an end. The mean free path of the molecules in the tube is much greater than that at ordinary densities, and they can accordingly move through long distances in the tube before their original motion is checked by collision with other mole- cules. It is assumed that the molecules of gas in the tube are attracted by the negative electrode, are charged negatively by it, and are then repelled. The phenomena which have been described are then due to their collision with other bodies or with the wall of the tube, or to their mutual electrical repul- sions and to the action between a moving quantity of electricity and a magnet. The experiments of Spottiswoode and Moulton, who showed that the same phenomena appeared at lower exhaustions, if the intensity of the discharge were increased, are in favor of this explanation. So is also the fact that the Crookes phenomena appear with a maximum intensity at a certain period during the exhaustion of the tube, while if the exhaustion be carried as far as possible, by the help of chemical means, they cease altogether and no current passes in the tube. The connection of these phenomena with the action of the radiome'^f r (§156) .is also at once apparent. SOUND. CHAPTER I. ORIGIN AND TRANSMISSION OF SOUND. 240. Definitions. — Acoustics has for its object the study of those phenomena which may be perceived by the ear. The sensations produced through the ear, and the causes that give rise to them, are called sounds. 241. Origin of Sound. — Sound is produced by vibratory movements in elastic bodies. The vibratory motion of bodies when producing sound is often evident to the eye. In some cases the sound seems to result from a continuous movement, but even in these cases the vibratory motion can be shown by means of an apparatus known as a manometric capsule, devised by Konig. It consists of a block A, Fig. 'j6, in which is a cavity covered by a membrane b. By means of a tube c illuminating gas is led into the cavity, and, passing out through the tube d, burns in a jet at e. It is evident that, if the membrane b be made to move suddenly inward or outward, it will compress or rarefy the gas in the capsule, and so cause the flow at the orifice and the height of the flame to increase or diminish. Any sound of sufificient intensity in the vicinity of the capsule causes an al- ternate lengthening and shortening of the flame, which, how- ever, occur too frequently to be directly observed. By mov- 23 354 ELEMENTARY PHYSICS. [243 ing the eyes while keeping the flame in view, or by observing the image of the flame in a mirror which is turned from side to side, while the flame is quiescent, it appears drawn out into a broad band of light, but when it is agitated by a sound near it, it appears serrate on its upper edge or even as a series of separate flames. This lengthening and shortening of the flame is evidence of a to-and-fro movement of the membrane, and hence of the sounding body that gave rise to the movement. If a hole be made in the side of an organ-pipe and the capsule made to cover it, the vibrations of the air-column within the pipe may be shown. By suitable devices the vibratory motion of all sounding bodies may be demonstrated. 242. Propagation of Sound. — The vibratory motion of a sounding body is ordinarily transmitted to the ear through the air. This is proved by placing a sounding body under the re- ceiver of an air-pump and exhausting the air. The sound be- comes fainter and fainter as the exhaustion proceeds, and finally becomes inaudible if the vacuum is good. Sound may, however, be transmitted by any elastic body. In order to study the character of the motion by which sound is propagated, let us suppose AB (Fig. "]"]) to represent .1 J / ^^ " '/' "" _ Ao a a d a a a S Fig. 77. a cylinder of some elastic substance, and suppose the layer of particles a to suffer a small displacement to the right. The effect of this displacement is not immediately to move forward the succeeding layers, but a approaches b, producing a conden- sation, and developing a force that soon moves b forward ; this in turn moves forward the next layer, and so the motion is transmitted from layer to layer through 'the cylinder with a 242] ORIGIN AND TRANSMISSION OF SOUND. 355 velocity that depends upon the elasticity (§ 76) of the sub- stance, and upon its density. This velocity is expressed by the formula V=a/-^, in which E represents the elasticity of the substance, and D its density (§268). Now, if we suppose the layer a, from any cause whatever, to execute regular vibra- tions, this movement will be transmitted to the succeeding layers with the velocity given by the formula, and, in time, each layer of particles in the cylinder will be executing vibra- tions similar to those oi a. If the vibrations of a be performed in the time t, the motion will be transmitted during one com- plete vibration of « to a distance s = vt, where v is the velocity of propagation, say to a', during two complete vibrations of a, to a distance 2s = 2vt, or to a", during three complete vibra- tions to a'", and so on. It is evident that the layer a' begins its first vibration at the instant that a begins its second vibra- tion, a" begins its first vibration at the instant that a' begins its second, and a its third vibration. The layer midway be- tween a and a' evidently begins its vibration just as a com- pletes the first half of its vibration, and therefore moves for- ward while a moves backward. This condition of things exist- ing in the cylinder constitutes a wave motion. While a moves forward, the portions near it are compressed. While it moves backward, they are dilated. Whatever the condition at a, the same condition will exist at the same instant at «', a" , etc. The distance aa' = a' a" is called a wave length; it is the dis- tance from any one particle to the next one of which the vibra- tions are in the same phase (§ 19). If the condition at a and a' be one of condensation, it is evident that at d, midway be- tween a and a' , there must be a rarefaction. In the wave length aa^ exist all intermediate conditions of condensation and rarefaction. These conditions must follow each other along the cylinder with the velocity of the transmitted motion, and they constitute z. progressive %vave moving with this veloc- 356 ELEMENTARY PHYSICS. [243 ity. If the vibratory motion with which a is endowed be com- municated by a sounding body, the wave is a sound-wave. \iy instead of a cylinder of the substance, we have an indefinite medium in the midst of which the sounding body is placed, the motion is transmitted in all directions as spherical waves about the sounding body as a centre. 243. Mode of Propagation of Wave Motion. — The mode of transmission of wave motions was first shown by Huyghens, and the principle involved is known as Huyghens principle. Let a (Fig. 78) be a centre from which sound originates. At the end of a certain' time it will have reached the surface mn. From the preceding discussion it is evident that each particle of the surface mn has a vibratory motion similar to that at a. Any one of those par- ticles would, if vibrating alone, be, like a, the centre of a system of spherical waves, and each of them must, therefore, be considered as a wave centre from which spherical waves jjjj proceed. Suppose such a wave to proceed from each one of them for the short dis- tance cd. Since the number of the element- ary spherical waves is very great, it is plain that they will coalesce to form the surface m'n' which determines a new position of the wave surface. In some cases the existence of these elementary waves need not be con- sidered, but there are many phenomena of wave motion which can only be studied by recognizing the fact that propagation always takes place as above described. 244. Graphic Representation of Wave Motion. — In order to study the movements of a body in which a wave motion exists, especially when two or more systems of waves exist in the same body, it is convenient to represent the movement by a sinusoidal curve, as described in § 19. Fig. 78. 244] ORIGIN AND TRANSMISSION OF SOUND. 357 Suppose the layer a (Fig. ']']') to move with a simple har- monic motion of which the amplitude is a and the period T, and let time be reckoned from the instant that the particles pass the position of equilibrium in a positive direction. A sinusoidal curve may be constructed to represent either the displacements of the various layers from their positions of equilibrium, or the velocities with which they are severally moving at a given time. To construct the first curve let the several points along OX (Fig. 79) represent points of the body through which the wave \6 c/ d \|e 'v/ 9/ / \ V- .^K MXi.. /"' / f'' h/.y V ly..y ^ Fig. 79. is moving. Let Oj/=^ a he the amplitude of vibration of each particle. The displacement of the particle at O at any instant t after passmg its position of equilibrium is j/ = a cos l-^ — -J, since when t is reckoned from the position of equilibrium € = — . Hence f= a sin —jt. If v represent the velocity of propagation of the wave, the particle at the distance x from the origin will have a displacement equal to that of the particle at O at the instant t, at an instant later than t by the time taken for the wave to travel over the distance x, or - seconds. Hence V its displacement at the instant t will be the same as that which existed at O, - seconds earlier. But the displacement at O, - seconds earlier is V S58 ELEMENTARY PHYSICS. [244 ■ 2Tt \t V I = a sin 27t [y-Vt)- ('°'> The quantity 7/7' equals the distance through which the move- ment is transmitted during the time of one complete vibration of the particle at O. Putting this equal to A, we have finally y = asirY27c (f-?)- (-3) Suppose t — o, and give to x various values. The corresponding values of J/ will represent the displacement at that instant of the particle the distance of which from the origin is x. For x = 0^ yt=o. Forx = l\j/ = -- a. For jr = |-\, j = o. For jr = |A, y =^ a. For jr — /I, j^ = o, etc. Laying off these values of x on OX and erecting perpendiculars equal to the corresponding values of y, we have the curve Obcde .... The above expression for j/ may be put in the form ^ = « sin 2 '(^> Hence, if any finite value be assigned to t, we shall obtain for y the same values as were obtained above for ^ = o, if we in- tX crease each of the values of x by -^. For instance, if t equal 245] ORIGIN AND 7'RANSMISSION OF SOUND. 359 \ T, we have y = o ior x = ^\, y = — a ior x — ^-A, etc., and the curve becomes the dotted line b'c'd' .... The effect of in- creasing t is to displace the curve along OX in the direction of propagation of the wave. The formula for constructing the curve of velocities is derived in the same way as that for displacements. It is 2na It x\ >.= -^cos2;r^^-^j. (104) Fig. 8o shows the relation of the two curves. The upper is the curve of displacement, and the lower of velocity. Fig. 8o. 245. Composition of Wave Motions. — The composition of wave motions may be studied by the help of the curves ex- plained above. If two systems of waves coexist in the same body, the displacement of any particle at any instant will be the algebraic sum of the displacements due to the systems taken separately. If the curve of displacements be drawn for each system, the algebraic sum of the ordinates will give the ordi- nates of the curve representing the actual displacements. In 36o ELEMENTARY PHYSICS. [245 Fig. 81 the dotted line and the light full line represent respec- tively the displacements due to two wave systems of the same period and amplitude. The heavy line represents the actual displacement. In I the two systems are in the same phase ; in II the phases differ by J, and in III by \, of a period. If both wave systems move in the same direction, it is evident that the conditions of the body will be continuously shown by suppos- ing the heavy line to move in the same direction with the III f h ^ ?? \ :/ Z^-' /'"'" ■"~~-v^ \J Vy vj Vy Fig. 81. same velocity. The condition represented in III is of special in- terest. It shows that two wave systems may completely annul \ fv \ /: r /I f / \ w \1 y /J \ A Fig. 82. each other. Fig. 82 represents the resultant wave when the periods, and consequently the wave lengths, of the two systems 245] ORIGIN AND TRANSMISSION OF SOUND. 361 are as 1:2. It will be noticed longer a simple sinusoid. In the same way the resultant wave may be constructed for any number of wave systems having any relation of wave lengths, am- plitudes, and phases. A very im- portant case is that of two wave sys- tems of the same period moving in opposite directions with the same velocity. In this case the two sys- tems no longer maintain the same relative positions, and the resultant curve is not displaced along the axis, but continually changes form. In Fig. 83, let the full and dotted lines in I represent, at a given in- stant, the displacements due to the two waves respectively. The re- sultant is plainly the straight line ab, which indicates that at that instant there is no displacement of any particle. At an instant later by \ period, as shown in II, the wave represented by the full line has moved to the right \ wave length, while that represented by the dotted line has moved to the left the same distance. The heavy line indicates the corresponding displacements. In III, IV, V, etc., the conditions at instants \, ■f, \, etc., periods later are repre- sented. A comparison of these that the resultant curve is no VII IX 3^2 ELEMENTARY PHYSICS. [246 figures will show that the particles at c and d are always at rest, that the particles between c and d all move in the same direc- tion at the same time, and that particles on the opposite sides oic ox d are always moving in opposite directions. It follows that the resultant wave has no progressive motion. It is a stationary wave. Places where no motion occurs, such as c and d, are called nodes. The space between two nodes is an inter- node or ventral segment. The middle of a ventral segment, where the motion is greatest, is an anti-node. It will be seen later that all sounding bodies afford examples of stationary waves. 246. Reflection of Waves.— When a wave reaches the bounding surface between two media, one of three cases may occur: (i) The particles of the second medium may have the same facility for movement as those of the first. The condition at the boundary will then be the same as that at any point pre- viously traversed, and the wave will proceed as though the first medium were continuous. (2) The particles of th'e second medium may move with less facility than those of the first. Then the condensed portion of a wave which reaches the boundary becomes more condensed in consequence of the restricted forward movement of the bounding particles, and the rarefied portion becomes more rare- fied, because those particles are also restricted in their backward motion. The condensation and rarefaction are communicated backward from particle to particle of the first medium, and con- stitute a reflected wave. It will be seen that, when the con- densed portion of the wave, in which the particles have a for- ward movement, reaches the boundary, the effect is a greater condensation, that is, the same effect as would be produced by imparting a backward movement to the bounding particles if no wave previously existed. In the direct rarefied portion of the wave the movement of the particles is backward, and the effect, at the boundary, of a greater rarefaction is what would 247] ORIGIN AND TRANSMISSION OF SOUND. 363 be produced by a forward movement of those particles. The effect in this case is, therefore, to reverse the motion of the particles. It is called reflection with change of sign. (3) The particles of _the second medium may move more freely than those of the first. In this case, when a wave in the first medium reaches the boundary, the bounding particles, instead of stopping with a displacement such as they would reach in the interior of the medium, move to a greater distance, and this movement is communicated back from particle to par- ticle as a reflected wave in which the motion has the same sign as in the direct v^ave. It is reflection without change of .ngn. The two latter cases are extremely important in the study of the formation of stationary waves in sounding bodies. 247. Law of Reflection. — Let us suppose a system of spherical waves departing from the point C (Fig. 84). Let mn be the intersection of one of the waves with the plane of the paper. Let AB be the trace of a plane smooth surface perpen- dicular to the plane of the paper, upon which the waves impinge, mo shows the position which the wave of which mn is a part would have occupied had it not been intercepted by the surface. From the last section it appears that reflection will take place as the wave mno '°' ^'■' strikes the various points of AB. In § 243 it was seen that any point of a wave may be considered as the centre of a wave system, and we may therefore take n' , n" , etc., the points of intersection of the surface AB with the wave mn when it occupied the positions m'n', m"n", etc., as the centres of sys- tems of spherical waves, the resultant of which would be the 3^4 ELEMENTARY PHYSICS. [247 actual wave proceeding from AB. With n' as a centre describe a sphere tangent to mno at o. It is evident that this will repre- sent the elementary spherical wave of which the centre is n' when the main wave is at mn. Describe similar spheres with n" , n'", etc., as centres. The surface tip, which envelops and is tangent to all these spheres, represents the wave reflected from AB. If that part of the plane of the paper below AB be re- volved about AB as an axis until it concides with the paper above AB, so will coincide with sp, s'o' with s'p', etc., and hence no with np. But no is a circle with C as a centre ; np is, there- fore, a circle of which the centre is C, on a perpendicular to AB .through C, and as far below AB as C is above. When, therefore, a wave is reflected at a plane surface, the centres of the incident and reflected waves are on the same line perpen dicular to the reflecting surface, and at equal distances from the surface on opposite sides. CHAPTER II. SOUNDS AND MUSIC. COMPARISON OF SOUNDS. 248. Musical Tones and Noises. — The distinction be- tween the impressions produced by musical tones and by noises is familiar to all. Physically, a musical tone is a sound the vibrations of which are regular and periodic. A noise is a sound the vibrations of which are very irregular. It may result from a confusion of musical tones, and is not always devoid of musi- cal value. The sound produced by a block of wood dropped on the floor would not be called a musical tone, but if blocks of wood of proper shape and size be dropped' upon the floor in succession, they will give the tones of the musical scale. Musical tones may differ from one another in pitch, depend- ing upon the frequency of the vibrations ; in loudness, depending upon the amplitude of vibration ; and in quality, depending upon the manner in which the vibration is executed. In regard to pitch, tones are distinguished as high or low, acute or grave. In regard to loudness, they are distinguished as loud or soft. The quality of musical tones enables us to distinguish the tones of different instruments even when sounding the same notes. 249. Methods of Determining the Number of Vibra- tions of a Musical Tone. — That the pitch of a tone depends upon the frequency of vibrations may be simply shown by hold- ing the corner of a card against the teeth of a revolving wheel. With a very slow motion the card snaps from tooth to tooth, making a succession of distinct taps, which, when the revolutions 3^6 ELEMENTARY PHYSICS. [249 are sufficiently rapid, blend together and produce a continuous tone, the pitch of which rises and falls with the changes of speed, Savart made use of such a wheel to determine the number of vibrations corresponding to a tone of given pitch. After regu- lating the speed of rotation until the given pitch was reached, the number of revolutions per second was determined by a simple attachment ; this number multiplied by the number of teeth in the wheel gave the number of vibrations per second. The siren is an instrument for producing musical tones by puffs of air succeeding each other at short equal intervals. A circular disk having in it a series of equidistant holes arranged in a circle around its axis is supported so as to revolve parallel to and almost touching a metal plate in which is a similar series of holes. The plate forms one side of a small chamber, to which air is supplied from an organ bellows. If there be twenty holes in the disk, and if it be placed so that these holes correspond to those in the plate, air will escape through all of them. If the disk be turned through a small angle, the holes in the plate will be covered and the escape of air will cease. If the disk be turned still further, at one twentieth of a revolution from its first position, air will again escape, and if it rotate continuously, air will escape twenty times in a revolution. When the rota- tion is sufficiently rapid, a continuous tone is produced the pitch of which rises as the speed increases. The siren may be used exactly as the toothed wheel to determine the number of vibra- tions corresponding to any tone. By drilling the holes in the plate obliquely forward in the direction of rotation, and those in the disk obliquely backward, the escaping air will cause the disk to rotate, and the speed of rotation may be controlled by controlling the pressure of air in the chamber. Sirens are sometimes made with several series of holes in the disk. These serve not only the purposes described above. 25°] SOUA'DS AND MUSIC. 367 but also to compare tones of which the vibration numbers have certain ratios. The number of vibrations of a sounding body may sometimes be de- termined by attaching to it a Hght stylus which is made to trace a curve upon a smoked glass or cylinder. In- stead of attaching the stylus to the sounding body directly, which is prac- ticable only in a few cases, it may be at- tached to a membrane which is caused to vibrate by the sound-waves which the body generates. A membrane re- produces very faithfully all the charac- teristics of the sound-waves, and the curve traced by the stylus attached to it gives information, therefore, ,not only in regard to the number of vibra- tions, but to some extent in regard to their amplitude and form. PHYSICAL THEORY OF MUSIC. 250. Concord and Discord. — When two or more tones are sounded together, if the effect be pleasing there is said to be concord; if harsh, discord. To understand the cause of discord, suppose two tones of nearly the same pitch to be sounded together. The re- sultant curve, constructed as in § 245, is like those in Fig. 85, which repre- sent the resultants when the periods of the components have the ratio 81 : Fig. 85. 80 and when they have 368 ELEMENTARY PHYSICS. [251 the ratio 16:15. The figure indicates, what experiment veri- fies, that the resultant sound suffers periodic variations in in- tensity. When these variations occur at such intervals as to be readily distinguished, they are called beats. These beats occur more and more frequently as the numbers expressing the ratio of the vibrations reduced to its lowest terms become smaller, until they are no longer distinguishable as separate beats, but appear as an unpleasant roughness in the sound. If the terms of the ratio become smaller still, the roughness diminishes, and when the ratio is f.the effect is no longer unpleasant. This, and ratios expressed by smaller numbers, as 1^, 1^, f , f, f, represent concordant combinations. 251. Major and Minor Triads. — Three tones of which the vibration numbers are as 4 : 5 : 6 form a concordant combination called the major triad. The ratio 10: 12: 15 represents another concordant combination called the minor triad. Fig. 86 shows the resultant curves for the two triads. 4:5:6 7\ rv^^-vA /^/--^^^ ^^^"^^"\A|/v-\^v\K/^>~v\ /v f^ Fig. 86. 252. Intervals. — The intervalhttwetr^ two tones is expressed by the ratio of their vibration numbers, using the larger as the numerator. Certain intervals have received names derived from the relative positions of the two tones in the musical scale, as described below. The interval f is called an octave; f, 2i fifth; f, 3. fourth; |, a major third; f, a 7ninor third. 253. Musical Scales. — A musical scale is a series of tones which have been chosen to meet the demands of musical com- position. There are at present two principal scales in use, each 253] SOUNDS AND MUSIC. 3^9 consisting of seven notes, with their octaves, chosen with refer- ence to their fitness to produce pleasing effects when used in combination. In one, called the major scale, the first, third, and fifth, the fourth, sixth, and eighth, and the fifth, seventh, and ninth tones, form major triads. In the other, called the minor scale, the same tones form minor triads. From this it is easy to deduce the following relations: Tone Number Letter Name do or ut re Number of vibrations Intervals from tone to tone. . MAJOR SCALE. I' 2' 123 4 5 6 7 8 9' C D E F G A B a D' J or ut re mi fa sol la si ut re m |m fm Jm fm |m 2m |m 1 ^ ^ f ¥ • f il __- MINOR SCALE. 123 4 5 6 7 8 q ABC D E F G A' B' la si ut re mi fa sol la si m fm fm fm fm |m |m 2m Jm % \% ^ 5 a 1 ¥ Tone Number Letter Name Number of vibrations , Intervals from tone to tone. . The derivation of the names of the intervals will now be apparent. For example, an interval of a third is the interval between any tone of the scale and the third one from it, count- ing the first as i. If we consider the intervals from tone to tone, it is seen that the pitch does not rise by equal steps, but that there are three difTerent intervals, f, i^, and ^f. The first two are usually considered the same, and are called whole tones. The third is a half-tone or semitone. It is desirable to be able to use any tone of a musical in- strument as the first tone or tonic of a musical scale. To per- mit this, when the tones of the instrument are fixed, it is plain that extra tones, other than those of the simple scale, must be provided in order that the proper sequence of intervals may be maintained. Suppose the tonic to be transposed from C to D. 24 370 ELEMENTARY PHYSICS. [253 The semitones should now come, in the major scale, between F and G, and C and D', instead of between E and F, and B and C. To accomplish this, a tone must be substituted for F and another for C. These are called F sharp and C sharp respec- tively, and their vibration numbers are determined by multiply- ing the vibration numbers of the tones which they replace by f f . The introduction of five such extra tones, making twelve in the octave, enables us to preserve the proper sequence of whole tones and semitones, whatever tone is taken as the tonic. But if we consider that the whole tones are not all the same, and propose to preserve exactly all the intervals of the transposed scale, the problem becomes much more difficult, and can only be solved at the expense of too great complication in the in- strument. Instead of attempting it, a system of tuning, called temperament, is used by which the twelve tones referred to above are made to serve for the several scales, so that while none are perfect, the imperfections are nowhere marked. The system of temperament usually employed, or at least aimed at, called the even temperam.ent, divides the octave into twelve equal semi- tones, and each interval is therefore the twelfth root of 2. With instruments in which the tones are not fixed, like the violin for instance, the skilful performer may give them their exact value. For convenience in the practice of music and in the con- struction of musical instruments, a standard pitch must be adopted. This pitch is usually determined by assigning a fixed vibration number to the tone above the middle C of the piano, represented by the letter A'. This number is about 440, but varies somewhat in different countries and at different times. In the instruments made by Konig for scientific purposes, the vibration number 256 is assigned to the middle C. This has the advantage that the vibration numbers of the successive octaves of this tone are powers of 2. CHAPTER III. VIBRATIONS OF SOUNDING BODIES. 254. General Considerations. — The principles developed in § 246 apply directly in the study of the vibrations of sound- ing bodies. When any part of a body which is capable of act- ing as a sounding body is set in vibration, a wave is propagated through it to its boundaries, and is there reflected. The re- flected wave, travelling away from the boundary, in conjunction with the direct wave going toward it, produces a stationary wave. These stationary waves are characteristic of the motion of all sounding bodies. Fixed points of a body often determine the position of nodes, and in all cases the length of the wave must have some relation to the dimensions of the body. 255. Organ Pipes. — A column of air, enclosed in a tube of suitable dimensions, may be made to vibrate and become a sounding body. Let us suppose a tube closed at one end and •open at the other. If the air particles at the open end be sud- denly moved inward, a pulse travels to the closed end, and is there reflected with change of sign (§ 246). It returns to the open end and is again reflected, this time without change of sign, because there is greater freedom of motion without than within the tube. As it starts again toward the closed end, the air particles that compose it move outward instead of inward. If they now receive an independent impulse outward, the two effects are added and a greater disturbance results. So, by properly timing small impulses at the open end of the tube, the air in it may be made to vibrate strongly. 