"'v '!'■■■; ; ■•':-'v^4«sei/ .•■•;-A:?;ih{JK'a ,■ ■' lir^r^tRKJ^i . I , '■■' ' ' ■'; i vmm^ 1 ' It '' !' f\ T I :,^' > J ' ^ 1 CORNELL UNIVERSITY LIBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 1891 BY HENRY WILLIAMS SAGE ^'If "[entary text-book of physics. 3 l'924'oT2"333"'435 Cornell University Library The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012333435 ELEMENTARY TEXT-BOOK OF PHYSICS. Peof. WM. a. ANTHONY, and Peof. CYRUS F. BRACKETT, Of Cornell University. Of Princeton University. REVISED BY Pbof. WILLIAM FRANCIS MAGIE, Of Princeton University. EIGMTE EDITION, REVISED. FIKST THOUSAND. NEW YORK: JOHN WILEY & SONS. London: CHAPMAN & HALL, Limited. 1897. Copyright, 1897, BT ■ JOHN WILEY & SONS. W^ ROBERT DRimHOND, ELECTROTYPER AND PRINTER, NEW lORK. PREFACE. The design of the authors in the preparation of this work has been to present the fundamental principles of Physics, the experi- mental basis upon which they rest, and, so far as possible, the methods by which they have been established. Illustrations of these principles by detailed descriptions of special methods of ex- perimentation 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 lecturer, who, in the perform- ance of his duty, will naturally be guided by considerations respecting the wants of his classes and the resources of his cabinet. Pictorial representations of apparatus, which can seldom be employed with advantage unless accompanied with full and exact descriptions, have been discarded, and only such simple diagrams have been introduced into the text as seem suited to aid in the demonstrations. By adhering to this plan greater economy of space has been secured than would otherwise have been possible, and thus the work has been kept within reasonable limits. A few demonstrations have been given which are not usually 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 demonstra- tion pertains may be complete, and that the student may be con- vinced of its validity. In the discussions the method of limits has been recognized Tvherever it is naturally involved ; the special methods of the cal- IV PKEFACE. cuius, however, have not been employed, since, in most institutions in this country, the study of Physics is commenced before the stu- dent is suflSciently 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. Bbackett. September, 1887. REVISER'S PREFACE. By the courtesy of the authors and publishers of this book, I have beien given an opportunity to make a rather extensive revision of it. The principal changes which have been made, besides such slight corrections or supplementary statements as seemed necessary, are, an entire rearrangement and enlargement of the mechanics, and the addition of a discussion of the kinetic theory of matter and of a treatment of magnetism and electricity by the method of tubes of force. The omissions have been largely of statements that would naturally be made by the lecturer or of demonstrations in which the results reached did not warrant the expenditure of time and trouble necessary to master them. I trust that I have adhered throughout to the original design of the authors. During the last few years I have been using with my classes Selby's "Elementary Mechanics of Solids and Fluids," and have availed myself in many places in the present revision of the sugges- tions which I received from that admirable book. The additions to the Magnetism and Electricity are based upon the treatment of the eubject by J. J. Thomson in his " Elementary Theory of Electricity aud Magnetism." W. F. Magie. Pkincbton UNrvEBSixy, February, 1897. CONTENTS. PAGE Intbodtiction 1 MECHANICS. Chapter I. Mechanics of Masses 10 II. Mass Attraction, 70 III. MoLECTjiiAK Mechanics, 83 IV. Mechanics op Fluids, 125 SOUND. Chapter I. Origin and Transmission op Sound, .... 149 II. Sounds and Music, 164 III. Vibrations op Sounding Bodies, 170 IV. Analysis op Sounds and Sound Sensations, . . 177 HEAT, Chapter I. Measurement op Heat, 186 II. Transfer op Heat, 201 III. Effects op Heat 206 IV. Thermodynamics, 243 MAGNETISM AND ELECTRICITY. Chapter I. Magnetism 259 II. Electricity in Equilibrium, 283 III. The Electrical Current, .... .312 IV. Chemical Relations op the Current, .... 333 V. Magnetic Relations op the Current, .... 841 VI. Thermo-electric Relations op the Current, . . 380 VII. Luminous Effects of the Current, .... 387 vii viii CONTENTS. LIGHT. PAGE: Chapter I. Propagation of Light, ...... 394 II. Bbflbction and Refraction, 405 III. Intbrferbncb and Diffraction, . . . • ■ 427 IV. Dispersion, 439 V. Absorption and Emission, 445 VI. Double Refraction and Polarization, .... 457 TABLEg. Table I. Units of Length 481 II. Aocelbration of Gravity, 481 III. Units op Work 481 IV. Densities of Substances at 0°, 482 V. Units of Pressure for g — 981 482 Vi. Elasticity, 482 VII. Absolute Density of Water, 483 VIII. Density of Mercury, 483 IX. Coefficients of Linear Expansion .... 483 X. Specific Heats — Water at 0° = 1, 484 XI. Melting and Boiling Points, etc 484 XII. Maximum Pressure of Vapor at Various Tempera- tures, 485 XIII. Critical Temperatures and Pressures in Atmos- pheres, AT THEIR Critical Temperatures, of Various Gases, 485- XIV. Coefficients of Conductivity for Heat in C. G. S. Units, 485 XV. Energy Produced by Combination of 1 Gram of Cer- tain Substances with Oxygen 486^ XVI. Atomic Weights and Combining Numbers, . . . 486 XVII. Molecular Weights and Densities of Gases, . . 486 XVIII. Electromotive Force of Voltaic Cells, . . . 487 XIX. Electro-chemical Equivalents, 487 XX. Electrical Resistance 487 XXI. Indices of Refraction 488 XXII. Wave Lengths op Light— Rowland's Determinations, 488 XXIII. Rotation op Plane of Polarization by a Quartz Plate, 1 mm. Thick, Cut Perpendicular to Axis, . 489 XXIV. Velocities of Light, and Values of «, . . . 489 Index, ••••-... 491 INTRODUCTION. 1. Divisions of Natural Science. — Everything which can affect our senses we call matter. Any limited portion of matter, how- ever great or small, is called a lody. All bodies, together with their unceasing changes, constitute Nature. Natural Science makes us acquainted with the properties of bodies, and with the changes, or phenomena, which result from their mutual actions. It is therefore conveniently divided into two principal sections, — Natural History and Natural Philosophy.. The former describes natural objects, classifies them according to their resemblances, and, by the aid of Natural Philosophy,, points out the laws of their production and development. The latter is concerned with the laws which are exhibited in the mutual action of bodies on each other. These mutual actions are of two kinds : those which leave the essential properties of bodies unaltered, and those which effect a complete change of properties, resulting in loss of identity. Changes of the first kind are called physical changes ; those of the second kind are called chemical changes. Natural Philosophy h.as, therefore, two subdivisions, — Physics and Chemistry. Physics deals with all those phenomena of matter which are not 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. lb is not possible, however, to draw sharp lines of demarcation between the variisus departments of Natural Science, for the sue- 2 ELEMENTARy PHYSICS. [§ 2 cessful pursuit of knowledge in any one of them requires 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 afPections of our minds. When a natural phenomenon arrests our attention, we call the result an observation. Simple observations of natural phenomena only in rare instances can lead to such complete knowledge as will suffice for a full understanding of them. An observation is the more complete, the more fully we apprehend the attending circum- stances. We are generally not certain that all the circumstances which we note are conditions on which the phenomenon, in a given case, depends. In such cases we modify or suppress one of the circumstances, and observe the effect on the phenomenon. If we find a corresponding 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 effect of the modification. We thus determine the conditions on which the phenomenon depends. In other words, we bring ex- periment to our aid in distinguishing between the real conditions on which a phenomenon depends, and the merely accidental cir- cumstances 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 before unobserved. Indeed, by far the greater part of our knowledge of natural phenomena has been acquired by means of experiment. To be of value, experi- ments must be conducted with system, and so as to trace out the whole course of the phenomenon. Having acquired our facts by observation and experiment we I 3] IKTRODUCTION. 3 seek to find out how they are related; that is, to discover the laws which connect them. The process of reasoning by which we discover such laws is called induction. As we can seldom be sure that we have apprehended all the related facts, it is clear that our inductions must generally be incomplete. Hence it follows that oonclusions 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 provision- ally as true, they enable us to predict phenomena before un- known. In conducting our experiments, and in reasoning upon them, we are often guided by suppositions suggested by previous experi- ence. If the course of our experiment be in accordance with our supposition, there is, so far, a presumption in its favor. So, too, in reference to our reasonings: if all our facts are seen to be con- sistent with some supposition not unlikely in itself, we say it thereby becomes probable. The term hypothesis is usually cm- ployed 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 demonstrated. Yet so great is the constancy of uniform sequence with which phenomena occur in accordance with the laws which we discover, that we have no doubt respecting their validity. When we would refer a series of ascertained laws to some common agency, we employ the term theory. Thus we find in the ■" wave theory " of light, based on the hypothesis of a universal ether of extreme elasticity, satisfactory explanations of the laws of reflection, refraction, difEraction, 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 distinctive characteristic is its persistence in whatever state of rest or motion it may happen to have, and the resistance which it offers to any 4 ELEMENTARY PHYSICS. [§ * attempt to change that state. This property is called inertia. It must be carefully distinguished from inactivity. 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 comparison. If we adopt arbitrary units of length, time, and mass (or quantity of matter), we can express the measure of all other quan- tities in terms of these so-caX\edi fundamental u7iits. 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 under- stood 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 of the length of a certain piece of platinum, declared to be a standard by legislative act, and preserved in the archives of France. This standard, called the metre, was designed to be equal in length to one ten-millionth of the earth's quadrant. The operation of comparing a length with the standard is often difficult of direct accomplishment. This may arise from the minuteness of the object or distance to be measured, from the dis- tant 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 in- direct methods, by which the difficulties are more or less com- pletely obviated. The vernier enables us to estimate small fractions of the unit of length with great convenience and accuracy. It consists of an 14] INTRO DUOTION. accessery piece, fitted to slide on the principal scale of the instru- ment to which it is applied. A portion of the accessory piece, ■equal to ii minus one or n plus one divisions of the principal scale, is divided into n divisions. In the former case, the divisions are numbered in the same sense as those of the principal scale ; in the latter, they are numbered in the opposite sense. In either case we can measure a quantity accu- rately to the one nth part of one of the primary •divisions of the principal scale. Fig. 1 will make the construction and use of the vernier plain. In Fig. 1, let 0, 1, 2, 3 ... 10 be the divisions on the vernier ; let 0, 1, 2, 3 . . . 10 be any set •of consecutive divisions on the principal scale. If we suppose the of the vernier to be in coincidence with the limiting point of the magni- tude to be measured, it is clear that, from the S'ig- 1- position shown in the figure, we have 29.7, expressing that magni- tude to the nearest tenth; and since the sixth division of the ver- nier coincides with a whole division of the principal scale, we have -j% of yVj or yf^, of a principal division to be added : hence the whole value is 29.76. The micrometer screw is also much employed. It consists of a -carefully cut screw, accurately fitting in a nut. The head of the screw carries a graduated circle, which can turn past a fixed line. This is frequently the straight edge of a scale with divisions equal in magnitude to the pitch of the screw. These divisions will then show through how many revolutions the screw is turned in any given trial; while the divisions on the graduated circle will show the frac- tional part of a revolution, and consequently the fractional part of the pitch that must be added. If the screw be turned through n revolutions, as shown by the scale, and through an additional fraction, as shown by the divided circle, it will pass through n times the pitch of the screw, and an additional fraction of the pitch 'determined by the ratio of the number of divisions read 6 ELEMENTARY PHYSICS. [§^ •1£ from on the divided circle to the whole number into which it is divided. The cathetometer is used for measuring differences of level. A graduated scale is cut on an upright bar, which can turn about a vertical axis. Over this bar slide two accu- rately 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 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 between which is sought, and the reading upon the scale is taken; the fixed piece is then undamped, and the telescope raised or lowered until the second point is covered by the cross-hairs, and the scale reading is again taken. The difference of scale reading is the difference of level sought. The dividing engine may be used for dividing scales or for 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 support. The table is carried along through any required distance deter- mined by the motion of the screw, and the cuts can be thus made at the proper intervals. Fig. 2. §4] INTRODUCTION. The same instrument, furnished with microscopes and access- ories, 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 Fig. 3. its name implies, it is primarily used for measuring the curvature 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 terminat- ing in points, which form the vertices of an equilateral triangle. The axis of revolution of the screw is perpendicular to the plane of the triangle, and passes through its centre. The screw ends in a point which may be brought into the same piano with the points of the legs. This is done by placing the legs on a truly plane sur- face, and turning the screw till its point is just in contact with the surface. The sense of touch will enable one to .decide with great nicety when the screw is turned far enough. If, now, we note the reading of the divided scale and also that of the divided head, and then raise the screw, by turning it backward, so that the given curved surface may exactly coincide with the four points, we can compute the radius of curvature from the difference of the two 8 ELEMENTAKY PHYSICS. [§5 Fig. 4. 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 srln of * me&u solar day. "We employ the clock, regulated by the pendulum or the chronometer balance, to indicate seconds. The clock, while sufficiently accurate for ordinary use, must for exact investiga- tions be frequently corrected by as- tronomical 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 sur- face, on which also are recorded the beginning and end of the in- terval 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 scien- tific work is the gram. It is equal to the one-thousandth part of a certain piece of platinum, called the kilogram, preserved 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 construction of which will be discussed hereafter. 7. Measurement of Angles. — Angles are usually measured by reference to a divided circle graduated on the system of division upon which the ordinary trigonometrical tables are based. A pointer or an arm turns about the centre of the circle, and the § 9] INTKODUCTION. 9 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 cirule equal to its radius. In terms of this unit, a semi-circumference equals TT = 3.141592. The radian, measured in degrees, is 57° 17' 44.8." 8. Dimensions of Units. — Any derived unit may be represented by the product of certain powers of the symbols representing 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 dimen- sio7ial 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] = U. In like manner, the dimensional equation for volume is [vol.] = U. 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 millimetre, milligram, and second is still sometimes used in Germany. MECHANICS. CHAPTER I. MECHANICS OF MASSES. 10. It is an obvious fact of Nature that material bodies move from one place to another, and that their motions are effected at different rates and in different manners. Continued experience has shown that these motions are independent of many of the characteristics of the bodies ; they depend on the arrangement and condition of surrounding bodies, and on the fundamental prop- erty of mattjer, called inertia. The science of Mechanics treats of the motions here referred to, and in a wider sense of those phe- nomena presented by bodies which depend more or less directly upon their masses. The general subject of Mechanics is usually divided, in ex- tended treatises, into two topics, — Kinematics and Dynamics. In the first are developed, by purely mathematical methods, the laws of motion considered m the abstract, independent of any causes producing it, and of any substance in which it inheres ; in the second these mathematical relations are extended and applied, by the aid of a few inductions drawn from universal experience, to the explanation of the motions of bodies, and the discussion of the interactions which are the occasion of those motions. For convenience, the subject of Dynamics is further divided into Statics, which treats of forces as maintaining bodies in equilibrium and at rest, and Emetics, which treats of forces as setting bodies in motion. 10 8 12] MECHANICS OF MASSES. 11 It has been found more convenient to neglect these formal dis- tinctions in the very brief presentation of the subject which will be given in this book. 11. Configuration and Displacement. — An assemblage of points may be completely described by selecting some one point as a point of reference and assigning to each of the others a definite distance and direction measured from this fixed point. Such a set of points is called a system of points, and the assemblage of distances and directions which characterize it is called its configuration. The motion of one or more of the points is recognized by a change in the configuration. The change in position of any one point, de- termined by the distance between its initial and final positions and the direction of the line drawn between those positions, is called the displacement of the point. Any particle in the system may be taken as the fixed point of reference, and the motion of the others may be measured from it. Thus, for example, high-water mark on the shore may be taken as the fixed point in determining the rise and fall of the tides; or, the sun may be assumed to be at rest in computing the orbital motions of the planets. We can have no assurance that the particle which we assume as fixed is not really in motion as a part of some larger system; indeed, in almost every case we know that it is thus in motion. As it is impossible to conceive of a point in space recog- nizable as fixed and determined in position, our measurements of motion must always be relative. 12. Composition and Resolution of Displacements. — If a point undergo two or more successive displacements, the final displace- ment is obviously given by the line joining its initial to its final position. This displacement is called the resultant of the others. If the point considered be referred to a point which is itself dis- placed relative to a third point taken as fixed, the motion of the moving point relative to the fixed point may be considered as re- sulting from a combination of the displacement of the first point, relative to the second point, and the displacement of the second point relative to the third or fixed point. These simultaneous 13. ELEMENTAKY PHYSICS. [§13 displacements are combined as if they were successive displace-' ments. Eepresenting them both by straight lines, of which the length measures the amount of the displacement, and the direction the direction of the displacement (Fig. 5), we apply the initial point of the second of these lines to the final point of the first and join the initial point of the first to the final point of the second. The line thus drawn is the resultant of the simultaneous displacements. The two displacements of which the resultant is thus obtained are called the components. 13. Vector Addition and Subtraction. — Any concept which is completely described when its magnitude and direction are given is called a vector. The sum of two vectors is the vector equivalent to them both, it is obtained by the rule just given for the compo- sition of two displacements, or by the following equivalent rule: Draw from any point the two straight lines which represent the vectors, and upon them construct a parallelogram; the diagonal of this parallelogram, drawn from the point of origin, is the resultant vector or the vector sum. Thus 00 (Fig. 5) is the resultant of OA and OB. This construction is called the parallelogram construction or the par- allelogram law. If more than two vectors are to be added, the resultant of two of them may be added to the third, the resultant thus obtained to the fourth, and so on until all the vectors have been combined. This addition is more easily made by drawing the vectors in succession, so that they form the sides of a polygon (Fig. 6), the initial point of each vector coinciding with the final point of the one preceding it. In general this polygon is not closed, and the line required to close it, drawn from the initial point of the first vector to the final point of the last, is' the sum of the vectors. This con- struction is called the polygon construction or the polygon law. The difference of two vectors is the vector which added to one ot the two will give the other. It is obtained by drawing from a given point the lines representing the vectors, and drawing a line § 14] MECHANICS OF MASSES. 13 from the final point of the subtrahend to the final point of the minuend. This line represents the vector difference of the two vectors. Thus AC (Fig. 5) is the difference between OG and OA. The same difference may be obtained by the following method : If two lines, equal in length, be drawn in opposite directions, they represent two vec- tor quantities which have the same mag- nitude but are affected with opposite signs. If, therefore, a vector be given which is to be subtracted from another, it may be replaced ^'<*- ^• by a vector of the same magnitude having the opposite direction, and the resultant obtained by adding this vector to the ono which serves as the minuend is the difference of the two given vectors. 14. Resolution and Composition of Vectors. — It is in many cases convenient to obtain component vectors which are equivalent to a given vector. If one component be completely given, the other is obtained by vector subtraction. If two components be desired, and their directions be given so that they and the original vector are in the same plane, their magnitudes may be determined by drawing from a common origin lines of indefinite length in the given direc- tions, drawing from the same origin the line representing the given vector, and drawing from its final point lines parallel to the given directions. The sides of the parallelogram thus constructed repre- sent the component vectors in these given directions. If three components be desired in three given directions not in the same plane, and so placed that the given vector does not lie in a plane containing any two of these directions, they may be found by constructing upon lines drawn in these directions a parallelepi- ped of which the diagonal is the given vector. This construction is most frequently used when the three directions are at right angles to one another. Representing the angles between them and the direction of the given vector by a, /?, y, the component vectors are proportional to cos a, cos /J, cos y If these three directions be the directions of the axes of a system of rectangular coordinates, these cosines are called the direction cosines of the vector. 14 ELEMENTAET PHYSICS. [§15 The composition of vectors is often conveniently effected by re- solving them in this way along the three coordinate axes; their components along each of these axes may then be added algebra- ically, and the vector obtained by combining the three sums is the required resultant vector. Thus if the vectors R^ 11,. . . Ji„ be given, making angles with the x, y, z-axes of which the cosines are A„ A^ . . . A„, fx^, ^,.../u„,v^,v,...v„, respectively, the sums of the components of these vectors along the axes are r^jR,M,+Ji,M, + ...+iinMn; > (1) The resultant vector is B == VX' + ¥' + Z% ■ and its direction cosines are X Y^ Z R' R' R' respectively. When only two vectors are given, they may be resolved along two axes in the plane of the vectors. In this case, if the angles made by the vectors R^, R, with the a;-axis be cp, ^, respectively, (Fig. 7,) the component sums are X = R^cos(p-\- R, cos -B, F =: i?, sin + R^ sin -B Fig. 7. 5. ] (3) The resultant vector is ^ = VX' + Y\ and the angle tp which it makes with the a;-axis is given by cos ^ = — or tan tfr = — . 15. Description of Motion.— If we observe a system of points in motion, we perceive not only the displacements of the points, but also that these displacements are in some way connected with the time required for their accomplishment. If we know the law of this connection, we may describe the motion at any desired instant, by the aid of certain derived concepts, which are now to be studied. If a variable quantity be a function of the time, it is usual in § 16] MHCHANICS OF MASSES. 15 Mechanics to call the limit of the ratio of a small change in that quantity to the time-interval in which it occurs the rate of change -of the quantity. This ratio is the differential coefficient of the quantity with respect to time. Other differential coefficients which occur in Mechanics, in which the independent variable is not the time, are sometimes spoken of as rates^ though not frequently. The motion of a point is described when we know not only the path along which it is displaced, but the rates connected with its displacement. 16. Velocity. — The rate of displacement of a point is called its velocity. If the point move in a straight line, and describe equal spaces in any arbitrary equal times, its velocity is constant. A constant velocity is measured by the ratio of the space traversed by the point to the time occupied in traversing that space. If s„ and s represent the distances of the point from a fixed point on its path at the instants t^ and t, then its velocity is represented by If the path of the point be curved, or if the spaces described by the point in equal times be not equal, its velocity is variable. The path of a point moving with a variable velocity may be approxi- mately represented by a succession of very small straight lines, which, if the real path be curved, will differ in direction, along which the point moves with constant velocities which may differ in amount. The velocity in any one of these straight lines is rep- o o resented by the formula v = f. As the interval of time t — t^ approaches zero, each of the spaces s — s^ will become indefinitely small, and in the limit the imaginary path will coincide with the S "~* s real path. The limit of the expression j will represent the velocity of the point 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 -j 16 ELEMENTARr PHYSICS. [§17 The practical unit of velocity is the velocity of a point moving, uniformly through one centimetre in one second. The dimensions of velocity are LT'\ Velocity, which is fully defined when its magnitude and direc- tion are given, is a vector quantity, and may be represented by a^ straight line. Velocities may therefore be compounded and re- solved by the rules already given for the composition and resolu- tion of vectors. 17. Acceleration. — When the velocity of a point varies, either by a change in its magnitude, or by a change in its direction, or by changes in both, the rate of change is called the acceleration of the point. Acceleration is either positive or negative, according as the velocity increases or diminishes. If the path of the point be a^ straight line, and if equal changes in velocity occur in equal times, its acceleration is constant. It is measured by the ratio of the change in velocity to the time during which that change oc- curs. If v„ and V represent the velocities of the point at the in- stants t^ and t, then its acceleration is represented by V — V, t: (4> If the path of the point be curved, or if the changes in velocity in equal times be not equal, the acceleration is variable. A variable acceleration in a curved path may always be resolved into two components,, one of which is tangent and the other nor- mal to the path. We will consider the case in which the path lies in a plane. Let A and B (Fig. 8) be two points in the path very near each other, from which normals are drawn on the concave side of the curve, meeting at the point 0, and ^making with each other the angle a. In Pig. 8. ^^^ li«"t, as a vanishes, the lines OA and OB become equal and are radii of curvature of the path at the § 17] MECHANICS OF MASSES. 17 point A. Draw the lines PQ and PH in the directions of the tangents at A and B, equal to the velocities v„ and v of the point at A and B respectively. The line QB is the change in the velocity of the point during the time in which it traverses the distance AB. Draw the line QS perpendicular to PE. The angle QPE, being the angle between the tangents at A and B, equals the angle a. In the limit, as a vanishes, v and «„ difEer by the infinitesimal tiB, and QS equals va. The line SB represents the change in the numerical magnitude of the velocity during the time t — t^, and the rate of that change, which takes place along the tangent to the path, is given by a. - i—i; (5> The line QS represents the change in velocity during the same time along the normal to the path. The acceleration along that yet normal is therefore ^ j. Now under the conditions assumed in AB these statements AB = ra, and ^ j = v, the velocity of the TtX point. Hence v = t r , and the acceleration along the normal ta the path is «n = p (6) • f the path be a straight line, the normal acceleration vanishes, and the whole acceleration is given by the limit of the ratio 2 r° = 77- if the path be a circle, and if the point move in it V ~~~ t f. at uniformly, the whole acceleration is given by — . The unit of acceleration is that of a point, the velocity of which changes at a uniform rate by one unit of velocity in one second. The dimensions of acceleration are LT'^. Acceleration is completely described when its magnitude and 18 ELEMENTAET PHYSICS. [§ 18 direction are given. Ifc is therefore a vector quantity and may be represented by a straight line. Two or more accelerations may be compounded by the rules for the composition of vectors. 18. Angular Velocity and Acceleration. — The angle contained by the line passing through two points, one of which is in motion, and any assumed line passing through the iixed point, will, in gen- eral, vary. The rate of its change is called the angular velocity of the moving point. If and 0„ represent the angles made by the moving line with the fixed line at the instants t and t^ , then the angular velocity, if constant, is measured by — 0„ ^ - T^- (^) If variable, it is measured by the limit of the same expression, -jT- = - _ , -, as the interval t — t„ becomes indefinitely small. The angular acceleration is the rate of change of angular velocity. If constant, it is measured by If variable, it is measured by the limit of the same expression, doo 00 — 00. -TT- — , _ , , as the interval t — t„ becomes indefinitely small. If the radian be taken as the unit of angle, the dimensions of angle become ;^;5j^g J — ~l — ^- Hence the dimensions of angular velocity are T'', and of angular acceleration T~'. If any point be revolving about a fixed point as a centre, its velocity in the circle is equal to the product of its angular velocity and the length of the radius of the circle. 19. Linear Motion with ■ Constant Acceleration. The space s -s„ traversed by a point moving with a constant acceleration a, during a time t-t^, is determined by considering that, since the acceleration is constant, the average velocity ■ ~^ "° for the time ^ 20] MECHANICS OF MASSES. 19 ^ — ^„ , multiplied by t —t„, will represent the space traversed ; hence s — s„ = ^"(^-0; (9) 2 -or, since -^ = — 4 —, we have, in another form, s-s, = v,{t - ^.) + i a{t - Q\ (9a) Multiplying equations (4) and (9), we obtain v' = V + 2a[s - s„). (10) "When the point starts from rest, v^—0; 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 5„ = 0, #„ = 0, and equa- tions (4), (9a), and (10) become v = at, s = ^af, and v' = 2as. Formula (9a) may also be obtained by a geometrical construction. At the extremities of a line AB (Fig. 9), equal in length to t — t„, erect perpendiculars ^C and BD, proportional to the initial and final velocities of the moving point. For any interval of time Aa, so short that the velocity 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 increas- ing by successive equal increments, is rep- resented by a series of rectangles, eb, fc, gd, *" ''''' «tc., described on equal bases, ab, be, cd, etc. If ab, be ... he 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 A BCD. This area, therefore, represents the space traversed by the point, having the initial velocity v„, and moving with the acceleration a during the time t — t„. But ABGD is equal to A C{t - ;;„) -f {BD — AG){t - tj -i-2 ; whence s - s„ = v„(t - Q + ia{t - Q\ (9a) 20. Angular Motion with Constant Angular Acceleration. — If a 20 ELEMENTAKT PHYSICS. [§ 81 point move in a circle its velocity is equal to the product of its angular velocity and the radius of the circle; its acceleration in the circle is equal to the product of its angular acceleration and the radius of the circle. If its angular acceleration be constant, the relations between the distance traversed by it in the circle, its velocity, its acceleration in the circle and the time are the same as those expressed in equations (9), (9a), (10). Substituting for these quantities their equivalents in terms of the angular magnitudes involved, we obtain the following relations among these angular magnitudes: 0-0„ = ^±^(;-O; (11) - 0. = «o(^ - ^o) + Mt- t^y ; (12) cj' = go/ + 2«(0 - 0„). (13) If the line describing the angle start from rest, co, = 0, and if we take the line in this position as the initial line from which to reckon and and A cos (p, A — (2a' + Sa" cos e)* ; and, by dividing the value of A sin — tan"^:; — ; . •' 1 + cos e The displacement thus becomes s + s' = a(2+2 cos e)* cos [cot - tan-> -^^BJ— V m\ ^ ' \ 1 + cos e/ ^ ' This equation is of great value in the discussion of problems in optics. The principle suggested by the result of the above discussion, that the resultant of the composition of two simple harmonic motions is a periodic motion of which the elements depend on those of the components, can be easily seen to hold generally. A very important theorem, of which this principle is the con- verse, was given by Fourier. It may be stated as follows: Any complex periodic function may be resolved into a number of simple harmonic functions of which the periods are commensurable with that of the original function. 22. Force. — When we lift or sustain a weight, stretch a spring, or throw a ball, we are conscious of a muscular effort which we designate as a force. Since no change can be perceived in the weight if it be suspended from a cord, or in the spring if it be held stretcTied by being fastened to a hook, and since the ball moves in just the same way if it be projected from a gun, we conclude that § 22] MECHANICS OF MASSES. 35 bodies can exert force on one another. This conclusion is not strictly justifiable, and our comparison of the action of one body ■on another to the action of our muscles may be only a convenient analogy. If we throw a weight by exerting a certain efEort for a short time, and then by exerting an equal efEort for a longer time, we find that the velocity acquired by the weight is greater in the latter case. If we apply different efforts for the same time in throwing the same weight, we find that the efEort which we are •conscious of as greater gives the weight a greater velocity than that -efEort which we are conscious of as less. We may substitute for the forces exerted by our muscles those forces which we have assumed by analogy to act between bodies. Eelying upon the uniformity -with which these forces act, as determined by universal experience, 'we can exhibit, more precisely than by the use of our muscular «fiEort, the relations which obtain between the force exerted and the motion caused by it. As our experiments increase in precision, and as one disturbing cause after another is eliminated, we find that the velocity acquired by a given body acted on by a given force increases m proportion to the time during which the force acts, or, as may be said, a constant force produces a uniform acceleration. Purther, if different forces act on the same body for the same time, the velocities produced are proportional to the forces. If F repre- sent the magnitude of the force, t the time during which it acts, v the velocity which the body acquires, and m a proportional factor, the results of these experiments may be embodied in the formula Ft = mv. (18) The factor m is called the mass or the inertia of the body. Since — measures the acceleration of the body, this equation is equiva- lent to F=ma. (19) The dimensions of force are MLT'^. The practical unit of force is the dyne, which is the force that 26 EL-EMENTAET PHYSICS. [§ 23 can impart to a gram of matter one unit of acceleration; that is to- say, one unit of velocity in one second. 23. Impulse.— The product Ft is called the impulse. If the force which acts upon the body vary during the time, the impulse is determined by dividing the time into intervals so small that the force which acts during any one of them may be considered con- stant, forming the product Ft for each interval, and adding those products. 24. Momentum. — The product mv is called the momentum of the body. It is sometimes defined as the quantity of motion of the body; in Newton's laws, which follow, the word "motion" is equivalent to momentum, when it designates a measurable quan- tity. 25. Laws of Motion. — The relation between force and accele- ration, which is embodied in the formula F = ma, was first per- ceived by Galileo, and illustrated by him by the laws of falling bodies. This relation may be expressed otherwise by the state- ment that the effect of a force on a body is independent of the motion of the body. Newton, who first formulated the funda- mental facts of mdtiou in such a form that they can be made the basis of a science of Mechanics, extended Galileo's principle by recognizing that when several forces act on a body at once the effect of each is independent of the others. Newton's Laws of Motion, in which the fundamental facts of motion are stated, are as follows : Law I. — Every body continues in its state of rest or of uni- iorm motion in a straight line, except in so far as it may be com- pelled by external forces to change that state. Law II.— Change of motion is proportional to the external force applied, and takes place in the direction of the straight line in which the force acts. Law III.— To every action there is always an equal and con- trary reaction; or, the mutual actions of any two bodies are always equal and oppositely directed. These laws cannot be applied, without some limitations or modi- § 36] MECHANICS OF MASSES. 27 fications, to all bodies. They are to be understood as applying tO' very small masses, for which we can neglect the velocity of rotation in comparison with the velocity of translation. Such a mass is called a particle. A particle may also be defined as a mass concen- trated at a point. Another definition will be given in | 37. These laws of motion are not immediately susceptible of proof ; they are abstractions, which can be illustrated but not proved by experiment. They cannot be referred to any more ultimate prin- ciples deduced from our observation of Nature, and are therefore to be considered as postulates upon which the science of Mechanics is erected. The question of their validity as expressions of the mode of motion of matter is one which lies outside the range of the purely physical study of the subject. 26. Discussion of the Laws of Motion. — (1) The first law is a statement of the important truths, 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 any deviation from this uniform rectilinear motion is caused by a force. 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 conceive 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 direc- tion 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 28 ELEMENTARY PHYSICS. [§ 37 further statement that more than one force can act on a body at the same time, leads directly to a most important deduction re- specting the combination of forces ; for the parallelogram law for the resolution and composition of velocities being proved, and forces being proportional to and in the same direction as the velocities which they cause in any given body, 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 velocity. This construction is called the ^parallelogram 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 pro- ■duce, motion, their mutual action is called a stress. If one body he 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 mature of stress, and that the two forces constituting the stress aro •equal and oppositely directed. 27. Constrained Motion. — One of the most interesting appli- ■cations of the third law is to the case of constrained motion. If the motion of a particle be restricted by the requirement that the particle shall move in a particular path, it is said to be constrained. If the velocity of the particle at a point in the path, at which the radius of curvature is r, be v, its acceleration v' toward the centre of curvature is -, and the force which must act r on it in that direction is — . However this force is applied, whether by a pull toward the centre or by a push or pressure from the body determining the path, or by the action of the forces •which bind the particle to others moving near it, the reaction of the particle will in every case be equal to -^—, and will be directed § 28] MECHANICS OF MASSES. 29 outward along the normal to the path. This reaction is sometimes called a centrifugal force. There are certain cases in which it may be treated as if it were a real force, determining the motion of a body. 28. Work and Energy. — If the point of application of a force F move through a distance s, making the angle a with the direction of the force, the product Fs cos a is defined as the toork done by the force during the motion. If the force or the angle between, the direction of the force and the displacement vary during the displacement, the work done may be found by dividing the path of the point into portions so small that F cos a may be considered constant for each one of them. By forming the product Fs cos a for each portion of the path, and adding all such products, the work done in the path is obtained. 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 position, and the amount of work is independent of the time taken to perform it. Both of these statements need to be made, because of our natural tendency to confound work with conscious effort, and to estimate it by the effect on ourselves. If work be done upon a particle which is perfectly free to move, its velocity will increase. In this case the force F is measured by ma, where m is the mass of the particle and a its acceleration. We may suppose that the particle has the velocity «„ when it enters upon the distance s, and that the distance s coincides with the direction of the force. Using equation (10), we then have Fs = mas = ^mv' — \mv'. (20) The product ^mv' is called the kinetic energy of the particle. The equation shows that the work done upon the particle by a constant force is equal to the kinetic energy which it gains during the motion. If the direction of the, motion or the magnitude of the force vary, we may divide the path into small portions, for each of which the force may be considered constant. Forming the equation just proved for each of these portions and adding the 30 ELBMBNTAEY PHYSICS. [§ 38 equations thus obtained, we obtain for this general case the same result as that already obtained for the special case. The forces introduced by constraints need not be considered, since they are always perpendicular to the path, and so do no work. When several forces act at a point, the work done by them during any small displacement of the point is equal to the work done by their resultant; for the sum of the projections of all the forces on the line of direction of the resultant is equal to the resultant, and the sum of the projections of each of these projec- tions upon the direction of motion or the projection of the result- ant upon the direction of motion is equal to the sum of the pro- jections of each force upon the direction of motion. If, then, several forces do work on a particle, the kinetic energy gained by the particle will be equal to Es cos a, where E is the resultant of the forces, and a the angle between its direction and the direction of the displacement s. Let us suppose that the forces are so related that E=0. Then the work done by one of the forces must be equal and opposite to that done by the others, the particle will move with a constant velocity, and no kinetic energy will be gained. If any of the forces against which work is done are such that they depend only upon the position of the particle in the field, the work that is done against these forces is equal to that which is done by them if the particle traverse the path in the opposite direction. Such forces are called conservative forces. Other forces, which are not functions of the position of the par- ticle only, but depend on its motion or some other property, are called non-conservative forces. When a particle acted on by con- servative forces is so displaced that work is done against those forces, it is said to have acquired jiotential energy. The measure of the potential energy acquired is the work done against the conservative forces. Energy is frequently defined as the capacity for doing work. The propriety of this definition is obvious in the case of potential energy; for the particle, acted on by conservative forces, and left free, will move under the action of these forces, and they will § 29] MECHANICS OF MASSES. 31 thereby do work. The particle possessing kinetic energy has also the capacity for doing work, for, in order to bring it to rest, the amount of work given by the formula Fs = imv^ must be done upon it. The unit of work and energy is the work done by a unit force upon a particle while it is displaced in the direction of the force through unit distance. , The dimensions of energy are MUT'', 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 and is 3iot a vector quantity. The practical unit of work and energy is the erg. It is the work done by a force of one dyne, in moving its point of application in the line of the force through a space of one cen- timetre : 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 Telocity. 29. Bodies, Density. — The particle with which we have been dealing hitherto has no counterpart in Nature. In our experi- «nce we have to deal with extended bodies or systems of bodies, and the description of their motions and of the way in which forces act on them is more complicated than the corresponding descrip- tion for the ideal particle. The notion of the particle is never- theless of great utility : we may in the first place consider bodies as composed of numbers of these particles or as being systems of particles; and, in the second place, we may to some extent de- scribe the motion of bodies by comparison with the motion of a particle. It is, however, often convenient to be able to represent the mass of a body as distributed continuously throughout its volume. In that case we make use of a special concept, the density. To define it we suppose the particles of the body so distributed that each unit volume in the body contains the same number of them. 32 ELEMENTARY PHYSICS. [§ 30 The density is then defined as the ratio of the mass of the body to its volume, or as the mass contained in a unit of volume. By sup- posing the mass of the body uniformly distributed throughout its volume, so that the ratio of mass to volume has the same value no- matter how small the volume is, we may represent the mass con- tained in any infinitesimal volume by the product of the density and the volume. The concept of density used in this way is an artificial one, and the validity of the results obtained by it is due to the fact that the particles constituting a body are so small that their distribution is practically uniform in a homogeneous body in. any jolume which can be examined by experiment. M The formula for density is D —-^ , and the dimensions are [£>] = ML'^- The unit of density is the density of a homogeneous body so constituted that unit of mass is contained in unit of volume. By using the hypothesis of a continuous distribution of matter in a body, we may define the density at a point in a body which is not homogeneous as the ratio of the mass contained in a sphere described about that point as centre to the volume of the sphere,, when that volume is diminished indefinitely. 30. Centre of Mass. — The ceyitre of mass of two particles is defined as the point which divides the straight line joining thfr particles into two segments, the lengths of which are inversely pro- portional to the masses of the particles at their extremities. Thus if A and B be the positions of the two particles of which the masses are ?»„ and m,, respectively, then the point C, lying on the line joining A and B, is the centre of mass if it divide AB so that m^. AC = lUf, . BC. The centre of mass of more than two particles is found by find- ing the centre of mass of two of them, supposing a mass equal to their sum placed at that centre, finding the centre of mass of this- ideal particle and a third particle, and proceeding in a similar way until all the particles of the system have been brought into com- bination. The final centre thus found is the centre of mass of the § 30] MECHANICS OF MASSES. 33 system. The point thus determined is independent of the order in which the particles are taken into combination; it is a unique point, and depends only on the positions of the particles and their masses. The centre of mass may be defined analytically as follows: Let the particles m^, m„ ... be referred to a system of rectangular coordinates. The coordinates $, r/, Z of the centre of mass are then given by the equations ^ _ m,a;, + rn^x, + ■ ■ • _ -^^^ "" 7J = m, -\-m^-{- . . . 2m ' in,y, + w,y, + . . ■ _ 2my _ '^i + "'a + • • • -Swi ' _ _ m,z, + m^z^ + . . . _ 2mz ~ w, + OT, + . . . — 2m ' (21) These equations are evidently consistent with the former defini- tion of the centre of mass, if we remember that if the line joining any two particles be projected on one of the axes, the segments into which it is divided by the centre of mass of the two particles will be in the same ratio after projection as before. Consider the two particles m^ and m^, and denote the coordinate of their centre of mass by 5. Then from the former definition of the centre of mass we have mX^ — xj — m^x^ — ^), from which 4 _ — !-_!_ZI — 2-2-. This demonstration can easily be extended to m^ + m, include all the particles of the system. If some of the particles of the system be in motion, the centre of mass will, in general, also move. Its velocity is determined by the velocities of the separate particles. Let 5„, »;„, C„, represent the co- ordinates of the centre of mass at the time t^ , while ^, tj, Q repre- sent its coordinates at a later time t. The component velocities of the centre of mass are then given by the limit of the ratios -3 ~, -^ 7^, -J — f^. Using the equations which define the coordinates of the centre of mass, we have : 34 ELEMENTARY PHYSICS. [§30 5„ ^ ^m{x-x,) / t- t. V - V, t- ■ t. C- -c. i 2m = 2 2m = 2- t-L 2m = 2- My - y.) .m{z — z„) t-t^ (22) The terms on the right are the components of momentum of the separate particles, and the equations express the law that the veloc- ity of the centre of mass of a system of particles is equal to the resultant obtained by compounding the momenta of the separate particles and dividing it by the sum of all the masses of the system. Representing the component velocities of the centre of mass by IJ, V, W, and those of the separate particles by u, v, w, the rule just given may be expressed by U2m = 2mu, V2m = 2mv, W2m = 2mw. It the velocities of some or all of the particles vary, the velocity of the centre of mass will in general vary also. Its acceleration depends upon the accelerations of the separate particles. Letting U and f/„,etc., represent the component velocities at the times t and t^, we may express the component accelerations of the centre of mass by U-U„ , m(u - u„) t- K t- -h V- y. 2m = 2 m{v t- -O t- K -t. w- t- 2m 2 m(w t- (23) The terms on the right represent the components of the forces which act on each particle of the system, and the equations express the law that the acceleration of the centre of mass of a system of particles is equal to the resultant of all the forces which act on the separate particles divided dy the sum of the masses of the particles. This law may be otherwise expressed by saying that the acceleration of the centre of mass is the same as that which would be given to § 31] MECHANICS OF MASSES. 35 a particle having a mass equal to the sum of all the masses if it ■were acted on by a force equal to the resultant of all the forces. Forces which act between particles belonging to the same system are called internal forces ; such forces do not afEect the motion of the centre of mass, for, by Newton's third law of motion, they al- ways occur in pairs, of which the two members are equal and oppo- site. They therefore contribute nothing to the resultant force, and so do not influence the acceleration of the centre of mass. If the only forces which act be internal forces, the acceleration of the centre of mass is zero and the momentum of the system remains constant. This principle is known as the conservation of mo- mentum. 31. Kinetic Energy of a System of Particles. — The kinetic en- ergy of a system of particles may also be expressed in terms of the Telocity of the centre of mass. Represent by u, v, w the compo- nents of velocity of each particle, by U, V, W the components of velocity of the centre of mass, and by a, b, c the components of Telocity of each particle relative to the centre of mass. We have then M,= Z7+a„ u,=:^ U+a„ . . . v,^V+b„ v,= V+i„... to^ = W+c,, w,=W+c,,... The kinetic energy of the particle wz, is im^{u' + v,'' + w'), ■and the kinetic energy of all the particles or of the system is the sum of the similar expressions obtained for each particle of the system. Substitute in the equation for the kinetic energy the val- ues of m", v', w'. We consider first the values of m'. We have u,' = n' + a.' + 2a, U, < = U' + a,' + 2a,U, ... Multiplying by ^m and adding, we obtain ^^mu' = iU'{m, + w, + . , . ) + im,a,' + Jm^ff/ + . . . + U\mfi^ -f m,a, + ...). Now since a„ a,, . . . are referred to the centre of mass as origin, and since in that case the coordinates of the centre of mass are zero, the sum m,a, + »",«, . . . must equal zero. If the expres- 36 ELEMENTARY PHYSICS. [§ 32 Bions for Jm.v,' + ^m,v,' . . . , im,w,'' + im^w,' ... be formed in a similar manner, and added to the expression just obtained, we have on the left the sum of the kinetic energies of the particles, and on the right the expression The first of these terms expresses the kinetic energy of a mass equal to the sum of all the masses moving with the velocity of th& centre of mass. The other terms express the kinetic energies of the separate particles moving with their velocities relative to the centre of mass. We therefore arrive at the following rule : The kinetic energy of a system of particles is equal to the kinetic energy of a mass equal to the sum of all the masses moving with the velocity of the centre of mass, plus the kinetic energies of the separate masses moving with their velocities relative to the centre of mass. 32. Work done by Forces on a System of Particles. Potential Energy. — The forces which act on the particles of a system may be classified as external and internal forces. The external forces arise from the action of bodies outside the system, the internal forces from action between parts of the system. If the resultant of all the forces which act on any one particle be considered as the force which acts on that particle, the particle will acquire kinetic energy, given by the formula Fs = ^mv" — ^mv', already estab- lished (§ 28). If, however, we consider the resultant of the ex- ternal forces acting on the particle as producing kinetic energy and doing work against the internal forces which act on the parti- cle, the work done by the former will be equal to the kinetic en- ergy gained by the particle plus the work done against the latter. If the internal forces be conservative, the work done against them can be recovered when the external forces cease to act. The action of the external forces in that case gives to each particle potential energy. In case the external forces equilibrate the internal forces for each particle, the velocities of the particles remain constant, no kinetic energy is gained, and the energy given to the system by the work done is wholly potential. In any case the energy gained % 34] MECHANICS OF MASSES. 37 by the system is equal to the work done on it by the external forces. If no external forces act on a system, its energy remains constant, however the velocities of the separate particles may ■change in consequence of the action of internal forces. A rigid body is. one in which the particles retain the same rela- tive positions. Whatever internal forces act between the particles, they are equilibrated by others due to the reactions in the system. The internal forces can therefore do no work, and the internal en- ergy of such a body is wholly kinetic energy. 33. Conservation of Energy.— The theorem stated in the last section is the simplest illustration of the general principle known as the conservation of energy. If no external forces act on a sys- tem, and if the internal forces be conservative, the sum of the ki- netic and potential energies of the system remains constant. In many operations in Nature, however, the internal forces are not all conservative, and the theorem just stated no longer holds true. Ex- periment has shown that when non-conservative forces act, other forms of energy are developed, which cannot as yet be expressed as the potential and kinetic energies of masses, and that if these forms of energy be taken into account, the sum of all the energies of the system remains constant so long as no external forces act on it. This principle is called the principle of the conservation of energy. It may be used as a working principle in solving ques- tions in mechanics, and finds a very wide application in all depart- ments of physical science. The evidence for it will appear in ■connection with many of the topics which are subsequently treated. 34. Systems to be Studied. — The description of the mot-ions of a system of particles which are free to move among themselves, and between which forces act, cannot in most cases be given. Cer- tain general theorems relating to this general case can be found, but the conditions which determine the individual motions of the particles are so complicated that they cannot be brought into a form suitable for mathematical discussion, and hence the motion of the system cannot be completely described. There are two ■cases, however, of very general character, in which, by the aid of 38 ELEMENTARY PHYSICS. [§ 35 certain limitations assumed for the system, we are able fully to de- scribe its motions. The first of these is that of a pair of bodies which act on each other with a force, the direction of which is in the line joining the bodies. This case, known as the problem of two bodies, may be completely solved. The problem of three bodies can be solved only approximately, under certain limitations as to the relative magnitudes of the bodies. The second case is that in which the system forms a rigid body. While no truly rigid bodies exist in Nature, yet the changes of shape which most solids under- go under the action of ordinary forces are so slight in comparison with their dimensions that in many cases we may consider such solids as rigid, and illustrate the theorems relating to rigid bodies by experiments made upon solids. We shall first examine the mo- tion of rigid bodies, and we shall limit ourselves to the case in which the motions of any one particle of the body always take place in one plane. By thus restricting the problem, it is possible to obtain the most essential facts connected with the motions of rigid bodies without the use of advanced mathematical methods. 35. Impact. — The changes in motion impressed upon bodies by their impact with others depend upon so many conditions that they present complications which render the discussion of them impossi- ble in this book. We will consider, however, the simple case of the^ impact of two spheres, the centres of which are moving in the same straight line. We call the masses of the two spheres m, and m^ and their respective velocities tt, and «,. The two spheres consti- tute a system for which the velocity of the centre of mass is given by (m, +m,)V= m^u^ + m^w,. (35) The bodies on impact are momentarily distorted, and a force arises between them tending to separate them, the magnitude of which depends upon the elasticity of the bodies. The velocity of the centre of mass will remain uniform, whatever be the forces act-, ing between the bodies, and the momenta of the two bodies relative to the centre of mass, both before and after impact, will be equal and opposite. Call the velocities of the bodies after impact v and § 35] MECHANICS OF MASSES. 