%. 1 Br^ -^^i.':' QforttBU Uniweraitg Slibrara Stliata, 97etn ^artt £.£MiJuJbzM. Cornell University Library arW3842 Elementary text-book of physics / 3 1924 031 362 779 olin.anx The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031362779 ELEMENTARY Text-Book of Physics. BY PROFESSOR WILLIAM A. ANTHONY, OF CORNELL UNIVEBSITY, AND PROFESSOR CYRUS F. BRACKETT, OF THE COLLEGE OF NEVr JERSEY. THIRD EDITION, REVISED AND ENLARGED. NEW YORK: JOHN WILEY & SONS, 15 AsTOR Place. 1887. H Copyright, 1887, By John Wiley & Sons. DBmnroHD & Ned, 1 to 7 Hague Street, New York. PREFACE. The design of the authors in the preparation of this work lias been 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 estabhshed. 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 ex- act 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 I 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 before 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. / FACE 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, 161 III. Effects of Heat, 168 IV. Thermodynamics, 205 MAGNETISM AND ELECTRICITY. Chapter I. Magnetism, 223 II. Electricity in Equilibrium, 246 III. The Electrical Current, 272 IV. Chemical Relations of the Current, .... 282 V. Magnetic Relations of the Current 297 VI. Thermo-electric Relations of the Current, . . 340 VII., Luminous Effects of the Current, . . . . 348 SOUND. Chapter L Origin and Transmission of Sound, il. Sounds and Music, III. Vibrations of Sounding Bodies, IV. Analysis of Sounds and Sound Sensations, V. Effects of the Coexistence of Sounds, . VL Velocity of Sound, 353 365 371 380 385 390 VI • CONTENTS. LIGHT. PAGE Chapter I. Propagation of Light, 396 II. Reflection and Refraction, 404 III. Velocity of Light 432 IV. Interference and Diffraction, . . ■ .* . . 436 V. Dispersion, 453 VI. Absorption and Emission, 462 VII. Double Refraction and Polarization, .... 474 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 foe g — 981, 499 VI. Elasticity, 499 VII. Absolute Density of 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 th^ir 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, XVI. Atomic Weights and Combining Numbers, XVII. Molecular Weights and Densities of Gases, XVIII. Electromotive Force of Voltaic Cells, . XIX. Electro-chemical Equivalents, XX. Electrical Resistanck XXI. Indices of Refraction, .... 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 w, . . . . 506 Index 507 503 503 503 504 504 504 505 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 wifh 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 tiie 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 ncit directly related to chemical 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 E.LEMENTARY PHYSICS. [2 It is not possible, however, to draw sharp lines of demarca- tion between the various departments of Naitural 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 sufifice 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 vy^hich 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 correspondr 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 experintent 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- 2] INTRODUCTION. 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 a? 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 /awj which connect them. The process of reasoning, by which we discover such laws ,is called induction. As we can seldom be sure that ^ve have apprehended all the related fac^s, 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 experience. If the course of our experiment be in accordance with our supposition, th^re 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 existence 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 ^ series of ascertaifted laws to some common agency, we employ the term theory. Thus we find in the " wave theory" of hght, 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 the 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] IN TROD UCTION. 5 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 difificult 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 verniei" 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 numbere,d 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 ... 10 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- fig. .. nitude to be measured, it is clear that, from the position shown in the figure, we have 29.7, expressing that magnitude ELEMENTARY PHYSICS. [4 to the nearest tenth ; and since the sixth division of the vej- ' nier coincides with a whole division of the principal scale, we have -j^ of -^j or -j-^, of a principd 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 point. This is frequently the zero point of a scale with divisions equal in magnitude to the pitch of the screw. These divisions will then show through how many revolutions the screw is turned in any given trial ; while the divisions on the graduated circle will show the fractional part of a revolu- <3i tion, and consequently the frac- tional part of the pitch that must be added. If the screw be turned through n revolutioijs, as shown by the scale, and through an additional fraction, as shown by, the divided circle, it will pass throiigh 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 cathetometer ^s 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 slide ' 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. 3. 4] IN TROD UCTION. 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 uncla,niped, 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. [5 tance determined by the motion of the 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 turn«d 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 diilerence 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 gg^oa of a mean solar day. We employ the clock, regulated by the pendulum or the chronometer balance, to indicate seconds. The clock, while suilSciently ac- FlG. 7] INTRODUCTION. curate for ordinary use, must for exact investigations be frcr 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 to 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. Measuremeat 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 circle 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 it = 3. 141 592. The radian, measured in degrees, is 57° 17' 44.8." 10 ELEMENTARY PHYSICS. t? 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- namics. In the first are developed, by purely mathematical methods, the laws of motion fconsidered in the abstract, inde- pendent of anj^ 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 motioris 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. 11. 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 [2 ELEMENTARY PHYSICS. [i2 •epresented by the help of the quantities of matter in its elementary volumes. The density of any substance is defined IS the hmit of the ratio of the quantity of matter in any volume Bvithin the substance to that volume, when the volume is dimin- shed indefinitely. In case the distribution of matter in the Dody is uniform, its density may be measured by the quantity if matter in unit volume. Since density is measured by a mass divided by a volume, :ts dimensions are ML ~ '. 12. Particle. — A body constituting a part of a material system, and of dimensions such that tbey may be considered infinitely small in comparison with the distances separating it [rom all other parts of the system, is called 2i particle. 13. Motion. — The change in position of a ftiaterial particle is called its' motion. It is recognized by a change in the config- 'iration of the system containing the displaced particle ; that is, by a change in the relative positions of the particles making jp the system. Any particle in the system may be taken as ;he fixed point of reference, and the motion of the others may 3e measured from it. Thus, for example, high-water mark on :he 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 [lave no assurance that the particle which we assume as axed is not really in motion as a part of some larger system ; indeed, in alinost every case we know that it is thus in motion. A.S it is impossible to conceive of a point in space recognizable IS 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 :ontinuous line or path. In all investigations the path maybe IS] ■ MECHANICS OF MASSES. 1 3 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 tfie space traversed by the particle to the time occupied in travers- ing that space. 11 s^ and s represent the distances of the par- ticle from a fixed point on its path at the instants /„ and t, then its velocity is represented by s — s„ ' ^ = TZrf ' (0 °> 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 varia- ble. The path of a particle moving with a variable velocity may he. 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 = - — T—r. As the interval of time t — t^ approaches zero, t '0 - each of the spaces s — s„ wi^ll become indefinitely small, and in the limit the imaginary path will coincide with the real path. s — s The limit of the expression - — ^ will represent the velocity of the particle along the tangent to the path at the time t = t^, or, as it is called, the velocity in the path. This limit is usually expressed by -^. The practical unit of velocityis the velocity of a body mov- ing uniformly through one centimetre in one second. The dimensions of velocity are LT~^. 14 ELEMENTARY PHYSICS. [i6 i6. Momentum. — ^The momentum of a body is its quantity of motion. This varies with its mass and its velocity jointly, and is measured by their product. ThuSj for example, a body weighing ten grams, and having a velocity of ten centimetres, has the same momentum as a body weighing one gram, and having a velocity of one hundred centimetres. The practical unit of momentum is that of a gram of matter moving with the unit velocity. The formula is mv, (2) where ih 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 Hne, 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 j/„ and v represent the velocities of the par- ticle at the instants t^ and /, then its acceleration is represented by 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 ... f _{ = ^ will represent the acceleration in the path at the time t = t^. This acceleration is due to a change of velocity in the path. It is not in all cases the total acceleration of the [7] MECHANICS OF MASSES. 15 particle.! As. will be seen in § 36, a particle moviiig along d :urve has an acceleration which is not due to a change of velocity in the path. , The practical unit of acceleration is that of a particle^ the ve- ocity 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 itioving with a con- stant acceleration f, during a time t — t„ is determined by ;onsidering that, since the acceleration is constant, the aver- ige velocity for the time t — t„ multiplied by t — t„ will represent the space traversed ; hence .-.„ = ^X^-/,); (4) Dr. since - = ^, we have, in another form, 2 2 s-s, = vst - o + m^ - t.r- ' (4) Multiplying equations (3) and (4), we obtain ■!;> = v: + 2f{s-s,). • .(5) When the particle stairts 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 s^ — o, t^ = o, and; equations (3), (4), and (5) become v —ft, s = ^ft^, and v^ — 2fs. Formula (4) may also be' obtained by a geometrical con- struction. At the extremities of a line AB (Fig. 5), equal in length to t — t„ erect perpendiculars A C and BD, proportional to the t ffl ^^ l6 ELEMENTARY PHYSICS. [l8 initial and final velocities of the irioving particle. For any in- 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 t^a-bcd, equal increments, is represented by a fig. s. series of rectangles, ^3, y^,^*/, etc., described on equal bases, ab, be, cd, etc. If ab, be . . . 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 — /„. But ABCD is equal to ^ C (^ — ?„) -\- {BD -AC){t-t,)-^2; whence s-s, = vlt - Q + \f{t - t:f. • (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 «, moye with a constant velocity relative to a third point a^, then the motion, in any fixed time, of «, relative to a, may be readily found. Represent the motion, in a fixed time, of «, relative to a^ (Fig. 6) by the line v„ and of a^ 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 ^B 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 a^ are con- stant or proportional to z/, and v„ bring the point a;, to some point c lying on the line joining A and /.. ^^^^1 t =f ^.. / 6 Fra. «. l8] s MECHANICS OF MASSES. 17 B. Therefore the diagonal AB of the parallelogram haying the sides I/, 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 v^ 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 obta,in a new resultant, and proceeding in this way till 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 Hne 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 front the origin represent the component motions in direction and amount. A motion may be resolved in three directions not in the same plane by drawing from the extremity of the line repre- senting the motion, taken as origin, lines in the three given direc- tions, and constructing upon those lines the parallelopiped of whicfh the line 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 th^ l8 ELEMENTARY PHYSICS. [19 line representing the motion, and Q, 0, and ^ the angles which it makes with the three axes, the components along those axes are a cos 6, a cos 0, and a cos ip. Two motions may be compounded by first resolving theni along two rectangular axes in their plane, and obtaining the resultant of the sums of their components along the axes. If a and b (Fig. 7) represent motions, a cos 0, b cos 6, a sin 0, b sin 6 are the resolved components of a and b along the axes. Let a cos <(> -\- b cos 6 = X and a sin <})-\-b smd = F; then the diago- nal of the rectangle, of which X and F.0. 7. ^ ^^^ sides, is R = {X^+ YJ ; or, since the angle between the resultant and the axis of X is known by Y—Xtan ip, it follows that j^ X Y ^ ~ coslb °^ siir?6' ^^ 's evident that this process may be 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 drav/n from the moving point to this diameter will have a simple harmonic motion. 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 inotion can be expressed in terms of the interval of time which has elapsed since the point last passed through the W] 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 eapsidered 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, , , , 2nt represented by a, and = —j^, where T is the period ; hence 271 1 s — a cos - (6) To find the velocity at the , Fig. 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 2na is V ^va. 4>; or, since V = —j^^ V = 2na . 2itt —^=- sm r' (;) 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 Q given by Eq. (7). 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 , . 2nd . ( 2itt , 2nAt\ V -\- Av ■= -=r sm —=- A — — ;;— 1 2Tta I 27tt 27tAt , 2Ttt . 2nAt\ — Ism -Y cos — 1- cos -Y sm—=^j. 2nAt As At approaches zero, cos —=. — approaches the limit unity, , . 2nAt , , , , . 2nAt , . , and sin —= — can be replaced by its arc „ ; making these changes, and transposing, Av ATt^a 2Ttt — — ■ CC\^ ' .\ At~ ' T^ T ' sJAv But in the limit where these changes .are admissible, -j- becomes --7- ; that is, the acceleration of the point. '' Hence the acceleration sought is 47r'' 27tt / = — 2^r«cos -^. (8) This for niula 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, accderation to the right of O is negative, and to the left of O positive. It is often necessary to reckon titne from some other posi tion than that of greatest plositive 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 circumfereqce 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 ^' in the circumference, the motion in the diameter is negative, and the epdch is the time ^required ^ for the moving-point to go from L through O to X' and back \oX. ' 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'KX. Choosing K in the circle, or L in the diameter, as the point from which time is to be reckoned, the angle equals angle 2i7tt KOB — angle KOX, or -jr — 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 l2nt S — a COS\^-yr -^)^ 27r . (27ri -=: a sm \p^ \ V — — -~ a sin V-j, — el; / = — yr « cos \^ — 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 BC = asivKp = a cos^0 — -j = a cos(-™ 1, the projection of B on the diameter O V also has a simple harmonic motion, differing in epoch from that in the' diameter TV 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 m 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 X = a cos(0 — e) and y = b cos 0'. If and ' are commensurable, that is, if 0' = «0, the curve is re-entrant. Making this supposition, X = a cos cos 6 -j- a sin sin e, and y ■=■ b cos «0. Various values may be assigned to a, to b, and to n. Let a equal b and n equal i ; then X = y cose-\-(a^ — /)* sin e ; I?] MECHANICS OF MASSES. 23 from which x^ —2xy cos e +7' cos' e = «' sin' e — f sin' e, or, x' — 2xy cos e -|- y = a° sin' e. This becqrnes, when e = 90°, jr* -f-y = a', the equation for a circle. When e = 0°, it becomes jr — j/ = 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 = 2, we obtain, as special cases of the curve, a parabola and a lemniscate, according as e = 0° or 90°. If a and b are unequal, and n r= i, we get, in general, an ellipse. If a. line in which a point is describing a simple harmonic motion be made to move in a direction perpendicular to itself, tljie moving-point will describe a harmonic curve, called also a sinusoid. It is a diagram of 2. 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 ■ ELMMENTARY physics. [1-9 the. displacement from the line of progress of any point on the wave is represented by = a cos \2n-~\. The displacement due to any other wave differing from the first only in the epoch is represented by Sj ^ a cos [^Tt-^ — ej. 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\ cos 27t -^ -\- cos 1 27? 7^ — ej r ^1 * , • t . ' = a cos 27r ^ -|- cos 27t -^ cos e -j- sm 2;i -~ sm e = a cos 2n -={\ -\- cos e) -|- sin 27r -^ sin e . If for brevity we assume a value A and an angle such that A cos = «(i + cos e), and A sin = a sin e, we may represent the last value o{ s -\- s^ by A cos \2n-7p — 0|. 19] MECHANICS OF MASSES. 25 From the two equations containing A, we obtain, by adding the squares of the values of A sin and A cos 0, A = {2a' + 2a' cos e)*; and, by dividing the value of A sin by that of A cos 0, we obtain sin e = tan-' : I -|- cos e" The displacement thus becomes / t sin e \ s + s,= «(2 + 2 cos e)i cos ^2;r ^ - tan"' flf^^J- ^9) This equation is of great value in the discussion of prob- lems in optics. The principle suggested by the result of the above discus- sion, that the resultant of the composition of two simple har- monic motions is a harmonic motion of which the elements depend on those of the components, can be easily seen to hold generally. Avery important theorem, of which this principle is the converse, was given by Fourier. It may be stated as follows : Any complex periodic function maybe resolved into a number of simple harmonic functions of which the periods are com- mensurable with that of the original. function. As an example, any wave not simple may be decomposed into a number of simple waves the lengths of which are to each 26. ELEMENTARY PHYSICS. [20 Other as \, \, \, etc. The number of these simple waves is, in general, infinite, but in special cases determinate both as to number and to period. 20. Force. — Whenever any change occurs, or tends to occur, in the momentum of a body, we ascribe it to a cause called z. force. Whenever motions of matter are effected by our direct personal effort, we are conscious, through our muscular sense, of a resistance to our effort. The conception of force to which this consciousness gives rise, we transfer, by analogy, to the interaction of any bodies which is or may be accompanied by change of momentum. The question whether this analogy is or is not valid, is not involved in a purely physical discussion of the subject. ' A force, in the physical sense, is the assumed cause of an observed change of momentum. It is known and measured solely by the rate of that change. If a body.be moving with any acceleriation whatever, the force acting on it is fully expressed by the product of the mass of the body into its acceleration. The formula for force is, therefore, The dimensions of force are MLT'^. As acceleration is always referred to some fixed direction, it follows that force is a quantity having direction. The product of the time during which a force acts by its mean intensity is called the impulse of the force. The practical unit of force is the dyne, which is the force that can impart to a gram of matter one unit of acceleration ; that is to say, one unit of velocity in one second. 22] MECHANICS OF MASSES. 2^ 21. Field of Force. — A field of force is a region such that a particle constituting a part of a mutually interacting system, placed at any point in the region, will be acted on by a force, and will move, if free to do. so, in the direction of the force. The particle so moving would, if it had no inertia, describe what is called a line of force, the tstngent to which, at any point, is the direction of the force at that point. The strength of field at a point is measured by the force developed; by unit quantity at that point, and is expressible, in terms of lines of force, by the convention that each line represents a unit of force, and that the force acting on unit quantity at any point varies as the number of lines of force which pass perpendicu- larly through unit area at that point. Eacli line, therefore, represents the direction of the force, and the number of lines passing through unit area, the strength of field. An assem- blage of such lines of force considered with reference to their bounding surface is called a tube of force. 22.. Newton's Laws of Motion. — We are now ready for the consideration of the laws of motion, first formally enun- ciated and successfully applied by Newton, and hence known by his name : Lex L — Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare. Lex II. — ^Mutationem motus proportionalem esse vi mo- trici impressae & fieri secundum lineam rectam qua vis ilia imprimitur. Lex III. — Actioni contrariam semper & aequalem esse reactionem ; sive corporum duorum actiones in se mutuo semper esse aequales & in partes- contrarias dirigi. The subjoined translations are given by Thomson and Tait : Law I. — Every body continues in its state of rest or of motion in a straight line, except in so far as it may be com- pelled by force to change that state. 28 .ELEM'kNTARY PHYSICS. [23 Law Il.^-Change of motion is proportional to force ap- plied, and takes place in the direction of the straight line in which the force acts. Law IIL — To every action there is always an equal and contrary reaction : or, the mutual actions of any two bodies are always equal and oppositely directed. 23. Discussion of the Laws of Motion. — (i) The first law is a statement of the important truths implied in our defi- nition of force, — that motion, as well as rest, is a natural state of matter ; that moving bodies,' when entirely free to move, proceed in straight lines, and describe equal spaces in equal times ; and that force is the cause of any deviation from this uniform rectilinear motion. That a body at rest should continue indefinitely in that state seems perfectly obvious as soon as the proposition is entertained ; but that a body in motion should continue to move in a straight line is not so obvious, since motions with which we are familiar are frequently arrested or altered by causes not at once apparent. This important truth, which is forced upon us by observation and experience, may, however, be presented so as to appear almost self-evident. If we con- ceive of a body moving in empty space, we can think of no reason why it should alter its path or its rate of motion in any way whatever. (2) The second law presents, first, the proposition on which the measurement of force depends; and, secondly, states the identity of the direction of the change of motion with the direction of the force. Motion is here synonymous with momentum as before defined. The first proposition we have "already employed in deriving the formula representing force. The second, with the further statement that- more than one force can act on a body at the same time, leads directly to a most important deduction respecting the com- 24] MECHANICS OF MASSES. 29 bination of forces ; for the parallelogram law for the resolution and composition of motions being proved, and forces being proportional to and in the same direction as the motions which they cause, it follows, if any number of forces acting simultaneously on a body be represented in direction and amount by lines, that their resultant can be found by the same parallelogram construction as that which serves to find the resultant motion. This construction is called the paral- lelogram of forces. In case the resultant of the forces acting on a body be zero, the body is said to be in equilibrium. (3) When two bodies interact so as to produce, or tend to produce, motion, their mutual action is called a stress. If one body be conceived as acting, and the other as being acted on, the stress, regarded as tending to produce motion in the body acted on, is a, force. The third law states that all interaction of bodies is of the nature of stress, and that the two forces constituting the stress are equal and oppositely directed. From this follows directly the deduction, that the total momentum of a system is unchanged by the interaction of its parts ; that is, the momentum gained by one part is counter- balanced by the momentum lost by the others. This princi- ple is known as the conservation of momentum. 24. Collision of Bodies. — If two bodies, m^ and m^, with velocities v^ and v^ in the same line, impinge, their velocities after contact are found, in two extreme cases, as follows : (i) If the bodies are perfectly inelastic, there is no tend- ency for them to separate, their final velocities will be equal, and their momentum will be equal to the sum of their sepa- rate momenta ; hence ■m,v^ ■\- m^v^ = {m, + m^)x, (i i) where: x is the velocity after impact. 30 ELEMENTARY PHYSICS. , [2^ (2) If the bodies are perfectly elastic, they separate with a force equal tO' that by which they are compressed. Let V represent their common velocity just at the instant when the resistance to . compression balances the impulsiv^e force. Then the change in momentum in each body up to this instant is m^v^ — v), or m^{v— v^ ; and the further change of momentum, by reason of the- elasticity of the bodies, is the same; whence the whole momentum lost by the one is 2mjj)^ — v) and that gained by the other 2mJ^v — v^. If x represent the final velocity of m^ we have the equation m^v^ — m^x = 2m^v^ — 2m^v, whence X = 2V ^ V^. In like manner, ily represent the final velocity of m^, we find y = 2v — V,. From the formula for inelastic bodies, which is applicable at the moment when both bodies are moving with the same velocity, whence, finally, ^__ {m,-m,)v,-{-2m,v, m^ -\-m, • ' .__ \^^ — fn;)v^ -\- 2m,v, (12) 25. Inertia.— The prihciple of equality of action and re- action holds equally well when we consider a single body as •26] MECHANICS OF MASSES. 31 acted on by a force. The resistance to change of motion offered by the inertia of the body is equal in amount and opposite in direction to the acting force. Inertia is not of itself a force, but the property of a body, enabling it to offer a resistance to a change of motion. 26. Work and Energy. — When a force causes motion through a space, it is said to do work. The measure of work is the product of the force and the space traversed by the body on which the force acts. The formula expressing work is therefore mfs. (13) The dimensions of work are MUT~'. In the defined sense of the term, no work is done upon a body by a force unless it is accompanied by a change of posi- tion, and the amount of work is independent of the time taken to perform it. Both of these statements need to be made, because of a natural tendency to confound work with conscious effort, and to estimate it by the effect on our system. A body may, in consequence of its motion or position with respect to other bodies, have a certain capacity for doing work. This capacity for doing work is its energy. Energy is of two kinds, usually distinguished as potential and kinetic. The former is due to the position of the body, the latter to its motion. 1 Since th^ potential energy of a body is due to the exist- ence of a force acting upon it, it is clear that, if the body be free to move, it will be moved by the force, and its potential energy will be diminished. Hence, in any system of bodies free to move, movements will occur until the potential energy of the system becomes a minimum. If a niass m be moving with a velocity v, its capacity for doing work may be determined from the consideration, that, 32 ELEMENTARY PHYSICS. [27 if the motion be opposed by a force F equal to mf, the mass suffers a negative acceleration /, and is finally brought to rest after traversing a space s in opposition to the force. From Eq. (5) we have ^ = —p. Multiplying both sides of this,equa- 2 tion by i^= mf, we have Fs = . But ^Fs is the work done by the body against the force F, and is, therefore, the capacity which the body originally had for doing work. ' This capacity — that is, the kinetic energy of the body — is then represented by the expression . The dimensions of energy s.r&MDT-", the same as those of work. Since the square of a length cannot involve direction; it follows that energy is a quantity independent of direction. The practical unit of work and energy is the erg. It is the work done against a force of one dyne, in moving its point of application in the line of the force through a space of one centimetre ; Or, it is the energy of a body so conditioned that it can exert the force of one dyne through a space of one centimetre ; Or, it is the energy of a mass of two grams moving with unit velocity. 27. Conservation of Energy. — The difference between the kinetic energy of a body at the beginning and at the end of any given path is equal to the work done in traversing that path. For, if we consider the mass m having an acceleration /, and moving through a space s, so small that the acceleration may be assumed constant, we have, from Eq. (5), v-' = v: + 2fs, where s replaces the .? — j„ of the equation. 27] MECHANICS OF MASSES. 33 Multiplying by ^m, we have \m'v' = ^mv' -\- mfs. Since any motion whatever may be divided into portions in which the above conditions hold true, it follows that we have finally, for any motion, ^v' — imv„' + m/,s,-{- m/,s, -f- . . . = ^v' -\- ^mfs. Since \mv^, or the initial kinetic energy, is a constant quantity, it follows that \mv'' — 2mfs, or the sum of the kinetic and potential energies, is a constant quantity for any body moving under the action of forces without collision with other bodies. In other words, a body, by losing potential energy, gains an equal amount of kinetic energy; and the kinetic energy, being used to .do work against acceleration^ places the body in a position where it again possesses its .original amount of potential energy. This statenient holds true for any body of a system made up of bodies moving, without collision, only under their mu- tual interactions. It follows therefore that the total energy of such a system remains constant. There are other forms of energy besides the potential and kinetic energies of masses. By suitable operations energy in any one form may be transformed into energy of any other form. The simplest example of such a transformation is the. simultaneous production of heat and loss of mechanical en- ergy by friction or collision. In any closed system, into which no energy enters, and out. of which no energy passes, the statement made above for the energy of a simple system of bodies holds true, if all forms of.' 3 34 ELEMENTARY PHYSICS. [28 energy in the system are taken into account. Whatever trans- formations of the energy within the system occur, its total amount remains constant. This principle, called the principle of the conservation of energy, can be demonstrated to hold for the mechanical interaction of bodies moving without collision, and has been established by experiment for operations involv- ing molecular and atomic interactions. It is a general prin- ciple, with which all known laws of the material universe are consistent. The principle of the conservation of energy is so well estab- lished and so universally accepted, that, where convenient, it has been used in the demonstrations of this book as a funda- mental principle. 28. Difference of Potential. — The difference of potential between two points in a field of force is measured by the work done by the forces of the field in moving a test unit of the quantity to the presence of which the force is due from one point to the other. If Vp— Vq represent the difference of potential between the points Pand Q, and if F represent the average force be- tween those points and s the distance between them, then the amount of work done in moving a unit from P to Q, and hence the difference of potential between P and Q, is rep- resented by Vp-Vq = Fs. From this relation we have F = Vp-Vq_ Vq-Vp If s become indefinitely small, in the limit F represents the Vp — Vj> dV force at the point P, and — — —j- becomes the :28] MECHANICS OF MASSES. 35 rate of change of potential at that point with respect to space, taken with the opposite sign. Hence we obtain a definition of Jiotential. It is a function, the rate of change of which at any point, "with respect to space, taken with the opposite sign, measures the force at that point. In the discussion which follows we deal with forces which vary directly as the product of the quantities acting, and in- versely as the squares of the distances which separate them. For convenience, these acting quantities will be called masses or quantities of matter. By the substitution of proper terms the theorems to be presented will hold equally well in all cases involving forces acting according to this law. If the field be due to the presence of a mass m, which repels the test unit, at a distance s, with a force expressed by km the difference of potential between two points, distant r and R from the mass m, is expressed by The symbol k represents the force with which two unit masses at unit distance repel one another. To obtain this formula in the simplest case, let us suppose a mass at the point O (Fig. 9) acting upon ^ ^^ km I LJ_ i_u a unit at Pwith a force 'fequal to ^^. If p,e. g. the unit be moved to Q the force at Q is -^ ; and the average force acting while the unit is moving in the path PQ, provided 36 ELEMENTARY PHYSICS. ' [28. this path be taken small enough, is km OP-OQ' The work done in moving the unit through PQ is (a^)^2 = -^op - OQ) = ta^g - j^). The work done in moving the unit through any other small space QR towards 5 is, similarly. The last value obtained by moving the unit from 5 to T is ^^\0T~~ OS/' The sum of these values, gives the work done in moving the unit through the space PT. It is evident that the amount of work done upon the unit to move it from P to 7" is independent of the path. For, if this were not so, it would be possible by moving from Pto T on one path, and returning from T' to P on another, ta accumulate an indefinite amount of energy ; which the principle: of the conservation of energy shows to be impossible. 38] MECHANICS OF MASSES. 37 Since the point T can be considered as on the surface of a sphere of which O is the centre, and since the force -?yj^ acts along TO perpendicular to that surface, and cannot, therefore, have a component tending to produce motion on that surface, there is no work done in moving the unit at T over the surface of the sphere to any other point X on it : it follows that the difference of potential between P and any other point at dis- tance r from O is the same as that between P and T. Whence Vr-V^^kJ^--^^. (14) Such a surface as the one described, to which the lines of force are perpendicular, is called an equipotential surface. If the point P be supposed to be at a distance from O so great that the force at that distance vanishes, it is then at zero potential. R becomes indefinitely large, and the absolute poten- tial z.\. 2" becomes km Vr=—. (15) This formula expresses the work necessary to move the unit against the repulsion of the mass at O up to the point T from an infinite distance. If the mass attract the unit, the work is done by the attraction upon the unit in so moving up to T, and the potential is negative. From the definitions, it is plain that the difference of poten- tial between P and T equals the difference of the potential energies of a unit at those points. If the potential of any point be due to the action of more than one mass, it is found by adding the potentials due to the 38 ELEMENTAR Y PHYSICS. [29. separate masses. If 2 be a summation sign indicating this operation, Eqs. (14) and (15) become Vp-VQ = :Skm^--'--^, (I4> and Vr = -2—. r 29. Theorems relating to Difference of Potential. — (i) The force at any point within a spherical shell of uniform' thickness and density is zero. For, if S represent the density of the shell, and if a^, and cd^ represent the volumes of the portions of the shell cut out by a cone having its apex at O (Fig. 10), then the force, if an attrac- kSad, tion, acting towards « is Oa' -, and towards Fig. 10. C IS Hence the efficient force tend- ing to produce motion, say towards a, is expressed by Mi^-^; cd,\ Oc'r Now, if ab^, cd„ be taken small enough, they will be frusta of similar cones, and, as a consequence, Od' 29] MECHANICS OF MASSES. 39 from which, since the density of the shell is uniform, ^(S-^)-- Since the whole surface of the shell may be cut by similar cones, for which similar equations will hold, the total force ex- erted by the shell on a unit within it becomes zero. This be^ ing so, it follows that the potential throughout the sphere is constant ; for no work is required to move the unit from one point to another in the interior. (2) The potential, and therefore the force at a point, due to the presence of a spherical shell of uniform density, depends only on the mass of the shell and on the distance of the point considered from the centre of the sphere. Let CKL (Fig. 11) represent a central section of the shell, of which O is the centre. Let d represent the mass of a portion of the shell having unit of area. The potential at B, due to the element of the sphere at Ky having an area represented by s, is ds V, BK' and the potential due to the whole sphere is the summation of that due to all the similar elements making up 40 ELEMENTARY PHYSICS. [29 the sphere. Take a point A on the line OB, such that OA . OB = R, where R is the radius of the sphere ; draw AK, produce it to L, and draw OK and OL. Now, if we represent the angle OKA by a, and the solid angle subtended by the element s, as AK" .oa seen from A,hy 00, we may express s in other terms as ; lience dAK'co ' ~ BK cos a Now, since, by construction, OB : R = R : OA, and the angle KOB is common to the two triangles KOA and BOK, these triangles are similar; hence AK _ R_ BK ~ OB' The value of the potential due to s may then be written ' COS a OB The value for the potential of the corresponding element at L is, similarly, ^"^'^■Esr^-oB-'^' Adding these values, we obtain J. , „ ,R (AK-\-AL\ ' ' " OB \ cos a J J!9] MECHANICS OF MASSES. 4I But AK-^AL KL cos a cos a hence we obtain, finally, = 20K=2R; V, + V„ = 2d-^^. Now the sphere may be divided into two portions, made up of elements similar to K and L, by a plane passing through A normal to OB. We obtain tHe whole potential, therefore, by summing all the potential values due to these pairs of elements; whence V= 2d^y^oa. The sum of all the elementary solid angles on one side of the plane from the point A in it is 2n; hence, finally, R'd _ m Tvhere m is the mass of the spherical shell. Since the force at the point B depends on the rate of change of potential at that point with respect to space, it varies in- versely as the square of the distance OB. Represent OB by /. m In the expression V=-j,\&t I change by a small increment Al, and denote the corresponding change in the potential by A V; then V+AV= "^ l+Al' Vl+VAl-\-lAV-\-A VAl = m. 42 ELEMENTARY PHYSICS. [29 If Al become indefinitely small, in the limit the product. A VAl may be neglected. We then have VAl-\-lAV = o, AV Al '■ dV V ~ dl ~ I = - m or This is the rate of change of potential at the point B, with respect to space, and, taken with the opposite sign, measures the force at that point. The force, therefore, at a point out- side a spherical shell of uniform density varies inversely as the square of its distance from the centre of the sphere. This result enables us to deal with spheres of gravitating matter, or spherical shells, upon which is a uniform distribution of electricity, as if they were gravitating or electrified points. (3) If in the last proposition we let / = ^, we obtain for the value of the force just outside the shell AnKd — ^r~ = A'^d. Since the force just inside the shell vanishes, in conse^ quence, as we have seen, of the equal and opposite actions of the portions of the sphere ab and acb (Fig. 12), and since the- total force at the point P outside the sphere is /i,nd, it follows that the force at P, due to ab, is 2nd. If the radius be taken large enough, ab may be considered as flat, and constituting a disk: hence the force at a point near a flat disk of density d is 2nd. Since the force at a point near one surface p,^ ^ of the disk is 2nd in one direction, and near the other surface 30J MECHANICS OF MASSES. 43 2nd in the other direction, it is clear that, in passing through the disk, the force changes by ^ttd. 30. Moment of Force. — 1'\\& moment of force z!oovX a point is defined as the product of the force and the perpendicular drawn from the point upon the line of direction of the force. The moment of a force, with respect to a point, measures the value of the force in producing rotation about that point. If momentum be substituted for force in the foregoing defi- nition, we obtain the definition of moment of momentum. In order to show that the moment of a force measures the value of that force in producing rotation, we will find the di- rection and amount of the resultant of two forces in the same plane acting on a rigid bar, but not applied at the same point. Let BD (Fig. 13) be the bar, Z'i^'and 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 C|/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 applied 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 1;hat plane, is called the centre of inertia. This point is perfectly definite for any system of particles. It fol- 33] MECHANICS OP 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 solid 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 ^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 lengths of their respective lever-arms. If /and /^ represent the lengths of the arms of the lever, and P and P^ the forces ap- plied to their respective extremities, then P/^ P/^. The principle of the equality of action and re-action 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 IP one of these forces may be considered ^'°- '♦• as taking the place of the fulcrum, and either 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 P is fulcrum and R power. (2) The Pulley is a f rictionless 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 eord 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 op 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 force •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 com- ponents Pcos and Psin perpendicular to and parallel with the plane, P sin is alone effective to produce motion. Con- sequently, a force P sin acting parallel to the surface will balance a force P, mak- ^'°" ''' ing an angle with the normal to the surface. If the plane be taken as the hypothenuse of a right-angled triangle ABC, of which the base AB is perpendicular to the line of direction of the force, then, by similarity of triangles, the angle BAC equals 0: whence the force obtained parallel \.q AC xs, equal to the force applied multiplied by the sine of the angle of in- clination of the plane. If the components of the force applied be taken, the one, as before, perpendicular to the plane AC, and the other parallel to the base AB, the force obtained parallel to AB is equal to the force applied multiplied by the tangent of the angle, of inclination of the plane. (4) The Wheel and Axle is essentially a continuously acting lever. 48 ELEMENTARY PHYSICS. [3S (5) The Wedge is made up of two similar inclined planes set together, base to base. (6) The Screw is a combination of the lever and the in- clined plane. The special formulas expressing the relations of the force applied to the force obtained by the use of these combinations, are deduced from those for the more elementary mechanical powers. It may be seen, in general, in the use of the mechanical powers, that the force applied is not equal to the force ob- tained. A little consideration will show, however, that the energy expended is always equal to the work done. Any arrangement of the mechanical powers, designed to do work, is called a machine. The more nearly the value of the work done approaches that of the energy expended, the more closely the machine approaches perfection. The elastic- ity of the materials we are compelled to employ, friction, and other causes which modify the conditions required by theory, make the attainment of such perfection impossible. The ratio of the useful work done to the energy expended is called the efficiency of the machine. Since in every actual machine there is a loss of energy in the transmission, the effi- ciency is always a proper fraction. 35. Angular Velocity. — The angle contained by the line passing through two points, one of which is in motion, and any assumed line passing through the fixed point, will, in gen- eral, vary. The rate of its change is called the angular velocity of the moving-point. If