372 ELEMENTARY PHYSICS. [255 If a continuous vibration be maintained at the open end of the tube, waves follow each other up the tube, are reflected with change of sign at the closed end, and returning, are reflected without change of sign at the open end. Any given wave a, therefore, starts up the tube the second time with its phase changed by half a period. The direct wave that starts up the tube at the same instant must be in the same phase as the reflected wave, and it therefore differs in phase half a period from the direct wave a. In other words, any wave returning to the mouth-piece must find the vibrations there opposite in phase to those which existed when it left. This is possible only when the vibrating body makes, during the time the wave is going up the tube and back, i, 3, 5, or some odd number of half-vibrations. By constructing the curves representing the stationary wave resulting from the superposition of the two systems of vibrations, it will be seen that there is always a node at the closed end of the tube and an anti-node at the mouth. When there is i half-vibration while the wave travels up and back, the length of the tube is \ the wave length ; when there are 3 half-vibrations in the same time, the length of the tube is f the wave length, and there is a node at one third the length of the tube from the mouth. If the tube be open at both ends, reflection without change of sign takes place in both cases, and the reflected wave starts up the tube the second time in the same phase as at first. The vibrations must therefore be so timed that i, 2, 3, 4, or some whole number of complete vibrations are performed while the wave travels up the tube and back. A construction of the curve representing the stationary wave in this case will show, for the smallest number of vibrations, a node in the middle of the tube and an anti-node at each end. The length of the tube is therefore \ the wave length for this rate of vibration. The vibration numbers of the several tones produced by an open tube are evidently in the ratio of the series of whole num- 256] VIBRATIONS OF SOUNDING BODIES. 373 bers I, 2, 3, 4, etc., while for the closed tube only those tones can be produced of which the vibration numbers are in the ratio of the series of odd numbers i, 3, 5, etc. It is evident also that the lowest tone of the closed tube is an octave lower than that of the open tube. This lowest tone of the tube is called the fundamental, and the others are called overtones, or harmonics. These simple relations between the length of the tube and length of the wave are only realized when the tubes are so narrow that the air particles lying in a plane cross-section are all actuated by the same movement. This is never the case at the open end of the tube, and the distance from this end to the first node is, there- fore, always less than a quarter wave length. 256. Modes of Exciting Vibrations in Tubes. — If a tun- ing fork be held in front of the open mouth of a tube of proper length, the sound of the fork is strongly reinforced by the vibration of the air in the tube. If we merely blow across the open end of a tube, the agitation of the air may, by the reaction of the returning reflected pulses, be made to assume a regular vi- bration of the proper rate and the column made to sound. In organ pipes a mouthpiece of the form shown in Fig. 87 is often em- ployed. The thin sheet of air projected against the thin edge is thrown into vibration. Those elements of this vi- bration which correspond in frequency with the pitch of the pipe are strongly reinforced by the action of the station- ary wave set up in the pipe, and hence the tone proper to the pipe is produced. Sometimes reeds are used, as shown in Fig. 87a. The air es- caping from the chamber a through the passage c causes the reed r to vibrate. This alternately closes and opens the passage, Fig. 87. Fig. 87a. 374 ELEMENTARY PHYSICS. [257 and so throws into vibration the air in the pipe. If the reed be stiff, and have a determined period of vibration of its own, it must be tuned to suit the period of the air column which it is intended to set in vibration. If the reed be very flexible it will accommodate itself to the rate of vibration of the air col- umn, and may then serve to produce various tones, as in the clarionet. In instruments like the cornet and bugle, the lips of the player act as a reed, and the player may at will produce many of the different overtones. In that way melodies may be played without the use of keys or other devices for changing the length of the air column. Vibrations may be excited in a tube by placing a gas flame at the proper point in it. The flame thus employed is called a singing flame. The organ of the voice is a kind of reed pipe in which little folds of membrane, called vocal chords, serve as reeds which can be tuned to different pitches by muscular effort, and the cavity of the mouth and larynx serves as a pipe in which the mass of air may also be changed at will, in form and volume. 257. Longitudinal Vibrations of Rods. — A rod free at both ends vibrates as the column of air in an open tube. Any dis- placement produced at one end is transmitted with the velocity of sound in the material to the other end, is there reflected with- out change of sign and returns to the starting point to be re- flected again exactly as in the open tube. The fundamental tone corresponds to a stationary wave having a node at the cen- tre of the rod. 258. Longitudinal Vibrations of Cords.— Cords fixed at both ends may be made to vibrate by rubbing them lengthwise. Here reflection with change of sign takes place at both ends,, which brings the wave as it leaves the starting point the second time to the same phase as when it first left it, and there must be, therefore, as in the open tube, i, 2, 3, 4, etc., vibrations 259] VIBRATIONS OF SOUNDING BODIES. 375 while the wave travels twice the length of the cord. The veloc- ity of transmission of a longitudinal displacement in a wire de- pends upon the elasticity and density of the material only. The velocity and the rate of vibration are, therefore, nearly independent of the stretching force. 259. Transverse Vibrations of Cords.— If a transverse vibration be given to a point upon a wire fastened at both ends, everything relating to the reflection of the wave motion and the formation of stationary waves is the same as for longitudinal displacements. The velocity of transmission, and consequently the frequency of the vibrations, are, however, very different. If the cord offer no resistance to flexure, the force tending to restore it to its position of equilibrium is entirely due to the stretching force. This, therefore, takes the place of the elas- ticity in the formula for transmission of longitudinal vibrations (§ 268). The mass of the cord per unit length takes the place of the density in the same formula. Thus we have the formula for the velocity y m where P is the stretching force and m the mass per unit length. The greatest time of vibration, the time required for the wave to travel twice the length of the string, is 7-=^ = .^^ (.05) and the number of vibrations per second is ^='t=TlsI-- (^°^) 376 ELEMENTARY PHYSICS. [260 Hence, the number of vibrations of a string is inversely as the length, directly as the square root of the tension, and inversely as the square root of the mass per unit length. These laws are readily verified by experiment. 260. Transverse Vibrations of Rods, Plates, etc. — The vibrations of rods, plates, and bells are all cases of stationary waves resulting from systems of waves travelling in opposite di- rections. Subdivision into segments occurs, but, in these cases, the relations of the various overtones are not so simple as in the cases before considered. For a rod fixed at one end, sound- ing its fundamental tone, there is a node at the fixed end only. For the first overtone there is a second node near the free end of the rod, and the number of vibrations is a little more than six times the number for the fundamental. A rod free at both ends has two nodes when sounding its fundamental, as shown in Fig. 88. The distance of these nodes from the ends is about f the length of --'" ' '- the rod. If the rod be bent, the nodes Fig. 88. approach the centre until, when it has assumed the u form' like a tuning-fork, the two nodes are very near the centre. This will be understood from Fig. 89. / Fig. 89. The nodal lines on plates may be shown by fixing the plate in a horizontal position and sprinkling sand over its surface. When the plate is made to vibrate, the sand gathers at the nodes 26i] VIBRATIONS OF SOUNDING BODIES. 177 and 'marks their position. The figures thus formed are known as Chladni's figures. 261. Communication of Vibrations. — If several pendulums be suspended from the same support, and one of them be made to vibrate, any others which have the same period of vibration will soon be found in motion, while those which have a different period \vill show no signs of disturbance. The vibration of the first pendulum produces a slight movement of the support which is communicated alike to all the other pendulums. Each move- ment may be considered as a slight impulse, which imparts to each pendulum a very small vibratory motion. For those pen- dulums having the same period as the one in vibration, these impulses come just in time to increase the motion already pro- duced, and so, after a time, produce a sensible motion ; while for those pendulums having a different period, the vibration at first imparted will not keep time with the impulses, and these will therefore as often tend to destroy as to increase the motion. It is important to note that the pendulum imparting the motion loses all it imparts. This is not only true of pendulums, but of all vibrating bodies. Two strings stretched from the same sup- port and tuned to unison will both vibrate when either one is caused to sound. A tuning-fork suitably mounted on a sound- ing-box will communicate its vibrations to another tuned to exact unison even when they are thirty or forty feet apart and only air intervenes. In this case it is the sound-wave generated by the first fork which excites the second fork, and in so doing the wave loses a part of its own motion, so that beyond the second fork, on the line joining the two, the sound will be less intense than at the same distance in other directions. Air columns of suitable dimensions will vibrate in sympathy with other sounding bodies. If water be gradually poured into a deep jar, over the mouth of which is a vibrating tuning-fork, there will be found in general a certain length of the air column for which the tone of the fork is strongly reinforced. From 378 ELEMENTARY PHYSICS. [261 the theory of organ pipes, it is plain that this length corresponds approximately to a quarter wave length for that tone. In this case, also, when the strongest reinforcement occurs, the sound of the fork will rapidly die away. The sounding-boxes on which the tuning-forks made by Konig are mounted are of such dimensions that the enclosed body of air will vibrate in unison with the fork, but they are purposely made not quite of the dimensions for the best resonance, in order that the forks may not too quickly be brought to rest. A membrane or a disk, fastened by its edges, may respond to and reproduce more or less faithfully a great variety of sounds. Hence such disks, or diaphragms, are used in instruments like the telephone and phonograph, designed to reproduce the sounds of the voice. The fhonograpk consists of a mouthpiece and disk similar to that used in the telephone, but the disk has fastened to its centre, on the side opposite the mouthpiece, a short stiff stylus, which serves to record the vibrations of the disk upon a sheet of tinfoil or wax moved along beneath it. The foil is wrapped upon a cylinder having a spiral groove on its surface, and upon its axle a screw thread of the same pitch works in a fixed nut so that, when the cylinder revolves, it has also an endwise motion, such that a fixed point would follow the spiral groove on its surface. To use the instrument, the disk is placed in position with the stylus attached adjusted to enter the groove in the cylinder and slightly indenting the foil. The cylinder is revolved while sounds are produced in front of the disk. The disk vibrates, causing the stylus to indent the foil more or less deeply, so leaving a permanent record. If now the cylinder be turned back to the starting-point and then turned forward, causing the stylus to go over again the same path, the indentations previously made in the foil now cause the stylus, and consequently the disk, to vibrate and reproduce the sound that produced the record. The sounding-boards of the various stringed instruments are 26l] VIBKATIONS OF SOUNDING BODIES. 379 in effect thin disks, and afford examples of the reinforcement of vibrations of widely different pitch and quality by the same body. The strings of an instrument are of themselves insuffi- cient to communicate to the air their vibrations, and it is only through the sounding-board that the vibrations of the string can give rise to audible sounds. The quality of stringed instru- ments, therefore, depends largely upon the character of the sounding-board. The tympanum of the ear furnishes another example of the facility with which membranes respond to a great variety of sounds. CHAPTER IV. ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 262. Quality. — As has already been stated, the tones of di ■ ferent instruments, although of the same pitch and intensity- are distinguished by their quality. It was also stated that tho quality of a tone depends upon the manner in which the vibra- tion is executed. The meaning of this statement can best be understood by considering the curves which represent the Fig. go. vibrations. In Fig. 90 are given several forms of vibration curves of the same period. Every continuous musical tone must result ^om a periodic vibration, that is, a vibration which, however complicated it may be, repeats itself at least as frequently as do the vibrations of the lowest audible tone. According to Fourier's theorem (§ 19), every periodic vibration is resolvable into simple har- monic vibrations having commensurable periods. It has been 262] ANALYSIS OF SOLWDS AND SOUND SENSATIONS. 381 seen that all sounding bodies may subdivide into segments, and produce a series of tones of which the vibration periods gener- ally bear a simple relation to one another. These may be pro- duced simultaneously by the same body, and so give rise to complex tones the character of which will vary with the nature and intensity of the simple tones produced. It has been held that the quality of a complex tone is not affected by change of phase of the component simple tones relative to each other. Some experiments by Konig seem to indicate, however, that the quality does change when there is merely change of phase. Fig. 91. In Fig. 91 are shown three curves, each representing a fun- damental accomp'anied by the harmonics up to the tenth. The Fig. 92. curves differ only in the different phases of the components relative to each other. 382 ELEMENTARY PHYS/CS. [263 Fig. 92 shows similar curves produced by a fundamental accompanied by the odd harmonics. 263. Resonators for the Study of Complex Tones. — An apparatus devised by Helmholtz serves to analyze complex tones and indicate the simple tones of which they are composed. It consists of a series of hollow spheres or cylinders, called resonators, which are tuned to certain tones. If a tube lead from the resonator to the ear and a sound be produced, one of Fig. 93. the components of which is the tone to which the resonator is tuned, the mass of air in it will be set in vibration and that tone will be clearly heard ; or, if the resonator be connected by a rubber tube to a manometric capsule (§ 241), the gas flame con- nected with the capsule will be disturbed whenever the tone to 265] ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 383 which the resonator is tuned is produced in the vicinity, either by itself or as a component of a complex tone. By trying the resonators of a series, one after another, the several compo- nents of a complex tone may be detected and its composition demonstrated. 264. Vowel Sounds. — Helmholtz has shown that the dif- ferences between the vowel sounds are only differences of quality. That the vowel sounds correspond to distinct forms of vibration is well shown by means of the manometric flame. By connecting a mouthpiece to the rear of the capsule, and singing into it the different vowel sounds, the flame images assume distinct forms for each. Some of these forms are shown in Fig. 93. 265. Optical Method of Studying Vibrations. — The vi- bratory motion of sounding bodies may sometimes be studied gs=^ ^rr^- jn Fig. 94. to advantage by observing the lines traced by luminous points upon the vibrating body or by observing the movement of a beam of light reflected from a mirror attached to the body. Young studied the vibrations of strings by placing the string where a thin sheet of light would fall across it, so as to illuminate a single point. When the string was caused to vibrate, the path of the point appeared as a continuous line, in consequence of the persistence of vision. Some of the results which he obtained are given in Fig. 94, taken from Tyndall on Sound. 384 ELEMENTARY PHYSICS. [265 The most interesting application of this method was made by Lissajous to illustrate the composition of vibratory motions at right angles to each other. If a beam of light be reflected to a screen from a mirror attached to a tuning-fork, when the tuning-fork vibrates the spot on the screen will describe a sim- ple harmonic motion and will appear as a straight line of light. If the beam, instead of being reflected to a screen, fall upon a mirror attached to a second fork, mounted so as to vibrate in Fig. 95. a plane at right angles to the first, the spot of light will, when both forks vibrate, be actuated by two simple harmonic mo- tions at right angles to each other and the resultant path will appear as a curve more or less complicated, depending upon the relation of the two forks to each other as to both period and phase (§ 19). Fig. 95 shows some of the simpler forms of these curves. The figures of the upper line are those produced by two forks in unison ; those of the second line by two forks of which the vibration numbers are as 2 : i ; those of the lower line by two forks of which the vibration numbers are as 3 : 2. CHAPTER V. EFFECTS OF THE COEXISTENCE OF SOUNDS. 266. Beats. — It has already been explained (§ 250) that^ when two tones of nearly the same pitch are sounded together, variations of intensity, called beats, are heard. Helmholtz's theory of the perception of beats was, that, of the little fibres in the ear which are tuned so as to vibrate with the various tones, those which are nearly in unison affect one another so as to increase and diminish one another's motions, and hence that no beats could be perceived unless the tones were nearly in unison. Beats are, however, heard when a tone and its octave are not quite in tune, and, in general, a tone making n vibra- tions produces 711 beats when sounded with a tone making 2n ± m, in ± in, etc., vibrations. This was explained in ac- cordance with Helmholtz's theory, by assuming that one of the harmonics of the lower tone, which is nearly in unison with n Fig. 96. the upper, causes the beats, or, in cases where this is inad- missible, that they are caused by the lower tone in conjunc- tion with a resultant tone (§ 267). An exhaustive research by Konig, however, has demonstrated that beats are perceived 25 386 ELEMENTARY PHYSICS. [266 when neither of the above suppositions is admissible. Figs. 96 and 97 show that the resultant vibrations are affected by- changes of amplitude similar to, though less in extent than, the changes which occur when the tones are nearly in unison. In Fig. 96, I represents a flame image obtained when two tones making n and n ± m vibrations respectively, are produced to- 15:16 iiiii^^ III ifi) 15:46 IV Fig. 97. gather, and II represents the image when the number of vibrations are n and 2« ± m. Fig. 97 shows traces obtained mechanically. In I the numbers of the component vibrations were n and n -\- m, m II and III n and 2« ±- m, and in IV n and 3« + m. In all these cases a variation of amplitude occurs during the same intervals, and it seems reasonable to suppose that those variations of amplitude should cause variations in intensity in the sound perceived. 267] EFFECTS OF THE COEXISTENCE OF SOUNDS. 387 Cross has shown that the beating of two tones is perfectly well perceived when the tones themselves are heard separately by the two ears ; one tone being heard directly by one ear, while the other, produced in a distant room, is heard by the other ear by means of a telephone. Beats are also perceived when tones are produced at a distance from each other and from the listener, who hears them by means of separate telephones through separate lines. In this case there is no possibility of the formation of a resultant wave, or of any combination of the two sounds in the ear. 267. Resultant Tones. — Resultant tones are produced by combinations of two tones. Those most generally recognized have a vibration number equal to the sum or difference of the vibration numbers of their primaries. For instance, ut,, making 2048 vibrations, and re^, making 2304 vibrations, when sounded together give utj, making 256 vibrations. These tones are ■only heard well when the primaries are loud, and it requires an effort of the attention' and some experience to hear them at all. Summation tones are more difficult to recognize than differ- ■ence tones, nevertheless they have an influence in determining the general effect produced when musical tones are sounded together. Other resultant tones may be heard under favorable ■conditions. As described above, two tones making n and n-\-in vibrations respectively, when m. is considerably less than n, give a resultant tone making m vibrations, but a tone making n vibrations in combination with one making 2« -(- m, 3« -(- m, or xn-\^ tn vibrations, gives the same resultant. This has sometimes been explained by assuming that intermediate re- sultants are produced, which, with one of the primaries, pro- duce resultants of a higher order. In the case of the two tones making n and in-\- m vibrations, for instance, the first differ- ence tone would make 2n -\- m vibrations. This tone and the •one making n vibrations would give the tone making n-{- m vibrations ; this tone, in turn, and the one making n vibrations 388 ELEMENTARY PHYSICS. [267 would give the tone making m vibrations. This last tone is- the one which is heard most plainly, and it seems difficult to admit that it can be the resultant of tones which are only heard very feebly, and often not at all. In Fig. 97 are represented the resultant curves produced in several of these cases. The first curve corresponds to two tones of which the vibration numbers are as 15: 16. It shows the periodic increase and de- crease in amplitude, occurring once every 15 vibrations, which, if not too frequent, give rise to beats (§250). If the pitch of the primaries be raised, preserving the relation 15:16, the beats become more frequent, and finally a distinct tone is heard, the vibration number of which corresponds to the num- ber of beats that should exist. It vi^as for a long time consid- ered that the resultant tone was merely the rapid recurrence of beats. Helmholtz has shown by a mathematical investigation that a distinct wave making m vibrations will result from the coexistence of two waves making n and n -\- m vibrations, and he believes that mere alternations of intensity, such as consti- tute beats, occurring ever so rapidly cannot produce a tone. In II and III (Fig. 97) are the curves resulting from two tones, the intervals between which are respectively IS:29(= 2 X 15 — i) and 15 :3i(= 2 X IS + i)- Running through these may be seen a periodic change corre- sponding exactly in period to that shown in I. The same is true also of the curve in IV, which is the resultant for two tones the interval between which is 15 :46(= 3X15 + i). In all these cases, as has been already said (§ 266), if the pitch of the components be not too high^ one beat is heard for every 1 5 vibrations of the lower component. Fig. 96 shows the flame images for the intervals n:n-\-m and n:2n -\- in. The vary- ing amplitudes resulting in m beats per second are very evident in both. In all these cases, also, as the pitch of the compo- 267] EFFECTS OF THE COEXISTENCE OF SOUNDS. 389 nents rises the beats become more frequent, and finally a re- sultant tone is heard, having, as already stated, one vibration for every 15 vibrations of the lower component. In Fig. 98 IdSJiS Fig, 98. are shown two resultant curves having three components of which the vibration numbers are as 1:15:29. In I the three components all start in the same phase. In II, when 15 and 29 are in the same phase, I is in the opposite phase. CHAPTER VI. VELOCITY OF SOUND., 268. Theoretical Velocity. — The disturbance of the parts of any elastic medium which is propagating sound is assumed, ' in theoretical discussions, to take place in the line of direction of the propagation of the sound, and to be such that the type of the disturbance remains unaltered during its propagation. The velocity of propagation of such a disturbance may be in- vestigated by the following method, due to Rankine. Let us consider, as in § 242, a portion of the elastic medium in the form of an indefinitely long cylinder. If a disturbance be set up at any cross-section of this cylinder (Eig. 99), which consists of a displacement of the matter in that cross-section in the direction of the axis of the cylinder, it will, by hy- pothesis, be propagated in the direction of the axis with a con- stant velocity V, which is to be determined. If we consider any cross-section of the cylinder which is traversed by the dis- turbance, the matter which passes through it at any instant will Fig. 99. have a velocity which may vary from zero to the maximum velocity of the vibrating matter, either positively when this ve- locity is in the direction of propagation of the disturbance, or negatively, when it is opposite to it. If we now conceive an imaginary cross-section A to move 268] VELOCITY OF SOUND. 39^ along the cylinder with the disturbance with the velocity V, the velocity of the particles in it at any instant will be always the same. Let us call this velocity Va- The velocity of the cross-section relative to the moving particles in it is then V — Va- If we represent by «?„ the density of the medium at the cross-section through which the velocity of the particles is I'a, which is the same for all positions of the moving cross- section, and if we assume that the area of the cross-section is unity, then the quantity of matter M which passes through the moving cross-section in unit time is If we conceive any other cross-section B to be moving with the disturbance in a similar manner, the same quantity of mat- ter M will pass through it in unit time, since the two cross- sections move with the same velocity and the density of the matter between them remains the same. Hence we have M= di {V — Vi), where di and v^ represent the qualntities at the cross-section B corresponding to those at the cross-section A represented by da and Va- Hence da{V — v^) = di,{V ~ Vi). Since this equation is true whatever be the distance between the cross-sections, it is true for that position of the cross-section B for which v^ = o, and for which dj = D, the density of the medium in its undisturbed condition. Hence we have M^ nV, da{V- Va) = DV, and Vada — D If the disturbance be small, the expression on the right is approximately the condensation per unit volume of the me- dium at the cross-section A, and the equation shows that the latio of the velocity of the matter passing through the cross- 392 ELEMENTARY PHYSICS. [269 section A to the velocity of propagation of the disturbance is equal to the condensation at that cross-section. Now, to eliminate the unknown expressions Va and da, we must find a new equation involving them. A quantity of mat- ter M enters the region between the two moving cross-sections with the velocity v^, and an equal quantity leaves the region with the velocity V/,. The difference of the momenta of the entering and outgoing quantities is M(va — v^. This differ- ence can only be due to the different pressures /« and/^ on the moving cross-sections, since the interactions of the portion of matter between those cross-sections cannot change the momen- tum of that portion. Hence we have Mi^Va -vt)^Pa-p6. If we for convenience assume vi, = o, we have pi = P, the pressure in the medium in its undisturbed condition. If we pa — P further substitute for v^ its value, we obtain MVz= da -r f;- da —JJ If the cjianges in pressure and density be small, the quantity Pa — P da^ f. equals E, the modulus of elasticity of the medium. If we further substitute for M its value VD, we obtain finally ^' = -D - ^=\/I- (^°«) 269, Velocity of Sound in Air. — In air at constant tem- perature the elasticity is numerically equal to the pressure (§ yj). The compressions and rarefactions in a sound-wave occur so rapidly that during the passage of a wave there is no time for the transfer of heat, and the elasticity to be considered, therefore, is the elasticity when no heat enters or escapes (§ 158). 