LIBRARY UNIVERSITY OF CALIFORNIA. Class A TEXT BOOK OF THE PRINCIPLES OF PHYSICS A TEXT BOOK OF THE PBINCIPLES OF PHYSICS BY ALFKED DANIELL, M.A. LECTURER ON PHYSICS IN THE SCHOOL OF MEDICINE, EDINBURGH MACMILLAN AND CO. 1884 vi PREFACE. will find that, after assimilating its contents, he is in some measure prepared for the reception of further information in the course of that wider reading and practical study to which I hope the following pages will be found fitted to serve as an elementary introduction. It is wholly beyond question that to him who desires to become a physicist, Practical Laboratory Work is absolutely essential. Thorough knowledge must be drunk in by the eyes and the ears, and absorbed by the finger- tips ; and the true use of a book of this kind is, I take it, not to replace practical work but to economise the labours of the student. This it may do by so furnishing his mind with a store of general principles, that when he comes to enter a physical laboratory he may there find around him, in the concrete form, a collection of pieces of apparatus, the construction and the action of which he is able, by the application of the principles already familiar to him, promptly and intelligently to comprehend. Bearing the necessary limitations of the usefulness of any mere book steadily before me, I have endeavoured, as far as possible, to simplify and generalise all descriptions of apparatus, and in the same way to simplify and generalise the accompanying diagrams ; and thus I have tried not only to adapt the work to the requirements of those who may use it as a stepping-stone to further attain- ments, but also to render it a suitable text-book for that larger circle of readers who, having no distinct desire to follow out the special study of physics, may yet wish to possess an elementary acquaintance with the modern aspect of natural philosophy. PREFACE. vii This book was primarily designed as a contribution Jo Medical Education, and as such I hope it may be found useful. That arrangement, which still prevails in some of our Universities, under which a student of medicine may even proceed to the degree of M.D. with- out any adequate knowledge of physics, is self -evidently opposed to common sense, and to the exigencies of physiological study and of medical practice. Such an anomaly cannot, it may be anticipated, endure much longer. Before many years are over it will be univer- sally acknowledged in practice, as it already is in theory, that knowledge of natural philosophy is an essential part of the mental equipment of the medical student and of the properly-trained medical man. The needs of the intelligent student of physiology have been kept con- stantly before my mind, as I hope those of my readers who are already physiologists will recognise ; but I have been careful not to make the book one suited for admis- sion only into a medical class-room; my aim has been to produce a work useful at once to the Student of Medicine, the Student of Science, and the General Eeader. The plan of the work is that of a gradual progression from the simplest to the more complex. No preliminary knowledge of physical principles is assumed, and every effort has been made to attain to absolute lucidity of expression, even though this be found occasionally to neces- sitate the frequent repetition of a single word in the course of a single sentence. While the reader is expected as he reads each page to remember the contents of the preced- ing pages, I trust that I have sufficiently carried out my viii PREFACE. intention of nowhere setting before him anything of the nature of an unsolved riddle, so far as that could be guarded against by my own efforts on his behalf. I have endeavoured to secure intelligible continuity throughout the paragraphs printed in larger type, and thereby to enable the reader on his first perusal to confine his attention to the more prominent portions of the text. However imperfectly its design may have been exe- cuted, I shall be glad if this work be found to contribute in any degree to the extension of that mode of teaching Natural Philosophy for the establishment of which we have come, directly and indirectly, to owe so much to the advocacy and example of Professors Thomson and Tait a mode of teaching under which the whole of Natural Philosophy is regarded as substantially a single science, in which scattered facts are connected and co-ordinated by reference to the principles of Dynamics and the great experimental Law of the Conservation of Energy. CONTENTS. PAGE INTRODUCTORY 1 CHAPTER I. TIME, SPACE, AND MASS. Standards of Time . . . . . . . .9 Standards and Dimensions of Space ...... 9 Standards of Mass 1-2 CHAPTER II. NOTIONS DERIVED FROM THE PRECEDING. Motion .......... 14 Velocity .......... 14 Acceleration . . . . . . . . .16 Momentum ......... 17 Force . . ' , . . . . . . . .17 Weight .... . . 19 CHAPTER III. MEASUREMENTS. Measurement of Length Line Measurement, p. 21 ; End Measurement . 24 Surface ........ 26 Volume . . . . . . .27 Time ........ 28 Mass . . . . . . 29 Force . . 29 CHAPTER IV. WORK AND ENERGY. .......... 34 Ener gy ... . . . . . . .35 Potential Energy ..... .36 CONTENTS. PAGE Kinetic Energy . . . . . . . .39 Conservation of Energy . . .40 Transformations of Energy ....... 41 Availability of Energy . . . . . . .44 Graphic representation of Energy .... 47 The Indicator Diagram . . . . . .48 CHAPTER V. KINEMATICS. GENERAL PROPOSITIONS . . . . . . .51 Direction, p. 51 ; Velocity, p. 52 ; Dimensions, p. 53 ; Simultaneous Motions, p. 54 ; Parallelogram of Velocities, p. 55 ; Triangle of Velo- cities, p. 56 ; Resolution of Velocity into Components, p. 57 ; Com- position of more than two velocities, p. 59 ; The Polygon of Velocities, p. 60 ; Reference to Axes, p. 60 ; Velocities not in one plane, p. 61 ; Change of Velocity ....... 62 Accelerated Motion, p. 63 ; Composition of uniform with Accelerated Motion ......... 65 Degrees of Freedom of a Particle, p. 66 ; Translation, p. 67 ; Rigid Body, p. 67 ; Centre of Figure, p. 67 ; Rotation, p. 68 ; Composition of Rota- tions, p. 69 ; Degrees of Freedom of a Rigid Body . . .69 Strain, p. 70 ; Shear . . . . . . .71 Circular Motion, p. 72 ; Curvature . . . . .73 SIMPLE HARMONIC MOTION AND WAVE-MOTION . . . -74 Simple Harmonic Motion ....... 74 Acceleration in S.H.M. proportional to Displacement, p. 76 ; Isochronous S.H.M.'s, p. 77 ; Projection of a S.H.M., p. 77 ; Harmonic Curve . 78 Composition of S.H.M.'s . . . . ... .80 Conversion of circular into reciprocating motion, p. 82 ; of reciprocating into circular ........ 82 Composition of S.H.M.'s at right angles of the same period, p. 83 ; of different period, p. 84 ; of different phase, p. 86 ; of non-commensurable periods ......... 88 Resolution of S.H.M. into rectangular components . . .89 Composition of S. H. M. with uniform motion . . , .90 Composition of S.H.M.'s in the same plane . . .90 Beats ......... 92 Rotation of plane of S.H.M. . . . . . .94 deposition of several S.H.M.'s, p. 95; Fourier's Theorem, p. 96; Tide calculating machine . . . . . .96 Oscillatory Movement of Systems of Particles . . . . .97 Waves, p. 97 ; Wave-length, p. 97 ; Velocity of propagation . . 98 Transversal vibrations of a cord, p. 99 ; Longitudinal vibrations of a cord ......... 103 Waves on a surface, p. 106 ; in a tridimensional substance, p. 107 ; Con- centric Waves, p. 108 ; Direction of the Wave, p. 108 ; Flat Wave- front, p. 108 ; Wave passing through an aperture . . .109 Reflexion of linear waves, p. 110 ; of a plane wave -front at a plane surface, p. 112; of a curved wave -front at a plane surface, p. 113; General Construction, p. 114 ; Problems . . . .115 CONTENTS. XI PAGE Transmission of a linear wave into a denser medium, p. 117 ; into a rarer medium ......... 118 Refraction of a plane wave at a plane surface, p. 119 ; General construction for refraction, p. 120 ; Refraction of a spherical wave at a plane surface, p. 121 ; Caustic by refraction, p. 122 ; Approximate foci . . 122 Rays as used in geometrical construction, p. 123 ; Ptolemy's Law, p. 125 ; Format's Law ........ 125 Superposition of Wave-motions on an indefinite cord, p. 125 ; nodes and loops, p. 126 ; Stationary vibrations ..... 126 Vibrations of a Cord whose extremities are fixed, p. 127 ; transversal, p. 127 ; longitudinal, p. 127 ; whose extremities are free, p. 128 ; a rod of which one extremity is free . . . . . .128 Vibrations of a membrane ....... 129 Interference . . . . . . . . .129 Propagation of waves along normals, p. 130 ; Effects of a screen, p. 131 ; Diffraction round a screen, p. 132 ; Wave traversing an aperture, p. 132 ; Relation of Wave-length to Diffraction-Fringes . . 133 Energy of S.H.M., p. 133; Energy of wave-motion .... 133 CHAPTER VI. KINETICS. GENERAL PROPOSITIONS relating to Force parallel to those relating to Telocity 134 Experimental proof of the Parallelogram of Forces . . .135 Centre of Figure . . . . . . . .136 Inertia of Matter, p. 136 ; Examples . . . .136 Momentum, p. 139 ; Impact of Inelastic Bodies, p. 139 ; Apparent loss of Energy on Impact, p. 140 ; Impact of Elastic Bodies, p. 140 ; Oblique Impact, p. 141 ; Energy in Impact of Elastic Bodies . . . 141 " Centrifugal Force" . . . . . . .141 Moment of Inertia, p. 144 ; Radius of Gyration, p. 144 ; Energy of a rotating body, p. 144 ; Centre of Oscillation, p. 145 ; Force constant in direction, rotational component varies . . . . .146 Accelerated Motion, p. 146 ; Problems ..... 147 The Principle of Moments . . . . . . .149 Couples, p. 150 ; Examples ....... 151 The Mechanical Powers . . . . . . .152 The Lever, p. 152 ; The Wheel and Axle, p. 154 ; Wheelwork, p. 155 ; The Inclined Plane, p. 155 ; The Screw, p. 156 ; The Wedge, p. 157 ; Pulleys, p. 157 ; The Bell-crank, p. 158 ; The Knee . . .158 FRICTION, p. 159 ; The Limiting Angle, p. 160 ; The Angle of Repo. ;, p. 160 ; Kiiietical Friction, p. 161 ; Influence of Duration of Contact, p. 161 ; Belting, p. 162 ; Resistances to Traction, p. 163 ; The Mechanical Powers, p. 163 ; Friction on a raindrop, p. 164 ; Rope wound round a post, p. 164 ; Rolling Friction ..... 165 CHAPTER VII. ATTRACTION AND POTENTIAL. Attraction, p. 166 ; Particular cases . . . . . .167 Convention as to Attraction - and Repulsion + . . . . 168 Xll CONTENTS. PAGE Potential Energy in the case of Kepulsion ..... 168 Work done by Repulsion . . . . . . .168 Potential Energy unexhausted at any given distance . . . 169 Direction of Movement . . . . . . .170 Potential .......... 170 Potential a condition at a point in space . . . . .170 Gravitation Potential . . . . . . .171 Absolute Zero of Potential, p. 171 ; Fields of Force in opposite conditions, p. 171 ; Continuity of Potential through Zero value, p. 171 ; Arbitrary Zero of Potential, p. 171 ; Analogy of Sea-level, p. 171 ; Equipotential Surfaces, p. 172 ; Motion parallel to these surfaces, p. 173 ; across them, p. 173 ; Work done in crossing them, p. 173 ; Distances between successive Equipotential Surfaces, p. 174 ; Free movement at right angles to them, p. 175 ; Lines of Force, p. 176 ; Tubes of Force, p. 176 ; Number of Lines of Force, p. 176 ; Systems of Surfaces and Lines, p. 176 ; Travelling of Equipotential Surfaces . . . .176 CHAPTER VIII. GKAVITATION AND THE PENDULUM. Law of Gravitation, p. 178 ; Cavendish's Experiment . . . . 179 Acceleration g due to Gravity . . . . . .179 Path of a Projectile . . . . . . . .179 Universal Gravitation, p. 180 ; Variations of g on the earth's surface, p. 180 ; Measurement of g ....... 181 Centre of Gravity, p. 182 ; of Two Masses, p. 182 ; of a System of Masses . 182 Overturning a body, p. 183 ; Work done . . . . .184 Equilibrium, stable, unstable, and neutral . . . . .185 Pendulum ... . . . . . . . 185 Simple Pendulum, p. 185 ; Isochronous Oscillations, p. 187 ; Work done, p. 187 ; Blackburn's Pendulum, p. 187 ; Compound Pendulum . 188 CHAPTER IX. MATTER. THE PROPERTIES OF MATTER . . . . . . .190 Essential properties Quantity, p. 190; Quality Elements, p. 191; Extension, p. 192 ; Impenetrability, p. 192 ; Indestructibility . . 192 General properties Weight, p. 193; Divisibility, p. 193; Porosity . 194 Contingent properties . . . . . . .194 Density, p. 194 ; Specific Densities, p. 194 ; Measurement of . . 195 THE STATES OF MATTER, p. 199 ; Solid and Fluid, p. 199 ; Rigid Solid, p. 199 ; Soft Solid, p. 200 ; Viscous and Mobile Liquid, p. 200 ; Viscosity .......:. 201 Gas, p. 202 ; Vapour, p. 205 ; The Critical State, p. 206 ; Radiant Matter 207 The Ether 207 Change of State ........ 209 THE CONSTITUTION OF MATTER ...... 211 Chemical Views . . . . . . . .211 Physical Views, p. 217 ; Atoms, p. 218 ; Vortex-Atoms . . . 219 CONTENTS. Xlll PAGE The Kinetic Theory .... .219 MOLECULAR FORCES .... . 225 In gases, p. 225 ; in liquids, p. 226 ; in solids . . . .227 CHAPTER X. SOLIDS. CONTINGENT PROPERTIES OF SOLIDS ... . . . . 229 Cohesion, p. 229 ; Hardness Softness, p. 230 ; Hardness Fragility, p. 230 ; Extensibility, p. 230 ; Ductility, p. 232 ; Compressibility, p. 232 ; Malleability, 232 ; Plasticity, p. 233 ; Rigidity Flexibility, p. 233 j Toughness, Brittleness .... Elasticity ....... .233 Resistance to Distortion, p. 233 ; Power of Restitution, p. 235 ; Restitution- pressure, p. 235 ; Perfect and Imperfect Elasticity, p. 236 ; Viscosity of Elastic Solids, p. 237; Fatigue of Elasticity, p. 237; Velocity of propaga- tion of a displacement, p. 238 ; Physiological Examples of Elasticity, p. 238 ; its Mechanical Advantages Strength of structures as depending on their form . . . 239 CHAPTER XL LIQUIDS. 1. MOLECULAR ACTIONS ....... 241 Cohesion, p. 241; Surface Tension, p. 241 ; its measurement, p. 244; Capil- larity ......... 245 Superficial Viscosity . ..... 247 Cohesion Figures . ....... 248 Solubility of Solids in Liquids, p. 248 ; Miscibility of Liquids . . 249 Imbibition . . . . . . . . .250 Diffusion, p. 250 ; Colloids and Crystalloids . .251 Osmosis ......... 252 2. THE STATICS OF LIQUID MASSES ...... 255 Pascal's Principle, p. 256 ; Hydraulic Press . . 256 Heavy Liquids, pressure within, p. 257 ; Communicating Vases . . 260 Archimedes' Principle, p. 261 ; Equilibrium .... 261 Measurement of Fluid Pressure . . . . . .261 Superficial, p. 261 ; interior, p. 263 ; variable .... 264 3. THE KINETICS OF LIQUID MASSES ...... 265 Streams .... ..... 265 Law of Continuity, p. 266; Forces producing flow, p. 266; Head of Water, p. 268 ; Torricelli's Law, p. 268 ; The Vena Contracta, p. 269 ; Ajutages, p. 270; Lateral diminution of pressure, p. 270; Energy of Jet, p. 271 ; Velocity of Outflow, p. 272 ; Recoil Resistances . . . . . . . . . 273 Surface Adhesion, p. 273 ; Surface Friction, p. 273 ; Eddies, p. 273 ; Vis- cosity . . . . . . . .273 Flow through uniform rigid tubes, p. 275 ; Constant flow, p. 275 ; Vari- able flow, p. 278 ; Interrupted flow . . . . .278 xiv CONTENTS. PAGE Flow through bent tubes ... .279 Flow in tubes not of uniform diameter . . . .279 Flow in branched rigid tubes ...... 280 Flow through capillary tubes ...... 281 Measurement of the pressure in a stream ..... 282 Measurement of the velocity of a stream . . . . .282 "Work done in keeping up a stream . . . . .284 Streams in elastic tubes ....... 285 Primary waves in elastic tubes, p. 286 ; The form of a simple pulse-wave, p. 287 ; Secondary waves in elastic tubes, p. 287 ; The form of the physiological pulse-wave, p. 287 ; Reflected waves in elastic tubes, p. 288 ; Amount of outflow from distensible elastic tubes . . . 288 CHAPTER XII. GASES. PROPERTIES OF GASES. Density, p. 289 ; Compressibility ... . 289 Tendency to indefinite expansion, p. 290 ; Air-pumps . . . 290 Absorption of gases by solids, p. 291 ; by liquids . . . 292 Diffusion of Gases, p. 294 ; Effusion, p. 294 ; Transpiration, p. 295 ; Mem- brane-Diffusion . . . . . . Diffusion of Gases from Liquids ...... THE STATICS OF GASES . . ... STREAMS OF GAS ........ Recoil, p. 298 ; Viscosity, p. 298 ; Measurement of Flow . THE PRESSURE OF THE ATMOSPHERE ...... General propositions ....... Suction, so-called, p. 303 ; examples ..... Liquid columns supported by atmospheric pressure Torricellian Vacuum, p. 305 ; Suspended Loops . Siphon, p. 308 ; Pump and force-pump, p. 309 ; Valves . Measurement of Atmospheric Pressure .... Variations in the Barometric Pressure ..... Standard Atmospheric Pressure ...... Gases passed into the Torricellian Vacuum .... CHAPTER XIII. HEAT. HEAT A FORM OF ENERGY . . . . . . .314 First Law of Thermodynamics . . . . . .317 Heat the lowest form of Energy, p. 317 ; Conclusions . . .317 Change of state a cause of absorption or liberation pf Energy in the form of Heat ......... 319 Change of state effected adiabatically ..... 322 EFFECTS OF HEAT .......'. Internal and External Work ...... Increase of kinetic energy of molecules ..... The Radiometer, p. 324 ; The Spheroidal State CONTENTS. XV PAGE Increase of Temperature . . . . . . .327 Absolute Zero . . . . . . .327 Specific Heat . . . . . . . .328 Atomic and molecular heat ...... 329 Specific heat at constant volume - . . . . 330 Specific heat at constant pressure ..... 330 The ratio of the two specific heats ..... 331 Work done in heating a gas , 331 Variations in specific heat ...... 332 The six thermal capacities . . . . . . 333 Inter-molecular work ...".... 333 Expansion . . . . . . , 333 Proof that there is no perfect gas (Joule's experiments, etc.) . 334 The Latent Heat of Expansion . 336 The Coefficient of Expansion, p. 337 ; Linear, p. 337 ; Cubical . 337 Examples of expansion by heat ..... 338 Applications, p. 338 ; Measurement . . . .339 Fusion . . . . . . . . . 342 Prof. Jas. Thomson's proposition as to melting ice . .343 Regelation. ....... 344 Boiling and Evaporation . . . . . .344 The boiling point at different pressures .... 345 Saturation pressure ...... 346 Measurement of vapour density ..... 347 Dewpoint, p. 348 ; Hygrometer, p. 349 ; Dew . . . 349 Changes of elasticity ....... 349 TRANSFORMATIONS OF HEAT . . . . . . .350 Work into Heat . . ... . . . .350 Heat into Work . . . . . . . .350 The nature of a Cycle ....... 351 Carnot's Cycle, p. 351 ; its steps ..... 352 The Reversibility of Carnot's Cycle . . . . .353 Carnot's principle . . . . . . . . 354 Carnot's function , . . . . .354 Efficiency of Carnot's reversible engine a maximum . . .354 Second Law of Thermodynamics ..... 354 Its various forms ....... 355 Degradation of Energy ...... 356 MEASUREMENT OF HEAT ....... 356 Temperature : methods of measuring it . , . . 356 Air thermometer ....... 857 Mercurial thermometer, p. 358 ; its construction, p. 358 ; its gradua- tion, p. 358 ; its sensitiveness, p. 359 ; testing, p. 359 ; different forms 360 Calorimetry . . . . - . . . . 360 The Method of Mixtures, p. 360 ; Latent-Heat Methods . . 361 TRANSFERENCE OF HEAT ....... 362 Conduction ......... 362 Conductivity, p. 362 ; steady flow ..... 362 Flow of heat and flow of temperature .... 363 Relative conductivities . . . . . . .364 Convection currents . . . . . . 3G4 Radiation ......... 365 Transport of Heat . . 365 XVI CONTENTS. CHAPTER XIV. SOUND - NATURE OF SOUND ........ 366 Sound, p. 366 ; Sounding bodies, p. 366 ; Sound-waves . . . 367 Characteristics of Sounds : Pitch, Loudness, Quality . . . 368 Noise ......... 370 Pitch : Physical Determination, p. 372 ; Musical Specification . . 373 Musical Intervals, p. 376 ; Transition . . . . 376 Loudness . . . . . ' . .381 Quality of Sound . . . . . . . .382 Analysis of Sound, p. 382 ; Resonators .... 383 Synthesis of Sound ....... 385 Complex Sound- Waves, p. 386 ; The Phonograph . . . 387 LAWS OF VIBRATION OF SOUNDING BODIES ..... 387 Transverse Vibrations of Strings, p. 388 ; The Monochord, p. 390 ; Experi- ments with, p. 391 ; Melde's Experiments, p. 393; Longitudinal Vibra- tions of Strings, p. 394 ; of Rods, p. 395; Transverse Vibrations of Rods, p. 395 ; Rotatory Vibrations of Rods, p. 396; Vibrations of Discs or Plates, p. 397; of Membranes, p. 398 ; of Bells, p. 398 ; Effect of Loading . 398 Free Vibrations . . . . . . . .398 Resonance ......... 398 Forced Vibrations . . . . . . . .400 MUSICAL INSTRUMENTS ........ 402 Singing and Sensitive Flames, p. 408 ; Trevelyan's Rocker, p. 409 ; Radiophony ........ 409 PROPAGATION OF SOUND ....... 409 In Solids, p. 409 ; in Liquids, p. 410; in Gases, p. 410; Beats, p. 411; Diffrac- tion, p. 412 ; Reflexion, p. 412 ; Refraction, p. 414 ; Interference, p. 414 ; Velocity of Sound, p. 414 ; Propagation according to the Kinetic Theory of Gas, p. 417 ; Doppler's principle . . . . .418 THE HUMAN EAR ........ 419 HARMONY AND DISSONANCE ....... 425 Differential Tones, p. 426 ; Summational Tones .... 427 VOICE VOWELS ........ 428 TRANSFORMATIONS of the Energy of Sound . . . . . 429 CHAPTER XV. OF ETHER-WAVES. PRELIMINARY ......... 431 NATURE OF RADIATION ..... . 431 Limits of Frequency, p. 432 ; Velocity and Wave-lengths . . . 433 Kinds of Radiation so-called : Heat, Light, Actinic "Rays" . . 433 Colour, p. 435 ; White Light, p. 437 ; The Spectrum . 438 Compound Coloured Light, p. 438 ; Complementary Colours . . 439 RADIATIONS OF A HOT BODY . . . . . . . 440 Exchange of Radiations, p. 441 ; Prevost's Law, p. 443 ; Stokes's Law . 444 Spectrum Analysis, p. 446 ; Linear Spectrum, p. 446 ; Band Spectrum . 447 CONTENTS. XV11 PAGE TRANSMISSION, REFLEXION, AND ABSORPTION . . 448 Absorption-bands, p. 450 ; Colour . . . . . .461 Blue colour of opalescent bodies ... . 453 FLUORESCENCE, PHOSPHORESCENCE, CALORESCENCE .... 454 SOURCES OF ETHER- WAVES . . . . . . 457 Vibrations of Molecules, p. 457 ; communicated to the Ether . . 458 PROPAGATION OF WAVES THROUGH THE ETHER . . . . 459 Ether -Vibrations transverse ...... 460 The Velocity of propagation, p. 461 ; Methods of measurement . . 461 Intensity at a place ........ 463 Direction of Propagation . ... 463 Plane-polarised, Circularly-, Elliptically-polarised Light, p. 464 ; Common Light ......... 465 Polariser, p. 465 ; Partially-Polarised Light . . . .466 Rotatory Polarisation . . . . . . .466 REFLEXION AND REFRACTION . . . . . . .466 Fresnel's Laws : for light whose vibrations are at right angles to the plane of incidence, p. 467 ; parallel to that plane, p. 469 ; Mixed Light, p. 470 ; Neumann and MacCullagh ..... 470 Plane of Polarisation, p. 471 ; Modification of Character of Light by Reflexion and Refraction . ... . . . 472 Mirrors . . . . . . . . . 472 Prisms, p. 477 ; Monochromatic Light, p. 477 ; Mixed Light, p. 478 ; Spec- trum, p. 478 ; Deviation without Dispersion, p. 479 ; Dispersion without Deviation, p. 480 ; Abnormal Dispersion . . .481 Lenses ........ 482 INTERFERENCE ......... 486 Bent Mirror, p. 486 ; Biprism, p. 487 ; Fringes, p. 487 ; Measurement of wave-length ........ 488 Colours of thin films . . . . . . .488 Iridescence . . . . . . . . 490 Shadow, p. 490 ; Camera Obscura . . . . . .491 Diffraction, p. 492 ; Diffraction-grating ..... 493 Twinkling of Stars . . . ' ..... 493 DOUBLE REFRACTION ...... . ,494 Uniaxial Crystals, p. 495 ; Mcol's Prism . . . . 499 Biaxial Crystals . . . . . . . . 499 Interposed Lamina between two prisms . . . . . 500 Circularly- and Elliptically-polarised Light .... 502 Determination of the character of a Beam of Light . . . 505 Colours produced by interposed film ..... 505 ROTATORY POLARISATION ....... 508 Soleil's Saccharimeter ....... 510 TRANSFORMATIONS of the Energy of Ether -Waves . . . .511 OPTICAL INSTRUMENTS . . . . . . . .512 The Eye, p. 512 ; The Microscope, p. 513 ; The Telescope, p. 513 ; The Opera Glass, p. 513 ; The Ophthalmoscope . . . .513 VISUAL PERCEPTION . . . . . . . .514 Perception of colour, p. 514; of form, p. 516 ; lustre . . .517 PAGE XVlll CONTENTS. CHAPTER XVI. ELECTRICITY AND MAGNETISM. GENERAL PHENOMENA of Electricity may be explained as those of Ether-stress 7l8 Quantity of Electricity, p. 519; Unit quantity, p. 519; Vitreous and Eesmous, p. 519 ; their mutual action Density Complementary Distribution Imaginary Electric Matter . Electric Force, p. 522 ; Field of Force Potential: Absolute Electric Potential, p. 523 ; Difference of Potential Travelling of the electric condition of a body Electromotive difference of potential Potential of Earth, p. 526 ; The Potential of a body ! Conductors and Non-Conductors : Dielectric Insulation .... Electrostatic and Electrokinetic "Free" and "Sound" Charges . Division of Charge Capacity, p. 529 ; Work spent in charging a conductor Induction Accumulator, p. 532 ; Specific Inductive Capacity, p. 533 ; Leyden Jar p. 534 ; Leyden Batteries, p. 534 ; Effects of Induction, p. 535 ; Electric Screens .... Electric Stress across Dielectric Dimensions of Electrostatic Measures OBSERVATION OF DIFFERENCE OF POTENTIAL Gold-leaf Electroscope, p. 537 ; Peltier's Electroscope, p. 538 ; Bohnen- berger's, p. 538 ; Sir W. Thomson's^ Quadrant Electrometer, p. 539 ; Coulomb's Torsion Balance Measurement of Difference of Potential PRODUCTION OF DIFFERENCE OF POTENTIAL Contact, p. 542 ; of conductors, p. 543 ; of non-conductors, p. 544 ; Fric- tional machines .... Chemical Action .... One-fluid cells and batteries, p. 549 ; Two-fluid, p. 551 ; others Friction of water against steam or air Evaporation, p. 554 ; Pressure, p. 554 ; Heat Electro-capillarity . Thermo-electricity . ... Thermo-electric Diagram, p. 556 ; Neutral point, p. 556 ; "Gaugaiii's Curves," p. 557 ; Temperature of Reversal . . 553 Atmospheric Electricity . 559 The Electrophorus, p. 560 ; Induction Electric Machines . . .561 ELECTRICAL CURRENTS . K i 001 Intensity and Resistance .... 561 Intensity, p. 561 ; Resistance, p. 561 ; Ohm's Law . . .552 Specific Conductivity, p. 562 ; Specific Resistance, p. 562 ; Table . 563 CONTENTS. XIX PAGE Value of Ohm . . . .564 Variable Conductivity . . . . . - . . . . 564 Reduced Resistance, p. 565 ; Intensity of a current equal throughout, p. 566 ; Dimensions of Intensity, Resistance, etc. . . . 566 Fall of Potential in a Conductor . . . . . .567 In a Homogeneous Conductor of uniform thickness, p. 567 ; Resistance and fall of potential in a Heterogeneous Conductor . . . 567 Flow through large Conductors . ; . . . . .571 Derived Currents ........ 572 Kirchhoffs Laws, p. 573 ; Wheatstone's Bridge . . . 573 Measurements of the constants of a battery . . , . . 574 The Energy of a Steady Current . . . . . . 574 Static Charge of a Conductor . . . . ; " .575 Variable Period . . . . . . . 576 Arrival curve, p. 577 ; Deep-sea cables ..*. . . . . 578 EFFECTS OF A CURRENT . . . -. . . . . 579 Production of Heat, p. 579 ; Peltier's effect, 580 ; Thomson's effect . 581 Production of Light . . . -.'... -., . 582 Geissler's Tubes . . , . . . < . . 583 Electrification of Radiant Matter ..... 584 Electrolysis, p. 585 ; Faraday's Laws of Electrolysis, p. 587 ; Nobili's and Guebhard's Equipotential Surfaces, p. 589 ; Polarisation of Electrodes, p. 590 ; Secondary Cells and Batteries, p. 590 ; Electrical Storage of Energy, p. 592 ; Regulation of a Current . . . . 592 THE DYNAMICAL PROPERTIES OF A CURRENT . . 593 Electromagnetic Field of Force . . . . . . 593 Lines of Force, p. 593 ; their Direction . . . . . 594 Mutual action of Currents, p. 594 ; Ampere's formula . . . 595 Field of force round a closed circuit ..... 596 Solenoidal system of currents, p. 598 ; the surrounding field of force . 598 MAGNETISM : General Phenomena . ' . . . . . 598 Terrestrial Magnetism, . . . . . . . 604 Magnetic Induction . . . . . . . 606 Electromagnets . . . . . . . . 609 Astatic Arrangements .... . 610 Nature of Magnetism . . . . . . .611 ELECTROMAGNETIC INDUCTION . . * . . .. . . 611 Equivalence of Circuits and Magnets . . . . .611 Secondary Currents, p. 613 ; Lenz's Law . . . . . 613 Self Induction . . . . , . . .615 Induction Coils . . . . . . . 615 Magneto-Electric Machines and Dynamo- Electric Machines . . .616 The transmission of Energy to a distance, p. 620 ; Galvanometer, p. 620 ; Telegraphy, p. 621 ; Interrupter for tuning-forks, p. 622 ; Telephone 623 Simultaneous Currents . . - . . . . 623 Differential Galvanometer, p. 624 ; Duplex Telegraphy, p. 624 ; Harmonic Telegraphy . . .' . . . ; .624 Reversed Action ......... 625 Electromotor, p. 625 ; utility . . . . . .625 XX . CONTENTS. PAGE THE NATURE OF THE MAGNETIC FIELD . . . . .... 628 Magnetic rotary polarisation, p. 628 ; Hall's Experiment, p. 628 ; Kerr's Experiment, p. 628 ; Maxwell's Theory . . . . . 629 Maxwell's Theory of Light . ..... 629 Measurement of v . . . . . . 630 ELECTROMAGNETIC MEASURE . . . . . . . 630 Dimensions . . . . . . . . . 630 Practical units : their derivation . . . . . 631 Measurement of Current Intensity . ." . . . .632. Galvanometers, p. 633 ; Electrodynamometers . . . 634 Langley's Thermic Balance . . ' . . . . 634 BIBLIOGRAPHY ........ 635 INDEX 639 CORRIGENDUM. PAGE 533, line 11 to end of paragraph, read : All gases have very nearly the same inductive capacity, whatever their chemical constitution, their temperature, or their density. If, however, their pressure be increased or diminished, the minute difference between their sp. ind. cap. and that of a vacuum is also increased or diminished in the same proportion ', and conversely, when a gas is employed as a dielectric, induction across it diminishes its pressure, the gas then adjusting itself so as to become rarer and consequently less inductive. DANIELL'S PRINCIPLES OF PHYSICS.] The reader will kindly take note of the following corrections, in addition to that on page xx : Energy. Page 41, line 32, for 772'43 read 772-55. Kinematics. Page 70, footnote ; the area of a circle = 7854 x diar. 2 = irr 2 . Last line of text, for read TT. Kinetics. Page 151, line 5, for Q read R. Attraction and Potential. Page 166, bottom of page ; the total attraction of one set of particles for another set. Page 171, lines 10 and 16, read 21,120,000 g or (21,120,000 x 32-2). Heat. Pages 316, 317. JOULE'S EQUIVALENT. Joule's revision of this physical constant gives him the value 773-467 foot-pounds at Higher Broughton, Manchester (lat. 53'28j'N.), at 120 feet above sea- level ; this reduced to sea -level and the latitude of Greenwich gives 772-55 as the number of foot-pounds of work necessary to raise the temperature of one pound of water from 60 F. to 61 F., the pound of water having been weighed in vacuo. In accordance with this value, the Calorie is equal to 41,593,010,000 ergs or 42 3 '9 8 5 kilogramme-metres. Page 320, line 29, for 34'362 and 343'62 read 34'462 and 344'62. Sound. Page 406, Problem ; the working is : C, = 32 vibrations = n. ' l> = &' etc - Page 426, last line, for component read compound. Electricity and Magnetism. Page 560, the Electrophorus, a misstatement : the potential is increased, the charge is not. The latter part of the paragraph, if rewritten, would run thus : " Work is now done from without in pulling the metallic plate away from the resin ; as the distance between the metal and the resin increases, the electrostatic capa- city of the electrophorus, considered as an accumulator, diminishes ; the potential therefore increases both on the metal and over the resin : the knuckle applied to the edge of the insulated metallic plate may now re- ceive a spark. When the metallic plate is next laid on the resin a new charge is induced in it, which may again become a charge of high poten- tial when the plate is removed. .Small original charges may thus induce successive charges of high potential." Page 544, line 9, for V Cu - V Zn read V Zn - V Cu . Page 564, line 19, for 0'9870 read 0'9873. Page 615, Self-Induction, some words lost in copying MS. Line 3 of the paragraph reads : " it does not flow on, and its intensity is rendered less for an instant than it would have been if the wire had been straight ; while, when it is suddenly broken, the current does not at once cease, but is, as it were, piled up, and its intensity and the potentials at the ends of the wire are rendered greater for an instant." Page 621, Morse Code: some points have been dropped on their way through the press. The corrections are D -- E I Z (Ameri- can form) A- 6 ALFRED DANIELL. EDINBURGH, March 20, 1884. THE PRINCIPLES OF PH 1NTKODUCTOKY. NATURAL PHILOSOPHY or PHYSICS may be briefly definecTas the Science of Matter and Energy. This definition is one which is obviously comprehensive enough to include within its range the whole of Chemistry and of Biology as well as of Chemical and Physiological Physics. Chemistry is in truth but a colony of facts closely related to one another, and classified by us on principles which depend almost entirely upon our ignorance of the fundamental nature of the relation between those apparently different forms of Matter which we know as the various Chemical Elements ; and the consummation of Chemistry, a full and accurate knowledge of the inner mechanism of all chemical reactions, would probably result in the absorption of all Chemistry in the wider Science of Molecular Physics. In the meantime the fundamental unity of the two nominally distinct sciences, Chemistry and Physics, is shown by the extent to which they overlap one another in the field of Chemical Physics. Physiology, again, or in a wider sense Biology, is concerned with the matter and the energy of living beings; and if it ever come to attain its highest ideal, even Biology must thereupon necessarily merge in Natural Philosophy. Already we see that while physiological research is steadily conquering the unknown, that which it succeeds in thoroughly explaining falls out of its grasp and comes to form a part of ordinary physical, or, it may in the meantime be, of ordinary chemical knowledge. We may now more amply define Natural Philosophy or Physics as the systematic exposition of the phenomena and properties of Matter and Energy in so far as these phenomena and properties can be stated in terms of definite Measurement and explained by reference to mechanical principles or Laws. B 2 THE PRINCIPLES OF PHYSICS. Here, again, we must admit that our definition is, in the present state of our knowledge, too ideal. A perfect and accurate physical knowledge even of the simplest actual phenomenon would imply absolute omniscience. Often we find that we can measure but not explain the phenomena of Nature ; and we find at the very outset of our exposition that we are compelled to confess entire ignorance as to the very nature of our subject- matter ; for we do not know what Matter is. To us the question, What is Matter ? What is the essential basis of the phenomena with which we may as physicists make ourselves acquainted ? is absolutely insoluble. Even if we became perfectly and certainly acquainted with the intimate structure of Matter, we would but have made a further step in the study of its properties ; and as physicists we are forced to say that while somewhat has been learned as to the properties of Matter, its essential nature is quite unknown to us. As little able are we to give any full and satisfactory answer to the question, What is Energy ? As a provisional statement we may say that Energy is the power of doing Work ; a rifle- bullet in motion, a coiled watch-spring, possess the power of doing work upon other bodies suitably arranged ; but plainly this power depends upon the relation into which the matter which is said to possess it is brought with reference to other matter, and it ulti- mately depends upon the position of one set of particles of matter with reference to other sets. If, then, Energy depends upon the relative position of particles of Matter, we are not able to explain its own essential nature, though we may acquire a considerable amount of information as to its very remarkable properties. These properties of Energy, those of Matter, their mutual relations, and the laws which govern these, constitute the subject- matter of Natural Philosophy ; and these properties and laws have been ascertained by observation, by measurement, and by judicious reasoning upon the data supplied by investigation. In the investigations upon which Natural Philosophy is founded, the guiding principle is a well grounded belief, based on the recorded experience of the human race, in the Constancy of the order of Nature. This does not mean that things are to continue for ever as they are at present. If a closed boiler containing water be heated to a certain temperature, the Constancy of Nature would not be interfered with by the consequent ex- plosion of the boiler; it would be seriously infringed if the boiler did not burst. So, again, if. volcanic eruptions thrust up INTRODUCTORY. 3 mountain ranges through a flat plain, as in the case of the Rocky Mountains, or if a crack in the earth's crust allow a flood of lava to flow over a wide region, as in the geological history of Idaho, such a cataclysm would appear to be an awful break in the uniformity of Nature ; yet if the earth's crust be so pressed up- wards that it can resist no further pressure, the Constancy of Nature is confirmed by its giving way. On this belief in the Constancy of Nature are based all rational calculations of eventu- alities, and all our arrangements from day to day, which are subject to the transpiry of facts unknown or unforeseen at the time when these arrangements were made. This belief finds formulated expression in the Law of Causality, which affirms that all effects have a sufficient cause ; so that if we observe a given phenomenon, we conceive ourselves entitled as the result of experience to enquire into its cause, and conversely to affirm that if there be no cause tending to produce change in the present condition of things in any particular respect, there will in that respect be no change. It is scarcely necessary here to investigate the meaning of the word Cause itself; it will be quite sufficient to point out that for us the relation of cause and effect is one of sequence, found to be in- variable if not interfered with by the intervention of circum- stances which render cases dissimilar. In similar cases, the same causes are observed to be followed by the same effects. It is plain, however, that the same effects are not always and neces- sarily the results of the same causes ; and when different causes are found to produce the same effects they are equivalent in effectiveness to, and may be substituted for, one another. Again, the principle may be stated that the cause is equiva- lent and in proper terms of measurement numerically equal to the effect produced by it. Apparent exceptions to this statement arise only when the problem is not of the extremely simple form in which one cause, and one cause alone, is brought into play. It is not, except in a loose popular sense, the heat of the spark which causes the explosion of a magazine and consequent destruc- tion of property ; it is not drawing the trigger which is the cause of the bullet's leaving the gun. The heat of the spark, the drawing of the trigger, is necessary as one cause out of several ; but the problem is not here so simple that these can be adduced as cases in which the effect is greater than the cause. They only point out an extended statement that the total effect produced is equivalent to the effective sum of the causes acting ; and when 4 THE PRINCIPLES OF PHYSICS. one of the causes acting is an arrangement of matter which is explosive or in unstable equilibrium, ready to topple over so as to assume a stable position, the effect produced, though apparently greater than the small disturbance which disarranged the unstably- balanced matter, must be traced not to it only, but to all the conditions and circumstances involved, including the unstable equilibrium, the antecedent cause of which may itself be sought for. If several causes act simultaneously, each produces only a part of the aggregate effect, and the total effect is equal to the sum of the acting causes. Under the name of Galileo's principle this is one of the fundamental truths of physics, and is thus enunciated : If a body be acted on by two or more Forces (force being meanwhile denned as any cause of motion), each of these forces acts independently, and produces its own effect without reference to the others, the total effect produced being ascertained by finding, in any appropriate way, the sum of the effects due to each force. A cannon-ball, for instance, fired from a height, is, as it passes through the air, under the influence of at least two forces or causes of motion : the force exerted upon the ball during the explosion sending the ball forwards, that of gravity con- tinuously drawing it downwards. If, for the sake of convenience, we neglect the resistance of the air, and enquire what would be the path pursued by a shot travelling in vacuo, we would find by making use of this principle that the position of the shot at any moment would be found by enquiring (1) How far outwards the shot would have been projected had there been no tendency to fall ; and (2) How far the ball would have fallen if gravity had alone acted on it. For any specified instant a point may in this way be found, which, being both so far outwards and so far down- wards, must be the position of the ball at the instant in question ; and by thus finding the position of the ball at several separate succeeding instants of time, we may find the curved path which a ball fired in vacuo would traverse. This principle of the inde- pendence of simultaneously -acting causes was an experimental discovery of Galileo's : before his time it was held as truth that one cause must cease to act before another can commence to do so ; and it was accordingly believed that when a projectile was shot into the air, the force of projection must be expended and dissipated before any tendency to fall to the earth could assert itself. Experimentation. When we learn that a certain pheno- INTRODUCTORY. O menon is due to a congeries of causes we may arrange matters so as to prevent one of the ordinarily -acting causes from pro- ducing its effect, and then we may observe in what respect the resultant phenomenon now produced differsvfrom that usually seen. Thus we may find the way in which a given cause acts. Again, we may directly arrange matters so that a given cause, and, as far as possible, that cause alone, shall act, and we may then observe what happens. The principle of the Constancy of Nature shows us that like causes will always produce like results ; and if we find that by ingeniously varied interrogations of Nature we have obtained as reply an assurance that certain causes are allied to certain effects, we rest assured that the same causes and the same effects will continue to be so allied. This assurance is the only basis of the art of Experimentation. By this art we become acquainted with the constant modes in which events follow one another in the material world, these modes being the Laws of Nature arbitrarily appointed, and only to be learned by us through the instrumentality of our own experimental enquiry, or else through attentive consideration of the varying phenomena of the universe, " experiments made at Nature's own hand." Newton's Laws of Motion or Axioms. If a body be at rest it will remain at rest : if in motion it will continue to move until stopped by friction or some external force. Here we find the word Force meaning not only that which causes motion, but also that which arrests motion. Experiment shows us that this is true in reference to bodies which are at rest, for they remain so if undisturbed ; but it also shows that among bodies which are in motion, it is only those which are moving in a straight line that retain their course unaffected when allowed to move unex- posed to the action of any disturbing cause. Bodies which are moving in curved paths, such as sling-stones, do not retain their curved paths when liberated, but continue their course in a straight line in that direction in which they happened to be moving at the iustant of release from constraint. Hence Newton, in his First Law of Motion, says, "Every body tends to persevere in its state of rest or of uniform motion in a straight line unless in so far as it is acted on by impressed force," and this is tersely expressed by saying that " Matter has Inertia." If a single force act upon a body which is at rest, the body will begin to move in a straight line ; and further, the greater the force, the more rapid will be the motion of the body acted upon. If the body be already in motion, the force acting upon it will b THE PRINCIPLES OF PHYSICS. cause it to move more rapidly or more slowly in the same straight line or else in a deflected course. Experiment shows that every force has a definite direction in which it tends to cause a body to move, whether that body be already under the action of other forces or not. Thus the words of Newton, in his Second Law of Motion, are: "Change of Motion is proportional to the impressed force, and takes place in the direction of the straight line in which the force acts." The word Motion in this law is now rendered Momentum (p. 17). The third of the Laws of Motion which Newton formulated as axiomatic is the following "To every Action there is always an equal and contrary Reaction; or the mutual actions of any two bodies are always equal and oppo- sitely directed." The truth of this statement is based upon experimental evidence, but its universal applicability is, after con- sideration, seen to be reasonable enough ; and in this sense Newton uses the word Axiom. When a shot is fired from a gun, if the gun be free to move there is con- siderable recoil, the shot moving forward and the gun backwards. If the gun be fixed to the ground, the shot is apparently the only thing which moves. If the shot were held fast and the gun were free to move, the gun would move backwards. In this case we see, then, that to the action which impels the shot forward there is a contrary reaction which impels the gun back- wards ; and in the sequel we shall learn what evidence there is for the state- ment that that reaction is equal to the action. When a man walks on firm ground, the action of his legs in locomotion tends to separate his body from the ground at each step. The action which tends to raise his body is contrary to the reaction tending to depress the earth, and at every step the earth is pushed down as a whole, or else if the soil be soft it yields locally and the foot sinks. Hence the difficulty experi- enced in getting out of boggy soil ; the soft mud yields under the foot at each effort made by the individual, so that every step causes him to sink more deeply. When a horse is loosely harnessed to a car, it may sometimes be observed that an inexperienced animal starts forward quickly ; but suddenly the traces tighten, the car is jolted forward, and the horse is jolted backwards. If a locomotive with a heavy train be suddenly started, it will be seen that its wheels may uselessly turn round ; it has given a sudden pull to the carriages, and their reaction upon it is equivalent to a backward pull given to a moving engine. The earth attracts the moon, and the moon attracts the earth. The former attraction mainly keeps the moon in her orbit, and the latter is one of the causes of tidal phenomena. When a' stone is thrown upwards from the earth, the earth is thrown back by recoil, and moves downwards to a very small extent as long as the stone continues to ascend : when the stone is at its highest point the earth is at its lowest, and as the stone falls the earth ascends to meet it. This is, of INTRODUCTORY. 7 course, not the result of direct observation, but is deduced by calculation from Newton's third law of motion, which is confirmed by all phenomena, terrestrial and astronomical, by which it can be put to the test. The next statement generally applicable is that of the Inde- structibility of Matter. This is, that Matter as we at present know it cannot be destroyed by any process with which we are acquainted. The limitations of, this statement should be borne in mind, for there is no scientific warranty for saying that Matter is absolutely indestructible, and more than one consideration indicates that the structure of Matter may be such as to denote that in its present form it has had a beginning and may have an end. Within our experimental knowledge, however, Matter cannot be destroyed : and when it apparently disappears, as when a candle is burned in the air, Chemistry charges itself with the explanation of that disappearance, and shows what new forms the matter has assumed. Another principle of the greatest possible use, and entirely the result of experiment, is that of the Indestructibility or the Conservation of Energy. Energy has been provisionally defined as the power of doing work ; and this doctrine states that this power of doing work may alter its form but is never destroyed : when a coiled watch-spring uncoils, the energy which it loses appears as Work done or as Heat or Sound, or any other of the forms into which, as we shall afterwards learn, Energy may s be transformed. The Energy of a body depends on the advantage which it possesses either of motion or of position : if it lose that advantage, that can only occur through some other body or bodies simultaneously acquiring either motion or an advantage of position. If Energy disappear in one form, it will reappear in one or several others, and none of it is ever lost, though it may assume such a form that it is no longer a power of doing work available to man, namely, the form of uniformly diffused Heat. This principle, which is so important that the whole of Natural Philo- sophy may be said to be a commentary on it, will be better understood when the laws of Energy have been discussed, as they will be at greater length in Chapter IV. A corollary to this principle takes the form of a statement of the belief that the Perpetual Motion is impossible : if the sum of the energy in the universe be constant, no machine in which this energy is employed in doing work, in which friction is overcome, in which sound is produced, and so on, can possibly go on for ever, for the reserve of energy at its disposal will ultimately be 8 THE PKINCIPLES OF PHYSICS. exhausted and become useless to that machine. Even the tides will ultimately cease, as the earth loses we know it is losing speed in its daily rotation round its own axis. It cannot be too strongly insisted on that these general principles, the Constancy of Nature, the Law of Causality, Galileo's principle, the three laws of Motion, the Indestructibility of Matter and of Energy, are of no value for us except in so far as they are supported by experimental evidence. They are grouped together here, for the statement of them is necessary for comprehension of the results which have been obtained through their aid. We are not here called upon to go through the steps by which they have been arrived at, but we must bear in mind that no a priori deduction of them by any metaphysical reasoning is for a moment admissible. The doctrine of the Conservation of Energy is very simple when stated as the result of experiment, and its simplicity has led to statements that the contrary is unthinkable, and that a belief in this doctrine is deeply grounded in the constitution of the mind of man ; but all conclusions derived from such reasoning must be regarded with suspicion, for we must take warning by the example of the ancients, who believed circular motion to be perfect and heavy bodies to fall faster than light ones, until experimental evidence was adduced to the con- trary. The truth of these principles must be proved by their perfect accord with the phenomena which we may actually observe, and by their enabling us to predict results of hitherto untried experiments which agree with those actually obtained. Exact science depends directly for its facts and indirectly for its principles upon experimental evidence, and the true place of speculative imagination in scientific work is the conception of new combinations of circumstances, and hence of new fields of experi- mental researches also the construction of Hypotheses which explain and co-ordinate observed facts, and which, when they are found to do this consistently, are raised to the rank of accepted Theories. CHAPTER I. TIME, SPACE, AND MASS. So far as man's knowledge of phenomena occurring around him has become accurate, it has been obtained by means of precise measurement ; and the fundamental units in terms of which every measurement must be executed are those of Time, Space, and Mass. The unit of Time is usually taken as one Second, and the time during which phenomena appear or are observed is reckoned in seconds, unless motives of obvious convenience cause it to be "reckoned in minutes, hours, days, years, or cen- turies. The second is usually a second of mean solar time that is to say, the ^ 6 ^ 00 th part of the average length of a solar day. The solar day is the period which elapses between the sun's crossing the meridian, or being situated directly south of a place, and the next occasion on which it crosses that line. The sidereal day, in the same way, is the interval between two successive transits of any fixed star. The sidereal days are shorter than the solar ; they are constant in length, while the solar day is not so. A clock can keep time with the stars, and keep good " sidereal time ; " but a clock of ordinary construction does not always indicate noon when the sun is highest in the heavens ; it is sometimes apparently 14^ minutes fast, and sometimes appears to be 16^ minutes slow. A good clock, however, is one which measures off and indicates as twenty-four hours a period of time equal to the average length of the solar day for a year or a century or an age, and such a clock is said to keep " mean solar time ;" while the Second used in physical measurements is the second as indicated by a clock such as this ; and it is the 31,558,150th part of a sidereal, the 31,556,929th part of a solar or tropical year. Space. When a single point moves it describes a line : if it travel by the shortest distance between two points, its path is a straight line; and a line is an example of space of one dimension. Movement and measurement may be effected in a forward or a 10 FUNDAMENTAL NOTIONS. [CHAP. backward direction along it, but as a line has neither breadth nor thickness there can be no other. Distance along a straight line may be measured in one direction arbitrarily chosen ; let this be, for instance, the direction from left to right; if, then, a point travel towards the right its motion is positive, if to the left, negative. If it move a inches to the right and then I inches to the left, its distance from the start- ing point becomes a I; while if it first go b inches to the left and then a to the right, its position will become I + a from that point ; and these two positions are the same, for a I = I -f a. Hence we learn that if a point move backwards and forwards by varying amounts along a line, it does not matter in what order it performs these operations : the spot ultimately arrived at will be the same in all cases. In order to effect measurements along lines, we require a standard of length. This is taken as the Foot or the Metre. The British standard yard, which is equal to three feet, is defined by law as " the distance between the centres of the transverse lines in the two gold plugs in the bronze bar deposited in the office of the Exchequer" at the temperature of 62 F. A number of authorised copies of this have been made and are deposited at the Eoyal Mint, the Eoyal Observatory at Green- wich, the New Palace at "Westminster, and under the care of the Eoyal Society of London. The metre is the distance, at the temperature of melting ice, between the end's of a platinum rod preserved in the Archives, and of which copies, to regulate French commerce, are preserved at the Ministere de 1'Interieure in Paris. It was originally intended to represent the ten-millionth part of the distance from the Equator to the Pole : the measurements of Delambre and Mechain, from which Borda made the standard metre according to a law of the French Eepublic passed in 1795, have been found not to be quite correct, for the earth's quadrant is now known to measure 10,000,880 metres. The metric system of measurement of length is decimal ; each metre contains 10 decimetres, 100 centimetres, or 1000 milli- metres : 1000 metres make a kilometre, which is, roughly speak- ing, about | (fffi) of a mile ; one metre is e( l ual to 39-37043196 inches, or 3 '2 8 087 feet; a decimetre very nearly corresponds to 4 inches (really 3'937043196); a millimetre is very nearly equal to the twenty-fifth of an inch. For the purpose of physical measurement it is customary and convenient to make use of the SPACE. 11 Centimetre"' (393*7043196 inch) as a unit of length. One English foot is equal to 30 '4 7972654 centimetres. A plane surface has length and breadth but no thickness, and is therefore said to be space of two dimensions. Two terms are always necessary for the precise statement of the position of an)' point on a surface. The position of a ship at sea is determined when its latitude and its longitude are known. The position of a point a on a plane surface is determined by choosing a fixed point 0. as the origin ; then two axes, Ox and Oy, are chosen, generally at right angles to one another ; aA is drawn parallel to Ox, and B parallel to Oy, and the point a is said to be situated at a distance OA along the axis of y, and OB along the axis of x. If a point lie at the same time three miles to the north and four miles to the west of a given place, its true position (at the distance of five miles) can be easily indicated on a chart. The symbols + and - are also used here to denote that the measurement is to one side or the other of the point assumed as the origin. Points to the right of O have a positive, points to the left a negative, value of Ox ; points above have a positive, points below a negative, value of Oy. Thus (Fig. 1) the Fig.l. point a has abscissa (or line cut off along the axis of x) OB, and ordinate (cut off along the axis of y) OA ; the point a" has abscissa - OB and ordi- nate + OA ; the point a'" has abscissa OB and ordinate - OA ; that at a"" has abscissa + OB and ordinate - OA. * It is worth remarking that a French ten-centime piece measures 3 centimetres j across, while a five-centime piece has a diameter of 2 centimetres. Similarly an 1 English halfpenny measures an inch, while a penny measures an inch and a fifth. 1 2 FUNDAMENTAL NOTIONS. [CHAP. The area of a surface may be measured if we fix upon a standard unit of area. The unit of length may be made use of in order to obtain this. If a square be constructed, one of whose sides is one foot or one centimetre, we shall have a unit- surface whose area is known as one square foot or one square centimetre ; and the areas of other surfaces may be measured by comparison with these standards. A solid has length, breadth, and thickness, and is said to occupy space of three dimensions. The position of any point in tridimensional space requires three numerical terms for its exact statement. The position of a balloon, for instance, will be definitely known if the latitude and longitude of the spot over which it stands and its height above that spot be ascertained. Three terms are also required to define the position of a star : the telescope has to move so much "in azimuth" round a vertical axis ; then so much in altitude ; and thirdly, the distance of the star in a straight line must be known. A cube whose side is one foot or one centimetre that is, a cubic foot or a cubic centimetre serves as the unit of volume. For convenience sake other units of volume are often chosen, such as the cubic inch, the cubic decimetre (otherwise known as the liquid measure, one litre), the cubic metre, and so forth. The remaining fundamental idea involving measurement is that of Mass, or quantity of Matter. The notion implied in this term is quite distinct from that of weight. The weight of a certain quantity of matter depends upon the presence and nearness of other matter which attracts it according to the well- known law of Gravitation. This may, and, even within our terrestrial observation, does vary ; the effect of gravity on a given mass that is to say, its Weight is greater as we near the Poles than it is at the Equator ; and the weight of a substance varies, therefore, according to local causes, while the mass or quantity of matter in it remains the same. Cceteris paribus, however, equal masses counterpoise one another in a balance, and we may define the unit of mass as that quantity of matter which will counterpoise in a balance a certain standard mass known as a standard Pound or Gramme. The British standard pound is a piece of platinum preserved in the same place as the standard yard, while authorised copies of it are preserved at the same institutions. The French standard is the Kilogramme (=1000 grammes), made of platinum, and preserved at the Archives in Paris. This is intended to have the i.J MASS. 13 same weight as a cubic decimetre of water at its temperature of maximum density that is, 3*9 C. Since a kilogramme contains a thousand grammes, and a cubic decimetre a thousand cubic centimetres, it follows that the gramme is intended to be equal to the mass of one cubic centimetre of water at 3'9 C. Comparison of the actual standards shows, however, that a litre of water weighs, at 3*9 C., 1*000013 kilogrammes, and a cubic centimetre of water at 3' 9 C. weighs therefore not 1 gramme, but T000013 grm. For most practical purposes the intended value may, however, be taken as correct. The British pound avoir- dupois weighs 7000 grains, while the standard kilogramme weighs, according to Prof. W. H. Miller, 15432-34874 grains, and the gramme 15'43234874 grains. It may be noticed that the British fluid ounce of water at 62 F. weighs one ounce avoirdupois ; that a pint of water (20 fluid ounces) weighs there- fore a pound and a quarter, and a gallon of water ten pounds. A French franc-piece weighs, when new, five grammes. CHAPTEE II. NOTIONS DERIVED FROM THE PRECEDING. WHEN a physical particle changes its position, it effects Motion. This Motion or Change of Position must be performed by passing along a definite continuous path continuous because it is not possible for any physical particle to occupy two consecutive positions without traversing the intermediate space. In this respect the path of a physical particle differs from many mathe- matical curves which abruptly end at one point and recommence their course at another. Obviously the path described by the moving particle may have any form, straight or curved and the shortest possible path between the initial and final positions is a straight line. A moving body may travel rapidly or slowly : the rate at which it travels along its path is called its rate of motion, its rate of change of position, its Velocity. The Velocity of a moving body may be stated in units of length per unit of time, e.g. feet per second; and a body is moving with Unit velocity when it moves one foot in a second, or one centimetre in a second. It will be observed that it is necessary for us to make consistent use of the British or of the metric units of measurement, and not to use them confusedly within the limits of the same problem. A body which moves sixty feet in five seconds has a velocity, evidently, of twelve feet per second. The velocity is equal to sixty divided by five that is, to the whole space traversed divided by the time occupied in the movement. In algebraical language this is expressed thus : v = ~ , where the velocity, space, and time are denoted by their initial letters. Multiplying both sides of this equation by t, we get vt = s ; the space traversed in a given time is equal to the velocity per second multiplied by the number of seconds. If the body move through equal spaces in equal times, the velocity is said to be uniform. CHAP, ii.] VELOCITY. 1-5 We are familiar with instances in which a body such as a railway train is said to be running at a certain time with a velocity of (say) thirty miles an hour. This indicates that if the train ran for a whole hour at the rate at which it was travelling at the instant of observation, it would at the end of an hour be thirty miles away from the point which it occupied at the beginning of it. But the train may possibly not have run more than a mile on the whole. The statement means, then, that during (say) a minute it ran half a mile, and that therefore during sixty minutes it might, at the same speed, have run thirty miles. But even during a minute it may have gained or lost speed, so as to render its motion not uniform but variable : the statement would be still more exact if we knew that in six seconds it ran the twentieth of a mile, or in one second the hundred-and -twentieth; for when the interval of time is very short, there is less possibility of variation during that interval, and the speed approximates more nearly to uniformity. Hence the variable velocity of any moving body at a particular instant is found by observing the amount of motion effected in a certain very short interval of time, and finding what movement would be effected in one unit of time if the velocity were to remain uniform during that period. If a body move over a certain space, s (say thirty feet), in time t (say ten seconds), the equation v : - = 3 feet per second, shows what the mean or average velocity is during the motion. The mean velocity of a train which travels fifteen miles in one hour is a quarter of a mile a minute, or ^ J^th part of a mile in a second, although during some seconds or minutes it may be travelling at the rate of sixty miles an hour, at others may be standing still, and at others may be actually going backwards. All velocities, mean and constant, uniform and variable, may be expressed in feet or in centimetres per second, and can, when so expressed, be compared with one another. All measurable velocities are Eelative ; we know nothing about Absolute velocities in space, for we have no standard of comparison. Problems. 1. If a body move 144 feet in 3 seconds, what will be its mean velocity ? Ans. 48 feet. 2. In the previous question : What will be its velocity during the second second if it travel 16 feet in the first second and 80 feet in the third ? Ans. 48 feet. 3. A body moves with a uniform velocity of 40 miles 1600 yards per 16 DERIVED NOTIONS. [CHAP. hour : what is its velocity in feet per second ; and how many feet will it traverse in 10 seconds ? Ans. 60 feet per second ; 600 feet. 4. A railway train explodes two detonating signals placed 011 the rails at a distance from one another of 176 feet; an interval of exactly 2 seconds elapses between the explosions. Compare the velocity during that interval with the mean velocity, which is indicated by the statement that the train takes an hour and a half to perform the journey between two stations 45 miles distant from one another. Ans. It is twice the mean velocity. 5. Which is the greater velocity, 40 miles an hour or 12 metres per second ? 6. A train travels 10 miles at a velocity of 20 miles per hour; then 4 miles at an average rate of 30 miles per hour; then 6 at a uniform rate of 40 miles per hour ; it takes 1 mile to come to rest, running at an average speed of 20 miles an hour ; it stands for 7 minutes : it starts and. runs for 20 minutes at the average speed of 21 miles an hour. What has been its mean velocity ? Ans. 32 feet per second. Acceleration. When the velocity of a moving body varies, the Eate of Change of Velocity is called its Acceleration. In popular language this word indicates increase of speed, but it is in this connection used to signify the rate of change, whether that change be an increase or a diminution of the velocity. If a body be moving at the rate of ten feet a second at the beginning of a certain second of time, and at the end of that second be found to be moving at the rate of eleven or of nine feet, it is said to have received during that second an acceleration, positive in the former case, negative in the latter, of one foot per second. Acceleration is usually indicated by the symbol a, and the Unit of acceleration is the acceleration observed when a body alters its speed by one unit of velocity every second. A body, then, undergoing a unit acceleration (in British units) has its speed increased or dimin- ished by one foot in one second, two feet in two seconds, and so forth. If a body move with velocity V, and at the end of t seconds with velocity V y , the total change of velocity during t seconds is V y V, the time during which this change is effected is t, and y y the acceleration per second is -+ . This is the mean t acceleration during the time t : and the acceleration may during that period t be uniform or variable, but an approximation to its value at any instant may be found by making the interval t as short as possible. Problems. 1. A body starts from rest under the influence of a force which produces acceleration a = 2 ft. : when will it have a velocity of 1000 feet per second ? Ans. At the end of the 500th second. n.] ACCELERATION. 17 2. A body travels at 12 feet per second ; in 10 seconds it is moving 7 feet per second : what is the mean acceleration ? Ans. ^ foot per second. 3. If in the last question the acceleration had been -f J foot per second, what would have been the rate of movement at the end of 10 seconds? Ans. 17 feet per second. 4. A body moves in the first second during which it is under observation through a space of 16 feet ; in the fourth second through 112 feet : what is the acceleration per second ? Ans. 32 feet, so that in consecutive seconds it moves 16, 48,80, 112 feet. 5. A body as it moves is made to record its own speed : it is found that at a certain instant it is moving at the rate of 1 1 2 feet a second ; after an interval of -^ second its velocity is 114 feet per second : what is its accel- V - V 114-112 oration I Ans. = 40 feet per second. 217 6. A particle moves at the mean rate of 10 centimetres per second ; the next second it moves at the mean rate of 8 centimetres per second ; the acceleration is constant : how far will it travel, and what will it do when it has come to rest ? Ans. It will go on for 5j seconds, will traverse 30*25 centimetres, and will return, arriving at every point on its previous path with the same speed as that with which it left it, and will retrace the 30*25 centimetres in another 5j seconds, passing the starting point with a reversed velocity of 10 centimetres per second. Momentum. When a body whose mass is m moves with velocity v, it is said that the total Momentum or Quantity of Motion is the product of these two terms, namely, mv. The greater the velocity or the greater the mass moved, the greater the quantity of motion. Force. When a body which is at rest is set in motion, or one which is in motion is accelerated (positively or negatively) or deflected from its straight course, we attribute these effects to impressed force, or simply to Force. This is sometimes denned as any Cause which tends to alter a body's state of rest, or of uniform motion in a straight line. It is better denned as it is by Xewtou, not as a cause, an existing reality of any kind, but simply as an observed Phenomenon, a measurable Action upon a body, under which the state of rest of that body, or its state of uniform motion in a straight line, suffers change. As to the nature of this action, we can say no more than that it is always associated with the presence of a second body, which possesses energy of motion or of position, but sacrifices it in acting upon the body whose state of rest or uniform motion is disturbed. The mutual action of two bodies is essential to the production of those effects which we commonly attribute to Force. We are quite familiar with the effects of Gravitation, but we cannot yet say that we know what the force of Gravitation is. Forces con- sidered as measurable Actions are measured by the Masses set in c 18 DERIVED NOTIONS. [CHAP. motion, and by the Velocities imparted to them in unit of time- that is to say, by their Accelerations ; and the equation F = ma enables us to measure any force F, in terms of a mass m and the acceleration a imparted to it, as found by observation. Force may also be measured as an Observed Kate of Change of Momen- tum. Acceleration a = rate of change of velocity v; F = ra x a = ra x rate of change of v = rate of change of mv = rate of change of momentum. By a convenient form of speech a given Force is said to act upon a given body and to impart to it a given acceleration. It must be constantly borne in mind, however, that a Force is not a physical entity, and can never be measured until we already know, absolutely or by comparison, the mass acted upon and the acceler- ation imparted to it ; and force may be increased or diminished by varying the arrangement of the bodies to whose mutual actions it corresponds, as in the case of the Hydraulic Press whose ordinary action presents an apparent increase of Force ; while if the action be reversed, Force seems to be destroyed. If a body weighing three pounds be set in motion so as at the end of one second to have had a velocity of four feet per second imparted to it, then F = ma = 3x4 = 12 poundals or British units of force. The same force would have imparted to a two-pound mass an acceleration of six feet, to a one-pound mass an acceleration of twelve feet, and to a twelve-pound mass it would have imparted in one second a speed of one foot per second. If the mass moved be a unit, and the acceleration imparted be unity, the product F = ma is also unity ; and hence the Unit of Force is to be defined as that under the action of which a unit of mass will come to move with unit velocity when it has been acted upon for one second. It will be observed that this definition of the Unit of Force is absolute, is not affected by variations in the force of gravity, and is hence everywhere the same. When we measure force in pounds or in grammes, since gravity varies from place to place, we introduce the unnecessary complication of a local variation in the value of our unit of force that force, namely, which is equal to the weight of a pound-mass or of a gramine-mass. If the unit of length chosen be the centimetre, and the unit of mass the gramme, the Unit of Force will in one second cause a gramme-mass to acquire a velocity of one centimetre per second; and the unit of Force so defined is called a Dyne. Any force may be stated to be equal to so many dynes. Problems. 1. A certain force acts upon m units of mass of matter, and at the end of a second that mass is found to be moving with a velocity of 32 feet per ii.] FORCE. 19 second : what velocity will be produced if the same force act upon B2m units of mass for the same period ? Ans. One foot per second. 2. How many dynes of force are required to set a mass weighing 50 kilogrammes in motion with a velocity of 1 2 metres per second, the force being supposed to act for precisely one second ? Ans. 60,000,000. 3. How many dynes are required to make a gramme-mass move with a velocity of 9 "81 metres per second, the force measured in dynes being sup- posed to act for precisely one second ? what, if it act for two seconds ? Ans. 981 dynes ; 490'5 dynes. 4. Compare the velocities produced by the action on masses of 2 kilo- grammes, 750 grammes, and 1 gramme, of forces measuring respectively 300,000, 112,500, and 150 dynes. Ans. All equal; 150 centimetres per second if the action endure for one second. 5. Equal forces act upon the masses specified in the last question : what will be the relative accelerations produced ? Ans. 3 : 8 : 6000. Weight. We have here to anticipate what will be after- wards more fully discussed, by defining the Weight of any given mass as the Action of Gravity upon it at a given spot. This is identical with a Force acting upon that mass ; and as every force may be represented as equal to ma, the weight of any given mass is the product of that mass into the acceleration produced by the action of gravity. Experiments show that every mass of matter acquires, if gravity act freely upon it for one second, a velocity of nearly 3 2 '2 feet (981 centimetres) per second. This accelera- tion is found to be independent of the nature and of the size of the falling body. The force acting on a falling gramme-mass is therefore ma = 1 grm. x 981 cm. = 981 dynes. The British unit of force (that force which, acting on a pound-mass for one second, produces an acceleration of one foot per second) is one poundal ; the weight of a pound-mass (since it produces in that mass an acceleration of 32-2 feet per second) is equal to 32-2 poundals; or the poundal, as a unit of force, is equal to the weight of ^7^ of a pound-mass. Hence it is said that the British unit of force is nearly equal to the weight of half an ounce. Fig.2. If a pound-mass be divided into two parts, one of which, A, weighs 31-2 1 g^Tcj "* and the other, B, ;r Ibs., and if the weight of the smaller be em- ployed to set both in motion, then the whole mass set in motion is 1 lb., 20 DERIVED NOTIONS. [CHAP. n. and the force acting is ^ of the weight of 1 lb., i.e. r^ of 3 2 '2 poundals that is, one poundal. Hence F= 1, m = 1 ; and since F = ?na, a, the accel- eration, must be unity, and at the end of a second the whole mass will be found to be moving at the rate of one foot per second. We can state, then, that any given force is equal to the weight of so many units of mass at a certain definite place. The observed acceleration, 981 (or 3 2 '2 according to the system employed), is not constant from place to place on the earth's surface, and so weights vary while masses do not. This variable number is, however, usually represented by the letter g, which denotes the local acceleration produced by gravity, and then the equation F = ma is transformed for convenience' sake into w (weight, or Force of gravity) = mg. CHAPTER III. MEASUREMENTS. IN the foregoing chapters we have become acquainted with the units of Space, Time, and Mass, and with those, derived from the preceding, of Velocity, Acceleration, Momentum, and Force ; and it is for us now to ascertain what principles are made use of in the various measurements effected in terms of these units. In the Measurement of Lengths two main methods are resorted to Line measurement (mesure it traits) and End measure- ment (mesure a bouts). The former is the method in habitual use among carpenters, who lay off so many feet and inches by the aid of their pocket-rule ; the latter is the method which they use when they measure the width of a cavity by means of a pair of callipers, which they open until it exactly fits the space. Line Measurement. The length of any line may be measured by a graduated scale, which may be, like the carpenter's pocket-rule, somewhat roughly graduated ; or instruments on the same principle may be made use of which are finely and very accurately divided. The measurement is effected by observing the nearest coincidence of the marks on the scale with the length of the object, and by then reading off the value on the scale. This is a familiar operation, but it will be observed that it depends on the accuracy of the sight. The eye requires to be held directly first over one then over the other end of the object to be measured, together with the corresponding part of the scale, for the true coincidences would be disturbed if the scale and object were looked at obliquely. It is found that in estimating measurements which are very small the eye gets confused ; and besides this, the difficulty of making accurately and very finely graduated scales increases with the accuracy required. Thus it happens that, while the ordinary 2 4 -inch rule divided into various fractions of an inch, or Wnitworth's very convenient 20-inch rule, decimally divided, or a measure divided to half or quarter millimetres, may be used for measurements involving differences of the hundredth of an inch, it is only with difficulty that they can be so applied, and it is much more convenient in cases involving minute measure- ments, such as observations of the height of the barometer or thermometer, to use a contrivance called a Vernier. This is a subsidiary scale which slides up and down past the main scale, and is differently divided from it. 22 MEASUREMENTS. [OHAP. There are two kinds of Vernier in use, those known as the Barometer- Vernier and the Sextant-Vernier, which require separate description. The Barometer-Vernier is thus graduated : a line is set off on the vernier equal to eleven divisions on* the main scale. This line is divided into ten equal parts. Each of these parts therefore is equal to 1 y 1 ^ division on the main scale. If the main scale be divided to tenths of an inch, the difference between a division on the main scale and one on the vernier is y^j- inch. Suppose the object measured to be more than 29 '5 inches and less than 29-6 inches on the scale. Fig.s. Barometer Sextant 30 The 10 A 5 zero of the vernier is laid, as exactly as possible, opposite to the point whose position is to be found, the ex- tremity of the object to be measured, or the height of the mercurial column in the barometer ; then on looking down the vernier it will be found that at some point there is a coincidence be- tween a graduation mark on the vernier and one on the main scale. The num- ber of that mark on the vernier is noted, and that is the figure required in the second place of decimals. For example, let the point A be above 29-5, below 29'6 incbesj the zero point of the vernier is brought opposite to it : the point 6 of the vernier coincides with a division of the main scale ; the length is 29-5& In the Sextant -Vernier, as it is called in this country, which is the form more usually found in instruments of foreign make, the divisions on the vernier run in the same direction as those on the main scale. A line is set off on the vernier equal to nine divisions of the main scale ; this is divided into ten parts, each of which is equal to nine-tenths of a division of the main scale. In the same way the vernier is moved until its zero point is brought opposite the end of the object to be measured, and the mark up the vernier which first coincides with a division-mark on the main scale gives the figure as before. Frequently, as in the sextant, a little magnifying glass is so placed that the zero point of the vernier may be by its aid brought more accurately opposite the object to be measured. When still greater accuracy is required, a microscope is so placed as to ensure the greatest possible completeness of coincidence between the zero point of the vernier and the end of the object to be measured, both of which are simultaneously brought into the centre of the field. The Cathetometer is an instrument whereby vertical heights are measured. It consists of a vertical rod on which a finely-graduated scale is engraved. This carries a sliding piece to which is attached a telescope. In this telescope is fixed a pair of spider threads or fine platinum wires arranged at right angles to one another, and so placed (in the focus of the eyepiece) as to be visible simultaneously with the object looked at through 29 10 in.] MEASUREMENT OF LENGTH. 23 the lenses. The telescope-carrier is placed in such a position on the vertical rod that the lower end of the object to he measured is seen, when looked at through the telescope, to coincide exactly with the point of crossing of the spider-threads in the field of view ; then it is slid up until the upper end of the object to be measured appears to coincide with the same point ; the distance along which the carrier has been slid along the vertical rod indicates the height of the object to be measured. Provision must be made in the construction of the apparatus for ensuring that the vertical rod is quite perpendicular to the horizon j'this is effected by making it stand upon three screws whose heights can be adjusted till a spirit-level indicate the base of the apparatus to be quite horizontal. It may be necessary to compare a standard measure with the length of a body which is very nearly of the same length as the standard. In this case, a microscope may be placed at each end. Coincidence as perfect as possible is established between the images of the object and the standard measure in the field of the first microscope. If, then, the coincidence is perfect in the field of the second microscope, the object is of precisely the same. length as the standard. This but rarely occurs, and the object in view frequently is to ascertain what the error amounts to. The second microscope is provided with spider threads in the focus of the eyepiece, and the end of the object is brought exactly under the apparent crossing-point of these threads ; then the microscope is moved along until the end of the standard appears to be in the same position ; the extent to which the microscope has been moved indi- cates the difference between the two lengths compared. Since, however, the amount to which the microscope has been moved may be exceedingly small and difficult to measure, the methods hitherto described may be insufficient in accuracy, and we have to resort to those more delicate devices which depend on the properties of the screw. The Screw, as will be seen on examination of any specimen, presents a spiral coiled round a cylinder. If a screw, having twenty threads to the inch, be inserted in a fixed body, and turned round exactly once, its point will have advanced the twentieth part of an inch. If the head of the screw be connected with a pointer fixed on it at right angles, which can indicate on a graduated circle the amount of rotation of the screw, there will be no difficulty, even with roughly made apparatus, in causing the screw to execute a rotation of half a circle, a quadrant, 45, 5, or even 1. If a screw, then, which has twenty turns to the inch be turned through one degree (^j- of a complete turn), its point will have advanced or been retracted by 3~5U" x "2TT = 7 2*0 o i ncn - But this is rough measurement. By making the head of the screw part of a large wheel with graduated circumference and using a fixed vernier, rotation of the screw to the extent of half-a-minute of arc can be easily observed, and this would correspond to onward motion on the part of the point of the screw of 7 6 ^ 6 inch. The principle of the screw thus enables us to detect and to measure very small quantities of motion. If the second microscope in the last paragraph be connected with a graduated screw of this kind, the amount of its motion, indicating the difference of length of the two objects measured, can be very exactly determined. In Sir Joseph Whitworth's measuring machines advantage is taken of another principle for producing and measuring very slight motion. The screw (twenty threads to the inch) is driven by a "worm wheel," a wheel bearing 200 teeth on its circumference : this is propelled by a tangent screw, a screw whose threads fit between the teeth of the worm wheel : 24 MEASUREMENTS. [CHAP. each turn of the tangent screw sends each tooth of the worm wheel forward into the position previously occupied by the tooth immediately before it that is to say, causes the worm wheel itself to revolve through the two-hundredth part of 360, and to press the point of the screw forward by the four-thousandth of an inch. But the tangent screw is itself driven by a wheel divided into 250 parts, so that if this wheel be turned round only one division, the tangent screw is rotated -^ of a turn, and the point of the "worm wheel" screw is thus pressed forward the 250th part of ^^j, i.e. the millionth part of an inch. If a screw be fixed at each end so that it can rotate but not progress, the " thread " of the screw will appear to travel when the screw itself is turned. If any object (the slide-rest of a lathe, or the like), have a female screw * cut in it, and be by means of that screw fitted upon a rotatory but otherwise fixed male screw ; and if it be then placed between guides so as to be free to move backwards and forwards along the fixed screw but in no other direction : if then the fixed screw be rotated, the object borne by it will travel along it in one direction or the other, according to the sense of the rotation. This mechanism will be thoroughly understood on looking at the traversing screw and slide-rest of a lathe. If the travelling carrier bear a pencil or a diamond, and mark paper or glass at equal intervals, as indicated by equal rotations of the driving wheel, we shall have a contrivance illustrating the main prin- ciple of the Dividing Engine which is used for graduating thermometer- tubes, etc. End Measurement. If a couple of rods, exactly ten feet in length, be placed on the ground end to end ; if then the first rod be taken up and care- fully laid down endways at the other end of the second ; and if the second be taken up and placed in the same way beyond the first and just in contact with it, and so on : then a very accurate setting off of any multiple of ten feet can be easily effected, provided that the rods themselves are exactly ten feet long. Measurement of a given length can also be thus effected : if there be an odd number of feet and inches, they can be measured by a set of smaller rods, or by an ordinary tape measure. In measuring or setting-off in this way, it is plain that we depend upon the sense of touch for the perception of the contacts set up between the ends of the rods. The sense of touch is found to give more satisfactory results in many ways than the sense of sight ; for if one object be intended to fit into another, and have a diameter g-oVo' inch less than what is exactly necessary, its fit will be perfectly loose. The eye could not perceive this directly without the intervention of lenses. The Callipers used by carpenters can be opened out so as exactly to fit into a cavity, or exactly to grasp an object. They are usually made so that the one end serves for inside, the other for outside measurement. They are useful in comparing the dimensions of objects which should be of the same size ; but it is difficult to take very accurate measurements off a scale with them. Gauges are made of known sizes, and the size of the object to be measured is compared with that of the gauge by trying the fit. If the gauge be made conical, then from the extent to which it penetrates a given aper- ture can the width of that aperture be determined. * A screw cut out of a solid mass, through which another screw, the " male" passes. In the ordinary nut and holt, the bolt bears the male screw, the nut the female. III.] MEASUREMENT OF LENGTH. 