39 V,. We then nave m,(M, - V) = m,{V - u,) and W2,( V - v^) = in^y^ — V). That these equations may both be true we must have f>- = rr — ^> ^'i experimental constant, called the coef- ficient of restitution. The coeificient e depends upon the elasticity of the bodies and their mode of impact. It has been shown by experiment to be always less than unity. From these equations we deduce «(«*. - M,) = t', - V,. ' (26) Combining this equation with the equation for the velocity of the centre of mass, we obtain for the velocities v, and v, after impact the equations V, = F -? — e(M, — M ) ; V+ -^ e{u,-u,). (27) The kinetic energy before impact equals im^uj' + ^m^u^. The kinetic energy after impact equals ^wi,«;,° + ^m^z;/. Substituting in this last expression the values just obtained for v^ and v^ and reducing, we obtain for the kinetic energy after impact (m, +wO(l-e')F ' e'(OT,M.' + m,u,') 3 "^ 3 • By subtracting this from the kinetic energy before impact we find that the loss of kinetic energy by impact is _3^ . LZ1\,, _ ^^).. (28) If the bodies are such that e = 0, or such that the velocities after impact are both equal to the velocity of the centre of mass, they are called iweZas^ic bodies; the kinetic energy lost by their collision is V-^- . ' „ — ^. If, on the other hand, e = 1, so that the m, + m, 2 velocities after impact relative to the centre of mass are equal to those before impact but of opposite sign, the bodies are called perfecily elastic bodies. In this case no kinetic energy is lost by 40 ELEMENTARY PHYSICS. [§ 36 the collision. These extreme values of e are never exhibited by real bodies, though the value « = may be closely approached in many instances. No body has a value of e that is even appreciably equal to 1, so that there is always a loss of kinetic energy by im- pact. The energy thus lost is transformed into other forms of energy, principally into heat. 36. Displacement of a Rigid Body. — Under the limitation that we have set, that the points of the body shall move only in parallel planes, it is manifest that the motion of the body is completely given if the motion of its section by any one plane be given. In describ- ing the displacement of a body under these limitations we need only describe the displacement of one of its sections by one of the planes in which the motion occurs. It is furthermore clear that the motion of this section will be completely described if the motion of any two points in it or of the line joining them be given. When a body is so displaced that each point in it moves in a straight line through the same distance, its displacement is called a translation. When the points of the body describe arcs of circles which have a common centre, its displacement is called a rotation. Any displacement of a body may be effected by a translation com- bined with a rotation. To show this, let AB (Fig. 11) represent the initial posi- tion of a line in the body, A'B' its final position. The transfer from the initial to the final position may be effected by a translation of the line AB to such a position that the point C, which may be any point in the body, coincides with the corresponding point C". Taking this point C" as the centre, a rotation through an angle 6, which is the same whatever point be chosen for C, will bring the line into its final position. While the angle of rotation is the same whatever point be chosen for O the translation which brings C into coincidence with C" will differ for different positions of C. § 37] MECHANICS OF MASSES. 41 If the line AB he rotated through the angle ff about any point in it, and if then another point in it be taken and the line rotated about that point through an angle ~ff, the result is a translation of the line AB. We may therefore substitute for a rotation about one point a translation and an equal and opposite rotation about another properly chosen point. By the following construction it is always possible to find a point in the plane in which AB moves, such that a pure rotation of AB about it will bring the body from its initial to its final posi- tion. Join AA', BB' (Fig. 12), and bisect the lines AA' and BB' at the points C and D. At those points erect perpendiculars which will intersect at the point 0. Join OA, OB, OA', OB'. By the geometry of the figure the triangles A OB and A' OB' are similar, and adding to their equal angles at the common angle A' OB, we have AOA' — BOB'. Hence a rotation through the angle AOA' = BOB' will transfer AB to A'B'. The perpendicular through may be called the axis of rotation. This construction fails when the initial and final positions ot AB are parallel. 37. Kinetic Energy of a Rotating Body. — Let r represent the ■distance of any particle of the body, of mass m, from the axis about which the body rotates, and co its angular velocity about that axis. Then the kinetic energy of this particle is ^mr^oo', and the kinetic ■energy of the rotating body is 100' Smr'- In § 36 we have shown that we may replace a rotation by a translation and a rotation of the same amount about another axis. Since velocities are measured by the displacements of the moving particle which occur in the same interval of time, it is also possible to replace an angular velocity by a velocity of translation and an equal angular velocity in the oppo- site sense about another axis. We choose for the new axis that passing through the centre of mass, at the distance R from the original axis. The velocity of the centre of mass Is then Rao. We represent by 1 the distance of the mass m from the axis passing 42 ELEMENTAKT PHYSICS. [§ 38 through the centre of mass. The kinetic energy of the body rotat- ing about this centre is -^SmV, and the kinetic energy of the whole body moving with the velocity of the centre of mass is ioa'R'2m. By § 31 we have ioo'^mr' = ioo'R'Sm + ^as'^mr. (29) When a rigid body is so small that its kinetic energy due to its rotation about its centre of mass is negligible in com- parison with that due to its translation, it is called a particle. This definition supplements that of § 35. 38. Moment of Inertia. — The expression ^mr' is called the moment of inertia of the body about the axis from which r is measured. The formula just obtained shows that the moment of inertia about any axis is equal to the moment of inertia about a parallel axis passing through the centre of mass plus the moment of inertia of a particle of which the mass is equal to the mass of the body placed at the centre of mass. The moment of inertia depends entirely upon the magnitude of the masses making up the body and their respective distances from the axis. If the mass of the body be distributed so that each ele- ment of volume contains a mass proportional to the volume of the element, the moment of inertia then becomes a purely geometrical magnitude, and may be found by integration. It is evident that it is always possible to find a length h such that k^2m = ^mr\ This length k is called the radius of gyra- tion of the body about its axis. 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 pendulum— which has not been proved— for the sake of clearness. The body of which the moment of inertia is desired is set oscillating about an axis under the action of a constant force. Its time of oscillation is, then, t = ni^ § 39] MECHANICS OF MASSES. 43 ■where j. is the moment of inertia and / a constant depending on the magnitude of the force. If, now, another body, of which the moment of inertia can be cal- culated, be Joined with the first, the time of oscillation changes to /7+7 t' = ny — ^ — , where /' is the moment of inertia of the body added. Combining the two equations, we obtain, as the value of the moment of inertia desired, I-fT^^' (30) 39. Rotation about a Fixed Point.— Suppose a body so condi- tioned that its only motion is a rotation about the fixed point (Fig. 13). Suppose the force i^ applied at a point in the body, which moves under the action of the force ^ - through the infinitesimal distance QR. This motion is a rotation about the point through the angle (b — %-r. The work done by the OQ ^ force during this rotation is Since, in the limit, when QR and Q8 are infinitesimal, the triangles QT> OP QRS and OPQ are similar, — = jj^, and hence W=F.OP.^ = F.OP. — for the hyperbola, ^ < — for the el- lipse, and — = - for the parabola. That is, for the three curves respectively, the velocity of the particle at a point in its path is greater than, less than, or equal to the velocity which it would ac- quire by falling to that point from an infinite distance under the action of the central force. The elements of the path may be obtained from these equa- 2v '« ' tions. The latus rectum of the parabola is — 2_o__ The semi- '^ m 58 ELEMENTARY PHYSICS. [§ 51 major axis a and the semi-minor axis h of the hyperbola and ellipse are given by 2a = '^, &' = 3" = ^^^- Hence C7 = ^, and V^ 7TI Til using this value of C in the original equation, we get ^ = - ± -^, the upper and lower signs holding for the hyperbola and ellipse respectively. In case the particle is moving in an ellipse, its periodic time T, or the time in which it traverses the ellipse, may be found in terms of the elements of the ellipse and the constant m. The area of the ellipse is Ttab, and since the areas swept out in equal times by the radius vector drawn to the particle are equal, the rate at which the area is swept out is given by -=-. But -^ also rep- resents this rate, so that -;=- — -~. Substituting in this equa- tion the value of 6 = '".p\~j ' "^^ S** i^^) "^ = ~^> °^' "^ - -f^- If, therefore, different particles revolve in ellipses about a. common centre of force in such a way that the squares of their periodic times are in the same ratio to the cubes of their semi-major axes, the constant m is the same for all of them. 51. The Problem of Two Bodies. — The problem of two bodies may be reduced to the problem of the action of a central force. For, suppose two particles to attract each other with a force given Ll97t by —5-, where pi and m are their masses and r the distance between them. The acceleration of the particle m, relative to the centre of mass, which will remain fixed in position, is given by ma — ^-^, or by a = -. The acceleration of the mass fx relative to the centre of mass is similarly -5-. If now an acceleration equal to ^ and oppo- site to it in direction be impressed on both particles, the particle m will remain fixed, and the particle fi will move relatively to it § 52] MECHANICS OF MASSES. 59 with the acceleration -^t — _ The path of the particle yu relative to the particle m will be therefore that due to a central force pro- ceeding from w and equal to -^^ — 5 '. The radius vector drawn to // from m will still sweep out equal areas in equal times, and the path of f^ will still be a conic section. If its path be an el- lipse, the periodic time will be given by y= - — ■ — r-; so that 47rV rpi • 52. Motion of Projectiles. — In the special case of central motion in which the distance of the centre from the moving particle is very great and the velocity of the particle small, the particle de- scribes a portion of an ellipse which differs very little from a para- -yS nyi /yy) bola. This may be seen at once from the equation — = — , for if r be very great and v small, the semi-major axis a must also be very great, and the path approaches the curve for which 2a is infinite, or a parabola. This is the path described by a particle moving near the surface of the earth under the earth's attraction. The force which acts on the particle is really variable and directed toward the earth's cen- tre, but within the limits of the path it may be considered constant and directed vertically downward. This motion was first discussed by Galileo in connection with his study of falling bodies. His method was as follows: Let us as- sume the rectangular coordinates x and y, of which x is horizon- tal and y vertical, drawn upward in the direction opposite to the acting force. Let a particle be projected from the origin in the plane of the axes with the velocity z) in a direction which makes with the a;-axis the angle a. The component velocities along the two axes are then v cos a and V sin a. At the end of any time t reckoned from the instant at which the particle leaves the origin, the displacement of the €0 ELEMENTARY PHYSICS. [§ 53 particle along the 2:-axis is vt cos a. If the force of gravity did not act on the particle -its displacement along the «/-axis in the same time would be vt sin a; but, since gravity acts, its real dis- placement along that axis is less than this by s = \gf, where g is the measure of the force or the acceleration of gravity, so that its displacement along the ?/-axis is vt sin a — Igf. The path of the particle, or the series of points which it occupies at successive in- stants, is found by eliminating t between the two equations for the two rectangular displacements. The equation of the path thus obtained is sin « 9 i ion\ y = « — -■ , , x\ (39) ^ cos a 2v cos a This represents a parabola passing through the origin. The axis ^ V^ COS (X is vertical, and the latus rectum is ^^ . If or = 0, or if the 2v' projection is horizontal, the equation becomes a;' = y, repre- senting a parabola with its vertex at the origin. When the body is projected above the horizontal plane, so that a lies between zero and -, it will attain its greatest height at the instant when its velocity along the y-a,xia becomes zero, or when t; sin a = gt. The time required for it to describe its whole path and return to the a;-axis is double this time or . Its range, or the distance between its starting-point and the point at which it again meets the a;-axis, is given by the product of this time and its v' v^ horizontal velocity w cos (y, or is - 2sinacosa = — sin Sa. The range is therefore a maximum when a = 45°. Since sin {tt — 2a') = sin 2a, the range is the same for projections at the angles a and 90° — a, or for projections equally inclined to the line bisecting the angle between the axes and on opposite sides of it. 53. Difference of Potential. The Potential. — Forces may arise from various causes. In any case they are only exhibited when § 53] MECHANICS OF MASSES. 61 they affect the motion of bodies and they may be considered, for purposes of mathematical representation, as acting between the par- ticles of the bodies. K field of force is a region in which a particle, constituting a part of a mutually interacting system, will be acted on by a force, and will move, if free to do so, in the direction of the force. The strength of field or the intensity in the field at a point is measured by the force which acts upon a unit quantity or test unit of that agent to which the force is due when placed at that point. The test unit is supposed not to affect the forces of the field. When the force acting on a particle depends only on the position of the particle the study of its effects is often very much facilitated by the use of a concept called the potential. To explain this con- cept and its relation to force, we begin with a definition of the dif- ference of potential. The diilerence of potential between two points in a field of force is equal to the work done by the forces of the field in moving a test unit from the one point to the other. If Vp — Vq represent the difference of potential between the points P and Q, and if F represent the average force between those points and s the distance between them, then the amount of work done by that force in moving a test unit from P to Q, and hence the difference of potential between P and Q, is represented by Vp-Vg^ Fs. From this relation we have ^^Z-zZ^^-E^nIf (40) If s become indefinitely small, in the limit F represents the force at the point P, and ^^ — - — -f— becomes the rate s el's of change of potential at that point with respect to space, taken with the opposite sign. Hence we obtain a definition of potential. It is a function, the rate of change of which at any point, with re- spect to space, taken with the opposite sign, measures the force at that point. Let the test unit be situated at the point A and be moved over 63 ELEMENTARY PHYSICS. [§ 54 any path to the point B. It is clear that if it be moved back over the same path from B to A the amount of work required to effect this motion will be equal and opposite to that done during the motion from A to B. This equality will also hold if the test unit be moved from 5 to ^ by any other path, provided the field of force is one which is nowhere interrupted by a region in which the force is not a function only of the position of the test unit, or is, as it is called, a singly connected region. The fields of force due to all forces known in Nature, except those caused by elec- trical currents, are singly connected regions. When the forces which act on the test unit at different points in its path are paral- lel, as in the case of gravity, this equality of the work done in carrying the test unit from one place to another over any path is obvious. If we assume the principle of the conservation of energy as a general principle, this equality may also be shown for fields in which the forces are not parallel ; for, if the work done in moving the test unit over one path between A and B were not equal to that done in moving it over any other path between the same points, an endless supply of work could be obtained by repeatedly moving the unit over a path in which the work done by the forces of the field is greater, and returning it to its starting-point by mo- tion over a path in which the work done is less. As this result is inconsistent with the principle of the conservation of energy, we conclude that the hypothesis from which it is deduced is untrue, and that the same amount of work will be done in moving the unit from the one point to the other, by whatever path the motion is effected. The difference of potential between the two points is therefore a function of their positions only. 54. Equipotential Surfaces and Lines of Force.— Let the test unit be moved from along the different paths OA, OB, etc. (Fig. 32), so that the same amount of work is done upon it in each of these paths. The surface drawn through the end points of these paths is called an equipotential surface ; as may be seen from the proposition just proved, it is a surface in which the test unit may be moved without doing any work upon it. Since the forces § 55] MECHANICS OF MASSES. 63 in the field will only do no work on the test unit when it is moved at right angles to their directions, it follows that the forces at the different points in an equipotential surface are normal to that sur- face. Draw the normals AP, BQ, etc., of such lengths that the work done in moving the test unit over them is the same. The surface drawn through their end points is again an equipotential surface. By repeating this process, the whole field of force may be mapped out by equipoten- tial surfaces. In the limit, as the lengths of the normals thus drawn become in- finitesimal, the successive normals will form continuous curves, everywhere nor- _, „„ mal to the equipotential surfaces which they cut. These curves, which represent the direction of the force at the points through which they pass, are called lines of force. If a small area be described on an equipotential surface, the lines of force which pass through its boundary will form a tubular surface, which will cut out corresponding areas on the other equi- potential surfaces. This tubular surface, with the region enclosed by it, is called a tube of force. 55. An Expression for DiflFerence of Potential. — In what follows we will for convenience assume that the test unit is a unit mass, and that the field of force is due to the presence of particles which attract the unit mass with forces that are proportional to their masses and vary inversely with the square of the distance. By the proper choice of units the force due to any one particle may be set €qual to — , where m is a constant proportional to the mass of the particle and r the distance between it and the test unit. Let the point (Fig. 23) be the point at which a particle m is placed, and let a unit mass traverse the path Pi?X under the action of the force -5 directed toward 0. When the particle is at 64 ELEMENTARY PHYSICS. [§ 55 P, the force acting on it is ^^; after it has moTed through the infinitesimal distance PB, the force acting on it at R is ^^. The work done upon it during this motion, equal to the product of the force and the line PQ, the pro- jection of its path upon the direction m . PQ' of the force, is greater than ' m . PQ and less than — tt-^; it may be shown UK that the work done during this dis- placement may be represented by ''"''■ Wi^mt- ^owPQ=OP-OQ = OP — on in the limit, so that the work done in this displacement equals \^p — „ „ ' = MyrB — 775) • ^^^ work done in travers- ing the following elements of the path, BS, ST, etc., is expressed by mijr-a— 7Jp)> ^[7yr~ TT^J' ^^^' '^^® work done in traversing the whole path from P to Xis the sum of these expressions, or mfyr^— jrpj. By thc definition of difference of potential, this expression is equal to the difference of potential between the points P and X, due to the force of which the centre is 0. If the point X lie at an infinite distance from 0, the work done by the force in moving the unit mass to that point equals — ^^p- This expres- sion is called the potential at 'the point P. It has been obtained on the supposition that the force at P is directed toward 0, or i& an attractive force. In this case the test unit at an infinite dis- tance possesses the potential energy B. In moving to P the forces of the field do upon it the work -y^, so that its potential energy at P IS ^ - -^. § 56] MECHANICS OF MASSES. 65 If the force at P and at the other points on the path be directed from 0, the work done in the successive elements of the path is numerically equal to the expressions already obtained, but is opposite in sign; so that the work done by such forces, as the test unit moves from P to X, is equal to »wf-yp — yf^X When the point X is at an infinite distance from 0, the work done in moving the test unit from P to X equals yjp. This is the poten- tial at the point P, due to a repulsive force with its centre at 0. In this case the test unit at an infinite distance has no potential 7)1 energy, so that ^^7, expresses its potential energy at P. 56. Flux of Force. Tubes of Force. — Still retaining the con- Tention that the forces of the field are due to mass attraction and follow the law of inverse squares, we will now prove certain pro- positions which are of great importance in the theories of gravita- tion, electricity, and magnetism. If in a field an area s be described so small that the force is the same for all points of it, the product of the area and the normal component of the force is called the elementary _/?Ma: of force over or through that area. We will show that the total flux of force,, that is, the sum of all the elementary fluxes, taken over a closed surface in the field which does not contain any masses is equal to zero. We consider first the flux of force arising from a mass m situated at the point 0. Let ABC (Fig. 24) rep- resent a closed surface not containing the mass m; draw a tube of force cutting this surface in the elements s and s' The forces due to the mass m at points in these areas will be -„ and ^7^ respec- ■*!<*• <'*• tively. We represent the angles between the common direc- tion of these forces and the normals to the elements s and s' 66 ELEMENTARY PHYSICS. [§ 56 drawn outward from the surface by a and a' respectively. The components of the forces ^ and -75- drawn outward normal to the surfaces s and s' are — cos a and —f^ cos a' respectively. Hence the flux of force through these elements is —^ s cos a -f ?7Z -75- s' cos a'. But s cos a and — s' cos a' are equal to the normal cross-sections of the tube of force at the distances r and r' from 0, the minus sign being inserted because one of the two cosines is negative; and since the tube of force is a cone, s cos a s' cos a' Hence the flux of force through these two elements, due to the mass at the point 0, is equal to zero. Since similar tubes of force may be drawn from the point so as to include all the elements of the surface ABC, and since to each pair of elements thus de- termined the same proposition applies, it follows that the total flux of force due to the mass m through the surface is equal to zero. The same proposition will hold for the flux of force due to any other particle situated outside the surface, and therefore holds true for any mass whatever situated outside the surface. The flux of force through a closed surface containing any number of particles is equal to 4:7rM, where M is the mass of all the particles. To prove this, let us consider a single particle m situated at the point 0. About this point describe a sphere of radius r. The force at each point of the sphere is -, and the total flux of force through the sphere is equal to this force multiplied by the area of the sphere, or to ^ Attt^ = inm. Now to prove a similar proposition for any closed surface enclosing the mass m, m describe about the point a sphere which is entirely enclosed by the surface. Since the region enclosed between this sphere and § 57] MECHANICS OF MASSES. 67 the surface contains no mass, the total flux of force through it equals zero. But the flux of force through the sphere equals — 4:7Tm, the minus sign being used because the normals to the sphere, when considered as bounding the region enclosing the mass and as bounding the region between it and the closed surface, have opposite directions. Therefore the flux of force through the closed surface must equal iTim. This proposition holds for each of the masses contained within the closed surface, so that, if the sum of these masses heM,the total flux of force through the closed surface is 4:7rM. Let us apply the theorem just proved to the region bounded by a tube of force of very small cross-section and by two rectan- gular cross-sections of the tube. Since the tube of force is every- where bounded by lines of force, and since, therefore, the force at a point on the tube has no component normal to the surface of the tube, the only parts of the closed surface under consideration which contribute to the flux of force through it are the two ■end cross-sections. Represent the areas of the two cross- sections by s and s', and the forces acting at them by i^and F' respec- tively. Then, since the total flux of force equals zero, we have Fs -\- F's' = ov Fs = — F's'. The minus sign appears because the force and the normal to the cross-section are in the same direc- tion at one end of the tube and in opposite directions at the other. If we confine our attention to the numerical value of the product Fs, we may say that the flux of force is the same for all cross-sections of the tube of force. This proposition, though here proved only for a tube of force of very small cross-section, mani- festly may be generalized for any tube of force whatever. 57. Special Cases. — It is sometimes important, especially in the study of electricity, to know the force which is exerted by a plane sheet of matter at a point near it. We call the quantity of matter which is enclosed by a unit area drawn on such a sheet the surface density of the sheet at the point where the area is taken; more strictly, the surface density is the ratio of the quantity of matter enclosed by the area to the magnitude of the area, as the area di- 68 ELEMENTARY PHYSICS. [§ 57 minishes indefinitely. If we consider an infinitely extended 'plane sheet, it is evident that the lines of force in the region near it are perpendicular to its surface. Take any small area on the surface of the sheet, and consider the closed surface bounded by the lines of force which pass through the boundary of that area and by two cross-sections taken parallel with the sheet on the opposite sides of it. The ilux of force through the sides of the surface thus formed is zero, because the lines of force lie in that surface. The only portions of the surface, therefore, which contribute to the flux of force, are the end cross-sections. Let s represent the area of each of these cross-sections, which are equal, F the force at one of them, and F' that at the other. If is the angle between the perpendicular and the new position of the thread. The force acting upon the bob at any point in the circle of which the thread is radius, if it be released and allowed to swing in that circle, varies as the sine of the angle between the perpendic- ular and the radius drawn to that point. If we make the oscillation so small that the arc may be substituted for its sine without sensi- ble error, the force acting on the bob varies as the displacement of the bob from the point of equilibrium. A body acted on by a force varying as the displacement of the body from a fixed point will have a simple harmonic motion about its position of equilibrium (§ 21). Hence it follows that the oscillations of the pendulum are sym- metrical about the position of equilibrium. The bob will have an amplitude on the one side of the vertical equal to that which it has on the other, and the oscillation, once set up, will continue forever unless modified by outside forces. The importance of the pendulum as a means of determining the value of g consists in this : that, instead of observing the space traversed by the bob in one second, we may observe the number of oscillations made in any period of time, and determine the time of one oscillation; from this, and the length of the pendulum, we can. calculate the value of g. The errors in the necessary observations and measurements are very slight in comparison with those of any other method. § 62] MASS ATTKA.CTIOK. 75 61. Formula for Simple Pendulum.— The formula connecting the time of oscillation with the value of g is obtained as follows : The acceleration of the bob at any point in the arc is, as we have seen, g sin 0, or gcp if the arc be very small. The acceleration in a simple harmonic motion is — go's = ^^s, where s is the displacement. Since the bob has a simple harmonic motion, we may set these two expressions for the acceleration equal, neglecting the minus sign, which merely expresses the fact that the acceleration is toward 4:7l' the centre of the path; hence g

and ' s' for every external stress in which its molecules are in equilibrium. Any change of the stress in the body is accompanied by a readjust- ment of the molecules, which is continued until equilibrium is- again established. 98. Strains. — The complete geometrical representation of the changes of form which occur when a body is strained is in general impossible, or at least exceedingly complicated. In the theory of elasticity it is generally possible to avail ourselves of a simplifica- tion in the character of the strain, which facilitates its geometrical representation, by assuming that the strain is such that a line in the body which was straight in its unstrained position remains straight after the strain: such a strain is called a homogeneous- strain. It may be shown, by an argument too extended for pre- sentation here, that in any case of homogeneous strain there are always three directions in the strained body, at right angles to one another, in which the only change produced by the strain is a change in length and not a change in relative direction. Thus, if the strained body be originally a cube, with its sides parallel to these three directions, the cube will strain into a rectangular parallelepiped. If the strained body be originally a sphere, it will strain into an ellipsoid, the three axes of the ellipsoid being the three directions already mentioned. These three directions are- called the principal axes of strain. The increase in length of a line of unit length by strain is. called its elongation. Evidently, from the description of the rela- tions of a homogeneous strain to the principal axes, the whole strain will be described if the elongations along the principal axes be given. Let us denote by e, , e, , e, the elongations, which may be either positive or negative, along the three principal axes. These elongations are assumed to be so small in comparison with the unit line that their squares or products may be negkcted. Then, in the examples just given, if a represent a side of the cube before strain and a' its volume, the increase in volume of the cube by the strain is given by a\l -h e,) (1 + «,) (1 + e,) - «' = «'(«, + e, + «,)> since- the products of the e'a may be neglected. Similarly, the sphere,. 110 ELEMENTABT PHYSICS. [§ 98 of which the radius is r, becomes by the strain the ellipsoid, of which the axes are r(l + «,), r{\ + ej, r(l + e,); the increase in volume of the sphere by the strain is therefore inr\l + e,)(l + ej (1 + e,) - inr' = inr^e^ + e, + e,). The quantity e, + e, + e, is called the coefficient of expansion of the body. Two cases of strain need to be specially examined — the pure expansion or dilatation, and the shear or shearing strain. A dilata- tion occurs if the three coefficients of elongation are equal; in this case the strained cube remains a cube, the strained sphere remains a sphere, and the change of volume in each case is 3e times the original volume. A shear occurs when one of the coefficients, say «3 , equals zero, and when e, equals — e,; in this case the expansion is zero. The shear may be defined from another point of view. For, consider a body subjected to a shear and suppose a section made in c c, D E D ^* ^y ^^^ plane containing the elonga- 7 tions e and — e: it is clear that the shear / will be completely described if Ve de- scribe the deformation of a figure in this plane. We select for this purpose a * ^ rhombus, ABDC, of which the diagonals AB and BC are so related that after the shear we have AD{l+e) = BO and BC{1 - e) = AD. If the rhombus produced by the shear be turned until one of its sides co- incides with AB, we shall have the original rhombus and the one produced by shear in the relation shown in Fig. 34. The new rhombus AC'D'B may manifestly be produced from the original rhombus by the displacement of all its lines parallel to the fixed base AB, each line 'being displaced by an amount proportional to its distance from the line AB. The ratio of this displacement to the distance of the displaced line from the base AB is called the DD' amount of the shear; that is, -^ is the amount of the shear. § 100] MOLECULAR MECHANICS. HI 99. The Superposition of Strains.— We will now show that two elongations, applied successively or simultaneously in the same direction, are equivalent to a single elongation equal tp their sum. This follows from the assumption already made, that the elonga- tions are so small that their squares or products may be neglected. For, suppose a line of unit length to receive the elongation e, ; its length becomes 1+e,. If it then receive the elongation e, , its length becomes 1 + e, + e,(l+«,) = 1 +e, + e,, because the" pro- duct e,e, may be neglected. This principle is called the principle ■of the superposition of strains. By its help we may show that a simple elongation may be pro- duced by the combination of a dilatation and two equal shears in planes at right angles to each other. In the case of a simple elon- gation, the elongations along the principal axes are e, 0, 0. Let Jis suppose a dilatation of which the elongations are -, -, -; a shear O O of which the elongations are 4— -^, 0; and a shear of which the o o 6 6 elongations are -, 0, — -. By the principle of the superposition of strains we find the elongations produced if these three strains be superposed by adding the three elongations along the three axes. Carrying out this operation we obtain e, 0, as the elongations pro- duced by the superposition, that is, the superposition of these three strains is equivalent to a simple elongation. Since all homogeneous strains may be produced by three simple elongations at right angles to each other, any homogeneous strain may be produced by a Combination of dilatations and shears. 100. Stresses. — If a body be maintained in equilibrium by forces applied to points on its surface, and if we conceive it divided into two parts, A and B, by an imaginary surface drawn through it, and if we assume, for the present, the molecular structure of matter, it is clear that the forces applied to the portion A of the body are in equilibrium with the forces which act between the molecules of A lying near the surface which divides it from B, and the mole- 113 ELEMENTAKT PHYSICS. [§ 100 cules of B lying on the other side of that surface. Similarly, th& forces which act on B are in equilibrium with the forces which act across the surface between the molecules of B and A. Let us con- sider any area s taken in the surface separating A and B. Repre- sent by F the sum of the molecular forces which act across that area. If the forces which act across different equal elements of the F area be equal, the ratio — is called generally the pressure per unit s area on the surface s, or, simply, the pressure on the surface. This pressure is positive if the force i^^be directed away from the portion of the whole body which is held in equilibrium, negative if directed toward that portion. It is plain, from the equality of action and reaction, that if this force be directed toward the portion A of the body, an equal force is directed toward the portion B at every point of the surface which separates A and B. The name pressure is frequently reserved for a negative pressur& in the sense just defined; when the pressure is positive, it is fre- quently called a tension. In case the force which acts across the surface between A and B vary from element to element of that surface, the pressure at a point of the surface is the limit of the F ratio - , when the area s is so drawn that its centre of inertia is o always kept at that point, and is diminished indefinitely. The forces acting across the surface separating A and B will, in general, make different angles with the surface at the different points of it. Similarly, the pressure which is substituted for the- forces makes different angles with the surface at different points. The pressure, being a vector quantity, like the force from which it is derived, may be resolved into components perpendicular to the surface and in the plane tangent to it. It is best, for the sake of greater generality in our statements, to consider the tangential component of pressure as resolved into two components, at right angles to each other in the tangent plane. These components are called respectively, the normal pressure and the tangential pres- sures. § 100] MOLECULAR MECHAU-ICS. Il3 To examine the relations which must hold among the compo- nents of pressure in different directions at any point within a body subjected to stress, we consider a small cube described in a body, and examine the relations among the pressures on its faces neces- sary to maintain it in equilibrium. We assume that no external forces act directly on the matter contained in the cube. In gen- eral, each of the faces of the cube will be subjected to a stress. This stress may be resolved into a normal component and two tan- gential components taken parallel with the sides of the face to which the stress is applied. Calling the normal components acting on two opposite faces P and P', those acting on another pair of opposite faces Q and Q' , and those acting on the third pair R and R' , we may ex- press the conditions that the centre of mass of the cube will not be displaced by the equations P=P', Q= Q', R — R'. Since the forces which act upon the cube are in equilibrium, and since their normal components maintain the equilibrium of the centre of mass, their tangential components give rise to couples, and these couples are also in equilibrium. These couples are ar- ranged as shown in Fig. 35, for those lying in the plane of one pair of faces. Since equilibrium exists, the two couples formed by the forces S and the forces /S'are equal, and therefore S = S', where S and ;S" may be used to denote the tangential pressures on the surfaces of the cube. Similar coup- les in equilibrium will act on the cube in two other planes at right angles with this one so that the whole set of pressures acting on the cube are the three normal pressures P, Q, R, and the three tangential pressures S, T, U. It may be shown, by an analytical method that need not be given, that if a small sphere be described about a point in the body and the pressures applied to its surfaces examined, there will be three radii at right angles to each other, at the extremities of which the pressures are normal to the surface of the sphere. These three directions are called the principal axes of stress. 114 ELEMElirTARY PHYSICS. [§ 101 The combination of tangential stresses which maintain equilib- rium may be considered from another point of view. For, if we ex- amine the triangular prism of which the cross-section is ABD (Pig. 35), and to which the tangential stresses S and S' are applied, it appears at once that equilibrium will obtain when a fbrce equal to the resultant of aS and aS', where a is the area of each of the square faces of the prism, is applied to the face of which AB is the trace. The area of this face is a V2, and if X represent the pres- sure on this face, the force applied to it is aXV2. But S equals S', and the resultant of aS and aS' is aS V2 ; whence JC=S. A similar pressure acts in the opposite direction upon the face of the similar prism ACB. These pressures are positive, that is, they are tensions which tend to separate the parts of the body to which they are ap- plied. If we compound the tangential stresses in another manner by taking as the element of the combination the stresses applied to the faces AD and A C, it is at once evident that they are equivalent to a negative pressure S upon the diagonal face CD. A similar pres- sure acts across the same face toward the other prism CBD. We may therefore consider the set of stresses constituting the couples in the plane A CBD as equivalent to a positive pressure or tension in the direction of one diagonal and a negative pressure in the direction of the other diagonal. This combination of couples, or its equivalent tension and pressure, is called a shea?-ing stress. 101. Superposition of Stresses. — Stresses, whether pressures or tensions, being vector quantities, are compounded like other vector quantities, and, in particular, when they are in the same line, are added algebraically. Suppose a cube so subjected to stress that equal and opposite pressures, which we will assume to be directed outward from the cube, act on two opposite faces, and that the other faces experience no stress. Such a stress is called a longitudinal traction. We will show that this form of stress may be obtained by the combination of a stress made up of equal tensions acting on each face of the cube, and of two shearing stresses. In Fig. 36 let P represent the value of the longitudinal §101] MOLECULAR MECHANICS. 115 traction. It may be considered as made up of three equal trac- p tions — -. Apply to each of the four other faces of the cube two p ■opposite stresses, each equal to — . Two of these pairs of stresses are represented in the figure. These stresses on the sides of the cube, being equal and opposite, are equivalent to no stress. It is evident that •«— the combination of stresses here described is equivalent to a tension — applied to «ach face P of the cube, to a shearing stress — acting in the plane of the figure. Fig. 36. and to a shearing stress — acting in the plane at right angles to the plane of the figure. Thus the longitudinal traction may be resolved into a tension uniform in all directions and two shearing stresses, all of the same numerical value. The uniform tension just employed is an example of a hydro- static stress. More generally, a hydrostatic stress is a stress which is normal to any surface element drawn in a body, whatever be its di- rection. The numerical value of a hydrostatic stress is the same in ■whatever direction the surface be drawn to which it is applied. To show this, we examine the relations of the pressures on the faces of the tetrahedron formed bypassing a plane through the points ABC taken infinitely near the point (Fig. 37) on lines drawn through that point in the directions of the three coordinate axes. Let I, m, n represent the di- rection cosines of the normal to the face ABC, and let a represent the area of this face ; the areas of the other faces are respectively equal to al, am, an. Let X, P, Q, R represent the pressures on the faces in the order mentioned : the forces acting on the faces are then Xa, Pal, Qain, and Ran. By the definition of hydrostatic 116 EJiEMENTART PHYSICS. [§ 102 stress these forces are normal to their respective faces, and the tetrahedron will be in equilibrium when the components of the force JT are equal respectively to the forces applied to the other faces ; that is, when Xa . Z = P.al, Xa .m — Q. am, Xa .n = R.an-^ that is, when X=P= Q=R. It has been stated that the stresses in a body may always be rep- resented by the combination of three longitudinal stresses at right angles to each other. Since a longitudinal stress may be replaced by a hydrostatic stress and two shearing stresses, it follows that any stress in a body may be replaced by a hydrostatic stress and a proper combination of shearing stresses. 102. Relations of Stress and Strain. Modulus of Elasticity. — When a body serves as the medium for the transmission of stress it experiences a deformation or strain, the type of strain depending upon the stress applied. The resistance offered by a body to de- formation is ascribed to its elasticity. If the body be deformed in a definite way by a given stress, and recover its original condition when the stress is removed, it is said to be perfectly elastic. If the deformation of a body do not exceed the limits within which it may be considered perfectly elastic, it may be proved by experiment that the strain is of the same type as the stress and proportional to it. This law was proved for certain cases by Hooke, and is known as Hoohe's Laxv. The ratio of the stress applied to the strain experienced by a unit of the body measures the elasticity of the substance com- posing the body. This ratio is called the modulus of elasticity of the body, or simply its elasticity; its reciprocal is the coefficient of elasticity. It is of course understood that the stress and strain are of the same type. Thus, for example, the voluminal elasticity of a fluid is measured by the ratio of any small change of pressure to the corresponding change of unit volume. The tractional elas- ticity of a wire stretched by a weight is measured by the ratio of any small change in the stretching weight to the corresponding change in unit length. Since all stresses may be reduced to hydrostatic stresses and S 1^^] MOLECULAE MECHANICS. 117 shearing stresses, and all strains to dilatations and shearing strains, the knowledge of the voluminal elasticity and of the elasticity ex- hibited during a shear, or the rigidity, is sufficient to describe the elasticity of the body under any form of stress. 103. Voluminal Elasticity.— Let a body of volume V be sub- jected to a uniform hydrostatic stress P, by which it undergoes a change of volume, given by v. From Hooke's law we know that, at least within certain limits of stress and consequent deformation, V and P are proportional. The dilatation or the change of the unit of volume is y. The modulus of elasticity in this case, or the vol- uminal elasticity of the body, is therefore — . The voluminal V elasticity is denoted by k. 104. Rigidity.— Let S be one of the tangential stresses which constitute a simple shearing stress, that is, a shearing stress of which the elements act in one plane; then the deformation pro- duced is a simple shear. The modulus of rigidity is measured by the ratio of the shearing stress to the amount of the shear (§ 98); it is denoted by n. The amount of the shear may be defined in a more convenient form as follows : Let us suppose that the rhombus A GDB (Pig. 38) has been strained by a simple shear into the , ■' '^ c c D E d' rhombus A C'D'B, and that this deformation \ — ] / is infinitesimal. The elongation of the diag- \ I ■ /y onal AD is then FD'. The triangle DFD' \ \ // is then an isosceles triangle, since the angle \ \ yy DFD' is a right angle, and the angle DD'F \/_ differs from half a right angle only by an in- * ^ finitesimal. Therefore i^lCVg = Z»Z»'. Now Pre- 38. AD, being the diagonal of a rhombus that is only infinitesimally dif- ferent from a square, is equal to BE V2; and therefore the amount of the shear, or ^^^ , equals , that is, equals twice the elonga- iion along the axis of the shear. 118 ELEMENTARY PHYSICS. [§ 10& The modulus of rigidity is therefore equal to half the tangen- tial stress ^S" divided by the elongation of unit length along the axis of the shear. 105. Modulus of Voluminal Elasticity of Gases. — Within cer- tain 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 afterwards fully proved by Mariotte. It is known, from its discoverer, as Boyle's law. Thus, if p and p' represent different pressures, v and v' the cor- responding volumes of any gas at constant temperature, then pv = p'if. (45/ 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. Let p and v represent the pressure and volume of a unit mass of gas at a constant temperature. A small increase Ap of the pres- sure will cause a diminution of volume Av; by Boyle's law we have the relation pv ^= {p -\- A'())(v — Av) =pv-\- vAp — pAv — ApAv. We may assume that the increment Ap is very small, in which case Jv will also be small; we may therefore, in the limit, neglect the product of these increments and obtain -~- = — . Now — is the Av v V Ap change of unit volume, and therefore — r-*' = ^ is the modulus of voluminal elasticity. The elasticity of a gas at constant tempera- ture is therefore equal to its pressure. 106. 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 § 106] MOLECULAR MECHANICS. 119 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 instru- ment was devised by Oersted, while not the best for accurate deter- minations, 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 im- mersed 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 downwards 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 com- pressed-air manometer (§ 124) 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 the change in the capacity of the flask, due to the pressure. Oer- sted assumed, since the pressure on the interior and exterior walls was the same, that no change would occur. Poisson, however, showed that such a change would occur, and gave a formula by which it might be calculated. By introducing the proper correc- tions. Oersted's piezometer may be used with success. A different form of the instrument, employed by Kegnaulfc, is, however, to be preferred. In it, by an arrangement of stopcocks, it is possible to apply the pressure upon either the interior or ex- terior wall of the flask separately, or upon both together, and in this way to experimentally determine the correction 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 120 ELEMENTAKY PHYSICS. [§ 107 regarded as incompressible. Thus, for example, the alteration of volume for sea- water by the addition of the pressure of one atmos- phere is 0.000044. The change in volume, then, at a depth in the ocean of one kilometre, where the pressure is about 99.3 atmos- pheres, is 0.00437, or about 5-^ of the whole volume. 107. Modulus of Voluminal Elasticity of Solids. — The modulus of voluminal elasticity of solids is believed to be generally greater than that of liquids, though no reliable experimental results have yet been obtained. The modulus, as with liquids, difEers for difPerent bodies. 108. Elasticity of Traction. — The iirst experimental determina- tions of the relations between the elongation of a solid and a tension acting on it were made by Hooke in 1678. Experimenting with wires of different materials, he found that for small tractions the elongation is proportional to the stress. It was afterwards found that this law is true for small compressions. The ratio of the stretching weight to the elongation of unit length of a wire of unit section is the modulus of tr actional elas- ticity. For different wires it is found that the elongation is pro- portional to the length of the wires and inversely to their section. The formula embodying these facts is SI , ,„. e = — , (46) where e is the elongation, I the length, s the section of the wire, S the stretching weight, and ju the modulus of tractional elasticity. The behavior of a body under traction may be examined in the following way: We assume for convenience that the traction is applied to the upper and lower faces of a cube with sides of unit length. As already shown, the traction P is equivalent to a hydro- P static tension — - and two shearing stresses equivalent to two tensions P . . . P — in the direction of the traction, and a pressure — in each of two directions at right angles to this and to each other. The hvdro- f 109] MOLECULAR MECHANICS. 131 P static tension causes an increase of volume given by -^ . This is P •equivalent to an elongation of each side of the cube equal to ^, .since the changes of form are supposed infinitesimal (§ 98). One -of the shears produces an elongation between the upper and lower p faces equal to ^ — , and a negative elongation or contraction equal to P — between one pair of the other faces. The other shear produces an equal elongation between the upper and lower faces and an equal contraction between the remaining pair of faces. The total elonga- tion between the upper and lower faces is therefore P (-^ + - — J, xind the total contraction between either pair of the other faces is given by p(A^-gi). Since the two shears involved in the longitudinal traction cause no change of volume, the change of volume experienced by the body is due to the hydrostatic tension alone. It is therefore equal p to —f-. A body under longitudinal traction will therefore experi- otC €nce an increase of volume unless it is practically incompress- p ible, that is, unless the ratio ^ is negligible. 109. Elasticity of Torsion. — When a cylindrical wire, clamped at one end, is subjected at the other to the action of a couple, the axis of which is the axis of the cylinder, it is found that the amount of torsion, measured by the angle of displacement 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 relation among these magnitudes may be shown to be represented by the formula r=-^., (47) nnr 123 ELEMENTARY PHYSICS. [§ 109 where t is the amount of torsion, I 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 Coulomb, who applied them in the construction of an apparatus called the torsion balance, 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. Arrangements 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 oftorsinn; 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 radians; then C= 0r. 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 the 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. The angular acceleration a of the oscillating body is given by the equation C = &t =^ la {% 39). Now, since every point in the body has a simple harmonic motion, in which its displacement is proportional to its acceleration, and since its displacement and ac- celeration are proportional respectively to the angular displacement r and the angular acceleration a, we may set a = -^r. Making § 111] MOLECULAR MECHAN'IOS. 123 this substitution, we obtain © = -^, or = ^, (48) if we observe the single instead of the double oscillation. The torsion balance may therefore be used to measure forces in absolute units. If the value of just obtained be substituted in equation (47), we obtain 30Z "inn n — — I = — rjT-- (49) 7ir rH ^ ' Since all these magnitudes may be expressed in absolute units, we may obtain the value of n, the rigidity, by observing the oscilla- tions of a wire of known dimensions, carrying a body of which the moment of inertia is known. 110. 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 origi- nal 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 expressing the relations of these magnitudes is f=^-^ (50) 111. 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 both the voluminal elasticity and the rigidity of solids to be prac- tically perfect for every kind of strain. If the strain be carried beyond the limits of perfect elasticity, the body is permanently de- formed. This permanent deformation is called set. 124 ELEMENTAKT PHYSICS. [§ 111 Upon these facts we may base a distinction between solids and fluids : a solid requires the stress acting on it to exceed a certain limit before any permanent set occurs, and it makes no difference how long the stress acts, provided it lies within the limit. A fluid, on the contrary, may be permanently 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 have no rigidity and 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 moMle ; 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 allowing intervals of time to elapse between the suc- cessive additions of small weights, 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 origi- nal form is not immediate, but gradual. If the wire carrying 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 elasticity 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 in- iernal work done among the molecules becomes transformed into heat. CHAPTER IV. MECHANICS OF FLUIDS. 112. Pascal's Law. — A ^perfect fixiid may be defined as a bodj which ofEers no resistance to shearing-stress. No actual fluids are- perfect. Even those which approximate that condition most nearly,. ofEer 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 containing vessel, the pressure at every point and in every direction is the same. This law was first stated by Pascal, and is known as Pascal's law. The truth of Pascal's law appears at once from what has been proved about hydrostatic stress (§ 101). For since the fluid ofEers no resistance to a shearing stress, the only stress within it on any surface must be perpendicular to that surface, and hence has the same value in all directions at a point. To compare the pressure at any two points we draw a line joining them, and, with it as an axis, describe a right cylinder with an infinitesimal radius, and through the two points take cross-sections normal to the axis. Then the pressures on the cylindrical surface being everywhere nor- mal to it, have no tendency to move it in the direction of its axis, and since it is in equilibrium, the presspres on its end surfaces must be equal. 135 126 BLEMENTAET PHYSICS. [§ 113 If a vessel filled with a fluid be fitted with a number of pis- tons 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. If n -\- 1 be the number of the pistons, the outward pressure on n of them, caused by the force applied to one, is up A. The fiuid will be in equilibrium when a pressure p is acting on unit area of each piston. It is plain that the same reasoning will hold if the area of one of the pistons be A and of another be nA. A pressure Ap on the one will balance a pressure of nAp on the other. This principle governs the action of the hydrostatic press. 113. Relations of Fluid Pressures due to Outside Forces. — If forces, such as gravitation, act on the mass of a fluid from with- out, Pascal's law no longer holds true. For, suppose the fluid to be acted on by gravity, and consider a cylinder of the fluid, the axis of which is vertical, and which is terminated by two normal cross-sections. The pressure on the cylindrical surface, being everywhere normal to it, has no efiect in sustaining the weight of the cylinder. The weight is sustained wholly by the pressure on the lower cross-section, and must be equal to the difEerence be- tween that pressure and the pressure on the upper cross-section. As the height of the cylinder may be made as small as we please, it appears that, in the limit, the pressure on the two cross-sections only differs by an inflnitesimal; that is, the pressure in a fluid acted on by outside forces is the same at one point for all direc- tions, but varies continuously for different points. If, in a fluid acted on by gravity, a surface be considered which is everywhere perpendicular to the lines of gravitational force, the pressure at every point in this surface is the same. To show this we draw a line in the surface between any two points of it, and construct around it as axis .a cylinder terminated at the chosen points by end-surfaces drawn normal to the axis. The pressures on the cylindrical surface,- being normal to it, occasion resultant § 113] MECHANICS OF FLUIDS. 127 forces which are everywhere in the opposite direction to the gravi- tational force and make equilibrium with it. The cylinder being in equilibrium, by hypothesis, the forces on the end surfaces, which alone can produce movement in the direction of the axis, must also be equal, and the pressures on those surfaces are therefore equal. Surfaces of equal pressure are equipotential surfaces; in small masses of liquid they are horizontal planes; in larger masses, such as the oceans, they are curved so as to be always at right angles to the divergent lines of force. The surface of separation between two fluids of different den- sities in a field in which the lines of gravitational force may be supposed parallel is a horizontal plane. For, take two points, a and c, in the same horizontal plane in the lower fluid, and from them draw equal vertical lines terminated at the points b and d, respectively, in the upper fluid. The horizontal planes containing a and c, t and d, respectively, are surfaces of equal pressure. Now -with these lines as axes construct right cylinders with the same small radius and terminated by equal cross-sections in the upper and lower horizontal planes. The pressures on the cylindrical surfaces, being everywhere normal to them, will have no effect in sustaining the weights of these cylinders. Their weights are sustained by the difference in pressure between the upper and lower cross-sections, and, since these cross-sections are in surfaces of equal pressure, the difference of pressure is the same for both cylinders, and the weights of the cylinders are therefore equal. By the construction the cylinders contain portions of both the fluids, and since these fluids are of different densities the weights in the cylinders can only be the same when each cylinder contains the same quantity of each fluid, that is, when the surface of separation between the fluids is parallel with the planes which contain the end cross-sec- tions. The surface of separation is therefore also a horizontal plane. This theorem may be extended so as to prove that the sur- face of separation between two fluids in any gravitational field is at right angles to the lines of gravitational force, or is an equipoten- tial surface. 138 ELEMENTABT PHYSICS. [§ 113 In an incompressible fluid or liquid the pressure at any point is proportional to its depth below the surface. For, the weight of a column of the liquid contained in a vertical cylinder, terminated by the free surface and by a horizontal cross-section containing the point, is manifestly proportional to the height of the cylinder j and this weight is sustained by the pressure on the lower end cross-sec- tion, which must therefore be proportional to the height of the cylinder. If the height of the cylinder be Ji and the area of its cross-section s, and if the density of the liquid be D, the weight of the column is Dshg. If p represent the pressure at the base, the upward forc& on the base i&ps; so that we have p = Dhg. (51) From the foregoing principles it is evident that a liquid con- tained in two communicating vessels of any shape whatever, will stand at the same level in both. If, however, a liquid like mercury be contained in the vessels, and if another liquid, like water, which does not mix with it, be poured into one of the vessels, the surface of separation will sink, and the free surface in the other vessel will rise to a certain point. If a horizontal plane be passed through the surface of separation between the two liquids, the pressures at all points of it within the liquids, in both vessels, will be the same. These pressures, which are due to the superincumbent columns of liquid in the two vessels, are given by Dgh and D'gh', and since they are equal, we have Dh = D'h'; that is, the heights of the two columns above the horizontal plane passing through the surface of separation are inversely as the densities of the liquids. There is nothing in this demonstration which requires us to- consider both the columns as liquid : one of them may be of any fluid, and equilibrium will obtain when the pressure exerted by that fluid on the surface of separation is equal to the pressure ex- erted by the column of liquid in the other vessel on the horizontal plane containing the surface of separation ; so that, if we know the § 114] MECHANICS OF FLUIDS. 139: density and the height of the liquid column, the pressure exerted by the fluid may be measured. 114. The Barometer.— The instrument which illustrates these principles, and is also of great importance in many physical inves- tigations, is the barometer. It was invented by Torricelli, a pupil of Galileo. The fact that water 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 " Nature abhors a vacuum." They must have been familiar with the action of pumps, 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 Galileo, however, that the first recorded observa- tions were made that the column of water in a pump rises only to a height of about 10.5 metres. Galileo failed to give the true ex- planation 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 760 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 mercury in the basin. The specific gravity of the mercury being 13.59, the pressure 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 Pas- cal, 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, 130 BLBMEN-TARY PHYSICS. [§ 114 The form of barometer first made by Torricelli is still often used, especially when the instrument is stationary, and is intended 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 dis- tance, added to the length of the screw, gives the height of the column. Other forms of the instrument are used, most of which are arranged with reference to convenient transportability. Various contrivances are added by means of which the column is made to move an index, and thus record the pressure on a graduated scale. All these forms are only modifications of Torricelli's original in- strument. The pressure indicated by the barometer is usually stated in terms of the height of the column. Mercury being practically in- compressible, this height is manifestly proportional 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 value of g at the place and the specific gravity of mer- cury, as well as the height of the column. The standard baro- metric pressure, represented by 760 millimetres 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 meas- ured 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 quantities of mercury, and boiling § 117] MECHANICS OF FLUIDS. 