269] VELOCITY OF SOUND. 393 If the ratio of the two elasticities be represented by y we have for the elasticity when no heat enters or escapes E = yP, and the velocity of a sound-wave in air at zero temperature is given by ^ D The coefificient y equals 1.41. P is the pressure exerted by a column of mercury 76 centimetres high and with a cross-section of one square centimetre, or 76 X 13-59 X 981 = 1013373 dynes per square centimetre. D equals 0.001293 grams at 0°, hence w — a/ I-4I X 1013373 v=y -JL_^i ii^ = 33240, 0.001293 -^-^ ^ or 332.4 metres per second. Since the density of air changes with the temperature, the velocity of sound must also change, li dt represent the den- sity at temperature t, and d^ the density at zero. 4 = i + kf from § 128. The formula for velocity then becomes ^=^■+4 This formula shows that the velocity at any temperature is the velocity at 0° multiplied by the square root of the factor of ex- pansion. 394 ELEMENTARY PHYSICS. [270 270. Measurements of the Velocity of Sound. — The ve- locity of sound in air has been measured by observing the time required for the report of a gun to travel to a known distance. One of the best determinations was that made in Holland in 1822. Guns were fired alternately at two stations about nine miles apart. Observers at one station observed the time of seeing the flash and hearing the report from the other. The guns being fired alternately, and the sound travelling in oppo- site directions, the effect of wind was eliminated in the mean of the results at the two stations. It is possible, by causing the sound-wave to act upon diaphragms, to make it record its own time of departure and arrival, and by making use of some of the methods of estimating very small intervals of time the velocity of sound may be measured by experiments conducted within the limits of an ordinary building. Thfe velocity of sound in water was determined on Lake Geneva in 1826 by an experiment analogous to that by which the velocity in air was determined. In § 255 and § 257 it is shown that the time of one vibration of any body vibrating longitudinally is the time required for a sound-wave to travel twice the distance between two nodes. The velocity may, therefore, be measured by determining the number of vibrations per second of the sound emitted, and measuring the distance between the nodes. In an open organ-pipe, or a rod free at both ends, when the fundamental tone is sounded the sound travels twice the length of the rod or pipe during the time of one complete vibration. If rods of different materials be cut to such lengths that they all give the same fundamental tone when vibrating longitudi- nally, the ratio of their lengths will be that of the velocity of sound in them. In Kundt's experiment, the end of a rod having a light disk attached is inserted in a glass tube containing a light powder strewn over its inner surface. When the rod is made to vibrate 270] VELOCITY OF SOUND. 395 longitudinally, the air-column in the tube, if of the proper length, is made to vibrate in unison with it. This agitates the powder and causes it to indicate the positions of the nodes in the vi- brating air-column. The ratio of the velocity of sound in the solid to that in air is thus the ratio of the length of the rod to the distance between the nodes in the air-column. LIGHT. CHAPTER I. PROPAGATION OF LIGHT. 271. Vision and Light. — The ancient philosophers, before Aristotle, believed that vision consisted in the contact of some subtle emanation from the eye with the object seen. Aristotle showed the absurdity of this view by suggesting that if it were true, one should be able to see in the dark. Since his time, it has been generally admitted that vision results from something proceeding from the body seen to the eye, and there impress- ing the optic nerve. This we call light. Optics treats of the phenomena of light. It is conveniently divided into two branches, Physical Optics, which treats of the phenomena resulting from the propagation of light through space and through different media, and Physioldgical Optics, which treats of the sense of vision. 272. Theories of Light.— At the time of Newton, light was generally considered to consist of particles which were not those of ordinary matter, projected from a luminous body, and exciting vision by their impact on the retina. This theory was strongly supported by Newton himself, who found in it plausible explanations of most luminous phenomena then known. But even in Newton's time phenomena were known which could only be explained by assigning to the luminiferous 274] PROPAGATION OF LIGHT. 397 particles very improbable forms and motions, and, since his time, facts have been discovered that are inconsistent with any emission theory. The undulatory theory, which is the one universally adopted, assumes that light is a wave motion in an elastic medium per- vading all space. All luminous bodies excite in this medium systems of waves which are propagated according to the same mechanical laws as those which govern wave systems in other media, some of which have been developed in § 19 and §§ 242— 245. The undulatory theory has stood well the test of ex- plaining newly discovered phenomena, and has moreover led to the discovery of phenomena not before known. The ob- jections to the theory are that it requires the hypothesis of a medium of the existence of which there is no direct evidence, pervading space, and requires us to ascribe to that medium properties unlike those of any body with which we are ac- quainted. A modified form of the undulatory theory, known as the electromagnetic theory of light, was proposed by Maxwell. It will be briefly presented after the facts connecting light and electricity have been considered. 273. Wave Surfaces. — In § 243 is explained the general mode of propagation of wave motion in accordance with Huyghens' principle. When light, emanating from a point, proceeds with the same velocity in all directions, the wave fronts are evidently concentric spherical surfaces. There are, however, many cases, especially in crystalline bodies, of un- equal velocities in different directions. In these cases the wave fronts are not spherical but ellipsoidal, or surfaces of still greater complexity. 274. Straight Lines of Light. — When a small screen A (Fig. 100) is placed between the eye and a luminous point, the luminous point is no longer visible. Light cannot reach the eye by the curved or broken line PAE, and is therefore 398 ELEMENTARY PHYSICS. [274 said to move in straight lines. This seems not to accord with Huyghens' principle which makes any wave froftt the resultant of an infinite number of elementary waves proceeding from the Fig. ; Fig. ioi. various points of the same wave front in one of its earlier posi- tions. It can, however, easily be shown that when the wave lengths are small, the disturbance at any point /"(Fig. ioi)is due almost wholly to a very small portion of the approaching wave. Let us consider first the case of an isotropic medium, in which light moves in all directions with equal velocities. Let mn be the front of a plane wave per- pendicular to the plane of the paper, moving from left to right or towards P. Draw PA perpendicular to the wave front, and draw Pa, Pb, etc., at such obliquities that Pa shall exceed PA by half a wave length, Pb exceed Pa by half a wave length, etc. We will designate the wave length by A. It is evident that the total effect at P will be the sum of the effects due to the small portions Aa, ab, etc. Since Pa is half a wave length greater than PA, and Pb half a wave length greater than Pa, each point of ab is half a wave length farther from P than some point in Aa : hence elementary waves from ^(^ will meet at Z' waves horn A a in the opposite phase. It appears, therefore, that the effects at Pol the portions ab and Aa are opposite in sign, and tend to annul each other. The same is true of be and cd. But the effects of Aa and ab may 274] PROPAGATION OF LIGHT. 399 be considered as proportional to their lengths. Hence, by computing the lengths, we can determine the resultant effect at P. \^tX.AP^x. From the construction, we have Ab^ */{x-^Vf -x" =4/25a:+1^ Ac = \\x + f A)^ -x'' = V^xX + IT^; Ad = V\x-\-2iy-x' = VaxX + 4^^; etc. = etc. For light the values of A. are between 0.00039 and 0.00076 mm., and if x be taken as looo mm., A" will be very small in comparison to xX and may be omitted. The above formulas then become, if VlcX. be represented by /, Aa = iVT; Ab = IV2; Ac = /I/3; Ad= IVa; etc. = etc., and the several portions into which the wave front is divided are Aa = I = i/; ab = 1{V2 — i) = 0.414/; be = l{Vz - V2) - 0.318/; cd = /{V4- V3) = 0.268/. 400 ELEMENTARY PHYSICS. [274 Taking now the pairs of which the effects at P are opposite in sign, we find Aa a little more than twice ab, while be and cd are nearly equal. It is evident also, that for portions beyond d, adjacent pairs will be still more nearly equal, and the effect at P, therefore, of each pair of segments beyond b almost van- ishes. The effect at P is then almost wholly due to that por- tion of Aa that is not neutraHzed by ab. But, taking the greatest value of X, Aa = VxX — 4/0.76 — 0.87 mm., a very small distance. Hence, under the conditions assumed, the effect at any point Pis due to that portion of the wave front near the foot of the perpendicular let fall from P on the wave front. It may be demonstrated by experiment that the portions of the wave be- yond Aa neutralize each other. Suppose a screen mn in the position shown in Fig. 102. The point P will be in shadow. If the darkness at P is due to interference as explained, light should be restored by suppressing the in- terfering waves. If a second screen be placed at m'n' so as to cut off the waves proceeding from points above b, waves from points between a and b will no longer be neutralized, and light should fall at P. To test this conclusion the edge of a flat flame may be observed through a narrow slit in a screen. In- stead of the narrow edge of the flame, a broad luminous surface is seen, in which the brightness gradually diminishes from the centre towards the edges. If we consider the wave front just entering the slit, it will be seen that elementary waves proceed from all points of it, and the slit being very narrow it is only in very oblique directions that pairs of these waves can meet in opposite phases. Hence, light proceeds in oblique lines behind the screen, and from our habit of locating visible objects back along the line of light entering the eye, the flame appears as a Fig. 275] PROPAGATION OF LIGHT. 401 broad surface. It will be seen by reference to Fig. loi that the elementary wave that first reaches P is the one to which the disturbance there is principally due. Other waves arriving later find there the opposite phase of some wave that has preceded them. When the velocity in all directions is the same, the first wave to reach P is the one that starts from the foot of a perpendicular let fall from Pan the wave front. Hence light is said to travel in straight lines perpendicular to the wave front. If, however, light does not move with equal velocities in all directions, the last statement is no longer true, as will be seen from Fig. 103. Here mn represents a wave front, proceeding towards Pin a medium in which the velocities -X \ Fig. 103, in different directions are such that the elementary wave surfaces are ellip- soids. The ellipses in the figure may be taken as sections of these ellipsoids. The wave first to reach P is not the one that starts from A at the foot of the perpendicular, but from A'. It is from A' that P derives its light, and the line of propagation is no longer perpendicular to the wave front. It is important to note that the deductions of this section apply only where A is small in relation to x, so that A' may be neglected in comparison with x\. With soundwaves this is not true, and if a computation similar to that given above for light-waves be made for sound, not omitting T\^, it will be seen why there are no definite straight lines of sound and no sharp acoustic shadows. 275. Principle of Least Time. — The above are only par- ticular cases of a law of very general application, that light in going from one point to another follows the path that requires least time. The reason is that values in the vicinity of a mini- mum change slowly, and there will be a number of points in the neighborhood of that point from which the light-waves are 26 402 ELEMENTARY PHYSICS. [276 propagated to the given point in the least time, from which waves will proceed to that point in sensibly the same time, and, meeting in the same phase, combine to produce light. It is also true that values change slowly in the vicinity of a maxi- mum, and there are cases where the path followed by the light is determined by the fact that the time is a maximum instead of a minimum. 276. Shadows. — An optical shadow is the space from which light is "excluded by an opaque body. When the luminous source is a point, or very small, the boundary between the light and shadow is very sharp. When the luminous source is large, there is a portion of the space behind the opaque body, called the umbra, which is in deep shadow, and surrounding this is a space which is in shadow with reference to one portion of the luminous source while it is in the light with reference to an- other portion. The space from which light is only partially ex- cluded is X^Q penumbra. Fig. 104 shows the boundaries of the umbra and penumbra. It is evident that the light di- FlG. 104. minishes gradually from the outer boundary of the penumbra to the boundary of the umbra. 277. Images by Small Apertures. — If light from a single luminous point pass through a small hole of any form, and fall on a screen at some distance, it produces a luminous spot of the same form as the opening. Light from several points will pro- duce several such spots. If the luminous source be a surface, the spots produced by the light from its several points will 277] PROPAGATION OF LIGHT. 403 overlap each other and form an illuminated surface, which, if the source be large in comparison with the opening, will have the general form of the source, and will be inverted. The illu- minated surface is an inverted image of the source. If a small opening be made in the window-shutter of a darkened room, images of external objects will be seen on the wall opposite. The smaller the opening, the more sharply defined, but the less brilliant, is the image. CHAPTER II. REFLECTION AND REFRACTION. 278. Law of Reflection. — In § 246, it is shown that when a wave passes from one medium into another where the parti- cles constituting the wave move with greater or less facility, a wave is propagated back into the first medium. It is shown in § 247, that when the surface separating the two media is a plane surface, the centres of the incident and reflected waves are on the same perpendicular to the sur- face, and at equal distances on oppo- site sides. Considering the lines to which, as shown in § 274, the wave propagation in the case of light is re- stricted, a very simple law follows at once from this relation of the Fig. JOS. incident and reflected waves. In Fig. 105, C and C represent the centres of the incident and re- flected waves w««, ok, CA, AB are the paths of the incident and reflected light. It will be evident from the figure that CA, AB are in the same plane normal to the reflecting surface, and that they make equal angles with the normal AN. CAN is called the angle of incidence, and NAB the angle of reflec- tion. Hence we may state the law of reflection as follows : The angles of incidence and reflection are equal, and lie in the same plane normal to the reflecting surface. It can easily be shown that light traverses the path CAB from C to B which fulfils these laws, in less time than it requires to traverse any other path by way of the reflecting surface. 279] REFLECTION AND REFRACTION. 405 279. Law of Refraction.— If the incident wave pass from the one medium into the other, there is in general a change in the wave front, and a consequent change in the direction of the hght. Let us first consider the simple case of a plane wave en- tering a homogeneous, isotropic medium of which the bounding surface is plane. Suppose both planes perpendicular to the , Fig. 106. plane of the paper, and let AB (Fig. 106) represent the intersection of the surface of the medium, and mn the in- tersection of the wave, with that plane. Let v represent the velocity of light in the medium above AB, and v' the ve- locity in the medium below it. Let m' be the position of the wave in the first medium after a time t. Then mo equals vt. As the wave front passes from mn to m'o, the points of the separating surface between n and are succes- sively disturbed, and become centres of spherical waves propa- gated into the second medium with the velocity z/'. The wave surface of which the centre is n would, at the end of time t, have a radius nn" = v't, such that — r, = — ,. Similarly, the nn V ■' ' wave from any other point, as s, would have a radius si' such st V , , . .... that —7 = —7, and the wave surface withm the second medium St V 406 ELEMENTARY PHYSICS. [279 is evidently the plane on" . As the direction of propagation is perpendicular to the wave front, op will represent the direc- tion of the light in the second medium. In the triangles nov! and non" we have nn' = no sin Aon' , and nn" = no sin Aon"; hence sin Aon' nn' v sin Aon" ~~ nn" ~ 1/' If we represent the angle of incidence moN by i, and the angle of refraction poN' by r, we have sin i V , . —. — = — , = u, a constant. ( loo) This constant is called the index of refraction. This is the expression of Snell's law of refraction. Here again the time required for the light to pass by mop from m in one medium to/ in the other is less than by any other path. We may now trace a wave thrbugh a medium bounded by plane surfaces. Suppose the wave front and bounding planes of the medium all perpendicular to the plane of the paper. sin % 1) We shall have as above for the first surface -: — = —,= «, sm r V , . 1 , r sin i' v' and for the- second surface -. — -. = -.-, = ix . sm r V If, as is often the case, the light emerge into the first me- dium, , sin i' v' I v" = v, and -: — - = - = -. (no) smr V fi If the bounding planes be parallel, i' = r, and we have sin r I sin r' ~ fji' 279] REFLECTION AND REFRACTION. 407 hence «"= r' , or the incident and emergent waves are parallel. If the two bounding planes form an angle A the body is called a prism. The wave incident upon the second face will make with it an angle A — r, and the emergent wave is found by the relation sin(^ — r) I sm r - or -■ , . ^ }x sin(^— r) =11. The direction of the emerging wave front may be found by construction. Draw Ai (Fig. 107) parallel to the incident wave. From some point B on AB describe an arc tangent to Ai; from the Bi same point with a radius — describe the arc rr.' Ar, tangent to rr, is the refracted wave front. From some point C on AC describe an arc tangent to Ar, and from the same point as cen- tre describe another arc r'r' with a radius ix X Cy. A tan- gent from A to r'r' is parallel to the emergent wave. It might be that A would fall inside the arc r'r' so that no tangent could be drawn. That would mean that there could be no emergent wave. The angle of incidence for which this occurs can readily be obtained from Eq. (no). We have sm t I . , —. — -, = — , or sm r sm r fx /< sm ? , Now the maximum value of sin ^' is i, which is reached when sin i' = — . Any larger value of sin i' gives an impossible value 4o8 ELEMENTARY PHYSICS. [279 for sin r' . The angle i' = sin " ' - is called the critical angle of the substance. For larger angles of incidence the light cannot emerge, but is totally reflected within the medium. Another construction for the front of the emergent wave is very instructive. Let AB, AC (Fig. 108), be the faces of the prism, and let Ai drawn through A be parallel to the front of the incident wave. With A as centre, and any radius, draw an arc im. From the same centre Ai with radius Ar = — describe another /< arc. From r draw rx parallel to AB and join Ax. Ax is parallel to the front of the refracted wave. For in the triangle Arx we have Fig. 108. sin Arx sin Axr sin irx Ax , — — ;: — — -7- = ;«, by construction. sm Axr Ar ■' Since irx equals the angle of incidence, Axr equals the angle of refraction. Now draw xr' parallel to AC, and Ar' is parallel to the front of the emergent wave. The angle r'Ar is the deviation that the wave suffers in passing through the prism. Suppose the prism to rotate about A and the angle of incidence to change in such a way that the condition of things may be always represented by rotating the angle rxr', of which the sides are parallel to the sides of the prism, around x. It is plain that the arc r'r will be longer or shorter as it crosses the angle more or less obliquely, and that its length will be a minimum when xr' and xr are equal — that is, when the line Ax bisects the angle at x and consequently the angle A of the prism. But the arc r'r may be taken as the measure of the 28o] REFLECTION AND REFRACTION. 4O9 angle of deviation r' Ar at its centre. Hence that angle is a minimum when it is bisected by Ax, and when, therefore, the angles of incidence and of emergence are equal. Considering that the path of the light is perpendicular to the wave front, the above construction shows that the deviation, when jx is greater than unity, is always toward the thicker portion of the prism. The case when emergence is no longer possible is also shown by the failure of xr' , parallel to AC, to cut the arc r'r. The critical angle is reached when xr' becomes tangent to r'r. If, in a prism of any substance, xr and xr' be both tangent to r'r, the angle of that prism is the greatest angle which will ad- rnit of the passage of light through the prism. If a beam of white light be allowed to fall upon a prism through a narrow slit, it will be refracted, in general, in accord- ance with the law already given. The image of the slit, how- ever, when projected upon a screen, appears not as a single line of white light, but as a variously colored band. This is due to the fact that the indices of refraction for light of different colors are different. Hence the index of refraction of a sub- stance, as ordinarily given, depends upon the color of the light used in determining it, and has no definite meaning unless that color is stated. 280. Plane Mirrors. — The wave on, represented in Fig. 105, is the same as would have come from a luminous point at C if the reflecting surface did not intervene. If this wave reach the eye of an observer, it has the same effect as though coming from such a point, and the observer apparently sees a luminous point at C . C is a virtual image of C. When an object is in front of a plane mirror each pf its points has an image symmetrically situated in relation to the mirror, and these constitute an image of the object like the latter in all respects, except that by reason of symmetry it is reversed in one direction. The reflected light may for all purposes be considered as 410 ELEMENTARY PHYSICS. [281 coming from the image. If it fall on a second mirror and be again reflected, a second image appears behind this mirror, the position of which is determined by considering the first image as an object. When two mirrors make an angle, an object between them will have a series of images, as shown in Fig, 109. .(45 and ^C represent the intersections of the two mir- rors with the plane of the paper, to which they are supposed perpendicular. O is the object. It will have an image produced by AB, the position of which is found by drawing Ob perpendicular to AB and making , mb = mO. The light reflected from AB proceeds as though b were the ob- ject, and falling on AC is again reflect- ed, giving an image at c'. Proceeding iromAC, it may suffer a third reflection from AB and give a third image at b". With the angle as in the figure none of the light can suffer' a fourth reflection, because after the third reflection the light proceeds as though originating at b", and b" is behind the plane of the mirror AC. Images c, b', and c" are produced by light which suffers its first reflection from AC. It is easy to show that all these points are equidistant from A, and hence are on the circumference of a circle of which A is the centre. If OAC were an even aliquot part of four right angles, c" and b" would coincide, and the whole number of images, including the object, would be the quotient of four right angles by the angle formed by the mirrors. This is the principle of the kaleidoscope. 281. Spherical Mirrors. — A spherical mirror is a portion of a spherical surface. It is a concave mirror if reflection occur on the concave or inner surface ; a convex mirror if it occur on the convex surface. The centre of the sphere of which the mirror forms a part is its centre of curvature. The 28l] REFLECTION AND REFRACTION. 411 middle point of the surface of the mirror is the vertex. A line through the centre of curvature and the vertex is tht principal axis. Any other line through the centre of curvature is a secondary axis. The angle between radii drawn to the edge of the mirror on opposite sides of the vertex is the aperture. To investigate the effects of reflection from a spherical surface, let us consider first a concave mirror. Let a light-wave ema- nate from a point L on the principal axis (Fig. 1 10). In general, J'- ^ 'A ^ % / 7 y ' CI % D2 /// Fig. no. different points of the wave will reach the mirror successively, and, considering the elementary waves that proceed in turn from its several points, the reflected wave surface may be con- structed as for a plane mirror. If the mirror were not there the wave front would, at a certain time, occupy the position aa. Drawing the elementary wave surfaces we have bb, the position at that instant of the reflected wave. Its form sug- gests that of a spherical surface, concave toward the front, and having a centre at some point / on the axis. If we assume it to be so, and try to determine by analysis the position of /, a real definite result will be proof of the correctness of our assumption. If bb be a spherical surface and / its centre, it is 412 ELEMENTARY PHYSICS. [281 plain that the disturbances propagated from the various points of bb will reach / at the same instant, and / will at that instant be the wave front. It is plain, too, that the time occupied by the wave in going from the radiant point to all points of the same wave front must be the same. Hence, in a homogeneous medium, the length of path to the various points of the wave must be constant, that is, in the case under consideration, LB -^ Bb must be constant for all points of the wave front bb. If / be a subsequent position of bb, it follows that LB -\- Bl must be constant wherever the point B is situated on the re- flecting surface. Draw B'D perpendicular to the axis of the mirror. Represent BD by y, AD by x, LA by /, lA by /', and CA by r. Then we have LB = V{p — xy -\-y', and y ^ (2r — x^x = 2rx — ;^r^ Hence follows LB = Vf — 2px +x^-\-2rx — x' = V/ + 2x{r —p). If the aperture be small, x will be small in comparison with the other quantities, and we may obtain the value of LB to a near approximation by extracting the root of the ex- pression found above and omitting terms containing the second and higher powers of x. We obtain ^^ = /+|(^-/) + -... In like manner we have lB=p'^j.{r-p')^..., whence LB -\-lB = p^ p' ^-{r - p)^ ^,{r — p'). 28l] REFLECTION AND REFRACTION. 413 When B coincides with A, the above value becomes/ -f/', and since upon our supposition all values of LB -(- IB are equal, we niust have from which we obtain r r and p' = ^^ 2/ As this is a definite value, it follows that, for the apertures for which the approximations by which the result was arrived at are admissible, the wave surface is practically spherical. Since the disturbances propagated from bb reach / simultaneously, their effects are added, and the disturbance at / is far greater than at any other point. The effect of the wave motion is concentrated at /, and this point is therefore called a focus. Since the light passes through /, it is a real focus. If / were the radiant point, it is clear that the reflected light would be concentrated at L. These two points are therefore called con- jugate foci. If we divide both sides of the equation — -j- — , = 2 by r, we have which is the usual form of the equation used to express the relation between the distances from the mirror of the conju- gate foci. 414 ELEMENTARY PHYSICS. [281 A discussion of this equation leads to some interesting results. Suppose p =^ ^ , then /' = ^^ ; that is, when the radiant is at an infinite distance from the mirror, the focus is midway between the mirror and the centre. In this case the incident wave is normal to the principal axis, and the focus is called t\\Q principal focus. Suppose/ = r; /'= r also. When ^12 121 p ^ ^r, /' = 00 . When / < 12 ,1 - > - and -> : p r p -r= a 2' p ' r p' r p negative quantity. To interpret this negative result it should be remembered that all the distances in the formulas were assumed positive when measured from the mirror toward the >■{ Fig. in. source of light. A negative result means that the distance must be measured in the opposite direction, or behind the mirror. Fig. iii represents this case. It is evident that the reflected wave is convex toward the region it is ap- proaching, and proceeds as though it had come from /. / is therefore a virtual focus. Either of the other quantities of the formula may have negative values, p will be. negative if waves approaching their centre / fall on the mirror. Plainly they would be reflected to Z at a distance from the mirror less than — , as may be seen from the formula If r be negative, the centre is behind the mirror. The mirror is then convex, and the formula shows that for all positive values oi p,p' is negative and numerically smaller than/. 