is made evident by the rotation of a fixed line on the earth's sur- 3S] MECHANICS OF MASSES. 53 face, cutting the invariable plane of oscillation at the point of equilibrium of the pendulum. The plane of oscillation of the pendulum consequently appears to rotate in the opposite di- rection with an angular velocity a? sin 0, and the angle swept out in any time t is oot 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 Fig. 18. by the help of the diagram (Fig. 18). The outermost ring rests in a frame, and turns on the points a, 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 equilibrium 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 rapid 54 ELEMENTARY PHYSICS. [3S. 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,. ^ 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 B °be the point of appHcation 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- B val of time t, the system acquires an an- FiG. 19. 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 t becomes indefi- nitely small, OQ also becomes indefinitely small, and the re- solved component Ox parallel to OR vanishes in comparison OQ Ox with OQ ; because from the triangles we have -^ = -^- The 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 changer 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 cf- based on the action of couples formed by these separate forces. p\ 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. S6 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 oo, its velocity is rw and its ki- netic energy is ^moo^'r'. The whole kinetic energy of the body is, therefore, ^w'Smr' ; and since we have assumed l-ta" to be •constant, ^mr" is proportional to the kinetic energy of the ro- tating body. If we can find a distance k such that ^k^af'Sm — ^oa'Smr', k is called the radius of gyration, and is the dis- tance at which a mass equal to that of the whole body must be concentrated to possess the same moment of inertia as the body possesses. The formula for moment of inertia is I = '2mr\ (18) and its dimensions are MD. 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 36] 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 which the moment of inertia is to be re- ferred, making the construction as in Fig. 21, we have r' = rt" + zrfi + ^/. Multiplying by the mass m., performing a similar operation for every particle of the body, and summing the results, we have The term 2'2mrfi on the right van- ishes, for we may write it zr/^mb ; and, since C is the centre of inertia, 2mb is zero (§ 33). Therefore Fig. 21. /^ = /+ Mr;. (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 tn' 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. 58 ELEMENTARY PHYSICS. [3« Their moments of inertia are m!^ r_ m' P_ in!_ J!^ and the moment of inertia of the half-rod is m'l" /'=-^(i+4 + 9-- • + «")• But (I + 4 + 9 . . . -|- «"), where n is indefinitely large, is — ^ hence / = . 3 Fig. If / equal the whole length of the rod, m the whole mass, and / the entire moment of inertia. /: 12 (20) (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 : MECHANICS OF MASSES. SQ. Suppose the half-plate to be divided into n rods, parallel tO' b' length : each rod will have a length / and a breadth -. . . ■ b' 2b' , , , eir distances from the axis are -, — , etc. Let m be the n n ,ss of the plate. The moment df inertia of each rod, with. pect to an axis passing through its centre of inertia and •pendicular to its length, is — X — ■ The moments of in- ia of the several rods about the parallel axis xx' are ■m 2n ' IP , b'\ mil' . b''\ ^ i the moment of inertia of the half-plate is r , m b'\ , , , ,, mil-" . b'\ m 2n d of the whole plate equals m-^—. (21). 12 A parallelepiped ot which the axis is xx' may be supposed be made up of an infinite number of plates, such as AB. i moment of inertia will be the moment of inertia of one ite multiplied by the number of plates ; or, if M is the mass the parallelopiped, its moment of inertia is f(^' + ^'). (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 y. 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 t, = n^. f ' where I, 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 t' - f 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. MECHANICS OF MASSES, 6 1 the law of its variation with the position of the body, may- determined by considering the path or orbit traversed by body and the circumstances of its motion. We shall illustrate this by a few propositions, selected on ount of their applicability in the establishment of the ory of universal gravitation. The proofs are substantially se given by Newton in the " Principia." Proposition I. — If the radius vector, drawn from a fixed point a body moving in a curve, describe equal areas in equal es, the force which causes the body to move in the curve lirected towards the fixed point. Let us suppose the whole time divided into equal periods, ing any one of which the body is not acted on by the force. vill, in the first period, move over a space represented by a light line, as AB (Fig. 24). In the second period, it would, inhindered, move over an equal space BD and in the same :. Let us suppose it, however, deflected by a force acting tantaneously at the point B. It will move in a hne BC such t, by hypothesis, triangle OBA — triangle OBC. Now, ,ngle ODB also = triangle OAB, therefore triangle OCB = ingle ODB, and CD is parallel to OB. Complete the paral- )gram CDBE ; then it is evident that the motion BC is npounded of the motions BD and BE ; and since forces are portional to the motions they asion, the force acting at^ is portional to BE, and is directed ng the line BO. If now the iods into which the whole time F1G24. divided become indefinitely small, in the limit the broken : ABC approaches indefinitely near to a curve, and the force ich causes the motion in the curve is always directed to the tre O. Proposition II. — If a body move uniformly in a circle, the ■62 ELEMENTARY PHYSICS. [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 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 Fig, 25. ac = ad' ae ' But ae = 2r\ ad represents the space traversed in the time t, and in the limit — represents v, the velocity in the circle. From the previous reasoning ac represents \ff ; whence 2 and mf = 2r' 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 MECHANICS OF MASSES. 63 ing on them will be inversely as the squares of their radii, 1 conversely. For, if T and T, represent the periodic times the two bodies moving in circles of radii r and r,, with ve- ities V and v^, then, by hypothesis, T:T. = 27cr 2nr, = r» : r/; ence w ence V -.v^ = r,^ : r*. v" V? f--f< - ^ • '; Corollary 11. — The relation of Corollary I. holds with refer- :e to bodies describing similar parts of any similar figures /ing the same centre. In the application of the proof, ivever, we must substitute for uniform velocity the uniform icription of areas ; and instead of radii we must use the tances of the bodies from the centre. The proof is as fol- ios: If D and D, represent the radii of curvature of the paths of : two bodies, R and R^ the distances of the bodies from the itre of force, then, by hypothesis, letting A represent the a described in one period of time, T : T, = ^ ■.'^- = Ri : R} ^ D^ '. D} m the similarity of figures. 64 ELEMENTARY PHYSICS. [37 Now hence and A:A, = vR\ v,Ri ; v:v, = Ri : R^ = D^ s Z», f-f, = -D--± = ^'--^- 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 chord Qv, cutting SP in x, and complete the parallelogram PRQx. From-Q draw QT perpendicular to SP. Also draw the diameter G^/'and its conjugate DK. The force which acts on the body, causing it to leave the tangent PR and. move in the Hne 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, Fig. 26/ whence 2Px f=z f 37] MECHANICS OP MASSES. 65 Again : the area described by the radius vector in the time 3 equal to in unit time, t is equal to '- ; and if A represent the area described 2 Equating these values of /, we obtain SP .QT' 2Px whence 8A' . Px I / = QT' ■ SP' From Proposition L, the value of A is constant for any Px part of the ellipse. We shall now show that -?ypi 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 .Pi: Qv" = PC : CD'. 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 = zPC. The last proportion then becomes PC .Pv: Qx" = PC : 2CD\ 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':Qr= CD':CB\ we obtain, finally, Px: QT'= AC:2CB'; Px that'is, since AC 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 -^. 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 generalization 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 law of gravitation. Some of the ancient philosophers had a vague behef 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. [3& 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 Huyghens 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 by the assumption of the existence of an attraction acting between the sun and the planets, and vary- ing inversely aa 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 \:^\e force of gravity were 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 attraction between the earth and a body near its surface, if we adopt a point on the earth's surface as the fixed point of reference, the acceleration of the body alone need be considered. Since the force acting upon it varies with its mass, and since its gain in momentum also varies with its mass, it follows that its acceleration will be constant, however its mass may vary. We may, therefore, obtain a direct measure of the 70 ELEMENTARY PHYSICS. \Z9 earth's attraction, or of the force of gravity i 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 centimeters. Its acceleration is there- fore 32.16 in feet and seconds or 980 in centimeters and seconds.. We denote this acceleration by the symbol g. 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 represents the distance traversed by the centre of gravity between the highest'and the lowest points of its arc, and ^Mg(t> 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, ^MRgcf^. Hence we have whence ^=-\/^- (^5> 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 -j^ 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 MASS ATTRACTION. 75 vity of the system were easily calculated, and the length of simple pendulum to which the system was equivalent was s obtained. (2) We may determine the length of the equivalent simple idulum directly by observation. The method depends upon principle that, if the axis of oscillation be taken as the s of suspension, the time of oscillation will not vary. The of of this principle is as follows : Let rand / — r represent the distances from the centre gravity to the axis of suspension and of oscillation re- ctively, m the mass of the pendulum, and / its moment inertia about its centre of gravity. Then, since the ment of inertia about the axis of suspension is / + mr', we 'e mr When the pendulum is reversed, we have _ /+ m{l - ry '~ m{l — r) From the first equation we have /= mr{l—r), which ue substituted in the second gives, after reduction, = /; that is, the length of the equivalent simple pen- ^\ um, at^d consequently the time of oscillation when pendulum swings about its axis of suspension, is the le as that when it is reversed^ and swings about its tner axis of oscillation. A pendulum (Fig. 28) so constructed as to take rantage of this principle was used by Kater in his ermination of the value of g; and this form is known. :onsequence, as Kater's pendulum. o 76 ELEMENTARY PHYSICS. [43 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 eqiiilibrium 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 -f-/, 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 /3. Then the moments of force about O are equal^ ; that is, {P+p)l. cos (« + >«) = PI. cos {a-l3)J^Wd. sin ft ; 42] MASS ATTRACTION. "Jl from which we obtain, by expanding and transposing, // cos a *^" ^ = (2i'+/)/sin«+fr^- (2^> 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 ft, and therefore /?, increases as the weight 2P of the load diminishes. As the angle a becomes less, the value of /? also increases, until, when A, O, and B are in the same straight line, it depends only on -^-, and is independ- Wd ent of the load. In this case tan ft 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 must 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 PHYSICS. [42 which moves on a screw, 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 MASS ATTRACTION. 79 n replaced by known weights till equilibrium is again estab- ed. It is manifest that the replacing weights represent weight of the body. If the error of the balance consist in the unequal length of arms of the beam, the true weight of a body may be ob- led by weighing it first in one scale-pan and then in the ler. The geometrical mean of the two values is the true ight ; for let l^ and /„ represent the lengths of the two arms of I balance, P the true weight, and P^ and P, the values of the ights placed in the pans at the extremities of the arms of gths /i and /„ which balance it. Then Pl^ = PJ^ and P/, = . : from which P= VP^,. 43. Density of the Earth.— One of the most interesting )blems connected with the physical aspect of gravitation is : determination of the density of the earth. It has been acked in several ways, each of which is worthy of consider- on. The first successful determination of the earth's density s based upon experiments made in I774by Maskelyne. He served the deflection from the vertical of a plumb-line sus- [ided near the mountain Schehallien in Scotland. He then lermined the density of the mountain by the specific gravity specimens of earth and rock from various parts of it, and culated the ratio of the volume of the mountain to that of ; earth. From these data the mean specific gravity of the ■th was determined to be about 4.7. The next results were obtained from the experiments of vendish, in 1 798, with the torsion balance already described, .e density, volume, and attraction of the leaden balls being own, the density of the earth could easily be obtained. The ue obtained by Cavendish was about 5.5. 8o ELEMENTARY PHYSICS. [43: Another method, employed by Carlini 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 g 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 R'D g = ^Tt—^k = \nRDk. itR 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 0.000000066 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 estabhshed, — 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

--^ ~- N, Al ( P s \ "\ ~ r m' ^ - y 1 vr 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 P 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 mn. 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 Fig. 32. — ABCD (Fig. 32), of which the part BCD is solid and fixed, and the part .4 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. [6r 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- angle (Fig. 33) on the surface. Let the length of its sides be represented by s and s^ respectively, and the radii of curvature of those sides by R and R,. Also let and 0, represent the angles in circular measure sub- tended by the sides from their respective centres of curvature. Now, a tension T for every unit of length acts normal 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 nornial to the normal at the point P, the centre of the rect- angle, we obtain for the parallel component 7>sin— , or, since ^, is a very small angle, Ts— or Ts—^. The opposite side gives a similar component ; the side s^ and the side oppo- site it give each a .component Ts,—^. The total force along the normal at P is then and since ss^ is the area of the infinitesimal rectangle, the force -62] MOLECULAR MECHANICS. 95 or pressure normal to the surface at P referred to unit of sur- face is ^U, + Rl' From a theorem given by Euler we know that the sum 1,1. . , . . , , ^ + "n IS constant at any pomt for any position of the rect- angular normal plane sections ; hence the expression we have obtained fully represents the pressure at P. If the surface be convex, the radii of curvature are positive, and the pressure is directed towards the liquid ; if concave,, they are negative, and the pressure is directed outwards. This pressure is to be added to the constant molecular pressure which we have already seen exists everywhere in the mass. If we denote this constant molecular pressure by K, the ex- pression for the total pressure within the mass is ^+^fe + i) R, ^ R) Tvhere the convention with regard to the signs of R, and R must be understood. For a plane surface, the radii of curva- ture are infinite, and the pressure under such a surface reduces toK. 62. Angles of Contact. — Many of the capillary phenom- ena appear when different liquids, or liquids and solids, are brought in contact with one another. It becomes, therefore, necessary to know the relations of the surface tensions and the angles of contact. They are determined by the following ■considerations : Consider first the case when three liquids meet along a line. 96 ELEMENTARY PHYSICS. [62 Fig. 34. Let O represent the. point where this line cuts a plane drawn at right angles to it. Then the ten- sion Tab of the surface of separation of the liquid a from the liquid b, ■ acting nbrmal to this line, is coun- terbalanced by the tensions T^c and The of the surfaces of separation of a and c, b and c. These tensions are always the same for the three liquids under similar conditions of temperature and purity. Knowing the value of the tensions, the angles which they make with * one another are determined at once by the parallelogram of forces ; and these angles are always constant. Similar relations arise if one of the liquids- be replaced by a gas. Indeed, some experiments by Bosscha indicate that capillary phenomena occur at surfaces of separation between gases. We need, therefore, in "the subsequent discussions, make no distinction between gases and liquids, and may use the general term fluids. If Tab be greater than the sum of Tac and Tbc, the angle be- tween Tac and Tbc becomes zero, and the fluid c spreads itself out in a thin sheet between a and b. Thus, if a drop of oil be placed on water, the tension of the surface of separation between the air and water is greater than the sum of the ten- sions of the surfaces between the air and oil, and between the oil and water ; hence the drop of oil spreads out over the water until it becomes almost indefinitely thin. In the case of two fluids in contact with a plane solid (Fig. 35), it is evident that when the system is in equilibrium, the surface of separation between the fluids a and b, making th& angle with the solid C, is Fig. 35. 63] MOLECULAR MECHANICS. 97 Tac = Tic + Tab COS d. The angle of contact is then determined by the equation cos d = J- ae ■'■be '■ab If Tac be greater than T^b + Tbc, the equation gives an im- possible value for cos d. In this case the angle becomes evanescent, the fluid d spreads itself out, and wets the whole surface of the solid. In other cases the value of 6 is finite and constant for the same substances. Thus, a droj^of water placed on a horizontal glass plate will spread itself over the whole plate ; while a small quantity of mercury placed on the ' same plate will gather together into a drop, the edges of which make a constant angle with the surface. 63. Plateau's Experiments. — The preceding principles will enable us to explain a few of the most important experi- mental facts of capillarity. A series of interesting results was obtained by Plateau from . the examination of the behavior of a mass of liquid removed from the action of gravity. His method of procedure was to place a mass of oil in a mixture of alcohol and water, carefully mixed so as to have the same specific gravity as the oil. The oil then had no tendency to move as a mass, and was free to arrange itself entirely under the action of the molecular forces. Referring to the equation of Laplace, already obtained, it is evident that equilibrium can exist only when the sum (-5- -|- -Q- ) is constant for every point on the surface. This is- manifestly a property of the sphere, and is true of no other finite surface. Plateau found, accordingly, that the freely floating mass at once assumed a spherical form. This result we had previously reached by another method. If a solid 7 98 ELEMENTARY PHYSICS. [63 body — for instance, a wire frame — be introduced into the mass of oil, of such a size as to reach the surface, the oil clings to it, and there is a break in the continuity of the surface at the points of contact. Each of the portions of the surface divided from the others by the . solid -then takes a form which fulfils the condition already laid, down, that (-5- + ^ j equals a con- stant. Plateau immersed a wire ring in the mass of oil. So long as the ring nowhere reached the . surface, the mass re- mained spherical. On withdrawing a portion of the oil with a syringe, that which was left took the form of two equal calottes, o» sections of spheres, forming a double convex lens. A mass of oil, filling a short, wide tube, projected from it at either end in a similar section of a sphere. As the oil was removed, the two end surfaces became less curved, then plane, and finally concave. Plateau also obtained portions of other figures which fulfil the required condition. For example, a mass of oil was made to surround two rings placed at a short distance from one another. Portions of the oil were then gradually withdrawn, when two spherical calottes formed, one at each ring, and the mass between the rings became a right cylinder. It is evident .that the cylinder fulfils the required condition for every point on its surface. t Plateau also studied the behavior of films. He devised a miJJture of soap and glycerine,, which formed very tough and durable films ; and he experimented with them in air. Such films are so light that the action of gravity on them may be neglected in comparison with that of the surface tension. If the parts of the frame upon which the film is stretched be all in one plane, the film will manifestly lie in that plane. When, however, the frame is constructed so that its parts mark the edges of any geometrical volume, the films which are taken up by it often meet. Any three films thus meeting so arrange <4] MOLECULAR MECHANICS. 99 themselves as to make angles of 120° with one another. This follows as a consequence of the proposition which has already been given to determine the equilibrium of surfaces of separa- tion meeting along a line. If four or more films meet, they always meet at a point. Plateau also measured the pressure of air in a soap-bubble, and found that it differed from the external pressure by an amount which varied inversely as the radius of the bubble. This follows at once from Laplace's equation. This measure- ment also gives us a means of determining the surface tension ; for, from Laplace's equation, the pressure inwards, due to the 2 outer surface, is TV, and the pressure in the same direction ,2 -due to the inner surface is also TV, for the film is so thin that we may neglect the difference in the radii of curvature of the two surfaces : hence the total pressure inwards is f ; and if this be measured by a manometer, we can obtain the value . of T. 64. Liquids influenced by Gravity. — Passing now to con- sider liquid masses acted on by gravity, we shall treat only a few of the most important cases. If a glass tube having a narrow bore be immersed perpendicularly in water, the water rises in the tube to a height inversely proportional to the diameter of the tube. This law is known as JuritCs law. ——— Let Fig. 36 represent the section of a tube of radius r ^'°' s^- immersed in a liquid, the surface of which makes an angle ^ ^ 100 ELEMENTARY PHYSICS. [64. with the wall. Then if T be the surface tension of the liquid, the tension acting upward is the component of this surface tension parallel to the wall, exdrted all around the circumfer- ence of the tube. This is expressed by 2nrTco% 6. This force, -for each unit area of the tube, is 27trTcos d Tir' The downward force, at the level of the free surface, making equilibrium with this, is due to the weight of the liquid column (§86). If we neglect the weight of the meniscus, this force per unit area, or the pressure, is expressed by hdg, where h is. the height of the column and d the density of the liquid. We- have, accordingly, since the column is in equilibrium, ' -^,T cos 8 = Mg; whence 2 T cos 9 h rdg ' and the height is inversely as the radius of the tube. If the liquid rise between two parallel plates of length /, separated by a distance r, the upward force per unit area is 2/ given by the expression ^T'cos ^, and the downward pressure- by hdg; whence , _ 2 Tcos ~ rdg ' «s] MOLECULAR MECHANICS. lOI and the height to which the liquid will rise between two such plates is equal to that to which it will rise in a tube the radius ■of which is equal to the distance between the plates. If the two plates are inclined to one another so as to touch along one vertical edge, the elevated surface takes the form of a rectangular hyperbola ; for let the line of contact of the plates be taken as the axis of ordinates, and a line drawn in the plane of the free surface of the liquid as the axis of abscissas, the elevation corresponding to each abscissa is in- versely as the distance between the plates at that point, and the elevations are therefore inversely as the abscissas : hence the product of any abscissa by its corresponding ordinate is a ■constant. The extremities of the ordinates then mark out a rectangular hyperbola referred to its asymptotes. 65. Liquid Drops in Capillary Tubes. — When a drop of liquid is placed in a conical tube, it moves, if the surfaces are •concave, towards the smaller, end ; if convex, towards the larger end. The explanation of these movements follows readily from the foregoing results. In case the surfaces are concave, letting d .(Fig. 37) be the angle of contact and a the angle of inclination of the wall of the tube to the axis, r and r, the radii of the tube at the extremities of the drop, r being the smaller of the two, then the expressions for the com- ponents of the tensions parallel with the axis acting in both cases outwards, are respectively Fig. 37. 2'JtrT ■Tff COS {Q — a), and -V COS (61-1- a). Tcr: 102 ELEMENTARY PHYSICS. [66 Of these two exprdssions the former is manifestly greater than the latter: hence the tendency of the drop is to move towards the smaller end of the tube. If we assume that the concave surfaces are portions of spheres, of which R and R^ are the respective radii of curva- ture, it follows that r = R cos {0 — a), and r^ = R^ cos {6-\- a); iT 2T hence the expressions for the tensions become -g- and -5-' These are the values of the tensions as determined by Laplace's; equation, and the movements of the drop might have been in- ferred directly from this equation by making the same assump- tion. If a drop of water be ifitroduced into a cyHndrical capillary tube of glass, and if the air on the two ends of the drop have unequal pressures, the concavities thereby become unequal, the one on the side of the greater pressure presenting the greater concavity. The drop so circumstanced offers a resist- ance to this pressure ; and it may, if the pressure be not too great, entirely counterbalance it. It is also evident, that, if several such drops be introduced successively, with intervening air-spaces, the pressure which they can unitedly sustain is equal to that which one can sustain multiplied by their number. Jamin found that, with a tube containing a large number of drops, a pressure of three atmospheres was maintained without diminution for fifteen days. 66. Movements of Solids. — In certain cases the action of the capillary forces produces movements in solid bodies partially immersed in a liquid. For example, if two plates, which are both either wetted or not wetted by the liquid, bfe partially immersed vertically, and brought so near together that the rise or depression of the liquid due to the capillary action begins, then the plates will move towards one another. In either case tills movement is explained by the inequality of pressure on the two sides of each plate. When the liquid rises between 68] MOLECULAR MECHANICS. ' IO5, the plates, the pressure is zero at that point in the column which lie's in the same plane as the free external surface. At every internal point above this the molecules gf the liquid are in a state of negative pressure or tension, and the plates are consequently drawn together. When the liquid is depressed between the plates, they are pressed together by the external liquid above the plane in which the top of the column between the plates lies. When one of the plates is wetted by the liquid arid the other not, the plates move apart. This is explained by noting, that, if the plates be brought near together, the convex surface at the one will meet the concave surface at the other, and there will be a consequent diminution in both the elevation and the depression at the inner surfaces of the plates. The elevation and depression at the outer surfaces remaining unchanged, there will result a pull outwards on the wetted plate and a pressure outwards on the plate which is not wetted ; and they will consequently move apart. Laplace showed,* howeyer, as the result of an extended discussion, that, though seeming repulsion exists between two plates such as we have just con- sidered, yet, if the distance between the plates be diminished beyond a certain value, this repulsion changes to an attraction. This prediction has been completely verified by the most care- ful experiments. 67. Porous Bodies. — Porous bodies may be considered as assemblages of more or less irregular capillary tubes- Thus the explanation of many natural phenomena — as the wetting of a sponge, the rise of the oil in the wick of a lamp — follows directly from the preceding discussion. DIFFUSION. 68. Solution and Absorption.— Many solid bodies, im- jnersed in a liquid, after a while disappear as solids, and are taken up by the liquid. This process is called solution. The I04 , ELEMENTARY PHYSICS. [69 quantity of any body which a unit quantity of a given liquid will dissolve at a given temperature, is called its solubility in that liquid at that temperature. The solubility of a given solid varies greatly for different liquids, in many cases being so small as to be inappreciable. Gases are also taken into solution by liquids. The process is usually called absorption. The quantity of gas dissolved in' any liquid depends upon the temperature, and varies directly with the pressure. The solubility of any gas at a given temperature and at standard pressure is called its coefficient of absorption at that temperature. Gases, in general, adhere strongly to the surfaces of solids with which they are in contact. This adhesion is so great, that the gases are sometimes condensed so as to form a dense layer which probably penetrates to some depth below the surface of the solid. The process is called the absorption of gases by solids. When the solid is porous, its exposed surface is greatly extended, and hence much larger quantities of gas are condensed on it than would otherwise be the case. When this condensa- tion occurs there is in general a rise of temperature which may be so great as to raise the solid to incandescence. Thus, for example, spongy platinum, placed in a mixture of oxygen and hydrogen, becomes so heated as to inflame it. 69. Free Diffusion of Liquids. — When two liquids which are miscible are so brought together in a common vessel that the heavier is at the bottom and the lighter rests upon it in a well-defined layer, it is found that after a time, even though no agitation occur, they become uniformly mixed. Molecules of the heavier liquid make their way upwards through the lighter ; while those of the lighter make their way downwards through the heavier, in apparent opposition to gravitation. Diffusion is the name which is employed to designate this phenomenon and others of a similar nature. When one of the liquids is colored, — as, for example, "70] MOLECULAR MECHANICS. I05 solution of cupric sulphate,— while the other is colorless, the progress of the experiment may easily be watched and noted. When both liquids are colorless, small glass spheres, adjusted and sealed so as to have different but determinate specific gravities between those of the liqui'ds employed, may be placed in the vessel used in the experiment, and will show by their positions the degree of diffusion which has occurred at any given time. 70. Coefficient of Diffusion. — Experiment shows that the -amount of a salt, in solution which at a given temperature passes, in unit time,, through unit area of a horizontal surface, ■depends upon the nature of the salt and the rate of change of concentration at that surface, — that is, the quantity of a salt that passes a given horizontal plane in unit time is kCA, where A is the area, C the rate of change of concentration, and k a coeflScient that depends upon the nature of the substance. By rate of change pf concentration is meant the difference in the quantities of salt in solution measured in grams per cubic centimetre, at two horizontal planes one centimetre apart, supposing the concentration to diminish uniformly from one to the other. It is plain, that, if C and A in the above expres- sion be each equal to unity, the quantity of salt passing in unit time is k. The quantity k, called the coefficient of diffu- sion, is, therefore, the quantity of salt that passes in unit time through unit area of a horizontal plane when the difference of concentration is unity. Coefficients of diffusion increase with the temperature, and are found not to be entirely independent ■of the degree of concentration. As implied above, the units of mass and length employed in these measurements are respectively the gram and the centi- metre ; but, since in most cases the quantity of salt that diffuses in one second is extremely small, it is usual to emp^loy the day as the unit time. I06 ELEMENTARY PHYSICS. [71, 71. Diffusion through Porous Bodies. — It was found by Graham that diffusion takes place through porous solids, such, as unglazed earthenware or plaster, almost as though the liquids were in direct contact, and that a very considerable, difference of pressure can Be established between the two faces, of the porous body while the rate of diffusion remains nearly- constant. 