131 iifter 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 completely 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 tapping 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. 115. Archimedes' Principle. — If a solid be immersed in a fluid, it loses in weight an amount equal to the weight of the fluid dis- placed. This law is known, from its discoverer, as Archimedes' principle. The truth of thjs law will appear if we consider the space in the fluid which is afterwards occupied by the solid. The fluid in this space will 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 differ- ence between the upward and the downward pressures on it must «till be the same as before,— namely, the weight of the fluid dis- placed by the solid; that is, the solid loses in apparent weight an amount equal to the weight of the displaced fluid. 116. 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 vertical, 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. 117. Specific Gravity.— Archimedes' principle is used to deter- mine the 'specific gravity of bodies. The specific gravity of a body 133 ELEMENTAKT PHYSICS. [§ 117 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. 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 dis- placed 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 counter- poised in water. Then the body, the specific gravity of which is sought, is attached to the sinker, and it is found that the equilib- rium is destroyed. To restore it, weights must be added to the same side. These, being added to the weight of the body, repre- sent 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 succes- sively 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 completely fills it is determined 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 solid is introduced into the bottle, which is then filled with water, and the weight of the joint contents of the bottle determined. The difference § 117] MECHANICS OF FLUIDS. 133 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 dis- placed by the solid. The ratio 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 hydrometers of con- stant weight and those of constant volume. The first consists usually of a glass bulb surmounted by a cylindrical 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 immersed in water; and, by the addition of weights in the pan, a fixed point on the stem is brought to coincide with the surface 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 hydro- gen 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 and 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 gi-avity desired. The weight of the flask used in i*is experiment must be very exactly determmed. The presence 134 ELEMENTAKT PHYSICS. [§ 118 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 con- sequent error is avoided either by performing the weighings in a vacuum produced by the air-pump, or by correcting the apparent 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. 118. Motions of Fluids. — If the parts of the fluid be moving relatively to each other or to its bounding-snrface, the circum- stances 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 uniform when each element 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 rotational or irrotational accord- ing as the elements of the fluid have or have not an angular veloc- ity about their axes. In all discussions of the motions of fluids a condition is sup- posed to hold, called the condition of continuity. It is assumed that, in any volume selected in the fluid, the change of density in that volume depends solely on the difference between the amounts of fluid flowing into and out of that volume. In an incompressi- ble fluid, or liquid, if the influx be reckoned plus and the efflux minus, we have, letting Q represent the amount of the liquid passing through the boundary in any one direction, 2Q = 0. The results obtained in the discussion of fluid motions must all be interpreted consistently with this condition. If the motion be such that the fluid breaks up into discontinuous parts, any results obtained by hydrodynamical considerations no longer hold true. § 119] MECHANICS OF FLUIDS. 135 If we consider any stream of incompressible fluid, of which the cross-sections at two points where the velocities of the elements are v^ and r, have respectively the areas J, and A^, we can deduce at once from the condition of continuity A^v^ - A.^v,. (52) 119. Velocity of Efflux.— 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 may call a filament.- Eepresent the velocity of the filament at B (Pig. 39) by v^ , and at G by v, and the areas of the cross-sections of the fila- ment at the same points by Ai and A. We have then, as above, A^v^ = Av. We assume ' that the flow has been established for a time sufficiently long for the motion to become steady. The energy of the mass contained in the filament between B and C is, therefore, constant. Let T", represent the potential or the potential energy of unit mass at B due to gravity, V the potential at C, and d the density of the liquid. The mass that enters at ^ in a unit of time or the rate at which mass enters at B is dA^v^. The rate at which mass goes out at is the equal quantity dAv. The energy entering at B is dA^v^(^v^^ -\-V^),th.e energy passing out at Cis dAv{iv' + V). If the pressures at B and C on unit areas be expressed hj p^ and p, the rate at which work is done at B on the entering mass by the pressure p, is p.A^v., and at C on the outgoing mass is pAv. This may be seen by considering the cross-section of the filament at C. The pressure p acting on each unit of area of that cross- section is equivalent to a force pA, and v is the rate at which the cross-section moves forward, so that pAv is the rate at which the pressure does work. The energy within the filament remaining constant, the incoming must equal the outgoing energy; therefore 136 ELEMENTARY PHYSICS. [§ 119 pAv + dAv{iv' +V) = p,A,v, + dA.v.ilv,'- + F,), whence, since A,v, = Av, we have | + i*^' + ^ = f + i^.'+ ^'• By using again the relation A^v^ = Av, this equation becomes iv\l-^^^{V^-V)+P^. (53) To apply equation (53) to the case of a liquid flowing freely into air from an orifice at G, we observe that the difference of potential (^V^—V) equals the work done in carrying a gram from C to 5 or equals g{h — A,), where h represents the height of the surface above G, and A, that of the surface above B. Further we have p^ — p^ + dgh^, where p^ is the atmospheric pressure. At the orifice p I A^\ equals <■ We have then ^v'\l - -j-\ = g{h — h;) + gh^ = gli, whence v'' = — ^--r^ . If, now, A becomes indefinitely small as 1- — a: compared with ^, , in the limit the velocity at C becomes ■0 =-- V2gh ; (54) that is, the velocity of efflux of a small stream issuing from an ori- fice in the wall of a vessel is independent of the density of the liquid, and is equal to the velocity which a body would acquire in falling freely through a distance equal to that between the surface of the liquid and the orifice. This theorem was first given by Torricelli from considerations based on experiment, and is known as Torricelli's theorem. Its demonstration is due to Daniel Bernoulli. We may apply the general equation to the case of the efflux of a liquid through a siphon. A siphon is a bent tube which is used to convey a liquid by its own weight over a barrier. One end of the siphon is immersed in the liquid, and the discharging end, which must be below the level of the liquid, opens on the other side of § U9J MECHANICS OF FLUIDS. 137 the barrier. To set the siphon in operation it must be first filled ■with the liquid, alter which a steady flow is maintained. A' In this ease, as before, we may set —— ^ = 0, v^= 0, p and p^ both =Pa> and ( F, — V) = gl, where I is the distance between the surface level and the discharging orifice. The velocity be- comes V = V'Zgl. The siphon, therefore, discharges more rapidly the greater the distance between the surface level and the orifice. It is manifest that the height of the bend in the tube cannot be greater than that at which atmospheric pressure would support the liquid. The flow of a liquid into the vacuum formed in the tube of an -ordinary pump may also be discussed by the same equation. The pump consists essentially of a tube, fitted near the bottom with a partition, in which is a valve opening upwards. In the tube slides a, tightly fitting piston, in which is a valve, also opening upwards. The piston is first driven down to the partition in the tube, and the enclosed air escapes through the valve in the piston. When the piston is raised, the liquid in which the lower end of the tube is immersed passes through the valve in the partition, rises in the tube and fills the space left behind the piston. When the piston is again lowered, the space above it is filled with the liquid, which is lifted out of the tube at the next up-stroke. To determine the velocity of the liquid following the piston, -we notice that in this case p, = Pa and ^ = if the piston move iipward very rapidly, ( F, — V) = — gh, where h is the height of the top of the liquid column above the free surface in the reser- voir, and — again = 0. We then have i^' = j^ ~ 9^- The velocity when }i = Q\%v— \ -j" • ^^^^n h is such that dgh — Pa, V — Q, which expresses the condition of equilibrium. The equation v = 'y -j" expresses, more generally, the velocity 13S ELBMENTAEY PHYSICS. [§11^ of efflux, through a small orifice, of any fluid of density d, from a region in which it is under a constant pressure p^, into a vacuum. Torricelli's theorem is shown to be approximately true by al- lowing liquids to run from an orifice in the side of a vessel, and measuring the path of the stream. If the theorem be true, this ought to be a parabola, of which the intersection of the plane of the stream and of the surface of the liquid is the directrix; for each portion of the liquid, after it has passed the orifice, will behave as a solid body, and move in a parabolic path. The equation of this path is found, as in § 52, to be a;" = y. Now by Torricelli's theorem we may substitute for v' its value '^gh, whence a;' = — ihy. In this equation, since the initial movement of the stream is sup- posed to be horizontal, the perpendicular line through the orifice being the axis of the parabola, and the orifice being the origin, Ji is the distance from the orifice to the directrix. Experiments of this kind have been frequently tried, and the results found to approximate more nearly to the theoretical as various causes of error were removed. When, however, we attempt to calculate the amount of liquid discharged in a given time, there is found to be a wider discre- pancy between the results of calculation and the observed facts. Newton first noticed that the diameter of the Jet at a short distance from the orifice is less than that of the orifice. He showed this to be a consequence of the freedom of motion among the particles in the vessel. The particles fiow from all directions towards the ori- fice, those moving from the sides necessarily issuing in streams in- clined towards the axis of the jet. Newton showed that by taking the diameter of the narrow part of the jet, which is called the vena contracta, as the diameter of the orifice, the calculated amount of liquid escaping agreed far more closely with theory. When the orifice is fitted with a short cylindrical tube, the in- terference of the different particles of the liquid is in some degree lessened, and the quantity discharged increases nearly to that re- quired by theory. §120] MECHAN^ICS OF FLUIDS. 139 120. Diminution of Pressure.— The Sprengel air-pump, an im- portant piece of apparatus to be described hereafter, depends for its operation on the diminution of pressure at points along the line of a flowing column of liquid. Let us consider a large reservoir filled with liquid, which runs from it by a vertical tube entering the bottom of the reservoir. Prom equation (53) the value of p, the pressure at any point in the tube, is p = p^ + (^i — 1^)<^ \Av (-9- The ratio -J-., may be set equal to zero. If h (Fig. 40) represent the Fig. 40. height of the upper surface above the point in the tube at which we desire to find the pressure, then ( F, — V) — gh. We then have j[> = ^„ -(- dgh — ^dv\ If the tube be always filled with the liquid, Av = A^v„, where A and A, represent the areas of the cross-sections of the tube at the point we are considering and at the bottom of the tube, and v and w, represent the corresponding velocities. Further, «„" = 2gh^ if 7t„ represent the distance from the upper surface to the bottom of the tube. We obtain, by sub- stitution, do A' P=Pa-\- dgi^o-^K). (55) If Ji equal -A-^n, we have p = Pa'> a^nd if ^t^ opening be made in the wall of the tube, the moving liquid and the air will be in equi- A ' librium. If h be less than -rr^c the pressure p will be less than Pa, and air will flow into the tube. Since this inequality exists when ^„ = J, it follows that, if a liquid flow from a reservoir down a cylindrical tube, the pressure at any point in the wall of the tube is less than the atmospheric pressure by an amount equal to the pressure of a column of the liquid, the height of which is equal to the distance between the point considered and the bottom of the tube. 140 ELMENTAKT PHYSICS. [§ 131 121. Waves. — "When a disturbance is set up at a point in the free surface of a liquid, it moves over the surface of the liquid as a wave or series of waves. Each wave consists of a crest or elevated portion and a hollow or depressed portion of approximately equal length, and the distance from a particle at the summit of one crest to a particle at the summit of the next succeeding crest, or the distance between particles in successive waves which are in the same condition of motion, is called a ivave length. A line which is drawn along the crest of any one wave or through the par- ticles in that wave which are in the same condition of motion, and which at every point is at right angles to the direction in which the wave is propagated, may be called the wave front. The formation of waves is explained by inequalities of hydro- rstatic pressure arising in the liquid if by any cause one part of it be elevated above the rest. H. and W. AVeber examined the peculi- arities of waves in water and the motions of the water particles in them by the aid of a long trough with glass sides; by immersing one end of a glass tube below the surface, raising a column of water in it a few centimetres high by suction, and allowing it to fall, they excited a series of waves which proceeded down the trough and could be examined through the sides. The motions of the particles in the wave were studied by scattering through the water small fragments of amber, which were so nearly of the same specific gravity as the water that they remained suspended without motion except during the passage of the wave, and took part in the motion ex- cited by the wave as if they had been particles of water. It was found that the wave motion was a form of motion transferred from ■one portion to another of the water, and did not involve a displace- ment of the particles concerned in it, — at least when the successive waves had the same wave length. In that case — which is the typi- cal one — the particles in the surface of the water described closed curves, which were elliptical or circular in form, the diameter of the circle being equal to the vertical distance between the crest and the hollow or the height of the wave. In the upper part of the «ircle the particle moved in the direction in which the wave was § 121] MECHANICS OF FLUIDS. 141 moving, in the lower part of the circle in the opposite direction^ The velocity of the wave was found to be dependent on its height and on the period of oscillation in the wave, and to be independent of the density of the liquid. The disturbance of the liquid by the wave is not merely on the surface, but extends to a considerable depth; as the depth increases the elliptic paths of the particles approach more and more closely to short horizontal lines. The theory of these waves is extremely complicated, and has not yet been satisfactorily worked out; but we can indicate in a general way their causes and the mode of their propagation. Imagine a small hillock of water elevated at some print in the surface, and consider a particle at the base of this hillock; the hydrostatic pressure arising from the elevated column near it will tend to move it upward and outward from the centre of the hillock. It will ac- cordingly begin to move in the upper half of its circular path and in the direction in which the wave is propagated ; the precise form of its path being determined by the changes of pressure which it experiences and by its inertia. Since the pressure which sets it in motion will be different for different heights of the hillock which gives rise to it, the velocity of the particles, and therefore also the velocity of the wave, will depend on the height of the wave, being greater as this is greater; the velocity of the wave is also greater as the wave length is greater. Since the pressure behind the particle and the inertia are both proportional to the density of the liquid, it is evident that the acceleration of the particle will be the same under similar circumstances, whatever be its density, s& that the velocity of the wave should not depend on the density of the liquid. The form of a wave is greatly modified by the character of the- channel in which it moves, on account of the motion of the parti- cles extending to a considerable depth, and on account of their viscosity. On the free surface of a large and very deep body of water the successive waves have the same form; the slope of the crest is a little steeper than the slope of the hollow, and its length is less than that of the hollow. As the depth decreases, the slope 142 ELEMENTARY PHYSICS. [§ 132 of the front of the crest becomes still steeper because of the re- straint which then is imposed upon the movement of the particles in the lower half of their paths, and at last the forward motion in the crest so much predominates that the wave curls over and breaks. 122. Vortices. — A series of most interesting results has been ob- tained by Helmholtz, Thomson, and others, from the discussion of the rotational motions of fluids. Though the proofs are of such a nature that they cannot be presented here, the results are so im- portant that they will be briefly stated. A vortex line is deflned as the line which coincides at every point with the instantaneous axis of rotation of the fluid element at that point. A vortex filament is any portion of the fluid bounded by vortex lines. A vortex is a vortex filament which has " contiguous to it over its whole boundary irrotationally moving fluid." The theorems relating to this form of motion, as first proved by Helmholtz, in 1868, show that, — (1) A vortex in a perfect fluid always contains the same fluid elements, no matter what its motion through the surrounding fluid may be. (2) The strength of a vortex, which is the product of its angu- lar velocity by its cross-section, is constant; therefore the vortex in an infinite fluid must always be a closed curve, which, however, may be knotted and twisted in any way whatever. (3) In a finite fluid the vortex may be open, its two ends termi- nating in the surface of the fluid. (4) The irrotationally moving fluid around a vortex has a mo- tion due to its presence, and transmits the influence of the motion of one vortex to another. (5) If the vortices considered be infinitely long and rectilinear, any one of them, if alone in the fluid, will remain fixed in position. (6) If two such vortices be present parallel to one another, they revolve about their common centre of. mass. (7) If the vortices be circular, any one of them, if alone, moves with a constant velocity along its axis, at right angles to the plane § 123] MECHANICS OF FLUIDS. 143 of the circle, in the direction of the motion of the fluid rotating on the inner surface of the ring. (8) The fluid encircled by the ring moves along its axis in the •direction of the motion of the ring, and with a greater velocity. (9) If two circular vortices move along the same axis, one fol- lowing the other, the one in the rear moves faster, and diminishes in diameter; the one in advance moves slower, and increases in diameter. If the strength and size of the two be nearly equal, the one in the rear overtakes the other, and passes through it. The two now having changed places, the action is repeated in- definitely. (10) If two circular vortices of equal strength move along the same axis toward one another, the velocities of both gradually de- crease and their diameters increase. The same result follows if one «uch vortex move toward a solid barrier. The preceding statements apply only to vortices set up in a perfect fluid. They may, however, be illustrated by experiment. To produce circular vortices in the air, we use a box which has one of its ends flexible. A circular opening is cut in the opposite end. The box is filled with smoke or with finely divided sal-ammoniac, resulting from the combination of the vapors of ammonia and hy- drochloric acid. On striking the flexible end of the box smoke- Tings are at once sent out. The smoke-ring is easily seen to be made up of particles revolv- ing about a central core in the form of a ring. With such rings many of the preceding statements may be verified. An illustration of the open vortex is seen when an oar-blade is drawn through the water. By making such open vortices, using a -circular disk, many of the observations with the smoke-rings may be repeated in another form. 123. Air-pumps. — The fact that gases, unlike liquids, are easily compressed, and obey Boyle's law under ordinary conditions of temperature and pressure, underlies the construction and operation of several pieces of apparatus employed in physical investigations. The most important of these is the air-pump. 144 ELEMENTARY PHYSICS. [§ 125 The working portion of the air-pump is constructed essentially like the common lifting-pump already described. The valves must be light and accurately fitted. The vessel from which the air is to- be exhausted is joined to the pump by a tube, the orifice of which is closed by the valve in the bottom of the cylinder. A special form of vessel much used in connection with the air- pump is called the receiver. It is usually a glass cylinder, open at one end and closed by a hemispherical portion at the other. The edge of the cylinder at the open end is ground perfectly true, so that all points in it are in the same plane. This ground edge fits upon a plane surface of roughened brass, or ground glass, called the plate, through which enters the tube which joins the receiver to the cylinder of the pump. The joint between the receiver and- the plate is made tight by a little oil or vaseline. The action of the pump is as follows : As the piston is raised,, the pressure on the upper surface of the valve in the cylinder is diminished, and the air in the vessel expands in accordance with Boyle's law, lifts the valve, and distributes itself in the cylinder, so that the pressure at all points in the vessel and the cylinder is the same. The piston is now forced down, the lower valve is closed by the increased pressure on its upper surface, the valve in the piston is opened, and the air in the cylinder escapes. At each successive stroke of the pump this process is repeated, until the pressure of the remnant of air left in the vessel is no longer sufficient to lift the valves. The density of the air left in the vessel after a given number of strokes is determined, provided there be no leakage, by the relations of the volumes of the vessel and the cylinder. Let V represent the volume of the vessel, and C that of the cylinder when the piston is raised to the full extent of the stroke. Let d and c?, respectively represent the density of the air in the vessel before and after one stroke has been made. After one down and one up stroke have been made, the air which filled th,e volume Fnow fills V -\- O. It follows that ^ = As this ratio is §123] MECHANICS OF FLUIDS. 145 constant no matter what density may be considered, it follows that, if dn represent the density after n strokes, V I -\v+c) ■ As this fraction cannot vanish until n becomes infinite, it is plain that a perfect vacuum can never, even theoretically, be obtained by means of the air-pump. If, however, the cylinder be large, the fraction decreases rapidly, and a few strokes are sufficient to bring the density to such a point that either the pressure is insufficient to lift the valves, or the leakage through the various joints of the pump counterbalances the effect of longer pumping. In the best air pumps the valves are made to open automati- FiG. 41. cally. In Fig. 41 is represented one of the methods by which this is accomplished. They can then be made heavier and with a larger surface of contact, so that the leakage is diminished, and the limit of the useful action of the pump is much extended. With the best pumps of this sort a pressure of half a millimetre of mercury is reached. The Sprengel air-pump depends for its action upon the princi- 146 ELEMENTARY PHYSICS. [§ 124 pie, discussed in § 120, that a stream of liquid running down a cylinder diminishes the pressure upon its walls. In the Sprengel pump the liquid used is mercury. It runs from a large vessel down a glass tube, into the wall of which, at a distance from the bottom of the tube of more than 760 millimetres, enters the tube which connects with the receiver. The lower end of the vertical tube dips into mercury, which prevents air from passing up along the walls of the tube. When the stream of mercury first begins to flow, the air enters the column from the receiver, in consequence of the diminished pressure, passes down with the mercury in large bub- bles, and emerges at the bottom of the tube. As the exhaustion proceeds, the bubbles become smaller and less frequent, and the mercury falls in the tube with a sharp, metallic sound. It is evi- dent that, as in the case of the ordinary air-pump, a perfect vacuum cannot be secured. There is no leakage, however, in this form of the air-pump, and a very high degree of exhaustion can be reached. The Morren or Alvergniat mercury-pump is in principle merely a common air-pump, in which combinations of stop-cocks are used instead of valves, and a column of mercury in place of the piston. Its particular excellence is that there is scarcely any leakage. The compressing-pump is used, as its name implies, to increase the density of air or any other gas within the receiver. The re- ceiver in this case is generally a strong metallic vessel. The work- ing parts of the pump are precisely those of the air-pump, with the exception that the valves open downwards. As the piston is raised, air enters the cylinder, and is forced into the receiver at the down- stroke. 124. Manometers. — The manometer is an instrument used for measuring pressures. One variety depends for its operation upon the regularity of change of volume of a gas with change of pres- sure. This, in its typical form, consists of a heavy glass tube of uniform bore, closed at one end, with the open end immersed in a basin of mercury. The pressure to be measured is applied to the surface of the mercury in the basin. As this pressure increases, the air contained in the tube is compressed, and a column of mer- § 136] MECHANICS OF FLUIDS. 147 If the disturbance be small, the expression on the right is ap- proximately the condensation per unit volume of the medium at the cross-section A, and the equation shows that the ratio of the velocity of the matter passing through the cross-section A to the velocity of propagation of the disturbance is equal to the conden- sation at that cross-section. Now, to eliminate the unknown quantities t»„ and da , we must find a new equation involving them. A quantity of matter if en- ters the region between the two moving cross-sections with the velocity Va, and an equal quantity leaves the region with the veloc- ity Vb. The difference of the momenta of the entering and out- going quantities is M{Va — Vt). This difference can only be due to § 135] ORIGIN AXD TEANSMISSIOK OF SOUND. 161 the different pressures p„ and p^ on the moving cross-sections, since the interactions of the portion of matter between those cross-sec- tions cannot change the Uiomentum of that portion. Hence we have M{v^ — vt) =Pa — Pb- If we for convenience assume v,, = 0, we have p^ = P, the pres- sure in the medium in its undisturbed cudition. If we further substitute for «„ its value, we obtain MV — d/-~ =-. If the n p changes in pressure and density be small, the quantity d^ ~f y: equals E, the modulus of elasticity of the medium. If we further substitute for Jf its value VD, we obtain finally F' = J or F=|/|. (60) 135. Velocity of Sound in Air. — In air at constant temperature the elasticity is numerically equal to the pressure (§ 105). The compressions and rarefactions in a sound-wave occur so rapidly that during the passage of a wave there is no time for the transfer of heat, and the elasticity to be considered, therefore, is the elasti- city when no heat enters or escapes (§ 313). If the ratio of the two elasticities be represented by y we have for the elasticity when no heat enters or escapes E = yP, and the velocity of a sound-wave in air at zero temperature is given by F = ^yr-. The coeflScient y equals 1.41. P is the pressure exerted by a column of mercury 76 centimetres high and with a cross- section of one square centimetre, or 76 X 13.59 X 981 = 1013373 dynes per square centimetre. D equals 0.001293 gram at 0°, hence /: / 1 A~\ \/ 1 C\'\ Q "^ f** Q X iuiao/a _ ggg^Q^ pj. 333 4 metres per second. 0.001393 Since the density of air changes with the temperature, the ve- locity of sound must also change. If dt represent the density at temperature t, and c?„ the density at zero, df = -. ° , , from § 311. 163 ELEMENTARY PHYSICS. [§ 136 The formula for velocity then becomes V — y ^[l + at). This formula shows that the velocity at any temperature is the velocity at 0° multiplied by the square root of the factor of expansion. 136. Measurements of the Velocity of Sound. — The velocity of sound in air has been measured by observing the time required for the report of a gun to travel a known distance. One of the best determinations was that made in Holland in 1833. Gruns were fired alternately at two stations about nine miles apart. Observers at one station observed the time of seeing the flash and hearing the report from the other. The guns being fired alternately, and the sound travelling in opposite directions, the effect of wind was eliminated in the mean of the results at the two stations. It is possible, by causing the sound-wave to act upon dia- phragms, to make it record its own time of departure and arrival, and by making use of some of the methods of estimating very small intervals of time the velocity of sound may be measured by experi- ments conducted within the limits of an ordinary building. The velocity of sound in water was determined on Lake Geneva in 1836 by an experiment analogous to that by which the velocity in air was determined. In § 144 and § 146 it is shown that the time of one vibration of any body vibrating longitudinally is the time required for a sound- wave to travel twice the distance between two nodes. The velocity may, therefore, be measured by determining the number of vibra- tions per second of the sound emitted, and measuring the distance between the nodes. In an open organ-pipe, or a rod free at both ends, when the fundamental tone is sounded the sound travels twice the length of the rod or pipe during the time of one complete vibration. If rods of different materials be cut to such lengths that they all give the same fundamental tone when vibrating longitudinally, the ratio of their lengths will be that of the velocity of sound in them. In Kundt's experiment, the end of a rod having a light disk at- tached is inserted in a glass tube containing a light powder strewn § 136] ORIGIN AND TRANSMISSION OF SOUND. 163 over its ianer surface. When the rod is made to vibrate longitudi- nally, the air-column in the tube, if of the proper length, is made to vibrate in unison with it. This agitates the powder and causes it to indicate the positions of the nodes in the vibrating air-column. The ratio of the velocity of sound in the solid to that in air is thus the ratio of the length of the rod to the distance between the nodes in the air-column. CHAPTBK II. SOUNDS AND MUSIC. COMPAKISON OF SOUNDS. 137. Musical Tones and Noises. — The distinction between 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 7ioise is a sound the vibrations of which are very irregular. It may result from a confusion of musi- cal tones, and is not always devoid of musical 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, depending 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 loto, 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 instru- ments even when sounding the same notes. 138. Methods of Determining the Number of Vibrations of a Musical Tone. — That the pitch of a tone depends upon the fre- quency of vibrations may be simply shown by holding the corner of 164 § 138] SOUNDS AND MUSIC. 165 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 are sufficiently rapid, blend together and produce a continuous tone, the pitch of which rises and falls with the changes of speed. Savart made use of such a wheel to determine the number of vibrations corresponding to a tone of given pitch. After regulating the speed of rotation until the given pitch was reached, the number of revolutions per second was determined by a simple attachment; this number multiplied by the number of teeth in the wheel gave the number of vibrations per second. The siren is an instrument for producing musical tones by puffs ■of air succeeding each other at short equal intervals. A circular disk having in it a series of equidistant holes arranged in a circle around its axis is supported so as to revolve parallel to and almost touching a metal plate in which is a similar series of holes. The plate forms one side of a small chamber, to which air is supplied from an organ bellows. If there be twenty holes in the disk, and if it be placed so that these holes correspond to those in the plate, ^ir 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 twen- tieth of a revolution from its first position, air will again escape, and if it rotate continuously, air will escape twenty times in a revo- lution. When the rotation 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 vibrations corresponding to any tone. By drilling the holes in the plate obliquely forward in the direction of rotation, and those in the disk obliquely backward, the escaping air will cause the disk to rotate, and the speed of rotation may be controlled by controlling the pressure of air in the chamber. Sirens are sometimes made with several series of holes in the disk. These serve not only the purposes described above, but also to compare tones of which the vibration numbers have certain ratios. 166 ELEMENTAKT PHYSICS. [§13» The number of vibrations of a sounding body may sometimes be determined by attaching to it a light stylus ■which is made to trace a curve upon a smoked glass or cylinder. Instead of at- taching a stylus to the sounding body di- rectly, which is practicable only in a few cases, it may be attached to a membrane which is caused to vibrate by the sound- waves which the body generates. A mem- brane reproduces very faithfully all the characteristics of the sound-waves, and the curve traced by the stylus attached to it gives information, therefore, not only in regard to the number of vibrations, but to some extent in regard to their amplitude and form. PHYSICAL THEOKY OF MUSIC. 139. Concord and Discord. — When two or more tones are sounded together, if the effect be pleasing there is said to be con- cord ; if harsh, discord. To understand the cause of discord, suppose two tones of nearly the same pitch to be sounded to- gether. The resultant curve, constructed as in § 132, is like those in Fig. 53, which represent the resultants when the periods of the components have the ratio 81 : 80 and when they have the ratio 16 : 15. The figure indicates, what experiment verifies, that the resultant sound suffers periodic variations in intensity. When these varia- tions occur at such intervals as to be read- ily distinguished, they are called beats. These beats occur more and more frequently as the numbers expressing the ratio of the Fig. 53. §143] SOUNDS AND MUSIC. 167 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 be- come smaller still, the roughness diminishes, and when the ratio is I the effect is no longer unpleasant. This, and ratios expressed by- smaller numbers, as f , |, |, |, f , represent concordant combinations. 140. Major and Minor Triads.— Three tones of which the vi- bration numbers are as 4 : 5 : 6 form a concordant combination called the major triad. The ratio 10 : 12 : 15 represents another concordant combination called the minor triad. Fig. 53 shows the resultant curves for the two triads. 4:5:6 Iff: 12:15 yv Fig. 53. 141. Intervals. — The interval 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 the musical scale, as de- scribed below. The interval f is called an octave; f , a fifth; f, a fourth; f , a major third; |, a minor third. 142. Musical Scales. — A musical scale is a series of tones which have been chosen to meet the demands of musical composition. There are at present two principal scales in use, each consisting of seven notes, with their octaves, chosen with reference to their fit- ness to produce pleasing effects when used in combination. In one, called the major scale, the first, third, and fifth, the fourth, sixth, and eighth, and the fifth, seventh, and ninth tones, form major triads. In the other, called the minor scale, the same tones form minor triads. Prom this it is easy to deduce the following relations : 168 ELEMEH^TAEY PHYSICS. [§ 143 MAJOK SCALE. 1' 2' ToneNumber 123456 7 89 Letter C DEFGABCD' Name do or ut re mi fa sol la si ut re Number of vibrations m |m {m |m fm |m ^m 2m |m Intervals from tone to tone.. f V fl I V I fl MINOR SCALE. ToneNumber 1234 5 6789 Letter AB C D E FGA'B' Name la si ut re mi fa sol la si Number of vibrations m |m fm |m fm fm |m 2m Jm Intervals from tone to tone. . I if V I If i V The derivation of the names of the intervals will now be appar- ent. For example, an interval of a third is the interval between any tone of the scale and the third one from it, counting the first as 1. 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, |, V^, and |f. The first two are usually con- sidered the same, and are called toliole tones. *rhe third is a half- tone or semitone. It is desirable to be able to use any tone of a musical instru- ment as the first tone or tonic of a musical scale. To permit this, when the tones of the instrument are fixed, it is plain that extra tones, other than those of the simple scale, must be provided in order that the proper sequence of intervals may be maintained. Suppose the tonic to be transposed from C to D. 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 respectively, and their vibration numbers are determined by multiplying the vibration numbers of the tones which they replace by ff . 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 ■■§ 142J SOUNDS AND MUSIC. 169 taken as the tonic. But if we consider that the whole tones are not all the same, and propose to preserve exactly all the intervals of the transposed scale, the problem becomes much more difficult, and can only be solved at the expense of too great complication in the instrument. 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 eve7i temperament, divides the octave into twelve equal semitones, and each interval is therefore the twelfth root of 2. With instru- ments in which the tones are not fixed, like the violin for instance, the skilful performer may give them their exact value. For convenience in the practice of music and in the construc- tion 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 0. This has the advantage that the vi- bration numbers of the successive octaves of this tone are powers of 2. CHAPTEE III. VIBRATIONS OF SOUNDING BODIES. 143. General Considerations. — The principles developed in § 133 apply directly in the study of the vibrations of sounding bodies. When any part of a body which is capable of acting as a sounding body is set in vibration, a wave is propagated through it to its boundaries, and is there reflected. The reflected 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. 144. Organ Pipes. — A column of air, enclosed in a tube of suitable dimensions, may be made to vibrate and become a sound- ing body. Let us suppose a tube closed at one end and open at the other. If the air particles at the open end be suddenly moved inward, a pulse travels to the closed end, and is there reflected with change of sign (§ 133). 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 inde- pendent 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. If a continuous vibration be maintained at the open end of the tube, waves follow each other up the tube, are reflected with 170 §144] VIBKATIONS OF SOUNDING BODIES. 171 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 tub© 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 mouthpiece 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, 1, 3, 5, or some odd number of half-vibrations. By con- structing 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 1 half-vibration while the wave travels up and back, the length of the tube is i 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 1,2, 3, 4, or some whole number of complete vibrations are performed while the wave travels up Ihe 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 numbers 1, 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 1, 3, 5, etc. It is evident also that the lowest tone of the closed tube is an octave lower than that of the open tube of the same length. 173 ELEMENTAKT PHTSICS. [§145 This lowest tone of the tube is called the fundamental, and the others are called overtones, or harmonics. These simple rela- tions 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 move- ment. This is never the case at the open end of the tube, and the distance from this end to the first node is, therefore, always less than a (quarter wave length. 145. Modes of Exciting Vibrations in Tubes. — If a tuning-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 vibration of the proper rate and the column made to sound. In organ pipes a mouthpiece of the form shown in Fig. 54 is often employed. The thin sheet of air projected against the thin edge is thrown into vibration. Those ele- ments of this vibration which correspond in frequency with the pitch of the pipe are strongly reinforced by the action ot the stationary wave set up in the pipe, and hence the tone proper to the pipe is pro- duced. Sometimes reeds are used, as shown in Pig. 54a. The air escaping from the chamber a through the passage c causes the reed r to vibrate. This alternately closes and opens the passage, 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 accommo- date itself to the rate of vibration of the air-column, and may then serve to produce various tones, as in the clarionet. In instruments like the cornet and bugle the lips of the player Fig. 54a. Fig. 54. § 148] VIBKATIONS OF SOUNDIUrG BODIES. 173 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 sing- ing flame. The organ of frhe 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. 146. Longitudinal Vibrations of Rods.— A rod free at both ends vibrates as the column of air in an open tube. Any displace- ment produced at one end is transmitted with the velocity of sound in the material to the other end, is there reflected without change of sign, and returns to the starting-point to be reflected again exactly as in the open tube. The fundamental tone corresponds to a stationary wave having a node at the centre of the rod. 147. 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, 1, 2, 3, 4, etc., vibrations while the wave travels twice the length of the cord. The velocity of transmission of a longitudinal displacement in a wire depends upon the elasticity and density of the material only. The velocity and the rate of vibration are, therefore, nearly independent of the stretching force. 148. 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. They depend on the stretching force or tension and on the mass of the cord per unit 174 ELEMENTARY PHYSICS. [§149 length. The number of vibrations is inversely as the length of the cord, directly as the square root of the tension, and inversely as the square root of the mass per unit length. 149. Transverse Vibrations of Rods, Plates, etc. — The vibrations of rods, plates, and bells are all cases of stationary waves resulting from systems of waves travelling in opposite directions. Subdivi- sion into segments occurs, but in these cases the relations of the various overtones are not so simple as in the cases before consid- ered. For a rod fixed at one end, sounding its fundamental tone, there is a node at the fixed end only. For the first overtone there is a second node near the free end of the rod, and the number of vibrations is a little more than six times the number for the funda- mental. A rod free at both ends has two nodes when sounding its funda- mental, as shown in Fig. 55. The distance of these nodes from the . ,. ends is about f the length of the rod. '-'^ ' ^- If the rod be bent, the nodes approach Fig. 55. 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. 56. '^ -n Jf m Fig. 56. 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 and marks their position. The figures thus formed are known as Chladni's figures. 150. Resonance. — If several pendulums be suspended from the § 150] VIBRATION'S OF SOUNDIlfQ BODIES. 175 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 diilerent period will show no signs of dis- turbance. The vibration of the first pendulum produces a slight movement of the support, which is communicated alike to all the other pendulums. Each movement may be considered as a slight impulse, which imparts to each pendulum a very small vibratory motion. For those pendulums having the same period as the one in vibration, these impulses come just in time to increase the mo- tion already produced, 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 support and tuned to unison will both vibrate when either one is ■caused to sound. A tuning-fork suitably mounted on a sounding- box will communicate its vibrations to another tuned to exact unison «ven when they are thirty or forty feet apart and only air intervenes. In this case it is the sound-wave generated by the first fork which €xcites the second fork, and in so doing the wave loses a part of its «wn 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. Such communication of vibrations is called resonance. Air-columns of suitable dimensions will vibrate in sympathy with other sounding bodies. If water be gradually poured into a •deep jar, over the mouth of which is a vibrating tuning-fork, there will be found in general a certain length of the air-column for which the tone of the fork is strongly reinforced. From the theory of organ pipes, it is plain that this length corresponds ap- proximately to a quarter wave length for that tone. In this case, also, when the strongest reinforcement occurs, the sound of i he fork will rapidly die away. The sounding-boxes on which the tr.n- 176 ELEMENTARY PHYSICS. [ §150 ing-f orks made by Konig are mounted are of such dimensions that the enclosed body of air will yibrate 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 wax is in the form of a cylinder mounted on an axle moved by a screw working in a fixed nut, so that when the cylinder revolves it has also an end- wise motion, such that a fixed point would follow a spiral track on its surface. To use the instrument, the disk is placed in position with the stylus attached and slightly indenting the wax. The cyl- inder is revolved while sounds are produced in front of the disk. The disk vibrates, causing the stylus to indent the wax 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 pre- viously made in the wax 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- m effect thin disks, and afford examples of the reinforcement of vibrations of widely different pitch and quality by the same body. The strings of an instrument are of themselves insufficient to com^ municate 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 instruments, 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. CHAPTEE IV. ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 151. Quality. — As has already been stated, the tones of differ- ent instruments, although of the same pitch and intensity, are dis- tinguished by their quality. It was also stated that the quality of a tone depends upon the manner in which the vibration is exe- cuted. The meaning of this statement can best be understood by Fm. 57. considering the curves which represent the vibrations. In Fig. 57 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 (§ 21), every periodic vibration is resolvable into simple harmonic vibrations having commensurable periods. It has been seon that all sound- ing bodies may subdivide into segments, and produce a series of tones of which the vibration periods generally bear a simple rela- 177 178 ELEMBNTAKY PHYSICS. [§152 tion to one another. These may be produced simultaneously by the same body, and so give rise to complex tones, the character of which will vary with the natnre and intensity of the simple tones produced. It has been held that the quality of a complex tone is not affected by change of phase of the component simple tones relative to each other. Some experiments by Konig seem to indi- cate, however, that the quality does change when there is merely change of phase. Fig. 58. In Fig. 58 are shown three curves, each representing a funda- mental accompanied by the harmonics up to the tenth. The curves differ only in the different phases of the components rela- tive to each other. Fig. 59 shows similar curves produced by a fundamental accom- panied by the odd harmonics. Fig. 59. 152. 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 ^ 153] ANALYSIS OF SOUNDS AND SOUND SBNSATIONS. 179 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 Pig. 60. sound be produced, one of 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 (§ 128), the gas flame connected with the capsule will be disturbed whenever the tone to which the resonator is tuned is produced in the vicinity, either by itself or as a component of a complex tone. By trying the resonators of a series, one after another, the several components of a complex tone may be detected and its composition demonstrated. 153. Vowel Sounds. — Helmholtz has shown that the differences between the vowel sounds are only dififerences 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 180 ELEMENTARY PHYSICS. [§ 154 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. 60. 154. Optical Method of Studying Vibrations. — The vibratory motion of sounding bodies may sometimes be studied to advantage Oo Fig. 61. by observing the lines traced by luminous points upon the vibrating body or by observing the movement of a beam of light reflected from a mirror attached to the body. Young studied the vibrations of strings by placing the string where a thin sheet of light would fall across it, so as to illuminate a single point. "When the string was caused to vibrate, the path of the point appeared as a continuous line, in consequence of the per- sistence of vision. Some of the results which he obtained are given in Fig. 61, taken from Tyndall on Sound. The most interesting application of this method was made by Lissajous to illustrate the composition of vibratory motions at right angles to each other. If a beam of light be reflected to a screen from a mirror attached to a tuning-fork, when the tuning-fork vibrates the spot on the screen will describe a simple harmonic motion and will appear as a straight line of light. If the beam, instead of being reflected to a screen, fall upon a mirror attached to a second fork, mounted so as to vibrate in a plane at right angles to the first, the spot of light will, when both forks vibrate, be actuated by two simple harmonic motions at right angles to each other, and the resultant path will appear as a curve more or less complicated, depending upon the relation of the two forks to each other as to both period and phase (§ 21). Fig. 62 shows some of § 155] ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 181 the simpler forms of these curves. The figures of the upper line are those produced by two forks in unison ; those of the second V- FiG. 63. line by two forks of which the vibration numbers are as 2 : 1 ; those of the lower line by two forks of which the vibration numbers are as 3 : 2. 155. Beats. — It has already been explained (§139) that, when two tones of nearly the same pitch are sounded together, variations of intensity, called heats, are heard. Helmholtz's theory of the perception of beats was, that, of the little fibres in the ear which are tuned so as to vibrate with the various tones, those which are nearly in unison affect one another so as to increase and diminish one another's motions, and hence that no beats could be perceived unless the tones were nearly in unison. Beats are, however, heard when a tone and its octave are not quite in tune, and, in general, a tone making n vibrations produces m beats when sounded with a tone making 2m ± m, 3« ± m, etc., vibrations. This was explained in accordance with Helmholtz's theory, by assuming that one of the harmonics of the lower tone, which is nearly in unison with the upper, causes the beats, or, in cases where this is inadmissible, that they are caused by the lower tone in conjunction with a resultant 182 ELEMENTAET PHYSICS, [§15& tone (§ 156). An exhaustive research by Konig, however, has de- monstrated that beats are perceived when neither of the above sup- positions is admissible. Figs. 63 and 64 show that the resultant n I i\MiWm\mmkkfMmmmmM^^ Fig. 63. vibrations are affected by changes of amplitude similar to, though less in extent than, the changes which occur when the tones are nearly in unison. In Fig. 63 I represents a flame image obtained in 15:46- Fig. 64. when two tones making n a.nd n ± m vibrations, respectively, are produced together, and II represents the image when the number § 156] AN'ALTSIS OF SOUNDS AND SOUND SENSATIONS. 183 of vibrations are n and 2n ± m. Fig. 64 shows traces obtained mechanically. In I the numbers of the component vibrations were n and w + m, in II and III n and 2n ± m, and in IV n and 3m + m. In all these cases & variation of amplitude occurs during the same intervals, and it seems reasonable to suppose that those varia- tions of amplitude should cause variations in intensity in the sound perceived. Cross has shown that the beating of two tones is perfectly well perceived when the tones themselves are heard separately by the two ears ; one tone being heard directly by one ear, while the other, produced in a distant room, is heard by the other ear by means of a telephone. Beats are also perceived when tones are produced at a distance from each other and from the listener, who hears them by means of separate telephones through separate lines. In this case there is no possibility of the formation of a resultant wave, or of any combination of the two sounds in the ear. 156. Resultant Tones. — Resultant tones are produced by com- binations of two tones. Those most generally recognized have a vibration number equal to the sum or difference of the vibration numbers of their primaries. For instance, ut^, making 2048 vibra- tions, and re„, making 2304 vibrations, when sounded together give utj, making 256 vibrations. These tones are only heard well when the primaries are loud, and it requires an effort of the attention and some experience to hear them at all. Summation tones are more difficult to recognize than difference tones, nevertheless they have an influence in determining the general effect produced when musical tones are sounded together. Other resultant tones may be heard under favorable conditions. As described above, two tones making n and n + m vibrations respectively, when m is considerably less than n, give a resultant tone making m vibrations; but a tone making n vibrations in combination with one making 2m + m, Sn + m, or xn + m vibrations, gives the same resultant. This has sometimes been explained by assuming that internnediate resultants are produced, which, with one of the primaries, produce resultants of a higher order. In the case of the two tones making n and 184 liLEMESTARr PHTSICS. [§ 156 3n + m vibrations, for instance, the first difference tone would make 2n + tn vibrations. This tone and the one making n vibrations would give the tone making n -\-m vibrations ; this tone, in turn, and the one making n vibrations would give the tone making m vibrations. This last tone is the one which is heard most plainly, and it seems difficult to admit that it can be the resultant of tones which are only heard very feebly, and often not at all. In Fig. 64 are represented the resultant curves produced in several of these cases. The first curve corresponds to two tones of which the vibration numbers are as 15 : 16. It shows the periodic increase and decrease in ampli- tude, occurring once every 15 vibrations, which, if not too frequent, give rise to beats (§ 139). If the pitch of the primaries be raised, preserving the relation 15 : 16, the beats become more frequent, and finally a distinct tone is heard, the vibration number of which cor- responds to the number of beats that should exist. It was for along time considered that the resultant tone was merely the rapid recur- rence of beats. Helmholtz has shown by a mathematical investi- gation that a distinct wave making m vibrations will result from the coexistence of two waves making n and n -\- m vibrations, and he believes that mere alternations of intensity, such as constitute beats, occurring ever so rapidly cannot produce a tone. In II and III (Fig. 64) are the curves resulting from two tones, the intervals between which are respectively 15: 39(= 3 X 15 — 1) and 15: 31(= 3 X 15 + 1). Eunning through these may be seen a periodic change corresponding exactly in period to that shown in I. The same is true also of the curve in IV, which is the resultant for two tones the intervalbetween which is 15: 46(=: 3x15+1). In all these cases, as has been already said (§ 155), if the pitch of the components be not too high, one beat is heard for every 15 vibra- tions of the lower component. Fig. 63 shows the flame images for the intervals n:n + m and n:2>i + m. The varying amplitudes resulting in m beats per second are very evident in both. In all these cases, also, as the pitch of the components rises the beats be- come more frequent, and finally a resultant tone is heard, having, as already stated, one vibration for every 15 vibrations of the lower § 156] ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 185 component. In Fig. 65 are shown two resultant curves having three components of which the vibration numbers are as 1 : 15 : 29. In I the three components all start in the same phase. In II, when 15 and 39 are in the same phase, I is in the opposite phase. HEAT. CHAPTEK I. MEASUREMENT OF HEAT. 157. General Effects of Heat. — Bodies are warmed, or their temperature is raised, by heat. The sense of touch is often suffi- cient to show difference in temperature; but the true criterion is the transfer of heat from the hotter to the colder body when the two bodies are brought in contact, and no work is done by one upon the other. This transfer is known by some of the efEects described below. Bodies, in general, expand when heated. Experiments show that different substances expand differently tor the same rise of tem- perature. Gases, in general, expand more than liquids, and liquids more than solids. Expansion, however, does not universally ac- company rise of temperature. A few substances contract when heated. Heat changes the state of aggregation of bodies, always in such a way as to admit of greater freedom of motion among the mole- cules. The melting of ice and the conversion of water into steam are familiar examples. Heat breaks up chemical compounds. The compounds of sodium, potassium, lithium, and other metals, give to the flame of a Bunsen lamp the characteristic colors of the vapors of the metals 186 § 159] MEASUREMENT OF HEAT. 187 which they contain. This fact shows that the heat separates the metals from their combinations. When the junction of two dissimilar metals in a conducting circuit is heated, electric currents are produced. Heat performs mechanical work. For example, the heat pro- duced in the furnace of a steam-boiler may be used to drive an en- gine. 158. Production of Heat. — Heat is produced by various proc- esses, some of which are the reverse of the operations just men- tioned as the effects of heat. As examples of such reverse opera- tions may be mentioned, the production of heat by the compression of a body which expands when heated; the production of heat during a change in the state of aggregation of a body, when the freedom of motion among the molecules is diminished; the pro- duction of heat during chemical combination; and the production of heat when an electric current passes through a junction of two dissimilar metals in an opposite direction to that of the current which is set up when the junction is heated. Heat is produced in general in any process involving the ex- penditure of mechanical energy. The heat produced in such processes cannot be used to restore the whole of the original me- chanical energy. The production of heat by friction is the best example of these processes. Further, an electric current, in a homogeneous conductor, gen- erates heat at every point in it, while if every point in the conduc- tor be equally heated no current will be set up. These cases are examples of the production of heat by non- reversible processes. 159. Nature of Heat. — Heat was formerly considered to be a substance which passed from one body to another, lowering the temperature of the one and raising that of the other, which com- bined with solids to form liquids, and with liquids to form gases or vapors. But the most delicate balances fail to show any change of weight wheh heat passes from one body to another. Eumford was able to raise a considerable quantity of water to the 188 ELEMENTARY PHYSICS. [§ 159 boiling-point by the friction of a blunt boring-tool within the bore of a cannon. He showed that the heat manifested in this experi- ment could not have come from any of the bodies present, and also that heat would continue to be developed as long as the borer continued to revolve, or that the supply of heat was practically inexhaustible. The heat, therefore, must have been generated by the friction. That ice is not melted by the combination with it of a heat substance was shown early in the present century by Davy. He caused ice to melt by friction of one piece upon another in a vac- uum, the experiment being performed in a room where the tem- perature was below the melting-point of ice. There was no source from which heat could be drawn. The ice must, therefore, have been melted by the friction. Eumford was convinced that the heat obtained in his experi- ment was only transformed mechanical energy; but to demonstrate this it was necessary to prove that the quantity of heat produced was always proportional to the quantity of mechanical work done. This was done in the most complete manner by Joule in a series of experiments extending from 1842 to 1849. He showed that, however the heat was produced by mechanical means, whether by the agitation of water by a paddle-wheel, the agitation of mercury, or the friction of iron plates upon each other, the same expendi- ture of mechanical energy always developed the same quantity of heat. Joule also proved the perfect equivalence of heat and elec- trical energy. These experiments prove that heat is a form of energy. Con- sistent explanations of most if not all of the phenomena of heat may be given if we assume that the molecules of bodies, and the atoms constituting the molecules, are in constant motion, that the temperature of a body varies with the mean kinetic energy of an atom, and that the heat in a body is the sum of the kinetic energies of its atoms. § 161] MEASUREMENT OF HEAT. 189 THERMOMETRY. 