282] REFLECTION AND REFRACTION. 415 282. Refraction at Spherical Surfaces.— The method of discussion which has been applied to reflection may be em- ployed to study refraction at spherical surfaces. Let BD (Fig. 112) be a spherical surface separating two transparent Fig. 112. media. Let v represent the velocity of light in the first medium, to the left, and v' the velocity in the second medium, to the right, of BD. Let Z be a radiant point, and mn a sur- face representing the position which the wave surface would have occupied at a given instant had there been no change in the medium, inn' the wave surface as it exists at the same instant in the second medium in consequence of the different velocity of light in it. Assume as before, in § 281, that m'n' is a spherical surface with centre /. We have Bm Bm! and LB-\-Bm _LB ,:^ V V V - = c. a constant for all points of mn. spherical surface m'n', we have If / be the centre of the 4l6 ELEMENTARY PHYSICS. [282 IB , Bm' ^, z/' + v' ~^' a constant for all points of m'n'. Taking the difference of the last two equations, and re- membering that Bm Bm' / v^ we obtain y =: C — C, V V a constant for all points of BD, and hence V LB rlB = a constant. V But — ; = M is the index of refraction of the second sub- V stance in relation to the first. Hence LB — filB = a constant = LA — filA. Using the notation of the last section, and substituting the values of LB and IB as there found, "except that/" is used instead of/', we have r fir whence we obtain 1 — ^J? — ^ ~ M> P P and ^__ = ^-. (112) 282] REFLECTION AND REFRACTION. A^7 If the medium to the right of BD be bounded by a second spherical surface, it constitutes a lens. Suppose this second surface to be concave toward / and to have its centre on AC. The wave /«'«', in passing out at this second surface, suffers a new change of form precisely analogous to that occurring at the first surface, and the new centre is given by the formula just deduced by substituting for p the distance of the wave centre from the new surface, and for /t the index of refraction of the third medium in relation to the second. If s represent the distance of / from the new surface, /*' the new index, and p' the new focal distance, we have //_£ _ ;<'- I I P' s r' ' If we suppose the lens to be very thin we may put s = p". If we suppose also that the medium to the right is the same as that to the left of the lens, u' is equal to — . Hence lA. I I 7 p" r' ' Multiplying through by n, we have I f^ _ ^ ~ f^ _ y"~l p'~y'~ '~V~ ~ ?~'- Eliminating/" between this equation and Eq. 112, we obtain -,-- = (/.- i)(---,j, (113) 27 41 8 ELEMENTARY PHYSICS. [282 which expresses the relation between the conjugate foci of the lens. It should be noted that r in the above formulas rep- resents the radius of the surface on which the light is incident, and r' that of the surface from which the light emerges. All the quantities are positive when measured toward the source of light. Fig. 113 shows sections of the different forms of lenses produced by combinations of two spherical surfaces, or of one plane and one spherical surface. An application of Eq. 113 will show that for the first three, which are thickest at the centre, light is concentrated, and for the second three diffused. The first three are therefore called converging, and the second three diverging, lenses. Let us consider the first and fourth forms as typical of the two classes. The first is a double convex lens. The r of Eq. 113 is nega- tive because measured from the lens away from the source of light. The second term of the formula has therefore a negative value, and /' is negative except when — > (/^ — i)(- \. If / = 00 , we have -: = o and -, = (/^ — i)( >), a negative quantity because r is negative. /' is then the distance of the principal focus from the lens, and is called the focal length of the lens. The focal length is usually designated by the sym bol /. Its negative value shows that the principal focus is on the side of the lens opposite the source of light. This focus is real, because the light passes through it. Eq. 113 is a little more simple in application if, instead pf making the algebraic 283] REFLECTION AND REFRACTION. 4I9 signs of the quantities depend on the direction of measure- ment, they are made to depend on the form of the surfaces and the cliaracter of tlie foci. If we assume that radii are positive when the surfaces are convex, and that focal distances are positive when foci are real, the signs of/' and r in Eq. 113 must be changed, since in the investigation p' is the distance of a virtual focus, and r the radius of a concave surface. The formula then becomes /'+) = (>'- 4 + ?)- ("4) To apply this formula to a double concave lens, r and r' are both negative; p' is then negative for all positive values of /. That is, concave lenses have only virtual foci. For a plano-convex lens (Fig. 113, 2), if light be incident on the plane surface, r=oo and j-, = (/. _ i)i _ 1. This gives positive values oi p' and real foci for all values of I , ,1 For a concavo-convex lens (Fig. 113, 6) the second member ■of the equation will be negative, since the radius of the con- cave surface is negative and less numerically than that of the convex surface. Hence /' is always negative and the focus virtual when L is real. 283. Images formed by Mirrors. — In Fig. 1 14 let ab rep- resent an object in front of the concave mirror mn. We know from what precedes that if we consider only the light incident 420 ELEMENTARY PHYSICS. [28 J not too far from c, the light reflected will be concentrated at some point a' on the axis ac at a distance from the mirror given by Eq. 114. a' is a real image; of a. ]^n the same way «m the various points of the wave front lying between A and B. First con- sider the point P on the perpendicular to AB at its middle point. AB is so small that the distances from P to each point of AB may be regarded as equal, or the time of passage of the light from each point of AB to P may be made 444 ELEMENTARY PHYSICS. [303 AcB Fig. 137, the same, by placing a converging lens of proper focus between AB and P. Then all the elementary waves from points of AB meet at P in the same phase, and the point P is illumijjated. Now consider a second point, P', in an oblique direction from C, Fig. 137, and suppose the obliquity such that the tiriie of passage from B to P' is half a vibration period less than the time of passage from C to P, and a whole vibration period less than the time of passage from A to P. Plainly the elementary waves from B and C will meet at P' in opposite phases, and every wave from a point between B and C will meet at P a wave in the opposite phase from some point between C and A. The point P is, therefore, not illuminated. Suppose another pbint, P" (Fig. 138), still further from P, such that .(4.5 may be divided into three equal parts, each of which is half a wave length nearer P" than the adjacent part. It is plain that the two parts Be and ca will annul each other's effects at P", but that the odd part Aa will furnish light. At a greater obliq- uity, .^.5 may be divided into four parts, the distances of which from "the point, taken in succession, differ by half a wave length. There being an even number of these parts, the sum of their effects at the point will be zero. Now let us suppose the point P to occupy successively all positions to the the normal. While the line joining P with Fig. 138. right or left of the middle of the aperture is only slightly oblique, the ele- mentary waves meet at P in nearly the same phase, and the loss of light is small. As /"approaches P' (Fig. 137), more and more of the waves meet in opposite phases, the light grows rapidly less, and at P' becomes zero. Going beyond P' 304] INTERFERENCE AND DIFFRACTION. 445 the two parts that annul each other's effects no longer occupy the whole space AB, some of the points of the aperture send to P waves that are not neutralized, and the light reappears, giving a second maximum, much less than the first in intensity. Beyond this the light diminishes rapidly in intensity until a point is reached where the paths differing by half a wave length divide AB into four parts, when the light is again zero. Theo- retically, maximum and minimum values alternate in this way, to an indefinite distance, but the successive maxima decrease so rapidly that, in reality, only a few bands can be seen. 304. Effect of a Narrow Screen in the Path of the Light. — It can be shown that the effect of a narrow screen is the complement of that of a narrow aperture ; that is, where a narrow aperture gives light, a screen pro- duces darkness. Let mn (Fig. 139) be a plane wave and AB a surface on which the light falls. If no obstacle intervene, the surface AB will be equally illuminated. The illumi- nation at any point C is the sum of the effects of all parts of the wave mn. Let the effects due to the part of the wave op be represented Fig. 139. by / and that due to all the rest of the wave by /'. Then the illumination at C is / + I\ equal to the general illumination on the surface. Let us now suppose mn to be a screen and po a narrow aperature in it. If the illumination at C remain un- changed, it must be that the parts mo andpn of the wave had no effect, and if, for the screen with the narrow aperture, we substi- tute a narrow screen at op, there will be darkness at C. If, how- ever, a dark band fall at C when op is an aperture, a screen at op will not cut off the light from C. That is, if C be illuminated when op is an aperture, it will be in darkness when op is a screen, and if it be in darkness when op is an aperture, it will be illuminated when op is a screen. 446 ELEMENTARY PHYSICS. [30s 305. Diffraction Gratings. — Let AB (Fig. 140) be a screen having several narrow rectangular apertures parallel and equi- distant. Such a screen is called a grating. Let the approaching waves, moving in the direction of the arrow, be plane and parallel to AB. Draw the parallel lines ab, cd, etc., at such an angle that the distance from the centre of a to the foot of the perpen- dicular let fall from the centre of the adjacent opening on ab shall be equal to some definite wave length of light. It is evident that an will contain an exact whole number of wave lengths, co one wave length less, "etc. The line mn is, therefore, tangent to the fronts of a series of elementary waves which are in the same phase, and may be considered as a plane wave, which, if it were ■ received on a converging lens, would be concentrated to a focus. If the obliquity of the lines be increased until ae equals 2A, 3A, etc., the result will be the same. Let us, however, suppose that ae is not an exact multiple of a wave-length, but some fractional part of a wave length, y'-j'^A for example. Let m be the fifty-first opening counting from a ; then an will be rVff-^ X SO = 49.5A. Hence the wave from the first opening will be in the opposite phase to that from the fifty-first. So the wave from the second opening will be in the opposite phase to that from the fifty-second, etc. If there were one hundred openings in the screen, the second fifty would exactly neutralize the effect of the first fifty in the direction assumed. Light is found, therefore, only in directions given by sin B n\ "d' (116) where n is a whole number, 6 the angle between the direction of the light and the normal to the grating, and d the distance from 305] INTERFERENCE AND DIFFRACTION. 447 centre to centre of the openings, usually called an element of the grating. Gratings are made by ruling lines on glass at the rate of some thousands to the centimetre. The rulings may also be made on the polished surface of speculum metal, and the same effects as described above are produced by reflection from its surface. 'Since the number of lines on one of these gratings is several thousands, it is seen that the direction of the light is closely confined to the direction given by the formula, or, in other words, light of only one wave length is found in any one direction. If white light, or any light consisting of waves of various lengths, fall on the grating, the light corresponding to different wave lengths will make different angles with AC, that is, the light is separated into its several constituents and pro- duces a. pure spectrum. Since different values of n will give different values of 6 for each value of A, it is plain that there will be several spectra corresponding to the several values of n. When n equals i the spectrum is of the. first order ; when n equals 2 the spectrum is of the second order, etc. The grating fur- nishes the most accurate and at the same time the most simple method of determining the wave lengths of light. Knowing the width of an element of the grating it is only necessary to measure 6 for any given kind of light. In this discussion it has been assumed that the light was normal to the surface of the grating. This need not be the case. Let AB (Fig. 141) be the intersection with the paper of a reflecting grating supposed perpen- dicular to it, mn an approaching wave front also perpendicular to the paper, and m"n" the reflected wave front constructed as in.§ 278. The line m"n" is a tangent to all the elementary waves that origi- nate in the surface AB in conse- Fig. 141. quence of disturbances produced by the passage of the wave 448 ELEMENTARY PHYSICS. [305 m'n' . The surface AB consists of a number of narrow, equidis- tant, reflecting surfaces separated by roughened channels. If the reflecting surfaces be considered infinitely narrow, each of them will be the centre of a system of waves due to the successive in- cident waves similar to mn which fall upon them. Since the number of the elements of the grating is finite there, will be a ■finite number of such wave systems. In the diagram one of these systems is represented about the centre d. Let us repre- sent by a, b, c, d, etc., the centres of these systems, such that the distances in"a, ab, be, cd, etc., are elements of the grating. Let us suppose the wave systems all represented, and draw in"n"' tangent to the wave front of which the centre is a, and which is one wave length behind the wave to which m"n" is tangent. The line in"n"' will be also tangent to waves of the systems of which b, c, d, etc., are the centres, and which are respectively two, three, four, etc., wave lengths behind the wave to which ni' n" is tangent. These elementary waves, differing by successive periods, are all in the same phase, and m"n"' may, therefore, be considered as constituting a plane wave front in which light of one particular wave length is pro- pagated in the direction dx. Represent by i the angle of inci- dence, by r the angle of reflection, by a the angle between the normal to the grating and the path of the diffracted light. Then i equals r, and if m"a equal j,.the radius of the elemen- tary wave having its centre at a, and tangent to m"n" , is s sin i, and of the elementary wave having the same centre, and tan- gent to m"n"', is s sin a. Hence, by hypothesis, we have s sin i — s sm a = A. Let us designate by /? the angle between the path of the in- cident and that of the diffracted light, and by 6 the .angle be- tween the path of the reflected and that of the diffracted light. If the grating be turned so that the path of the reflected light n coincides with dx, its normal will turn through the angle - and 305] INTERFERENCE AND DIFFRACTION. 449 will bisect the angle/?. Hence we have « = — -I — .and a=— — — . ° 2 2 2 2 Substituting these values in the equation for A we obtain A. = 2JCOs— Sin—. (.117) 22 ^ -^ Hitherto the spaces from which the elementary waves pro- ceed have been considered infinitely narrow, so that only one system of waves from each space need be considered. In prac- tice, these spaces must have some width, and it may happen that the waves from two parts of the same space may cancel each other. Let the openings, Fig. 142, be equal in width to the opaque spaces, and let the direction am be taken such that ae equals 2 A.. Then ae' equals \\, or the waves from one half of each opening v are opposite in phase to those from the other half, and there can be no light in the direction am. In general, if d equal the width of the opening, there will be interference and light will be destroyed in that direction for which fig. 142. sin 6 = —-, if the incident light be normal to the grating. Let /represent the width of the opaque space. Then d -\- f ^ s, and light occurs in the direction given by sin = -j-r—>> Pro- vided that the value of given by this equation does not satisfy the first equation also. If d equal/, we have sm o — -y-TT — —5- d-\-f 2d When n is even, sin B becomes 2\ _ A, /)A _ 2A, 2d^d' 2d~~d' ^^'^■' 29 450 ELEMENTARY PHYSICS. [306 and satisfies the equation sin d = -J, which expresses the condition under which light is all de- stroyed. Hence in this case all the spectra of even orders fail. Moreover, the spectra after the first are not brilliant. When/" equals 2d the spectrum of the third order fails. It may be shown that whatever be the relative widths of the transparent and opaque spaces, one may be substituted for the other without altering the result. In Fig. 143 let nc rep- resent an opening and cd an opaque por- tion. Let us assume that cd equals 2^c, and let ad be the path of the diffracted light giving the spectrum of the first order; then we have ae = X and ae' = JA. Now let ac become the opaque portion and cd the opening. We will then have z/i — JA. Fig. 143- Each of the elementary waves from points between c and i will be half a wave length behind a correspond- ing wave from some point between d and /, so that the waves coming from ci and d/ annul one another, and tj is the only efficient portion of the opening cd. This portion y is equal to the former opening ac. Since the effect of the grating is that of one opening multiplied by the number of openings, it is plain that in this case it is indifferent whether the openings are of the width ac or cd. 306, Measurement of Wave Lengths. — To realize prac- tically the conditions assumed in the theoretical discussion of the last section, some accessory apparatus is required. It has been assumed that the wave incident upon the grating was plane. Such a wave would proceed from a luminous point or line at an infinite distance. In practice it may be obtained by 3o6] INTERFERENCE AND DIFFRACTION. 451 3h=^J> illuminating a very narrow slit, taking it as the source of light, and placing it in the principal focal plane of a well-corrected converging lens. The plane wave thus obtained passes through the grating, or is reflected from it, and is received on a second lens similar to the first, which gives an image either on a screen or in front of an eyepiece, where it is viewed by the eye. The general construction of the apparatus may be inferred from Fig. 144. It is called the spectrometer. .<4 is a tube carrying at its outer end the slit and at its inner end the lens, called a collimating lens. CD is a horizontal graduated circle, at the centre of which is a table on which the grating is mounted, and so adjusted that the axis of the circle lies in its plane and parallel to its lines. In using a reflecting grating the collimating and observing telescopes may be fixed at a constant angle with each other which may be determined once for all in making the ad- justments of the instrument. This angle is the angle /J of ^ 305. To determine this angle the grating is turned until light thrown through the observing telescope upon the grating is reflected back on itself. The position of the graduated circle is then read. The difference between this reading and the reading when the grating is in such a position that the reflected image of the slit is seen in the telescope is the angle -. If the grating be now turned until the light of which the wave length is required is observed, the angle through which it is turned B from its last position is the angle -. If the width of an element of the grating be known, these measurements substituted in Eq. 117 give the value of A. Fig, 144. 452 elementa:ry physics. [307 Wave lengths are generally given in terms of a unit called a tenth metre; that is, i metre X 10- '°- The wave lengths of the visible spectrum Ije between 7500 and 3900 tenth metres. Langley has found in the lunar radiations wave lengths as long as 170,000 tenth metres, and Rowland has obtained photo- graphs of the solar spectrum in which are lines representing wave lengths of about 3000 tenth metres. Instead of the arrangement which has been described, Rowland has devised a grating ruled on a concave surface, and is thus enabled to dispense with the collimating lens and the telescope. 307. Phenomena due to Diffraction. — The colors exhibited by mother-of-pearl are due to diffraction effects produced by the striated surface. Luminous rings are sometimes seen closely surrounding the sun or moon, due to small globules of vapor or particles of ice in the upper atmosphere. Similar rings may be seen by looking at a small luminous source through a plate of glass strewn with lycopodium powder. CHAPTER V. DISPERSION. 308. Dispersion. — When white hght falls upon a prism of any refracting medium, it is not only deviated from its course but separated into a number of colored lights, constituting an image called a spectrum. These merge imperceptibly from one into another, but there are six markedly different colors: red, orange, yellow, green, blue, and violet. Red is the least and violet the' most deviated from the original course of the light. Newton showed by the recomposition of these colors by means of another prism, by a converging lens, and by causing a disk formed of colored sectors to revolve rapidly, that these colors are constituents of white light, and are separated by the prism because of their different refrangibilities. To arrive at a clear understanding of the formation of this spectrum, let us suppose first a small source of homogeneous light L (Fig. 145). If this light fall on a converging lens from a point Fig. 14s. at a distance from it a little greater than that of the prin- cipal focus, a distinct image of the source will be formed at the distant conjugate focus /. If now a prism be placed in the path of the light, it will, if placed so as to give the minimum 454 ELEMENTARY PHYSICS. [309 deviation, merely deviate the light without interfering with the sharpness of the image, which will now be formed at /' instead of at /. If the source L give two or three kinds of light, the lens may be so constructed as to produce a single sharp image at / of the same color as the source, but when the prism is in- troduced the lights of different colors will be differently deviated and two or three distinct images will be found near /'. If there be many such images, some may overlap, and if there be a great number of kinds of light varying progressively in refrangibility, there will be a great number of overlapping images constituting a continuous spectrum. 309. Dispersive Power. — It is found that prisms of dif- ferent substances giving the same mean deviation of the light deviate the light of different colors differently, and so produce a longer or shorter spectrum. The ratio of the difference be- tween the deviations of the extremities of the spectrum to the mean deviation may be called the dispersive power of the sub- stance. Thus if d' , d" represent the extreme deviations, and d d' - d" the mean deviation, the dispersive power is -3 — . ■r V. 0,1 .sin Arx In § 279 we find the equation —. — -5 — ■ = fi, and referrmg to sm jfxXT Fig. 108 we may set sin Arx = sin {Axr -\- xAr). From the discussion of § 279 it appears that when the prism is in the position of minimum deviation, the angle Axr equals half the A refracting angle of the prism, or — , and the angle xAr equals half the deviation, or -. Hence we obtain . A+d sm 2 . A ' sin — 2 (118) 3io] D/sPE/isiojv. 455 A+d • or when A is small, 2 M- ^ , 2 from which d=A{M-i). Hence we obtain d' - d" A{pi' - I) - A{M" - _L)_ M' - M" A{fA. — i) M— I where /<' and /i" are the refractive indices for the extreme colors, and /n the index for the middle of the spectrum. 310. Achromatism. — If in Newton's experiment of recom- position of white light by the reversed prism the second prism be of higher dispersive power than the first, and of such an angle as to effect as far as possible the recomposition, the light will not be restored to its original direction, but will still be deviated, and we shall have deviation without dispersion. This is a most important fact in the construction of optical instruments. The dispersion of light by lenses, called chromatic aberration, was a serious evil in the early optical instruments, and Newton, who did not think it possible to prevent the dis- persion, was led to the construction of reflecting telescopes to remedy the evil. It is plain, however, from what has been said above, that in a combina- tion of two lenses of different kinds of glass, one converging and the other diverging, one may correct the dispersion of the other within certain limits, while the combination still acts as a converging lens forming real images of Fig. 146. objects. Fig. 146 shows how this principle is applied to the 456 ELEMENTARY PHYSICS. [310 correction of chromatic aberration in the object-glasses of tele- scopes. Thus far nothing has been said of the relative separation of the different colors of the spectrum by refraction by different substances. Suppose two prisms of different substances to have such refracting angles that the spectra produced are of the same length. If these two spectra be superposed, the extreme colors may be made to coincide, but the intermediate colors do not "coincide at the same time for any two substances of which lenses can be made. Perfect achromatism by means of lenses of two substances is therefore impossible. In practice it is usual to construct an achromatic combination to superpose, not the extreme colors, but those that have most to do with the brilliancy of the image. The indistinctness due to chromatic aberration, existing even in the compound objective, may be much diminished by a proper disposition of the lenses of the eyepiece. Fig. 147 shows the negative or Huyghens eyepiece. Let A be the objective of a telescope or microscope. A Fig. 147. point situated on the secondary axis ov would, if the objective were a single lens, have images on that axis, the violet nearest and the red farthest from the lens. If the lens could be per- fectly corrected, these images would all coincide. By making the lens a little over-corrected, the violet may be made to fall beyond the red. Suppose r and v to be the images. B and C are the two lenses o^f the Huyghens eyepiece. B is called the field-lens, and is three times the focal length of C. It is placed 31 1] DISPERSION. 457 Fig. 148. between the objective and its focal plane, and therefore prevents the formation of the images rv, but will form images at r'v' on the secondary axes o'r, o'v. If everything is properly propor- tioned, r'v' will fall on the secondary axis o"R of the eye-lens C at such relative distances as to produce one virtual image at R V. It will be noted that the image r' is smaller than would have been formed by the objective. The magnifying power of the in- strument is therefore less than it would be if the lens C were used alone as the eyepiece. This loss of magnifying power is more than counterbalanced by the increased distinctness. Fig. 148 shows the Ramsden or positive eyepiece. The aid it gives in correcting the residual errors of the objective is evident from the figure. 311. The Rainbow. — The rainbow is due to refraction and dispersion of sunlight by drops of rain. The complete theory of the rainbow is too abstruse to be given here, but a partial explanation may be given. Let O, Fig. 149, represent a drop of water, and SA the paths of the incident light from the sun. The light enters the drop, suffers refraction on entrance, is reflected from the interior surface near B, and emerges near C, as a wave of double curvature of which mn may be taken as the section. Of this wave the part near/, the point of inflection, gives the maximum effect at a distant point, and if the eye be placed in the prolongation of the line CE per- pendicular to the wave surface, light will be perceived, but at a very little distance above or below CE there will be darkness. The direction CE is very nearly that of the minimum deviation Fig. 149. 