25 The Spherometer consists of a disc of metal with graduated circum- ference. This is supported on three equal legs, which are furnished with hard steel points, equidistant, and rounded off so as not to pierce any object on which the instrument is set. In the axis of it is a screw, the steel point of which is also rounded, and forms a fourth foot. Any instrument which stands on three feet is certain to be steady, because three points are always in some one plane : one which stands on four feet will only be steady if the point of the fourth foot is exactly in the same plane as the other three. If it be above this plane the instrument does not rest on it at all ; if it be below this plane the instrument can never stand on more than three feet at a time, and may be rocked from one set of three to another. If the sphero- meter be set upon a piece of glass, it will stand steadily ; if the central screw be turned so as to bring down the fourth foot, the instrument will be easily rocked if it be brought down too far. The hand in perceiving and the ear in hearing this rocking, just at its commencement, concur in detecting very small motions of the screw just at that part of its movement. By means of a pointer attached to the head of the screw the exact position of the screw which corresponds to the commencement of rocking can be observed on the graduated scale. Suppose the thickness of a piece of microscopic cover glass is to be determined. It is placed under the fourth foot. This central foot of the spherometer is brought down upon it until the whole rocks ; the central screw is then raised until the rocking ceases ; it is turned back again till it just commences, and, as before, the position of the screw corresponding to the commencement of rocking can be observed by means of the pointer and the graduated scale. l the pointer had stood at 75 when the instru- ment stood on the plain glass, and at 3 when the central point was on the piece of thin glass, the difference of position of the pointer coresponds to 72, or ^g^ of the circumference ; and if the screw itself have twenty turns to the inch, the thickness of the glass is -/-$ x^ = TTTTT inch. The curvature of a lens may be determined by this instrument, for if the lens ABD be placed u.nder a _ spherometer, Fig. 4 shows that the amount of curva- ture depends upon the length of the line DE, and it is known that the radius R, of the sphere of which the lens may be considered a part, is related to the line DE A^ 1 ^B (represented by a), and the distance I between the equidistant tripod feet, by the formula 2 2E = gj- + I. In Whit worth's Measuring Engine a bar representing the unit of length is placed between two jaws, which are made to move towards one another so as, without pressure, just to grasp it : they are then separated from one another, and the standard unit removed : the bar to be measured is placed instead of it, and the jaws are again brought together so as to grasp it in the same way. The jaws are brought together by fine screw adjustments, such as those previously described, so that the difference of the millionth part 20 MEASUREMENTS. [CHA.P. Fig.5. of an inch in two bars of metal can be detected. The precise position at which the jaws grasp objects without pressure is determined by a plane piece of metal, which is included along with them between the jaws, with its edges in a vertical plane. If the grasp be too loose, this piece of metal can be moved freely, and will fall back when lifted and let go ; if the grasp be top tight, this metal plane cannot be moved ; if it be exact, the metal plane can be raised, and will remain in any position in which it may be placed. Another plan by which an alteration in the length of a bar may be determined is the Optical. The end A of a bar AB rests against a strong framework at B, so that any alteration in its length may only affect the position of the point A. At A the bar is in contact with a lever CD, jointed at E, and bearing a mirror at D. A lamp at X casts a ray of light on the mirror ; this is reflected to a screen SS'. If BA alter in length, or if another bar of slightly dif- ferent length be substituted for it, the bar CD assumes another position, and the spot of light on the screen SS' is deflected. From the amount of deflection may be calculated the alteration in length of the bar BA. Good linear measurement, in whatever way effected, ought to present an error less than y^J^ %, or one-millionth of the whole. . Measurement of Surface. If a surface be bounded by straight lines at right angles to one another, the parallelogram may be measured by the product of two adjacent sides : if it be of any other form bounded by straight lines, it can be broken up into triangles, and its area be found by the rules of trigono- metry: if its boundary be a regular curve, its area can generally be found : but if the sur- iX face be bounded by an irregular curve, the de- termination of the. area involves the following principle. Let the figure YXO be bounded by the two rectangular straight lines OY and OX, and the curve ABCDX. Find its Fig.G. O abode III.] MEASUREMENT OF SURFACE. Fig.7. area. Draw a series of lines parallel to OY ; these will cut the curve in the points A, B, C, D, E, and so forth. Then the area YXO is divided into a number of narrow parallelograms, OYAa, AafrB, B&cC, etc. Each of these is equal to the product OY x Oa, aA x ab, etc. : these being all found and added together give the area of the surface. If now the surface be completely bounded by an irregular curve, as in Fig. 7, the Y area ABCDEA is first found by the above method, then the area A&CDEA. The differ- ence between these re- presents the area of the curved surface ABC b. This method is very diffi- cult in actual practice, but all the mathematical methods of integration are based upon this prin- ciple. For actual work a convenient means of measurement of surface, which gives very fair results and which is specially useful in those cases in which mechanical contrivances have registered their own performances on paper, is the follow- ing : The paper on which the curve is drawn is laid on a flat board, and the outline of the surface very carefully traced* by a sharp- pointed penknife, so as to cut out the part of the paper bounded by that outline : this is then weighed and its weight compared with that of a standard area, say a square inch of the same paper. This method is not unexceptionable, but it often gives a very useful approximation to the value required. An instrument called a planimeter is also used for this purpose. Measurement of Volume. The volume of a substance may often be found by calculation from its form if that be a known geometrical figure ; but the volume of a mass of irregular figure is best ascertained by the rough method of immersing it in water or any liquid which will not affect it, and by observing how much more bulk the whole now occupies than the water alone had done. 2 8 MEASUREMENTS. [CHAP. If, for instance, a piece of metal be placed with three fluid ounces of water in a measure, and if the whole measure exactly four fluid ounces, the piece of metal must occupy exactly the same bulk as one fluid ounce or ^L- gallon of water; that is, since a gallon of water occupies 277*274 cubic 277-274 inches, or 3'466 cubic inches ; and so for fractional parts of the units of liquid measure. Measurement of Time. It is not possible or necessary to do more in treating of this than to suggest one or two leading principles. A simple water-dropper, consisting of a vessel of water in the bottom of which there is a minute hole, through which the water falls, drop after drop, into a dish, was used anciently under the name of the Clepsydra. The water which fell through was kept in the lower vessel : the amount there ac- cumulated, or equally the loss of level in the upper vessel, indi- cated approximately the lapse of time. It was found, however, that the flow of water from a vessel of this description was far from uniform. The use of Wheel work set in motion by some constantly acting force was a fruitful suggestion : setting the wheels to indicate the amount of their own rotation by means of pointers connected with their axles was a plan early adopted ; the train of wheelwork was set in motion by a falling weight ; but these wanted some regulating contrivance by which the motion might be rendered uniform. A heavy flywheel was adapted to the mechanism, but without the desired result being fully attained ; and it was only after Galileo's observation of the fact that the Pendulum oscillates from side to side in almost exactly equal periods of time, whether its arc of oscillation be great or small, that it was suggested that this property of the pendulum might be rendered available for regulating clockwork. This was effected by Huyghens ; and the action of all pendulum clocks, however various the trains of wheelwork, depends on their regulation by an isochronously i.e., in equal times oscillating pendulum. The simplest mode in which this regulation may be effected is the following: One of the wheels of the train of mechanism bears on its circumference an appropriate number of teeth. The descent of the weight would, if there were no pendulum attached, cause the mechanism to run on continuously until the weight had run down to its lowest possible point ; but at every stroke of the pendulum one of the teeth of the wheel is caught and the farther progress of the wheelwork arrested. The isochronism of the oscillations of the pendulum is not sustained ; variations in the external temperature cause changes in.] MEASUREMENT OF TIME. 29 in the length of the pendulum, and hence in its rate of motion. The contrivances by which compensation is made for this cause of error, so that the rate of oscillation is maintained practically uniform, will be explained under Heat. The measurement of small intervals of time is of great im- portance. A tuning-fork, if a writing-point be attached to it, will, when vibrating, describe wavy lines on a piece of smoked glass or paper drawn under the writing-point. If the tuning- fork vibrate 400 times per second, the time taken to draw each wave on the paper must be the four -hundredth part of a second; and if any other phenomenon be so produced and arranged as to record its own performance by a line on the paper or the glass, parallel to the wavy line of the tuning-fork, its duration may be estimated by counting the number of recorded vibrations of the tuning-fork to which that duration corresponds. Measurement of Mass. Masses are compared with one another by means of the Balance. The accurate and expeditious use of a delicate balance involves attention to certain practical rules, which will be found set forth in Kohlrausch's Physical Measurements. Measurement of Force. There are four main methods of measuring any force. These may be stated as 1. Direct Observation of Mass and Acceleration. 2. Direct Counterpoising. 3. Indirect Counterpoising. 4. The Method of Oscillations. The first, the method of direct observation of the mass moved and the acceleration imparted to it by the force to be measured, is based on the equation F = ma ; and if m the mass and a the acceleration be known, F, the Force acting, can easily be found. This method presents, however, serious practical difficulties in the observation of the acceleration produced. One important problem to be solved by this method is the determination of the force with which gravity acts upon a unit mass of matter at any place. The equation F = ma shows that if we use a unit mass, F = a, and we need only find the acceleration produced. This is effected roughly by Attwood's Machine. In this the weight of one gramme is used as the force which sets a larger mass in motion. If it set only its own mass in motion, a velocity is acquired so great as not to be easily observed : if this limited force set a larger mass in motion, the speed acquired is less, varying inversely F as the mass, for a = If a gramme in falling set a mass of 100 grammes (including its own substance) in motion, it can only acquire a velocity 30 MEASUREMENTS. [CHAP. one-hundredth that which it would have acquired if it had falleii alone. The essential part of Attwood's machine consists of a wheel over which two masses are suspended. Let these masses be 49^ arid 50^ grammes. The total mass set in motion is 100 grammes, and the force acting is the excess in weight of the heavier mass over the lighter that is, 50 J - 49 J =~the weight of one gramme. Let this gramme not be a fixed part of the heavier mass, but merely a piece of wire which can be removed by making the weighted mass fall through a metal ring. A pendulum which beats seconds regulates a time- piece ; attached to the wheel work of the timepiece is an " excentric," which works a lever; this lever, at a pre-arranged instant, pushes or pulls away a little plate which supports the heavier mass ; this mass finds itself suddenly exposed to the action of gravity ; the weight of the little gramme-load imparts to the whole mass a certain velocity ; the ring is placed at such a position as to catch the wire exactly at the end of one second, this being indicated by the sound of the timepiece and pendulum coinciding with the click of the wire upon the ring which catches it as it falls. Thereafter there is no force acting, and the mass of 99 grammes continues to move uniformly according to the first law of motion. Its speed can be observed by comparing the distance it travels with the ticking of the timepiece. This is done by placing a little plate to receive the falling body. A slight sound will be made by the falling body touching this plate. If this sound and that of the pendulum coincide the plate is in the right position ; if not, that position must be found by a process of trial and error. It is found that if a pair of masses of 49 J grammes each be suspended over the pulley, and one of them be loaded with one gramme so that the whole mass to be moved weighs 100 grammes ; if .the overweight be taken off at the end of one second by a ring ; if the balanced masses be allowed to move onward with their then acquired velocity for one second ; if a plate be so adjusted under the ring as to check this motion precisely at the end of a second it is found that that plate must be 9 "81 centimetres below the ring. This shows that the force acting (the weight of one gramme), acting for one second, is able to impart to a mass of 100 grammes a velocity of 9 '81 centimetres per second. Hence by the equation F = ma, F, the weight of one gramme, is equal to 100 grammes x 9'81 centimetres = 981 dynes. This method can give no more than an approximation to the value required. Much greater accuracy is attained by the use of the Pendulum. The time of oscillation of a pendulum, as we shall afterwards learn, varies in- versely as the square root of the force of gravity at the place where the in.] MEASUREMENT OF FORCE. 31 observation is made. The time of the oscillation of any pendulum can be very accurately known by observing the time taken to perform a certain sufficiently large number of oscillations, and dividing that time by the whole number of oscillations. From this observation can be deduced the local acceleration of gravity. Measurement of Force by Direct Counterpoising. In an ordinary balance whose arms are perfectly equal, the force with which gravity acts on the mass in one pan is equal to that with which it acts on the mass in the other. For one of these we may substitute another force of any kind but of equal amount. If, for instance, we use a balance with glass pans, we may lay one of the glass pans on the surface of mercury and determine what mass must be put in the other pan to pull the first from the mercury. Let this be 47 grammes, and the area of the glass pan 25 square centimetres. Then a force equal to the weight of 47 grammes is necessary to pull 25 square centimetres of the sur- face of glass away from mercury that is, 1'88 gramme per square centimetre ; and the force of adhesion between mercury and glass is for every square centimetre equal to the weight of 1-88 grammes that is, a force of 1*88 X 981 = 1844*28 dynes. A soap film tends to contract. If we find how much weight must be suspended on a soap film to prevent it from contracting, the force of contraction will be equal to the weight of the mass which it supports, and that force can hence be measured in absolute units. Measurement of Force by Indirect Counterpoising. Let us suppose that we have access to a standard unit of mass. This is hung upon a spiral or spring of steel wire. It is observed to lengthen the spring by a certain measured amount. If another mass be hung upon the same spiral, and if the lengthening pro- duced be the same, the inference is that the action of gravity upon the second mass is equal to that on the first, and hence,- if the two observations be made at the same place, that the second mass is itself equal in quantity to the first. This is the principle of the Spring Balance. Different known weights may be sus- pended on such a spiral, and the elongations produced may be recorded on a scale attached to the instrument. If a mass of unknown amount be attached to the spiral, its weight may be found by reading on the scale the number of standard pounds and ounces, etc., requisite to produce the same distortion as the unknown mass causes when hung upon the spring. 32 MEASUEEMENTS. [CHAP. The instrument known in one form as a spring balance is known in another as a Dynamometer. The form of the steel spring used is quite independent of the principle involved, which is that if two forces produce equal distortions in a body, these forces must be equal to one another. If a man can pull a spring out two inches, and if 200 Ibs. must be hung on the spring to pro- duce the same distortion, the man's pull is equal to the weight of 2 Ibs. ; similarly the force required to pull the spring out two inches is equal to that which must be exerted to raise a weight of 2 Ibs. ; and these can be translated, when we know the local acceleration of gravity, into forces measured in absolute units. If he can give it a blow which will compress it for a moment to the same extent as a weight of 140 kilogrammes placed on it would do, the force of his blow is equal to the weight of 140 kilogrammes that is, 140,000 x 981 = 137,340,000 dynes. If he can, by closing his hand firmly, distort a spring to a certain extent, it can easily be ascertained ^what amount of weight acting on the spring is capable of pro- ducing the same distortion. This is usually done beforehand, and the instrument is provided with a graduated scale which indicates what amounts of weight at the place where the in- strument is made, be it remembered correspond to the various readings of the pointer. When his flexor muscles contract so as to force the pointer of the dynamometer to indicate 84 kilo- grammes, the distortion produced by them is equal to that pro- duced by the weight of 84 kilos at Paris if the instrument has been made there that is, since W = mg, 84,000 grammes x 981 cm. = 82,404,000 dynes. Illustrations of this principle abound. The attraction of magnetism may be measured in a similar way. Let a magnet attract a piece of iron, which is attached to a spiral, to such an extent that the spiral is lengthened, say one inch, when the magnet is at a distance of a tenth of an inch from the iron. It is found that, say, 2 Ibs. 3 oz. must be hung on that spiral to produce the same distortion ; the magnetic attraction is equal to the local weight of a mass of 2 Ibs. 3 oz. This is an undesirable method from the practical point of view, but it shows how magnetic and other attracting forces can be compared with forces whose absolute amounts we know. If an electromagnet can hold ten pounds of iron, but cannot support ten pounds and a grain, the force of attraction is equal to the weight of ten pounds ; for instead of the magnetic attrac- Hi.] MEASUREMENT OF FORCE. 33 tion, we might have used, in order to prevent the ten-pound-mass of iron from falling, another ten-pound-mass connected with it by a cord passed over a pulley. If we take a bar of metal, suspend it on centres at each end, fix it firmly at one end so as to prevent that end from rotating, and hang a known weight over the side of that end which is free to rotate, we find that the bar is twisted ; this effect is measurable. Whatever other force will produce the same effect must be equal to the known weight which caused it. If the body to be twisted be a glass or silk fibre, the amount of force required to twist it is small. To twist such a fibre through a certain number of degrees, a certain fractional number of grammes' weight must be applied at unit-distance from the centre. If an electric attraction be applied to a body suspended by such a silk fibre, the suspended body is attracted, the suspending fibre may be twisted ; to produce the observed torsion or twist, a certain number of grammes' weight must be applied ; the electric attrac- tion can be stated to be equal to the weight of so many grammes, and therefore to so many absolute units of force. The fourth method is that of oscillations. If a magnet be brought near another magnet it oscillates from side to side. If it be brought near a stronger magnet it oscillates more fre- quently. It can be proved that the velocities produced vary as the square root of the forces causing the oscillations. Hence we count the number of oscillations in a given period in two cases, and the ratio of their squares is the ratio of the two forces. If, for instance, a magnetic needle oscillate fifteen times a minute in the presence of a magnet A, and sixty times in the presence of a magnet B ; the forces acting in the two cases are as the square of 15 is to the square of 60, or as 1 to 16. In this way we are able to compare the forces acting under the given condi- tions, but we do not learn the absolute amount of either. That must be ascertained by one of the other methods previously discussed. CHAPTEE IV. WORK AND ENERGY. Work. When a force " acts upon " a body, and that body moves in the direction of the force, that force is said to Do Work, and the work said to be done ~by it is measured by the product of F, the force acting, into s, the space through which the body has moved Work = Fs = mas. For example : Steam exerts on the piston of a cylinder a force or pressure of, say, 30 Ibs. per square inch ; the area of the piston is, say, 30 square inches ; the whole pressure exerted is thus equal to the weight of 900 Ibs. The piston is thrust through, say, 16 inches. The work done is 900 Ibs. x 1 J ft. = 1200 foot-pounds at each stroke. Similarly, when one force is resisted by another, which acts through a certain space, in a direction opposed to that of the force resisted, the product of the force resisted into the space traversed represents the work said to be done against the force resisted. When a ten-pound mass is raised ten feet against gravity, the work done against gravity is equal to the product of the space traversed into the force resisted i.e., 10 ft. x wt. of 10 Ibs. = 100 foot-pounds. In this case Work = Fs as usual ; but F, the force resisted, is the weight of a mass m, and therefore F = mg : consequently the work done Fs = mgs. If Fs = 1, we have the Unit of Work. This is the case when F = 1 and s = 1 ; that is, a unit of work is done when a body acted on by unit force moves through a unit distance. In centifeiaum second-gramme measures the unit of work is done by raising ^ T gramme (mass whose weight at Paris = 1 Dyne) to the height of one centimetre. This is the Erg. The erg is, however, a very small unit of work, and for many purposes it is convenient to use the Megalerg, which is equal to 1,000,000 Ergs, and would therefore be the amount of work done in raisino- CHAP, iv.] WORK. 35 ^ =10'19 grammes through one metre; or the Ergten, 10 10 981 or 10,000,000,000 Ergs; or the Watt, 10 r or 10,000,000 Ergs. In British measures 32*2 units of work are done in raising a pound-mass through one foot. Such units of work are called foot-poundals. British engineers are in the habit of using the foot-pound (the work spent in raising one pound one foot) as a unit of work. This would be satisfactory if foot- pounds were equal over the whole earth, but g, the accelerating force of gravity, varies. Hence the foot-pound is from place to place a variable measure, and has to be reduced for each place to absolute units of work by the equation Work = Fs = Weight x s = M<7, and the foot-pound is equal to g foot-poundals. The foot-pound is equal to 13,562,691 Ergs. The kilogramme-metre, or French engineers' unit of work, is 1000 grms. x 100 cm. x g = 98,100,000 Ergs. Any amount of work may be specified as the product of two numbers, a force and a space. These may vary, but if they have the same product the amount of work done is the same. A pound raised 100 feet, 100 pounds raised one foot, fifty pounds raised two feet, four pounds raised twenty-five feet, all represent the same amount of work, namely, 100 foot-pounds, it being here assumed that the force of gravity is uniform within heights of 100 feet. The mean rate of doing work is the whole work done in a given time divided by the time. If an engine can raise 1,980,000 pounds one foot in an hour, its mean rate of doing work is 33,000 foot-pounds per minute, or 550 foot-pounds per second. This particular mean rate is known by Engineers as a Horse- power; and an engine of one-horse power can do this amount of work. A horse can, according to General Morin, do 26,150 foot- pounds per minute, and a labourer from 470 (lifting earth with a spade) to 4230 per minute. The French horse-power (cheval- vapeur) is 75 kilogrammeters, or 7,500,000 g = 7,357,500,000 Ergs per second. If a man weighing 14 stone run upstairs at such a rate as to gain 3 feet in vertical height every second, his muscular system is doing every second the work of carrying 196 Ibs up 3 feet, i.e., 588 foot-pounds. If this could be kept up for a minute, 60 x 588 = 35,280 foot-pounds would be done, and the man would be in the case supposed undergoing an exertion which for the moment would be much greater than a horse can keep up, and seventy- five times that which a continuously-toiling labourer, lifting earth with a spade, can sustain ; and even under the most favourable circumstances, a labourer, raising his own weight merely, can only keep up one-eighth of this effort. Energy. When a body weighing ten pounds is raised ten feet, and prevented by a catch from falling, the work done upon 36 WORK AND ENERGY. [CHAP. it 100 foot-pounds can be recovered "by permitting it to fall upon a train of mechanism. If the mechanism were per- fect, the work would be so restored that another ten-pound mass might be projected by it to a height of ten feet, a fifty-pound mass to a height of two feet, and so on. The fact that we cannot obtain perfect mechanism does not affect the principle. The body at a height has therefore a power of doing work, equal to the work done upon it in lifting it. In this case the power of doing work has been conferred upon a body by the separation of it from the earth against the action of gravity : as it remains in its elevated position, there is a stress, or pull, or attraction, tending to draw it down, and it is only in virtue of this stress that it has any power of doing work. If the earth and the elevated body ceased to attract one another, the body would, if liberated, not fall down, and would not restore the 100 ft.-lbs. of work spent upon it. We know that the work done in raising a mass m through a height s against gravity is mgs : the energy stored up in the body is therefore equal to mgs, and is seen to depend on the mass of the body, the height at which it is placed, and the local accelerative effect of gravity. Energy, or power of doing work stored up in this, way, is called Potential Energy, or Static Energy, or Energy' of Position, or Energy of Stress. As an example of Potential Energy we may take that stored up in a mill-pond. The number of units of Energy in such a pond may be found by taking the product of the quantity of water in it and the average height at which it is placed, and multiplying that product by the local value of g. A small quantity of water at a great height may obviously have the same amount of energy Stored up in it as a larger quantity at a lesser height. If the question be put, How much work could be got by appropriate mechanism from the rise and fall of the tide ? we consider (1) the total amount of water carried into the area which can be brought within the range of the mechanism, (2) the average height to which it rises, and (3) the local value of g. "We have also energy stored up in such bodies as watch- springs. Work is done upon them in distorting them, and pro- ducing a movement, not of their mass as a whole, but a relative displacement of their parts. This work is restored and utilised in producing movement of the mechanism attached. Here we find that the spring, when its form is altered, tends to resume its original form, and exerts a pressure or a pull upon any object so placed as to prevent its doing so. This pressure is transformed iv.] POTENTIAL ENERGY. 3 "7 into a moving force if the bodies pressed upon can move ; if not, the pressure is continuously applied without producing move- ment, and such a pressure or pull is called a Stress. In popular language a stress is called a strain, as when it is said that a bridge or wire is subjected to too great a strain, and breaks or snaps. Properly the word strain means deformation of a body. "When a watch-spring is distorted, the distortion or strain persists, and its whole mass is in a condition of stress, and tends to restore the work done upon it. If we look at our previous example of the earth and a stone lifted from its surface, we see that the phenomenon is on the large scale one of the same order. The earth and the stone together constitute a system: when this is deformed by pulling the stone away from the earth, the system tends to return to its original form, and there is a stress between the earth and the stone, which continues until the stone is allowed to fall back to the earth. If the stone had been connected with the earth by a band of indiarubber, we would have seen the indiarubber to be stretched or under stress, and would easily see that if the stone were liberated, it would be pulled back towards the earth ; but the question is, What is under stress in the actual case ? for there is no visible con- necting cord between the stone and the earth. If we could state what this was, we should be able to arrive at the cause of gravi- tation. As it is our knowledge ceases. That there is some medium, and that it may be under stress, is a theory necessary for the exposition of Electricity, of Light, of Magnetism, and of Heat; but we are as yet not entitled to say that stress in this medium is the cause of gravitation. Work may be done, then, in altering the relative configuration of a system, whether this consists of large masses or of smaller particles. If this, system be what is known as a "conservative system," in which a stress may be established depending upon the configuration, it will tend to return to its original form, and to restore the work done upon it. If its relation to surrounding objects be such that it cannot so return, it will be under stress, and will continue under stress until its relations to surrounding objects have become such as to permit it liberty of restitution ; then, at the first opportunity, it will restore the work done upon it. The change in its relations to surrounding objects necessary to render this restitution possible may be very small ; for example, a heavy mass may be prevented from falling by a very small catch, but when the catch is removed 38 WORK AND ENERGY. the "body falls. The cause of the body falling is not simply the release of the catch, but also the previously existing conformation of the distorted system. Similarly, the ingredients of gunpowder have a tendency to combine i its particles are chemically separate, but chemically attract one another, and there- fore possess potential energy ; the application of a very small amount of heat, as by a spark, liberates these particles, which can rush together and form new and stable compounds, which have no longer any tendency to alter their chemical constitution, being no longer under the same stress, having no longer the same potential energy. As it happens that in this special case the new and stable compounds formed are mainly gaseous at the ordinary temperature and pressure, the products of combination occupy a much larger bulk than the original gunpowder, and the result is an explosion. The spark only pro- duces its own small effect ; the previous arrangement of the particles of the powder is responsible for the rest. Cases in which energy is stored up in mechanical arrangements abound. The air-gun consists of a volume of air which has been, by work done upon it, compressed into a small bulk, and which tends to return to its original dimensions. When permitted to do so, it suddenly expands, and may be made to restore the work done upon it in propelling bullets. When a clock is wound up by pulling up the weights, work is done upon the system ; this is restored by the whole system returning to its original form, the weights descending to their lowest position. It takes a definite number of days or hours to do this, according to the mechanical arrangements devised. The work done in bending a bow is swiftly restored as the bow returns to its original form, and may be spent in imparting motion to the arrow. A non-conservative system is one in which, when the system is deformed, there is no stress established tending to re- store the original arrangement. Such a system is exemplified by a gun and bullet. When the bullet has left the gun, Newton's first law applies, according to which the bullet tends to go straight on at a uniform rate, unless acted on by impressed forces. The bullet forms a part of two systems, one conservative and the other non-conservative; its motion will necessarily be that due to its relations to both. Let it be fired obliquely upwards : in virtue of its separation from the earth, with which it forms a conservative system, a stress is established which brings it back to some part of the earth's surface : in virtue of its separa- tion from the gun it does not tend to return to the barrel of the gun, but goes on until it is stopped. The question, What causes one system to be conservative, another not to be so ? is scarcely to be answered at present. The presumption is that a body if set in motion will, according to the first law of motion, travel onwards in a straight line and with uniform velocity, unless acted on by impressed forces; in other words, that all systems are non-conservative. A shot fired vertically upwards iv.] POTENTIAL ENERGY. 39 should, according to this law, pass on in the same direction with- out ceasing ; but experience shows that it does return, that some impressed force does act upon it, and this, which is another expression for the attraction of gravitation, is at present not explained. Similarly, the particles of a distorted spring undoubt- edly form a conservative system ; stress is established between them : but the explanation of this fact would imply a knowledge of the constitution of those particles and of their actions upon one another which we do not yet possess. Kinetic Energy. Power of doing work is possessed also by bodies which are in Motion. If, for instance, a rifle bullet be received on an appropriate mechanism, the jolt suffered by the instrument might be utilised in producing a certain amount of work. Or otherwise, the bullet, in whatever direction flying, might, by a cord passed over a pulley, be attached to a weight which it pulled up. The simplest case of this problem is, How far can a shot fired from a rifle carry itself vertically upwards, in virtue of the power of doing work possessed by it because it is in motion? It is known (see p. '179) that a body travelling upwards against gravity, and . passing a certain point with a V 2 velocity V, will rise to a height -^ . The power of doing work possessed by the bullet in virtue of its motion (its Kinetic Energy, or Energy of e Motion, or Actual Energy) is com- V s petent then to raise its own mass m through a space s = -~- against gravity whose local acceleration is g. The work done is V 3 mV 2 mgs = mg-^~ = . The kinetic energy, then, of a body moving in any direction with velocity V depends only on its mass and on its velocity not at all on the local intensity of gravity. When the bullet arrives at the top of its course it has no velocity, and therefore no kinetic energy ; but it will easily be seen that if it be caught when " at the turn," it can be retained on a ledge, and will possess potential energy. This we know how to express as mgs. The kinetic energy which the bullet has lost it still retains under the form of potential energy. If it be allowed to fall, it will lose its potential energy, and will have acquired the original velocity V as it passes the point of observa- tion. Let us suppose a body weighing 10 Ibs. to leave the ground, starting upwards with a velocity of 64*4 feet per second ; let 40 WORK AND ENERGY. [CHAP. fj 32*2. Then the body will ascend , or . ^ = 64'4 feet. The body whose mass ra= 10 will rise 64*4 feet, and if caught at the instant when it comes to rest will have a potential energy of 644 foot-pounds. The absolute value of this amount of energy depends on the local force of gravity, but as g is taken =3 2*2, the potential energy may be expressed absolutely as 2 1,7 3 6 '8 foot-poundals. The kinetic energy which the body pos- sessed at the moment of starting was ^- = - = 21, 736*8 foot-poundals, measured directly and irrespectively of the local force of gravity. Hence the kinetic energy lost by the bullet in ascending is exactly equal to the potential energy gained by it. At any intermediate point, where it has less velocity but some potential energy, it will always be found, in the case supposed, that the sum of the kinetic and potential energies is 21,736*8 foot-poundals. The one kind of energy, the potential, is trans- formed into another, the kinetic, and there is in the system (earth and stone) neither gain nor loss of energy during the transforma- tion. This is the simplest case of a widely applicable principle, that of the Conservation or Indestructibility of Energy. This principle is, that if a system of bodies have a certain amount of energy in one form, it must retain that energy in one form or another unless it come into such relations with other bodies as, together with them, to form a larger system in which the energy becomes differently distributed ; and if the system be so large that there is no other body with which it can enter into such relations that is, if the system which possesses the energy be the whole universe that system cannot gain or lose energy by sharing with other bodies, and hence the total amount of Energy in the Universe is invariable and is numerically constant. If we take the instance just discussed, that of the earth, the bullet, and the gun pointed vertically upwards, these three bodies possessed before the explosion a certain amount of energy, potential in the gunpowder : just as the bullet left the gun, kinetic in the bullet : when the bullet was detained at the summit of its course, potential between the bullet and the earth, but always equal in amount, the same number of foot-poundals. When the kinetic energy was being transformed into potential, work was being done in establishing a state of stress. During this period the bullet and the earth were relatively moving, and the acceleration associated with the transformation of one kind of energy into another is attri- iv.] CONSERVATION OF ENERGY. 41 buted to a force acting during that period. Force is associated with a variation in the rate of change of the configuration of a system under which the energy in that system is altered in its distribution and form, and is said to act only as long as that variation continues. The word force is limited to the case in which some movement of masses or of particles is produced, varied, or checked : what is popularly known as the force tending to bring a spring back to its original form, but not actually doing so, is a stress. Stresses may obviously be measured in the same way as the forces which produce them. A spring is pressed upon by a certain weight ; it bends to a certain extent : it is then caught by a ratchet, and the weight is removed. The force necessary to cause the given bending of the spring is known, being equal to the weight of the mass employed : the stress established in the spring is equal to the force used. The upward pressure on the ratchet must be equal to the weight removed. The opposite extremity of the spring imposes an equal and oppositely- directed pressure on its support. Transformations of Energy. Energy, however, assumes other forms than the two discussed. If the bullet in the case adduced be allowed to fall on the ground, it falls more and more rapidly until it regains its original velocity, and therefore its whole kinetic energy. But this bullet may suddenly strike the ground and lose all its kinetic energy : it has already lost all its potential energy ; what has become of the energy of the system ? We find that the bullet and the part of the earth on which it has fallen are warmed, and we learn from a wide induction of similar cases that Heat is one of the forms of Energy. It is proved to be so by the observation that the same amount of work, if entirely spent m producing heat, will always produce the same amount : 77 2'$& foot-pounds of work were found by Joule to correspond to an amount of heat capable of raising the temperature of a pound of water from 60 to 61 F. The heat possessed by a body is explained as being the motion of its particles. Just as a swarm of insects may remain nearly at the same spot while each individual insect is energetically bustling about, so a warm body is conceived as an aggregation of particles which are in active motion while the mass as a whole has no motion. Heat is therefore a form of kinetic energy : and the more heat is imparted to a body the greater is the kinetic energy of each particle. If M represent the average weight of the particles, and V their 42 WORK AND ENERGY. [CHAP. average velocity, - represents the average kinetic energy of each 2 particle; and the sum of all the masses multiplied by half the square of the average velocity represents the intrinsic kinetic energy of the mass. The words "sum of" are expressed by the symbol 2. Hence, Intrinsic Kinetic Energy = 2 ^- = JV 2 2 M V 2 , where m is the whole mass. When a bullet possessing actual energy of motion impinges on a target there is a certain amount of heat obtained, and the bullet may be partly fused : there is also a flash of Light and a certain amount of Sound. These things are also forms of Energy. Light is partly kinetic, partly potential energy of wave-motion in that Ether whose existence throughout space seems to be a necessary hypothesis. So also is Eadiant Heat, such heat as streams to us from the sun. When a tuning-fork is made to vibrate, work is done upon it in giving it in the first place a distorted form. Its arms swing like pendulums, but their vibration gradually dies away and the energy of vibration of the fork becomes converted into the partly kinetic, partly potential energy of vibration of the air that is, into Sound ; and ultimately it is converted into uniformly-diffused Heat. Energy may appear, then, as Energy of Mechanical Position or Motion, as Heat, as Light, as Sound, and again as that of Electrical or Magnetic condition ; and a great part of our work is to study the modes in which the various forms of Energy are transformed and redistributed, and the forces and the phenomena attributed to forces which are associated with these transformations and redistributions. A few other examples of Transformation of Energy may here be added. A man, ascending a stair, gains some potential energy : it is found (Him) that he is perceptibly cooler for a moment. The heat of his body has been partly transformed into potential energy. Of course the exertion of his muscles and the excitement of his circulation cause him to become warm immediately afterwards. When he comes downstairs he sacrifices the potential energy which he had possessed when upstairs in virtue of his elevated posi- tion, and which he might conceivably have utilised by dropping himself out of the window on an appropriate machine placed on the pavement. This energy is not lost, for he is (Hirn) perceptibly warmer at the bottom of the stair than he had been at the top. At every step downstairs he had arrested his own fall, and had consequently converted a part of his potential energy first into kinetic energy and then into heat. When a quantity of water is decomposed by an electric current, the iv.] TRANSFORMATIONS OF ENERGY. 43 electric current is diminished and work is done in tearing asunder a certain number of particles of oxygen and hydrogen. These separated atoms tend to fall together again and form the stable compound, water. The mixture of oxygen and hydrogen thus formed by "electrolysis" possesses potential energy of chemical separation. When a light is applied to the mixture, a process of recombination commences, and the whole of this potential energy is sacrificed as such, but appears in the form of heat, light, and sound, and may in an appropriate gas-engine be partly spent in doing mechanical work. The heat and light produced by combustion and by chemical combina- tions in general are forms of energy obtained by transformation of the potential energy which the particles had previously possessed in virtue of their chemical separation and chemical affinity. Under certain circumstances this potential energy may not be transformed into heat or light, but, as in the galvanic battery, into the energy of a current of electricity, which may in its turn be made to do work, be transformed into heat, into light, into sound, or be spent in setting up magnetic condition, and so on. When an engine goes round without doing work it is heated. When it does work it is not heated to the same extent, and the researches of Him have shown that the amount of work done is exactly equivalent to the difference in the heat produced. The energy of an engine is derived from the heat evolved by the com- bustion of the coal. The coal of the furnace and the oxygen of the air rush together and sacrifice their energy of chemically-separate position, which was originally obtained by the action of the chlorophyll in the coal-producing plants. When a plant is exposed to sunlight it has the power, by means of the chlorophyll or colouring matter of the leaves, of breaking up carbonic dioxide, C0 2 , of evolving part of its oxygen in the free form, and of depositing the carbon in a less oxidised form in its own tissues. The work thus done by the plant in tearing asunder the constituents of C0 2 it is enabled to do by the energy supplied to it in the form of Light and Heat radiated from the Sun. The Sun's radiant energy has next to be accounted for. This is not derived from combustion, for the sun would last but a comparatively short time if its energy were derived from any such source : its radiation of energy seems to correspond to 500 horse-power from every square yard, and such an enormous outflow would soon exhaust the store of energy if the sun were merely a huge fire : it is said that it could not last more than about 8000 years. It has been suggested that the meteorites which fall into the sun in great numbers are capable of accounting for the sun's energy ; of the thicken- ing of the sun due to this cause a very small amount corresponds to a very large amount of energy. Those meteorites which strike our own earth's atmosphere are retarded and greatly heated in their course through the upper regions of the air. If they be small enough they are entirely broken up, and their dust, characteristically ferruginous, settles down on the surface of the earth, and may be recognised in the dust collected from some specially favourable spots, such as glaciers, roofs, and snowy wastes. The kinetic energy lost by a meteorite falling upon the earth becomes distributed between it and the earth in the system of which the meteorite becomes a part, and this contributes to the total energy possessed by the earth ; while its material goes to increase the earth's mass. In the same way, the meteorites which fall 44 WORK AND ENERGY. [CHAP. on the sun must produce a flash of light, some heat, and a slight thickening of the sun. It has also been suggested that a very slight shrinking of the sun's mass would evolve a large amount of energy, its particles not being so far from one another after this contraction in bulk. If the meteorite theory be accepted, the question arises, How did the meteorites get their energy of motion ? This would relegate us to the con- sideration of the universe as a system of masses and particles containing as a whole a fixed quantity of energy : and this would bring us to the problem of the origin of this system. Availability of Energy. When a certain amount of energy has been spent in rubbiug a button, the button is perceptibly warmed. The heat produced is exactly equal to the work done y2 in rubbing. It is m , wbere m is the hot mass and V the average velocity of its moving particles. All this we know. If a little time elapse, the button is no longer perceptibly warm : it has shared its heat with surrounding objects : their particles have been induced to oscillate more rapidly. Heat has thus a tendency to become uniformly diffused. It is then no longer available to man for doing work. It ceases to be power of doing work as far as he is concerned ; but none the less do the particles of a hot body set in motion the particles of a cooler body, and the energy which has thus been imparted to these they can in their turn share with the particles of other cooler bodies. The temperature of a hot body tends uniformly to diffuse itself through the whole material universe. In every transformation of energy we find that some energy is wasted through conversion into Heat, the result, direct or indirect, of friction, noise, flashes of light, and so on. This heat is presently distributed pretty uniformly among the surrounding objects, and can no more be made use of by us for the sake of producing work. A large quantity of the Energy of the Universe must have already assumed this relatively- useless condition, and in the course of time the whole of the Energy in the Universe will have assumed it. The Energy of the Universe is a constant amouut : some of it is available, some is non-available : the former is in every phenomenon somewhat diminished but never increased : the non-available energy is constantly increasing : hence, as Tait puts it, the " Entropy " (available energy) of the Universe tends to zero. Clausius uses the word entropy to signify the non-available energy, and points out that in this sense the entropy tends to increase to a maximum. If with this clue we trace back the history of the Energy of iv.] AVAILABILITY OF ENERGY. 45 the Universe we find, as we go back, less and less of the total Energy of the Universe to have become non-available. On going back we arrive at a period when none of the total energy had become non-available. But in every phenomenon there is dis- sipation in this way of some part of the total energy of a system. Hence we find that we are forced to realise a precise instant before which there were no phenomena such as those with which we are acquainted, and since which the phenomena due to the relations of matter and energy which are within our knowledge have been occurring : while in the future we have to contemplate a moment at which the whole physical universe will have run itself down like the weights of a clock, and after which an inert, uniformly- warm mass will represent the whole material order of things. The only way of escape from this conclusion is to lay emphasis on the fact that one part of the total Energy of the Universe is unavailable to man, and to suggest that at some time a state of things may supervene, as a result of which the molecular motion which is implied in a state of uniformly- diffused heat may be so arranged and directed as once more to produce a state of things such that particles may become aggregated into masses, in which all the particles may move on the whole in the same direction. This is what Clerk Maxwell's " Demon " is pleasantly imagined to do ; he separates those particles which he prevents from going in one direction from those which he allows to go in another, so that ere long, without expending any work, he has the particles divided into two groups, moving in opposite directions. This is interesting, but it is not pretended that it is any other than a speculation. There is one case in which the principle of the conservation of energy is not as yet definitely established. This is in the domain of Physiology, but the words of Clerk Maxwell may in this con- nection be quoted. "It would be rash to assert that any experiments on living beings have as yet been conducted with such precision as to account for every foot-pound of work done by an animal in terms of the diminution of the intrinsic energy of the body and its contents ; but the principle of the Conservation of Energy has acquired so much scientific weight during the last twenty years that no physiologist would feel any confidence in an experiment which showed a considerable difference between the work done by the animal and the balance of the account of Energy received and spent." * " Conservation of Force " an erroneous phrase. There is now no warranty for this expression. It was originally a translation of the * Nature, vol. zix. p. 142. 46 WORK AND ENERGY. [CHAP. German Erhaltung der Kraft, where Kraft, meaning strength or force, was used in 1847 by Professor Helmholtz, for want of a better term, to indicate what is now in the English language rigorously named Energy. Forces are of the same order as pressures exerted, pounds' or grammes' weight, resistance overcome ; forces may be represented by lines which indicate their magnitude and direction. Energy is of the same order as work accomplished, as pounds or grammes lifted or resistance overcome through a certain number of feet or centimetres, and it may be represented by areas which are inde- pendent of direction. The hydraulic press apparently creates force, and if its action be reversed, force disappears ; but the work done upon it must be the same as the work done by it, and though there is no Conservation of Force, yet there is strict Conservation of Energy in this as in all those other mechanical contrivances in which force is altered in amount. We have seen that Energy may be represented by Fs, the product of force resisted through space s ; by mgs where mass m is raised through space s against gravity whose local acceleration is raV 2 g : by when a mass m has a velocity V imparted to it. 2 We shall further see that Energy may be represented by the product CV of a charge C of Electricity into a numerical quantity V, called Electric Potential ; by the product qp of a quantity q of fluid forced into a space against an average pressure p ; by the product of a chemical affinity (which is equal to the work done in separating the atoms of an equivalent of a chemical com- pound) into the number of electro-chemical equivalents which enter into combination ; and in other similar ways. These things, however, will find their explanation in due place. Problems. 9 1. Energy is power of doing work: this depends on r ; a body 2i moving with a certain velocity v t can pierce a plank of thickness t ; if it moves with velocity v u what thickness can it pierce ? Ans t = t ( " ) \*,/ ' 2. A shot travelling at the rate of 700 feet a second is just able to pierce a 2 -inch board. _What velocity is required to pierce a 3 -inch board? ~~ Ans. 700 x 7-7- = 857-42 feet. v * 3. A shot travelling at a certain rate can bury itself 1 feet in sand : how far could a shot travelling with double that speed bury itself ? Ans. 40 feet. 4. If a mass of 154'486 pounds be allowed to fall 10 feet, but in its fall be inade to set a train of mechanism in action, and if that mechanism do no other work than to stir up a pound of water with a paddle, how much will the water thus stirred up be warmed 1 Ans. 2 F. iv.] MEASUREMENT OF ENERGY. 4V 5. If a locomotive weighing 5000 kilogrammes run at the uniform rate of 10 metres per second round a circular railway whose radius is 2 kilo- metres, what will be its kinetic energy ? Ans. m = 5,000,000 grins.; v = 1000 cm. per second ; ^- =2,500000,000000 Ergs, or 250 Ergtens. The energy does not depend on the form of the path traversed, but only on the velocity at each instant along that path ; for kinetic energy is independent of direction. Graphic Representation of Energy. The representation of work by the product Fs (force acting into the space through which it acts) finds its graphical equivalent in the representation of work done as a rectangular Area, the product of two lines, of which one represents the force acting and the other the space through which a body has been moved. If any instrument can be devised which will mechanically describe such an area, the amount of work done by a moving body can be recorded ; such an instrument is a Dynamometer. This name is applied to the apparatus in which an elastic spring is deformed, the extent of its deformation show- ing, by comparison with that produced by a given weight, the amount of Force acting on the instrument. The same name has, however, been given to instruments designed to record not only the force acting on the spring at any given instant, but also the whole Energy spent in producing the deformation, and measured by a simultaneous record of the force acting and of the space through which it has acted. If a distorted spring have a writing-point attached to it, as the distortion of the spring varies the pencil will move backwards and forwards in one line ; if a piece of paper be held against the writing-point as it travels back and fore, the tracing produced is not instructive, for it is simply a line traced over and over. If the paper be drawn past the writing-point at a uniform rate, the line is a curve from which may easily be deduced the mean value of the deforming force during the whole time of observation. If, however, the paper be moved not uniformly but at a varying rate, proportioned at every instant to the space passed through by the moving body (that is, to the rate of change of deformation of the spring), then there are two factors recorded in the same tracing first, the amount of space passed through (this being indicated by the amount of paper unrolled under the writing-point in a given period of time) ; and second, the force which has acted in producing deformation (this being recorded by the oscillations of the writing-point attached to the deformed spring). If the writing-point thus attached to the spring be supposed 48 WORK AND ENERGY. [CHAP. to draw the curve ABCDEF of Fig. 9, the various parts of the line give rise to the following discussion. The line Oabcd shows the various spaces traversed by the body set in motion ; the lines a A, Z>B, cC, etc., show the various pressures at successive instants of Fig.9. time. The condition of affairs is more easily realised if we consider a cylinder the steam in which pushes a piston. Then the expansion of the steam is correlated to the movement of the piston, which may be represented by distances along the line Ox. Then ab denotes the expansion of the steam as its volume increases from Oa to 06, A or 5B its pressure, and hence the work done by the steam during the increase of volume ab is represented by the rectangle AB&. When the steam expands still further, so that its in- crease in volume is Ic, the pressure or force acting is again con- stant, and the work done is represented by the rectangle 1C. Next, when the volume Oc becomes Od } the pressure sinks from cC to dD ; the average value of the pressure is -^ (cC -f dD), and the work done (or average pressure X space cd) = -J (cC + dD) cd. This is the area of the figure cCDd, which accordingly represents the work done during the increase cd of volume. The area De is made up of numerous rectangles, and if these are made suffi- ciently numerous the line DE is a curved line. The area com- prised between the curved line DEF, the ordinates dD and /F, and the abscissa df, represents the work done by the steam during expansion of the steam from the volume Od to the volume Of. The area- aA~Ffa represents the work done during expansion of the steam from volume Oa to volume Of. The Indicator-Diagram. Since the time of Jaines Watt engineers have been accustomed- to make their engines record their own working by the iv.] MEASUREMENT OF ENERGY. 49 method just discussed. In the Indicator-Diagram, as the curve traced out is called, the two factors which it is desired to record are the space traversed, which is measured by the amount of movement of the piston, and the force acting, which is the pressure of the steam in the cylinder. The former factor, the space moved through hy the piston, is determined by making the piston or any part of the machinery upon which it directly acts set in operation the mechanism that unrolls the paper upon which the record is to be preserved. This paper is drawn over the writing -point at a rate depending on the velocity of the piston : hence the space traversed by the piston during successive intervals of time is proportional to the amount of paper which is drawn under the writing-point. The pencil is borne by the piston of a small side cylinder attached to the main cylinder. The steam is let into this, and presses the little ^piston out- wards ; it presses it against a spring until the resistance offered by that spring prevents further propulsion. If the pressure were constant, the little piston would remain at the same level ; but as the pressure of the steam varies, the position of the piston also varies, as it lies between the opposing spring and steam. Its displacement, then, measures the pressure of steam in the cylinder, which is the second factor. If the pencil be drawn over the pencil-point when no steam has access to the side cylinder, a smooth line is drawn, the Line of No Pressure; if the steam be allowed to enter the side cylinder, the divergences of the line then produced from the line of no pressure measure the variations of the pressure of the steam. While the machine is working steam is not allowed to enter the side cylinder until the apparatus is ready to record. The small piston is conse- quently at rest. "While the piston of the main cylinder is moving in one direction, paper is rolled over the pencil-point ; as the piston slackens speed the paper also slackens in speed ; as the piston stops, the paper does the same ; as the piston travels in the opposite direction, the paper travels in the oppo- site way at a proportionate rate. As long as the steam is not admitted into the side cylinder, the paper travels backwards and forwards over the pencil, and the same straight line is traced and retraced. The steam is admitted to the side cylinder for the space of one complete oscillation of the main piston, and the pencil-point itself travels in accordance with the varying pressure of steam during that period. The curve traced thereby is composed of two parts. The one ACB is produced during expansion. The work done by Fig. 10. the steam in the cylinder during its expansion is the area ACB&a if Oa& be the line of no pressure. When the piston has finished its stroke it, and therefore the paper, stands for an instant at rest. Then the piston is pressed against the steam either by other steam or by the atmosphere, and E 50 WORK AND ENERGY. [OHAP. iv. the paper is drawn backwards. Work is thus done against the steam during the backward stroke, and it is represented by the area BDAafr, where BDA is the line recorded. The difference between the areas ACB&a and BDAa& represents the excess of work done by the steam over that done against it : hence the total work done by the engine is represented by the area of the surface ACBD traced out by the pencil in the formation of the so-called Indicator-Diagram. In some cases it is sufficient to know the mean force acting. In such cases the space traversed being a known quantity, the energy can be determined if the mean force alone be recorded. For example, if the mean force required to pull a vehicle be found, it is a very simple matter to multiply the recorded mean force by the space traversed in order to find the total amount of energy expended. In Marey's investigations (Trav. du Ldborat., 1875) into the comparative total work expended on a vehicle, according as an elastic spring is or is not placed between it and the draught animal, a capsule was so arranged that the air in it suffered irre- gular compressions and rarefactions, corresponding to the irregular jolts between the animal and the car. The writing -point, set in movement by the correspondingly-irregular oscillations of one of the walls of the capsule, which was flexible, described irregular lines when there was no elastic intermediary, and more regular ones, nearer the line of no disturbance, when there was such an intermediary introduced; from these it was found that the mean force, and therefore the total energy expended in the two cases were in the ratio of about 4 to 5, showing that the use of an elastic spring between a draught animal and the vehicle which it draws results in an economy of labour amounting to about 25 per cent. CHAPTER V. KINEMATICS. To the part of Science which deals with Motion, considered per se and without reference either to the force producing it or to the body moved, is given the name of Kinematics. The nature of the questions discussed under this title is essentially mathema- tical ; and though no great acquaintance with mathematical methods is presumed in the reader of this volume, it will be necessary to assume in him a certain amount of knowledge of elementary geometry and algebra. GENERAL PROPOSITIONS. Direction. There cannot be Motion without Direction ; we cannot think of a body or a point as moving, and yet not moving in any direction. If it move at all, it must either move so as to travel constantly in the same direction, in which case it is moving in a straight line; or else the direction of its motion must change as it proceeds from point to point of the path traversed, so that the body travels in some kind of curved line. In the great majority of those curves which possess physical interest as being those in which bodies actually do move, it is possible to draw at any point of the curve a straight line known as the Tangent to the curve at that point. The Tangent to the Circle at any point is familiar enough, and is easily understood to be a straight line at right angles to a radius connecting the centre of the circle with that point of the circumference at which the tangent is to be oraw^n ; and the characteristic property of the line as a tangent is that it touches the circle without cutting it. Tangents may in a similar way be drawn to most curves,* so as at any determined point to touch but not to cut the curve, unless * For a valuable discussion of certain anomalous cases, which do not find a place here, see Clifford's Dynamic, pp. 45, 46. 52 KINEMATICS. [CHAP. the curve changes its curvature beyond the point at which the tangent touches it. If a circle be drawn on a very large scale, and a tangent be drawn to it at any point chosen, it will be found that the larger the scale the more nearly will the circle appear to coincide with the tangent at the point of contact. This can easily be seen by actually drawing such a figure. In fact, if a circle be drawn on a very great scale, any very little part of its circum- ference will appear to be practically straight. Of course it is not straight, but by drawing the circle sufficiently large, and by diminishing the size of the little part of the circumference considered, the approximation to perfect straightness in the little part or "element" considered may be rendered as close as may be desired. Such a circle may, then, be considered as a polygon, having an infinite greater, that is, than any definite assignable number of sides, the length of each of which is indefinitely small, and each of which coincides for an infinitesimal distance with the tangent which is drawn past it. What is true of a large circle is true of a small one, and hence motion in a circle may be considered as motion round a polygon of an indefinite number of sides, whence the following proposition. As a body or point moves round a circle, the direction of its motion is that of the tangent at each successive instant. Similarly, the direction of motion of a body which travels in any other curve is, at each successive instant of time, the same as the direction of the tangent to the curve at the point of the curve momentarily occupied by the moving body. Velocity. We have already anticipated some kinematical statements in discussing the velocity of a moving body. This was denned as the distance passed over in a unit of time by a body in motion ; and if we consider, not the moving body, but the motion itself, we may say that one of the necessary properties of pure Motion is Velocity. It is not possible to think of Motion without thinking of a corresponding Eate of motion, so long as Space and Time are related to Motion in the way in which experience shows them to be ; and the idea of Eate of Movement is as necessary a constituent of the idea of Motion as is that of Direction. Velocity may be uniform or variable. The measurement of uniform velocity is simple enough ; and it has already been ex- plained on what principle the measurement of variable velocity is based. Whether the direction of motion be constant or variable whether the moving particle travel in a straight line or in a curve the principle involved is always the same, namely, that the velocity of a moving particle is the length of path traversed by it in a unit of time, or the length of path which would have V.] VELOCITY. 53 been traversed by it during a unit of time if the speed had remained uniform during that period. In the case of motion in curved paths, there arise subsidiary mathematical difficulties in the estimation of the precise length of the path traversed, but these only arise in the determination of the value of one of the terms of the formula v = - , and do not affect the validity of the formula itself. Velocity may be otherwise defined as the relation of change of position to change of time. If at a certain instant of time the moving point be at a distance s from a fixed point chosen as a standard of reference, then, as time goes on, the position of the body, and therefore the value of s, changes. Let the time during which motion is going on be 8t, a very small element of time, and the corresponding change of position be 8s, then this relation of the change of position to the change of time may be expressed by the 8s fraction sr , which, when 8t is chosen sufficiently small, becomes the function Ot j ds familiar to students of the Differential Calculus as -57 This change of s in dt accord with the passage of time may be very advantageously represented as s, where the dot above the letter indicates " the value of the change in unit of time of " the quantity expressed by the letter over which the dot is placed, when the unit of time chosen is very small. This is the notation employed by Newton in his Fluxions, s being the change or Fluxion of s. If the velocity (s) itself change, this change of velocity in unit of time which we otherwise know under the name of Acceleration would be represented by the symbol s. So if the acceleration (s) itself varied, the change of accelera- tion per unit of time would be represented by the symbol 's. Such a number of dots as this rarely occurs in physical problems. We may here make use of previous discussions to bring together some of the symbols used to express frequently-recurring terms. Distance of particle from point of origin = s. Velocity, . v = s = v Acceleration, a = change of v = v = s = Force, / = ma = mv = ms. Energy = \ mv 1 = J ra (s) 2 = Kinetic energy. Do. = fs = mas = mss = Potential energy. Dimensions. It is sometimes important to consider these quantities from another point of view, to wliich reference may be made here in passing. Distance of a particle from a point of reference may be represented by a straight line wliich is measurable in units of Length. Unit-distance would be represented by unit of length : distance, therefore, is said to be of one dimension in length, and a distance may be represented by the symbol [L]. Velocity is a length (space traversed) divided by a time, and its dimensions are said to be p~q- Acceleration, the velocity acquired per unit 54 KINEMATICS. [CHAP; of time, is a velocity divided by a time, and its dimensions are said to be U . p=-j or k^ - Force is a mass x acceleration, and its dimensions are accordingly Tip^ Energy, if we take the expression , has the dimen- or rrp^-i 5 while if we take the expression was, it is found [LI [ML 2 ] to have the dimensions [M] F^T [L] or pjT 2 -| > tne same result - [M] I m Simultaneous Motions. If a particle have by any means two separate independent motions communicated to it simul- taneously, each will produce its own effect, and the total move- ment of the particle can be found by any process of summation which may be found mathematically appropriate. It will always be found as the result of experiment on bodies which may be taken to represent particles, that if the motions imparted to the particle be themselves constant in velocity and direction, the result of their concurrence is a single motion in a straight line with a single velocity and direction. The single motion which is produced as the result of the concurrence of two motions is called their Resultant, and they with regard to the resultant are called its Components. If a ship travel from west to east and a man on board also walk from west to east, the speed of the ship and the speed of his walking will have to be added together to find the rate at which he is moving eastwards : if the ship travel from west to east and he walk along the ship from east to west, the difference between his own speed and that of the ship is the rate at which he is travelling eastwards. In the latter case the result may be positive i.e. he is really going eastward ; negative i.e. he is really going westward ; or zero, in which case he has no move- ment at all, the ship carrying him east just as much as he walks to the west, so that he is really beating time in the same place. If a steamer travel to the east and be at the same time carried to the north by a current, the path traversed by the steamer will be a line which is the diagonal of a parallelogram whose sides represent the eastward and northward velocities respectively. The steamer will describe this diagonal line in the same time as it would have taken to have steamed or to have drifted along one or the other side of the parallelogram if the steaming or the drifting respectively had been the only cause of its movement. Hence, to find the resultant of two simultaneous velocities, the rule is : Construct a parallelogram whose adjacent sides represent v.] PARALLELOGRAM OF VELOCITIES. 55 in magnitude and direction the velocities produced, and the diagonal which lies between these adjacent sides represents the resultant velocity. If the lines AB, AC (Fig. 11), represent in direction and, on any conventional scale, the magnitude of the velocities simul- taneously imparted to the point A, the point A will move along the line AD to the point D in the same time that, under the influence of the velocity 'AB alone, it would have taken to reach B, or, under the influence of the velocity AC, to reach the point C. The actual construction of the diagram representing the resultant of any two velocities is an easy matter : the calculation of the value of the resultant c D that is, of the length of the diagonal -involves a little geo- metrical working. When the two components are at right angles to one another, we resort to Eucl. I. 47, which shows that in a right- angled triangle ACD, of which the right angle is at B, AD 2 = AC 2 + CD 2 . In the parallelo- F i*.i2. gram Fig. 1 1 (a) it is plain that AD 2 = AC 2 + CD 2 ; but CD = AB ; hence AD 2 = AC 2 + AB 2 ; or in words, the square of the resultant is equal to the sum of the squares of the two components, if these be at right angles to one another. Again, if they be not at right angles to one another, they must make either an acute or an obtuse angle. In the former case we resort to Eucl. II. 12, which shows that if the parallelogram be drawn (Fig. 13) and the side AC be produced so far that a line DE can be drawn at right angles to it from the point D, the equation AD 2 = AC 2 + CD 2 + 2AC'CE is true ; this enables us to find the value of AD if we know that of AC and AB (which is equal to CD), and if we can find that of CE. In the latter case, where the Pigr.is. Fig.H. angle BAG between the components AB and AC is obtuse (Fig. 14), Eucl. II. 13, shows us that if we drop a perpendicular DE from D upon the base AC, the equation AD 2 = AB 2 + AC 2 - 2 AC 'EC holds good, and enables us to find the value of AD. 5 6 KINEMATICS. [CHAP. Problems. 1. A person on board a ship which is going eastwards walks back and fore at the rate of 4 miles an hour relative to the ship : the ship is travelling at the rate of 12 miles an hour. What is his absolute velocity when he is walking forward ? what when he is going aft ? what is his average absolute velocity ? Ans. 16 miles an hour ; 8 miles an hour; 12 miles an hour. 2. A point moves with velocity a eastwards and velocity b westwards simultaneously. What is its eastward velocity ? Ans. a b. 3. Interpret the result if b is greater than a. Ans. The eastward velocity = a - b ; this is negative : the velocity must therefore be westward and = b a. 4. Interpret the result if b = a. Ans. The eastward velocity = a b = o : or the body is absolutely at rest. 5. If in a railway carriage compartment a man walk across at the rate of 5 miles an hour while the train goes forward at the rate of 12 miles an hour, what will have been his real path and velocity relative to the railway line underneath ? Ans. In Fig. 1 1 (a), if AB = 5 and AC = 12: the real path is AD, which has a value of 13. 6. To the same particle are imparted a velocity of 12 and one of 6 feet per second in directions which stand to one another at an angle of 60 : what are the directions and the amount of the resultant velocity ? Ans. In Fig. 1 3, if AB represent the velocity of 6 feet per second, and AC on the same scale that of 12 feet, the angle BAG being one of 60, then the line AD will indicate the direction of the resultant movement, and the equation AD 2 = 12 2 + 6 2 + 2 x 12 x CE (CE being seen, since the triangle CDE is half an equilateral triangle, to be equal to half CD that is, to 3), or 144 + 36 + 72 = 252, shows that AD = \/252i = 15'822. 7. If in the last question the angle between the directions of the velocities had been 120, what would the resultant velocity have been? .4ns. In Fig. 14, if AB be 6 and AC 12, the angle BAG being 120, AD 2 = 12 2 + 6 2 - 2 x 12 x CE= 144 + 36 - 72 = 108 ; whence AD= VT08 = 10-392 feet per second. 8. What would have been its direction ? In the same figure, 14, the triangle ADC has its sides AC = 12, CD = 6, and the angle DCA = 60. By trigonometry we find that the angle DAG is 35 16', and hence the direction of the resultant motion is inclined to those of its components AC and AB at the angles of 35 16' and 84 44' respectively. Triangle of Velocities. If the figures just made use of be reduced to their simplest necessary elements, it will be seen that there is no need to describe a complete parallelogram in order to find the line which would be its diagonal. The three sides of a triangle are quite sufficient to express the relation between two component velocities and their resultant, and for the determina- tion of the resultant of two velocities the rule may be thus stated : Take a starting point ; from it draw a line representing in mag- nitude and direction one of the component velocities ; from the point thus arrived at that is to say, the end of the line thus v.] TRIANGLE OF VELOCITIES. 57 drawn draw another line similarly representing the second com- ponent velocity. The third side may now be laid down, and the problem is reduced to the form which in trigonometry is simple enough, namely Given two sides of a triangle and the angle between them, to find the third side, and the angles which it makes with the two sides given. In the triangle ABC (Fig. 15), if the sides AB and BC represent the component velocities in amount and direction, AC in the same way represents their Fig.15. resultant ; and it will be observed that if the sides of the triangle be taken consecutively in the " cyclical " order, AB, BC, CA, the direction of the resultant is in this diagram opposed to that of the components. Resolution of a Velocity into Components. The converse proposition is one of very general utility. In the former case, by Composition, that single resultant was found which was the effect of two simultaneously-imparted movements. A single movement may, conversely, be considered as the resultant of two component movements which we may wish to find. The process of finding them is known as the Resolution of a motion into its components. A given pair of velocities can only have one resultant, for if two sides of a triangle be fixed, there is no scope for variation in the position or length of the third side ; but if the resultant be given, it can be resolved into components in an indefinite number of ways, for there can be an infinite number of triangles made by supplying two sides when only one side is determinately fixed. Hence the question how to resolve a velocity into its components, set in this vague way, never arises ; but the question how to resolve a velocity into its components in certain fixed directions is of constant occurrence. Such a question is generally solved by construction in the following way : Let AB (Fig. 16) be a line indicating the direction and rate of movement of a particle. It is required to know what are the corresponding components in two directions, LM, NO, arbitrarily chosen or deter- mined by the conditions of the problem. Draw lines from the extremities of the line AB parallel to the directions assigned. In this way a parallelogram will be formed in which AD, CB will 58 KINEMATICS. [CHAP. Fig.ie. be parallel to LM, and DB, AC to NO. In this parallelogram AB represents the single motion whose components are to be found : the length of AD or CB represents the proportionate value of the component parallel to LM, and DB or AC the proportionate value of that parallel to NO. Hence the problem is solved. Very little practice enables one to dispense with drawing a com- plete parallelogram, and to find the components by constructing either the triangle ABC or ABD. If numerical values are required, we can find them by means of the known angles which the directions of LM and NO make with that of AB : the value of these two angles together with the numerical value of AB give by trigonometry the numerical values of AD and DB, which represent the components. If we wish to resolve a single velocity into components at right angles to one another, the process is precisely the same, LM and NO being drawn at right angles to one another. A modified form of the problem which we very often encounter is Given a velocity in a certain direction, what is the value of its component in another assigned direction ? This is solved by the following construction : Let AB (Fig. 17) represent the given velo- city and LM the direction of the required component of AB. Draw through A a straight line, AC, indefinite in length, but parallel to LM. From the other extremity, B, of the line AB, draw a line BD at right angles to AC, cutting it in the point E. AE is the component required in the direction parallel to LM. Problems. 1. A velocity of 30 feet per second : what is the value of its component in a direction which makes with its own an angle of 60 ? Ans. In Fig. 17, if the angle ABD be 60, the line AB may represent the velocity given ; EB represents its component at an angle of 60 with it. The triangle BE A is Fig.17. v.] KESOLUTION OF A VELOCITY INTO ITS COMPONENTS. 59 half an equilateral triangle, and EB is half of AB ; it represents therefore a component velocity of 15 feet per second. 2. A velocity of 20 feet per second : what is the value of its component whose direction makes with its own an angle of 30 ? Am. In the same figure, if the angle ABE be 30, and AB represent the velocity 20 feet per second : in such a triangle EB : AB : : / : 1, and the value of the com- ponent is * /-- x 20 = v/300 = 17'32 feet per second. 3. A velocity of 60 feet a second : what is the value of its component at an angle of 45 ? Ans. I x 60 = 42 '42 feet per second. 4. A velocity v in a certain direction : what is its component at right angles to that direction ? Ans. It has none. Composition of more than Two Velocities. If more than two velocities be imparted to a body the resultant is always, if they themselves be uniform in amount and direction, a single uniform motion in a straight line. If the several velocities imparted be all in the same plane, their resultant may easily be found by finding the resultant of any two of them, compounding the result- ant thus obtained with any other of the velocities imparted, and so on, till all the velocities have been taken into consideration, and the final resultant obtained. Let the several velocities which are imparted to a particle be represented by the lines AB, AC, AD, AE, AF (Fig. 18), all in one plane. It is required to find Fig.18. their resultant. The resultant of AB and AC is AL ; the result- ant of AL and AD is AM ; the resultant of AM and AE is AN ; that of AN and AF is AO, the final resultant of the five veloci- 6 KINEMATICS. [CHAP. ties AB, AC, AD, AE, AF. It does not matter in what order they are compounded ; it may be left as an exercise for the reader to show that the same result is always obtained whatever be the order followed. The Polygon of Velocities. If, in the last diagram, the figure ABLMNOA be traced out, it will be seen that it is a poly- gon whose sides represent the various velocities and the resultant : for these sides are AB, BL ( = AC), LM (= AD), MN (= AE), NO ( = AF), and AO, which represents the Eesultant. Hence the method of finding of the resultant of any number of forces in the same plane may be exemplified as follows : Take a starting-point K (Fig. 19) ; from K draw the line KP, representing AB in mag- (=AB) nitude and direction ; from P draw PQ, representing AC ; from Q QE, representing AD; from E ES, representing AE; from S ST, representing AF ; then join KT. KT represents the Eesultant sought. It will be seen that the direction of the resultant is opposed to that of the other sides of the polygon taken in cyclical order. The rule, then, for the composition of a number of velocities in the same plane is Construct a polygon with lines representing them (it being a matter of indifference in what order they are taken, or whether they cross one another or not), and if there be a side missing complete it ; it will represent the magnitude of the resultant, and its direction will be opposed to that of the other constituent sides taken in cyclical order. If the two points K and T coincide, then the line KT has no value, there is no resultant motion, and the result of the simultaneous velocities is a state of rest. Reference to Axes. It is often as convenient, or more so, first to resolve each velocity into two components, which are made parallel to arbi- trarily chosen axes. Let the same velocities, AB, AC, AD, AE, AF, be sup- v.] COMPOSITION OF VELOCITIES. 61 posed as in the previous paragraphs. Through the point A (Fig. 20) draw axes of x and y at right angles to each other. Resolve each velocity into its components parallel to these axes. AB is resolved into Ab and A/? ; AC Flg.20 / into Ac and Ay, and so on. The value of the resultant is found after summing up with reference to each axis separately. The total result with reference to the axis of x is (Ab + Ac Ad-Ae + A/), which has a value, say, + Ar. In the axis of y the total result is ( A/3 + Ay + AS Ac A<), which has the aggregate value, say, -f Ap. The resultant therefore is to be drawn from A to a point R, which has co-ordinates, x + Ar, y + Ap. Velocities not in one Plane. The same essential principles apply here as in the preceding paragraphs. In the case of a railway train travel- ling at the same time northwards, westwards, and upwards, the motion, while it may be represented by a straight line, is the resultant of three com- ponents at right angles to each other. The proposition in three dimensions, which corresponds to that known as the parallelo- gram of velocities in bidimensional space (in a plane), is called the parallelipipedon of velo- cities. If the three velocities, Ax, Ay, Az (Fig. 21), at right angles to each other be compounded, the resultant is expressed by a line drawn from A to the opposite angle of that parallelipipedon, of which Ax, Ay, Az measure the length, breadth, and thickness. If Ax, Ay, Az be at right angles to one another, AR 2 = (Ac 2 + Ay 2 + Az 2 ), while A if they be not at right angles to one another, AR is the diagonal of an oblique prism. Any rectilineal velocity may be resolved into three components in an indefinite number of ways, for there may be an infinite number of prisms constructed on a given diagonal line, but there can only be one way of 62 KINEMATICS. [CHAP. resolving such a movement into components if these must be at right angles to one another, or if the directions of two of them be assigned. The Polygon of Velocities also applies when the component move- ments are not restricted to one plane, for a so-called "gauche polygone" or " twisted polygon," may be realised, no three of whose contiguous sides are in the same plane ; the only essential criterion of such a polygon is that it shall be continuous and closed. If such a polygon whose sides represent velocities be realised, but be incomplete or "unclosed," the missing side represents the Eesultant, and the direction of the resultant opposite to that of the rest of the sides taken in cyclical order and its magnitude are found in the same way as if the polygon had been restricted to a plane surface. The method of reference to axes, illustrated by Fig. 20, is of special use when extended to tridimensional space. Of a number of velocities in different directions in space, each may be resolved into three components, parallel to the axes of x, of y, and of z, and the resultant is found after summing up the effects produced with reference to each of these axes respectively. Change of Velocity. This phrase is sometimes employed, as when the statement is made that a certain velocity has been changed into another, and the question is asked, What has been the " Change of Velocity ?" Another way of stating the same is A known component and an unknown one have produced a given resultant : what was the value of the unknown component ? This is easily solved if the direction of motion have not changed ; while if the direction have also changed, the question is answered by the aid of the triangle of velocities ; the two sides being known, the third side is easily found. Parallelograms, etc., of Acceleration. What is true of simultaneous velocities imparted in general is true of velocities simultaneously imparted in unit of time that is, of Accelerations, and hence, if a body receive two accelerations, these must be com- pounded in exactly the same way as two velocities. So every one of the geometric propositions just laid down with reference to velocities finds its exact counterpart in a proposition relating to accelerations, and we thus have such propositions as the Parallelogram, the Triangle, the Polygon, the Parallelipipedon of Accelerations. Problems. 1. If the same particle be simultaneously affected by a northward velocity of 10 feet per second, an eastward of 8, one towards the S.W. of 7, to the W. of 8, to the S.E. of 5, and to the N.E. of ^,ifind the resultant movement, and show that it does not matter in what Border the components are resolved. 2. If the axes of x and y be drawn at right angles to one another through the common point A ; if then the point fe be supposed to be simultaneously affected by velocities represented by thj)/ blowing lines, viz. (a) one drawn v.l ACCELERATION. 63 making an angle of 15 with Ax, and of suoh a length as to represent a velo- city of 10 metres per second ; (b) one making an angle of 45 with Ax, and representing a speed of 1 5 metres per second ; (c) one making an obtuse angle of 120 with Ax, and representing 8 metres per second ; and (d) one at an angle of 195 with Ax, and representing a rate of 12 metres per second. Find the resultant velocity (1) by the polygon, and (2) by reference to axes. 3. If a body moving 10 miles an hour northward come to move at the same rate southward, what is the change of velocity ? Ans. 20 miles an hour. 4. If a body be moving with a velocity 4 miles an hour northward, and is after some time found to be moving at the same rate eastward, what is the change of velocity ? Ans. 4 x v/2, acting towards the S.E. ; "the hypo- tenuse of a right-angled triangle. 