160. Temperature. — Two bodies are said to be at the same temperature when, if they be brought into each other's presence, no heat is transferred from one to the other. A body is at a higher temperature than the other bodies around it when it gives up heat to them. The fact that it gives up heat may be shown by its change in volume. A body is at a lower temperature when it re- ceives heat from surrounding bodies. It is understood, of course, in what is said above, that one body has no action upon the other, or that no work is done by one body upon the other. 161. Thermometers. — Experiments show that, in general, bodies expand, and their temperature rises progressively, with the appli- cation of heat. An instrument may be constructed which will show at any instant the volume of a body selected for the purpose. If the volume increase, we know that the temperature rises; if the volume remain constant or diminish, we know that the tempera- ture remains stationary or falls. Such an instrument is called a thermometer. The thermometer most in use consists of a glass bulb with a fine tube attached. The bulb and part of the tube contain mer- cury. In order that the thermometers of different makers may give similar readings, it is necessary to adopt two standard tem- peratures which can be easily and certainly reproduced. The tem- peratures adopted are the melting-point of ice, and the temperature of steam from boiling water, under a pressure equal to that of a column of mercury 760 millimetres high at Paris. After the instru- ment has been filled with mercury, it is plunged in melting ice, from which the water is allowed to drain away, and a mark is made upon the stem opposite the end of the mercury column. It is then placed in a vessel in which water is boiled, so constructed that the steam rises through a tube surrounding the thermometer, and then descends by an annular space between that tube and an outer one, and escapes at the bottom. The thermometer does not touch the water, but is entirely surrounded by steam. The point 190 ELEMENTARY PHYSICS. [§ 161 reached by the end of the mercury column is marked on the stem, as before. The space between these two marks is then divided into a number of equal parts. While all makers of thermometers have adopted the same stand- ard temperatures for the fixed points of the scale, they differ as to the number of divisions between these points. The thermometers used for scientific purposes, and in general use in Prance, have the space between the fixed points divided into a hundred equal parts or degrees. The melting-point of ice is marked 0°, and the boiling- point 100°. This scale is called the Centigrade or Celsius scale. The Reaumur scale, in use in Germany, has eighty degrees between the melting- and boiling-points, and the boiling-point is marked 80°. The Fahrenheit scale, in general use in England and America, has a hundred and eighty degrees between the melting- and boil- ing-points. The former is marked 32°, and the latter 212°. The divisions in all these cases are extended below the zero point, and are numbered from zero downward. Temperatures below zero must, therefore, be read and treated as negative quan- tities. A few points in the process of construction of a thermometer deserve notice. It is found that glass, after it has been heated to a high temperature, and again cooled, does not for some time return to its original volume. The bulb of a thermometer must be heated in the process of filling with mercury, and it will not return to its normal volume for some months. The construction of the scale should not be proceeded with until the reservoir has ceased to con- tract. For bhe same reason, if the thermometer be used for high temperatures, even the temperature of boiling water, time must be given for the reservoir to return to its original volume before it is used for the measurement of low temperatures. It is essential that the diameter of the tube should be nearly uniform throughout, and that the divisions of the scale should rep- resent equal capacities in the tube. To test the tube a thread of mercury about 50 millimetres long is introduced, and its length is I 162] MEASUREMENT OF HEAT. 191 measured in different parts of the tube. If the length vary by more than a millimetre, the tube should be rejected. If the tube be found to be suitable, a bulb is attached, mercury is introduced, and the tube sealed after the mercury has been heated to expel the air. When it is ready for graduation, the fixed points are determined; then a thread of mercury having a length equal to about ten de- grees of the scale is detached from the column, and its length measured in all parts of the tube. By reference to these measure- ments, the tube is so graduated that the divisions represent parts ■of equal capacity, and are not necessarily of equal length. If such a thermometer indicate a temperature of 10°, this means that the thermometer is in such a thermal condition that the volume of the mercury has increased from zero one tenth of its total expansion from zero to 100°. There is no reason for sup- posing that this represents the same proportional rise of tempera- ture. If a thermometer be constructed in the manner described, using some liquid other than mercury, it will not in general indi- cate the same temperature as the mercurial thermometer, except at the two standard points. It is plain, therefore, that a given frac- tion of the expansion of a liquid from zero to 100° cannot be taken as representing the same fraction of the rise of temperature. 162. Air-thermometer. — If a gas be heated, and its volume kept constant, its pressurs increases. For all the so-called permanent gases — that is, those which are liquefied only with great difficulty — the amount of increase in pressure for the same increase of tem- perature is found to be almost exactly the same. This fact is a reason for supposing that the increase of pressure is proportional to the increase of temperature. There are theoretical reasons, as will be seen later, for the same supposition. An instrument constructed to take advantage of this increase in pressure to measiye temperature is called an air-thermometer. A bulb so arranged that it may be placed in the medium of which the temperature is to be determined, is filled with air or some other gas, and means are provided for maintaining the volume of the gas constant, and measuring its pressure. For the reasons given above. 193 ELEMENTAEY PHYSICS. [§ 153 the air-thermometer is taken as the standard instrument for scien- tific purposes. Its use, however, involves several careful observa- tions and tedious computations. It is therefore mainly employed as a standard with which to compare other instruments. If we make such a comparison, and construct a table of corrections, we may reduce the readings of any thermometer to the corresponding readings of the air-thermometer. 163. Limits in the Range of the Mercurial Thermometer.— The range of temperature for which the mercurial thermometer may be employed is limited by the freezing of the mercury on the one hand, and its boiling on the other. For temperatures below the freezing-point of mercury alcohol thermometers may be used. For the measurement of high temperatures several different methods have been employed. One depends upon the expansion of a bar of platinum, another upon the variation in the electric resistanca of platinum wire, another upon the strength of the electric current generated by a thermo-electric pair, another on the density of mer- cury vapor. The special devices used in applying these methods need not be considered here. CALOEIMETBT. 164. Unit of Heat. — It is evident that more heat is required to raise the- temperature of a large quantity of a substance through a given number of degrees than to raise the temperature of a small quantity of the same substance through the same number of degrees. It is further evident that the successive repetition of any operation by which heat is produced will generate more heat than a single operation. Heat is therefore a quantity the magnitude of which may be expressed in terms of some unit. The unit of heat gen- erally adopted is the heat required to raise the temperature of one kilogram of water from zero to one degree. It is called a calorie. It is sometimes convenient to employ a smaller unit, namely^ the quantity of heat necessary to raise one gram of water from zero to one degree. This unit is designated as the lesser calorie or the § 1G6] MEASUEEMENI OF HEAT. 193 gram-degree. It is one one-thousandth of the larger unit. It may, therefore, be called a millicalorie. The fact that heat is energy enables us to employ still another unit. It is that quantity of heat which is equivalent to an erg. This unit is called the mechanical unit of heat. According to the determination of Griffiths, a calorie contains about 41,983,000,000 mechanical units. 165. Heat reciuired to raise the Temperature of a Mass of Water. — It is evident that to raise the^temperature of m kilograms of water from zero to one degree will require m calories. If the temperature of the same quantity of water fall from one degree to- zero, the same quantity of heat is given to surrounding bodies. Experiment shows, that if the same quantity of water be raised to different temperatures, quantities of heat nearly proportional tO' the rise in temperature will be required : hence, to raise the tem- perature of m kilograms of water from zero to t degrees requires mt calories very nearly. This is shown by mixing water at a lower temperature with water at a higher temperature. The temperature of the mixture will be almost exactly the mean of the two. Regnault, who tried this experiment with the greatest care, found the tempera- ture of the mixture a little higher than the mean, and concluded that the quantity of heat required to raise the temperature of a kilogram of water one degree increases slightly with the temperature ;: that is, to raise the temperature of a kilogram of water from twenty to twenty-one degrees, requires a little more heat than to raise the temperature of the same quantity of water from zero to one degree. Eowland found, by mixing water at various temperatures, and also by measuring the energy required to raise the temperature of water by agitation by a paddle-wheel, that, when the air thermo- meter is taken as a standard, the quantity of heat necessary to raise the temperature of a given quantity of water one degree diminishes slightly from zero to thirty degrees, and then increases to the boiling- point. 166. Specific Heat. — Only one thirtieth as much heat is required to raise the temperature of a kilogram of mercury from zero to one 194 ELEMENTARY PHYSICS. [§ 167 degree as is required to raise the temperature of a kilogram of water through the same range. In order to raise the temperatures of other substances through the same range, quantities of heat peculiar to each substance are required. The quantity of heat required to raise the temperature of one kilogram of a substance from zero to one degree is called the specific heat of the substance. If the temperature of one kilogram of a substance rise from t, to t, the limit of the ratio of the quantity of heat required to bring about the rise in temperature to the difEerence in temperature, as that diSerence diminishes indefinitely, is called the specific heat of the substance at temperature t. If we represent the quantity of heat by Q, the limit of the ratio — — -jj expresses this specific heat. The specific heats of substances are generally nearly constant between zero and one hundred degrees. The mean specific heat of a substance between zero and one hundred degrees is the one usually given in the tables. The measurement of specific heat is one of the important objects of calorimetry. 167. Ice Calorimeter.— 5^ac^'s or Wilcke's ice calorimeter con- sists of a block of pure ice having a cavity in its interior covered by a thick slab of ice. The body of which the specific heat is to be determined is heated to t degrees, then dropped into the cavity, and immediately covered by the slab. After a short time the tem- perature of the body falls to zero, and in so doing converts a certain quantity of ice into water. This water is removed by a sponge of known weight, and its weight is determined. It will be shown, that to melt a kilogram of ice requires 80 calories; if, then, the weight of the body be P, and its specific heat c, it gives up, in falling from t degrees to zero, Pet calories. On the other hand, if p kilo- grams of ice be melted, the heat required is &0p. Therefore Pet ~ 80/?; whence 80» ' = W (61) MEASUREMENT OF HEAT. 195 Bunsen's ice calorimeter (Fig. 66) is used for determining the specific heats of substances of which only a small quantity is at hand. The apparatus is entirely of glass. The tube B is filled with water and mercury, the latter extending into the graduated capillary tube C. To use the apparatus, alcohol which has been artificially cooled to a temperature below zero is passed through the tube A. A layer of ice forms around c the outside of this tube. As water freezes, it expands. This causes the mercury to advance in the capillary tube G. When a sufficient quantity of ice has been formed, the alcohol is removed from A, _ the apparatus is surrounded by melting snow or ice, and a small quantity of water is introduced, which soon falls in tem- perature to zero. The position of the mercury in C is now noted; and the sub- stance the specific heat of which is to be Fig. 66. determined, at the temperature of the surrounding air, is dropped into the water in A. Its temperature quickly falls to zero, and the heat which it loses is entirely employed in melting the ice which surrounds the tube A. As the ice melts, the mercury in the tube G retreats. The change of position is an indication of the quantity of ice melted, and the quantity of ice melted measures the heat given up by the substance. The number of divisions of the tube G cor- responding to one calorie can be determined by direct experiment. 168. Method of Mixtures. — The method of mixtures consists in bringing together, at different temperatures, the substance of which the specific heat is desired and another of which the specific heat is known, and noting the change of temperature which each undergoes. The water calorimeter consists of a vessel of very tbm copper or brass, highly polished, and placed within another vessel upon non- conducting supports. A mass P of the substance of which the specific heat is to be determined is brought to a temperature t' in a suitable bath, then plunged in water at a temperature t, con- 196 ELEMENTAKY PHYSICS. [§ 168 tained in the calorimeter. The whole will soon come to a common temperature 0. The heat lost by the substance is Fc{t'—&) calories. The heat gained by the calorimeter is the sum of that gained by the water and that gained by the materials of which the calo- rimeter is constructed. If p represent the mass of water, and p' the water equivalent of the calorimeter, or the mass of water which will rise by the same temperature interval as the calorimeter vessel does on the introduction of a given quantity of heat, the total heat gained by the calorimeter is {p -\-p'){ff — t); and hence Pc{t' - ff) ^ {p + PW - t), (62) from which c may be determined. The water equivalent p' is de- termined by experiment. There is a source of error in the use of the instrument, due to the radiation of heat during the experiment. This error may be nearly eliminated by making a preliminary experiment to determine what change of temperature the calorimeter will experience; then, for the final experiment, the calorimeter and its contents are brought to a temperature below the temperature of the surround- ing air, by about half the amount of that change. The calorimeter will then receive heat from the surrounding medium during the first part of the experiment, and lose heat during the second part. The rise of temperature is, however, much more rapid at the begin- ning than at the end of the experiment. The rise from the initial temperature to the temperature of the surrounding medium occu- pies less time than the rise from the latter to the final temperature. The gain of heat, therefore, does not exactly compensate for the loss. If greater accuracy be required, the rate of cooling of the calorimeter must be determined by putting into it warm water, the same in quantity as would be used in experiments for determining specific heat, and noting its temperature from minute to minute. Such an experiment furnishes the data for computing the loss or gain by radiation. To secure accurate results the body must be transferred from the bath to the calorimeter without sensible loss of heat. I 170] MEASUREMENT OF HEAT. 197 169. Method of Comparison. — The method of comparison con- sists in conTeying to the substance of which the specific heat is to be determined a known quantity of heat, and comparing the con- sequent rise of temperature with that produced by the same amount of heat in a substance of which the specific heat is known. In the early attempts to use this method, the heat produced by the same flame burning for a given time was applied successively to difEerent liquids. A more exact method was the combustion, within the calorimeter, of a known weight of hydrogen. The best method of obtaining a known quantity of heat is by means of an electrical current of known strength flowing through a wire of known resist- ance wrapped upon the calorimeter. 170. Method of Cooling. — The method of cooling consists in noting the time required for the calorimeter, in a space kept con- stantly at zero, to cool from a temperature t' to a tem- perature t, when empty, when containing a given weight of water, and when containing a given weight of the substance of which the specific heat is sought. The thermo-calorimeter of Eegnault, represented in Fig. 67, is an example. It consists of an alcohol thermometer, with its bulb A enlarged and made in the form of a hollow cylinder, inside of which- the substance is placed. The thermometer is warmed, and then placed in a vessel surrounded by melting ice. It radiates heat to the sides of the vessel,'and the column of alcohol in the tube falls. Let X be the time occupied in falling from the division n to the division n' when the space B is empty. Let the times occupied in falling between the same two divi- sions, when the space B contains a mass P of water, and when it contains a mass P' of the substance of which the specific heat c' is sought, be respectively x' and x". Fig. 67, Let M be the water equivalent of the instrument. We then have M ^ M + P ^ M + ^'c' gjjj^g^ ^^j^^gj. the conditions of the ex- X %' x" periment, the heat lost per second must be the same in each case. \y 198 ELEMENTARY PHYSICS. [§ ^'''1 Eliminating M, we obtain ir^e-:)- '^^' 171. Determination of the Mechanical Equivalent of Heat.— It has been stated that whenever heat is produced by the expenditure of mechanical energy, the quantity of heat produced is always pro- portional to the quantity of mechanical energy expended. The mechanical equivalent of heat is the energy in mechanical units, the expenditure of which produces the unit of heat. Heat applied to a body may increase the motion of its mole- cules; that is, add to their kinetic energy. It may perform inter- nal work by moving the molecules against molecular forces. It may perform external work by producing motion against external forces. If we could estimate these effects in mechanical units, we might obtain the mechanical equivalent of heat. But the kinetic energy of the molecules cannot be estimated, for we do not know their mass or their velocity. We must, therefore, in the present state of our knowledge, resort to direct experiment to determine the heat equivalent. In one of the experiments of Joule, already referred to, a paddle-wheel was made to revolve, by means of weights, in a vessel filled with water. In this vessel were stationary wings, to prevent the water from acquiring a rotary motion with the paddle-wheel. By the revolution of the wheel the water was warmed. The heat so generated was estimated from the rise of temperature, while the mechanical energy required to produce it was given by the fall of the driving weight. Joule repeated this experiment, substituting mercury for the water. In another exper- iment he substituted an iron plate for the paddle-wheel, and made it revolve with friction upon a fixed iron plate under water. Joule expressed his results in kilogram-metres — that is, the work done by a kilogram in falling under the force of gi-avity through one metre. He stated the mechanical equivalent of one calorie, in this unit, to be 423.9, from the experiments with water; 425.7, from those with mercury; and 426.1, from those with irott § m-] MEASUREMENT OF HEAT. 199 plates. He gave the preference to the smallest value, and it has been generally accepted as the mechanical equivalent. This me- chanical equivalent is called Joule's equivalent, and is represented by /. In absolute units, according to the later determinations of Griffiths, it is about 41,982,000,000 ergs per calorie. Eowland has repeated Joule's experiment with water; but he caused the paddle-wheel to revolve by means of an engine, and de- termined the moment of the couple required to prevent the revolu- tion of the calorimeter. Fig. 68 shows the apparatus. The shaft -o -Wo= FiQ. 68. of the paddle-wheel projects through the bottom of the calorimeter, and is driven by means of a bevel-gear. The vessel A is suspended from C by a torsion wire, and its tendency to rotate balanced by weights attached to cords which act upon the circumference of a pulley D. By this disposition of the apparatus he was able to ex- pend about one half a horse-power in the calorimeter, and obtain a 200 ELEMENTARY PHYSICS. [§ 171 rise of temperature of 35° per hour; while in Joule's experiments the rise of temperature per hour was less than 1°. These experi- ments give, for the mechanical equivalent of one calorie at 5°, 429.8 kilogram-metres; at 30°, 436.4 kilogram-metres. Several other methods have been employed for determining the mechanical equivalent. The concordance of the results by all these methods is sufficient to warrant the statement that the ex- penditure of a given amount of mechanical energy always produces the same amount of heat. An experiment to determine the mechanical work done by the expenditure of a known quantity of heat was executed by Hirn. By the help of Regnault's measurements of the heat of vaporization Hirn was able to calculate the amount of heat which entered the cylinder, during the operation of a steam-engine, with the steam from the boiler, and by direct measurements he determined the amount of heat which left the cylinder during the operation of the engine and entered the condenser. So long as the engine was run- ning without doing any external work, he found that these amounts of heat were appreciably equal; when the engine was made to do work, less heat passed from the cylinder into the condenser than had entered it from the boiler. A comparison of the amount of heat lost with the work done by the engine showed the same ratio between heat and work as that determined by Joule. Hirn's ex- periments were on so large a scale and the sources of error and the difficiilties connected with the experiments were so numerous, that the number obtained by him for the mechanical equivalent of heat is of no great value. His experiments are, however, X>t very great in- terest because, while the experiments of Joule and of all the others who have worked on the problem prove the convertibility of work into heat, those of Hirn alone have proved the converse converti- bility of heat into work. CHAPTER II. TRANSFER OF HEAT. 172. Transfer of Heat. — In the preceding discussions it has been assumed that heat may be transferred from one body to another, and that if two bodies in contact be at different temperatures, heat will be transferred from the hotter to the colder body. In general, if transfer of heat be possible in any system, heat will pass from the hotter to the colder parts of the system, and the temperature of the system will tend to become uniform. There are three ways in which this transfer is accomplished, called respectively convection, con- duction, and radiation. 173. Convection. — If a vessel containing any fluid be heated at the bottom, the bottom layers become less dense than those above, pro- ducing a condition of instability. The lighter portions of the fluid rise, and the heavier portions from above, coming to the bottom, are in their turn heated. Hence continuous currents are caused. This process is called convection. By this process, masses of fluid, al- though fluids are poor conductors, may be rapidly heated. Water is often heated in a reservoir at a distance from the source of heat by the circulation produced in pipes leading to the source of heat and back. The winds and the great currents of the ocean are con- vection currents. An interesting result follows from the fact that water has a maximum density (§ 190). When the water of lakes cools in winter, currents are set up and maintained, so long as the surface water becomes more dense by cooling, or until the whole mass reaches 4°. Any further cooling makes the surface water lighter. It therefore remains at the surface, and its temperature 201 202 ELEMENTAKT PHYSICS. [§ 174 rapidly falls to the freezing-point, while the great mass of the water remains at the temperature of its maximum density. 174. Conduction. — If one end of a metal rod be heated, it is found that the heat travels along the rod, since those portions at a distance from the source of heat finally become warm. This proc- ess of transfer of heat from molecule to molecule of a body, while the molecules themselves retain their relative places, is called con- diiction. In the discussion of the transfer of heat by conduction it is as- sumed as a principle, borne out by experiment, that the flow of heat between two very near parallel planes, drawn in a substance, is pro- portional to the difference of temperature between those planes, or that the flow of heat across a plane is proportional to the rate of fall of temperature across that plane. 175. Flow of Heat across a Wall. — The simplest body in which the flow of heat can be studied is a wall of homogeneous material bounded by two parallel infinite planes, one of which is kept at the temperature t' and the other at the temperature t ; we repre- sent the distance between the planes or the thickness of the wall by d. We suppose that the flow of heat across this wall has con- tinued so long that it has become steady, or that the tem- peratures at all points have assumed flnal values. Manifestly the temperature at all points in any plane parallel with the faces of the wall is the same, and the same amount of heat passes across any one such plane as passes across any other. We conclude therefore by the fundamental principle assumed (§ 174) that the rate of change of temperature across each plane in the wall is the same, or that the change of temperature through- out the wall from one face to the other is uniform; the rate of change of temperature is therefore given by -, where it has been assumed that t' is the higher temperature. If d' represent the distance of any plane in the wall from the hotter surface, the fall of temperature between it and the hotter surface is (f — t)— ^ d* § 177] TRANSFER OF HEAT. 203 d' and the temperature of the plane is t' — {t' — t)-j. The confirma- tion by experiment of this law of temperature distribution in a wall is a warrant for our assumption of the fundamental princi- ple of the flow of heat. 176. Conductivity. — If, now, we consider a prism extending across the wall, bounded by planes perpendicular to the exposed surfaces, and represent the area of its exposed bases by A, the quantity of heat which flows in a time T through this prism may be represented by Q = E*^^AT, (64) where X is a constant depending upon the material of which the wall is composed. K is the conductivity of the substance, and may be defined as the quantity of heat which in unit time flows through a section of unit area in a wall of the substance whose thickness is unity, when its exposed surfaces are maintained at a difference of temperature of one degree; or, in other words, it is the quantity of heat which in unit time flows through a section of unit area in a substance, where the rate of fall of temperature at that section is unity. In the above discussions the temperatures t' and t are taken as the actual temperatures of the surfaces of the wall. If the colder surface of the wall be exposed to air of temperature T, to which the heat which traverses it is given up, t will be greater than T. The difference will depend upon the quantity of heat which flows, and upon the facility with which the surface parts with heat. 177. Flow of Heat along a Bar. — If a prism of a substance have one of its bases maintained at a temperature /, while the other base and the sides are exposed to air at a lower temperature, the con- ditions of uniform fall of temperature no longer exist, and the amount of heat'which flows through the different sections is no longer the same: but the amount of heat which flows through any section is still proportional to the rate of fall of temperature at that 204 ELBMEN-TART PHYSICS. [§ ITO section, and is equal to the heat which escapes from the portion of the bar beyond the section. 178. Measurement of Conductivity. — A bar heated at one end furnishes a convenient means of meas- uring conductivity. In Fig. 69 let AB represent a bar heated at A. Let the ordinates aa', W , cc', represent the excess of temperatures above the tem- B perature of the air at the points from which they are drawn. These temper- atures may be determined by means of thermometers inserted in cavities in the bar, or by means of a thermopile. Draw the curve a'b'c'd' . . . through the summits of the ordinates. The inclina tion of this curve at any point represents the rate of fall of tem- perature at that point. The ordinates to the line b'm, drawn tangent to the curve at the point b', show what would be the tem- peratures at various points of the bar if the fall were uniform and at the same rate as at b'. It shows that, at the rate of fall at V, the bar would at m be at the temperature of the air; or, in the length bm, the fall of temperature would equal the amount repre- sented by bb'. The rate of fall is, therefore, t— • If Q represent the quantity of heat passing the section at b in the unit time, we have, from § 176, Q = E X rate of fall of temperature X area of section. Q is equal to the quantity of heat that escapes in unit time from all that portion of the bar beyond b. It may be found by heating a short piece of the same bar to a high temperature, allow- ing it to cool under the same conditions that surround the bar ^5, and observing its temperature from minute to minute as it falls. These observations furnish the data for computing the quantity of heat which escapes per minute from unit length of the bar at different temperatures. It is then easy to compute the amount of heat that escapes per minute from each portion, be, cd, etc., of the bar beyond b; each portion being taken so short that its tempera- § 183] TKANSFER OF HEAT. 205 ture throughout may, without sensible error, be considered uniform and the same as that at its middle point. Summing up all these quantities, we obtain the quantity Q which passes the section b in the unit time. Then E=. ^ . rate of fall of temperature at b X area of section 179. Conductivity diminishes as Temperature rises. — By the method described above, Forbes determined the conductivity of a bar of iron at points at different distances from the heated end, and found that the conductivity is not the same at all temperatures, but is greater as the temperature is lower. 180. Conductivity of Crystals. — The conductivity of crystals of the isometric system is the same in all directions, but in crystals of the other systems it is not so. In a crystal of Iceland spar the con- ductivity is greatest in the direction of the axis of symmetry, and equal in all directions in a plane at right angles to that axis. 181. Conductivity of Non-homogeneous Solids. — De la Eive and De Candolle were the first to show that wood conducts heat better in the directioii of the fibres than at right angles to them. Tyndall, by experimenting upon cubes cut from wood, has shown that the conductivity has a maximum value parallel to the fibres, a minimum value at right angles to the fibres and parallel to the annual layers. Feathers, fur, and the materials of clothing are poor conductors because of their want of continuity. 182. Conductivity of Liquids. — The conductivity of liquids can be measured, in the same way as that of solids, by noting the fall of temperature at various distances from the source of heat in a column of liquid heated at the top. Great care must be taken in these experiments to avoid errors due to convection currents. Liquids are generally poor conductors. 183. Radiation. — We have now considered those cases in which there is a transfer of heat between bodies in contact. Heat is also transferred between bodies not in contact. This is effected by a process called radiation, which will be subsequently considered. CHAPTER III. EFFECTS OF HEAT. 184. The Kinetic Theory of Heat. — In order to describe more easily certain of the effects of heat, it is advantageous to have an idea of the theory by which they are explained. This theory, the kinetic theory of heat, asserts that the molecules of all bodies are in constant motion, and that the heat of a body is the kinetic energy of its molecules. The idea that heat consists of the motion of the least parts of matter was introduced into science by New- ton, of course with a very imperfect knowledge of the facts. The apparently unlimited production of heat by mechanical work led Eumford and Davy, more particularly the latter, to assert the equivalence of heat and motion. This theory was afterwards dis- placed for many years by the influence of the French school of physicists, who considered bodies, at least in their mathematical discussions, as assemblages of stationary particles, and heat as a separate substance. It was revived by Mohr, who showed its very general applicability in the explanation of ordinary heat phenomena. Since the discovery of the conservation of energy, the reasons in its favor have been very much strengthened and its foundations securely laid by the complete success attained with it in explaining the laws of gases. We will use this theory in its general form in the description of some of the effects of heat, and will discuss it more fully in § 221 seq. SOLIDS AND LIQUIDS. 185. Expansion of Solids.— When heat is applied to a body it increases the kinetic energy of the molecules, and also increases the 206 § 185] EFFECTS OF HEAT. ^ 207 potential energy, by forcing the molecules farther apart against their mutual attractions and any external forces that may resist ex- pansion. Since the internal work to be done when a solid or liquid expands yaries greatly for different substances, it might be expected that the amount of expansion for a given rise of temperature would vary greatly. In studying the expansion of solids, we distinguish linear and mluminal expansion. The increase which occurs in the unit length of a substa,nce for a rise of temperature from zero to 1° C. is called the coefficient of linear expansion. Experiment shows that the expansion for a rise of temperature of one degree is very nearly constant between zero and 100°. Kepresent by Z„ the distance between two points in a body at zero, by l^ the distance between the same points at the temperature i, and by a the coefficient of linear expansion of the substance of which the body is composed. The increase in the distance Z„ for a rise of one degree in tem- perature is al^ , for a rise of i degrees atl,. Hence we have, after a rise in temperature of t degrees, I, = ?„(1 + at). (65) The binomial 1 + at is called the factor of expansion. In the same way, if k represent the coefficient of voluminal ex- pansion, the volume of a body at a temperature t will be F, = F„(1 + M); (66) and if d represent density, since density is inversely as volume, we have For a homogeneous isotropic solid, the coefficient of voluminal expansion is three times that of linear expansion j for, if the tem- perature of a cube, with an edge of unit length, be raised one degree, the length of its edge becomes 1 -f «, and its volume 1 ^ 3a _|_ Sa" + a'. Since a is very small, its square and cube 208 ELEMENTARY PHYSICS. [§' 186 may be neglected; and the volume of the cube after a rise in tem- perature of one degree is 1 + 3ar. 3a; is, therefore, the coefficient of voluminal expansion. 186. Measurement of Coefficients of Linear Expansion. — Coeffi- cients of linear expansion are measured by comparing the lengths, at different temperatures, of a bar of the substance the coefficient of which is required, with the length, at constant temperature, of another bar. The constant temperature of the latter bar is secured by immersing it in melting ice. The bar the coefficient of which is sought may be brought to different temperatures by immersing it in a liquid bath; but it is found better to place the bar upon the instrument by means of which the comparisons are to be made, and leave it for several hours exposed to the air of the room, which is kept at a constant temperature by artificial means. Of course several hours must elapse between any two comparisons by this method, and its application is restricted to such ranges of temper- ature as may be obtained in occupied rooms; but within this range the observations can be made much more accurately than would be the case when the bar is immersed in a bath, and it is within this range that an accurate knowledge of coefficients of expansion is of most importance. 187. Expansion of Liquids.— In studying the expansion of a liquid, it is important to distinguish its absolute expansion, or the real increase in volume, and its apparent expansion, or its increase in volume in comparison with that of the containing vessel. To determine the absolute expansion, some method must be used which does not require a knowledge of the expansion of the vessel containing the liquid. The method used by Eegnault in determining the absolute expansion of mercury consisted in compar- ing the heights of two columns of mercury at different tempera- tures when they were so adjusted as to give the same pressure. The apparent expansion is determined by filling a vessel of known volume with the liquid at one temperature, and by measur- ing the amount of the liquid which runs out when the temperature is raised. This method was also used by Regnault in his study of § 189] EFFECTS OF HEAT. 209 the expansion of mercury. The vessel which he used was a glass bulb furnished with a capillary tube. It was filled with mercury at a known temperature, and its volume determined by the weight of the mercury contained in it and the specific gravity of mercury. It was then heated to another known temperature, and the mer- cury which ran out was collected and weighed. From these data the apparent expansion of mercury in glass could be determined. When the absolute expansion of mercury is known the knowl- edge of its apparent expansion in glass enables us to determine the absolute expansion of glass also. If the apparent expansion of mercury be known, and if we assume that its expansion is proportional to the rise of tempera- ture, we may evidently use the amount of mercury which runs out when the bulb is heated as a measure of its change of temperature. The instrument just described is therefore called a weight ther- mometer. 188. Determination of Voluminal Expansion of Solids. — The "weight thermometer may be used to determine the coefficient of voluminal expansion of solids. For this purpose, the solid, of which the volume at zero is known, must be introduced into the bulb by the glass-blower. If the bulb containing the solid be filled, with mercury at zero, and afterward heated to the temperature t, it is evident that the amount of mercury that will overflow will depend upon the coefficient of expansion of the solid, and upon the coefficient of apparent expansion of mercury. If the latter has been determined for the kind of glass used, the former can be deduced. By this means the coefficients of voluminal expansion of some solids have been determined; and the results are found to verify the conclusion, deduced from theory (§ 185), that the voluminal coefficient is three times the linear. 189. Absolute Expansion of Liquids other than Mercury. — The weight-thermometer may also serve to determine the coefficients of expansion of liquids other than mercury; for, if the absolute expan- sion of glass has been found as described above, the instrument may be filled with the liquid the coefficient of which is desired, and the 210 ELEMJENTABT PHYSICS. [§ 190 apparent expansion of this liquid found exactly as was that of mer- cury. The absolute coeflScient for the liquid is then the sum of the coeflBcient of apparent expansion and the coefficient for the glass. 190. Expansion of Water. — The use of water as a standard with which to compare the densities of other substances makes it neces- sary to know, not merely its mean coefficient of expansion, but its actual expansion, degree by degree. This is the more important since water expands very irregularly. The best determinations of the volumes of water at different temperatures are those of Mat- thiessen. The method which he employed was to weigh in water a mass of glass of which the coefficient of expansion had been pre- viously determined. Water contracts, instead of expanding, from 0° to 4°. At 4° it is at its maximum density, and from that temperature to its boiling- point it expands. 191. EfiFect of Variation of Temperature upon Specific Heat. — It has already been stated (§ 166) that the specific heat of bodies changes with temperature. With most substances the specific heat increases as the temperature rises. For example, the true specific heat of the diamond At OMs 0.0947 At 50° is 0.1435 At 100° is 0.1905 At 200° is , 0.2719 192. Effect of Change of Physical State upon Specific Heat. — The specific heat of a substance is not the same in its different physical states. In the solid or gaseous state of the substance it is generally less than in the liquid. For example : Jlean Specific Heat. Solid. Liquid. Gaseous. Water 0.504 1.000 0.481 Mercury 0.0314 0.383 Tin 0.056 0.0637 Lead 0.0314 0.0403 Bromine 0.1129 0.0555 § 193] EFFECTS OF HEAT. 211 193. Dulong and Petit's Law. Atomic Heat.— In their study of the specific heats of a number of chemical elements which are solid &t ordinary temperatures, Dulong and Petit found that the product of the specific heat by the atomic weight of the element was ap- proximately a constant quantity. Further researches, especially those of Kopp, have confirmed this statement as a general law for all solid elements. The constant number to which the product of the specific heat and the atomic weight approximates is ordinarily given as 6.4 when the specific heat is measured in calories, though this is probably a little too high. The deviations from this number presented by different elements are rather large, amounting in many cases to as much as 5 per cent. If masses of the different elements be taken which are propor- tional to their atomic weights, these masses will contain the same numbers of atoms. The heat required to raise one of these masses -one degree in temperature is therefore the same for all such sub- stances. This statement is of course true only within the limits of accuracy with which the, different substances conform to Dulong and Petit's law. The experiments of P. Neumann and Eegnault showed that a similar law applies to compounds of solids which are -of the same chemical constitution; that is, which contain the same number of atoms in the molecule. For all such bodies the product ■of the specific heat and the molecular weight is a constant ; this constant is different for the different classes of substances — that is, for those substances which have different numbers of atoms in the molecule. But if the constant obtained for each class of substances be divided by the number of atoms in the molecule of that class, the quotient is approximately the same constant, 6.4, as that ob- tained for the elements. By applying this law to compounds in which one of the elements is a substance, like hydrogen, which •cannot be examined directly in the solid state, the atomic heat of that substance may be calculated. It is found that the atomic heats of certain substances, notably hydrogen, carbon, oxygen, nitrogen, and silicium, deviate very widely from the constant with which the other atomic heats approximately agree. 312 ELEMENTAKY PHYSICS. [§ 19i The elementary gases obey a similar law with considerable ex- actness; the constant given by the product of their specific heats at constant pressure and their atomic weights is about 3.4. The following table will illustrate the law of Dulong and Petit. The atomic weights are those given by Clarke. Specific Heat Product of Specific Elements. of Atomic Weight. Heat and Atomic Equal Weights. Weight. Iron 0.114 55.9 6.372 Copper 0.095 63.17 6.001 Mercury.... 0.0314 (solid) 199.71 6.128 Silver 0.057 107.67 6.137 Gold 0.0339 196.15 6.453 Tin 0.056 117.7 6.591 Lead 0.0314 306.47 6.483 Zinc 0.0955 64.9 6.198 194. Fusion and Solidification. — When ice at a temperature below zero is heated, its temperature rises to zero, and then the ice begins to melt; and, however high the temperature of the medium that surrounds it may be, its temperature remains constant at zero so long as it remains in the solid state. This temperature is the melting-point of ice, and because of its fixity it is used as one of the standard temperatures in graduating thermometric scales. Other bodies melt at very different but at fixed and definite tempera- tures. Many substances cannot be melted, as they decompose by heat. Alloys often melt at a lower temperature than any of their con- stituents. An alloy of one part lead, one part tin, four parts bis- muth, melts at 94°; while the lowest melting-point of its constitu- ents is that of tin, 328°. An alloy of lead, tin, bismuth, and cad- mium melts at 62°. If a liquid be placed in a medium the temperature of which is below its melting-point, it'will in general begin to solidify when its temperature reaches its melting-point, and it will remain at that temperature until it is all solidified. Under certain conditions, however, the temperature of a liquid maybe lowered several degrees below its melting-point without solidification, as will be seen below. § 196] EFFECTS OF HEAT. 213 195. Change of Volume with Change of State.— Substances are generally more dense in the solid than in the liquid state, but there are some notable exceptions. Water, on solidifying, expands; so that the density of ice at zero is only 0.9167, while that of water at 4° is 1. This expansion exerts considerable force, as is evidenced by the bursting of vessels and pipes containing water. 196. Change of Melting- and Freezing-points. — If water be en- closed in a vessel sufficiently strong to prevent its expansion, it cannot freeze except at a lower temperature. The freezing-point of water is, therefore, lowered by pressure. On the other hand, substances which contract on solidifying have their solidification hastened by pressure. The lowering of the melting-point of ice by pressure explains some remarkable phenomena. If pieces of ice be pressed together, even in warm water, they will be firmly united. Fragments of ice may be moulded under heavy pressure, into a solid, transparent mass. This soldering together of masses of ice is called regelation. If a loop of wire be placed over a block of ice and weighted, it will cut its way slowly through the- ice, and regelation will occur be- hind it. After the wire has passed through, the block will be found one solid mass, as before. The explanation of these phe- nomena is, that the ice is partially melted by the pressure. The liquid thus formed is colder than the ice; it finds its way to points of less pressure, and there, because of its low temperature, it congeals, firmly uniting the two masses. "Water, when freed from air and kept perfectly quiet, will not form ice at the ordinary freezing-point. Its temperature may be lowered to —10° or —13° without solidification. In this condition a slight jar, or the introduction of a small fragment of ice, will cause a sudden congelation of part of the liquid, accompanied by a rise in temperature in the whole mass to zero. A similar phenomenon is observed in the case of several solu- tions, notably sodium sulphate and sodium acetate. If a saturated hot solution of one of these salts be made, and allowed to cool in a closed bottle in perfect quiet, it will not crystallize. Upon opening 314 ELEMENTARY PHYSICS. [§ 197 the bottle and admitting air, crystallization commences, and spreads rapidly through the mass, accompanied by a considerable rise of temperature. If the amount of salt dissolved in the water be not too great, the solution will remain liquid when cooled in the open air, and it may even suffer considerable disturbance by foreign bod- ies without crystallization; but crystallization begins immediately upon contact with the smallest crystal of the same salt. 197. Freezing-point of Solutions. — It has been long known that the freezing-point of a solution of salt and water is lower than that of pure water. The relation of the lowering of the freezing-point to the concentration of the solution was investigated by Blagden, who found that for dilute solutions the lowering of the freezing- point was proportional to the concentration. This matter has been investigated by Raoult, who established some most important gen- eralizations. Raoult showed that, for indifferent solutions, that is, for solutions which are not electrolytes, provided they are very di- lute, the lowering of the freezing-point is very closely proportional to the concentration; its amount differs for different solvents. He fur- ther showed that, for any one solvent, the lowering of the freezing- point is the same whatever be the dissolved substance, provided that the solutions are equimolecular, that is, contain the same num- ber of molecules of the dissolved substance in unit volume of the solution. It may be shown on theoretical grounds that the change in the freezing-point depends upon the osmotic pressure, the freez- ing-point, the heat of fusion, and the density of the solution. Solutions which are electrolytes, or are not indifferent, also ex- hibit a lowering of the freezing-point proportional to the concentra- tion, but the amount of change is greater than in indifferent solutions. This difference is explained by assuming a partial or complete dis- sociation of the molecules of the dissolved substances into their constituent ions (§ 285). 198. Heat Equivalent of Fusion.— Some facts that have ap- peared in the above account of the phenomena of fusion and solid- ification require further study. It has been seen that, however rapidly the temperature of a solid may be rising, the moment fusion. § 199] EFFECTS OF HEAT. 215 begins the rise of temperature ceases. "Whatever the heat to which a solid may be exposed, it cannot be made hotter than its melting- point. When ice is melted by pressure its temperature is lowered. When a liquid is cooled, its fall of temperature ceases when solidi- fication begins; and if, as may occur under favorable conditions, a liquid is cooled below its melting-point, its temperature rises at once to the melting-point, when solidification begins. Heat, there- fore, disappears when a body melts, and is generated when a liquid becomes solid. It was stated (§ 159) that ice can be melted by friction; that is, by the expenditure of mechanical energy. Fusion is, therefore, work which requires the expenditure of some form of energy to ac- complish it. The heat required to melt unit mass of a substance is the heat equivalent of fusion of that substance. When a substance solidifies, it develops the same amount of heat as was required to melt it. As will be shown later at greater length, the absorption of heat which occurs when a solid is melted is explained by supposing that it is used in doing work against the forces which determine the direc- tion of the molecules in the solid and in increasing the kinetic energy of molecular translation. 199. Determination of the Heat Equivalent of Fusion. — The heat equivalent of fusion may be determined by the method of mixtures (§ 1 68), as follows : A mass of ice, for example, represented by P, at a temperature t below its melting-point, to insure dryness, is plunged into a mass P' of warm water at the temperature T. Eepresent by the resulting temperature, when the ice is all melted. If p represent the water equivalent of the calorimeter, (P' -\- p) {T ~ 0) is the heat given up by the calorimeter and its contents. Let c represent the specific heat of ice, and x the heat equivalent of fusion. The ice absorbs, to raise its temperature to zero, Ptc calories; to melt it, Px calories; to warm the water after melting, PO calories. We then have the equation Ptc + Pe + Px = {P' + p){T - 6), (68) from which x may be found. 216 ELEMENTARY PHYSICS. [§ 200 Other calorimetric methods may be employed. The best ex- periments give, for the heat equivalent of fusion of ice, very nearly eighty calories. VAPORS AND GASES. 200. The Gaseous State. — A gas may be defined as a highly compressible fluid. A given mass of gas has no definite volume. Its volume varies with every change in the external pressure to which it is exposed. A vapor is the gaseous state of a substance which at ordinary temperatures exists as a solid or a liquid. 201. Vaporization is the process of formation of vapor. There are two phases of the process : evaporation, in which vapor is formed at the free surface of the liquid; and ebullition, in which the vapor is formed in bubbles in the mass of the liquid, or at the heated surface with which it is in contact. 202. Evaporation. — If a liquid be enclosed in a vessel which it does not entirely fill, the space above the liquid begins at once to be occupied by the vapor of the liquid. The presence of the vapor can be detected in many ways, some of which are applicable only in special cases. Those which are always applicable are the meas- urement of the increased pressure due to the vapor and the con- densation of the vapor into the liquid state after isolating it from the mass of liquid beneath it. The process of forming vapor in this way is evaporation. Evaporation goes on continually from the free surfaces of many liquids, and even of solids. It increases in rapidity as the temperature increases, and ceases when the vapor has reached a certain density, always the same for the same tem- perature, but greater for a higher temperature. It goes on very rapidly in a vacuum; but it is found that the final density of the vapor is no greater, or but little greater, than when some other gas. is present. While evaporation is going on, heat must be supplied to the liquid to keep its temperature constant. Evaporation may be readily explained on the kinetic theory (§ 184) on the supposition that, in the interaction of the molecules, the motion of any one may be more or less violent, as it receives § 203] EFFECTS OE HEAT. 217 motion from its neighbors or gives up motion to them. At the ex- posed surface of the substance the motion of a molecule may at times be so violent as to project it beyond the reach of the molec- ular attractions. If this occur in the air, or in a space filled with .any gas, the molecule may be turned back, and made to rejoin the molecules in the liquid mass; but many will find their way to such a distance that they will not return. They then constitute a vapor -of the substance. As the number of free molecules in the space .above the liquid increases, it is plain that there may come a time when as many will rejoin the liquid as escape from it. The space is then saturated with the vapor. The more violent the motion in the liquid, that is, the higher its temperature, the more rapidly the molecules will escape, and the greater must be the number in the space above the liquid before the returning will equal in number the outgoing molecules. In other words, the higher the tempera- •ture, the more dense the vapor that saturates a given space. If the .space above a liquid be a vacuum, the escaping molecules will at first meet with no obstruction, and, as a consequence, the space will be very quickly saturated with the vapor. The presence of another vapor or a gas impedes the motion of the outgoing molecules, and •causes evaporation to go on slowly, but it has very little influence upon the number of molecules that must be present in order that those which return may equal in number those which escape. Since ■only the more rapidly moving molecules escape, they carry ofE more than their share of the heat of the liquid, and thus the temperature will fall unless heat is supplied from without. 203. Pressure of Vapors. — As a liquid evaporates in a closed space, the vapor formed exerts a pressure upon the enclosure and upon the surface of the liquid, which increases so long as the quantity of vapor increases, and reaches a maximum when the space is saturated. This maximum pressure of a vapor increases with the temperature. When evaporation takes place in a space filled by another gas which has no action upon the vapor, the pressure of the vapor is added to that of the gas, and the pressure of the mixture is, therefore, the sum of the pressures of its constituents. The law 218 ELEMENTARY PHYSICS. [§ 204 was announced by Dalton that the quantity of vapor which satu- rates a given space, and consequently the maximum pressure of that vapor, is the same whether the space be empty or contain a gas. Eegnault has shown that, for water, ether, and some other sub- stances, the maximum pressure of their vapors is shghtly less when air is present. 204. The Vapor Pressure of Solutions. — The pressure of the saturated vapor formed from an indifferent solution, or one which is not an electrolyte, is always less than the vapor pressure of the pure solvent. Kaoult discovered that the diminution of vapor pressure is proportional to the concentration, provided the solutions are very dilute and that, for any one solvent, the diminution of vapor pressure is the same, whatever be the dissolved substance, provided the solutions are equimolecular, .that is, contain the same number of molecules in equal volumes of the solutions. It may be shown on theoretical grounds that the diminution of vapor pressure depends upon the density of the vapor and the osmotic pressure and density of the solution. Solutions which are not electrolytes, or which are not indiffer- ent, exhibit a diminution of vapor pressure proportional to the concentration, but the amount of change is greater than in indiffer- ent solutions. This difference is explained by assuming a partial or complete dissociation of the molecules of the dissolved substances into their constituent ions (§285). 205. Ebullition. — As the temperature of a liquid rises, the pressure which its vapor may exert increases, until a point is reached where the vapor is capable of forming, in the mass of the liquid, bubbles which can withstand the superincumbent pressure of the liquid and the atmosphere above it. These bubbles of vapor, escaping from the liquid, give rise to the phenomenon called ehul- lition, or boiling. Boiling may, therefore, be defined as the agita- tion of a liquid by its own vapor. Generally speaking, for a given liquid, ebullition always occurs at the same temperature for the same pressure ; and, when once commenced, the temperature of the liquid no longer rises, no § 306] EFFECTS OF HEAT. 219 matter how intense the source of heat. This fixed temperature is called the boiling-point ot the liquid. It differs for different liquids, and for the same liquid under different pressures. That the boiling-point must depend upon the pressure is evident from the explanation of the phenomenon of ebullition above given. Substances in solution, if less volatile than the liquid, raise the boiling-point. While pure water boils at 100°, water saturated with common salt boils at 109°. The material of the containing vessel also influences the boiling-point. In a glass vessel the tem- perature of boiling water is higher than in one of metal. If water be deprived of air bj long boiling, and then cooled, its tem- perature may afterwards be raised considerably above the boiling- point before ebullition commences. Under these conditions the first bubbles of vapor will form with explosive violence. The air dissolved in water separates from it at a high temperature in minute bubbles. Into these the water evaporates, and, whenever the elastic force of the vapor is sufficient to overcome the superin- cumbent pressure, it enlarges them, and causes the commotion that marks the phenomenon of ebullition. If no such openings in the mass of the fiuid exist, the cohesion of the fiuid, or its adhesion to the vessel, as well as the pressure, must be overcome by the vapor. This explains the higher temperature at which ebullition com- mences when the liquid has been deprived of air. 206. Spheroidal State. — If a liquid be introduced into a highly heated capsule, or poured upon a very hot plate, it does not wet the heated surface, but forms a fiattened spheroid, which presents no appearance of boiling, and evaporates only very slowly. Boutigny has carefully studied these phenomena, and made known the fol- lowing facts: The temperature of the spheroid is below the boil- ing-point of the liquid. The spheroid does not touch the heated plate, but is separated from it by a non-conducting layer of vapor. This accounts for the slowness of the evaporation. To maintain the liquid in this condition the temperature of the capsule must be much above the boiling-point of the liquid; for water it must be at least 200° C. If the capsule be allowed to cool, the temperature- 220 ELEMENTAKT PHYSICS. [§ 207 will soon fall below the limit necessary to maintain the spheroidal state, the liquid will moisten the capsule, and there will be a rapid ebullition with disengagement of vapor. If a liquid of very low boiling-point, as liquid nitrous oxide, which boils at - 88°, be poured into a red-hot capsule, it will assume the spheroidal state; and, since its temperature cannot rise above its boiling-point, water, or even mercury, plunged into it, will be frozen. 207. Production of Vapor in a Limited Space.— When a liquid is heated in a limited space the vapor generated accumulates, in- creasing the pressure, and the temperature rises above the ordinary boiling-point. Cagniard-Latour experimented upon liquids in spaces but little larger than their own volumes. He found that, -at a certain temperature, the liquid suddenly disappeared ; that is, it was converted into vapor in a space but little larger than its own volume. It is supposed that above the temperature at which this ■occurs, which is called the critical temperature, the substance can- not exist in the liquid state (§ 223). 