458 ELEMENTARY PHYSICS. [313 produced by the drop with one internal reflection. It is also the direction in which the angle of emergence equals the angle of incidence. The direction CE corresponds to the minimum deviation for only one kind of light. If this be red light, the yellow will be more deviated, and the blue still more. To see these colors the eye must be higher up, or the drop lower down. If the eye remain stationary, other drops below O will send to it the yellow and blue, and other colors of the spectrum. Since this effect depends only on the angle between the directions SA and CE, it is clear that a similar effect will be received by the eye at E from all drops lying on the cone swept out by the revolution of the line CE and all similar lines drawn to the drops ^'°- '5°' above and below the drop O, about an axis drawn through the sun and the eye, and hence parallel to SA. This cone will trace out the primary rainbow having the red on the outer and the blue on the inner edge. The secondary bow, which is fainter, and appears outside the primary, is pro- duced by two reflections and refractions as shown in Fig. 1 50. 312. The Solar Spectrum. — As has been seen (§ 308) solar light when refracted by a prism gives in general a continuous spectrum. Wollaston, in 1802, was the first to observe that when solar light is received upon a prism through a very narrow opening at a considerable distance, dark lines are seen crossing the otherwise continuous spectrum. Later, in 1814-15, Fraun- hofer studied these lines, and mapped about 600 of them. That these may be well observed in the prismatic spectrum it is im- portant that the apparatus should be so constructed as to avoid as far as possible spherical and chromatic aberrations. The slit must be very narrow, so that its images may overlap as little as possible. The most important condition for avoiding spherical aberration is that the waves reaching the prism should be plane waves, since all others are distorted by refraction at a plane sur- 313] DISPERSION. 459 face. Fig. 1 5 1 shows the disposition of the essential parts of the apparatus known as the spectroscope. S is the slit, which may be considered as the source of light. C is an achromatic lens, called a collimating lens, so placed that 5 is in its principal focus. The waves emerging from it will then be plane. These will be deviated by the prism, and the waves representing the different colors will be separated, so that after passing through the second lens O these different colors will each give a separate M\t\-\m;'l Fig. 131. image. These images may be received upon a screen, or ob- served by means of an eyepiece. Sometimes a series of prisms is used to cause a wider separation of the different images. If the images at F be received on a sensitive photographic plate, it will be found that the image extends far beyond the visible spectrum in the direction of greater refrangibility, and a thermopile or bolometer will show that it also extends a long distance in the opposite direction beyond the visible red. The solar radiations, therefore, do not all have the power of exciting vision. Much the larger part of the solar beam manifests its existence only by other effects. It will be shown that, physi- cally, the various constituents into which white light is separated by the prism differ essentially only in wave length. 313. Spectrum Analysis. — If, in place of sunlight, the light of a lamp or of any incandescent solid, such as the lime of the oxyhydrogen light or the carbons of the electric lamp, illuminate the slit, a continuous spectrum like that produced by sunlight is seen, but the black lines are absent. Solids and liquids give in general only continuous spectra. Gases, however, when incan- 460 ELEMENTARY PHYSICS. [313 descent give continuous spectra only very rarely. Their spectra are bright lines which are distinct and separate images of the slit. The number and position of these lines differ with each gas employed. Hence, if a mixture of several gases not in chemical combination be heated to incandescence, the spectral lines belonging to each constituent, provided all be present in sufficient quantity, will be found in the resultant spectrum. Such a spectrum will therefore serve to identify the constituents of a mixture of unknown composition. Many chemical com- pounds are decomposed into their elements, and the elements are rendered gaseous at the temperature necessary for incan- descence. In that case the spectrum given is the combined spectra of the elements. A compound gas that does not suffer dissociation at incandescence gives its own spectrum, which is, in general, totally different from the spectra of its elements. The appearance of a gaseous spectrum depends in some de- gree on the density of the gas. When the gas is sufficiently compressed, the lines become broader and Ipse their sharply defined edges, and if the compression be still further increased the lines may widen until they overlap, and form a continuous spectrum. Some of the dark lines of the solar spectrum are found to coincide in position with the bright lines of certain elements. This coincidence is absolute with the most perfect instruments at our command, and not only so, but if the bright lines of the element differ in brilliancy the corresponding dark lines of the solar spectrum differ similarly in darkness. The close coincidence of some of these lines was noted as early as 1822 by Sir John Herschel, but the absolute coinci- dence was demonstrated by Kirchhoff, who also pointed out its significance. Platcing the flame of a spirit lamp with a salted wick in the path of the solar beam which illuminated the slit of his spectroscope, Kirchhoff found the two dark lines corre- sponding in position to the two bright lines of sodium to be- come darker, that is, the flame of the lamp had absorbed from 313] DISPERSJON. 461 the more brilliant solar beam light of the same color as it would itself emit. The explanation of the dark lines of the solar spectrum is obvious. The light from the body of the sun gives a continuous spectrum like that of an incandescent solid or liquid. Soniewhere in its course this light passes through an atmosphere of gases which absorbs from the solar beam such light as these gases would emit if they were self-luminous. Some of this absorption occurs in the earth's atmosphere, but most of it is known to occur in the atmosphere of the sun itself. By comparison of these dark lines with the spectra of various incandescent substances upon which we can experiment, the probable constitution of the sun is inferred. CHAPTER VI. ABSORPTION AND EMISSION. < 314. Effects of Radiant Energy.— It has been stated that the solar spectrum, whether produced by means of a prism or by a grating, may, under' certain conditions, give rise to heat, light, or chemical changes. It was formerly supposed that these were due to three distinct agents emanating from the sun,, giv- ing rise to three spectra which were partially superposed. Numerous experiments show, however, that, at any place in the spectrum where light, heat, and chemical effects are pro- duced, nothing which we can do will separate one of these effects from the others. Whatever diminishes the light at any part of the spectrum diminishes the heat and chemical effects also. Physicists are now agreed that all these phenomena are due to vibratory motions transmitted from the sun, which differ in length of wave, and which are separated by a prism, because waves differing in length are transmitted in the sub- stance of the prism with different velocities. The effect pro- duced at any place in the spectrum depends upon the nature of the surface upon which the radiations fall. On the photo- graphic plate they produce chemical change, on the retina the sensation of light, on the thermopile the effect of heat. Only those waves of which the wave lengths lie between 3930 and 7600 tenth metres affect the optic nerve. Chemical changes and the effects of heat are produced by radiations of all wave lengths. 315] ABSORPTION AND EMISSION. 463 To produce any effect the radiations must be absorbed ; that is, the energy of the ethereal vibrations must be imparted to the substance on which they fall, and cease to exist as radiant energy. The most common effect of such absorption is to gen- erate heat, and there are some surfaces upon which heat will be generated by the absorption of ethereal waves of any length. Langley, by means of the bolometer, has been able to measure the energy throughout the spectrum, and has shown the exist- ence of lines like the Fraunhofer lines, in the invisible spectrum below the red. He has demonstrated the existence, in the lunar spectrum, of waves as long as 170,000 tenth metres, or more than twenty-two times as long as the longest that can excite human vision. 315. Intensity of Radiatibns.— The intensity of radiations can only be determined by their effects. If the radiations fall on a body by which they are completely absorbed and con- verted into heat, the amount of heat developed in unit time may be taken as the measure of the radiant energy. Let us suppose the radiations to emanate from a point equally in all directions, and represent the total intensity of the radiations by E. Let the point be at the centre of a hollow sphere, of which the radius is r, and represent by I the intensity of the radia- tions per unit area of the sphere. Then, since the surface of the sphere equals 47rr^ we have E — OfTir'I, and ^=7^^- ('^9) That is, the intensity of the radiation upon a given surface is in the inverse ratio of the square of its distance from the source. 464 ELEMENTARY PHYSICS. [316 Fig. 152. If the surface is not normal to the rays, the radiant energy it receives is less, as will ap- pear from Fig. 152. l^etadhe a surface the normal to which makes with the ray the angle 6; then ad will receive the same quantity of radiant en- ergy as a'd', its projection on the plain normal to the ray. But a'b' equals ad cos 0; and if / represent the intensity on a'd', and /' the intensity on ad, we have /'=/cos6'; or, the intensity of the radiations falling on a given surface is proportional to the cosine of the angle made by the surface and the plane normal to the direction of the rays. 316. Photometry. — The object ol photometry is to compare the luminous effects of radiations. It is not supposed that the radiations which fall on the retina are totally absorbed by the nerves that impart the sensation of light. The luminous effects, therefore, depend on the susceptibility of these nerves, and can only be compared, at least when different wave lengths are concerned, by means of the eye itself. The photometric comparison of two luminous sources is effected by so placing them that the illuminations produced by them respectively, upon two surfaces conveniently placed for observation, appear to the eye to be equal. If E and E' represent the intensities of the sources, / and /' the intensities of the illuminations pro- duced by them on surfaces at distances r and r' , the ratio be- tween these intensities, as was seen in the last section, is E^ I _ r' _ Er^ r" 3I0J ABSORPTION AND EMISSION. 4^5 and when / and /' are equal, Er'' = £V», or E E = (120) That is, when two luminous sources are so placed as to produce equal illuminations on a surface, their intensities are as the squares of their distances from the illuminated surfaces, Ricmford's phototneter consists of a screen in front of which is an upright rod. The luminous sources are so placed that the rod casts two shadows near together upon the screen, and are adjusted at such distances that these shadows are apparently equal in intensity. In Foucaulf s photometer the screen is of ground glass, and in place of the rod a vertical partition is placed in front of and perpendicular to the middle of the screen. The luminous sources are so placed that one illuminates the screen on one side of the partition, and the other on the other. The parti- tion may be moved to or from the screen until the two illumi- nated portions just meet without overlapping. In Bunsens photometer the sources to be compared are placed on the opposite sides of a paper screen, a portion of which has been rendered translucent by oil or parafifine. When this screen is illuminated upon one side only, the translucent portion appears darker on that side, and lighter on the other side, than the opaque portion. When placed between two luminous sources, both sides of it may, by moving it toward one or the other, be made to appear alike, and the translucent portion almost invisible. The light transmitted through this portion in one direction then equals that transmitted in the opposite direction ; that is, the two surfaces are equally illumi- nated. 30 466 ELEMENTARY PHYSICS. [317 317. Transmission and Absorption of Radiations. — It is a familiar fact that colored glass transmits light of certain colors only, and the inference is easy that the other colors are ab- sorbed by the glass. It is only necessary to form a spectrum, and place the colored glass in the path of the light either before or after the separation of the colors, to show which colors are transmitted, and which absorbed. By the use of the thermopile or bolometer, both of which are sensitive to radiations of all periods of vibration, it is found that some bodies are apparently perfectly transparent to light, and opaque to the obscure radiations. Clear, white glass is opaque to a large portion of the obscure rays of long wave length. Water and solution of alum are still more opaque to these rays, and pure ice transmits almost none of the radiations of which the wave lengths are longer than those of the visible red. Rock salt transmits well both the luminous and the non-luminous radiations. On the other hand, some substances apparently opaque are transparent to radiations of long wave length. A plate of glass or rock salt rendered opaque to light by smoking it over a lamp is still as transparent as before to the radiations of longer wave length. Selenium is opaque to light, but trans- parent to the radiations of longer wave length. This fact ex- plains the change of its electrical resistance by light, but not by non-luminous rays. Carbon disulphide, like rock salt, transmits nearly equally the luminous and non-luminous rays ; but if iodine be dissolved in it, it will at first cut of? the lu- minous rays of shorter wave length, and as the solution be- comes more and more concentrated the absorption extends down the spectrum to the red, and finally all light is extin- guished, and the solution to the eye becomes opaque. The radiations of which the wave lengths are longer than those of the red still pass freely. Black vulcanite seems perfectly opaque, yet it also transmits radiations of long wave length. 319] ABSORPTION AND EMISSION. 4^7 If the radiations of the electric lamp be concentrated by means of a lens, and a sheet of black vulcanite placed between the lamp and the lens, bodies may be still heated in the focus. 318. Colors of Bodies.^Bodies become visible by the light which comes from them to the eye, and bodies which are not self-luminous must become visible by sending to the eye some portion of the light that falls on them. Of the light which falls on a body, part is reflected from the surface ; the re- mainder which enters the body is, in general, partly absorbed, and the unabsorbed portion either goes on through the body, or is turned back by reflection at a greater or less depth within the body, and mingles with the light reflected from the sur- face. In general the surface reflection is small in amount, and the different colors are reflected almost in the proportion in which they exist in the incident light. Much the larger portion of the light by which a body becomes visible is turned back after penetrating a short distance beneath the surface, and contains those colors which the substance does not absorb. This deter- mines the color of the object. In a few instances there seems to be a selective reflection from the surface. For example, the light reflected from gold-leaf is yellow, while that which it trans- mits is green. 319. Absorption by Gases. — If a pure spectrum be formed from the white light of the electric lamp, and sodium vapor, obtained by heating a bit of sodium or a bead of common salt in the Bunsen flame, be placed in the path of the beam, two narrow, sharply defined dark lines will be seen to cross the spectrum in the exact position that would be occupied by the yellow lines constituting the spectrum of sodium vapor. Gases in general have an effect similar to that of the vapor of sodium ; that is, they absorb from the light which passes through them distinct radiations corresponding to definite wave lengths, which are always the same as those which would be emitted by the 468 ELEMENTARY PHYSICS. [320 gas were it rendered incandescent. It has been seen already (§313) that the Fraunhofer Hnes of the solar spectrum are thus accounted for. 320. Emission of Radiations. — Not only incandescent bodies, but all bodies at whatever temperature they may be, emit radiations. A warm body continues to grow cool until it arrives at the temperature of surrounding bodies, and then if it be moved to a place of lower temperature, it cools still further. To this process we can ascribe no limit, and it is necessary to admit that the body will radiate heat, and so grow cooler, what- ever its own temperature, if only it be warmer than surrounding bodies. But it cannot be supposed that a body ceases to radiate heat when it comes to the temperature of surrounding bodies, and begins again when the temperature of these is lowered. It is necessary, therefore, to assume that all bodies at whatever temperature are radiating heat, and that, when any one of them arrives at a stationary temperature, it is, if no change take place within it involving the generation or consumption of heat, receiving heat as rapidly as it parts with it. This is called the principle of movable equilibrium of temperature. We know that if a number of bodies, none of which are generating or consum- ing heat otherwise than in change of temperature, be placed in an inclosure the walls of which are maintained at a constant-- temperature, these bodies will in time all come to the tempera- ture of the inclosure. It can be shown that, for this to be true, the ratio of the emissive to the absorbing power must be the same for all bodies, not only for the sum total of all radiations, but for radiations of each wave length. For example, a body which does not absorb radiations of long wave length cannot emit them, otherwise, if placed in an inclosure where it could only receive such radiations, it would become colder than other bodies in the same inclosure. This is only a general statement of the fact which has been already stated for gases, that bodies ab- sorb radiations of exactly the same kind as those which they emit. 320] ABSORPTION AND EMISSION. 469 Since radiant energy is energy of vibratory motion, it may be supposed to have its origin in the vibrations of the molecules of the radiating body. In § 156 it was shown that the various phenomena of gases are best explained by assuming a constant motion of their molecules. If these molecules should have definite periods of vibration, remaining constant for the same gas through wide ranges of pressure and temperature, this would fully explain the peculiarities of the spectra of gases. In § 261 it was seen that a vibrating body may communicate its vibrations to another body which can vibrate in the same period, and will lose just as much of its own energy of vibration as it imparts to the other body. Moreover, a body which has a definite period of vibration is undisturbed by bodies vibrat- ing in a period different from its own. This explains fully the selective absorption of a gas. For, if a beam of white light pass through a gas, there are, among the vibrations constituting such a beam, some which correspond in period to those of the molecules of the gas, and, unless the energy of vibration of these molecules is already too great, it will be increased at the expense of the vibrations of the same period in the beam of light. Hence, at the parts in the spectrum where light of those vibration periods would fall, the light will be enfeebled, and those parts will appear, by contrast, as dark lines. In solids and liquids, the molecules are so constrained in their movements that they do not vibrate in definite periods. Vibrations of all periods may exist ; but if in a given case there were a tendency to one period of vibration more than to an- other, it is evident that the body would transfer to or receive from another, that is, it would emit or absorb, vibrations of that period more than of any other. Furthermore, a good radiator is a body so constituted as to impart to the medium around it the vibratory motion of its own molecules. But the same pecu- liarity of structure which fits it for communicating its own motion to the medium when its own motion is the greater, fits 47° ELEMENTARY PHYSICS. [321 it also for receiving motion from the medium when its own motion is the less. Theory, therefore, leads us to the conclu- sion which experiment has established, that at a given temper- ature emissive and absorbing powers have the same ratio for all bodies. 321. Loss of Heat in Relation to Temperature.— The loss of heat by a body is the more rapid the greater the differ- ence of temperature between it and surrounding bodies. For a small difference of temperature the loss of heat is nearly pro- portional to this difference. This law is known as Newton's law of cooling. For a large difference of temperature the loss of heat increases more rapidly than the difference of tempera- ture, and depends not merely upon this difference, but upon the absolute temperature of the surrounding bodies. An ex- tended series of experiments by Dulong and Petit led to a for- mula expressing the quantity of heat lost by a body in an in- closure during unit time. It is e t Q = m{i.oo77) (1.0077 — i)' where represents the temperature of the inclosure, t the dif- ference of temperature between 'the inclosure and the radiating body, both measured in Centigrade degrees, and m a constant depending on the substance, and the nature of its surface. 322. Kind of Radiation as Dependent upon Tempera- ture. — When a body is heated we may feel the radiations from its surface long before those radiations render the body visible. If we continue to raise the temperature, after a time the body becomes red hot ; as the temperature rises still further it becomes yellow, and finally attains a white heat. Even this rough ob- servation indicates that the radiations of great wave length are the principal radiations at the lower temperature, and that to these are . added shorter and shorter wave lengths as the tern- 322] ABSORPTION AND EMISSION. 4/1 perature rises. Draper showed that the spectrum of a red-hot body exhibits no rays of shorter wave length than the red, but that as the temperature rises the spectrum is extended in the direction of the violet, the additions occurring in the order of the wave lengths. At the same time the colors previously existing increase in brightness, indicating an increase in energy of the vibrations of longer wave length as those of shorter wave length become visible. Experiments by Nichols on the radia- tions from glowing platinum show that vibrations of shorter wave length are not altogether absent from the radiations of a body of comparatively low temperature, and he was led to believe that all wave lengths are present in the radiations from even the coldest bodies, but are too feeble to be detected. With gases, as has been seen, the radiations are apparently confined to a few definite wave lengths, but careful observa- tions of the spectra of gases show that the lines are not defined with absolute sharpness, but fade away, although very rapidly, into the dark background. In many cases the existence of ra- diations may be traced throughout the spectrum, and it is a ques- tion whether the spectra of gases are not after all continuous, only showing strongly marked and sharply defined maxima where the lines occur. In general, increase of temperature does not alter the spectra of gases except to increase their intensity, but there are some cases in which additional lines appear as the temperature rises, and a few cases in which the spectrum under- goes a complete change at a certain temperature. This occurs with those compound gases which suffer dissociation at a cer- tain temperature, and at higher temperatures give the spectra of their elements. When it occurs with gases supposed to be elements it suggests the question whether they are not really compounds, the molecules of which at the high temperature are divided, giving new molecules of which the rates of vibra- tion are entirely different from those of the original body. 472 ELEMENTARY PHYSICS. [323 323. Fluorescence and Phosphorescence. — A few sub- stances, such as sulphate of quinine, uranium glass, and thallene, have the property, when illuminated by rays of short wave length, even by the invisible rays beyond the violet, of emit- ting light of longer wave length. Such substances dixe. fluorescent. The light emitted by them, and the conditions favorable to their luminosity, have been studied by Stokes. It appears that the light emitted is of the same character, covering a considerable region of the spectrum, no matter what may be the incident light, provided this be such as to produce the effect at all. The light emitted is always of longer wave length than that which causes the luminosity. There is another class of substances which, after being ex- posed to light, will glow for some time in the dark. These are phosphorescent. They must be carefully distinguished from such bodies as phosphorus and decaying wood, which glow in consequence of chemical action. Some phosphorescent sub- stances, especially the calcium sulphides, glow for several hours after exposure. 324. Anomalous Dispersion. — As has been already stated, there is a class of bodies which show a selective absorption at their surfaces. The light reflected from such bodies is comple- mentary to the light which they can transmit. Kundt, follow- ing up isolated observations of other physicists, has shown that all such bodies give rise to an anomalous dispersion ; that is, the order of the colors in the spectrum formed by a prism of one of these substances is not the same as their order in the diffraction spectrum or in the spectrum formed by prisms of substances which do not show selective absorption at their surfaces. Solid fuchsin, when viewed by reflected light, appears green. In solution, when viewed by transmitted light, it ap- pears red. Christiansen allowed light to pass through a pristn formed of two glass plates making a small angle with each other, and containing a solution of fuchsin in alcohol. He 324] ABSORPTIOiY A .YD EMISSION. 473 found that the green was almost totally wanting in the spec- trum, while the order of the other colors was different from that in the normal spectrum. In the spectrum of fuchsin the colors in order, beginning with the one most deviated, were violet, red, orange, and yellow. Other substances give rise to anomalous dispersion in which the order of the colors is dif- ferent. In order to account for these phenomena, the ordinary theory of light is extended by the assumption that the ether and molecules of a body materially interact upon one another, so that the vibrations in a light- wave are modified by the vibra- tions of the molecules of a transparent body through which light is passing. This hypothesis, in the hands of Helmholtz and Ketteler, has been sufficient to account for most of the phenomena of light. CHAPTER VII. DOUBLE REFRACTION AND POLARIZATION. 325. Double Refraction in Iceland Spar.— If refraction take place in a medium which is not isotropic, as has been assumed in the previous discussion of refraction, but eolo- tropic, a new class of phenomena arises. Iceland spar is an eolotropic medium by the use of which the phenomena re- ferred to are strikingly exhibited. Crystals of Iceland spar are rhombohedral in form, and a crystal may be a perfect rhom- bohedron with six equal plane faces, each of which is a rhombus. Fig. 153 represents such a crystal. At A and Xare two solid angles formed by the obtuse angles of three plane faces. The line through A making equal angles with the three edges AB, AE, AD, or any line parallel to it, is an optic axis of the crys- tal. 326] DOUBLE REFRACTION AND POLARIZATION. 475 Any plane normal to a surface of the crystal and parallel to the optic axis is called -a^ principal plane. If such a crystal be laid upon a printed page, the lines of print will, in general, ap- pear double. If a dot be made on a blank paper, and the crys- tal placed upon it, two images of the dot are seen. If the crystal be revolved about an axis perpendicular to the paper, one of the images remains stationary, and the other revolves around it. The images lie in a plane perpendicular to the paper, and parallel to the line joining the two obtuse angles of the face by which the light enters or emerges. The entering and emerging light is supposed in this case to be normal to the surfaces of the crystal. If the crystal be turned with its faces oblique to the light, the line joining the images will, in certain cases, not lie parallel to the line joining the obtuse angles of the faces. If the distances of the two images from the observer be carefully noticed it will be seen that the stationary one appears nearer than the other. If the obtuse angles A and X be cut away, and the new surfaces thus formed at right angles to the optic axis be polished, images seen perpendicularly through these faces do not appear double. By cutting the crystals into prisms in various ways its indices of refraction may be measured. It is found that, of the two beams into which light is, in general, divided in the crystal, one obeys the ordinary laws of refraction, and has a refractive index 1.658. It is called the ordinary ray. The other has no constant refractive index, does not in general lie in the normal plane containing the incident ray, and refrac- tion may occur when the incidence is normal. It is the extra- ordinary ray. The ratio between the sines of the angles of in- cidence and refraction varies, for the Fraunhofer line D, from 1.658, the ordinary index, to 1.486. This minimum value is called the extraordinary index. 