72. Diffusion through Membranes. — If the membrane through which diffusion occurs be of a type represented by ani- mal or vegetable tissue, the resulting phenomena, though in some respects similar, are subject to quite different laws. Col- loid substances pass through the membrane very slowly, while crystalloid substances pass more freely. It is to be noted that the membrane is not a mere passive medium, as is the case with the porous substances already considered, but takes, an active part in the process ; and consequently one of the liquids frequently passes into the other more rapidly than would be the case if the surfaces of the liquids were directly in con- tact. An explanation of these ' facts follows if we suppose that diffusion of a liquid through a continuous membrane can occur only whenthe liquid is capable of temporarily uniting with the membrane, and forming a part of it. Diffusion would then oc- cur by the union of the liquid with the membrane on one face,, and the setting free of an equal portion on the other. If the membrane separate two crystalloids, it often happens- that both substances pass through, but at different rates. In accordance with the usage of Dutrochet, we may say there is endosmose of the liquid, which passes more rapidly to the other liquid, and exosmose of the latter to the former. The whole process is frequently called osmosis. If the membrane be stretched over th6 end of a tube, into which the more rapid current sets, and the tube be placed in a vertical position, the liquid will rise in' the tube until a very considerable pressure is. 74] MOLECULAR MECHANICS. I07 attained. Dutrochet called such an instrument an endosmo- meter. Graham made use of a similar instrument, which he called an osmometer, by means of which he studied, not only the ac- tion of porous substances, such as are mentioned above, but al^o that of various organic tissues ; and he was able to reach quantitative results of great value. Pfeffer has more recently made an extended study of the phenomena of osmosis, espec- ially in those aspects relating to physiological phenomena. He has shown that colloid membranes produced by purely chemi- cal means are even more efficient than the organic membranes employed by Graham. 73. Dialysis. — Upon the principles just set forth Graham has founded a method of separating crystalloids from any col- loid matters in which they may be contained, which is often of great importance in chemical investigations. .The apparatus employed by Graham consists of a hoop, over one side of which parchment paper is stretched so as to constitute a shallow basin. In this basin is placed the mixture under investigation, and the basin is then floated upon pure water contained in an outer vessel. If crystalloids be present, they will in due time pass through the membrane into the water, leaving the colloids behind. The process is often employed in toxicology for sep- arating poisons from ingesta or other matters suspected of containing them. I* is called dialysis, and the substances that pass through are said to dialyse. 74. Laws of Diffusion of Gases. — Gases obey the same elementary laws of diffusion as liquids. The rate of diffusion varies inversely as the pressure, directly as the square of the absolute temperature, and inversely as the square root of the density of the gas. A gas diffuses through porous solids ac- cording to the same laws. An apparatus by which this may be conveniently illustrated consists of a porous cell, the open end of which is closed by a stopper, through which passes a I08 ELEMENTARY PHYSICS. [73 long tube. This is placed in a vertical position, with the open end of the tube in a vessel of water. If, now, a bell-jar con- taining hydrogen be placed over the porous cell, hydrogen passes into the cell more rapidly than the air escapes from it : the pressure inside is increased, as is shown by the escape of bubbles from, the end of the tube. If, now, the jar be removed^, diffusion outward occurs more rapidly than diffusion inward; the pressure within soon becomes less than the atmospheric* pressure, as is shown by the rise of the water in the tube. ', ELASTICITY. 75. Stress and Strain. — When a body is made the medium for the transmission of force, the application of Newton's third law shows that there is a stress inithe medium. This stress is always accompanied by a corresponding change of form of the body, called a strain. In some bodies equal stresses applied in any direction pro- duce equal and similar strains. Such bodies are isotropic. In others the strain alters with the direction of the stress. These bodies are eolotropic. According to the molecular theory of matter, the. form of a body is permanent so long as the resultant of the stresses act- ing on it from without, with the interior forces existing be- tween the individual molecules of the Ibody, reduces to zero. The molecular forces and motions are such that there is a cer- tain form of the body, for every external stress in which its molecules are in equilibrium. Any change of the stress in the body is accompanied by a readjustment of the molecules, which is continued until equilibrium is again established. If the stress tend only to increase or diminish the distance between the molecules, it is called a tension or a pressure re- spectively ; if it tend to slide one line or sheet of molecules past another tangentially, it is called a shear or a shearing-stress. 76] MOLECULAR MECHANICS. IO9 All stresses can be resolved into these two forms. The cor- responding changes of shape are called dilatations, compressions, and shearing-strains. The term pressure is used with several different meanings. In order to most clearly present these, /we will consider a right cylinder, transmitting a stress in the direction of its axis. The stress itself is often called the total pressure upon the cylinder. If we concfeive the cylinder to consist of a great number of elementary cylinders of small cross-section, and if the total pressure upon any one of them, as here defined, be to the total pressure on the' whole cylinder as the cross-section of the ele- mentary cylinder is to the cross-section of the whole cylinder, then it is said that the pressure on the cross-section is uniform, and th.Q pressure on an area in that cross-section is defined as the product of the total pressure on the cylinder into the ratio of that area to the cross-section of the cylinder. Further, \.\^& pressure at a point, in a direction normal to the cross-sec- tion, is defined as the ratio of the pressure on an area, taken in the cross-section with its centre of inertia at the point, to that area, when the area is diminished indefinitely. This definition may at once be generalized. The pressure in any given direc- tion at a point in a medium transmitting stress in any manner whatever, is the ratio of the pressure on any area, taken' nor- mal to the given direction and with its centre of inertia at the point, to that area, when the area is diminished indefinitely. In case a stress exists between two bod.ies, which acts nor- mally across a common surface of contact, the term pressure is also used to denote this stress, and the pressures on an area and at a point in the surface of contact, are defined exactly as above. 76. Modulus of Elasticity.— If, for a given amount of stress between certain limits, a body be deformed by a definite amount, which is constant so long as the stress remains con- stant, and if, when the stress is removed, the body regain its no ELEMENTARY PHYSICS. [77 original condition, it is said to be perfectly elastic. Any body only partially fulfilling these conditions is said to be imperfectly elastic. The definition of elasticity in its physical sense, as a prop- erty of bodies, has been already given. It is measured by the rate of change, in a unit of the body, of the stress with respect to the strain. Thus for example, the voluminal elasticity of a fluid is measured by the limit of the ratio of any small change of pressure to the corresponding change of unit volume. The tractional elasticity of a wire under tension is measured by the liniit of the ratio of any small change in the stretching-weight to the corresponding change in unit length. This ratio is called the modulus of elasticity, ox simply the elasticity of a body, and its reciprocal 'Cc^^ coefficient of elasticity. 77. Modulus of Voluminal Elasticity of Gases. — Within certain limits of temperature and pressure the volume of any gas, at constant temperature, is inversely as the pressure upon it. This law was discovered by Boyle in 1662, and was after- wards fully proved by Mariotte. It is known, from its discov- erer, as Boyle's law. Thus, if / and p, represent different pressures, v and v^ the corresponding volumes of any gas at constant temperature, then I I p :p, = -•• -', whence pv = p,v,. (27) Now, p/v, is a constant which may be determined by choosing any pressure p^ and the corresponding volume v^ as standards : hence we may say, that, at any given temperature, the product pv is a constant. The limitations to this law will be noticed later. 77] MOLECULAR MECHANICS. Ill If we draw the curve marked out by a point having its ■ordinate and abscissa so related that xy equals a constant, we ■obtain a rectangular hyperbola referred to its asymptotes. Let X represent the'volume and y the pressure of a quantity of gas- Then this curve shows the relation of pressure and volume in all their combinations. Draw the lines as in Fig. 38, letting AC, JD, represent volumes differing only by a small amount. We must first show that AE is numeri- •cally equal to the modulus of elasticity. The CG ratio -g^ is the voluminal compression per -unit volume for the increment of pressure GD GD: hence, by definition, tstt is the modulus AC of elasticity. But, from similarity of . tri- angles, AE : GD = AC : CG. Hence we have Fig. 3S. GD AE = jyT- = the modulus of elasticity. Al: Now, since, by construction, the rectangles AB and /K are equal, and the rectangle AK is common to them, the rectangles J^G and CiTare equal, and CG:DG=GA: GK. By similar triangles, "whence CG:DG= CA -.AE; GA:GK=CA: AE. 112 ELEMENTARY PHYSICS. [7$: Now, if the increment of pressure be made indefinitely- small, so that in the limit D arid C coincide, the Hne CE be- comes a tangent to the curve, and GA, GK, are respectively equal to CA, CB. CB therefore equals AE frdm the last pro- portion: hence, in the case of a gas obeying Boyle's law, the modulus of elasticity is numerically equal to the pressure. The discussion of the experimental facts in connection with the elasticity of gases, and the explanation of the apparatus- founded "upon it, will be resumed in a future chapter. 78. Modulus of Voluminal Elasticity of Liquids. — When liquids.are subjected to voluminal compression, it is found that their modulus of elasticity is much greater than that of gases, For at least a limited range of pressures the modulus of elasticity of any one liquid is constant, the change in volume being proportional to the change in the pressure. The modulus- differs for different liquids. The instrument used to determine the modulus of elasticity of liquids is called a piezometer. The first form in which the , instrument was devised by Oersted, while not the best for ac- curate determinations, may yet serve as a type. The liquid to be compressed is contained in a thin glass, flask, the neck of which is a tube with a capillary bore. The flask is immersed in water contained in a strong glass vessel fitted with a water-tight metal cap, through which moves a piston. By the. piston, pressure may be applied to the water, and through it to the flask and to the liquid contained in it. The end of the neck of the small flask is inserted down- wards under the surface of a quantity of mercury which lies at •the bottom of the stout vessel. The pressure is registered by means of a compressed-air, manometer (§ 96) also inserted in the vessel. When the apparatus is arranged, and the piston depressed, a rise of the mercury in the neck of the flask occurs,, which indicates that the water has been compressed. , An error may arise in the use of this form of apparatus from. 8o] MOLECULAR MECHANICS. 1 1 J 'the change in the capacity of the flask, due to the pressure. Oersted assunjed, since the pressure on the interior and exterior walls was the same, that no change would occur. Poisson, how- ever, showed that such a change would occur, and gave a formula by which it might be calculated. By introducing the proper corrections, Oersted's piezometer may be used with success. A different form of the instrument, employed by Regnault, is, however, to be preferred. In it, by an arrangement of stop- cocks, it is possible to apply the pressure upon either the interior or exterior wall of the flask separately, or upon both - together, and in this way to experimentally determine the cor- rection to be applied for the change in the capacity of the flask. It is to be noted that the modulus of elasticity for liquids is so great; that, within the ordinary range of pressures, they may be regarded as incompressible. Thus, for example,, the alteration of volume for sea-water by the addition of the pres- sure of one atmosphere is 0.000044. The change in volume, then, at a depth in the ocean of one kilometre, where the pres- sure is about 99.3 atmospheres, is 0.00437, oi" about -g^ of the whole volume. 79, Modulus of Voluminal Elasticity of Solids. — The modulus of voluminal elasticity of solids is believed to be gen- erally greater than that of liquids, though no reliable experi- mental results have yet been obtained. The modulus, as with liquids, differs for different bodies. 80. Shears.^A strain in which parallel planes or sheets of molecules are moved tangentially over one another, each plane being r P' \ displaced by an amount proportional \ / \ to its distance from one of the planes \ / ^ assumed as fixed, is called a shear. \ / To illustrate this definition, let uS ^ consider a parallelopiped, of which F1G.39. the cross-section made at right angles to its sides is a rhombus, 8 114 ELEMENTARY PHYSICS. [8l and let ABDC in the diagram (Fig. 39) represent that cross- section. If the rhombus ABDC be deformed so as to become ABD^C^, that deformation is a simple shear. It is plain that a simple shear is equivalent to an extension in lines parallel to AD, and a contraction in those at right angles to AD. The directions AD and CB are called the principal axes of the shear. The amount of the shear is the displacement of the planes per unit of distance from the fixed plane ; that is, -r^ EB is the amount of the shear. The stresses that give rise to a simple shear can plainly be conceived of as consisting of two equal couples, the forces comprising which act tangentially upon parallel planes which are moved over one another, and make equal angles with the axes of the shear. The foirces making, up these couples may be compounded two and two, a and b, a^ and b^ (Fig. 40), making up, a tension normal to the dimin- ished axis ; a^ and b, a and ^,, making up a pressure normal to the increased axis. These stresses are measured per unit of area of the undeformed sides or sections of the solid. The resistance offered by a body to a shearing-stress is called its rigidity, and the ratio of a very small change in the stress to the corresponding increment in the amount of the shear is called the modulus of rigidity. 81. Elasticity of Tension. — The first experimental deter- minations of the relations between the elongation of a solid and the tension acting on it were made by Hooke in 1678. Experimenting with wires of different materials, he found that for small tensions the elongation is proportional to the stress. It was afterwards found that this law is true for small com- pressions. 82] MOLECULAR MECHANICS. IlJ The ratio of, the stress to the elongation of unit length of a wire of unit section is the modulus of tr actional elasticity. For different wires it is found that the elongation is proportional to the length of the wires, and inversely to their section. The formula embodying these facts is where V is the elongation, / the length, s the section of the wire, 5 the stress, and fi the modulus of tractional elasticity. A method of expressing the modulus of elasticity, due to Thomas Young, is sometimes valuable. " We may express the elasticity of any substance by the weight of a certain column of the same substance, which may be denominated the modu- lus of its elasticity, and of which the weight is such that any addition to it would increase it in. the same proportion as the weight added would shorten, by its pressure, a portion of the substance of equal diameter." For example, considering a cubic litre of air at o° C. and 760 millimetres of mercury pres- sure, and calling its weight unity, *e find, from the fact that the weight of one litre of mercury is 105 17 times that of a litre of air, that the pressure of the atmosphere upon a square decimetre is 79929 units. If we conceive the air as of equal density throughout, this pressure is equivalent to the weight of a column of air one square decimetre in section and 7992.9 metres high. The weight of this column is the modulus of elasticity for air ; for we know, by Boyle's law, that if the column be altered in length, and its weight' therefore cor- respondingly altered, the volume of the cubic litre of air under consideration will also alter inversely. The height of such a column of air as we have assumed is called the height of the homogeneous atmosphere. 82. Elasticity of Torsion. — When a cylindrical wire, clamped at one end, is subjected at the other to the action of a Il6 ELEMENTARY PHYSICS. [8* couple the axis of which is the axis of the cylinder, it is found that the amount of torsion, nleasured by the angle of displace- ment of the arm of the couple, is proportional to the moment of the couple, to the length of the wire, and inversely to the fourth power of its radius. It also depends on the modulus of rigidity. The formulated statement of these facts is = — ' (29). nr whei-e is the angle of torsion, / the length, r the radius of the- wire, C the moment of couple, and n the modulus of rigidity. No general formula can be found for wires with sections of variable form. , The laws of torsion in wires were first investigated by Cou- lomb, who applied them in the construction of an apparatus of great value for the measurement of small forces. The apparatus consists essentially of a small cylindrical wire, suspended firmly from the centre of a disk, upon which is cut. a graduated circle. ' By the rotation of this disk any required, amount of torsion . may be given to the wire. On the other extremity of the wire is fixed, horizontally, a bar, to the ends, of which the forces constituting the couple are applied. Ar- rangements are also made by which the angular deviation of this bar from the point of equilibrium may be determined. When forces are applied to the bar, it may be brought back to its former point of equilibrium by rotation of the upper disk. Let represent the moment of torsion; that is, the couple which, acting on an arm of unit length, will give the wire an amount of torsion equal to a radian, C the moment of couple acting on the bar, r the amount of torsion measured in ra- dians ; then C=®r. «2] MOLECULAR MECffAN/CS. 117 We may find the value of in absolute measure by a method of oscillations analogous to that used to determine g with the ■ pendulum. A body of which the moment of inertia can be determined by calculation is substituted for the bar, and thd time T of one of its oscillations about the position of equilibrium observed. Since the amount of torsion is proportional to the moment of couple, the oscillating body has a simple harmonic motion. If a represent the amplitude of oscillation of any particle at distance r from the axis of rotation, we have a = rr. • The velocity of the particle at the point of equilibrium is then 27ta and the angular velocity of the body, therefore, equals 2nr The kinetic energy of a body rotating about a centre is ^loo' ; and the kinetic energy of the body considered, at the point of equilibrium, is, therefore, ' i/4!Ll The potential energy due to the torsion of the wire is ^©r', since ^©r is the average moment of couple, and r the distance through which this couple acts. These expressions are neces- sarily equal : hence or Il8 ELEMENTARY PHYSICS. [8$ We may use a single instead of a double oscillation, when we may write the formula This apparatus was used by Coulomb' in his investigation of the law of electrical and magnetic actions. It was also employed by Cavendish, as has been already noticed, to determine the constant of gravitation. 83. Elasticity of Flexure.— If a rectangular bar be clamped by one end, and acted on at the other by a force normal to one of its sides, it will be bent or flexed. The amount of flexure — that is, the amount of displacement of the extremity of the bar from its original position — is found to be proportional to the force, to the cube of the length of the bar, and inversely to its breadth, to the cube of its thickness, and to the modulus of tractional elasticity. The formula there- fore becomes 84. Limits of Elasticity. — The theoretical deductions and empirical formulas which we have hitherto been considering are strictly applicable only to perfectly elastic bodies. It is found that the voluminal elasticity of fluids is perfect, and that within certain limits of deformation, varying for different bodies, we may consider the elasticity of solids to be practically perfect for every kind of strain. If the strain be carried beyond the limit of perfect elasticity, the body is permanently deformed.. This permanent deformation is called set. Upon these facts we may base a distinction between solids. and fluids : a solid requires the stress acting on it t'o exceed a certain limit before any permanent set occurs, and it makes no- 84] MOLECULAR MECHANICS. II9 difference how long the stress acts provided it lie within the limits. A fluid, on the contrary, may be deformed by the slightest shearing stress, provided time enough be allowed for the movement to take place. The fundamental difference lies in the fact that fluids offer no resistance to shearing stress other than that due to internal friction or viscosity. A solid, if it be deformed by a slight stress, is soft ; if only by a great stress, is hard or rigid. A fluid, if deformed quickly by any stress, is mobile ; if slowly, is viscous. It must not be understood, however, that the behavior of elastic solids under stress is entirely independent of time. If, for example, a steel wire be stretched by a weight which is nearly, but not quite, sufficient to produce an immediate set, it is found that, after some time has elapsed, the wire acquires a permanent set. If, on the other hand, a weight be put upon the wire somewhat less than is required to break it, by al- lowing intervals of tinie to elapse between the successive ad- ditions pf small weiglits, the total weight supported by the wire may be raised considerably above the breaking-weight. If the weight stretching the wire be removed, the return to its original form is not immediate, but gradual. If the wire car- rying the weight be twisted, and the weight set oscillating by the torsion of the wire, it is found that the' oscillations die away faster than can be explained by any imperfections in the elas- ticity of the wire. These and similar phenomena are manifestly dependent upon peculiarities of molecular arrangement and motion. The last two are exhibitions of the so-called viscosity of solids. The molecules of solids, just as those of liquids, move among themselves, but with a certain amount of frictional resistance. This resistance causes the external work done by the body to be diminished, and the internal work done among the mole- cules becomes transformed into heat. CHAPTER IV. MECHANICS OF FLUIDS. 85. Pascal's Law. — A perfect fluid xf\z.Y be defined as a body which offers no resistance to shearing-stress. No actual fluids are perfect. Even those which approximate that condi- tion most nearly, offer resistance to shearing-stress, due to their viscosity. With most, however, a very short time only is needed for this resistance to vanish ; and all mobile fluids at rest can be dealt with as if they were perfect, in determining the conditions of equilibrium. If they are in motion, their viscosity becomes a more important factor. As a consequence of this definition of a perfect fluid, follows a most important deduction. In a fluid in equilibrium, not acted on by any outside forces except the pressure of the con- taining vessel, the pressure at every point and in every direc- tion is the same. This law was first stated by Pascal, and is known as Pascal's law. The truth of Pascal's law appears, if, in a fluid fulfilling the conditions indicated, we imagine a cube of the fluid to become solidified. Then, if the law as just stated were not true, there would be an unbalanced force in some direction, and the cube would move, which is contrary to the statement that the fluid is in equilibrium. If a vessel filled with a fluid be fitted with a number of pistons of equal area A, and a force Ap be applied to one of them, acting inwards, a pressure Ap will act outwards upon the face of each of' the pistons. These pressures may be balanced by a force applied to each piston. Yin -\- \\>^ the number of the pistons, the outward pressure on n of them, caused by the force applied to one, is npA. 86] MECHANICS OF FLUIDS. 121 The fluid will be in equilibrium when a pressure^ is acting on unit area of each piston. It is plain that the same reason- ing will hold if the area of" one of the pistons be A and of an- other he nA. A pressure Ap on the one will balance a pres- sure of nAp on the other. This principle governs the action of the hydrostatic press . 86. Relations of Fluid Pressures due to Outside Forces. — If forces, such as gravitation, act on the mass of a fluid from without, Pascal's law no longer holds true. For suppose the, cube of solidified fluid to be acted on by gravity ; then the pressure on the upper face must be less than that on the lower face by the weight of the cube, in order that the fluid may still be in equilibrium. As the cube may be made as small as we please, it appears that, in the limit, the pressure ■on the two faces only differs by an infinitesimal ; that is, the pressure in a fluid acted on by outside' forces is the same at one point for all directions, but varies continuously for different points. The surface of a fluid of uniform density acted on by grav- ity, if at rest, is everywhere perpendicular to the lines of force ; for, if this were not so, the force at a point on the surface could be resolved into two components, one normal and the other tangent to the surface. But, from the nature of a fluid, the tangential force would set up a motion of the fluid, which is contrary to the statement that the fluid is at rest. If a su'r- face be drawn through the points in the field at which the pressure is the same, that surface will be perpendicular to the liries of force. For, consider a filament of solidified fluid lying in the surface ; its two ends suffer equal and opposite pressures ; hence, since by hypothesis the fluid is in equilibrium, the force acting upon it, due to gravity, can have no component in the -direction of its length, and is perpendicular to the surface in which it lies. Surfaces of equal pressures are equipotential surfaces. In 122 ELEMENTARY PHYSICS. [87 small masses of fluid, in which the lines of force due to gravity- are parallel, these surfaces are horizontal planes. In larger masses, such as the oceans, they are curved to correspond to the divergence of the lines of force from the centre of the earth. In a liquid the pressure at a point is proportional to its depth below the surface of the liquid. For, imagine two rec- tangular prisms of solidified liquid with bases which are equal and coincident with the surface of the liquid, and with heights such that the one is n times the other. From .the fact that liquids are practically incompressible, the weight of. these prisms acting downwards is proportional to their volumes^ and hence to their heights. Since the liquid is in equilibrium, these weights are balanced by the upward pressures on their lower bases. These pressures are therefore proportional to the heights of the prisms, or to the depths of the surfaces to which they are applied. From the foregoing principles, it is evident that a liquid contained in two communicating vessels of any shape whatever will stand at the same level in both. If one, however, be filled with a liquid of different density from that in the other, equi- librium will be established when the depths are inversely as the densities of the liquids. 87. The Barometer. — The instrument best adapted to il- lustrate these principles, and also of great importance in many physical investigations, is the barometer. It was invented by Torricelli, a pupil of Galileo. The fact that Ajs^ater can be raised in a tube in which a complete or partial vacuum has been made was known to the ancients, and was explained by them, and by the schoolmen after them, by the maxim that " TMature-abhors a vacuum." They must have been familiar with the action of pujnps, for the force-pump, a far more complicated instrument, was invented by Ctesibius of Alexandria, who lived during the second century B.C. It was not until the time of Gahleo, however, that the first recorded observations were made that 87] MECHANICS OF FLUIDS. 1 23: the column of water in a pump rises only to a height of about 10.5 metres. Galileo failed to give the true explanation of this fact. He had, however, taught that the air has weight ; and his pupil Torricelli, using that principle, was more successful. He showed, that if a glass tube sealed at one end, over 76a millimetres long, were filled with mercury, the open end stopped with the finger, the tube inverted, and the unsealed end plunged beneath a surface of mercury in a basin, on withdrawing the finger the mercury in the tube sank until its top surface was about 760 millimetres above the surface of the mercufy in the basin. The specific gravity of the mercury being 13.59, ^^ weight of the mercury column and that of the water column in the pump agreed so nearly as to show that the maintenance of the columns in both cases was due to a common cause, — the pressure of the atmosphere. This conclusion was subsequently verified and established by Pascal, who requested a friend to observe the height of the mercury column at the bottom and at the top of a mountain. On making the observation, the height of the column at the top was found to be less than at the bottom. Pascal himself afterwards observed a slight though distinct diminution in the height of the column on ascending the tower of St. Jacques de la Boucherie in Paris. The form of barometer first made by Torricelli is still often used, especially when the instrument is stationary, and is in- tended to.be one of precision. In the finest instruments of this class a tube is used which is three or four centimetres in diameter, so as to avoid the correction for capillarity. A screw of known length, pointed at both ends, is arranged so as to move vertically above the surface of the mercury in the cistern. When an observation is to be made, the screw is moved until its lower point just touches the surface. The distance between its upper point and the top of the column is measured by means of a cathetometer; and this distance added to the length of- the screw gives the height of the column. 124 ELEMENTARY PHYSICS. [^8 Other forms of the instrument are used, most of which are arranged with reference to convenient transportability. Vari- ous contrivances are added by means of which the column is made to move an index, and thus record the pressure on a graduated scale. AH these forms are only modifications of Torricelli's original instrument. The pressure indicated by the barometer is usually stated in terms of the height of the column. Mercury being practi- cally incompressible, this height is manifestly proportiona 1 ,to the pressure at any point in the surface of the mercury in the cistern. The pressure on any given area in that surface can be calculated if we know the va|lue of g at the place and the spe- cific gravity of mercury, as well as the height of the column. The standard barometric pressure, represented by 760 millimer tres of mercury, is a pressure of 1.033 kilograms on every square centimetre. It is called a pressure of one atmosphere ; and pressures are often measured by atmospheres. In the preparation of an accurate barometer, it is necessary' that all air be removed from the mercury : otherwise it will collect in the upper part of the tube, by its pressure lower the top of the column, and make the barometer read too low. The air is removed by partially filling the tube with mercury, which is then boiled in the tube, gradually adding small quan- tities of mercury, and boiling after each addition, until the tube is filled. The boiling must not be carried too far; for there is danger, in this process, of expelling the air so com- pletely that the mercury will adhere to the sides of the tube, and will not move freely. For rough work the tube may be filled with cold mercury, and the air removed by gently tap- ping the tube, so inclining it that the small bubbles of air which form can coalesce, and finally be set free at the surface of the mercury. 88. Archimedes' Principle. — If a solid be immersed in at fluid, it loses in weight an amount equal to the weight of the 90] MECHANICS OF FLUIDS. I25 fluid displaced. This law is known, from its* discoverer, as Archimedes' principle. The truth of this law will appear if we consider the space occupied by the solid as filled with the fluid. The fluid in this space will then be in equilibrium, and the upward pressure on it must exceed the downward pressure by an amount equal to its .weight. The resultant of the pressure acts through the centre of gravity of the assumed portion of fluid, otherwise equilibrium would not exist. If, now, the solid occupy the space, the difference between the upward and the downward pressures on it must still be the same as before, — namely, the weight of the fluid displaced by the solid ; that is, the solid loses in apparent weight an amount equal to the weight of the displaced fluid. 89. Floating Bodies. — When the solid floats on the fluid, the weight of the solid is balanced by the upward pressure. In order that the solid shall be in equilibrium, these forces must act in the same line. The resultant of the pressure, which lies in the vertical line passing through the centre of gravity of the displaced fluid, must pass through the centre of gravity of the solid. Draw the line in the solid joining these two centres, and call it the axis of the solid. The equilibrium is stable when, for any infinitesimal inclination of the axis from the ver- tical, the vertical line of upward pressure cuts the axis in a point above the centre of gravity of the solid. This point is called the metacentre. , 90. Specific Gravity. — Archimedes' principle is used to de- termine the specific gravity of bodies. The specific gravity of a body is defined as the ratio of its weight to the weight of an equal volume of pure water at a standard temperature. The specific gravity of a solid that is not acted on by water may be determined by means of the hydrostatic balance. The body under examination, if it will sink in water, is suspended from one scale-pan of a balance by a fine thread, and is weighed* 126 ELEMENTARY PHYSICS. [90 It is then immersed in water, and is weighed again. The difference between the weights in air and in water is the weight of the displaced water, and the ratio of the weight of the body- to the weight of the displaced water is the specific gravity of the body. If the body will not sink in water, a sinker of unknown weight and specific gravity is suspended from the balance, and counterpoised in water. Then the body, the specific gravity of which is sought, is attached to the sinker, and it is found that the equilibrium is destroyed. To restore it, weights must be added to the same side. These, being added to the weight of the body, represent the weight of the water displaced. The specific gravity of a liquid is obtained by first balancing in air a mass of some solid, such as platinum or glass, that is not acted on chemically by the liquid, and then immersing the mass successively in the liquid to be tested and in water. The ratio of the weights which must be used to restore equilibrium in each case is the specific gravity of the liquid. The specific gravity of a liquid may also be found by means of the specific gravity bottle. This is a bottle fitted With a ground glass stopper. The weight of the water which com- pletely fills it is determinpd once for all. When the specific gravity of any liquid is desired, the bottle is filled with the liquid, and the weight of the liquid determined. The ratio of this weight to the weight of an equal volume of water is the specific gravity of the liquid. The same bottle may be used to determine the specific gravity of any solid which cannot be obtained in continuous masses, but is friable or granular. A weighed amount of the soHd is introduced into the bottle, which is then filled with water, and the weight of the joint contents of the bottle deter- mined. The difference between the last weight and the sum of the weights of the solid and of the water filling the bottle is the weight of the water displaced by the solid. The ratio -90] MECHANICS OF FLUIDS. 1 27 of .the weight of the solid to the weight thus obtained is the specific gravity of the solid. The specific gravity of a liquid may also be obtained by means of hydrometers. These are of two kinds,— ^the hydro- meters of constant weight and those of constant volume. The first consists usually of a glass bulb surmounted by a cylindri- cal stem. The bulb is weighted, so as to sink in pure water to some definite point on the stem. This point is taken as the zero ; and, by successive trials with different liquids of known specific gravity, points are found on the stem to which the hydrometer sinks in these liquids. With these as a basis, the divisions of the scale are determined and cut on the stem. The hydrometer of constant volume consists of a bulb weighted so as to stand upright in the liquid, bearing on the top of a narrow stem a small pan, in which weights may be placed; The weight of the hydrometer being known, it is im- mersed in water ; and, by the addition of weights in the pan, a fixed point on the stem is brought to coincide with the sur- face of the water. The instrument is then transferred to the liquid to be tested, and the weights in the pan changed until the fixed point again comes to the surface of the liquid. The sum of the weight of the hydrometer and the weights added in each case gives the weight of equal volumes of water and of the liquid, from which the specific gravity sought is easily obtained. The specific gravity of gases is often referred to air or to -hydrogen instead of water. It is best determined by filling a large glass flask, of known weight,, with the gas,, the specific -gravity of which is to be obtained, and weighing it, noting the temperature arid the pressure of the gas in the flask. The weight of the gas at the standard temperature and pressure is then calculated, and the ratio of this weight to the weight of the same volume of the standard gas is the specific gravity •desired. The weight of the flask used in this experiment must 128 ELEMENTARY'PHYSICS. ' [01 be very exactly determined. The presence of the air vitiates all weighings performed in it, by diminishing the true weight of the body to be weighed and of the weights employed, by an amount proportional to their volumes. The consequent error is avoided either by performing the weighings in a. vacuum produced by the air-pump, or by correcting the appar- ent weight in air to the true weight. Knowing the specific gravity of the weights and of the body to be weighed, and the specific gravity of air, this can easily be done, 91. Motions of Fluids. — If the parts of the fluid be inov-^ ing relatively to each other or to its bounding-surface, the cir- cumstances of the motion can be determined only by making- limitations which are not actually found in nature. There thus arise certain definitions to which we assume that the fluid under consideration conforms. ' The motion of a fluid is said to be um/orm when each ele- ment of it has the same velocity at all points of its path. The motion is steady when, at any one point, the velocity and direction of motion of the elements successively arriving at. that point remain the same for each element. If either the velocity or direction of motion change for successive elements,, the motion is said to be varying. The motion is further said to be rotationat or irrotational according as the elements of the fluid have or have not an angular velocity about their axes. In all discussions of the motions of fluids a condition is supposed to hold, called the condition of continuity. It is as- sumed that, -in any volume selected in the fluid, the change -of density in that volume depeftids solely on the difference between the amounts of fluid flowing into and out of that volume. In an incompressible fluid, or liquid, if the influx be, reckoned plus- and the efflux minus, we have, letting Q repl-esent the amount of the liquid passing through the boundary in any one direc- tion, ^Q = o. The results obtained in the discussion of fluid. 