5. If a body moving at the rate of 10 feet a second is found after some time to be travelling at the same rate, but in a direction inclined at an angle of 60 to its former one, what is the change of velocity ? Ans. 10 feet per second, making, with the original component and the resultant, an equilateral triangle. Accelerated Motion. If a body be moving at a rate which increases or decreases with the time, its velocity is said to be accelerated. The acceleration is said to be positive when the velocity of the motion is increased, negative when it is dimin- ished. It is measured by the amount of increase or decrease of the velocity per unit of time. If a particle be at a certain initial instant moving at a rate V, and if its acceleration be db a, then its various rates of motion are At the initial instant . . "V. At the end of one second . -. V db a. At the end of two seconds . V 2 a. At the* end of t seconds . . Y at. Hence we arrive at a general equation expressing the relation between v the velocity acquired at the end of t seconds, V the original velocity, and a the acceleration, namely, v = V at, (1.) in which the + or the - sign is used according to the positive or V the negative character of the uniform acceleration a. It is supposed that the acceleration is uniform, and hence the average velocity during any interval of time is the arithmetical mean between the velocity (V) at the commencement and the velocity (V db at) at the end of the interval ; that is to say, it is equal to half their sum or 1{ (V) + (V at) } = (v + - ) . This \ 2 ' being the average velocity during the interval, the space traversed 64 KINEMATICS. [CHAP. will be found by multiplying the average velocity by the time, and hence we have (s being the space traversed) V* (2.) From equations (1) and (2) we may eliminate t* and thus obtain a third equation ^ = V 2 2as, (3.) which expresses the relations between the original and acquired velocities, the space passed over, and the acceleration. All elementary problems concerning accelerated movement in one direction, which give a sufficient number of terms to enable a conclusion to be arrived at, can be solved by the aid of these equations. Problems. 1. If the velocity at the initial instant be 10 feet per second, and the acceleration be + 2 feet per second, what will be the speed at the end of 1 3 seconds ? Ans. Here, by equation (1) (v being the unknown term), v = V + ai=10 + (2x 13) = 36. 2. If the acceleration be - 2 feet per second, what will be the velocity ? Ans. v=10 (2x1 3)= -16; that is, 16 feet per second, in a direction opposed to the original velocity. 3. If the terminal velocity be 20 feet per second, the acceleration be 4 feet a second, and the initial velocity 4 feet per second, what was the time spent in acquiring the ultimate speed ? Ans. Here, by equation (1) (t being the unknown term), 20 = 4 + 4t, whence t = 4. 4. A body falls with accelerated velocity; its acquired velocity is 100 feet per second, its acceleration is 10 feet per second, and it has been gaining speed for 8 seconds. What was the initial velocity ? Ans. By equation (1), V being the unknown term, 100 = V + (10 x 8), whence v = 20. 5. A body falls from rest : its velocity increases by 32'2 feet per second. What will be its speed at the end of 5 seconds ? Ans. By equation (1), v being the unknown term, and V = 0, v = + (32-2x5) = 161 feet per second. 6. What space will have been traversed, the terms remaining as in the last question ? Ans. By equation (2), s being the unknown quantity, and V = 0, s = + 32 ' 2 2 X 25 = 402-5 feet. 7. What time will a body take to fall 502-5 feet if it be thrown down from a cliff at the initial rate of 20 feet per second, and if the acceleration of * From equation (1): t- db^~ V . a Substitute this value for t in equation (2) V a v.] ACCELERATED MOTION. 65 a falling body be 32-2 feet per second ? Ans. Here, by equation^ the unknown term, 502 '5 = 20+ 16'li 2 , a quadratic : whence t = 5 8. If its initial velocity had been 20 feet per second upwards, how long" would it take to fall 1 Ans. Here the acceleration is opposed to the original velocity, and equation (2) becomes 502'5 = 20* - 16'H 2 , whence t = 6'24 seconds. 9. What speed is acquired by a falling body if it start from rest and fall 1610 feet? Here v is unknown, V== and a = 32*2. By equation (2), v =322 feet per second. 10. If a body start with initial velocity V, and the acceleration be a, what will be the space traversed in the first, in the second, in the third, in the fourth seconds respectively ; and what will be the space traversed in 4 a 3a 5a 7a 16a seconds ? Ans. V + - ; V + ; V + -^ ; V + ~2~ 4V+ ~2~' Composition of uniform with accelerated motion. If a particle be affected with both Fig.22. a uniform and an accelerated motion, and if these be in the J\ same straight line, we have simply the problem dealt with T\ by the aid of the last three equations. "When, however, the uniform and the acceler- ated velocities are not along the same line, but are in direc- tions inclined 'to one another, the resultant must be found by a geometrical or an alge- braical process of composition of velocities. If, for the sake of fixing our ideas, we con- sider such a motion as that of a projectile fired horizontally from a gun placed on a height, we see that the ball is affected with two simultaneous motions, the one horizontal and uniform, the other vertically downwards and accelerated. If we con- sider the positions reached by such a body in successive equal intervals of time, we find that while it passes forward, by reason of its horizontal component, over spaces varying directly as the F \ \ \ \ 66 KINEMATICS. [CHAP. time, the amount of its vertical drop due to the downward acceler- ated component is proportional (Equation 2, where V = 0) to the square of the time during which it has been in motion ; so that if we separately find its various positions at the end of successive small intervals of time, we can draw a line joining these positions, which line we find to be a curve known as a Parabola (Fig. 22). If, again, we consider that the one movement is in the axis of x, and is uniform, so that at the end of time t the horizontal com- ponent motion has carried the body along the axis of x a distance x = vt; while the vertical fall represents the distance y along the axis of y at which the body is situated at the end of the same time t, so that according to equation (2) above, y = ^ at 2 ; then we find" 5 ' 5 " that in time t (whatever this time be) the body moves to a position such that its vertical distance y from the starting point bears to x 2 , the square of its horizontal distance from the starting point, a constant ratio, or in symbols Jcx 2 = y, which is recognised as " the Equation to " a Parabola. This indicates that in order to preserve the given relation between the values of x and 3/, the path of the body as it moves from point to point must be in a curve known as a parabola. It is stated that the body will move in a, parabolic not in the parabolic path ; this is because there is an indefinite number of parabolic paths possible, there being an infinite number of parabolic curves (just as there may be an infinite variety in the forms of a jet of water expelled from a fire-engine), which resemble each other in having some constant proportion between the values of the one co-ordinate and of the square of the other, but which differ in the numerical value of that ratio. Degrees of Freedom of a Particle. If a particle be free to move in any direction in space, it is said to have three " degrees of freedom," because it may move in tridimensional space ; it may move, e.g. (1) up or down, (2) for- wards or backwards, or (3) to the right or left ; or more generally, it may move in the direction of any of the three axes arbitrarily chosen at right angles to one another, by reference to which we agree to specify any given direction in space, or it may move in any other direction, motion in which may be con- sidered as the resultant of simultaneous motions in the three directions assumed as axes of reference. If the particle be restricted to a surface, it cannot move in a direction at right angles to that surface, and is accordingly said to have one degree of freedom less ; it has now two degrees of freedom, * This is an instructive example of a method frequently in use. One consideration leads us to the equation x=vt: another to the equation y = %at 2 : the question is, what law governs the relations of x and y ? The two equations are combined in any way so as to represent x as some multiple (or other " function ") of y, and also, if possible, so as to eliminate a letter common to both equations. Here the value of t( = ) derived from the first equation may be substituted for t in the second, thus making it y = a ( \ , whence x* -~ y =. const. v.] DEGREES OF FREEDOM OF A PARTICLE. 67 for it may travel along the surface in two main axial directions (e.g. (1) for- ward or backward, (2) to the right or left), or in any direction derived from the combination of these. If the particle be restricted to two surfaces, on both of which it must lie at the same time, it can lie nowhere but on the line in which these two surfaces cut one another, and it has now only the one de- gree of freedom implied in the possibility of moving (backward or forward) along this line. If the particle be restricted to three surfaces which cut one another in a point, the particle cannot leave that point without leaving one or other of the surfaces ; its position is definitely fixed, and it has no degree of freedom to move in any direction. Translation. If there be a system of separate particles, all of which are affected with equal and parallel velocities, each particle will move in such a way as to retain its relative position with regard to its fellow-particles, and the system will move as a whole, undergoing no deformation, just as a com- pany of soldiers, all the constituent units of which march in the same direction and at the same rate, retains its formation. If a straight line be drawn be- tween any two of these particles when the system is in its initial position, it will be found that the line drawn between the same particles after such move- ment will always remain parallel to its former position, and will be unaltered in length. Motion in which every such line remains parallel to all its pre- vious positions is called Simple Translation. If we study the motion of such a straight line or of the particles between which it lies, we shall have complete knowledge of the positions of the various particles of the system, if that system be restricted to a plane surface. If the system be not restricted to a plane surface, then it is possible that though one line and all lines parallel to it may continue to be parallel to their former positions, the whole system may have rotated round one of these lines as round an axis ; and hence in this case it is necessary, before the motion of the system can be said to be a motion of simple translation, that not a line only, but any plane through the system or, which amounts to the same thing, every line in any such plane should retain parallelism to its initial position. Rigid Body. This is an ideal, not physically realisable. A rigid body may be regarded as a system of particles which may move as a whole with reference to surrounding objects, but in which there can be no displacement of its particles with reference to one another. Centre of Figure. There is in every body of any shape whatsoever a point occupying a position which may be described as the average of all the respective positions of the several particles of the body. A body suspended in the air somewhere to the N.W., for example, will have a point, situated in most instances within it, uniting in itself a situation at an average distance to the N., one to the W., and one upwards from the point chosen as the point of origin ; and this is the Centre of Figure. Not only with respect to the planes chosen as those of reference is this point the centre of figure, and its distance the average of the several distances of all the particles, but it has this property with reference to any plane whatsoever. The centre of figure of a straight line or linear body is its middle point ; the centre of fignre of a circle is its centre ; the centre of figure of a sphere, of an ellipsoid, of a spheroid, is equally obvious ; that of a hollow spherical shell is the centre of the corresponding solid sphere, and is therefore not 68 KINEMATICS. [CHAP. within the substance of the shell ; that of a parallelogram is the point at which the diagonals cross one another. That of any regular plane figure is obtained by dividing it into numerous thin strips and bisecting these ; by joining the point of bisection a line is drawn in which the centre of figure must lie. These two lines will cross one another in some point, and the point where they do so is the only point which lies in both the lines, and is the centre of figure. This holds good only when the lines thus containing the centre of figure are straight ; if they be not so the construction fails, and we may modify one of the experimental methods described under the Centre of Gravity, further on. The importance of the Centre of Figure lies in this : that if a rigid body be subject to translation without rotation, the motion of the body may be quite effectively studied by considering the movement of the centre of figure, and, on the other hand, if a rigid body be subject to accelerations whose result- ant passes through the centre of figure, the whole rigid body will participate in the movement of its centre, and there will be translation : while if the resultant of parallel accelerations do not pass through the centre, there will be rotation. Rotation takes place when a straight line drawn through a moving body or system of particles does not con- tinue to be parallel to its previous direc- tions in space. Let us suppose the moving system to be restricted to a plane surface. Then a determinate line AB, arbitrarily _ chosen in the body, may move so that its ultimate position is A'B' (Fig. 23). Ob- viously the line AB, and with it the system, has rotated round the point 0. Again, the relative positions of the same line may be AB and A'B' in Fig. 24. In this case a point may be found,* round which the line AB has rotated so as to acquire its new position A'B'. If the lines AB and A'B' be very nearly parallel to one another, con- struction will show that the point O is at a great distance. When the lines AB and A'B' are perfectly parallel, the point O is at an inde- finitely great, an infinite distance. Thus we see that though it is con- venient to regard translation and rota- tion as distinct forms of motion, yet translation may be considered as a limiting case of rotation effected round an infinitely distant centre. Any translation of AB into the parallel position A'B' may be resolved into a succession of rotations first round one extremity as A, then to an equal extent but in an opposite direction round B : this is easily verified by con- struction. If the line AB cannot be brought into coincidence with the line * Join AA' ; bisect AA' in L ; draw LO at right angles to AA'. Join BB' ; bisect BB' in M ; draw MO at right angles to BB'. LO and MO will intersect in ; is the point required. Join OA, OA', OB, OB' ; the triangles OAB and OA'B' are equal in every respect. v.] ROTATION. 69 A'B' by a single pair of such rotations, a sufficient number of pairs of rotations will certainly effect this. Composition of Rotations. If by reason of a rotation round the point 0, the line AB be brought into the position A'B' ; if it be then rotated round the point 0' so as to assume the position A"B" : the two rotations can be compounded into one round a point O", which is found directly by com- paring the initial and final positions AB and A"B", without reference to the intermediate position A'B'. If a solid body, of which one point is fixed, move in any way whatso- ever, the result is the same as if it had revolved round some definite axis passing through that point. Any movement under this condition is equiva- lent to a single rotation. If a body be subject to two or three simultaneous rotations round axes which meet in a fixed point, the resultant movement is rotation round a single axis, which is found by a construction precisely the same as that of the parallelogram or the parallelipipedon of velocities : the sides of the figure represent in direction the axes round which the rotations occur, and in length the amount of angular velocity ;* the diagonal obtained represents in the same way the axis and the angular velocity of the resultant rotation. The most indeterminate motion of a rigid body may always be resolved into the same motion as that of a screw in its nut, namely, a Kotation and a Translation. As the body continues to move, the axis of the imaginary screw may change its direction in space ; but when considered at and for the space of any particular very small instant, it may be regarded as fixed, and is then called the Instantaneous Axis. As limiting cases, the translation may = o, when there is Simple Kotation ; or the angular velocity of rotation may = o, in which case we would have Simple Translation. The ultimate position attained may always be reached by means of a single translation and a single rotation round some axis. Degrees of Freedom of a Rigid Body. When a rigid body is absolutely free to move in any direction in space, it is said to have six degrees of freedom. These are (1) three degrees of freedom of translation, like those of a simple particle ; and (2) three degrees of freedom of rotation round three axes arbitrarily chosen at right angles to one another. Any such body may move, for example, (1) upwards or downwards, (2) to the N. or S., * When a body moves in a circular path, its velocity in that path may be measured by the length of the path traversed divided by the time, as usual ; or it may, in many respects, more conveniently be expressed in terms of angular velocity. Here the path is measured, not directly in terms of its own length, but with reference to the angle which it subtends and to the length of the radius. The unit angle is that angle (57'29578 = 57 0< 17''44"'8 nearly) which is subtended by a part of the circum- ference equal in length to the radius. Hence the circumference = 360 = ITT x unit angle, w being equal to 3 '1416. Unit angular velocity is that under which a particle travelling in a circle whose radius = \ would itself describe a path = 1 ; that is, unit angular velocity is that of a rotating body which traverses the unit angle in unit of time. If the radius be r and the angle traversed be 6, the part of the circumference passed over is r0, and if this be accomplished in time t, the velocity of a particle on the circumference is r ; that of a particle nearer or further from the centre is propor- tionately less or greater ; while the angular velocity of all the particles of a rotating \ wheel is the same, namely, 70 KINEMATICS. [CHAP. (3) to the E. or W., or it may rotate round (4) a vertical axis, (5) an axis lying N. and S., or (6) an axis lying E. and W. Any rotation not round these axes, or any translation not in the direction of these axes, may be resolved into its components, round or parallel to them ; and as any change of position whatsoever may be produced by a single translation and a single rotation, any motion whatsoever may be effected by a body which has these six degrees of freedom. If one point in a rigid body be fixed, there can be no translation, and three degrees of freedom are thus lost ; the body has, however, unlimited freedom of rotation round any axis passing through the fixed point, and thus retains three degrees of freedom. If a line in the body be fixed in position, there can be no translation, and there can be no rotation except round this fixed line, and so there can be only one degree of freedom, which corresponds to that rotation. If a surface (or, which amounts to the same thing, if three points) in the body be fixed in position, there can be neither translation nor rotation, and the rigid body has no freedom. If a point in the body be restricted to motion along a given line, there can only be one translation, but there may be any rotation, and so the rigid body has four degrees of freedom. When a given line in the body must coincide with some part of a line assigned in space, there can be only one translation that along the line assigned, and one rotation that round the line ; and here we find the rigid body to have two degrees of freedom. If a point in the body be restricted to a given surface, the only motion which is impossible is translation in a direction at right angles to the surface, and hence the body has in this case five degrees of freedom. If a line in the body be restricted to a given surface, one translation is impossible, as in the previous instance, and there are two rotations possible, the one round the line which is restricted to the surface, and the other round an axis at right angles to the surface : in this case there are accordingly four degrees of freedom. If three points in a body be restricted to a surface, there can be rotation round an axis at right angles to the surface, and there can be trans- lation in any direction along the surface but not away from it, so that in this case we have three degrees of freedom. Strain. When a body is not rigid, its particles may so move with reference to one another that their displacement produces deformation, and such relative motion of the particles of which a boHy is made up is called a Strain of the body. Suppose a circular plate to be expanded uniformly, as a disc of iron is when heated ; the radius will enlarge in the ratio of (say) 1 to a ; the area of the plate increases in the ratio 1 : a 2 . The linear expansion is the difference between the initial and the final length of the radius, i.e., r (a - 1) where r is the original radius, and is hence proportional to (a - 1). If the body have contracted, a is less than 1, and a - 1 is negative; hence the linear ex- pansion is negative. The superficial expansion is the difference between the areas before and after the strain, viz. < ^- (ar) 2 - /c r 2 I = - (a 2 - 1); * * TT = 3 -14159 : the .relation of the circumference of a circle to its diameter. The area of a circle = 7854 x*di$4: JT r 2 . A % v.] STRAIN. 71 hence the superficial expansion is proportional to (a 2 - 1). A square similarly affected has its sides and area increased or diminished in the same ratios : so would a parallelogram or any other plane figure, if the linear ex- pansion were the same in all directions. Again, suppose a globular body to be thus uniformly expanded, it increases in size and becomes a larger globe : if its radius increase in the ratio 1 : a, its bulk will increase in the ratio 1 : a 3 , and its cubical expansion will be proportional to a 3 - 1. A cube, a parallelipipedon, or any other solid figure, would under the same circum- stances become a larger or smaller cube, parallelipipedon, etc., whose sides and bulk would bear a similar ratio to their original dimensions. Suppose a square to be unequally dilated or contracted along axes parallel to its sides, the square will become a parallelogram. If a circle, it will become an ellipse ; if an ellipse, it will become an ellipse of another form. As a circle is an ellipse of a particular form whose length (its major axis) is equal to its breadth (its minor axis), any ellipse may be converted by a strain into a circle, if its axes be in due proportion lengthened or shortened. If the expansions along the two rectangular axes be in the ratios 1 : a and 1 : /?, the area of the resultant parallelogram, ellipse, or circle, will be to that of the original figure in the ratio a/3 : 1. If the body be a cube and be unequally expanded in directions parallel to its sides, it becomes an unequal-sided parallelipipedon. If the several linear expansions be 1 : a, 1 : ft and 1 : y, the bulk of the parallelipipedon will increase in the ratio 1 : a/3y. A sphere strained in the same way will become an ellipsoid, a so-called " strain-ellipsoid " : any ellipsoid will become an ellipsoid of a different form, and may become that particular kind of ellipsoid known as a sphere, an ellipsoid whose three axes are equal to one another. It will be seen on drawing any of the figures just described that any two parallel lines drawn through the body in its original form will be parallel after the strain. In this kind of strain, called Homogeneous Strain, there are always three axes, which were at right angles to one another in the original position of the body, and continue to be so after the strain. These are the axes of the strain-ellipsoid into which an imaginary sphere existing in the body would be transformed by the strain. Shear. If a body be so distorted that one plane passing through it is fixed while others move past this at rates propor- tional to their distances from it ; if, for example, the body ABCD (Fig. 25) be so distorted that CD remains in its original position, Fig.25. \ P B A' P 1 B' while the line AB travels into the position AB', the body is being sheared, or is undergoing the kind of deformation or strain known as a Shear. 72 KINEMATICS. [CHAP. The greater the distance of any plane in the body from the plane passing through CD, the greater will be its displacement from its original position ; and a body so sheared we may conceive as made up of an indefinite number of indefinitely thin layers which relatively move by slipping over one another. A shear is measured by the amount of relative motion between two non-distorted planes which are situated at a unit distance from one another, and which remain parallel. In Fig. 25 the amount of the shear of ABCD is the ratio of the displacement AA' or PP' to PC, the shortest possible line drawn between the two parallel planes AB and CD, and vertical to them PP' both ; that is, it is equal to 11- = tan PCF if PC = 1, or to PC x tan PCP' -T C if PC have any other value. Circular Motion. If a body move in a circular path, as, for example, a stone whirled in a sling, its motion at every instant is compounded of a tangential motion and a motion towards the centre. If it pass the point A in Fig. 26 with such a velocity that it would in a unit of time, if not drawn A towards the centre, have reached the point C, and igr.26. ^ | --^y if it be at the end of that interval found at the point D, it is evident that the acceleration towards the centre must have produced in unit of time the change of position represented by the line CD. In other words the whirling sling-stone is constantly being drawn in from the tangential path, which, in virtue of its inertia, it would at every instant naturally take ; and a planet in its orbit is con- stantly falling towards the sun, but does not pro- ceed straight towards it, for the resultant of its tangential and its centripetal motions is an elliptical path which is approxi- mately circular. If the line CD be prolonged through the centre to the point E, Eucl. III. 36 shows that CD'CE = AC 2 . If v represent the velocity of the body in the direction AC, v t the average velocity in the direction CD, and r the radius of the circle, we thus find v t (2r + v) = v 2 (1.) 2ry, + vf = v 2 . If the unit of time taken be sufficiently small, the square of the small quantity v t will be so small as to be negligible, and the equation (1) will become 2v,r = v 2 . (2.) But v / is the average velocity of fall from the direction AC during the instant in question, and hence the velocity at the end of the interval is 2v, ; this is the velocity acquired in unit of time, and hence - the acceleration towards the centre is a = 20,.. Hence the equation (2) may be written v 2 ar v 2 or a = ; the Acceleration towards the centre of the circular path in which a body is moving is numerically equal to the Square of its Velocity at any instant divided by its Radius of curvature. If a body be travelling in any other curve, there can at every instant be found a circle a part of the circumference of which coincides to an indefinite approximation with the curve at the instant. v.] CURVATURE. 73 Curvature. Any curve may be considered as made up of successive elements, each of which approximately coincides with a part of the circum- Figr.27. ference of a particular " osculating " circle, which may always be found. For each element of the curve, the radius of the corresponding osculating circle whose circumference coincides with that element, and which would have the same tangent is called the instantaneous radius of curvature ; and as the curve passes from point to point the osculating circle may be changed in respect of its radius or its centre. Thus in an ellipse, near the extremity of the major axis, the osculating circle of curvature is smaller than it is near the end of the minor axis, as is shown in Fig. 27. Accordingly, 74 KINEMATICS. [CHAP. if a body move in a curved path, its acceleration at every instant towards the instantaneous centre of curvature is numerically equal to the square of the instantaneous tangential velocity divided by the instantaneous radius of curvature, p. But in a curve, the Curvature is the angle through which the tangent sweeps round per unit of length of the curve, and this varies inversely as the radius, as may be seen on comparing the circles in Fig. 28. The radius of I is twice that of II : the length AA' = Cc' is supposed to be a unit of length. In I the tangent AB has swept round into the position A'B' through an angle 9 : the tangent CD has swept through twice as great an angle, the length of circular path traversed being the same : wherefore the curvature (as above defined) of the circle II is twice that of the circle I, and that of any circle is inversely as the radius : and since curvature and acceleration towards the instantaneous centre both vary inversely as the radius, they are propor- tional to one another, and therefore the acceleration of a body moving in a curved path is directed towards the instantaneous centre of curvature, and is equal to the product of the square of the instantaneous velocity into the v 2 I curvature, a = ; - = c ; .'. a = v 2 c, where c is the curvature. Hence a comet turning sharply round the sun, the curvature of its path being very great, has a very great acceleration inwards. SIMPLE HARMONIC MOTION AND WAVE-MOTION. Motion in a circle may be practically effected by a heavy ball suspended by a string, and set to swing in a circular path. A pendulum set to swing in this way goes by the name of the " Conical Pendulum." If the path of the bob of the so-called conical pendulum be looked at from above, it appears circular : if looked at from a point somewhat to one side, it appears elliptical : as the eye approaches the level of the plane in which the bob travels, its path appears to be an ellipse comparatively long and narrow ; and as the eye is placed exactly on a level with that plane, the bob appears to travel backwards and forwards in a straight line. In a similar way, the satellites of Jupiter, which travel round that planet pretty nearly in the plane of the Ecliptic,^ and therefore astronomically on a level with ourselves, seem to travel backwards and forwards in lines nearly straight. The bob of the conical pendulum and the satellites of Jupiter appear to move very slowly at the end of their apparently linear courses. This is because the moving body is really travelling either towards the eye of the observer or away from it at the time * The bodies which make up the solar system may be said in a rough way to be situated in a plane fixed in space, and called the Ecliptic, from which they do not very widely depart. Objects moving in their respective orbits in this plane may appear to pass and obscure i. e. eclipse one another, like so many ships at sea. v.] SIMPLE HARMONIC MOTION. when it appears to be at the end of its swing. When it is travel- ling right across the field of view, when it is in the middle of its apparently linear path, it seems to travel rapidly. Just in the same way a railway train seems to be moving very much faster when it runs right across the field of view than when it is coming or going round a curve, and is seen not broadside but end-on. If we represent the circle in which the body is moving by the circle QAR, and its apparent linear path by the line QR, and if we represent a certain num- ber of positions of the body in the circle by the points A, B, C, D, etc., we may define the apparent motion of the body in the straight line QR by finding the points a, b, c, etc., to which the respective positions of the F body in the Circle of Refer- ence correspond. This is done by drawing lines Aa, B&, etc., at right angles to the line QR. It will be easily seen that if QA, AB, BC, Co, etc., be equal to one another, the corresponding lines drawn along QR are longer near the centre of that line than near its ends ; and these represent the spaces apparently traversed in equal intervals of time. A representation of the relative values of these lines Q, ab, etc., is obtained as follows. If the line QR be taken as the axis of X, a line OA, joining the centre and the corresponding position A of the particle, may be supposed to sweep round into the successive positions OB, 00, OC, etc. As it does so it forms an increasing angle with the line OQ. Then the lengths of the lines Oa, O&, Oe, etc., bear to one another the ratios of the cosines of the angles QOA, QOB, etc. These angles will, if the corresponding motion in the circle of reference be uniform, depend directly on the time. If the angle swept through in unit of time be #, that swept through in time t is tO : hence, if the starting-point in time be that instant at which the body is at the point Q, the apparent distance of the body from the point O will be proportional to cos t0, and, if a be the radius, will be equal to a cos tO. Hence x = acostB. When tO = o, since cos = 1, x = a : when tO = 90, since cos 90 = 0, x = o : when t6 = 180, since cos 180 = - l, //A\\A /K/fa r- Wrk i^/i ''\Y\\ v&l \ \ M\//yy/ \/%f u w\ N -h \V A V-f .., v ---x II III A \7 / \ \ / A ' "V m A/R /-\" cK/H A |v V^ Mf \^ W^M%A./W'A JTi IK / \ F \ whole-number ratio than that of equality, similar phenomena occur ; at any one instant the curve resembles the corresponding 94 KINEMATICS. [CHAP. Fig.46. compound harmonic curve, but alternately waxes and wanes ; the deficiency of amplitude occurring once for each complete oscillation gained by the more rapidly vibrating body. Thus oscillations whose frequencies are 500 and 751 per second give one beat or period of relatively small amplitude during each 751 of the more rapid vibrations that is, for every occasion on which it gains one oscillation on that number, 750, which would make the ratio of frequencies exactly the ratio 2:3. If the oscillations had been 500 and 75 OJ per second, there would have been a beat every two seconds. If the periods of the S.H.M.'s be non- commensurable, the resultant curve approximates in form to that of the nearest commensurable ratio, and successively assumes forms nearly resembling those assumed by that curve when the difference be- tween the phases of its components gradually changes. As a particular case of the composition of harmonic motions we may take the following problem, which is of importance in the Theory of Light. A particle is acted upon by two simultaneous circular vibrations. These, consi- A ~ dered singly, act in oppo- site directions, as in Fig. 46. Let them be identical in period and in amplitude. Let them be resolved into components at right angles to one another, which lie respectively in the lines AB and CD. Let the S.HJVf.'s in AB coincide in phase, while those in CD differ in phase by half a period. In such a case the one circular vibration resembles the other, as an object resembles its image in a mirror situated in a plane parallel to AB ; and the S.H.M.'s in CD will neutralise one another, while those in AB will reinforce one another ; so that the result will be a S.H.M. in the plane AB, and of double amplitude. / If, however, the compo- Fig.47. / nents in CD do not differ in phase by half a period, the same considerations of sym- metry do not apply in refer- ence to the plane of AB ; and the resultant motion cannot be a S.H.M. in that plane. If the circular mo- tions be identical in period and amplitude, there must be some plane in respect to which the circular motions are symmetrical, as in v.] COMPOSITION OF SIMPLE HARMONIC MOTIONS. 95 Fig. 47 ; the resultant motion being a S.H.M. of identical period parallel to that plane and of double amplitude. If the two circular motions differ in period, they will continuously differ in relative phase, and the resultant will be a S.H.M. of double amplitude effected in a plane which is constantly changing a plane at right angles to that of the paper, and rotating round 0, the point of rest in Figs. 46 and 47. Hence a S.H.M. can always be con- sidered as compounded of two circular oscillations ; and if one of these be re- tarded or accelerated, whether suddenly or continuously, the plane of the S.H.M. will be rotated, suddenly or continuously. Composition of several S.H.M.'s in the same Plane. This may be geometrically effected by the same method as that em- ployed in the construction of the curves of Fig. 45, viz., by drawing separately the Harmonic Curves corresponding to each S.H.M., and adding from point to point all the respective displacements indicated by each of these curves. Fig. 48 shows the harmonic curves corre- sponding to five S.H.M.'s, each of which is drawn so as to represent its proper phase, period, and am- plitude relatively to the others. The resultant curve is periodic that is, the complex form is repeated at regular intervals if the periods of the component S.H.M.'s be commensurable ; it cannot be if they are not so. In all cases of a body affected by several simultaneous S.H.M.'s, in which the component S.H.M.'s have been in the same line, the real resultant motion of the particle may be studied by finding the complex harmonic curve produced by compounding these S.H.M.'s 9 6 KINEMATICS. [CHAP, with a uniform movement, and on this curve laying a card in which a slit is cut which is laid at right angles to the axis of the curve. The card is then moved uniformly along this axis, the slit being kept at right angles to it. At any one moment only one point of the resultant curve can be seen in the slit, if that slit be made narrow enough. As the card is moved along, this point appears to move up and down in the slit with greater or less regularity, and the way in which it so moves is the way in which the body really moves when affected with the given simultaneous S.H.M.'s in the same line. Fourier's Theorem. The great variety in the forms of the resultant curves drawn to illustrate the previous discussion will prepare the reader to accept the positive statement that by properly choosing a number of harmonic curves, their amplitudes, their periods, and their phases, and by compounding these, any Peri- odic Curve of any complexity may be built up, provided that the curve required never goes off to an infinite distance from the axis. Conversely, any complex periodic motion must be compounded of and may be resolved into a definite number of S.H.M.'s of definite periods, definite amplitudes, and definite phases. In order, how- ever, that such motion may be periodic that is, that the complex resultant motion may accurately repeat itself at regular intervals it is necessary that the periods of the component S.H.M.'s should be exactly commensurate; for if they were not so, the resultant motion could not exactly repeat itself, and would not be periodic. Granted, however, the periodicity of the complex motion and the limitation mentioned above as to the form of its curve, Fourier's theorem states that any such motion is compounded of a definite number of commensurate S.H.M.'s ; and this is true not only of motion represented by the curve, but also, with wider interpretation, of any phenomenon which the curve may represent. Tide Calculating Machine. A number of wheels may have pre- arranged velocities imparted to them by being separately connected with cranks, of which a number are actuated by the same clockwork, through the intervention of toothed wheels. Thus the mechanism of Fig. 33 may be multiplied and any number of S.H.M.'s simultaneously produced. If the extremities of the oscillating rods be connected by a tense and flexible cord, that cord will be drawn upon or tightened to an extent depending at each instant upon the position of the wheels. One extremity of this cord being fixed to an immovable point, the other may be connected with a spring, and the varying distortions of that spring will indicate the varying tensions of the cord. To some point of the spring a pencil or pen may be attached, and under this writing-point a piece of paper may be unrolled at a rate propor- tioned to the velocity of the clockwork. When the mechanism is set in v.i FOURIER'S THEOREM. motion there is recorded upon the unrolling paper a curve wl the summation of all the S.H.M.'s which are being executed. The' adjustment consists in adjusting for each wheel the position of the pi works in the slot this regulating the amplitude, and also in modifying the angular position of each crank so as to represent the appropriate epoch of each S.H.M. at the moment of starting. A machine of this kind enables a curve to be drawn after preliminary adjustment, which represents the height of the tide for every moment of a considerable period, such as a year, without the aid of further calculation than that involved in deducing from astronomical considerations, and from the tidal record of a place, a knowledge of the com- ponent S.ELM.'s, their respective periods, amplitudes, and epochs. Oscillatory Movement of Systems of Particles. In Fig. 49, A, B, C, D, etc., represent a number of particles in a linear series at equal distances. In the lower part of the figure the same particles are seen, each describing a circular path in the plane of the paper. The point B is represented as being J period behind A, C an equal amount behind B, and so on. If a line be drawn through the positions of A, B, C, D, etc., at a given instant, it will be seen that the system of particles has assumed for that instant the form of a line more or less resembling the curve of sines. If such an interval of time be now supposed to elapse that each particle has moved forward through, say, J of the circumference of its circular path, a line similarly drawn through the then position of the particles will present exactly the same form, but it will lie, as shown by the dotted line in Fig. 49, at a little distance ( = BD) from its previous position. A similar result is obtained after the lapse of any other interval of time. So long as the particles con- tinue to move in their respective circular paths, so long will there apparently be a Travelling of a Wave- Form along the system of particles. Particles whose position is intermediate to those shown as equidistant will be found to occupy intermediate posi- tions on the same line, which presents no abrupt angles but is a continuous curve ; and so, if we roughly represent a linear system of particles by a chain or cord, we can understand how it is pos- sible for a wave-form to run along such a cord while its component material particles never separate themselves by more than a certain distance from the mean positions round which they oscillate. Wave -Length. Such a wave-form is seen to consist of a successive series of parts which resemble one another. In Fig. 49 H 9 8 KINEMATICS. [CHAP. the particle B and the particle J are seen to be at the same time in their position of maximum displacement in the same direction, and the form assumed by the system between B and J is "repeated beyond J and behind B. This distance between B and J is called the Wave-length, the distance between a point on one wave and a similarly situated point on the next wave. The point of maximum displacement in one direction may be called, from the analogy of waves on the surface of the sea, the Crest of the wave ; the point of maximum displacement in the opposite direction may in the same way be called the Trough, and the wave-length may be defined as the distance between crest and crest, or that between trough and trough. Each wave is in this case like its successor and its predecessor, and an observer stationed at a fixed point near the cord would perceive a succession of similar waves passing him. When the wave-length is great, a smaller number of waves will pass him in a given time than when the wave-length is small ; twice the wave-length, half as many waves pass ; half the wave-length, twice as many waves ; thus the number of waves arriving at a given point in a given time is inversely proportional to the w^ave length. Velocity of Propagation. If the particles perform their individual revolutions in, say, half the time supposed to be taken by those represented in Fig. 49, and if the amount by which B lags behind A be still the same fraction (-J-) of the time taken by each particle to perform a complete revolution, shortened though that be, the form assumed by the cord will be the same as in Fig. 49 ; the wave-length will be the same, but the wave will travel twice as fast. The velocity of propagation of the wave would vary directly as the velocity of oscillation of each particle. If, however, the -retardation of B behind A, of C behind B, and so forth, be independent of the rapidity of motion of these particles, if, that is to say, the retardation be for a period of time which depends only on the distance between the particles, then, whatever the rate of oscillation of the particles, the wave will travel at the same rate, but the wave-length may vary, and the form of the wave may vary with it. This is illustrated by Fig. ease 5 0, in which the particles are represented as moving, for example's sake, twice as fast as those shown in Fig. 49 ; while the interval of v.J PROPAGATION OF WAVES. 99 time by which B is delayed in its path as compared with A is exactly the same in the two figures, and therefore in Fig. 5 twice as large a fraction of the circumference expresses the difference of phase between A and B. In Fig. 49 the difference of phase between A and B is assumed to be represented by a difference of 45 in their positions on their respective circles ; in Fig. 50, the movement being twice as rapid, and B being retarded by the same interval of time as in Fig. 49, A must assume a position twice as far in advance of B, i.e., 90. The results shown in this case by drawing the wave in angular outline, are (1) that there are twice as many waves, the wave- length being half as great as in the previous figure ; and (2) that the speed of travelling of the wave-form is the same in both cases ; for if a sufficient time be supposed to have elapsed to permit each particle to have performed half a revolution (this corresponding to the tune allowed in Fig. 49 for the accomplishment of a quarter revolution), and if the then position of the chain of particles be investigated, the wave -form will be found to have travelled for- wards through a space which is the same as in Fig. 49. If, then, the relative retardation of the particles be independent of the speed of the particles, the rate of propagation of the wave-form will be constant, and the only effect of a change in the rate of the oscillation of the particles will be a change in the wave-length and in the corresponding curved form assumed by the chain, and in the number of waves which pass any given point during a given interval of time. The circular form is not a necessary attribute of the path of each particle : the path may be elliptical with a similar result, the difference being one in the form of the resultant wave. The two limiting cases are of great interest. These are (1) the case in which the ellipse is reduced to a straight line at right angles to the chain of particles ; and (2) that in which it is reduced to a straight line in the same direction as that chain. The former gives rise to Transversal vibrations; the latter to Longitudinal vibrations. Transversal Vibrations. Each particle effects a S.H.M. in Fig.51. a direction at right angles to the chain of particles. In Fig. 51 are shown (a) the series of particles unaffected by vibration ; (b) 100 KINEMATICS. [CHAP. the same particles affected by transversal vibrations, executing S.H.M.'s, the phases of which differ to an equal extent in equidistant particles. The form assumed by the system of particles in this case is exactly the curve of sines. It is an easy matter to show in this case, as in those previously discussed, that as the particles perform their several S.H.M.'s the wave-form travels along the cord. Composition of Transversal Vibrations. It is quite pos- sible for an indefinite cord or series of particles, such as the one considered in these paragraphs, to have several wave -motions running along it simultaneously, each producing its own effect, and the total effect of their united action should be traceable by some process of composition analogous to our previous composition of simultaneous movements. There are two main cases to be considered (1) that in which the vibrations are in the same plane, and (2) that in which they are not in the same plane. Transversal Vibrations in the same plane : their Com- position. Since the effect of transversal vibrations on an inde- finite straight cord is to cause it to assume the form of the curve of sines, and since each vibration acts independently in this sense, the effect of compounding such movements is reduced to exactly the same problem as has been considered on pages 90-96, and there illustrated. If, for instance, we refer to Fig. 48, the resultant wave on a perfectly flexible and extensible cord on which the five wave -systems there represented were simultaneously travelling, would, for the instant at which the phases happened to be as there shown, assume the form there drawn ; but when that wave had travelled a little way along the cord, the relative phases of the component transversal vibrations would have altered, and the wave would thus continuously alter its form from instant to instant, returning, however, nearly to the form shown in Fig. 48, as often as the same coincidence of phases recurred. Their Resolution. If a changing wave-form run in this way along a cord, and if the same form recur at regular intervals, the wave-form passing at every recurrence through the same changes, then Fourier's theorem applies, and the most complex phenomenon of this kind may be analysed or resolved into a number of separate waves whose periods are commensurable, running simultaneously along the cord. Transversal Vibrations not in the same plane : their Composition. Here we have to consider two cases (1) that in which the vibrations are at right angles to one another, and (2) v.] TRANSVERSAL VIBRATIONS. 101 that in which they are not so. The latter case differs from the former only in the form of the curve described by each particle. When the simultaneous vibrations are in planes at right angles to one another, the motion of each particle is confined to a plane at right angles to the line of the cord. In the plane in which it moves, each particle describes paths such as those exemplified in figures 35-40. In these figures AB may be takeo as representing the amplitude and direction of the S.H.M. in one plane, and CD as representing the amplitude and direction of the S.H.M. in the plane at right angles to it. If the periods of the component vibrations be not the same, the different particles of the cord will be in different phases of S.H.M., and the form of their respective paths will differ ; and as the compound wave runs along the cord, the path of each particle will pass successively, and in the way exemplified in Fig. 41, through all those forms which are possible as the result of the rectangular composition of S.H.M.'s whose periods are those belonging to the wave-motions which are to be compounded. If the periods of the vibrations to be compounded be the same, the points at which the respective curves of sines cross the line of mean positions will be the same for each vibration. Hence, if one particle describe an ellipse or circle, all the particles which are in motion describe circles or ellipses similar in form, though differing in size or in the direction of motion ; while, as the wave runs along, any given particle will describe an ellipse or circle which alternately enlarges, diminishes, vanishes, and reappears, first enlarging and then diminishing, but in the reversed direction, and then vanishes to reappear, recommencing the cycle in the original direction. This kind of movement may be roughly realised by taking a rope, fixing it at one end, and rapidly rotating the hand which holds the free end. The rope assumes, and may by practice and dexterity be caused to retain, a form in which there are a certain number of fixed points, the number of which may increase with the rapidity of movement of the hand. On each side of these steady points the particles of the rope are describing circles or ellipses in opposite directions. If this condition instead of being steady in its position travelled along a rope of indefinite length, it will, on considera- tion, become plain how a particle would rotate first in one direction and then in the other, and how, while rotating in each direction, the extent of motion is at first small, increases to a maximum, and then wanes away till it vanishes, again to reappear. If the rectangular vibrations which have to be compounded be more than two in number, the problem of finding the approxi- 102 KINEMATICS. [CHAP. mate path at any instant is precisely that of compounding several S.H.M/s. Point by point the independent displacements produced by each S.H.M., whatever the plane of that S.H.M., must be added together and the resultant points joined. Resolution of Transversal Vibrations in general. In the paragraph illustrated by Fig. 42, it has been seen that any S.H.M. may be resolved into any two others in directions at any angles to each other. The only case of other than theoretical interest in its application to the doctrine of transversal vibrations is that in which the S.H.M. is resolved into two components at right angles to each other, these components then being of the same period and phase, and their amplitudes and directions being represented by the sides of a rectangle of which the diagonal may in the same respects represent the original S.H.M. If a simple transversal vibration in one plane were prevented from taking place in a direction at right angles to that plane, such prevention would be superfluous, and the vibration would not be affected. If it were prevented from taking place in the plane in which it is actually occurring, obviously the vibration would cease. If, again, it were hindered by some cause which prevented any movement in a given plane inclined to the plane of vibration at an angle intermediate to these extremes, then a reference to Fig. 42 will enable us to see that if a vibration in a plane passing through AB be prevented from effecting any movement relative to a plane passing through yy^ the result ,is as if the vibration in the plane AB had been broken up into two components of the same phase and period as the original one, and executed in the planes xx f and yy t respectively, and the latter of these then extinguished ; there is thus left only a vibration in the plane xx^ the amplitude of which, as compared with that of the original vibration in the plane passing through AB, depends on the angle between AB and xx f being proportional to the cosine of that angle. Let now the plane yy t rotate round the centre ; when its position is at right angles to AB, xx t sweeps round so as to coincide with AB, and there is neither diminution of the vibration in amplitude nor change in its direction ; when yy t coincides with AB, xx t is reduced to nothing, and the vibration is completely stopped : between these limits there is an indefinite number of positions of yy t , and an indefinite number of corresponding values of ampli- tude of the resultant vibration in xx t as that plane sweeps round at right angles to yy r What is thus true of one transversal vibration is true of each of all those which may be running simultaneously along the cord ; v.] TRANSVERSAL VIBRATIONS. 103 and as the effect of inhibiting vibration parallel to the plane of yy / (Fig. 42) is to restrict it to that of xx /} whatever the original direction of AB, so, if a number of wave-motions affect the same cord simultaneously, and if the cord be restricted from executing any vibrations whatsoever parallel to a certain plane yy t , the result will be a more or less complex vibration restricted to the plane XK t at right angles to yy f . If we can suppose such a cord along which a number of waves are running to be passed through a slot in a thick wall, so that all vibrations in a direction at right angles to this slot are completely prevented, and if then the same cord be passed through another such slot placed at right angles to the first, all vibration whatsoever will be prevented in the part of the cord which lies beyond the second slot, and that part of the cord will be at rest. If, however, the second slot be inclined at any other than a right angle to the first, a certain amount of vibration will pass through, which will be executed in the plane of the second slot, and the amplitude of which will be proportional to the cosine of the angle between the two slots ; in this way the more nearly the two slots coincide in direction the greater the amplitude of that vibration which affects the cord beyond the double obstruction. This is understood to be the kind of fact which underlies some of the phenomena of Polarised Light. Longitudinal Vibrations. The other limiting case of vibrations of a linear series of particles spoken of on page 99, was that in which each particle performed a S.H.M. in the direc- tion of the line of particles. The result is represented in Fig. 5 2, in which a indicates the primitive position of the particles Fig. 52. when at rest ; and b, e, d, their positions after equal intervals of time when oscillating in this way. Each particle has its amplitude of vibration, may be in a certain phase, and have a certain period of oscillation : the wave -motion runs along the vibrating cord : several wave-motions of this kind may simul- taneously affect it ; and a complex longitudinal wave-motion may be analysed into simple wave-motions, as in the preceding para- graphs we have seen that we may analyse a complex transversal wave into its components. The study of this kind of vibration 104 KINEMATICS. [CHAP. is, however, greatly facilitated by giving an arbitrary representa- tion to the form of the wave. If the disturbance represented in Fig. 5 2 & be indicated by drawing lines at right angles to the line of primitive position, each of these lines having a vertical height, positive or negative, equal to the horizontal displace- ment of the corresponding particle forwards or backwards in the line of the cord ; by joining the extremities of these ordinates we shall produce, as has been done in Fig. 5 2 e, a curved line which is the curve of sines or harmonic curve, simple or com- pound, with which we are already familiar. The interpretation of such a curved line would, however, be different in the case of a longitudinal vibration from that of the same form in the case of transversal vibrations. In the latter case the curve represents the actual form assumed by the cord : in the former case, that of the longitudinal vibrations, it only indicates from point to point the extent of the departure of the corresponding particle from its primitive position. It is of great importance to observe that in a longitudinal vibration, where there is the greatest displace- ment, there is the least crowding together of the line of particles, the least density, and, vice versd, that in the centre of each spot of maximum density or compression is one particle which occupies its original position, and thus is situated at the point of least- i.e., zero displacement. We have hitherto supposed the vibration to be permanent, and to be kept up by a continuous and periodic movement of each of the particles of a linear body. Let it be now supposed that there is some relation between the particles of this linear body, such that when one particle is displaced it executes a S.H.M., and, in some way exerting Force upon them, it induces its neighbours to commence executing similar S.H.M.'s following its own at intervals of time, and therefore with differences of phase, corresponding to their respective distances from it. The result will be as shown in Fig. 53. There the cord is first seen undisturbed; then the particle A being disturbed moves in S.H.M., and sets the follow- ing particles B, C, etc., in motion. On comparing b and c it will be seen that the Wave-Form travels along or is propagated, the "Wave-front" travelling onwards with a velocity equal to that of the permanent wave described in the preceding paragraphs. If the disturbed particle A do not oscillate continuously, but travel once merely through a complete S.H.M., the figure 53 / shows that a single wave is propagated, leaving at rest the part of the cord which it has traversed, and continually displacing fresh v.] LONGITUDINAL VIBRATIONS. 105 particles if the cord be of indefinite length. If the motion of each particle be not completely extinguished after the execution Fig.53 H of one exact S.H.M., but dwindle away with diminishing ampli- tude, the wave is not single but is followed by a certain number of shallower waves which presently die away. If the wave-motion be kept up by a continuous supply of energy, there may be a continuous succession of waves, equidistant and following one another at equal intervals of time. The distance between any point in one wave and a precisely similar point in its predecessor or successor is the wave-length of the wave ; the distance tra- versed by a given wave during one second is the velocity of pro- pagation of the wave, and this velocity, divided by the length of each wave, plainly gives the number of waves which pass a given spot during a second of time, or, in other words, the Frequency of the undulation ; while the reciprocal of this number gives the length of time taken by a single complete wave in passing a given spot, and therefore denotes the period of a single oscillation. If A. be the wave-length, v the velocity of propagation, and n the fre- quency or number of waves per second, r the period of each wave, v = nX = r In these cases of propagation of wave-motion along a linear body the wave-front implicates only one particle, and its form is accordingly a single point. The amplitude of the wave as it travels along will be constant if no energy be lost by the way. 106 KINEMATICS. [CHAP. Waves on a surface. On a surface there may run from a starting point waves of compression and rarefaction in the plane of the surface, or waves of vibration transverse to that plane. If one point be disturbed the disturbance is propagated in all direc- tions. If this be equally in all directions the wave-front will be circular; if the material be such that the velocity of propagation in one direction differs from that at right angles to it, the result will be an elliptical wave-front. In Fig. 54 is shown a wave in a membrane (an ideal solid which has length and breadth but in- definitely small thickness), whose structure is such that the dis- turbance is propagated equally in all directions ; and at I, m, n, three points are shown which themselves act as centres of disturb- ance, and the wave-front is propagated to I', m', n', the whole still retaining its circular form. In the same way each point on an elliptical wave-front may act as the centre of an elliptical dis- turbance which keeps up the propagation of an elliptical wave. Fig.54 In Fig. 54 it will be seen that the lines II ', mm', nn f , are parts of radii of the circles ; and these lines are hence at right angles to the circles at I, m, n, and also at Z', m', n'. The normal (i.e., a line perpendicular to the tangent) to the wave-front at any point at any instant is also normal to the corresponding part of the wave-front at any succeeding instant if the medium be iso tropic, i.e., if the velocity of propagation be equal in all directions. If we know the form of the wave-front at any instant, and desire to learn its form after any given v.] WAVES ON A SURFACE. 107 interval of time, this may be done by drawing normals to the original wave-front equal to each other, and of lengths corres- ponding to the time indicated, and by joining their extremities. In this case we simply obtain a circle surrounding a circle, but we shall soon come upon cases in which the result is not so extremely simple. When the initial disturbance is single the wave which is pro- duced is also single. The energy imparted to the system by the single disturbance remains a fixed quantity. As the circular wave progressively increases it acts . upon material whose mass increases with the radius, the energy imparted to each particle varies inversely as the radius, and the amplitude of movement of each particle in its S.H.M. varies inversely as the square root of the radius. In this way the further the wave has travelled from its centre of disturbance the shallower it becomes. At two instants of time the respective radii of the circular wave are r and r t , and the maximum velocities of the particles affected v and v t . Let ra be the amount of mass affected per unit of length of the wave-front. The whole mass set in motion is at the two successive instants 27rr.m and 2nr / .m. But the energy is constant, and STTTW. = Whence But the amplitude of a S.H.M. is proportional to the velocity with which the particle passes its mean position ; for the velocity in the circle of refer- ence, which is equal to the velocity at the middle point of the S.H.M., is 2?r x radius of circle 2?r T = x amplitude, whence the amplitude = V x Time r ZTT' and therefore varies as the velocity. The amplitudes therefore vary inversely as the square root of the radius of the wave. Waves propagated in a tridimensional substance. The solid figure whose surface is everywhere at equal distances from its centre is a globe or sphere. If a disturbance at a point be propagated with equal velocities in all directions in space, the form of the wave-front will be spherical. If the velocities be unequal, the wave-front will be ellipsoidal or spheroidal. As the distance from the centre increases, the amplitudes of oscillation of the particles will diminish, for they vary inversely as the distance from the centre of disturbance. When the radius of the spherical wave changes from r to r f , the mass set in motion becomes greater in the proportion of r 2 to r f 2 ; but the energy, which is equal to 27rrV, or Zirrfvf, remains constant. ' Therefore rV = r/v^ 2 , 108 KINEMATICS. [CHAP. 75 7* * and ' } or the velocity of the particles varies inversely as the radius of the wave-front. But the amplitude varies as the velocity, and therefore varies inversely as the radius or the distance of the wave-front from the centre of disturbance ; and the energy of motion of each particle that is, the Intensity of its vibration varies as the square of the amplitude, and therefore inversely as the square of the radius. This corresponds to the statement that the intensity of Light varies inversely as the square of the distance from the illuminating point. Concentric Waves. In all these cases, if the primitive dis- turbance be repeated at regular intervals, the wave system will assume the form of equidistant and concentric waves of circular, elliptical, spherical, ellipsoidal, spheroidal form, as the case may be. If the primitive disturbance be repeated at irregular inter- vals, the waves will still be concentric but not equidistant, and they will arrive at any point in an order of irregular sequence exactly reproducing the irregularity of the central disturbance. If the central particle be affected by complex periodic disturb- ances, these will be, as regards the period and the phase as well as the relative, but not the absolute, amplitude of every com- ponent motion, faithfully reproduced in the motion of any particle of the system affected by the resultant complex wave-motion ; and this motion of such a particle may in many experimental instances be taken cognisance of by an observer. Direction of the wave-front. When a wave is said to be at a certain time and place travelling in a certain direction, it is meant that the normal to the wave-front, a straight line drawn at right angles to the wave-front i.e., at right angles to its tan- gent or tangent plane takes the direction said to be that of the wave itself. In Fig. 54 the lines qr and st indicate the directions of the circular wave at the points q and s. Fig.55 Flat wave-front. The nearer the centre of disturbance the v.] FLAT WAVE-FRONT. 109 more marked the convexity of the wave'-front ; the further the centre the flatter the wave-front : when the centre of disturbance is very far the wave-front may for any small area be regarded as approximately plane, just as any small portion of the surface of a very large sphere may be. If, again, all the points of a plane surface act as centres of disturbance, then in the immediate proximity there may be a flat- fronted wave. If the disturbed surface be represented in section by AB in Fig. 55, the wave-front will be flat opposite its centre. Modifications in the Form of a flat "Wave-Front. Through the upper and cooler layers of the atmosphere waves travel with less rapidity than they do through the lower. A flat wave-front is thus distorted, its upper part travels with the least velocity, and the wave-front comes to converge upwards. Points on a level with the point of disturbance may remain unaffected, for the wave-front is, in the main, restricted to its normals, and the sound ascends. If, however, there be a movement (such as that due to wind) in which the upper strata move more rapidly than the lower, it is not difficult to see that the wave-front may come to bear down, and that sound-waves may thus appear to travel with the wind, and to be best heard at certain distances, which depend upon the speed of the wind. Wave passing through an aperture. If a flat wave im- pinge upon an ob- stacle containing an orifice, it will in part be propagated through that orifice. If the orifice lead into the lumen of a cylindrical tube whose diameter is the same as that of the orifice, no lateral expansion of the wave-front is possible in that tube. If there be no such tube, there may or there may not be expansion of the wave beyond the orifice. Such expansion, if it take place at all, will take place in the way shown in Fig. 56. The disturbed particles in the aperture act as centres of disturbance to those lying beyond. There is a curious proposition, the nature of the proof of which will be indicated further on, that if the aperture through Fig.56 110 KINEMATICS. [CHAP. which a wave passes be small in comparison with the wave-length, there will be expansion of the wave -front, such as that shown in Fig. 56 ; that if the aperture be wide in comparison with the wave-length, the wave will only travel in the direction of all the lines drawn normal to that part of it which passes through the aperture, the wave therefore travelling with a corres- pondingly limited amount of expansion or none at all; while for conditions intermediate there will be a certain amount of ex- pansion beyond the limitation indicated by the normals. Fig.57. \\ /// When a wave - motion passes through an aperture relatively wide, then the cases are three : - (a.) The wave-front may "be flat, as in Fig. 57a, in which case it does not ex- pand : this is the condition of a "parallel beam" of light. (6.) It may be convex, as in Fig. 576, in which case the wave - front expands, being limited by the nor- mals pq, rs. This is the condition of a "diver- gent beam" of light. (c.) It may in some cases be concave, in which case the wave -front first con r tracts and then expands : this is the condition of a convergent beam or "convergent pencil of rays" passing through a "Focus." When a .wave - front passes through a focus, ex- actly or approximately, the energy, constant in amount, is distributed over a com- paratively small field, and the intensity of disturbance is at the focus correspondingly great. Reflexion of Linear Waves. If a linear longitudinal wave of compression be incident, or impinge, on an obstacle so firm that the first particle of it at which the wave arrives does not move, there is then produced between this first particle of the obstacle and the nearest particle of the vibrating cord (which we REFLEXION OF LINEAR WAVES. Ill Fig.5S. shall represent as particle i) a Compression which results in part- icle i rebounding at a rate equal to that with which it struck the obstacle that is, with velocity v f but in the opposite direc- tion, and in its then meeting the next particle, ii, as it comes up with velocity v /r We may here borrow a proposition from the theory of Elasticity, which shows that if two equal elastic bodies meet one another and rebound, they will do so with exchanged velocities. Particle i, meeting particle ii, exchanges velocities with it; i acquires velocity v yy and returns towards the obstacle ; ii acquires v r Particle i strikes the obstacle with velocity v u and rebounds ; in the meantime particle ii has acquired velocity v yyy by exchange with particle Hi. When particles i and ii again meet, i is impelled towards the obstacle with velocity v y/y and the backward velocity v u is imparted to particle ii. So on : the particle i successively strikes and rebounds from the obstacle with each successive velocity, v /t v /fl v ///t v yyy/ , etc. ; at the same time the backward speed v y f == *^<-^ is transferred successively to all the particles ii, iii, iv, etc., and is followed by the successive velocities v /r v //r v wj , etc. The consequence is that just as the end of the wave is being dashed against the obstacle, a wave -front exactly like the original one is travelling away from the obstacle at the distance of one wave-length. The " re- flected wave " has travelled through the incident wave, and then, becoming clear of it, travels alone, equal to the incident wave in wave- length, in period, in phase, and in amplitude, but op- posed in direction. In Fig. 58 a single wave is shown running along a cord * against a fixed obstacle AB. Beyond the obstacle thin lines show the course which the "wave would have 112 KINEMATICS. [CHAP. taken had the cord not been interrupted. To the left of AB light dotted lines indicate the course of the reflected wave. The light dotted lines are seen to be of exactly the same form as the thin lines within AB, but turned sharply in the reverse direction at the surface of impact. The reflected wave is then a direct continuation of the incident wave in all but direction. If of incident waves there be a succession, simple, complex, regular, irregular, all these peculiarities will be faithfully repro- duced in the reflected wave. What has been said of a wave longitudinal and commencing with a compression may be easily modified so as to become generally applicable to the explanation of any kind of linear undulatory disturbance. If the obstacle stand fast, it is a matter of indifference whether it be composed of matter whose particles lie more or less closely together. Special consideration of the reflexion of waves traversing space of two dimensions may be omitted. Reflexion of a plane wave-front at a plane surface. If a plane wave- front meet a plane surface, any section through the wave and surface will present a condition such as that shown in Fig. 5 9. The line AB represents the wave-front advancing ; CD represents the surface on which it impinges. Every point in the wave-front acts as a centre of disturbance. Thus the wave-front advances parallel to its former plane forms. After the lapse of a certain time the whole of the wave-front has impinged on the obstacle : what is then its condition ? The part of the wave cor- responding to the particle A would have travelled as far as A' if there had been no obstacle. After reflexion it has travelled to a corresponding extent in some direction tending away from the surface that is, to some point on the circumference of a circle, the centre of which is at A, and the radius of which is AA'. So v.] REFLEXION OF WAVES. 113 the part of the wave indicated by 5 would have reached &'; the line tt>' crosses the surface at E ; that part of the wave is re- flected to a distance limited by a circle whose centre is the point E on the surface, and whose radius is the distance, EZ/, between the surface and the position at whicli the wave-front would have arrived if there had been no obstacle. By drawing a sufficient number of circles in this way we see that the aggregate disturb- ance produces a plane wave-front A"B' receding from the surface CD. If, as it approached the plane surface, it had been parallel to that surface, it would retrace its path. If it had approached the surface obliquely, so that the direction of the wave makes an angle 6 with the normal to the surface, the direction of the receding wave will make an equal angle 6 with the normal, but on the other side of it. This is expressed by saying that the Angle of Incidence, 6, is equal to the Angle of Eeflexion; these being understood to be angles made between the direction of the wave and the normal to the surface, or, what amounts to the same thing, between the plane of the wave and the plane of the surface. Fig.60. Jf The same proposition may be otherwise demonstrated. In Fig. 60 let AB represent the direction of a wave, and CD the reflect- ing plane surface. Every movement of the vibrating body with reference to the direction AB may be re- solved into two, these being referred to the axes By and Bx. On reflexion, the com- ponent in yE has its direc- tion reversed ; that in the direction xB is not thus in- terfered with. After reflexion, on recompounding the components, the resultant is found to be a disturbance similar to the original one, but in the direction BA', while the angle ABy is equal to the angle ?/BA'. Reflexion of a curved wave-front at a plane surface. If a wave-front be curved, it may be considered as consisting of a very large number of very small plane surfaces. To each of these a normal may be drawn ; each such normal indicates the direc- tion of the corresponding part of the wave-front ; the angle which each such normal makes with the reflecting surface when its own part of the wave-front strikes the obstacle is the angle of inci- dence ; to this angle the angle of reflexion for that part of the wave-front must be equal. Let the convex spherical wave-front AB strike the surface I 114 KINEMATICS. [CHAP. Fig.61. CD. Each part of the wave-front is reflected at its own angle. The result is the reflexion of a convex wave which is of the same form as AB would have assumed in the time, but which travels in the op- posite direction. Such a wave is, in effect, exactly such a wave as would have travelled from the point 0' as far behind the reflecting surface as is in front of it. In the same figure, if the directions be reversed, so that a concave wave- front travels towards the reflecting surface, converg- ing upon 0', it will, when reflected, become convex and will recede from the reflector, converging upon 0, which is as far on the one side of CD as the point 0' upon which it had originally been converging is on the other. General construction of a reflected wave. Let AB be a wave-front, and CD a reflecting surface, both of any form. Draw normals to AB of such lengths that they may all cut CD ; from these normals cut off equal portions and join the extremities of these portions ; the line EF is thus obtained, which represents the form that the wave would have assumed but for the reflecting obstacle. Draw a number of circles ; the centre of each of these is a point at which one of the normals to AB cuts CD ; the radius is the distance along the normal in question from the sur- face CD to the surface EF. These circles have a common tangent, the curved line GH, which indicates the form of the reflected wave. Normals drawn to this, of equal length, may indi- cate the form of the reflected wave at any subsequent instant ; if these be drawn backwards, all the previous positions, real or apparent, may be investigated (Fig. 62). This is the general construction ; but it very frequently leads to difficul- ties where different parts of the wave cross one another. In most cases, however, the following method is effective. Consider a part of an incident wave and the point at which it impinges ; from the point of incidence draw v.] EEFLEXION OF WAVES. 115 a line indicating the direction in which that part of the incident wave will Figr.62. -~v^ ^^., H be reflected. Find to what distance behind the reflecting surface the incident wave would have travelled Fi ^ in a given time if there had been no obstacle ; mea- sure off along the direc- ' tion of the reflected wave a distance equal to this ; repeat this operation for several parts of the wave- front ; join all the points i thus obtained. This gives c/ the form of the reflected wave at the time chosen. Equal distances measured forwards or backwards upon the normals to this wave will give the form of the wave at instants subsequent, and its true or hypothetical form at instants previous. The following are examples : Problems. 1. Let the reflecting surface be a paraboloid : let the advancing wave- front be plane. The focus of the parabola is at F (Fig. 63). We may choose three instants for consideration. 116 KINEMATICS. [CHAP. a. That at which the whole wave-front would have arrived at 0. Each part of the wave-front is reflected as shown in the diagram. The part which would have taken the course aa' is turned into the direction aF ; bb' into 6F ; cc' to cF ; and so on. aF = aa ; bb' == 6F ; cc' = cF. Tne wave-front is reduced to a point ; it is at that instant passing through the focus F. b. Any instant at which the wave-front having passed the point A would not yet have reached the point ; the reflected wave is spherical and con- cave, converging on F. c. Any instant at which the wave would have reached a plane further away from the reflecting surface than a'O ; the wave is spherical, divergent from the centre F. 2. In the same figure, the wave-front is one which starts from F as a centre ; it meets the paraboloid reflector ; it is reflected with a plane wave- front. 3. The reflecting surface is spherical and concave, the incident wave- front flat. In Fig. 64 the reflecting surface is represented by the line ES. The wave travels from right to left and meets KS. Each part of the front Fi 64 of the wave is turned back at its own angle of re- flexion : the wave-front be- comes convergent. It does not, however, converge on any one point ; it is not spherical. The figure shows that there is a curved line, a " caustic by reflexion," in which lie all the foci of all the separately considered parts or elements of the wave- front. In each wave the ele- ment reflected from the outer part of the surface KS will sooner come to focus than that reflected from the centre of that surface ; hence if the wave be single a spot of maxi- mum disturbance will appear to run along each limb of the caustic and to disappear in a diverging wave at its apex. A succession of waves will keep the whole of the caustic in a state of maximum dis- turbance.* 4. A reflecting mirror a segment of a sphere : a centre of disturbance midway between the centre of the sphere and the reflecting surface. With * Take a long narrow strip of bright tinplate, bend it into a semicircle, place it on its side on a sheet of paper in the sunlight, exposing the concavity to the sun ; a brilliantly-illuminated caustic curve will be seen on the paper. The form of this curve may be varied by altering that given to the tinplate. O-L-fc? r lT^ 9~ ^J V.] KEFLEXION OF WAVES. ^ ^ 117 ruler and compass draw the form of the reflected wave as in Fig. 6<~ ^ mately plane at its centre. 5. A complete spherical surface used as a reflector ; a centre of dSras^<" J * ance at the centre of the sphere : a divergent spherical wave produced. Prove that after reflexion this becomes a convergent spherical wave, converg- ing on and passing through the same centre, and then repeatedly reflected and alternately converging 'on and diverging from the original point of dis- turbance. 6. Prove that a spherical wave starting from one focus of an ellipse con- verges after reflexion on the other focus. Hence prove that if the centre of disturbance be at one focus, and if it be surrounded by a complete ellipsoidal reflecting surface, a wave passes back and fore between the foci, alternately converging and diverging, first at one, then at the other focus. Transmission of a linear wave into a denser medium. If, as in Fig. 65, the particles Fig 65 be more closely placed in B ^ * m ? than in A, B is the denser medium. A linear wave-motion travels to the right in A; it arrives at P. It meets a relative obstruction. P, the first particle of the dense substance, is more resisted than the pre- ceding particles set in motion by the wave. The wave is not entirely obstructed and goes on into B ; but there is to some extent the production of a reflected wave in A. This reflected wave, like that of Fig. 58, is of the same phase and period, and of the same wave-length as the original wave. It cannot be of the same amplitude, for some of the energy of the wave-motion has been spent in setting up a wave in B. The wave-motion propagated along B must necessarily be of the same period as that in A, for the particles in B must move in unison with those of A, which impel them ; it must be of the same phase, for it is the direct continuation of the wave in A. Since the particles are more crowded together in B, a less distance corresponding to the same number of particles, a wave cannot propagate itself in B so far in a given time as it can in A, for its doing so, still retain- ing the same wave-length, would imply its setting a greater mass in motion. This would, however, require a greater amount of energy. If the latter be definite in amount, as it must be, the wave-length and the speed of propagation must be less in the denser medium.* As to the relative amplitudes of the respective vibrations, the original, * To avoid misconception it may be remarked here that in concrete cases, while the density of a body is a powerful factor in determining the velocity of vibration in a given body, this also depends greatly on the peculiar molecular properties the elasticity special to each substance. 118 KINEMATICS. [CHAP. the reflected, and the transmitted, the amplitudes of the two latter taken together are not necessarily equal to that of the first, but in every case the energy of vibration of the original wave is equal to the sum of the energies of the reflected and the transmitted waves. If in Fig. 6 5 a wave beginning with compression be supposed to run through B towards the left, when it comes to the particle P which may be considered as the last of B or the first of A that particle, meeting less resistance than its predecessors in B had encountered, plunges into the rarer medium and sets up in A a wave depending on the original wave in B for its period and phase, but of greater amplitude; and the wave in A will also have a greater wave-length than that in B, for a reason the con- verse of that stated in the last paragraph. The effect on the denser body is, however, singular. The particle P, plunging away from the rest of the particles of B, produces in that part of B a dilatation which is propagated backwards, and there then travels in B a reflected wave, agreeing with the incident wave in period and in wave-length, necessarily not in amplitude, and opposed inphase. A maximum compression arriving at P causes that particle to yield to the greatest extent, and to produce a maximum dilatation in B ; hence, when the incident wave produces a maxi- mum compression among all the particles of A in the neighbour- hood of P, P itself starts a wave in B, commencing with a maximum dilatation, and the incident and reflected waves are not continuations of one another as in Fig. 58, but there is loss of half a wave-length* A comparison of the diagrams in the preceding discussion shows that in every case where the medium in which the wave has travelled is the same, the space traversed by every part of the wave, reflected or not reflected, or sooner or later so affected, must necessarily be the same in a given time ; and hence, count- * This curious result has an interesting bearing on the Conservation of Energy. Both the amplitude and the length of the wave in the rarer medium are greater than in the denser. The body B, if the metaphor may be allowed, finds itself to have done more work on the body A, and therefore to have transmitted more energy to it than it had intended : the particle P has compromised the body B by giving a greater dash forward than was expected. Under the circumstances, matters are adjusted by the propagation of a wave of opposite phase in the body B. If each of these waves, the reflected and the transmitted, be regarded separately as containing so much energy, the sum of their energies may appear to exceed that of the original wave. The whole vibrating matter must, however, be regarded as forming one system. In this system, a compression in one wave, and a dilatation in another, produce a relative motion amounting only to their difference ; and this is the true motion of the system, the energy corresponding to which is equal to the energy of the original vibration. T.J REFKACTION OF WAVES. 119 ing from any initial condition to any final wave-front form, the space traversed before reflection -f- that traversed after it = a con- stant quantity for a given time, and that for every element of the wave-front ; or, as it is often expressed, the " incident ray " + the " reflected ray " = constant for the whole wave. Refraction of a plane wave at a plane surface. If the incident wave strike the plane surface simultaneously at all parts of its own front, it will simply pass more slowly through the denser medium, while a reflected wave is sent back ; but if it strike it obliquely, there are some changes in the wave, which result from one part of it being hampered in the retarding sub- stance while the rest is still moving with comparative rapidity in the rarer medium. In Fig. 66 let AB be the wave-front in the rarer medium; CD the surface separating the denser from the rarer medium; A'B' a position at which the wave-front would have arrived if it had not encountered the denser substance. The lines &&', cc t dd', etc., are normals to the incident wave-front, meeting the line CD in I", c y d\ etc. The angle BAB' between the wave-front and the surface, or iOn between the direction of the incident wave and the normal to the surface, is called the angle of incidence. Let us suppose that the velocity of propagation in the denser medium is ^ of that in the rarer. Then with centres A, V, c", d", etc., and radii = -f- AA', f &"&', cY, etc., draw circles. The line A"B', which is their common tangent, indicates the position of the wave-front at the end of the time during which it would have advanced to A'B'. The wave has been rendered somewhat 120 KINEMATICS. broader, and has changed its direction. The angle AB'A" or n'Or is called the "angle of refraction." In the figure, AA" : A A' : : 2 : 3 ; but in the two triangles AB'A', AB'A"-, we see that AA" : AA' : : sin L AB'A" : sin L AB'A'. Sin L AB'A" sin ang. refr. ' Sin L AB'A' sin ang. incid. . . vel. in denser med. velocity in denser medium velocity in rarer medium Generally, if = a fraction, vel. in rarer med. sin ang. incidence = /A x sin ang. refraction, and ^ is called the "index of refraction" of the denser substance as compared with the rarer one. This formula shows that in Fig. 6 6 if iO indicate the direc- tion of the incident, Or that of the refracted wave; nOri the normal to the plane refracting surface ; if a circle be drawn with centre and any radius, the lines iO and Or will cut it in I and E ; from I and K draw lines at right angles to nri, as in the figure. These lines always bear to one another, whatever the angle of incidence, the same ratio as the Velocities in the respective media, and this law defines the relation of the angle of refraction to the angle of incidence. An equivalent construction is given under "Light," Fig. 182. Refraction of a wave at a surface : General construction. Let AB be an advancing wave-front, CD the bounding surface / / ; r of a denser medium. Let the assumption be made that each several element of the wave-front, as long as it is in the same medium, travels mainly in the direction of the normal drawn to it. In this way the whole wave-front is always simply related to all its previous forms, all the parts of it having at any instant travelled along their respective normals to an equal extent during any given interval of time ; and a line once normal to the wave- V.] KEFKACTlON OF WAVES. 