326. Explanation of Double Refraction.— In § 279 it was seen that the index of refraction of a substance is the reciprocal of the ratio of the velocity of light in the substance to its 4/6 ELEMENTARY PHYSICS. [327 velocity in a vacuum. It is plain, then, that the velocity of light for the ordinary ray of the last section is the same for all directions, and, if light emanate from a point within the crystal, the light, following the ordinary laws of refraction, must proceed in spherical waves about that point as a centre, as in any single refracting medium. The phenomena presented by the extra- ordinary light in Iceland spar are fully explained by assuming that the velocities in different directions in. the crystal are such as to give a wave front in the form of a flattened spheroid, of which the polar diameter, parallel to the optic axis, is equal to the diameter of the ordinary spherical wave, and the equatorial Fig. 154. diameter is to its polar diameter as 1.658 is to 1.486. From these two wave surfaces the path of the light may easily be de- termined by construction by methods already explained in § 279, and exemplified in Fig. 154, in which ic represents the di- rection of the incident light, and co and ce the ordinary and extraordinary rays respectively. 327. Polarization of the Doubly Refracted Light. — If a second crystal be placed in front of the first in any of the ex- periments described in the last section, there will be seen in general four images instead of two; but if the second crystal be turned, the images change in brightness, and for four positions of the second crystal, when its principal plane is parallel or at 327] DOUBLE KEFRACTION AND POLARIZATION. 47/ right angles to the principal plane of the first, two of the images are invisible, and the other two are at a maximum brightness. If one of the beams of light produced by the first crystal be intercepted by a screen, and the other allowed to pass alone through the second crystal, the phenomena presented are easil}' followed. If the principal planes of the two crystals coincide, only one image is seen. If the second crystal be now rotated about the beam of light as an axis, a second image at once ap- pears, at first very faint, but increasing in brightness. The origi- nal image at the same time diminishes in brightness, and the two are equally bright when the angle between the principal planes is 45". If the angle be 90° the first image disappears, and the second is at its maximum brilliancy. As the rotation is continued the first image reappears, while the second grows dim and disappears when the angle between the principal planes is 180°. These changes show that the light which emerges from the first crystal of spar is not ordinary light. Another experi- ment shows this in a still more striking manner. Let the extra- ordinary ray be cut off by a screen, and the ordinary ray be received on a plane unsilvered glass at an angle of incidence of 57". When the plane of incidence coincides with the principal plane of the spar, the light is reflected like ordinary light. If the mirror be now turned about the incident ray as an axis, that is, so turned that, while the angle of incidence remains unchanged, the plane of incidence makes successively all pos- sible angles with the principal plane of the crystal, the re- flected light gradually diminishes in brightness, and when the angle between the plane of incidence and the principal plane of the crystal is 90° it fails altogether. If the rotation be con- tinued it gradually returns to its original brightness, which it attains when the angle between the same planes is 180°, and then diminishes until it fails when the angle is 270?. The ex- traordinary ray presents the same phenomena except that the reflected light is brightest when the angle between the planes is 478 ELEMENTARY PHYSICS. [327 90° and 270°, and fails when that angle is 0° and 180°. Beams of light after double refraction present different properties on different sides, and are said to be polarized. The explanation must, of course, be found in the character of the vibratory motion. In the polarized beam it is plain that the vibrations must be transverse; for if the light were the result of longitudinal vibrations, or even of vibrations having a longitudinal com- ponent, it could not be completely extinguished for certain azimuths of the second crystal or of the glass reflector. The difference between ordinary and polarized light is explained if we assume that in both the vibrations of the ether particles take place at right angles to the line of propagation of the wave, and that in ordinary light they occur successively in all azimuths about that line, and may be performed in ellipses or circles as well as in straight lines, while ih polarized light they occur in one plane. In the ordinary ray in Iceland spar the vibrations are in a plane at right angles to the optic axis. In the extraordinary ray they are in the plane containing the optic axis and the ray. The equation v = \ ~ holds for tran vibrations, if by ^ be understood the modulus of rigidity of the medium. If we assume that the modulus of rigidity at right angles to the optic axis is a minimum, and along the optic axis a maximum, and varies between these two directions according to a simple law, all the phenomena of double refrac- tion and polarization in the crystal are. accounted for. If a crystal be cut so as to present faces parallel to the optic axis, and if light enter along a normal to one of these faces, the vibrations, which previous to entering the crystal were in all azimuths, are resolved in it in two directions, that of great- est and that of least elasticity, or parallel to and at right angles to the optic axis. The wave made up of vibrations parallel to the optic axis is propagated with the greater sverse 327] DOUBLE REFRACTION AND POLARIZATION. 479 velocity. In this case the two wave fronts continue in parallel planes, and upon emergence constitute apparently one beam of light. If the incidence be oblique and in a plane at right angles to the principal plane, the two component vibrations are still parallel to and at right angles to the optic axis, but refraction occurs which is greater for the ray of which the vibrations are in the direction of least elasticity. If the inci- dence be oblique and in the principal plane, it is evident that there may be a component vibration at right angles to the optic axis, but the other component, since it must be at right angles to the ray, cannot be parallel to the optic axis, and therefore cannot be in the direction of greatest elasticity in the crystal. The second component is, however, in the direc- tion of greatest elasticity in the plane of vibration, which direc- tion is at right angles to the first component. In general, if a ray of light pass in any direction within the crystal, the line drawn at right angles to that direction and to the optic axis, that is, at right angles to the plane determined by the ray and the optic axis, is in the direction of least elasticity. One of the component vibrations is in that direction. A line drawn at right angles to the ray and in the plane formed by it and the optic axis is in the direction of the greatest elasticity to which any vibration giving rise to that ray of light can corre- spond. In that direction is the second component vibration. The two component vibrations are therefore always at right angles. One of the components is always at right angles to the optic axis, and hence in the direction of least elasticity. The light resulting from this component always travels with the same velocity whatever its direction, and hence suffers re- fraction on entering the crystal or emerging from it, according to the ordinary law for single refraction. The other component, being in the plane containing the ray and the optic axis and at right angles to the ray, may make all angles with the optic axis from 0° when it is in the direction of maximum elasticity and is 480 ELEMENTARY PHYSICS. [32S propagated with the greatest velocity, to 90" when it is in a direc- tion in which the elasticity is the same as that for the other component, and the entire beam is propagated as ordinary light. Light for which vibrations occur in all azimuths will, on enter- ing the crystal, give rise to equal components, but light already polarized will give rise to components the intensities of which are determined by the law for the resolutions of motions. When its own direction of vibration coincides with that of either of the components, the other component will be zero, and only when its vibrations make an angle of 45° with the compo- nents can these components be equal. The varying intensi- ties of the two beams into which a polarized beam is divided by a second crystal are thus explained. 328. Polarization by Reflection. — Light reflected from a transparent medium is found in general to be partially polarized, and for a certain angle of inci- dence the polarization is perfect. This angle is that for which the reflected and refracted rays are at right angles. In Fig. 155 let xy Fig. 155. " represent the surface of a trans- parent medium, ab the incident, fc the reflected, and bd the re- fracted ray. If the angle cbd=<^o°, we have r -\- i= 90° also ; sin i , sin i ■ tt 1 and since u = -. — , we have w = . = tan t. hlence the '^ sm r cos t angle of complete polarization is given by the equation tan i — jx. The fact embodied in this equation was discovered by Brewster, and is known as Brewster's law. The angle of com- plete polarization is called the polarizing angle. The plane of incidence is the plane of polarization. The vibrations of polar- ized light are at right angles to the plane of polarization. In the transmitted ray is an equal amount of polarized light the vibrations of which are in the plane of incidence. 329] DOUBLE REFRACTION AND POLARIZATION. 48 1 If a beam of ordinary light traverse a transparent medium, in which are suspended minute solid particles, the light which is reflected from them is found to be partially polarized. The maximum polarization is found in the light reflected at right angles to the beam. The plane of polarization of the polarized beam is the plane of the original beam and the beam which reaches the eye of the observer. 329. Polariscopes. — In experimenting with polarized light we need a polarizer to produce the polarized beam, and an analyzer to show the effects of the polarization. A piece of plane glass, reflecting hght at the polarizing angle, is a simple polarizer. Double refracting crystals, if means be employed Fig. 156. to suppress one of the beams into which the light is divided, are excellent polarizers. Tourmaline is a double refracting crystal which has the property of being more transparent to the extraordinary than to the ordinary ray. By grinding plates of tourmaline to the proper thickness, the ordinary ray is com- pletely absorbed, while the extraordinary ray is transmitted. The best method of obtaining a polarized beam is by the use of a crystal of Iceland spar in which, by an ingenious device, the ordinary ray is suppressed, and the extraordinary transmitted. Fig. 156 shows how this is accomplished. AB is a crystal of considerable length. It is divided along the plane ^5 making an angle of 22° with the edge AD and perpendicular to a prin- cipal plane of the face AC. The faces of the cut are polished and the two halves cemented together again by Canada balsam 31 482 ELEMENTARY PHYSICS. [339 in the same position as at first. In Fig. 157, which is a section through ACBD of Fig. 156, ab represents the direction of the light which is incident upon the face AC. It is separated into the two rays o and e. Since the refractive index of the balsam is intermediate between the ordinary and extraordinary in- dices of the spar, and since the angle DAB is so chosen that Fig. 157. the ray strikes the balsam at an angle of incidence greater than the critical angle, the ray o is totally reflected. The ray e, on the other hand, having a refractive index in the spar less than in the balsam, is not reflected, but continues through the crystal. A crystal of Iceland spar so treated is called a Nicol's prism, or often simply Nicol. The Foucault prism is simila . to the Nicol, except that the two halves after polishing are not cemented together, but are mounted with a film of air between. The total reflection of o now occurs at a much less angle of incidence. The section AB is, therefore, much less oblique, and a shorter crystal serves for the construction of the prism. It will be observed that the section AB must be so made that the angle of mcidence of c shall be greater, and of e less, than the corresponding critical angle. Since the two critical angles are nearly the same, but little variation in the angle of incidence of o and e is permissi- ble, and the Foucault prism is, therefore, only useful for par- allel rays. A pair of Nicol's prisms, mounted with their axis coincid- ing, serve as a polariscope. The first Nicol transmits a single 330] DOUBLE REFRACTION AND POLARIZATION. 483 beam of polarized light the vibrations of which are in the prin- cipal plane. When the principal plane of the second Nicol co- incides with that of the first this light is wholly transmitted through it. If the second Nicol or analyzer be turned about its axis, whenever its principal plane makes an angle with the direction of the vibrations, these are resolved into two com- ponents, one in and the other at right angles to the principal plane. The latter is reflected to one side and absorbed, and the former is transmitted. As the angle between the two prin- cipal planes increases, the transmitted component diminishes in intensity, until when this angle becomes 90° it disappears entirely. In this position the polarizer and analyzer are said to be crossed. 330. Effects of Plates of Doubly Refracting Crystals on Polarized Light. — If a plate cut from a doubly refracting sur- face so that its faces are parallel to the optic axis, or at least not at right angles to it, be placed between the crossed polar- izer and analyzer, if the principal plane of the plate coincides with, or is at right angles to, the plane of vibration, no effect is perceived. But if the plate be rotated so that its principal plane makes an angle with the plane of vibration, the motion may be considered to be resolved into two components, one in, and the other at right angles to, the principal plane of the plate, and these two components on reaching the analyzer are again resolved each into two others, one in, and the other at right angles to, the principal plane of the analyzer. The vibra- tions in the principal plane of the analyzer are transmitted through it, and hence, in general, the introduction of the plate restores the light which the crossed polarizer and analyzer had extinguished. It is ea.'i) to see that the restored light will be most intense when the principal plane of the plate makes an angle of 45° with the plane of vibration of the polarized ray. It is not to be understood that in the plate there are two separate beams of light, in one cf which one set of particles is 484 ELEMENTARY PHYSICS. [330 vibrating in one plane, and in the other another set in another plane. What really takes place is that each particle in the path of the light describes a path which is the resultant of the two components spoken of above. Let ab, Fig. 1 58, be a plate of Iceland spar, and cd the direction of its optic axis. Suppose the path of the light perpendicular to the plane of the paper, and ef to represent the direction of the disturbance produced by the d entrance of a plane polarized wave. A motion in the direction of ef is compounded of two motions, one along the axis, and the other per- pendicular to it. In the propagation of this motion to the, next particle, the motion in the direction of the optic axis will begiri a little Fig. 158. sooner than that at right angles because of the greater elasticity in the former direction, and this difference becomes greater as the light is propagated into the plate. This is equivalent to a change in the relative phases of two vibra- tions at right angles, and this causes the path of a vibrating particle to change from the straight line to an ellipse. The result is, therefore, that, when the initial disturbance has any direction except in or at right angles to the principal plane of the plate, the motion of the vibrating particles within the plate becomes elliptical, the ellipses changing form as the dis- tance from the front surface of the plate increases. It is en- tirely admissible, however, in the discussion of the problem to substitute for the actual motion its two components, as was done above. It remains to consider what is the effect of the retardation or change of phase of one of the components with respect to the other. It will be remembered that in the analyzer each ray from the plate is again resolved into two components, and that two of these components are in the principal plane of the ana- lyzer and are transmitted. These two components will evi- dently differ in phase just as did the two motions from which 330] DOUBLE REFRACTION AND POLARIZATION. 485 they were derived, and since they are in the same plane their resultant is represented by their algebraic sum. If they differ in phase by half a period their algebraic sum will be zero, and no light will be transmitted by the analyzer. This will occur for a certain thickness of the interposed plate. If the light experimented upon be white, it may occur for some wave lengths and not for others. Hence, some of the constituents of white light may fail in the beam transmitted by the analyzer, and the image of the plate will then appear colored. A study of the resolution of the vibrations for this case shows that, of the two beams formed in the analyzer, one contains just that portion of the light that the other lacks ; hence if the analyzer be turned through 90°, the image will change to the comple- mentary color. In Fig. 159, let ab represent the plane of the vibrations in the polarized ray, and let cd and ef represent the two planes of vibration of the rays in the in- terposed plate. At the instant of entering the plate, the primary vibration and its two compo- nents will have the relation shown in the figure. The two components are then in the same phase. As the movement penetrates the plate, one com- ponent falls behind the other, and the relation of their phases changes, until, with a retardation of ^"^' '^* one wave length, the phases are again as in the figure. Sup- pose the thickness of the plate such that this retardation occurs for some constituent of white light. After leaving the plate the relative phases of the components remain unchanged and the constituent in question enters the analyzer as two vibra- tions at right angles and in the same phase. In Fig. 160, let oe and od represent the two components, and xx and yy the two planes of vibration in the analyzer. 0^ will give the components om and on, and od the components oin' and on'. Since the com- ponents om and om' annul one another, the color to which they correspond is wanting in the light resulting from vibrations in / , , \ 486 ELEMENTARY PHYSICS. [331 the plane xx, while since the components on and o«' are added, this color is found in full intensity among the vibrations in the plane yy. For light of other wave lengths, the relative retardation is "T* different, but for each vibration ,''' j period, the component in the di- i rection xx combined with that in TO the direction yy represents the ''' i "d total light for that period in the \y beam entering the analyzer; that Fig. 160. is, the total effect of vibrations in the direction xx combined with that of vibrations in the direc- tion yy must produce white light, and one effect must, there- fore, be the complement of the other. Let us suppose the plate thick enough to cause a retarda- tion equal to a certain number of wave lengths, which we will assume to be ten, of the shortest waves of the visible spec- trum. Since the longest waves of the visible spectrum are about twice the length of the shortest, they will suffer a retar- dation of five wave lengths. Other waves will suffer a retar- dation of nine, eight, seven, and six wave lengths. But, as was seen above, a retardation of one or more whole wave lengths of any kind of light causes extinction of that kind of light in the beam transmitted by the crossed analyzer. In the case considered the transmitted beam will lose six kinds of light distributed at about equal distances along the spectrum. The light remaining will consist of the different colors in about the same proportions as they exist in white light, and the beam will therefore be white but diminished in intensity. Hence, when a thick plate is interposed between the crossed polarizer and analyzer the restored light is white. 331. Elliptic and Circular Polarization. — In the last sec- tion, in discussing. the effects of a thin plate, we considered the two components of the vibratory motion propagated from it. It 331] DOUBLE REFRACTION AND POLARIZATION. 487 was stated that the real motion of the vibrating particles was in general elliptical. Let us consider more fully the real mo- tion. Let us suppose that the light is light of one wavelength only, and that, as before, the principal plane of the plate makes an angle of 45° with the plane of vibration of the incident light. In Fig. 161 let jy represent the original plane of vibration, and ab and cd the planes of maximum and minimum elasticity in the plate. As already explained, the first disturbance as the light enters the plate is in the direction yy\ but as the disturbance is propagated into the plate, each disturbed particle receives an impulse first of all in the direction cd of greatest elasticity, then in other directions between cd and ab, and finally in the direc- tion ab. From this results an elliptical or- y \* bit with the major axis in the direction yy. To determine this orbit exactly it is only necessary to take account of the time that elapses between the impulse in the direction cd and the corresponding impulse in the direction ab. It is suffi- cient to consider any particle as actuated by two vibratory mo- tions in the directions cd and ab at right angles, and differing in phase. In Fig. 161, one side of the rect- angle represents the greatest displace- ment in the direction cd, and the other side the displacement occurring at the «-• same instant in the direction ba. The point r will represent the actual position of the vibrating particle. Constructing now the successive displacements of the particles in the directions cd and ba and combining these, we have the elliptical path as shown. As the light penetrates farther and farther into the plate the relative phases of the two vibrations change continually, and the ellipse passes through y Fig. 161. .d y Fig. 162. 488 ELEMENTARY PHYSICS. l33' all its forms from the straight line yy to the straight line xx at right angles to it and back to the straight line yy. The direction of the path of the particle in the surface of the plate as the light emerges will be the direction of the path of all the particles in the polarized beam beyond the plate. If the component vibrations be in the same phase, that is, if they reach their elongations in the directions ba and cd (Fig. 162) at the same instant, the resultant vibration is in the line yy and the hght is plane polarized exactly as it left the polarizer. This will occur when the retardation of light in the plane of ba with respect to that in the plane of cd is one, two, or more whole wave lengths. When the retardation is one half, three halves, or any odd number of half wave lengths, the phases of the two vibrations are as shown in Fig. 163, and the resultant is a plane polarized beam the vibrations of which are at right angles to those of the beam from the polarizer. A case of special interest is shown in Fig. 164, in which the difference of pha.se is one fourth a period, and the result- ant vibration is a circle. Fig. 163. A difference of three fourths will give a circle also, but with the rotation in the opposite direction. A plate of such thickness as to produce a retardation of one quarter of a wave length will give a circular vibration, and the beam issuing from the plate is then circularly polarized. Its peculiarity is that the two beams into which it is divided by a double refracting crystal are always of the same intensity, and no form of ana- lyzer will distinguish it from ordinary light. Quarter wave plates are often made by splitting sheets of mica until the re- quired thickness is obtained. r i X Fic. 164. 334] DOUBLE REFRACTION AND POLARIZATION. 4^9 332- Circular Polarization by Reflection.— It has been seen that light reflected from a transparent medium at a cer- tain angle is polarized, and that an equal amount of polarized light exists in the refracted beam. Light totally reflected in the interior of a medium is also polarized, and here, there being no refracted beam, the two components exist in the reflected light, but so related in pha;se that the Hght is elliptically polar- ized. Fresnel has devised an apparatus known as FresneVs rhomb, by means of which circularly polarized light is obtained by two internal reflections of a beam of light previously polar- ized in a plane at an angle of 45° with the plane of incidence. 333. Effect of Plates Cut Perpendicularly to the Axis from a Uniaxial Crystal. — A crystal, such as Iceland spar, which has but one optic axis, is called a uniaxial crystal. Polar- ized light passing perpendicularly through a plate cut from such a crystal perpendicularly to its optic axis suffers no change. If, however, the plate between the crossed polarizer and analyzer be inclined to the direction of the beam, light passes through the analyzer. It is generally colored, the color changing with the obliquity of the plate. If a system of lenses be used to convert the polarized beam into a conical pencil and the plate be placed in this perpendicular to its axis, the central ray of the pencil will be unchanged, but the oblique rays will be resolved except in and^at right angles to the plane of vibration, and there will appear beyond the analyzer a system of colored rings surrounding a dark centre, and intersected by a black cross. If the analyzer be turned through 90°, a figure comple- mentary to the first in all its shades and tints is obtained : the black cross and centre become white, and the rings change to complementary colors. 334. Biaxial Crystals. — Most crystals have two optic axes or lines of no double refraction, instead of one. They are biaxial crystals. Their optic axes may be inclined to each other at any angle from 0° to 180°. The wave surfaces within 49° ELEMENTARY PHYSICS. [335 these crystals are no longer the sphere and the ellipsoid, but surfaces of the fourth order with two nappes tangent to each other at four points where they are pierced by the optic axes. Neither of the two rays in such a crystal follows the law of or- dinary refraction. The outer wave surface around one of the points of tangency has a depression something like that of an apple around the stem. By reference to the method already, employed for constructing a wave front, it will be seen that there may be such a position for the incident wave that, when the elementary wave surfaces are constructed, the resultant wave , will be a tangent to them in the circle around one of these de- pressions where it is pierced by the optic axis. Now since the direction of a ray of light is from the centre of an elementary wave surface to the point of tangency of that surface and the resultant wave, we shall have in this case an infinite number of rays forming a cone, of which the base is the circle of tangency. In other words, one ray entering the plate in a proper direction may be resolved into an infinite number of rays forming a cone, which will become a hollow cylinder of light on emerging from the crystal. This phenomenon is called conical refraction. It was predicted by Hamilton from a mathematical analysis of the wave propagation in such crystals. If a plate be cut from a biaxial crystal perpendicular to the- line bisecting the angle formed by the optic axes, and placed between the polarizer and analyzer in a conical pencil of light, there will be seen a series of colored curves called lemniscates, resembling somewhat a figure 8. The existence of this phe- nomenon was also predicted and the forms of the curves in- vestigated by mathematical analysis before they were seen. 335. Double Refraction by Isotropic Substances when Strained. — A piece of glass between the crossed polarizer and analyzer, if subjected to forces tending to distort it, will restore the light beyond the analyzer and in some cases produce chromatic effects. Unequal heating produces this result, and 336] DOUBLE REFRACTION AND POLARIZATION. 