921 MECHANICS OF FLUIDS. I2g motions must all be interpreted consistently with this condition. If the motion be such that the fluid breaks upi into discontinu- ous parts, any results obtained by hydrodynamical considera- ations no longer hold true. If we consider any stream of incompressible fluid, of which, the cross-sections at two points where the velocities of the ele- ments are f, and z*, have respectively the areas A^ and ^„ we- can deduce at once from the condition of continuity A,v, = A„v„. (32) Fig. 41. 92. Velocity of Efiflux. — We shall now apply this principle to discover the velocity of efflux of a liquid from an orifice in the walls of a vessel. Consider any small portion of the liquid, bounded by stream lines, which we maycalla^/aww^. Represent the velocity of the filament at B (Fig. 41) by v^, and at C by V, and the areas of the, cross-sections 'of the elements at the same points by A^ and A . We have then, as above, A^v^ = -^ v. We assume that the flow has been estab- lished for a time sufiiciently long for the motion to become steady. The energy of the mass contained in the filament be- tween B and C is, therefore, constant. Let V^ represent the potential at B due to gravity, V the potential at C, and d the density of the liquid. The mass that enters at .5 in a unit of time is .. dA,ii,. The mass that goes out at C is the equal quantity dAv. The. energy entering at B is dA,vi\v^ + FO, 13° ELEMENTARY PHYSICS. [92 the energy passing out at C is dAv{^ + V). If the pressures at B and C on unit a:reas be expressed by /j and /, the work done at B on the entering mass by the pressure. /j \s p^A^v^, and at C on the outgoing mass is pAv. The energy within the filament remaining constant, the incom- ing must equal the outgoing energy ; therefore pAv + dAv^W ■^V)= pA.v. + dA,vl\v,^ + V^, whence, since A^v^ = Av, we have We may write this equation i=i'.(i+^). (56) lii=- 273°, °\ 273/ 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 T whence T = \,?-^-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 thq 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 the 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 completely 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, /*= 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 (§ Tj) 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 supposfe the gas to be compressed, and no heat to escape, it is plain that if the volume dimin- o g c ish from OC to OG, the pressure will F'°' s^- become greater than GD; suppose it to be GM. If a number 13 194 ELEMENTARY PHYSICS. [159 of such points as J/ 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 adiabatic ' line. It evidently makes a greater angle with the horizontal than the isothermal. 159. Specific Heats of Gases. — In § 156 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 teinperature, must be supplied from the source of heat. It has been seen that the loss of energy resulting fropi 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 these 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 (§ iii). 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 p'roduce IS9] EFFECTS OF HEAT. I95 .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 Jiave been determined or ehminated. The specific heat of a gas at constant volume is generaliy determined from the ratio between it and the specific heat at constant pressure. The first determination of this ratio was accompHshed 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 t 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 C-o the specific heat at constant volume. Now let the gas expand, at the constant temperature ^ + 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. .^ if represents the increase in volume of the gas during this process. The same increase in volume would have occurred had the gas beeii 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 ffF"„, where V^ repre- sents the volume at zero. Hence we have Ad == aV„ and the work done during the expansion is PAd = PoiV„. The heat absorbed, therefore, in raising the temperature of the gas one degree at constant pressure is Q = d, + P^V„. Cp represents the specific heat of the gas at constant pressure, measured in 196 ■ ELEMENTARY PHYSICS. [iS* mechanical units. The ratio of the two specific heats is ^=i + ^/'«F,. (S8> 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^, whicL would lower the temperature -^ . Pa F„ degrees, since the ab- straction of a quantity d, of heat would lower the temperature one degree. Represent this change of temperature by 6. Re- , membering that the supposed change of volume was a V„ which aVt equals -— — -, and that the original volume was Vt, it is seea a that the change of — ; — : in unit volume would cause a fall in ° 1 -\- at terriperature of ^degrees. Substituting d iovj^PaV^ in Eq.. r (58), we have ji =. i -\- B. It is the object of the experiment to find 0. The method of Clement and Desormes is as follows:. 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 be /f — h, H representing the height of the barometer, and h 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 the 159] -EFFECTS OF HEA T. 197 atmospheric pressure, compresses the air originally in the flask, and raises its temperature. The volume of the air becomes \ — (p, where its original volujne is taken as unity and repre- sents its reduction ; and, if there were no change of tempera- ture, the pressure wpuld be ■:~~d>- ^^ *^^ temperature in 'crease Q' d.egrees, and become / + 6*', the pressure will be H-_h l±o^_n = H, (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 ^LuA^H-h'. {66) I — From these two equations we have ^-^-^ ft'- (^ + ^^)^' a Suppose, now, the change of volume had been ^ . ^^ , then the change of temperature woyld have been 6 ; and, since change of volume is proportional to change of temperature, we have ^ i-\-at hence 6'^^ l + at 198 ELEMENTARY PHYSICS. [160 or, substituting the values of and B', we have h' H-h! h! H-k! '^ h-h'- h-h!' Now we have shown that Cp a = 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 -\- /it), M representing a very small increment of tempera- ture. The heat consumed will be CpAt, and the increase of volume aV^M. 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- C 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 cfvAt = ^^' (^^) l6o] EFFECTS OF HEAT. 1 99 where Eh is the elasticity under the condition that no heat enters or escapes. If; now, the heat produced by compression be allowed to escape, there will remain the quantity Cv^t, and the increment of pressure will be reduced to Sp ■=. ^p-^ . This is the increase of pressure that will occur if the gas be compressed by the amount a. V^^t without change of temperature ; hence WAt = ^-' <'3) where Et is the elasticity for constant temperature. Dividing (62) by (63), we have • Ap Ek_ aV,At Ap _Ap_ Cp Et~ Sp ~ Sp- C-~ Cr,' 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 I ELEMENTAR Y PlfYSICS. [l6i pressed and rarefied, the compressions and rarefactions occur- ring in such rapid succession that there is no time for any tra:nsfer of heat. If Eq. (64) be applied to air, the E becomes E;„ 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, Ek can be computed. In § yj it is shown that Et is numerically equal to the pressure ; hence we have th;e 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 a't 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 attract 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 :i64] EFFECTS OF HEAT. 201 same amount of heat. This is the heat equivalent of vaporiza- tion. When a vapor condenses into the liquid 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 heat 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, dnd 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 temperatui-e 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 700 milli- metres pressure. At the pressure / of the vapor, and tem- perature t of the air at the time of th6 experiment, the same space would contain 1293.2 X^X:^, 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^Xj^,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, arid, 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. [168 covered surface of the box. A thermometer plunged in the €ther 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. 66 gives the absolute humidity. 168. Relative Humidity. — The amount of moisture that the air may contain depends upon its tem,perature. The damp- mess 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 humid- ity is the ratio of the amount of moisture in the air to that which would be required to 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 air. 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 (§ 1 14). 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 c6ndition at the end as at the beginning, internal work may have been done, or internal energy expend- 206 ^ ELEMENTARY PHYSICS. [170 ed, which would increase or diminish the work apparently de- veloped from the heat. To develop the second law of thermodynamics, we makd 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 priiaciple,. 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 fundanjental 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^, and perform a cer- tain amount Woi mechanical work. If it be perfectly reversible, Jl^ 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 ff. 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 £ 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 Whe the work it performs, and iv the work B performs, \5 will, from its reversibility, by the performance upon it of the Vfork 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 ^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] THERMOD 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 JF—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 ifalse ; 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, perfprm more work than ,a rever- sible engine. It follows that all reversible engines, whatever the working substance, have the same efificiency. 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 which presents the greatest advantage fpr the study. Since of all substances the properties of gases are laest kriown^, 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 ^ to ^ ; where t repre- sents the temperature .of the source, and 8 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, q -b an c This is the first operation. For the , Fig. 53. second operation, let the piston rise, and the volume increase 208 ELEMENTARY PHYSICS. [170. from b to c at 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 /. For the third operation let the piston still ascend, and the volume increase from Oc to C^ with- out loss or gain of heat until the temperature falls from t to 0, 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 Q of the refrigerator. The volume becomes Oa and the pressure aA, as at the beginning. DA is the isother- mal line for the temperature 0. 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<2:^ is evi- dently the arfea ABba. In the same way it is shown that when the gas expands from ^ to c it 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 A BCD 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 m^ist 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 maybe 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 area ABCD, has disappeared. Of the "heat withdrawn from the source, then, , , , . area ABCD . , . , ^, . only the fraction i^T^r-r- is converted into work. This ■' 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 B, ab- sorbs a quantity of heat represented by h, the same as it gave up when compressed, and performs work represented hy 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 xipon 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 whil^ the quantity of heat k 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 =^ Be X be ^ gi X be. «\lso BCcb = ^^ X be. Then we have H — h _ area ABCD _ gi y^be _ gi H ~ area BCcb ~ ghx be~ 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 G. These pressures are proportional to the absolute temperatures (§ 1 56) ; that is, if t and 9 are tempera- tures on the absolute scale, gh ~ t' and gi H-h t-9 m gh~ H ~ t ' hence area ABCD t — d In another form the result contained in Eq. i^G) may be written , h 6 170] THERMOD YNAMICS. • 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 B. The second takes the heat h from the refrigerator of the first, and gives to its own refrigerator the heat ^,, working between the temperatures Q and ^,. The operation of the others is similar ; then, from Eq. 68, we have H~ t' h ~ 6' K K e„' h ~ ' multiplying, we obtain -— V — V — V — _ V — V — ' \/ " or and hn _Gn^ H~ t H — hn t — 0„ H ~ t ' Hence it appears that, in a perfect heat-engine, the heat con- 212 ELEMENTARY PHY.SICS. [171 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 — 0°, or when the refrigerator is at the abr 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 thermodynainics. , 171. Absolute Scale of Temperatures. — K-n. absolute scale of temperatures, formed upon the assumed properties of a perfect gas, has already been described (§ 156). No such sub- FiG. 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 I7i] THERMODYNAMICS. 213 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 y5/?' be the isothermal line for the temperature t^ of melting ice, and let bb' be an isothermal line for an intermediate tempera- ture. Let Bfi, B'I3', 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'j3 Represents Carnot's cycle; and the heat given to the refrigerator at the temperatiire /„, 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'^'/i ; or, the heat given to the refrigerator is equal to i — area B^': hence J ~ i ' and t — t, area B ft' Now , if ^ — the volume at tem- perature i. 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- charging the steam at the proper times to impart to the piston a reciprocating motion, which may be converted into a circulaf 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 cylinder, 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-pdmp." In non-condensing engines the stpam escapes into the open air. In this case the. 173] THERMODYNAMICS. 21? temperature of the refrigerator must be considered at least as high as that of saturated steam at the atmospheric pressure, or about 100°, and the temperature of the source must be taken as that of saturated steam at the boiler pressure. Applying the expression for the efificiency (§ 1 70), e = — --, 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 ^ — ^ is a much larger quantity for condensing than for non-condensing engines. The gain of efificiency 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 temjSerature 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. [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 iii 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 1 74] THERMOD YNAMICS. 2 1 7 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 aif 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 fjirnish 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. .t — 6 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 Glass. 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 transpbrted 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 waterypower, is, therefore, derived from solar energy. The ocean currents also possess energy due to their motion,,, 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 ■potentlsl 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 heaty. and was the first to attempt a measur.ement of it. He com- 174] THERMODYNAMICS. 219 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 intp 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 treadVheel to revolve, performed work, which produced heat in addition to that dtie 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 non-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 woul'd 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. 22 1 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 estirnated 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 generate 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, sa 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 knowlefdge 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. 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 outsidci force they remain perjnanently in contact with it. While in contact v^ith 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. [17& 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 the 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 ma,gnet, 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 principle 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 extremi- i78] 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 mkgnet 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 ^the horizontal intensity of the earth's magnetism, by k^ and ^j the force in the region occupied by the susperided niag- 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 /jf = tt y tf?>> ^^j — ^^ T~TJr* ■tilVl hyti t^^ = ^rj-iTf- From such equations, by elimination of If, the Values of A, and A, were obtained, and were 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 upoh 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 fft and m, represent the strengths of two magnet poles, r 15 226 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- — ^. 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, [a = MLT-- 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 \ml^ = MiL^T— and [^p{=MiL-^T-\ 180. Distribution of Magnetism in a Magnet. — If we con- ceive of a single row of magnetic molecules with their uillike 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 surface 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- button 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 [7] = -^^-=^4 — =MiL-iT-K 228 ELEMENTARY PHYSICS. [l8r Coulomb showed, by a method of oscillations similar to that described in §178, that the magnetic force at different points along a straight bar magnet gradually increases from the mid- dle of- the bar, where it is inlperceptible, 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 hnes 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 the 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 ih.^ 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 fromh: FiG. 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 y = K(/ + (^ -/)=)* ~ (/-f (^-f//)*) I8i]' MAGNETISM. 229 This expression expanded gives 2mlx ^mPx ^mrx' 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 75 cos ft (7i) Since cos 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==x, the poten- tial becomes ^-J. + ^+--- (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. [181. 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 pole of the second magnet by m^, the lengths of the two magnets by 2/ and 2j/ 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 »2^ is Fig. 56. f^{x-iY The cosine of the angle made by this force with the x axis is X — I (y' \(^_ ^\n» - Hence the component of this force along the X axis is mm^{x — /) Hence the component along the x axis of the whole force on the pole m^, due to the first magnet, is mm J J . X — I x_^ I , .(/ + (^ - i)y ~ (/ + (F^nW' 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 / I , 2/' 3/\ l82] MAGNETISM. 23 1 t An equal and oppositely directed component acts upon the other pole — m^ of the second magnet. Hence the moment of couple due to the action of the first magnet upon the sec- ' end is / I 2/" % -fK Smm,fy[-,-{-^-^). (74) If jf be such that 37' = 2/", or if the ratio of the lengths of the two magnets used be i : Vi-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 Vi.S, the numerator of the term having x^ in the denominator is made very small, and is eliminated by the method 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- noted by the symbol/. The dimensions of the strength of a magnetic shell follow at once from this definition. We have {j'\ equal to the dimen- sions of intensity of magnetization multiplied by a length. Therefore {j^ = MiLiT'K We obtain first the potential of such a shell of infinitesi- 232 ELEMENTARY PHYSICS. [182 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 ^'^' \y axis. Let a represent the area of th© ^,''^ 1 shell, supposed infinitesimal, 2/ the °(ir """ thickness of the shell, and d the mag- ■ ^'°-57- • 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 maybe neglected; ^ We have ' „ if , . 2ald V= —rcos 6 = — — cos a. r r Now a cos 6 is the projection of the area of the shell upon a plane through the origin normal to the radius vector r, and, ft "* since a is infinitesmal, 5 — is the solid angle eo bounded by the lines drawn from P to the boundary of the area a. The potential then becomes V= ialdoo =^ jw, since 2ald 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 "Sjoo. If the shell be of uniform strength, the potential due to it becomes j'2oo 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 J^- (75): It is independent of the form of the shell, and dependent* only 183] MAGNETISM. 233 on the form of its boundary. 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 — znj. 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 47r;'. This result is of im- portance 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 asaumed by the axis of a magnetic needle suspended to move freely in a horizontal plane, iand 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 ot in a field in which ' the horizontal intensity is repiresented 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 334 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 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 a or mHa, if a be always very small. Since a varies between and 0, the average force sufficient in turning the needle is \mH(t>. 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 Icjt, 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, Hence if we write 2ml = M, the magnetic moment of the mag- 183] MAGNETISM. . 23$ 4?rY > net, we obtain MH = -r™- ; or if we take the time of oscillation T 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, which 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 ifMm,y\—-^-\ — A, where P represents the small numerator of the correction term. This correction can be made very small in practice by giving to the , 236 ELEMENl^AliY PHYSICS. [183 magnet, as already explained, lengths in the ratio of i to ^1.5. The opposing equal couple is im^Hy sin 0, where represents the angle of deflection from the magnetic meridian. We have then 4Mmj\^-^ + ^J = 2my^JI sin 0, or -^ + ^ = - -^sin \i-^y (7;) P is determined by measuring the angles 4> and 0^ for two dif- ferent distances x and x^. The equations containing the results of these measurements are M_ H and = ij;* sin 0(1 — ^j From these equations the value of P is found to be equal to ^x' sin

o when F^y F„; the cube moves from a place of stronger to a place of weaker magnetic force. y The subject may be looked at from a different point of view. The coefificient of induced magnetization k is negative in all diamagnetic bddies, but its numerical value is small. It has never been found to be numerically greater than — in ■ 1 8s] MA GNE TISM. 243 diaiiiagnetic bodies. In such bodies, therefore, the value of }x, the magnetic permeability, is less than i, though never negative. When k vs, o, II equals i, and for paramagnetic bodies jx is greater than i. The ratio of the force within the substance of which the magnetic permeability is }i to that in vacuum, in ' N which it is supposed to be placed, is -vr = i + A'"^k = /x. If the convention of §2i be used, by which the strength of a field of force is represented by the number of lines of force passing perpendicularly through unit area, it is evident that when a paramagnetic body in which /* > i and Ny F ii brought into the field, the lines of force are converged into the body. When a diamagnetic body is in the field the lines of force are deflected from it. As may be easily seen, a paramagnetic body of permea- bility jx, surrounded by a medium also paramagnetic, but of permeability /z, > /z^, will act relative to the medium as a dia- magnetic body. The condition of any body of which the permeability is less than that of the medium in which it is im- mersed is like that of a weak magnet between the ends of two stronger ones, all three being magnetized in the same direc- tion. The movements of both paramagnetic and diamagnetic bodies may be rqughly illustrated by the movements of. bodies immersed in water, which rise or sink according as their specific gravities are less or greater than the specific gravity of water. 185. Changes in Magnetic Moment. — When a magnet- izable body is placed in a powerful magnetic field, it often receives, temporarily, a more intense magnetization than it can retain when removed. It is said to be saturated, or magnetized to saturation, when the intensity of its magnetization is the greatest which it can retain when not under the inductive a,ction of other magnets. The coercive force of steel is much greater than that of any other substance; the intensity of magnetization which it can retain is, therefore, relatively very 244 ELEMENTARY PHYSICS. [l86 great, and it is hence used for permanent magnets. It is found that the coercive force depends upon the quality and temper of the steel. Changes of temperature cause corresponding changes in the magnetic moment of a magnet. If the temperature of a mag- net be gradually raised, its magnetic moment diminishes by an amount which, for small temperature changes, is nearly pro- portional to the change of temperature. The magnet recovers its original magnetic moment when cooled again to the initial temperature, provided that the temperature to which it was raised was never very high. If it be raised, however, to a red heat, all traces of its original magnetism permanently disap- pear. Trowbridge has shown that, if the temperature of a magnet be carried below the temperature at which it was originally magnetized, its magnetic moment also temporarily diminishes. Any mechanical disturbance, such as jarring or friction, which increases the freedom of motion among the molecules of a magnet; in general brings about a diminution of its magnetic moment. On the other hand, similar mechanical disturbances facilitate the acquisition of magnetism by any magnetizable body placed in a magnetic field. l86. Theories of Magnetism. — It has .been shown by mathematical analysis that the facts of magnetic interactions and distribution are consistent with the hypothesis, which we have already made, that the ultimate molecules of iron are themselves magnets, having north and south poles which attract and repel similar poles in accordance with the law of magnetic force. Poisson's theory, upon which most of the earlier mathematical work was based, was that there exist in each molecule indefinite quantities of north and south magnetic fluids, which are separated and moved to opposite ends of the molecule by the action of an external magnetizing force. Weber's view, which is consistent with other facts that Pois- i86] MAGNETISE. 245 son's theory fails to explain, is that each molecule is a magnet, with permanent poles of constant strength, that the molecules of an iron bar are, in general, arranged so as to neutralize one another's magnetic action, but that, under the influence of an external magnetizing action, they are arranged so that their magnetic axes lie more or less in some one direction. The bar is then magne'tized. On this hypothesis there should be a limit to the possibible intensity of magnetization, which would be reached when the axes of all the molecules have the same direction. Direct experiments by Jqule and J. Miiller indicate the existence of such a limit. An experiment of Beetz, in which a thin filament of iron deposited electrolytically in a strong magnetic field becomes a magnet of very great intensity, points in the same direction. The coercive force is, on this hypothesis, the resistance to motion experienced by the molecules. The facts that magnetization is facilitated by a jarring of the steel brought into the magnetic field, that a bar of iron or steel after being removed from the magnetic field retains some of its magnetic properties, that the dimensions of an iron bar are altered by magnetization, the bar becoming longer and dimin- ishing in cross-section, and that a magnetized steel bar loses its magnetism if it be highly heated, are all facts which are best explained by Weber's hypothesis. CHAPTER II. ELECTRICITY IN EQUILIBRIUM. 187. Fundamental Facts.— (i) If a piece of glass and a piece of resin be brought in contact, or preferably rubbed together, it is found that, after separation, the two bodies are attracted towards each other. If a second piece of glass and a second piece of resin are treated in like manner, it is found that the two pieces of glass repel each other and the two pieces of resin repel each other, while either piece of glass attracts either piece of resin. These bodies are said to' be electrified or charged. All bodies may be electrified, and in other ways than by contact. It is sufificient for the present to consider the single example presented. The experiment shows that bodies may be in two distinct and dissimilar states of electrification. The glass treated as has been described is said to be vitreously or positively electrified, and the resin resinously or negatively elec- trified. The experiment shows also that bodies similarly elec- trified repel one another, and bodies dissimilarly electrified at- tract one another. (2) If a metallic body, supported on a glass rod, be touched by the rubbed portion of an electrified piece of glass, it will become positively electrified. If it be then joined to another similar body by means of a metallic wire, the second body is at once. electrified. If the connection be made by means of a damp linen thread, the second body becomes electrified, but not so rapidly as before. If the connection be made by means of a dry white silk thread, the second body shows no signs of electrification, even after the lapse of a considerable time. Bodies are divided according as they can be classed with the i87] ' ELECTRICITY IN EQUILIBRIUM. 247 metals, damp linen, or silk, as good conductors, poor conductors, and insulators. The distinction is one of degree. All con- ductors offer some opposition to the transfer of electrification, and no body is a perfect insulator. A conductor separated from all other conductors by insu- lators is said to be insulated. A conductor in conducting con- tact with the earth is said to he. grounded or joined to ground. During the transfer of electrification in the experiment above described the connecting conductor acquires certain pfoperties which will be considered under the head of Electri- cal Currents. (3) If a positively electrified body be brought near an insu- lated conductor, the latter shows signs of electrification. The end nearer the first body is negatively, the farther end posi- tively, electrified. If the first body be removed, all signs of electrification on the conductor disappear. If, before the first body is removed, the conductor be joined to ground, the posi- tive electrification disappears. If now the connection with ground be broken, and the first body removed, the conductor is negatively electrified. The experiment can be carried out so as to give quantita- tive results, in a way first given by Faraday. An electrified body, for example a brass ball suspended by a silk thread, is introduced into the interior of an insulated closed metallic vessel. The exterior of the vessel is then found to be electri- fied in the same way as the ball. This electrification disap- pears if the ' ball be removed. If the ball be touched to the interior of the vessel, no change in the amount of the external electrification can be detected. If, after the ball is introduced into the interior, the vessel be joined to ground by a wire, all external electrification disappears. If the ground connection be broken, and the ball removed, the vessel has an electrifica- tion dissimilar to that of the ball. If the ball, after the ground connection is brpken, be first touched to the interior of the 248. ELEMENTARY PHYSICS. [iS^r vessel and then removed, neither the ball nor the vessel is any longer electrified. ' A body thus electrified without contact with any charged body is said to be electrified by induction. The above-men- tioned facts show that an insulated conductor, electrified by induction, is electrified both positively and negatively at once, that the electrification of a dissimilar kind to that of the in- ducing body persists, however the insulation of the conductor be afterwards modified, and that the total positive electrifica- tion induced by a positively charged body is equal to that of the inducing body, while the negative electrification can ex- actly neutralize the positive electrification of the inducing body. The use of the terms positive and negative is thus justified, since they express the fact that equal electrifications of dis- similar kinds are exactly complementary, so that, if they be superposed on a body, that body is not electrified. These two kinds of electrification may then be spoken of as opposite. If the glass and resin considered in the first experiment be rubbed together within the vessel, and in general if any appa- ratus which produces electrification be in operation within the vessel, no signs of any external electrification can be detected. It is thus shown that, whenever one state of electrification is produced, an equal electrification of the opposite kind is also produced at the same time. Franklin showed that, by the use of a closed conducting vessel of the kind just described, a charged conductor intro- duced into its interior and brought into conducting contact with its walls is always completely discharged, and the charge is transferred to the exterior of the vessel. This procedure furnishes a method of adding together the. charges on any number of conductors, whether they be charged positively or negatively. It is thus theoretically possible to increase the charge of such a conductor indefinitely. l88] ELECTRICITY IN EQUILIBRIUM. 249 (4) If any instruxnent for detecting forces due to electrifica- tions be introduced into the interior of a closed conductor charged in any manner, it is found that no signs of force due to the qharge can be detected. The experiment was accurately executed by Cavendish, and afterwards tried on a large scale by Faraday. It proves that within a closed electrified con- ductor there is no electrical force due to the charge on » the conductor, or that the potential due to the electrical forces is uniform within the conductor. 188. Law of Electrical Force. — If two charged bodies be considered, of dimensions so small that they may be neglected in comparison with the distance between the bodies, the stress between the two bodies due to electrical force is proportional directly to the product of the charges which they contain, and inversely to the square of the distance between them. If Q and Q, represent two similar charges, r the distance between them, and k a factor depending on the units in which the charges are measured, the formula expressing the repulsion between them is Coulomb used the torsion balance (§ 82) to demonstrate this law. At one end of a glass rod suspended from the torsion wire and turning in the horizontal plane is placed a gilded pith ball, and through the lid of the case containing the apparatus can be introduced a similar insulated ball so arranged that its centre is at the same distance from the axis of rotation of the suspended system, and in the same horizontal plane, as the centre of the first ball. This second ball may be called the carrier. To prove the law as respects quantities, the suspended ball is brought into equilibrium at the point afterwards to be occu- pied by the carrier ball. The carrier ball is then charged and 250 ELEMENTARY PHYSICS. [189 introduced into the case. When it comes in contact with the suspended ball, it shares its charge with it and a repulsion ensues. The torsion head must then be rotated until the sus- pended ball is brought to some fixed point, at a distance from the carrier which is less than that which would separate the two balls in the' second part of the experiment if no torsion were brought upon the wire. The repulsion is then measured in terms of the torsion of the wire. The charge on the carrier is then halved, by touching it with a third similar insulated ball, and, the charge on the suspended ball remaining the same, the repulsion between the two balls at the same distance' is again observed. If the case be so large that no disturbing effect of the walls enters, and if the balls be small and so far apart that their inductive action on one another may be neg- lected, the repulsion in the second case is found to be one half that in the first case. In general the problem is a far more difficult one, for the distribution on the two spheres is not uniform. That portion of the distribution dependent on the ' induction of the balls can be calculated, but the irregularities of distribution due to the action of the walls of the case and other disturbing elements can only be allowed for approxi- mately. The law as respects distance is proved in a somewhat simi- lar way. The repulsions at two different distances are meas- ured in terms of the torsion of the wire, the charges on the two balls remaining the same. The same corrections must be introduced as in the former case. 189. Distribution. — The law of electrical force has been stated in terms of the charges of two bodies. We may, how- ever, consider electricity as a quantity which has an existence independent of matter and which is distributed in space. The fact cited in § 187 (4) shows that this distribution must be looked on as being on the surfaces of conductors and not on their interiors. If we define surface density of electrification igp] ELECTRICITY IN EQUILIBRIUM. 251 at any point on the surface of a charged conductor as the limit of the ratio of the quantity of electricity on an element in the surface at that point to the area of the element as that area approaches zero, we may measure quantities of electricity in terms of surface density. The surface density of electricity is usually designated by u. If the law of electrical force hold true not only for charges on bodies but also for quantities of electricity on the surface . elements of a conductor, it is evident, from the fact that within an electrified conductor there is no electrical force, that its surface density of electrification must be proportional at every point on its surface to the thickness at that point of a shell of matter which is so distributed on that surface that there is no force at any point enclosed by the surface. The distribution on a charged sphere may, from symmetry, be assumed uniform. The fact that there is no electrical force within a charged sphere is then, from § 29 (i), consistent with the law of electrical force which has been given ; and since the means of detecting elec- trical force, if there were any, within a charged conductor are very delicate, this fact affords a strong corroborative proof of the law- The determination of the distribution of electricity on irreg- ularly shaped conductors is in general beyond our power. If • we consider, however, a conductor in the form of an elongated egg, it can be readily seen that, in order that there may be no electrical force within it, the surface density at the pointed end must be greater than that anywhere else on its surface. In general, the surface density at points on a conducting surface depends upon the. curvature of the surface, being greater where the curvature is greater. Thus, if the conductor be a long rod terminating in a point, the surface density at the pointed end is much greater than that anywhere else on the rod. 190. Unit Charge. — The law of electrical force enables us to define a unit charge, based upon the fundamental mechanical units. 