121 front is, if produced, always normal to it as long as it travels in the same medium. Then a number of these normals to the incident wave, such as aa in the figure, are drawn equal to each other, and of length sufficiently great to enable the surface to cut them all. The extremities of these equal normals are joined; in this way a curve EF is produced which indicates the form that the incident wave would have assumed had it travelled thus far in the original medium. Lines normal to AB are also normal to EF. We see seg- ments of these normals, such as da, cut off between CD and EF. 1 velocity in denser The fraction = -, - : must now be known. From IJL velocity in rarer da cut off da", which is to da as : 1 ; with d as a centre, and M da" as radius, draw an arc of a circle through a". Treat similarly all da"s fellow-normals. A number of arcs are thus obtained, to which the common tangential curve must be drawn. This gives the form of the refracted wave-front. This having been obtained, normals may now be drawn to it ; these will not in general coincide with aa and its fellows. By measuring off equal distances along these normals to the refracted wave-front, all the future forms of the refracted wave and all its apparent past forms may be ascertained. Refraction of a spherical wave at a plane surface. Let a spherical wave whose centre is at F strike the plane surface KS and enter a denser medium. The wave would at a certain instant have arrived, say at a'Vc'd'. According to the preceding con- struction, lines aa", W , etc., are cut from aa, W, etc., to which they respectively bear the constant ratio 1 : p. Arcs are drawn with centres a, &, c, d, etc., and radii aa", W, cc", dd", etc*. : the common tangent BCD a curved line is found : this gives the form of the refracted wave in the denser medium; it is hyperbolic. Normals may now be drawn to this, by means of which the future and the apparent past forms of the wave may be traced out. In the denser substance, as it travels onward it retains the hyperboloid form ; and if the normals JB, cC, etc., be traced back- wards, and such equal lines as BB', CC ; , etc., measured off along them, all those hypothetical wave -fronts may be drawn from which the wave-front as it travels through the denser medium pre- sents the appearance of having been developed. On tracing back 122 KINEMATICS. [CHAP. far enough we find that the wave appears to be developed from a wave-front convergent not upon the centre F, but through a caustic the apex of which is at G, where OG is to OF as p, : 1. Flg.68. Problem. A spherical convergent -wave meets the plane surface of a refracting substance. By construction show that the wave converges through a caustic, the distance of whose apex from the surface is less than that of the original centre in the ratio of 1 : ft. Passage of a wave through a parallel-sided sheet of a denser substance. At its entrance into the denser substance, the wave -front becomes hyperbolic; at exit every part of the wave resumes the direction which had pertained at the instant of the first refraction to that part of the wave-front from which it had been developed, and thus the wave-front again approxi- mately, but not exactly,* resumes its original spherical form. In Fig. 68 the part of the wave-front which passes through a is, when the wave approximately resumes its spherical form, again refracted, so that that element assumes a direction parallel to its original direction Fa. Approximate Foci. In Fig. 68, if of the hyperbolic wave * An object seen through a pane of glass is never as distinct as the same seen through the intervening air alone. v.] FOCI. 123 in the denser medium only a limited portion in the centre be considered, it will be found to approximate very closely to a small arc of a circle whose radius is Go. We have already learned to express this by saying that its radius of curvature is Go. If accordingly the incident wave -front be narrow, the refracted wave-front appears to have diverged from a group of points in the immediate neighbourhood of G, the apex of the caustic; and the narrower the incident wave, or the less its divergence, the more nearly will it appear to come from a single point, the very apex of the caustic itself. Similarly in Fig. 64, the narrower the incident wave is in comparison with the breadth of the curved reflecting-surface, the more nearly will the reflected wave converge on the very apex of the caustic, the centre of the sphere. Consequently these points, the apices of the caustics, are approximate foci for comparatively narrow-fronted waves. Utility of the idea of "rays" in geometrical construction. All the preceding discussions have been grounded on consider- ation of the various forms assumed by the wave-front : we have shown that any form may be developed from any of its pre- decessors by taking each point in that predecessor as a centre of disturbance, and drawing equal circles of appropriate radii which indicate the extent to which the disturbance has travelled ; then of these circles the common tangential line denotes the developed form of the wave-front : we found that in simple cases every line normal to any wave was normal to all those developed from it, and to all those forms through which it had passed ; in all this it being supposed that the medium was one in which the velocity of transmission was the same in all directions. We also made an assumption that when the form of the ori- ginal wave was complex and the medium isotropic, the same law applied; that the maxim once a normal always a normal was true ; that there was no lateral expansion of the wave-front beyond the limits indicated in a diagram by lines set down to represent such normals. The assumption is approximately true only in special cases in general there is lateral expansion of the wave-front beyond such limits ; but on reference to what was said in connection with Figs. 56 and 57, we are reminded that it is possible for us to conceive a point of disturbance on a wave- front as one in a wide aperture; and hence if the wave-length be very small, the wave-front propagated from each little element of the wave -surf ace travels along the normal to that element. In any case this is never an absolute statement, and there is 124 KINEMATICS. [CHAP; always more or less lateral divergence ; but as a first approxima- tion of sufficient value for most purposes, it may be said that iri an isotropic medium, where the velocity of propagation is equal in all directions, the wave-front is developed from all its prede- cessors along their common normals ; that this is nearly the case when the wave-length is comparatively short : but the greater the proportionate length of the wave, the more lateral expansion is there, and the less able would we be to find the form of the wave-front at any moment by exclusively considering the normals to its previous forms. If, however, the wave-length be compara- tively short, we may, by considering the normals only, erect in a diagram the scaffolding on which the form of the wave-front may be constructed. In reflexion, for instance, as in Fig. 64, normals may be drawn to the front of the incident wave ; the reflected wave is of such a form that each normal to it makes, with the corresponding normal to the reflecting surface, an angle equal to that made by a normal to the incident wave. But the wave-front itself might have been omitted from the diagram, and the same results as regarding focus and caustic would have been obtained. Then attention might be fixed on the normal lines, and on the way in which these change their direction on reflexion or refraction. They might be treated as if they were physical entities, and might receive special names. This has actually happened. The imaginary straight line drawn at right angles to the wave -front at any point has been called a "ray" passing through that point. Each ray is straight, for the normals preserve always the same direction; and of all wave-motion^such as light which does not expand perceptibly beyond the limits laid down for it by its normals, as in Fig. 5 7, the progress is described by saying that its rays travel in straight lines as long as it is in the same medium. This mode of expression has both advantages and disadvantages. It leads to the assumption that a divergent wave -front is a divergent "pencil" of rays, each of which is somehow distinct from its fellows ; it leads to these rays being conceived as themselves reflected, refracted, etc.; it isolates the physics of those phenomena those of Light in which waves approximately follow their normals only from those in which this approximation is much less complete, as in the case of Sound. On the other hand, it presents certain advantages ; it simplifies diagrams ; it enables any problem to be reduced to its simplest elements by an absolute rejection of all lateral disturbances, and of the effects produced by any parts of the wave other than those v.] KAYS. 125 at the points of the fronts crossed by the normals ; and it gives results which in the theory of Light are, up to a certain point, of sufficient accuracy. This advantage persists, however, up to a certain point only, and, as a whole, a habit of referring the phenomena of wave-motion to the form of the wave-front is to be preferred, though hereafter we shall make free use of the device of reference to rays whenever it is found convenient to do so. The law can easily be verified that the path traversed by every element of the wave in the rarer medium, together with fi x that traversed in the denser, is a constant quantity. Ptolemy's Law. If a ray pass from A to B, striking some point of a reflecting surface in its course, and being then reflected to B, there is no path from A to B wd any point of the mirror which is so short as that traversed by the ray under the law 'of reflexion. Format's Law. If a ray pass from A in one medium to B in another, there is between these points no path which, the relative velocities in the two media being taken into account, could be traversed in so short a time as that actually traversed under the law /j, sin ang. refr. == sin ang. incid. If the construc- tion be attempted, it will be seen that the actual law allows the ray the greatest possible proportion of time in the rarer medium. Superposition of simultaneous wave -motions on an in- definite cord. "We shall here simply discuss the single case in which two equal waves travel in opposite directions on the same cord. In Fig. 69 are seen two waves approaching one 126 KINEMATICS. [CHAP. another on a cord of indefinite length ; they meet at A, and pass through one another. Where crest meets crest and trough trough, as at A, the amplitude is doubled ; where crest meets trough, as at B and C, there is no movement as long as the waves are passing through' one another : B and C are, during this mutual interfer- ence of the waves, non- vibrating or "nodal" points, between which the cord vibrates. If the waves had been equal, but opposite in phase, so that crest met trough at A, then A, the point of meeting, would have been a nodal point. The two nodal points, B and C, are seen to be at a distance of half a wave-length from each other. If the waves meeting each other had been indefinitely numer- ous, the interference would occur in every region of the indefinite cord, and there would be an indefinite number of nodal points, half a wave-length distant from each other. In Fig. 7 such waves are seen running on an indefinite cord, and the nodal points, where crest meets trough, are marked with crosses. Fig.TO. Cord of definite length. In Fig. 70 let us limit our atten- tion to a part of the string comprised between two nodal points, say between and X. Within this limited part of the string we observe two waves running in opposite directions, the crest of one meeting the trough of the other at four points : there are five loops, each of which is equal to half a wave-length ; the centres of the loops, the points of greatest vibration, are the points A, B, C, D, E ; the wave-length is here equal to f of OX. Nodes and Loops. If the points and X in Fig. 70 had been the ends of the string, and if it had been possible in OX to establish two waves each of wave-length = J- OX, these meeting one another so that trough always coincided with crest, the neces- sary result would be a continuous vibration of the cord in five segments, marked off by four non- vibrating points, and these segments would always be in opposite phases of vibration. The vibration would in such a case be said to be Stationary Vibration. This is precisely the case where a wave running from to X meets its own reflexion returning from X to 0. If the wave- v.] NODES AND LOOPS. 127 length be f , f , f , f , f , etc. of the length of the cord OX, the result of the composition of the wave and its reflexion will be a stationary vibration, in which the string will vibrate in the first case as a whole, or in the others in two, three, four, five, etc. vibrating segments or Loops, separated by non-vibrating points or Nodes. Vibrations of a cord whose extremities are fixed. A string acted on by any force tending to bring the particles back to their mean position, and varying as the displace- ment a criterion, as we have seen, of harmonic motion will thus enter into vibrations of a type obeying Fourier's law, and in the general sense any periodic disturbance of such a cord will be compounded of vibrations such as those shown in Fig. '71. These simultaneous vibrations will, as regards amplitude, be independent of one another, and will also from moment to moment necessarily differ in their relative phase. The whole motion is, however, periodic. If a point situated in the loop of any one of these harmonic components be held fixed, the corresponding oscillation is pre- vented. If the centre of a vibrating string be touched, the oscilla- tions corresponding to the whole string, to one-third, to one-fifth, etc. all the odd components are suppressed, and only the even components those, namely, which already have a node at the point fixed are allowed to go on. If the string be touched at J of its length from the end, all vibrations except those correspond- string string string ,, _ ,_ ing to -, -, -, etc. cease ; these still continue, for the o u y effective fixing of their nodes does not affect them. Similarly, if the string be held steady at a point J of the string-length from the end, the 4th, 8th, 12th, 16th, etc. components remain un- affected, while all the rest are stopped. Longitudinal vibrations of a string or rod for a rod acts in this case like a bundle of parallel strings whose ends are held fixed obey the same principles as transverse vibrations. Fourier's law holds good ; and if any point be held steady, those component vibrations which have a node of displacement at the point held steady, and those components only, will remain un- 128 KINEMATICS. [CHAP. affected. In longitudinal vibration, where there is the greatest displacement there is least actual change of density ; and at the extremities of a rod fixed at both ends, and at the nodes, while there is no displacement, there is a maximum change of density. If longitudinal vibrations occur in a string or rod, or in a cylindrical mass of gas such as the air in an open organ-pipe - which is free at both extremities/it is plain that at the free ends there can be no change of density, but that there is freedom of movement ; hence each extremity must, as regards displacement, be the centre of a loop. ..The component vibrations which make up the Fourier-motion in such a case are such as those shown in Fig. 72. In this case, as well as in the preceding, all the components, even and Fig.72. ,,* .,, , odd, are possible, and the wave-length of the slowest or funda- mental vibration is equal to twice the length of the rod or string vibrating longitudinally. In these cases we see that the wave-length of the fundamental as well as of the concomitant vibrations is determined by the length of the vibrating string or rod itself. We have seen that v = ; A, the wave-length, may easily be found from the length of the rod ; r, the period, may be measured by acoustical or graphic methods ; these being experimentally known, we may find 17, the velocity of propagation of an undulatory disturbance in a vibrating string or rod. If the central particle of the system of Fig. 72, vibrating longi- tudinally, be held fixed, those vibrations (2, 4, 6, etc.) are sup- Fig . 73 pressed which have not their nodes at the _ centre of the rod. Thus only the odd com- ponents are left; but the rate of these is unaffected. If now one half of the rod were removed altogether, we would have remain- ing a rod fixed at the one end, free at the other. This rod would have component vibrations, as shown in Fig. 73. A rod thus vibrating longitudinally will have a funda- - , PC, Pd, etc v Fig.76. v.] INTERFERENCE OF WAVES. 131 differ from one another successively by half a wave-length. The disturbance caused at P by the movement of N# is to some extent counteracted by that derived from db, aided by that from be, counteracted by that from cd, and so on. But N# is greater than db, db than be, be than cd, and so on. On the other side of N the circumstances are similar ; and thus on the whole the disturbance of P, which is due to the wave-front on both sides of N, is positive. If on the other hand the point P' be considered, it will be seen that if the wave-front be wide enough in comparison with the wave-length, the disturbances radiating from it interfere with one another ; for points on the wave-front can always be chosen and set off in pairs, differing in distance from P 7 by half a wave- length ; and consequently there is no disturbance produced at any such lateral point as P' by the wave-motion at N, and the wave- front travels along the normals without any lateral expansion. The narrower the wave-front or the greater the wave-length, the greater will be the difficulty in this construction, and the greater will be the lateral divergence of the wave. The wave- front must be at least one wave-length in breadth before this construction begins to become possible. Effects of a screen. In the figure 76 we may neglect the influence on P of the part of the wave-front beyond d, for the extent to which it disturbs P is very small If a screen were thrust between the wave-front and the point P, so as to cut off the influence of the part cd, the disturbance of P would be increased : if the screen come to b, the motion of P will be less than at first : if it come to a, it will be greater than if there were no screen : if to N, it will be less, being about one-half of the original amount. If the screen be pushed still further, P will, in the same way, be in more or less active motion according to the position of the screen. The waves therefore pass round the edge of the screen, producing fringes of alternate maximum and minimum motion. The possibility of this result depends on the smallness of the wave-length. Effect of a very small screen. If a wave-front be inter- rupted by a very small screen placed so as to allow the wave motion to pass it all round its edge, Fig. 77 shows that dis- turbances passing round this obstacle produce, even behind the screen, hyperbolic fringes of maximum disturbance, between which there may be traced hyperbolic fringes of minimum disturbance where crests coincide with troughs ; and that, further, even 132 KINEMATICS. [CHAP. beyond the shadow of the screen, there are hyperbolic fringes of maximum and minimum disturbance. The centre of the shadow of the screen is a spot of a maximum disturbance. Flg.77. Wave traversing an aperture. The same kind of figure shows that a wave whose wave-length is small, passing through an aperture, gives fringes of maximum and minimum disturbance beyond the edge of the aperture. A very curious result is, that a wave-front passing through an aperture may produce no move- ment in a point situated immediately opposite its centre, if there be interference between the waves proceeding from the edges and those from the central regions of the aperture. v.] INTERFERENCE OF WAVES. 133 Relation of the wave-length to Fringes. The position of the fringes depends on the wave-length. The smaller the wave- length, the nearer to one another will the fringes be. If the incident wave, diverging from a point, be compound, each component will form its own set of fringes without reference to the others, and any particle within the field of fringes may be stationary as regards one of the component vibrations, while at the same time it is affected by the others. Energy of S.H.M. If a S.H.M. be wholly in one plane, its energy is wholly kinetic as the moving body passes its middle point ; it is equal to \ mv 2 where v is the velocity. If another S.H.M. of equal amplitude and period be compounded with the first, the energy is now doubled. In circular motion the two S.ELM.'s differ by J period ; when the energy of the one is wholly potential, that of the other is wholly kinetic, and vice versa. The energy of a circular movement of velocity v, when the moving body is attracted towards the centre by a force which varies inversely as the displace- ment from the body, is mv 2 ; of which half is kinetic, half potential. The Energy of a S.H.M. is proportional to the Square of its Ampli- tude. The angular velocity is constant, for S.H.M.'s are isochronous ; the velocity in the circle of reference varies as the radius, i.e., as the amplitude ; the Energy varies as the square of the velocity with which the body execut- ing the S.H.M. passes the midpoint ; this velocity is the same as the velocity in the circle of reference ; therefore the Energy varies as the square of the amplitude. Energy of wave-motion. The energy of a wave-motion is equally divided between the potential and the kinetic forms. Let us first consider a linear wave : at the crest and at the trough the whole energy is potential ; midway between crest and trough the energy of the particles as they pass through their mean position is wholly kinetic ; elsewhere all the particles are symmetrically and continuously losing potential and gaining kinetic energy, or gaining potential while losing kinetic energy ; the gains must be equal to the losses, for there is no change in the type of vibration, and no change in the amount either of potential or of kinetic energy (friction being imagined absent). Hence the whole energy of the wave is cLivided into two equal moieties, kinetic and potential in their respective forms. In a circular wave the kinetic energy is, under the same supposition, invariable in its absolute amount, and the potential energy bears to it the same symmetrical fixed ratio of equality. So for tridimensional waves. CHAPTEE VI. KINETICS. GENERAL PROPOSITIONS relating to the possible forms of motion find their parallel in those relating to the forces which produce motion. The formula / = ma, already established, shows that every force is measured by the motion which it produces. In this way we see that the truth of the propositions entitled the Parallelogram of Velocities and that of Accelerations involves that of a similar proposition in regard to Forces. Two forces acting on a single particle produce the same result as would be produced if a single force were acting on it, represented in mag- nitude and direction by the diagonal of a parallelogram of which the two single forces are represented in the same respects by the adjacent sides. This is the proposition of the " Parallelogram of Forces." In the same way we have a Parallelipedon of Forces, just as we have one of velocities. Hence if two component forces be at right angles to one another, the square of the resultant force will be equal to the sum of their squares. Eefer to Fig. 11; let it be desired to apply force to a particle lying at A, so as to make it move in the direction AC : the available force is represented in magnitude and direction by the line AD : then obviously this force cannot exert its full effect in the direction AC ; it is only its effective component in that direction that can produce any such effect : this component is represented by the line AC. The other component AB, at right angles to AC, can produce no such effect. This is an example of the Resolution of Forces. Plainly, we may resolve any force into two components at any angle to one another, just as we- may so resolve a velocity. In tridimen- sional space we may resolve a force into three components. Suppose, then, such a question as the following : A pull is made in the direction AD ; this pull is designed to draw an object through a tube whose direction is AC : what proportion do the component effective in pulling down the object and the lateral pressure on the walls of the tube respectively bear to the force CHAP. VI.] PARALLELOGRAM OF FORCES. 135 applied ? If Fig. 11, a, were drawn with the proper angle be- tween the direction of application of the force, and the line along which the object is to be drawn : then the effective component would be represented by AC, and the component producing lateral pressure by AB. But AB = AD sin ; AC = AD cos 6 : or AB : AD : : sin : I AC : AD : : cos : 1. Hence, if AD be taken as unity, AB and AC may easily be found, if the angle 6 be known, by finding the value of sin 6 and cos 6 in a table of trigonometrical ratios. If AD have any other value than unity, the values of sin 6 or cos 6 derived from the tables must be multiplied proportionately. Problem. A force of 50 Ibs. is applied to a solid body drawn down a fixed canal which the solid body exactly fits ; the angle which the direction of traction makes with the axis of the canal is 29. What is the effective component available in pulling the solid body ? and what is the pressure produced by the traction-force on the walls of the canal ? In Fig. 11, a, if be 29, AD represents the force applied equivalent to the weight of a 50-lb. mass ; AC represents the effective component = AD x cos ; AB represents the component at right angles to AC that is, the detrimental pressure = AD x sin 9. But cos 29 = '8746197 ; sin 29 = -4848096. The effective component is thus '8746197 of the force applied i.e. y it is equal to 43'731 Ibs.; the detrimental pressure is 24'2405 Ibs. It seems rather surprising that the effective and lateral com- ponents together should appear to be so much greater than the original force applied. But we have already seen that there is no law of the Conservation of Force, though there is a law of the Conservation of Energy. The principle of the Conservation of Energy is main- tained, for the en- ergy imparted in a given time is pro- portional to squares of the forces acting, and the squares of AB and AC are together equal to the square of AD. Experimental proof of the Par- allelogram of Forces. A cord, two masses of 12 and 13 Ibs. respectively, a pulley, and a dynamometer, are arranged on a beam as shown in Fig. *78. The string can be adjusted so that ' 136 , KINETICS. [CHAP. the angle BAG may have a wide range of values : for every position there is a corresponding stress on the dynamometer. When AB and AC are at right angles to one another, the spring of the dynamometer is pulled out to exactly the same extent as it would have been by the weight of a 5-lb. mass. The dynamometer may be replaced by a 5-lb. mass suspended over a pulley; in that case the cord would so adjust itself that the angle A would be a right angle. Here 5 2 + 12 2 = 13 2 , or 25 + 144 = 169; and the law is confirmed. In a similar way, the propositions known as the Triangle of Velocities, the Polygon, etc., are replaced in Kinetics by the Triangle, the Polygon, etc., of Forces ; and the resultant force is the missing side of the triangle or polygon, of which all the sides except the missing one represent the various forces acting, in precisely the fashion already studied under Kinematics. Centre of Figure. "We have already seen that a rigid body, of which the several particles are subject to accelerations which are equal and parallel to one another, moves as if it were con- centrated at its centre of figure, which is subject to a single acceleration. A body may thus be acted upon by parallel forces affecting its particles, the result being the same as if a single force had acted at the centre of figure; while conversely, if a single force act at the centre of figure, the result is to impart parallel and equal accelerations to all the particles, and thereby to effect a translation of the body. Hence most of the proposi- tions of Kinematics, which describe the motion of a single point, may be transferred to Kinetics, not only as relating to the move- ments of single particles, but also as relating to translation of material bodies. Inertia of Matter. If in any system of bodies there be no force acting, the formula / = ma ^ shows that a = 0, that there is no acceleration : hence if there be no force acting, there is no change in the speed with which a body is moving, or in its state of rest, as the case may be; in other words, Matter has Inertia. This is Newton's first law of motion, and it appears to be here derived from the formula ; but it will be remembered that the formula was itself derived by implication from that law. Examples of Inertia. Examples of this abound. Collisions between ships and between trains which do- not stop if there be not sufficient retarding-force at command ; trains passing stations when their speed is great and the rails are slippery ; a person falling off the stern of a boat or the back of a car when the vi.] INERTIA OF MATTER. 137 vehicle makes a sudden movement forwards in which his body does not participate; the onward motion retained by a rider when his horse stops under him ; the jerk received by a horse suddenly starting in order to set in motion a heavy waggon; when the waggon is running, if the horse suddenly stop, he is bruised, for the massive waggon does not stop at once ; a grey- hound chasing a hare is carried forward and cannot stop or turn his path instantly at the spot where the hare doubled or turned abruptly from her course ; the inertia of the dust of a carpet, when the carpet is beaten the carpet moves forwards at each blow, but the dust remains, and is thus separated from the carpet and blown away by the wind ; the inertia of the dust when it is shaken off a book the book and dust are made to describe together a rapid movement in the air the book is suddenly arrested by a smart blow, while the dust does not stop but moves onwards ; the inertia of the snow which in the same way is kicked off one's boots the boot is suddenly stopped, but the snow goes on, and is thus shaken off; the inertia of loose grain cargo in a ship it acquires a certain velocity when the ship rolls, and does not stop when the ship arrives at its normal limit, but pours on so as sometimes to make the ship roll beyond the limits of safety ; the oscillations of mercury in an ordinary barometer at sea, the mercury being jerked up by each roll of the ship ; the inertia of the mercury in a mercury manometer used to investigate fluid pressure the variations in the height of the mercurial column being greater than the real variations in the pressure, for the mercury does not stop moving when the fluid pressure ceases to rise or fall ; the inertia of water in house water-pipes if it be set to run and then suddenly stopped the water is compressed against itself and a violent jerk is produced, which is utilised in the hydraulic ram ; the inertia of water in the case of the water-supply of the loco- motive engines of passenger express trains on the L. and N.W. Railway system the engine puts down a tube the lower end of which acts as a scoop for the water, which tends to remain in its trough on the ground between the rails and at rest relatively to the ground, but this being equivalent to a backward movement relatively to the engine, the water slips up the inclined tube into the tender if the train be moving at sufficient speed. There are some further remarkable consequences of the inertia of matter : a body may be struck or pressed so suddenly that it stands practically at rest during the time that the blow is being 138 KINETICS. [CHAP. spent on it, and it may be crushed or broken by such a blow. A grain of corn or a granule of gold-quartz, if thrown up into the air and struck a blow by an iron bar moving at the rate of about 180 feet a second, will be crushed by compression, and will be by a succession of such blows very effectively pulverised. Milling machinery has been constructed on this principle. A bullet in a gun, though free to move onwards, is crushed against itself before it fairly starts, so that the soft lead is moulded into the grooves of the rifle-barrel by the rapidly -applied pressure due to the explosion of quick-burning gunpowder. Another example is afforded by that instrument with, the aid of which M. Kosapelly* has investigated the movements of the larynx during the emission of sounds. A heavy mass of metal is suspended in a light frame- work which, is tied over the larynx : as this mass cannot at once participate in the rapid movements which the vibration of the larynx communicates to the light framework, it forms a kind of fixed point, and the light framework, as it vibrates in contact with the skin over the larynx, may strike it a series of blows ; these may cause an electric current to be alternately made and broken ; the number and frequency of these interruptions may be registered on an appropriate recording-instrument. Further, the inertia of matter is a property of retaining what- ever motion an object has, and that in a plane fixed in space without reference to the movements of surrounding objects, unless these are so connected with it as to be able to affect its motion. A hammock retains its position in space independently, in the main, of the roll and rock of the ship. The statement would be approximately accurate that the hammock does not swing in the ship, but that the ship swings enclosing the hammock, which may for any short period of time be regarded as moving onward in space with the average velocity of the ship, but independent of it. A long and heavy pendulum set to swing in one plane, and connected by a very slender attachment to the roof of the building in which it is suspended, will swing in the same plane in absolute space though the earth rotate under it : the apparent result is, that the plane in which the pendulum swings gradually alters its aspect, so that the pendulum swings successively in every possible direction. The real state of the case is not that the heavy pendulum alters its direction of oscillation, but that the earth rotates or has a component of rotation under the pendulum. If a heavy wheel be set in motion, it will in the same way, if it can rotate for a sufficiently long time, show the same phenomenon, for it tends to continue to rotate in the same plane in space. * Tray, du Laboratoire de M. Marey, 1876. vi.] MOMENTUM AND IMPACT. 139 Momentum. The product MV of M, the mass of a moving body, into V, its velocity, is called the Momentum of the body. If a shell explode, its fragments form a system of bodies moving at different velocities. The average velocity of the centre of figure of the whole system is, however, unchanged: some frag- ments travel with a greater, some at a less, velocity than that with which the shell had travelled before the explosion; but the mass M is unchanged though differently arranged, the mean velo- city of the system is the same as that of the original shell, and thus the momentum of the whole system is the same after explo- sion as before it. Impact. If there be two inelastic bodies of masses m f and m jt respectively, of which the first moves with velocity v r while the second is at rest : if the moving one, whose momentum is m/Oj, strike the other, it will divide its momentum with that other; it itself will travel more slowly while the other is set in motion ; but the two travel together with a common velocity V. The whole mass moving with this new velocity V is (ra, + m /7 ) ; its momentum is equal to the original mv / ; hence the velocity V may be found by stating this equality of momenta in the form of the equation (m i + mj V = m / v l ... V = m ' V ' m, t + m a If the mass m u be large in comparison with m /t the velocity V is much less than v r If a man lie with an anvil on his chest, and the anvil be struck a blow with a hammer relatively not too heavy, the person lying down, if he can support the anvil, will not be much affected by the blow, for the movement imparted to the anvil will be slow as compared with that of the hammer. Let the two inelastic masses be m i and m /y , moving with the respective velocities v t and v //} and together moving after impact with the velocity V ; the respective momenta of the masses before impact were m t v t and m a v a ; that of the conjoined mass after, impact is (m, + mj V. Hence mp. + m^ = (m,+ mj V. V = m ' V ' + m> ' ?) " . (1 ) It was found experimentally by Newton that, in such a case, the momentum lost by one body was equal to that gained by the other. To express this algebraically, if M represent the momentum gained by one and. lost by the other m,V - m i v i = M. (2.) *m .. -.. TT "it* /*) \ u u si JJJ-* \***/ From either of these, with the aid of equation (1) we find 140 KINETICS. [CHAP. M = Apparent loss of Energy. In this case the kinetic energy after impact (/ \ 9 f ~~* i *, .*. \9 \ i n -f / 97Z- 1) ~r~ 771 1? \ i (77Z' V "t" Wb V ) \ i.e. y J mass x V 2 = A (m. + m y ) ( '-i '!- } = \ L-LJ. s-eL } \ m / + m a J m, + m a / is less than the sum of the kinetic energies before impact (which were J mp? and J m M v^). This is not true if v t be equal to v a , but in that case the two bodies would be travelling in the same direction with equal speed, and the one could not overtake and strike the other. The energy, which has appar- ently disappeared, has assumed the form of Heat. Impact of Elastic Bodies. "We may here anticipate a state- ment of the nature of Elasticity so far as to say that a perfectly elastic body, possessed of a certain amount of kinetic energy, and striking a perfectly rigid body, will rebound, and it will be found that the body possesses as much kinetic energy after the impact as before it; for it leaves the rigid body with a velocity equal to that with which it approached it. The mass and the velocity being numerically unchanged, the momentum is numerically equal after impact to that before it ; but as it is no longer mv but m x ( v) = mv, it has become negative, and has therefore altered by an amount equal to 2mv. If the body be imperfectly elastic, so that the velocity is not completely regained, it is found ex- perimentally that it returns with a certain fraction, X, of its original velocity (this fraction, X, being called the coefficient of restitu- tion), and the change of velocity is not 2v but (1 +\) v ; and its momentum has become negative and = m\v, so that it has changed by the amount (1 + X) mv. If two masses > m l and m y/ , moving with velocities v, and v a , and formed of such material that the coefficient of restitution between them is A, strike one another,. they will, after impact, travel with velocities V, and V y/ . The momentum gained by one is equal to that lost by the other ; but it is not equal, as it is in the case of inelastic bodies where A = 0, simply to - " " ~ , but to (1 + A) x that quantity. This equality of momenta is expressed by the equations (1) Gained by m, ; m,V, - m,v t = (1 + A) m ' m (v ~ v <\ ... T , , Tr ,, ^.mm(v v] (2) Lost by m jt ; m a v a - m M V M = (1 x A) ' " v " - . m t + m u Whence _ + + A m. (.. - ,)_ vi. ] IMPACT. 141 Take, as a particular instance, the case in which the elasticity is perfect or the restitution complete (i.e., A=l); and the balls which strike one another are of equal weight, so that m / = m a ; then V / = v u and V,, = v r Two equal and perfectly elastic balls striking one another directly in the line joining their centres exchange their velocities, and that whether they meet or overtake one another. Oblique Impact. If a ball strike a rigid surface obliquely, its motion relative to that surface may be resolved into two com- ponents : one parallel to it, which is not affected by the impact ; the other at right angles to it, which, after impact, will be wholly or in part restored in the reverse direction. Eeference to Fig. 6 will show that if X = 1, the angle of reflexion will be equal to the angle of incidence ; while if X be less than unity, the angle of reflexion will be proportionately less acute to an extent easily determined by construction. If the oblique impact be between two balls, the investigation is based on similar principles. Take a line joining their centres of figure ; in a direction at right angles to this line the motion is unaffected by the impact, and the component in this direction will be the same after impact as before it ; in the line joining the centres of figure, the velocities V, and V may be found as in the preceding discussion. By compounding the component velocities after im- pact, the resultant velocities and their direction may be found. In practice, as the two balls are passing one another in contact, friction between their sur- faces causes a relative delay of one aspect of each, and causes rotation of the balls ; the energy necessary for this is taken from that theoretically available for the direct translational movements of the balls as wholes. Energy in impact of Elastic Bodies. In the case of per- fectly elastic bodies, the energy after impact is equal to that before it; \ w,V/ + \ mj' =\ mtf + $ m A 2 . If X be less than 1, there is apparent loss of Energy, which has assumed some other form than that of motion of the mass. If a horse with loose traces rush forward and jolt a car, the energy which disappears is wasted in the form of heat, or deleteriously spent in disintegration of the materials of the car, or in bruising the animal. " Centrifugal Force," so-called. It has been already shown that when a body describes a curved path, there is an acceleration towards the centre = , where v is the tangential velocity in the curve, and r the instantaneous radius of curvature. If the path v 2 be a circle of radius R, the acceleration is constantly g. The component force drawing it from the tangential path is therefore 142 KINETICS. [CHAP. one which produces an acceleration towards the centre = g-, and is itself equal to -^~. Suppose a stone of mass m to be whirled round like a slingstone by a string, but in a perfectly circular path. The tangential velocity at any instant may be resolved into two -y 2 components, one along the circle, one away from it = -^ ; the latter is nullified by the tension of the string. The tension of the string must = ^5- units of force. If the string snap, both components of the tangential velocity come freely into joint and simultaneous play, and the body moves on in the resultant straight line. The two components of the tangential velocity produce different results : the one freely manifests itself as motion in the curved path, the other tends to communicate itself to the string, and corresponds to a stress which is set up in the string by the action of its molec- ular forces. The string is thus under stress it is stretched, it is under tension, and this is the same as would be established by a n force = -np- units. If it were cut or snapped, it would cease to exercise tension, and the stone, flying off along the tangent to the point where it happened to be at the instant of snapping, would then obey Newton's First Law of Motion. It does not fly straight from the centre, and hence there is no " centrifugal force " counteracted by the tension of the string ; the tension of the string does, however, balance the centrifugal com- ponent of the tangential velocity at every instant. Any string will snap if force be applied to it beyond a cer- tain limit. If a string be just so strong that N grammes of matter may be suspended on it without its snapping, it can survive the application of 9 8 IN" units of force. If this string be used to whirl a slingstone of mass m, it will snap unless the velocity v be Q such that -TJ- is less than 9 8 IN that is, v must be less than If the velocity be greater than this, the string will snap. As the velocity increases, its centrifugal component in- creases, and requires a greater force or reaction to be exerted in a direction towards the centre in order to bend the path into the same curve in a shorter time. In the same way, if a fly stand on the rim of a rotating wheel, the tension between the foot of vi.] " CENTRIFUGAL FORCE." 143 the fly and the rim of the wheel may become so great that the fly cannot hold on and is hurled off. When a grindstone or fly- wheel is rotated too rapidly, the molecular forces of cohesion cannot keep the particles together against their tendency to fly off. If the earth rotated on its axis seventeen times as fast as it does, the attraction of gravitation, the effect of which is even now masked to some extent by the rotation of the earth, would only just be able, at the equator, to keep bodies from flying off its* surface. When a drop of oil is suspended in a mixture of spirit and water, so that it is free to assume any form, and a motion of rotation is communicated to it, the globular drop assumes the form of an oblate spheroid, and bulges at its equator ; for particles at its original equator have, when set in motion, a greater velocity than those nearer its poles. For the same reason the earth itself has assumed the form of an oblate spheroid. In the trundling of a wet mop, when the drops fly off because they do not adhere so firmly as to be able to develope tension sufficient to cause them to retain their position in the rotation of a steam governor, the balls of which fly asunder as the speed of the engine increases, thereby actuating an appropriate train of mechanism which to a greater or less extent shuts off the steam we find examples of this phenomenon. If a man were placed on a revolving table, with his feet towards the centre, the blood in his body would be urged towards his head ; and this has actually been proposed as treatment in bloodlessness of the brain. When a circular cylindrical vessel containing water is rotated on its axis, the water is heaped up towards the sides of the vessel. If the speed exceed a certain limit, the water will be hurled over the sides of the vessel, and if the supply of water and the rotation be continuous, an engine may expend its energy in thus continu- ously lifting water against gravity. Mr. Siemens has made use of this principle in the construction of a governor for machinery ; when the engine goes too fast it begins to spend energy in pro- ducing this , current of water. The form of the surface of water thus produced is always parabolic. If a particle move in a circle whose radius is r with angular velocity , the actual space traversed by it in time, t , is rut ; its velocity per second is ro> ; the force necessary to impart this velocity in a second is mroo the Energy mr 2 o> 2 of its movement is . 144 KINETICS. [CHAP. Moment of Inertia. If a mass M and a point in space, internal or external to the mass, be considered in relation to each other, the name Moment of Inertia is given, and the symbol I is usually assigned to the numerical quantity which is obtained by summing up in appropriate units the products of the mass m of each particle of the mass, and the square of its corresponding distance d. This operation generally requires the aid of the Integral Calculus, but the resultant sum, 2md 2 , is a numerical quantity, and is always positive. Radius of Gyration. Suppose a uniform disc of radius 1 ft. to rotate round its centre under the action of a given force, it rotates with less angular velocity than it would assume if the same force were applied to the same matter gathered nearer the centre ; for / = mrto, and if r increase, to diminishes. When / is constant it rotates with greater angular velocity than that which it would assume if the matter were gathered near the circumfer- ence. Between these two limits there must be a distance from the centre such that if the whole mass were concentrated there, the angular velocity acquired under the influence of a given force would be the same as that actually assumed by the disc. This distance is the radius of gyration with reference to that point. In the case of a disc, its distance from the centre is 702 ( = =0 X the radius. In the same way a linear body of length I \ V X 2/ suspended at one end rotates with the same angular velocity as if its mass were accumulated at a distance = =r from the point of suspension. If V3 the linear body were poised on its centre, its radius of gyration would be I ~j=- ' A parallelogram of sides a and &, poised at its centre, would have a x/a 2 + b 2 radius of gyration = g . The criterion of the radius of gyration is, that if the whole mass M were placed at the distance Jc (k = rad. of gyr.) from the point of suspension, it would have with reference to it the same Moment of Inertia as that actually possessed by the physical mass in question. Hence Mfc 2 = I = 2rf. Energy of a rotating body. The energy of a particle in rotational TOT^tO 2 movement is - ; that of a system of particles each at its own distance r 2i mr 2 to 2 to 2 and with its own mass m must be 2 - = Zmr 2 ; to, the angular velocity, being the same in all particles of a rotating body. But 2mr 2 = I, the to 2 moment of inertia : therefore the energy of a rotating body is I, O r, where to 2 k is the radius of gyration, = M& 2 . Suppose a heavy body to be suspended at a point, and then, that point of suspension retaining its fixed position, to swing down so far that its centre of figure sinks through a height h, acquiring an angular velocity to. The kinetic energy acquired by the body considered as rotating round the point of sus- to 2 pension is I ; the potential energy lost by descent of the mass m through height h is mgh. These are equal. VI.] ENERGY OF KOTATION. 145 or I = mgh. w 2m// A. But I = mk z where k is the radius of gyration. A flywheel in motion possesses a large amount of kinetic energy ; and if an obstacle be placed in the way of the engine, the engine cannot be stopped by it unless the flywheel can be arrested also : this would involve the sudden exercise of a very great force ; hence an engine with a heavy flywheel rapidly rotating can over- come a very great resistance, and in this way, for ordinary resist- ances, it is prevented from manifesting any very great irregularity of motion. w 2 If a flywheel whose energy is M& 2 were called upon to expend n unite of energy in overcoming a certain resistance, the energy in it after doing so would be ( MA; 2 n j , and a new angular velocity o> / would be assumed, such that afi*- ($*-..) If a flywheel have a heavy rim, and the spokes be thin, the radius of gyration is practically the distance between the centre of the wheel and the middle of the thickness of the rim. Centre of Oscillation. In Fig. 79 let A be the point of suspension Fig.79. of a body, B its centre of figure or of mass (centre of gravity), k the length of the radius of gyration of the mass with reference to the point of suspen- sion A : then there is in the same straight line with A and B, and on the opposite side of B from A, a point C called the Centre of Oscillation, which has the following properties : (1.) The body may be swung upon A or upon C indifferently, and in either case it will oscillate pendulum- wise with equal rapidity. (2.) The body thus suspended at A or at C will oscillate at the same rate as an ideal simple pen- dulum of the length AC. (3.) This body will, if struck at C, oscillate round A without producing any pressure on the supporting axis at A. (4.) Though the support at A were withdrawn as, for instance, if the body float submerged in water yet if the point C were struck, the point A would remain at rest, and all the part of the body lying above A would move in a direction opposite to that in which C is struck. For every point C at which a body may be struck, or for every centre of percussion, there is a corresponding point A on the other side of the centre of figure L 146 KINETICS. [CHAP. through which passes an axis of spontaneous rotation, round which the body rotates. If the lower part of any object be suddenly pulled for- wards, the upper part will move backwards. This property is found applied in the jaw of echinoidei ; " the upper end of each of the five jaws is suddenly tilted outwards, and the lower, the tooth-bearing ends, are tilted together. k 2 (5.) The distance AC is equal to when the body is suspended at A, AB 2 k being the radius of gyration in this case ; or to L - when suspended at C, CB k 4 being the radius of gyration in this case. related that = *-. AB CB The radii of gyration are so Force causing rotation constant in direction. If a body be caused to rotate by a force whose direction is the same or nearly so throughout the movement, the effect of the force varies greatly. In Fig. 80 let AC, AD, AE, AF, be successive positions of a rod rotating round A, and acted upon by a force applied at the extremity remote from A, and always parallel to the lines Gc, Dd, E0. In the position AF the effect produced by the force is a maximum, for two reasons : (1) that the component effective in producing rotation is there the greatest; and (2) that the force is applied with the great- est " leverage," or so as to have the greatest possible " moment." In this way the forearm moves with the greatest swiftness at the middle of flexion. Accelerated Motion. The discussion of the accelerated motion of a particle moving with constant acceleration i.e., under the continuous influence of a constant force has already led us to the formulae =V at, (i.) s =V*^-, (ii.) v z = V 2 2as, (iii.) where V represents the velocity of a particle at the beginning, and v that at the end of time t, s the space traversed, and a the acceleration (positive or negative, as the case may be) per unit of time. The most familiar examples of this kind of movement are those in which bodies exposed to the constant influence of gravity fall with constantly-increasing speed. vi.] ACCELERATED MOTION. 147 Problems. 1. A' train travelling at 50 miles an hour comes into collision with a fixed obstacle and is abruptly stopped : the passengers receive a blow. What height must one fall in order to receive a similar blow ? Ans. 50 miles an hour = 73-3 feet per second. A body falling from rest (V = 0) acquires a speed of 73-3 per sec. in falling 83*51 feet, for (iii.)- s; .' s = 83*51. A blow in a collision at 50 miles an hour is equivalent to a blow received in consequence of a fall of 83*51 feet; for a body which has fallen 83*51 feet is in consequence travelling at 50 miles an hour at the instant. 2. A ball weighing 5 ounces is hurled upwards. It is supposed that while it is in the hand it is swung through 4 feet : the thrower during this swing continuously exerts an accelerating pressure on it This pressure must be equivalent to the effort which would be put forth in raising some definite weight in the same position of the body. What is this weight if the ball rises 100 feet ? The ball leaves the hand with an unknown velocity V ; it rises through a space s=100 feet against a force (gravity), the negative acceleration (a) produced by which is 32*2 feet per second ; it comes for an instant to rest (v = 0) at the top of its course. 2 = V 2 2a*. (iii.) = V 2 -(2x 32*2 x 100). V 2 = 6440. V = 80-25. The question thus becomes Under the influence of a force / acting through 4 feet, a velocity 80*25 feet per second is imparted to a mass. m = ^ Ib. Find /; or, since / = ma and ra = y 5 ^, find a. i? 2 = V 2 +2as (iii.) ; V = ; a is positive. ^ = 205. 6440 = 2ax4 = 8a. a = 805 feet. f=ma= 226 T 9 ^ British units of force. But wt. of 1-lb. mass = 32-2 Brit units of force. 226- 9 7r /= wt of S Ib.-masses = 7 jf Ibs. The effort then is the same as that put forth in raising a weight of 7 Ibs, 13 oz. 3. A shot is fired vertically from a gun whose barrel is 30 inches long : it rises half a mile. Compare the acceleration of a body falling under the influence of gravity with that under which the bullet acquires such velocity in the space of 30 inches. First find the velocity of the shot as it leaves the gun. In its course in the air (friction being meanwhile entirely neglected) it commences with the unknown velocity V, traverses space * = 2640 feet against gravity which pro- duces acceleration a= - 32*2, and comes to rest (v = Q) for an instant. V 2 = V 2 205. (iii.) _ =V 2 - (2 x 32-2x2640). V= \/170016 = 412'3. In the gun-barrel a velocity v, 412-3 feet per second, is acquired within a 148 KINETICS. [CHAP. space s=2j feet. Here v = 412'3 ; V = ; s = 2*5 ; a is unknown, but is positive. V 2 = v2+2a. (iii.) (412*3) 2 =2ax2j. 170016 = 5a. a = 34003'2 feet per second. If the barrel were long enough to expose the projectile to the influence of such an accelerating force for a whole second, this velocity would be acquired, and the barrel would be 17001*6 feet long. But gravity produces an acceleration of 3 2 '2 feet per second. Hence the acceleration due to the gunpowder is greater than that of gravity in the ratio of , or 105*6 : 1. The force exerted by the powder = ma = 34003'2m, and in the case of an ounce-bullet would be equal to 2125*2 British units of force. 4. With what "force" will a 10-lb. mass falling 100 feet strike at the end of its course ? As it stands, this question is devoid of sense, for it does not specify the time during which this force acts on impact. If the body struck were rigid, and the falling mass perfectly elastic, it would rise on rebounding to an equal height of 100 feet. During the rebound it must have come to rest. Let us arbitrarily assume it to have taken ^THru" sec - * come to rest, and an equal period ^Vcr sec< to 8 a ^ n i ts upward initial velocity : this upward initial velocity is 80*25 feet per second, for v 2 = V 2 -2as. = V 2 -6440. V= 80*25. The question thus becomes What is the mean pressure between the body which has fallen and that on which it falls, if a speed of 80*25 feet per second can be arrested or developed by it in YTTUTT sec - The answer is Since v = at : v = 80*25 ; t = ^-^\ a =160,500; and /= ma = 1,605,000 British units of force = the wt. of 1 ' 605 > 000 = 4984*47 Ibs. 32S-2 5. A ball weighing 10 Ibs. falls from a height of 100 feet on a rigid floor. It is flattened to the extent of -$ inch, measured in the direction of its motion : it recovers its form and rebounds. What is the time taken to bring the ball to rest, and what is the mean pressure between the ball and the floor on which it falls ? Here a velocity of 80*24954 ft. per sec. is arrested in the space of -^ inch : what is the retarding acceleration a ? what is the corresponding pressure ft what is the time t ? <-2 = V 2 -2as. (iil) = (80-24954) 2 - (2a x ^ inch.) = 6440 - (2a x ^ foot) a = 6440x36 = 1,159,200 feet per second. /= ma = 1,159,200 x 10 = 11,592,000 British units of force = the weight of - = 360,000 Ibs. = mean pressure. 32*2 Again, v = at. (i.) 80*24954 = 1,159, 200*. t 14,444 vi.j THE PRINCIPLE OF MOMENTS. 149 The Principle of Moments. In Fig. 81 a linear body is poised at the point F ; at A there acts a force equal to 1 units, Fig.81. at B a force equal to 1 unit. The former, acting alone, would turn the bar round F through an angle 0, and the work done at A by the force F ( = Is) = 10 units x Aa 10 AF sin 6. The latter force applied at B would turn the bar round F through an angle, say <, and would in producing rotation do work = 1 x FB sin = 0. If the one force raise as much as the other depresses every point of the bar, there is equilibrium. As regards rotation round F, the forces are of opposite sign ; they are accordingly -f 1 0, tending to produce positive rotation in the direction opposed to that of the hands of a watch, and 1, tending to produce an opposite rotation. Hence, if there be equilibrium, the work done by force = -f 1 acting at A, and that done by force 1 acting at B are together = 0. (10 x AF sin ff) + (1 x BF sin 0) = 0. (1.) (lOxAF)-f (lxBF) = 0; or, generally, if the forces be P and Q, (PxAF) + (QxBF) = 0; or P:Q :: BF : AF. The forces which balance one another are inversely proportional to their distances from the fixed point F. Thus a smaller force acting at a greater distance can balance a greater force acting at a less distance. The Importance of the greater force with reference to the point F is exactly the same as that of the smaller force which has the advantage of greater dis- tance, or greater " leverage " or " purchase." This Importance of a force not passing through a point is called the Moment of that force round that point. It is equal to the amount of the force x the distance from the point to the line of application of the force. The shortest distance from a 150 KINETICS. [CHAP. point to a line is well known to be a line drawn from the point to the line in question, at right angles to the latter. In Fig. 81 the moments of the forces round the point F are respectively 10 x AF and 1 X BF. In Fig. 82 the forces acting at A and at B are not parallel ; their lines of application are AE and BE ; the dis- tances of these lines from F are FC and FD at right angles to AE and BE : the moments of the respective forces round F are Force A X dis- k * tance CF, and Force B x distance B\ FD ; and if the forces are to produce equal and contrary rotational effects round F, so that there may be rest and statical equilibrium, their moments must be equal and of opposite sign, so that their sum = 0. This is the Principle of Moments. If in Fig. 81 the forces at A and B acted at the ends of an immovable rod, there would be a reaction = 11 units spread over the whole extent of the rod, but more intense near the point A: if the rod were held fast only at one fixed point F, all the reaction (=11 units) is concentrated at that point, if it be such a point that there is no tendency to rotation round it i.e., if the moments round the point of resistance = ; if these be not = 0, the pressure on it is still 1 1 Ibs., but the energy is partly spent in producing rotation round that point. If the reaction pass through the point round which the moments = 0, there is neither translation nor rotation, and hence the three- forces are in equi- librium : these are, 1 units at A, 1 unit at B, parallel to the former, and 11 units at F, parallel but in the opposite direction. Thus Fig. 83 is established as indicating the conditions of 5 equilibrium of two parallel forces, P and G Fig.ss. Q ; a third, E, equal to their sum, must I act in the opposite direction at F, a point Q round which their moments vanish or are together equal to zero. If the two con- ditions (1) that P + Q = E, and (2) R that the moments round F be equal, be satisfied, there will be statical equilibrium : if the former be not satisfied there will be translation; if the latter there will be rotation ; if both be violated there will be both translation and rotation. Couples. Now let the point of application of the force Q 8 vr.] COUPLES. 151 be shifted to the right : the force P must increase in order that its moment may remain equal to that of Q. If Q be transferred to an indefinite distance, the force P would have to become indefinitely great in order to balance it. If the force Q were absent, P and ^jould not balance one another, but would turn the body round a point G between A and F, determined accord- ing to the relative importance or moment of P and E with respect to it by the principle of moments, so that P x PG- = E x GF. Two forces which are in this way not directly opposed to one another, but give rise to a rotation, are called a Couple. A couple may be balanced by a single force : P and E are balanced by Q. In this case P and Q have opposite moments round F ; E has no moment round F, its own point of action. There is equilibrium here between P, Q, and E ; their moments round F are together = 0. So are their moments round any other point, as may be easily proved. In general, whatever point is con- sidered, if there is to be no rotation round that point, the sum of all the moments of all the forces acting round that point must be equal to 0. The moment of a couple is the numerical sum of the moments of its two components round the point G ; the two components concur in producing rotation round that point* Examples of Couples. Examples abound in the muscular and osseous system. Such are the elbow joint, where the triceps pulls the olecranon process backwards, and the reaction of the articular surface of the humerus against the sigmoid cavity of the ulna constitutes the other member of the couple ; the jaw in lateral chewing, where the external pterygoid muscle may pull one side of the jaw forward, while the result of the action of the hinder fibres of the opposite temporal muscle, together with the corre- sponding muscles below the jaw, is to pull the opposite side of the jaw back- wards ; the weight of the head when a person stands in a very erect position is equivalent to a force acting along a line passing through a point a little behind the occipital condyles, and this, together with the reaction between the atlas and the occipital condyles, forms a couple which is equilibrated by the muscles of the front of the neck ; the same weight of the head when it bends forward a little passes along a line a little in front of the condyles, and it forms with the reaction of the atlas a couple, which is balanced by the contraction of the muscles of the back of the neck : when these contractions slacken, as when a person is falling asleep, the head is rotated by the couple on a transverse axis, and it drops forward or backward according to the position in which it happens to be at the time when muscular contraction ceased to balance its weight. The action of the two hands on the handles of a copying-press is that of a couple : one pulls, the other pushes. * Numerous examples may be found discussed in Hermann Meyer : Die Statik u. Mechanik des menschl. Knochengerustes ; Leipzig, Engelmann. 152 KINETICS. [CHAP. The Mechanical Powers. The principle of moments or what is essentially the same thing the principle that the work done "by or on a machine = 0, or that, on the whole, there is no accumulation of work in a machine, is the key to the explanation of the action of many of the Simple Machines or so-called Mechanical Powers. The work done by a simple machine is equal to that done upon it, and upon the machine itself there is no work done. This is, of course, not strictly accurate ; but simple machines are supposed, in the first instance and for the sake of theory, to be themselves without weight, and to work without friction. The Lever. This is a bar of any substance, rigid enough to retain its form under the pressures applied to it. If the pressure P be applied at A, the fixed point or ful- crum be at F, and another pressure Q at Fig - 84 - , B, then round F the moments (P x AF) + Q (Q X BF) = 0. If AF be shorter than BF, the downward pressure, P at A, which can be balanced by another downward pressure, Q at the point B, is greater than Q, and there is a mechanical advantage in employ- ing a smaller pressure Q to balance a greater pressure P. As an example of this take a crowbar : the man's strength is exerted at B ; the fixed point is at F, and the weight of the body to be lifted acts downwards at A. Suppose the lever to be 42 inches long, the point F to be 2 inches from A, and the man's strength, which is competent to raise 56 Ibs., to be exerted at B : then (AF x P) + (BF x Q.) = 0. 2P = 40 x 56. P= 1120 lbs. = 10 cwt. That is to say, the resultant pressure at P, due to the pressure Q at B, is numerically equivalent to 1 cwts., and acts in the direc- tion opposed to the weight of the mass to be lifted. A pressure at B of 5 6 Ibs. will sustain the 1 cwts. at A ; one a little greater will lift it. On the other hand, a pressure of 1120 Ibs. at P can only balance a weight of 5 6 Ibs. at Q : here there is great mechanical disadvantage. An example of this occurs in the case of forceps, with blades relatively long : the pressure which they can exert at the tip is relatively small, for vi.] THE LEVER. each blade of the forceps is a lever supported at the hinge.' scissors do not cut so well at the point as near the joint. But what is gained or lost in pressure is lost or* range of movement ; for the work done ( = Fs) by the one arnf is always the same as that done upon the other. Whatever the special arrangement of the two forces and the reaction, the principle is always the same, that the lever is studied at the instant of equilibrium, when round every point the sum of the moments = ; then an excess either of the pressure or of the resistance, beyond their proportions at that instant, will cause rotation round the fulcrum. There is a popular division of levers into three classes, which it is well to explain ; Fig. 84 illustrates them all. Class I. The fixed point is at F in Fig. 84. A crowbar, a handspike, a pair of forceps, scissors, or shears, a poker, a common balance, all these have the fulcrum or fixed point between the point of application of the force and that of the resistance. Class II. If the fulcrum be at A in Fig. 84, and the force be applied at B, the resistance overcome at F is numerically greater than the pressure exerted at B, as is found by taking moments round A. AF x R = (AB x Q). Examples of this are furnished by nut-crackers, where the resisting nut is nearer the hinge than the hand is ; by the oar of a boat, in which the force is applied at the handle while the tip of the oar is approximately at rest, and the resistance of the boat is overcome between these ; by a claw hammer used for extracting nails, where the fulcrum is at the end of the claw, the force is applied through the handle, and the resisting head of the nail is between these points ; by a wheelbarrow, in which the fulcrum is at the axle of the wheel, the raising force is applied at the handle, and the resistance to be overcome is the weight of the substance in the barrow between the handle and the wheel Class III. This is the same as the second class, except that the Force and the Resistance have changed places. As example of this we find that in a pair of tongs for sugar or for coal, in which the fixed point is at the hinge or the flexible end, the resistance is encountered near the end, and the force is encountered between these points. The pressure that can be applied by such an arrangement is comparatively feeble, while to produce any given pressure the force applied must be proportionately very great. This is seen in opening a gate by pressing on it near the hinges ; a considerable force has to be exerted. The whole mass of the gate may be conceived as concen- trated at a distance = O f its length from the hinge. The force applied, multiplied by its short distance from the hinge, must give a product equal to that of the necessary pressure into the greater distance. Such an arrange- ment, in which force is sacrificed in order to gain amplitude of movement, is of ordinary occurrence in the muscular system. The biceps is inserted into the radius at a point about one-sixth of the distance between the axis of rotation of the elbow joint and the centre of the palm of the hand. In order to raise a pound-mass in the hand, that muscle, if it acted alone, would 154 KINETICS. [CHAP. have to exert a force which would directly lift 6 Ibs. ; but, on the other hand, the forearm has a range and rapidity of movement which it would not have had had the muscles been inserted in the position of greatest mechanical advantage, not to mention the inconvenience of having muscles extending from prominence to prominence of the skeleton like the rigging of a ship. The pectoral muscle of a bird, the deltoid muscle of man, his glutei muscles, actuate conspicuous examples of osseous levers of the third order. Problems. 1. Two porters bear a burden, 56 Ibs. in weight, by means of a bar of such length that the distance between shoulder and shoulder is 70 inches. The weight is suspended from a point 40 inches from the shoulder of one of the porters. What is the pressure on the shoulder of each respectively ? Ans. This is a case of Fig. 84, in which the weight of the burden corresponds to R, and the upward shoulder-reactions to P and Q. There is no tendency to rotation round F, which is relatively fixed ; hence the reactions at A and B must be such that their moments round F are equal. Hence the two ' equations, P + Q = 56 and 40 P = 30 Q, give P = 24, Q = 32. The porter nearer to the burden carries 32 Ibs., the one further from it carries 24 Ibs. 2. A nut-cracker 6 inches long has a nut in it an inch from the hinge. The hand exerts a pressure of 4 Ibs. : what is the stress on the hinge ? Ans. The nut, as long as it does not yield, affords a fixed point : the stress on the hinge = the weight of 24 Ibs. The Wheel and .Axle. The lever, when it has done work and raised a burden against resistance, moves into a position where the leverage and the effective component are both dimin- ished, as seen in Fig. 80, if the force retain, or nearly retain, its original direction. If by any means matter could be so arranged that a lever would, when it had moved out of its position of greatest advantage, be replaced in the most favourable position by another lever, to which the burden and the force applied will be shifted, the apparatus thus constructed would in some respects be more useful than a simple lever. This criterion is satisfied as regards levers of the first order by the Wheel and Axle. This consists of a large wheel or cylinder and a small one, both on the same axis, and capable of rotating together on that axis. Each wheel may, if solid, be regarded as consisting of an infinite number of spokes. One of these spokes in the larger wheel, and one running in the opposite direction from the centre in the smaller wheel, together make up, when they are for an instant at right angles to the lines of application of the force and the resistance, a lever in the most favourable position. As soon as this has left the position of greatest advantage by reason of rotation of the system, its place is at once taken by another. A larger weight hung on the smaller wheel and a smaller weight hung on the larger one will balance one another, if their vi.] THE WHEEL AND AXLE. 155 moments round the axis of rotation be equal. The weights may be replaced by a force applied at the circumference of the larger wheel, and a larger resistance balancing this at the margin of the smaller wheel. This is the principle of the capstan and the winch the former used on ships for raising the anchor, the latter in use for drawing buckets up wells. In the former the spokes of the larger wheel are few, while the smaller takes the form of a cylinder or drum ; in the latter the smaller takes the same cylindrical form, while the larger consists virtually of only one spoke, the handle, which is turned through all successive positions in a circle. The wheel and axle is a statical instrument as long as its moments round the axis = ; but when one of the moments is greater than the other, there is rotation. Wheelwork. If a force be applied to the first wheel of a chain of wheelwork, so that it acquires an angular velocity, o>, and if the last wheel of the chain have, in consequence of this, an angular velocity a> v the force which the last wheel can exert is, as compared with that which the first wheel alone might exert when running with angular velocity o>, as &> : &> r The principle holds good, whatever the nature or complication of the mechanism which intervenes between the first and the last wheel. In a crane or in a lathe arranged for metal-cutting, we see the wheelwork so devised that the last axis moves very slowly, and with a corre- sponding power of overcoming resistance. The Inclined Plane. The mechanical advantage of this machine depends on the principle of the resolution of a force into its components. When a body is pushed up an inclined plane by a force just sufficient, and no more, to prevent it from moving down the plane in obedience to gravity, there is equilibrium between three forces viz., this Force, acting along the slope of the incline, the Weight of the body acting vertically, and the Eeaction between the body and the surface of the plane, acting at right angles to the latter. These three forces can be represented by the sides of a right- angled triangle, in which the Hypotenuse : Height : Base as Weight of body : Force pushing up plane : Eeaction. If the push be applied horizontally, the three stresses in equilibrium which are the weight of the body downwards, the reaction at right angles to the surface of the plane, the push up 156 KINETICS. [CHAP. the plane applied horizontally will have the relation of the sides of a right-angled triangle, in which Hypotenuse : Height : Base as Keaction : Horizontal Force : Weight of body. Velocity of fall down a frictionless inclined plane. If a body fall down an inclined plane, the potential energy lost by it in virtue of its vertical descent h is mgh ; there is none gained or lost in virtue of horizontal motion, in which there is no work done by or against gravity. The kinetic energy acquired is . These must be equal ; hence v = \/2gh, the same speed as would have been acquired by a vertical fall. In the latter case, however, the direction of motion would have been directly downwards ; in the former it is in the direction of the plane. The reaction of the plane has not modified the speed of the fall ; it has modified its direc- tion. The speed at which the body is moving down the plane after effecting a vertical descent h is thus the same velocity as that which it would have acquired if it had fallen vertically through that height h. But it has travelled through a greater space in order to attain this speed. The acceleration down the plane is therefore smaller ; and a body rolling down a smooth slope of 1 in 20 would take a greater time to reach the bottom than it would take to fall vertically through an equivalent height, in the ratio of 20 : 1. If the body rolled down a succession of inclined planes, or down a curve, the same velocity would be acquired : the reaction of the curve alters the direction but not the speed. The Screw. In Fig. 85, across the rectangular parallelogram Fig.85. Aaa f ~R are drawn equidistant lines fytf, bb', cc' ', dd f , etc., at right angles to Aa and Bo/. The lines ab f , be', cd f , etc., are drawn as there shown. If the surface Aaa'B be wrapped round a cylinder whose circumference is equal to AB, the line Aa will coincide with Ba', and the lines ab', be', cd' ', etc., will form a continuous spiral line abed round the cylinder, and will trace out the form of the thread of a screw whose pitch is ab, the distance between the equidistant lines ab', bc f , etc. Hence the thread of a screw is seen to correspond to a narrow inclined plane wrapped round a cylinder. A screw is usually used as a mechani- cal power for the sake of moving a body through a small space with great force, as in the copying press : in this the mechanical advantage is due to two causes : (1) in accordance with the principle of moments, vi.] THE SCREW. 157 the resistance borne by the thread of the screw is at a less dis- tance from the axis of rotation than are the points of application of the forces to the handle ; and (2) the resistance directly en- countered by the point of the screw has merely a component effective along the thread of the screw, this component varying with its pitch. In fact, the screw is an inclined plane in which the force is applied horizontally, the resistance vertically, and the reaction at right angles to the thread ; and we have seen that in this case the Force : Resistance overcome : : Height of incl. plane : its Base. Hence, the less the pitch of the screw the greater the number of turns to the inch as well as the greater the leverage of the handles, the greater the mechanical advantage that can be derived from its use. Problem. What is the mechanical advantage which can be obtained in a copying- press of the following construction : Effective radius of the arms 1 2 inches screw 1 J inches thict pitch inch ? Ans. The hands move through 1 inch, while the point of the screw descends -^-g inch. .*. F 1 = 128F. The Wedge. A wedge, as seen in Fig. 86, is practically a double inclined plane movable between resist- ances. During a blow there are at work (1) the driving force acting downwards through the centre of AB, (2) a reaction at right angles to AC, and (3) one at right angles to BC ; these latter being through the point of contact, or, if there be con- tact over the whole of AC and BC, through the centre of these lines. These must cross in a point if the equilibrium, which subsists the instant before the wedge commences to move, be con- sidered ; and they must be represented by the sides of a triangle. Eound the point at which they meet the moments = 0. Pulleys. These well known objects are wheels, solid or spoked, mounted in a framework or block, which is either mov- able or fixed to a beam or other solid attachment. The simplest use of a pulley is to change the direction of application of a force applied to a cord. The tension of the cord on one side of a pulley would, if there were no friction, be equal to that on the other side of it, while the motion of the cord on the one side of the pulley is in any case equal to that on the other side, what- ever be the size of the pulley, and whatever be the amount of the 158 KINETICS. [CHAP. flexure to which the cord is subjected. A single pulley thus produces no mechanical advantage if it simply serve this pur- pose ; but if this pulley be itself movable against a resistance if, for instance, a heavy mass be suspended from it, while the other end of the cord is attached, say, to the roof a movement of the suspended mass through one inch would correspond .to the pulling in of two inches of cord, and the hand exerting the force would move through a space twice as great as that traversed by the pulley. Thus, by the intervention of a cord, one end of which is fixed, and of a single pulley round which the cord is bent, a resistance 2 may be overcome by a force just greater than 1. This principle of reduplication of a string round a pulley is taken advantage of, and practically turned to use in combinations of pulleys, in any of which the mechanical advantage is the numerical ratio of the amount of string pulled out to the corresponding movement of the body pulled upon. The Bell-crank. If in Fig. 87 the rigid body ABC, which can rotate round A, have a force applied to it at C, its tendency to rotate round A may cause motion at B against a re- sistance R The principle of moments c shows us that, whatever the ornamental "" shape of the crank, the relation of the resistance K overcome to the force P exerted depends on the relative lengths of the effective arms AB and AC. The Knee. In Fig. 8 8 if two bars be jointed at 0, and their ends A and B be confined to a given straight line CD, a move- ment of the hinge athwart Fig. 88. tne direction of the line CD. corresponds, especially when AO and OB are nearly in the same straight line, to a relative motion of A and B, which is proportionately very small. Hence A and B are thrust asunder with a force greater than that which acts upon the hinge and presses it into its central position. This contrivance is used in some copying-presses, hand printing-presses, and railway-ticket endorsing machines. It is seen in the human knee : when the leg is straightened out a vigorous thrust upwards and forwards is given to the body, and a corresponding one downwards and back- wards to the earth on which the foot presses. THE KNEE. 159 A wire stretched between two points, and loaded by a weight or by the pressure of the wind, is a knee whose action is reversed. It tends to pull together the two supports to which it is fixed ; and if there were any move- ment of these supports, it would be small in comparison with the correspond- ing movement of the centre of the wire. Thus the force acting upon the supports and resisted by them is greater than that acting upon the wire itself. FKICTION. If a mass M be supported on a table T, and the weight of a mass F be employed to pull it towards the edge, the body M will Fipr-89. not commence moving unless the mass of F bear a certain propor- tion to that of M. This proportion, a fraction less than unity, is the coefficient of statical friction, is usually represented by the symbol ft, and has to be experimentally found. F : M : 1, or F = Experiment has shown that the coefficient of friction depends upon (1) the nature of the substances of which M and T consist, (2) the smoothness or roughness of their surfaces, (3) the presence or absence of lubricating substances oil, soap, blacklead between them. The coefficient of friction is the same between the same two substances, whatever the mass of M that is to say, the mass F must bear to the mass M always the same ratio ; thus the friction itself the resistance to sliding between M and T, overcome by the weight of F varies directly with the weight of the mass to be moved, and therefore varies directly with the pressure between the surfaces. Again, the mass M may be distributed in any way, and the contact between M and T may be by a surface large or small. If the area of contact be diminished, the pressure on each unit of area will be increased, and therefore the friction 160 KINETICS. [OHAP. on each unit of area will be proportionately greater; but the number of units of area over which this resistance is exerted is correspondingly smaller, so that the mass F, the weight of which is just competent to pull the mass M towards the edge, remains the same. Hence the friction is independent of the area of contact between two given masses. Limiting Angle. In Fig. 8 9 we may consider the equilibrium subsisting at the instant before the mass M begins to slide on T. The mass M is at rest under (1) its own weight acting vertically, (2) the force F acting horizontally, and (3) the reaction E between M and T. This is inclined at an angle 6 to the vertical. The horizontal component is to the vertical as tan 6 is to 1. But horiz.-compon. F ., . ~ . - = = jj, -, .'. /A = tan 6 = the coefficient of vert.-compon. M statical friction. Any force applied to M, or any set of forces whose resultant acts on it in a line making with the normal to the surface be- tween M and T an angle less than 0, will not produce sliding. If there were no friction, any force applied to M in a direction differing in the least degree from the normal would have a com- ponent which would produce sliding ; but friction makes it neces- sary that a force should be wide of the normal to the surfaces of contact by something more than the limiting angle 6 before sliding can be caused by it. If a flat piece of wood be placed on a table, and pressed against the table by a stick held at right angles to it, it will not slip : the stick may be in- clined somewhat, and still it will not slip : when the stick is inclined more than a certain amount the piece of wood commences to slip on the table. If M be pressed against T by a force P acting in a line which makes any angle with the normal, we resolve it into two components : one at right angles to the surface, = P cos <, produces pressure between these surfaces ; the other, P sin , is the component which tends to produce sliding. If Psin< fr sluing component Qr ^ ua ^ P cos (f> pressure sliding ; if it be greater than p (i.e., if < be greater than #), sliding will occur. If there be no sliding, the component which tends to produce it sets up a condition of stress between the particles of the two bodies, and thus a reaction is set up by molecular forces, equal and opposite in that direction. The amount of this reaction depends on the molecular conditions of the sub- stances, and is only to be determined by experiment. Angle of Repose. Suppose a mass to be placed on a table, and the table to be tilted up until the mass is just about to begin to slide. At that moment there is equilibrium. To what angle can the table be tilted up ? Let //, be the coefficient of friction vi.] FRICTION. 161 between the mass and the table. The one will then slip on the other if a force be communicated between them in the direction of a line making, with the normal to the surfaces of contact, an angle greater than 6. In Fig. 90 are shown two positions of the table T bearing the mass M. In both the dotted lines indicate the limiting angle 0; in both the weight W of the mass M acts vertically downwards. The equilibrium, then, is between (1) the pressure pro- duced between the mass and the table, at right angles to the latter, / \ U- --''\ = W cos ; (2) the sliding com- ponent, = W sin < ; and (3) the reaction of the table, equal and opposite to W, the weight of the body. In the first case of Fig. 90 the reaction between the body and the table falls within the limiting angle, and there is no sliding : the sliding component is less than /j, x the pressure. In the second case the limit is reached ; the sliding component is just equal to fju x the pressure, and is just able to balance the friction. If the table were tilted up any further, sliding would occur. But in the second case, it is easy to show that the angle , to which the table has been tilted, is itself equal to 6, the limiting angle. Hence the angle 6 is also called the Angle of Eepose. Upon the coefficient of friction depend in this way the angles at which heaps of sand, of grain, and the like, will adjust themselves when poured out and allowed to find their own position. This angle 6 has, then, three properties : (1) the Coefficient of Statical Friction, //, = tan 6 ; (2) 6 is the Angle of Eepose ; (3) it is the Limiting Angle. Kinetical Friction. After slipping has commenced, the acceleration produced by a given force is less than it would have been if there had been no friction. The force is apparently diminished in a certain proportion : it has lost a fraction k of its amount ; energy is wasted, but not destroyed, in the form of heat. The " coefficient of kinetical friction," k, is, however, not so great as //,, the coefficient of statical friction. Influence of Duration of Contact. When two bodies have been in contact for a long time, the particles of each develop such relations to one another that p, the coefficient of statical friction, increases with the duration of contact; it is more difficult to make a body slide on another with which it has been long in con- M 162 KINETICS. [CHAP. tact than on one on which it has freshly been placed. When one surface slides rapidly on another, the particles seem to have no time to assume such relations, and the coefficient of kinetical velocity is comparatively small ; at very great velocities it is still smaller than at ordinary velocities ; but when the velocity of sliding is very small, the condition approximates to one of relative rest, the coefficient of kinetical friction approximates to that of statical friction, and a larger proportion of energy disappears at very low speeds than at high. Friction, then, is not a force : it is a resistance or reaction ; it corresponds to the absorption [of energy by its transforma- tion into a molecular kinetic or potential form : in the former case the energy absorbed may perhaps assume the form of the energy of electrical condition, but ultimately it takes that of heat ; in the latter case it corresponds to a stress between the particles of the bodies,, to pull which asunder requires a certain amount of force. But Friction acts and may enter into cal- culations as if it were a Force, never coming into action unless force be applied, always tending to prevent slipping, and either proportional to the pressure between the bodies in contact, if there be rest, or to the velocity of motion, if there be relative motion. Belting-. There is a very interesting and familiar case in which friction serves as a means for the transmission of energy that is, transmission by machine-belting. A rotating wheel has a belt tightly drawn over it, as also over a second wheel, not too near. The belt must be tight, so that there may be more pressure between the leather and the iron. If the wheel be very small or the motion be very rapid, the pressure between the leather and the iron may be lessened by the inertia of the belt, which tends to pass the wheel and to be carried on. The friction is proportional to the pressure between the wheel and the belt, for the relation of the wheel to the belt is practically one of rest, though the surfaces in contact are changed from instant to in- stant. There being no slipping, the friction is statical, and is proportional to the pressure. Though the belt and the wheel do not move relatively to one another, they move relatively to surrounding objects, and the belt is set in motion. If the second wheel be free to rotate on an axis, portion after portion of its rim tends to remain at rest relatively to the leather, and the second wheel is set in motion round its axis. The tension of the belt is greater nearer the driving power than it is on the other side of the wheel driven. This is because energy has been taken up in preventing relative motion of the belt and the driven wheel, or another mode of expressing the same thing in producing absolute motion of the latter. This difference of tension is equivalent to a force directly applied to the rim of the wheel. Thus the kinetic energy of the driving wheel is in part imparted to the leather belt and the rotating wheel ; these come to form a part of the same system with the driving wheel ; the latter cannot rotate so fast when it is vi.] FRICTION. 