49 ^ a long tube made to vibrate longitudinally shows it when the light crosses it near the node. Pieces cut from plates of un- annealed glass exhibit double refraction when examined by polarized light. Indeed, the absence of double refraction is a test of perfect annealing. 336. Effects of Plates of Quartz. — A quartz crystal is uni- axial, and gives an ordinary and an extraordinary ray, but is un- like Iceland spar in that the extraordinary wave front in it is a prolate spheroid and lies wholly within the spherical ordinary wave, not touching it even where it is pierced by the optic axis. The effects due to plates of quartz in polarized light differ very greatly from those due to Iceland spar or selenite. If a plate of quartz cut perpendicular to the axis be placed in a beam of parallel, homogeneous, plane polarized light at right angles to its path, the light is, in general, restored beyond the analyzer, and is unchanged by the rotation of the quartz through any azimuth. If the analyzer be rotated through a certain angle, depending on the thickness of the quartz plate, the light is extinguished. It is evident that the plane of polar- ization has simply been rotated through a certain angle. Light of a different wave length would have been rotated through a different angle. A beam of white polarized Hght, therefore, has the planes of polarization of its constituents rotated through different angles, and the effect of rotating the analyzer is to quench one after another of the colors as the plane of polari- zation for each is reached. The result is a colored beam which changes its tint continuously as the analyzer rotates. The best explanation of these phenomena was given by Fresnel. It is found that neither of the two beams from a quartz crystal is plane polarized. The polarization is in gen- eral elliptical, but becomes circular for waves perpendicular to the axis of the crystal, the motion in one ray being right-- handed and in the other left-handed. Each particle of ether in the path of the light within the ci-y-'-al is actuated at the same 492 ELEMENTARY PHYSICS. [337 time by two circular motions in opposite directions. Its real motion is in the diameter which bisects the chord joining any two simultaneous hypo- thetical positions of the particle in the two circles. In Fig. 165 let /"and Q represent these two simultaneous positions. It is plain that the two components in the direc- tion AB have the same value and are added, while those at right angles to AB are equal and opposite and annul each other. So long as the two components retain the same relation as that assumed, the real motion of the particle is in the line AB. But in the quartz plate one of the motions is propagated more rapidly than the other, and another particle farther on in the path of the light may reach the point P in one of its circula'r vibra- tions at the same time that it reaches Q' in the other. This will give CD as its real path, and the plane of its vibration has been rotated through the angle BOD. When the light finally emerges from the plate its plane of vibration will have been rotated through an angle which is proportional to the thick- ness of the plate and depends upon the wave length of the light employed. A plate of quartz one millimetre in thick- ness rotates the plane of polarization of red light corresponding to Fraunhofer's line B, 15° 18', of blue light corresponding to the line G, 42° 12'. Some specimens of quartz rotate the plane of polarization in one direction, and some in the oppo- site. Rotation which is related to the direction of the light as the directions of rotation and propulsion in a right-handed screw is said to be right-handed, and that in the opposite direction is left-handed. 337. Artihcial Quartzes.— Reusch has reproduced all the effects of quartz plates by superposing thin films of mica, each film being turned so that its principal plane makes an angle of 45° or 60°, always in the same direction, with that of 339] DOUBLE REFRACTION AND POLARIZATION. 493 the film below. If a plane polarized wave enter such a com- bination, an analysis of the resolution of the vibration as it passes from film to film will show that the result is equivalent to that of two contrary circular vibrations, one of which is propagated less rapidly than the other. This helps to establish Fresnel's theory of the rotational effects of quartz. 338. Rotation of the Plane of Polarization by Liquids. — Many liquids rotate the plane of polarization, but to a less amount than quartz. A solution of sugar produces a rotation varying with the strength of the solution, and instruments called sac- charimeters are made for determining the strength of sugar solu- tions from their effect in rotating the plane of polarization. In these instruments the effect is often measured by interposing a wedge-shaped piece of quartz, and moving it until a thickness is found which exactly compensates the' rotation produced by the solution. 339. Electromagnetic Rotation. — Faraday discovered that when polarized light passes through certain substances in a magnetic field, the plane of polarization is rotated through a certain angle. The experiment succeeds best with a very dense glass consisting of borate of lead, so placed that the light may traverse it along the lines of magnetic force, in the field pro- duced by a powerful electromagnet. The amount of rotation is proportional to the difference of magnetic potential between the two ends of the glass. The direction of rotation, as was shown by Verdet, is generally right-handed in diamagnetic media, and left-handed in paramagnetic media. It also depends upon the direction of the lines of force, and is therefore re- versed by reversing the current in the electromagnet. It fol- lows, also, that if the light, after traversing the glass with the lines of force, be reflected back through the glass against the lines of force, the rotation will be doubled. It is important to note that this is the reverse of the effect produced by quartz, 494 ELEMENTARY PHYSICS. [339 solutions of sugar, etc., which rotate the plane of polarization in consequence of their own molecular state. When light of which the plane of polarization has been rotated by passage through such substances is reflected back upon itself, the rota- tion produced during the first passage is exactly reversed during the return, and the returning light is found to be polarized in the same plane as at first. In the magnetic field the effect is as though the medium which conveys the light were to rotate around an axis parallel to the lines of force, and to carry with it the plane of vibration. Evidently the plane of vibration would be turned through a certain angle during the passage of the light through the body, and would be turned still ^further in the same direction if thfe light were to return. An illustration may be drawn from the movement of a boat rowed across a current. If we row at right angles to the current, the boat is carried downward, and lands on the opposite shore below the point of starting. If then we row back, still at right angles to the current, the boat on reaching the shore from which it started is farther down the stream. On the other hand, in moving across a still lake, we might find ourselves compelled to take an oblique course on account of rocks or other permanent obstacles. If so, we should, on returning,be compelled to retrace our path, and would land at the point of starting. When we remember that iron becomes magnetic by the effect of currents of electricity flowing in conductors around it, and that Ampfere conceived that a permanent magnet consists of molecules surrounded by electric currents, all in the same direction, it is easy to imagine that the magnetic field is a re- gion where the ether is actuated by vortical motions, all in the same direction, and in planes at right angles to the lines of magnetic force. Such a motion would account for the rota- tional effects of the magnetic field upon polarized light. Not only glass but most liquids and gases exhibit rotational 34°] DOUBLE REFRACTION. AND POLARIZATION. 495 effects when placed in a powerful magnetic field ; and Kerr has shown that when light is reflected from the polished pole of an electromagnet, its primitive plane of polarization is rotated when the current is passed, in one direction for a north pole, and in the opposite direction for a south pole. 340. Maxwell's Electromagnetic Theory of Light. — In Maxx^ell's treatment of electricity and magnetism, he assumed that electrical and magnetic actions take place through a uni- versal medium. In order to determine whether this medium may not be identical with the luminiferous ether, he investigated its properties when a periodic electromagnetic disturbance is supposed to be set up in it, such as would result from a rapid reversal of electromotive force at a point, and compared them with the observed properties of the ether, on the assumption that light is an electromagnetic disturbance. He showed that such a disturbance would be propagated through the medium in a way similar to that in which vibrations are transmitted in an elastic solid. He showed further that if light were such a disturbance, its velocity in the ether should be equal to v, the ratio of the electrostatic to the electromagnetic system of units. Numerous measurements of the velocity of light and of this ratio show that they are very nearly equal. He also showed that the indices of refraction of transparent media should be equal to the square roots of their specific in- ductive capacities. This relation may be deduced as follows : We may suppose electrical and luminous effects to be trans- mitted through the dielectric by means of the ether within it, and farther suppose electrical effects in the medium, and there- fore its specific inductive capacity, to be proportional to dis- placements produced in the ether in it by electrical forces. Other things being equal, a displacement is inversely propor- tional to the elasticity of the medium. The velocity of propa- gation of a disturbance is directly proportional to the square root of the elasticity, if the density of the ether remain constant, 49^ ELEMENTARY PHYSICS. [340 and the index of refraction for light is inversely as the velocity of propagation. Hence the index of refraction is equal to the square root of the specific inductive capacity. To illustrate this let us suppose the specific inductive capacity of a dielectric to be 2. This means that a given electric force produces in the ether in that substance twice the displacement which it would produce in the ether in air. Hence the elasticity of the' ether in that substance is one half as great as in air, the velocity of propagation of light in it will be to the velocity in air as i : S/'2, and the index of refraction will be ^2. Measurements of indices of refraction and specific inductive capacities have shown that the relation which has been stated holds true in many cases. Hopkinson has shown, however, that it does not hold true for animal and vegetable oils. The theory leads to the conclusion that the direction of pro- pagation of the electrical disturbance and the accompanying magnetic disturbance at right angles to it is normal to the plane of these disturbances. By making the assumption, which is justified by Boltzmann's measurements upon sulphur, that an eolotropic medium has different specific inductive capacities in different directions, Maxwell shows also that the propagation of the electrical disturbance in a crystal will be similar to that of light. It has also been shown that the electrical disturbance will be reflected, refracted, and polarized at a surface separating two dielectrics. Lastly, Maxwell concludes that, if his theory be true, bodies which ar'e transparent to the vibrations of the ether should be dielectrics, while opaque bodies should be good conductors. In the former the electrical disturbance is propagated without loss of energy ; in the latter the disturbance sets up electrical currents, which heat the body, and the disturbance is not pro- pagated through the body. Observation shows that, in fact, solid dielectrics are transparent, and solid conductors are opaque, to radiations in the ether. Maxwell explains the fact that 340] DOUBLE REFRACTION AND POLARIZATION. 497 many electrolytes are transparent and yet are good conductors by supposing that the rapidly alternating electromotive forces which occur during the transmission of the electrical disturbance act for so short. a time in one direction, that no complete sepa- ration of the molecules of the electrolyte is effected. No elec- trical current, therefore, is set up in the electrolyte, and elec- trical energy is not lost during the transmission of the disturb- ance. 29 TABLES. TABLE I. Units of Length. Foot = 30.48 cm. log. 1.484015 Inch ~~ 2.54 cm. Units of Mass. log. 0.404830 Pound = 453-59 grams. log. 2 . 656664 Grain = 0.0648 grams. log. 8.811575 TABLE n. Acceleration of Gravity §■ = 980.6056 — 2.5028 cos 2/— 0.000003/2, where /is the latitude of the station and h its height in centimetres above the sea level, f at Washington = 980.07 I ^ at Paris = 980,94 f at New York = 980.26 | ^ at Greenwich = 981.17 TABLE in. Units of Work. Kilogram-metre = Foot-pound = Watt 100,000^ ergs. 13,825^ ergs. I 355 X 10' ergs, log 7.13200, when g = 980. Units of Rate of Working. = 10' ergs per second. Horse-power = 550 foot pounds per second. = 746 Watts. Unit of Heat. Lesser calorie (gram-degree) = 4.16 X lo' ergs. TABLES. 499 TABLE IV. Densities of Substances at o°. The densities of solids given in this table must be taken as on^y approx- imate. Specimens of the same substance differ among themselves to such an extent as to render it impossible to give more precise values. Aluminium 2.6 Brass 8.4 Copper 8.g Gold 19.3 Glass (crown) 2.5 to 2.7 Hydrogen 0.0000895 Ice .0.918 Iron (wrought) 7.6 to 7.8 " (cast) 7.2107.7 Lead 11. 3 Mercury 13- 596 Platinum. ; 21.5 Silver lo. 5 Zinc 7-1 TABLE V. Units of Pressure for g ■= 981. Grams per sq. cm. Dynes per sq. cm. Pound per square inch 70-3i 6.9 X lO* T inch of mercury at 0° 34-534 3-388 X 10* I millimetre of mercury at 0° 1-3596 1333-8 1 atmosphere (760 mm.) 1033.3 1.0136X10' I atmosphere (30 inches) 1036. 1 .0163 X 10' TABLE VI. Elasticity. If / is the force in dynes per unit area tending to extend or compress a di , , ... dp body, the linear elasticity is ^ and the volume elasticity is ~. dp. df dl dv Glass 6.03X10" 4.15X10" Steel 2.I4XI0" 1.84X10" Brass 1.07X10" Mercury 3.44X10" Water 2.02X10'° goo ELEMENTARY PHYSICS. TABLE VII. Absolute Density of Water at f in Grams per Cubic Centimetre. t°. Density. O 0.999884 I 0.999941 2 O.9999S2 3 1.000004 4 1. 000013 5 1.000003 t°. Density. 7 0.999946 8 0.999899 9 0.999837 10 0.999760 15 0.999173 20 , . . 0.998272 6 0.999983 30 0.995778 100 0.95866 /". Density. 40 0.99236 50 0.98821 60 0.98339 70 0.97795 So 0.97195 90 0.96557 TABLE VIII. Density of Mercury at t°. Water at 4° being i. t°. Density. log. o 13-5953 r.13339 10 13-5707 1.13260 i°. Density. log. 20 13.5461 I.I3182 30 13-5217 I.I3IO3 TABLE IX. Coefficients of Linear Expansion. ' Temperature. Aluminium 16° to 100° Brass , o to 100 Copper o to 100 German silver o to 100 Glass o to 100 Iron 13 to 100 Lead o to 100 Platinum o to 100 Silver o to 100 Zinc o to 100 dV CoefiHcients of voluminal expansion, -3- = %a. at dl 0.0000235 0.0000188 0.0000167 0.0000184 0.0000071 0.0000123 0.0000280 0.0000089 0.0000194 0.0000230 TABLES. 501 TABLE X. Specific Heats — Water at 0° = i. Solids and Liquids, Aluminium 0.212 Brass 0.086 Copper 0.093 Iron 0.112 Lead 0.031 Mercury 0.033 Platinum 0.032 Silver 0.056 Water (0° to 100°) i .005 Zinc 0.056 Gases and Vapors at Constant Pressure. Air 0.237 I Hydrogen 3.410 | Ratio, -i- ■ Nitrogen o . 244 Oxygen 0.217 .404. TABLE XL L Melting Points. IL Boiling Points. IH. Heats of Liquefaction. IV. Heats of Vaporization. V. Maximum Pressure of Vapor at o' in Millimetres of Mercury. I. II. III. IV. v. Ammonia .. —33.7° .. 294 at 7.8° 3344 Carbon dioxide —65° —78.2 .. 49.3 at 0° 27100 Chlorine .. —33-6 .. .. 4560 Copper 1200 Lead 325 .. 5.9° .. ' .. Mercury — 39 357 2.8 62 0.02 Nitrous oxide, N3O .. —105 .. .. 24320 Platinum 1780 .. 27.2 Silver 1000 .. 21. 1 Water o 100 80 537 4.6 Zinc 415 .. 28.1 502 ELEMENTARY PHYSICS. TABLE XII. Maximum Pressure of Vapor of Water at Various Temperatures in (I.) Dynes per Square Centimetre, (II.) Millimetres of Mercury. Temp. I. II. — 20" 1236 — 10° 2790 0° 6133 4.6 10 12220 9.2 20 23190 17.4 30 42050 31.5 40 73^00 54.6 50 1.226X10' 96.2 Temp. I. 60° 1.985 X 10" 80 100 10.14 120 19.88 140 - 36.26 160 62.10 180 100.60 200 156. 4.729 X 10' X io> X 10' X 10^ X 10° X 10' X 10' II. 149. 355- 760. 1491. 2718. 4652. 7546. 11689. TABLE XIII. Critical Temperatures (T') and Pressures in Atmospheres (/"), at their Critical Temperatures, of Various Gases. T. Hydrogen —174. Nitrogen —124. Oxygen —105. p. T. P. 99- , Carbon dioxide. . . 30.9 77 42. Sulphur dioxide . . 155-4 79 49. TAB LI S XIV. Coefficients of Conductivity for Heat {K) in C. G. S. Units, in WHICH Q IS GIVEN IN LeSSER CaLORIES. Brass 0.30 Copper i.ii Glass 0.0005 Ice 0.0057 Iron 0.16 Lead 0.08 Mercury 0.015 Paraffin 0.00014 Silver 1.09 Vulcanized india-rubber 0.00009 Water 0.0015 TABLES. 503 TABLE XV. Energy Produced by Combination of i Gram of Certain Substances WITH Oxygen. Gram-degree of Heat. Energry in ergfs. Carbon, forming CO 2141 8.g8 X lo'" COa 8000 3.36X10'^ Carbon monoxide, forming COj.. 2420 1.02 X 10" Copper, CuO 602 2.53X10'" Hydrogen, HaO 34000 1.43 X lo'^ Marsh gas, COj and HjO 13100 5 -50 X 10" Zinc, ZnO 1301 5.46X10'" TABLE XVL Atomic Weights and Combining Numbers. Atomic Weight. Combining Number. Aluminium 27.04 9.01 Copper 63.18 (cupric) 31-59 " " (cuprous) 63.18 Gold 196.2 65.4 Hydrogen i . i . Iron 55-88 (ferric) 18.63 " " (ferrous) 27.94 Mercury 199.8 (mercuric) 99.9 " " (mercurous) 199.8 Nickel 58.6 29.3 Oxygen 15-96 7-98 Platinum 194-3 64.8 Silver 107.7 107.7 Zinc 64.88 32.44 TABLE XVU. Molecular Weights and Densities of Gases. Simple Gases. Atomic Weight. Sp. gr., H = i. Mass in i litre. Chlorine, CI, 70-75 35-37 3-i67 Hydrogen, Hs 2.00 i.oo 0.0895 Nitrogen, Nj 28024 14.012 1.254 Oxygen, O2 31927 15-96 ^-429 504 ELEMENTARY PHYSICS. Carbonic oxide, CO Carbonic dioxide, COj. . Hydrochloric acid, HCl. Vapor of water, H2O Atmospheric air Compound Gases. Atomic Weight. Sp.gr., «■=.. Mass in litre. 27.937 14-97 1. 251 43-90 21-95 1.965 36-376 i8.i88 1.628 17.96 8.98 0.804 1.293 TABLE XVIII. Electromotive Force of Voltaic Cells. Daniell i.ivolt. | Grove i. 88 volt. | Clark... 1.435 volt at 15°. Electromotive force of Clark cell for any temperature t is i.435[i - o.ooo77(<- 15)]. TABLE XIX. Electro chemical Equivalents. Grams per second deposited by the electromagnetic unit current, Hydrogen, 0,0001038. To find the electro-chemical equivalents of other substances, multiply the electro-chemical equivalent of hydrogen by the combining number of the sub- TABLE XX. Electrical Resistance. Absolute resistance R in C. G. S. units of a centimetre cube of the substance. Temperature coefficient, a. Rt = i?o(i -f- oit). J?„. «. Aluminium 2889 Copper 1611 0.00388 German silver 20763 0.00044 Gold 2041 0.00365 Iron 9638 Mercury 9434° 0.00072 Platinum 8982 0.00376 Platinum silver, 2 Pt. i Ag 24190 0.00031 Silver 1580 0.00377 Zinc 5581 0.00365 TABLES. 50s Carbon (Carrfe's electric light) 3.9 X 10' Glass at 200° 2.23X 10'' Gutta-percha, at 24° 3 ■ 46 X 10^' " 0° 6.87X 10" Selenium, at 100° 5 • 9 . X 10" Water, at 22° 7.u X io'° Zinc sulphate + 23 HaO x . 83 X lo'" Copper sulphate + 45 H2O 1.91X 10'° TABLE XXI. Indices of Refraction. Soft crown glass. .. Index. I . 5090 I. 5180 1.5266 Dense flint glass 1.6157 1.6289 1-6453 Rocksalt 1.5366 1.5490 1-5613 Diamond 2 . 47 Amber 1-532 Kind of Light. A E G B E G A E G D D Iceland spar. Quartz Ordinary Index. 1.658 1-544 Index. Canada balsam 1.528 Water I-33I 1-336 1-344 Carbon disulphide i . 6 14 1.646 Air at 0°, 760 mm.. 1.684 1.00029 1.000296 1.000300 Kind of Light. Red B E H A E G A E H Kind of Light. D Extraordinary Index. 1.486 1-553 TABLE XXII. Wave Lengths of Light — Rowland's Determinations. Fraunhofer's line A (edge), 7593 . 975 tenth metres. B " 6867.382 C " 6562.965 Dx " 5896.080 D, " 5890.125 E " 5270.429 b 5183-735 F " 4861.428 G " 4307.961 So6 ELEMENTARY PHYSICS. TABLE XXIII. Rotation of Plane of Polarization by a Quartz Plate, i mm. thick, CUT perpendicular to Axis. A i2°.fe68 B i5°-746 C i7°-3i8 Ds 21°. 727 E 27°. 543 F 32°-773 G 42°. 604 H 5i°-i93 TABLE XXIV. Velocities of Light. Cm. per Sec. Micnelson, 1879 2.99910 X 10" Michelson, 1882 2.99853 X 10" Newcomb, 1882 2.99860 X 10" Cm. per Sec. Foucault, 1862 2.98000 X lo"> Cornu, 1874 2.98500X10'° Cornu, 1878 2.99990 X io"> The Ratio between the Electrostatic and Electromagnetic Units. Cm. per Sec. Weber and Kohlrausch 3 . 1074 X 10'° W.Thomson 2.825 X lo'" Maxwell 2.88 X 10'" Ayrton and Perry 2.98 X 10'° J.J. Thomson 2.963 X 10'" Cm. per Sec. Exner 2.920X10"' Klemencic 3 . 018 X 10'" Himstedt 3.007 X 10'° CoUey 3.015 X lo"> INDEX. Aberration, spherical, 426; chromatic, 455 Aberration of fixed stars, 432 Absolute temperature, zero of, 191; scale of, 212 Absorption, 103; coefficient of, 104; of gases, 104; of radiant energy, 463; of radiations, 466; by gases, 467; relation of, to emission, 470 Acceleration, 14; angular, 49 Accelerations, composition and resolutions of, 16 Achromatism, 455 Acoustics, 353 Adhesion, 87 Adiabatic line, 194 Aggregation, states of, 85 Air-pump, 137; receiver of, 137; plate of, 138; theory of Sprengel, 134; Spren- gel, 139; Morren, 140 Airy, determination of Earth's density, 80 Alloys, melting points of, 176 Ampere, relation of current and magnet, 273; mutual action of currents, 311; equivalence of circuit and magnetic shell, 314; theory of magnetism, 315 Ampere, a unit of electrical current, 309 Amplitude of a simple harmonic motion, 18; of a wave, 23; its relation to in- tensity of light, 438 Analyzer, 481 Andrews, critical temperature, 184; heat of chemical combination, 2or. Aneroid, 141 Angles, measurement of, 9; unit of, 9 Animal heat and work, 218 Antinode, 362 Aperture of spherical mirrors, 411 Apertures, diffraction effects at, 443 Archimedes, his principle in hydrostatics, 124 Aristotle, his theory of vision, 396 So8 INDEX. Astatic system of magnetic needles, 317 Atmosphere, homogeneous, 115; pressure of, 123; how stated, 124 Atoms, 85 Attraction, mass or universal, 67; constant of, 80 Avenarius, experiments in thermo-electricity, 342 ; thermo-electric formula, 345 Axis of rotation, 52; of shear, 114; of floating body, 125; magnetic, 224, 228; of spherical mirror, 411; optic, of crystal, 474, 489 Balance, 76; hydrostatic, 125 Barometer, 122; Torricellian form of, 123; modifications of, 124; preparation of, 124 Beam of light, 422. Beats of two tones, 385; Helmholtz's theory of, 385; Konig's theory of, 385; Cross's experiment on, 387 Beetz, his experiment on a limit of magnetization, 245 Berthelot, heat of chemical combination. 202 Berzelius, his electro-chemical series, 287 Bidwell, view of Hall effect, 316 Bifilar suspension, 268, 321 Biot, law of action between magnet and electrical current, 298 Biot and Savart, action between magnet and electrical current, 297 Bodies, composition of, 85; forces determining structure of, 87; isotropic and eolotropic, 108 Boiling. See Ebullition, 182 Boiling point, 182 Bolometer, depends upon change of resistance with temperature, 279; used to study spectrum, 459 Boltzmann, specific inductive capacity of gases, 264 Borda, his pendulum, 74; his method of double weighing, 78 Bosscha, capillary phenomena in gases, 96 Boutigny, spheroidal state, 183 Boyle, his law for gases, no; limitations of, 141; departures from, 185 Bradley, determined velocity of light, 433 Breaking weight, iig Brewster, his law of polarization by reflection, 480 Cagniard-Latour, critical temperature, 183 Calorie, 151; lesser, 151 Calorime'ter, Black's ice, 153; Bunsen's ice, 153; water, 155; thermocalorime- ter of Regnault, 157; water equivalent of, 156 Calorimetry, 153; method of fusion, 153; of mixtures, 155; of comparison, 156; of cooling, 157 INDEX. 509 Camera obscura, 427 Capacity, electrical, 255; unit of, 256; of spherical condenser, 258; of plate condenser, 260; of freely electrified sphere, 260; of Leyden jar, 261 Capacity, specific inductive, 257; relation of, to index of refraction, 264, 495; relation of, to crystallographic axes, 264 Capillarity, facts of, 89; law of force treated in, go; equation of, 94; in gases, 96; Plateau's experiments in, 97 Carlini, determination of Earth's density, 80 Carlisle, his apparatus for electrolysis of water, 283 Carnot, his engine, 206; his cycle, 207 Cathetometer, 6 Cavendish, experiment to prove mass attraction, 6g; determination of Earth's density, 79; determined force in electrified conductor, 249; discovered specific inductive capacity, 257 Caustic curve, 426; surface, 426 Central forces, propositions connected with, 60 Centrobaric bodies, 45 Charge, unit, electrical, 252; energy of electrical, 262 Chemical affinity measured in terms of electromotive force, 286 Chemical combination, heat equivalent of, 201 Chemical separation, energy required for, 218; gives rise to electromotive force, 285 Chladni's figures, 377 Christiansen, anomalous dispersion in fuchsin, 472 Circle divided, 9 Circuit, electrical, equivalence of, to magnetic shell, 300, 305; direction of lines of force due to, 310 Clark, his standard cell, 293; its electromotive force, 293; his potentiometer, 334 Clarke, his atomic weights used, 176 Clausius, his principle in thermodynamics, 206; his theory of electrolysis, 289 Clement and Desormes, determination of ratio of specific heats of gases, 195 Coercive force, 223 Cohesion, 87 CoUimating lens, 451, 459. Collision of bodies, 29 Colloids, 86; diffusion of, 107 Colors of bodies, 467; produced by a thin plate of doubly refracting crystal in polarized light, 485 ; by a thick plate, 486 Colors and figures produced by a thin plate of doubly refracting crystal in polarized light, 489, 490, 491 Comparator, 8 Compressibility, 84 5IO INDEX. Compressing- pump, 140 Compressions, log Concord, musical, 367 Condenser, electrical, 256; spherical, 258; plane, 260 , Conduction of electricity, 246 Conductivity for heat, 164; measurement of, 165; changes of, with temperature, 167; of crystals, 167; of non-homogeneous solids, 167; of liquids, 167 Conductivity, specific electrical, 278 Conductors, good, 247; poor, 247 Contact, angles of, 95 Continuity, condition of, 128; for a liquid, 128 Convection of heat, i6i r Copernicus, his heliocentric theory, 67 Cords, longitudinal vibrations of, 374; transverse vibrations of, 375 Cornu and Bailie, determination of Earth's density, 80 Coulomb, his laws of torsion, 116; his torsion balance, 116; law of magnetic force, 225; distribution of magnetism, 228; law of electrical force, 249 Coulomb, a unit of quantity of electricity, 252 Counter electromotive force. 279; general law of, 279; of decomposition, measure of, 285; of polarization, 291; of electric arc, 348 Couple, 44: moment of, 44 Critical angle of substance, 408 Critical temperature, 183 Crookes. invented the radiometer, 192; his tubes, 351; explanation of phe- nomena in tubes, 352 Cross, experiment on beats, 387 Crystal systems, 86 Crystalloids. 86; diffusion of, 107 Crystals, conductivity of, for heat, 167; specific inductive capacity of, 264; elec- trification of, by heat, 264; optic axis of, 474; principal plane of. 475; vary- ing elasticity in, 478; varying velocity of light in, 479; effects of plates of, on polarized light, 483, 486, 489, 490, 491; uniaxial, 489; biaxial, 489; optic axes of biaxial, 489 Ctesibius, invented force-pump, 122 Cumming, reversal of thermo-electric currents, 342 Current, electrical, 275; effects of, 272; electrostatic unit of, 275; strength, 275; strength depends on nature of circuit, 276; set up by movement of a liquid surface, 296; electromagnetic unit of, 307; practical unit of, 309; direction of lines of force due to, 298, 310; mutual action of two, 310; Ampere's law for the mutual action of, 311; deflected in a conductor by a magnet, 315; measured in absolute units, 321; Kirchhoff's laws of, 331 Current, extra, 325 INDEX. 5 1 1 Current, induced electrical, 321; quantity and strength of, 323; measured in terms of lines of force, 323; discovered by Faraday, 324; Lenz's law of, 325; Faraday's experiments relating to, 325 Cycle, Carnot's, 207; illustrated in hot-air engines, 216 Dalton, his law of vapor pressure, 181 Daniell's cell, 291; electromotive force of, calculated, 292 Dark lines in solar spectrum, explanation of, 469 Davy, his melting of ice by friction, 145; his electrolysis of caustic potash, 283 Declination, magnetic, 233 Density, n Density, magnetic, 227 Depretz and Dulong, measurement of animal heat by, 219 Dew point, 203; determination of, 203 Dialysis. 107 Diamagnet, distinguished from paramagnet, 239, 242 Diamagnetism, 237; explanation of, by Faraday, 237; by Thomson, 238; on Ampere's theory, 315; by Weber, 315 Diaphragm, vibrations of, 378 Dielectric, 256; strain in, 263 Diffraction of light, 443; at narrow apertures, 443; at narrow screens, 445; grating, 446; phenomena due to, 452 Diffusion, 103; of liquids, 104; coefficient of, 105; through porous bodies, 106; through membranes, 106; of gases, 107 Dilatability, 84 Dilatations, 109 Dimensional equation, 10 Dimensions of units, 10 Dip, magnetic. 233 Discord, in music, 367 Dispersive power of substance, 454 Dispersion, normal, 409. 