208. Liquefaction. — Only a certain amount of vapor can exist at s. given temperature in a given space. If the temperature of a space .saturated with vapor be lowered, some of the vapor must condense into the liquid state. It is not necessary that the temperature of the whole space be lowered; for when the vapor in the cooled por- tion is condensed, its pressure is diminished, the vapor from the warmer portion flows in, to be in its turn condensed, and this con- tinues until the whole is brought to the density and pressure due to the cooled portion. Any diminution of the space occupied by a saturated vapor at constant temperature will cause some of the vapor to become liquid, for, if it do not condense, its pressure must increase; but a saturated vapor is already at its maximum pressure. If the vapor in a given space be not at its maximum pressure, its pressure will increase when its volume is diminished, until the maximum pressure is reached; when, if the temperature remain constant, further reduction of volume causes condensation into the liquid state, without further increase of pressure or density. This statement is true of several of the gases at ordinary temperatures. I 310] EFFECTS OF HEAT. 321 Otilorine, sulphur dioxide, ammonia, nitrous oxide, carbon dioxide,, and several other gases, become liquid under sufficient pressure. Andrews found that at a temperature of 30.9S° pressure ceases to liquefy carbon dioxide. This is the critical temperature for that substance. The critical temperatures of oxygen, hydrogen, and the other so-called permanent gases, are so low that it is only by methods capable of yielding an extremely low temperature that they can be liquefied. By the use of such methods any of the gases may be made to assume the liquid state. In the case of hydrogen, however, the low temperature necessary for its liquefaction has only been reached by allowing the gas to expand from a condition of great condensation, in which it had already been cooled to a very low point. The first successful attempts to condense these gases were made by Cailletet and Pictet, working independently. The best work on the subject has been done by Olszewski, who has succeeded in obtaining large quantities of liquid oxygen, nitrogen,, and hydrogen, and in freezing nitrogen. 209. Pressure and Density of Saturated Gases and Vapors.— It has been seen that, for each gas or vapor at a temperature below the critical temperature, there is a maximum pressure which it can exert at that temperature. To each temperature there corresponds a maximum pressure, which is higher as the temperature is higher. A gas or vapor in contact with its liquid in a closed space will exert its maximum pressure. The relation between the temperature and the corresponding maximum pressure of a vapor is a very important one, and has been the subject of many investigations. The vapor of water has been especially studied, the most extensive and accurate experi- ments being those of Regnault. 210. Pressure and Density of Non-saturated Gases and Vapors. — If a gas or vapor in the non-saturated condition be maintained at constant temperature, it follows very nearly Boyle's law (§ 105), If its temperature be below its critical temperature, the product of volume by pressure diminishes, and near the point of saturation the departure from the law may be considerable. At this point 222 ELEMENTARY PHYSICS. [§ 211 the pressure becomes constant for any further diminution of vol- ume, and the gas assumes the liquid state. The less the pressure and density of the ^as, the more nearly it obeys Boyle's law. 211. Gay-Lussac's Law. — It has been stated already that gases expand as the temperature rises. The law of this expansion, called, after its discoverer, Gay-Lussac's law, is that, for each increment of temperature of one degree, every gas expands by the same con- stant fraction of its volume at zero. This is equivalent to saying that a gas has a constant coefficient of expansion, which is the same for all gases. Let Fo , Ft represent the volumes at zero and t respectively, , and a the coefficient of expansion. Then, the pressure remaining constant, we have Vt = F„(l + at). (69) If d„ , dt represent the densities at the same two temperatures we have, since densities are inversely as volumes. Later investigations, especially those of Eegnault, show that this simple law, like the law of Boyle, is not rigorously true, though it is very nearly so for all gases and vapors which are not too near their points of saturation. The common coefficient of expansion is a = 0.003666 = -^^ very nearly. 212. Boyle's and Gay-Lussac's Laws. — From the law of Boyle we have, for a given mass of gas, if the temperature remain constant, Vpp = Vj,,p' = volume at pressure unity, where Vp , Vp, represent the volumes at pressure p and p' respectively. From the law of Gay-Lussac we have, if the pressure remain constant, Fj = fX^^ = f IT^/ I^ ^he temperature and pressure both vary, we have l-^at 1 + at" y^^l § 313J EFFECTS OF HEAT. 223 that is, if the volume of a given mass of gas be multiplied by the corresponding pressure and divided by the factor of expansion, the quotient is constant. Let us represent this constant by C and write ^j for a and v for Fpt. Then we have ^^J^ ^ = _ = ^, where ^ is a constant. If the temperatures of the gas be reckoned from a zero point which is 273° below the melting-point of ice, or the zero of the centi- grade thermometer, we may set 273 + if = T, where T is the tem- perature reckoned from the new zero, and have finally pv = RT (72) as the equation which embodies Boyle's and Gay-Lussac's laws. The temperature T is called the temperature on the scale of the air-thermometer, and the zero from which it is reckoned is called the zero of the air-thermometer. For reasons which will subse- quently appear, it is also called the absolute temperature, and its aero the absolute zero. 213. Elasticity of Gases. —It has been shown (§ 105) that the •elasticity of a gas obeying Boyle's law is numerically equal to the pressure. This is the elasticity for constant temperature. But when a gas is compressed it is heated (§158); and heating a gas increases its pressure. Under ordinary conditions, therefore, the ratio of a small increase of pressure 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 compression 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 (Pig. 70) be a curve representing the relation between volume and pressure for constant temperature, of which the abscissas represent volumes and the ordinates pressures. Such a curve is called an isothermal line. It is plain that to each tem- perature must correspond its own isothermal line. If, now, we suppose the gas to be compressed, and no heat to escape, it is plain 224 ELEMENTARY PHYSICS. [§ 3M that if the volume diminish from OC to OG, the pressure will become greater than GD; suppose it to be GM. If a number of such points as M be found, and a line be drawn through them, it will repre- sent the relation between volume^ and pressure when no heat enters- or escapes. It is called an adia- latic line. It evidently makes a greater angle with the horizontal than the isothermal. The tangents to these lines at the point of intersection, being the ratios of the changes of pres- sure to the same changes of volume under the conditions repre- sented by those lines are proportional to the elasticity at constant temperature, or the isothermal elasticity Ef, and to the elasticity when no heat is allowed to enter or escape, or the adiabatic elas- ticity E^, respectively. 214. Specific Heats of Gases. — The amount of heat necessary to raise the temperature of unit mass of a gas one degree, while the- volume remains unchanged, is called the specific heat of the gas at constant volume. The amount of heat necessary to raise the tem- perature 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 constant pressure. 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. 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 reasonable rise of temperature. The various sources of error pro- duce effects which are sometimes as great as, or even greater than, the quantity to be measured. It is beyond the scope of this work I 215] EFFECTS OF HEAT. S25 to describe in detail the means by which the effects of the disturb- ing causes have been determined or eliminated. The specific heat of a gas at constant volume is generally de- termined from the ratio between it and the specific heat at constant pressure. The first direct determination of this ratio was accom- plished by Clement and Desormes. It is now most commonly de- termined from the velocity of sound (§§ 135, 216). 215. Work Done by the Expansion of a Gas.— It was shown by Joule that when a gas expands without doing external work, its temperature remains practically constant. His expei'iment consisted in allowing gas compressed within a reservoir to flow into another reservoir in which a vacuum had been made. The reservoirs were immersed in the water of a calorimeter; it was found that in these circumstances the expansion of the gas was not attended either by the evolution or absorption of heat. As the gas had done no ex- ternal work during the expansion, this proved that its energy remained unchanged. The energy of a gas is therefore a function of its temperature alone. If the temperature of a unit mass of gas be raised 1° while its volume is kept constant, the quantity of heat C„, the specific heat at constant volume, must enter the gas. If its temperature be raised by the same amount while it is allowed to expand under con- stant pressure and to do work IF by that expansion, a quantity of heat G^, the specific heat at constant pressure, must be used. Since the gas is at the same temperature at the end of each of these opera- tions, its energy must be the same m both cases, and the difference between the quantities of heat employed, or Op — C„, must be equal to the work W done by the expansion. The experiments of Joule and Thomson, which proved that the experiment of Joule just described was not sufficiently sensitive to yield an exact result, and that the temperature of a gas really falls slightly when it expands without doiug external work, do not seriously invalidate the conclusion just drawn; they merely prove that some internal work is done in the gas during its expansion. This internal work is so small in amount that it may be neglected in most cases. 226 ELEMENTAKT PHYSICS. [§ 216 216. Ratio of the Elasticities and of the Specific Heats of a Gas. —The ratio of the two principal specific heats of a gas is the same as the ratio of its two principal elasticities. To show this, construct an adiabatic line and an isothermal line (Fig. 71) intersecting at the point ; from that point draw a line parallel with the axis of volumes and take a point A on that line very near the point 0. Through that point draw a line parallel with the axis of pressures, intersecting the isother- ^'^- '^^- mal and the adiabatic lines at B and respectively. OA is the diminution of volume, Av, caused by an increase of pressure AB — Sp if the compression is isothermal, or by the increase of pressure AC- Jpit the compression is adiabatic. From the definition of elasticity (§ 102) we have the equations Et - -^^, K- ^^ , and iience ^^ - ^^. Ap We will now determine the value of the ratio —■ in terms of the o'p principal specific heats. For convenience we assume that we are dealing with a unit mass of gas. The diminution of volume Av at constant pressure sets free the quantity of heat C^ . At, where At is the change of temperature that occasions the change of volume; the point A then represents the condition of the gas. The gas may be brought into this same condition by an adiabatic compression from to C, during which no heat either enters or leaves the gas, and by a diminution of pressure AC ^ Ap while the volume is con- stant, caused by the abstraction of the heat produced by the com- pression. The heat which must be abstracted from the gas in order that it shall attain the condition denoted by ^, is to the heat that must be abstracted to cause the diminution of pressure BA = Sp in the ratio of Ap to Sp. The heat which must be ab- stracted to cause the diminution of pressure BA = Sp at constant volume is C„ . At, where At has the same value as before, since the fore be set equal, so that we have Op/It ~ C^-~At, and hence § 317] EFFECTS OF HEAT. 227 -change of temperature is that experienced in passing from the isothermal OB to the isothermal which passes through A. The heat abstracted to produce the diminution of pressure Ap is there- in fore C„ . -^ . M. Now the internal energy of the gas in the con- -dition represented by A depends only on its temperature and is independent of the way in which that condition is reached. The work done on the gas in its change from to ^ does depend on the way in which the change is effected, but the difference between the work done on it during the first operation and that done on it during the second operation is an infinitesimal of the second order, represented by the area OCA {§ 232), and may be neglected. The quantities of heat abstracted during the two operations may there- Ap dp dp- c; - Et' ^^^' by the equation already obtained. It has been shown that the Telocity 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 = y -=■ In the progress of a sound-wave the air is alternately compressed and rarefied, the compressions and rarefactions occurring in such rapid succession that there is no time for any transfer of heat. If this equation 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, E,^ can be computed. In § 105 it is shown that Et is numerically equal to the pressure; hence we have the values of the two elasticities of air, and, as seen above, their ratio is the ratio of the two specific heats of air. 217. 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 evapora- 228 ELEMENTAKT PHYSICS. [§ 218 tion. By promoting evaporation, intense cold may be produced. In a vacuum, water may be frozen by its own evaporation. If a liquid be heated to a temperature above its ordinary boiling-point under pressure, relief of the pressure is followed by a very rapid evolution of vapor and a rapid cooling of the liquid. Liquid nitrous oxide at a temperature of zero is still far above its boiling- point, and its vapor exerts a pressure of about thirty atmospheres. If the liquid be drawn off into an open vessel, it at first boils with extreme violence, but is soon cooled to its boiling-point for the atmospheric pressure, about — 88°, and then boils away slowly, while its temperature remains at that low point. By liquefying nitrogen and then allowing it to evaporate under low pressure, Olszewski obtained the temperature of — 220° C, and by allowing liquid hydrogen to boil under atmospheric pressure, — 243.5° C. was reached. 218. Heat Equivalent of Vaporization. — It is plain that the for- mation 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 attraction, at the expense of the energy of the molecules that remain behind. A quantity of heat disappears when a liquid evaporates; and experiment demonstrates, that to evaporate a kilo- gram of a liquid at a given temperature always requires the same amount of heat. This is the heat equivalent of vaporization. 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 vaporization is determined by passing the vapor at a known temperature into a calorimeter, there condensing it into the liquid state, and noting the rise of tempera- ture in the calorimeter. This, it will be seen, is essentially the method of mixtures. Many experimenters have given attention to this determination ; but here, again, the best experiments are those of Eegnault. He determined what he called the total lieat 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 § 221] EFFECTS OF HEAT. 229 of saturated vapor at the pressure chosen, and then convert it wholly into steam. The result of his experiments give, for the heat equiva- lent of vaporization of water at 100°, 537 calories. That is, he found that by condensing a kilogram of steam at 100° 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°. 219. Dissociation. — It has already been noted (§ 157), that, at high temperatures, compounds are separated into their elements. To effect this separation, the powerful forces of chemical affinity must be overcome, and a considerable amount of energy must be consumed. 220. 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 determine, because the process is usually com- plicated by changes of physical state. We may cause the union of carbon and oxygen in a calorimeter, and, bringing the products of combustion 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 chemical energy must have been consumed in that process. The heat meas- ured is the available heat. The best determinations of the available heat of chemical union have been made by Andrews, Favre and «Silbermann, and Berthelot. THE KINETIC THEORY OF HEAT. 221. Molecular Motion. States of Matter.— The continued pro- duction of heat by the expenditure of mechanical work proves that heat is not a substance, and suggests that it must be in some way dependent on motion. It has been seen that such phenomena as expansion and fusion may be explained on the hypothesis that the molecules of a body move more rapidly when the body is heated. 230 ELEMENTARY PHYSICS. [§ 221 The emission of light or, in general, of radiant energy from a body affords a demonstration of the existence of some motion in those parts of a body which are so small that the motion cannot be directly perceived by ordinary observation ; for we can explain radiance only as a motion in a medium through which it travels, and it is evident . that this motion cannot be due to the mere presence of a sub- stance, but must be set up by the motion of matter. We may first apply the kinetic theory to the distinction between solids, liquids, and gases. Each molecule of a solid is supposed to be retained within a certain small region by the action of the surrounding molecules and to move within that region. The phenomenon of crystallization leads us to think that molecules in a solid have certain determinate forms and an arrangement in the body; their motions, therefore, are such that they do not overstep the limits of this arrangement, and we think of their motion as vibratory, using the word vibratory in a rather loose sense. The molecules of a liquid have no fixed position in the mass, but are free to move from one point to another; they are in very close- proximity to one another, as appears from the phenomena of capil- larity, and exert considerable forces on one another. The chief difference between solids and liquids consists in the absence in the latter of any definite arrangement ; we may think of the molecules of a liquid as rotating and as gliding past each other, and can characterize their motion as rotatory. The great increase in vol- ume exhibited on the change of a mass of liquid into vapor shows that the molecules of a vapor or gas are farther apart than those of a liquid. They are so far apart that their mutual actions due to • molecular forces have very little influence on their motions, except during the excessively short period within which any two of them come close together or undergo an encounter. A molecule of a gas is therefore thought of as moving in a succession of short rec- tilinear paths, the direction of which is in general changed at each encounter. "We may therefore characterize the motion in a gas as translatory. The consideration of this translatory motion is sufii- § 231] EFFECTS OF HEAT. 331 cient to explain most of the laws of gases, though to explain others a rotation or something equivalent to it must be assumed. The characteristics of the molecular motion assumed in the kinetic theory may be best explained by considering the motion in a gas. Let us suppose that a very large number of material par- ticles is distributed uniformly throughout the region contained within a closed vessel, and that velocities are given to these mole- cules at a certain instant in various directions. If we further sup- pose that these molecules act on each other only by collision or by forces which are effective only when two molecules are extremely near each other, it is plain that the paths of the molecules thus assumed will in general be short straight lines, changing in direc- tion with every encounter between two molecules. It is also evi- dent that, no matter what the initial velocities were, they will not be maintained for any length of time, but that the velocity of any one molecule will change at each encounter, and that the velocities of the molecules in the mass will speedily acquire values ranging from zero to a very great or practically infinite velocity. It is also plain that very few molecules will possess these extreme velocities at any one time, and that most of them will possess velocities which do not depart far from a certain mean. An obvious con- dition to which the velocities must conform is that the kinetic energy of all the molecules in the mass must remain the same at all times, it being assumed that no energy enters the mass from without and that the encounters do not involve the loss of kinetic energy. It was shown by Clausius, and afterwards more rigorously by Maxwell, that the distribution of velocity among the molecules may be deduced by the theory of probabilities. Some idea of it may be got from the distribution of shots iu a target; if a rifleman shoot at a target a great many times, and if the distance of the shots from the centre of the bull's eye be measured, these distances conform to the same law of distribution. It is clearly infinitely improbable that any one of the shots will strike the exact centre of the bull's eye, and also infinitely improbable that any one will be sent directly away from the target, and it is very highly improb- 333 ELEHENTAKT PHYSICS. [§ 322 able that any one will miss the target entirely; the vast majority of the shots will meet the target, and their distances from the centre will lie around a certain average distance. Similarly, it is extremely , improbable that any molecule of a gas will have a velocity far exceeding the average; the great majority of them will have veloci- ties which lie around a certain mean velocity. The law of distri- bution of velocities among molecules of liquids and solids is not known, but it probably possesses the essential characteristics of the law for gases. When a gas is heated, all but a very small part of the heat which enters it is uoed in increasing the kinetic energy of the molecules; this is not true for solids and liquids, because, when they are heated, work is done against their molecular forces which does not appear as kinetic energy. The kinetic energy of the molecule is the sum of the kinetic energy due to the motion of its centre of mass or to its translation, and of the kinetic energy due to its motion relative to its centre of mass. This latter energy may be thought of as due either to rotation about the centre of mass or to the vibrations of the atoms constituting the molecule. We will subsequently prove that the temperature of a gas is proportional to the kinetic energy of its molecules. It is therefore natural to assume that the measure of temperature is some part of the kinetic energy of the molecule. The most consistent explanation of all the effects of heat can be reached by supposing that the energy of atomic vibration or of molecular rotation is directly proportional to the temperature measured on the absolute scale (§ 212). The total kinetic energy of the molecules of a body measures the heat in the body. 222. Kinetic Theory of Gases. — The foundation of the theory of matter now under discussion is the linetic theory of gases. In this theory a perfect gas consists cf an assemblage of free, perfectly elastic molecules in constant motion. Each molecule moves in a straight line with a constant velocity, until it encounters some other molecule, or the side of the vessel. The impacts of the molecules I 223] EFFECTS OF HEAT, 233 upon the sidos of the vessel are so numerous that their effect is that of a continuous constant force or pressure. The entire independence of the molecules is assumed from the fact that, when gases or vapors are mixed, the pressure of one is added to that of the others ; that is, the pressure of the mixture is the sum of the pressures of the separate gases. It follows from this, that no energy is required to separate the molecules j in other words, no internal work need be done to expand a gas. This was ■demonstrated experimentally by Joule (§ 215). The action between two molecules, or between a molecule and a solid wall, must be of such a nature that no energy is lost; that is, the sum of the kinetic energies of all the molecules must remain constant. Whatever be the nature of this action, it is evident that ■when a molecule strikes a solid stationary wall it must be reflected back with a velocity equal to that before impact. If the velocity be resolved into two components, one parallel to che wall and the ■other normal to it, the parallel component remains unchanged, while the normal component is changed from + u, its value before impact, to — u, its value after impact. The change of velocity is therefore 2m, and if 6 represent the duration of impact, the mean acceleration is -^, and the mean force of impact p = fn-rr, where m represents the mass of the molecule. Since the effect of the impacts is a continuous pressure, the total pressure exerted upon unit area is equal to this mean force of impact of one molecule multiplied by the number of molecules meeting unit area in the time 0. To find this latter factor, we sup- pose the molecules confined between two parallel walls at a dis- tance s from each other. Any molecule may be supposed to suffer reflection from one wall, pass across to the other, be reflected back to the first, and so on. Whatever may be the effect of the mutual collisions of the molecules, the number of impacts upon the surface considered will be the same as though each one preserved its rec- tilinear motion unchancred, except when reflected from the solid walls. The time required for a molecule moving with a velocity u 234 ELEMEJfTAKY PHYSICS. [§ 232 to pass across the space between the two walls and back is — ; and the number of impacts upon the first surface in unit time is -^. Consider the molecules contained in a rectangular prism, with bases of area a in the walls. These molecules must be considered as moving in all directions and with various velocities. But the velocity of any molecule miiy be resolved in the direction of three rectangular axes, one normal to the surface and the other two par- allel to it, and the effect upon the walls will be due only to the normal components. Let us single out for examination a group of molecules which have a normal velocity that lies near the value m, , and let w, represent the number of such molecules in unit volume. Then the number of such molecules within the prism considered is w,sa. The number of impacts made by them in unit time on one 01 the walls is n,sa . jr- = ~r-, and in the time o is ' ' -. ' 2s 2 2 Hence the total pressure which they exert on the area a is 2m, n,au,6 j j •. ■ i m-jr . '_-,' - = mn^u'a, and on unit area is inn^u^. Now the total pressure on unit of area is the sum of the pres- sures due to all the i groups into which the molecules of the gas may be divided, or p — m^n^u^ ■\- n^u^ + • • • «!»,■"). If we repre- sent by n the number of molecules in unit volume and by u the mean velocity given by nu'' = n^u' -\- n^u^ + . . . «,? ^t! ^iid there- JJb JJa fore H^ > Eg, and also h^ > ^b- The result of these combined operations is that no work is done by the engines, and that the source receives heat while the refrigerator loses heat. This conclu- sion is contrary to Clausius's principle and must be rejected, as inconsistent with the operations of Nature. We conclude, there- fore, that no engine can have an efficiency greater than that of the reversible engine. It follows as a corollary that the efficiency of all reversible engines is the same. 231. Absolute Scale of Temperatures. — Since the efficiency of all reversible engines is the same and is a maximum, it is mani- festly indiflEerent what material is used in the working body; in- deed, since the demonstration just given does not involve as essential the particular mode of doing work assumed in the con- struction of the diagram, it is also indifferent in what way the working body changes its dimensions and does work. The work done depends only on the heat received from the source and on the 248 ELEIIENTART PHYSICS. [§ 231 temperature of the source and refrigerator, and the efiBciency de- pends only on the temperatures of the source and the refrigerator, or is a function of these temperatures. If the temperatures be represented on any conventional scale, the form of this function may be found by experiment; on the other hand, the assumption of a form of this function will determine a scale of temperatures. The proposal to form such a scale, which is dependent only on the efficiency of the reversible engine, and is therefore independent of the properties of any particular body, was made by William Thomson. The scale of temperatures which is most convenient for applica- tion in thermodynamics, and which is so distinguished by its sim- plicity from all others that might be formed that it is called distinctively the absolute scale of temperatures, is formed by as- suming that the efficiency of a reversible engine is equal to the ratio of the difference of temperature between the source and the refrigerator and the temperature of the source, that is, by assuming This assumption may also be stated in the form h_R H~ S' (80) The maximum efficiency of an engine is attained when all the heat which is received from the source is transformed into work, so that no heat is transferred to the refrigerator; on the scale of temperatures just assumed this condition is attained when i? = 0. This zero is an absolute and not an arbitrary zero. It depends on the general properties of bodies, and not on the particular proper- ties of any one body. It is the lowest temperature attainable in Nature, for, if it were possible to have a refrigerator at a lower temperature than this, the efficiency of an engine working with that temperature as the temperature of its refrigerator, would be § 232] THERMODTSAMICS. 249 greater than unity. This temperature is therefore called the abso- lute zero. The length of the degree on the absolute scale may be deter- mined by designating the difference of temperature between two bodies by an arbitrarily chosen number and by measuring the effi- ciency of an engine working between the temperatures of those bodies. The most convenient assumption to make is that the abso- lute difference between the temperature of boiling water and the temperature of melting ice is 100 degrees. The temperature inter- vals or degrees on the scale thus formed are very nearly those of the Centigrade scale. 232. Relation of the Absolute Temperature to the Temperature of the Air Thermometer. — Let us assume that a substance exists which obeys perfectly the laws of Boyle and Gay-Lussac; such a substance is called a perfect gas. We wish to show that the tem- peratures indicated' by the expansion of a perfect gas, used as a thermometric substance, will be those of the absolute scale. We must first prove that the work done by the expansion of a gas is equal to the area included between the lines representing its changes of pressure and volume, the two ordi nates representing its extreme pressures and the horizontal line of zero volume. The proof of this proposition does not depend on the properties of a perfect gas, and the proposition holds in all cases in which the body ■does work by expanding under a hydrostatic pressure which is the same at all points of its surface. Let us select a small area s on the surface of the body. The pressure p is applied to all points of the surface, and the force which acts on the area s is therefore ps. Let the body expand slightly, so that the area s is displaced along its normal through the distance n. The work done in displacing the area ,s is psn, and the work done in expanding the whole body is 2psn = p2sn, since p is the same for all points on the surface. Now 2sn is equal to the increase in the volume of the body, or to dv. The work done during the small expansion is therefore pdv. This expansion will, in general, involve an infinitesimal change in the pressure; but if the process here described be repeated for each 250 ELEMENTARY PHYSICS. [§ 232 infinitesimal increment of volume, the sum of all the terms pdv •will equal the total work done by the ezpansion of the body. Now let us consider the area hBCc standing under the line BC (Fig. 73). This area may be conceived of as made up of a series of infinites- imal rectangles, the heights of which are the ordinates of the successive points of ^~)? the line BC, and the bases of which are i\ successive elements taken along the line ./i|° he. If dv represent the length of one of I j these elements, and p the corresponding ~c d ordinate, the area of the infinitesimal rect- PwJ. 73. angle determined by them is pdv. The sum of such areas for the expansion indicated by the line BC is the area hBCc; and since 'S.pdv represents the work done, the area hBCc also represents the work done during the expansion of the body in the way indicated by the line BC. Now to demonstrate the relation between the temperatures in- dicated by the perfect gas thermometer and those of the absolute scale, let us suppose an engine m which the working body is a perfect gas, and let us suppose that the changes in pressure and volume experienced by the working body during the cycle are so small that the portions of the isothermal and adiabatic lines which bound it are straight, and that the cycle is a parallelogram. This cycle is represented by the area ABCD (Fig. 73). We may assume as the result of the experiments of Joule that when a gas expands at constant temperature, no internal work is done upon it, or that the heat which enters it is entirely spent in doing ex- ternal work. Produce DA to e; then the parallelogram ABCD is equal to the parallelogram eBCf, and this parallelogram represents the work done in the cycle by the gas acting as the working body. The work done during the expansion from B to C, which is equal to the heat received during that expansion, is represented by the area hBCc. Let g be the middle point of the line BC; the perpendicular gh will bisect the line ef at i. The area hBCc = ic . gh, and the area eBCf= he . gi. Therefore the efficiency of I 232] THEUMODYNAMICS. 251 ^BCf , qi ,, the engine, or ^jj^^, equals ^. Now gh represents the pressure of the gas at the temperature t of the source, when its volume is Ohy and gi represents the diminution of pressure caused by a fall of temperature to 0, the temperature of the refrigerator, when the volume is kept constant. The eflaciency of the engine is therefore V^ — . And since the efiSciency is also given by — ^^— -, where S Pt b and R are the temperatures of source and refrigerator on the ab- solute scale, — ^ — = ^ — i-* ov- =^. We know, from the experi- ments of Gay-Lussac, that if t and 6 be measured on the Centigrade scale, and if p„ represent the pressure of the gas at the Centigrade- zero on the condition that the volume is constant, p^ =^o(l + oit) and pe =i'o(l + «^)> where a = --- is the coefficient of expansion. Using these values in the above equation we obtain -= = * S !-{■ at 273 A- 6 „^„ , . If the pressure or volume of a gas, the two being inter- changeable by Boyle's law, be used as a measure of its temperature,,. the pressure or volume and the temperature will always be directly proportional, provided the zero of temperature be taken at — 273°' Centigrade; this temperature is the zero of the perfect gas thermom- eter. Prom the equation just obtained it is clear that the absolute- scale of temperatures is the same as the one given by the perfect gas thermometer, and that the absolute zero is the zero of the per- fect gas thermometer. No gases conform precisely to the laws of Boyle and Gay-Lussac-,, and consequently no gas thermometer can be constructed which will accurately indicate the absolute scale of temperatures. Never- theless, some gases depart only slightly from the conditions of a perfect gas, and the temperature determinations given by thermom- eters in which such gases are employed may be converted by suit- able corrections into the corresponding absolute temperatures. 253 ELEMENTARY PHYSICS. [§ 233 233. The Steam-engine. — The steam-engine in its usual form consists essentially of a piston, moving in a closed cylinder, which is provided with passages and valves by which steam can be ad- mitted and allowed to escape. A boiler heated by a suitable fur- nace supplies the steam. The valves of the cylinder are opened and closed automatically, admitting and discharging the steam at the proper times to impart to the piston a reciprocating motion, which may be converted into a circular motion by means of suita- ble 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 cir- culation of cold water. Here the steam is condensed into water; and this water, with air or other contents of the condenser, is re- moved by a pump. In non-condensing engines the steam escapes into the open air. In this case the 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-pres- Cf T> sure. Applying the expression for the efficiency (§ 231), e — — - — , S it will be seen that, for any boiler-pressure which it is safe to em- ploy in practice, it is not possible, even with a perfect engine, to convert into work more than about fifteen per cent of the heat used. In the condensing engine the temperature of the refrigerator may be taken as that of saturated steam at the pressure which ex- ists in the condenser, which is usually about 30° or 40° : hence S — Risz, much larger quantity for condensing than for non-con- densing engines. The gain of efficiency is not, however, so great as would appear from the formula, because of the energy that must be expended to maintain the vacuum in the condenser. 234. Hot-air and Gas Engines. — Hot-air engines consist essen- tially of two cylinders of different capacities, with some arrange- ment 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 be- § 2^4:] THERMODYNAMICS. 253: tween the two cylinders, of which the smaller may be considered as a supply-purap for taking air from outside 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. Ou the re- turn stroke this air is expelled into the external air, still hot, but at a lower temperature than it would have been had it not ex- panded and performed work. This case is exactly analogous to- that of the steam-engine, in which water is forced, by a piston work- ing in a small cylinder, 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 supplied 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 Eider " compression-engine " are interesting as realizing an approach to Oarnot's cycle. These engines, like those described above, consist of two cylin- ders 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, alternately expands and con- tracts, moving the piston, and performing 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 circulation of water. The cylinders communicate freely with each other. The pistons are connected to cranks set on an axis, so as to make an angle of nearly ninety degrees with each other. Thus both pistons are moving for a short time in the same direction twice during the revolution of the axis. At the instant that the small piston reaches the top of its stroke, the large piston will be near the bottom of the cylinder, and descending. The small piston now descends, as well .254 BLEMBNTAET PHYSICS. [§ 235 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 smaller con- tinues. The air is rapidly transferred to the larger heated cylinder, .and expands while taking heat from the highly heated surface. After the small piston has reached its lowest point there is a short time during which both the pistons are rising and the air expanding, -with but little transfer from one cylinder to the other, and with a relatively small absorption of heat. When the descent of the large piston begins, the small one still rising, the air is rapidly trans- ferred to the smaller cylinder: its volume is diminished, and its heat is given up to the cold surface with which it is brought in con- tact. The completion of this operation brings the air back to the condition 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 per- formed without transfer of heat, and the second and fourth not with- out change of temperature, — still furnish an example of work done by heat through a series of changes in the working substance, which brings it back, at the end of each revolution, to the same -condition as at the beginning. Gas-engines derive their power from the force developed by the combustion, within the cylinder, of a mixture of illuminating gas and air. As compared with steam-engines, hot-air and gas engines use the working substance at a much higher temperature. S—B is, therefore, greater, and the theoretical efficiency higher. There are, however, practical diflBculties connected with the lubrication of the sliding surfaces at such high temperatures that have so far pre- vented the use of large engines of this class. 235. Sources of Terrestrial Energy. — "Water flowing from a higher to a lower level furnishes energy for driving machinery. The energy theoretically available in a given time is the weight of the water that flows during that time multiplied by the height of § 335] THEEMODTNAMICS. 255 the fall. If this energy be not utilized, it develops heat by friction of the water or of the material that may be transported by it. But water-power is only possible so long as the supply of water con- tinues. The supply of water is dependent 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 multi- plied 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-potver, like that of water-power, 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 largest part of the energy employed by man for his purposes is derived from the combustion of wood and coal. This energy exists as the potential energy of chemical combination of oxygen with 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 radia- tions. 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 combines, by proc- esses 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 combustion of the same substances elsewhere. Lavoisier assumed that such chemical combinations were the source of animal heat, and was the first to attempt a measurement of it. He compared the heat developed with that due to the formation of the carbon dioxide exhaled in a given 256 ELEMENTARY PHYSICS. [§ 235 time. Despretz and Dulong made similar experiments with mofe 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 exhaled. These and similar experiments, although not taking into ac- count 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 transformed energy of food. Eumford, 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 fo,od as fuel for a steam-engine. Mayer, in 1845, held that an animal is a heat-engine, and that every motion of the animal is a transformation into work of the heat developed in the tissues. Hirn, in 1858, executed a series of interesting experiments bear- ing upon this subject. In a closed box was placed a sort of tread- mill, which a man could cause to revolve by stepping from step to step. He thus performed work which could be measured by suit- able apparatus outside the box. The tread-wheel could also be made to revolve backward by a motor placed outside, when the man de- scended 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 fiist experiment the man remained in repose; in the sec- ond he performed work by causing the wheel to revolve; in the third the wheel was made to revolve backward, and work was per- formed upon him. In the second experiment the 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 chemical 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 § 336] THBEMODYKAMICS. 357 was developed; while in the third case the motor, causing the tread-wheel to revolve, ;performed work, which produced heat in ad- dition to that due to the chemical action. It has been thought that muscular energy is due to the waste of the muscles themselves; but experiments show that the waste of nitrogenized material is far too small in amount to account for the energy developed by the animal; and we must, therefore, conclude that the principal source of muscular energy is the oxidation of the non-nitrogenized material of the blood by the oxygen absorbed in respiration. An animal is, then, a machine for converting the potential en- ergy of food into mechanical work : but he is not, as Mayer sup- posed, 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 are, therefore, the trans- formed energy of the sun. The tides are mainly caused by the attraction of the moon upon the waters of the earth. If the earth did not revolve upon its axis, or, rather, if it always presented one face to the moon, the elevated waters would remain stationary upon its surface, and fur- nish 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 tides comes from the rotation of the earth upon its axis; and a part of the energy of the earth's rota- tion is, therefore, being dissipated in the heat of friction it causes. 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 considered as the rem- nants of the original energy of the earth. 236. Energy of the Sun — It has been seen that the sun's rays 258 ELEMENTARY PHYSICS. [§ 337 are the source of all the forms of energy practically available, ex- cept that of the tides. It has been estimated that the heat re- ceived by the earth from the sun each year would melt a layer of ice over the entire globe a hundred feet in thickness. This repre- sents 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-huiidred-millionth of the heat that leaves the sun. Notwith- standing this enormous expenditure of energy, 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 particles, 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 condensation of these masses continues to generate heat. 237. Dissipation of Energy. — It has been seen that only a frac- tion of the energy of heat is available for transformation into other forms of energy, and that such transformation is possible only when a difference of temperature exists. Every conversion of other forms of energy into heat puts it in a form from which it can be only partially recovered. Every transfer of heat from one body to another, or from one part to another of the same body, tends to equalize temperatures, and to diminish the proportion of energy available for transformation. Such transfers of heat are continually taking place; and, so far as our present knowledge goes, there is a tendency toward an equality of temperature, or, in other words, a uniform molecular motion, throughout the uni- verse. If this condition of things were reached, although the total amount of energy existing in the universe would remain unchanged, the possibility of transformation would be at an end, and all ac- tivity and change would cease. This is the doctrine of the dissipa- tion of energy to which our limited knowledge of the operations of Nature leads us; but it must be remembered that our knowledge is very limited, and that there may be in Nature the means of restor- ing the differences upon which all activity depends. MAGNETISM AND ELECTRICITY. CHAPTBE L MAGNETISM. 238. Fundamental Facts. — Masses of iron ore are sometimes found which possess the property of attracting pieces of iron and a few other substances. Such masses are called natural 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 proper- ties of magnetism. If pieces of iron or steel be brought near a steel magnet, they are attracted by it, and unless removed by an outside force they remain permanently in contact with it. While in contact with the magnet, the pieces of iron or steel also exhibit 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 difference is not fully 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 magnetic condition disturbed is said to be affected by magnetic induction. In an ordinary bar magnet there are two small regions, near the ends of the bar, at which the attractive powers of the magnet 259 260 ELEMENTAKT PHYSICS. [§ 239 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 horizontal plane, the axis assumes a direction which is approximately nortli and south. The pole toward the north is usually called the north or positive pole; that toward the south, the south or negative pole. If two magnets be brought near together, it is found that their like poles repel and unlike poles attract one another. If the two poles of a magnet be successively placed at the same distance from a pole of another magnet, it is found that the forces- exerted are equal in amount and oppositely directed. The direction assumed by a freely suspended magnet shows that the earth acts as a magnet, and that its north magnetic pole is situated in the southern hemisphere. If a bar magnet be broken, it is found that two new poles are formed, one on each side of the fracture, so that the two portions are each perfect magnets. This process of making new magnets by subdivision of the original one may be, so far as known, con- tinued 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. 239. Law of Magnetic Force. — By the help of the torsion bal- ance, the principle of which is described in §5 109, 253, and by us- ing very long, thin, and uniformly magnetized bars, in which the poles can be considered as situated at the extremities. Coulomb showed that the repulsion between two similar poles, and the at- traction between two dissimilar poles, is inversely as the square of the distance between them. A more exact proof of the same law was given by Gauss, who calculated the action of one magnet on another on the assumption. % 240] MAGNETISM. 261 of the truth of the law, and showed by experiment that the action calculated was actually exerted. All theories of magnetism assume that the force between two magnet poles is proportional to the product of the strengths of the poles. The law of magnetic force is then the same as that upon which the discussion of potential and of flux of force was based. The theorems there discussed are in general applicable in the study of magnetism, although modifications in the details of their appli- •cation occur, arising from the fact that the field of force about a magnet is due to the combined action of. two dissimilar and equal poles. If m and m' represent the strengths of two magnet poles, r the •distance between them, and k a factor depending on the units in which the strength of the pole is measured, the formula bxpressing the force between the poles is k — —, 240. Definitions of Magnetic ftuantities. — 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 another 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, ~ I =MLT-\h:om which the dimensions of a magnet pole are [m] = M*L^T -'. 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 uniformly and longitudinally magnetized magnet into the distance between its poles is called its magnetic moment. 263 ELEMENTAEY PHySICS. [§ 241 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. Since any magnet may be divided into small magnets, each of which is uniformly magnetized, and for which by this definition a particular value of the intensity of magnetization can be found, it is clear that the magnetic condition of any magnet can be stated in terms of the intensity of magnetiza- tion of its parts. The dimensions of magnetic moment and of intensity of mag- netization follow from these definitions. They are respectively [ml] = M^L-T-' and [^H = M^L'^ T-\ 241. Distribution of Magnetism in a Magnet.— If we conceive of a single row of magnetic molecules with their unlike poles in contact, we can easily see that all the poles, except those at the ends, neutralize one another's action, and that such a row will have a free north pole at one end and a free south pole at the other. If a magnet be thought of as made up of a combination of such rows of different lengths, the action of their free poles may be represented by supposing it due to a distribution of equal quan- tities of two imaginary substances, called north and south magnet- ism. This distribution will be, in general, both on the surface and throughout the volume of the magnet. If the magnet be uni- formly magnetized, the volume distribution becomes zero. Thfr surface distrilution of magnetism will sometimes be used to express the magnetization of a magnet, by the use of a concept called the magnetic density. It is defined as the ratio of the quan- tity of magnetism on ai element of surface to the area of that ele- ment. The magnetic density thus defined has the same numerical value as the intensity of magnetization which measures the real distribution. 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 inten- sity of magnetization is -p or — . If, now, for the pole m we sub- to S § 242] MAGNETISM. 363 stitute a continuous surface distribution over the end of the bar, then — is also the density of that distribution. The dimensions of magnetic density follow from this definition. They are [f] == ^^^^ = ilf^i-^r-'. Coulomb showed, by oscillating a small magnet near different parts of a long bar magnet, that the magnetic force at different points along it gradually increases from the middle of the bar, where it is imperceptible, to the extremities. This would not be the case if the bar magnet were made up of equal straight rows of magnetic molecules in contact, placed side by side. With such an arrange- ment 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 contiguous rows makes such an arrangement, how- ever, impossible. In the earth's magnetic field, in which the lines of magnetic force may be considered parallel, a couple will be set up on any magnet, so magnetized as to have only two poles, due to the action of equal quantities of north and south magnetism distributed in the magnet. The points at which the forces making 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. 242. Action of One Magnet on the Other.— The investigation of the mechanical action of one mag- net on another is important in the construction of apparatus for the measurement of magnetism. (1) To determine the potential of a short bar magnet at a point distant from it, let JSTS (Fig. 74) g- represent the magnet of length 21, the poles of which are of strength m, and let the point P be at a distance r from the centre of the magnet, taken as origin. 264 ELEMENTARY PHYSICS. [§ 243 Let the angle POiV equal 6 and draw the perpendiculars NQ and OR to PS. Then, in the limit, if SN is very small in com- parison with OP, we have PN= r - Ar and PS=r-]- Z(r, where Ar is a small length equal to SR = L cos 0. The potential at P due to the pole at iV^ is ^ ^^ = m [-+ -^j, since z/r is very small in comparison with r. Similarly the potential at Pdue to the pole at ^ is — — = —m{ 5- V The potential at P due to the r -\- Ar \r r j magnet is therefore 2mAr 2ml cos M cos 6 . . ^X- = r' = —r^—' ^^^' where M is the magnetic moment of the magnet. We may consider the magnetic moment as projected upon the line r by multiplica- tion by cos d; the formiila shows that the potential at any point due to a short magnet is equal to the projection of the magnetic mo- ment upon the line joining the centre of the magnet with the point, divided by the square of the length of that line. The maximum value of the potential due to the magnet, for a M given value of r, is p^-, where R represents the assigned value of r. If we set -rp^ = — '—^ we obtain r' — R' cos 6 as the equation of R r the equipotential surfaces at a considerable distance from the small magnet. When i? = co , it determines an equipotential surface of zero potential, for which, for every finite value of r, we have COS 6^ = 0, and d = ~. The plane passing through the centre of the magnet and perpendicular to its axis is therefore an equipotential surface of zero potential. Since r = whenever cos ^ = 0, whatever be the value of R, all the other equipotential surfaces pass through the point 0; they are in general ovoid surfaces surrounding the poles. The lines of force of the magnet arise at the north pole and pass perpendicularly through all these surfaces to the south pole. § 243] MAGNETISM. 265 (2) The force due to a sliort lar magnet in any direction may- be determined by determining the rate of change of its potential in that direction. It is not, however, important to determine this iorce in the general case : it will be sufficient to determine it for points in the line of the axis of the magnet. Let the length of the magnet N8 (Pig. 75) be represented by •2Z and the distance from its centre to the point F by r. Then the force at Pig 75 F due to the pole at N, and directed away from the magnet, is , ^, and the force due to the pole at S, and directed toward the magnet, is . ,.^ . Now we may write m m / 1 ,2A . , ^^TTT)^ = r' - 2lr "^ ^ Ir' +?/' ^^^^^ ^ ^^ very small in compari- son with r, and similarly . ,,, = »i f-^ A. The force at P ■due to the magnet and directed away from it is, therefore, (3) In the construction of apparatus used in the measuring of magnetic quantities it is important to know the moment of couple set up by one magnet on another. We will determine this for the particular case in which both the magnets are small in comparison with the distance between their centres, and in which the centre of one is situated on the prolongation of the axis of the other. We will call the magnet, the axis of which lies in the line joining the centres, the first magnet, and the other the second magnet, and will examine the couple exerted on the second magnet by the first. Under the limitations made as to the size of the magnets, we may assume that the forces exerted by the first magnet on the poles of the second are the same as if the poles of the second magnet lay in the prolongation of the axis of the first magnet, and that they are the same for any position of the second magnet (Fig. 76). 266 BLBMBNTAEY PHYSICS. f§ 243' We designate by m' the pole of the second magnet, by IV its length, and by ^ the complement of the angle made by its axis with the line joining the cen- s N /^ tres of the magnets. On these assumptions, the force acting on the north pole of the second s'^ '0 %m' M Fig. 76. magnet is — 5 — , and the force acting on its 2971' M south pole is =— . These two forces constitute a couple with an arm 2V cos 6, and the moment of this couple is Am'l'M cose 2MM' cos ,„„. p — ~p ' y^^i where M' represents the magnetic moment of the second magnet. 2MM' This moment of couple varies from — 5 — if the magnets are at right angles to each other, to zero if they are in the same straight line. 243. The Magnetic Shell. —A magnetic shell may be defined as an infinitely thin sheet of magnetizable matter, magnetized 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 contiguous. The product of the in- tensity of magnetization at any point in the shell into the thick- ness of the shell at that point is called the strength of the shell at that point, and is denoted by the symbol/. Since we may substitute for the magnetic arrangement an imag- inary distribution of magnetism over the surfaces of the shell, we may define the strength of the shell as the product of the surface- density and the thickness of the shell. The dimensions of the strength of a magnetic shell follow at once from this definition. We have [/] equal to the dimensions of intensity of magnetization multiplied by a length. Therefore [/] = M^L^T-\ We obtain first the potential of such a shell of infinitesimal § 243] MAGNETISM. 267 area. Let the origin (Fig. 77) be taken half-way between the two- faces of the shell, and let the shell stand perpendicular to the x axis. Let a rep- -'"^T resent the area of the shell, supposed in- ,''^'' 'y finitesimal, 21 the thickness of the shell, ,,-''' | and d the intensity of magnetization. °\B'' ~ ' The volume of this infinitesimal magnet Fig- "7. is 2al, and, from the definition of intensity of magnetization, Zald is its magnetic moment. The potential at the point ^ is then given by equation (81), since Hs very small. We have F= -3- cos 6^ = -^^ cos (9 r r" Now a cos is the projection of the area of the shell upon a plan& through the origin normal to the radius vector r, and, since a is- Ct COS infinitesimal, 5 — is the solid angle 00 bounded by the lines drawn from P to the boundary of the area a. The potential then becomes V= 2ldoa =joo, since 2ld is what has been called the strength of the shell. The same proof may be extended to any number of contiguous areas making up a finite magnetic shell. The potential due to such, a shell is then ^joo. If the shell be of uniform strength, the poten- tial due to it becomes y^ftj, 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 therefore j£l. (84) It does not depend on the form of the shell, but only on the angle subtended by its -contour. At a point very near the positive face of a flat shell, so near that the solid angle subtended by the shell equals 27C, the potential is 27ij; at a point in the plane of the shell outside its boundary, where the angle subtended is zero, the poten- tial is zero ; and near the other or negative face of the shell it is — 27tj. 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 ■268 ELEMENTARY PHYSICS. [§ "244 Jace is 4:7rj. This result is of importance in connection with elec- trical currents. 244. 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 magnetic relations of the earth are of importance in navigation and geodesy. The princi- pal magnetic elements are the declination, the dip, and the horizontal intensity. The declination is the angle between the magnetic meridian, or the direction assumed by the axis of a magnetic needle suspended to move freely m a horizontal plane, and the geographical meridian. The dip is the angle made with the horizontal by the axis of a magnetic needle suspended so as to turn freely m a vertical plane •containing the magnetic meridian. The horizontal intensity is the strength of the earth's magnetic field resolved along the horizontal line in the plane of the magnetic meridian. A magnet pole of strength m in a field in which the horizontal intensity is represented by H is urged along tliis 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 yH'\ = -J m The horizontal intensity can be measured relatively to some .assumed magnet as standard, by allowing the magnet to oscillate 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 for- mula 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 suspension, it can be shown, as in § 60, that it is describing a simple harmonic motion. If cp represent the angle made by the magnet with the magnetic meridian, the moment of couple acting on the magnet is § 244] MAGNETISM. 369 giyen by MH sin