25? ELEMENTARY PHYSICS. [191 Let there be two equal and similar positive charges concen- trated at points unit distance apart in air, such that the repul- sion between them equals the unit of force. Then each of the charges is a unit cliarge, or a unit quantity of electricity. . With this definition of unit charge, it may be said that the force be- tween two charges is not merely proportional to, but equals, the product of the charges divided by the square of the dis- tance between them. The factor k in the expression for the force between two charges becomes unity, and the dimensions of — r^ are those of a force. If the charges be equal, we have ^ ' MLT-\ m Hence [<2] = M^I}T ~ ' are the dimensions of the charge. This equation gives the charge in absolute mechanical units, and by means of it all other electrical quantities may be expressed in absolute units. It is at the basis of the electrostatic sy stern of electrical measurements. The practical unit of charge or quantity is called the cou- lomb. It is the quantity of electricity transferred during one second by a current of one ampere (§ 218). 191. Hlectrical Potential.— The electrical forces have a po- tential similar to that discussed in § 28. The unit quantity of positive electricity is taken as the test unit. Since [§ 187 (4)] the potential at every point of a charged conductor is the same, the surface of the conductor is an equipotential surface. The potential of this surface is often called the potential of the conductor. A conductor joined to ground is at the potential of the earth. It will be shown (§ 195) that the potential of the earth is not appreciably modified when a charged conduc- tor is joined to groun,d. All conductors, moreover, however they may afterwards be charged, are when uncharged at the igi] ■ * ELECTRICITY IN EQUILIBRIUM. 253 potential of the earth. For these reasons it is usual to take the potential of the earth as the fixed potential or zero from )vhich to reckon the potentials of electrified bodies. The po- tential of a freely electrified conductor and of the region about it is thus positive when the charge of the conductor is positive, and negative when it is negative. A conductor joined to ground is at zero potential. The difference of potential between two points is equal to the work done in carrying a unit quantity of electricity from one point to another. We then have the equation QiVi — V) = work. I Hence follows the dimensional equation [F^ — F] = — r-5 = M^I}T~^, the dimensions of difference of poten- M^L^T - ^ ' tial in electrostatic units. If any distribution of a charge exist on a conductor, which is such that the potential at all points in the conductor is not the same, it is unstable, and a rearrangement goes on until the po- tential becomes everywhere the same. The process of rear- rangement is said to consist in a flow of electricity from points of highet to points of lower potential. On this property of electricity depends the fact that a closed conducting surface completely screens bodies within it from the action of external electrical forces. For, whatever changes in potential occur- in the region outside the closed con- ductor, a redistribution will take place in it such as to make the potential of every point within it the same. Electrical force depends on the space rate of change of potential, and not on its absolute value. Hence the changes without the closed conductor will have no effect on bodies within it. Further, any electrical operations whatever within the closed conductor will not change the potential of points outside it. For, whatever operations go on, equal amounts of positive and negative electricity always exist within the conductor, and hence the potential of the con- ductor remains unaltered. Hence electrical experiments per- 254 ELEMENTARY PHYSICS. [191 formed "within a closed room yield results which are as valid as if the experiments were performed in free space. The advantage gained by the use of the idea of potential in discussions of electrical phenomena may be illustrated by a statement of the process of charging a conductor by induction described in § 187 (3). To fix our ideas, let us suppose that the field of force is due to a positively electrified sphere, and that the body to be charged is a long cylinder. When this cylinder, previously in contact with the earth and therefore at zero potential, is brought end on to a point near the sphere, it is in a region of positive potential, and is itself at a positive potential. If we consider the original potentials at the points in the region now occupied by the cylinder, it is easily seen that the potential of points nearer the sphere was higher than that of those more remote. When the cylinder is brought into the field, therefore, the portion nearer the sphere is temporarily ' raised to a higher potential than the portion more remote. The difference of potential between these portions is annulled by a flow of electricity from the points of higher potential to those of lower potential at a rate depending on the cohductiv- ity of the cylinder. The end of the cylinder nearer the sphere is negatively charged, the end more remote is positively charged, and the two charged portions are separated by a line on the surface, called the neutral line, on which there is no charge. If the cylinder be now joined to ground, a flow of electricity takes place through the ground connection, and it is brought to zero potential. The potential of the cylinder is therefore everywhere lower than the original potentials of the points in the region which it occupies. This necessitates a negative charge distributed over the whole cylinder. In other words, the earth and the cylinder may be considered as forming one conductor' charged by induction, in which the neutral line is not within the cylinder. 192] . ELECTRICITY IN EQUILIBRIUM. 255 If the ground connection be broken the electrical relations are not disturbed. If the cylinder be now removed to a region of lower potential against the attraction of the sphere, work will be done against electrical forces, which reappears as electri- cal energy. The poteritial of the cylinder is lowered, and, if it be again connected with the earth, work will be done by a flow of electricity to it. The fact that there is no electrical force within a closed electrified conductor of any shape permits some extensions of the theorems of § 29. Some small portion of the surface of any electrified conduc- tor may be considered a plane relatively to a point situated just outside it. Represent the surface density of electricity on that plane by cr. It was proved (§ 29) that the force due to such a plane is 2n(j, if we substitute c for the corresponding factor d. Now, just inside the conductor the force is zero. This results from the equilibrium of the force due to the plane portion and that due to the rest of the conductor. The force due to the rest of the conductor is therefore 2itcT. At a point just putside the conductor these two forces act in the same direction. Hence the total force due to the conductor at a point just outside it is the sum of the two forces, or \n(7. From the preceding proposition follows at once a deduction as to the pressure outwards on the surface of an electrified conductor due to the repulsion of the various parts of the chat-ge for one another. Select any small portion of the sur- face of the electrified conductor of area a. The force on unit quantity acting outward from the conductor at a point in- that area due to the charge of the rest of the conductor is 2;ro-. This force acts on every uiiit of charge on the area!. The force on the area acting outwards is then 2na&\ or the pressure at a point in the area referred to unit of area is 2wcr'. This quan- tity is often called the electric pressure, 192. Capacity. — The electrical capacity of a conductor is defined to be the charge which the conductor must receive to 256 ELEMENTARY PHYSICS. [192 raise it from zero to unit potential^ while all other conductors in the field are kept at zero potential. This charge varies for any one conductor in a way which cannot be always definitely determined, depending upon the medium in which the con- ductor is immersed and the position of other conductors in the field. When the charged conductor is in very close prox- imity to another conductor which is kept at zero potential, the amount of charge needed to raise it to unit potential is very great as compared with that required when the other conduc- tor is more remote. Such an arrangement is called a condenser. If the charge on a conductor be increased, the increase in po- tential is directly as that of the charge; Hence the capacity C is given by dividing any given charge on a conductor by the potential of that conductor, or ' O c=f. (78) The practical unit of capacity is the farad, which is the ca- pacity of a conductor, the charge ort which is one coulomb (§ 190) when its potential is one volt (§ 228). This unit is too great for convenient use. Instead of it a microfarad, or the one-millionth part of a farad, is usually employed. This equation gives the dimensions of capacity. Measured in electrostatic units, they are Capacity, therefore, is of the dimensions of a length. In the theory of Faraday, which has been adopted and de- veloped by Maxwell, electrification is made to consist in an arrangement or displacement of the insulating medium, called by him the dielectric, surrounding the electrified conductor. 193] ELECTRICITY IN EQUILIBRIUM. 2$^ This displacement, beginning at the surface of the electrified conductor, continues throughout the dielectric until it termi- nates at the surfaces of other conductors. The electrification of the charged conductor is the manifestation of this displace- ment at one face of the dielectric, that of the surrounding con- ductors the manifestation of the displacement on the other face. The one charge cannot exist without an equal and op- posite charge on surrounding conductors, as was experiment- ally proved by Faraday's experiment already described in § 187 (3). It is therefore necessary, in considering the capacity oi any conductor, to take account of the medium in which it is immersed, and of the arrangement of surrounding conductors. 193. Specific Inductive Capacity. — The fact that the capacity of a condenser of given dimensions depends upon the medium used as the dielectric was first discovered by Caven- dish, and afterwards rediscovered by Faraday. The property of the medium upon which this fact depends is called its specific inductive capacity. The specific inductive capacity of . vacuum is taken as the standard. If Q represent the charge required to raise a condenser in which the dielectric is vacuum to a potential V, then if another dielectric be substituted for vacuum, it is found that a different cTiarge Qi is required to raise the potential to V. The ratio -- = ^ is the specific in- ductive capacity. Since C^ = -W and C=^ are the capacities of the condenser with the two dielectrics, it follows that C; = CK, (79) where C is the capacity with vacuum as the dielectric. The specific inductive capacity K is always greater than unity. Some dielectrics, such as glass and hard rubber, have a high 17 258 ELEMENTARY PHYSICS. [194 specific inductive capacity, and at the same time are capable of resisting the strain put upon them by the electric displace- ment to a much greater extent than such dielectrics as air. They are therefore used as dielectrics in the construction of condensers. 194. Condensers. — The simplest condenser, one which ad- mits of the direct calculation of its capacity, and from which the capacities of many other condensers may be approximately calculated or in- ferred, consists of a conducting sphere surrounded by another hollow concentric conducting sphere which is kept always at zero potential by a ground connection. For convenience we assume the specific Fig. 58. inductive capacity of the dielectric sepa- rating the spheres to be unity. Let the radius of the small sphere (Fig. 58) be denoted by R, that of the inner spherical surface of the larger one by R^ ; let a charge Q be given to the inner sphere by means of a conducting wire passing through an opening in the outer sphere, which may be so small as to be negligible. This charge Q will induce on the outer sphere an equal and opposite charge, — Q. Since the distribution on the surface of the spheres may be aWumed uniform, the poten- tial at the centre of the two spheres, due to the charge on the inner one, is -„-, and the potential due to the charge of the outer sphere is — ^. Hence the actual potential V at the centre, due to both charges, is R R-'^K RR, I' 194] ELECTRICITY IN EQUILIBRIUM. 259 Hence the capacity is ^ Q RR In order to find the effect of a variation of the value of R, divide numerator and denominator by R, and write c= ^ R' '-R, Now, if R, be greater than R by an infinitesimal, the fraction R "H" is less than unity by an infinitesimal, and the capacity of ■«■/ the accumulator is infinitely great. It becomes infinitely small if R be diminished without limit. The presence of any finite charge at a point would require an infinite potential at that point, which is of course impossible. The existence of finite charges concentrated at points, which we have assumed some- times in order to more conveniently state certain laws, is therefore purely' imaginary. If electricity is distributed in space, it is distributed like a fluid, a finite quantity of which never exists at a point. If R^ increase without limit, C becomes more and more nearly equal to R. Suppose the inner sphere to be surrounded not by the outer sphere but by conductors disposed at unequal distances, the nearest of which is still at a distance R^ so great TO that ^ may be neglected in comparison with unity. Then if the nearest conductor were a portion of a sphere of radius Ri concentric with the inner sphere, the capacity of the inner sphere would be approximately R. And this capacity is evi- dently not less than that which would' be due to any arrange- 26o ELEMENTARY PHYSICS. [194 ment of conductors at distances more remote than R^. There- fore the capacity of a sphere removed from other conductors by distances very great in comparison -with the radius of the sphere is equal to its.. radius i?. -This value R is often called the capacity of 2, freely electrified sphere. Strictly speaking, a freely ekctrified conductor cannot exist ; the term is, however, a convenient one to represent a conductor remote frorp all other conductors. A common form of condenser consists of two flat conduct- ing disks of equal area; placed parallel and opposite one another. The capacity of such a condenser may be calculated from the capacity of the spherical condenser already discusged. Let d represeht the distance R^ — i? between the two-spherical surfaces. Let A and A^ represent the area, of the surfaces of the two spheres of radius R and R,. Then we have K.= — and ie; = — '. The capacity of the spherical condenser may then be written \fAA, 47td If R^ and R increase indefinitely, in such a manner that R^ — R always equals d, in the limit the surfaces become plane and A becomes equal to A^. The capacity therefore equals — -5. Since the charge is uniformly distributed, the capacity of any portion of the surface cut out of the sphere is proportional to the area .S of that surface, or ' I9S] ELECl^RICITY IN EQUILIBRIUM. 261 This value is obtained on the assumption that the distribution over the whole disk is uniform, and the irregular distribution at the edges of the disk is neglected. It is therefore only an approximation to the true capacity of such a condenser. The so-called Leyden jar is the most usual form of con- denser in practical use. It is a glass jar coated with tinfoil within and without, up to a short distance from the opening. ' Through the stopper of the jar is passed a metallic rod fur- nished with a knob on the outside and in conducting contact with the inner coating of the jar. To charge the jar, the outer coating is put in conducting contact with the ground, and the knob brought in contact with some source of electrification. It is discharged when the two coatings are brought in conduct- ing contact. When the wall of the jar is very thin in compari- son with the diameter and with the height of the tinfoil coat- ing, the capacity of the jar may be inferred from the preceding propositions. It is approximately proportional directly to the coated surface, to the specific inductive capacity of the glass, and inversely tt) the thickness of the wall. 195, Systems of Conductors. — If the capacities and poten- tials of two or more conductors be known, the potential of the system formed by joining them together by conductors is easily found. It is assumed that the connecting conductors are fine wires, the capacities of which may be neglected. Then the charges of the respective bodies may be represented by Cj^^, CjK, . . . C„V„, and the capacity of the system by the sum C^ _|_ C", + • • • ^»- Hence F„ the potential after connections have been made, is In the case of two freely electrified spheres joined up 262 ELEMENTARY PHYSICS. [igfi together by a fine wire, we have C, = 7?„ and C, = R^, where R^ and R^ represent the radii of the spheres. Hence we have „_ R.V.^R.V. *"- R^^R, ' When /?, is very great compared with ^3, we obtain ' ■p Unless Fj is so great that the term „" f^2 becomes appreci- able, the potential of the system is appreciably equal to the original potential of the larger sphere. Manifestly the same result follows if R^ represent the capacity of any conductor relatively small compared with the capacity of the large sphere. This proposition justifies the adoption of the potential of the earth as the standard or zero potential. 196. Energy of Charge. — In order to find the work done in charging a conducting body to a given potential, we will consider all surrounding bodies as being kept by ground con- nections at zero potential. Then if an infinitesimal charge be given to the body, previously uncharged and at zero potential, the work done is that which would be done if the charge were brought from infinity to a point of potential o ; that is, the work = o. The charge q raises the potential of the body so that it q becomes v^ = j\. If then another infinitesimal charge q be given to the body, the work done is equal to qv^ or |v, and the potential is raised to v^ ~ ~, So also the work done when 197] ELECTRICITY IN EQUILIBRIUM. 263 the («-f-«)th charge is given to the body is qvn, and the (ft I ,., * \q potential becomes ^ T. . The total work done is then W= ?(2'i + 2', +...«'«) = 1^(1 + 2 + .. . n) = ^-^•^ = 10^, (82) where nq= Q and V— v„. When the charges q are infinitesi- mal, 2 is equal to the sum of all the charges given to the bbdy. Hence the work done in raising a body from zero potential to potential V is. equal to one half the charge multiplied by the potential of the body. 197. Strain in the Dielectric. — An instructive experiment illustrating Faraday's theory that the electrification of a con- ductor is due to an arrangement in the dielectric surrounding it, may be performed with a jar so constructed that both coat- ings can be removed from it. If the jar be charged, the coat- ings removed by insulating handles without discharging the jar, and examined, they will be found to be almost without charge. If they be replaced, the jar will be found to be charged as before. The jar will also be found to be charged if new coatings similar to those removed be put in their place. This result shows that the true seat of the charge is in the dielectric. The experiment is due to Franklin. That the arrangement in the dielectric is of the nature of a strain is rendered probable by the fact, first noticed 'by Volta, that the volume occupied by a Leyden jar increases slightly when the jar is charged. Similar changes of volume were ob- served by Quincke in fluid dielectrics as well as in different solids. Another proof of the strained condition of dielectrics' is found in their optical relations. It was discovered by Kerr 264 ELEMENTARY PHYSICS. [197 that dielectrics previously homogeneous become ' doubly re- fracting when subjected to a powerful electrical stress. Max- well has shown, from the assumptions of his' electromagnetic theory of light, that the index of refraction of a transparent dielectric should be proportional to the square root of its specific inductive capacity. Numerous experiments, among which those of Boltzmann on gases are the most striking, show that this predicted relation is very close to the truth. It has further been shown that the specific inductive capac- ity of sulphur has different values along its three crystallo- graphic axes. This is- probably, true also for other crystals. Some crystals, while being warmed, exhibit on their faces positive and negative electrifications, which are reversed as the crystals are cooling. This fact, while as yet unexplained, is probably due to temporary modifications of molecular ar- rangement by heat. If a jar be discharged and allowed to stand for a while, a second discharge can be obtained from it. By similar treat- ment several such discharges can be obtained in succession.' The charge which the jar possesses after the first discharge is called the residual charge. It does not attain its maximum immediately, but gradually, after the first discharge. The attainment of the maximum is hastened by tapping on the wall of the jar. This phenomenon was ascribed by Faraday to an absorption of electricity by the dielectric, but this explana- tion is at variance with Faraday's own theory of electrification. Maxwell explains it by assuming that want of homogeneity in the dielectric admits of the production of induced electrifica- tions at the surfaces of separation between the non-homogene- ous portions. When the jar is discharged the induced electri- fications within the dielectric tend to reunite, but, owing to the want of conductivity in the dielectric, the reunion is gradual. After a sufficient time has elapsed, the alteration of the electrical state of the dielectric has proceeded so far as to 198] ELECTRICITY IN EQUILIBRIUM. 265 sensibly modify the field outside the dielectric. The residual charge then appears in the jar. 198. Electroscopes and Electrometers. — An electroscope is an instrument to detect the existence of a difference of electri- cal potential. It may also give indications of the amount of difference. It consists of an arrangement of some light body or bodies, such as a pith ball suspended by a silk thread, or a pair of parallel strips of gold-foil, which may be brought' near or in contact with the body to be tested. The movements of the light bodies indicate the existence, nature, and to some extent the amount of the potential difference between the body tested and surrounding bodies. An electrometer is an apparatus which gives precise measure- ments of differences of potential. The most important form is the absolute or attracted disk electrometer, originally devised by Harris, and improved by Thomson. The essential portions of the instrument (Fig. 59) are a large fiat disk B which can be put in con- c ducting contact with one of the two '■ g bodies between which the difference of fig. 59. potential is desired ; a similar disk C, in the centre of which is cut a circular opening, placed parallel to and- a little distance above the former one ; a smaller disk A with a diameter a little less than that of the opening, which can be placed accurately in the opening and brought plane with the larger disk; and an arrangement, either a balance arm or a spring of knowh strength, from which the small disk is suspended, and by means of which the force acting on the disk when it is plane with the surface of the larger disk can be measured. The three disks can be conveniently styled the attracting disk, the guard ring, and the attracted disk. The position of the at- tracted disk when it is in the plane of the guard ring is often called \hs. sighted position. The guard ring is employed' in order that the distribution on the attracted disk may be uniform. 266 ELEMENTARY PHYSICS. [19& To determine the difference of potential between the at- tracted and attracting disks, we consider them first as forming a flat condenser. If we represent by Q the quantity of eleg^ tricity on the attracted disk, hy V and V^ the potentials of the , attracted and attracting disks respectively, by d the, distance between them, and by S the area of the attracted disk, then, as has been shown in § 194, the capacity of such a condenser is Q V,-V~ iptd' Now from the nature of the condenser, and in consequence of the regular distribution due to the presence of the guard ring, we have „ = cr, the surface density on either plate, whence V — V cr = — ^ — -j—. The surface density cannot be measured, and must be eliminated by means of an equation obtained by ob- servation of the force with which the two disks are attracted. The plates are never far apart, and the force on a unit charge due to the charge on the lower one may be always taken in the space between the plates as equal to 2;ro- (§ 191). Every unit on the attracted disk is attracted with this force, and the total attraction, which is measured by means of the balance or spring, is F= 2ncfS. Substituting this value of in terms which are all measurable in absolute units. In Thom-) 198] ELECTRICITY- IN EQlMiSBJilUM, 267 son's form of the electrometer the attracted disk is leept at a. high constant potential V; the attracting disk is brought to the potential Vi of one of the two bodies of which the differ- ence of potential is desired, and the position of the attracting disk when the attracted disk is in its sighted position is noted. The attracting disk is then brought to the potential V^ of the other body, and by a micrometer screw the distance is measured through which the attracting disk is moved in order to bring the attracted disk again into its sighted position. This meas- urement can be made with much greater precision than the measurement of the distance between the two plates. The formula is easily deduced from the one already given. In the first observation we have ii^ the second,' F,-r=V^; K-v=dy^; whence 1/ — 77- — (A — //NvL- K-V, = {d,-d,)\/^, (84) and d^ — d^ is the distance measured. Thomson's quadrant electrometer is an instrument which is not used for absolute measurement, but being extremely sensi- tive to minute differences of potential, it enables us to compare them with each other and with some known standard. The conistruction of the apparatus can best be understood from 268 ELEMENTARY PHYSICS. [199 Fig. 60. Fig. 60. Of the four metallic quadrants which are mounted ■ on insulating supports, the two marked P and the two marked iVare respectively in '■ conducting contact by means of wires. The body C, technically called the needle,' is a thin sheet of metal, suspended sym- metrically just above the quadrants by two parallel silk fibres, forming what is known as a bifilar suspension. When there ' is no charge in the apparatus, the axes, of symmetry of the needle lie above the spaces which separate the quadrants. To use the apparatus, the needle is maintained at a high, constant potential, and the two points, the difference of poten- tial between which is desired, are joined to the pairs of quad- rants P and N. The needle is deflected from its normal posi- tion, and the amount of deflection is an indication of the ' difference of potential between the two pairs of quadrants. 199. Electrical Machines. — Electrical machines may be divided into two classes : those which depend for their opera'' tion upon friction, and those which depend upon induction. The frictional machine, in one of its forms, consists of a circular glass plate, mounted so thaf it can be turned about an axis, and a rubber of leather, coated with a metal amalganiy ' pressed against it. The rubber is mounted on an insulatirig support, but, during the operation of the machine, it is usually joined to ground.^ Diametrically opposite ig placed a row of metal points, fixed in a metallic support, constituting what is ' technically called the comb. The comb is usually joined to an accessory part of the machine presenting an extended metallic ' surface, called the prime conductor. The prime conductor is carried on an insulating support. When the plate is turned, an electrical Reparation is pro- duced by the friction of the rubber, and the rubbed portion of' ' the plate is charged positively. When the charged portion of* 199] ELECTRICITY IN EQUILIBRIUM. 269 the plate passes before the comb, an electrical separation oc- curs in the prime conductor due to the inductive action of the plate, a negative charge passes from the comb to neutralize the positive charge of the plate, and the prime conductor is charged positively. Since accessions are received to the charge of the prime conductor as each portion of the plate passes the comb, it is evident that the potential of the prime conductor will continuously rise, until it is the same as that of the plate, or until a discharge takes place. The fundamental operations of all induction machines are presented by the action of the ilectrophorus, an instrument in- vented by Volta in 177 1. It consists of a plate of sulphur or rubber, which rests on a metallic plate, and a metallic disk mounted on an insulating handle. The sulphur is electrified negatively by friction, and the disk, placed upon it and joined to ground, is charged positively by induction. When the ground connection is broken and the disk lifted from the sulphur, its positive charge becomes available. The process is precisely similar to that described in §191. It may evidently be repeated indefinitely, and the electrophorus may be used as apermanent source of electricity. It is evident that a charged metallic plate may be substi- tuted for the sulphur in tlie construction of an electropho- rus, provided that the disk be not brought in contact with it, but only near it. A plan by which this is realized, and at the same time an imperceptible charge on one plate is made to develop an indefinite quantity of electricity of high potential, is shown in Fig. 61. A^ and , Fig. si. A^ are conducting plates, called inductors. In front of them 270 ELEMENTARY PHYSICS. [199 two disks B^ and B^, called carriers, are mounted on an arm so as to turn about the axis E. Projecting springs b^ and b^ at tached to these disks are so fixed as to touch successively the pins Z>, and Z>„ connected with the plates A^ and A^, and the pins C"i 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 Z?, and C^, and suppose the plate ^1 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 ^4, andyi, 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 ipiprovement 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. 271 ^(. — ~^~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 "^vhich they terminate serve the same purpo'se 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 1 79 1 Galvani of Bologna published an account of some experiments made two years before, which opened a new de- partment of electrical scien,ce. He showed that,, if the lumbaf 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 agitaited 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 the 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 20o] 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 solution 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 sevferal 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 the same as that of a uniform 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 ELEMENTAR Y 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 t^ie 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- i taic battery maintains a permanent difference of potential. 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 \E\ = M^L^T~ ', 202] THE ELECTRICAL CURRENT. 275 202. Electrostatic Unit of Current.^— Let us denote the potentials ^t the two points i and 2 in the circuit by V^ and F",, and let Fj be greater than F, ; then if, in the time t, 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 QiV^ — 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 inheating 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 <2(f;-fo = ^a or H=^{y,-v^. The transfer of electricity in the circuit is called the electrical current, and the rate of transfer — = Z 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 utiit of current. The system of units here used is the electrostatic system. The dimensions of current strength in the electro- 276 ELEMENTAR Y PHYSICS. [203 static system are obtained from the equation above. We have [/]=-—= M^L^T-^, 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, Ohiiis law is expressed in its sim- plest form by IR=V,- V,. (86) The quantity R is called the resistance of 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 these 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 m = \^—~\ = L-'T. 203] THE ELECTRICAL CUkRENT. 277 To generalize Ohm's law for the whole circuit, let us con. aider a special circuit which may serve as a type. It shall consist 'o^ a voltaic cell contain- ing acidulated water, in which are immersed a zinc and a platinum plate, pined 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 is Vz-\- Z/L, \i 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 Vl^ and t)y the elementary form of Ohm's law, already con- sidered we have Fig. 62. / = Vz + Z/L - Vl 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+ F/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 j^Z/L_±L/P±P/Z^ Rl -\- Rp -\- Rz 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 /=§. ■ (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 proper- tional to. its cross-section and inversely as its lengtli. 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 points at potentials V^ and V^, Ohm's law for this special case can 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 = I{V^ — V^) by the help of Ohm's law, we obtain H^PR. (89) The heat developed in a homogeneous portion of any cir- cuit is equal to the square of the current in the circuit multi- pHed 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 iby chemical action, or by the Peltier effect, occurs at non-hqmogeneous 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 FRt by Joule's law, and partly as other work, which in every case is proportional to /, and can be set equal to I A, where A is a factor which varies with the particular work done. Then we have lEt — I'Rt -\- lA, whence R . ' (90)' 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 thait of the original circuit. If then, in a circuit containing no impressed electromotive force, or in which £ = 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 ■ -^ 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 RELATIOJTS 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 ao7] CHEMICAL RELATIONS OP 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 cathpde 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 Tyater 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. [20? 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 tlie 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. 28$ 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 ivoltameter 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 currents The second form depends for its indications on the evolu tion of gas, the volume of whith is measured. The water vol- tameter is a type, and is the form especially used. The gases evolved are either collected together, or t'he hydrogen alone is collected. The latter is preferable, because oxygen is more easily absorbed by water than hydrogen and an error is thus introduced when the oxygen ig 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 d the heat evolved by the combination of a 286 ELEMENTARY PHYSICS. \pill unit mass of this ion with aa 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 le will represent the number of electro-chemical equivalents evolved, and led will represent the energy expended, which appears as chemical sepa- ration and mechanical work. This is equal to I A ; 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 tirne t, so that equal quantities of the ion considered are evolved in equal ' A ed ^^ A , •' times, we have — = — JN ow, — represents the counter-elecr tromotive force set up in the circuit by electrolysis. Hence the electromotive force set up irf the electrolytic process may be measured in terms of heat units ; or, since these heat units are measures of chemical afifinity, 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 B. 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 B. , 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 212] 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 difficulties are met with at once. The foundation of all the present theories is found m 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 this 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 solu'tiori 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 towardsi 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 OP 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 A • ' than -T-, 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 riot 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, then, 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, liquid 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 Daniell's 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 B, 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 1.95 • 10° C. G. S. electromagnetic units. The secondary cell of Planti is an example of a cell made directly by electrolysis, as has been assumed in the preliminary discussion. The electrodes are both lead plates, and the elec- 214] CHEMICAL RELATIONS OF THE CURRENT. 