453; anomalous, 472 Dissociation, 201 ; heat equivalent of, 201 Distribution of electricity on conductors, 251 Dividing engine, 7 Divisibility, 84 Double refraction,, in Iceland spar, 474; explanation of, 475; by isotropic sub- stances when strained, 490 Draper, study of spectrum in relation to temperature, 471 Drops, in capillary tubes, loi; Jamin's experiments on, 102 Dulong and Petit, law connecting specific heat and atomic weight, 176; formula for loss of heat by radiation, 470 512 INDEX. Dutrochet, his definition of osmosis, io6 Dynamics, ii Dynamo-machine, 328 Dyne, 26 Ear, tympanum of, 379 Earth, density of, 79 Ebullition, 180, process of, 182; causes affecting, 182 Edlund; study cf counter electromotive force of electric arc, 348 Efflux through narrow tubes, 88; of a liquid, 129; quantity of, 134 Elasticity, 84, 108; modulus and coefficient of, no; voluminal, of gases, no; of liquids, 112; of solids, 113; perfect, 118; of tension, 114; of torsion, 115; of flexure, 118; limits of, 118 Elasticity of gases, no, 193; at constant temperature, 193; when no heat enters or escapes, 193: ratio of these, 198; determined from velocity of sound, 199 Electric arc, 348; counter electromotive force of, 348 Electric discharge, in air, 348; in rarefied gases, 350 Electric pressure, 255 Electrical convection of heat, 347 Electrical endosmose, 288; shadow, 349 Electrical machine, 268; frictional, 268; induction, 269 Electricity, unit quantity of, 252; flow of, 253, 274 Electrification by friction, 246; positive and negative, 246, 248 Electro chemical equivalent, 284 Electrode, 282 Electrodynamometer, 321 Electrolysis, 282; bodies capable of, 282; typical cases of, 283; influenced by secondary chemical reactions, 283; Faraday's laws of, 284; theory of, 287; modified by outstanding facts, 288; Clausius' view of, 289 Electrolyte, 282 Electromagnet, 314 Electromagnetic system of electrical units, basis of, 307 Electrometer, 265; absolute, 265; method of use of, 267; quadrant, 267; capil- lary, 294 , Electromotive force, 274; measured by difference of potential, 274; means of setting up, 280; measured in heat units, 286; a measure of chemical affinity, 286; of polarization, 291; theories of, of voltaic cell, 293; due to mo- tion in magnetic field, 321; measured in terms of lines of force, 323; de- pends on rate of motion, 323; electromagnetic unit of, 326; practical unit of, 326; at a heated junction, 341; required to force spark through air, 349 Electromotive force, counter. See Counter electromotive force, 279, etc. INDEX. 513 Electromotive forces, compared by Clark's potentiometer, 334 Electrophorus, 269 Electroscope, 265 Electrostatic system of electrical units, basis of, 252 Elements, chemical, 85; electro-positive and electro-negative, 286 Emission of radiant energy, 468; relation of, to absorption, 470 Endosmometer of Dutrochet, 107 Endosmose, 106 Endosmose, electrical, 288 Energy, 31; potential and kinetic, 31; and work, equivalence of, 32; unit of, 32; conservation of, 32; of fusion, 179; of vaporization, 200; sources of terres- trial. 217: of sun, 221; dissipation of, 221 Engine, efficiency of heat, 205; reversible, 206; Carnot, 206; efficiency of re- versible, 206, 209, 210, 211; steam, 214; hot-air, 215; gas, 215; Stirling, 216; Rider, 216 Eolotropic bodies, 108 Epoch of a simple harmonic motion, 21 Equatorial plane of a magnet, 224 Equilibrium, 29 Equipotential surface, 37 Erg, 32 Ether, 85; luminiferous, 397; velocity of, in moving body, 435; interacts with molecules of bodies,- 473; transmits electrical and magnetic disturbances, 495 Ettinghausen, view of Hall effect, 316 Evaporation, 180; process of, 180 Exosmose, 106 Expansion, of solids by heat, 168; linear, 168; voluminal, 168, 173; coefficient of, 168; factor of, 169; measurement of coefficient of, 169, 173; of liquids by heat, 170; absolute, 170, 174; apparent, 170; of mercury, absolute, 170; apparent, 171; of water, 174; of gases by heat, 185; coefficient of, 186; heat absorbed and work dune during, 194 Extraordinary ray, 475 : index, 475 Eye, 427; estimation of size and distance by, 428 Eye-lens or eye-piece, 431; negative or Huyghens, 456; positive or Ramsden, 457 Farad, a unit of electrical capacity, 256 Faraday, discovery of magnetic induction in all bodies, 237; explanation of this, 237; experiment in electrical induction, 247; on force in electrified body, 249; theory of electrification, 256; theory illustrated, 263; explana- tion of residual charge, 264; showed that discharge of jar can produce ef- 33 SH INDEX. fects of current, 274; nomenclature of electrolysis, 282; voltameter, 285; division of ions, 286; theory of electrolysis, 287; chemical theory of electro- motive force, 293; electromagnetic rotations, 306; induced currents, 324; effect of medium on luminous discharge, 350; electromagnetic rotation ot plane of polarization, 493 Favre and Silbermann, studied heal of chemical combination, 202; verified con- nection of electromotive force and heat units, 286; value of heat equivalent, 292 Ferromagnet. See Paramagnet, 237 Field of force, 27; strength of, 27 Filament, in a fluid, 129 Films, studied by Plateau, 98; interference of light from, 441 Fizeau, introduced condenser in connection with induction coil, 329; deter- mined velocity of light, 433; velocity of light in a moving medium, 435 Flexure, elasticity of, 118, Floating bodies, 125 Flow of heat, 162; across a wall, 162; proportional to rate of- fall of tempera ture, 163; along a bar, 165 Fluid, body immersed in a, 125; body floating on a, 125 Fluids, distinction between solids and, 119; mobile, viscous, 119; perfect, 120 Fluids, motions of. See Motions of a fluid, 128 Fluorescence, 472 Focal line, 425 Focus, of spherical mirror, 413; real, conjugate, 413; principal, 414; virtual, 414 Force, 26; unit of, 26; field of, line of, tube of, 27; defined by potential, 34, within spherical shell, 38; outside sphere, 41; just outside a spherical shell, 42; just outside a flat disk, 42; moment of, 43 Force, capillary, law of, 90 Force, electrical, in charged conductor, 249; law of, 249; screen from, 253; just outside an electrified conductor, 255 Force, magnetic, law of, 224; due to bar magnet, 230; within a magnet, 238; between magnet and current element, 297, 298; between magnet and long straight current, 299; due to magnetic shell, 304 Forces, composition and resolution of, 29; resultant of parallel, 43; central, 60 Forces, determining structure of bodies, 87; molecular, 87, 108; of cohesion, 87; of adhesion, 87 Foucault, his pendulum, 52; determined velocity of light, 434; his prism, 4S2 Fourier, his theorem, 25 Franklin, complete discharge of electrified body, 248; experiment with Leyden jar, 263; identity of lightning and electrical discharge, 350 Fraunhofer, lines in solar spectrum, 458 Freezing point, change of, with pressure, etc., 177 INDEX. 5 1 5 Fresnel, interference of light from two similar sources, 439; his rhomb, 489; explanation of rotation of plane of polarization by quartz, 491 Friction, laws of, 88; coefficient of, 88; theory of, 89 Fusion, 176; heat equivalent of, 178; energy necessary for, 179; determination of heat equivalent of, 179 Galileo, the heliocentric theory, 68; measurement of gravity, 70; path of pro- jectiles, 81; weight of atmosphere, 123 Galvani, discovered physiological effects of electrical current, 272 Galvanometer, 316; Schweigger's multiplier, 316; sine, 317^ tangent, 318 Gas, definition of, iSo Gases, S5; absorption of, 104; diffusion of, 107; elasticity of, no; liquefaction of, by pressure, 142, 1S4; departure of, from Boyle's law, 185; coefficient of expansion of, 186; pressure of saturated, 186 Gases, kinetic theory of. See Kinetic theory of gases, 188 Gauss, theory of capillarity, gl Gay-Lussac, law of expansion of gases by heat, 185 Geissler tubes, 351 Gilbert, showed Earth to be a magnet, 233 Graham, his osmometer, 107; method of dialysis, 107 Grating, diffraction, 446; element of, 447; pure spectrum produced by, 447; nor- mal 'and oblique incidence, 447; with irregular openings, 450; wave lengths measured by, 450; Rowland's curved, 452 Gravitation, attraction of, 67 Gravity, centre of, 45 Gravity, measurement of, 69; value of, 70 Grotthus, theory of electrolysis, 287 Grove, his gas battery, 291 Grove's cell, 292 Gyration, radius of, 56 Gyroscope, 53 Hall, deflection of a current in a conductor, 31S Halley, theory of gravitation, 68 Hamilton, prediction of conical refraction, 490 Harmonic tones of pipe, 373 Harris, absolute electrometer, 265 Heat, effects of, 143; production of, 144; nature of, 144; a form of energy, 145; unit of, 151; mechanical unit of, 151; mechanical equivalent of, 158; Joule's determination of, 158; Rowland's, 159; transfer of, 161; convection of, 161: internal, of Earth, a source of energy, 221; developed by the electri- cal current, 273, 275; generated by absorption of radiant energy, 463 Sl6 INDEX. Heat, conduction of. See Flow of, 162 Heat, atomic, 175 HelmholtZ; vortices, 135; theory of solar energy, 221; law of counter electro- motive force, 280; theory of capillary electrometer, 295; resonators, 382; vowel sounds, 383 ; theory of beats, 385 ; interaction of ether and molecules of bodies, 473 Herschel, study of spectrum, 460 Hirn. work done by animals, 2ig Holtz, elecf.rical machine, 270 Hooke, theory of gravitation, 63 Hopkinson, relation between index of refraction and specific inductive ca- pacity 496 Horizontal intensity of Earth's magnetism, 233; measurement of, by standard magnet, 234; absolute, 235 Humidity, absolute, 202; relative, 204 Huyghens, theorems of, on motion in a circle, 68; views of, respecting gravita- tion, 68; principle of wave propagation, 356 Hydrometer, 127 Hydrostatic balance, 125 Hydrostatic press, 121 Hygrometer, AUuard's, 203 Hygrometry, 202 Ice, density, of, 177; melting point of, used as standard, 176 Iceland spar, 474; wave surface in, 476 Images, formed by small apertures, 402; virtual, 409; by successive reflection, 410; by mirrors, 419; by lenses, 421; geometrical construction of, 422 Impenetrability, 4 Impulse, 26 Incidence, angle of, 406 Inclined plane, 47 Induced magnetization, coefBcient of, 239 Induction coil, 328; condenser connected with, 329 Induction, electrical, 247 Induction, magnetic, 223, 237, definition of, 239 Induction ol currents, 321 Inertia, 4, 30, centre ot, 44; moment of, 56 [nsulatoi, electrical, 247 Interference of light, cause ol propagation in straight lines, 397; from two simi- lai sources, 436, experimental realization of, 439; from thin films, 441 inir.rnode, '^bn, Intel v«ti&, 36b INDEX. 5 1 7 Ions, 2S2; electro-positive and electro-negative, 286; arrangement of, by Fara- day, 286; by Berzelius, 287; wandering of the, 288 Isothermal line, 193 Isotropic bodies, 108 Jamin. drops in capillary tubes, 102 Jolly, determination of Earth's density, 80 Joule, equivalence of heat and energy, 145, 205; mechanical equivalent of heat, 158; expansion of gas without work, 188; limit of magnetization, 245; law of heat developed by electrical current, 279; electromotive force in heat units, 286; development of heat in electrolysis, 288 Jurin, law of capillary action, 99 Kaleidoscope, 410 Kater, his pendulum, 75 Kepler, laws of planetary motion, 67 Kerr, optical effect of strain in dielectric, 263; rotation of plane of polarization by reflection from magnet, 495 Ketteler, interaction of ether and molecules of bodies, 473 Kinematics. 11 Kinetics. 11 Kinetic theory of gases, 188 Kirchhoff, laws of electrical currents, 331; spectrum analysis, 460 Kohlrausch, value of electro-chemical equivalent, 292 Konig, A., modification of surface tension by electrical currents, 295 Konig, R., manometric capsule. 353; pitch of tuning-forks made by, 370; boxes of his tuning-forks, 378; quality as dependent on change of phase, 381 ; investigation of beats, 385 Kundt, experiment to measure velocity of sound, 394; anomalous dispersion, 472 Lang, counter electromotive force of electric arc, 348 Langley, his bolometer, 279: wave lengths in lunar radiations, 452 Laplace, theory of capillarity, 91 Lavoisier, measurement of animal heat, 218 Least time, principle of, 401 Length, unit of, 4; measurements of, 5 Lenses, 417; formula for, 417; forms of, 418; focal length of, 418; images formed by, 421, optical centre of, 421; thick, 423; of large aperture, 423; aplanatic combinations of, 427; achromatic combinations of, 455 Lenz, law of induced currents, 325 5 1 8 INDEX, Le Roux, experiments in thermo-electricity, 342; electrical convection of heat in lead, 347 Lever, 46 Leyden jar, capacity of, 261; dissected, 263; volume changes in, 263; residual charge of, 264 Light, agent of vision, 396; theories of, 396; propagated in straight lines, 397; principle of least time. 401; reflection of, 404; refraction of, 405; ray of, beam of, pencil of, 422 Light, velocity of, determined from eclipses of Jupiter's satellites, 432; from aberration of fixed stars, 432; by Fizeau, 433; by Foucault, 434; by Michel- son, 434; in moving medium, 435 Light, electromagnetic theory of, 495 Lightning, an electrical discharge, 350 Lines of magnetic force, positive direction of, 304; measure of strength of field in, 309; relation of, to moving magnetic shell or current, 309 Lippmann, electrical effects on capillary surface, 294; capillary electrometer, 296; production of current by modification of capillary surface, 296 Liquefaction, 184; of gases, by pressure, 184 Liquids, 85; modulus of elasticity of, 112 Lissajous, optical method of compounding vibrations, 384 Loudness of sound, 365 Machine, 48; eflSciency of, 48; electrical, 268; dynamo- and magneto-, 328 Magnet, natural, 223; bar, relations of, 228 Magnetic elements of Earth, 233 Magnetic force. See Force, magnetic, 224 Magnetic inductive capacity, 239 Magnetic shell, 231; strength of, 231; potential due to, 232; equivalence of, to closed current, 300 Magnetic system of units, basis of, 226 Magnetism, fundamental facts of, 223; distribution of, in magnet, 227; deter- mination of, 228; theories of, 244; Ampere's theory of, 315 Magnetization, intensity of, 226 Magneto-machine, 328 Magnifying glass, 430 Magnifying power, 429 Manometer, 140 Manometric capsule, 353 Mariotte, study of expansion of gases, no Maskelyne, determination of Earth's density, 79 Mass, 11; unit of, 9 Masses, comparison of, 9 INDEX. 519 Matter, i; constitution of, 84 Matthiessen, expansion of water, 174 Mayer, views concerning work done by animals, 219 Maxwell, proposed unit of time, 83; coefficient of viscosity of a gas, 89; defini- tion of magnetic induction, 239; theory of electrification, 256; explanation of residual charge, 264; relation between specific inductive capacity and index of refraction, 264, 495; suggested test of Weber's theory of diamag- netism, 315; measurement of v, 337; force on magnet due to moving elec- trical charge, 338; electromagnetic theory of light, 397, 495 Mechanical powers, 46 Melloni, use of thermopile, 341 Melting point of ice, 176; of alloys, 176; change of, with pressure, 177 Mercury, expansion of, by heat, 170, 171 Metacentre, 125 Michelson, determined velocity of light, 434 Michelson and Morley, velocity of light in moving medium, 435 Micrometer screw, 6 Microscope, simple, 430; compound, 430 Mirrors, plane, 409; spherical, 410; images formed by, 419; of large aperture, 423; not spherical, 424 Modulus of elasticity. See Elasticity, no; Young's, 115 Molecular action, radius of, 90 Molecule, 84; structure of, 87; kinetic energy of, proportional to temperature, 191; mean velocity of, 193 Moment, of force, 43; of momentum, 43; of couple, 44 Moment of inertia, 56; of rod, 57; of plate, 58; of parallelopiped, 59; experi- mentally determined, 60 Moment of torsion, 116; determination of, 117 Moment, magnetic, 226; changes in, 243; depends on temperature, 244; on mechanical disturbance, 244 Momentum, 14; conservation of, 29; moment of, 43 Motion, 12; absolute angular, 12; simple harmonic, 18; Newton's laws of, 27; in a circle, 61; in an ellipse, 64 Motions, composition and resolution of, 16; of simple harmonic, 22; of a fluid, 128; optical method of compounding, 384 MUller, J., limit of magnetization, 245 Newton, laws of motion, 27; central forces, 61; law of mass attraction, 58; quantity of liquid flowing through orifice, 134; theory ol light, 396; inter- ference of light from films, 442; composition of white light, 453; chromatic aberration, 455; law of cooling, 470 Nichols, study of radiations, 471 S20 INDEX. Nicholson and Carlisle, decomposition of water by electrical current, 273 Nicol, prism, 482 Node, 362 Noise, 365 Objective, 431 Ocean currents, energy of, 218 Oersted, his piezometer, 112; relation between magnetism and electricity, 273 Ohm, law of electrical current, 276, 277 Ohm, a unit of electrical resistance, 330; various values of, 330; determination of, 330 Optic angle, 429; axis of crystal, 474, 489 Optics, 396 Ordinary ray, 475; index, 475 Organ pipe, 371; fundamental of, 373; harmonics of, 373; mouthpiece of, 373; reeds used with, 373 Osmometer, Graham's, 107 Osmosis, 106 Overtones, of pipe, 373 Parallelogram, of motions, etc., 16; offerees, 29 Particle, 12 Pascal, pressure in liquid, 120; pressure modified by gravity, 121; barometer, 123 Path, 12 Peltier, heating of junctions by passage of electrical current, 273; effect, 274, 340 Pencil of light, 422 Pendulum, Foucault's, 52; simple, 70; formula for, 71; physical, 72; Borda's, 74; Kater's, 75 Penumbra, 402 Period, of a simple harmonic motion, 18; of a wave, 23 Permeability, magnetic, 239 Pfeffer, study of osmosis, 107 Phase, of a simple harmonic motion, 19 Phonograph, 378 Phosphorescence, 472 Photometer, Rumford's, Foucault's, Bunsen's, 465 Photometry, 464 Piezometer, Oersted's, 112; Regnault's, 113 Pitch of tones, 365; methods of determining, 365; standard, 370 Plants, secondary cell of, 292 Plateau, experiments of, in capillarity, 97 INDEX. 521 Plates, rise of liquid between, 100; transverse vibrations of, 376 Poggendorff, explanation of gyroscope, 55 Poisseuille, friction in liquids, 88 Poisson, correction for use of piezometer, 113; theory of magnetism, 244 Polariscope, 481, 482 Polarization, of an electrolyte, 287; of cells, 291 Polarization of light, by double refraction, 476; by reflection, 480: plane of, 4S0; by refraction, 480; by reflection from fine particles, 481; elliptic and circular, 4S6; circular by reflection, 489; rotation of plane of, by quartz, 491; by liquids. 493; in magnetic field, 493 Polarized light, 478; explanation of, 478; effects of plates of doubly refracting crystals on, 483, 486, 488, 489, 491 Polarizer, 481 Polarizing angle, 480 Pole, magnetic, 224, 228 ; unit magnetic, 226 Poles, of a voltaic cell, 290 Porous body, 103 Potential, difference of, 34; absolute, 35, 37; within spherical shell, 38; outside sphere, 39. Potential, electrical, in a closed conductor, 249, 252; of a conductor, 252; zero, positive, and negative, 253; of a system of conductors, 261; difference of, measured, 267 Potential, magnetic, due to bar magnet, 228; due to magnetic shell, 232; of a closed circuit is multiply -valued, 306; illustrated by Faraday, 306 Potentiometer, Clark's, 334 Pressure, 108, 109; in a fluid, 120; modified by outside forces, 121; surfaces of equal, 121; diminished on walls containing moving liquid column, 134 Principal plane of crystal, 475 Prism, 407 Projectiles, path of, 81; movement of, in circle, 82 Properties of matter, 4 Pulley, 46 Pump, 132; air, 137; compressing, 140 Quality of tones, 365, 380; dependent upon harmonic tones, 380; upon change of phase, 381 Quarter wave plates, 488 Quartz, effects of plates of, in polarized light, 491; imitation of, 492 Quincke, change in volume of dielectric, 263; electrical endosmose, 288; move- ments of electrolyte, 288; theory of electrolysis, 289 Radiant energy, effects of, 462; transmission and absorption of, 466; emission of, 468; origin of, 469 $22 INDEX. Radiation, 167; intensity of, as dependent on distance, 463; on angle of in- cidence, 464; kind of, as dependent on temperature, 470 ' Radicals, chemical, 85 Radiometer, 192 Rainbow, 457; secondary, 458 Ratio between electrostatic and electromagnetic units, 336; a velocity, 336, physical significance of, 337, 338 Ray of light, 422 Rayleigh, electromotive force of Clark's cell, 293 Reeds, in organ pipes, 373; lips used as, 374; vocal chords as, 374 Reflection, of waves, 362; law of, 363; of light, law of, 404; total, 408; of spherical waves, 424; selective, 467; polarization of light by, 480 Refraction of light, law of, 405; angle of, 406; dependent on wave lehgth, 409; at spherical surfaces, 415; polarization of light by, 480; conical, 490 Regelation, 177 Regnault, his piezometer, 113; expansion of mercury, 170; extension of Du- long and Petit's law, 176; modification of Dalton's law, 182; modification of Gay-Lussac's law, 185; pressure of water vapor, 186; total heat of steam, 201 Resistance, electrical, 276, 329; depends on circuit, 276; of homogeneous cyl- inder, 278; specific, 278; varies with temperature, 279; units of, 329; boxes, 331; measurement of, 332; of a divided circuit, 333; used to measure tem- perature, 149. Resonator, 382 Reusch, artificial quartzes, 492 Reuss, electrical endosmose, 288 Rider, hot-air engine, 216 Rigidity, 114; modulus of, 114 Rods, longitudinal vibrations of, 374; transverse vibrations of, 376 Roemer, determination of velocity of light, 432 Rotation of plane of polarization by quartz, 491; right-handed and left-handed, 492; by liquids, 493; in magnetic field, 493; explanation of, 494; by reflec- tion from magnet, 495 Rotational coefficient, Hall's, 316 Ro\*land, mechanical equivalent of heat, 159, 205; magnetic permeability, 239; force on magnet due to moving electrical charge, 337; measurement of ■v, 338; photographs of solar spectrum, 452; curved grating, 452 Ruhmkorff's coil, 328 Rumford, relation of heat and energy, 145; views concerning work done by animals, 219 Saccharimeter, 493 Saturation of a magnet, 243 INDEX. 523 Savart, his toothed wheel, 365 Scales, musical, 36S; transposition of, 369; tempered, 370 Schonbein, chemical theory of electromotive force, 293 Schweigger, his multiplier, 316 Screens, diffraction effects at, 445 Screw, 48 Seebeck, thermo-electric currents. 340; thermo-electric series, 341 Self-induction, 325 Set, 118 Shadows, optical. 402 Shear, 108, 113; amount of, 114; axis of, 114 ,^ Shearing stress, 108; strain, 109 Shunt circuit, 334 Siphon, 131 Siren, determination of number of vibrations by 366 Smee's cell, 291 Snell, law of refraction, 406 Solenoid, 314 Solidification, 176 Solids, 85; structure of, 86; crystalline, amorphous, 86; movements of, due to capillarity, 102; distinction between fluids and, 119; soft, hard, 119 Solubility, 104 Solution, 103 Sound, 353: origin of, 353; propagation of, 354; theoretical velocity of, 390; velocity of, in air, 392; measurements, 394 Sounding boards, 378 Specific gravity, 125; determination of, for solids, 125; for liquids, 126; for gases, 127; correction for temperature, 174 Specific gravity bottle, 126 Specific heat, 152; mean, 153; varies with temperature, 175; with change of state, 175 Specific heat of gases, 194; at constant volume, 194; at constant pressure, 194; ratio of these. 195; determination of, at constant pressure, 195; rela- tion to elasticities, 198 Specific inductive capacity. See Capacity, specific inductive, 257 Spectrometer, 451; method of using, 451 Spectroscope, 459 Spectrum, pure, 447; produced by diffraction grating, 447; of first order, etc., 447; formed by prism, 453; solar,- 453, 458; dark lines in, 458; study of, 459; of solids and liquids, 459; of gases, 460, explanation of, of a gas, 469; characteristics of, 471; of gases which undergo dissociation, 471 Spectrum analysis, 459 524 INDEX. Spheroidal state, 183 Spherometer, 8 Spottiswoode and Moulton, electrical discharge in high vacua, 352 Sprengel, his air-pump, 139; theory of, 134 Statics, II Steam, total heat of, 201 Stirling's hot air engine, 216 Stokes, study of fluorescence, 472 Strain, 108 Stress, 29; in medium, 108 Substances, simple, compound, 85 Sun, energy of, 221 Surface density of electrification, 251 Surface energy of liquids, 93 Surface tension of liquids, 91; relations to surface energy, 93; modified by electrical effects, 294 Tait, experiments in thermo-electricity, 342; thermo-electric formula, 345 Telephonic transmitters and receivers, 327 Telescope, 430, 431; magnifying power of, 431 Temperament of musical scale, 370 Temperature, 146; scales of, 147; change of, during solidification, 178; critical, 183, 184; absolute zero of, 191; absolute, 212; movable equilibrium of, 468; radiation of heat dependent on, 470 Tension, 108; elasticity of, 114 Thermodynamics, laws of, 205 Thermo-electric currents, 340; how produced, 342; reversal of, 342 Thermo-electric diagram, 342 Thermo-electric element, 341 Thermo-electric power, 343 Thermo-electrically positive and negative, 341 Thermometer, 146; construction of, 146; air, 149; limits in range of, 149; weight, 149, 172; registering, 150 Thermopile, 341 ; used to measure temperature, 149 Thomson, vortices, 135; absolute scale of temperature, 214; theory of solar energy, 221; treatment of magnetic induction, 238; magnetic permeability, 239; absolute electrometer, 265; quadrant electrometer, 267; law of coun- ter electromotive force, 280; contact theory of electromotive force, 293; measurement of v, 336; thermo-electric currents in non-homogeneous cir- cuits, 342; thermo-ele;ctric power a function of temperature, 345; the Thomson effect, 347; electromotive force required to force spark through air, 349 INDEX. 525 Thomson effect, 345 Tides, energy of, 220 Time, unit of, S; measurements of, 8, 9; Maxwell's proposed unit of, 83 Tones, musical, 365; differences in, 365; determination of number of vibra- tions in, 365; whole and semi-, 369; fundamental, 373; analysis of compl.x, 382; resultant, 387 Tonic, 369 Torricelll, barometer, 122; experiment of, 123; theorem for velocity of efflux, 131; experiments to prove, 133 Torsion, amount of, 116; moment of, 116 Torsion balance, 116, 249 Transmission of radiations, 466 Triad, major, 368; minor, 368 Tubes, rise of liquid in capillary, 99; drops in capillary, lOi Tuning-fork, 376; sounding-box of, 378 Umbra, 402 Units, fundamental and derived, 4 ; dimensions of, 10; systems of, 10 Vacuum tube, electrical discharge in, 350 Vapor, 180; saturated, 180; pressure of, 181; production of, in limited space, 183; departure of, from Boyle's law, 185; pressure of saturated, 186; pres- sure of water-, 186; in air, determination of, 202; pressure of, 202 Vaporization, energy necessary for, 200; heat equivalent of, 200 Velocity, 13; angular, 48; constant in a circle, 61 Velocity of efflux of a liquid, 129; into a vacuum, 133 Velocity, mean, of molecules of gas, 193 Velocities, composition and resolution of, 16; of angular, 49 Vena contracta, 134 Ventral segment, 362 Verdet, electromagnetic rotation of plane of polarization, 493 Vernier, 5 Vertex of spherical mirror, 411 Vibrations of sounding bodies, 371; modes of exciting in tubes, 373; longi- tudinal, of rods, 374; of cords, 374; transverse, of cords, 375; of rods, 376; of plates, 376; communication of, 377; of a membrane, 378; optical method of studying, 383 ; velocity of propagation of, 390 Vibrations, light, transverse to ray, 478; relation to plane of polarization, 480; elliptical and circular, 486 Viscosity, 88; of solids, 119 526 INDEX. Vision, ancient theory of, 396; Aristotle's view of, 396 Visual angle, 429 Vocal chords, 374 Volt, a unit of electromotive force, 326 Volta, change in volume of Leyden jar, 263; electrophorus, 269; contact differ- ence of potential, 272; voltaic battery, 273; heating by current, 273; con- tact theory of electromotive force, 293 Voltaic cells, 2go; polarization of, 291; theories of electromotive force of, 293; arrangements of, 335 Voltaic cells, kinds of : Grove's gas battery, 290; Smee's, 291; Daniell's, 291; Grove's, 292; Plantfe's secondary, 292; Clark's, 293 Voltameter, weight, 285; volume, 285 Volume, change of, with change of state, 177 Vortex, in perfect fluid, 135; line, 135; filament, 135; properties of a, 136; strength of, 136; illustrations of, 137 Vowel sounds, dependent on quality, 383 Water, specific heat of, 151; maximum density of, 161, 174; expansion of, by heat, 174; on solidification, 177 Water-power, energy of, 218 Wave, simple, 23; compound, 24, 359; propagation of, 354; length, 355; pro- gressive, 355 Wave, sound, 356; moeje of propagation of, 356; graphic representation of, 356; displacement in, 358; velocity of vibration in, 359; stationary, 361; reflec- tion of, 362; in sounding bodies, 371 Wave, light, surface of, 397; relation of, to the direction of propagation, 401; emergent from prism, 407, 408; measurement of length of, 440, 450; values of lengths of, 452; surface of, in uniaxial crystals, 476; in biaxial crystals, 490 Weber, theory of magnetism, 244; theory of diamagnetism, 315; his electro- dynamometer, 321 Weber and Kohlrausch, measurement of v, 336 Wedge, 48 Weighing, methods of, 78 Weight of a body, 70 Wheatstone, his bridge, 331 Wheel and axle, 47 Wiedemann, electrical endosmose, 288 Wind power, energy of, 218 WoUaston, dark lines in solar spectrum, 458 Work, 31; and energy, equivalence of, 32; unit of, 32 INDEX. 527 Wren, theory of gravitation, 6S Wright, connection of electromotive force and heat of chemical combination, 286 Young, theory of capillarity, 91; modulus of elasticity, 115; optical method of studying vibrations, 3S3; interference of light from two similar sources, 439