0, so that the induction is greater than the magnetic force of the field, the resultant magnetic force within the body is less than the magnetic force of the field, because the poles induced in the body act in the opposite sense to the force of the field. 248. Energy in a Magnetic Field. — On the view we are now taking, that the actions between magnets are due to a condition of the medium which occupies the field, it is natural to suppose that the energy of a set of magnets is distributed in the field. We will find a law for this distribution, which associates the energy with the tubes of induction. The energy of the system is manifestly equal to the work that would be required to construct that system. We will first show that this may be expressed, in terms of the magnet poles and of the potentials of the places occupied by them, by the formula 2'|m V. 274 ELEMENTARY PHYSICS. [§ 248 We assume that whatever bodies are in the field are of such a character that their magnetization is proportional to the magnetiz- ing force; on this assumption, the potential at any point and the magnitude of the poles vary in the same proportion. Let m„ ni,, m„ represent the values of the respective poles, and r„ F„ F„ the potentials at the places occupied by them in the final condition of the field. Each of these poles may be conceived of as an assemblage of a great number n of small poles, each equal to-. If we think of the region occupied by the field as originally free from magnets, its energy after the magnets are present in it will be equal to the worls done in forming the magnet poles by the suc- cessive addition of such elementary poles. Let the field be free from magnetism, and let the quantities of magnetism —' -—> —> be brought to the points which the separate poles occupy in the final condition of the field ; since the potentials at those points are originally zero, no work will be done in this operation. The presence of these poles causes a rise of potential throughout the field, and the potentials at the places occupied by the poles become — ij. — , ... — -■ Let elementary poles similar to those already intro- duced be brought to their respective places in the field; the work mV done on any one of them is — j-j and the work done on them all is 2 — r- By this increase in the quantities at the poles the potentials become %—, 2—, 2^^. This operation is repeated until m quantities have been brought to each pole, so that the poles are in their final condition and the potential has everywhere its final value. The work done in bringing up the w* elementary pole to its place is -(w— 1) — ; the work done in forming the field is there- . ^fl + 2 + 3 + . . . + (w - 1)\ ^^ ^^ fore ^[ — ! ! '—^ — ^^—^ ']mV. Now § 348] MAGNETISM. 375 l + 2 + 3 + ... + (w-l) _ (w - l)n _ 1/ 1\ _ 1 if n be supposed to be very large. The work done in forming the magnetic field is therefore 2im V. Now, to show how this energy may be distributed in the field, we may consider any one of the magnets which give rise to the field as being the origin of 47r/a unit tubes of induction, the magnets being thought of as bar magnets. The energy of this magnet is, by the previous proposition, equal to hn(V„- F^), where V„ and Fj are the potentials at the places occupied by its poles; the pole m is equal to la (§ 341). The difference of potential V,. — V^ equals 2RAI, where R is the force along a line of force in the field pass- ing outside the magnet from its north to its south pole, and Al is an element of that line, the summation being extended over the whole line. If, therefore, we suppose each unit tube of induction which proceeds from the magnet to contain an amount of energy equal to RAl 2— — , the energy contained in the bundle of tubes belonging to the magnet will equal the energy of the magnet. We may therefore consider the energy of the magnet as distributed throughout the field, in such a way that each unit length of a unit tube of induc- tion contains 5— units of energy. The tubes of induction here con- sidered are those which exist outside the magnets. It has already been shown that the number of tubes of induction which pass through unit area is equal to the induction, or that N — F — (1 -f 4:7ik)R = fiR. Hence the energy in unit length of a tube JSF of induction may be expressed by -q— • The energy in unit volume of the field may be determined by considering a small cylinder of length I and cross-section s placed in the field with its end surfaces normal to the lines of in- duction. The number of tubes of induction which pass through the end surfaces is iVJs = /^Rs, and the energy contained in the length I of each of these tubes is^^ = -^. The energy contained 276 ELEMENTARY PHYSICS. [§ 249 in the cylinder is therefore —^^ — ^ — -, and the energy contained m unit Tolume is — r— = — — • 249. Paramagnetism and Diamagnetism. — It was discovered by- Faraday that all bodies are affected when brought into a magnetic field: some of them, such as iron, nickel, cobalt, and oxygen, are attracted by the magnet setting up the field ; others, such as bis- muth, copper, most organic substances, and nitrogen, are repelled from the magnet. The former are said to he ferromagnetic or paramagnetic, the latter diamagnetic. The most obvious explanation of these phenomena, and the one adopted by Faraday, is to ascribe them to a distribution of the in- duced magnetization in paramagnetic bodies, in an opposite direc- tion from that in diamagnetic bodies. If a paramagnetic body be brought between two opposite magnet poles, a north pole is induced in it near the external south pole, and a south pole near the external north pole. The magnetic separation is then said to be in the di- rection of the lines of force. According to this explanation, then^ the separation of the induced magnetization in a diamagnetic body is in a direction opposite to that of the lines of force. In other words, if a diamagnetic body be brought between two opposite magnet poles, the explanation asserts that a north pole is induced in it near the external north pole, and a south pole near the exter- nal south pole. One of Faraday's experiments, however, indicates that the dif- ferent behavior of bodies of these two classes may be due only to a more or less intense manifestation of the same action. He found that a solution of ferrous sulphate, sealed in a glass tube, behaves, immersed in a weaker solution of the same salt, as a paramagnetic body; but, when immersed in a stronger solution, as a diamagnetic body. It may from this experiment be concluded that the direc- tion of the induced magnetization is the same for all bodies, and that the exhibition of diamagnetic or paramagnetic properties de- pends, not upon the direction of induced magnetization, but upon § 349] MAGNETISM. 277 the greater or less intensity of magnetization of the surrounding medium. Faraday discovered that many bodies while in a vacuum exhibit diamagnetic properties. In accordance with this explanation, we must conclude that a vacuum can have magnetic properties. It seemed to Faraday unlikely that this should be the case, and he therefore adopted the explanation which was first given. As it has since been shown that the ether which serves as a medium for the transmission of light, and which pervades every so-called vacuum, is also probably concerned in electrical and magnetic phenomena, there is no longer any reason for the opinion that the possession of magnetic properties by a vacuum is inherently improbable. To classify bodies as paramagnetic or diamagnetic, we examine the energy existing in them when placed in a magnetic field. We will first assume that Jc, the coefficient of magnetization, is so small that the resultant force in the region occupied by the body is not appreciably changed by the presence of the body. The value of k for vacuum is assumed to be zero, and for air it is very slightly dif- ferent from zero ; hence the value of n for air may be set equal to 1. Before a body is brought into the field, the energy per unit volume in the space finally occupied by it is ^ ; the energy per unit volume in the same space when the body is brought into the field is -5—. The increase of energy caused by the introduction of the body, on the assumption we have made that the field is not ■disturbed by the body, or that N remains the same after the introduction of the body as it was before, is — - -], and this 07l\ }A, I is positive or negative according- as yw is less or greater than 1. Now a body free to move will move so as to diminish its potential •energy, and therefore a body for which yu > 1 will move so as to make N" as large as possible, or will move from a weaker to a stronger part of the field. Such a b@dy is called a paramagnetic body. On the other hand, a body *f or which n nearest of which is still at a distance R' so great that ^, may be R neglected in comparison with unity. Then if the nearest conductor were ii portion of a sphere of radius R' concentric with the inner sphere, the capacity of the inner sphere would be approximately R. And this capacity is evidently not less than that which would be due to any arrangement of conductors at distances more remote than R'. Therefore 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 R. This value R is often called the capacity of a freely electrified sphere. Strictly speaking, a freely electrified conductor cannot exist ; the term is, however, a convenient one to represent a conductor remote from all other con- ductors. A common form of condenser consists of two flat conducting 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 discussed. Let d represent the § 360] ELECTKICITY IN EQUILIBRIUM. 295 distance R' — R 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 R' = j^ and R" = ~. The capacity V^AA' of the spherical condenser may then be written =-. If R' and 47rrf 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 -— 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 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 approxima- tion to the true capacity of such a condenser. The so-called Ley den jar is the most usual form of condenser in practical use. It is a glass jar coated with tin-foil within and without, up to a short distance from the opening. Through the stopper of the jar is passed a metallic rod furnished 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 coat- ings are brought in conducting contact. When the wall of the jar is very thin in comparison with the diameter and with the height of the tin-foil coating, 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 to the thickness of the wall. 260. Systems of Conduetors. — If the capacities and potentials of two or more conductors be known, the potential of the system 396 ELEMENTARY PHYSICS. ■ [§ 361 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 C^ V^, C\V,, . . . C„ F„ , and the capacity of the system by the sum 0^ -\- C, -\- . . . C„. Hence V, the potential after connections have been made, is In the case of two freely electrified bodies joined up together by C V -\- C V a fine wire, we have V — ' „' ^° — -. When C, is very great Q compared with C^, we obtain V = F, + ^ F,. C Unless Fj is so great that the term -=^ F^ becomes appreciable, the potential of the system is appreciably equal to the original po- tential of the larger body. The capacity of the earth, being equal to its radius, is very great in comparison with the capacity of any body used in our experiments, and hence the potential of the earth is not changed when it is connected with a charged body. This proposition justifies the adoption of the potential of the earth as the standard or zero potential. 261. Electroscopes and Electrometers. — An electroscope is an in- strument used to detect the existence of a difference of electrical 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 ex- istence, nature, and to some extent the amount of the potential dif- ference 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, § 261] ELECTKICITY IN EQUILIBRIUM. 397 and improyed by Thomson. The essential portions of the instru- ment (Fig. 80) are a large flat disk B, which can be put in conducting contact with one c ■of the two bodies between which the differ- B ence of potential is desired ; a similar disk Fig. 80. 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 known 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 attracted disk when it is in the plane of the guard ring is often called the sighted j^osition. The guard ring is employed in order that the distribution on the at- tracted disk may be uniform. To determine the difference of potential between the attracted and attracting disks, we consider them first as forming a flat con- denser. If we represent by Q the quantity of electricity on the at- tracted disk, by Fand F, the potentials of the attracted and attract- ing disks respectively, by d the distance between them, and by S the area of the attracted disk, then, as has been shown in § 259, the O S capacity of such a condenser is y _ y = 4^- Now from the nature of the condenser, and in consequence of the regular distribu- tion due to the presence of the guard ring, we have ^ = '''j the V — V surface density on either plate, whence cr = ^^^ - The surface density cannot be measured, and must be eliminated by means of an equation obtained by observation 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 298 ELEMENTARY PHYSICS. [§262 always taken in the space between the plates as equal to 27r(r (§ 57). Every unit on the attracted disk is attracted with this force, and the total attraction, which is measured by means of th& balance or spring, is i^ = ^na^S. Substituting this value of a in the former equation,' we get V (93) which gives the difference of potential between the two plates in terms which are all measurable in absolute units. Thomson's quadrant electrometer is an instrument which is not used for absolute measurement, but being extremely sensitive to minute diflerences of potential, it enables us to compare them with each other and with some known standard. The construction of the apparatus can best be understood from Fig. 81. Of the four metallic quadrants which are mounted on insulating supports, the two marked F and the two marked N are respec- tively in conducting contact by means of wires. The body C, technically called the needle, is a thin sheet of metal, suspended symmetrically Fio- 81. 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, con- stant potential, and the two points, the difference of potential be- tween which is desired, are joined to the pairs of quadrants F and iV. The needle is deflected from its normal position, and the amount of deflection is an indication of the difference of potential between the two pairs of quadrants. 262. Electrical Machines. — Electrical machines may be divided into two classes: those which depend for their operation upon friction, and those which depend upon induction. The frictional machine, in one of its forms, consists of a circu- § 263] ELECTRICITY IN EQUILIBRIUM. 399 lar glass pMe, mounted so that it can be turned about an axis, and a rtibber of leather, coated with a metal amalgam, pressed against it. The rubber is mounted on an insulating support, but, during the operation of the machine, it is usually joined to ground. Dia- metrically opposite is 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 con- ductor. The prime conductor is carried on an insulating support. When the plate is turned, an electrical separation is produced by the friction of the rubber, and the rubbed portion of the plate is charged positively. When the charged portion of the plate passes before the comb, an electrical separation occurs in the prime con- ductor 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 pre- sented by the action of the electrophorus, an instrument invented by Volta in 1771. 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 fric- tion, 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 § 256. It may evidently be repeated indefinitely, and the electro- phorus may be used as a permanent source of electricity. It is evident that a charged metallic plate may be substituted for the sulphur in the construction of an electrophorus, 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 imper- .300 ELEMENTARY PHYSICS. [§262 ■<='fl «eptible charge on one plate is made to develop an indefinite quantity of electricity of high potential, is shown in Fig. 82. A^ and A, are conducting plates, called inductors. In front of them two disks B, and B^, called carriers, are mounted on an arm so as to turn about the axis E. Projecting springs 6, and b^ at- tached to these disks are so fixed as to touch successively the pinsi), Fio- 82. and D^, connected with the plates A^ and A„ and the pins C, 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 carriers so placed that B^ is between B^ and C^, and suppose the plate A^ to be at a slightly higher potential than the rest of the machine. The carrier B, is then charged by induction. When the carriers are turned in the direction of the arrows, and the carrier 5, makes contact with the pin C\ it loses 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 negative charge of the carrier 5,, when con- tact is made at D,. The plate A, is then negatively charged. The carrier B, at its contact 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 A^ and A, is greater than it was at first. When, after suflBcient repetition of this process, the difference of potential has become sufficiently 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 § 264] ELECTRICITY IN EQUILIBRIUM. 301 action of the machine. This improvement is usually effected by- attaching disks of tin-foil at equal distances from each other on one face of a glass -wheel, so that, as the wheel revolves, 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 are re- placed by a revolving glass plate, and the two metallic inductor plates by a fixed glass plate. In the fixed plate are cut two open- ings, diametrically opposite. Near these openings, 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 armatures nearest the openings- are the combs of two prime conductors. Opposite the other end& of the armatures, and also in front of the revolving wheel, are twa other combs joined together 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 sur- face of the revolving plate performs the functions of the carriers, in the induction machine already explained. The armatures take the place of the inductors, and the points in which they terminate- serve the same purpose as the contact points in connection with the inductors. The explanation of the action of this machine is, . in general, similar to that already given. The effect of the combs joined by the cross-bar is equivalent to joining to ground that por- tion of the outside face of the revolving plate which is passing^ under them. 263. Energy of a System of Charged Bodies. — If the charge on a body be changed, the potential at every point in the field changes in the same proportion. To obtain the energy of a systera of charged bodies we may apply the method used in § 248 to ob- tain the energy of a system of magnets. If Q represent the charge of one of the bodies and V its potential, the energy of the system is given by ^2QV. 264. Strain in the Dielectric. — An instructive experiment illus- 302 ELEMENTAKT PHYSICS. [§ 264 trating Faraday's theory that the electrification of a conductor is due to an action in the dielectric surrounding it, may be performed with a jar so constructed that both coatings can be remoTed from it. If the jar be charged, the coatings 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 observed by Quincke in fluid dielectrics as well as in difEerent solids. Another proof of the strained condition of dielectrics is found in their optical relations. It was discovered by Kerr that dielec- trics previously homogeneous become doubly refracting when sub- jected to a powerful electrical stress. Maxwell has shown, from the assumptions of his electromagnetic theory of light, that the index of refraction of a transparent dielectric should be propor- tional to the square root of its specific inductive capacity. Numer- ous experiments, among which those of Boltzmann on the index of refraction of light in gases and those of Hertz and others on the index of refraction of electromagnetic waves in solids and liquids are the most striking, show that this predicted relation is very close to the truth. It has further been shown that the specific inductive capacity of sulphur has difEerent values along its three crystallographic axes. This is probably true also for other crystals. Some crystals, while being warmed, exhibit on their faces posi- tive 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 arrangement by heat. If a jar be discharged and allowed to stand for a while, a second § 265] ELECTRICITY IN EQUILIBRIUM:. 303 discharge can be obtained from it. By similar treatment 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, hut gradu- ally, 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 di- electric, but this explanation 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 electrifications at the surfaces of separation between the non-homogeneous portions. When the jar is discharged the in- duced electrifications 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 sensibly modify the field outside the dielectric. The residual charge then appears in the jar. This explanation is supported by the fact that no residual charge remains when the dielectric is a fluid. 265. Tubes of Electrical Force. — If we admit that the nature and condition of the dielectric between conductors determine the charge upon them, an admission which the facts of specific induc- tive capacity and those cited in the last section render necessary, we must conclude that the hypothesis of electrical charges acting on each other directly at a distance, which we have used up to this point, is an artificial one, and that a more accurate representation of the real state of an electrical field will be had by assuming the action between the electrified bodies to be due to an action in the dielectric. We cannot explain the relation between electricity and the condition of the dielectric which will cause the actions observed between the electrified bodies, but we can show that these actions are consistent with certain conditions in the dielectric which are mechanically possible. Let us consider a positively charged conductor A which is evervwhere surrounded with other conductors. We may assume 304 ELEMENTARY PHYSICS. [§ 366 that these other conductors are at any distance from A, and that they are at the common potential zero. They are then equivalent to a single conductor £ surrounding the conductor A. Lines of force start from every point of A and pass to corresponding points of £. Mark out a small area on the surface A, and consider the closed surface formed by the lines of force passing through the contour of that area and surfaces dravrn in the dielectric just out- side the conductor A and just outside the conductor B. This closed surface is a tube of force, and if F^ and F^ represent the forces acting at the two cross-sections of the tube at A and B re- spectively, and s^ and s^ represent the areas of those cross-sections, we have (§ 56), F^Sj^ = — FbSb, the forces being considered as directed along the normals drawn outward from the conductors. Since the force within the conductors vanishes, the force just outside the surface of A is F^ = '^tkt^, and that just outside the surface of B is Fb = 4:7i(Tg. Using these values for F^ and F^, we have (^aSa — — o-B^B- Now these products are equal to the quantities of electricity present on the areas s^ and s^, so that we have Qa = ~ qs- The charges at the two ends of the tube of force are therefore equal and of opposite sign. Since the tubes of force which proceed from A either extend to infinity or end on con- ductors, the charges on those conductors are never greater than the total charge on A. If, as we have assumed, the conductors B com- pletely surround A, the charges on B are equal to the charge on A. If we divide the surface of A into areas upon each of which a unit charge of electricity is present, and erect tubes of force upon those areas, the dielectric will be mapped out by those tubes. Such a tube may be called a unit tube or a Faraday tube, in accordance with the proposition of J. J. Thomson. 266. Electrical Forces explained by Tubes of Force.— The strength of the field at any point in the dielectric is inversely as the area of the normal cross-section of the unit tube of force at that point. For, by § 56, the product Fs is constant throughout the tube. At the surface of the conductor from which the tube starts, F is equal to 47rcr and Fs = iTtas — i7r, since s is the cross- § 366] ELECTRICITY IN EQUILIBRIUM. 305 section of the unit tube at the charged surface, and crs is therefore equal to unity. The force F at any point in the dielectric is there- 4:7t fore equal to — , or is inversely as the cross-section of the tube. If s we represent by N the number of unit tubes which pass through a unit area of an equipotential surface, and if we assume that the force is appreciably constant over this area, we have JV = ~ and F= 4:7tN, that is, the force at any point in the field is proportional to the number of unit tubes which pass perpendicularly through a unit area at that point. In the discussion up to this point we have assumed that the medium between the two conductors has the specific inductive capacity or dielectric constant unity. If the dielectric constant be not unity, but some other number, say K, the difference of poten- tial between A and B that will be produced by a given charge on A is less than that which will be produced when the dielectric con- stant is unity, in the ratio of 1 to E. The general expression for the force in the field is therefore F = ~^- The electric pressure or force on unit area of the surface of the conductor, when the conductor is surrounded by a medium of which the dielectric constant is K, is given by —^ . This may be seen at once by applying the proof of § 256 to this case, remembering that, as has just been shown, the force outside the conductor is given by —^. Now on the view here taken, that the electrical forces are due to actions in the dielectric, this pressure should not be looked at as the result of the repulsions of the various elements of charge on the conductor, but rather as the result of some action in the dielectric. This action must be a pull or tension on the surface of the conductor. Since cr represents the number of unit tubes which proceed from unit area of the conductor, this pull is equal to ^ applied to the end of each unit tube, or since the force 306 ELEMENTAEY PHYSICS. [§ 267 just outside the surface is equal to — ^, the pull on the end of F each unit tube is also given by •^. The forces which act upon electrified bodies may therefore be considered as arising from tensions in the unit tubes, provided these tensions are not mechanically impossible. It may be shown that a medium in which such tensions exist is not in equilibrium unless pressures nuaaerically equal to the tensions, and at right angles to them, act throughout the medium. In order, therefore, that we may adequately represent the electrical field by the aid of unit tubes, we must assume a tension in each of these tubes along the lines of force and a pressure in every direction at right angles to it of the same numerical value. The tensions tend to shorten the tubes, the pressures to repel them from one another. All the forces which act between electrified bodies may be explained in terms of these actions between the tubes of force. 267. Energy in the Dielectric. — The tension on the cross-section F of the unit tube at any point m the field is also — , where F repre- sents the force at that point. To show this, it is sufficient to suppose one of the equipotential surfaces around the charged body replaced by a conductor maintained at the potential of that surface. The distribution in the field between the two conductors will then be the same as before. By reasoning similar to that already employed, it is seen at once that the force on the surface of the new conductor which carries unit charge, or the pull on the end of a unit tube at F that surface, is given by -, where P is the force at a point in the end of the unit tube. No restriction has been made as to the par- ticular equipotential surface chosen to be replaced by a conductor, and thus it appears that the tension or pull on the cross-section of F the tube of force is everywhere equal to — , where F is the force at a point in that cross-section. To find the tension or pull across unit area normal to the lines § 267] ELECTEICITT IK EQUILIBEIUM. 307 of force, we notice that if JV represent the number of tubes of force which pass through unit area drawn at a certain point in the field normal to the lines of force, and if s represent a small area normal to the lines of force, then JSfs represents the number of tubes of force which pass through that area, and the tension on that area is iFJis. Now we have seen that in any iield in which the dielec- AttJSF trie constant is K, F =— =- . Hence, substituting for N and di- 27cN' KF' Tiding by s, we have the tension on unit area given by — j^- ^:z-— — . -ft 07t On the view which we are now taking it is natural to consider the work done in charging bodies in a field as expended in modify- ing the dielectric or in setting up unit tubes in it. We will examine on this supposition the distribution of energy in the electrical field. It has already been proved (§ 263) that the energy of a system of charged bodies is equal to \^Q V, where Q is the charge and V the potential of each body. Let us consider a tube of force starting from a body at potential F, and proceeding to another body at po- tential Fj ; the charges at the ends of these tubes of force are equal and of opposite sign. The energy of the first conductor due to the por- tion of its charge we are now considering, which may be called q, is ^^F, ; the energy of the second conductor due to the correspond- ing equal charge is \q F,. The energy, therefore, due to the charges associated with a tube of force is \q{ F, — F,). All charges in the field may be associated in this way in pairs, and the total energy of the field expressed by ^2q( F, — FJ. Now F, — F, measures the work done by the electrical forces in moving a unit charge from the first conductor to the second. If F represent the force at any point in a tube of force and Al an element of length of the tube, the product FAl represents the work done in moving the unit charge along that elemeht, and the total work done in moving over the length of the whole tube is 2FAI = F, — F,. The energy associated with the whole tube is therefore ^q2FAl, and if we assume the unit length so small that the force does not appreci- ably vary within it, this energy may be considered as distributed 308 ELEMENTARY PHYSICS. [§ 268 throughout the tube in such a way that each unit of length of th& tube contains the energy ^qF. If the tube be a unit tube so that g = I, each unit of length of this tube will have in it a quantity of energy equal to ^F. To find the energy in unit volume of the dielectric, we consider a small cylinder, its height I being taken along the lines of force and its base s normal to them. The number of unit tubes which pass through the base is JVs. Since the energy in unit length of each of these tubes is ^F, and since therefore the energy in the length L is ^Fl, we have the energy in the volume Is equal to ^FNls, or the 4:71 N energy in unit volume equal to ^FN. Now we have F = — =- ,, so that the energy in unit volume is — v^- = -^ — -. By comparing this result with the value obtained for the tension across unit area it appears that the tension across unit area and the energy of unit volume are numerically equal. They both vary from point to point in the dielectric, depending upon the electrical force at each point. Unless the force is appreciably constant for all points of a finite region, the actual tension across a unit area and the actual energy of unit volume will not be given accurately by these expressions: they are more strictly the limits of the ratios be- tween the tension and the area on which it acts, and the energy and the volume containing it. 268. Forces on Electrified Bodies. — It has already been stated that the stresses between charges may be represented by supposing that the tubes of force exert a tension along the lines of force and an equal pressure in all directions perpendicular to the lines of force, or as may be said, the lines of force tend to diminish in length and to repel each other. This mode of conceiving the stresses between charged bodies may be illustrated in some simple cases without the aid of diagrams of lines of force. The lines of force around a uniformly charged sphere are radial and the tubes of force are similar cones; if the sphere be charged positively, the force is directed outward from it, and if charged negatively, is § 268] ELECTRICITY IN EQUILIBRIUM. 309 directed toward it. When two such spheres, one charged positively and the other negatively, are brought near each other, the tubes of force in the region between them to some extent coincide, so that the number of tubes of force which pass through unit area in the region between them is greater than that passing through the same area when only one of the spheres is present. On the other hand, the tubes in the region outside both the spheres counteract each other, and the number of tubes of force which pass through unit area in this region is less than when only one of the spheres is present. It may be seen thus roughly, and a diagram of the actual tubes of force in the field shows clearly, that the number of tubes of force which proceed from unit area of either one of the spheres on the surfaces confronting each other is greater than the number which proceeds from unit area on the outer surfaces. The tensions tending to draw the spheres together are thus greater than the tensions tending to separate them, and the spheres therefore appear to attract each other. If the two spheres which are brought near each other have similar charges, the tubes of force in the region between them are opposed to each other and the number of tubes of force in that region is therefore diminished, "while in the region outside the two spheres their tubes of force partly coincide and the number is increased. The tension is therefore greater on the outer surfaces of the spheres, and they are pulled apart or appear to repel each other. In these explanations no account has been taken of the inductive action of one sphere on the other. We may use the results obtained in the last section in the dis- cussion of the forces which act upon a body originally uncharged, having a dielectric constant K and brought into an electrical field set up in a medium of which the dielectric constant is different from K; for convenience, we will assume it to be unity. Let us assume that the body to be brought into the field is small and represent its volume by v. Now, before the body is brought into the field the energy in the volume afterwards occupied by it is %TtN'v. The energy in the same volume, after it is occupied by ihe body, is ^^^^. Now we know by experiment that K is always 310 ELEMENTARY PHYSICS. [§ 269 greater than unity, so that the introduction of such a body into the field involves a loss of energy, and this loss of energy is greater as N is greater. Bodies tend to move so as to make their potential energy a minimum, and the given body will therefore move from a place of weaker to a place of stronger electrical force. This con- clusion is reached on the supposition that the electrical field is not modified by the presence of the body — a supposition which can be made only when K is very nearly equal to unity. When E is not nearly equal to unity, the potential is not only diminished by the movement of the body from a place of weaker to a place of stronger electrical force, but also by the movement into it of the tubes of force; for a unit tube of force is associated with less energy in a medium of which the dielectric constant is E than in the medium of which the dielectric constant is unity, and the potential energy of the field is therefore diminished by a crowding of the tubes of force into the given body. This process cannot go on indefinitely so that the body includes all the tubes of force of the field, for as some of them enter the body others outside of it are lengthened and their energy is thereby increased. The concentration of the tubes in the body ceases, therefore, when the loss of energy due to their entrance into the body is balanced by the gam of energy due to the lengthening of those outside the body. A conductor may be looked on as a body having a dielectric constant K = cc. There is no electrical force within a conductor, and the energy lost by the field in consequence of a conductor being introduced into it is — — , where v is the volume of the con- ductor. This loss of energy is greater as F is greater, and the conductor therefore tends to move from a place of weaker to a place of stronger electrical force. There will also be a diminution of potential energy due to the concentration of tubes of force upon the conductor ; the conductor disturbs the electrical field and con- centrates the tubes of force upon it in a way similar to that of the body just described, but to a greater extent. 269. Cause of the Stress in the Dielectric— The theory that thfr § 369] ELECTEICITY IN EQUILIBRIUM. 311 electrical forces are due to stresses in the medium between the electrified bodies serves very well in expressing the results of experiment, but it gives no information about the origin or cause of the stresses in the medium. Faraday, who originated the theory, apparently thought that they arise from the electrification by induction of the separate particles or molecules of the medium in such a way that they resemble, so far as their external action is concerned, the magnetic molecules in Weber's theory of magnet- ism. This view was not held consistently even by its author, and cannot be accepted if we remember that electrical actions take place through vacuum. Maxwell conceived of electricity as dis- tributed everywhere in space, and considered the charging of a body as a displacement of the electricity in the region around it in one sense if the body is charged positively, in the opposite sense if charged negatively. Conductors ofEer no resistance to such a displacement, but in dielectrics the displacement is resisted by an action which Maxwell called electrical elasticity. This mode of describing the electrical field is satisfactory so long as the field is considered in equilibrium, but becomes difiicult of application when movements of charges occur in the field. J. J. Thomson has shown that all the phenomena of the electrical field may be described in terms of the motions or interactions of tubes of force, one of which is supposed to be connected with each atom of matter in the field. Thomson gives no mechanical explanation of the properties which these tubes must be assumed to have, only saying that " the analogies between their properties and those of the tubes of vortex motion irresistibly suggest that we should look to the rotary motion in the ether for their explanation," CHAPTER III. THE ELECTRICAL CURRENT. 270. Fundamental EflFects of the Electrical Current. — In 1791 Oalvani of Bologna published an account of some experiments made two years before, which opened a new department of electrical science. He showed that, if the lumbar nerves of a freshly skinned frog be touched by a strip of metal and the muscles of the hind leg by a strip of another metal, the leg is violently agitated when the two pieces of metal are brought in contact. Similar phenomena had been previously observed when sparks were passing from the conductor of an electrical machine in the vicinity of the frog prep- aration. He ascribed the facts observed to a hypothetical animal elec- tricity or vital principle, and discussed them from the physiological standpoint; and thus, although he and his immediate associates pursued his theory with great acuteness, they did not affect any marked advance along the true direction. 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 elec- trical potential 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 the first class, this statement is no longer true. There exists then in the circuit a preponderating difference of potential in one direction. 312 § 271] THE ELECTKICAL CUERBNT. 313 Volta described in 1800 an arrangement for utilizing these proper- ties of bodies for the production of continuous electrical currents. 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 dif- ferent potentials, and they could be used to produce the effects ob- served by Galvani. The effects were heightened, and especially the difference of potential between the two terminal wires was increased, ■when several such cups were used, the copper of one being joined to the zinc of the next, so as to form a series. This arrangement was called by Volta' the galvanic battery, but is now generally known as the voltaic lattery. 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 Eoyal 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 between electricity and magnetism. He showed that a magnet brought near a wire joining the terminals of a battery is deflected, and tends to stand at right angles to the wire. His discovery was at once fol- lowed up by Ampere, who showed that, if the wire joining the ter- minals be so bent on itself as to form an almost closed circuit, and if the rest of the circuit be so disposed as to have no appreciable influence, the magnetic potential at any point outside the wire will be similar to that due to a magnetic shell. In 1834 Peltier showed that, if the terminals of the battery be joined by wires of two different metals, there is a production or an absorption of heat at the point of contact of the wires, depending upon which of the wires is joined to the termihal the potential of which is positive with respect to the other. This fact is referred to as the Peltier effect. 271. The Electrical Current.— In 1833 Faraday showed con- 314 ELEMENTARY PHYSICS. [,§ 271 clusively that if a Leyden jar be discharged through a circuit, it will momentarily produce thermal, chemical, and magnetic effects which are similar to those just described as produced continuously by the voltaic battery. The discharge of the jar may be variously represented. So long as electricity is considered as a fluid or substance, it is easiest to think of the discharge as the transfer of electricity from a place of higher to a place of lower potential, or rather as the equalization of potential by the transfer of equal and opposite quantities in op- posite senses, and to explain the continuous effects produced by the voltaic battery by ascribing them to a current, or continuous trans- fer of electricity around the circuit. This view is capable of rep- resenting most of the phenomena of steady or permanent currents, but it is less successful in representing the phenomena of variable currents. If we consider electrical phenomena as due to actions in the dielectric, we may obtain a more adequate representation of the discharge and also of all the phenomena of the current by the use of the unit tubes of force described in § 265. We may obtain some idea of the connection of these tubes with the current if we examine their behavior during the discharge of a condenser. To make the discussion as simple as possible, we suppose the condenser to be made of two equal plates A and B; their potentials are V^ and Vb, V^ being the greater. The lines of force origi- nate at A and pass to B in the manner shown in Pig. 83. This figure has been roughly copied from the one given by J. J. Thom- son. Let Q represent that part of the charge on A to which corresponds an equal and opposite charge on B: the number of unit tubes of force which pass from A to B will then be given by Q. Now let us join ^ to 5 by a conductor C, which for the sake of simplicity shall coincide with the direction of the lines of force. No tube of force can exist within a conductor, and those which were present in the volume which the conductor Fig. 83. § ^''IJ THE ELEOTKICAL CURRENT. 315 occupies immediately disappear. For our purposes the manner of this disappearance of the tubes of force is of no consequence : it may be described as a shrinking of the tubes in the conductor, so that the ends which leave the plates of the condenser move toward the middle of the conductor and at last meet. The disappearance of these tubes from the conductor relieves those lying immediately around it of the lateral pressure which maintained them in equilib- rium, and they are accordingly driven into the conductor and in turn disappear. This process is continued until all the tubes of force have disappeared. The discharge of the condenser may there- fore be represented as the lateral movement of the tubes of force originally in the field and their disappearance in the conductor. The discharge is not really so simple as it is here supposed to be. We have supposed the process to cease when the difference of potential between the two plates becomes zero, but this is in fact not the case: the discharge is really oscillatory, the difference of potential being alternately positive and negative, rapidly diminishing in numerical value until it disappears. In order to account for this, something analogous to inertia must be ascribed to the tubes of force. While the discharge is going on, a magnetic field exists around the conductor. If the discharge be thought of as being merely the transfer of- charge along the conductor, there is no apparent mechanism connecting the discharge with the magnetic force, but on the view now being presented the magnetic field may be thought of as due in some way to the movement through the dielectric of the tubes of force. If the discharge pass through a compound body, capable of decomposition by it, a portion of that body will be resolved into two constituents. On the view that the discharge is a mere transfer of charge, these constituents must be supposed to serve as carriers of that charge, but this view cannot represent the connection between these constituents and their charges, nor the conditions which enable them to give up their charges. On the other hand, by supposing each unit of the constituent to be invari- ably associated with a tube of force, we may describe the discharge through such a chemical compound in terms of the changes which 316 ELEMENTARY PHYSICS. [§ 372 take place in the tubes of force, in a manner consistent with what we know of their nature. Thus this latter view furnishes a more adequate representation of the discharge than the older and simpler view. We haye already seen (§ 267) that the energy contained in each unit tube is equal to one half the difEerence of potential between its ends. Since Q represents the number of unit tubes which pass between the plates, the energy of the field is ^Q{V^— Vg); after the discharge this energy has entered the conductor. If an arrangement be efEected by which the difEerence of poten- tial between A and B is kept constantly equal to ( F^ — Vg), the work done by the transfer of Q units of charge, and therefore the energy lost by the disappearance of Q unit tubes of force, is Q{Va— Vb)- Let W represent the energy lost by such a continu- ous discharge or current in unit time, and t the time in which Q tubes of force disappear. Then Wt = Q^V^— Fg), and PF=|(F^-F^). (94) The ratio -j is represented by / and called the ctirrent strength or simply the current in the conductor. It may be variously con- sidered as the rate of transfer of charge between the conductors, or as the rate at which the unit tubes of force are destroyed. 272. Electrostatic Unit of Current. — Let us denote the poten- tials at the two points 1 and 2 in a circuit by F, and F, , and let F, be greater than F, : then if, in the time t, a quantity of elec- tricity equal to Q passes through a conductor joining those points from potential F, to potential F,, the energy expended is Qi.v.-v.). If the conductor be a single homogeneous metal or some analo- gous substance, and if no motion of the conductor or of any exter- nal magnetic body take place, the energy expended in the conductor is transformed into heat. If we suppose this transformation to go on at a uniform rate, and denote the heat developed in unit time by H, we may substitute H for W in equation (94), and have § 374] THE ELECTRICAL CURKENT. 317 H=J{V,-V,). (95) If heat and difference of potential be measured in absolute units, this equation enables us to determine the absolute unit of current. The system of units here used is the electrostatic system. The di- mensions of current strength in the electrostatic system are obtained from this equation. We have [7] = ^ = M^L^T-^. 273. Electromotive Force. — We may consider the current as an operation by which energy is transformed in the conductor, either by the transfer of electricity through it or by the entrance of tubes of force into it and their disappearance within it. In the example before us, we have assumed that the transfer or the movement of the tubes was due to some cause which set up a difference of poten- tial between parts of the conductor. This condition is not neces- sary for the maintenance of a current; in certain circumstances energy may be expended in a conductor, without the existence of a finite difference of potential between neighboring parts of the con- ductor. The power of establishing and sustaining the conditions which make a continuous expenditure of energy possible is called electromotive force. The energy expended in unit time in a circuit carrying the current I, and in which the electromotive force is E, is W=IK (96) In a circuit containing a voltaic battery, if two points on the circuit outside the battery be tested by an electrometer, a differ- ence of potential between them will be found. If the circuit be broken between the two points considered, the difference of poten- tial between them becomes greater. This maximum difference of potential is the sum of finite differences of potential supposed to be due to molecular interactions at the surfaces of contact of different substances in the circuit, and is the measure of the electromotive force of the battery. 274. Ohm's Law.— In § 252 it was remarked that a body is distinguished as a good or a poor conductor by the rate at which it 318 ELEMENTART PHYSICS. [§ 274 will equalize the potentials of two electrified conductors, if it be used to connect them. Manifestly this property of the substances forming a circuit will influence the strength of the current in the circuit. It was shown on theoretical considerations, in 1827, by Ohm that in a homogeneous conductor which is kept constant the current varies directly with the difEerence of potential between the terminals. If R represent a factor, constant for each conductor. Ohm's latv is expressed in its simplest form by IE= F, - F,. (97) The quantity E is called the resistance of the conductor. If the difEerence of potential be maintained constant, and the con- ductor 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 E with each change in the conductor. The quantity E is therefore a function of the nature and materials of the conductor, and does not depend on the current or the difEerence of potential between the ends of the conductor. Since it is the ratio of the current to the differ- ence of potential, and since we know these quantities in electro- static units, we can measure E in electrostatic units. From the dimensions of /and ( F, — V„) we may obtain the dimensions of R. They are in electrostatic units [E] = [ ^' J ^" ~\ =^L-'T. Since the difEerence of potential in equation (97) is the measure of the electromotive force in the conductor considered, it is natural to extend the relation therein expressed to the whole circuit, in which the current is maintained by the electromotive force E. The expression of Ohm's law for the whole circuit is IE = K (98) This relation cannot in every case be experimentally verified, but in many cases in which the electromotive force may be directly and accurately calculated its validity has been demonstrated. 276] THE ELBCTBICAL CUKEENT. 319 275. Specific Conductivity and Specific Resistance.— If two points be kept at a constant difference of potential, and be joined by a homogeneous conductor of uniform cross-section, it is found that the current in the conductor is directly proportional to its cross-section and inversely to its length. The current also depends npon the nature of the conductor. If conductors of similar dimen- sions, 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 conduct- ing 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 re- sistance of the conducting material, S the cross-section, and I the length of a portion of the conductor of uniform cross-section between two points at potentials F, and V^, Ohm's law for this special case is presented in the formula Ip (99) The specific resistance is not perfectly constant for any one material, but varies with the temperature. In metals the specific resistance increases with rise in temperature ; in liquids and in car- bon it diminishes with rise in temperature. Upon this fact of change of resistance with temperature is based a very delicate in- strument, called by Langley, its inventor, the iolometer, for the measurement of the intensity of radiant energy. 276. Joule's Law. — If we modify the equation H = I {V, — V,) by the help of Ohm's law, we obtain H = PR. (100) The heat developed in a homogeneous portion of any circuit is equal to the square of the current in the circuit multiplied by the resistance of that portion. This relation was first experimentally proved by Joule in 1841, and is known as Joule's law. It holds true for any homogeneous circuit or for all parts of a circuit which are 330 ELEMENTAKY PHYSICS. [^ 277 homogeneous. The heat which is sometimes evolved by chemical action, or by the Peltier effect, occurs at non-homogeneous portions, of the circuit. 277. Counter Electromotive Force in the Circuit. — In many cases the work done by the current does not appear wholly as heat devel- oped 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 decomposi- tion of chemical compounds, in producing movements of magnetic bodies 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 appears 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 t, which is such that, during it, the current is constant, is lEL It appears partly as heat, which equals FEf by Joule's law, and partly as other work, which experiment proves is in every case proportional to /, and can be set equal to lA, where ^ is a factor which varies with the par- ticular work done. Then we have IJSt = PEt + lA, whence F ^ R (101) It is evident from the equation that E — j is an electromotive force, and that the original electromotive force of the circuit has been modified by 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 coimter electromotive force — . The separated con- t stituents of the chemical compound, the moved magnet, the heated junction, are all sources of electromotive force which oppose that § 278] THE ELECTRICAL CURRENT. 331 of the original circuit. If then, in a circuit containing no im- pressed electromotive force, or in which E = Q, there be brought an arrangement of uncombined chemical substances which are capable of combination, or if in its presence a magnet 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 cur- z ^®^*, ^^ IR' ^^^ ^^ *^®®® methods may then be used as the means of generating a current. The first gives the ordinary bat- tery currents of Volta, the second the induced currents discovered by Faraday, and the third the thermoelectric currents of Seebeck. This demonstration fails when applied to the case of the induc- tion of one current by another, in consequence of the changes pro- duced in both by their mutual interactions. The correct demon- stration in this case can only be reached by the aid of the dynamical equations of the electromagnetic field. 278. Poynting's Theorem. —On the view that the current con- sists of the disappearance of tubes of force in the conductor, the energy developed in the circuit enters it from the dielectric. By choosing a very simple case we may determine the rate at which this energy moves through the dielectric and into the conductor;. We will suppose the current maintained in a very long straight. cylindrical wire stretched between two parallel and very large planes, which are kept at the potentials F, and F^. In such an arrangement the tubes of force are cylindrical, passing perpendicu- larly between the two plates and parallel with the conductor join- ing them; the electrical force in such a field is everywhere the same. Now consider a plane parallel with the plates, and describe in it a circle having any radius r with its centre at the centre of the wire. Let iV represent the number of tubes of force which pass through unit area in this plane, and v the velocity with which these tubes of force pass through the circumference of the circle of radius r Then the number of tubes which pass through this circum- ference in unit time is 2nrvN, and since the current is supposed to 322 ELEMENTAKT PHYSICS. [§ 278 be steady, this number is the same whatever be the radius of the circle; it therefore expresses also the number of unit tubes which enter the wire in unit of time. We have already seen that the energy carried into the conduc- tor by Q unit tubes, when the difference of potential is maintained constant, is equal to Q(V^ — V,), that is, is twice the energy asso- ciated with these tubes when at rest. It has also been shown that F the energy in unit length of a unit tube at rest is -, and F there- fore measures the energy carried into the conductor by unit length of each unit tube. In the case before us, the energy transferred in one second through a cylindrical surface of unit height and of radius r, concentric with the wire, is '27TrvNF. Now, on the view of the current here taken, the number of unit tubes which dis- appear in one second is equal to the current strength, so that 2nrvN = I. The energy introduced through the cylindrical sur- face is therefore FI. Since in this case the difference of potential equals the electrical force multiplied by the length of the wire, the energy introduced into the whole wire is J( V, — FJ. The energy which passes through unit area of the cylindrical surface is ^ — . It may be shown that the magnetic force due to the current at the 27 distance r is P = — , and hence the energy which passes through FP unit area may also be represented by . The example here given is a special case of a general theorem due to Poynting. This theorem asserts that the energy expended in the current enters the conductor from the dielectric, passing at right angles to the lines of electrostatic force and the lines of mag- netic force, and that the amount of energy which passes perpendic- ularly through unit area is proportional to the electrostatic force and to the magnetic force. CHAPTER IV. CHEMICAL RELATIONS OF THE CURRENT. 279. Electrolysis. — It has been already mentioned that, in cer- tain cases, the existence of an electrical current in a circuit is accompanied by the decomposition into their constituents of chem- ical compounds forming part of the circuit. This process, called electrolysis, must now be considered more fully. It is one of those treated generally in § 277, in which work other than heating the circuit is done by the current. That work is done by the decom- position 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 combination is always attended with the evolution of heat or the appearance 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 decom- position. Those bodies which exhibit electrolysis are always such as have considerable freedom of motion among their molecules. Ordinarily, they are liquids or solids in solution or fused. The discharge through gases is also probably accompanied by electrolysis. Bodies which can be decomposed were called by Faraday, to whom the nomenclature of this subject is due, electrolytes. The current is usually introduced into the electrolyte by solid terminals called electrodes. The one at the higher potential is called the positive electrode or anode; ihQ other, the negative electrode, or cathode. The two constituents into which the electrolyte is decomposed are 323 324 ELEMENTARY PHYSICS. [§ 279 called ions. One of them appears at the anode and is called th& 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 Nichol- son and Carlisle, was soon modified by Carlisle in a way which is. still generally in use. Two platinum electrodes are immersed in water slightly acidulated with sulphuric acid, and tubes are ar- ranged above them so that the gases evolved can be collected sepa- rately. When the current is passing, bubbles of gas appear on the electrodes. When they are collected and examined, the gas which appears at the anode is found to be oxygen, and that which appears at the cathode to be hydrogen. The quantities evolved are in the- proportion to form water. This appears to be a simple decomposi- tion of water into its constituents, but it is probable that the acid in the water is first decomposed, and that the constituents of water are evolved by a secondary chemical reaction. An experiment performed by Davy, by which he discovered the elements potassium and sodium, is a good example of simple elec- trolysis. 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 deposited on the cathode. This is the type of a large number 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 quan- tity of cupric sulphate. This is a case similar to the electrolysis of acidulated water, in which the simple decomposition of the electro- lyte is modified by secondary chemical reactions. If two copper electrodes be immersed in a solution of cupric sulphate, copper will be removed from the anode and deposited on the cathode, without any important change occuring in the charac- ter or concentration of the electrolyte. This is an example of the § 280] CHEMICAL RELATIONS OF THE CURRENT. 335 «pecial case in which the secondary reactions in the electrolyte exactly balance the work done by the current in decomposition, so that on the whole no chemical work is done. 280. Faraday's Laws.— The researches of Faraday in electroly- sis developed two laws, which are of great importance in the theory of chemistry as well as in electricity : (1) 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 electrolyzed is proportional to the current strength. (2) If the same current be passed through different electro- lytes, the quantity of each ion evolTcd is proportional to its chemi- cal equivalent. The chemical equivalent is the weight of the radical of the ion in terms of the weight of the atom of hydrogen, divided by its valency. 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 electrolyte is equal to the number of units of electricity which pass through the elec- trolyte in the given time. The electro-chemical equivalents of different ions are propor- tional to their chemical equivalents. Thus, if zinc sulphate, cu- pric sulphate, and argentic chloride be electrolyzed 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 de- posited are in proportion to the chemical equivalents, 32.6 parts of zinc, 31.7 parts of copper, and 108 parts of silver. Faraday's laws may also be stated in another form, in which the word "ion " has a different meaning. The process of electroly- sis consists in the separation of each molecule of the electrolyte into its constituent radicals. Each of these radicals is called an ion. If the valency of the radical be 1, the ion is called a univa- lent ion; if it be w, the ion is either called an w-valent ion or m-uni- 336 ELEMBNTAEY PHYSICS. [§ 381 valent ions. To illustrate, we know that when hydrogen is evolved from hydrochloric acid, HOI, its ion is univalent. Now when it is evolved from water, H,0, we may either consider the H^ as a hivalent ion or as two univalent ions. Similarly we may consider the as a bivalent ion or as two univalent ions, though it can never be actually broken up into two such ions. We may consider a molecule, then, as made up either of two M-valent ions or of 3w univalent ions. The weight of each of the re-valent ions may be measured in terms of the weight of the hydrogen atom taken as a unit, and is the molecular weight of the ion. This weight divided by the valency n is the weight of the univalent ion. It may be called the ionic weight. Now the passage of a current through different electrolytes evolves their constituents in amounts proportional to their molec- ular weights divided by their valencies. It therefore evolves the ions in proportion to their ionic weights, or it evolves the same number of univalent ions in each electrolyte. Faraday's two laws may therefore be summed up in the statement that the number of univalent ions evolved by a current in any electrolyte is propor- tional to the quantity of current. By this mode of considering electrolysis, we are led to the conclusion that each pair of univalent ions liberated during elec- trolysis is associated with a pair of charges numerically equal and of opposite sign. These charges are called ionic charges. An w-valent ion is associated with n ionic charges. If we use the con- ception of tubes of force, each positive univalent ion may be con- sidered as the origin of a tube of force which terminates on a negative ion. Since the ionic charges are all equal, these tubes may be taken as unit tubes, which are no longer defined arbitrarily, but are based upon a constant of Nature. 281. The Voltameter.— These laws were used by Faraday to establish a method of measuring current by reference to an arbi- trary standard. The method employs a vessel containing an elec- trolyte in which suitable electrodes are immersed, so arranged that the products of electrolysis, if gaseous, can be collected and meas- § 383] CHEMICAL EBLATIONS OE THE CURKBNT. 337 ured, or, if solid, can be weighed. This arrangement is called a voltameter. If the current strength be desired, the current must be kept constant in the voltameter by suitable variation of the resistance in the circuit during the time in which electrolysis is going on. Two forms of voltameter are in frequent use. In the first form there is, on the whole, no chemical work done in the electrolytic process. The system consisting of two copper electrodes and cupric sulphate as the electrolyte is an example of such a voltameter. The weight of the copper deposited on the cathode measures the current. The second form depends for its indications on the evolution of gas, the volume of which is measured. The water voltameter is a type, and is the form especially used. The gases evolved are either collected together, or the hydrogen alone is collected. The latter is preferable, because oxygen is more easily absorbed by water than hydrogen, and an error is thus introduced when the oxygen is measured. 