293 trolyte dilute sulphuric acid. When a current is passed through the cell, the oxygen evolved on the anode combines with the lead to form peroxide of lead, which coats the surface of the electrode. When the cell is inserted in a circuit, a cur- rent is set up, the peroxide is reduced to a lower oxide, and the metallic lead of the other plate is oxidized. The Latimer-Clarke standard cell is of great value as a standard of electromotive force. As it polarizes at once if a current pass, through it, it should never be joined up in a closed circuit. The positive pole consists of pure mercury, which is covered by a paste made by boiling mercurous sul- phate in a saturated solution of zinc sulphate. The negative pole consists of pure zinc resting on the paste. Contact with the mercury is made by means of a platinum wire. As no gases are generated, this cell rtay be hermetically sealed against atmospheric influences. Accoriiing to the measurements of Rayleigh,the electromotive force of this cell, is very constantly 1.43s • 10' C. G. S. electromagnetic units at 15° Cent. 214. Theories of the Electromotive Force of the Voltaic Cell. — The plan followed in the preceding discussions has rendered it unnecessary for us to adopt any theory to ex- plain the cause of the electromotive force of the voltaic cell. The different theories which have been advanced may . he ■classed under one of two general theories, the contact theory and the chemical theory. On the contact theory, as advanced by Volta and supported by Thomson and others, the difference of potential which exists between two heterogeneous substances in contact is due to molecular interactions across the surface of contact, or, as it is commonly stated, is due merely to the contact. The chemical theory, as advodated by Faraday and Schonbein, holds that the difference of potential considered cannot arise unless chemical action or a tendency to chemical action exist at the surface of contact. Numerous experiments have shown that the sum of all the 294 ELEMENTARY PHYSICS. [215 differences of potential at the surfaces of contact of the various substances making up any voltaic cell is equal to the electro- motive force of that cell. This is true even when the cell is formed solely of liquid elements. On the contact theory this electromotive force is due merely to the several contacts, while the chemical actions of the cell begin only when the circuit is made, and supply the energy for the maintenance of the cur- rent. The quantity of heat produced at a junction of dis- similar substances by the passage of a current (§ 233) is such as to show, however, that the differences of potential thus measured are not the true differences of potential due to the contact of the substances tested, but must depend in part upon the action of the air or other medium by which these substances are surrounded. The supporters of the chemical theory point to this fact as evidence that the chemical action of the medium is concerned in the production of the difference of potential observed. On either theory it is clear that the energy maintaining the current must have its origin in the chemical actions which go on in the voltaic cell. 215. Capillary Electrometer. — It has been stated that a difference of potential exists between a metal and a fluid elec- trolyte in contact with it. There will then exist on the sur- faces of the metal and the . electrolyte in contact with it such an electrical distribution as exists in a charged condenser of which the plates are very near together. One arrangement by which the effects due to this distribu- tion may be observed was devised by Lippmann. It consists of a vertical glass tube, drawn out at its lower end in a capil- lary tube. The capillary tube dips into dilute sulphuric acid, which rests on mercury in the bottom of the vessel containing it. Mercury is poured into the vertical tube until its pressure is such that the capillary portion of the tube is nearly filled with it. When the mercury in the vessel is joined with the 21S] CHEMICAL RELATIONS OF THE CURRENT. 295 positive pole of a voltaic cell, and that in the tube with the negative pole, the meniscus in the capillary tube moves up- ward, in the sense in which it would move if its surface tension were increased. This movement may best be explained by the help of the theory of electrolysis given by Clausius (§ 212). So long as there exists an electromotive force in the circuit, posi- tive and negative ions will be released on their respective elec- trodes. If we assume that they are associated with the trans- fer of electricity in the circuit in such a way that it is trans- ferred from th,em to the electrodes, such a movement of the ions would give rise to a modification of the distribution on the surfaces of contact. In the case now under consideration the charge on the meniscus is in part neutralized by the charge transferred with one of the ions.' The true surface tension of the surface of separation between the mercury and the liquid is, on ^ this theory, lessened by the presence of the electrical charge on the surfaces of contact, owing to the interaction of the parts of the charge in a manner similar to that described in § 191. If, therefore, any dirriinution of this charge occur, a seeming increase of the surface tension will be observed. On this theory the true surface tension of the surface of separation is the value observed when the mercury and liquid are at the same potentials^ and this value is a maximum. The experi- ments of A. Konig and Helmholtz show that such a maximuni value exists in a manner consistent with the theory. The arrangement described can manifestly be used to pro- duce the effects just discussed only when the electromotive force introduced into the circuit is less than that required to cause active decomposition of the electrolyte. If any suitable electromotive force be introduced into the circuit, the theory here given assumes that the transfer of the ions goes on until the diJEferences of potential on the surfaces of contact are such as to counterbalance the introduced electromotive force. The' mercury column then comes to rest. 296 ELEMENTARY PHYSICS. [215 Lippmann constructed an apparatus similar to the one de- scribed, with the addition of an arrangement by which pressure can be applied to force the end of the mercury column in the capillary tube back to the fixed position which it occupies when no electromotive force is introduced into the circuit. He found that when small electromotive forces were introduced, the pressures required to bring the end of the column back to the fixed position were proportional to the electromotive forces. He hence called this apparatus a capillary electrometer. Lippmann also found that if the area of the surface of separation between the mercury and the liquid in the capillary tube were altered by increasing the pressure and driving the mercury down the tube, a current was set up in a galvanome- ter inserted in the circuit, in . a sense opposite to that which would change the area of the 'meniscus back to its original amount. CHAPTER V. MAGNETIC RELATIONS OF THE CURRENT. 2l6. Biot's Law. — Very soon after tlie discovery by Oer- sted of the fact that a magnet was acted upon by an electrical ^:urrent brought near it, Biot and Savart instituted a series of -experiments to determine the law of the force between a mag- net and a current. They suspended a short magnet by a silk fibre, and so modified the earth's magnetic field near it, by means of magnets, that the suspended magnet pointed in any azimuth with equal freedom. A current was then passed through a long vertical wire near the magnet. It was observed that the magnet placed itself so that its poles were equally •distant from the wire. The directions of the north pole and of the current were related as the motions of rotation and of pro- pulsion in a right-handed screw. Then the magnet was set in oscillation, and the times of oscillation determined, when the -current was at different distances from the magnet, and when ■different currents were set up in the wire. From the first ob- servation it follows that the force exerted between a magnet pole and a current is normal to the plane passing through the n s ■ \ \ c Fig. 63. / / 'Current and the magnet pole. For, suppose the current rising vertically out of the paper at C (Fig. 63), and suppose that it 298 ELEMENTARY PHYSICS. [216. acts on the north pole of the magnet ns with a force repre- sented by na, making any angle with the line nC. It is as- sumed as probable that the force on a south pole placed at « would be oppositely directed to na. The angle which the force sb, acting on the south pole. J, makes with the line sC will then be jr — 0. Now the magnet is in equilibrium, hence the moments of the components of these forces at right angles to the magnet must be equal. The components are respectively na sin (^ — 0) and sb sin (ip — {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 (rp — {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 , , /= 11 • (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 , bet-ween 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. 2X6] MAGNETIC RELATIONS OF THE CURRENT. 299 the current, is consistent with this law. For siniplicity we will consider an infinitely long straight current. Let the magnet pole m be at the point /"(Fig. 64). Let QR be the current Fig. 64. 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 ^^y^ • -^. In the limit, as QR becomes indefinitely small, the triangles QRS and POR become similar. Hence QS equals — -^ — , and the ex- miQS PR pression for the force becomes —ppr- 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 /'{/equals -5H- Using these values, the expression „ be- comes -pTy- There will be a similar expression for the force due to any other element. The total force due to the whole current will be equal to the constant factor -^Ty. multiplied by 300 ELEMENTARY PHYSICS. [217 the sum of all the projections corresponding to b, for the infinite current, is manifestly 2PU = 2PO. 2mi total force is -575 ; or, it is inversely as the distance PO be This sum, Hence the PO 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- aj >"-"-!»= 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 2s, 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 force on it due 217] MAGNETIC RELATIONS OF THE CURRENT. 3OI to one side, 2s, is, as stated in Eq. (90), proportional to the 'length 2j, 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. ' c^" + (■^ — ^yy' This sine is expressed by }^-n — T'T~r ~K¥I- The total force ^ ' \x -\-y -\- (z — s)f , , . , mi2s(x' -{-{z — j')')* ^, . , ; due to the element is then , , , ~ , . ^r^. 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 forcq by , ,. j _ ,y.y The expression for the component along the x axis then becomes 2mis{z —s') . {^-}-f + {^-s')T We will expand (z — s')' in the denominator, reject the term /", remembering that the sides are indefinitely small, and write for brevity ^tr" +y + ■2^" = ^"^ We then have this component expressed by -r^ 7^- Similar expressions hold for the components due to each of the other sides, with the difference 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 s', rejecting all terms containing the second or higher powers of s. 302 ELEMENTARY PHYSICS. [217 2mis(z ^ s') ., ,^. , , ^ ~ {/' - 2zsy ^ ~ ^^"^^ ~ '^^^^ ~ ' + 3-f ^^ - ; , 2mzs(z-]-s') , .,,,., , V + (r' + 2^/)t "= + ^^"^^ + -^ )^^~ ' " ^^^'' ~ '^ ' 2mis'{y—s) ... ., , \ - (r' - 2j ;^ ^ ~ 2M«/(7 - j)(r-3 + 3Jj/r-s) ; , 2mis'{y-\-s) , ■ ,, , ^, . * + (r' + 2jj)J "" + ^'^"^■^ + '^^'' ~ ' ~ ^'y'' ~ '^■ 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- /jy' + 3^" 2 \ pression — /^iss'^ ; ,j. The term in parenthesis 3/ + 32''' — 2^" r' — ■^x' ^, can be written 1 = ^—- The factor dss' is r r equal to a, the area of the rectangle. The force along the x axis is then finally 4-^) ^^""x? - -^ ' (92) For the component along the y axis we have to consider only the forces due to the sides 2s', 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 which is in the positive direction of y, are 4- 2mzs'-r-„ rr ■ (r=-2^j)4 ,v and — 2mis' , ,. .^ . The sum of these components is 4mtss'-^-;- = mta—r. (gza) , 817] MAGNETIC RELATIONS OF THE CURRENT. 303 Similarly the total component along the z axis is mta-^. (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 is _/'«?. In that discussion the con- vention was made that the positive face of the shell was turned toward the positive direction of the x axis. We then have oa, the solid angle subtended by the shell as seen from the point P, equal to a cos 6 ax x ' = -, = a r' ~ r" -"(j;»+/ + ^»)r The potential at the point P is then X. ^~^\x'^f^^Y 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 Ax ; then the potential will take a small increment A V. We will have JT ^ AIT ■ x-\-Ax \{x + Axr^f + z')i' and as Ax becomes indefinitely small, V4- AV=ja , , I — -j-rs. ' (^ + 2xAx)i 304 ELEMENTARY PHYSICS. , [ziy Expanding this expression, rejecting all terms containing the second or higher powers of Ax, we obtain From this we have further AV .(I 3^ Ax -J'^\i^~ r*> In the limit, as Ax becomes indefinitely small, this is the rate of change of potential along the x axis at the point {xyi). 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 — — ^ ). Similarly the forces along the ^ and z axes can be found to be respectively mja — ^ 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 Isut 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 DELATIONS OF THE CURRENT. 305 alent circuit is such that its direction is related to the positive direction of the lines 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 j\ 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. 66. 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 under the influence of the circuit equivalent to the mag- netic shell, it would move, as in the case of the shell, along the line of force from the positive to the negative face of the circuit, and in so doing would do work equal to ^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 infinitesimal 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 ^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 CO, 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 /\,nni , so that the total work done, or the total difference of potential between the two points, is V,-V= i{Go, -00 + 4nn), (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 12I8] MAGNETIC RELATIONS OF THE CURRENT. 3O7 opposite side dips in mercury contained in a circular trough 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 notations 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 proved as fol- lows. Conceive a circular plane magnetic shell of strength/ to be set up normal to the x axis, with its centre at the origin. Then at the centre ;tr = o, and from §217 the component of force along 'PPl'JfT the X axis due to each element of the shell is numerically-^. Now divide the circle up into any very great number of circular rings by striking circles about its centre with radii differing by the small distance d. The elementary areas making up any one of these rings are all at the same distance from the centre, 308 ELEMENTARY PHYSICS. [2l5 and the force along the x axis due to the whole circle can be found by summing the areas of the rings-divided by the cubes of their respective radii. Select the ring the radius of which is one half that of the circle, and call that radius /. The radii of the rings distant nd from this middle one are l-\-nd and / — nd, and the areas of these rings are 2nd{J --f- nd^ and 27td{l — nd). The forces due to them are 75-; — ^Ca and ^ ' {l-{-nd) ZTtmJd _., . , .fd 2nd'\ jj -jy- These expressions are equal to 2itmj \j.^ — —tt') Id 2nd'\ and 27rm/\ji-{-—j^j if we neglect the higher powers of d.. \itmid I The force due to both rings is — j^ — . As there are -7 such pairs, the force due to the whole shell is — -, — . The force due to a circuit equivalent to the magnetic shell is — -, — . Since / = -, where R is the radius of the circle, the force along the 2ni'in X axis equals — ^— . If we adopt the second definition of unit current, arid use Biot's formula for the action of a current 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 ^^55. The sum of all the elements of the circle is 2nR. Hence the force on this definition is also — 5 — . K 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 systen^. Zig] MAGNETIC RELATIONS OF THE CURRENT. 309 In practice another unit of current, called the ampere, is used. It is equal to io~^ 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^L^ T~^. 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 lines 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 soHd 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 ma,ny 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 thd 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 th6 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 Hnes. 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 fe- 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 tTiey be placed so that unlike faces are opposed, they will move towards one 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- 221] MAGJSTETIC RELATIONS OF THE CURRENT. 31I circuit, be brought near one of the vertical sides of the square, that side will move towards tfie 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 opp'osite 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 |p 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 1 1 1 1 frame by the points a and b, upon which the ^'°' ^'' frame is supported. It is evident that the two halves ol the: frame tend to face in opposite directions in the earth's mag- 312 ELEMENTARY PHYSICS. [221 netic field, so that there is no tendency of the frame as a whole to face in any one direction ratfier than any other. If a long straight wire be placed near to one of the extreme vertical sides of tlje 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 Ampfere'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 a,n 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 321] 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 ^,^, 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 m^de 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 attractiori between two current elements. ' It is ii' ds ds' [t, cos e — 3 cos B cos G'\. (94) 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 electro-magnetic units, r the distance between the cur- rent elements, e the angles made by the two elements with one another, Q and Q' the angles made by ds and ds' with r ox 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 d =■ 6' ■=. o\ therefore the force given by the equation is — 5 — _ Since this is negative it expresses a repulsion. 222. Solenoids and Electromagnets. — Ampere 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] MAGNETIC RELATIONS OF THE CURRENT. 3IS 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. The direction of the per- manent current and its amount differ under the same circum- stances for different metals. The coefiScient which represents the amount of the Hall effect in any metal is called th6 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 d^ue 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 tipon 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 rlegative 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 thercr fore called an astatic system. When 9, current passes in the wire, however, the lines of force due to the current form closed curves psissing through the coil, and both needles tend to turn in the same direction. Since the earth's field offers almost np resistance to this tendency, an astatic system will indicate the presence of very feeble currents. The apparatus h^re 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 nee,dle 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 Hml sm 0, where H represents the horizon- tal, component, ml the magnetic moment of the magnet, and the angle made by the plane of the coils with the magnetic meridian. The couple due to the current is, by Biot's law, proportional to the current. It may then be set equal to kfnil, where /^ is a constant factor depending upon the dimensions of the galvanometer. Since these two couples are equal, we have the equation Fig. 68. i = -T" sm 0, (95) With the same galvanometer, then, different currents are pro- 3l8 ELEMENTARY PHYSICS. [225 portional to the sines of the angles made with the magnetic meridian by the plane of the coils when the needle lies in that plane. If i be greater than -r-, the equilibrium supposed in this explanation cannot occur. The tangent galvanometer is that form of galvanometer which is commonly used to measure current in electromagnetic units. It can best be discussed by considering first the action of a single circular current of strength i upon a magnet pole situated at any point on the normal to the plane of the circle drawn from its centre. The force due to any current element j (Fig. 6^ upon the magnet pole m, at the distance /, is, by Biot's law, -j^. This force tends to move the pole m. at right angles to the plane containing s and the line joining s and m. If we repre- ^ sent by 6* the angle between the line k joining s and m and the x axis, the com- \ ^ . tnis \ '■ 4 ponents of this force become —j^ sin 6 I ^^ • I ^ s fftZS •-i-"-f > im along the x axis, and -r-^- cos d normal to \ I \l ^^^ ^ ^^^^' "^^^ equal element S', dia- \ / " metrically opposite s, also gives rise to V_>' mis' . „ , ^""^ Fig. 69. *^° components, —r^ sm 6 along the x axis, which is added to the similar component due to s, mis' and -j7- cos 6 normal to the x axis, which is opposed to and annuls the similar component due to s. Every other similar pair of elements will give rise to two similar components along the X axis, and will annul one another's action normal to the X axis. The total force on m will then be a force ajong the X axis equal to the sum of all the components along that axis. 325] MAGNETIC RELATIONS OF THE CURRENT. 3I9 fPtZS '2,7tfilt,T ■or -S-TT sin B. This equal^ — j^ — sin ^, where r is the radius •of the circle. Since j- = sin 0, this force may be written 2Ttmir' ZTtmir' {x^ + ry If the circular coil considered be set vertical in the plane -of the magnetic meridian, and a short magnetic needle be , -.mounted at the point m, so as to turn in the horizontal plane, the needle will be deflected from the meridian, and will rest in equilibrium between the force due to the current and that due to the earth's magnetism. If the needle be so short that the dis- tance of its poles from the x axis may be neglected, the formula just obtained will give the force upon its poles. Let / repre- sent the half length of the needle (Fig. 70), its angle of ■deviation from the magnetic meridian, and d the •distance from its centre to the plane of the coil. Then ^ — / sin and ' d -\- / sin represent the •distances of the magnet's poles from the plane of the coil. The . forces acting on these poles are then ,(^+(^_/sin0y)* ^""^ (75+(^+/sin0)')r F.c,,p. H another precisely similar 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 \Tcmit^l cos , Anmir^l cos and (r» + (rf - / sin 07)^ """ (^" + (^+/sin0)=)l' 320 ELEMENTARY PHYSICS. [225: both tending to turn the magnet in the same direction. The. ^^^t°'^^r"+(^±/sin0)=)t are equal to {f _|i ^») - 5 q: ^{f + ^?) - « (2^/sin ± /' sin' ) + V-(^' + fl?")-*^7»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, Syrmir'/ cos

. tan 0. . (c7) 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 iixed 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 bifilar 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 inovement 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 magndt, would assist the actual motion of the mag- net. This current is called an induced current. From the 322 ELEMENTARY PHYSICS. \ii2.6 equivalence 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 ^.ny 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 {(oa^ — 00 -\- 47in), and the work done on a magnet pole m, in moving it from one point • to another, is mi{w^ — ta -f- /\.nri). In the demonstration of § 206 we may substitute »z((»^—03-|-4zr«) for A, and, provided the change in potential be uniform, we , . ' • mioo, — w -\- 47tn) , 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(00, — Q0-\-A7tTl) form. Then the limit of the expression ^^^ ■ -, as t becomes indefinitely small, is the electromotive force during that interval. The current strength due to this electromotive force is . _ m{oo^ — (» + 4'F«) 226] ' MAGNETIC SELATl'oNS OF THE CURRENT. 323 If the induced current be steady, the total quantity of electricity flowing in the circuit is expressed by 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, dependsj 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 § 219. 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 circuity 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. ' [23(S moving part of the circuit, if it be supposed to move normal to the lines of force, is related to the direction of motioq 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 z, 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 is 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 fact 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 1831. 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 withdrawh 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 propulsioh 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 currerlts 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 327] 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 Lena'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 s,econdary circuit is independent of the materials consti- tuting either circuit, rfnd 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 electromor 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. 326 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 maa^ to one where it is moo, 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- rent is —'-- -. If the movement take place m unit time, and if m {go^ — go) also equal unity, the electromotive force in the circuit is defined to be imii electromotive force. The expression m {w^ — w) 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 hnes 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 -=5 = [^] ; whence \n\ = Mi Z* T~^. Since the electromo- tive force is measured by the rate of change of the number of lines of force we have [^] = -:= = M^Li 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 magrietic 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 movements of 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 elementarV 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. Whetl 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-eiectrical 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 apoweirful 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 magn,etSj' 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 thfe electro- 830] -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. T,he 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 circuitj 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 ciirrent 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 th^at 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 tha,t 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 currertt in the galvanometer. If it move with 33° 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 ohn. 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 th.Ms contains 10 C.G.S. elec- tromagnetic units. The dimensions of resistance in the elec- tromagnetic system are [r\= L-^ J = LT~\ 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 temperance of freezing." The legal ohm contains 1.0112 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 between 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 coiiventions 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 aU the currents rneeting 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 suin, taken around any number of branches forming a closed circuit, ©f 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 33^ • ' 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 showji in Fig. 71..J In the branch 6 is a voltaic cell with an electromotive force .£. 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 Kirchhofi's first law the sum of the currents meeting at the point C is i^ + h + K = o, and of those meeting at the point D is i^ + i^ -\- \ = o. By the second law, the sum of the products zr in the circuit ABC is i^r^ -\- i,r^ -|" h^i, = o, and in the circuit VBC is i^r^ -)- i^r, + h^^ — 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 z^ is zero, and these equa- tions give the relations, t, = — i^, % = — i^, i^r^ = — Vsi hr^^ —i^r^. Frpm these four equations follows at once a relation betweein the resistances, expressed in the equation r^r, = rjr^. (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 Associationt bridge the method can be carried to a high degree of accuracy. In this form of the bridge, the portion vmrked ACB (Fig. 71) is a straight cylindrical wire, along which the end of the branch CD 231] 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 cbnductors 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 7-4. 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 ^, and that no electromotive force exists in those portions. Then the difference of potential between.^ and^ is V^—Vb=^ h^i = Vi = ^s^s- We have also by Kirchhoff's first law — ^^ = ii-{- h + *s- ^y t^^ combination of these equations we obtain -».=(>'.-»:.)(^+^_+^). (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 cdls, depends for its action on the principles here discussed. It consists of aspiralof 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 con- 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 ,231] MAGNETIC RELATIONS OF THE CURRENT. 33$ 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 V^, the current in the spiral by i, and the resistances of the lengths of wire considered by r, and r„ we have The rules for joining up sets of voltaic cells in circuits so as to afccomplish any desired purpose may be disqussed by the Same riiethod. 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. \i m 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 circuit — , and for the resistance of the circuit s A -• The m, m current in the circuit is therefore i = — - — ; Two cases ,ms-f-nr may arise which are common in practice. The resistance s of the external circuit may be so great th,at, in comparison with )n''s, tir may be neglected. In that case i is a maximum when m= 1, that is, when the cells are arranged tandem, or in series, with their unlike poles connected. On the other hand, if n^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. [232 arranged abreast, or in multiple arc, with their hke 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 thfe 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 ^'jCan be measured in several ways. The first method, used by Weber and Kohlrausth, 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 [y] = [«' T] = M^L^. The dimensions of quantity in the electro- static system are [(2] = M^L^ T~^. The ratio of these dimen- — = 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 z/ by a comparison of an electromotive sions is 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 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 w;hich 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 3-10'° centimetres in a second. This velocity agrees very closely with the velocity, of light. , The physical significance of this quantity v may be underr 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. 333] . 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 direction of movement, in unit time, is — The force due to such a current on a magnet may be calculated. Conversely, if the force on the magnet be observed, and the surface density (T and the velocity x be also measured, the value of v may be calculated. The probability gf 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 tr. If we imagine another such surface near the one already considered, the repulsion be- tween them due to their opposite charges is iTt(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 siirfaces, of which the strengths in the surface are both ELEMENTARY PHYSICS. [23,9 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 accprdingly 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 th^ 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 exhaustibn 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 ^yith the action of the radiometer (§1 56) .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 blocks. Fig. 76, < 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 fig. ^e. 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 sufficient intensity in the vicinity of the capsule causes an al- ternate lengthening arid shortening of the flame, which, how- ever, occur too frequently to be directly observed. By mov- 23 354 ELEMENTARY PHYSICS. [242 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 vibration^ 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. jf) to represent AS , Fig. 77. a cylinder of some elastic substance, and siippose 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 /E the formula V=t^ y:, 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 of a. If the vibrations of a be performed in the time t, the motion will be transmitted during one com- plete vibration of a 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 thfe 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 mption. 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' , ■^ i.-- ^ ^ ^ /'"' \j L--1 vj Vy Fig. Si. same velocity. The condition represented in III is of special in- terest. It shows that two wave systems may completely annul \ K \ A. — R^ r /I / V 1 / \ \ ^ w ^ J V] \ «i 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 III IV are as 1:2. It will be noticed that the resultant 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 I 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 viii line indicates the corresponding displacements. In III, IV, V, €tc., the conditions at 'instants \, ■f, \, etc., periods later are repre- sented. A comparison of these ' pio. 83. curve IS no VI VII IX e 362 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 -vfrill be seen later ; that all sounding bodies afford examples of stationary waves. .;, 246. Reflection of Waves. — When a wave reaches thq.<, 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 the second medium may move with less.i 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 becorties 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; injparting 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, wh^n 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- I tide as a reflected wave in which the motion has the same sign as in the direct wave. It is reflection without change of sign. The two latter cases are extremely important in the study of the formation of stationary waves in sounding bodies. 247. Law of Reflection. — Let lis 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 pap6r. 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 npt 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 ^5. 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 Fig. 84. 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 np, 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 -45 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 366 ELEMENTARY PHYSICS. [249 are sufficiently rapid, blend together and p'Voduce 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 thos,e 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 aif in the chamber. Sirens are sometimes made with several series of holes in the disk. These serve not only the purposes described above, so] SOUNDS AND MUSIC. 367 lut also to compare tones of which the vibration numbers have ertain ratios. The number of vibrations of a ounding body may sometimes be de- ermined by attaching to it a light tylus which is made to trace a curve pon a smoked glass or cylinder. In- tead of attaching the stylus to the punding body directly, which is prac- icable only in a few eases, it may be at- ached to a membrane which is caused vibrate by the sound-waves which he body generates. A membrane re- roduces very faithfully all the charac- sristics of the sound-waves, and the urve traced by the stylus attached to : gives , information, therefore, not nly in regard to the number of vibra- ions, but to some extent in regard 3 their amplitude and form. PHYSICAL THEORY OF MUSIC. 250. Concord and Discord.— i/hen two or more tones are sounded jgether, if the effect be pleasing there said to be concord; if harsh, discord. o understand the cause of discord,' ippose two tones of nearly the same; itch to be sounded together. The re- dtant curve, constructed as in § 245, like those in Fig. 85, which repre- :nt the resultants when the periods Fig, 8s.' the components have the ratio 81 : 80 and when they have 368 ELEMENTARY PHYSICS. [2SI 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 eflect is no longer unpleasant. This, and ratios expressed by smaller numbers,, as |-, I", f, f, \, represent concordant combinations. 251. Major and Minor Triads. — Three tones of which the vibration numbers are as 4 : S : 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:S:6 lffil2:15 Fig. 86. 252. Intervals. — The 2«^^rz/«/ between 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 tlie musical scale, as described below. The interval \ is called an octave;: f , a fifth; 1^, a fourth; \, a major third; f , a minor 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. 369 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 eighty, 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 : MAJOR SCALE. 1' 9 Tone Number i 2 3 4 5 6 7 8 Letter CDE F GA BCD Name do or ut re mi fa sol la si ut re Number of vibrations m |m f m |m |m f m ^m 2m \m Intervals from tone to tone.. f J^ ^ | i^ 9. ^s MINOR SCALE. ' Tone Number. , i 2 3 4 5 6 7 8 9 Letter A B C D E F G A' B' Name la si ut re mi fa sol la si Number of vibrations m |m |m |m fm f m |m 2m |m Intervals from tone to tone.. f if V I tI I ¥ 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 different intervals, \, 1^, and ^. 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 extr^ 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|. 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. J^ut if we consider that the whole tones are not all the same, and propose to preserve exactly all the intervals of tjie transposed; scale, the problem becomes much more diflficult, 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 temperament, 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 pf 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.i 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 tubs, 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 tim- 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. Fig. 37. Fig. Sia. 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 t he p assage. 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-, umh, 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 trahsmitted 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 haying a node at the cen- tre bf 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. 37S while the wave travels twice the length of the cord. The veloc- ity of transmission of a longitudinal displacement in a wir.e 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 ihe 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 fprmula. Thus we have the formula for the velocity 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 r=^ = .i^ , (.05) . and the number of vibrations per second is 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 system? 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 time* 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 | 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. 377 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 will 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 €ach 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 vibtations 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 soundrwave 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 dirnensions 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 ior 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 phonograph 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 end^Vise 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] VIBRATIONS OF SOUNDING BODIES. S79 in effect thin disks, and afford examples of the reinforcement of vibrations of widely different pitch and quaHty 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 stat'ed, the tones of dif- ferent instruments, although of the same pitch and intensity, are distinguished by their quality. It was also stated that the 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. 90. vibrations. In Fig. 90 are given several forms of vibration curves of the same period. Every continuous musical tone must result^ from 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)1 every periodic vibration is resolvable into simple har- monic vibrations having commensurable periods. It has been 262] ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 38 1 seen that all sounding bodies may subdivide into segments, and produce a series of tones pf which the vibration periods gener- ally bear a simple relation to each other. 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 pf 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 accompanied by the harmonics up to the tenth. The Fig. gz. curves differ only in the different phases of the components relative to each other. 382 ELEMENTARY PHYSICS. [263 Fig. 92 shows similar curves produced by 'a fundamental accompanied by the odd harmonics. 