163 driving a second wheel as it can when not doing so, the same energy being .supplied to it ; and thus energy is transmitted. The efficiency of belting is greatly increased when the rim of the wheel is lined with leather, hair side outwards, the hair side of the belt being inwards. Resistance to Traction. In moving a heavy body along a frictionless level path, no work is done by or against gravity. The force required to se't the body in motion is proportional to its mass and to the velocity imparted in a unit of time. F = ma. But kinetic friction would apparently diminish the force by a fraction k. Thus it is equivalent to a retarding force of kF. But as F is proportional to the mass set in motion, the kinetic friction, which is proportional to the force acting, must also be proportional to the mass moved. Thus the friction is proportional to the mass, and therefore to the weight of a vehicle. It is said, then, that the friction overcome by the engine of a railway train is equal to a resistance of so many pounds to each ton of the train's weight. In this way, a train of 100 tons may be pulled along a level road by an engine which overcomes a resistance of 50 Ibs. per ton, and thus the work to be done by the engine is the same as it would have been had the load been 5000 Ibs. drawn vertically upwards. The resistances are said to be so many pounds per ton, nothing being said of the velocity. This is because, within wide limits, the coefficient, k, of kinetical friction is practically independent of the velocity ; though, when the speed becomes very small, the waste of energy occasioned by friction becomes pro- portionately large, the converse holding good at high speeds. When a horse draws a cart up a hill the slope of which is 1 (vertical rise) in 10 (road-length), the principle of the inclined plane shows us that the animal has to put forth force corresponding to lifting a weight of ^ the mass moved ; but further, it has to overcome the frictional resistances, which are equivalent to an extra weight. When a man walks his knee is straightened, and his body is projected forwards and upwards at each step. The impulse may be resolved into two components : one upward, which may raise the centre of gravity of the body about an inch or an inch-and-a-quarter ; one forward, which has to overcome the intermittent resistances introduced by the stoppage occurring at the end of each step, when the foot of the opposite side strikes the ground. This is an intermittent frictional resistance. If there were no friction between the wheels of a railway train and the rails of the railroad, there would be slipping but no progress. Friction between the wheels and the rails friction propor- tional to the weight of the vehicles has the effect of preventing slipping ; but (as in the case of belting) this corresponds to the maintenance of a state of rolling adhesion, under which each wheel is turned round and rolls upon the rail The mechanical powers, when friction is taken into account, give rise to several problems ; but the physical principle underlying the whole subject is the same, that friction acts in the same way as a force opposed to sliding, and that it is proportional to the pressure or to the actual velocity of sliding, as the case may be. As an example let us take this question : A copying-press is pressed hard down on the copying-book ; the hands are removed ; the book remains under pressure ; why does the screw not come up ? The reaction of the book has a component up the line of the thread of the screw ; this would tend to send up the screw, but it is counterbalanced by friction, acting as a resistance in the opposite direction down the thread. 164 KINETICS. [CHAP. It may be left to the reader to show (1) that the better the screw is oiled, the less able will it be to retain its hold ; and (2) that a screw of too large a pitch (one the turns of whose thread are too far apart) may fail to hold the book down. The upward pressure is resolved into two components, of which the one along the thread of the screw must not be greater than ju, x the com- ponent at right angles to the thread ; if it be greater than this the screw will slip upwards in its nut. Problem. If an unsymmetrical wedge have a force applied at the centre of one side, and its reactions at the centres of its other sides (the ends being neglected), and if there be friction : prove that there is a point round which the sum of the moments = ; that that point is not the centre of figure ; and that the moments round the centre of figure will indicate that there is a rotation of the whole wedge in such a sense that the centre of figure is carried forward under the action of the driving force further than is that point round which the wedge rotates. Friction on a raindrop. A raindrop falling in vacuo through a height h would acquire a velocity v = \/2gh= 8*249 ^h feet per second. It's starting point might easily be so distant that a blow from a raindrop travel- ling under these circumstances would be fatal to any living being struck by it. But at every instant of its course it is subject to kinetic friction tending to reduce its velocity at the instant ; at the same time it is subject to the accelerating force of gravity : and thus there must be a certain velocity at which the retardation of friction and the acceleration due to gravity will balance one another, and the drop, if it once attained this speed, would retain it, and fall with a constant velocity. This happens in the case of the raindrop, and also in the case of a stone or granule falling in deep water. Friction of a rope round a post. This is familiar in the example of a rope passed round a post on a quay in order to hold fast a ship. If any little part or element of the rope be considered, it will be seen that the friction is proportional to the pressure of that part of the rope on the post, and to a certain extent it tends to prevent slipping ; in this it partly counter- acts the tension of the rope ; the tension communicated to the end of the element of the rope further from the applied force is less in consequence of this than it would have been if there had been no friction. If we trace out in this way, along the rope, the gradual diminution of tension, we find that the tension, after a complete turn of the rope round the post, dwindles down to a constant fraction of the original tension. Between a flexible rope and c wood this constant fraction is about -| ; hence a force of 1 Ib. could prevent .a force of 9 Ibs. from pulling a flexible rope round a post round which it had been passed so as to form a complete turn. After two turns the tension becomes -^ ; after ten turns it becomes -^ = 3 486 784 401 - Hence a man exerting a force of 1 Ib. at the end of a rope wound ten times round a post would be able to resist a pull of about one-and-a-half million tons. Of course this is not attained in practice, because no ropes are thoroughly flexible, and none are strong enough to stand such stresses ; but a perfectly flexible rope would diminish tension in this manner without reference to the diameter of the post round which it is wrapt. vi.] FRICTION. 165 Boiling Friction. When a ball is set to roll on ice it goes further than it can on a wooden floor ; farther on that than on a carpet ; further on a carpet than on grass. The rotation is, however at length stopped. To produce a rotation a couple, or something equivalent to a couple, is needed ; to stop rotation a single force is competent. The greater the moment of the force opposed to the rotation, the sooner will the rotation be stopped. The rotation of the ball is stopped by the resistance of friction. This is equal to a small force acting at the surface of the ball, and bearing a constant ratio to the pressure produced by its weight. This ratio is very small. The resist- ance, then, to a wheel rolling along the ground is much less than the resist- ance to the same object pressing against a brake. It is very much easier to move the trunk of a tree by setting it on logs which roll on the ground and on which it rolls, than it is to drag it along the ground. There is less friction at a well-oiled hinge or well-lubricated joint than there would be in any other contrivance used for transferring a given mass from one position to another. If a wheel, instead; of having its axle supported in bearings, have it supported on a couple of pairs of friction-wheels which are free to rotate, the axle as it turns does not rub against a fixed bearing, but the friction -wheels yield and rotate, so that the rotating axle is supported by surfaces which travel at the same rate with it, and the friction is accord- ingly very smalL CHAPTEK VII. ATTRACTION AND POTENTIAL. WHEN one body forms with others a system, a conservative system, which will be put under stress when the bodies are removed from contact with one another, these bodies are said to be Attracted towards one another; and if they be fixed in such a position that the stress of the system is permanent, the condi- tion is one of statical equilibrium. When a spring is drawn out and fixed by a catch there is equilibrium between the recoil of the spring and the molecular forces within the catch, which resist its deformation ; when a heavy stone is placed on a wooden table, if the table be strong enough to support the stone, there will be equilibrium between the weight of the stone and the resistance to crushing offered by the wooden support. If the spring be re- leased it will fly back : if the supporting table be removed the stone will fall. In the former case there is a visible medium, the spring, the elasticity of which comes into play; in the latter the elastic medium, which tends to relieve itself of stress by pulling the stone towards the earth, if such a medium be present, is not a visible one. Let us consider attraction, which, whatever be its cause, obeys the particular law that a mass M and a mass m are attracted by each other with a force which depends on each mass directly, and on the square of the distance between them inversely. Then Mm . Mm / -p\ that is,/ = k-jp-' In a system of particles of this kind we must further assume and experience warrants us in so doing that every particle is connected with every other particle fry an independent attrac- tion; then the total attraction of one/particlec for another set of particles has to be found by a process of summation. To effect this summation the aid of the Integral Calculus has" in general CHAP, vii.] ATTRACTION. 167 to be called in; the process is, however, of this kind: The mass and the distance of every particle from every other being taken into account, the force of attraction between each particle and every other is to be separately found, and the whole is then to be summed up. In simpler cases, either of the mutually at- tracting masses may be considered as acting at its centre of figure ; then the mean distance between the two masses is the distance between their centres of figure. Attraction in particular cases. (1.) A hollow shell, whose thick- ness is infinitesimal, attracts an external particle as if all its mass were gathered at its centre. Its area is 47tr 2 ; the amount of mass per unit of surface (its " density ") = . (3.) An attracting shell of any thickness has no action whatsoever on a heavy particle contained within it. For every area of the shell on one side of the particle which may attract it in one direction, there is another on the other side attracting it in an opposite direction ; and the one exactly balances the other, for what advantage the one area may have in size the other exactly makes up for in proximity. Thus it is not possible to find any area of the sphere the attracting effect of which on the particle within the sphere is not exactly counterbalanced by the opposing effect of another area. The particle attracted equally in every direction remains at rest. If a point be just outside a shell, the attraction just outside = k . 47nr ; just inside it = ; and thus the particle, if it pass through the shell, passes from a field where the attraction on a unit particle is k . 4?ro- to another where the attraction differs from its former value by k . 47r etc. and if these differences vii.] EQUIPOTENTIAL SURFACES. 173 each represent one unit of work, a set of equipotential surfaces thus obtained is called a " System of Equipotential Surfaces." Motion parallel to Equipotential Surfaces does not involve work done either by or against the attracting or repelling force. Motion across Equipotential Surfaces, from one surface to another, implies movement from a place where the potential has one value to a spot where it has another. A unit-mass moving from the second to the first surface in Fig. 91, loses potential energy = V 2 Vj : on a mass m the repelling force would do work = m x (V 2 Vj). A mass m moving from equipotential surface No. 10 to 15 in a system of such surfaces, whatever be their form, would have work = 5m units done upon it against the repelling force. The work done would be the same whatever be the points of the respective surfaces between which the motion is effected. Any transference of a particle from one equipotential surface to another may be effected by a vertical translation from the one to the other, which involves work, compounded with a translation along the second equipotential surface, which involves none. The work done by a transference of a particle from a point A on one equipotential surface to a point B on another is also always the same, by whatever path the transference be effected, provided always that there is no friction. The most complex path may be resolved into so much movement at right angles to the equipotential surfaces, which implies work done by or against the forces, and so much parallel to them, which consumes or liberates no energy. This may also be proved by a reductio ad dbsurdum. If in Fig. 9 1 there were two possible paths between A and B, one of which, ACB, corresponded to a units of work done by a unit-mass of matter traversing it, while the other, BDA, corresponded to a greater amount, 6 units, of work ; then it would be possible to cause a body to fall from B to A down the path BDA, corresponding to the greater work, and by falling to pull directly or indirectly a mass equal to its own up the easier path ACB ; it would itself acquire kinetic energy corresponding to energy = b a ; the body thus pulled up along ACB might in its turn fall down the path BDA, and raise along the path ACB the mass which had previously traversed the path BDA, again with gain of energy equal to 6 - a. Thus the circuit might be kept up with con- tinuous gain of energy, and this contrivance might be utilised as a perpetual motor ; but this is an impossibility ; therefore there is an equal expenditure or liberation of energy, so far as the attracting or repelling forces are concerned, in effecting a transference along every possible path between any two given points in space. If A and B be two equal particles, A attracting and B repel- 174 ATTRACTION AND POTENTIAL. [CHAP. ling external particles, the space surrounding A will be a region of negative potential, while the potential of the neighbourhood of B is positive. Over a plane symmetrically situated with respect to A and B the potential will be zero, while the attraction will carry bodies across the equipotential surfaces to the point A. Distance between Concentric Equipotential -Surfaces. In a system of concentric spherical equipotential-surfaces, the dis- tance between every pair of these surfaces is proportional to the square of their mean distance (i.e., of the geometrical mean) from the centre of the single attracting or repelling mass. Two concentric spherical equipotential surfaces whose potentials are V and (V + l), and whose respective radii are R and R 1 ; we wish to find the value of R - R 1 . V = k ~, and (V+ 1) = fcjgj whence R - R 1 = * y (V + 1 ) " Uro" ' Thus, if the equipotential surfaces be those surrounding the earth, over which the potential due to Gravitation is constant, and if the distances between the surfaces be such that transfer of a pound-mass from any one surface to the next one represents a foot-pound of work done : then, at the distance of one earth's- radius from the centre of the earth that is to say, on the surface of the earth the distance between two equipotential surfaces is one foot ; twice as far from the centre that is, 40 miles (nearly) from the surface of the earth the distance is 4 feet ; and the same amount of work which would raise a pound-mass through one foot near the surface of the earth would, at a height of 4000 miles, raise it 4 feet ; and similarly, at a height of 8000 miles, it would raise it 9 feet, and so on. Thus at a very great distance exceed- ingly long paths would be traversed by a pound-mass as the result of doing a single unit of work on it. It follows that if the equipotential surfaces be chosen at equal distances from one another, the amount of work corresponding to the removal of a mass from one surface to the next is in the inverse ratio of the square of the mean distance of the two surfaces from the attracting mass. Two concentric spherical equipotential surfaces whose potentials are V and V p and whose radii are R and R + 1 ; V = km ; Vj = km - ; A pound-mass at a distance of 240,000 miles ( = 60 radii nearly) from the earth's centre will be attracted by the earth with a force which bears to VII.] EQUIPOTENTIAL SURFACES. 175 the attraction at the earth's surface the proportion of : = 1 : 3600. Hence, to move a pound-mass through one foot that is, from an equipotential surface by any path to any point on an equipotential surface one foot distant from it at a distance of 240,000 miles, or, roughly, at the distance of the moon, would involve the expenditure of approximately g^oo foot-pound of work. Free movement always at right angles to Equipotential Surfaces. Whatever be the form of the equipotential surface, it always happens that a body placed on such a surface and free to move, will tend to move under the influence of the attracting or repelling forces in a direction at right angles to that surface. This is because the forces of attraction or repulsion can have no component tending to produce motion in any direction along a surface of equal potential, or parallel to it. Lines of Force. Thus, if the equipotential surfaces be con- centric spheres, as those of Fig. 91, the body repelled from O will travel along radial lines such as are exemplified by the dotted lines in that figure. When the equipotential surfaces have a more complex form, the lines along which a body tends to travel are more complex, as is shown in Figs. 241 and 242. These lines, always at right angles to the equipotential surfaces which they cross, are called Lines of Force. Space in the neighbourhood of an attracting or repelling body may be conceived to be pervaded by a system of Lines of Force, along which bodies will move if free to do so. The work done on a particle thus set in motion by any attraction or repulsion is the product of the mean force by the space traversed; the latter must be measured along the line of force which is the body's actual path. Tubes of Force. Suppose AB to be a portion of an equipotential surface : lines of force pass through the equipoten- tial surface : some of these lines graze the edge of the area AB; these cut off an area A'B' from another given equipotential surface. The space comprised between these equipotential areas and the marginal lines of force is called a Tube of Force. This space may be supposed to be filled with a bundle of lines of force, extending from AB to A'B'. < Tubes of force have this property, that so far as the area A'B' cut by them 176 ATTRACTION AND POTENTIAL. [CHAP. from one equipotential surface is greater than the area AB cut off from another, so does the intensity of the attracting force over any unit of area diminish ; so that if * and s be the respective areas of AB and A'B', and F, F' the re- spective forces per unit of area acting across these equipotential areas, the product, force x area, is constant, and Fs = FY. Thus the force at the level A'B' is less than that at AB in inverse proportion to the relative magni- tude of the area A'B' cut off by a tube of force. Number of Lines of Force. The forces at any two points may be compared by stating the relative numbers of the lines of force which pass through units of area of those equipotential sur- faces which pass respectively through each of the points compared ; the fewer these lines the less the local intensity of the force. Thus in Fig. 91 the lines of force which cross the outer spheres are less numerous per unit of area of the sphere than those which cross the inner spheres, and the force there is correspondingly less. Systems of Surfaces and Lines. The space in the neigh- bourhood of an attracting or repelling mass or system of masses may thus be mapped out by a system of equipotential surfaces and lines of force. The system may be so constructed that (1) the work done in passing from one equipotential surface to the next is always the same, one unit of work, and (2) the lines of force are drawn in just such numbers that at a place where the force is equal to unity, one line of force passes through the corre- sponding equipotential surface in each unit of area of that surface. Then the following advantages are secured : (1.) The potential at any point in the field of space surround- ing the attracting or repelling mass or masses is found by deter- mining on which imaginary equipotential surface that point stands. (2.) If unit -length of a line of force cross n equipotential surfaces, the mean force along that line along the course of that part of it is equal to n units ; for the difference of potential of the two ends of that part of the line of force n; it is also equal to FS, because it represents numerically a certain amount of work ; but S = 1 ; whence n = F. (3.) The force at any point of the field corresponds to the extent to which the lines of force are crowded together; and thence it may be determined by the number of lines of force which pass through a unit of area of the corresponding equipotential surface, that area being so chosen as to comprise the point in question. Travelling of Equipotential Surfaces. Suppose that by some means the repelling mass in Fig. 9 1 were increased, then the equipotential surfaces in that figure would approach one another : vii.] EQUIPOTEXTIAL SURFACES. 177 if the mass were diminished the surfaces would separate from each other. If the increase or diminution of the mass were con- tinuous, the equipotential surfaces would appear to move continu- ously together or apart, as the case might be. Again, suppose a body of such a length as to be able to extend from one equipotential surface to another, at a mean distance E : let the system be, e.g., that of Fig. 91 : there will be between the two extremities of the body a unit difference of potential Now bring this body nearer to the repelling mass : let it rest at the -p mean distance : it now extends across so many equipotential surfaces that the difference of potential between points of space near its two ends would be four units. Generally, any movement of a body across the surfaces of equal potential alters the relative difference of potential between its two extremities. If a body approach the repelling mass, the difference between the potentials at its two ends thus increases ; the equipotential surfaces which cross it become nearer to each other, and thus appear to travel along the body : the body approaching the repel- ling mass meets and cuts more surfaces than it quits. Hence, if a body be exposed to the attracting or repelling influence of a given mass or force, or system of masses or forces, an increase of the attraction or repulsion, or a movement of the body acted on towards the masses attracting or repelling, will tend to increase the difference between the potentials of the ex- tremities of that body ; while a diminution of the attraction or repulsion, or a withdrawal from it of the body acted on, will tend to diminish the difference of the potentials at the extremities of the body. N CHAPTEE VIII. GRAVITATION AND THE PENDULUM. Law of Gravitation. Every particle of matter in the Universe is attracted by every other particle with a force varying directly as the mass of each particle, and inversely as the square of the distance between them. We have already seen that the Weight of a body is a synonym for the Force with which it is attracted by the earth. The law just enunciated indicates that the weight of a double mass is twice that of a single mass, and so on. This seems .a truism ; but it is an experimental result, not a truism, that the weight of a mass of lead is equal to that of an equal mass of wood. This might have been otherwise. The mass of a given piece of wood is known to be equal to that of a certain piece of lead by the experimental fact that equal forces acting on each for equal times produce equal velocities : E = ma. Now a piece of iron and a piece of cork whose masses are thus found to be equal, will, if placed in the neighbourhood of a magnet, be found to be by -no means affected by equal accelerations towards the magnet; yet they are both equally attracted by the earth, have the same weight in the balance, and, if caused to fall through a vacuum (the friction of the air being thus removed), are found to fall with con- currently equal velocities. Again, a heavy and a light mass of any substance fall at the same rate through a vacuum. It was long believed that the heaviest bodies fall fastest ; but Galileo experimentally disproved this. The attraction of the earth for a large mass is greater than for a small one, but the mass to be moved increases in the same proportion as the attraction ; and thus the acceleration produced is the same in all cases, and is independent of the amount as well as of the substance of the falling mass. CHAP, viii.] MEASUREMENT OF GRAVITATION. 179 Cavendish's Experiment. This was a direct measurement of the attraction of masses for one another. Light balls were poised on a rod and their position carefully noted : large balls of lead were carefully brought near them : the light balls were attracted by the heavy masses and their displacement measured. Great experimental precautions were necessary, such as the observation of the position of the balls with a telescope placed at a distance, the avoidance of draughts of air and of vibrations, etc. ; the result showed that if lead balls had been employed as large as the earth, the attraction of such balls would have been greater than the actual attraction of the earth in the ratio of 11 '3 5 to 5 '6 7 : but lead is 11*35 times as heavy as water; hence the earth as a whole is 5*67 times as heavy as an equal bulk of water, or the density of the earth is 5 '6 7. If two masses be respectively ra and m 1 , and their distance d, the attrac- , , mm 1 . , mm 1 , , . ._. tion between these masses oc - or / = k , where A; is a constant. The a 2 a- earth being approximately spherical attracts falling bodies of mass m as if ita own mass, M, were gathered together at the centre, about 637,000,000 cm. from the surface. Its mass M is 6140,000000,000000,000000,000000 grammes. A mass = 1 gramme is attracted by the earth with a force , Mm , (6-14 x 10 27 ) x 1 T = 981 dynes: hence / = 981 = fc _ = k ^ - J - - . d 2 (637,000,000)* = 981 x ( 63 7 QQO >OQ) 2 = i 614 x 10 25 15,430,000 Accelerated Motion under Gravity. A body free to fall in would be subject to constant acceleration of about 981 cm. or 3 2 '2 feet that is, of g units of length per second, and its movement would be described by the three formulae of page 146; the + sign being used when the attraction of gravity acts in the same sense as the original velocity V ; the ^ sign when it acts in an opposite sense. When bodies fall through the air there is friction between the air and the falling body. This is found to vary as the radius of the sphere if the falling body be a very small sphere ; and generally it increases with, but is not proportionate to, the surface exposed. Thus a feather which presents much surface falls more slowly than a similar feather rolled into a ball. Path of a Projectile. We have already seen that combina- tion of a uniform rectilinear movement with an accelerated move- ment not in the same direction results in movement in a parabolic path. This is the theoretical course of a bullet flying in vacua,. 180 GRAVITATION AND THE PENDULUM. [CHAP. but the actual course of a shell or bullet in the air differs widely from this on account of friction, its path being at first somewhat straight and ending with a somewhat sudden and rapid fall. If a shot were fired horizontally in vacua at such a rate V, about 26,077 feet per second, that g, the attraction downwards, = 32'2 ft. per second, "\7*2 * would be , r being the distance of the earth's centre from the bullet's path, the shot would never fall to the ground, but would travel round the earth at the level of the gun's mouth. Any object travelling past the earth with a velocity V such that at the V 2 distance r the acceleration due to gravity g = , will travel round the earth and not cease to do so ; one travelling at a greater velocity will pass the earth, being deflected by it from its path ; one travelling at a less velocity will fall in towardsTthe earth till r diminishes so far that V 2 becomes = rg. The surface of the earth may be reached before this limit is attained ; if not, the body would, in vacuo, travel round the earth at a distance r from the V 2 centre such that r . g Universal Gravitation. The fact of terrestrial gravitation and many of its laws were well known before Newton's time ; he stated the law of gravitation as a universal one : " Gravitatem in corpora universa fieri," etc. Principia, Bk. III. Prop. vii. and Corol. 2. The moon makes a revolution round the earth in about 2,360,000 seconds in an orbit whose mean radius is 59*964 times the earth's equatorial radius. V 2 The formula g = shows that this corresponds to an actual fall of the moon towards the earth of - foot per second : this, compounded with the tangential velocity at every instant, keeps the moon in its. orbit. This acceleration, due to the attraction of the earth, is of 32*2 ft. ; thus the moon is under the influence of terrestrial attraction which obeys the law that / oc . Newton made similar deductions from other astronomical phenomena, particularly those of the satellites of Jupiter, and ultimately asserted the universality of the law of gravitation. The moon attracts the water of the sea, and thus produces a lunar tide on the side of the earth nearest it ; it also pulls the earth away from the water on the further side, and produces a lunar tide on the further side. The sun produces similar tides -f as great. These two sets of tides may concur or partly counteract one another. Variations of the acceleration of gravity on the earth's surface. At the equator the acceleration of gravity is at the sea-level 978*1028 cm. per second: at the pole it would be VIIL] VARIATIONS OF GRAVITY. 181 983*1084 if the law of variation in accessible regions of the earth's surface were obeyed there. This law is, that at any place whose latitude is \, the acceleration of gravity is (980/6056 2-5028 cos 2X '000003A), where h is the height of the observing station measured in centimetres. This diminution of gravity equal masses weighing less, and therefore distorting spring-balances less, in regions nearer the equator is due to two concurrent causes : (1) That the mean equatorial radius is greater than the polar ; the polar radius is 635,639,000 cm. ; the longest equatorial radius (from lat. 14- 23' E. to lat. 165' 37'W.)is 637,839,000 cm. ; the shortest equatorial radius, at right angles to the former, is 637,792,000 cm...* (2) Th< rotation of the earth. If the earth came to rest, the actfeleratioi^ of gravity would be increased by or 3 '39 08 cm. ^per > second, and the weight of 289 bodies would be increased in the ratio of 288 to- 289. If the earth rotated 17 (= \/289) times as fast as it does, loose objects would, at the equator, fly off its surface. The acceleration due to gravity is in Paris 980'94, at Greenwich 981-17, at Manchester 981*30, at Edinburgh 981-54 cm. per second. The velocity of rotation at the equator is 456,510 cm. per second : whence = 3-3908. r Local Variations. In the neighbourhood of a high range of mountains a plumb-line inclines towards the mountains. The ebb and flow of water in the Firth of Forth affects the apparent latitude of Edinburgh by about 14 4 00 degree, for when the water is at high tide, plumb-lines are inclined towards it, and the mercury used as a means of producing perfectly level mirrors is, in the vessels containing it, heaped up towards the mass of sea- water. At sea the effect of gravity is less than it is on land, because the mass of water under the spring-balance is lighter than a corresponding amount of rock would have been. The depth of the sea may be determined by a graduated instrument of the nature of a spring-balance sufficiently sensitive to take account of these variations. Measurement of the Local Force of Gravity. The force of gravity must, like all other forces, be measured by its effects. This may be done directly by Attwood's machine, already de- scribed. Observation of a single fall cannot, however, give accurate results, and the value of g is best determined by the oscillations of a pendulum. There are at the basis of this determination four main facts : (1) that a pendulum of a given length will oscillate through small arcs in equal times, of whatever substance it be made this last 182 GRAVITATION AND THE PENDULUM. [CHAP. experimental result being due to Newton; (2) that the relation between I the length of a pendulum, t its time of oscillation, and g the local acceleration of gravitation, is given by the formula 47T 2 / 9- ~f~> presently to be proved ; (3) that the length I of a simple pendulum may be very accurately observed, for in practice it is equal to the distance between two points on a solid rod, called a compound pendulum ; and (4) that the time of one oscillation may be very accurately observed by counting the number of oscillations in a sufficiently long period of time. Hence g can be found to any nicety. Centre of Gravity. The earth is approximately spherical, and bodies on the surface have all their particles drawn approxi- mately towards its centre. But the centre is so distant that, within -the limits of ordinary terrestrial objects, the gravitation forces acting on the several particles of a body are nearly parallel to one another, and their resultant acts on the Centre of Figure. This centre is called the Centre of Gravity of a body attracted by the earth. The centre of gravity of any plane figure may be found by cutting that figure out in cardboard, and suspending the card first from any one point and then from any other. A line drawn vertically downwards from the first point of suspension when the body is suspended from it, and another drawn in the same way from the second point of suspension, will cross one another at the centre of gravity. . Whatever be the form or the arrangement of matter in a body, if it be suspended from any point arbitrarily chosen, the centre of gravity is in a line vertically drawn through the point of suspension vertically here meaning at right angles to the free horizontal surface of liquid at that place. If the centre of gravity be found by two suspensions, the vertical lines drawn from any other points of suspension will all pass through the same centre- Centre of Gravity of Two Masses. In Fig. 93 the two bodies, Fi 93 A and B, whose masses are m and m,, will have their centre of gravity at a point C, which, is (m mi} - ?' determined by the equation m x AC = m / x BC. / 3 The whole mass m + m / may, as regards other bodies, be considered as if it were aggregated at the point C. Centre of Gravity of a System of Masses. This is found by taking account of each seriatim. In Fig. 94 let the bodies be A, B, C, D, E, VIII.] CENTRE OF 'GRAVITY. 183 Fig.94. whose respective masses are ra y , m yy , m yy/ , m yyyy , and m^ First the centre of gravity of any two, say A and D, is found at O ; A and D are supposed to be replaced by a mass m / + ra //y/ at 0. Next the centre of gravity between another of the masses, say E, and the imaginary mass m y + m /y// at is found A to be at O y ; at this point O /? there is supposed to be placed a body whose mass is m / + ra y//y + m^ In the same way, the centre between this imaginary mass and another, say m yy at B, may be found at O y/ . Finally the centre between this and the mass m y/y is found at (),, and this is the last operation, for, as regards external masses, the system, ABODE, acts as if it were a mass (m y + m y/ + m yy/ + m yy/y + m p ) concentrated at the point O y// ; this point is therefore the centre of gravity of the system. The same point will be found whatever the order in which the masses are considered. Fig.95 Overturning a body. Let ABCD be a block of material supported on a base CD. How great a force applied at E, in the direction EF, is necessary to overturn the block ? The question is really one of moments round C, for if the force along EF prevail over the weight of the block, it will do so by turning it over the point C. From that point C, CG is the shortest line drawn to meet the line EF, and CH is the shortest distance to the line MH, along which the force of gravity may be considered to act. At the instant when overturning is just going to commence, the moments round C 'must be equal, and CG x force along EF = CH x wt. of body. Therefore, if the force along EF be greater CH than pTr x weight of body, the body will be overturned. The greater CH is, the greater must be the force exerted along EF in order to overturn the body ; the smaller CH is, the less need that force be. When CH = i.e., when the centre of gravity M is vertically above C any force, however small, will upset the block ABCD ; while, if H be on the other side of C, the block cannot stand unless propped up. In this way a body resting on a wide base is less easily upset than one standing on a narrow ; one in which a vertical line drawn from the centre of gravity falls out- side the base of support cannot stand unsupported ; while one in which the centre of gravity stands over the very edge of the base of support is upset by the least disturbance. A microscope, then, ought for the sake of steadiness to have a wide base ; 184 GEAVITATION AND THE PENDULUM. [CHAP. and since a tripod stand is the most steady form of support, for reasons already stated, instruments of this class should be supported on broad tripod stands. An old man using a staff widens his basis of support by virtually converting his two legs and the staff into a broad tripod stand. It amounts to the same thing whether the base of an object be relatively broad or its centre of gravity be relatively low. If the centre of gravity be relatively high or the base relatively narrow, as in the case of young animals learning to walk, chil- dren learning to walk, persons learning to move on skates, or on stilts, or on a narrow rail, or rope, or wire, or a bicycle, or a per- son standing on one foot or on his heels, a relatively small dis- placement of the object will readily cause the centre of gravity to be placed vertically over a point beyond the base of support ; then the object, if it be not propped up, or if the centre of gravity be not brought over the base of support, or the base of support brought up under it, will topple over. If, on the other hand, the base be relatively wide, or the centre of gravity be relatively low, as in the case of a lampstand loaded at its base, the task of upsetting such an object is greater, since the centre of gravity is in such a case less easily induced to pass to a position vertically beyond the base. Curious positions may be assumed by objects when they are so balanced that the centre of gravity is vertically over some point in the basis of support. A man's centre of gravity is at a point about the front of his last lumbar vertebra. If he carry a burden, in order to bring the centre of gravity of his body and the load jointly over the narrow basis of support furnished by his feet (heels, and balls of great toes, and lines joining these), he must stoop ; if the burden be towards the front of the body, as in the case of obese persons, the gait becomes very erect. When a body is suspended from a point in its upper part, its centre of gravity swings down into the lowest position possible. Work done in overturning a body. If there be rotation round the point C of Fig. 95, so far that M, the centre of gravity of the body, comes to be immediately over that point, and over- turning is effected, the centre of gravity is raised through a cer- tain height h. The weight of the body, mg, x that height, A, is the work which must be done before overturning can be accom- plished. Angle of Overturning. If the force applied at E in Fig. 95 be applied in a direction too nearly parallel to BD, its moment may be too small (its arm CG being too short) to produce overturning. At a certain viii.] OVERTURNING. 185 definite angle, BEK, there will be equilibrium, the arm CG being of exactly such length as will make the moment of EF equal to that of the weight of the body. If, then, this angle BEK be less than 0, the angle of repose, the body will overturn before sliding ; if BEK be greater than 6, the body will slide before overturning. Equilibrium, Stable, Unstable, and Neutral. If the centre of gravity be so situated in a body that work has to be done in disturbing it that is to say, if the centre of gravity be already at the lowest possible position the equilibrium is stable. If a ball lie in a bowl, work must be done in order to effect any displacement of it, for no displacement can be effected without raising the centre of gravity of the ball, and thus imparting potential energy to it. When the ball is let go, it rolls back and oscillates in the bowl till it comes to rest. The same thing is seen in a swing, a cradle, a rocking-horse, a pendulum, or a ship well ballasted, which are all in Stable Equilibrium; in the last case the oscillations somewhat resemble those of a pend- ulum whose point of suspension and whose length both vary. If a body have its centre of gravity placed above its point of support, so that any displacement lowers its centre of gravity, then the body already has potential energy, which it is disposed to convert into kinetic by the fall of its centre of gravity to the lowest possible point. Hence in bodies thus in Unstable Equilibrium, a very slight disturbance may cause a very great displacement, disproportionate to the disturbance, but depending on the potential energy stored in the system. In this case are boats in which people stand, high chairs in which children are seated, cars which are heavily loaded atop, deck-loaded ships, and, in short, everything which is " topheavy." When no work is done upon or by an object, as far as the attracting forces are concerned, when it is displaced, the equi- librium is neutral. A uniform sphere may be displaced and assume a new position without either raising or depressing its centre of gravity. A sphere floating in water is in neutral equilibrium ; a plank floating in the usual way is in stable, while a plank floating with its edges vertical is in unstable equilibrium. Simple Pendulum. This is an ideal It is a heavy part- icle suspended by a weightless cord. An approximation to a simple pendulum is obtained by suspending a small bullet by a very thin wire. The length of this pendulum is the distance between the point of suspension and the centre of the bullet. 186 GKAVITATION AXD THE PENDULUM. [CHAP. If in Fig. 9 6 a simple pendulum of length I = AC be repre- sented as displaced from the vertical position through the angle 0, m being the mass of the bob, and mg conse- Fig.96. quently its weight, when the bob is at C, the force of gravity may be resolved into two com- ponents : one == mg sin 6 in the direction of the tangent at C, and tending to bring the bob towards the middle line with acceleration = g D sin 0; and a radial component = mg cos 0, which renders the cord tense. The displace- ^ ment of the bob, the distance between B and C measured along the arc, is equal to Id. Now, as long as 6 is small, some 2 or 3 at most, sin 6 and the angle are nearly equal, and the tangential acceleration = g sin 6 bears to the displacement 16 an almost constant ratio, 9 s ^ n ^ 9 for Yn (approximately) y. But we know that when a body it/ I after displacement is subject to a force tending to bring it back, which produces an acceleration proportional to the displacement, the result is a S.H.M. ; and thus a pendulum very slightly dis- placed oscillates in S.H.M. In S.H.M. the angular velocity oo in the circle of reference is equal /acceleration at any point of the S.H.M. V dislacement at tha displacement at that point. That is, /? and energy = Jlw, 2 = _ w, 2 . The kinetic energy acquired is equal to the potential energy lost, whence to, 2 = mgl (cos 9 cos 9). 2 w y 2 = -j- (cos 9 t - cos 9). (2.) VIIL] COMPOUND PENDULUM. 189 If the simple pendulum were of such a length as to oscillate at the same rate as the compound one that is, if w, the angular velocity of the one = a> y , that of the other, we find from (1) and (2) that 2 J^ (cos B - cos 0) =*L (cos - cos 6). Hence the compound pendulum oscillates at the same rate as a theoretical k 2 simple pendulum of length h If a body of any form be suspended at a certain point and be found to oscillate at a certain rate ; if another point in the body be found after trial at which the body being suspended will oscillate at the same rate, then the distance between these inter- changeable points of suspension is the true length of the ideal simple pendulum oscillating at the observed rate. CHAPTEE IX. MATTER. THE essential nature of matter its substratum is unknown to us ; we only know matter by those of its properties which we perceive by our senses. These properties are subject to our direct observation and to our study, and from them we may infer as to the constitution of matter much which we cannot directly perceive. THE PROPERTIES. OF MATTER. Some of these properties of matter are general, so that if they were other than they actually are, the nature of our universe would be totally different ; and thus, in relation to the matter of the existent universe, these properties are with sufficient appro- priateness said to be essential. For instance, all experience leads us to say that matter must necessarily exist in definite or measurable quantity ; and, since quantity of matter is expressed briefly by the word Mass, we say that every body must have a definite mass, for it is to us, with our range of ideas, impossible to conceive of a definite body having a physical existence, but consisting of an indefinite quantity of matter. There are many bodies of which we do not definitely know the mass, but every body must have some definite mass, great or small. If the mass of a body be great, the body is said to be massive ; if its mass be small, it is usually said to be light, though that adjective is properly antithetical to heavy, a perfectly distinct idea. A massive gate is difficult to move, not because it is heavy, for gravity does not affect the horizontal swing of a gate on its hinges, except indeed by affecting the friction at the hinges ; it is difficult to move because its mass ra is great, and since / = ma,. to produce a given acceleration a, if the mass m be large, the force applied must be great, and would in theory be equally con- siderable were the gate and its hinges removed to a region where CHAP, ix.] ESSENTIAL PROPERTIES OF MATTER. 191 the effect of gravity vanished, and the gate had therefore not even a feather's weight. As regards quality of matter, experience shows us that every substance with which we are acquainted is made up of one or other or more or fewer of about seventy different kinds of matter. These kinds of matter are called elements. They are considered to be distinct kinds of matter, and are called separate elements simply as a confession of our relative experimental impotence, and of our complete failure up to this time to break up any one of them into simpler substances, or to build any one up by any synthetic process. A piece of brass may be by analytic pro- cesses resolved into its component copper and zinc, and ,when copper and zinc are fused together in proper proportions, brass of a similar quality may be made ; but no one has broken up either zinc or copper into any simpler components, neither have these metals been made by causing any simpler substances to combine. These metals are, then, elements ; and the substances which they form by entering into combination with other elements, as well as the circumstances under which these combinations are effected, form the subject-matter of the science of Chemistry. The descrip- tion of any given substance chlorine, nitrogen, calcium as an element is thus seen to be entirely provisional. The experience of 1807 may possibly be repeated when least expected. Before that date lime, soda, and potash were enumerated in the list of elements, though, from their strong likeness to metallic oxides, it was vehe- mently suspected that they were really not elements at all, but oxides. When Sir Humphry Davy brought the galvanic ba'ttery of the Eoyal Institution to bear upon masses of these substances he resolved them into oxygen and into metals never seen till then ; and thus the list of elements suffered a profound modification. Xow evidence of a speculative character based (Mendelejeff and Xewlands) upon the remarkable relations existing between the chemical properties of the elements and their atomic weights, and of a directly observational character, based (Lockyer) upon the results of spectrum analysis as applied to the stellar bodies which seems to show that many elements are decomposed by intense heat into simpler elements lends support to a belief which is rapidly gaining ground that all the elements differ from one another only in their intimate structure, and have a common basis which may possibly be hydrogen ; or in other words, that all the elements are structural modifications of one form of Matter. Thus even the alchemist's dream of the transmutation 192 MATTEK. [CHAP. of metals cannot now be treated with such unmitigated contempt as it received thirty or even ten years ago, though it may con- tinue to be a dream to the realisation of which no approach is possible on account of the necessary limitations of our experi- mental appliances. In reference to space, every mass of matter must at any in- stant occupy a definite volume of space : it must have some Extension in tridimensional space it must have dimensions expressible in terms of length, breadth, and thickness. As a natural consequence every mass of matter must have some definite form, whether that form be imposed on it by surrounding matter or not whether, like a solid rock, it have a form of its own, or, like water in a basin, it have a form which depends partly on the form of the vessel containing it, or, like gas confined in a gaso- meter, its form as well as its volume depend on that of the vessel in which it is enclosed. Among the properties of matter which are said to be essential we usually find mentioned that known as impenetrability. This means that two masses of matter cannot occupy the same space. In view of the peculiar phenomena attending the solution of sub- stances in water a very large quantity of different salts being soluble in water without materially increasing its bulk we cannot state this absolutely. But matter is believed to be composed of minute masses called molecules, and of these it is true that two cannot coincide in position. But these are not in contact even in solids, and so a body is always free to shrink in size as, for instance; when it is cooled down or compressed because the dis- tance between its molecules is capable of diminution ; and thus a quantity of water, which is not a continuous substance, may receive between its own molecules a number of molecules of other substances, and so form a solution, without entirely sacrificing that freedom of movement past one another which its own molecules possess that is, without entirely losing its fluidity. The impene- trability of matter is true, then, of molecules, not necessarily of masses. If a certain bulk of metallic potassium contain 45 atoms or half-molecules of potassium, an equal bulk of caustic potash will contain 70 atoms of potas- sium and 140 of hydrogen and oxygen. In respect of time : Lapse of time brings about no change either in the quantity of matter or in its quality that is, matter is indestructible both in regard to its total quantity and to the quantity of each element. Such is the ordinary belief; the IX.] ESSENTIAL PROPERTIES OF MATTER. 193 former statement is in accord with the universal experience of Chemistry ; but he would be bold who, from the experience of mankind on the surface of the earth, should venture to deny that in the interior of this planet there may even now, as the earth is cooling, be an increase taking place in the quantity of the more complex at the expense of the more simple elements ; not to speak of the positive probability which spectrum analysis lends to a belief that this kind of action is actually going on in the fixed stars. Be that as it may, within our experimental range of know- ledge there is no transmutation of elements and no destruction or creation of matter. Matter changes its forms and its combina- tions incessantly, but it can always be traced up by chemical analysis. A closed glass tube containing oxygen and powdered charcoal weighs exactly the same after the charcoal has been induced to burn in the oxygen, and thus to disappear, as it did before that action ; the gaseous carbonic acid produced is equal in total weight, and therefore in mass, to the sum of the carbon and oxygen which composed it. The general properties of matter. By a distinction which is somewhat arbitrary the preceding properties are said to be essential, while those of inertia, weight, divisibility, and porosity are said to be general, because found to be possessed by all matter. The statement that inertia is a general property of matter simply means that Xewton's first law of motion is a universal result of experiment. All bodies possess weight at the earth's surface and within experimental or observational limits. A mass placed on the earth's surface is attracted by the earth, by the sun, the moon, the planets of the solar system, and in a less degree the attrac- tion being so small that we have no direct evidence of its exist- ence by the distant fixed stars ; the resultant differs so very little from the direct attraction of the earth that the latter alone may be considered as pulling the mass downwards towards its centre ; but this is only a first approximation, for a more careful discus- sion of the intensity and direction of the resultant force enables us to explain the phenomena of the Tides. All masses are divisible ; the only question which here emerges is that as to indefinite divisibility. Is a given mass say of chalk divisible to infinity, or would we after division effected with sufficient frequency obtain a small mass of chalk which, if further divided, would be no longer chalk, but might perhaps be broken up into lime and carbonic acid ? The facts 194 MATTER. [CHAP. of chemical equivalence, as ascertained by the balance, seem to be susceptible of no natural explanation other than that matter is made up of such ultimate particles, and hence matter is con- cluded not to be infinitely divisible. We shall recur to this subject. All matter is porous or possesses porosity. Hydrogen gas leaks through white-hot iron under pressure ; cold water can be pressed through iron, as may be seen in Bramah's hydraulic press, or through lead, as in Francis Bacon's famous experiment, in which he took a shell, of lead rilled with water and compressed it ; the water oozed through the lead and stood in drops and beads on the surface of the shell. Contingent properties of matter. Some of the properties of matter are contingent, and depend on the particular kind of matter considered and on the surrounding circumstances. As examples, we may take the facility with which a body is heated, the rate at which heat can run along it, the ease or difficulty with which light can pass through it, and so on. The quantity of matter per unit of space is defined as the density of the mass filling that space. Thus a gramme of water occupies a cubic centimetre, and according to the C.G.S. or centimetre-gramme-second system of measurement, the density of water is = 1. More accurately (Kupffer), 1 cub. cm. 1 c.c. water at 3 '9 C. weighs 1-000,013 standard grammes, and its density is 1 '00 0,0 13. In the same way the density of lead is 11*5, because 1 c.c. weighs 11*5 grammes. In general, if m be the mass contained in volume V and p the density, m Y = p, or m = V/o. Every kind of matter, simple or compound, has a special density of its own ; thus a given bulk of lead contains 11 '5 times as much mass as the same bulk of water. Water is taken as the standard of density; its specific density is said to be 1,. though sometimes, in estimating the density of fluids, it is, in order to avoid decimal fractions, reckoned as 1000. In the same way, the specific density of lead is 11 '5, and those of all substances may be experimentally found and recorded in a table of specific densi- ties, or, as it is more commonly called, a table of specific gravities. These are experimentally found by taking advantage of the fact that weight is proportional to mass ; w mg. The piece of lead which will occupy a given space not only contains 11^ times as IX.] DENSITY. 195 much mass, but also weighs 11^ times as much as the quantity of water which would fill the same space. Thus the density of a body, as compared with that of water, presents a ratio its specific density which is numerically identical with the ratio its specific gravity of its weight to that of an equal bulk of water. If m and in i be the masses of equal volumes of the body and of water, V these equal volumes ; m densitv of the bodv V m Sp. density = ^ ^ = = . density of water m t m t V weight of the body mg m P- gr vi y we jg nt ot - e q ua i ^ik O f W ater ~~ rn t g ~ ?/, " Hence sp. density and sp. gravity are numerically identical ratios. Sketch of the Experimental Methods of finding the Specific Density of Bodies. (a) Solids. 1. Weigh the body in air (properly in vacuo) ; measure its bulk by dipping it (suspended by a thin string) into water contained in vessel A, and observing the rise of level in that vessel ; take it out : out of a known quantity of water in vessel B pour enough water into vessel A to produce an equal rise of level in the vessel A ; find the weight of the water that has been poured out of vessel B. Then the weight of the body -=- the wt. of the equal bulk of water poured out of B = the sp. density of the body. The practical objection to this method is, that the body when taken from vessel A removes some of the water. 2. Weigh the body in air ; put it in a vessel a " specific-gravity flask " marked distinctively at a certain level ; fill with water alone up to the marked level ; weigh. Empty the vessel and fill with water alone up to the mark ; weigh. Then -weight of vessel, body and water up to level, = V + B + w. vessel and water alone up to level = V + W. . . wt. of water which replaces the body is W - w, and wt. of body is B. wt. of body B wt. of equal bulk of water W w 3. Take advantage of the following proposition in Hydrostatics : A body suspended in a liquid is buoyed up by that liquid to such an extent as to diminish in apparent weight by an amount equal to the weight of the bulk of the liquid which it may be considered as displacing. If a body of exactly the same density as water be suspended in water, it will neither sink nor wt. in air 1 rise ; its apparent weight will = : its sp. gr. = , : = = 1. loss in water 1 If it be more dense it will sink, but slowly, for while its mass is unaltered, the force acting on that mass is apparently diminished. If it be less dense than water, it will rise ; its weight will appear to be less than nothing, negative. 196 MATTER. [CHAP. A cork of volume v and density '8 will have a mass v x -8 = -8r. An equal bulk of water would have a mass v x 1 = v. The weight of this mass of water would be mass x g = vg ; the weight of the mass of cork is similarly 'Svg. The apparent weight of the mass of cork will be "Srg - vg = '2vg ; i.e., its downward acceleration will be negative, or the cork will move upwards. This upward accelerating force, = '2vg, acting on a mass -8v y will produce an / /\ '2vg I acceleration ( since a = ) = - - = -q and but for friction in the water \ Wl/ oV 4 the cork would rise with an upward acceleration of 8 '05 feet per second. Therefore weigh a body in air ; suspend it by a fine thread from the pan of a balance, so that it just sinks wholly into water. It will appear to weigh less. Then the sp. gr. = ' , or, accurately, apparent loss of wt. in water If the body be lighter than water, attach to apparent loss of wt. in water it a piece of heavy substance say, lead of known weight and known density (lead = 11 -5). Then the weight of the lead is 11 -5 Vg; that of the light substance is pV,g; together they weigh 1 1'5 "Vg + pV,ff. In water they weigh 1 0*5 Vg + (p - 1) V,(/. The loss of the lead alone must be V#, or r- of its weight ; whence the loss of the light body in water is easily determined by difference ; and its density, the fraction - 2 found. loss in water 4. If the solid be soluble or be otherwise acted on by water, some other fluid is made use of, such as naphtha or turpentine, the specific density of which is known. If the specific density of the solid with reference to the liquid, found by any of the above mentioned methods, be - , and the sp. d. of the liquid with reference to water be , then the product of these w specific densities = - x = - is the sp. d. of the solid as compared with water. l w w (b) Of Liquids. 1. Fill a weighed vessel up to a certain mark with the liquid : it weighs so much : by difference find the weight of the liquid alone. Empty the vessel and repeat the process with water ; the water which fills the vessel up to the same mark weighs so much. Then the sp. gr. of wt. of given bulk of liquid, the liquid = ~ wt. of same bulk ot water. 2. A body immersed in water appears to lose of its weight ; immersed X in the fluid to be tested it appears to lose of its weight. This body is x times (x being a whole number or a fraction) as dense as water, y times as dense as the fluid. The density of the fluid is to that of water as the apparent loss in the fluid is to the apparent loss in the water that is, y x as x : y ; and the sp. d. of the fluid is - 3. By the use of Hydrometers, Alcoholometers, and the like. The prin- ciple of these instruments is the following : A body which floats in water IX .] DENSITY. 197 without being wholly submerged is in equilibrium under the action of two f orces (i) the weight of the whole body ; (2) the buoyancy of the water, equal to the weight of the part of the water displaced. Thus ice floats in water with j^ 1 ^ of its bulk submerged. ' If the volume of a mass of ice be v and its specific density />, the weight of the mass is vpy, and the weight of the mass of water equal in volume to the submerged part of the mass of ice is -918v x g. These are equal ; vpg = -QlSvg ; whence p = '918, the specific density of ice as compared with water = 1. Thus the sp. d. of a body float- part immersed . ing in water = This method might be used to determine whole volume the sp. d. of solids were it not for practical difficulties of measurement, which, for the sake of explanation, we here suppose overcome. If a mass of ice be placed in chloroform, it will float with "6125 of its mass submerged ; the sp. gr. of ice in reference to chloroform is "6125. The problem comes to be this : Water is ^f/^- times as heavy as ice, chloroform is VYW* ti mes as heavy as i ce What are the relative densities of water and chloroform ? Chloroform is heavier than water in the ratio of *- : VVVo -, or 9180 : 6125, or 1-497 : 1 ; that is, its sp. d. is T497. We see, then, that the comparative sp. d. of liquids could be ascertained by finding to what depth bodies floating in them will sink, if we could perform the necessary measurements with the requisite accuracy. But instruments which have been graduated by the in- strument-maker, who observes and marks on them with more or less care the depth to which they sink in various liquids of known specific gravity, and marks the corresponding position by proper figures, are in common use for promptly ascertaining the density of various liquids. They are usually made with large bulbs, generally loaded with mercury or lead-shot, in order that they may float vertically, and they have a narrow stem which is graduated. They are caused to float in the liquid whose density is to be found : the gradu- ated stein stands more or less out of the liquid ; the figure upon it which corresponds most nearly with the general level of the surface of the liquid is read off and recorded. It is convenient in effecting such readings to arrange a piece of black paper to serve as a background, and to place the liquid to be tested in a glass vessel. Rousseau's densimeter bears at its summit a little cavity. In this a cubic centimeter of the fluid is placed : according to the depth to which this makes the instrument sink in water is the density of the fluid deter- mined, according to the graduation performed beforehand by the instrument maker. Fahrenheit's areometer consists of a similar instrument provided at its summit with a little platform or pan. It is placed in water at 3'9 C., and loaded by small additional weights placed in the pan, until the areometer sinks in the water just so far that the level of the water coincides with a certain mark on the instrument. Then the sum of the known weight of the areo- meter -f the additional weights put on = W = the weight of the bulk of water displaced. It is removed, dried, and placed in the liquid to be tested, and again loaded till it stands at the same level as before. The weight of the in- strument + the weights now added = W / = the weight of an equal bulk of the W liquid to be tested. Then , the ratio of these weights, is the specific density of the liquid in question. If, for example, the instrument weigh 800 grs. and float in water at the proper level when loaded with 200 grains ; loaded with 80 grains it floats at the same level in an aqueous solution of 198 MATTER. [CHAP. ammonia: the total weights are 1000 and 880; the density of ammonia solution is '880. Nicholson's areometer, which is sometimes used for determining the density of solids, is a modification of these instruments. It bears two platforms one at the summit, out of the water, and a lower one in the water. The body whose density is to be found is placed on the upper pan along with weights just sufficient to sink the areometer to a certain mark. The body is transferred to the lower pan, beneath the level of the water. More weights must be placed in the upper pan to cause the areometer to sink to the same level ; these weights are equal to the apparent loss of weight suffered by the body when placed in the water on the lower pan. Then weight of body r-p- = density of the solid, additional weights 4. By specific-gravity bulbs. Bulbs are sold which are known to float without rising or sinking in liquids of the sp. gr. marked in numbers upon them. A number of them are thrown into the liquid ; those which bear too high a number sink, those which are too light rise ; the one exactly corre- sponding, if there be one, is at rest anywhere in the fluid. (c) Of Gases. The density of a gas is found by an application of the same principles as those employed in determining that of a liquid. A copper or glass vessel, as light as is consistent with adequate bulk and strength, has the contained air extracted from it by means of a good air-pump ; it is then weighed empty ; it is very slowly filled with the gas and left for some time in communication with a reservoir of it, its stopcock is closed, and the vessel thus filled with the gas is again weighed : the weight of the volume of gas which fills the vessel is thus ascertained. It is again emptied by means of the air-pump, and then air is allowed to enter it. After standing for some time in order to acquire the temperature of the room, it is again closed and weighed : thus the weight of that volume of air which fills the vessel is found. Then the weight of the gas -f- wt. of equal vol. of air = density of the gas in reference to air as a standard. It is convenient for many purposes to take the rarest gas, that is, the least dense gas, Hydrogen, as a standard of density, and then we say that the specific density or sp. gr. of Hydrogen is 1, that of air 14*47, that of Oxygen 16, and so on. A cubic centimetre of Hydrogen weighs '0000895682 grammes, a cubic cm. of air weighs (at Paris) '0012932 grammes at the freezing point of water and at the barometric pressure of 760 mm. of mercury. A cub. cm. of the lightest liquid known liquefied acetylene at the freezing point of water weighs *45 gramme; the same volume of water weighs, 1 gramme at 3'9 C., or more accurately it weighs 1*000013 standard platinum grammes ; the same volume of lithium, the least dense solid metal, weighs '5936 grammes, and of hammered platinum weighs 21 '2 5 grammes. Thus we see that, bulk for bulk, solid platinum is nearly 240,000 times as dense as gaseous hydrogen. For the estimation of Vapour-density, see p. 347. IX.] MATTER. 199 THE STATES OF MATTER. We know that in popular language there are said to be three states of matter, the solid, the liquid, and the gaseous. Closer observation shows us that these merge into one another, and that, in addition to these, matter exists in conditions more recondite ; thus we shall have to pass in review the following conditions : (1) Eigid solid, (2) soft solid, (3) viscous liquid, (4) mobile liquids, (5) vapour, (6) critical state, (7) gas, (8) radiant matter. These may be all classified under the two heads of Solid and Fluid, which are not, however, separated from each other by any distinct line of demarcation. A perfect solid is an ideal body which, when acted upon by stress, if it do become deformed, becomes deformed to a definite extent, and then retains its newly -acquired form for an indefinite period of time, as long, that is, as the same stress ,.,..,,, . deformation is continuously applied to it ; thus the ratio = deforming force - = const. A steel spring becomes stretched by a weight stress hung upon it : it is stretched to a definite extent, and then retains the same length for any length of time, or rather it appears to do so when we do not use sufficiently accurate measurements ; if it did so perfectly, steel would be a perfect solid. A Fluid, on the other hand, is a substance which, if continuously acted upon by deforming force or stress, continuously yields with continuously increasing deformation or strain ; and thus the ratio - is not stress constant, but increases with the lapse of time. The stress so applied to a fluid in order to deform it must not be one which is applied equably over its whole surface in other words, it must not be a " Hydrostatic stress " but must act more effectively on one part of the mass of the fluid than on another. A rigid solid is one which, when a stress is applied to it, experiences no deformation, no strain ; and therefore in a perfectly rigid solid the fraction = 0, and continues so for an in- stress definite period of time. This is an ideal; no substance is absolutely rigid ; but we may form a sufficient idea of a rigid solid by considering an anvil on which a nail is placed: the 200 MATTER. [CHAP. anvil appears undeformed it appears to be absolutely rigid. There is, then, no body absolutely solid, no substance absolutely rigid ; the bodies which we call rigid solids are found to yield, and to yield continuously if we experiment upon them in sufficiently small masses (such as thin wires), and act upon them with sufficient forces for adequate periods of time. In practice a body is said to be solid if its deformation remain practically or sensibly constant for a long time ; it is said to be rigid if the deformation produced by a given stress be exceedingly small, or, in other words, if the fraction , the coefficient of rigidity, be very large. A strain soft solid is one in which the coefficient of rigidity is very small that is, it is a solid in which a small stress produces a great deformation. A mass of jelly is very easily deformed by stress ; but if a moderate stress, such as the weight of a caraway seed or currant, be applied to jelly even of very thin consistency, the deformation produced by it is approximately constant ; the jelly does not flow over the small load, and is therefore a true solid, though it is soft. It would seem at first sight rather an abuse of terms to declare that thin jelly is a solid, and that such substances as sealing-wax, pitch, and cobbler's wax are fluids ; yet these latter are fluids because they flow, because they suffer continuous de- formation under the action of a continuous stress. A stick of sealing-wax supported at its ends yields continuously to its own weight ; a mass of sealing-wax or pitch will flow down hill ; a cake of cobbler's wax of a definite form will soon lose its sharp- ness of outline ; a cake of this material placed in water, with bullets on it and corks under it, will be traversed by the bullets at the rate of about a quarter of an inch per month, and by the corks at a somewhat slower rate ; the wax slowly flows round them as they sink or rise under the influence of their relative weights, just as water would much more rapidly do. These sub- stances are fluids, then ; but their flow is extremely slow, or, in other words, their viscosity or resistance to flow is extremely great. Treacle is another example of a viscous fluid; strong syrup, weak syrup, very weak syrup, cold water, alcohol, hot water, ether, are examples of liquids whose flow is successively more rapid, whose viscosity is less ; and the last mentioned are said to be mobile liquids, though perfect mobility, the perfect absence of viscosity, is an ideal attribute not possessed by any actual fluid. ix.] VISCOSITY. The Coefficient of Viscosity. In Fig. 99 let volume of fluid comprised between two planes passing throi and let a shearing force represented by arrows act on the fluid for unit of time. The defor- c. ination, since the substance is fluid, goes on in- creasing; let AcdE represent the form assumed through the different layers (imagined separate as in Fig. 25) slipping over one another, the lowest, AB, being at rest. If the velocity im- parted to each stratum be proportional to its distance from AB, the ratios of their displacements II', mm', nri, oo, to their II' mm nri oo respective distances, Am, An, etc., will be equal ; i.e., = - = = = p-r- = tan < ; or, in other words, this is a proper Shear. This slipping of one imaginary stratum over another is retarded as by friction, for each stratum rubs against or is delayed by the one next to it ; and thus the retardation acts as a force would do opposed to that which tends to produce movement i.e., to the shearing force and it has to be measured over every unit of surface over which the retardation is effected. The shear- ing force itself is measured as a force exerted parallel to the planes passing through AB and CD (and therefore tangential to the surfaces of the different strata, in the general case), and acting on each unit of area of these surfaces. The ratio shearing force J^_ = the Coefficient of Vis . shear per unit of time tan cosity. When < = 0, there is no deformation in unit of time, and the viscosity is infinite ; when < is very small, the deformation is very small in unit of time that is, it is very slow, and the viscosity is very great. When the viscosity is very small, the deformation produced in unit of time is very great, or the obstruction offered to flow is very small. If the distance AC = 1, and the displacement of CD in unit of time (i.e., its velocity) be also = 1, then tan < = 1, and the shearing force is equal to the coefficient of viscosity ; whence the coefficient of viscosity may be measured like the coefficient of friction, in terms of a force that is, by " the tangential force on the unit of area of either of two horizontal planes at the unit of distance apart, one of which is (relatively) fixed, while the other moves with the unit of velocity, the space between being filled with the viscous material " (Maxwell). Relation between Rigidity and Viscosity. The measure of Eigidity in the ideal perfect solid, the ratio of the deforming force to the total deformation produced by it, or the fraction . remains con- Total Strain stant for any period of time during which the deforming force applied remains constant, and this ratio or fraction is independent of the unit of time. The measure of viscosity in a fluid, on the other hand, is the ratio of the deform- ing force to the deformation produced in a unit of time. Whether the unit of time chosen be great or small, the numerical value of the fraction Stress applied x Time t ^ *--*.' f j 1 ~f - ought to be the same. Yet it is found that 11 an Shear produced in time t exceedingly small unit of time be chosen, the coefficient of viscosity may have some experimental value different from that found when the unit of time is larger. If, for instance, Canada balsam be stirred with a spatula, it 202 MATTER. [CHAP. will be found (Maxwell) to be doubly refracting. This shows that it is for the instant, and locally, in a condition like that of a stretched or a compressed solid, and that, though it is liquid, if sufficient time (time of relaxation) be allowed it, yet under the action of impulsive forces its coefficient of vis- cosity is so high that it practically behaves after the fashion of a solid. There is no substance perfectly solid, and hence when an apparently solid substance in a suitable form for convenience, that of a wire is exposed to a constant distorting stress, though its form is found to be affected to a certain extent (say by lengthening or by twisting), and it appears soon to reach the limits of its distortion, which may then be measured, the apparent rigidity of the substance being thus ascertainable ; yet in general it will be found that the prolonged application of a constant stress induces a constant slowly-increasing additional distortion, the substance then acting as a fluid of exceedingly great viscosity. Further, we now know by the experience of manufacturing industry that lead and even iron, when exposed to sufficient pressure, can be made to flow slowly ; and the operation of wire- drawing involves among the particles of the metal drawn, as they pass through the aperture in the plate, a relative movement which is similar to that of the particles of a flowing fluid. When the viscosity of a fluid is infinite, there is no difference between that fluid and a rigid solid. It is supposed by some that the matter at the centre of the earth approximates to this condi- tion. If the earth have a crust about 25 or 30 miles thick floating on a melted magma, at increasing depths the pressure (which near the centre must amount to about 45,000,009 Ibs. per square inch) would compress the liquid so much that, though melted, it would have a viscosity so extreme that the mass would have the same relation to extraneous forces as a very rigid solid body would have. The earth would, in consequence of this, comport itself as if it had a solid nucleus floating in and merging by gradations of relative softness into a thin liquid layer on which the crust floats. Substances in the solid and liquid form are broadly dis- tinguished from those in the gaseous condition by the two following characteristics. In the first place, the former have a free surface, while the latter cannot permanently retain a free bounding surface independent of the vessel containing them. Secondly, solids and liquids tend (apart from volatilisation) to assume a definite limited bulk and density; while gases always tend to assume an infinite volume and a correspondingly ix.] GASES. 203 small density. If any definite quantity of a gas be confined within a limited space, it will always fill that space and press against the sides of the containing vessel ; and it will subject that vessel to stress or pressure which the vessel must be strong enough to withstand. Thus a gas, with its tendency to indefinite expansion, has elasticity of volume: work has to be done in order to compress it, and when compressed it tends to restore the work done upon it. Under ordinary circumstances gases are prevented from expanding to an indefinite extent by the pressure of the mass of air which lies upon the earth's surface, which is drawn down towards and pressed upon that surface by the force of gravity, and which consequently compresses itself and all objects at the earth's surface with an " atmospheric pressure " of about 15 Ibs. per square inch. Hence a closed flask containing air is subject to two equal and opposite stresses, whose resultant is nil ; the air in the flask tends to burst it outwards ; the air external to it tends to make it collapse : between the two pres- sures the flask has no need for strength. If, however, the air be in any way extracted from the flask, the external pressure will alone act, and the flask may collapse. In a steam boiler the internal pressure, the tendency of the steam to expand, is greater than the external pressure per square inch, and the boiler must be of sufficient strength to provide for the difference. The pressure exerted by a confined gas is equal over the whole internal surface of the vessel containing it ; and a pres- sure of a Ibs. per sq. in. exerted on a piston forced into a cylinder of gas is communicated to every sq. in. of the inner surface of the cylinder, for the face of the in-moving piston is at any instant a part of the inner surface of the cavity containing the gas ; and if any other part of the wall of the cavity e.g., the face of a second piston be at the same time movable outwards, the excess of internal pressure will cause it to yield. If its area be the same as that of the in-moving piston, the out-moving piston would be acted on by an equal total pressure, and would move to exactly the same extent as the in-moving one. There is an apparent paradox in the statement that a single- inch piston pressed in with a pressure of 60 Ibs. per square inch will communicate a pressure of 60 Ibs. to every square inch of a cylinder, however large ; but there is no law of Constancy or Conservation of Force : there is one of Conservation of Energy, which in this instance is rigorously respected. The work that can be done by a second piston allowed to move outwards is 204 MATTEK. [CHAP. (friction not being considered) equal to that done by. the compress- ing piston, whatever be their respective areas : the range of move- ment and the area vary inversely : a smaller second piston may be pushed through a larger space than the first, a larger one through a smaller space. In this way machinery may be driven at the distance of a mile : air is driven into a tube' ; at the other end of the tube there is a cylinder connected with the tube, and by the alternating admission of the pressure to one and the other face of a piston movable in this cylinder, the piston is caused to move, and the energy is thus applied at a distance. If pressure p will keep a certain quantity of gas in a space v, pressure 2p is found to be required to confine within an equal space twice as much gas : pressure xp will keep x times the quantity of gas within the same space. The densities in these examples are in the ratio 1 : 2 : x, and the pressures are pro- portional to the densities (the same kind of gas being supposed to be used throughout) or to the quantity of gas forced into a given space. The pressures being proportional to .the density acquired under their action, p oc p ; but m vp ; m . . p oc or pv oc TO, v or for a given mass, pv = c ; or p oc , the usual form of Boyle's law. This statement may be otherwise expressed in the form of Boyle's law, that the pressure exercised by a given mass of gas varies inversely as the volume of the space within which it is confined ; or that the space occupied by a given quantity of gas varies inversely as the pressure. Thus, if a quantity of gas occupying one cubic foot at a pressure of 15 pounds per square inch were compressed by a piston forced down with an addi- tional pressure of 15 Ibs. per square inch, the total pressure being doubled the volume would be halved ; if, on the other hand, the pressure were diminished to half by the piston being pulled out with a force equivalent to 7l> Ibs. per sq. in., the volume of the gas contained in the cylinder would be doubled. There are no perfect gases which absolutely obey this law at all temperatures and pressures; but a substance at a temperature and pressure far removed from those at which it will be con- densed into a liquid approximates to this condition. Now suppose that a quantity M of a liquid is placed in a vessel which it does not completely fill, every other substance, such ix.] VAPOURS. 205 as air, being removed from the vessel, which is then closed. The liquid does not fill the whole vessel ; it has a free surface. Above this free surface there is a space, which becomes filled with part of the liquid substance, volatilised into a gaseous form. Let the quan- tity of liquid which has assumed the gaseous form be represented by y : the remainder, M y, is still in the liquid form. The pro- portion volatilised ( ^ of the whole j depends on the temperature as well as on the space which has to be filled. At another tem- perature some different proportion will be volatilised. When the liquid is heated this proportion rapidly increases. But as we have already seen, the pressure exerted by a confined gas depends directly on the amount of it. Hence in this case the pressure exerted rapidly rises as the temperature rises. Gas having this relation to the liquid form of the same substance, confined with it in a vessel otherwise empty, and in contact with it, is called the vapour of that substance ; if it be compressed or cooled, it partly condenses into liquid. Even though not in contact with the liquid, if the gaseous form of a substance be compressed or cooled just so far that any further condensation or cooling will cause the deposition of some of it in the liquid form, it is said to be a vapour. In some cases a vapour condenses directly into a solid (Arsenious acid). The term vapour is often applied in a wider sense to the gaseous form of a liquid or solid substance as, for instance, ether- vapour, chloroform -vapour, the vapour of arsenious acid; and then those vapours which are on the point of condensation are called saturated vapours, while those which can suffer a certain amount of compression or cooling without condensation are called unsaturated vapours. In this sense (with the exception, as yet, of pentafluoride of phosphorus) all gases are unsaturated vapours; for they can all be condensed by the simultaneous application of sufficient cold and sufficient pressure. Oxygen has been condensed ; at a pressure of 300 atmospheres, and at the tem- perature of 39 C. (Cailletet), there is no condensation, but when the gas is liberated it becomes foggy ; according to Pictet it is liquefied at 320 atmospheres and 140 C. ; and then, upon allowing a jet of this liquid to escape into the air, the escaping jet of liquid oxygen becomes extremely cold and is partly solidi- fied, while the remaining oxygen in the vessel becomes cloudy. Hydrogen treated according to Cailletet's method gives a mist. Pictet evolved it under a pressure of 650 atmospheres from a 206 MATTER. [CHAP. mixture of formiate and hydrate of potash ; the hydrogen, on being chilled by a freezing mixture to 140 C., and allowed to issue in a jet, appeared as a steel-blue stream of liquid, the light reflected from which, being partly polarised, revealed the presence of solid particles in the liquid, while the tube of exit became blocked with solid hydrogen. Ozone is with comparative readi- ness condensed by Cailletet's method to a bright-blue liquid. The term vapour is used in still another sense that is, a gas at such a temperature that by the application of pressure alone it can be condensed into a liquid. In this sense carbonic acid below 30*92 C. is a vapour; above that temperature it is properly a gas, for no amount of pressure alone will liquefy it. The Critical State. When a liquid and its vapour are together in a tube otherwise unoccupied and are exposed to heat, there arrives a temperature at which the singular phenomenon of a blending of the liquid and gaseous (or vaporous) states is observed. If a capillary tube, for instance, be filled with liquid CO 2 and slightly heated, some of the carbonic anhydride will escape : the tube may be sealed up, and will then contain nothing but liquid C0 2 and the saturated vapour of C0 2 . If they be heated to 30*92 C., and if there be sufficient C0 2 present to pro- duce a pressure above 73 atmospheres, the free surface of the liquid becomes blurred and merges into the superjacent gas : above this temperature the tube is full of what appears to be nothing but gas : if cooling be permitted there is a flickering seen in the tube, and the liquid and the gas again separate. Some say that the liquid and the gas mutually dissolve each other ; others (Ramsay) point out that the liquid C0 2 rapidly becomes lighter, while the confined gas becomes heavier, at high temperatures, and that at the critical temperature and pressure these two states meet and become undistinguishable. If the C0 2 gas be exposed to a temperature above 30*92 C. and be subjected to any pressure above 73 atmospheres, it will still be a gas : allow it to cool, the pressure being kept up, and it will be a liquid after it passes 30*92 C. ; and yet the transition is unobservable. If pressure and temperature be allowed to fall together, the flickering already mentioned is produced. If the liquid originally fill the tube and then be heated to the critical temperature, the tube becomes filled with gas, but the precise mode of transition from the one state to the other cannot be observed. If the tube thus containing gaseous C0 2 at a IX.] CRITICAL STATE OF MATTER. 