282. Measure of the Counter Electromotive Force of Decomposi- tion.— In the general formula developed in § 377 the quantity lA represents the energy expended in the circuit which does not ap- pear as heat developed in accordance with Joule's law. In the present case it is the energy expended during electrolysis in de- composing chemical compounds and in doing mechanical work. In many cases the mechanical work done is not appreciable; but when a liquid like water is decomposed into its constituent gases, work is done by the expansion of the gases from their volume as water to their volume as gases. In many cases some of this energy is also used in keeping the temperature of the electrolyte constant. These cases occur when the electromotive force developed varies with the temperature. In case no such variation with the temperature occurs, we may calculate the electromotive force developed in terms of heat. Let e represent the electro-chemical equivalent of one of the ions, and the heat evolved by the combination of a unit mass of this ion with 338 ELEMENTARY PHYSICS, [§ 283 an equivalent mass of the other ion, in which is included the heat equivalent of the mechanical work done if the state of aggregation change. Then / will represent the number of electro-chemical •equivalents evolved in unit time, and leBt will represent the en- ergy expended in the time t, which appears as chemical separation and mechanical work. This is equal to I A; whence A = eOt.^ 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 A bring both constituents to their normal condition. Wow, - repre- sents the counter electromotive force set up in the circuit by elec- trolysis. Hence the electromotive force set up in the electro- lytic process may be measured in terms of heat units. It often is the case that the two ions which appear at the elec- trodes are not capable of direct recombination, as has been tacitly assumed in the definition of 0. 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 9. The theory here presented is abundantly verified by the experi- ments of Joule, Favre and Silbermenn, Wright, and others. The extension of the theory to cases in which the electromotive force varies with the temperature was made by Helmholtz. 283. Positive and Negative Ions. — Experiment shows that cer- tain 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 elements. Faraday divided all the ions into these two classes, and thought that every compound capable of electrolysis was 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. There :§ 384] CHEMICAL KELATIONS OF THE CURRENT. 339 is no reason to believe that such a rigorous arrangement of the ions can be made. 284. Grotthus's Theory of Electrolysis.— The foundation of all the present theories of electrolysis is found in the theory published by Grotthus in 1805. He considered the constituent ions of a mole- cule as oppositely electrified to an equal amount. When the cur- rent 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 mole- cule moves toward the positive electrode 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 is freed at the cathode. This process is repeated indefinitely so long as the current passes. Faraday modified this view, in that he ascribed the arrangement 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's theory. He showed that an electrical stress exists in the electrolyte by means of fine silk threads im- mersed in it. These arranged themselves along the lines of electri- cal stress. Other phenomena, however, show that Grotthus's hypothesis can only be treated as a rough illustration of the main facts. 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 migration of the ions are also at variance with Grotthus's theory. If the electrolysis of a copper salt, in a cell with a copper anode at the bottom, be ex- amined, it will be found that the solution becomes more concen- trated 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 330 ELEMENTARY PHYSICS. [§ 285 by Grotthus's 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. It is an objection against Grotthus's theory, and indeed against Thomson's method given in § 282 of connecting chemical affinity and electromotive force, that, on those theories, it would require an electromotive force in the circuit greater than ^, the counter electro- motive force in the electrolytic cell, to set up a current, and that the current would begin suddenly, with a finite value, after this electromotive force is reached. On the contrary, experiments show that the smallest electromotive force will set up a current in an electrolyte aud even maintain one constantly, though the cur- rent strength may be extremely small. 285. The Dissociation Theory of Electrolysis. — The foundations of a more satisfactory theory of electrolysis were laid by Clausius, who proceeded from the view with which he had become familiar by his study of the kinetic theory of gases, that the molecules of all bodies are in constant motion. He assumed that the collisions of the molecules of the electrolyte occasionally caused a separation of some of the molecules into their constituent ions, and that the province of the electromotive force in the electrolyte was to direct the motion of these ions toward their respective electrodes. A considerable extension of Clausius's theory has been made by Arrhenius and de- veloped by Ostwald and others, in which the leading idea is, that the molecules of an electrolyte in solution are always separated to a greater or less extent into their constituent ions. In many cases, and always in very dilute solutions, the separation, according to this view, is complete. This theory is called the dissociation theory of electrolysis. The ions, however, are not in the condition of the constituent parts of a molecule which have been dissociated at a high temperature (§ 219), but possess certain peculiar electrical and chemical properties. It has been proposed to call their condition in solution ionization. This term certainly possesses advantages, but § 385] CHEMICAL KELATIOIfS OF THE CUKHENT. 331 it has not yet come into common use, and we will therefore retain the term dissociation. We have already seen that a current in an electrolyte may be considered as the transfer of charges on the moving ions. If the ions in solution be dissociated from each other, and if the effect of the electromotive force in the circuit be merely directive, it is plain that the quantity of current transferred will depend on the relative velocity with which the ions move past each other in the solution as well as on their number. Starting with this conception, we will show that the conductivity of an electrolyte is proportional to the sum of the velocities of its ions. The discovery of this fact by Kohlrausch laid the foundation for the dissociation theory. Let us suppose a series of electrolytic cells, e'ach one of which is a cubical box with sides of unit length, and so arranged that a current passes in them between two opposite faces which serve as electrodes. The column of the electrolyte between the electrodes is then one centimetre long and has a cross-section of one square centimetre. Let the electrolytes used in these cells be prepared by dissolving in equal volumes of the same solvent masses of the substances to be decomposed which are proportional to the sums of the ionic weights of their constituent ions (§ 280). Equal volumes of these solutions will then contain the same number of univalent ions. If a current be sent through the series of cells containing these solutions, the same number of univalent, ions will be liberated in each. The difference of potential between the terminals of the cells will be in general different for each of them. We have from Ohm's law the relation 7= ^ (F, — FJ, where the current / is the same for each cell and the difference of potential F, — F, and the conductivity k (§ 275) different for the different cells. Now con- sider a cross-section in one of the cells parallel with the electrodes ; let u and v represent the velocities of the ions evolved in this cell. Let 2M represent the number of univalent ions in the cell, and let c represent the ionic charge. Now the relative velocity of the ions which pass through the cross-section taken in the cell isu-^v; the 332 ELEMENTAET PHYSICS. [§ 285 number of ions which pass through that cross-section in unit time in both directions is therefore M{u -\- v) and the quantity of elec- tricity carried through with them in both directions is cM{u -f- v). But this quantity is equal to the current strength /, and therefore cM{u -\-v)^k{V,- l\) ■,oxu-\-v = ^^^' ~j- ^^\ Now cM is the = sm a, where a is the angle between the element I and the direction of the magnetic force H. Substituting these values, the resultant force on the ele- ment is found to be equal to iH sin a. In the special case in which the magnetic field is due to a sin- gle magnet pole of strength m, we have H = ^^, where r is the distance from the pole to the element of the circuit. The force § 396] THE MAGKETIC EELATIONS OF THE CUKEENT. 351 exerted by a magnet pole on an element of the circuit is therefore — sin a, and this force urges the element to move at right angles to itself and to the line joining it with the magnet pole. Since the action between the pole and the circuit is mutual and the work done dependent only on their relative displacements, the force which each element of the circuit exerts on the pole is also equal to mil . -1.1 , , —3- sm a, ana tends to urge the pole to move at right angles to the plane containing it and the element' of the circuit.- This action on the magnet pole is the same as that deduced by Biot from his study of the force between a pole and a long straight current. We will apply this theorem to determine the force due to a circular current on a magnet pole placed at a point on the line drawn normal to the plane of the circuit through its centre. The force on the circuit, and therefore the force on the pole, has been shown to be equal to ['2RI sin d cos (p. In the case before us H = -^, where m js the strength of the magnet pole and R the distance from the pole to the circuit. Since the elements of the circuit are symmetrical with respect to the pole, the force on the pole is along the line joining it to the centre of the circuit. The angle 6 therefore equals — and sm 6' = 1; the angle (f) is the angle 3 r between the radius of the circle and R ; and cos cp = j^, where r is the radius of the circle. The sum of all the elements of the circuit equals the circumference of the circle, or 2nr. The force on the , ^ 27tni.tr'' pole IS therefore equal to — pr~- If the magnet pole be placed at the centre of the circle, so that R = r, the force on it becomes -^^ . Let the radius of the circle ' r be the unit length, or one centimetre; the force acting on the magnet pole is then 27rmi, and if the magnet pole be the unit pole. 352 ELEMENTARY PHYSICS. [§ 297 the force is 2m. If therefore the force exerted be equal to 27t, i will be equal to unity. We have thus arrived at another defini- tion of unit current from the point of view of Biot's law. The unit current is defined to be that current which, flowing in a circle of unit radius, will exert upon a unit magnet pole at its centre a force equal to 2;r dynes. 297. Ampfere's Law for the Mutual Action of Currents. — The mutual action of two currents may also be considered as arising from forces between the elements of the currents. It was from this point of view that the action of currents was first investigated by Ampere. While the results obtained by him were not a unique solution of the problem, and must be regarded only as an artificial representation of the action between currents, they are yet of great interest. Without attempting to deduce Ampere's law, we will briefly consider the experiments upon which his deductions were based. Ampere's method consists in submitting a movable circuit or part of a circuit carrying a current to the action of a fixed circuit, and in so disposing the parts of the fixed circuit that the forces arising from different parts exactly annul one another, so that the movable circuit does not move when the current in the fixed circuit is made or broken. In the first two of his experiments the mov- able circuit consists of a wire frame of the form shown in Fig. 86. ^ The current passes into the frame by the points ia a and b, upon which the frame is supported. It is evident that the two halves of the frame tend to face in opposite directions in the earth's mag- netic field, so that there is no tendency of the frame as a whole to face in any one direction rather than any other. If a long straight wire be Fig. 86 placed near to one of the extreme vertical sides of the frame and a current be sent through it, that side will move towards the wire if the currents in it and in the wire be in the same direction, and will move away from the wire if the currents be in opposite directions. i 1 t I 1 § 397] THE MAGNETIC RELATIONS OP THE CURRENT. 353 If now this wire be doubled on itself, so that near the frame there are two equal currents occupying practically the same posi- tion, but in opposite directions, then no motion of the frame can be observed when a current is set up in the wire. This is Ampere's first case of equilibrium. It shows that the forces due to two cur- rents, identical in strength and in position, 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 com- pared 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 conductor. To obtain the third case of equilibrium, a wire, bent in the arc of a circle, is arranged so that it may turn freely about a vertical axis passing through the centre of the circle of which the wire forms an arc, and normal to the plane of that circle. The wire is then free to move only in the circumference of that circle, or in the direction of its own length. Two vessels filled with mercury, so that the mercury stands above the level of their sides, are brought under the wire arc, and raised until conducting contact is made between the wire and the mercury in both vessels. A current is then passed through the movable wire through the mercury. Then if any closed circuit whatever, or any magnet, be brought near the wire, it is found that the wire remains stationary. The deduction from this observation is that no closed circuit tends to displace an element of current in the direction of its length. In the fourth experiment three circuits are used, which we may call respectively A, B, and C. They are alike in form, and the dimen- sions of B are mean proportionals to the corresponding dimensions of A and C. B is suspended so as to be free to move, and A and are placed on opposite sides of B, so that the ratio of their dis- tances 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, 354 ELEMENTARY PHYSICS. [§ 298 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 equilibrium. 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. Ampere made the assumption that the action between two current elements is in the line joining them. Prom the four cases of equilibrium he then deduced an expression for the attraction between two current elements. It is — *^'(2 cos 6 - 3 cos e cos 6'). (103) In this formula ds and ds' represent the elements of the two cir- cuits, i and i' the strength of current in those circuits measured in electromagnetic units, r the distance between the current elements, e the angles made by the two elements with one another, and 6' the angles made by ds and ds' with r or r produced, the direction of the two elements being taken in the sense of their respective currents. 298. 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 circuit, 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 similar magnetic shells arranged per- pendicularly 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. § 299] THE MAGNETIC KELATIONS OF THE CTJKKENT. 355 Such an arrangement is called a soletioid. The poles of a solenoid of very small cross-section are situated at its ends, and it is equiv- alent 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 combina- tion is called an electromagnet. Since the strength of the magnetic field Taries with the strength of the current in the solenoid, aud with the number of layers of wire wrapped around the iron core, the magnetization of auy bar of iron whatever may be raised to its maximum by increasing the current and the number of turns of wire. 299. Ampere's Theory of Magnetism. — Ampere based upon these facts a 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 hypothesis with regard to the origin or the permanency of these currents. The theory agrees with Weber's hypothesis that magnetization consists in an arrange- ment of magnetic molecules. Ampere's theory admits of an 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 (§ 306) 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 magnetization of a body magnetized by induction, by increasing the strength of the field beyond a certain limit. No such effect has as yet been observed. We may state the facts of magnetism in a way which is more in accordance with our view that the current is the result of actions in the medium by saying that each magnetic molecule is the origin of a certain definite number of tubes of induction. The existence 356 ELEMENTAKT PHYSICS. [§ 300 of these tubes is supposed to be connected with the peculiar motions which characterize the molecule of the magnetic body. Diamagnet- ism would then be explained by supposing a similar motion enforced upon the molecules of the other bodies in the field to an extent in each which depends upon the nature of the body. 300. The Hall Effect. — Hitherto it has been assumed that when currents interact, it is their conductors alone which are affecte^, and that the currents in the conductors are not in any way altered. Hall has, however, discovered a fact which seems to show that cur- rents may be displaced in their conductors. If the two poles of a voltaic battery be joined to two opposite 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, and if a magnet pole be brought opposite the face of the cross, a per- manent current will be indicated by the galvanometer. The same effect appears in the case of other metals. The direction of the permanent current and its amount differ under the same circum- stances for different metals. The coefficient which represents the amount of the Hall effect in any metal is called the rotational 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 (§ 316), it is argued by Bidwell, v. Ettingshausen, and others that the Hall effect is due to thermoelectric action. 301. Currents in a Magnetic Field Due to Inequalities of Temper- ature. — If a thin strip of bismuth be placed in a magnetic field so that the magnetic force is normal to its surface, and if ono of the edges of the strip be kept at a higher temperature than the other and the two ends of the strip joined by a wire in which a galvanom- eter is inserted, a continuous current will flow through the circuit. The direction of this current changes when the direction of the flow of heat changes or when the magnetic field is reversed. The strength of the current is different in different metals. These facts were discovered by v. Ettingshausen. Conversely, if a current be sent through the strip of bismuth placed in the magnetic field, there § 303] THE MAGNETIC BELATIONS OF THE CURREKT. 357 will be a flow of heat across the strip and the temperatures ol its ■edges will difEer. This difference of temperature is due to a cooling of one edge of the strip. These effects are not reversible in the sense in which the Peltier effect and the thermoelectric effect are reversible, but ,J. J. Thomson has shown that they are consistent with each other. 302. 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 instrument called the Schweigger's multiplier. It consists of a flat spool upon which an insulated wire is wound a number of times. The plane of the coils is vertical, and usually also coincides with the plane of the magnetic meridian. A magnetic needle is suspended in the interior of the spool. When a current is passed through the wire, the needle is deflected from the magnetic meridian. Usually, in order to make the indications of the apparatus more sensitive, a combination of two needles is used. They are joined rigidly together, so that when suspended the lower one hangs in the interior of the spool, and the other in the same plane directly above the spool. These needles are magnetized so that the positive end of one is above the negative «nd of the other. If they are of nearly equal strength, such a com- bination will have very little directive tendency in the earth's mag- netic field. It is therefore called an astatic system. When a current passes in the wire, however, the lines of force due to the current iorm closed curves passing through the coil, and both needles tend to turn in the same direction. Since the earth's field offers almost no resistance to this tendency, an astatic system will indicate the presence of very feeble currents. The apparatus here described is no longer used to measure currents, but only to detect their pres- ence and direction. The tangent galvanometer is that form of galvanometer which is commonly used to measure electrical currents in electromagnetic units. We will consider it only in one of its simplest forms. In 358 ELEMENTARY PHYSICS. [§ 302 this form it consists of a circular conductor set up in the earth's magnetic field, so that its plane is parallel with the lines of force, and having a small magnet placed at its centre. The magnet )s free to swing in the horizontal plane. If the current i be sent through the circuit, the couple which it will exert on the magnet, on the sup- position that the magnet is so short that the force at its poles is the same as that at its centre, is cos cp (§ 396), where M repre- sents the magnetic moment of the magnet, and

This iirst law is evident, because, after the current has become steady, there is no accumulation of electricity at the junctions. 2. The sum, taken around any number of branches forming a ■closed circuit, of the products of the currents in those branches and their respective resistances is equal to the sum of the electromotive forces in those branches. This law can easily be seen to be only a modified statement of Ohm's law. These laws may be illustrated by their application in a form of apparatus known as Wheatstone's Iridge. The circuit of the AVheat- stone's bridge is made up of six branches. An end of any branch meets two, and only two, ends of other branches, as shown in Fig. 87. In the branch 6 is a voltaic cell with an electromotive force E. In the branch 5 is a galvanometer which will indicate the presence of & current in that branch. In the other branches are conductors, the J rlG.o/. resistances of which may be called respectively r,, r„ r^, »\. From Kirchhoff's first law the sum of the currents meeting at the point C is i, + \ -j- i^= 0, and of those meeting at the point B \si -\- i -\- ii= 0. By the second law, the sum of the products ir in the circuit ^DC is i,r,+ i>3-f «:,r,= 0, and in the circuit DBG is ir -\- i r^-{- i^r^=0, since there are no electromotive forces in those circuits. If we so arrange the resistances of the l)ranches 1, 3, 3, 4 that the galvanometer shows no deflection, the current i. is zero, and these equations give the relations i = — i„ i — _ ^ l^ — ^ir,ir = — i^r^. From these four equations 362 ELEMENTARY PHYSICS. [§ 305 follows at once a relation between the resistances, expressed in the equation ^,r, = r^Vy (105) If, therefore, we know the yalue 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 Wheat- stone's bridge is of great importance in practice. By the use of a form of apparatus known as the British Association Iridge the method can be carried to a high degree of accuracy. In this form of the bridge, the portion marked AGB (Fig. 87) is a straight cylin- drical wire, along which the end of the branch CD is moved until a point G is found, such that the galvanometer shows no deflection. The two portions of the wire between C and A, and C and B, are then the two conductors of which the resistances are r, and r^, and these resistances are proportional to the lengths of those portions (§ 275). The ratio of r^ to r, is therefore the ratio of the lengths of wire on either side of C, and only the resistance of r, need be known in order to obtain that of r,. It is often convenient in determining the relations of current and resistance in a network of conductors to use Ohm's law direct- ly, and consider the difference of potential between the two points on a conductor as equal to the product ir. When a part of a cir- cuit 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 suppose the divided circuit (Fig. 88) made up of only three portions, 1, 3, 3, meeting at the points A and B, and that no electromotive force exists in those portions. Then the difference of potential be- tween A and B\sV^-Vb- i,r, = i,r, = i,r^. We have also by Kirchhoff's first law - t, = t, + i, +i,. By the combination of these equations we obtain -i.= iV.-V.)^+'-y-\ (106) § 306] THE MAGKETIC KELATIONS OF THE CURRENT. 363 The current in the divided circuit equals the difference of po- tential between A and B multiplied by the sum of the reciprocals of the resistances of the separate portions. If we set this sum equal to -, and call r the resistance of the divided circuit, 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 por- tions 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. The rules for joinijig up sets of voltaic cells in circuits so as to accomplish any desired purpose may be discussed by the same method. Let us suppose that there are n cells, each with an elec- tromotive force e and an internal resistance r, and that the c! "I'lual resistance of the circuit is s. If m be a factor of n, and if we join up the cells with the external resistance so as to form a divided cir- cuit of m parallel branches, each containing — cells, we shall have for the electromotive force in such a circuit — , and for the resist- ance of the circuit s-\-—,. The current in the circuit is therefore mne . . Two cases may arise which are common in practice, m's -f nr The resistance s of the external circuit may be so great that, in comparison with m's, nr may be neglected. In that case i is a maxi- mum when m = l, that is, when the cells are arranged tandem, or in series, with their unlike poles connected. On the other hand, if m'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 arranged abreast, or in multiple arc, with their like poles in con- tact. 306. Induced Currents.— It was shown in § 277 that the move- ment of a magnet in the neighborhood of a closed circuit will give rise, in general, to an electromotive force in the circuit, and that 364 ELEMENTARY PHYSICS. [§ 306 the current due to this electromotive fores will be in the direction opposite to that current which, by its action upon the magnet, would assist the actual motion of the magnet. This current is called an induced current. From the equivalence between a magnetic shell and an electrical current, it is plain that a similar induced current will be produced in a closed circuit by the move- ment near it of an electrical current or any part of one. Since the joining up or breaking the circuit carrying a current is equivalent to bringing up that same current from an infinite distance, or removing it to an infinite distance, it is further evident that similar induced currents will be produced in a closed circuit when a circuit is made or broken in its presence. The demonstration of the production of induced currents in § 277 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 thd magnetic field due to a closed current is (§ 290) i[£l' — £1 -{■ Ann), and the work done on a magnet pole m, in moving it from one point to another, is mi{D,' — f 2 + inn). In the demonstration of § 277 we may substitute m{£2' — £1 + 4t7in) for A, and, provided the change in potential be uniform, we obtain at once the expres- m(n' -n-\- iTtn) ^ ,, , , . „ sion for the electromotive 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 uniform. Then the limit of the expres- in(n' — n -\- Ann) ^^°^ f > as t becomes indefinitely small, is the •electromotive force during that interval. The current strength due to this electromotive force is •/ _ _ m{n' -n + Ann) rt § 306] THE MAGNETIC EBLATIONS OF THE CURRENT. 365- If the induced current be steady, the total quantity of electricity flowing in the circuit is expressed by i't = — '"•(-Q' — D.-\- imi) r The total quantity of electricity flowing in the circuit depends,, 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 electromotive force due to the movement of the magnet, and consequently the current strength,, depends, on the other hand, upon the rate at which the potential changes with respect to time. A more general statement of the mode in which induced cur- rents are produced may be given in terms of the changes in the number of tubes of induction which pass through the circuit. When the number of tubes of induction which pass through a, circuit is altered, an electromotive force is induced in the circuit which is proportional to the rate of change of the number of tabes of induction. This law may be easily proved, as in the special case already considered, if the change in the number of tubes of induc- tion be produced by a movement of magnet poles or their equivalents, and not by changes in other currents in the field ; in case there are other currents in the field, the interactions between them introduces- conditions which cannot be discussed by elementary methods. The law, however, is a perfectly general one, and holds for all cases in which the tubes of induction passing through the circuit change in number. While we cannot, by elementary methods, determine exactly the laws of the production of an induced current in a circuit by changes in the currents in neighboring circuits, we may yet form some idea of the induced current by considering the magnetic field about the circuits. Suppose that a current traverses circuit 1 and that there is no current in circuit 2 ; circuit 2 encloses a number of tubes of induction due to the current in circuit 1. If the current in circuit 1 be suddenly interrupted, these tubes of induction are removed from circuit 2, and from the dynamical principle that a change is resisted by the non-conservative forces to which it gives rise, there will arise in circuit 2 a current tending to maintain the tubes; 366 BLEMBNTAEY PHYSICS. [§ 306 within it. If the two circuits are parallel, this current will be in the same sense as that in circuit 1. The current induced in circuit 2 gives rise to tubes of induction which enter circuit 1, and their entrance into circuit 1 is resisted by a current tending to repel them from circuit 1, or to set up tabes of induction in the opposite sense. Thus there will be a small current in circuit 1 in the oj^posite sense to that originally in it and the current in circuit 1 will therefore diminish more rapidly than if circuit 2 were not present. On the other hand, if neither circuit carries a current, and a carrent be suddenly impressed on circuit 1, the tubes of induction to which it gives rise will enter circuit 2, and will be resisted by a momentary current in circuit 3 tending to repel them, or to set up tubes of induction in the opposite sense. Thus the induced current in circuit 2 in this case, if the two circuits are parallel, is in the oppo- site sense to that in circuit 1. This current in circuit 2 will in turn set up tubes of induction which enter circuit 1 and are there resisted by a momentary small current which will be in the same sense as that impressed upon circuit 1. Thus the presence of circuit 2 will temporarily increase the current m circuit 1. 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 pritnary, the second the secondary, circuit. 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 with- drawn from the secondary coil, the galvanometer was deflected. The currents indicated were related to the direction of motion of the positive magnet pole, as the directions of rotation and propul- sion in a left-handed screw. The direction of the induced currents I 307] THE MAGNETIC BELATIOKS OF THE CUEKENT. 367 in .these experiments is, easily seen to be in accordance with the law- above stated. ■ A simple statement, known as Lenz's law, which enables us to determine the sense of an induced current produced by the motion of a magnet or a circuit, is as follows: When an induced current is produced, it is always in such a sense as to oppose the action which produces it. This is equivalent to the statement that the induced current tends to oppose the change in the number of tubes of induction which pass through the circuit. The case in which an induced current in the secondary circuit 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 experiments of Paraday and others show that the total quantity of electricity induced when the primary circuit is made is exactly equal and oppo- site to that induced when the primary circuit is broken. They also show that the electromotive force induced in the secondary circuit is independent of the materials constituting either circuit, and is proportional to the current strength in the primary circuit. These results are consistent with the formula already deduced for the induced current. 307. Currents of Self-induction. — If the current in a circuit be changed, the number of tubes of induction which pass through the circuit will vary, and an induced current will be set up in the circuit. If there be originally no current in the circuit and if an electromotive force be suddenly impressed upon it, so that the cur- rent which finally exists in the circuit is i, the number of tubes of induction developed through the circuit as equal to Li (§ 293). Let t be the time required for the current to rise to its full value ; then the average electromotive force induced in the circuit by the increase in the number of tubes of induction which pass through it will be — , and the average current will be — r. The total current t rt due to this induced electromotive force is therefore—, and is op- posed, in sense, to the current impi-essed upon the circuit. If the circuit be suddenly broken, the same expression represents the total 368 ELBMENTAEY PHYSICS. [§ 30S induced current due to the loss of the tubes of induction which pass, through the circuit ; this current is in the same sense as the current of the circuit. Since by Ohm's law i = -, where e is the electromotive force impressed upon the circuit, the average electromotive force is eL in both these cases -- . Now t, the time required for the current rt to rise from zero to its full value, or to sink from its full value to zero, is very small, and the average electromotive force of induction may be much larger than the electromotive force of the circuit. When the current is made, this induced electromotive force diminishes the electromotive force of the circuit; so that the current is established gradually and not instantaneously. The time required to establish the current depends upon the resistance and self-in- duction of the circuit. When the circuit is broken, the electro- motive force of induction is in the same sense as that of the circuity and produces a momentary current which is much greater than the steady current of the circuit. The induced electromotive force is frequently so high as to cause the current to leap across the gap formed where the circuit is broken, and to give rise to a spark at that gap. The induced current thus formed is often called the extra current or the current of self-induction. It should be noted that the induced electromotive force is proportional to the coefficient of self-induction of the circuit. The establishment of a current in the circuit may therefore be retarded and the extra current at the break may be increased by so arranging the circuit as to increase its coefficient of self-induction; while by so winding the circuit that its coefficient of self-induction is reduced to a minimum these effects may be almost entirely avoided. A wire doubled on itself, and coiled so that a current in it always passes in opposite directions, through immediately contiguous portions of the wire, will mani- festly have a very small coefficient of self-induction ; such a coil is called a non-inductive coil. 308. Alternating Currents. — If the electromotive force in a cir- cuit be made to vary, especially if it be made to change in sense, the tubes of induction which pass through the circuit will also vary^ § 309] THE MAGNETIC RELATIONS OF THE CURRENT. 369 and the current in the circuit will vary in a way dependent not only on the variations in the electromotive force, but also on the currents produced by induction. The case of the greatest interest and importance is that in which the electromotive force varies periodically; in this case the current also varies periodically. It may be shown, by a method which cannot be given here, that the maximum value of the current is never as great as that deduced from the maximum electromotive force on the supposition that the current follows Ohm's law. The formula which expresses the maxi- mum value of the current is , , ,,^ , , where e is the maximum /T72 electromotive force and T the period of the alternation. The de- nominator of this expression is a quantity of the same order as resistance, but it involves, besides the resistance of the circuit, its coefficient of self-induction and the period. In case is very large in comparison with r', the current has its maximum value at the time when the electromotive force is zero, and is zero when the electromotive force is a maximum. The theory farther shows that the rate of propagation of the electrical disturbance along the con- dactor is a function of the period of the alternation, being less when the period is greater. When the period is infinitesimal, or in general when it is very small, the velocity is equal to the velocity Vy the ratio between the electrostatic and the electromagnetic units (§ 311), or to the velocity of light. The currents developed in the conductor, by rapid alternations of electromotive force, are not the same for all parts of the cross-section of the conductor, but diminish from the outside of the conductor inwards. For very rapid alter- nations the currents exist only in a small layer near the surface of the condactor. These deductions of theory have been fully con- firmed by experiment. 309. Apparatus employing Induced Currents.— The production of induced currents by the relative movements of conductors and magnets is taken advantage of in the construction of pieces of 370 ELEMENTARY PHYSICS. [§ 309 apparatus which are of great importance not only for laboratory use but in the arts. The telepho7iic 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. AVhen the coils of two such instruments are joined in circuit by conducting wires, any disturbance of the iron diaphragm in front of one coil will change the magnetic field near it, and a current will be set up in the circuit. The strength of the magnet 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 m motion by sound waves due to the voice, the induced cur- rents, and the consequent 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. Other transmitters are generally used, in which the diaphragm presses upon a small button of carbon. A current is passed from a battery through the diaphragm, the carbon button, and the rest of the circuit, including the receiver. When the diaphragm moves, it presses upon the carbon button, and alters the resistance of the circuit at the point of contact. This change in resistance gives rise to a change in the current, and the diaphragm of the receiver is moved. The telephone serves in the laboratory as a most delicate means of detecting rapid changes of current in a circuit. The various forms of magneto-electrical and dynamo-electrifcal machines are too numerous and too complicated for description. In all of them an arrangement of conductors, usually called the ar7na- hire, is moved in a powerful magnetic field, and a suitable arrange- ment is made by which the currents thus induced may be led ofi and utilized in an outside circuit. The magnetic field is sometimes established by permanent magnets, and the machine is called a magneto-machine. In most cases, however, the circuit containing the armature also contains the coils of the electromagnets to which the magnetic field is due. When the armature rotates, a current starts in it, at first due to the residual magnetism of some part of § 309] THE MAGNETIC KELATIONS OF THE CUEKENT. 371 the machine: this current 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 proportionally to the current which produces it. Such a machine is called a dynamo-machine. By suitable arrangements of the con- ductors which lead the current from the machine, either direct or alternating currents may be obtained. The induction coil, or Ruhmhorff's coil, consists of two circuits wound on two concentric cylindrical spools. The inner or primary oircuit 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. Withm the primary circuit is a bundle of iron wires, which, by its magnetic action, increases the electromotive force of the induced ourrent 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 secondary coil, and as this is very great the electromotive force of the induced current greatly exceeds that of the primary current. The electromotive force of the induced current set up when the primary circuit is broken is further heightened by a device proposed by Fizeau. To two points in the primary circuit, one on either side of the point where the circuit is broken, are joined the two surfaces of a condenser. When the circuit is broken, the extra current, if the condenser be not intro- duced, forms a long spark across the gap, and so prolongs the fall of the primary current 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 con- denser is introduced, the extra current is partly spent in charging the condenser, 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. 373 ELEMENTARY PHYSICS. [§ 310 310. Determination of the Unit of Resistance. — If the circuit considered in § 306 move from a point where its potential relative to the magnet pole is mfl' to one where it is mO,, provided that the magnetic pole do not pass through the circuit, and that the move- ment be so carried out that the induced current is constant, the mfp.' -O.) ,„ electromotive force of the induced current is . It « f the movement take place in unit time, and if m{(l' —£1) also equal unity, the electromotive force in the circuit is the unit electromo- tive force. The expression »i((l' —CI) is equivalent to the change in the number of tubes of induction 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 tubes of induction passing through the circuit in the positive direction increase or diminish by unity, at a uniform rate, the electromotive force induced is unit electromotive force. This definition is consistent with the one given in § 303. For, the energy of a circuit carrying the current (', due to the field in which it is placed, equals iN, and the change of this energy in unit time is the energy expended in the circuit in that time. But ]^' — jSf N' — N . this change in energy is { , and — — is the electromo- t t tive force, so that ie represents the energy expended in unit time. A simple way in which the problem can be presented is as follows: Suppose two parallel straight conductors at unit distance apart, joined at one end by a fixed cross-piece. Suppose the circuit to be completed by a straight cross-piece of unit length which can slide freely on the two long conductors. Suppose this system placed m 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 perpen- dicular to itself along the parallel conductors, the electromotive force set up in the circuit will be the unit electromotive force, and if it move with any other velocity v, the electromotive force will be equal to v. § 311] THE MAGNETIC EELATIONS OF THE CURRENT. 373 If we now insert a galyanometer in the fixed cross-piece, and suppose the resistance of all the circuit except the sliding piece to be negligible, and moye the sliding piece at such a rate that the -current in the galyanometer is unity, we have the resistance of the sliding piece determined from the velocity with which it moves. For, by Ohm's law, i= -, and since i = 1 and e = v, we have r ' r — V. Such an arrangement as that here described is of course impos- sible in practice, but it embodies the principle of the method actually used to determine the unit of resistance by the Committee of the British Association. In their 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. 311. Ratio between the Electrostatic and Electromagnetic 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 electromagnetic unit of any electrical - and — 7 = , a negative quantity. To interpret ^ 2 p r p r p this negative result it should be remembered that all the distances in the formulas were assumed positive when measured from the Fig. 103. mirror toward tlje source of light. A negative result means that the distance must be measured in the opposite direction, or behind the mirror. Fig. 103 represents this case. It is evident that the reflected wave is convex toward the region it is approaching, and proceeds as though it had come from I. I is therefore a virtual focus. Either of the other quantities of the formula may have neo-ative values, p ^^^^ ^f^- nes-ative if waves approaching their 412 ELEMENTARY PHYSICS. [§ 337 centre I fall on the mirror. Plainly they would be reflected to L r at a distance from the mirror less than -j 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 p, p' is negative and numerically smaller than p. 337. Refraction at Spherical Surfaces. — The method of discus- sion which has been applied to reflection may be employed to study xefraction at spherical surfaces. Let BD (Fig. 103) be a spherical Pig. 103. surface separating two transparent media. Let v represent the velocity of light in the first medium, to the left, and v' the velocity in the second medium, to the right, of BD. Let i be a radiant point, and mn a surface 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, and I the point where the .prolongation of Bm' backwards cuts the axis. We will investigate the conditions which must be fulfilled in order that the refracted wave shall appear to proceed from a point on the axis, or shall be a spherical wave. In order that this should be the case, the time occupied by the light in travelling from the point I on the axis with the velocity § 337] KEFLECTIOIf AND EEFEACTIOJST. 413 which it has in the second medium should be the same for all points on the refracted wave. This time is given by t'= -"^ = ^^ "^ ^'^' ■ ° •' v' v' we are to investigate the conditions that this shall be the same for all points on the refracted wave. The time occupied by the light in travelling from L to m' is LB , Bm' ^, . , . „ — — I ^ = 6, a constant for all points on the refracted wave. Subtracting from this the expression for t\ we have C —t' = J, and the condition that t' should be constant is V V therefore that — ■- 7 is constant. Since — , = u, we may write the expression which should be constant LB — jxlB = LA — yu I A. Using the notation of the last section, and substituting the val- ues of LB and IB as there found, except that f" is used instead of J)', we have as the condition that t' is a constant, whatever be the value of X within the limits set to it, the equation m P+^ir -P) - /^(/' -^^Ar-P")) =P T UT From this we obtain ji — l — fx, P V and 4-i = ^l. (110) p" p r Hence the point at the distance p" from the centre of the refract- ing surface is the centre of a spherical refracted wave. If the medium to the right of BD be bounded by a second spherical surface, it constitutes a lens. Suppose this second sur- face to be concave toward I and to have its centre on AG. 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 sur- face, and for fx the index of refraction of the third medium in rela- tion to the second. Tf .9 renresent the distance of I from the new 414 ELEMENTARY PHYSICS. [§ 337 surface, n' the new index, and 'p' the new focal distance, we have // _ 1 _ n' —\ p' s r' ' If we suppose the lens to be very thin, we may put s = p". If ■we suppose also that the medium to the right is the same as that to the left of the lens, //' is equal to — . On these suppositions Multiplying through by /<, we have 1 — /i _ fi — \^ Eliminating p" between this equation and equation (110), we obtain }'-^=(^-i)(f-p)' (111) •which expresses the relation between the conjugate foci of the lens. It should be noted that r in the above formulas represents 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. 104 1 ;" 1 p' p" L-1 - -" Multi 1 yu y p' shows sections of the different forms of lenses produced by com- binations of two spherical surfaces, or of one plane and one spherical surface. An application of equation (111) 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 con- verging, and the second three diverging, lenses. Let us consider the first and fourth forms as typical of the two classes. The first § 337] REFLECTION' AND KEFBACTION. 415 is a double-convex lens. The r of equation (111) is negative because measured from the lens away from the source of light. The sec- ond term of the formula has therefore a negative value, and p' is negative except when ->(;/_ i)L ^j. If p = oo, we have - = and -7 = (/i — 1) ( ^j, a negative quantity because r is negative, p' 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 symbol/. 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. Equation (111) is a little more simple in application if, in- stead of making the algebraic signs of the quantities depend on the direction of measurement, 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 dis- tances are positive when foci are real, the signs of p' and r in that equation must be changed, since in the investigation js' is the dis- tance of a virtual focus, and r the radius of a concave surface. The formula then becomes l+l=(;U-l)g+l). (112) p p ' \r r j 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. 104, 3), if light be incident on the plane surface, r = CO and -, = (a« - l)p- + -. This gives positive values of p' and real foci for all values of i < (;U - 1)4. p ^ 'r For a concavo-convex lens (Fig. 104, 6) the second member of the equation will be negative, since the radius of the concave surface is negative and less numerically than that of the convex 416 ELEMENTARY PHYSICS. [§ 338 surface. Hence p' is always negative and the focus -virtual whea L is real. 338. Images formed by Mirrors. — In Fig. 105 let ab represent an object in front of the concave mirror mn. We know from what precedes that if we consider only the light incident near c, the light reflected will be concentrated at some point a' on the axis ac at a distance from the mirror given by equation (109). a Y ^^^^^^ X ^\ ^^-^S^ ^y^ aV^ -4 Fig. 105. a' is a real image of a. In the same way h' is an image of I. If axes were drawn through other points of the object, the images of those points would be found in the same way. They would lie between a' and I', and a'b' is therefore a real image of the object. It is inverted, and lies between the axes ac, hcl, drawn through the extreme points of the object. The ratio of its size to that of the object is seen from the similar triangles abC, a'l'C, to be the ratio of the distances from C. From equation (109) we ob- . . p' r r — p' tain — = = ^— . p 2p — r p — r Since r — }/ and p — r are respectively the distances from (t u V 7)' w' the centre of the image and object, we have — =- = ~ = -t- ; or, ab p — r p 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 object. When the object is between the principal focus and the mirror, the image is virtual § 340] REFLECTION- AND REFRACTION. 417 and larger than the object. Convex mirrors produce only virtual images, which are erect and smaller than the object. 339. Images formed by Lenses.— Let us suppose an object in front of a double-convex lens, which may be taken as a type of the converging lenses. The point c (Fig. 106) will have an image at the conjugate focus on the principal axis, a and I will have im- 6 Pig. 106. -J'ar ages on secondary axes drawn through those points respectively, and a point called the optical centre of the lens. So long as these secondary axes make but a small angle with the principal axis, de- finite foci will be formed at the same distances as on the principal axis, and an image a'h' 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 ~ -\- — — Iia. — 1)[- 4- — ] = ~ shows that p p '\r r j f when p increases p' diminishes, and conversely. It shows also that when p is le.ss 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. 340. 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 position of this point is determined as follows: In Kg. 107, let C, Che 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 (§ 334), and suffers no deviation. 418 ELEMEKTAET PHYSICS. [§ 341 If we draw AB, cutting the axis at 0, the triangle CAO, C'BO CA CO CA are similar, and ^7^ = ^jtq. But ^^, being the ratio of the CO radii, is constant for all parts of the surfaces, hence -^tq must be constant, or all lines such && AB must cut the axis at one point 0. is the optical centre, and light passing through it is not devi- ated by the lens. 341. 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 perpendicular 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 a pencil. A plane wave surface perpendicular to the axis of a lens is con- verted by the lens into a spherical wave surface with its centre at the principal focus. The rays perpendicular to the plane wave sur- face 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 L, Fig. 108, be a con- verging lens, and AB wa. object. Let 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, which is the image of A, to find the position of the image it is only necessary to draw two such rays and find their intersection. The ray through the optical centre is not deviated, and the straight line A A' represents both the incident and emergent rays. The ray AL may be consid- FiG. 108. Fig. 109. § 343] EEFLECTIOK AlfD REFRACTION. 419 ered 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 in- tersection -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 correspond- ing emergent rays. The inter- section of these will determine the image. The rays most con- venient to use are the ray through the optical centre and the ray parallel to the axis or through the principal focus. Fig. 109 gives another example of an image determined by construction. 342. Thick Lenses. — "When a lens is of considerable thickness, the formula derived in § 337 does not give the true position of the conjugate foci. A formula involving the thickness of the lens may be derived without difficulty, 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 tliese planes involves a discussion which does not come within the scope of this book. 343. Mirrors and Lenses of Large Aperture.— The equations derived in §§ 336, 337 are only approximations, applying with suf- ficient 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 p', and p' will, therefore, not have a definite value. In other words, the re- flected or refracted wave is not spherical, and there is no one point I where the light will be concentrated. Surfaces may, however, be constructed which will, in certain particu- lar cases, produce by reflection or refraction perfectly spherical waves. If we desire to find a surface such that light from L (Fig. Fig. 110. 110) is concentrated by reflection at I, we remember that the sum LB + Bl must be constant, and that this 420 ELEMENTARY PHYSICS. [§ 343 is a property of an ellipse witli foci at L and J. If the ellipse 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 corre- sponding wave becomes a plane perpendicular to LI, and we must have LB -\- BO (Fig. Ill) con- stant, a property of the parabola. A parabolic mirror will therefore concentrate at its focus in- cident light moving in paths parallel to its axis, or will reflect incident light diverging from its Hi. focus in plane waves perpendicular to its axis. Mirrors and lenses having surfaces which are not spherical are seldom made because of mechanical difficulties of construction. It becomes necessary, therefore, to consider how the disadvantages arising from the use of spherical surfaces of large aperture for re- flecting 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 concentric with the other focus. Let Fig. 112 represent a plane section 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 re- flected 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 FaF', and hence in the triangle FaF" Fa Fe wehave-^=-p-. If d move toward c, F a increases and Fa diminishes. Hence, from the above proportion, F'e must increase and B'e 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 there- § 343] REFLECTION AND REFRACTION. 431 fore 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 <3entre at F', and are also sections of wave surfaces reflected 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 conceived 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 surfaces of curved tubes of which the axis is the arc described by the point F'. The line described by i^' is a 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 F'F" and finally coincides withi?"', 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 ^i^ between the points x and y. The light coming from F and reflected from a small portion of the spherical surface around h, the 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 i and the middle of the focal line through F' is the axis of the re- 432 ELEMENTAEY PHYSICS. [§ 344 fleeted pencil. It will intersect the axis of the mirror between x and y. If a plane be passed through the point of intersection per- pendicular 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 surfaces 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 confined within narrow limits of normal incidence. When reflection or refraction takes place at two or more surfaces in succession, the aberration of one may be made to partially correct the aberration of the other. For instance, when the waves incident 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 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. 344. 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 photog- rapher's camera is a box in one side of which is a lens so adjusted as to form an image of external objects on a plate on the opposite side. The relation deduced in § 339 serves to determine the size of §^^5] KBFLEOTION AND KEFRAOTIOST. 423 the image which a given lens will produce, or the focal length of a lens necessary to produce an image of a certain size. 345. The Eye as an Optical Instrument.— The eye, as may be seen from Pig. 113, which represents a section by a horizontal plane, is a camera obscura. a is a transparent membrane called the cornea, behind 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 en- tire posterior cavity of the eye. The aqueous humor, crystalline lens, and vitreous humor con- stitute 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 ex- pansion of the optic nerve. The light that forms the image upon it excites the ends of the nerve, and, through the nerve-fibres lead- ing to the brain, produces a mental impression, which, partly by the aid" of the other senses, we have learned to interpret as the characteristics of the object the image of which produces the im- pression. 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 changing the curva- ture 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 sen- sitive. 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 424 ELEIIENTAEY PHYSICS. [§ 345 is much more distinct than the other parts. How small this most sensitive area is, can be judged by carefully 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 control 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 sensitive 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 produced outward meet at the point. For objects at a moderate distance we have learned to associate a particular muscular effort with a par- ticular distance, and our judgment of such distances depends mainly on this association. The angle between 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. 114, when ab is an object and I 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. b Fig. 114. It is plain, too, that an object of twice the size, at twice the dis- tance, would subtend 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. The visual angle is the apparent size of the object. A less precise estimate of distance can be made with a single §347] REFLECTION AND REFRACTION. 425 Bje, probably from the perception of the effort required to get the object clearly focussed on the retina. 346. Magnifying Power.— To increase the apparent size of an object, and so improve our perception of its details, we must in- crease 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 limit to the near approach, due to our inability to see distinctly very near objects. Certain optical instruments serve to increase the visual anglcj and so improve our vision. Instruments for examining small objects, and increasing the 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 distance are telescopes. In both cases the ratio of the visual angles, as the object is seen with the instrument, and with- out it, is the magnifying power. 347. The Magnifying-glass. — Fig. 115 shows how a converg- ing lens may be employed to magnify small objects. The point a of an object just inside the principal focus F of the lens A is the origin of light-waves which, after passing through the lens, are changed to waves having a centre a' (§ 337) which, when the lens Fig. 115. Fio- 116. is properly adjusted, is at the distance of distinct 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 436 ELEMENTAKT PHYSICS. [§ 348 Oa" , when the angle subtended is a"OV'. The ratio of these angles 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. 348. The Compound Microscope. — A still greater magnifying power may be obtained by first forming a real enlarged image of the object (§ 339) and using the magnifying-glass upon the image, as shown in Fig. 116. The lens A is called the objective, and E is called the eye-lens or ocular. As will be seen in § 359, both A and E often consist of combinations of lenses for the purpose of correcting aberration. 349. Telescopes. — If a lens or mirror be arranged to produce 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 Pig. 117. Suppose the nearest point from which the object can be observed by the naked eye to be the centre of the lens 0. A 6 oj Fig. 117. The visual angle is then A OB = aOb, while the visual angle for the image is aEb. Since these angles are always very small, we have — ^ = ^ 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 in- creased in the ratio of the distance of distinct vision to the focal length of the magnifying-glass. The magnifying pov'^r of the combination is therefore the ratio of the focal length of the object- glass to the focal length of the eye-glass. A concave mirror may be substituted for the object-glass for producing the real image. CHAPTER III. INTERFERENCE AND DIFFRACTION. 350. 