1 263. 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 26s] ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 383 which the resonator is tuned is produced in the vicinity, either ty 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 »=€ smr V f* If the bounding planes be parallel, i' = r, and we have sin r _ I sin r' ~ fi' .279] REFLECTION AND REFRACTION. 407 hence ?"= r' , or the incident and emergent waves are parallel. If the two bounding planes form an angle .(4 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) sin r' sm r I -or . ,, . r /< sin(^— r) =/t. 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 At; 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 /< 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 ,7 sm t sinr' — , or sm r fA, sm t . 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 the substance sin " ^ - is called the critical angle of 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 pris'm, and let Ai drawn through A be parallel to the front of the incident wave. With A as centre, and any radius, •f draw an arc im. From the same centre Ai with radius Ar H- 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 sin Arx sin irx sin Axr sin Axr Ax -J- = /t, by construction. 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 jr. 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. 409 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 p. 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 XT' , .parallel to A C, 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- mit 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 F"ig. 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 of 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 4IO ELEMENTARY PHYSICS. [28t 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. ^5 and yl 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 falhng on ^C is again reflect- ed, giving an image at d . Proceeding from A C, 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 V , 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 cif'cumference 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 \kit 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. Fig. iio. 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 elenlentary 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 gpme 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 -f- Bl must be constant wherever the point B is situated on the re- flecting surface. .Draw BD perpendicular to the axis of the mirror. Represent BD by y, AD by x, LA by /, lA by p', and CA by r. Then we have LB = \/{p — xf -\-y^, and y = (2r — x)x — 2rx — x''. Hence follows LB — Vp' — 2px -\- ^ -\- 2rx — jr* = '\/p'-\-zx{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 Z5 = /+|(r-/) + .,.. In like manner we have /■5=/+|(r-/) + ..., whence LB + IB -p +p' + %■ - p) +^X'' - /)• 28l] REFLECTION AND REFRACTION. 413 When B coincides with A, the above value becomes/ -j-/', and since upon our supposition all values of LB -\- IB are equal, we must have from which we obtain r . r p+p' = - and p'= ^^ 2p — r 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 ariy 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 cori- r r jugate foci. If we divide both sides of the equation -r-\--, = 2 by r, we have * -4- -7 = -, (in) p~ p r' ^ ' 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/ = 00, then ■/' = s^r ; 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 CciSS&A.\!i\Q principal focus. Suppose/ = r; /'= r also. When / -.TTi r I 2 ,121 t=\r, /' = oo. When/<-, - > - and -, = --^ = a 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. hi; 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. / 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 r 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 of /, /' 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. tnediap Let v represent the velocity of light in the first *> medium, to tlie 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, m'n' 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 mln' is a spherical surface with centre /. We have Bm _ Bm' V V and Z5 + LB Bm _ a constant for all points of mn. spherical surface m'n', we have If / be the centre of the 4l6 ELEMENTAR'Y PffYSICS. \2i^ IB . BW _ V a constant for all points of m'n'. Taking the difference of the last two equations, and re- membering that we obtain r = C — C. a constant for all points of BD, and hence 1) LB jlB = a constant. V V But -7 = /I is the index of refraction of the second sub- stance in relation to the first. Hence LB — fdB = a constant = LA — /zM. Using the notation of the last section, and substituting the values of L^ and IB as there found, except that/" is used instead of/', we have P + j{ r-p) - J^p" ^-j,{r - p")) = p- /I/', whence we obtain -: — -777 = i — w, P P and ___ = ^. („2) 282] REFLECl^ION AND REFRACTION. 417 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 m'n' , 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 /< the index of refraction of the third medium in relation to the second. If s represent the distance of / from the new surface, /t' the new index, and p' the new focal distance, we have /*' \ _jj! — 1 If we suppose the lens to be very thin we may put s = /". 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 I I /< I _ fi~ p' p" r' ' Multiplying through by /«, we have I M _ ^ — M _ /*— I p p T r Eliminating/" between this equation and Eq. 112, we obtain j,-i = (;._i)(i-i), (113), 27 4l8 ELEMENTARY PHYSICS. [2S2 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 cominations 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. 1 1 3 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)(~ ?)• \l p = 00 , we have - = o and —, =.{y, — 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 f. 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 of making the algebraic 383] REFLECTION AND REFRACTION. 419 signs of the quantities depend on the direction of measure- ment, they are made to depend on the form of the surfaces and the character of the 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 /' is the distance of a virtual focus, and r the radius of a concave surface. The formula then becomes j>+| = (/^-i)(7 + p). (114) To apply this formula to a double concave lens, r and r' are both negative ; p' is then negative for all positive values of p. That is, concave lenses have only virtual foci. For a plano-convex lens (Fig. 113, 2), if light be incident on the plane surface, y=oo and j; = (;* - i)- - -. This gives positive values of/' and real foci for all values of 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 p' 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. [28s. Fig. 114. 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. In the same way b' is an image of b. If axes were drawn through other- points of the object, the im- ages of those points would her found in the same way. They would lie between a' and b' , and a!b' is therefore a real image of the object. It is inverted, and lies between the axes ac, bd, drawn through the extreme points of the object. The ratio of its size to that of the ob- ject is seen from the similar triangles abC, a'b'C; to be the. ratio of. the distances from C. From Eq. in we obtain /'_ r _r—p' p 2p —r~ p — r' Since r —p' and p — r are respectively the distances from- the centre of the image and object, we have a'b' _ r-p' p' ab p — r p' or, the image and object are to each other in the ratio of their respective distances from the mirror. As the object approaches, the image recedes from the mirror and increases in size. At the centre of curvature the image and object are equal, and when the object is within the centre and beyond the principal, focus the image is outside the centre and larger than the ob- ject. "When the object is between the principal focus and the- mirror, the image is virtual and larger than the object. Con- vex mirrors produce only virtual images, which are erect and smaller than the object. 285] REFLECTION AND REFRACTION. 421 284. Images formed by Lenses. — Let us suppose an ob- ject in front of a double convex lens, which may be taken as a type of the converging lenses. The point c (Fig. 115) will have an image at the conjugate focus on the principal axis, a and b will have images on secondary axes drawn through those points respectively, and a point called the optical cen- ^'°- "s- tre of the lens. So long as these secondary axes make but a small angle with the principal axis, definite foci will be formed at the same distances as on the principal axis, and an image a'V will be formed which will be real and inverted, or virtual and erect, according to the distance of the object from the lens. The formula ^ + x. = (/'-i)(^4-p) = ^, / ' / /' shows that when/ increases/' diminishes, and conversely. It shows also that when p is less than /, p' is negative, and the image virtual. It is plain from the figure that the sizes of image and object are in the ratio of their distances from the lens. Diverging lenses, like diverging mirrors, produce only virtual images smaller than the object. 285. Optical Centre. — It was stated in the last section that the secondary axes of a lens pass through a point called the •optical centre. The location of this point is determined as fol- lows: In Fig. 116, let C, C be the centres of curvature of the two surfaces of the lens, and let CA and C'B be two parallel radii. The tangents at A and B are also parallel, and light entering at B and emerging at A is light passing through a medium with parallel surfaces (§ 279), 422 ELEMENTARY PHYSICS. [286 and suffers no deviation. If we Ax^yfi AB, cutting the axis at 0^ CA CO CA the triangles CA O, C'BO are similar, and -^^p^ = 'riTf -^^^ 'HK* being the ratio of the radii, is constant for all parts of the sur- CO faces, hence -t^tt] must be constant, or all lines such as AB must cut the axis at one point O. O is the optical centre, and light passing through it is not deviated by the lens. 286. Geometrical Construction of Images. — For the geometrical construction of images formed by curved surfaces, it is convenient to use, in place of the waves themselves, lines per- pendicular to the wave front, which represent the paths which the light follows, and are called rays of light. These rays, when perpendicular to a plane wave surface, are parallel, and an assemblage of such rays, limited by an aperture in a screen, is called a beam. When the rays are perpendicular to a spherical wave surface, they pass through the wave centre, and constitute ^pencil. A plane wave surface perpendicular to the axis of a lens is converted by the lens into a spherical wave surface with its centre at the principal focus. The rays perpendicular to the plane wave surface are parallel to the axis, and after emergence must all pass through the principal focus. Conversely, rays emanating from the principal focus emerge from the lens as rays parallel to. the axis. Also, rays emanating from any focus must, after emerging from the lens, meet at the conjugate focus. Let Z, Fig. 117, be a converging lens, and AB an object. Let O be the optical centre, and F the principal focus. Since all the rays from A must meet, after emerging from the lens, at the conjugate focus, whrch is the image of A^ to find the position of the image it is only necessary to draw two such rays Fig. 117. 288] REFLECTION AND REFRACTION. 425. and find their intersection. The ray through the optical centre is not deviated, and the straight line^^' represents both the in- cident and emergent rays. The ray^Z may be considered as one of a group parallel to the axis. All such rays must, after passing through the lens, pass through the principal focus. LA', passing through F, is therefore the emerging ray, and its intersection with AA' locates the image of A. Hence, to construct the image of a point, draw from the point two incident rays, and determine the , corresponding emergent rays. The intersec- tion of these will determine the image. The rays most conve- nient to use are the ray through f*-- the optical centre and the ray ''^'°- "*■ parallel to the axis or through the principal focus. Fig. 118 gives another example of an image determined by construction. 287. Thick Lenses. — When a lens is of considerable thick- ness, the formula derived in § 282 does not give the true posi- tion of the conjugate foci. A formula involving the thickness of the lens may be derived without difiSculty, but for practical purposes it is usual to refer all measurements to two planes,, called the principal planes of the lens. The determination of the position of these planes involves a discussion which does not come within the scope of this book. 288. Mirrors and Lenses of Large Aperture. — The ;equations derived in §§281, 282, are only approximations, ap- plying with sufficient exactness to mirrors and lenses of small aperture. But for large apertures, terms containing the higher powers of x cannot be neglected, x will not disappear from the expression of /', and /' will, therefore, not have a definite value. In other words, the reflected or refracted wave is not spherical, and there is no one point / where the light will be concentrated. Surfaces may, however, be constructed which will, in certain particular cases, produce by reflection or refrac- 424 ELEMENTARY PHYSICS. [288 tion perfectly spherical waves. If we desire to find a surface such that light from L (Fig. 1 19) is con- centrated by reflection at /, we remem- ber that the sum LB -\- Bl must be constant, and that this is a property of ^'°' "^' an ellipse with foci at L and /. If the fellipse be constructed and revolved about LI as an axis, it will generate a surface which will have the required property. If one of the points L be removed to an infinite distance, the corresponding wave becomes a plane perpendicular to LI, and we must have LB-\-BC (Fig. 120) constant, a property of the parabola. A parabolic mirror will therefore concentrate at its focus incident light moving in paths parallel to its axis, or will reflect incident light diverging from its focus in plane waves perpendicular to its axis. Mirrors and lenses having surfaces which ^'°- "°- are not spherical are seldom made because of mechanical diffi- culties of construction. It becomes necessary, therefore, to consider how the disadvantages arising from the use of spheri- cal surfaces of large aperture for reflecting or refracting light may be avoided or reduced. We will consider first the case of a spherical mirror. It was shown above that light from one focus of an ellipsoid is reflected from the ellipsoidal surface in perfectly spherical waves concen- tric with the other focus. Let Fig. 121 represent a plane sec- tion through the axis of an ellipsoid, and Fca a small incident pencil of light proceeding from the focus F. F'ac is a section of the reflected pencil. It is a property of the ellipse that the normals to the curve bisect the angles formed by lines to the two foci. The normal ae bisects the angle FaP , and hence in Fa Fc the triangle FaF' we have -^~- = "sr- ^ Fa F'e ,-288] REFLECTION AND REFRACTION. i;Zt) If a' move toward c, Fa increases and Fa diminishes. Hence, from the above proportion, Fe must increase and Fe diminish ; or, the successive normals as we approach the minor axis cut the major axis in points successively nearer the centre of the •ellipse. The normals produced will therefore meet each other at n beyond the axis. If ac be taken small enough it may be considered the arc of a circle of which an, en are radii and n the centre. It is therefore a meridian section of an element ■of a spherical surface of which Fn is an axis. Sections of wave surfaces reflected from the ellipsoid have their centre at F , and are also sections of wave surfaces re- flected from the elementary spherical surface. Evidently the :same would be true for any other meridian section passing through FA of the sphere of which the elementary surface forms a part, and the form of the wave surfaces may be con- ceived by supposing the whole figure to revolve about FA as an axis. The arc ac describes a zone of the sphere, s, s, r, r, describe wave surfaces, and F describes a circumference having its centre on FA. The wave surfaces are portions of the sur- faces of curved tubes of which the axis is the arc described by the point F. The line described by F is 2, focal line, and all the light from the zone described by ac passes through it, or does so very approximately. If ac be taken nearer to A on the sphere, F' approaches the axis along the curve FF and finally 426 ELEMENTARY PHYSICS. [288 coincides with F" , the focus conjugate to F. F'F" is a caustic curve, which, when the figure revolves about the axis AF, describes a caustic, surface. It will be noted that all the light from the zone described by ac passes through the axis AF be- tween the points x and y. The light coming from F and re- flected from a small portion of the spherical surface around b, thfc middle point of ac, is then concentrated first in a line through F' at right angles to the paper, and again into the line xy in the plane, of the paper. Nowhere is it concentrated into a point. A line drawn through b and the middle of the focal line through F' is the axis of the reflected pencil. It will in- tersect the axis of the mirror between x and y. If a plane be passed through the point of intersection perpendicular to the axis of the pencil, its intersection with the pencil will be like an elongated figure 8, which may be considered as a focal line at right angles to the axis of the pencil, and in the plane of the paper, and therefore at right angles to the focal line . through F . Between these two focal lines there is a section of least area, nearly circular, which is the nearest approach to an image of F produced by an oblique incidence such as we' have been considering. If refraction instead of reflection had taken place at ac, a result very similar would have been obtained for the refracted pencil. This failure of spherical reflecting or refracting sur- faces to bring the light exactly to a focus is called spherical aberration. In order to obtain a sharp focus, therefore, if only a single spherical surface be employed, the light must be con- fined within narrow limits of normal incidence. When reflec- tion or refraction takes place at two or more surfaces in succes- sion, the aberration of one may be made to partially correct the aberration of the other. For instance, when the waves in- cident upon a double convex lens are plane, the emerging waves are most nearly spherical when the radius of the second surface is six times that of the first. Two or more lenses may 290] REFLECTION AND REFRACTION. 427 be so constructed and combined as to give, for sources of light at a certain distance, almost perfectly spherical emerging waves. Such combinations are called aplanatic. The same term is applied to single surfaces so formed as to give by re- flection or refraction truly spherical waves. SIMPLE OPTICAL INSTRUMENTS. 289. The Camera Obscura. — If a converging lens be placed in an opening in the window-shutter of a darkened room, well-defined images of external objects will be formed upon a screen placed at a suitable distance. This constitutes a camera obscura. The photographer's camera is a box in one side of which is a lens so adjusted as to form an image of ex- ternal objects on a plate on the opposite side. The relation deduced in §284 serves to determine the size of the image which a given lens will produce, or the focal length of a lens necessary to produce an image of a certain size. 290. The Eye as an Optical Instrument. — The eye, as may be seen from Fig. 122, which represents a section by a horizontal plane, is a camera obscura. « is a ^, transparent membrane called the cornea, be- hind which is a watery fluid called the aqueous humor, filling the space between the cornea and the crystalline lens. Behind this is the vitreous humor, filling the entire posterior cavity of the eye. The aqueous humor, crys- talline lens, and vitreous humor constitute a system of lenses, equivalent to a single lens of ^'°- "^■ about two and a half centimetres focus, which produces a real inverted image of external objects upon a screen of nervous tissue called the retina, which lines the inner surface of the posterior half of the eyeball. The retina is an expansion of the optic nerve. The light that forms the image upon it excites the 428 ELEMENTARY PHYSICS. [290 ends of the nerve, and, through the nerve-fibres leading to the brain, produces a mental impression, which, partly by the aid of the other senses, we have learned to interpret as the charac- teristics of the object the image of which produces the impres- sion. For distinct vision the image must be sharply formed on the retina ; but as an object approaches, its image recedes from a lens, and if, in the eye, there were no compensation, we could see distinctly objects only at one distance. The eye, however, adjusts itself to the varying distances of the object by chang- ing the curvature of the front surface of the crystalline lens. There is a limit to this adjustment. For most eyes, an object nearer than fifteen centimetres does not have a distinct image on the retina. We may here consider the means by which we estimate the distance and size of an object. The retina is not all equally sensitive. The depression at b, called the yellow spot, is much more sensitive than the other portions, and a minute area in the centre of that depression is much more sensitive than the rest of the yellow spot. That part of an image which falls on this small area is much more distinct than the other parts. How small this most sensitive area is, can be judged by care- fully analyzing the effort to see distinctly the minute details of an object. For instance, in looking at the dot of an i, a change can be detected in the effort of the muscles that con- trol the eyeball, when the attention is directed from the upper to the lower edge of the dot. The eye can then be directed with great precision to a very small object. The line joining the centre of the crystalline lens with the centre of the sensi- tive spot may be called the optic axis ; and when the attention is directed to any particular point of an object, the eyeballs are turned by a muscular effort, until both the optic axes pro- duced outward meet at the point. For objects at a moderate distance we have learned to associate a particular muscular effort with a particular distance, and our judgment of such 89X] REFLECTION AND REFRACTION. 429 distances depends mainly on this association. The angle be- tween the optic axes when they meet at a point is called the , optic angle. Our estimate of the size of an object is'based on our judgment of its distance, together with the angle which the object subtends at the eye, called the visual angle. In Fig. 123, when ab is an object, and / the crystalline lens, a is the visual angle. It is plain that the size of the image on the retina is proportional to the visual angle. It is plain, too, that Fig. 123. an object of twice the size, at twice the distance, would sub- tend the same visual angle and have an image of the same size as ab. Nevertheless, if we estimate its distance correctly we shall estimate its size as twice that of ab ; but if in any way we are deceived as to its distance, and judge it to be less than it really is, we underestimate its size. • Most persons underes- timate heights, and hence underestimate the sizes of objects high above them. The visual angle is the apparent size of the object. 291. Magnifying Power. — To increase the apparent size of an object, and so improve our perception of its details, we must increase the visual angle. This can be done by bringing the object nearer the eye, but it is not always convenient or possible to bring an object near, and even with objects at hand there is a Hmit to the near approach, due to our inability to see distinctly very near objects. Certain optical instruments serve to increase the visual angle, and so improve our vision. Instruments for examining small objects, and increasing the 430 ELEMENTARY PHYSICS. [292 visual angle beyond that which the object subtends at the nearest point of distinct vision by the unaided eye, are called microscopes. Those used for observing a distant object and enlarging the visual angle under which it is seen at that dis- tance are telescopes. In both cases the ratio of the visual angles, as the object is seen with the instrument, and without it, is the magn ifying power. 292. The Magnifying Glass. — Fig. 124 shows how a con- , verging lens may be employed to magnify small objects. The point a of an object just inside the principal focus F of the lens " ^ is the origin of light-waves which, after passing through the lens, are changed to waves hav- ing a centre a' (§ 282) which, when the lens is properly ad- FiG. 134. '■■ justed, is at the distance of dis- tinct vision. Waves coming from b enter the eye as though from b'. The object is therefore distinctly seen, but under a visual angle a' Ob', while, to be seen distinctly by the unaided eye, it must be at the distance Oa", when the angle subtended is a" Ob". The ratio of these an- gles is very nearly that of Oa" to OF. Hence the magnifying power is the ratio of the distance of distinct vision to the focal length of the lens. 293. The Compound Mi- croscope. — A still greater mag- nifying power may be obtained by first forming a real enlarged image of the object (§ 284) and using the magnifying glass upon the image, as shown in Fig. 125. 294] REFLECTION AND REFRACTION. 43 1 The lens A is called the objective, and E is called the eye-lens or ocular. As will be seen in § 310, both A and E often consist of combinations of lenses for the purpose of correcting aberration. 294. Telescopes. — If a lens or mirror be arranged to pro- duce a real image of a distant object, either on a screen or in the .air, we may observe the image at the distance of distinct vision when the visual angle for the object is enlarged in the ratio of the focal length of the lens to the distance of distinct vision. This will be plain from Fig. 126. Suppose the nearest Fig. 126. point from which the object can be observed by the naked eye to be the centre of the lens O. The visual angle is then AOB = aOb, while the visual angle for the image is aEb. Since these angles are always very small, we have aEb _ Oc aOb ~ Ec very nearly. But when AB is at a great distance, Oc is the focal length of the lens. By using a magnifying glass to ob- serve the image, the magnifying power may be still further increased in the ratio of the distance of distinct vision to the focal length of the magnifying glass. The magnifying power of the combination is therefore the ratio of the focal length of the object-glass to the focal lengl^h of the eye-glass. A con- cave mirror may be substituted for the object-glass for produc- ing the real image. CHAPTER III. VELOCITY OF LIGHT. 295. Velocity Determined from Eclipses of Jupiter's Moons. — Roemer, a, Danish astronomer, was led to assume a progressive motion for light in order to explain some apparent irregularities in the motions of Jupiter's satellites. A few ob- servations of one of Jupiter's moons are sufficient to determine the time of its eclipses for months in advance. If these observa- tions be made when the earth and Jupiter are on the same side of the sun, and the time of an eclipse occurring about six months later, predicted from them, be compared with the observed time of that eclipse, it is found that the observed time is about l6|- minutes later than the predicted time. This discrepancy is explained if it is assumed that light has a progressive motion and requires i6f minutes to cross the earth's orbit, for the dis- tance of the earth from Jupiter in the second case is about the diameter of its orbit greater than in the first. 296. Aberration of the Fixed Stars. ^The apparent direc- tion of the light coming from a star to the earth, that is, the apparent direction of the star from the earth, is the. resultant of the motion of the light and the motion of the earth. As the motion of the earth changes direction the apparent direction of the star will change also, and the amount of that change will depend on the relation betuveen the velocity of light and the change in the velocity of the earth in its orbit, understanding by change of velocity change in direction as well as in amount. This apparent change in the position of the stars is called aberra- , 297] VELOCITY OF LIGHT. 43$ tion. Knowing its amount corresponding to a known change in the earth's motion, we may compute the velocity of light. This method was first employed by Bradley. 297. Fizeau's Method'. — Several methods have been em- ployed for measuring the velocity of light by determining the time required for it to pass over a small distance on the earth's surface. In the form of experiment devised by tizeau, a beam of light is allowed to pass out through a small hole in the shut- ter of a darkened room to a distant station where it is reflected back on itself. It returns through the opening and produces an image of the source. A toothed wheel is placed in front of the opening in such a position that, to pass out or back, the light must pass through the spaces between the teeth. If the wheel revolve slowly, as each space passes the opening in the shutter light will pass out, and returning from the distant sta- tion will enter through the space by which it made its exit. An image of the source will therefore be visible whenever a space passes the opening, and in consequence of the persistence of vision this image will appear continuous. , Since it takes time for ,the light to go to the distant station and back, it is possible to give to the wheel such a velocity that when the light which passed out through a given space returns, it will find the ad- jacent tooth covering the opening, so that no image of the source can be seen. If the velocity of rotation be sufficiently increased, the linage again comes into view when the light can enter through the space following that by which it emerged. A still further increase of velocity may cause a second extinc- tion of the image. The experiment consists in determining ac- curately the velocities for which the several extinctions and reap- pearances of the image occur. A high degree of accuracy can- not be attained because the extinction of the image is not sud^ den. It disappears by a gradual fading away, and reappears by a gradual brightening. For quite a range of velocity the image cannot be seen at all. 38 434 .ELEMENTARY PHYSICS. [298 298. Foucault's Method. — Foucault's method depends . upon the use of the revolving nairror as a means of measuring a very small interval of time. Foucault's experiments were re- peated with some modification by Michelson in 1879 ^'^^ again in 1882. The general theory of the experiment may be under- stood from the following brief description. Let S (Fig. 1 27) be a narrow slit, m a mirror which may revolve about an axis in its own plane, L a lens, and m' a second mirror. Light from a source behind 6' passes through the , slit, falls on m, is re- flected, when »« is in a suitable position, through the lens Z, > ' Fig. 127. and forms an image at S'. S and S' are conjugate foci of the lens, and by so placing the lens that 5 shall be a little beyond the principal focus, S' may be removed to as great a distance as desired. The mirror m! is perpendicular to the axis of the lens, and at such a distance that the image S falls upon its surface. It is evident that any light reflected back from m' through L will return to the conjugate focus S, whatever the po- sition of the mirror m' , so long as it sends the light in such a di- rection as to pass through L both going and returning. If now the mirror m be given a rapid rotation clockwise, light passing through L will return to find mm ■3. changed position, and the image will be displaced from 5 to some point S" to the left of 5'. Knowing the displacement SS" and the number of rotations of the mirror per second, the time required for light to pass from m to S and back is determined. The value of the velocity 399] VELOCITY OF LIGHT. 435 -of light, as determined by Michelson in 1879, is 299,910, and in 1882, 299,853, kilometres per second. 299. Influence, upon the Velocity of Light, of the Motion of the Medium through which it Passes. — Fizeau showed by ■experiment in 1859 that a moving transparent body increases or diminishes the velocity of light passing through it, not by its •own velocity, but by a fraction of its own velocity, expressed «" — I /. . by ~ — , where n is the index of refraction. This result was n confirmed by experiments of Michelson and Morley in 1 886. The result follows if we suppose the change of velocity of light in a medium to be due solely to change of density of the ether. Remembering that the velocity of propagation of wave motion in any medium is i/ = y — , and that the velocity in a medium of which the index of refraction is n, is -th as great as that in a vacuum, it may be seen at once that the density of the ether A C E D Fig. 128. in such a medium must be «" times as great as that in a vacuum. In Fig. 128 let AC he a. body, o."' which the index of refraction is n. Let the body move forward so as to occupy the position CD. The ether occupying the space CD and having a density I must in the body have a density n', and hence must occupy a space ED, which is — times CD. The ether in AC must, there- fore, move forward through a distance C£, while the body moves through a distance CD. But CE equals CD X ^i — - j) ^» J or CD X 5 — . Hence the ratio of the velocity of the ether n ^ . I to the velocity of the body is 5 — . CHAPTER IV. INTERFERENCE AND DIFFRACTION. 300. Interference of Light from Two Similar Sources.-^ It has already been shown that the disturbance propagated to any point from a luminous wave is the algebraic . sum of the disturbances propagated from the various elements of the wave. The phenomena due to this composition of light-waves are called interference phenomena. Let us consider the case in which two elements only are Fig. 129. ''' ' efiticient in producing the disturbance. Let A and B (Fig. 129) represent two elements of the same wave surface separated by the very small distance AB. The disturbance at m, a point on a distant screen mn, parallel with AB, due to these two ele- ments, is the resultant of the disturbances due to each sepa- rately. The light is supposed to be homogeneous, and its wave length is represented by A.. When the distance mB — inA equals f A, or any odd multiple of l-A, there will be no disturbance at m. Take mC = mB, and draw BC. mCB is an isosceles triangle ; but since AB is very 300] INTERFERENCE AND DIFFRACTION. 437 small compared to Om, the angle at C may be taken as a right angle; the triangle ACB, therefore, is similar to Osm, and we have AB Om Os -Ty=. = — = — very nearly. AC sm sm ^ ' Represent sm by x, Os by c, AB hy b, AChy nX i^,, where n is any number. Then we have ^ = V- ("5) If n be any even whole number, the values of x given by this ■equation, represent points on the screen mn at which the waves from A and B meet in the same phase and unite to produce light. If n be any odd whole number, the corresponding values of X represent points where the waves meet in opposite phases, and therefore produce darkness. It appears, therefore, that starting from s, for which w = o, we shall have darkness at dis- tances b ' b ' ^ '^*^'' ^Xc p-c jpu \ and light at distances Xc 2\c ^Xc o, -y, -^, -J-, etc. From Eq. (115), we have 2bx 43^ ELEMENTARY iPHYSICS. [30& Since \iih. is the number of wave lengths that the wave front •from B falls behind that from A, ^nT, where /'represents the period of one vibration, is the time that must elapse after the wave from A produces a certain displacement before that from B produces a similar displacement. The expression — ^ — = nn: is, therefore, the difference in epoch of the two wave systems. Substituting ttTt for e in Eq. (9), we have „ , , , ,, /2?r# , sin uTt \ o = J + J, = fl(2 + 2 cos n7[)i cos —^ tan ~^-^, ], ' ' ^ ' ' \ 7^ I + COS njTj, Now the intensity of light for a vibration of any given period is proportional to the energy of the vibratory motion. It is, therefore proportional to the square of the maximum velocity,yf and this is proportional to the square of the amplitude. To- find the relative intensities of light at different points, we may suppose t in the second parenthesis above to have such a value- as shall render the cosine unity, when 6' 2= a(2 -|- 2 cos tiTty = A is the amplitude of the vibratory motion for any given value of n. Substituting for n its value and squaring, we have A^ = a^\2 -\- 2 cos — T-'^'jj in which A" is proportional to the intensity of the illuminatioifc at distamces x from s. When 2bx 300] INTERFERENCE AND DIFFRACTION. 439 its cosine is i, and A' is a maximum and equal to 4^'. As x increases A^ diminishes, until — =r-;r = n, in which case A^ = o. A^ then increases until it becomes again a maximum, when 2bx ck n = 2n. In short, if AB (Fig. 130) represent the line mn of Fig. 129, the ordinates to a sinuous curve like abc will represent the intensi- ties of the light along that line. The phenomena described above may be realized experi- mentally in several ways. Young admitted sunlight into a J g darkened room throXJgh a small hole /\ /\ r\ /\ /\ in a window-shutter. It fell upon A — B a screen in which were two small ^'°- '3°- holes close together, and, on passing through these, was received upon a second screen. Light and dark bands were observed upon this screen, the distances of which from the central barid were in accordance with theory. Fresnel received the light from a small luminous source upon, two mirrors making. a very large angle, as in Fig. 131, The light, reflected from each mirror proceed- ed as though from the image of the seurce produced by that mirror. The reflected light, therefore, con- sisted of two wave systems, from two precisely similar sources A and B. Light and dark bands were formed in accordance with theory. In order that .the experi* ment may be successfully repeated reflection must take place s \ -'■^ A,-' A/ f^~ A — . ,_ ""flS w ^>'" \ '^^ 440 ELEMENTARY PHYSICS. [301 from the front surface of each mirror only, the angle made by the mirrors must be nearly 180°, and the reflecting surfaces must meet exactly at the vertex of the angle. Two similar sources of light may be obtained als5 by sending the light through a double prism, as shown in Fig. 1 32. Light from A proceeds after passing through the prism as from the two virtual images a and a' . A divided lens, Fig. 133, serves the same purpose., The light from A is concentrated in two real images a and a! , from which proceed two wave systems as in the previous cases. What are really seen in these cases, when the source of light is white, are iris-colored bands instead of bands of light and darkness merely. Fig. 132. Fro. 133. When the light is monochromatic,, the bands are simply alter- nations of light and darkness, the distances between them being greatest for red light, and least for blue. From Eq. (i i S) it ap- pears that, other things being equal, x varies with A,, hence we must conclude that the greater distance between the bands in- dicates a greater- wave length; that is, that the wave length of red light is greater than that of blue. 301. Measurement of Wave Lengths. — Data may be ob- tained from any of the above experiments for the determination of the wave length of light. From Eq. (115) we have A.= 2bx en 302] INTERFERENCE AND DIFFRACTION. 44I where c, b, and x are distances to be measured. The distance X is the distance from j to a point w«, the centre of a light band, and n equals twice the number of dark bands between s and m. It is not necessary to consider the details of the apparatus, and the adjustments necessary for making these measurements. It is sufificient to show, in a general way, how the distance x can be measured. Instead of a screen, a lens or combination of lenses, called a positive eyepiece, is placed in the path of the light, and the observer looks through it towards the luminous source. This eyepiece has a spider-line stretched in front of it, which is seen magnified when the bands are observed, and lens and spider-line are arranged to be moved laterally by a microm- eter screw. By this movement the spider-line may be brought to coincide with the baiids in succession, and the. distances measured by the number of revolutions of thre screw. Better methods than this of measuring wave lengths will be found de- scribed in § 306. 