207 high pressure be locally cooled, there is local condensation and flickering. This temperature, 30*92 C. for carbonic acid, is called the Critical Temperature. If the temperature of the gas be above 30*92 C., no pressure can condense it into a liquid; if it be just below that point, a pressure of 73 to 75 atmospheres is competent to effect its liquefaction. Water filling a sufficiently strong boiler might be exposed to a low red heat, 720'6 C., and would then be transformed into a gas exercising such enormous pressures as to make its experimental determination excessively difficult ; yet it is known to present this phenomenon at that temperature. As for the gases, oxygen, hydrogen, and the like, whose con- densation has only recently been effected, their critical tempera- ture is very low in the thermometric scale, and exceeding cold is a necessary condition of their condensation under pressure. At the critical temperature, matter under sufficient pressure is in the critical state ; if heated a little more it is undoubtedly gaseous ; if allowed to cool a little it is undoubtedly liquid ; and if the pressure be kept up the transition is unrecognisable. By no optical test can the liquid just below the critical temperature and the gas just above that temperature be distinguished. If a solid be dissolved in a liquid, and the whole be heated under sufficient pressure to a temperature above the critical point, the liquid is now gas, and yet the solid remains dissolved in it (Hannay). Iodide of potassium, for instance, or chlorophyll, if dissolved in alcohol and treated in this way, will remain in solution in gaseous alcohol at 350 C. There is thus under varying pressures perfect continuity of the liquid and the gaseous state at all temperatures above the critical point; and this critical point is, in reference to each liquid, to be found by experiment. ,, When gaseous matter has been rarefied to a very great degree it assumes remarkable properties, of which the most striking is that such exceedingly rarefied gas or ultragaseous matter can be induced as we shall see under Electricity, p. 584 to exercise pressure specially on localised areae of the walls of the containing vessel, and by this concentrated pressure to produce mechanical and luminous effects characteristic of the so-called Eadiant Matter. The Ether. We have already said that we can know matter only by those of its properties which we perceive by 208 MATTER. [CHAP. means of our senses. The existence of any form of matter is to us only an inference from the phenomena to which it gives rise ; and if a large group of phenomena find their best or their only explanation in the assumed existence of a form of matter of an unfamiliar kind, the evidence for its existence is of exactly the same character as that on the ground of which we believe ourselves entitled to assert the existence of any kind or form of matter whatsoever. The phenomena of Light are best explained as those of undulations ; but undulations even in the most extensive use of the term, as- signifying any periodic motion or condition whose periodicity obeys the laws of wave-motion must be propagated through some medium. Heat while passing through space presents exactly the same undulatory character and requires a medium for its propagation. Electrical attraction and repulsion are explained in far the most satisfactory way by considering them as due to local stresses in such a medium. Current electricity seems due to a throb or series of throbs in such a medium when released from stress. Magnetic pheno- mena seem due to local whirlpools set up in such a medium. And the assumption of the existence of a single medium, with properties of great simplicity, will explain these varied phenomena and even co-ordinate them : thus the crest or the trough of a light- wave or a heat-wave is a point of maximum displacement or maximum tension exactly the condition of the medium during the persistence of electric attraction or repulsion, that is, the Electrostatic condition ; the middle point of the wave is changing its position or condition with rapidity exactly the condition of the medium during the passage of a current, the Electrokinetic condition ; thus light and heat are explicable as electromagnetic disturbances of rapidly-alternating character ; and this leads to the conclusion, sustained by experiment, that the velocity of light should be equal to the rate of propagation of an electric disturbance through a medium of this kind. We are led to infer therefore that there is such a medium, which we call the Luminiferous Ether, or simply the Ether ; that it can convey energy ; that it can present it at any instant partly in the form of kinetic, partly in that of potential energy ; that it is therefore capable of displacement and of tension ; and that it must have rigidity and elasticity. Calculation leads us to infer that its 936 density is (Clerk Maxwell) 10 oo,000000,OOOOOO t OOOOOO that f water, or equal to that of our atmosphere at a height of about ix.] THE ETHER. 209 210 miles, a density vastly greater than that of the same atmosphere in the interstellar spaces ; that its rigidity is about 10QQ QOQQOO ^ a ^ ^ s ^ ee ^ nence that & * s eas ily displaceable by a moving mass ; that it is not discontinuous or granular ; and hence that as a whole it may be compared to an impalpable and all-pervading jelly, through which Light and Heat waves are con- stantly throbbing, which is constantly being set in local strains and released from them, and being whirled in local vortices, thus producing the various phenomena of Electricity and Magnetism ; and through which the particles of ordinary matter move freely, encountering but little retardation if any, for its elasticity, as it closes up behind each moving particle, is approximately perfect. Change of State. Work must be done upon a solid in order to convert it into a liquid : energy must in some form be imparted to it. This form may be that of Heat, directly applied so as to fuse the solid. In such a case a definite amount of the energy imparted in the form of Heat apparently disappears (see Latent Heat, p. 324) for it does the mechanical work of liquefying 4 He solid. If the liquid again assume the solid form, as in freez- ig, the process is reversed : the energy absorbed during liquefac- tion gradually reappears in the form of heat, which must be dissipated before the freezing can become complete. If a solid body simply assume the liquid form without having external heat or other energy applied to it, the absorption of some of the heat of the body itself results in a cooling of the substance, as in the case of a freezing mixture, where certain chemical salts being dissolved in cold water, the resultant solution is found to be extremely cold. Again, where two chemical elements combine, their combina- tion is generally attended with heat, the elements losing their potential energy of separation. The supply of an equivalent amount of energy from without is necessary in order to reverse the process of combination that is, to effect chemical separation or decomposition. When the processes of chemical combination and liquefaction go on together the product of the combination of elements of which one or both are solid being itself liquid the result may be that the cooling effect of the latter action exceeds the heating effect of the former ; thus, in the union of carbon and sulphur to form carbon-disulphide, which is a liquid, the absorption of heat due to liquefaction is greater than the evolu- tion of heat due to combination, and the action stops unless heat p 210 MATTER. [CHAP. be supplied from without. On the other hand, in the combina- tion of quicklime with water to form slaked lime, we find much heat evolved partly due to the chemical combination, partly to the liquid water assuming a solid form. The transformation of a solid into a gas in a like manner involves the expenditure of heat or some other form of energy in performing the mechanical work of volatilisation. Snow evaporates in a cold high wind ; arsenic trioxide under ordinary atmospheric pressures is, without melting, volatilised by heat, while, if a sufficient pressure be applied, it melts before volatilis- ing. Dr. Carnelley finds that in a similar way ice, if heated under an exceedingly small pressure, may be rendered very hot (180 C.), and will volatilise freely, yet without melting, unless the pressure be allowed to exceed a certain low maximum, which he calls the critical pressure, this being very low for water, very high for arsenic trioxide. A sheet of metal may be dissipated in vapour by an electric discharge, part of the energy of which becomes spent in producing this mechanical effect. Again, in chemical combination we often see the conversion of solids into gases. Carbon and oxygen combine to form carbon monoxide ; of the heat which is evolved by the union of the elements, a large part is absorbed in rendering the solid carbon gaseous. If the CO produced be in its turn burned so as to form C0 2 , none of the heat of combination of oxygen and carbon-monoxide is absorbed in doing mechanical work of this kind, and the amount of heat evolved in the second stage of the production of C0 2 is greater than that evolved in the first. Conversely, where two gases produce a solid, as chlorine and sulphuretted hydrogen do, the amount of heat liberated is determined not only by the amount of energy absorbed in decomposing H 2 S, and by the amount liberated by the union of H 2 and C1 2 , but also by the fact that the sulphur is deposited in the solid form. If a liquid were exposed to an indefinite and perfect vacuum, it would evaporate at once at any temperature above the absolute zero. If it be exposed to an imperfect vacuum, it will still evaporate readily, but not so readily as before, for its vapour must be able to force its way from the liquid and against the superincumbent pressure. If the pressure be great, the amount of heat which must be applied to the liquid in order to enable it to overcome this resistance and to enter into ebullition I is also greater, and the Boiling-point of a liquid increases with the pressure. ix.] CHANGE OF STATE. 211 Let a liquid be supposed heated in a vessel provided with a piston, by means of which pressure can be exercised on the contents of the vessel ; the vessel being supposed of any sufficient length. The liquid is heated and converted into vapour ; the vapour forces out the piston, and the external air pushes it in ; the piston rests when the external and internal pressures are equal. If we press home the piston, the vapour is partly condensed : to retain it in the gaseous form we must simultaneously apply a stronger heat. This process may be supposed continued till at a certain high temperature (the " critical temperature ") and great pressure we have the whole of the liquid evaporated, and its vapour compressed into the same space as the original liquid. If expansion be altogether prevented, this process is con- tinuous, and the temperature at which water can be wholly converted into vapour under such circumstances is 720'6 C. Liquids are, as a rule, more bulky than the corresponding solids ; hence fusion, which involves expansion, obeys the same law as evaporation, which also involves expansion ; it is hindered by pressure, and the fusing point, like the boiling point, is raised by pressure. A few liquids water, cast-iron are denser than their solids. In such a case, an increase of pressure may be said to predispose the particles to set into the more compact and denser form, the liquid form, and fusion is facilitated by pressure. Thus the melting point of ice is lowered, that of most other solids raised, by pressure. Change of state involves, then, either an absorption or a liberation of energy ; and the amount of energy which must be supplied to a body in order to enable it to undergo a change of state depends on the pressure which tends to resist or to favour such a change, as well as on the intrinsic energy which it already There is no known means of effecting any transformation of matter in any of its ordinary forms into the Ether, or vice versd. THE CONSTITUTION OF MATTER. The question as to whether Matter is or is not infinitely divisible has been made the basis of much acute speculation ; but it is only within this century that any serious proof has been adduced in favour of an atomic theory, or theory according to which matter is considered as made up of indivisible particles. According to this view, matter consists of particles or Atoms, each of which it is impossible with our present appliances to divide, and the division of which, if it were possible, would probably result in the subversion of our ideas as to the apparently funda- mental nature of some of the properties of matter. Chemical Views. The probability of this atomistic view was raised almost to the rank of certainty by the researches of successive chemical investigators. It was first found that as a rule every definite chemical substance in a state of purity had 212 MATTEE. [CHAP. always the same constitution ; that an analysis effected for one- pure sample of, .say, oxide of lead, was applicable to every other pure sample of the same substance. Hence the law of Fixity of Proportions in chemical compounds. But it was remarked that the same substances often unite in different proportions to form compounds possessed of essentially different properties. Carbon and oxygen thus unite to form two well-known compounds, of which the percentage compositions are respectively : Carbon . . . 42'85 Carbon . . . 27'27. Oxygen . . . 57*14 Oxygen . . . 72-72. Analytical results tabulated in this way are not very instructive ; 42-85 but if the second example be multiplied by ^ H, we find the Zil'Jjf respective ratios to become Carbon. . . 42*85 \ Carbon ... 42'85. Oxygen . . . 57*14 J Oxygen . . . 114^28. or in round numbers, Carbon ... 3 Carbon . . 3. Oxygen ... 4 Oxygen . . 8. Here we find that the same quantity of carbon, united in the one compound (carbon-monoxide) with a certain quantity of oxygen is in the other (carbonic anhydride) united with twice the quan- tity of that element. From a large number of instances of this kind was evolved the Law of Multiple Proportions that the same substances may form a series of different compounds by uniting in several fixed proportions which bear a whole-number ratio to each other. Nitrogen and oxygen thus form five com- pounds, in which the nitrogen and oxygen are present in the respective ratios of 14 : 8, 14 : 16, 14 : 24, 14 : 32, 14 : 40 ; and in this case the quantities of oxygen, united with a fixed quantity (14) of nitrogen, bear to one another the relative ratios of 1:2:3:4:5. Iron has two oxides, in which the iron and the oxygen bear to one another the respective ratios of 28 : 16 and 28:24; here the quantities of oxygen, united with the same quantity of iron, bear to one another the ratio 2:3. Then, further, the law of Chemical Equivalence was formu- lated : chemical quantities, which are equivalent to the same thing as regards power of doing chemical work or forming chemical ix.] ITS CONSTITUTION. . 213 compounds, are equivalent to one another. One part by weight of hydrogen will combine with eight of oxygen (7'98165'00l75); so will 108 parts of silver (10*7-896). 108 pts. of silver and 1 of hydrogen are mutually equivalent, for they can both do the same chemical work they can enter into combination with 8 pts. of oxygen ; and they are both equivalent to the 8 pts. of oxygen with which they can combine. If now it be found that 1 pt. by wt. of hydrogen can combine with 3 5 '5 (35 '4 78) pts. by wt. of chlorine, then the law asserts that the equivalent quantity, 108 pts., of silver should also, in its turn, be able to combine with an equal quantity, 35'5 pts., of chlorine. This law is a generalisation, based upon facts determined by the aid of the balance, and independent of theory ; and this law of equivalence, so based, though it be too sweeping a generalisation to be now ac- cepted in its full sense, yet did useful service in its day in enabling tables of Equivalent Numbers or of Combining Proportions to be drawn up, and a system of Chemical Formulae to be devised, based upon these equivalents. According to this system, the com- position of water was symbolised as HO ; this symbol might be read in words as one equivalent of hydrogen and one of oxygen united to form one equivalent of water. The symbol of hydrogen peroxide was HO 9 ; one equivalent (1 pt. by wt.) of hydrogen com- bined with two equivalents (2x8 16 pts. by wt.) of oxygen. When such facts as these were known, a reasoned explana- tion of them was sought. None that offered was so plausible as Dalton's atomic theory, a revival of the old hypothesis of Leucip- pus, Democritus, and Lucretius, that matter consists of atoms, coupled with the proposition that the atoms of the different elements have different relative weights. According to this view the smallest mass of water must consist of an atom of hydrogen and another of oxygen, their relative atomic weights being 1 and 8 ; and these are connected as one might couple together a ball of wood and one of lead. More complex substances were pro- duced by the union of a greater number of such atoms as, for instance, H0 2 , 1ST0 5 , (KO, HO), etc. ; and the symbolic formulae were then used to denote the relative number of such atoms entering into the formation of compound substances. But it was found that the system of formulae based upon the facts of equivalence did not work well when made to signify the relative numbers of atoms united to form a compound. The equivalent number for carbon was 6, because that quantity of carbon was equivalent (in carbonic oxide) to 8 of oxygen, which 214 MATTER. [CHAP. quantity was in its turn equivalent to the standard unity of hydrogen. In marsh gas 6 pts. by wt. of carbon are found to be combined with 2 of hydrogen i.e., with two equivalents ; hence the formula, according to this system, must be CH 2 . It is known, however, that one fourth of the hydrogen can be replaced by half an equivalent (1*7|- pts. by wt.) of chlorine, forming CH^Cl^ : an expression intelligible though cumbrous when read in the language of equivalents, but absurd when read in terms of the atomic theory. This last formula had accordingly to be modified to CgHgCl ; and then the original marsh gas had to be supposed invariably to enter into reactions as 2CH 2 , or else its formula must be modified to C 2 H 4 . The latter is the more natural sup- position. It was pointed out (Gerhardt) that throughout the whole of the chemistry of the carbon compounds similar reason- ing shows that if the atomic weight of carbon be 6, the atoms always appear in reactions in even numbers ; whence the infer- ence is obvious that the at. wt. of carbon must be 12, and the proper formula of marsh gas is CH 4 . In a similar way it was shown that the assumption that the atomic weight of oxygen, as well as its equivalent number, is 8, leads to the invariable appear- ance of 2 , or of an even number of oxygen atoms in every equa- tion ; whence the atomic weight of oxygen must be 1 6 ; and the Atomistic formula for water, as distinguished from the Equivalence- formula, must be H 2 0. The atomistic formulae now in use do not directly make use of the idea of equivalence : they denote the number of atoms of which the Molecule a fruitful idea, due to Awogadro is made up. The symbol H 2 0, for instance, sig- nifies a molecule of water, made up of two atoms of hydrogen (at. wt. = 1) and one of oxygen (at. wt. = 1 6). When attention was first directed to this mode of representation, it was found to be entirely in accord with the half -forgotten researches of Gay Lussac on the relative volumes of gases which enter into combination. He found that one volume (say 1 cubic cm.) of oxygen and two (2 cub. cm.) of hydrogen unite to form two volumes (2 cub. cm.) of water- vapour. The atomistic equation, on the other hand, is O + 2H = H 2 ; that is, one atom of oxygen unites with two of hydrogen to form a molecule of water. These two statements are closely parallel ; and the molecule H 2 formed occupies in the state of water -vapour the same space as the original two atoms of hydrogen. Similarly it had been found that the electric spark decom- posed 2 c.c. NH 3 into 1 c.c. 1ST and 3 c,c. H. The equation is ix.] MOLECULES. 215 = N + H 3 One molec. One atom. Three atoms. 2 c.c. 1 c.c. 3 c.c. Here again the molecule of the compound, NH 3 , occupies the same space as two atoms of hydrogen. So forth ; the general rule is that the molecule of any com- pound in the gaseous state occupies the same space as two atoms of free hydrogen. Hence we may provisionally establish a general rule, subject to excep- tions further to appear : If in a chemical equation relating to gases we write " 2 vols." under every complete molecule, and " 1 voL" under every atom of any element entering into or resulting from the reaction in the free state, we learn the relative volumes of the gases concerned in the reaction. Thus, if alcohol-vapour be burned in oxygen, Alcohol-vapour. Oxygen. Carbonic anhydride. "Water-vapour. C 2 H 6 + 60 2C0 2 + 3 H 2 O. One molecule. Six atoms. Two molecules. Three molecules. 2 vols. 6 vols. 4 vols. 6 vols. Thus a system of equations based on the atomic theory is found readily to give important information beyond what it was designed to give. This lends probability to the system. The Molecule of a compound substance is the smallest mass that can exist in the free state. If we could break up a molecule we would sever it into its constituent atoms as HC1 into H and Cl but we would destroy the substance on which we operated, as such. A molecule of hydrochloric acid contains 2, one of protoplasm from 700 to 1000 atoms. What is the condition of elementary substances in the free state ? Here such equations as the following come to our aid : CuH + HC1 = CuCl + HH, Ag 2 + H 2 2 = Ag 2 + 00 + H 2 ; and we learn that the molecule, even of an elementary substance, consists of two atoms, and we find by experiment that it occupies, like the compound molecules already discussed, the same space as two atoms i.e., one molecule of hydrogen. All molecules, simple as well as compound, are thus seen each to occupy the same space ; and conversely, the same space must be occupied by an equal number of molecules of whatever kind they be. This is the ex- tremely important law known by the name of Awogadro. All gases (at the same temperature and pressure) consist, within equal volumes, of equal numbers of molecules. This is a general law, and its direct consequence is that the 216 MATTER. [CHAP. specific gravity of every gas, at a given temperature and pressure, as compared with that of hydrogen under the same conditions, is the relative weight of a molecule of the gas as compared with the molecular weight ( = 2) of hydrogen. Thus the molecular weight of alcohol, C 2 HIO, is24 + 6+16-46; that of hydrogen = 2 ; the single molecule of alcohol is twenty -three times as heavy as that of hydrogen, and accordingly the density of alcohol-vapour is twenty-three times that of hydrogen, other things temperature and pressure being equal. There are some apparent exceptions. Mercury - vapour which, if two atoms formed its molecule, would have a molecular weight of 400 and a sp. gr. of 200, has a sp. gr. of only 100 ; hence its molec. wt. (twice its sp. gr.) is only 200, and its molecule contains only one atom. Cadmium and zinc have also monatomic molecules when in the state of vapour. Phosphorus and arsenic vapours have, on the other hand, an excessive sp. gr. ; that of phos- phorus is 62, and its molec. wt. must be 124, but its at. wt. is only 31 ; hence its molecule must contain four atoms. The molecule of arsenic is also tetratomic, while that of ozone is triatomic. Hence, to provide for these exceptional instances, we must revise the rule provisionally laid down, and adjust it as follows : To find the relative volumes of gases entering or leaving a reaction, modify the equation, so that it repre- sents no free gaseous atoms, but only complete gaseous molecules ; then under every complete gaseous molecule write " 1 vol." Thus H 2 + O H 9 becomes 2H 2 + 2 2H 2 O. Two molecs. One molec. Two molecs. 2 vols. 4- 1 vol. = 2 vols. The equation C 2 H 6 + 60 200, + 3H 9 becomes C 2 H 6 + 30 2 2C0 2 + 3H 2 0. One molec. Three molecs. Two molecs. Three molecs. 1 vol. + 3 vols. 2 vols. + 3 vols. Again, the equation 3CaP 2 6 + 100 = Ca 3 P 2 8 + 10CO + 4P becomes 3CaP 2 6 + IOC = Ca 3 P 2 8 + 10CO + P 4 . Ten molecs. One molec. 10 vols. 1 vol. Another order of exceptions is presented in cases of Dissociation or Thermolysis. When NH 4 C1 is volatilised, its vapour has half the sp. gr. in- cated by the above theory ; in other words, it occupies twice the theoretical volume. This is because a molecule of NH 4 C1 is in reality split up into separate molecules of NH 3 and HC1 (which may be partly separated by diffu- sion), each of which occupies the whole space that the original single molecule would have been able to fill if it had not been decomposed by the heat applied. Similarly, a molecule of calomel volatilised occupies twice its normal volume ; for instead of a single molecule of Hg 2 Cl 2 we have, as the result of dissociation, a molecule of HgCl 2 , and another molecule, complete though ix.] MOLECULES. 217 monatoinic, of mercury, each of these molecules independently taking up as much space as the original HggC^ would have occupied if it had not been decomposed. By these apparent exceptions Awogadro's law is thus confirmed. Upon this basis has been erected the modern science of Chemistry, one of the leading auxiliary ideas in which is that of the Atomicity of an atom the number of atoms of hydrogen which an atom of any substance in question can combine with or replace. Whether the special manner of thought and expression of parti- cular chemists be or be not adopted, the theoretical chemist can hardly express himself without making some use of the well-known Graphic Formulae by means of which the relations of the atoms in a molecule may be indicated or suggested. Yet this mode of repre- sentation is sadly deficient, although exceedingly useful and sug- gestive. It gives a factitious representation in one plane of a statical condition of the molecule : it does not account for the energy possessed by a molecule in virtue of any one arrangement of its atoms, as compared with that possessed by a molecule of an isomeric compound in virtue of another disposition of atoms of the same kind and number ; and, indeed, it scarcely touches as yet at any point the physical molecule or atom with which perfect knowledge would presumedly show it to be identical. Still the attempt is being made to bridge over the gap as, for example, by the researches of Le Bel and Yan't Hoff, who trace out such relations as those between symmetry of the molecule, as in the case of propi- CH 3 onic acid, H C H, and the absence of rotary power as affecting the COOH plane of polarisation, on the one hand, and on the other between graphic asymmetry of the molecule, as in the case of lactic acid, CH 3 HO C H, and the possession of this rotary power. But, on the COOH whole, Chemistry and Physics, which should be parts of one dynamical science of matter and energy, are still separated by a wide gap, and one great stride which the Science of the future has to take is that of assimilating the theories of the physical and the chemical molecules, and thereby stepping over this gulf. Physical Views. Physicists have been obliged, independ- ently of chemists, to develop mechanical theories of the molecule or the atom, as they have indifferently termed it. That such a 218 MATTER. [CHAP. thing does exist is manifest to them on several grounds. Not to speak of compressibility and porosity of matter as showing that it does not entirely fill space, we learn from Cauchy's investigations that if light be a wave-motion, there would be no dispersion, no prismatic colours of the speatrum, if the glass of the dispersing prism were continuous or were of a granular structure with inde- finitely small grains. According to him matter must be distinctly granular, whether it be discontinuous or not, and its granulations must not be greatly less in diameter than about I OO Q O of the wave-length of the shortest wave of light i.e., about 2000 1 OOOIT mm., or about 500000000 mc ^- ^ir William Thomson finds that there must be from 200 to 600 molecules in one wave-length. He also finds by his Electrometer that plates of copper and zinc exert a certain measurable attractive force upon one another. An indefinite number of plates would multiply this attraction to an indefinite amount ; and if such plates were allowed to come together, the heat given out and representing their potential energy of sepa- ration would be indefinite, and they would combine, after the manner of gunpowder. The energy observed to be given out in the form of heat during the formation of brass by the fusion together of copper and zinc is not indefinite, but measurable : it corresponds to the mutual attraction of a number of plates more numerous than 10,000000, and less so than 4000,000000, to the milli- metre. Hence copper and zinc could not be made into plates thinner than this, and plates of this tenuity would be only one molecule thick. A soap film cannot be stretched beyond a cer- tain thickness without volatilising rather than yielding to further extension at the same temperature ; this limit appears to be reached when a thickness of 200000000 mm - ^ as ^ een attained. Further considerations derived from the kinetic theory of gases lead to the conclusion that a cubic cm. of solid or liquid contains a number of molecules which, though exceedingly large, is limited ; and the distance between these is a quantity of the same order as those above mentioned. From these considerations Sir William Thomson concludes Thomson and Tait, Natural Philosophy, vol. i. part 2, App. F, 1883, and Nature, July 1883 that if a drop of water were magnified to the size of the earth, the molecules or granules would each occupy spaces greater than those filled by small shot, less than those occupied by cricket balls. But this tells us nothing about the nature of the atoms or molecules. It would at first" sight be natural to conceive them as hard balls, but this would not explain their elasticity and mutual ix.] MOLECULES. 219 action: Faraday regarded them as "centres of force;" Macquorn Kankine as nuclei, each surrounded by an atmosphere in which there are whorls and currents of a complicated character. The most interesting hypothesis is that of Sir William Thom- son, who supposes each Atom of matter to be a Vortex-ring in the universal Ether. The ether itself we do not directly perceive ; but this hypothesis would render our perception of matter a phenomenon of exactly the same order as that of light or radiant heat, viz., a perception of Matter as a Mode of Motion in the Ether. If one look at a smoke -ring blown from a cannon, from a locomotive -engine chimney, from a tobacco-pipe, the lips of a smoker, or from an exploded bubble of phosphuretted hydrogen, it will be seen that the whole of the matter of -.the ring is in a state of rotation round an axis disposed in a circular form, and having no free ends. This is a Vortex-King ; and such is that motion in the ether which is supposed to constitute a vortex -atom. A rotating ring of this kind in an imperfect fluid such as air must be the result of friction ; but in a perfect fluid it could only originate by a special creation of some kind. Such a vortex-atom in a perfect fluid would have the following properties : it could move about in the fluid ; its volume would be invariable ; it would be indestructible ; if struck by another it would be indivisible, but would present perfect elasticity, for though for the moment dis- torted, it would recoil and oscillate through its mean form : it would thus be capable of harmonic vibration, as the spectroscope shows the particles of matter to be ; it would be capable of changes of form, becoming narrow and thick, or wide and thin ; and it is practically the only form of motion in the ether which could remain in or near the same mean position, and at the same time be capable of being compounded with movements of translation. This kind of atomic structure would also account for what Tolver Preston calls the open structure of matter, which allows light, electric and magnetic stresses, and the action of gravity, to be transmitted through it. These properties of the vortex-ring ex- plain well many of the observed properties of matter ; but we not only have the chemical atom and atomicity, but also physical mass and gravitation to explain before we can form a full theory of the inner structure of the Molecule. The Kinetic Theory. The next question is, Do these mole- cules remain at the same spot/rotating round it, or oscillating in its vicinity ? or have they, in addition to whatever intrinsic motion they may be possessed of, a motion of translation ? The pheno- \ 220 MATTER. [CHAP. mena of Diffusion help us to arrive at a conclusion on this subject. If a solution of a coloured salt be placed in a vessel, and a layer of a colourless solution be laid upon the coloured stratum, the whole being left at rest for some weeks and protected from all disturbance, the plane of demarcation between the strata becomes blurred, the strata ultimately mix, and the solution becomes uniform. This can only occur through a gradual travelling of the coloured solution into the colourless one, and vice versd. Again, if a jar of hydrogen and a jar of oxygen be brought into communication with one another, even though the former be uppermost, the gases will perfectly mix in a short time. This shows that the hydrogen rapidly travels into the oxygen, and vice versd. The particles of matter, therefore, cannot be at rest, but must be in perpetual relative motion; and this is the Kineiic Theory of Matter. Chemical analogy also illustrates this position. If steam be passed over red-hot iron filings, the iron takes the oxygen, and hydrogen passes off ; if, on the other hand, hydrogen be passed over oxide of iron, it forms water- vapour, and reduced metallic iron is left behind. These actions, apparently so contradictory, are explained thus : There is a molecular agitation and a continued process of decomposition and recomposition of chemical molecules ; the chemical atoms of iron, oxygen, and hydrogen are constantly changing their partners and forming new molecules ; and in the first instance any mole- cules of hydrogen, in the second any molecules of steam, that happen to be formed are carried off in the current of gas which passes through the appar- atus. The particles even of one and the same substance appear to be in this ceaselessly restless state of decomposition and recomposition : when the sub- stance is heated, the molecules are easily broken up, but are not so easily formed again, whence we have the phenomena of Thermolysis or Dissociation ; but even at ordinary temperatures the atoms associated within the molecules break asunder, and must seldom happen to meet each other again. Agitation and break-up thus occurring within the molecules are incompatible with rest, and must necessarily be associated with violent translatory movements of the whole molecules. In a gas, then, we must figure to ourselves a very large number of physical atoms, moving about with great velocity, striking one another and the sides of the containing vessel. Then the energy of any given quantity of gas, so far as that is due to movements of translation, will depend on the aggregate mass m and on the average velocity V ; and it will be -| mV 2 . If we consider a cube of unit-volume, filled with any gas, and take any one internal face of it ; that face is struck by particles travelling at an aver- age velocity u in a direction at right angles to that face, or having an average component of velocity = u in that direction, and having therefore a certain aggregate momentum. This momentum with which the particles strike the ix.] THE KINETIC THEORY. 221 wall during a unit of time must be equal to the counter pressure exerted by the walls of the vessel ;* the pressure p exerted on unit-area of the walls of the vessel by the gas is therefore equal to the momentum of the particles impinging on it in unit of time. But what is the amount of this momentum ? It is the product of the number of particles which strike the wall in a unit of time into the momentum of each. 1. The number of the striking particles If the gas contain N particles per unit of volume, and these move towards the wall struck by them with an average rate u per second, the number of particles which must strike the wall in a unit of time is Nw. 2. The average momentum of each The mass M of each particle is the same ; the average velocity is u ; the average momentum of each particle is M%, The momentum with which the wall is struck is thus N#M# ; and this = p, the pressure on the wall. But the cube is one of unit-volume ; the aggregate mass of the gas is NM = vp = p ; whence p = pu?. Next, what is the average velocity u in any one direction ? The average velocity V (which does not depend on direction) is, if resolved into compon- ents u t v, w, at right angles to one another, But , v, w are equal to one another, for there is no difference between one direction and another in respect of velocity : whence V = vT**, and = V 2 But p = pu 2 = p for unit-volume ; o V 2 V 2 wV 2 pv = pv = m = . = the molecular kinetic energy of the gas whose mass m is confined within volume v. We know that at the same temperature the pressure is equal in all gases ; in equal volumes of different gases pv must be the same ; and hence the molecular kinetic energy ( = pv J must be equal in equal volumes of all gases. If two gases have the same Temperature, the particles have the same mean molecular energy ( _ j of translation. -r-rO If the aggregate kinetic energy of translation - - be equal in equal 2 volumes of two gases, and if at the same time the molecular energy of the molecules of each be equal (their temperatures being equal), it follows that * This momentum would be lost to the gas within the cube were the particles conveying it not prevented by the counter pressure of the wall from escaping: this loss -r- time during which it would have been effected is the rate of change of momentum prevented during that time : this prevention is effected by the pressure, which is a force, and may therefore be measured as a rate of change of momentum in the opposite sense. 222 MATTER. [CHAI>. the number of molecules must be equal in the equal volumes of the two gases, and hence Awogadro's law is true in the physical as well as in the chemical sense, being a direct deduction from the kinetic theory. If there be two gases whose respective densities at equal temperatures and pressures are p and />,, Awogadro's law shows that these unequal masses are divided among equal numbers of molecules : hence the mass of each single molecule must be proportioned to the density of its gas ; for if M and M / be the respective masses of single molecules of the respective gases, M = eY a ndM,= ^ ; N JS whence M : M, : : p : p r Thus the Molecular Kinetic Theory of Gases explains the pressure on the sides of the vessel containing a gas : it explains the tendency of gases to indefinite expansion : it explains Heat as the energy of molecular agitation ; equality of temperature as equality of the mean energy of agitation in the several molecules. It also arrives at Awogadro's law, and explains the numerical identity of ratio existing between the relative weights of the several kinds of molecules and the specific densities of the corre- sponding aggregate gases. The equation p = ^ given above yields V = / , by means of which V p V, the mean velocity of movement of the particles of any gas, may be found. Thus for hydrogen p, the pressure, is equal to the weight of say 76 cm. of mercury (density = 13*596), or to 1033'3 grms.-mass resting on every square cm. But the weight of 1033*3 grms. is mg ; 1033 '3 grms. x 981 cm. = 1013667*3 dynes. Again, p, the density of hydrogen, is -0000895682 grammes per cubic cm. Hence A /-^ = 184260 cm. V p or 1842'6 metres per second, the average velocity of the particles of hydrogen. Hence also the mean velocities of gases vary inversely as j= ; or, which V p is an equivalent statement, the mean velocities of the particles of gases vary inversely as the square root of the molecular weight : whence oxygen atoms have one -fourth the velocity of hydrogen atoms because they are sixteen times as heavy. This is the law governing the relative speed with which the different components of a gaseous mixture will travel through a membrane. The kinetic theory also informs us that when we double the number of molecules in a given space we double the number of molecules which strike the walls, and accordingly we double the pressure ; or in other words, the pressure varies directly as the density of a given quantity of gas, this being another form of Boyle's Law. IX.] THE KINETIC THEORY. 223 It also tells us that if we mix a particles of one gas, b particles of another, c of a third, and so on, the average kinetic energy of all the particles being the same, or soon becoming equalised, the pressure produced by the a molecules of the first gas is proportional to their number, the pressure produced by the second gas is proportional to &, and so forth ; or in other words, that in a mixture of gases the pressure produced by each com- ponent of the mixture is independent of the rest, and depends only on the amount of such component which is present in the mixture (Dalton's Law). Again, when the temperature is increased, the energy of the particles is increased ; each particle strikes both oftener and harder; the pressure experienced by the walls of the vessel therefore varies as the square of the velocity, and is proportional to the molecular energy of the particle that is, to the absolute amount of heat-energy possessed by it. This if the volume be kept constant ; but if the pressure be kept constant and the volume allowed to increase, then the volume varies as the "absolute temperature" (see p. 328). (Charles's Law, often attributed to Gay Lussac). The kinetic theory of gases also explains how it is that when a stream of gas passes through air, its progress is retarded by " viscosity ; " rapidly-moving particles of the gas travel laterally into the air; slowly-moving particles of the air travel into the gas, and thus its progress is hampered. Similarly the viscosity of a gas will bring to rest a current set up within its own substance ; but this viscosity is independent of the density in gases. The theory also explains the conduction of heat in gases ; rapidly -moving particles by collision part with some of their energy to others, which in their turn enter into collision with those beyond them : and we have already seen it explain the diffusion of gases. These molecules, thus travelling with such great velocities and entering into a practically infinite number of collisions with one another, can never travel very far in an undisturbed path. At the ordinary temperature and pressure the mean free path of the molecules of hydrogen, which have the longest trajectory, seems to be about 20000 mm., or a tenth part of the average length of a wave of light (Maxwell) ; 10 ^ OO mm. (Crookes). The diameter of molecules is not the same in the case of all elements, but is on the average ab jut 100O 8 OOOO mm. Thus the smallest 224 MATTER. [CHAP. visible organic particle (^oVo mm - diar.) wi ^ contain about 30,000,000 atoms, which may be arranged in about 40,000 mole- cules. The number of molecules in a cubic cm. at the ordinary temperature and pressure is about 19,000000,000000,000000 (Maxwell), 1000,000000,000000,000000 (Crookes), not more than 6000,000000,000000,000000 (Sir W. Thomson).* These numbers are arrived at by using a proposition formulated by Clausius, that 8 x free path of each molecule Whole space filled by the molecules Diameter of each molecule Their real aggregate volume From this the real aggregate volume, which does not differ very widely from that of the corresponding liquid, is found, and, if divided by the cubic space occupied by each molecule, gives the number of molecules. When gas is rarefied the number of molecules in a given space is diminished. Let us suppose that the rarefaction is carried on so far that only one particle out of every original million is left in the space exhausted. The pressure is one- millionth of its original amount ; but any molecule once in motion has one-millionth its former chance of encountering any other molecule, and consequently its average free-path is magnified a millionfold. The mean path would then be (Crookes) 10 ^ 0o mm. x 1,000,000 = 100 mm., or about 4 inches. By means of a good Sprengel pump exhaustion to the hundred-millionth of an atmosphere can be attained, and the mean free-path of the gas so rarefied would be about 33 feet. In our atmosphere at a height of 210 miles the single molecules are relatively so few (1000 to the cubic cm.), that each molecule might travel through a uniform atmosphere of that density for 60,000,000 miles with- out entering into collision ; beyond a height of 300 miles the atmosphere is so rare (less than one molecule per cubic foot) that the particles might freely travel through such an atmosphere from one fixed star to another ; while in the fields of space, at distances practically infinite from the earth or any other star, the number of cubic miles containing a single molecule would be represented by the figure 1 followed by 314 cyphers. * While there is no confusion as to the principles at issue, there is still a dis- cussion in active progress as to the numerical details ; and the reader who wishes to study the subject fully is recommended attentively to peruse all the articles by Prof. Clerk Maxwell (Encyclop. Britann. ; Nature; Philos. Mag.; Trans. Roy. Societies London and Edin., Cambridge Phil. Soc.\ and by Dr. Crookes (Transactions and Proceedings Hoy. Soc. London), to which he can get access, together with the liter- ature to which he mil there find reference. IX.] THE KINETIC THEORY. 225 This opens up to us an extraordinary view of the nature of our atmos- phere. We must, though the process cannot be rapid, for each particle rising from the earth is retarded by gravity and falls back towards the earth, constantly be losing particles of nitrogen and oxygen as we are dragged through space, and we may constantly be picking up new ones. If we entered regions of space in which there were no particles fit to make up OUT losses, it would be an interesting question how short a time would suffice altogether to deprive us of our atmosphere. The region of space through which the earth is at present travelling contains much benzene vapour with ethyl-hydride and other alcohol-deriva- tives. Thus our ideas on the subject of the constitution of matter have undergone a profound modification. Matter is discontinuous in the highest degree, for it consists of separate particles or mole- cules which are mutually non-interpenetrable ; and the special properties of the different states of matter depend on the number of molecules which are contained within a given space, as well as on the energy of movement which is possessed by each ; and each particle is susceptible not only of translation as a whole, but also of vibration or rotation, and may besides be in a state of vortex-motion, upon the continuance of which its continued existence may depend. MOLECULAR FORCES. Hitherto we have conducted our reasoning on the implied assumption that the molecules had no mutual action, and we have arrived at results such as Boyle's law, Dalton's law, and others, which we have deduced from theory. Now we must con- firm our theory by reference to facts, and we find this assumption overruled by such material discrepancies as the following. Boyle's law, though obeyed on the whole, is disobeyed by every gas when the pressure is so high or the temperature so low that con- densation is not far off: this departure, though not extensive, is significant. All gases just about to become condensed are, except in the single case of hydrogen, more easily compressed than the law would indicate. Dalton's law is departed from by a mixture of gases condensible with difficulty : such a mixture is found to be even less condensible than the component gases, and the critical temperature is lowered. Charles's law is not obeyed throughout the whole range of experimental pressures and temperatures ; at a high pressure any increment of heat produces a disproportionately large increment of pressure. In fact, gases obey these laws only when their pressure is Q 226 MATTEE. [CHAP. very feeble and their temperature at the same time high above the critical temperature that is, when the molecules are com- paratively far from one another. At ordinary temperatures and pressures the particles do affect one another not merely by mutual impact or mutual gravitation, but also by mutual actions or molecular forces effectively coming into play when the particles are at exceedingly small distances from one another. When the pressure is small, the free path is comparatively long, and the molecules are mutually removed from each other's influence : and the higher the rate at which the particles are moving, the less will be the proportionate effect of the molecular forces ; or in other words, the higher the temperature the less appreciable will be the effect of intermolecular action. When a gas is being compressed into a liquid we know that in the first place, in all gases except hydrogen, the particles are attracted slightly towards one another, and also that there is on the other hand a practical repulsion from one another caused by their energetic movement. We further find, however, that though the particles become approximated with relative ease while lique- faction is approaching, yet when the liquid state has been attained, and even before it has been attained, repulsion takes the place of attraction ; the liquid when formed offers a relatively enormous resistance to compression. This is well seen in the case of car- bonic anhydride merging insensibly from the gaseous into the liquid state ; just before ceasing to be a gas it is very compress- ible, just after becoming a liquid it is relatively very slightly so. In liquids the molecules are within the spheres of one an- other's action. This accounts for the viscosity of all, even of the most mobile liquids : the particles detain one another by their mutual attraction, and a flowing liquid is thus hindered in its flow by molecular friction. Molecular action also accounts for the fact that a stream of liquid has a certain tenacity and will not readily break : such is the condition of a stream of liquid in a siphon. Again, it explains why, under ordinary circumstances, the effects of molecular attraction are strikingly manifest in liquids only at the surface, and in the form of Surface Tension. In the interior of a mass of fluid each particle is free to adjust its mean position under the influence of the surrounding molecules ; the mean position which it assumes is that in which it is acted on equally on all sides, and there is then nothing to render the mutual attractions manifest. At the surface of a liquid mass, however, if it be a free surface, the particles can only be acted ix.] MOLECULAR FORCES. 227 upon by others lying internal to them. The result is, as is shown in Fig. 100, a system of forces acting at right angles to the free surface of the fluid, and tending to pl reduce that free surface to the least possible area. We may, indeed, con- sider a drop of water as consisting of a quantity of water enclosed in a super- ficial skin of water which is under tension, and whose particles attract one another into the least possible super- ficial area; and since of all surfaces the sphere has the greatest content for the least area, the superficial film may be said to mould the drop to the spherical form, which in the case of falling raindrops is approxi- mately perfect, as is shown by the rainbow. To these surface- tensions are also due the important phenomena of Capillarity. Many of the properties of solids are also due to molecular forces. Such are toughness, hardness, and the like, which may be grouped under the generic name Strength, of Materials ; these depending probably, in part at least, on the proximity of the particles to one another. The molecular grouping of molecules is also very important, though very little can be said about it ; but upon it depend not only the crystalline or amorphous condition of a substance and in part its strength, but also that stable or un- stable equilibrium upon which the phenomena of elasticity or the properties of such things as Rupert's drops depend. These last consist of little masses of fused glass dropped into cold water ; the outside is suddenly chilled and solidified while the interior is still in a state of fusion. The internal mass has to accommodate itself as it best can to the dimensions of the outer skin ; it does so under tension, but the moment that the relations are disturbed by breaking off the narrow end, or even by the slightest scratch, the whole flies to powder : it is in a state of unstable equilibrium, and the slightest displacement precipitates a downfall of the whole arrangement. In the same way the slightest scratch in the interior of a large glass tube, especially if it have local variations of thickness, even such a scratch as is produced by the fall of a crystal of quartz or a rub with the end of an iron wire, will often shiver the tube ; for which reason no rough treatment should be internally applied to such tubes with any metal harder than copper. This state of internal tension accounts for the danger in the use of cast-iron in structures. 228 MATTER. [CHAP. ix. Many solid masses have, however, their particles so arranged as to form conservative systems, which tend to restore any work done on them, and consequently are in stable molecular equili- brium ; the details of the molecular grouping are unknown, but in a perfectly elastic body, or practically in any solid body within its Limits of Elasticity, any displacement among the molecules produces a restitution-pressure equal and opposite to the distorting force or stress ; and it is observed that as a general rule the dis- tortion is proportional to the distorting force (Ut tensio sicut vis, " Hooke's Law"), and hence the restitution-pressure is proportional to the distortion. This elasticity may in solids be observed more or less perfectly to obtain, whether the distortion be that of form or of volume, while liquids have elasticity of volume alone, never of form. To the same order of Molecular Forces must be attributed the effects of cohesion between masses or particles of the same sub- stance, and of adhesion between those of different substances ; and also the phenomena of chemical affinity, the potential energy of chemical separation, and the liberation of energy attend- ant on chemical combination. CHAPTEE X. SOLIDS. THE special properties of solids are due to the relative contiguity of their molecules. Their definite free surface is due to the mutual attraction of their molecules, and is retained in virtue of the same forces which in the aggregate manifest themselves as causes of cohesion, tenacity, etc., and the result of which is that a solid can persist under the action of a stress not evenly applied that is, of a stress which is not hydrostatic. Cohesion is the mutual attraction of the particles of a solid for one another, and is measured by the amount of force which must "be applied in order to overcome it. The term cohesion is generally applied to the mutual attraction of particles of the same substance, adhesion to that of different substances. When two pieces of white-hot iron or platinum are brought in contact they cohere by welding. When a piece of silver and a piece of plati- num are brought in contact at 500 C. they adhere. If metals in the state of dust be mixed and exposed to a pressure of 7000 atmospheres they will form a firm metallic mass, and will even combine and form an alloy. Cohesion is manifested by two sur- faces of glass, which, if ground exceedingly smooth and placed in contact, will cohere firmly ; and the well known Barton's cubes are little cubes of metal polished so smoothly that mere apposition causes them to cohere, the force of cohesion being so great that a string of a dozen may be supported in the air by this mutual attraction alone. Common graphite is ground to powder and puri- fied by boiling with nitric acid and chlorate of potash : it is then washed and dried ; the powder is placed in a mould and exposed to extreme pressure produced by a hydraulic press ; after com- pression the black powder is found to have been converted into a solid mass of coherent pencil-graphite, which may be sawn into strips and used for pencils. If a leaden bullet be cut into two 230 SOLIDS. [CHAP. with a sharp and heavy knife, the two halves will cohere firmly if pressed together by their bright surfaces. Hardness Softness. A body is said to be harder than another when it can be used to scratch the latter but cannot be scratched by it. In this sense the diamond is the hardest of all solids. The scratching body must not have too sharp a point, for this would prove a pin to be harder than glass, which is not the case. Hardness is a property that cannot be measured. All that we can do is to make up a list of substances in their relative order of hardness, and to express the hardness of any particular substance by stating its place in that series. The standard series, due to Mohl, is the following : 1. Green laminated Talc. 2. Crystallised Gypsum. 3. Transparent Calcspar. 4. Crystalline Fluorspar. 5. Transparent Apatite. 6. Pearly cleavable Felspar (Adularia.) 7. Trans- parent Quartz. 8. Transparent Topaz. 9. Cleavable Sapphire. 10. Diamond. Flint scratches quartz with difficulty, but is easily scratched by topaz : hence its hardness is set down as 7 '2 5 on this arbitrary scale. The rapidity of movement of the attacking sub- stance is a matter of practical importance : thus the sand-blast (a stream of sand rapidly blown from a tube) is capable of cutting through rocks and even through steel with relatively great rapid- ity ; and the same result is seen in the mechanical operation of filing. Mr. Edison finds that platinum wire may be rendered as hard as steel pianoforte wire by heating in vacuo, keeping up the vacuum, and gradually increasing the temperature. The particles of platinum have all air removed from their interstices, and they cohere very firmly by welding. Hardness Fragility. This is a distinct use of the word Hardness. In this sense the diamond possesses little hardness, for if struck a blow with a hammer it flies to pieces. Extensibility Inextensibility. Some substances can, like indiarubber, be extended greatly by the application of a stretching force : others, like baked clay, very little. The ratio of the extension produced to the extending force is the " extensibility ; " the former being measured by the ratio of the increase in length to the original length. The general rule is that a wire made of any substance, the area of whose transverse section is s square cm., and whose length is I cm., suffers, when stretched by a force of F dynes, an absolute elongation e, which is (1) pro- portional to the length of the wire, (2) proportional to the stretching force x.] EXTENSIBILITY. 231 applied, (3) inversely proportional to the transverse section s, and (4) pro- portional to a special coefficient E, the Coefficient of Extensibility, a small fraction, which must be determined experimentally for each substance. Thus The following coefficients of extensibility (reduced to C.G. measures) are given partly on the authority of Unwin, partly -on that of Wertheim. A rod of each substance whose transverse section is 1 square cm. is stretched by a force of one dyne. The proportionate extensions (measuring the coefficients of extensibility) are, in the respective cases of Cast Steel tempered, one (2520,000000 x 981)th of the whole length. Wrought Iron . . (2000,000000 x 981)th Copper . . . . (1050,000000 x 981)th Wood ..'.. (10,000000 x981)th Leather. , . . (175,000 x 981)th Fresh hone . . . (230,466000 x 981)th Tendon. . . . (16,341,000 x 981)th Nerves . . . . (1,890,000 x 981)th Living muscle at rest (95,000 x 981)th Arteries . . . (5,200 x 981)th What weight would be necessary in order to double the length of a piece of steel bolt 1 sq. cm. in sectional area, if the strain could be effected without rupture ? The elongation e would be equal to the original length, e = Z, I = ~EFl; .'. F = = (2520,000,000 x 981) dynes. If the sectional area TTn7 ' were 1 mm., F = = - ' - = (25,200,000 x 981) dynes = 2520.000000 X 981 the weight of 25,200000 grammes or 25,200 kilogrammes. French engineers are in the habit of reducing these inconveniently large physical constants by expressing extensibility in terms of the number of kilos, weight which would be required to double the length of a bar whose sectional area is one square millimetre : the resultant numbers are 1 ^ of those obtained when the extensibility is measured in terms of the number of grammes' weight which would be required to double the length of a bar whose sectional area is one square centimetre. . Muscles are more extensible when they are in a state of contraction than when they are at rest ; and if a muscle when loaded by a certain weight be stimulated to contraction, the mere effort to contract may so diminish the resistance to extension or increase the extensibility that the contracting effort may be more than counterbalanced by the mechanical stretching of the muscle produced by the weight hanging upon it, and the overloaded muscle may actually stretch when stimulated to contract. Muscles also become a little less resistant or more extensible under the action of a given load shortly after death. There is no substance of which wires or rods could be loaded 232 SOLIDS. [CHAP. with indefinite weights, or even with such weights as would double the length : there is for each substance a special limit of tenacity or cohesion, when extension can go no further, and the rod is ruptured. This breaking weight measures the cohesion, and the following table represents, according to Wertheim, the number of grammes which must be hung on a rod of 1 sq. cm. section in order to break it Bone . . ; . .' .'. . 800,000 Tendon . ,' . . 625,000 Nerve 135,100 Veins . . '"." . . 18,500 Arteries . "v . . 13,700 Muscle . . , '" 4 > 500 Thus a nerve whose section is ^ sq. cm. could bear a stretching force equal to the weight of 33*9 kilogrammes or over 5 stone ; but the danger of stretching an artery or a vein by mistake is obvious. There is a great difference in the breaking weight of the same tissue in persons of different age and habit. Wertheim found that the fibula of a young man of thirty had a breaking weight of 1,503,000 grammes per centimetre, while that of the same bone in an old man of seventy-four was reduced to 432,500. Ductility. Some metals can be drawn through fine apertures in a drawplate, and wires can thus be formed : other metals are incapable of this, for they snap. The order of ductility is Gold, Silver, Platinum, Iron, Copper, Palladium, Aluminium, Zinc, Tin, Lead. Platinum wires of exceeding tenuity, such as are adapted to the eye-pieces of microscopes for micrometric work, are made by constructing a thick silver bar with a thin platinum core, drawing this out to an extreme fineness, and dissolving off the silver by steeping the drawn wire in nitric acid. Compressibility follows the same laws as extensibility. Within narrow limits the coefficients of compressibility and of extensibility have the same value. Excessive compression leads to crushing, and each substance has its own crushing weight, found by experiment on masses of determinate size. Malleability, the property of yielding to the hammer without breaking at the edges. Gold can be hammered out into leaves extremely thin. A half square-inch of gold of the thickness of letter paper is hammered out to 8 1 square inches ; each square inch of this thin sheet is again hammered out into 81 square inches, of which each one is in its turn again hammered out to 81 inches. Antimony, on the other hand, flies to powder at the first blow of the hammer. x.] PROPERTIES. Plasticity. Some solids can be moulded, as le mould. Rigidity Flexibility. In every rod undergoing flexion, if this be due to a weight suspended from a free end, there must be a certain extension of the upper aspect of the rod, a compression of the lower, and a Neutral Line between, which retains its original length. If the flexure be due to a weight pressing down the middle of the rod which is supported at its extremities, the exten- sion is in the lower aspect of the rod, the compression in the upper. In the former case a cut in the upper aspect would weaken the rod ; in the latter the same effect would only be pro- duced by a cut on the lower aspect. Flexion may bring about compression and extension beyond the range of the breaking or crushing strengths, and the body may thus be broken. If this occur before there has been any perceptible flexion, the body is said to be brittle : if it allow a considerable range of flexion it is said to be tough it bends much before breaking. The crystal- line or granular or fibrous structure of a substance has much to do with the extent to which it can withstand stress. Tin is, for instance, very brittle; wrought -iron axles, which are at first fibrous and very tough, are subject to a molecular rearrangement facilitated by vibration, and become crystalline and brittle. Elasticity. " Elasticity is the property in virtue of which a body requires force to change its bulk or shape, and requires a continued application of the force to maintain the change, and springs back when the force is removed; and if left at rest without the force, does not remain at rest except in its previous bulk and shape " (Sir William Thomson). There are two properties, Resistance and Restitution, which must concur in any given body before it can be said to be elastic. The coefficient of resistance to extension, K, is the reciprocal of the coefficient of extensibility ; i.e., K = -g- ; and it is the Force acting fraction - . Proportionate elongation produced TV 1 From the equation e = E , substituting for E, and supposing s and s K to be both unity, we get K = F. Whence K is the number of units of force which would have to be applied to a rod (of the substance whose coefficient of resistance is K) whose sectional area = 1 sq. cm. in order to double its 234 SOLIDS. [CHAP. length, if such doubling were possible. A steel rod of 1 sq. cm. section is lengthened gggodooooo bv a wei g nfc of l gramme i.e., by a force of 981 dynes. A force of one dyne would lengthen it by - - . The J 981 X 2,520,00.0,000 resistance of steel is accordingly - -=2,472120,000000 981 x 2,520,000,000 dynes in C.G.S. measures. This measures the amount of force which would be necessary to produce an elongation equal to the original length of the rod. To produce any less proportionate elongation , the force necessary, F, is El. The "Work done in producing extension is the product of the average resistance overcome into the space through which it is overcome. If K be the coefficient of resistance, and a rod be exposed to a tensile force F, stretching will go on until the ultimate resistance arrived at is in equilibrium with the stretching force F. When this is the case, K = F. The average resistance encountered by the tensile force is ^ K = ^ F. L The space through which the resistance is overcome is e. The work done is resistance x space = ^ Fe. The coefficient of resistance to compression is within small limits equal to that of resistance to extension, and obeys the same laws. Torsibility of a body is measured in the simplest case that of a rod or wire in terms of the angle through which a unit of force, applied at the distance of 1 cm. from the axis of the rod or wire, can twist it. The resistance to torsion is the reciprocal of this angle. The work done in producing torsion is the average resistance (or half the " twisting moment") multiplied by the angle of twist. If the pressure p be applied at the end of a lever r, the twisting moment is pr : if the angle through which the wire is twisted be w, the work done is Jjww. The coefficient of resistance to shear is the same as the coefficient of rigidity, which is a constant independent of the compressibility. The work done in producing a shear may be specified in two ways as the product of the average resistance (i.e., of half the force applied) into the amount of the shear, or as the force applied into the amount of displacement of the centre x.] ELASTICITY. 235 of figure of the mass sheared. Either of these pairs of terms gives the same product, ^ F . PC . tan PGP' (see Fig. 25). Power of Restitution. These resistances are not the only criterion of Elasticity. A body may resist extension, compression, torsion, shear, and yet not be elastic. In order that it may be perfectly elastic, it must have all the following properties : (1.) It must offer a definite resistance to distortion. (2.) The distortion is not permanent, #nd if the distorting stress be removed, the distorted body springs back to its original form or bulk. (3.) The disturbing stress must be continuously applied in order to keep up the distortion. (4.) As long as the distorting stress is kept up, there is a counter-stress or restitution-pressure (P) developed and sustained in the elastic substance. As this holds the deforming pressure (F) in check, and is in equilibrium with it, it must be numerically equal to it ; P = F. (5.) The restitution-pressure does not become diminished by lapse of time. The restitution-pressure P is ultimately equal to tlie deforming force F : but F = K _, whence P = K , and the restitution-pressure is pro- portionate to the displacement. If the distortion _ = l, P = K, whence P = , and the restitution-pressure or restitution-force then exercised E is represented by a number, the Coefficient of Restitution (or "coefficient of elasticity"), which is equal to . . . or to r-r^ If - Extensibility Compressibility I have any other value than unity, the restitution-pressure is K -. Under any given distortion within the limits of restitutive power, the restitution -pressure is equal to the product of the coefficient of restitution into the distortion; the co- efficient of restitution being numerically identical with the reciprocal of the compressibility or of the extensibility. It is usual to profess to measure the elasticity of a solid by a " coefficient of elasticity" which is stated to be equal to the resistance to distor- tion. There is an equality, a numerical identity, between the resistance to distortion and the. coefficient of restitution (upon which the amount of restitution-pressure depends), provided that any one system of units is strictly adhered to, and that the body is perfectly elastic. It seems, however, strange to set up a method 236 SOLIDS. [CHAP. of measuring elasticity based on a tacit fundamental assumption that the bodies dealt with are perfectly elastic. If there be two bodies, of which one has a low the other a high coefficient of restitution, and if the same displacement be effected in both, the restitution-pressures in the two substances differ in the same ratios as their respective coefficients : and in these two bodies the relative amounts of work stored up in the form of tensional or potential energy also differ in the same ratio. Elasticity thus presents two aspects, the Statical and the Dynamical. On liberation of a strained body, the whole of the energy stored up in it may be restored in the kinetic form. This restitution may be due to the solid body being a con- servative system of particles, a sjnall displacement amongst which acts as a disturbance of masses in stable equilibrium : by such a displacement an aggregate force is called into action which tends to produce restoration to the original form or bulk. In an elastic body the greater the displacement or distortion the greater the restitution-pressure, and that in direct proportion (Hooke's law). Perfect and Imperfect Elasticity. A body is perfectly elastic when any given stress produces no permanent set or deformation, restitution being always complete. It is imperfectly elastic when it does permanently retain such a set. It is said to be strained beyond the Limits of Elasticity when it is so far strained that it retains such a set : it is said not to be strained beyond the limits of elasticity when it is not deformed so far that it cannot exactly return to its original form or bulk. When the limits of elasticity are narrow, as in the case of lead (which, though exceedingly easily bent so as to take a permanent set, can yet be induced to enter into vibration, and must therefore be elastic within narrow limits), the body is said to be " imperfectly elastic," or to possess little elastic toughness. When it can suffer distortions within wide limits without taking up a perma- ment set, it is said to have great elastic toughness ; and a body which has infinitely wide limits of elasticity is said to be perfectly elastic. There is no body perfectly elastic, but any body may within the limits .of its elasticity be considered as a perfectly elastic body. In popular language a body is said, like indiarubber, to be very elastic when it has great elastic toughness i.e., when it can be distorted through wide ranges without taking up a permanent set ; but this use of the word should be discouraged in favour of that use in which it is made to signify conjoined powers of x.] ELASTICITY. 237 resistance to deformation, and of restitution of shape, of bulk, and of work done upon the elastic object. The elastic toughness exemplified in a Toledo sword-blade must be distinguished from the ordinary ultimate toughness or breaking toughness ; the former may be much less than the latter. When a body is distorted, not beyond the limits of elasticity, and liberated, the work done upon it is restored. The body exactly regains its original form or bulk, but at the moment of complete restitution the energy possessed by the body (if perfectly elastic) has wholly assumed the kinetic form, and the body passes rapidly, if it be free to do so, through its mean form or bulk, and suffers an equal distortion or alteration of volume in the opposite sense. Again a restitution-pressure is developed, and the body swings back through its mean "position. This is repeated, and thus we have vibrations produced as the result of elasticity. The force bringing back every particle towards the mean position is proportional to the distance from that mean position, and this is the criterion of harmonic motions. Hence in a solid body, which is a system of particles, any displacement sets up an intramole- cular restitution -pressure, which results in harmonic motion (Fourier-motion) of the separate particles, and the particles of a disturbed tuning-fork or stretched string may execute harmonic vibrations, particles equidistant from one another generally assuming equal differences of phase in their respective S.BLM.'s. The whole body executes, like a pendulum, isochronous vibrations, as, for example, the vibrating mainspring of a watch. Viscosity of Elastic Solids. When an elastic body has entered into vibration it appears more or less rapidly to lose its energy ; its vibrations wane away. This waning away is due to the " viscosity " of the solid : the energy of vibration becomes converted into heat. The amplitude of each successive oscilla- tion bears to that of the one immediately preceding a constant ratio. If the ratio between the first and second oscillations be 1 : v, the third will be v 2 , the Tith will be v n ~ l . On account of this viscosity a tuning-fork cannot be made of lead or zinc, the vibra- tions of which too rapidly die away. This " viscosity " is what is frequently understood by the term imperfect elasticity : the restitution of form or bulk may be perfect, but that of energy is not, for some of it is dissipated in the form of Heat. Fatigue of Elasticity. When a tuning-fork is kept (as by an electromagnetic arrangement, p. 622) continuously vibrating for a long time, it stops almost instantaneously when the exciting 238 SOLIDS. [CHAP- cause is removed. The steel requires periods. of rest: if it be kept continuously vibrating it has a tendency to become viscous, and to return comparatively slowly to its mean form after each displacement. ,-- Velocity of propagation of a wave -motion = A / The V p restitution-pressure developed in consequence of a displacement varies as K, the coefficient of restitution ; the acceleration produced by the restitution- pressure varies as the restitution-pressure ; the velocity in the circle of refer- ence (in S.H.M.) varies as the square root of the acceleration ; the velocity of propagation varies as the velocity in the circle of reference : therefore the velocity of propagation varies as the square root of K. Given the same elastic pressures and the same work done upon two bodies whose respective densities are p and p /t the energy being equal, the respective velocities produced must vary inversely as the square roots of p and p r /~K~ Hence v cc / ; and it can be shown that no multiplier intervenes, and V p that v = y |. This is the case of a sound-wave travelling through a solid substance. A sound-wave is propagated through steel of sp. density = 7*85, with velocity . >< 2 - 7 52 8 5 - 000 ' 000 = x / 31 4,919,745. 223 =561177 cm. per second = 5611*77 metres per second. The property of Elasticity is not inconsistent with brittleness : glass has very narrow limits of pliability, and is accordingly brittle, but within these limits it is eminently elastic. Physiological Examples of Elasticity. The whole ligamentous system affords examples, and many of the bones also possess this property. The ligaments of the elastic arch of the foot, the vertebral ligaments, and the intervertebral discs acting against the down-dragging weight of the viscera ; those ligaments which by their very molecular constitution (however this may be accounted for) are always on the stretch, such as the elastic liga- ment of the eye, the filled arteries, the ligaments of the symphysis pubis ; the combined flexion and twist of the ribs in inspiration and their elastic resti- tution in expiration ; the ligaments of the lamellibranch shell, the tracheae of insects, all furnish examples of Elasticity. The Mechanical Advantages of Elasticity. These can be studied in a well-hung vehicle with light springs. Any sudden jolt or jar is not communicated to the body of the vehicle with its original abruptness, but gives rise to a wave -motion which lifts the body of the carriage and allows it to oscillate till it returns to relative rest. If a person jump, landing on his feet, the shock is partly broken by the elastic arches of the feet ; but further, before it reaches the brain it has to pass through a suc- cession of elastic discs, the ultimate effect of whose intervention x.] ELASTICITY. 239 is a gradual and not an abrupt arrest of the downward movement of the head. Were it not for this the brain would be ruptured by exceedingly small leaps. Work is directed by elastic intermediaries so that it can become useful and not harmful. Jolts and jars which, as we have seen under Momentum, involve the disappearance of Energy in doing harmful mechanical work are converted into smooth wave-motions. Thus energy is saved, as well as mischief prevented. If a person run over a gravelly road with a heavy vehicle attached to his person by a non-elastic cord, he will feel greatly bruised. If he interpose a steel spring or a thick piece of indiarubber between himself and the vehicle, the pain is infinitely lessened, and the actual energy expended is about 25 per cent less (Prof. Marey and M. Him) ; work has not been spent in jolting and jarring himself and the vehicle. The use of elastic intermediaries suggests itself in all cases where jolts of any kind would be injurious. Compare the effects of an involuntary move- ment made by a patient whose limbs are under extension by a weight and non-elastic cord with the effect of the same movement when a light spring intervenes between the limb affected and the extending weight The spring persists and keeps up the tension, but it yields to the momentary twitch ; the weight rises a little and sinks back to its former position. Strength of structures as depending on their form. This is the special study of the Engineer. Here we can only state a few principles. Galileo found that a given weight of material if disposed in solid bars presents less resistance to crushing or bending than the same material arranged in the form of tubes, provided that the walls of these tubes be not excessively thin. The following table is from Weisbach's Engineering Mechanics : Resistance to Breaking. Resistance to Crushing. Massive cylinder, rad. = 4 Mass = irr 2 lp = 16-n-lp Hollow ; radii 5 and 3 Mass = 257rfy> - 97rA.p = IQirlp ( 1000 1700 1000 2125 Massive cylinder, rad. = 5 Hollow, radii 5 and 4 5 and 3 1000 870-40 590-40 1000 870-40 590-40 240 SOLIDS. [CHAP. x. Hence, mass for mass, the hollow tube is stronger: diameter for diameter, the solid is the stronger. The strongest tube for all purposes has the relative radii 11 and 5. Examples of this kind of structure we find in the hollow stems of plants, in the feathers of birds, in the long bones of the human body. Economy of material may be carried still further by the adoption of the lamellar or trabeculated structure. We see in lattice-girders how the judicious arrangement of struts which sup- port each other makes a structure strong enough for all practical purposes, though very light, often much stronger than if it were burdened with the stress due to its own excessive weight, which, if it were solid, it would have to support. In the spongy structure of bones we find a similar arrangement. In the upper end of the femur we find a disposition of horizontal, vertical, and oblique trabeculse, which together form a rigid triangular framework support- ing the weight of the body. In the astragalus we have a comparatively light and porous structure, but the trabeculae are so arranged as to resist and to distribute the downward pressure of the body and the compressing pressure exerted by those bones, the os calcis and the scaphoid, which abut against it in the arch of the foot. CHAPTEK XL OF LIQUIDS. THIS chapter may be divided into three parts treating of (1) Molecular Actions, (2) the Statics of Liquid Masses, (3) the Kinetics of Liquid Masses. 1. MOLECULAR ACTIONS. Cohesion. If a ring of iron wire be dipped into a solution of soap, it will be seen on taking it out that the cohesion of the liquid causes a film to be formed which remains stretched across the ring. The force of cohesion is also manifested by a drop of water hanging from a glass rod. Such a drop may be gradually increased in size till at a certain maximum its weight overcomes its cohesion, the water breaks asunder, and the drop falls. A thin rod of glass or wire of metal may similarly be fused at the end, and the fused drop may be increased in size by continued fusion until the molecular forces can no longer counteract its weight. A little water on the end of a thick glass rod will retain a piece of paper placed in contact with it even though some grams' weight be suspended from the paper. Surface Tension. It has already been remarked that the molecular forces are most strikingly manifest at the surface of a liquid, and that every liquid may be regarded as bounded by a superficial skin or film, which behaves like a stretched membrane, and which in time reduces the contained fluid to that form which gives to the greatest cubical content the least superficial area. The tension of this superficial film is the Surface Tension of the fluid. A raindrop, a shot falling from a shot-tower, assume the globular form because the sphere is the simplest geometrical form which fulfils these conditions. A bubble of air in water assumes a spherical form not perfectly so on account of the re- sistance to its ascent. The most convenient way of studying the K 242 OF LIQUIDS. [CHAP. various forms assumed by masses of fluid under the influence of surface tension is to relieve them of the effect of gravity by float- ing them in liquids of their own specific density, with which they will not readily mix. A mixture of alcohol and water is made of the same specific density as olive oil. Masses of olive oil placed in this fluid will neither rise nor sink, but will assume the globular form. If they be not free to assume the globu- lar form, but have limiting conditions imposed upon them,* they may assume geometrical forms of great interest, all having the smallest superficial area possible under the given conditions. If, for example, into such a globular mass of oil, an oiled circular disc of iron be suspended, having a diameter greater than that of the mass, the mass of oil will spread over each face of the disc, and will form on each side of it a segment of a larger sphere. If such a disc be brought up to the globular mass by one face only, the oil will not pass round the edge of the disc. The curvature of the segments of spheres produced may be modified by adding or removing oil by means of a pipette. If a short oiled cylinder with open ends be put into the dilute alcohol, and a mass of oil inserted by means of a pipette, the oil will fill the cylinder and form a kind of biconvex lens of oil ; by means of a pipette oil may be taken from the mass till it becomes a biconcave lenticular mass ; or if the operation be stopped at the right instant, a plane-ended cylindrical mass of oil will be obtained. If a circular ring be immersed in a large mass of oil, and some of the oil be then removed, the mass will assume a lenticular form. If a little iron framework be constructed, representing in outline the sides (one inch) of a cube, and hung into a mass of oil which is then gradually removed, the mass of oil will have part of its periphery determined by the iron framework, and will assume the appearance successively of a cube with convex sides, of a cube with plane sides, of a cube with concave sides. But we can study the effect of surface tension to even greater advantage when we diminish the mass of a fluid without decreas- ing the area of the superficial film. This we can do by using soap films or collodion films. Soap Films. Plateau's method. 1 part fresh moist Marseilles soap (much better, pure oleate of soda) cut into very small pieces ; dissolve with moderate heat in 40 parts by weight of distilled water. Filter through moderately-thin filter paper. Mix thoroughly 15 volumes of this solution with 1 1 volumes of Price's glycerine ; let the mixture stand for seven days. On the eighth day cool down to 3 C. for six hours ; a considerable deposit is formed. Filter through porous paper, but take the precaution of placing in each filter along with the solution a little closed stoppered bottle full of ice. This will prevent the re-solution of the precipitate formed by cold. The first filtrate is turpid, but this must be refiltered till it is limpid. Films and bubbles made with this solution last for eighteen hours if kept under a glass shade in very slightly moist air. * Refer to an exceedingly charming work by M. Plateau, Statique dcs Liqutdes soumis aux settles Forces moUculaires, a treasure-house of experiments devised by a savant afflicted with total blindness. xr.] SURFACE TENSION. 243 Collodion Films. (Gernez). Ether 89 parts by weight, absolute alcohol 5.V, photographic gun-cotton 5^ ; dissolve. Decant. To 100 parts by volume of the clear solution add 70 to 100 parts of pure castor oil. This mixture is tenacious enough to permit the use of frameworks 8 cm. in diameter. If a roughened ring be dipped into either of these mixtures, and taken out, a plane film will be found stretched across it. A pipette (whose point has been dipped into the same mixture) may be employed to blow bubbles and place them on this film, and then to enlarge or diminish these bubbles. Such films and bubbles stretch themselves into the most singularly beautiful forms when iron frameworks forming the complete angular periphery of cubes, pyramids, cylinders, and so forth, are substituted for the roughened ring above described ; and these forms may be infinitely varied by modifying the size of the bubbles placed on the films, or by breaking with a hot needle the connection of the film with one or more of the peripherical bounding lines ; in the latter case the most beautiful skew surfaces are formed, all presenting the least area possible under the limiting conditions. If on a simple film stretched over a ring, a piece of silk thread (moistened beforehand with the same solution) be laid in such a way that the thread crosses itself at some one point on the film, and if the film be pierced inside the loop of thread, the loop will fly open and form a perfect circle : for the rest of the film -tends to occupy as small an area as possible. If a drop of alcohol be laid within the loop, the loop flies open in the same way ; although the film is not broken, yet its surface tension is diminished. If a shallow dish containing mercury be tilted over, so that the mercury is on the point of pouring out ; if then some of the mercury be drawn over so as to start a flow, the stream will drag the mercury out of the dish. If a piece of camphor be placed on clean water f it partly dissolves in the water. A strong solution of camphor has less superficial tension than a weak solution, which in its turn has less tension than pure water. At any part of the surface where the solution happens to be more dilute, there the weaker solution, - having the greater tension, pulls the camphor towards itself. The camphor accordingly flies about the surface of the water. This is generally prevented by allowing the finger to touch the water, unless the finger be beforehand specially purified ; so slight a trace of fatty matter communicated to the water prejudices its surface tension to so great an extent. If a drop of ether be sus- 244 OF LIQUIDS. [CHAP. pended by a glass rod close to a thin layer of water, the rest of the water is observed to flee from the spot ; the surface tension of the rest of the water is unchanged, but just under the drop of ether the tension is diminished by absorption of the ether- vapour. A thin layer of water into the centre of which a drop of alcohol is thrown leaves the alcohol for a similar reason. If a glass of strongly alcoholic wine be tilted so as to moisten the side of the glass, the film of wine left will gradually lose some alcohol, and becoming more aqueous it will have a greater superficial tension than the wine ; it will pull itself up the sides of the glass and gather into drops. Measurement of Surface Tension. A soap or collodion film lias two surfaces, and if the film be not too thin, these are independent of one another. Consequently the tension of a film is double that of a single surface of the same fluid and of the same area. The tension of a film can be measured directly. A little framework, consisting of a transverse bar AB, and two grooved slips CD and EF, allows the Fig.ioi. ^ piece of wire GHIJ to slip freely up and down the grooves. The wire is pushed home, and a quantity of the liquid is applied between HI and CE. The little pan X may be loaded with sand until the wire HI is pulled out to a certain distance Cp from AB. When it is in that position, the film has an area CE.Cp. The weight of the total mass suspended on the film and the tension over the area CE . Cp are equal to one another. If the total weight applied be increased, the area assumed by the film is increased in direct proportion. The total weight F is distributed over the breadth CE ; whence, if T represent the superficial tension across unit of length of CE, then F = T . CE. The energy of the film is the work done upon it ; force F has pulled the film through a space Cp : the energy is F x Cp = T . CE . Cp. The energy may also be represented as the product of S (the energy per unit of area) x CE . Cp (the area) = S . CE . Cp. Hence T = S The energy per unit of area (which in the case of a soap film is 54*936 ergs per sq. cm.) is numerically equal to the tension across each unit of length (which in the same case is 54-936 dynes per cm.). These are independent of the form of the film, and depend only on its actual area, not on its curvature. For each single surface T and S, as found by experiment on films, must be halved. This Modulus of Superficial Tension, T, is, in all the instances which we have considered, that of a surface between the liquid and. the sur- rounding air. In the case of pure and per^cly clean water and air, this modulus is equal for each surface to 81 '96 173 dynes per sq. cm t , very nearly XL] SUKFACE TENSION. 