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 phenom- ena due to this composition of light-waves are called interference phenomena. Let us consider the case in which two elements only are efficient 'in producing the disturbance. Let A and B (Fig. 118) represent twa Mn elements of the same wave surface sep- arated by the very small distance AB. The disturbance at m, a point on a \^ distant screen mn, parallel with AB, Fig. 118. due to these two elements, is the re- sultant of the disturbances due to each separately. The light is supposed to be homogeneous, and its wave length is represented by;i. When the distance mB — m4. equals ^X, or any odd multiple of \X, there will be no disturbance at m. Take mC = mB, and draw BC. mCB is an isosceles triangle; but since AB is very small compared to Om, the angle at G may be taken as a right angle; the triangle ACB, therefore, is similar to Osm, and w& have — = — = — very nearly. Represent sm by x, Os by c, AC sm sm ABhj b,AChj n X i^, where n is any number. Then we have z = ^. (113) 437 428 ELEMEJfTAKT PHYSICS. [§ 350 If n be any even whole number, the values of x given bj this equa- tion 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 z represent points where the waves meet in opposite phases, and therefore pro- duce darkness. It appears, therefore, that starting from s, for -Im/' -*-^^ §Ji/^ which w = 0, we shall have darkness at distances ^j-, -7-, -^ , , ,. , , ,. , „ Ac 2Xc 3Ac ^ etc., and light at distances 0, -r-, -j-, -j-, etc. 26a; From equation (113) we have n — -y- Since J?iA.is the number •of wave lengths that the wave front from B falls behind that from A,knT, where T represents the period of one vibration, is the time that must elapse after the wave from A produces a certain displace- ment before that from B produces a similar displacement. The ex- pression — ^^ — = nTt is, therefore, the difference in epoch of the • two wave systems. Substituting nn for e in equation (17), we have a , > ,n , n \i ('^'^t j. . sin 7nr \ JS = s + s = a(2 -+- 2cos«7r)i cos -7^^ tan"\ . Now ^ \ I 1 + cos UTT/ the intensity of light for a vibration of any given period is propor- tional to the mean energy of the vibratory motion, and this can be shown to be proportional to the square of the amplitude. Substi- tuting in the expression for the amplitude the value of n and squaring, we have A' = aM 2 + 2 cos -^ttL in which A' is propor- tional to the intensity of the illumination at distances x from s. 2bx When — T- Tf = 0, its cosine is 1, and A" is a maximum and equal to 4a'. As X increases A^ diminishes, until ^r- ;r = ;r, in which case CA A'' = 0. A' then increases until it becomes again a maximum, 36a; when ^-TT = 2;r. In short, if AB (Pig. 119) represent the line mn of Fig. 118, the ordinates to a sinuous curve like abc will represent the intensities of the light along that line. § 350] INTERFERENCE AND DIFFRACTION. 43& The phenomena described above may be obtained experiment- ally in several ways. Young admitted sunlight into a darkened room through a small hole in a win- , dow-shutter. It fell upon a screen /\ /\ /\ /\ /\ in which were two small holes close a b 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 band 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. 120. The light reflected from each mirror proceeded as though from the image of the source produced by that mirror. The reflected light, therefore, consisted of two wave systems, from two precisely similar sources A and B. Light and dark Fig. 120. , . , „ , . ^ , .^, bands were formed m accordance with theory. In order that the experiment may.be successfully repeated reflection must take place from the front surface of each mirror only^ the angle made by the mirrors must be nearly 180°, and the reflect- ing surfaces must meet exactly at the vertex of the angle. Two similar sources of light may be '^'' obtained also by sending the light through a a. double prism, as shown in Fig. 131. Light ^^ from A proceeds after passing through the ^^^ ^^^ prism as from the two virtual images a and a'. A divided lens, Fig. 122, 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 ar» really seen in these cases, when the source of light is white, are iris- colored bands instead of bands of light and darkness merely. When the light is monochromatic, the bands are simply alternations of light and darkness, the distances between them being greatest for red light, and least for blue. From equation (113) it appears that. 430 ELEMENTAEY PHYSICS. [§ 351 other things being equal, x varies with il; hence we must conclude that the greater distance between the bands indicates a greater Fig. 122, wave length; that is, that the wave length of red light is greater than that of blue. 351. Measurement of Wave Lengths. — Data may be obtained from any of the above experiments for the determination of the 26a; wave length of light. From equation (113) we have A = ^, where c, h, and x are distances that can be measured. The distance x is the distance from s to a point m, the centre of a light band, and M equals twice the number of dark bands between s and m. Better methods than this of measuring wave lengths will be found described in § 355. 352. Interference from Thin Films. — Thin films of transparent 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. 133) 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 (§ 132). The light entering the film is partly reflected at the lower surface without change of sign, and re- turning partly emerges at the upper surface. It is there compounded with FiQ- 123. the wave at that moment reflected. Let us suppose the light homogeneous, and the thickness of the film such that the time occupied by the light in going through it and returning is the time of one complete vibration. The returning wave will be in the same phase as the one at that moment entering. §3o2J INTBKFEKENCE AND DIFEEACTION. 431 and, therefore, opposite in phase to the wave then reflected. The reflected and emerging waves destroy each other, or would do so if their amplitudes were equal, and the result is that, apparently, 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 occu- pied in going through the film and returning is the period of one vibration, or any whole number of such periods, of that constitu- ent. The remaining constituents produce a tint which is the ap- parent color of the film. Similar phenomena are produced by the interference of that portion of the incident light which is transmitted directly 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 transmitted light is a maximum. This theory must be slightly modified on account of the internal reflections in the film. The light which enters the film and is re- flected does not all pass out in the reflected beam, but part of it is again sent through the film to the other surface, when it is again divided, so that the reflected and transmitted beams both contain light that has been several times reflected. The theory shows that the reflected beam is totally extinguished when the thickness is that indicated by the elementary theory, and that the transmitted beam is never totally extinguished, but merely passes through a mini- mum intensity. This conclusion is confirmed by observation. 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 at any point could be deter- mined when the radius of the sphere and the distance from the 432 ELEMENTAKY PHYSICS. [§353 point of contact were known. With this arrangement 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, the rings were alternately light and dark. Let ae^ (Fjg. 124) be the radius of the first dark ring, and denote it by d, and let r represent the radius of curvature of the lens. The thickness^ d' be — ef, which may be denoted bya;, is a; = ■ 2r- Since x is very small in comparison with 2r, this becomes z— —. This distance for the first dark ring, when 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 yTj-gVro inches, which corresponds to a wave length of about i^J^nr inches,, or 0.00057 mm. 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. 353. Effects Produced by Narrow Apertures. — It has been seen (§ 325) that cutting off a portion of a light-wave by means of screens, thus leaving a narrow aperture for the passage of the light,, prevents the interference which confines the 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. 125) represent the aperture, and inn one position of the approaching wave.^ To determine the effect at any point we must consider the elementary waves pro- ceeding from the various points of the wave front lying between A and B. First consider 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 ^i^ may be regarded as equal, or A CB 'p Fig. 125. §353] INTERFERENCE AND DIFFRACTION. 433 AcB Pig. 1)36. the time of passage of the light from each point of J 5 to P may he made the same, by placing a converging lens of proper focus be- tween 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, P', in an oblique direction from C (Fig. 126), 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 G to P', and a whole vibra- tion 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 G will meet at P' a wave in the opposite phase from some point between G and A. The point P' is, therefore, not illuminated. Suppose another point, P" (Fig. 127), still further from P, such that AB 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 obliquity, AB may be divided into four parts, the distances of which from the point, taken in succession, differ by half a wave length. There being an even number of these parts, the sum of their effects at the point will be zero. Now let us suppose the point P to occupy successively all positions to the right or left of the normal. While the line joining P with the middle of the aperture is only slightly oblique, the elementary waves meet at P in nearly the same phase, and the loss of light is small. As P approaches P'^ (Pig. 126), more and more of the waves meet in opposite phases, the light grows rapidly less, and at P' becomes zero. Going beyond P' the two parts that annul each other's effects no longer occupy the whole space A fi,some of the points of the aperture send to P waves Fig. 127. 434 ELEMBKTART PHYSICS. [§ 354 that are not neutralized, and the light reappears, giving a second maximum, much less than the first in intensity. Beyond this the light diminishes rapidly in intensity until a point is reached where the paths differing by half a waye length divide AB into four parts, when the light is again zero. Theoretically, maximum and minimum values alternate in this way, to an indefinite distance, but the successive maxima decrease so rapidly that, in reality, only a few bands can be seen. 354. 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 produces darkness. Let mn (Fig. 128) 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 illumination at any Ib point is the sum of the effects of all parts of Fig. 138. ^j^g wave 7)m. Let the effects due to the part of the wave op be represented by Z and that due to all the rest of the wave by I '. Then the illumination at C is I-\-l', equal to the general illumination on the surface. Let us now suppose mn to be a screen and fyo a narrow aperture in it. If the illumination at C remain unchanged, it must be that the parts mo and pn of the wave had no effect, and if, for the screen with the narrow aperture, we substitute a narrow screen at op, there will be darkness at C. If, however, 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 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 illumi- nated when op is a screen. 355. Diffraction Gratings.— Let AB (Fig. 139) be a screen hav- ing several narrow rectangular apertures parallel and equidistant. Such a screen is called a grating. Let the approaching waves, mov- ing in the direction of the arrow, bo plane and parallel to AB, and let the points a, c, etc., be the centres of the apertures. Draw §355] INTEKFBREKCE AND DIFFKACTION'. 435 Fig. 129. the parallel lines al, cd, etc., at such an angle that the distance from the centre of a to the foot of the perpendicular let fall from the centre of the adjacent opening on ab shall be equal to some definite wave length of light. It is evident that an will contain an exact whole number of wave lengths, co one wave length less, etc. The line mn is, therefore, tangent to the fronts of a series of elementary waves which are in the same phase, and may be considered as a plane wave, which, if it were received on a converging lens, would be concentrated to a focus. If the obliquity of the lines be increased until ae equals 2A, 3A, etc., the result will be the same. Let us, however, suppose that ae is not an exact multiple of a wave length, but some frac- tional part of a wave length, JJjA, for example. Let m be the fifty-first opening counting from a ; then an will be ^^-^X x 50 = 49. 5A. Hence the wave from the first opening will be in the oppo- site phase to that from the fifty-first. So the wave from the second opening will be in the opposite phase to that from the fifty-second, etc. If there were one hundred openings in the screen, the second fifty would exactly neutralize the effect of the first fifty in the direction assumed. Light is found, therefore, only in directions given by sin d = —r, (114) where w is a whole number, d the angle between the direction of the light and the normal to the grating, and d the distance from centre to centre of the openings, usually called an element ot the grating. Gratings are made by ruling lines on glass at the rate of some thousands to the centimetre. The rulings may also be made on the polished surface of speculum metal, and the same effects as described above are produced by reflection from its surface. Since the number of lines on one of these gratings is several thousands, it is seen that the direction of the light is closely confined to the 436' ELEMENTARY PHYSICS. [§ 355 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 differ- ent angles with A C, that is, the light is separated into its several constituents, and produces a pure spectrum. Since different values of n will give different values of d for each value of X, it is plain that there will be several spectra corresponding to the several values of n. When n equals 1 the spectrum is of the first order ; when n equals 3 the spectrum is of the second order, etc. The grating furnishes the most accurate and at the same time the most simple method of determining the wave lengths of light. Know- ing the width of an element of the grating, it is only necessary to measure 6 for any given kind of light. Hitherto the spaces from which the elementary waves proceed have been considered infinitely narrow, so that only one system of waves from each space need be considered. In practice, these spaces must have some width, and it niay happen that the waves from two parts of the same space may cancel each other. Let the openings, Pig. 130, be equal in width to the opaque spaces, and let the direction am be taken such that ae equals 2A. Then ae' equals ^X, or the waves from one half of each opening are opposite in phase to those from the other half, and there can be no light in the direction am. In genera], if d equal the width of the opening, there will be interference, and light will be destroyed in that direction for which sin = '^, if the incident light be normal to the grating. Let f represent the width of the opaque space. Then d -\-f = s, and light occurs in the direction given by sin 6 — , - ., provided that the value of given by this equation does not satisfy the first equation also. If d equal /, we have sin B = -^^. = |^. When n is even. §356] INTBRFEREKCE AND DlFFfiACXION. 437 sm f becomes g^ = ^; 3^ = "j. etc., and satisfies the equation . „ nX . . , sm ^ = -^j which expresses the condition under which light is all destroyed. Hence in this case all the spectra of even orders fail. Moreover, the spectra after the first are not brilliant. When / equals 2d the spectrum of the third order fails. 356. Measurement of Wave lengths.— To realize practically 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 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 in- ferred from Pig. 131. It is called the spectrometer. ^ is a tube carrying at its outer end the slit and at its inner end the lens, called a collimating lens. OD is a horizontal gradu- ated circle, at the centre of which is a table on which the grating is mounted, and so adjusted that the axis of the circle lies in its plane and parallel to its lines. In using a reflecting grating the collimating and observing telescopes may be fixed at a constant angle 3/S with «ach other, which may be determined once for all in making the adjustments of the instrument. 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 Fig. 131. 438 ELEMENTARY PHYSICS. [§ 356 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 p. 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 0. If the width s of an element of the grating be known, these measurements substituted in the equation X = 2s cos 13 sin 6 (115) give the value of X. Wave lengths are generally given in terms of a unit called a tenth metre ; that is, 1 metre X 10~'°. 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 YiOfiOd tenth metres, and Kowland has obtained photographs of the solar spectrum in which are lines representing wave lengths of about 3000 tenth metres. Instead of the arrangement which has been described, Eowland has devised a grating ruled on a concave surface, and is thus en- abled to dispense with the collimating lens and the telescope. CHAPTER IV. DISPERSION. 357. Dispersion. — When white light falls upon a prism of any refracting medium, it is not only deyiated from its course but separated into a number of colored lights, constituting an image called a spectrum. These merge imperceptibly from one into an- other, 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. 132). If this light fall on r con- verging lens from a point at a distance from it a little greater Fig. 133. than that of the principal focus, a distinct image of the source will be formed at the distant conjugate focus I. If now a prism be placed in the path of the light, it will, if placed so as to give the minimum deviation, merely deviate the light without interfering with the sharpness of the image, which will now be formed at V instead of at 1. If the source L give two or three kinds of 439 440 ELEMENTAEY PHYSICS. [§ 358 light, the lens may be so constructed as to produce a single sharp image at I of the same color as the source, but when the prism is introduced the lights of different colors will be differently de- viated, and two or three distinct images will be found near V. If there be many such images, some may overlap, and if there be a great number of kinds of light varying progressively in refrangi- bility, there will be a great number of overlapping images con- stituting a continuous spectrum. 358. Dispersive Power. — It is found that prisms of different 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 between the devia- tions of the extremities of the spectrum to the mean deviation may be called the dispersive power of the substance. Thus if d', d" represent the extreme deviations, and d the mean deviation, d' — d" the dispersive power is ^ . 359. Achromatism. — If in Newton's experiment of recomposi- tion 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 m 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 dispersion, 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 combination of two lenses of different kinds of glass, one converging and the other di- verging, one may correct the dispersion of the other within certain limits, while the combina- FiG. 133. tion still acts as a converging lens forming real images of objects. Fig. ]33 shows how this principle is applied to §359] DISPERSION. 441 the correction of chromatic aberration in the object-glasses of tele, scopes. Thus far nothing has been said of the relative separation of the difEerent colors of the spectrum by refraction by different sub- stances. Suppose two prisms of different substances to have such refracting angles that the spectra produced are of the same length. If these two spectra be superposed, the extreme colors may be made to coincide, but the intermediate colors do not coincide at the same time for any two substances of which lenses can be made. Perfect achromatism by means of lenses of two substances is there- fore 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. 134 shows the negative or Huygens eyepiece. Fig. 134. Let A be the objective of a telescope or microscope. A 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 perfectly cor- rected, these images would all concide. 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 Huygens eyepiece. B is called ih^ field-lens, and is three times the focal length of C. It is placed between the ob- jective and its focal plane, and therefore prevents the formation of the images rv, but will form images at r'v' on the secondary axes » Cornu, 1874 3.98500 x 10'» Cornu, 1878 2.99990 X 10'» The Ratio between the Electrostatic and Electromagnetic Units. Cm. per Sec. Weber and Kohlrausch 3.1074 X 10'» W. Thomson 3.835 X 10'" Maxwell 3.88 X lO'" Ayrton and Perry 2.98 X 10'» J. J. Thomson 3.963 X 10" Cm. per Sec. Exner 3.920x10'" Klemencic 3.018 X lO'" Himstedt 3.007 X 10'" CoUey 3.015 X 10'» INDEX. Aberration, spherical, 422 : chromatic, 440 Aberration of fixed stars, 402 Absolute temperature, 223, 235, 247; zero of, 223, 235, 249; scale of, 228, 247 Absorption, 102 ; coefficient of, 102 ; of gases, 103 Absorption of radiant energy, 445 ; of radiations, 448 ; by gases, 449 ; relation of, to emission, 451 Acceleration, 16 ; angular, 18 ; composition and resolution of, 17 Achromatism, 440 Acoustics, 149 Adhesion, 86 Adiabatic line, 224 Aggregation, states of, 84 Air-pump, 143 ; receiver of, 144 ; plate of, 144 ; theory of Sprengel, 139 ;■ Sprengel, 145 ; Morren, 146 Airy, determination of earth's density, 81 Alloys, melting-point of, 212 Amp6re, relation of current and magnet, 313 ; relation of current and lines of force, 341; equivalence of circuit and small magnet, 342; and magnetic shell, 342, 354 ; mutual action of currents, 352 ; theory of magnetism, 355 Ampere, a unit of electrical current, 345 Amplitude of a simple harmonic motion, 18 ; its relation to intensity of light, 428 Analyzer, 464 Andrews, critical temperature, 221; heat of chemical combination, 229 Aneroid, 147 Angle, measurement of, 8 ; unit of, 9 Animal heat and v?ork, 255 Anode, 323 Antinode, 157 Aperture of spherical mirrors, 409 Apertures, diffraction eflfects at, 432 Archimedes, principle in hydrostatics, 131 Aristotle, theory of vision, 394 491 492 INDEX. Arrhenius, theory of dissociation in solutions, 107, 333 ; theory of electrolysis, 880 Astatic system of magnetic needles, 357 Atmosphere, pressure of, 130 ; how slated, 130 Atom, 84 ; nature of, 86 Atomic heat, 211 Attraction, mass or universal, 70 ; constant of, 83 Avenarius, experiments in thermo-electricity, 383 ; thermo-electric formula, 384 Avogadro's law, a consequence of the kinetic theory of gases, 235 Axis of rotation, 50; of strain, 109 ; of stress, 113 ; of floating body, 131; mag- netic, 260, 263 ; of spherical mirror, 409 ; optic, of crystal, 457, 471, 472 Balauce, 78 ; hydrostatic, 133 Barometer, 139 ; Torricellian form of, 130 ; modifications of, 130 ; preparation of, 130 Beam of light, 418 Beats of two tones, 166, 181; Helmholtz's theory of, 181; K5nig's theory of, 183 ; Cross's experiment on, 183 Beetz, experiment ou a limit of magnetization, 280 Bernoulli, velocity of efflux, 136 Berthelot, heat of chemical combination, 339 Berzelius, electro-chemical series, 328 Bidwell, view of Hall effect, 356 Bifilar suspension, 298, 358 Biot, law of action between magnet and electrical current, 342 Biot and Savart, action between magnet and electrical current, 341 Black's calorimeter, 194 Blagden, freezing-point of solutions, 314 Bodies, composition of, 84 ; forces determining structure of, 85 ; isotropic and eolotropic, 108 Body, 1; rigid, 37; displacement of rigid, 40 ; energy of rotating, 43 ; motion of free, 45 ; motion of rigid, in three dimensions, 50 Boiling. See Ebullition, 216 Boiling-point, 319 Bolometer, depends upon change of resistance with temperature, 319 ; used to study spectrum, 444 Boltzmann, distribution of energy in a gas, 336; specific inductive capacity of gases, 303 Borda, pendulum, 77; method of double weighing, 80 Bosscha, capillai-y phenomena in gases, 96 Boutigny, spheroidal state, 319 Boyle, law for gases, 118, 233; limitations of, 147; departures from, 231- a consequence of the kinetic theory of gases, 335 Bradley, determined velocity of light, 402 INDEX. 493 Breaking weight, 134 Brewster, law of polarization by reflection, 463 British Association bridge, 363 British Association, experiment to determine unit of resistance 373 Bunsen, calorimeter, 195 ; photometer, 447 Cagniabd-Latour, critical temperature, 330 Cailletet, condensation of gases, 331 Calorie, 193 ; lesser, 193 Calorimeter, Blacls's ice, 194 ; Bunsen's ice, 195 ; water, 195 ; thermocalorime- ter of Regnault, 197; water equivalent of, 196 Calorimetry, 194 ; method of fusion, 194 ; of mixtures, 195; of comparison 197; of cooling, 197 ' Camera obscura, 433 Capacity, electrical, 391; unit of, 393 ; of spherical condenser, 393 ; of freely electrified sphere, 394 ; of plate condenser, 395 ; of Leyden jar, 395 Capacity, specific inductive, 393 ; relation of, to index of refraction, 303, 379, 477, 478 ; relation of, to crystallographic axes, 303 Capillarity, facts of, 90 ; law of force treated in, 91; equation of, 95 ; in gases, 96 ; Plateau's experiments in, 97 Carlini, determination of Earth's density, 81 Carlisle, apparatus for electrolysis of water, 334 Carnot, cycle, 344 ; engine, 344 ; theorem, 347 Cathetometer, 6 Cathode, 333 Cauchy's formula for dispersion, 479 Caustic curve, 431 ; surface, 431 Cavendish, experiment to prove mass attraction, 73 ; determination of Earth's density, 81; on force in electrified conductor, 385; specific inductive capacity, 393 Centimetre, 4 Central forces, propositions connected with, 54-69 ; proportional to the radius vector, 55 ; proportional to the inverse square of the radius vector, 56 Centrobaric bodies, 73 Charge, unit, electrical, 388; energy of electrical, 301 Chemical affinity measured in terms of electromotive force, 328 Chemical combination, heat equivalent of, 339 ; energy of, 355 Chemical separation, energy required for, 339; gives rise to electromotive force, 337 Chladni's figures, 174 Christiansen, anomalous dispersion in fuchsin, 455 Circle, divided, 9 Circuit, electrical, direction of lines of force due to, 341; equivalenee of, to magnetic shell, 343, 349. See Current, electrical. Claris, standard cell, 336; its electromotive force, 336 494 INDEX. Clausius, kinetic theory of gases, 231; principle in thermodynamics, S46 ; theory of electrolysis, 330 Clement and Desormes, determination of ratio of specific heats of gases, 325 Coercive force, 259 Cohesion, 86 Collimating lens, 487, 444 Collision of bodies, 38 Colloids, 85; diffusion of, 104 Colors of bodies, 449 ; produced by a thin plate of doubly refracting crystal in polarized light, 467; by a thick plate, 469 Colors and figures produced by a thin plate of doubly refracting crystal in polarized light, 467, 469, 472, 473, 474 CompS,rator, 7 Component of vector, 12 Compressibility, 83 Compressing pump, 146 Concord in music, 166 Condenser, electrical, 292 ; spherical, 293 ; plane, 294 ; discharge of, 314 ; os- cillatory discharge of, 375 Conduction of electricity, 283 Conductivity for heat, 203 ; measurement of, 204 ; changes of, with tempera- ture, 205 ; of crystals, 205 ; of non-homogeneous solids, 205 ; of liquids, 205 Conductivity, molecular, 332 Conductivity, specific electrical, 319 ; in electrolyte dependent on ionic veloc- ities, 331 Conductors, good, 284 ; poor, 284; systems of, 295; opacity of, 477 Configuration, 11 Conical refraction, 473 Conservation of energy, 37 Contact, angle of, 96 Contiuuity, condition of, 134; for a liquid, 135 Convection of heat, 201 Cooling, Newton's law of, 453 Copernicus, heliocentric theory, 70 Cords, longitudinal vibrations of, 173; transverse vibrations of, 173 Cornu and Bailie, determination of Earth's density, 82 Coulomb, laws of torsion, 123; torsion balance, 122; law of magnetic force, 260; distribution of magnetism, 363; law of electrical force, 286 Coulomb, a unit of quantity of electricity, 288 Counter electromotive force, 320; general law of, 320; of decomposition, measure of, 327; of polarization, 335; of electric arc, 387 Couple, 44; moment of, 44 Critical angle of substance, 408 Critical temperature, 220, 221, 239 INDEX. 495 Crookes, the radiometer, 337; tubes, 390; explanation of phenomena in tubes 391 Cross, experiment on beats, 183 Crystal systems, 85 Crystalloids, 85; diffusion of, 104 Crystals, conductivity of, for heat, 305; specific inductive capacity of, 303; electrification of, by heat, 303; optic axis of, 457; principal plane of, 457; varying elasticity in, 461; varying velocity of light in, 463; effects of plates of, on polarized light, 466, 469, 471, 473, 473; uniaxial, 471; biaxial, 473; optic axes of biaxial, 473 Ctesibius, force pump, 139 Cumming, reversal of thermo-electric currents, 383 Current, electrical, 313; effects of, 313; represented by movement of tubes of force, 314; electrostatic unit of, 316; strength, 316; strength depeiuds on nature of circuit, 317; sustained by energy from dielectric, 331; set up by movement of a liquid surface, 339; magnetic field of, 341; direction of lines of force due to, 341, 346; electromagnetic unit of, 344; practical unit of, 345; energy of, in magnetic field, 345; energy of, in its own field, 346; mutual energy of two, 346; mutual action of two, 346, 348; motion of, in a magnetic field, 347; action of, on magnet pole, 349; Ampere's law for the mutual action of, 353; deflected in a conductor by a magnet, 356; due to inequalities of temperature, 356; measured in absolute units, 358; Kirchhoff's laws of, 360; alternating, 368 Current, extra, 368 Current, induced electrical, 363; quantity and strength of, 364; measured in terms of lines of force, 365; discovered by Faraday, 366; Lena's law of, 367; Faraday's experiments relating to, 367; of self-induction, 367 Cycle, 344; Carnot's, 344; illustrated in hot-air engines, 353 Dalton, law of vapor-pressure, 318 Daniell's cell, 335 Daik lines in solar spectrum, explanation of, 451 Davy, melting of ice by friction, 188; conception of heat as motion, 305; elec- trolysis of caustic potash, 334 Declination, magnetic, 368 Deformation, 108 De la Rive and de CandoUe, conductivity of wood, 305 Density, 31 Density, magnetic, 263; electrical, 287 Despretz and Dulong, measurement of animal heat by, 356 De Vries, osmotic pressure, 106 Dialysis, 105 Diamagnet, distinguished from paramagnet, 376, 278 Diamagnetism, 376; explanation of, by Faraday, 376; on AmpSres theory, 355; by Weber, 355 496 INDEX. Diaphragm, vibrations of, 176 Dielectric, 392; strain in, 301; energy in, 306; stress in, 310 Dielectric constant, 392. See Capacity, specific inductive. Diffraction of light, 432; at narrow apertures, 432; at narrow screens, 434; grating, 484 Diffusion, 108; of liquids, 103; coefficient of, 103; through porous bodies, 104; through membranes, 104; of gases, 107 Dilatability, 83 Dilatations, 110 Dimensional equation, 9 Dimensions of units, 9 Dip, magnetic, 268 Discord, in music, 166 Dispersion, normal, 408, 439; anomalous, 455 Dispersive power of substance, 440 Displacement, 11; composition and resolution of, 11 Dissociation, 339; heat equivalent of, 229 Dissociation, in solutions, 107; freezing-point of solutions, 314; vapor-pressure of solutions, 318; in electrolytes, 330; theory of electrolysis, 330 Distribution of electricity on conductors, 387 Dividing engine, 6 Divisibility, 83 Oouble refraction in Iceland spar, 457; explanation of, 458; by isotropic sub- tances when strained, 473 Draper, study of spectrum in relation to temperature, 454 Dulong and Petit, law connecting specific heat and atomic weight, 211; thi& law a consequence of the kinetic theory of gases, 237, 841 j formula for loss of heat by radiation, 453 Dntrocbet, definition of osmosis, 104 Dynamics, 10 Dynamo-machine, 371 Dyne, 25 Ear, tympanum of, 176 Earth, density of, 80 Ebullition, 216; process of, 318; causes affecting, 319 Eddy, mean energy of vibrating body, 240 Edlund, study of counter electromotive force of electric arc, 387 Efflux through narrow tubes, 89; of a liquid, 135; quantity of, 138 Elasticity, 88, 116; modulus and coefficient of, 116; perfect, 116; voluminal,- 117; voluminal, of gases, 118; of liquids, 118; of solids, 120; of traction, 120; of torsion, 131; of flexure, 133; limits of, 133 Elasticity of gases, 118, 223; at constant temperature, 223; when no heat enters or escapes, 333; ratio of these, 326; determined from velocity of sound, 227 Electric arc, 387; counter electromotive force of, 387 INDEX. 497 Electric discharge, in air, 387; iu rarefied gases, 389 Electric pressure, 291 Electrical convection of heat, 386 Electrical double-sheet, 337 Electrical endosmose, 340: shadow, 388 Electrical force. See Force, electrical. Electrical machine, 298; frictional, 298; induction, 299 Electricity, fundamental facts of, 283; unit quantity of, 288; flow of 289 314, 316 Electrification by friction, 283; positive and negative, 283, 285; by induction, 284; explanation of electrification by friction, 338 Electrified bod}', forces on, 308 Electro-chemical equivalent, 325 Electrode, 328 Electrodynamometer, 358 Electrolysis, 323; bodies capable of, 323; typical cases of, 324; influenced by secondary chemical reactions, 324; Faraday's laws of, 325; Grotthus's theory of, 329; dissociation theory of, 330; Clausius's view of, 330 Electrolyte, 323 Electromagnet, 355 Electromagnetic system of electrical units, basis of, 344 Electromagnetic waves, 376; similarity of, to light waves, 478 Electrometer, 296; absolute, 296; quadrant, 298; capillary, 339 Electromotive force, 317, 359; measured by difference of potential, 317; means of setting up, 321; measured in heat units, 328; a measure of chem- ical afilnity, 328; of polarization, 335; theories of, of voltaic cell, 387; electromagnetic unit of, 859; practical unit of, 359; due to motion in mag- netic field, 372; measured in terms of tubes of force, 372; depends on rate of motion, 372; at a heated junction, 380; required to force spark through air, 388 Electromotive force, counter. Bee Counter electromotive force-, 320 Electrophorus, 299 Electroscope, 296 Electrostatic system of electrical units, basis of, 288 Elements, chemical, 84, electro-positive and electro-negative, 328 Elongation, 109; how produced. 111 Emission of radiant energy, 451; relation of, to absorption, 453 Endosmometer. Dutrochet's, 105 Eadosmose, 104 Endosmose, electrical, 340 Energy, 29. 30; kinetic, 39; potential, 30; and work, equivalence of, 39; unit of, 31; conservation of, 37; of fusion, 215; of vaporization, 227; sources of terrestrial, 254; of sun, 257; dissipation of, 258; electrical, in dielectric, 306; expended in a circuit, 317 Engine, thermodynamic, 343; efficiency of heat, 344; Carnot, 244; reversible. 498 IKDEX. 246; efficiency of reversible, 247, 248; steam, 252; hot air, 253; gas, 252; Stirling, 253; Rider, 253 Eolotropic bodies, 108 Epoch of a simple harmonic motion, 22 Equatorial plane of a magnet, 260 Equilibrium, 28, 44; of free body, 45 Equipotential surface, 62 Erg, 31 Ether, 84; luminiferous, 396; interacts with molecules of bodies, 456; transmits electrical and magnetic disturbances, 395, 476; theories of, 396 Ettingshausen, view of Hall effect, 356 ; currents due to inequalities of tem- perature, 856 Evaporation, 216; process of , 216 Ewing, magnetic hysteresis, 279; limit of magnetization, 280; theory of mag- netization, 281 Exosmose, 104 Expansion, 111; of solids by heat, 206; linear, 307; voluminal, 307, 209; coeffi- cient of, 307; factor of, 207; measurement of coefficient of, 208, 209; of liquids by heat, 208; absolute, 208, 309; apparent, 208; of mercury, absolute, 308; apparent, 208; of water, 210; of gases by heat, 333; coefficient of, 222; heat absorbed and work done during, 225; work done by pressure during, 249 Extraordinary ray, 458; index, 458 Eye, 423; estimation of size and distance by, 424 Eye-lens or eye-piece, 426; negative or Huygens, 441; positive or Bamsden, 443 Earad, a unit of electrical capacity, 292 Faraday, magnetic induction in all bodies, 276 ; explanation of this, 376 ; experiment in electrical induction, 884; on force in electrified body, 285; specific inductive capacity, 392; theory of electrification, 302, 811; theory illustrated, 302; explanation of residual charge, 303; discharge of jar can produce effects of current, 813; nomenclature of electrolysis, 328; laws of electrolysis, 835; voltameter, 326; divisionof ions, 328; theory of electrolysis, 339; chemical theory of electromotive force, 337; electromagnetic rotations, 344; induced currents, 366; effect of medium on luminous discharge, 389; electromagnetic rotation of plane of polarization, 475 Favre and Silbermann, heat of chemical combination, 229; connection of electromotive force and heat units, 328 Fedderseu, oscillatory discharge, 376 Ferromagnet. See Paramagnet, 376 Field of force, 61; strength of, 61 Filament, in a fluid, 135 Films, studied by Plateau, 98; interference of light froji, 430 Fitzgerald, voi tex ether, 396 ISTDEX. 499 Jizeau, introduced condenser in connection with induction coil, 371; deter- mined velocity of light, 403 Plexure, elasticity of, 133 Floating bodies, 131 Flow of heat, 303; across a wall, 303; proportional to rate of fall of tempera- ture, 302; along a bar, 303 Fluid, body immersed in, 131; body floating on, 131 Fluids, distinction between solids and, 134; mobile, viscous, 134; perfect, 135 Fluids, motions of. See Motions of a fluid, 134 Fluorescence, 454 Focal line, 431 Focus, of spherical mirror, 411; real, conjugate, 411; principal, 411; virtual, 411 Forbes, measurement of conductivity, 305 Force, 34 ; uuit of, 35 ; centrifugal, 39 ; conservative, 30 ; internal, 35, 86 ; external, 36; moment of, 43; field of, 61; defined by potential, 61; line of, 63; tube of, 63; flux of, 65; near a plane sheet, 68; within spherical shell, 68; outside a spherical shell, 69 Force, capillary, law of, 91 Force, electrical, in charged conductor, 285 ; law of, 286 ; screen from, 389 ; just outside an electrified conductor, 391; tubes of, 303; unit or Faraday tube of, 304 ; representation of, by tubes of force, 304 ; on bodies in the electrical field, 308 Force, magnetic, law of, 360; due to bar magnet, 265; unit tube of, 370; within a magnet, 371; lines of, 373; between magnet and long straight current, 341; between magnet and current element, 343, 349; due to circular cur- rent, 351 Forces, composition and resolution of, 28; resultant of parallel, 47; central, 54 Forces, determining structure of bodies, 85; molecular, 85, 108; of cohesion, 86; of adhesion, 86 Foucault, pendulum, 52; velocity of light, 403 Fourier, theorem, 34 Franklin, complete discharge of electrified body, 285; experiment with Leyden jar, 303; identity of lightning and electrical discharge, 389 Fraunhofer, lines in solar spectrum, 443 Freezing- point, change of, with pressure, etc., 313; of solutions, 314 Fresnel, assumption of transverse vibrations, 395 ; elastic solid ether, 396 ; interference of light from two similar sources, 439; rhomb, 471; explana- tion of rotation of plane of polarization by quartz, 474 Fresnel and Arago, interference of polarized light, 460, 463 Friction, 88; laws of, 88; coefiicient of, 89; of solid in fluid, 89; theory of, 90 Fusion, 213; heat equivalent of, 314; energy necessary for, 315; determination of heat equivalent of, 315 Galileo, relation of force and mass, 26; path of projectiles, 59; the heliocentric theory, 70; measurement of gravity, 73; weight of atmosphere, 139 500 INDEX. Galvani, physiological effects of electrical current, 31S Galvanometer, 357; Schweigger's multiplier, 357; tangent, 357 Gas, definition of, 216; perfect, 249 Gases, 84, 330; absorption of, 102; diffusion of, 107; elasticity of, 118; lique. faction of, by pressure, 147, 230; departure of, from Boyle's law, 321; pressure of saturated, 231; coefficient of expansion of, 233; formula con- necting pressure, volume, and temperature of, 222 ; elasticities of, 233 ; specific heats of, 224; van der Waal's theory of, 237; spectra of, 450 Gases, kinetic theory of. See Kinetic theory of gases, 231 Gauss, theory of capillarity, 92; proof of law of magnetic force, 260 Gay-Lussac, law of expansion of gases by heat, 222 Geissler tubes, 390 Gilbert, showed Earth to be a magnet, 268 Graham osmometer, 105; method of 'dialysis, 105 Gram, 8 Gram-degree, 193 Grating, diffraction, 434; element of, 485; pure spectrum produced by, 436; with irregular openings, 436; wave lengths measured by, 437; Rowland's curved, 438 Gravitation, attraction of, 70 Gravity, centre of, 73 Gravity, measurement of, 73; value of, 73 Griffiths, mechanical equivalent of heat, 193 ' Grotthtis, theory of electrolysis, 339 Grove, gas batteiy, 334 Gyration, radius of, 42 Gyroscope, 53 Hall, deflection of a current in a conductor, 356 Halley, theory of gravitation, 71 Hamilton, prediction of conical refraction, 473 Harmonic tones of pipe, 173 Harris, absolute electrometer, 396 Heat, effects of, 186; production of, 187; nature of, 187; a form of energy, 188; unit of, 193; mechanical uuit of, 193; mechanical equivalent of, 198; Joule's determination of, 198; Rowland's, 199; Hiin's 300; transfer of, 201; con- vection of, 201; conduction of, 202; internal, of Earth, a source of energy, 257; developed by the electrical current, 313, 319; generated by absorption of radiant energy, 445 Heat, atomic, 311 Heat, kinetic theory of, 206, 339 Helmholtz, vortices, 143; resonators, 178; vowel sounds, 179; theory of beats 181; theory of solar energy, 358; law of counter-electromotive force, 320- electromotive force of cell, 328; electrical double-sheet, 338; modiflc'ation INDEX. 501 of surface tension by electrical currents, 339; explanation of electrical endoBmose, 340; interaction of ether and molecules of bodies, 456 Henry, oscillatory discharge, 376 Herschel, study of spectrum, 450 Hertz, relation of index of refraction and specific inductive capacity, 303; experiments on electromagnetic waves 376, 478; passage of cathode dis- charge through aluminium, 391 Him, mechanical equivalent of heat, 300; work done by animals, 356 Hittorf, migration of ions, 833 Holtz, electrical machine, 301 Hooke, theory of gravitation, 71; law of elasticity, 116, 130 Hopkinson, relation between index of refraction and specific inductive capac- ity, 477 Horizontal intensity of Earth's magnetism, 368; measurement of, by standard magnet, 368; absolute, 369 Hot-air engine, 353 Huygens, theorems of, on motion in a circle, 71; views of, respecting gravita- tion, 71; principle of wave propagation, 151; theory of light, 394; eye- piece, 441 Hydrometer, 183 Hydrostatic balance, 133 Hydrostatic press, 136 Hydrostatic stress, 115 Hysteresis, magnetic, 379 Ice, melting-point of, used as standard, 313; density of, 313 Iceland spar, 457; wave surface in, 458 Images, formed by small apertures, 401; virtual, 408; by mirrors, 416; by lenses, 417; geometrical construction of, 418 Impact, 38 Impenetrability, 4 Impulse, 26 Inclined plane, 48, 49 Induced magnetization, coeflBcient of, 373 Induction coil, 871; condenser connected with, 371 Induction, electrical, 284, 390 Induction, magnetic, 259, 373, 373; tubes of, 372 Induction of cuiTcnts, 363 Inductive capacity, magnetic, 273 Inertia, 4, 35; moment of, 43 Instantaneous axis, 50 Insulator, electrical, 284; transparency of, 477 ': Intensity in a field of force, 61 '\ Interference of light, cause of propagation in straight lines, 396; from two 503 INDEX. similar sources, 427; experimental realization of, 429; from thin films, 430 Internode, 157 Intervals, 167 Ionic weight, 326; charge, 326; velocities, 331 Ions, 334, 335; electro-positive and electro-negative, 328; arrangement of, by Faraday, 328; by Berzelius, 328; migration of, 829 Isothermal line, 223 Isotropic bodies, 108 Jolly, determination of Earth's density, 82 Joule, equivalence of heat and energy, 188; mechanical equivalent of heat, 198; expansion of gas without work, 225, 283; limit of magnetization, 280; law of heat developed by electrical current, 319; electromotive force in heat uuits, 328; development of heat in electrolysis, 829 Joule and Thomson, expansion of gas without work, 325 Jurin, law of capillary action, 99 Kater, pendulum, 77 Kepler, laws of planetary motion, 70 Kerr, optical effect of strain in dielectric, 302; rotation of plane of polarization by reflection from magnet, 476 Ketteler, interaction of ether and molecules of bodies, 456 Kinematics, 10 Kinetics, 10 Kinetic theory of heat, 206, 229; explanation by it of properties of bodies, 239- Kinetic theory of gases, 231, 232 KirchhofE, laws of electrical currents, 360; spectrum analysis, 450 Kohlrausch, measurement of ionic velocities, 331 KSnig, A., modification of surface tension by electrical currents, 339 Konig, R., manometric capsule, 149; pitch of tuning-forks made by, 169; boxes of his tuning-forks, 175; quality as dependent on change of phase, 178; investigation of beats, 182 Kopp, atomic heat, 211 Kundt, experiment to measure velocity of sound, 162; anomalous dispersion,. 455 Lang, counter electromotive force of electric arc, 387 Langley, bolometer, 319; wave lengths in lunar radiations, 438, 446 Laplace, theory of capillarity, 92; equation of capillarity, 96 Lavoisier, measurement of animal heat, 355 Least time, principle of, 400 Lenard, the cathode discharge, 391 Length, unit of, 4; measurements of, 4 Xenses, 413; formula for, 414; forms of, 414; focal length of, 415; images formed by, 417; optical centre of, 417; thick, 419; of large aperture, 419; aplanatic combinations of, 432; achromatic combinations of, 440 INDEX. 503 Lenz, law of induced currents, 367 Le Roux, experiments in thermo-electricity, 382; electrical convection of heat in lead, 386 j Lever, 47, 49 ' Leyden jar, capacity of, 295; dissected, 303; volume changes in, 303; residual charge of, 303 Light, agent of vision, 394; theories of, 394; propagated in straight lines, 396; principle of least time, 400; reilection of, 404; refraction of, 406; lay of, beam of, pencil of, 418; characteristics of common, 464 Light, velocity of, determined from eclipses of Jupiter's satellites, 403; from aberration of fixed stars, 402; by Fizeau, 403; by Foucault, 403; by Michelson, 404 Light, electromagnetic theory of, 476 Lightning, an electrical discharge, 389 Lines of magnetic force, positive direction of, 348 Lippmann, electrical effects on capillary surface, 338; capillary electrometer, 389; production of current by modification of capillary surface, 339 Liquefaction, 330; of gases, by pressure, 330 Liquids, 84, 330; modulus of elasticity of, 118 Lissajous, optical method of compounding vibrations, 180 Lorentz, shrinkage of molecules with rise of temperature, 340 Loudness of sound, 164 Machinb, 48; efficiency of, 49; electrical, 398; dynamo- and magneto-, 370 Magnet, natural, 359; bar, relations of, 363; couple between two-bar, 265 Magnetic elements of Earth, 268 Magnetic force. See Force, magnetic, 360 Magnetic induction, 359; axis, 360, 263; pole, 260, 363; moment, 261; den- sity, 262; field, 270; inductive capacity, 373; permeability, 373; field, energy in, 273 Magnetic shell, 266; strength of, 266; potential due to, 267; equivalence of, to closed current, 343 Magnetic system of units, basis of, 361 Magnetism, fundamental facts of, 359; distribution of, in magnet, 263; deter- mination of, 263; theories of, 280; Ampere's theory of, 355; theory of, described by tubes of force, 355 Magnetization, intensity of, 263; changes in, 378 Magneto-machine, 370 Magnifying-glass, 425 Magnifying-power, 435 Manometer, 146 Manometric capsule, 149 Mariotte, study of expansion of gases, 118 Maskelyne, determination of Earth's density, 81 3, 35; unit of, 8; centre of, 32 504 INDEX. Masses, comparison of, 8 Matter, 1, 3; constitution of, 83; kinetic theory of, 206, 229; states of, 229 Matthiesseu, expansion of water, 209 Mayer, views concerning work done by animals, 256 Maxwell, coefficient of viscosity of a gas, 90; kinetic tbeoiy of gases, 231; law of distribution of energy in gases, 235; molecular constants, 242; relation between specific inductive capacity and index of refraction, 302, 477; ex- planation of vesidual charge, 303; theory of electrificatiou, 311; relation of current and lines of force, 341; suggested test of Weber's theory of dia- magnetism, 355; measurement of v, 374; force on magnet due to moving electrical charge, 375; electromagnetic theory of light, 476 Mechanics, 10 Mechanical powers, 46 Melloni, use of thermopile, 381 Melting-point of ice, 212; of alloys, 212; change of, with pressure, 313 Mercury, expansion of, by heat, 209 Metaceutre, 131 Micbelson, velocity of light, 404 Michelson and Morley, difference of path of interfering light, 464 Microfarad, 292 Micrometer screw, 5 Microscope, simple, 425; compound, 426 Migration of ions, 329; constant, 833 Mirrors, plane, 408; spherical, 409; images formed by, 416; of large aperture, 419 Modulus of elasticity. See Elasticity, 116 Mohr, kinetic theory of heat, 205 Molecular forces, 86; action, range of, 91, 242; motion, 229. See Kinetic theory. Molecule, 83; structure of, 86; kinetic energy of, proportional to temperature, 235; mean velocity of, 236; dimensions of, 241 Moment, of force, 44; of couple, 44; principle of, 44 Moment of inertia, 42; experimentally determined, 42 Moment of torsion, 122; determination of, 122 Moment, magnetic, 261; changes in, 278; depends on temperature, 279; on mechanical disturbance, 280 Momentum, 26; conservation of, 35 Motion, 11; description of, 14; linear, with constant acceleration, 18; angular, with constant angular acceleration, 19; in a circle, 17, 23; simple har- monic, 20; Newton's laws of, 26; constrained, 28: in an ellipse, 57 Motions, composition and resolution of simple harmonic, 23; of a fluid, 134; optical method of compounding, 180. See Displacement, 11 MuUer, J., limit of magnetization, 280 Mutual induction, coefficient of, 347 INDEX. 505 ITbknst, migration of Ions, 333 Neumann, atomic heat, 211 Hewton, laws of motion, 26; law of mass attraction, 71; description of atom, 86; quantitj' of liquid flowing through orifice, 138; theory of light, 400; interference of light from films, 431; composition of white light, 439; chromatic aberration, 440; law of cooling, 453 Kichols, study of radiations, 454 Nicliolsou aud Carlisle, decomposition of water by electrical current, 313 Nicol prism, 465 Node, 157 Noise, 164 Non-inductive coil, 368 Objbctite, 426 Ocean cuireuts, energy of, 255 Oersted, piezometer, 119; relation between magnetism and electricity, 313 Ohm, law of electrical current, 318 Ohm, a unit of electrical resistance, 360; various values of, 360; determination of, 372 Olszewski, condensation of gases, 221 ; low temperatures obtained by, 228 Optic angle, 424; axis of crystal, 457, 471, 472 Optics, 394 Ordinary ray, 458; index, 458 Organ pipe, 170; fundamental of, 172; harmonics of, 172; mouthpiece of, 172; reeds used with, 172 Oscillation, axis of, 77 Oscillatory discharge of condenser, 375 Osmometer, Graham's, 105 Osmosis, 104, 105 Osmotic pressure, 106; laws of, 106 Ostwald, theory of electrolysis, 330 Overtones, of pipe, 1 72 Pakallelogram, of vectors, 12; of forces, 28 Paramagnet, 276 Particle, 27, 42 Pascal, pressure in fluid, 125; pressure modified by gravity, 126; barometer, 129 Peltier, heating of junctions by passage of electrical current, 313 ; eflfect, 313, 380 Pencil of light, 418 Pendulum, Foucault's, 52; simple, 74; formula for, 75; physical, 75; Borda's, 77; Kater's or reversible, 77 Penumbra, 401 Percussion, centre of, 45 Period, of a simple harmonic motion, 20 506 INDEX. Permeability, magnetic, 273 Pfeffer. study of osmosis, 105; of osmotic pressure, 105; laws of osmotic pressure, 106 Phase, of a simple harmonic motion, 30 Phonograph, 176 Phosphorescence, 454 Photometer, Bunsen's, 447 Photometry, 447 Pictet, condensation of gases, 221 Piezometer, Oersted's, 119; Regnault's, 119 Pitch of tones, 164; methods of determining, 164; standard, 169 Plante, secondary cell of, 336 Plateau, experiments of, in capillarity, 97 Plates, rise of liquid between, 100; transverse vibrations of, 174 Poisseiillle, friction in liquids, 89 Poisson, correction for use of piezometer, 119; theory of magnetism, 280 Polariscope, 465 Polarization of cells, 335 Polarization of light, by double refraction, 459; by reflection, 463; plane of, 463; by refraction, 463; by reflection from fine particles, 463; elliptic and circular, 469; circular by reflection, 471; rotation of plane of, by quartz, 473; by liquids, 475; in magnetic field, 475 Polarized light, 460; explanation of, 460; interference of, 463; effects of plates of doubly refracting crystals on, 466, 469, 471. 473, 473 Polarizer, 464 Polarizing angle, 463 Pole, magnetic, 260, 263; unit magnetic, 361 Poles, of a voltaic cell, 334 Polygon of vectors, 13 Porous body, 103 Potential, difference of, 61; its relation to force, 61; in a field of force varying inversely with the square of the distance, 63 Potential, electrical, in a closed conductor, 286, 289; of a conductor, 289; zero, positive, and negative, 289; of a system of conductors, 295; difference of, measured, 297; contact difference of, 312 Potential, magnetic, due to bar magnet, 263; due to magnetic shell, 267; of a closed circuit is multiply- valued, 343; illustrated by Faraday, 344 Poynting, theorem, 321 Pressure, 113; in a fluid, 125; modified by outside forces, 136; surfaces of equal, 126 ; on surface of separation, 127 ; proportional to depth, 138 ; diminished on walls containing moving liquid column, 139 Prevost, law of exchanges, 451 Principal plane of crystal, 457 Prism, 407 Problem of two bodies, 58 INDEX. 507 Projectiles, 59 Properties of matter, 4 Pulley, 47, 49 Pump, 137; air, 143; compressing, 146 Quality of tones, 164, 177; dependent upon harmonic tones, 178; upon change of phase, 178 Quarter wave plates, 471 Quartz, effects of plates of, in polarized light, 473; imitation of, 475 Quincke, range of molecular action, 342; change in volume of dielectric, 303; electrical endosmose, 340; movements of electrolyte, 340 Radian, 9 Eadiunt energy, effects of, 445; transmission and absorption of, 448; emission of, 451 ; origin of, 453 Radiation, 205 ; intensity of, as dependent on distance, 446 ; on angle of in- cidence, 446 ; kind of, as dependent on temperature, 453 Radicals, chemical, 84 Radiometer, 237 Rainbow, 442; secondary, 443 Ramsden, eye-piece, 442 Rankine, theoretical velocity of sound, 159 Raoult, freezing-points of solutions, 214; vapor pressure of solutions, 218 Ratio between electrostatic and electromagnetic units, 873 ; a velocity, 373 j physical significance of, 375; equal to velocity of light, 477 Ray of light, 418 Rayleigh, electromotive force of Clark's cell, 386 Reeds, in organ pipes, 173; lips used as, 172; vocal chords as, 173 Reflection, of waves, 157; law of, 158; of light, law of, 405; total, 408; at spherical surfaces, 409; of spherical waves, 419; selective, 449; polarization of light by, 468 Refraction of light, law of, 406; index of, 407; dependent on wave length, 408; at spherical surfaces, 413; polarization of light by, 463; conical, 473 Regelation, 213 Regnault, piezometer, 119; specific heat of water, 193; thermocalorimeter, 197; expansion of mercury, 208; extension of Dulong and Petit's law, 211; modification of Dalton's law, 318; pressure of water vapor, 331; modifica- tion of Gay-Lussac's law, 332; total heat of steam, 328 Reinhold and Riicker, range of molecular action, 243 Residual charge, 303 Resistance, electrical, 318, 359; depends on circuit, 318; of homogeneous cyl- inder, 319; specific, 319; varies with temperature, 319; electromagnetic unit of, 360; practical unit of, 360; boxes, 360; measurement of, 362; of a divided circuit, 363; determination of unit of, 373 Resonance, 174 508 INDEX. Resonator, 178 Restilution, coeflBcient of, 39 Resultant of vectors, 11 Reusch, artificial quartzes, 475 Reuss, electrical eudosmose, 340 Reynolds, laws of diffusion of gases, 108 Rlieostat, 360 Rider, hot-air engine, 253 Righi, transmission of electromagnetic waves through wood, 378 Rigidity, 117; modulus of, 117, 123 Rods, longitudinal vibrations of, 173; transverse vibrations of, 174 Roemer, determination of velocity of light, 402 Rontgen, the Rontgeu radiance, 392; theories of, 392 Rotation, 40; of body about a fixed point, 43 Rotation of plane of polarization by quartz, 473; right-handed and left-handed, 475; by liquids, 475; In magnetic field, 475; explanation of, 476; by reflec- tion from magnet, 476 Rotational coefiicient. Hall's, 356 Rowland, specific heat of water, 193; mechanical equivalent of heat, 199; force on magnet due to moving electrical charge, 375 ; measurement of ®, 375: photographs of solar spectrum, 438; curved grating, 438 Ruhmkorff's coil, 371 Rumford, relation of heat and energy, 187; conception of heat as motion, 206; views concerning work done by animals, 256 Saccharimetbr, 475 Sarasin and de la Rive, velocity of electromagnetic waves, 376 Saturation of a magnet, 278, 281 Savart, toothed wheel, 165 Scales, musical, 167; ^transposition of, 168; tempered, 169 Solicinbein, chemical theory of electromotive force, 337 Scliweigger, multiplier, 357 Screens, diffraction effects at, 434 Screw, 48 Second, 8 Seebeck, thermo-electric currents, 380; thermo-electric series, 381 Self-induction, coefficient of, 346; current of, 367 Set, 123 Shadows, optical, 401 Shear, 110, 117; amount of, 110, 117 Shearing strain, 110; stress, 114 Shunt circuit, 363 Siphon, 136 Siren, determination of number of vibrations by, 165 Smee's cell, 335 INDEX. 509 Snell, law of refraction, 407 Solenoid, 354 Solidification, 312 Solids, 84, 230; structure of, 85; crystalline, amorphous, 85; movements of, due to capillarity, 101; distinction between fluids and, 124; soft, hard, 124 Solubility, 102 Solution, 102; equimolecular, 106; indifferent, 106; isotonic, 106 Soand, 149; origin of, 149; propagation of, 150; theoretical velocity of, 159; velocity of, in air, 161; measurements, 162 Sounding boards, 176 Specific gravity, 131; determination of, for solids, 132; for liquids, 132; for gases, 183 Specific gravity bottle, 132 Specific heat, 193; mean, 194; varies with temperature, 210; with change of state, 210 Specific heat of gases, 224; at constant volume, 224; at constant pressure, 224; determination of, at constant pressure, 224; ratio of these, 225; relation to elasticities, 226 Specific inductive capacity. See Capacity, specific inductive, 292 Spectrometer, 437; method of using, 437 Spectroscope, 443 Spectrum, pure, 436; produced by difEraction grating, 436; of first order, etc., 486; formed by prism, 489; solar, 443; dark lines in, 443; study of, 445; of solids and liquids, 445; of gases, 450; explanation of, of a gas, 452; characteristics of, 454; of gases which undergo dissociation, 454 Spectrum analysis, 449 Spheroidal state, 219 Spherometer, 7 Spottiswoode and Moulton, electrical discharge in high vacua. 391 Sprengel, air-pump, 145; theory of, 139 Statics, 10 Steam, total heat of, 228 Steam-engine, 252 Stirling's hot-air engine, 253 Stokes, study of fluorescence, 455 Storage cells, 386 Strain, 108, 109; homogeneous, 109; principal axis of, 109; superposition of, 111 Stress, 28; in medium, 108, 111; superposition of, 114; hydrostatic, 115 Substances, simple, compound, 84 Sun, energy of, 257 Surface density, 67; of magnetism, 262; of electrification, 287 Surface energy of liquids, 94 Surface tension of liquids, 92; relations to surface energy, 94; modified by electrical effects, 839 510 - IKDEX. Sutherland, shrinkage of molecules with rise of temperature, 240 System of points, 11 Tait, experiments in thermo-electricity, 382; thermo-electric formula, 385 Telephonic tiausmitters and receivers, 370 Telescope, 425, 426; magnifying power of, 426 Temperament of musical scale, 169 Temperature, 189; scales of, 190; change of, during solidification, 213; critical, 220, 221, 239; absolute zero of, 223, 235, 248; absolute, 223, 247, 249; kinetic measure of, 232; absolute, relation of, to temperature of air-ther- mometer, 249; movable equilibrium of, 451; radiation of heat dependent on, 453 Tension, 112 Thermodynamics, first law of, 243; second law of, 246 Thermo-electric currents, 380; how produced, 881; reversal of , 882 Thermo-electric diagram, 382 Thermo-electric element, 381 Thermo electric power, 382 Thermo-electrically positive and negative, 381 Thermometer, 189; construction of, 189; air, 191, 223; limits in range of, 192; weight. 209; zero of air, 223 Thermopile, 381 Thomson, J. J., tubes of force, 304; theory of electrical field, 311; currents due to inequalities of temperature, 357; explanation . of discharge in rarefied gases, 391 , Thomson, Sir Wm. (Lord Kelvin), theory of vortex atom, 87; vortices, 142; absolute scale of temperature, 248; theory of solar energy, 258; absolute electrometer, 297; quadrant electrometer, 298; law of counter electromo- tive force, 820; contact theory of electromotive force, 337; measurement of v, 374; oscillatory discharge, 376; thermo-electric c\irrents in non- homogeneous circuits, 382; thermo-electric power a function of temper- ature, 384; the Thomson effect, 386; electromotive force required to force spark through air, 388; gyroscopic model of ether, 396; estimates of rigidity and density of ether, 396 Thomson effect, 886 Tides, energy of, 257 Time, 3; unit of, 8; measurement of, 8 Tones, musical, 164; difEerences in, 164; determination of number of vibra- tions in, 164; whole and semi-, 168; fundamental, 172; analysis of com- plex, 178; resultant, 183 Tonic, 168 Torricelli, barometer, 129; experiment of, 129; theorem for velocity of efSux, 136; experiments to prove, 138 Torsion, amount of, 121; moment of, 122 Torsion balance, 122, 286 INDEX. 511 Tourmaline, 464 Traction, longitudinal, 114; elasticity of, 120 , Translation, 40 Transmissiou of radiations, 448 Triad, major, 167; miaor, 167 Troutou, polarization of electromagnetic waves, 379 Trowbridge, changes in intensity of magnetization, 379 Tubes, rise of liquid in capillary, 99 Tubes of force, 63; relation of force to cross-section of, 67 Tuning-fork, 174; sounding-box of, 175 Tyndall, conductivity of wood, 205 Umbra, 401 Units, fundamental and derived, 4; dimensions of, 9; systems of, 9 Vacuum tube, electrical discharge in, 389 Yan der Waals, theory of a gas, 237; pressures in gases, 242 Tapor, 216; saturated, 217; pressure of, 217; pressure of, over solutions, 218; production of, in limited space, 220; departure of, from Boyle's law, 221; pressure of saturated, 221 Taporization, 216; energy necessary for, 227; heat equivalent of, 228 Vector, 12; composition and resolution of, 13 Velocity, 15; angular, 18; composition and resolution of, 16 Velocity of efflux of a liquid, 135; into a vacuum, 137 Velocity, mean, of molecules of gas, 234, 241 Velocities, composition and resolution of, 16; of angular, 51 Vena contracta, 138 Ventral segment, 157 Verdet, electromagnetic rotation of plane of polarization, 475 Vernier, 4 Vertex of spherical mirror, 409 Vibratiojs of sounding bodies, 170; modes of exciting, in tubes, 172; longi- tudinal, of rods, 173; of cords, 173; transverse, of cords, 173; of rods, 174; of plates, 174; communication of, 174; of a membrane, 176; optical method of studying, 180 Vibrations, light, transverse to ray, 460, 463; relation to plane of polarization, 461; elliptical and circular, 469 Viscosity, 88; of fluids, 89; of gases, 90; of solids, 124 Vision, ancient theory of, 894; Aristotle's view of, 394 Visual angle, 424 Vocal chords, 173 Volt, a unit of electromotive force, 359 Volta, change in volume of Leyden jar, 302; electrophorus, 299; contact differ- ence of potential, 312; voltaic battery, 313; heating by current, 813; con- tact theory of electromotive force, 337 512 IKDEX. Voltaic cells, 334; polarization of, 335; theories of electromotive force of, 337; arrangements of, 363 Voltaic cells : Grove's gas battery, 334; Smee's, 335; Daniell's, 335; Plante's secondary, 386; Clark's, 336 Voltameter, 336; weight, 337; volume, 337 Volume, change of, with change of state, 313 Vortex, in perfect fluid, 143; line, 143; filament, 142; properties of a, 142; strength of, 143; illustrations of, 148 Vortex atom, theory of, 87 Vowel sounds, dependent on quality, 179 Water, specific heat of, 198; maximum density of, 301, 210; expansion of, by heat, 310; on solidification, 213 Water-power, energy of, 355 Wave, on surface of liquid, 140; studied by H. and W. Weber, 140 Wave, sound, 151; mode of propagation of, 151; graphic representation of, 153; displacement in, 153; velocity of vibration in, 154; stationary, 156; reflec- tion of, 157; in sounding bodies, 170 Wave, light, surface of, 396; relation of, to the direction of propagation, 399; emergent from prism, 407, 408; measurement of length of, 430, 437; values of lengths of, 438; surface of, in uniaxial crystals, 458; in biaxial crystals, 472 Weber, theory of magnetism, 280; equivalence of circuit and small magnet, 342; theory of diamagnetism, 355; electrodynamometer, 358. Weber, H. aud W., study of waves, 140 Weber and Kohlrausch, measurement of v, 373 Wedge, 48 Weighing, methods of, 80 Weight of a body, 73 Wheatstone's bridge, 361 Wheel and axle, 48 Wiedemann, electrical endosmose, 340 Wilcke's calorimeter, 194 Wind-power, energy of, 255 Wollaston, dark Hues in solar spectrum, 443 Work, 39; and energy, equivalence of, 39; unit of, 31; principle of, 42 Wren; theory of gravitation, 71 Wright, connection of electromotive force and heat of chemical combination. 328 Young, theory of capillarity, 92; optical method of studying vibrations, 180; interference of light from two similar sources, 429 #> |i«sji;i;^«::yj m.u.:.