302. Interference from Thin Films. — Thin films of trans- parent substances, such as the wall of a soap-,bubble or a film of oil on water, present interference phenomena when seen in a strong light, due to the interference of waves reflected from the two surfaces of the film. Let AA, BB (Fig. 1 34) be the surfaces of a transparent film. Light falling on AA is partly reflected and partly transmitted. The reflection at the upper surface takes place with change of sign (§ 246). The light entering the film is partly reflected at the lower surface without change of sign, and returning partly emerges at the upper surface. It is there compounded with the wave at , that moment reflected. Let us suppose ^'"^ '3*- the light homogeneous, and the thickness of the film such that the time occupied by the light in going through it and return- ing is the time of one complete vibration. The returning wave will be in the same phase as the one at that moment entering. 442 ELEMENTARY PHYSICS. [302- and, therefore, opposite in phase to the wave 'then reflected- The reflected and emerging waves destroy each other, or would^ do so if their ampHtudes were equal, and the result is that, ap- parently, no light is reflected. If the light falling on the film be white light, any one of its constituents will be suppressed when the time occupied in going through the film and returning is the period of one vibration, or any whole number of such periods, of that constituent. The remaining constituents ' pro- duce a tint which is the apparent color of the film. Similar phenomena are produced by the interference of that portion of the incident light which is transmitted direetly through the film, with that portion which is transmitted after undergoing an even number of internal reflections. Since these reflections occur without change of sign, the thickness of the film for which the reflected light is a minimum is that for which the transmit- ted light is a maximum. Newton was the first to study these phenomena. He placed a plane glass plate upon a convex lens, of long radius, and thus formed between the two a film of air, the thickness of which f ,— — , ^^ ''•^y point could be determined j, --^ when the radius of the sphere and the distance from the point of con- tact were known. With this ar- rangement Newton found .a, black spot at the point of contact, and surrounding this, when white light was used, rings of different colors. When homogeneous light was used. Fig- '35- the rings were alternately light and dark. Let ab (Fig. 135), be the radius of the first dark ring, and denote it by d. The thickness be = ef, which may be denoted by X, is d' X ■=■ . 2r — X '303] INTERFERENCE AND DIFFRACTION. 443 Since x is very small in comparison with 2r, this becomes X ■= — . 2r This distance for the first dark ring, 'tvhen the incident light is normal to the plate, is equal to half the wave length of the light experimented upon. Newton found the thickness for the first dark ring ^ , ^^ ^ ^ inches, which corresponds to a wave length of about, 44^0 fl inches, or 0.00057 ''^''^- This method affords a means of measuring the wave lengths of light, or, if the wave lengths be known, we may determine the thickness of a film at any point. ' 303. Effects Produced by Narrow Apertures.— It has been seen (§ 274), that cutting off a portion of a light-wave by means of screens, thus leaving a narrow aperture for the pas- sage of the light, prevents the interference which confines th^ light to straight lines, and gives rise to a luminous disturbance within the geometrical shadow. This phenomenon is called diffraction. Let us consider the aperture perpendicular to the plane of the paper, and an approaching plane wave parallel to the plane of. the' aperture. Let AB (Fig. 136] represent tbe aperture, and mn one position of the approach- ing wave. To determine the effect ^ at any point we must consider the elementary waves proceeding fromv 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 ACS 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 illuminated. Now consider a second point, /", in an oblique direction from C, Fig. 137, and suppose the obliquity such that the time of passage from B to P' is half a vibration period less than the time of passage from C tp 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 Cand A. The point P is, therefore, not illuminated. Suppose another point, P' (Fig. 1 38), still further from P, such that .(4^ 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, AB may be divided into four parts, the distances of which from the point, takpn 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 right or left of the normal. While the line joining P with ■ 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 P 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 /" Fig. 138. 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, Igiving 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 minirtium values alternate in this way, to an indefinite distance, but the s'uccessive maxima decrease so rapidly that, in reality, only a few bands can be seen. 3P4. 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 / + /', equal to the general illumination on the surface. ' Let us now suppose mn to be a screen arid pa a ' narrow aperature in it. If the illumination at C remain un^ changed, it must be that the parts mo Sindpn of the wave had no effect, and if, for the screen with the narrow aperture, we substi- tute a narrow screen atop, 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. 44^ ELEMENTARY PHYSICS. [305 305. Diffraction Gratings. — Let AB (Fig. 140) be a screen, having several narrow rectangular apertures parallel and equi- distant. Such a screen is called , ^ T grating. Let the approaching waves, yip^ moving in the direction of the arrow, //\\>\ be plane and parallel-to AB. Draw \\\^^ the parallel lines ab, cd, etc., at such ™^0\\\^ ^" angle that the distance from the bJ ^\\\. centre of a to the foot of the perpen- ^\ dicular let fall from the centre of the Fig. 140. 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 ccnsidered as a plane wave, which, if it were received on a converging lens, would be concentrated to a focus. If the obHquity of the lines be increased until ae equals 2/1, 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, -^T^ for example. Let m be the fifty-first opening counting from a ; then an will be ■j^A, X SO = 49.SA.. 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 opienings 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 e =■ -J, (116) ■where « is a whole number, d the angle between the direction of the light'and the normal to the grating, and d the distance from 305] INTERFERENCE AND DIFFRACTION. 447 'i iientre 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 inade on the polished surface of speculum metal, and the san^e 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 A C, that is, the light is separated into its several constituents and pro- "duces 2. pure spectrum. Since different values of n will give •different values of d 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 \!a.& 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 Q 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 m"a, ab, be, cd, etc., are elements of the grating. Let us supposethe wave systems ah represented, and draw m"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 m"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 m" 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 pathof the diffracted light. Then i equals r, and if m"a equal s, the radius of the elemen- tary wave having its centre at a, and tangent to ■ni"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 sin « = A. Let us designate by j^ 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 * ■ 2 30S] INTERFERENCE AND DIFFRACTION. 449 * B Q R fi will bisect the angfle/J. Hence we have ^ = — \ — , and a =— . ° 22 22 Substituting these values in the equation for A. we obtain A = 2s cos - sm — . (117) 2.2 ^ '■' Hitherto the spaces from which the elementary waves pra- 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 X. Then «/ equals ^X, or the waves from one half of each opening "f 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 fis. 142. sin 6 — — , if the incident light be normal to the grating. Let d /represent the width of the opaque space. Then d -\- f = s, and light occurs in the direction given by sin 6 = -—- pro- vided that the value of 6 given by this equation does not satisfy the first equation also. If d equal /, we have . nX- nX d-\-f 2d When « is even, sin B becomes 2X X 4A. 2X 29 4S0 ELEMENTARY PHYSICS. [306 and satisfies the equation • a ^^ 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 f 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 ac rep- resent an opening and cd an opaque por- tion. Let us assume that cd equals ■^ac, and let ab 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 ik = ^X. Fig. 143. Each of the elementary waves from points between c and z will be half a wave length behind a correspond- ing wave from some point between d and j, so that the waves coming from ci and dj annul one another, and tj is the only efficient portion of the opening cd. This portion i; 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 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. A\%2l 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 a3fis 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 yS 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 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 ELEMENTARY PHYSICS. [307 Wave lengths are generally given in terms of a unit called a tenth metre; that is, i metre X io~'°. The wave lengths of the visible spectrum lie 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 light 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. 145. 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 -j — . T „ ^ , , .sin Arx In § 279 we find the equation —. — ^ — ' = /*> ^^^ referring to sin jcixr 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 refracting angle of the prism, or — , and the angle xAr equals d half the deviation, or -. Hence we obtain 2 . A-[-d sin /* = ^; (118) sin — 2 aio] £)ISPEIiSION. 455 2 or when /4 is small, /t = A ' 2 from which d=A{jJi— i). Hence we obtain d' - d" _ Ajjji! - I) - A{fi" - I) _ fi' ^ fii' d ~ A{m — i) ~ f* — i ' where fi' and jn" are the refractive indices for the extreme colors, and /< 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 45,6 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 angfes that the spectra produced are of the same length. If these twp 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 of the Huyghens eyepiece. B is called the field-lens, ■axidi. is three times the focal length of C. It is placed 3"] DISPERSION. 4S7 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 RV. It will be noted that the image / 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 fig. 148. 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 458 ELEMENTARY PHYSICS. [31? 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 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 Fig. 150. 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. 150. 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. 151 shows the disposition of the essential parts of the apparatus known as the spectroscope. S is the slit, which may be ednsidered 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\^\Alt'i Fig. 151. 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 th^ 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. \%1% 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 etnployed. 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 sufificient 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 temperatiire necessary for incan- descence. In that case the spectrum given is the combined spectra of the elements. A compound gas that do?s 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 lose 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 cprresponding 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. Placing 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] DISPERSION. 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. Somewhere in its course this light passes througlf 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, buj; 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, Hght, or chemical changes. It was formerly supposed that these were dne 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. 3IS] 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 Hnes 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 Radiations.— 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- T^erted 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 / the intensity of the radia- tions per unit area of the sphere. Then, since the surface of the sphere equals 47rr\ we have E = 47rrV, and I=£^. (It9) 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. [3i« 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. Let «^ be a surface the normal to which makes with the ray the angle Q ; then ab will receive the same quantity of radiant en- ergy as a'b' , its projection on the plain normal to the ray. But a'b' equals ab cos ; and if / represent the intensity on a'b' , and /' the intensity on ab, we have /'=/cos(9; 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 oi 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^ E'r'' 3i6] ABSORPTION AND EMISSION. 465 and when / and /' are equal, Er" z= E'r\ 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. Rumf or d' s photometer 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 Bunsetis 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 paraffine. 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 tp 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 off 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. 46^ 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 becoriie visible by sending to the eye some portion of the hght 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 ai'e 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 lines of the solar spectrum are thus accounted for. 320. Emission of Radiations. — Not only do incandescent bodies enait radiations, but all bodies at whatever temperature they may be. 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 tetnperature, 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 thie 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- harity of structure which fits it for communicating its own motion to the medium when its own motion is the greater, fits 470 ELEMENTARY PHYSICS. [32r 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 Ne-Wton'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 • (2 = ^(1.0077) (1.0077 — 1)> where B represents the temperature of the inclosure, i the dif- ference of temperature between the inclosure and the radiating body, both measured in Centigrade degrees, and in 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 reiider 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 tem- 322] ABSORPTION AND EMISSION. /^Ti. perature rises. Draper showed that the spectrum of a red-hot body exhibits no fays 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 iof 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 riot 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 cgnfined to a few definite wave lengths, but careful observa- tions of the spectra of gases show that the lines are ijiot 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 contiguous, 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 giv^ the spectra of their elements. When it occurs with gases supposed to be ' elements it suggests the question whether they are not really j compounds, the molecules of which at the high temperature iare divided, giving new molecules of which the rates of vibra- tion are entirely different from those of the original body. 472 ELEMENTARY PHYSICS. [323 1 1 323. 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 ^x& 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 complex 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 f uchsin, 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 prism formed of two glass plates making a small angle with each other, and containing a solution of fuchsin in alcohol. He 324] ABSORPTION AND 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. Ctystals 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. t Fig. 153 represents such a crystal. At A arid .^are two solid angles formed by the obtuse angles of three plane faces. The line through A making equal angles witK the three edges AB, AE, AD, or any line parallel to- it, is an optic axis of the crys- tal. ^'J-^**^ 326] DOUBLE REFRACTION AND POLARIZATION. 475 Any plane normal to a surface of the crystal and parallel to the optic axis is called 2l 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. Thfe 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 D«uble Refraction.— In § 279 it was seen that the index of refracti^of a substance is the reciprocal of the ratio of the velocity dfKght in the substance to its 476 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 iri any singk refracting medium. The phenomena jJresented 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-i methods already explained in § 279, and exemplified in Fig. 1 54, 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, ^yhen its principal plane is parallel or at 327] DOUBLE REFRACTION AND POLARIZATION. 477 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 easily _ 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;,d;injinishes until it fails when the angle is 270°. The ex- traordinary ray presents the same phenomena except tjbat the refle<:ted 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 bo(:h 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 v/| axis and the ray. The equation v = \/ -=r holds for transverse vibrations, if by E 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 ela,sticity, 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 327] DOUBLE REFRACTION AND POLARIZATION. 479 ■ velocity. In this case the two wave fronts continue in parallel planes, and upon emergence cpnstitute 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 eviiient 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 48q ELEMENTARY PHYSICS. [328 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 it3 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- -v 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 represent the surface of a trans- parent medium, ab the incident, be the reflected, and bd the re- fracted ray. If the angle cbd=L 90°, we have r + ^ = 90° also ; sin i . sin i Fig. 155. and since ;< = -. — , we have /< = tan t. Hence the sin r cos z angle of complete polarization is given by the equation tan i = fi. 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 light 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. [329 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 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 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 NicoVs prism, or often simply a Nicol. The Foucault prism is similai 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 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 o 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 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 bther 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 easj 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 vibratioo of the polarized ray. It is not to be understood that in the plate there are two separate beams of light, in one of which one set. of particles is 434 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 ^to represent the direction of the disturbance produced by the 0, 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 begin 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. 48$ 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 cf^ 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, oe will give the components om and on, and od the components am! and on'. Since the com- ponents om and ow! annul one another, the color to which they correspond is wanting in the light resulting from vibrations in '^ ;if % 486 ELEMENTARY PHYSICS. [331 the plane xx, while since the components on and on! 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 ^. different, but for each vibration /' j period, the component in the di- i rection xx combined with that in m 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 genera,l elliptical. Let us consider more fully the real mo- tion. Let us suppose that the light is light of one wave length 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 letj/jK 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- 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 sufifi- 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 Fig. 162. 488 ELEMENTARY PHYSICS. [331 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 light 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 i shown in Fig. 164, in which the difference of phase is one fourth a period, and the result- Fig. 163. ant vibration is a circle. 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 ,a length will give a circular vibration, and the beam issuing from the plate is then circularly polarized. Its peculiarity is that j the two beams into which it is divided by a double refracting crystal are always of the same intensity, and no form of ana- \j lyzer will distinguish it from ordinary light. Quarter wave plates are often made x y Fig. 164. by splitting sheets of mica until the re- quired thickness is obtained. 334] DOUBLE REFRACTION AND POLARIZATION. 489 332. 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 t■w^D components exist in the reflected light, but so related in phase that the light is elliptically polar- ized. Fresnel has devised an apparatus known as Fresnel's 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 <:ross. 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. 491 a long tube made to vibrate longitudinally shows it when th& 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 bearn of white polarized light, 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 v 'es 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 Hght within the crystal 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 -Pand 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 circular 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. Artificial 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 sanje 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 the 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 340] 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 Maxwell'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, 49.6 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 : '(/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 are 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. 49T many electrolytes are transparent and yet are good conductors by slipposing 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. I. 48401 5 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 II. Acceleration of Gravity. g- = 980.6056 — 2.5028 cos 2/— 0.000003.4, where / is the latitude of the station and h its height in centimetres above the sea level. J- at Washington = 980.07 I g at Paris = 980.94 g- at New York = 980.26 | ^ at Greenwich = 981.17 TABLE III. Units of Work. Kilogram-metre = 100,000^ ergs. Foot-pound = 13,825^ ergs. = 1-355 X 10' ergs, log 7.13200, when g = 980. Units of Rate of Working. Watt = 10' ergs per second. Horse-power = 550 foot-pounds per second. = 746 Watts. Unit of Heat. Lesser calorie (gram-degree) = 4.16 X 10' ergs. TABLES. 499 TABLE IV. Densities of Substances at o°. The densities of solids given in tliis table must be taken as only 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 8.9 Gold 19.3 Glass (crown) 2.5 to 2.7 Hydrogen 0.0000895 Ice 0.918 Iron (wrought) 7-6, to 7.8 " (cast) 7.2 to 7.7 Lead 11 .3 Mercury i3 '596 Platinum. 21.5 Silver 10 . 5 Zinc 7-1 TABLE V. Units of Pressure for g = 981, Grams per sq. cm. Degrees per sq. cm. Pound per square inch.. 7o-3i 6.g X To" I inch of mercury at 0° 34-534 3-388 X 10* I millimetre of mercury at 0° 1-3596 '333 • 8 1 atmosphere (760 mm.) 1033-3 1.0136 X io« X atmosphere (30 inches) 1036 . 1 . 0163 X lo" TABLE VI. Elasticity. If p is the force in degrees per unit area tending to extend of compress a di , ... dp tody, the linear elasticity is ^, and the volume elasticity is ^-. dp. * di dv Glass 6.03X10" 4.15X10" Steel 2. 14X10" 1.84X10" Brass 1.07 X 10" Mercury -•-• 3-44 X io'» Water 2.02 X io>« Joo ELEMENTAR V PHYSICS. TABLE VII. Absolute Density of Water at t" in Grams per Cubic Centimetre. t^. Density. O 0.999884 1 0.999941 2 0.999982 3 1.000004 4 1. 000013 5 1.000003 f. Density. 7 0.999946 8 0.999899 9 0.999837 10 0.999760 15- 0.999173 20 0.998272 0.999983 30 0.995778 100 f. Density. 40 o. 99236 50 0.98821 60 0.98339 70 0.97795 80 0.97195 90 0.96557 0.95866- TABLE Vin. Density of Mercury at t°. Water at 4° being 1. if". Density. log. o 13-5953 I-I3339 10 13-5707 1.13260 f. Density. log. 20 13.5461 1.13182- 30 13-5217 I-13103, TABLE IX. Coefficients of Linear Expansion. ~ dl Temperature. " = -j:- Aluminium 16° to 100° 0.0000235 Brass o to 100 0.0000188 Copper o to 100 0.0000167 German silver o to 100 0.0000184 Glass o to 100 0.0000071 Iron 13 to 100 0.0000123 Lead o to 100 0.0000280 Platinum o to 100 0.0000089 Silver o to 100 0.0000194 Zinc o to TOO 0.0000230 dV Coefficients of voluminal expansion, -^ = 3a. TABLES. SOI yABLE X. Specific Heats— Water at o* = i. Solids and Liquids, Aluminium 0.212 Brass 0.086 Copper 0.093 Iron o. 112 Lead 0.031 Mercury 0.033 Platinum 0.032 Silver , 0.056 Water (0° to 100°) 1.005 Zinc 0.056 Gases and Vapors at Constant Pressure. Air o. 237 Hydrogen 3-4io Nitrogen 0.244 Oxygen 0.217 G> Ratio, —•= 1.404. TABLE XL 1. Melting Points. IL Boiling Points. IIL 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 .. •• 456o Copper 1200 Lead 325 •• 5-9° Mercury —39 357 2-8 62 0.02 Nitrous oxide, NaO — 105 . . . . 24320 Platinum 1780 .. 27.2 Silver 1000 . . 21 . i Water o 100 80 537 4.6 Zinc 415 •• 28.1 502 ELEMENTARY PHYSICS. TABLE Xyi. Maximum Pressure of Vapor of Water at Various Temperatures in (I.) Dynes per Square Centimetre, (II.) Millimetres of Mercury. Temp. ~-2o° I. II. 1336 — 10° , 2790 0° 6133 . 4.6 10 12220 9.2 20 33190 17.4 30 42050 31.5 40 73200 54.6 50 1. 226X10' 96.2 Temp. I. 60° 1.985 X 10' 80 ...,, . 4.729 X 10' 100 10.14 120 19.88 140 ■•• .a6-26 160 62.10 180 100.60 200 156. X I0» X lo^ X 10' X 10* X 10= Xio' II. 149. 355- 760., 1491. 2718. 4652. 7546. 11689.^ TABLE XIII. Critical Temperatures (y) and Pressures in Atmospheres (/■), THEIR Critical Temperatures, of Various Gases. T. Hydrogen — 174. Nitrogen —124. Oxygen —105. AT p. 99. 42. 49. Carbon dioxide . . . Sulphur dioxide . . T. 30.9 155-4 P. 77- 79 TABLl i 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 o . 0005 Tee 0.0057 Iron 0.16 Lead 0.08 Mercul^r 0.015 Paraffin 0.00014 Silver ■. 1.09 Vulcanized iildia-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. Energy in ergs. Carbon, forming CO 2141 8.98X10'* COa '8000 3.36X10" Carbon monoxide, forming COa. . 2420 1.02 X lo" Copper, CuO 602 2.53X10'* Hydrogen, HaO 34000 1.43 X 10'' Marsh gas, COa and HiO 13100 5.50X10" 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 igg.S (mercuric) 99.9 " " (mercurous), 199.8 Nickel 58.6 29.3 Oxygen 15.96 ' 7.98 Platinum I94'3 64.8 Silver 107.7 107.7 Zinc 64.88 32'44 TABLE XVIL Molecular Weights and Densities of Gases. Simple Gases. Atomic Weight, Chlorine, CI3 70.75 Hydrogen, Ha 2.00 Nitrogen, Na 28.024 Oxygen, O, 31927 Sp.gr.,.fir=j. Mass in I litre,- 35-37 3.167 1. 00 o.o89i 14.012 1.254 15.96 1.429 S04 ELEMENTARY PHYSICS. Compound Gases. Atomic Weight Carbonic oxide, CO 27-937 Carbonic dioxide, CO2 43-90 Hydrochloric acid, HCl 36 . 376 Vapor of water, HjO 17-96 Atmospheric air Sp. gr., H=i. Mass in litre. 14.97 1. 251 21.95 1.965 18.188 1.628 8.98 0.804 1.293 TABLE XVIII. Electromotive Force of Voltaic Cells. Daniell i.ivolt. | Grove 1.88 volt. |Clark... 1.435 voltat 15° Electromotive force of Clarlc 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- stance. TABLE XX.. Electrical Resistance. Absolute resistance R in C. G. S. units of a centimetre cube of the substance. Temperature coefficient, a. Rt — Ro(i -\- act). -ffo. Aluminium 2889 Copper 1611 0.00388 German silver 20763 0.00044 Gold 2041 0.00365 Iron ^ 9638 Mercury 94340 0.00072 Platinum 8982 0.00376 Platinum silver, 2 Pt. i Ag 24190 0.00031 Silver 1580 0.00377 Zinc 5581 0.00365 TABLES. 505 Carbon (Carrfe's electric light) 3.9 X 10' Glassat2oo° 2.23X 10" Gutta-percha, at 24° , 3-46X 10'' " " " 0° 6.87X10=* Selenium, at 100° , 5-9 X.ip'.'. Water, at 22° 7-P X io'.° Zinc sulphate + 23 HjO 1.83X.IP" Copper sulphate + 45 HsO i.giX 10'°, TABLE XXI. Indices of Refraction. Index. Soft crown glass i . 5090 I. 5180 I . 5266 Dense flint glass 1.6157 1.6289 1.6453 Rock salt 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 i . 528 Water 1.331 1-336 Carbon disulphide.. Air at 0°, 760 mm.. Kind of Light. D D 1-344 1. 614 1.646 1.684 1.00029 1.000296 1.000300 Kind of Light. Red B E H A E G A E H 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 Di " 5896.080 D, " 5890.125 E " 5270.429 b " 5183-735 F " 4861.428 G ' 4307.961 5o6 ELEMENTARY PHYSICS. TABLE XXIII. Rotation of Plane of Polarization by a Quartz Plate, i mm. thick,. CUT perpendicular to Axis. A i2°.668 B I5-.746 C. i7°-3i8 Dj 21°. 727 E 27°. 543. F 32°-773 G 42°. 604 H 51°. 193. TABLE XXIV. Velocities of Light. Cm. per Sec. Michelson, 1879 2.99910 X lo'" Michelson, 1882 2.99853 X lo'" Newcomb, 1882 2.99860 X 10'° Foucault, 1862 2.98000 X lo'o The Ratio between the Electrostatic and Electromagnetic Units, Cm. per Sec. Weber and Kohlrausch 3 . 1074 X lo*" 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. Cornu, 1874 2.98500XIO,"'' Cornu, 1878 2.99990X10*° Youngand Forbes, 1880 3.01382X10'*^ Cm. per Sec. Exner ..■.,.... 2.920 X lo'"' 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, igi; scale of, 212 Absorption, 103; coeiBcient 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, So 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, 201. 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; KSnig's theory of, 385; Cross's experiment on, 387 Beetz, his experiment on a limit of magnetization, 245 Berthelot, heat of chemical combination, 202 BerzeliiTs, his electro-chemical series, 287 Bidwell, view of Hall effect, 316 Bililar suspension, 268, 321 Biot, law of action between magnet and electrical current, 29S 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 Boutigny, spheroidal state, 183 Boyle, his law for gases, no; limitations of, 141; departures from, 185 Bradley, determined velocity of light, 433 Breaking weight, 119 Brewster, his law of polarization by reflection, 480 Cagniard-Latour, critical temperature, 183 Calorie, 151; lesser, 151 Calorimeter, Black's ice, 153; Bunsen's ice, 153; water, 155; thermocalorime- ter of Regnault, 157; water equivalent of, 156 Calorimerty, 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, 90; equation of, 94; Plateau's experiments in, 97 Carlini, determination of Earth's density. So Carlisle, his apparatus for electrolysis of water, 283 Carnot, his engine, 206 ; his cycle, 207 Cathetometer, 6 Cavendish, experiment to prove mass attraction, 69; 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 ia 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 5 lO INDEX. Compressing pump, 140 Compressions, 109 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, l6i 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, Il6; 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 Gumming, 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; Ampfere'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. SII 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, 11 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 definitiod of osmosis, io6 Dynamics, n Dynamo-machine, 328 Dyne, 26 Ear, tympanum of, 379 Earth, density of, 79 Ebullition, 180, process of, 182; causes affecting, 182 Edlund: study of counter electromotive force of electric arc, 348 EfHux 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 Electrical machine, 268; frictional, 268; induction, 269 Electricity, unit quantity of, 252; flow of, 253, 374 Electrification by friction, 246; positive and negative, 246, 248 Electro-chemical equivalent, 284 Electrode, 282 Electrodynamometer, 321 Electirolysis, 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 c?Il, 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, 3491 Electromotive force, counter. See Counter electromotive force, 279, etc. INDEX. 513 Electromotive forces, compared by Clark's potentiometer, 334 Electrophorus, 26,9 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,-22l 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; luininiferous, 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 done 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; j)ositive 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- 33 514 INDEX. fects of current, 274; nomenclature of electrolysis, 282; voltameter, 285J division of ions, 286; tiieoryof electrolysis, 287; chemical theory of electro- motive force, 293; electromagnetic rotations, 306; induced currents, 324; effect of medium on luminous discharge, 350; electromagnetic rotation of plane of polarization, 493 Favre and Silbermann, studied heat 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 Fpcal 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 Ijar 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, 482 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, interferencfe 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, 180 Gases, 85; absoirption of, 104; diffusion of, 107; elasticity of, no; liquefaction of, by pressure, 142, 184; 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, l88 Gauss, theory of capillarity, 91 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, 6g; value of, 70 Grotthus, theory of electrolysis, 287 Grove, his gas battery, 291 Grove's cell, 292 Gyration, radius of, 56 Gyroscope, 53 Hall, defleAion of a current in a conductor, 315 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, i6l; internal, of Earth, a source of energy, 221; developed b;)f 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 Him. work done by animals, 219 Holtz, electrical machine, 270 Hooke, theory of gravitation, 68 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, I2I Hygrometer, AUuard's, 203 Hygrometry, 262 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, coefficient of, 239 Induction coil, 328; condenser connected with, 329 Induction,, electrical, 247 Induction, magnetic, 223, 237; definition of, 239 ' Induction of currents, 321 Inertia, 4, 30, centre of, 44; moment of, 56 Insulatoi, electrical, 247 Interference of light, cause of propagation in straight lines, 397; from two simi- lar sources, 436, experimental realization of, 439; from thin films, 441 Iniernode, 362 < Intel vais, 368 INDEX. 517 Ions, 282; 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; electromotiv^e force in heat units, 286 ; development of heat in electrolysis, 288 Jurin, law of capillary action, 99 y 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 K6nig, 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 5l8 INDEX. Le Roux, experiments in thermo-electricity, 342; electrical convection of lieat 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, 2g6; production of current by modification of capillary surface, 296 Liquefaction, 184; of gases, by pressure, 184 Liquids, 85; modulus of elasticity of, ii2- Lissajou, optical method of comppunding vibrations, 384 ^Loudness of sound, 365 Machine, 48; efficiency 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, g INDEX. 5 19 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 w, 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 parallelepiped, 59; experi- mentally determined, 60 Moment of torsion, ii6; 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 MuUer, J., limit of magnetization, 245 Newton, laws of motion, 27; central forces, 61; law of mass attraction, 68; quantity of liquid flowing through orifice, 134; theory of light, 396; inter- ference of Jight from films, 442; composition of white light, 453; chromatic aberration, 455;/, law of cooling, 470 Nichols, study of radiations, 471 , 520 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 Ohmj 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, I2 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, lo^ Phase, of a simple harmonic motion, 19 Phonograph, 378 i 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 1 Plateau, experiments of, in capillarity, 97 IND^X. 521 Plates, rise of liquid, between, 100; transverse vibrations of, 376 Poggendorff, explanation of gyroscope, 55 Poisson, correction for use of piezometer, 113; theory of magnetism, 244 Polariscope, 481, 482 Polarization, of an electrolyte, 287; of cells, 2gi Polarization of light, by double refraction, 476; by reflection, 4B0; plane of, 480; by refraction, 480; by reflection from fine particles, 481; elliptic and circular, 486; circular by reflection, 489; rotation of plane of, by quartz, 491; by liquids, 493 ; in rnagnetic field, 493 Polarized light, 478; explanation of, 478J 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, I2t; 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; upoif 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, 28S; move- ments of electrolyte, 288; theory of electrolysis, 289 1 Radiant energy, effects of, 462; transmission and absorption of, 466; emission of, 468 ; origin of, 469 522 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, iti 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 length, 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 beat of steam, 201 Resistance, electrical, 276, 329; depends on circuit, 276; of homogeneous cyl- inder, 278; specific, 278; varies with temperature, 279; i(nits 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 Rowland, 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, 368; 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; stAacture 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, 35a 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, io8; elasticity of, 114 Thermodynamics, laws of, 205 Thermo-electric currents, 340; hovy 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, X49; 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-electric power a function of temperature, 345; the Thomson effect, 347; electromotive force required to force spark through air, 349 INDEX. 525 Thomson efifect, 345 Tides, energy of, 220 Time, unit of, 8; measurements of, 8, 9; Maxwell's proposed unit of, 83 Tones, musical, 365; differences in, 365; determination of number of vibra- tions in, 365; wliole and semi-, 369; fundamental, 373; analysis of compl.-x, 382; resultant, 387 Tonic, 369 Torricelli, barometer, 122; experiment of, 123; theorem for velocity of efflux, 131; experiments to prove, 133 Torsion, amount pf, 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, lOl Tuning-fork, 376; sounding-box of, 378 Umbra, 402 Units, fundamental and derived, 4; dimensions of, 10; systems of, 10 VACUlfM 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; pf rods, 376; of plates, 376; communication of,, 377J of a membrane, 378; optical metliod 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, 290; 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-; mode 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, thSory 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, 68 Wright, connection of electromotive force and heat of chemical combination, 286 Young, theory of capillarity,. gi; modulus of elasticity, 115; optical method of studying vibrations, 383; interference of light from two similar sources, 439 H^^/'fo/^'- iT/-'*^ "J!££i mimmiMKmm