245 three times the superficial tension of a single surface of soap solution in con- tact with air. The tension of the bounding surface separating olive oil and air is 36-8856 dynes ; that of the surface between olive oil and water is 20-56176 dynes per cm. At the meeting place of oil, water, and air, these three surfaces meet ; the tension of the water-air surface decidedly preponder- ates, and the edge of an oil-drop floating on water is drawn out indefinitely. If a drop of water be placed on chloroform the respective tensions being water-air 81-96173, chloroform-air 30*6072, and chloroform-water 29-5281 its surface tension (water-air) at first preponderates and pulls it into a drop. When water, air, and clean glass are placed in contact there is again a triplet of tensions, the resultant of which pulls the water over the glass, which is thus wetted by the water. The water tends to stand, so that its surface makes a certain angle with the glass ; this is the angle of capillarity, 180- bet ween -water and glass, 45*30 between mercury and glass. In the case of water this angle is such that the upper surface of water in contact with glass is concave ; in the case of mercury the upper surface is rendered convex. On the curvature of the upper surface, thus determined, depends the direction in which the contractile tension of the superficial film acts. The concave surface of water tends to contract and become flat, and it does so in proportion to the curvature imposed on it. The narrower a capillary tube is, the greater is the curvature of the surface of any liquid standing in it, and therefore the greater is the contractile tendency of that surface. The effect of this tendency is, in the case of water, to exert an upward pull, to neutralise to some extent the downward effect of gravity, and to support a column of water in the tube. Hence water stands at a higher level in a narrow tube whose lower open end is dipped in water than it does in a wider one ; and the height of the column supported is inversely proportional to the radius of the tube at the spot where the curved surface of the liquid is situated. Mer- cury under similar circumstances stands at a lower level. This tendency of a curved surface to exert traction or pressure on a fluid may be seen in a conical capillary tube ; if a drop of water be introduced, the smaller concave surface will pull the drop towards the apex ; if a drop of mercury, the smaller convex surface will push the mercury from the apex. Capillary phenomena are thus phenomena of surface tension ; and it is futile to explain the rise of sap in plants or the passage of fluids through minute vessels by "capillary attraction" when there is no free surface. An experiment which may, on the other hand, illustrate these movements, consists in oiling the interior of an open capillary tube, filling it with water and dipping the end of the tube in oil ; the attraction of the sides of the tube for oil will cause the oil to run along the tube and to drive out the water ; this, however, is not an exclusively capillary phenomenon. If two plates of clean glass be set to stand vertically and 246 OF LIQUIDS. [OHAP. parallel to one another in a shallow dish of water, the water will rise up the sides of each to a height half that which it would attain in a tube whose diameter is equal to the distance between the plates. If the two plates have two vertical edges in contact, the liquid will rise indefinitely where they are in contact ; at other parts it rises to a height inversely proportional to the local distance between the plates, and it thus presents the outline of an equilateral hyperbola. Just as the surface of a liquid is raised against a fixed plane of clean glass, so a floating vessel of clean glass may by surface tension be pulled down so as to lie more deeply in the liquid than its specific gravity would lead us to expect. A floating hydrometer always gives an abnormally low reading on this account ; it is pulled into the liquid, so that the liquid appears to be lighter than it really is. If a little vapour of ether be poured on the surface of the liquid so as to diminish the surface tension, the hydrometer rises. If the water have any grease on its surface, the same effect follows. If the hydrometer be greasy, it is repelled and stands abnormally high in the liquid. Hence great confusion and in- accuracy may result from films of grease 011 the glass or on the fingers of the manipulator. Objects which are wetted by the liquid in which they float are thus apparently attracted by it; those which are not so are apparently repelled. Two wetted objects floating on water seem to attract one another ; two objects floating on a liquid which does not wet them both seem also to attract one another. This may be seen by throwing upon the surface of water a number of wooden balls, of which some are smoked with lampblack, while others are purified first with soap and water, then with pure water; the smoked balls approach each other, the clean ones approach each other, but the clean balls appear to avoid the smoked ones. We may mention another consequence of surface tension. A jet of water issuing from a rectangular orifice is most acted upon by surface tension at its narrow edges. These are pressed together ; they meet, and when they do so, spread out laterally ; the same action is repeated, and the whole jet is a succession of flat segments at right angles to one another. At first sight such a jet seems to have a screw form. The distances at which molecular forces act are not immea- surably small. Quincke found that while water stands against glass at one angle, against silver at another angle of capillarity, XL] SUPERFICIAL TENSION AND VISCOSITY. 247 yet against glass coated with silver it stands at such an angle as to show that the influence of the glass is felt through the silver when the layer of silver is less than "000,005 cm. thick; this thickness being one-tenth of the average length of a wave of light. Superficial Viscosity. This is a property of the superficial film of liquids which is independent of the surface tension. If a magnetic needle be so adjusted as to have its lower surface in contact with the surface of a solution of saponine, it will remain in any position in defiance of the directive force of the earth's magnetism. On the surface of most other fluids it will move into the magnetic meridian, but the whole superficial film of the liquid will move with it, as may be shown by strewing lycopodium over the surface. The superficial film is, as a rule, exceedingly viscous as compared with the interior mass ; it is consequently hard to move or to break. If a liquid have great superficial viscosity and small surface tension (as in the case of soap and water), a bubble rising through the liquid may raise the surface film, which the tension is not able to break : the bubble may therefore persist. If a wire ring, bearing a soap film, be swept rapidly through the air, the air may fill and stretch the film, and separate part of it in the form of a complete bubble. A bubble rising with very great rapidity through a fluid may tear off some of the viscous superficial film and form, a complete bubble : this is seen when a mixture of olive oil and strong sulphuric acid is vigorously stirred. Pure water ^-has great surface tension, which is able to overcome the superficial viscosity. Thus pure water will not froth. Some liquids, such as a solution of gum arabic or of albu- men, will froth when shaken, but their superficial viscosity is not sufficiently great to enable bubbles to be blown with them. Alcohol, sulphuric ether, sulphide of carbon, and some other liquids, haVe a superficial viscosity less than their internal viscosity, and consequently, when alcohol is mixed with a superficially viscous liquid, it neutralises its relative superficial viscosity, and frothing is rendered impossible. Hence the practice of adding a few drops of spirit in order to check frothing in pharmaceutical operations. To this toughness of the superficial film, the floating of an oiled needle or the walking of an insect on water must be in part ascribed. The depth of the dimple produced by the needle is not sufficient to account, by displacement, for the support afforded to so heavy a body : the superficial tension is diminished by the oil : the tenacity of the surface film plays its part in supporting the needle. To the same cause we must attribute the smoothing of 248 OF LIQUIDS. [CHAP. the surface of a rough sea when oil is poured upon it : the new surface has great superficial tenacity and small superficial tension, and is not readily broken up into surf. A bubble in bursting does so with an audible sound : it scatters particles of its substance and of the contained gas to a height of three or four feet ; this happens during the effervescence of sewage which is undergoing fermentation. Cohesion Figures. If the surface of a tumblerful of salt water (j- teaspoonful to the tumbler) be touched with a pen not too full of ink, the ink will, in falling through the liquid, assume very remarkable vortical movements. A shower of rain falling on a troubled sea produces similar vortex rings, which are carried down into regions of comparative stillness, and moderate the tur- bulence of the water by equalising its distribution of momentum. The forms assumed by drops of water, or of mercury falling on a flat surface, at the instant when they spread out and break are very remarkable, and may be seen when the spreading drops are momentarily illuminated by the electric spark. The edge of the spreading drop breaks up into thinner and thicker nodes and loops which vibrate : very roughly the result may be seen in a cooled splash of candle wax. Solubility of Solids in Liquids. When a solid is dissolved in a liquid, work is done in overcoming its cohesion. This co- hesion is overcome by the adhesion between the solid and the liquid. Ice put into sulphuric acid has its superficial particles successively torn off, and a mass of dilute sulphuric acid (which on account of liquefaction assumes a low temperature unless heat be supplied) is produced. Such union may or may not be asso- ciated with a play of chemical affinities ; in the case of ice and sulphuric acid there is a tendency to the production of definite hydrates of sulphuric acid, the formation of which is accompanied by the evolution of a certain amount of heat. If sulphate of mag- nesia be placed in water it will be dissolved to a certain limited extent; if the salt be added in excess above this limit it will not be dissolved ; when this limit has been reached the solution is a saturated solution. This limit is expressed by the co- efficient of solubility, a number indicating the quantity of solid which can be dissolved and remain in solution in unit-mass of the liquid at the particular temperature for which the coefficient is or ought to be specified. A saturated solution can dissolve no further quantity of the same salt, for the adhesion of such a solu- tion to the salt is no longer greater than the cohesion of the salt XL] SOLUTION. 249 itself. If the cohesion of the salt be lessened by heat, more may be dissolved ; and as a general rule, with but few exceptions hydrate of lime, sulphate of soda, phosphate of magnesia salts are more soluble in hot than in cold water. The adhesion of a liquid to the solid which it holds in solution may be relatively great or feeble ; and its relative amount may be indicated, though not measured quantitatively (1) by a high or low coefficient of solubility ; (2) by the amount of energy which must be imparted to the molecules in order, by boiling, to tear the water away from the salt, or, in other words, by the high or low boiling point of a saline solution ; (3) by the relative effect of charcoal filters in retaining the salts of a saline solution while allowing the water to pass, a property made use of in the analysis of poisons ; and sometimes (4) by the detachment of the liquid from the solid by a stronger molecular attraction, as in the case of iodide of starch, a solution of which is precipitated by acetate of potash, the water leaving the iodide of starch and adhering to the salt. When a saturated solution is cooled, the adhesion between the liquid and the solid diminishes, the coefficient of solubility dimi- nishes, and the solid segregates in a separate form : thus hot satu- rated solutions may be set aside to cool, and on cooling they cry- stallise, the materials dividing into crystals of the salt and an ordinary cold saturated solution of the same. Sometimes, as in the case of sulphate of soda, such a solution (though cooled down to a temperature at which it cannot permanently retain all the salt which it holds in solution) does not crystallise, but forms a supersaturated solution. Such a solution is in a state of un- stable molecular equilibrium, and the instant it is touched with a crystal of the same salt or of an isomorphous substance, or by the dust of the air containing the same substance, or by an oil (especially if somewhat oxidised), or by a bubble of gas soluble in the liquid, or when it is exposed to the least vibration, the whole molecular arrangement topples over, and the excess of salt assumes the solid form with great evolution of heat. A similar delay in solidification occurs in the case of melted phosphorus, which can be kept fluid at 10 C. (its solidification point being 44 *2 C.) for weeks, especially if the water lying above it contain a trace of potash hydrate or of nitric acid. The slightest shake or contact with a piece of phosphorus determines solidification. Miscibility of Liquids. If a bottle be filled with oil and water, and shaken, the layers separate as soon as the disturbance ceases. Alcohol and water treated in the same way mutually 250 OF LIQUIDS. [OHAP. dissolve each other, and mix perfectly in any proportions. Ether and water will each take up a certain proportion of the other, and will separate into two layers, the one a saturated aqueous solu- tion of ether, the other a saturated ethereal solution of water; these two fluids are miscible only in certain proportions. Very often, as in the case of alcohol and water, there is a contraction of volume and evolution of heat, there being some potential energy of separation somehow liberated by the approximation of mutually attracting molecules of the different substances. Imbibition. Porous objects, such as a lump of sugar, blotting paper, a heap of sand, a sponge, a lamp wick, absorb liquids with a rapidity which depends on the nature of the porous substance itself and on that of the liquid absorbed, and which is greater if the materials be warm. This takes place by reason of an attraction between the solid and the fluid (which Chevreul called affiniU capillaire), and heat is evolved when this attraction is satisfied, as in the case of a wetted rope, which rises in tempera- ture from 2 to 10 C. (part of this effect being due to the con- current shrinkage of the rope). When a porous body which has thus taken up a quantity of liquid is subjected to pressure, the whole of the liquid can by no means be squeezed out ; some water still remains, which can be evaporated away. Imbibition will fill the pores of a solid with a fluid, but will not set up a permanent current in those pores unless, as in the case of a lamp-wick, there be constant removal of the fluid at one extremity of the porous object while imbibition goes on at the other. Diffusion Jar-diffusion.* If a phial, filled to within say half-an-inch of the top with a saline solution, be placed in a jar; if water be poured into the jar so as to surround the phial, and if more water be cautiously added till the phial is covered with a layer of water of about half-an-inch in depth, the whole being set aside in a quiet place, the solution in the phial will diffuse into the surrounding water. The quantity of substance diffused into the water in a given time depends (1) on the length of that time ; (2) on the quantity of substance in the phial solution, being (within narrow limits) proportional to its strength ; (3) on the temperature, being greater at a high temperature ; (4) on a Co- efficient of Diffusibility special to each substance. Other things being equal, urea and salt diffuse twice as fast as sugar, sugar twice as fast as gum arabic, gum arabic more than four times as fast as egg-albumen. * See Graham's Chemical and Physical Researches. XL] DIFFUSION. 251 Sugar travels as far in a column of water in two days as albumen in four- teen. The following numbers indicate the relative times necessary for the process of diffusion to convey in water through equal distances equal amounts of the several substances : Hydrochloric acid, 1 ; chloride of sodium, 2*33 ; sugar, 7 ; sulphate of magnesia, 7 ; albumen, 49 ; caramel 98 (Graham). The rate of diffusion of all substances is increased by moderate heat, but in those substances whose coefficient of diffusibility is small, it is more in- creased by heat than it is in those substances which are already very diffusible. Hence the greatest proportionate differences in diffusion-rates are found in the coldest solutions. Some liquids, such as water and sulphuric acid, ether and chloroform, mercury and molten lead, diffuse into one another with considerable rapidity. t If a mixture be placed in the diffusion-phial, the approximate rule is that each component of the mixture is diffused out at its own rate, and independently of the others. There is, however, a departure from strict adherence to this rule in the sense that the ordinary differences of diffusibility are exaggerated in such a mixture. If the phial contain a double salt, such as alum, diffu- sion may effect chemical decomposition : sulphate of potash and sulphate of alumina are separated, the former being diffused more rapidly. A high boiling-point of any solution (which indicates adhesion of water to the salt dissolved) is associated with rapid diffusibility of the same salt ; but there is no close relation between the rapidity of diffusion of a salt and its solubility. Colloids and Crystalloids. On surveying a number of objects which have a wide range of relative diffusibilities, we see that at one end of the scale we have such things as urea and chloride of sodium, and at the other such things as starch, gum, gelatine, albumen. The former are bodies of rapid diffusibility, have generally a certain chemical stability and a crystalline form, and are called Crystalloids* The latter are bodies of slow dif- fusibility, have very probably large molecules compounded of groups of their simplest molecules, have in general (with rare exceptions, such as the blood-crystals) the non-crystalline amor- phous glue-like character which gives them the name of Col- loids, and are for the most part in a state of unstable equilibrium when in the moist condition. Colloids have great cohesion, and adhere firmly to other colloids : thus isinglass heated with acetic acid forms a cement which adheres firmly to glass ; and when they dry they tend to contract firmly,. so that a strong solution of gum arabic, drying in a test tube, will sometimes break the tube. They often, when drying up, extrude their contained water, and form clots, on the surface of which the water presents 252 OF LIQUIDS. [CHAP. itself in drops. Colloids also in many instances possess the power of taking up alcohol or olein in the room of their water of constitution. This property is possessed even by such a sub- stance as colloid silicic acid. An animal tissue, which is in great part composed of colloids, may have its water expelled and replaced by alcohol by dint of repeated washing in that liquid. Colloids being very slightly diffusible are tasteless ; they do not reach the nerve-ends. For the same reason they are very indigestible e.g., gelatine unless peptonised ; peptones being, by exception, diffusible though other- wise colloidal. If a layer of pure jelly be laid on a layer of jelly charged with salts and also with caramel, the salts will diffuse into the upper layer of solid jelly nearly as fast as if it were pure water ; the caramel will not travel at all. If a film of colloid matter (starched paper) be placed between a saline solution containing colloid matters and a mass of pure water, the colloid septum will offer little obstruction to the passage of the salts into the water, but will prevent the colloid matter from passing through. Col- loid matter is thus impervious to the diffusion of other colloids, but does not hinder the diffusion of crystalloids. Diffusion through Membranes Osmosis. If three layers of liquid, chloroform, water, ether, be placed in a closed bottle and set aside, it will be found that in course of time the ether travels into the chloroform, but that the water does not to any appreciable extent allow the chloroform to pass into the ether. The ether dissolves to some extent in the water and diffuses through it ; it is removed from the water by the chloroform : step by step the water passes the whole of the ether through its own substance. A thin caoutchouc membrane lying between alcohol and water allows the alcohol to pass through it into the water ; but the reverse passage of water into the alcohol is barred. If an organic septum be used it is wetted, and the water passes into the alcohol. If hydrochloric acid and water be separated by an animal membrane, the hydrochloric acid passes through in greater quantity : both fluids wet the membrane, the hydrochloric acid is most attracted. Hence molecules can travel through a septum devoid of perceptible pores as well as through one in which pores exist. If the membrane employed be porous we have the process of Osmosis. The membrane becomes penetrated by that one of the two liquids (" liquid A ") for which the walls of its pores XL] OSMOSIS. 253 have the greater attraction or affinity. When each small capil- lary column of the liquid A comes at the further surface of the membrane into contact with the liquid B, the molecules of liquid B diffuse into it. Thus the end of the column of liquid A comes more to resemble that liquid B, which is less attracted by the walls of the pore, and it is extruded from the pore and pushed into liquid B. This process is continuous, and a stream of liquid A is drawn through each pore of the membrane into liquid B. Down the centre of the stream there is, however, a backward diffusion -current of molecules passing from the liquid B. This happens if the pores be wide enough to allow the centre of the stream to be comparatively out of reach of the immediate influence of the walls of the channels, an influence which we have seen to extend to a distance of ^ o OQ mm. or 2 5 o^) o i ncn - If the liquid stream be not too rapid, these molecules will make their way against it into liquid A. If the channels be very nar- row the liquid stream is relatively accelerated ; thus the ratio between the amount of water that passes through a porous mem- brane into a saline solution and the amount of salt that passes in the opposite direction is increased by diminution of the pores. This ratio is called the Endosmotic Equivalent. It is not a constant, but depends on the nature of the membrane, and even with the same membrane it differs according to its thickness or state of freshness, and may be increased by tanning with tannin or chromic acid, which diminish the size of the pores. Thus for a membrane on one side of which is dry common salt, on the other side water, if the membrane be a piece of cow's pericardium, for every grain of salt which passes into the water, 4 grains of water pass into the salt ; with a piece of cow's bladder, the endosmotic equivalent is 6. If on one side of an animal membrane there be placed a strong solution of sulphate of magnesia and on the other a quantity of blood serum, the fluid of the blood serum will pass into the saline solution, taking some albumen with it, and some sulphate of magnesia will pass into the blood serum. (Milne-Edwards.) The mechanical structure of the membrane has a marked in- fluence on the process ; thus water will pass more readily inwards through frogskin, more readily outwards through eelskin. The matters already moistening the membrane also affect the rate of transmission ; thus albumen more readily passes through a membrane previously moistened with alkalies. If between alcohol and water there be arranged a membrane previously soaked in oil, the membrane cannot be wetted, and the alcohol now passes into the water. 254 OF LIQUIDS. [CHAP. If the saline solution be in a state of movement relative to the membrane, the particles are drawn away from the membrane, and the diffusion-stream is hindered ; if the water into which the salts are passing be constantly renewed, the molecular diffusion is accelerated. Heat increases the rapidity of Osmosis. An electric current (the " electrodes " being on opposite sides of the membrane) has the singular effect of, as it were, pushing the liquid bodily through the membrane towards the negative electrode. Even gelatine and the fatty matters of milk can be thus driven through a membrane. If a mixture of different substances be exposed to osmosis through a porous membrane, the colloids will remain or will pass through in very small quantities, the crystalloids pass through freely. This is the basis of the process of Dialysis. Various mechanical arrangements for carrying out dialysis suggest them- selves : a phial with the bottom cut off or a wide glass tube, over the lower end of which a piece of membrane is stretched; the material to be dialysed being placed in this, and the whole sus- pended in water. The most convenient arrangement in many respects is a piece of parchment paper (the leaks in which are stopped with albumen coagulated by heat) laid upon a wooden ring, into which a smaller ring is thrust so as to form a dish with a membranous bottom ; this is floated on a mass of water, and the substance to be dialysed placed in a thin layer on the dish. The crystalloids (strychnine, etc.) pass into the water, the colloids (mucus, etc.) remain in the dish. This method is peculiarly applicable to the separation of poisons from animal matters. If the mixed solution exert pressure upon the membrane, colloids as well as crystalloids may be found to pass in consider- able quantities through that membrane along with the fluid forced through by the pressure. If peroxide of iron be dissolved in a solution of perchloride of iron, and the whole be then dialysed, the chloride of iron will pass through the mem- brane, leaving the colloid oxide of iron behind in solution. Neutral Prussian blue (as used in microscopical work) is also a colloid, and may be purified in the same way : so is sucrate of copper, a soluble compound of copper oxide with sugar, which is reduced on heating. Albumen may also be obtained in a relatively pure form by separating it by dialysis from the greater part of the salts that it may contain. If the membrane used be the gastric or intestinal membrane, taken after death, it is found that curare or snake poison will not pass through it, while they are absorbed readily by the dermis or by serous membranes. They seem not to wet the former ; hence the selective absorption of poisons has a certain physical basis. XL] OSMOSIS. 255 Absorption by the dermis is seen to be a physical process ; the walls of the vessels both lymphatic and venous are known to be physically permeable to osmose, and the salt being placed on a vascular region is quickly absorbed, the osmose being accelerated by the flow of liquid in the vessels. Substances brought in contact with the pulmonary epithelium are also very rapidly absorbed. Lymph acts towards blood as water does towards a saline solution, and the tendency of osmotic action is to carry the fluids of the body into the blood stream. Repletion of the vessels checks this tendency. Adhesion between water and oil is greatly increased if a little alkali be dissolved in the water. When the mucous membrane is covered with bile it has much more affinity for oil globules, which are, besides, each endowed by emulsionising with an aqueous or soapy covering, which makes them act like minute masses of water, and enables them not to experience any relative repulsion when carried with the rest of the aqueous stream. Osmose is thus related to capillary affinity and to diffusion, but it bears no exact numerical relation to either of these, for it depends on the relation between the pores and the solid parts of the membrane, upon the nature of the material (colloidal or other- wise) of the membrane, upon the width of the pores, upon the temperature and electrical condition, upon the mutual action of the fluids, and in physiological cases (Milne-Edwards, Physiologic, tome V) it seems to depend on the influence of the nervous system. 2. THE STATICS OF LIQUID MASSES. Liquids are incapable of resisting a change of shape when acted on by force which is not equally applied over the whole surface, and they flow when thus acted on unless supported on all sides. All soft masses which cannot in the aggregate permanently resist a change of shape are practically liquids and are subject to hydrostatical laws. It is often convenient in discussing the equilibrium of liquids "to imagine little elements of the liquid, floating in and forming part of the liquid, to be solidified or otherwise to become separ- ately recognisable while not altering their other relations to the surrounding mass. Then, if the liquid as a whole is at rest, each of these little elements of mass must also be at rest. This being so, the forces acting on each little element of mass must be in equilibrium, and their resultant must be nil. This can only occur (since each fluid element is subject to pressure on all sides, as may be understood by considering the rush of fluid from all sides that would occur if the little element of mass were sud- denly annihilated) if the pressure on all sides be equal ; and since 256 OF LIQUIDS. [CHAP. the element may be reduced to a material point, the proposition follows that at any point in a liquid the pressure in all directions is equal. The pressure at any point of the surface of a liquid at rest must be at right angles to the surface. If it were not so, it must be oblique ; being oblique, it would be resoluble into a component at right angles and one parallel to the surface. The latter could not fail to act, the surface being that of a liquid ; hence the liquid would not be at rest ; whence there is no such component, and the pressure is at right angles to the surface. Conversely, when a liquid is at rest, the pressure which it exercises on the vessel con- taining it is at right angles to the walls of the vessel, for the walls of the vessel coincide in aspect with the surface of the liquid. If in a liquid at rest, expressly supposed to be neither under the influence of gravity nor subjected to rotation, two elements were imagined to be in contact, and yet to be subject to different pres- sures, there would at the point or surface of contact be a relative difference of pressures which would necessarily cause movement of the liquid ; but the liquid is supposed to be at rest ; hence there can be no difference between the pressures of any two contiguous elements, and the pressure throughout a weightless liquid at rest is everywhere the same (Pascal's Principle). If pressure be applied from without to some of the particles of a liquid, and if that liquid be free to change its shape, it will do so ; if it be not free to flow, the particles pressed on will press against contiguous particles, and these against their neighbours ; thus the pressure becomes equalised throughout the whole of the liquid. This is the principle of the so-called Transmissibility of Fluid Pressures. The pressure applied to any area of the surface of a liquid not free to flow becomes equally felt over every equal area of the surface. If a wide cylinder with a piston whose area is a square inches be placed in communication by a tube with another cylinder, narrower, and provided with a piston whose area is b square inches, and if both cylinders and the communicating tube be completely filled with water, a pressure p applied to the smaller piston will produce an equal pressure p on every & square inches of the sur- face of the fluid, and therefore on every I square inches of the larger piston and a proportionately greater pressure - p on the whole surface (a sq. in.) of the greater piston. This is the prin- ciple of the Hydraulic Press, by which a smaller force p acting T? -) XI.] HYDRAULIC PRESS. ^J s\ 257 on a smaller piston may produce a greater force p on & surface of a larger piston ; and as the area a may bear any pro- portion to the area b, the force obtained may bear any proportion to the force applied. The principle of the Conservation of Energy holds good, however ; the volume of water remains constant, and if the smaller piston move through a space s, the larger piston moves through a space - 5. The work done upon the smaller a piston, force x space = j>s; that done by the larger piston, _ p x - s, gives the same product. b a An analogous action takes place in an aneurism. A small aperture of communication with the artery allows the arterial blood-pressure to be com- municated to the whole interior of the aneurismal sac ; the total pressure exerted is very great, the rate of distension comparatively slow. If the action of a hydraulic press be reversed, a great total pressure applied to the larger piston will have the effect of pro- ducing only a small pressure on the inner surface of the small piston. A small resistance applied to the smaller piston will have the effect of checking the onward motion of the larger piston under the influence of the powerful force. If a bladder full of water be connected with a narrow upright glass tube, heavy weights placed on the bladder will be able to uphold only a very small quantity of liquid in the tube, this arrangement being in fact a hydraulic press worked backwards. If the tube be shortened down so as to form simply the neck of the bladder, the total ex- pulsive pressure exerted by the bladder upon the contents of the neck may seem to be very small when compared with the total pressure exerted over the walls of the bladder upon the whole contents. Here we have apparent destruction of force. Heavy Liquids. Let us suppose a cylindrical vessel, filled with liquid, to stand upon a plane base ; the area of the base is a sq. cm. ; the height of the liquid is b cm. ; the density of the liquid is p ; and the local acceleration of gravity is g. The quan- tity of matter standing on the base is abp, and the weight of that mass is abpg. The total pressure on the base is therefore abpg, and the pressure per unit of area of the base is bpg. If the unit of area on which the pressure is to be found be not horizontal, it must be considered to lie at an average depth equal to the depth of its centre of figure. Then the pressure on s 258 OF LIQUIDS. [CHAP. a plane of unit-area, chosen anywhere in the fluid and looking in any direction, is equal to the product of pg into the vertical dis- tance I between the surface of the liquid and the centre of figure of that plane ; and if the plane have any area a other than unity, the pressure is the product of the area a x I, the vertical depth of the centre of figure, x p, the density, x g. For all points in the same horizontal layer the depth 5 is the same, and therefore in a heavy fluid the pressure is the same throughout the same indefinitely-thin horizontal layer. The lateral pressure on the rim of the stratum is equal to the vertical pres- sure at that level i.e., Ipg per unit of area. Fig. 102. B In Eig. 102 let A, B, C represent three vessels each having a base whose area is a square inches, and filled with water to a height of I inches. The pressure on the base is the same ( = abpg) in all the cases, though the weights of the masses of water differ greatly. In the first case the lateral pressure against the walls of the cylinder produces a reaction which has no vertical component and does not affect the pressure on the base. In the second we may isolate a cylinder of the fluid in the fluid ; the lateral parts of the fluid have a certain weight : the walls of the vessel are exposed to a certain pressure which is equal to the product of their area x the depth of their centre of figure [.-] into/o^. This pres- sure may be resolved into a horizontal and a vertical component, to each of which the corresponding reactions of the walls of the vessel are equal and opposite : the one resists outward yielding, the other supports the weight of the fluid. It will be found that the upward reaction of the sloping walls of vessel B is exactly equal to the weight of the fluid overlying them ; the walls support the whole weight of the lateral masses. In vessel C the reaction of the walls of the vessel may be found in the same way and resolved into horizontal and vertical components. The latter acts downwards upon the fluid, and will be found to be precisely equal to the weight of that quantity of fluid that would lie vertically above the base if the column of fluid were perfectly cylindrical and of the height 6, but which, owing to the form of the vessel, does not so lie. The pressure on the base of a vessel containing liquid depends XL] PRESSURE IN HEAVY LIQUIDS. 259 on the height (b) of the liquid and the area (a) of the vessel, the density p of the liquid, and g the local acceleration of gravity, but does not depend on the actual weight of the liquid employed. The pressure P = abpg. If a flask filled with water be fitted with a cork in which a long narrow tube is fixed upright, a very small quantity of water poured into the tube will be competent to burst the flask This proposition that the same amount of water may pro- duce widely-differing amounts of pressure on the vessel in which it is contained, these amounts depending on the form of that vessel is said to be a Hydrostatic Paradox ; the only paradoxical element about it is, however, its discrepancy with a certain uninformed intuitional belief in the Conservation of Force. A slack bag containing liquid, and set to rest upon a plane surface, exerts a pressure upon that surface which is equal to the product of the area of contact x the height of the centre of gravity of the liquid. So for semi-fluid masses. When the surface of a liquid is exposed to the atmospheric pressure of 7 6 mm. or 7 6 cm., it bears on each sq. cm. of surface the weight of 7 6 cub. cm. of mercury, or 1033'3 grammes; this is equal to (1033'3 x 981) dynes: or if the barometer stand at x cm., the pressure on each sq. cm. of surface is (13'596ic x 981) dynes. This number of absolute units of force per sq. cm. may be expressed by the symbol II. Then the total atmospheric pressure on area a is all. The liquid pressure on the area a at the mean depth b would have been abpg if there had been no pressure at the surface. When the atmospheric pressure acts at the surface of a liquid, the total pressure on any plane, whose area is a and whose mean depth below the surface is b, amounts to (all + abpg). When the human body (as in ordinary circumstances) has the head in the highest position, the blood in the head is exposed to the ordinary atmospheric pressure. If the head be downwards, the pressure on the blood vessels of the head is increased by the weight of the column of blood in the inverted body, and hence there is congestion. If the body float submerged in a liquid of its own sp. density, head up, the pressure on the blood vessels^of the head is the ordinary atmospheric pressure increased by the weight of the column of water immediately overlying the head ; but if "the head be suspended, though the increased depth cause a correspondingly-increased external pressure on the head, yet the equally-increased internal pressure of blood balances this effect, and there is no congestion. This may be illustrated by a loop of thin indiarubber-tubing filled with water : suspended in air, the depending part is distended : suspended in water, it is relieved from distension. 260 OF LIQUIDS. [CHAP. Pig.ios. Communicating Vases. " Water seeks its own level." If there be two communicating vessels containing the same liquid, the lowest part of the communicating channel may be considered as a common base : its area is a. Eegarding it as the base of vessel A (Fig. 103), we see that the pressure on it must be abpg, and the height of the liquid in A is I; regarding it as the base of vessel B, the pressure (which must be the same, for the liquid is at rest) is equal to ab^pg : whence b = b^ the height of the liquid in the two vases must be the same, and the level must be the same in two communicating vases, whatever be the shape of the communication, so long as the communication -pipe is con- tinuously filled with liquid. This implies that sufficient time for assuming equilibrium is allowed. If the fluids in the two communicating columns be not of the same sp. density, the effect is an inequality in the heights of the columns, which vary inversely as the sp. density. The two pressures are abpg and ob^g ; these are equal ; . *. bp = b-^p v or b : ^ : : p 1 : p. In Fig. 104 the column of water ab and that of mercury cd balance one another because they produce an equal pressure on the base e. If a U-tube contain water, of which that in one limb is heated while that in the other remains cool, the liquid in the hotter limb will stand at a higher level than that in the cooler. The relative specific densities of fluids may be estimated by methods based on this principle. The accuracy of the " water-level " may be inter- fered with by capillarity. If both limbs of a U-tube be narrow, but unequally so, the liquid will stand at -'L a greater height in the narrower limb. If a U-tube be taken of which the narrower limb is the shorter, the quantity of water placed in the tube may be regulated so as to afford the following three conditions : (1) The shorter limb filled with water, the upper surface of which is concave, and the water standing at a lower level in the wider tube ; (2) The shorter limb com- pletely filled with water the upper surface of which is plane, and the concave surface of the water in the wider tube at nearly the same level, but a little higher ; (3) The shorter limb completely filled with water the upper surface of which is convex, and the water standing at a higher level in the wider tube, its surface being concave. Fig. 104. XL] COMMUNICATING VASES. 261 Every liquid tends to set the whole of its surface at right angles to the force of gravity. When a cylindrical vessel containing a liquid is rotated round its longitudinal axis, the surface of the liquid assumes a parabolic form which is maintained constant as long as the rotation is uniform. Thus the form of the free surface of liquids is affected by gravity, by molecular forces, and by rotation. Archimedes's Principle. If an element of mass of a liquid be supposed to be solidified, this will not affect its equilibrium in the midst of the fluid of which it had previously formed part ; it will neither rise nor sink. Even though its nature be altered, provided that it does not become either lighter or heavier, it will neither sink nor rise : it has apparently lost its weight. If it become heavier than the liquid it will sink ; if it become lighter it will rise. Gravity has no effect in making a body rise or sink in a liquid except in so far as there is a difference between the density of the liquid and that of the body suspended in it. This leads to Archimedes' principle: "A body suspended in a fluid apparently loses as much weight as is equal to the weight of the mass of fluid which it displaces." The application of this principle to the study of sp. density we have already seen. A body lighter than water may be loaded with just so much mass as will sink the light body without that additional mass itself entering the liquid ; the whole will then float, the lighter body displacing a bulk of water equal to its own bulk; the weight opposing the buoyancy of the water being the weight of the body plus that of the load placed on it ; and the ratio weight of body , , . , , = sp. density of the floating body. weight of body + load That a body, even though sufficiently light to float, tends to sink in water until the weight of the water displaced becomes equal to the weight of the whole body, may be shown by a very simple experiment. Take two similar phials, two small elastic bands, and four nails which must not be too heavy. With these may be constructed a couple of rough models representing a person with his arms kept down by his side, and a person whose arms are elevated above his head. On putting these models into water the difference of floating capacity will be very obvious. Measurement of Pressure. The pressure to which the surface of a liquid is exposed can always be measured by the height of the liquid column which that pressure can support. If 262 OF LIQUIDS. [CHAP. in Fig. 105 the water contained in the cylinder AB be exposed to a certain pressure communicated by a piston at A, and if a Fig. 105. Writing Point side tube (a piezometer tube) placed at C be in communica- tion with the liquid, water will rise in the tube until there is equilibrium. This equilibrium is between the pressure of the fluid in AB (together with the atmospheric pressure acting through A), tending to push upwards the column of water CD, and the weight of that column which (together with the atmospheric pressure acting on D) tends to make it sink back into the cylinder. The outward pressure P exerted by the liquid on the orifice C must be equal to the weight of the column CD. The latter is (if the area of the orifice at C be ab, and H the height of the column) equal to g x the mass of the column = ab . Hp#. As this is distributed over the area ab, on every unit of area of the surface of the fluid its amount must be tt.pg. Whence P = Hpg. Let us suppose that the column CD is 13*596 cm. high ; the pressure per unit of surface is Hpg = (13*596 x 1 x 981) dynes per sq. cm. If at B a U-tube (a manometer tube) be fixed, containing in its bend a quantity of mercury, the mercury will stand at the same level in both branches as long as the internal pressure and the external are equal ; but if the internal pressure be increased, the mercury will be depressed in the branch nearer the cylinder, and will rise in the other. In the case supposed it will sink through J cm. in the nearer and rise through ^ cm. in the further limb : a difference of 1 cm. of mercury being thus established. This column of mercury is that whose weight balances the internal pressure : its weight is (1 cub. cm. x 13*596 x 981) dynes, acting upon every square centimetre. Hence The pressure on the surface of the liquid in the cylinder, AB of Fig. 105, may be equally well represented in brief phraseology as a pressure of 13*596 cm. of water, or one of 1 cm. of mercury. XL] PRESSURE IX LIQUIDS. 263 Exploration of the pressure in the interior of a sta- tionary liquid mass. In Fig. 105 let there be an aperture in the walls of the cylinder at O ; through this aperture pass a tube which exactly fits it. The inner end of this tube is furnished with a flexible and elastic cap. The outer end is connected directly or by means of indiarubber or, better, of leaden tubing first with a stopcock (the bore of which is the same as that of the tube), and then with a manometer tube. Before the tube is passed through the orifice the level of the mercury in the manometer must be adjusted. This is done while the stopcock is open, by pouring mercury into the manometer tube and bringing it to an exact level by the addition or subtraction of mercury in the outer limb ; the stopcock is then closed, and the tube adjusted with its elastic closed end in the body of the cylinder. The stop- cock is then opened ; the pressure of the fluid in the cylinder on the indiarubber cap (if it differ from the atmospheric pressure) alters the shape of tfre_ cap, and the mercury in the manometer assumes a difference of levefWhich indicates the pressure in the interior of the cylinder. If\^he cap be so small that it is collapsed by a given pressure P, it cannot be used to record pressures of greater amount than P. This defect can be remedied either by using a larger cap or else by using capillary manometers of uniform bore, in which the displacement of a very small quantity of mercury (and therefore a small compression of the indiarubber cap) will serve to indicate high differences of pres- sure. If the cap be at all inflated before it is inserted within the cylinder, the elastic recoil of the cap adds an un- known quantity to the internal fluid pressure, and the readings of the instrument are untrustworthy, unless special contrivances are made use of for ascertaining the exact effect due to this cause. Figure 1 5 F shows the essential parts of another instrument by which the pressure in the cylinder AB may be measured ; it is " substantially identical with Bourdon's Steam Gauge. A hollow tube of elastic metal, having an elliptical cross -section bent into the shape of a Q, and filled with liquid (alcohol, glycerine, water, or oil), suffers changes of shape under the influence of changes of pressure in the contained fluid. "When the internal pressure increases, the Q straightens out ; when it decreases it becomes more curved. The pressure increasing, the cross-section tends to become more circular (the circle being a figure of greatest area for least circumference) : the surface 264 OF LIQUIDS. [CHAP. and the mean curvature are constant ; the curvature across the tube increasing, that along the tube diminishes, and the tube straightens out. Such a tube is continuous with a box or cavity containing liquid, which may in its turn be continuous with the liquid of the cylinder when the surface pressure has to be found, or may be connected merely with an indiarubber cap like that inserted as an explorateur in orifice of the same figure. For physiological work this principle is applied in Pick's Federinano- ineter, in which the Q-tube is filled with alcohol, and the tubes which intervene between it and those blood vessels in which the blood pressure has to be determined are filled with a solution of bicarbonate of soda of a sp. gr. of 1-083 ; this being (Cyon) the strength of solution which most markedly checks any tendency to coagulation. A given amount of bend of the Q-tube ma y be interpreted as signifying exposure to a certain amount of pressure, if the instru- ment be previously graduated by finding the relation between certain known pressures and the distortion produced by them. The instrument S in Fig. 105 is the sphygmoscope of Marey. The tube a is closed by an elastic cap which projects into the lumen of the wider tube I ; a and its cap are filled with liquid, which is continuous with that of the cylinder ; the pressure within the cylinder forces the fluid into the cap until the elas- ticity of the cap and the pressure of the liquid are in equilibrium : the air in the tube 6 is compressed, and the pressure is com- municated to a manometric capsule or other registering apparatus, the displacement of the lever of which may be made by pre- liminary graduation to indicate the absolute value of the pressure in AB. The instrument M is the manometre me'tallique in- scripteur of Marey. An elastic metallic capsule filled with liquid, which is continuous with that of the cylinder AB, plays in this instrument a part which in principle is exactly the same as that of the elastic cap in the sphygmoscope. Measurement of Variable Pressure. If the pressure in the cylinder AB of Fig. 105 be variable as, for example, if the piston A oscillate the various manometers represented in the figure will give oscillating readings. The manometers at B or 0, and the piezometer at C are subject in action to the defect that when a single momentary increase of pressure produces a rise of the liquid or of the mercury in the column, the column does not return promptly to its mean position when the additional pressure is taken off, but oscillates like a pendulum for a period of time XI.] PRESSURE IX LIQUIDS. 265 more or less protracted, until at length friction and viscosity bring it to rest If the piston A oscillate, its movements are not faithfully reproduced by the oscillations of the mercury manometer, for the latter depend on (1) the weight of the column of liquid lifted at each displacement from the mean position; (2) the variations of internal pressure tending to make the column assume new mean positions ; and (3) on friction ; and they are the result of the composition of two sets of oscillations, the one due to the variations of pressure in AB (and agreeing with these variations in period, but not in form or amplitude or phase), while the other set, the pendulum-oscillations of the manometer-column (which may even overpower the former if the mass of mercury be great or if the tubes be wide and offer little resistance), are due to the inertia of the mercury, but vanish if the frictional resistance be very great The oscillations of the mercury may be checked by making one part of the manometer-tube capillary (Marey's mano- metre compensateur), or by interposing a stopcock (Setschenow) the orifice of which can be narrowed till all oscillations are cut off, the instrument then recording merely the slow variations of mean pressure. Tick's instrument F is damped (prevented from oscillating in virtue of its own elasticity) by connecting with the writing levers a disc immersed in glycerine, as shown in the figure : the viscosity of the glycerine causes all secondary oscillations rapidly to die away. The result is that the Federmanometer is very trustworthy as a recorder of the general form of the variations of pressure in AB. The sphygmoscope and the metallic inscriptor, not having much inertia to combat, render accurately the general form of the variations of pressure, especially if in the liquid surrounding the elastic capsules in the latter instrument there be lightly packed a number of bits of sponge to check elastic vibra- tions of the capsules; but all the different forms of pressure- indicators with the exception of those shown at 0, C, and B, require preliminary graduation before their indications can be held to denote the absolute value of the pressures ; and further, this preliminary graduation must be frequently repeated. 3. THE KINETICS OF LIQUID MASSES. Streams. When a liquid flows in a stream, its particles do not become separated from one another to any perceptible extent, and the liquid usually preserves its mean density. The liquid 266 OF LIQUIDS. [CHAP. moves as a whole and has inertia, as may be seen in a rapid and full stream leaping over a chink into which a slow or meagre stream would be pulled by gravity. This principle is sometimes made use of in order to prevent an excess of rain-water entering drain-pipes ; a sloping gutter has chinks in it, opening into the drainage system : when the gutters become flooded, the water rushes over these chinks, and the comparatively pure water is directed elsewhere than into the sewage, the excessive dilution of which may be considered as a commercial evil. When once a stream-flow has been set up, it can in general be maintained by the maintenance of the supply of liquid and of the propelling force. In any stream there may be drawn a series of imaginary lines which represent the direction of movement of the elements of liquid through which they pass. These lines are called Stream- lines, or Lines of Flow. As long as a stream retains the same breadth and form, these lines may be considered parallel to one another ; if the stream widen out they diverge ; if it contract they converge. Such lines are shown in Fig. 106, which repre- sents a stream of frictionless liquid flowing either from A towards C, or from C towards A. Fig.106. Law of Continuity. If we consider successive sections, equal or unequal, taken across a liquid stream, .it is plain that the amount of liquid which crosses each section during any given interval of time is equal in each case : otherwise there would be congestion at some part of the stream. In Fig. 106 the amount of liquid which crosses A or C in a second must be equal to that which crosses B: in other words, the Amount j of Flow across all sections of a liquid stream is the same. This may be other- wise expressed by saying that at any part of a stream the velocity varies inversely as the area of section at that part: if the stream be broadened out so as to have a tenfold cross-section, its velocity is decreased to one-tenth. The statement of this law is due to Leonardo da Vinci. Forces producing flow. -If a perfect liquid, exercising no intramolecular friction and no friction on the walls of the canal xi.] FLOW OF LIQUIDS. 26*7 or tube conveying it, were once set in motion (say in a close circuit or circular tube) it would go on moving without the con- tinued application of force. The energy actually expended on the liquid in producing its movement would remain in the fluid- mass, the velocity of which would consequently remain unaltered. Such a liquid might be exposed to a severe hydrostatic stress as, for example, if such a perfect closed stream were contained in a continuous flexible tube exposed to the weight of a mass of liquid in which it was deeply immersed at the one level and yet the flow would not be affected. A hanging loop of tubing containing a circulating liquid, of which the lower part is exposed to a greater pressure than the upper, will present a turgidity of the lower part of the loop if the tubing be distensible, while if it be rigid there will be no expansion of the stream ; in the latter case the flow will not be affected ; in the former the expansion of the stream affects the local velocities, and therefore the distribution of the energy of the system, but the mean velocity will remain constant. In the case of a suspended loop of distensible tubing the indirect effect of gravity is thus to diminish the velocity of the lower part and to increase that of the upper part both of the descending and of the ascending parts of the stream ; but on the amount of flow it produces no effect. Flow, on the one hand, and hydrostatic pressure uniformly applied, on the other hand, are thus seen to be perfectly distinct conceptions, and in a perfect fluid they might be independent of one another ; but in any physical fluid viscosity and friction come into play, and flow can only be kept up by maintaining a difference of hydrostatic pressure within the fluid considered as a whole, from end to^end. As it is kept up, so is it started : a column of liquid in equilibrium may be made to flow by locally increasing the pressure or by locally diminishing it. If the pressure at a point A be p A , and that at a point B be p Rl the difference of pressures between these two points is p A p s . The difference of pressure per unit of distance is ?L_??. The force producing the flow depends .A. J3 on this ratio ; - and the greater this ratio, the greater is the velocity pro- duced. Small velocities are associated with small differences of pres- sure ; or, in other words, with relatively great distances between points whose difference of pressure is equal to any predetermined quantity, say a unit of force. When the' velocity is great, the 268 OF LIQUIDS. [CHAP. points between which the difference of pressure is unity are relatively near to one another. The theory of Flow from this point of view resembles that of Potential : surfaces of equal pres- sure correspond to equipotential surfaces ; Lines of Flow or Stream- Lines correspond to Lines of Force. If within a stream there be local differences of velocity, where the velocity increases there will be a local fall of pressure. If in an existing stream the velocity be by any means increased locally or generally, while no energy is imparted, the pressure is diminished correspond- ingly : a larger proportion of the total energy is kinetic ; the potential energy, and therefore the pressure, is diminished. When the liquid is driven through a long uniform tube there is at the orifice of inflow a certain initial pressure ; at the other, the orifice of outflow, there is no pressure at all. If the liquid be driven by an equal force through a shorter tube, the pressure vanishes in the same way, but does so more rapidly, and since a greater difference of pressure per unit of length is associated with greater velocity the velocity is greater than in the longer tube. The shorter the tube the greater the velocity, other things being equal. The shortest tube possible would be a plain aperture in the side of the vessel from which the liquid issues. In this case the liquid at once assumes the greatest velocity which it can acquire under the action of a given pressure. " Head of Water." In the case of a vessel containing water which passes out through an aperture, the pressure driving the particles through the orifice is the hydrostatic pressure on that orifice; it is therefore equal (if the area of the orifice be ab and the height of the surface of the liquid above the centre of figure of the orifice be H) to ab . H . pg. The height H is known as the Head of the liquid producing the pressure, and the "Head of Water " is a term familiar to hydraulic engineers. Torricelli's Law. If v be the constant velocity of outflow of a stream passing out of a vessel under the pressure of a constant head H, v = \/2gK. If the aperture be in the sides of the vessel, the liquid issues with velocity v at right angles to the walls of the vessel ; its horizontal velocity becomes combined with a new downward fall due to gravity, and the liquid travels in a parabolic path, forming a continuous parabolic Jet. The form of the parabola indicates the proportion between v and g ; and thus v is found to differ very little (one per cent) from Torricelli's value, v = y/2^H. It is somewhat greater the more convex the wall of the vessel. The amount of outflow per unit XT.] FLOW OF LIQUIDS. 269 of time is not, however, the product of the area of the aperture into the velocity ; it is only about ^^ of that amount. The Vena Contracta. The issuing jet may be observed (especially when it is directed upwards) not to be perfectly cylindrical, but to diminish in diameter from the aperture to a spot called the vena contract a, whose position is sometimes somewhat difficult to define. This conical form is due to the fact that the onward flow of liquid is not confined to that part of the fluid which exactly faces the aperture, but that the lateral parts of the liquid converge on the orifice ; thus the most external stream-lines of the jet, which are at first tangential to the wall of the vessel, assume a direction at right angles to this wall, changing their direction gradually, and therefore presenting a curved form as shown in Fig. 107. From the vena contracta onwards the jet is approximately cylindrical, and presently breaks up into drops, which (especially if any vibration affect the vessel from which the jet issues) are found to be oscillating in form, each becoming alternately a pro- late and an oblate spheroid. The unaided eye cannot perceive these separate drops, but recognises the vein as continuous though troubled. When, however, the jet is instant- aneously illuminated by the electric spark, and its momentary shadow upon a screen observed, the existence not only of these separate drops, but also of others of a smaller size occupying intermediate positions, may be demonstrated with ease; for the instantaneous impression on the retina persists for the sixth part of a second, and the shadow of the jet appears stationary on the screen. The jet may also be looked at through a Stroboscopic Disc, a rotating disc provided with equidistant narrow apertures. Through each aperture a glimpse is caught of the jet in a certain position. If the rate of rotation of the disc be properly adjusted, each successive glimpse is caught just when each falling drop has had its place taken by its successor; and thus, on the whole, under such a succession of glimpses the jet appears to be stationary. This phenomenon is one of free fall in the air, for the break-up into drops depends greatly on surface tension ; a liquid cylinder of excessive length and with a free surface first assumes an undulating contour, and then breaks up into separate vibrating drops, as Plateau has shown. The vibra- tions of liquids in tubes are therefore not to be explained as phenomena of this kind. 270 OF LIQUIDS. [CHAP. Ajutages. The amount of outflow from an aperture in the wall of a vessel is greatly influenced by the form of the ajutage or mouthpiece through which the liquid passes. This may be made so as to present the same form as the jet itself, and if it be prolonged just as far as the vena contracta, the amount of outflow becomes equal to the product, area x v x time ; not because the outflow is itself altered, but because the area of the orifice of amount of outflow outflow is reduced so as to become equal to - , v. t the terms of this ratio being unaltered. If the ajutage project inwards, the outflow and the velocity are materially diminished. If the ajutage project outwards, being cylindrical, the cylinder, if its walls be wetted by the liquid, is completely filled by it, the jet is cylindrical, and the outflow is greater than when there is no such ajutage. The liquid is drawn towards the sides of the cylinder, and conversely, the sides of the cylinder are drawn towards the liquid. Hence there is no pressure exerted on the walls of the tubular ajutage ; on the contrary, there is suction, and if any part of the walls of the tube be mobile, it will be drawn into the stream. Lateral diminution of pressure. If through a tube of the form shown in Fig. 108 there pass a current of liquid in the branch AB under a pressure which is barely sufficient to keep up a stream filling the tube, the mutual attraction of the walls of AB and the liquid will put the liquid in AB in a state of tension and diminish the pressure in AB. In the side tube CD a certain column of liquid can be supported in consequence of the diminished pressure in AB. If this rise to the point C, the upper layers of the column DC will be constantly carried off by the stream BA, and thus a stream is set up in the direction DC. If the pressure in the main pipe AB be too great, liquid will be driven down CD. The former action is by some considered as explaining the flow of lymph up the thoracic duct. The same kind of suction-effect may be perceived in the older forms of washhand-basins connected with. a house drain-pipe by a simple bent tube or trap ; a downrush of liquid along the main pipe produces a deficiency of pressure, which allows the atmospheric pressure communicated through the XL] FLOW OF LIQUIDS. 271 basin to drive the liquid which seals the trap into the drain pipe, and thus to leave a channel patent to the entry of sewer gas. The peculiarly beautiful forms presented by jets under various circum- stances are described and figured by Savart in the Annales de Chimie et de Physique, vols. 54 and 55. If two vessels having an aperture in each of the same size, shape, and at the same level, be so arranged that these apertures are exactly opposite one another and close together : if liquid be poured into the one vessel, it will run into the other. The vessels may then be removed to a certain distance from one another, and the liquid will continue to pass from the one vessel into the other, through a tube formed of its own superficial film, until the same level is nearly attained ; then the liquid begins to flow out of both vessels, and the two jets, meeting, spread out into a sheet which is driven back and fore between the two orifices as the liquid in the one or the other vessel stands for the moment at the higher leveL Energy of Jet. If Torricelli's law held perfectly good, that v = v / 2^H, the velocity would be the same as if every par- ticle had fallen from the surface of the liquid to the orifice, and had passed out of the orifice with a velocity due to its fall through the height H. This would be absolutely the case were it not for friction and viscosity. If the level of liquid be main- tained constant by a continued supply, the velocity is constant. At the instant when the whole of the original liquid has passed out through the orifice, the experiment may be stopped. The liquid which has passed out has conveyed with it energy = 1 mi? = mgUL if Torricelli's law be true. At the commencement of the experiment it had potential energy /'mass m at an average HA H height of \ of mg only. It has therefore gained energy TT m 9 "FT- This ener gy nas been lost by the water which has Zi H replaced it, and sunk from the surface to an average depth of -^ TT below the surface, thus losing potential energy mg without any compensating gain of energy in any other form. The same principle is illustrated in the following experiment. One cork of a Woulffs bottle completely filled with water is fitted with a piece of glass tube drawn out so as to form a jet ; the other cork admits a tube leading from a vessel containing mercury ; the mercury is caused to fall into the bottle. Some of the water which already fills the bottle is driven out with great velocity in a thin stream. The mercury sinking through the water loses energy proportional to its density (nw/H = vpgH) ; the water forced out acquires this energy, and hence has a great velocity imparted to it. 272 OF LIQUIDS. [UHAP. Velocity of Outflow. Torricelli's law shows that the velocity with which a liquid issues through an aperture varies as the square root of the head of water or the pressure; or that the head of water necessary to produce a certain velocity in a free stream of liquid, subject to no resistances, is proportional to the square of the velocity. Any pressure exerted on a liquid may be measured as Head of the same liquid or of water. Example. The pressure at the bottom of a column of water 1033-3 cm. deep is equal, per sq. cm., to the weight of 1033-3 grammes. The atmos- pheric pressure on the surface of a liquid is equal to the same. Hence the atmospheric pressure is equal to that of a head of water of 1033*3 cm. Water at a depth of 1033*3 cm. under a water-surface exposed to the atmosphere is exposed to as much pressure as if it lay at a depth of 2066*6 cm. of water under a free surface not exposed to the atmosphere. In a vessel filled with water, 1033*3 cm. deep, provided with an aperture in its lower surface, this aperture communicating with a vacuum, and the upper surface of the liquid communicating with the atmosphere, the velocity of outflow will, according to Torricelli's law, be A/2 x 981 x 2066*6 = 2013*6 cm. per sec., while, if the lower aperture were also in communication with the atmosphere, the effective head of water would be 1033*3 cm. and v = A/ 2 x 981 x 1033*3 = 1423*8 cm. per sec. The pressure produced by compression, as in pressing home a syringe, the negative pressure produced by rarefaction, as in pull- ing up the handle of a syringe, may all be measured in the same way. In general, when a fluid is acted upon by a number of forces represented as heads of the same fluid by H, H y , H /x , etc., the velocity of outflow of the fluid through an orifice is (H + H, + H,, + eto)_ The speed of outflow, ^2gIL, does not depend upon p, the density ; all liquids ether and mercury issue with equal velocities under the action of equal heads of their own substance. Recoil. The law of action and reaction perfectly applies to liquid jets and to the vessels from which they issue. The hydraulic tourniquet is. an example : a cistern containing water and capable of rotating on an axis : pipes ending obliquely issue from its sides : water runs out of these pipes : and by reaction they are driven backwards. Since they are not fitted to an im- movable cistern, but to one free to rotate, the whole rotates, and thus the contrivance may be used to convey water-power, the water constantly running into the rotating cistern, and running out of the obliquely-set exit pipes. It must be easier for a horse to pull a water-cart out of which water is XL] FLOW OF LIQUIDS. 273 issuing in jets than it is to pull one in which there is no recoil to aid the animal. Resistances. When a fluid stream passes through a tube or a channel it experiences different retarding resistances, which con- vert energy of motion into heat, and of which the following are the chief : Surface Adhesion, Surface Friction, Inequalities of the Surface of the bounding solid, Eddies, and Fluid Viscosity. " Surface Adhesion. If a liquid wet the walls of the tube or channel through which it passes, the layer of liquid which is in contact with the walls does not change except by molecular diffusion and exchange. It remains in situ while the liquid flows past ; in other words, there is infinite friction between this layer and the walls wetted by it. This gives great scope to the viscosity of the fluid, though it itself, when once the flow has been set up, does not directly cause any waste of energy. While the walls are being wetted there is a slight liberation of heat, due to the satisfaction of the mutual molecular attractions between the liquid and the walls. Surface Friction. If the liquid do not wet the tube through which it passes, the surface of the moving liquid and the walls of the vessel rub against one another, and energy is lost in overcoming this friction. This friction is greatly increased by roughnesses on the walls of the tubes or channels, which cause little eddies or whirlpools. Eddies are produced when a moving fluid is subjected to unsymmetrical retardations. The cases in which eddies, whirl- pools, vortex-rings, rolling and tumbling water, and the like, are produced are extremely numerous. Water flowing in a tube which suddenly widens or suddenly narrows generally presents such eddies at the point of sudden enlargement or contraction. Viscosity. When a disc or cylinder suspended in a fluid is caused, by twisting the supporting wire or wires, to enter into oscillations, it is found that the oscillations soon die away ; though they continue isochronous, their amplitude diminishes ; and the amplitudes of any two oscillations stand to one another in a constant proportion. If the disc or cylinder be wetted by the fluid, the layer immediately in contact with the solid remains in contact with it ; this film, moving with the solid, sets in motion the film next in contact with it, and that in its turn sets the next in motion. Each film goes through a displacement some- what less extensive and more retarded than the one gone through by the film which sets it in motion. Continuous rotation of the T 274 . OF LIQUIDS. [CHAP. disc or cylinder would in time cause the whole fluid to rotate ; but the influence of an oscillating disc travels a very short distance, for half-an-inch away from the disc the fluid remains undisturbed. Within this small distance the liquid performs oscillations which in period resemble those of the oscillating disc, but which in amplitude are less, and in phase more retarded, the greater the distance from the disc. This lagging behind on the part of the liquid has the effect of dragging on the disc and of gradually bringing it to rest. If the disc be wetted the retardation is independent of the nature of the material of the disc, for there is no velocity lost by friction between the solid and the liquid. If the disc be not wetted, there is distinct friction (external friction) in addition to the viscosity (internal friction). The Coefficient of Viscosity serves as the means of measur- ing the viscosity of a substance. We have already seen that it is equal numerically to the force which is necessary to maintain a flow of one layer of one unit-area past another of the same area with a relative velocity of one unit, the distance between the layers being unity, and the space between them continu- ously filled with the viscous substance. If F be the force required to keep up the flow of two layers past each other, their area being each a, their respective distances from a plane of re- ference being r t and r u and their distance from each other therefore r t - r a ; if their respective velocities be v t and v at and their relative velocity v t - v u ; and if the coefficient of viscosity be rj, -n V V F = d -L - _ . In the case of water at 15*5 C. this coefficient is 0*0131 (Meyer), while that of air which obeys the same laws is '000360, both expressed in C.G.S. units. Though the density of air is y^jth that of water, its viscosity is as much as -g^th that of water. For brass, 77 is about 300,000,000. Moist air is more viscous than dry air : so is hot air more viscous than cold air. Hot water is less viscous than cold. Most saline solutions are more viscous than water, saltpetre solution being an excep- tion. Most saline solutions are more viscous the more concen- trated they are, saltpetre solution being again an exception (Meyer). The experimental determination of the coefficient y by means of observa- tions made with the aid of an oscillating disc involves much mathematical computation, and it is often quite sufficient to record the so-called Loga- XI.] VISCOSITY OF LIQUIDS. 275 rithmic Decrement or log. dec. special to each liquid. Let us suppose that the oscillating disc or cylinder first turns through the angle o> ; that at the next oscillation its deviation from its mean position is ^^ o06 = 790,02* cm. pff rfv x 981 = 7900*2 metres. If a barometer on the floor stand at 76 cm., the same barometer raised to the height of 1 metre should stand at a height of 76 cm., less mm., a perceptible diminution. 10*4 The pressure does not diminish regularly with the height, as it would in an ocean of incompressible fluid. The lower strata of the air are compressed, and therefore, to set up a given differ- ence of pressure, a shorter vertical ascent among them is sufficient than is necessary among the higher strata. Each stratum differs from the one below it in two respects : (1) it has fewer strata above it; (2) it is therefore less com- pressed, and for equal mass has greater volume. If we imagine the whole atmosphere to be divided into 7900*2 strata, the lowest of them all, which bears the weight of 789 9 '2 strata, will be 1 metre thick ; the next, which bears the superincumbent weight 7899*2 of 7898*2 strata, will have a thickness of 1 metre x -r- ; < 7898*2 the next stratum will have a thickness greater than this in the ., 7898*2 /, 7899*2\ 7898*2 7899*2 ratio ; i.e., it will be ( 1 X ) x ; 7897*2' V 7898*2/ 7897*2 7897'2 ' 7899*2 and the Tith layer will be metres thick. 7900-2ri Altitudes as indicated by the Barometer. If x be the vertical height between two stations, H the height of the barometer at the lower station observed at temperature , and H y the height of the barometer at the higher station at the temperature t', A being the latitude ; then x = 18393 . (1 + -002837 cos A) log -[ ( 1 + 2 ilY H \ loooy (Laplace's Formula). 312 OF GASES. [CHAP. Variations in the barometric pressure occur from moment to moment, as the atmospheric ocean is disturbed by currents, driven in whirlpools, varied in thickness by superficial waves or by expansion (due to heat) and lateral overflow. When any spot has a low pressure, there is a tendency for the surrounding air to rush in from all sides towards that spot, the centre of depression; the greater the difference of pressure between two places i.e., the steeper the barometric gradient the greater will be the tendency to an inflow of air towards the centre of de- pression. This tendency is so modified by the rotation of the earth from west to east (in a direction opposed to the apparent move- ment of the sun) that the flow does not take place directly towards the centre, but round it in a circular storm whose direction is in the northern hemisphere opposed to, in the southern the same as that of, the hands of a watch (Dove's Law of Storms). Correction for pressure. Variations in the barometric pressure render it necessary in measuring quantities of gas by volume to make a correction for pressure, and to reduce the gas to standard pressure i.e., to state what the volume would have been had the atmospheric pressure at the time of measure- ment been 76 cm. Boyle's law teaches us that the volume varies inversely as the pressure. If therefore the pressure on gas measured as x cub. cm. at 76*1 cm. had been, not 76*1 but 76'0 cm., the volume of that gas would have been greater under the less pressure in the ratio of 76'1 to 76*0. The general rule is, that a volume of gas measured at the pressure h cm. must be multiplied by in order to reduce it to the standard pressure. 76 Standard Atmospheric Pressure. In many modern books, instead of a pressure of 76 cm. mercury, or 103 3 '2 9 6 cm. water, or 1,013,663-376 dynes, the standard atmospheric pressure is taken as 1,000,000 dynes per sq. cm. Gases passed into the Torricellian Vacuum. If a bubble of gas be passed into the Torricellian vacuum, it will expand so as to fill it; further, it will exert pressure on the top of the column of mercury it will therefore depress that column ; the extent to which it depresses the column measures the pressure which it exerts upon the mercury : conversely, that depression measures the pressure of the mercury upon it, and therefore indicates the pressure under which it itself assumes its actual volume. xii.] ATMOSPHERIC PRESSURE. 313 Let a barometer tube whose cross area is J sq. in., and whose free internal height is 34 inches, have standing in it a column of 30 inches of mercury. Pass a cubic inch of air (measured under a pressure of 30 inches) through the mercury into the four-inch-long Torricellian vacuum. It would exactly fill that vacuum, exerting a pressure of 30 inches on the top of the mercury. This is impossible. The gas expands ; it depresses the mercury through x inches : it is then subjected to a pressure of x inches of mercury as compared with the atmospheric pressure of 30 inches under which it was measured. Its volume is 30 now accordingly increased to 1 cub. in. x . The length of tube occupied 30 120 by this volume is x 4 = inches ; of these 4 inches were already x x taken up by the vacuum. The actual depression is therefore ( - - 4 j ; but this depression is x itself. Hence x = ( 4 ) ; or x = 9vL 5 ; and the \ x J mercury will stand in the tube at a height of 20-85 inches. CHAPTER XIII. HEAT. Heat is a form of Energy. It would, perhaps, indeed be more correct to say that we designate under the one name Heat two totally distinct forms of Energy. The one of these is the energy of a wave-motion in the Ether, passing from a hot body to surrounding objects across the intervening space, as from the sun to our earth, or from a hot fire to the colder objects upon which it shines: this we call Radiant Heat. The other form is a confused oscillatory disturbance of the particles of a body : in virtue of this molecular movement a body may appear to our cutaneous sense of heat (a sense quite distinct from that of touch) to be more or less hot or warm ; or in the converse case it may, on account of the small amount of this movement, appear to be relatively cool or cold. The latter form of heat may be called Sensible Heat, or Heat simply, and of it we shall proceed to treat in this chapter. It is the only form of heat for the percep- tion of which we have special sense-organs. We do not directly perceive the undulations of radiant heat by our senses : when the sun shines on us heat-waves strike the skin, throw it into vibra- tions, and the sensible heat of the skin, not the radiant heat of space, affects the appropriate nerve- ends. When we touch a hot body it communicates its oscillations to the nervous system : when we approach a hot body we become indirectly sensible of the radiant undulations into which it is throwing the surrounding ether. Thus we may state that our sense of heat is our power of perception of the confusedly-vibrating condition of a body ; and that the more pronounced this condition of agitation, the hotter will a body appear. A hotter body may be readily supposed and rightly so if we confine our attention to bodies formed of the same substance to have in it a greater amount of Heat than a colder one. And a hotter body can become cold, a colder body can become warm : heat can be supplied to bodies, or they can be CHAP, xiii.] HEAT A FORM OF ENERGY. 315 deprived of it; heat can be gained or lost by material bodies. The primitive interpretatation of this was that heat was a sub- stance, a fluid, the so-called Caloric, invisible, imponderable ; that a piece of hot iron was a kind of temporary union of cold iron with this subtle imponderable fluid. When a piece of metal was rubbed it became warm : the reason assigned was that Caloric was squeezed out of it, like water out of a sponge. But this material theory of heat became untenable when it was shown that there was absolutely no limit to the amount of sensible heat which might be so produced by the friction of a trifling amount of metal ; the amount of water that might be boiled, for example, by heat produced in this way depended only on the mechanical power available (Rumford). The heat evolved by friction as, for instance, in metal boring or turning is practically limitless. Even two masses of ice, caused to rub against one another, melt (Davy) a fact which leads the material theory of heat into helpless con- fusion. Water was admitted to be ice plus caloric ; if, then, ice with its caloric rubbed or squeezed out of it and lost that is to say, ice minus caloric become water, how can the theory stand ? Plainly heat is not material : it is the energy imparted to the system it is equal to the work done upon it ; and we find that heat and the other forms of energy are reciprocally convertible. When a body is sensibly hot its particles are in an active state of motion. The particles strike one another and rebound ; the more rapidly they do so, the greater is the mean velocity of the particles, and the greater is the kinetic energy of the whole mass ; but it is impossible that the energy of the mole- cules should be entirely due to such a movement of translation. They are not material points, and they have if not in solids or in liquids, yet certainly in gases six degrees of freedom ; when they strike each other they not only rebound but they also spin ; to the energy of translation must be added one of rotation. Further, the molecules are made up of atoms : atoms are not stationary in the molecule, but may be so violently agitated as to leave it altogether, and thus to give rise to the phenomena of chemical decomposition by heat ; part of the energy of a heated body is due to intra- molecular atomic oscillations. Lastly the ether entangled in a mole- cule is also set in vibration, and absorbs some energy which appears as kinetic energy of ether- vibrations. The sum of these is found, by the agree- ment of experimental results with calculations based on the hypothesis that such is the law, to be proportional on the average an average not perceptibly departed from for any appreciable interval of time to the kinetic energy of translation alone. The convertibility or identity of heat with energy is independ- ent of the inner mechanism of the moving molecules which possess it ; and it is confirmed by instances from all sides. The energy of work which is apparently wasted in friction 316 HEAT. [CHAP. becomes Heat : the heating of a locomotive brake, the ignition of a lucifer match, the heat evolved during the mechanical operations of metal boring or turning, the heat found in a body which has received a sudden blow or a sudden distortion, all these prove the proposition. If work be done in driving a paddle in water, no work being done other than that of churning the water, when the operation is over the work appears to have been wasted and to have disap- peared ; but the energy is not destroyed ; it exists in the water in the form of heat. If 7 72 "4 3 foot-pounds of work (measured at Manchester, Joule) be expended in churning a pound of water, the temperature of that w^ater will be raised by 1 F., from 60 F. to 61 F. ; a similar rise of 1 C. in a kilogramme of water will be effected by the expenditure of 42 3 '43 7 kilogrammetres or 41,573,025,475 ergs of work. Hence the water at the base of Niagara Falls ought (setting aside the 'effect of evaporation and of cooling or heating by the air) to be about -J- F. higher in temperature than at the top, for the vertical fall is 161 feet. Hence also the sailor's maxim that the sea is warmed by a storm. When in a steam-engine at work the steam at its entrance to the cylinder from the boiler is compared with that which goes to the condenser, it is found that the latter is colder. The difference of heat is found to be equivalent to the work which the engine has done ; and if the engine do no work, then the energy which has not been converted into work remains as heat, and the engine becomes heated (Hirn). When a quantity of gas or of liquid is forced through a tube, as in Fig. 109, the potential energy of the system before the flow is started is greater than the kinetic energy of the outflowing stream. If the resistance be so great that the velocity of outflow is practically null, the whole of the worlf done on the fluid is spent in heating it."* The work done is equivalent to the heat * This must be done at a pressure corresponding to a certain definite head H of the same fluid. The fluid is found to rise in temperature by x C. ; a head of cm. TT would cause it to rise by 1 C. ; a vertical free fall of cm. would cause it, if abruptly stopped, to rise in temperature by 1 C. ; the amount of energy correspond- ing to such a fall would be mg ergs ; this energy in the form of heat, mg SC 36 TT ergs, would heat a mass ra of the fluid through 1 C. ; g ergs would heat one TT -I gramme of the fluid through 1 C. ; . g . - ergs (k being the specific heat of XIIL] FIRST LAW OF THERMODYNAMICS. / 317/ produced. We are now able to state the First Law OP Thermodynamics. Heat, being a form of Energy, can be measured in ergs. This law is usually stated in a somewhat different form. An arbitrary unit of heat is chosen, and designated a Calorie : this is the amount of heat which is required to raise the temperature of one kilogramme of water from C. to 1 C. This quantity of heat is found to be 41,573,025,475 ergs. This last number, 41,573,025,475ergs is the "Mechanical Equivalent of Heat," or "Joule's Equivalent:""" it should perhaps be called the Me- chanical Value of the Conventional Unit of heat, the Calorie. The first law is, then, that one Calorie is equal to 41, 573, 025, 4 75 ergs. Another unit of heat has been proposed, the Electromagnetic Unit, or 10,000,000 ergs ; this is the amount of heat developed in one second in an electrical circuit whose resistance is one Ohm when a current passes whose intensity is one Ampere. (See p. 575). Heat is energy, and it is the lowest form of energy. It may be said to have no organisation, but to depend on undirected and blind activity of molecules, which dash hither and thither. When in any action energy is liberated which is not guided by the environment into any specialised form, it manifests itself as heat ; and when energy is spent in doing work, the equivalent of which appears in no other form, it then appears as heat. This statement is widely applicable and important. / Work done upon a dynamoelectric machine whose circuit is complete appears in the first place as the energy of an electric current : if no exterior work be done, the system as a whole becomes heated. A voltaic cell can do exterior work : if it do none, the current being allowed to circulate uselessly, the whole of the energy liberated during the chemical combination appears as heat in the circuit. Heat being a form of energy, many propositions relating to it- are merely special cases of propositions relating to energy. If a certain number of bodies be arranged in a system A whose potential energy depending on the arrangement of the bodies in the system is P A ; if the same bodies can be arranged in other systems B, C, D, whose respective potential energies (less the fluid, p. 328) would heat one gramme of water through 1 C. ; 1000 ? ergs of 3C/C energy in the form of heat would raise the temperature of one kilogramme of water through 1 C. * Joule's Equivalent in its original form was a number (772) which denoted the number of foot-pounds of work found to be equivalent to the heat necessary to raise 1 Ib. of water through 1 F. 318 HEAT. [CHAP. than that of the former) are P B , P c , P D , etc. : then the transforma- tion of the more highly-stressed system A into a less-stressed system B, if this be brought about by a rearrangement of its con- stituent bodies, involves a liberation of energy equal to P A P B . If in this case the system A be converted into the system B with- out doing any exterior work, the whole of the energy liberated appears in the form of heat ; and, numerically expressed, the heat thus liberated is equal to the work W which would have had to be done upon the system B in order to convert it into the system A, if that operation had been effected ; that is, the heat so liber- ated is equal to W P A P B . A gramme of hydrogen and eight grammes of oxygen form a system (system A) which after explosion may be converted into nine grammes of water vapour of the same volume and temperature (system B). The former is con- verted into the latter without doing any exterior work. Much energy is liberated in the form of heat, and though the absolute values of P and P A B are unknown, their difference is found (see Calorimetry) to be an amount of energy equal to 34-462 Calories, or 1,432,689,600000 ergs, or 1,432689'6 megergs. The potential energy which the mixture loses when its particles clash together and combine is the energy of chemical separation. A mixture of explosive gases may be made to yield up some of this energy in the form of work, as in the modern gas-engine ; if no work be done, and if there be no other transformation, the whole of it must appear in the form of heat. Chemical combination is thus often attended with the evolution of heat. One gramme of carbon burned in oxygen yields 80 '8 Calories or 3,359,100,500000 ergs ; 1 gramme of carbonic oxide yields 24'03 Cals., (24-31 Cals., Andrews); 1 gramme of marsh-gas, 13'063 Cals.; 1 gramme of dry albumin, 49'98 Cals. ; urea, 2'206 Cals. ; fat, 9'096 Cals. ; starch, 3'9012 Cals., or 162,184,686,973 ergs per gramme (Frankland). When copper or antimony is dropped into chlorine it takes fire, and a chloride is formed : heat is evolved. In some instances the converse is true ; work has to be done upon separate elements in order to force them directly or indirectly to combine : and when their compound decomposes heat is evolved. Carbon and sulphur will only combine when they are kept hot by an external source of heat : they must be forced to combine. Nitrous oxide (N 2 O) evolves heat when it is decomposed into nitrogen and oxygen ; and hydrogen dioxide (H 2 2 ) evolves heat when it is decomposed by contact with platinum. A change from the condition B to the condition A (which possesses more potential energy) cannot be effected unless there be energy added ab externo, or else unless some of the kinetic energy of the body, if it have any, assume the potential form ; in the latter case the body may lose sensible heat, and may become cold. When a chemical decomposition is effected by heat, if heat had been evolved during the formation of the compound, heat must be continuously XIIL] HEAT A FORM OF ENERGY. 319 supplied to do the work of decomposition. The heat supplied has the effect of throwing the molecule into such agitation that the mutual affinity of the atoms cannot retain them in union. This is the process of Dissociation or Thermolysis. At moderately-high temperatures the atoms reunite with others which they encounter ; at very high temperatures (3000 C. in the case of oxygen and hydrogen) no such reunion is possible, and the de- composition is complete. Thus the proportion of decomposed to apparently undecomposed material varies with the temperature. The process is favoured by one or more of the resultants of dissociation being gaseous. After dis- sociation the separated elements again contain potential energy equal to the heat expended upon them, and upon cooling they may recombine with the evolution of this energy in the form of heat, which is gradually lost. In general, every Change in the State or Condition of a body or a system of bodies is associated* with a Change in the Intrinsic Potential Energy of the body or the system : and this change is accompanied and manifested either by the liberation of Energy in some form, useful or useless e.g., work or heat or else by the disappearance of Energy which is spent in producing the change of state, and is either taken in ab externo or is transferred from the kinetic energy already possessed by the body, as is shown in the ordinary case by that body becoming cold. Thus, if a quantity of air in a cylinder be suddenly compressed by means of exterior work done upon it, it becomes hot : if the piston be allowed to return, the air cools down to its former temperature ; but if it be kept com- pressed until it has assumed the temperature of surrounding objects, and be then allowed to drive the piston out against atmospheric pressure, it becomes very cold, for it obtains the energy required to do the work of driving out the piston at the expense of its own heat. If the system A disengage x units of energy (as heat, or in any other form) on being let down to the condition B, and if the same system A disengage y units when it acquires the condition C, then the system B, on being let down to the condition C, will dis- engage energy y x. Conversely, if B and C respectively require energy x and y to enable them to become converted into the system A, the system C requires energy y^x in order to enable it to become the system B. The relative amounts of chemical energy in organic compounds may be estimated by finding the amount of heat which they evolve when they are burned so as to form carbonic anhydride, water (and nitrogen). What is the intrinsic energy of Acetic Acid ? 60 grammes of acetic acid are found (Berthelot) to disengage on combustion 210 Calories of * This general conclusion is subject to the qualification that the change of state or condition must be a real one, not one which consists in a mere replacement of the particles occupying a given position by others physically similar, or by a mere change of the direction in which similar parts of the substance lie. 320 HEAT. [CHAP. heat : 3*5 Calories or 145,505-6 megergs per gramme. The total intrinsic potential energy of acetic acid we do not know ; the number given indicates the total amount available on combustion with oxygen. Its elements, 24 grammes of carbon, 4 of hydrogen, and 32 of oxygen, yield on combustion 326 Calories. The difference between the " Combustion -equivalent " of acetic acid and that of its elements that is, 116 Calories is the total amount of energy lost by these substances when they pass through the changes (what- ever be the number, the nature, or the order of these) in which they pass from the state of free elements to that of acetic acid. Oil of lemons, turpentine, and terbene, which have the same chemical constitution, seem to have a different intramolecular arrangement, for on combustion they evolve different amounts of heat. This shows that the potential energy of the molecules is different in each case. Amorphous sulphur kept in a solution of sulphuretted hydrogen becomes octohedral sulphur with absorption of heat. When zinc is dissolved in sulphuric acid, a certain amount of energy is liberated and heat is evolved : when zinc is amalgamated with mercury, it becomes cold unless heat be supplied : when amalgamated zinc is dissolved in sulphuric acid it evolves more heat than unamalgamated zinc does, and that by an amount exactly equal to that absorbed during amalgamation. The absolute amount of energy liberated or absorbed during any change of state is independent of the rate at which the change is effected. A slow change of state (as in the processes of decay or of the oxidation of the tissues of an animal) evolves the same amount of heat as a rapid change ; the temperature in the former case is lower than in the latter, because the lapse of time allows a more equable distribution of the heat. Thus, a gramme of hydrogen and 8 grammes of oxygen will evolve enough heat to raise 34'362 kilos, of water in temperature by 1 C., or 343 '62 kilos by y 1 ^ C. : this it will do whether the combination be explosive or gradual, as when the gases are induced slowly to combine by the presence of rolled platinum. The final condition of the water-vapour must be the same in both cases : if this be not borne in mind the amounts of energy evolved during combination will appear to differ in the two cases by an amount equal to the work which must be done in order to convert the one final state into the other. When a system of bodies passes from one state to another it is a matter of indifference what the intermediate changes have been so far as concerns the absolute amount of energy liberated or absorbed ; the system A may have assumed the conditions C, D, etc., and that in any order; but the amount of energy liberated depends only on the initial state A as compared with the final state B. If it had been otherwise, the perpetual motion might be realised ; for it might be possible to effect a change from A to B by one series of transforma- tions, and to effect the reverse operation by another series, such that the one series of changes would evolve more energy than the converse one consumed, and the result would be a repeated restoration of the status quo, associated with , xiii.] HEAT A FORM OF ENERGY. \^ 321 W- a perpetual supply of energy, available for useful work, and crafe^ out of nothing. ^^ The energy absorbed by a system during a given change oi state is exactly equal to that which is liberated when the change is reversed. It is assumed in this statement that no exterior work is done through the instrumentality of the change of state. The potential energy of every system of bodies always tends to diminish as far as possible. Every system which possesses potential energy thus tends to lose it ; its potential energy tends to become kinetic, and, if it assume no other form, to take the unspecialised form of heat. In any system which undergoes spon- taneous transformation, the transformation tends to take such a course that the heat evolved by it shall be a maximum.* If there be any such transformation possible, which would be accompanied by the evolution of heat, that is a necessary transformation, and sooner or later it will take place, directly or indirectly, the poten- tial energy of the system being added to the unavailable heat of the universe. In many cases a single change of state may be analysed into several others. ' The heat- value of the total change is equal to the sum of the heat- values of the separate component changes. Thus when a piece of sodium is put into water the following changes occur simultaneously : (1) decomposition of water into free atoms of hydrogen and oxygen ; (2) reduction of hydrogen to the gaseous state ; (3) coalescence of atoms of hydrogen to form molecules ; (4) exterior work done by the hydro- gen escaping against atmospheric pressure ; (5) combination of sodium with oxygen and hydrogen atoms to form sodium hydrate ; (6) solution of sodium hydrate in water. Each of these changes has its own heat- value, positive or negative, according as it involves the evolution or the absorption of a certain amount of energy. The combustion of 8 grins, oxygen with 1 grm. H. yields 34 '462 Cals. heat. The same quantity of the same elements combining in the nascent state yields 5 4 '6 2 3 Cals. Hence the heat evolved during the combustion of one gramme of hydrogen is the resultant of an absorption of energy (20'161 Cals.) due to the break-up of the gaseous molecules into atoms, and an evolution (5 4 '6 2 3 Cals.) due to the combination of these atoms in the formation of water-molecules. When a gas is dissolved in water there are two effects : (a) liquefaction of gas with evolution of heat ; (6) satisfaction of chemical affinity between * Berthelot's Mecanigue Chimique contains the very interesting and important results of sixteen years' experiments on this aspect of chemical physics. M. Berthelot has thrown great light on the subject, and he gives very complete references to the papers of Thomsen and others who have studied chemical reactions from this point of view. Y 322 HEAT. [CHAP. the water and the gas, with the evolution of still more heat. When a solid is dissolved in water the liquefaction of the solid causes the absorption of heat (as in freezing mixtures), while the latter effect still causes its evolution. When glacial acetic acid is dissolved in water, the absorption of heat caused by imparting greater fluidity to the acetic acid overpowers the evolution of heat due to chemical union. When two or more changes of state occur concurrently, it may be that some of these changes are accompanied by the libera- tion, some by the absorption of heat, and that these changes exactly compensate each other ; the result being that on the whole there is neither absorption nor liberation of energy in the form of heat. For example, a gas, while expanding against the atmospheric pressure (from whatever cause), does work in lifting the atmos- phere ; if it increase in volume alone, without undergoing any change in its temperature, energy must be supplied to it in order to enable it to do this work ; if it diminish in tempera- ture without suffering any change in its volume, it must neces- sarily lose heat ; if, on the other hand, it undergo both these changes increase in volume and diminution of temperature concurrently, it is possible that these two changes may be so adjusted that the body, while it undergoes the double change, neither loses heat nor acquires energy from without. Such expansion is called (Eankine) adiabatic expansion expansion during which the substance neither gains nor loses heat by conduction or radiation to or from surrounding objects. This is a kind of operation which could only be perfectly realised in practice if the expansion were infinitely rapid ; but any gas suddenly expanded is thus chilled. Conversely, adiabatic con- traction of volume is associated with increase of temperature. We have hitherto regarded any single change .of state as a possible antecedent cause of the liberation or of the disappear- ance of heat. We shall now change our standpoint, and consider the effects (including change of state or of condition) produced ~by the increase of heat in a body or by its withdrawal. EFFECTS OF HEAT. The principal effects of an increase of heat in a body may be the following : A. Internal Work. a. Increase of the kinetic energy of the molecules of xin.] EFFECTS OF HEAT. 323 the body an increase of the sensible heat of the body ; i.e., an increase of temperature. b. Inter molecular work work done by or against molecular forces change of volume, change of cohesion, change of elasticity, etc. c. Intramolecular work work done within each several molecule production of intramolecular vibra- tions. d. Chemical work, intermolecular and intramolecular. B. External Work. Work done by or on a body as it expands or diminishes in bulk.. These effects are not necessarily all produced by the action of heat upon any substance. There may, as in the following example, be no external work done when a body is heated ; the whole energy imparted to the body being spent upon the internal accumulation of energy in the form of heat. Water at 3'4 C. if heated to 4'4 C. first contracts and then returns to its original dimensions. On the whole there is in this case no external work done. Neither is there any work done in giving the particles a new position in opposition to the intermolecular forces,* nor is there any chemical effect. The whole heat imparted may thus be held to be spent in raising the temperature by 1 C. When a bar of iron is heated in a vacuum there are two effects : (1) increase of temperature ; (2) expansion of the iron, which represents work done against the molecular forces. When the same bar is heated in air, there is added a third effect, viz., the thrusting aside of the surrounding air by the expanding bar, in consequence of which exterior work is done during expan- sion. In a bar of iron the exterior work done in this way is very small, and the interior work done predominates so largely that the exterior work may for many purposes be neglected. When a mass of gas is heated, the work which is done in expanding the gas itself is appreciably null, for this is one of the Characteristics of gases ; if there be any work done during the expansion, it is all exterior. The effects in this case are two : <1) the increase of temperature ; (2) exterior work done in over- coming the exterior (atmospheric or other) pressure. When water above 3 '9 C. is heated it expands. The effects are (1) increase of temperature ; (2) work done in separating the This statement is only approximately true, for there are physical differences of viscosity and the like between water at 3 '4 C. and water at 4 '4 C. 324 HEAT. [CHAP. molecules ; (3) a small amount of work done against the external pressure. When water between and 3*9 C. is heated it contracts. The effects are (1) increase of temperature ; (2) intermolecular work ; (3) a small amount of work done by the external pressure. When a piece of caoutchouc is heated it contracts ; when pulled it expands and assumes the dimensions proper to a lower temperature, intra- molecular energy is set free, and the caoutchouc becomes warm. A piece of metal suddenly extended becomes cool. When ice at C. is heated the whole energy imparted to it is expended in producing the following results: (1) Fusion, with contraction of volume (intermolecular work work spent in producing a new arrangement of the molecules) ; (2) A slight amount of work done ~by the exterior pressure on the body. The latter may be for most purposes neglected ; if we do so we may say that all the heat supplied to the ice is spent in doing the interior work of liquefaction, and that none of it is spent in producing an increase of temperature. When, therefore, a piece of ice is heated it melts, but it does not rise in temperature until it has been wholly melted. The water produced has a temperature of C., and it does not begin to rise in temperature until the ice has entirely disappeared : when this has occurred,, the continued action of heat causes the water to rise in tem- perature. A kilogramme of ice at C. absorbs 80-025 (Bunsen) Calories of heat, and becomes a kilogramme of water at C. Conversely, a kilogramme of water at C. must continue to lose heat until it has parted with 80 '02 5 Calories before it can become a kilogramme of ice at C. ; whence we observe that water does not freeze throughout at the instant of the ther- mometer's touching the freezing point. It was obvious that a kilo, of water differed from one of ice in somehow possessing 80-025 Cals. of heat ; but this was not sensible to the thermometer j hence the heat so possessed by the water was said to be hidden or Latent Heat. "We now know that it is not heat of any kind ; it is latent or potential energy ; work must be done upon ice in order to convert it into the more highly-stressed condition of water : water differs from ice at the same temperature in possessing more potential energy. Direct increase of the kinetic energy of the particles of a heated gas is demonstrated by the Eadiometer. If a surface be heated, a molecule of gas striking against it is heated ; it leaves the hot surface with a velocity greater than that with which it had approached it. If the surface be fixed, the gas in front of it is driven away from it by the bombardment XIIL] THE RADIOMETER. 325 of the molecules which have touched the hot surface, and on their return strike their fellow-molecules ; in front of the hot surface the gas is therefore under a greater pressure than it would have been had the surface been cold. If the hot surface be not fixed, this increase of pressure has reaction being equal and contrary to action a tendency to drive that surface backwards. If the hot surface be the front aspect of a disc, the back of which is by some means kept colder than the front, and if this, disc be suspended in a gas, the heat of the front surface increases the pressure towards the front, and the gas flows round to the back of the disc. Thereafter the disc is struck on the hotter surface by fewer molecules with greater velocities, on the colder surface by a greater number of molecules with lesser velocities ; there is thus compensation ; the result is that the disc is equally pressed upon in front and on the back ; it does not move. Let us now suppose that the particles recoiling from the heated surface do not meet other molecules, but impinge on the walls of the vessel. A layer of particles in such a condition is called a Crookes's layer. This will occur in two cases (1) when the gas is so rarefied that the mean free path of the molecules exceeds the distance between the hot surface and the walls of the vessel ; (2) when, whatever the density of the gas, the opposite wall is so near the hot surface that the distance between them is less than the actual mean free path of the molecules. These conditions, which are substantially identical, may concur : there may be both rarefaction of the gas and approximation of the opposed surfaces. In such a case there is no flow of gas from the hotter surface towards the colder one : each molecule which strikes the hotter surface and rebounds with a greater speed adds independently to the recoil which the hotter surface suffers, and if the hotter sur- face be movable, it is driven backwards. If it be not movable, the particles which rebound from it strike the opposite wall of the containing vessel, and that wall has a tendency to move forward. There is yet another case : if the rarefaction of the gas be extreme, the particles which strike the heated surface are few in number or none at all, there is little or no recoil, and there is no movement set up when the rarefaction is carried too far. The disc of which we speak a disc of which one face is kept hotter than the other may be a disc covered on the one side with some heat-absorbent material such as lampblack, the other face being whitened. When radiant heat or light falls upon the disc, 326 HEAT. [CHAP. even in an equally -lighted field, the blackened side becomes hotter. If such a disc be suspended vertically by two threads, it will diverge slightly from the perpendicular. Such a disc may be attached to the end of a counterpoised rod, the whole being suspended by two threads : the effect of heat or light is to twist the suspending threads to a certain extent. If the suspensory arrangement be replaced by a pivoting one, we have the Radio- meter. A globe of glass, in which a vacuum is made, carries a vertical needle axially fixed, on the summit of which is poised a rotating vane consisting of light rods, to the extremities of which discs are affixed, each similarly blackened on one side. Such an instrument placed in light, even a uniformly-lighted field, has the black sides of its discs more heated than the unblackened sides, and if the radiance be of sufficient energy the vane rotates. Moonlight is too weak to produce this effect : a candle will make a sensitive radiometer rotate ; a paraffin lamp without a globe will at close quarters make the vane fly round so fast as -to be invisible. If a radiometer be floated in water, and if the vane be so constructed one of its spokes being a magnet that a powerful magnet in the neighbourhood can hold it motionless, when the radiometer is exposed to light the bulb itself will rotate in the water in which it floats. The radiometer is a machine in which heat (generally derived from the transformation of light into heat) is directly converted into the energy of work. The less the distance l^etween the discs and the walls of the bulb, the greater will be the effect, and the faster will the vane rotate, provided that the rarefaction is less complete than that which gives the greatest effect. Too complete a rarefaction is not an advantage, for it leaves an insufficient supply of working molecules. When the distance between the disc and the opposite wall is excessively small, the vacuum need not be very good ; indeed the effect of repulsion may be made manifest even in the open air. When a drop of water is placed upon a very hot iron it assumes the so-called Spheroidal State ; it does not wet the hot iron, but gathers itself into a drop, which rapidly evaporates and alters the local conditions of its surface tension so as to present an appearance of varying scroll-work on its surface, while the drop oscillates so as to present the form of rosette and other patterns, these being due to the formation of nodes and vibrating xni.] SPHEROIDAL STATE. 327 loops. The drop may be of very considerable dimensions several ounces in weight. It does not touch the iron ; there is an intervening layer of aqueous vapour on which it floats ; through the space intervening between the drop and the hot solid the light of a candle may be seen. This layer of aqueous vapour is a " Crookes's layer ; " particles strike the heated surface, re- bound, and strike the liquid, thus maintaining a clear space between the metal and the drop. Ether and small drops of bromine float in the same way on the surface of hot water. A lump of carbonate of ammonia thrown into a red-hot platinum crucible assumes the Spheroidal State superficially, but does not melt. The hand can be safely immersed in melted metal if it be not too dry, and if the immersion be effected with a certain degree of prompt deliberation ; a Crookes's layer intervenes between the hand and the metal. When liquid sulphurous acid is dropped into a white-hot platinum crucible it sinks greatly in temperature on account of its rapid evaporation and its slow reception of heat across the Crookes's layer ; if a little water be added to it the water freezes. Ice can thus be produced in a white-hot platinum crucible. A similar Crookes's layer is formed if a quantity of solid carbonic dioxide be lightly placed on the tongue; the extreme cold (__80 C.) is not felt. When the hot solid-body cools down, the Crookes's layer dis- appears, the liquid suddenly comes in contact with the solid still relatively hot, and the liquid explodes in vapour. This occurs in the case of water and iron at about 180 C. Increase of Temperature. We have freely made use of the term temperature because it is a term in common use, and not likely, so far as we have used it, to lead to ambiguity. We have still to defer the consideration of thermometry ; but we must now consider increase of temperature as directly due to increase of the molecular kinetic energy of a body. When we double the molec- ular kinetic energy of a hot body we double its temperature. Observe that it is not asserted that we double the temperature when we double the total energy of a body : some may disappear in doing work, as latent heat. This implies that there must be some point of Absolute Zero of temperature, independent of the conventions of Fahrenheit, Celsius, and others, afterwards to be explained a point of absolute cold, beyond which no cooling is conceivable. We have already seen that in a perfect 'gas one in which 328 HEAT. [CHAP. there is no complication due to inter-molecular forces the pres- sure is proportional to the kinetic energy ; the temperature is, or may by definition be held, proportional to the kinetic energy ; it follows that the temperature is proportional to the pressure. It is found that in all gases the pressure diminishes by about -^^ for each centigrade degree of cooling, the temperature of C. being the starting point, and the volume being maintained con- stant. If a gas could be cooled down in this way to 273 C. (a feat unachieved), it would have no pressure and therefore no temperature, for it would have no kinetic energy, no heat. The Absolute Zero of temperature is therefore 273 C. (or more accurately 273'72 C.), and the Absolute Temperature of a body whose temperature, as measured by the centigrade ther- mometer (see p. 358), is x C., is (273 -f x) Abs. ; thus the boil- ing point of water, 100 C., is 373 Abs. Specific Heat. The heat-energy of a molecule of hydrogen is equal to that of a molecule of oxygen ; but the latter weighs sixteen times as much as the former, and a mass of hydrogen contains sixteen times as many molecules as an eqiial mass of oxygen under similar physical conditions. Hence a given mass of hydrogen at a given temperature possesses sixteen times as much heat-energy as an equal mass of oxygen at the same tem- perature. To produce a given rise in the temperature of a mass of hydrogen we must supply sixteen times as much heat as we would find necessary to produce an equal rise of temperature in an equal mass of oxygen; the specific heat or thermal capacity of hydrogen is sixteen times that of oxygen. In general, the lighter the molecules of which a substance is made up, the more numerous must they be in a given mass, and the higher the specific heat of the substance, i.e., the more heat must be expended upon it in producing a given rise of temperature. For reference and comparison a standard of specific heat is necessary ; the specific heat of water is chosen as the standard. One Calorie will raise the temperature of one kilogr. of water from C. to 1 C. ; the specific heat of water is unity. The relative specific heat of any other substance is, accordingly, the fractional number of Calories of heat required to raise one kilogr. of the substance from C. to 1 C. The law here indicated that the specific heat varies inversely as the atomic weight is based on the assumption that a mass of a heated substance behaves like a group of isolated molecules xiii.] SPECIFIC HEAT. 329 which have no action on one another. It is not surprising, accordingly, to find, when heat supplied to a body is spent not only in raising the temperature of a body, but also in doing in- ternal and external work, that the law is only approximately obeyed. Still the approximate obedience is sufficiently striking to have caused Dulong and Petit to enounce it as a law, and as such it bears their name. It has been utilised as one means among others of ascertaining the atomic weight of different elements. For the formula sp. heat oc we may substitute sp. heat = '- at. wt. at. wt.' or sp. heat x at. wt. = const. This constant product, which bears the name of Atomic Heat, is about 6 '4 ; the metals, phosphorus, sulphur, may be said to form a group in which it varies from 5 '8 6 to 6 '9 3. Divergences from the average value are most marked in the case of solid bodies. In the case of carbon, silicon, and boron at ordinary temperatures the product is small, being about 3'3 ; but at higher temperatures the specific heat of these sub- stances increases so that the product 'rises to about 5*5. The Molecular Heat of a compound is approximately equal to the sum of the atomic heats of its component elements : this rule applies with tolerable accuracy to gaseous compounds formed without condensation ; solid and liquid compounds, and even gaseous compounds whose formation from their elements is accompanied by condensation, compounds which in some sense approximate to the liquid state, depart from it to a marked degree. This product the atomic heat of elements, the molecular heat of compounds has the following physical meaning. Of any substance whose atomic or molecular weight we know we may take a number of grammes numerically equal to the atomic or molecular weight; for example, 35*5 grammes of chlorine, 16 grammes of marsh gas ; we may call such a quantity the gramme-atom or the gramme-molecule of the substance. The Atomic Heat or the Molecular Heat of a substance is the number of Calories of heat necessary to raise the temperature of a gramme-atom or of a gramme-molecule of the substance through 1 C. The atomic heat of elementary substances is approxi- mately the same another form of Dulong and Petit's law. The specific heat of a substance determines the temperature which it will assume when a definite quantity of heat is supplied to it or liberated in it. Thus when 1 kilogr. of hydrogen and 8 of oxygen are exploded together, 34,462 Cals. of heat are liberated. If we could assume the action to be in- 330 HEAT. [CHAP. stantaneous, we might assume that none of the heat is lost. The 34,462 Gals, would then be divided among 9 kilogrammes of water- vapour whose sp. heat is -475 ; the temperature attained would be ' = 8061 -3 C. above 9 x '475 the initial temperature. This case is instructive as showing the influence^of dissociation ; for when a temperature of 3000 C. is actually attained, further combination becomes impossible, and the action is arrested, but not wholly, for it is gradually completed pari passu with the loss of heat by conduction or by radiation. If, however, the exploding mixture be allowed to expand, doing external work, the temperature of 3000 may never be attained, and the action may suffer no such check. Dissociation is easy at low pressures. Hence at low pressures the combustion of a candle is incomplete, and its flame is smoky. Where a substance while being heated is not allowed to expand, there is probably no internal work done ; neither is there any external work done; all the heat supplied is applied in raising the temperature. The specific heat in this case is specially known as the specific heat at constant volume. If, however, the substance be allowed to expand while it is being heated, an external pressure being maintained, both external land internal work are done, and in order to effect a given increase of temperature more heat -energy is required than in the former case. The specific heat at constant pressure is therefore greater than that at constant volume, and it is found in gases to exceed it in the proportion of 1*4058 : 1. The ratio ( - = k ) of the specific heat at constant pressure to that at constant volume may be found in two ways. I. Theoretical considerations lead to the conclusion (see Baynes's TJiermo- dynamics, p. 134) that if a gas suddenly exchange its pressure p, its density p, and its absolute temperature t, for others p,, p^ t t , the ratio of the specific heats being k log 1L log log p - , p , p , pt log t- log t- log Z-L. Ps Pi Pf Whence if two of the changes p to p it p to p it t to t tJ can be found, the value of k may be calculated. The experimental adiabatism necessary is very difficult to ensure ; yet Rb'ntgen has performed the following series of opera- tions upon known quantities of gas and determined the value k = 1-4053. 1 . Gas in a reservoir at a pressure p exceeding the atmospheric, at density p, and temperature t Abs. 2. Open a stopcock : air rushes out of the reservoir till the pressure p falls to II, the atmospheric pressure. 3. Immediately close the stopcock. The air within the reservoir is at pressure II, but has been cooled by doing external work during expansion ; it comes to the same temperature as surrounding objects that is, again t Abs. : xin.] SPECIFIC HEAT. 331 it now has the pressure p t and the density />,, which can be found at leisure, and the above formube applied. IL From the velocity of sound in air. This is 33,200 cm. per sec. /rF Newton's law of the velocity of propagation of waves is that v = / . The coefficient of elasticity K is equal numerically to the pressure in a gas if the temperature be constant ; . . v / = / , if the pressure be the at- mospheric. For air p = 0-012,932 grins, per cub. cm. at C. and 76 cm. bar. pr. at Paris ; 11 = 1,013,660 dynes per sq. cm. Hence, according to Newton's law, v = /- 1>013>6 f- = 27997 cm. per sec, but in fact it is found to be 33,200 cm. There is here to all appearance a material divergence from Newton's law ; but it is corrected when we observe that the assumption that the temperature is constant is unfounded ; that a travelling wave of sound subjects the air to adiabatic compression adiabatic * because the heat has not time to become diffused ; that the elasticity of air so compressed is greater than that of air maintained at a constant temperature ; that the ratio of these two elasticities of a gas is otherwise known to be the same as the ratio of their specific heats at const, pr. and at const, vol. ; and therefore that the coefficient of elasticity in the formula should have been, not K the elas- /7j ticity at const temp., but Ic x K. Whence v = / ; 33,200 = 27,997 *J k ; k = 1-40622. The mean value of & is thus 1'4058. In a perfect gas one whose molecules did not act upon one another the specific heat at constant volume would be quite independent of the temperature or of the pressure. In air the specific heat is sensibly constant at all temperatures between - 30 C. and + 225 C., and at pressures from 1 to 10 atmospheres. We shall see that this justifies us in relying upon the indications of the air thermometer. In a perfect gas the pressure at constant volume and the volume under constant pressure would both vary directly as the temperature. The general law is pv oc t, or pv = Ktf, where R is a constant ; this con- stant is otherwise known to be numerically equal to the difference between the two specific heats. When a gas is compressed it becomes heated that is, pro- vided that external pressure have produced the compression, and added energy to the gas by doing work upon it. * If the heat produced had time to become diffused, or if, as might be the case in excessively slow vibrations or rare gases, the gas had time to flow round the vibrating object, so that it could not become compressed or evolve heat, the speed of yg" 332 HEAT. [CHAP. When a gas "is allowed to expand it becomes cool that is, provided it expand against external pressure and sacrifice energy by doing external work. The work done upon or by the gas appears or is lost as heat. The rise of temperature may be calculated, on the express assumption that there is no internal work done an assumption approximately but not perfectly true (Joule) by dividing the whole work done on the gas (measured in terms of Calories of heat) by the mass and by the specific heat of the gas. Saturated vapour behaves in this regard in a peculiar manner. If work be done upon saturated steam at any temperature below 789'8 Abs. (516*8 C.), the heat evolved causes the vapour to become a superheated vapour, and heat must be parted with in order to allow the steam to remain saturated. Conversely, if saturated steam below 516'8 C. be allowed to expand, doing external work while no heat is supplied to it, it loses energy, loses latent heat, and is partly condensed ; and it does not fall in temperature as much as it would do if it were a perfect gas, expanding to the same extent, for the liquefaction of the vapour liberates heat. Thus an expanding saturated vapour, like steam, liberates more energy and can do more work than an expanding gas. Above 5 16*8 C. a sudden adiabatic expansion of saturated steam would, on the other hand, produce evaporation of water in contact with it ; and compression would produce condensation. The above facts are comprised in the equation S being the specific heat of satu- rated steam, t the abs. temperature S = (42,136,000 - ** S = { (42,!36,000 - 38.280.000.000) ^^ | Calories per Below 789 '8 Abs. S is negative ; above that temperature it is positive. The vapour of bisulphide of carbon acts at ordinary temperature like that of water below 516'8 C. ; that of ether, on the other hand, is rendered cloudy by compression even at ordinary temperatures. The specific heat of substances is not perfectly constant at all temperatures : whence the necessity of the qualification "from to 1 C." This want of constancy is, among gases, most remarkable in those which are most condensible ; but among solids and liquids the variations of specific heat are still more remarkable, and indicate differences in the amount of internal work associated with changes of temperature at different temperatures, this internal work being done in effecting changes in the density, the intermolecular stresses, the allotropic form, and so on. xiii.] SPECIFIC HEAT. 333 The specific heat of a body may be expressed by the fraction Increment of heat supplied (measured in Calories) to unit-mass Increment of temperature produced where both the increments are very small : if an amount of heat SH produce &1T a change of temperature 8t, the specific heat is ; and this is one of the Thermal capacities of a body, of which six may be distinguished. ' 1 . Specific thermal capacity per unit-increase of temperature at constant pressure ; the amount of heat required to raise the temperature of unit- mass by 1 C., the pressure being constant. This is called the Specific Heat at constant pressure, K. 2. Specific thermal capacity per unit-increase of pressure effected, the tem- perature being constant. This has no special name. 3. Specific thermal capacity per unit-increase of pressure effected by heat, at constant volume. This has no other name. 4. Specific thermal capacity per unit-increase of volume, the pressure being constant. 5. Specific thermal capacity per unit-increase of temperature at constant volume (the Specific Heat at constant volume, K). 6. Specific thermal capacity per unit-increase of volume, the temperature being constant. This is called the Latent Heat of expansion, I. Under No. 5, heat = k units supplied to unit-mass, produces a unit-increase of temperature ; a rise of temperature i is produced by heat = lei units. Under No. 6, in the same way, heat = Iv produces an increase of volume v. We commit no sensible error if we suppose that when the temperature and volume both vary, the amount of heat which must be supplied to a unit-mass of substance is found by simple addition, and is equal to ki + Iv. Internal Work. If any substance were a perfect gas, heat imparted to it would to no extent be spent in doing internal work against intermolecular forces. There is, however, no such perfect gas, as we shall now show. If our physical gases were perfect gases we would find 1. That the amount of heat evolved on compressing a gas would be exactly equal (when measured in ergs) to the work done in compressing the gas. If the original pressure and volume be p , v , and the new pressure be p t , the work done is p v log : a conclusion deduced from the two equations Po f -Jt**; (1) work done = / pdv ; and (2) p Q v = p t v t (Boyle's law). 2. That when a gas expands doing external work, the gas loses energy ; and that a perfect gas would in this way lose heat exactly equal in amount to the external work done, and would accordingly sink in temperature. 334 HEAT. [CHAP. A vessel, A, of compressed air (Fig. 125) is provided with an exit tube furnished with a stopcock : the extremity of the exit tube dips under water Fig.125. Fig.i26. in a bell jar B. The stopcock is opened ; air flows out ; it replaces the water in the bell jar : in so doing it forces water down against the atmospheric pressure : it thus does work ; $ie air remaining in A becomes cold (Joule). A similar vessel of compressed air (Fig. 126) ; the ex- tremity of the exit tube communicates with the open air. The stopcock is opened ; air flows out ; it thrusts aside the air immediately surrounding the orifice ; the air within A thus does work against the atmospheric pressure : the air remaining in A becomes cold. 3. That if a stream of a perfect gas were checked, the whole kinetic energy lost by the gas would appear as heat in it. The heating eifect of checking a stream of gas may be readily shown (Verdet) by pinching a rapidly-issuing jet of air between the finger and thumb, or by partly blocking it with the finger-tip. In the same way a jet of high-pressure steam when liberated into the free air suddenly expands and partly condenses into scalding droplets ; then a little way further on, by reason of friction against the air and of intermolec- ular friction, it is deprived of its momentum, and is heated so far as to become superheated or gaseous steam ; still further on it again becomes opaque. If the vessel A of Figs. 125 and 126 be connected with another in which a vacuum has been produced, the air in A loses energy and is cooled. The part of the gas which first arrives in B is heated by compression exer- cised by the part which arrives afterwards ; the latter is also heated by having its motion checked : the temperature in B thus becomes higher than the original temperature. 4. That expansion of a perfect gas would not, if no external work were done, affect its temperature : for no internal and no external work Bfcjng done, the amount of kinetic energy possessed XIII.] JOULE'S EXPERIMENTS. 335 by the gas would remain unaltered, and the temperature would be unchanged. There is no gas whose temperature remains unaffected under such circumstances; therefore there is no perfect gas. Fig.i27. ' -r -TTI-J^L- = Compressed .^Vacuum, N Air.A. * I] ^r B. = The apparatus of Fig. 127 being immersed in a large vessel of water, the stopcock is opened ; the air in A is cooled, that in B is warmed ; the amount of heat-energy gained by B is equal to that lost by A : the water surrounding A and B (which must be stirred) is not on the whole perceptibly cooled or warmed. This experi- ment, made by Joule, was believed to show that air did behave approximately as a perfect gas ; for the temperature of the water, and therefore the average temperature of the whole gas in A and B, remains unchanged after opening the stopcock. The objection to this experiment is, that a rise or fall of temperature in the gas, even though by no means insignificant, would under such circum- stances be imperceptible. The mass of water surrounding the vessels A and B cannot be made much less than about 7 kilogrammes : the specific heat of water is high, that of air is low ; and, besides, it is desirable that the experiment be continuous, and that the effects, if there be any, be accumu- lated. Hence a new form of the experiment was devised by Joule and Thomson. A tube obstructed by a diaphragm with a narrow orifice takes the place of the vessels A .and B. Air is forced from A, Fig. 128, towards B. The pressure within A is greater than that within B ; the gas which passes ig.128. into B ultimately becomes simply the same gas with a larger volume : it cannot become cooler by reason merely of its thrusting the exterior air at C out of the tube, for it simply acts as a buffer between the air in A and the exterior air at C, and the exterior work which it does is equal to that done upon it. If the air were a perfect gas, the temperature at B would be the same as that at A. It is found in such apparatus to vary from spot to spot on account of eddies ; these must be got rid of. This is done by substituting for the diaphragm with the single opening a porous plug of graphite or of cotton wool. It is then found that air isnota perfect gas; the temper- ature in B is a little lower than that in A. Energy has been consumed in doing internal work probably in separating the particles of the gas to the extent, when the pressures in A and B both differ little from the atmospheric pressure, of about :r fr f tne whole work spent upon the gas in forcing it through the plug. The proportion of the total energy spent in doing internal work varies from substance to substance, and from condition to condition. In carbonic dioxide, at a pressure varying little either in A or B from the 336 HEAT. [CHAP. atmospheric pressure, it amounts to about T T T ; in air at a presssure in A of 19 atmospheres, it amounts to as much as ^ of the whole. In the case of hydrogen, curiously, there is a slight increase of tempera- ture : the expanded gas has more kinetic energy than the unexpanded gas : energy is liberated when hydrogen expands ; its particles seem to repel one another. In gases the amount of heat which disappears during expan- sion in doing internal work is generally small in proportion to the external work done against the atmospheric pressure : in solids and liquids the internal work done is relatively much greater. When a substance is heated and rises in temperature without being allowed to expand, so much heat is absorbed during a given rise of temperature ; when expansion is permitted, an additional supply of heat is required. The Latent Heat of Expansion of a substance may thus be found by difference. When temperature varies, the volume being constant, the heat, H, supplied to a unit-mass is equal to Id; when the pressure is kept constant and expansion is allowed, the heat supplied, H /5 is equal to lei 4- Iv ; whence TT TT l = '. -. H y is equal to the specific heat at constant pressure, whence _ (K - k) i The latent heat of expansion of gases is tolerably easy to find, for it is possible to heat a gas without permitting it to expand ; but the estimation of that of solids and of liquids passes beyond the bounds of our experimental powers. To ascertain that of water between C. and 100 0., for instance, it would be neces- sary to compare the amounts of heat required to heat a certain mass of water from to 100 C. when it is free to expand and when it is prevented from expanding : but the latter investigation would require the application of a pressure of 8772 atmospheres. In the same way wrought-iron heated through 15^ Fahr. exercises a pressure of 1 ton per square inch. The latent heat of expansion of a substance is, as a numerical coefficient, the number of units (Calories) of heat required to effect unit-expansion in that substance that is, to double its volume and which disappears in doing that work. The work of expansion is always associated with that of raising the temperature ; only in idea can we form an abstract conception of the amount of heat required to effect a certain ex- pansion while the temperature is supposed to remain unchanged. Temperature and volume vary simultaneously, and the physical XIH.] EXPANSION. 337 constant known as the coefficient of expansion states numerically the relation between these associated effects of heat. A substance whose volume is v o at a temperature t o assumes a volume v t at the temperature t t ; the change of temperature is t t - t o ; the proportionate V V V V change of volume is -^ - ; the quotient ^ _ > is the coefficient of ex- pansion. If t t - t o = 1, the coefficient of expansion is -' The coefficient of expansion of any substance is the ratio between the increase of volume which it. undergoes when its tem- perature is raised by 1 C. and its original volume. If a cube of volume v assume volume , its side v^T becomes -v/zT; its o I v-y \/v coefficient of linear expansion is therefore ^ , or approximately J -i. 2 . Thus, if a body measuring a cubic foot on being heated 1 V o assume a volume of 1-0003 cub. ft., the side of the cube (1 foot) has become nearly T0001 linear foot. Since we have in general to deal with expansions proportion- ately very small, we may say that the coefficient of linear ex- pansion the proportionate increase in length, breadth, or thick- ness per degree centigrade is equal to one-third the coefficient of cubical expansion. If I be the coefficient of linear expansion of a body whose length at t o is L q , the length of the body at the temperature t t is L y = L o + L o (t, - * o ) or L o (1 + I (t - t) ). In this equation there are five terms, any four of which being known, the fifth can be found. In some cases many crystalline bodies the coefficients of linear dilatation are not equal in all directions. Crystals have three axes, in the directions of which the coefficients of expansion are not always equal to one another; thus the angles of crystals are often modified by changes of temperature. Substances be- longing to the regular system have the coefficients equal in the three axial directions, and they preserve similarity of figure when heated ; dimetric crystals have two axial coefficients equal, the third different; trimetric crystals have all three coefficients un- equal. Take plates of gypsum, cut parallel to the prismatic axis : cement them together so that the direction of the axis of one plate forms a right angle with that of the other. Heat till the cement is melted ; allow to cool. The unequal contraction in cooling will warp the whole (Fresnel). In the case Z 338 HEAT. [CHAP. of tliis substance a contraction in one direction is associated with expansion in two others. Indiarubber and iodide of lead, as well as water between and 3 0< 9 C., contract when heated : their coefficient is negative. In some substances (zinc and iron) the coefficient of expansion slowly alters with lapse of time. When a hollow body such as a flask or a thermometer-bulb is heated, it expands almost exactly as if it were solid : a glass tube expands as if it were a glass rod. It follows that when a hollow body is heated, its internal cavity increases in volume in the same proportion as it would if it had been occupied by a solid the same as that which surrounds it. Examples of Expansion by Heat. Bodies which, when cold, exactly fill certain apertures, will not enter them when they are warmed. Eailway rails are not laid in exact contact ; allowance must be made for their summer expansion and winter contraction. In designing lattice-girders for bridges, the same necessity must be taken into account. Eailway distance-signals are controlled by rods, which differ considerably in length at night and by day ; provision must be made for tightening them up, or the reverse. If the neck of a stoppered bottle clasp the stopper too tightly, it may be loosened by causing the neck to expand while the stopper does not do so ; this may be effected by winding a string round the neck and pulling it backwards and forwards so as to produce heat by friction ; the neck is heated before the stopper itself is affected. Glass suddenly heated expands superficially while the inside is still cool ; the stress set up may break the glass ; hence the thinner a flask the less risk there is of its cracking when it is heated. A cart-wheel tire is fitted on when it is hot; when it cools down it contracts and holds the rim, spokes, and hub firmly together : if it be originally too small it may break itself by its own contractile tension. The lead on a roof expands by day and contracts at night ; gravity aids the one and checks the other tendency ; the lead creeps down. The same theory has been applied to glacier movement. Applications of Expansion. The Compensation -pendulum is a pendulum constant in length, whatever be the temperature. A simple bar of metal would, by its variations in length, produce oscillations irregularly unequal, the clock going slow in summer, fast in winter. In order to correct this, the bob of the pendulum is suspended from a framework of bars of iron and brass so arranged that expansion of the bars of iron tends to depress the bob : that XIIL] EXPANSION. 339 of the bars of brass tends to raise it ; by proper adjustment these effects compensate one another. The bob itself is sometimes made of a tube containing quick- silver ; the expansion of the suspending bar tends to lower the centre of gravity of the pendulum : that of the mercury tends to raise it ; a proper adjustment of the quantity of mercury in the bob produces sensibly accurate compensation. Sometimes the rod of a pendulum bears a transverse bar, which is loaded at each end with heavy masses. This transverse bar consists of strips of different metals ; in weather warmer than the average the lower strips expand most, distort the bar, raise the heavy masses, and thus raise the centre of gravity of the whole pendulum : in colder weather the reverse effect is obtained, for the lower strips contract most. Measurement of Coefficients of Expansion. In solids the coefficient of linear expansion is found by direct observation. A. bar is heated to a known temperature ; its original length and temperature are known. The elongation of the bar may be mea- sured by a traversing bar with micrometer, or by the method of Fig. 5, or by the expansion of the bar in a tube pushing out a piece of porcelain, which can move outwards but cannot return. The first-mentioned method is by far the least liable to error. I -I t. - t is the coefficient of linear dilatation. The coefficient of cubical expansion may be found by multi- plying the coefficient of linear expansion by 3 ; or, better, by finding the different specific densities of the solid at different temperatures. The mass ( = ) remaining the same, p and p being the densi- \ 9 / ties, the volumes v ( = ) and v ( = ) are easily found ; and (t - t) \ P/ V /M being the difference of temperatures, (^ ^ . - - j,the coefficient of cubical expansion can be computed. If a solid be heated in a flask with a narrow orifice, com- pletely filled with mercury, the mercury expands, the flask expands, and so does the solid immersed in it. The absolute expansion of the mercury is previously known, that of the glass vessel must be known, and the amount of mercury which would fall out of the flask if the flask were completely filled with mer- 340 HEAT. [CHAP. cury and heated to the same degree is already known ; when the solid is immersed in the mercury, a different quantity of mercury escapes from the flask when heated ; the difference is due. to the difference of dilatation between mercury and the immersed solid : the coefficient of expansion of the immersed solid can thus be calculated. In liquids the expansion may be found by observation of the apparent increase of bulk undergone by a liquid contained in a flask. The width of the neck may be ascertained by the addi- tion of known weights of mercury : an apparent rise of the liquid in the neck may be interpreted as corresponding to so many cubic cm. apparent increase of bulk. But it is important to bear in mind that the cavity of the flask also expands, and that the real expansion of the liquid is the sum of the expansion of the cavity of the flask, and the apparent expansion of the liquid in the neck. If the liquid have the same cubical coefficient as the glass, there would be neither a rise nor a fall in the neck ; if it have a less rate of expansion than glass it will sink in the neck, and will then apparently contract ; only when it has a greater coefficient of expansion than the glass will it rise in the neck, and thus under such circumstances manifestly appear to expand. When a thin glass-flask filled with water is suddenly heated it expands before the water contained in it has had time to become heated, and the liquid in the first place appears to shrink into the flask. Then the liquid becomes heated and rises in the neck of the flask. The expansion of a liquid may also be found by observing its density at different temperatures. This may be done by means of separate observations. It may also be done by the observa- tion of the simultaneous heights of a hotter and a colder column F ig . 129 . of the same liquid, which balance one an- other in a U-tube. The heights are re- ciprocally proportional to the densities, and thus it is easy to find the coefficient of expansion per degree centigrade. Tig. 129 shows that each limb of the U-tube is maintained at a constant temperature by surrounding baths (of water, mercury, oil, etc.) whose temperatures are known. The heights of the columns may be measured by means of a cathetometer. The absolute expansion of mercury is by this method found to be (between 36 C. and 100 C.) almost exactly -5^5 of its total bulk per XIIL] EXPANSION. 341 degree centigrade; above 100 C. it increases rapidly with the temperature. The total amount of expansion is thus not quite proportional to the rise of temperature. In gases the coefficient of expansion is nearly uniform, about -g-ii-j for every degree centigrade. Not quite uniform ; for all gases are not necessarily in the same physical condition merely because they are at the same temperature, for some may be near, others far from, their point of condensation ; and the volume of gases is not exactly proportioned to their absolute temperature. The coefficient of expansion in gases may be determined by direct observation, the volume being allowed to vary, while the pressure is maintained constant during a given change of temper- ature ; or inferentially by observation of the increase in pres- sure exercised by a gas when its volume is kept constant during a given change of temperature, coupled with the assumption that Boyle's law is perfectly obeyed, and that the volume and the pressure bear an exact inverse ratio to one another. The latter method, as we shall see, is more valuable in thermometry than in the determination of the actual coefficient of expansion of a gas. If we assume Boyle's law and Charles's law to be both true, we have the equation ^ = const. If the same quantity of gas change in pressure, volume, or temperature, again %. = const. Hence ^ = ^-'. This enables i , t t , us to solve, to a first approximation, such problems as the following : Fifteen litres of air at C. and 761 mm. bar. pr. are heated to 10 C. while the barometer sinks to 759 mm. : what volume does the air assume ? pv 761 x 15 pv t 759 x v t T " 273 " T~ 283 /761 283 1C \ . Whence = x x 15 htres. Again, 15 litres of air at C. (273 Abs.) and 762 mm. Hg. pressure are, when they are heated to an unknown temperature and exposed to a pressure of 1000 mm. Hg., doubled in volume : what is the unknown temperature ? p> _ pfr 762 x 15 _ 1000 x 30 T = ~T ' 27ir~ ~T ; t, = . - ??. 273 = 716-5 Abs. = 443'5 C. We may combine with these equations the two following propositions : 1. The specific density of a gas is numerically equal to half its molecular weight. 2. One gramme of hydrogen measures 11 '1646 litres at C. and 760 mm. bar. pr. 342- HEAT. [CHAP. Problem. Fourteen litres of carbonic acid are measured at 10 C. and 759 mm. pressure : what is their mass ? First reduce the 14 litres to the volume which they would occupy at C. and 760 mm. bar. pr. i.e., / 273 759\ ... ( 14 x - x - ) litres. V 283 760/ 1 44 Each litre of carbonic acid at C. and 760 mm. weighs - . x - grammes. The whole weighs 273 759 1 44\ 1 4 x - x - x - x ) grammes. 283 760 11-1646 2/ Problem. What bulk is occupied by 20 grammes of ammonia gas at 15 C. and 740 mm. bar. pr. ? One gramme of hydrogen occupies at and 760 a bulk of 11*1646 litres; at 15 C. and 740 mm. it would have a volume of (11-1646 x f|-| x yf-g-) litres ; but ammonia gas has a sp. density = y ; hence 20 grammes of ammonia occupy a bulk (20 x Ax 11-1646 x! 8 !x litres. \ 17 273 740/ It may be left as an exercise to the student to find what corrections should be applied to reduce the apparent weight of a substance weighed in air at a given temperature to the real weight at a standard temperature, say C., the coefficients of expansion of air, of the counterpoising weights, and of the substance weighed, being supposed to be known. Fusion. Heat sometimes operates liquefaction of solid bodies. The temperatures at which fusion is effected differ widely: the fusing point of mercury ( 40 C.) and that of platinum, which can only be fused by the oxyhydrogen blowpipe or the electric arc, may be taken as examples. In general there is expansion during fusion ; in such event there may be a small amount of work done against external pressure. If the external pressure be increased the amount of heat-energy that must be supplied in order to effect this external work in addition to the internal work of fusion is proportionately increased. The temperature of fusion is thus in most cases raised by increase of pressure. In the cases of water, antimony, cast-iron, the freezing point is lowered by pressure, because these substances expand when they freeze. Tables of melting points therefore denote the melting points of substances at the atmos- pheric pressure. XIII.] FUSION. 343 We may here state the reasoning by which it was predicted * that an increase of pressure would be found to lower the melting point of ice ; though some of the steps will not be understood until after we have considered Carnot's cycle of operations and his " perfect engine." A cylinder with a square base, 1 cm. square, ffl Fig.iso. contains 1 gramme of water i.e., 1 cubic cm. S is a source of heat at C. ; R is a refrigerator at a temperature TC., very slightly below C. 1. The cylinder is kept upon the source S until the water assumes the temperature C. We now have 1 cub. cm. water at C. 2. Put the cylinder on the refrigerator R ; keep it there until the water is wholly frozen to ice at C. We now have 1-0908 cub. cm. ice at C. (the sp. density of ice being '91674, Bunsen). Work has been done during expansion ; the piston has been thrust upwards through '0908 cm. against an external pres- sure p dynes per sq. cm. ; the work done is 0*0908 p ergs. Put the^cylinder again on the source : the temperature of the source is supposed to be by an infinitely small amount higher than that of the ice. In course of time the ice melts ; now we again have 1 cub. cm. of water at C. It has had heat imparted to it equal to the latent heat of fusion of 1 cubic cm. of water that is, -080,025 Calories = 3,326,881,364 ergs. This amount of heat has been absorbed from the source at C. ; heat has been lost to the refrigerator at T C. The piston returns to its normal position, as we have seen, and the whole contrivance, perfectly imaginary, is a " perfect engine," with ice or water as its working substance, provided that r has a certain value to be deduced from the equation Work done difference of temp, between source and refrigerator Heat absorbed absoL temperature of source 0-0908^ T 273' 3,326,881,364 T 273* _ 273 x 0-0908 ~ 3,326,881,364^" When the external pressure p changes by II = 1,013,663 dynes per sq. cm. that is, when it changes by an amount equal to one atmosphere T changes by '0076 C. This means that such an engine is reversible, and its operation is theoretically perfect when the freezing operation is conducted at a tempera- ture lower than C. by an amount equal to -0076 C. for every additional atmosphere-pressure suffered by the freezing water. If the freezing could occur at a higher temperature than this, there would be production of work by the expanding ice accompanied by a withdrawal of heat from the source insufficient to account for it, and the perpetual motion would become pos- sible. When a piece of ice is placed in contact with another, both being at C., a very slight pressure will, by lowering the melting point, cause a certain * By Prof. James Thomson ; experimentally confirmed by his brother, Prof. Sir William Thomson. 344 HEAT. [CHAP. quantity of ice at the point of contact to melt. When the pressure is relieved the mass solidifies and becomes continuous ice. Ice is not without a small degree of viscosity, and it can slowly flow ; but at temperatures between C. and about - J C. it can be driven through narrow passages by the above process of Re gelation, for when crushed the fragments are relieved of pressure and reunite, again to be crushed and forced onwards. To the small viscosity of ice and to the process of crushing or regelation is to be ascribed the flow of glaciers. Sometimes the fusion-point of a mixture is below that of its ingredients. A mixture of common salt with about 2-^ parts of crushed ice melts at about 18 C. or Fahr.: above this temperature it is liquid ; and when ice and salt are mixed, the result is very cold liquid brine. When the pavements in snowy weather are cleared by means of salt, the brine thus formed being at a temperature of Fahr., or at "thirty -two degrees of frost," penetrates the shoe-leather and chills the feet of pedestrians, while it refuses to dry, the salt being hygroscopic that is, having a great affinity for water. Boiling or ebullition is a rapid process of reduction of a liquid to vapour. Evaporation is thus distinguished from ebullition ; in evaporation particles fly from the surface and mingle with the particles of gas or vapour already existing in the neighbourhood of the surface of the liquid, and drive or repel only a certain proportion of them away from the surface : in boiling, the particles which fly from the surface bombard the surrounding particles so hotly as to drive them all from the neighbourhood of the surface of the boiling liquid, and to take their place. Thus the vapour of a boiling liquid has to exert a pressure which is just a little greater than the atmospheric, 'or, in general, the exterior pressure, whatever that may happen to be ; the vapour of an evaporating liquid exerts a pressure which is only a fractional part of the atmospheric or exterior pressure. Besides, the process of evaporation is restricted to the exterior free surface : that of boiling occurs both at this surface and at the internal surface of bubbles in the interior of the liquid. A liquid may be heated to a temperature above its boiling point, and if there be no bubbles formed, no point at which the action may preferably start, the whole liquid may become over- stressed, like a Eupert's drop, and when it does give way and form vapour, it may do so explosively. This kind of explosive boiling may be observed when water void of air is heated, or when drops of water are suspended in a mixture of light and xiii. ] BOILING. 345 heavy oils of the same specific density as water and then heated, or when water is heated in a glass vessel, especially if it have been carefully cleaned with sulphuric acid. In the last case the surface of the vessel is very uniform, and there is no sharp point or roughness at which a bubble may commence : thus the temper- ature rises above the boiling point until it is brought down by a sudden outburst of vapour, and bumping ensues. There is less of this in a smooth-metal vessel than in a glass one ; still less in a rough-metal vessel ; still less where jagged pieces of platinum or stone have been immersed in the liquid to be boiled. The process of boiling depends to a great degree for its regularity on the presence of air-bubbles : we may sometimes see that water when long boiled ceases to evolve bubbles, and evaporates only at the surface with an occasional outburst of steam. A bubble of air or vapour produced in the interior of a hot liquid is increased in size by molecules escaping into it from the surrounding liquid ; if the temperature of these molecules, their energy, their velocity, their pressure, be such that they can expand the bubble against the surrounding pressure, 'the bubble enlarges and rises. If we artificially produce bubbles in the interior of a heated liquid, as when we electrolyse hot water, the liquid boils very rapidly at the electrodes, where gaseous oxygen and hydrogen are being given off. The boiling point at different pressures. The greater the external pressure to be overcome, the greater must be the energy, and therefore the greater the temperature, of the rising vapour. The temperature of ebullition and the external pressure are not directly proportional to one another, but are found experimentally, and recorded in tables such as those great tables of Eegnault's, to be found in his Relation des Experiences. At mountain heights the atmospheric pressure is less and the boiling point is lower; thus at Quito, at a height of 9540 feet, water boils at 90-1 C. If a flask containing water, boiling at 100 C., be corked and set aside until it has cooled, say, to 90 C., and if the upper part of the flask, the part containing the vapour of water, be suddenly cooled by cold water dashed upon it, the vapour in it will be partly condensed, and a partial vacuum will be formed : the water will find itself at a temperature of 90 C. under a pressure of less than 525-45 mm., and it will again begin to boil: the water is thus seemingly induced to boil by the application of cold to the flask con- taining it. If a cryophorus tube, Fig. 131, of which both bulbs are half filled with water, have one bulb immersed in a freezing 346 HEAT. [CHAP. mixture, the vapour in the cold bulb is condensed ; the vapour in the tube is pushed into the cold bulb by the uncompensated pressure of particles rising from the Flg ' 131 ' N liquid in the warmer bulb ; this pro- cess is continuous ; work is continu- ously done in maintaining the flow of vapour, which is as continuously condensed ; the liquid in the warmer bulb continuously evolves vapour, and does so so rapidly, the pressure being small, as to boil ; it continuously does work, but receives no energy ; it cools and ultimately freezes, even while evaporating. Boiling and evaporation may thus involve not only the giving of momentum to particles of the liquid, but also external work done against resistances; and during evaporation there may be cooling due not only (1) to the latent heat of evaporation absorbed in producing change of state, but also (2) to the external work which is done by the evaporating body work which generally takes the form of thrusting aside the external air. Examples of cooling due to evaporation are: The cooling of the skin by perspiration or by a draught of air, even though the air be warmer than the skin ; a dog cooling himself by panting with his tongue exposed ; a porous water-cooler or alcarraza, the evaporation at the surface of which cools the contained water ; the practice in some hot countries of cooling a room by throwing water over the floor ; the cooling undergone by a liquid which is being rapidly evaporated, as, for example, the rapid cooling of sulphurous anhydride or of ammonia, which is effected in the course of the process of artificial ice-making by the rapid evaporation of the liquefied gases under a powerful air-pump ; the cooling of a jet of liquefied carbonic acid when allowed to escape into the air, so that the substance is in part solidified. The latent heat of evaporation of steam is X = (33011,504000 - 33,200000 0) ergs per gramme, where the temperature of ebullition is Abs. At 994-32 Abs. or 720'6 C., A = 0, and this temperature is for steam the critical temperature, beyond which there is no change of state when liquid water becomes water-vapour. Saturation -Pressure. In the case of every vapour we find that for each particular temperature there is a maximum density ; if we compress the vapour beyond this density, a portion of it will be liquefied. If we allow it to expand, then provided that the temperature is kept constant, and that the vapour is kept in contact with its own liquid a portion of the liquid will be evaporated ; thus the density is maintained constant and the vapour is kept saturated. Each volatile liquid has its own saturation pressure for each temperature, this being the pres- sure necessary to bring the vapour to its maximum density. V XIII.] SATURATION-PRESSURE OF A VAPOUR. 347 A vapour which is not saturated may by compression, exerted until the pressure of the vapour is equal to the saturation-pressure, be made saturated, and by further pressure will be caused partly to condense. The saturation-pressure of any vapour at any temperature is the same as the pressure at which the corresponding liquid boils at that temperature. Even in contact with ice, water-vapour has a saturation- pressure, and evaporation will go on until this pressure is attained. A strong wind blowing over a snowfield may remove much of the snow by true evaporation without liquefaction. Saturated steam in contact with ice at C. has a pressure p = (107-2 + (6255 x r080^)} dynes per sq. cm. (Regnault.) As a general rule each component of a mixture of gases exercises its own pressure, and is not affected by the others which accompany it. Yet this rule is not absolute ; for if we heat in a flask a certain quantity of air alone, we find that it exerts a certain pressure ; a certain quantity of water- vapour introduced alone into a vacuum would exert a certain pressure ; but when both the water-vapour and the air are introduced into the same vessel, the joint pressure falls somewhat short of the sum of the several pressures, and thus it is shown that there is an attractive action between water and air. Vapours at variable pressures and temperatures generally obey Boyle's law with tolerable regularity until the pressure comes up to about -f^ of the saturation pressure, and that whether they be alone or mingled with air or other vapours. Fig.132. Measurement of Vapour Density at different temperatures. a. By measurement of the pres- sure exercised by the vapour of liquid at a series of known temperatures. This is effected by the arrange- ment sketched in Fig. 132. The mean temperature of boiling is indi- cated by four thermometers, two in the liquid, two in the vapour : the vapour is condensed in A and re- turned to the flask : the pressure is measured by a manometer. The use of this method depends on a tacit assumption that Boyle's law is obeyed throughout all ranges of temperature ; but this method is not applicable except at low temperatures 348 HEAT. [CHAP. and low pressures ; for at high pressures the vapour assumes abnormally small volumes as it approaches its saturation-pressure. /?. By measurement of the volume occupied by a known weight of fluid, or by measurement of the weight of vapour which can occupy a known volume. The first of these methods is that of Gay Lussac. A tube filled with mercury is inverted like a Torricellian barometer in a vessel of mercury, and has a Torricellian vacuum at its upper part ; the whole is immersed in a bath of liquid kept at a definite temperature. A little bulb containing a known quantity of the liquid to be vapourised is passed up into the tube ; being heated it bursts ; the vapour occupies a certain volume of the tube ; the mercury stands at a certain height in the tube. The mercury stands at a different height in an ordinary barometer ; the difference of readings indicates the pressure exercised by the vapour. Its weight is known, its volume, and its temperature. A series of observations is made at different bath-tempera- tures. It is difficult to ensure that all the substance is liquefied. The second method is that of Dumas. A bulb with a long-drawn neck is filled with liquid and immersed in a heated bath. The liquid in the bulb violently rushes out in the vapourous state through the narrow neck ; this ceases and equilibrium is set up ; the bulb is filled with vapour at the temperature of the bath. The end of the neck is then sealed by a blowpipe- flame ; the whole is removed, cooled, weighed. This gives the weight of bulb + vapour; already the weight of the bulb, its volume, the bath temperature, are known ; the density of the vapour occupying the bulb at the temperature of the bath can be thus found. At high temperatures bulbs of porcelain or iron, and baths of mercury-vapour, sulphur- vapour, or zinc- vapour, may be used (Deville and Troost). The density of saturated vapour. Fairbairn and Tate found the density of saturated steam by introducing into a recipient of known capacity and devoid of air a known quantity of water, and by measuring the temper- ature at which the whole of the water was evaporated. The measurement of the pressure of unsaturated vapour, if it present itself alone, is simply the measurement of gaseous pressure, and calls for no further remark. The measurement of the pressure exercised by an unsaturated vapour which forms one of the components of a mixture is in one case that of Aqueous Vapour in the Air a matter of importance. A numerical example will illustrate this. If water be exposed to a pressure of 9'16 mm. of mercury ( = 0-01205 atrnos.), it will boil at 10 C. ; if water- vapour of such density (supposed constant) that it exerts a pressure of 0*01205 atmos. be exposed to a temperature above 10 C., it will be unsaturated ; at 10 C. it will be saturated ; at any temperature below 10 C. it will be in part con- densed. 10 C. is, then, the condensation temperature for aqueous vapour of this pressure of '01205 atmos., just as the latter is the saturation pressure for aqueous vapour at a temperature of 10 C. If, now, we take moist air containing aqueous vapour and air in the proportion of '01 205 to 0'98795, at the ordinary atmospheric pressure: at any temperature above 10 C. it will not deposit moisture ; at 10 C. it will begin to do so. 10 C. is the condensing temperature or Dewpoint for air containing this pro- portion of moisture. To other proportions of moisture other dewpoints correspond ; these can be found in any table of the boiling points of water at different pressures. Hence, if we can find the temperature at which air xiir.] HYGROMETER. . 349 containing aqueous vapour begins to deposit moisture, we can by reference to such tables find the proportion of aqueous vapour in the air. This is effected by a Hygrometer. The essential part of a hygrometer is a glass or, better, a smooth silver surface which can be cooled down until- the moisture of the air begins to deposit as a film upon it, and whose temperature at the instant of the dimming of its brightness can be accurately ascertained. The surface may be fashioned into a bulb : this bulb may contain ether ; the bulb may be cooled by blowing through, and thus rapidly evaporating the ether ; the temperature at the instant of dimming of the surface can be read off on a thermometer whose lower end is dipped in the evaporating ether. The whole may be allowed spontaneously to become warmer ; as it does so the film disappears : the temperature at which this occurs is noted. The film is again caused to appear and disappear ; by dint of repetition a mean point between the highest temperature of appearance of the film and the lowest temperature of its disappearance is obtained, which is the dewpoint required. Another method for ascertaining the dewpoint one for doing so by a single observation is the following : If a thermometer bulb be by any means kept cool by evaporation being covered with a wet piece of linen which dips in water, or the like the bulb is cooled ; the extent of cooling depends on the rapidity of evaporation : the rapidity of evaporation depends on the humidity of the air that is, on the ratio between the amount of aqueous vapour actually present in the air, and that which might be present at the temperature of the air if the air were saturated with moisture. The less the humidity of the air, the greater will be the evaporation, and the greater will be the difference between the readings of a thermometer kept cool in this way and those of a thermometer subjected to normal circumstances. Tables have been constructed in which, for each reading of the " dry bulb " and of the "wet bulb," the corresponding percentage of aqueous vapour in the air is recorded. Dew. When, on a clear night, the earth, stones, plants, etc., become cool by free radiation, their temperature may sink below the condensation temperature proper to the particular percentage of aqueous vapour in the air. When the temperature thus sinks below the dewpoint, the moisture of the air is partly deposited in the form of dew ; and the more highly charged with moisture the air had become during the day, the earlier and the heavier is the deposit of dew at night. Changes of Elasticity. Among other changes of state effected by heat it is important to note the change produced in the elasticity of a body. If a gas t>e kept at the same temperature, its elasticity its resistance to a small compression is numerically equal to the pressure which it exerts. If a gas be compressed in such wise as to lose no heat-energy, it becomes heated, exercises more pressure, and offers more resist- ance to compression, and is therefore more elastic ; its elasticity 350 HEAT. [CHAP. under adiabatic compression is greater than that at constant tem- perature in the ratio of 1*4058 to 1. TRANSFORMATIONS OF HEAT. Transformation of Work into Heat may be effected directly by the agency of friction, or indirectly by the transformation of kinetic energy into the energies of noise, light, electrical condition, which are in their turn converted into heat. Even the conver- sion, apparently direct, by the agency of friction may be due in the first place to the generation of local electrical currents or stresses, the energy of which is afterwards converted into heat. Transformation of Heat into "Work. We know that work done = fs, the product of the average force acting into the space through which it has acted. From our previous discussions of the Indicator - Diagram we understand that the work done by any substance during expansion can be represented by the area pp'v'vp (Fig. 133), where Op, Op' represent the original and final pressures, Ov and Ov' the original and final volumes. The work is positive, done by the expanding substance (steam, air, etc.) if the expansion be positive, from Ov to Ov' ; negative, done upon it if the expan- sion be negative, as from Ov' to a less value Ov. Fig.133. BBUB1 ; r!i.: L . Fig.134. Where work is done both by and upon the working substance, as in Fig. 134, the negative work p'p'v'v'p' being subtracted from the positive work vp'v'vpp, there is left an area pp'p ff v"v, which represents the work done. If the curve pp'p" be complicated, the total work done may be found by dissecting the figure ; any complex operation may be resolved into a number of simple ones of which each produces its 6wn effect ; the work done is found by a process of summation of positive and negative areas. When the working substance returns to its original volume and pressure XIII.] CARXOT'S CYCLE. 351 Fig.135. as in Fig. 135, the shaded area again indicates the amount of work done by the working substance, just as if in Fig. 1 34 the line p"v" had been made to coincide with pv. The work is positive if the change of pressure and of volume have been effected in the direction of the arrows ; negative if effected in the contrary sense. Such an operation is a Cycle. The advantage of studying the amount of work done by a working substance oper- ating in a cycle is that we are not called upon to take the internal work into account. The body returns at the end of the opera- tion to its primitive condition, and there is no balance of work done either by or against internal forces. Into the consideration of a cycle we in- troduce an assumption that it is possible for a working substance to return to the same condition as regards pressure and volume at the original temperature ; this might not have been true as regards any actual substance, though it is theoretically true as regards perfect gases ; it is, however, actually true as regards physical gases, for the elasticity of gases is perfect. We must choose some particular kind of cycle for our ideal operations ; that to be explained is the one best adapted for the study of the relations between work and heat, and was devised in its primitive form by Sadi Carnot ; it is hence known as Carnot's cycle. If a gas expand at constant temperature, we know by Boyle's law that the pressure and the volume vary inversely ; this law can be expressed graphically by an equilateral hyperbola, for in that curve xy = const. The pressures and volumes at different temperatures are represented by points Fig.136. Fig.137. on different" hyperbolas. Imagine those of Fig. 136 to represent portions of these curves for temperatures 6^ Abs. and # 2 Abs. for a given mass of substance. This substance, at the temperature 2 and pressure p v will have the volume v : to the pressure p 2 at the same temperature corresponds volume 352 HEAT. [CHAP. v 2 ; if the temperature be O l and pressure _p' 1} the volume will be not v, but v t , a point on the higher hyperbola, on the line the so-called Isothermal line corresponding to the temperature 6^ Abs. Expansion of a gas involves a more rapid fall of pressure when it is effected adiabatically than when effected at constant temperature, for the gas cools down : the Adiabatic lines, which express the relations between pres- sure and volume when heat is neither supplied nor allowed to escape, slope more steeply than the jsothermal lines for the same substance. Fig. 137 represents these lines, and shows the relations between the pressures and volumes of a substance starting from conditions p l9 1> 15 # 1} and p l} v, $ 2 , which correspond to those of the previous figure. Let us now superpose the two figures 136 and 137, and we obtain Fig. 138, and are now prepared to understand Carnot's cycle. Fig.138. ' The steps of Carnot's cycle 1. Starting with our working substance at the condition p lt v v 0^ Abs. (point A), we allow it to expand at the temperature 9^, this temperature being maintained constant. Eunning through successive pressures and volumes represented by successive points, on the isothermal line 6 V it assumes a pressure, say, p^ and volume v^ (point B). Work is done equal to P-.V-. log ^L = the area ABv 1 V 1 . This work is *1 done at. the expense of heat-energy supplied to the working -substance from an external source. 2. Starting from the condition p^ t v^, 0^ (point B), we allow the work- ing substance to expand adiabatically, until it assumes the temperature # 2 and the corresponding condition p 2 ' t v 2 ', # 2 (point C). BC is a part of the adiabatic line passing through B and cutting the 2 isothermal in C. Work equal to the area BGt^'t^' is done by the expanding substance, but at the expense of its own heat-energy, for. no heat is supplied to it. 3. The substance is now compressed until it assumes the condition p 2 , v^i $ 2 that is, until it runs from C so far up the isothermal line $ 2 as to meet at D an adiabatic line, which passes through the original point A. "Work is done equal to the area CDv^v^ = p 2 v 2 1 ~ kut it is done upon XIIL] CARNOT*S CYCLE. 353 the working substance, for that substance is compressed, and heat to a corre- sponding amount is lost by the working substance, passes to surrounding objects, and may be wasted by conduction and radiation into all the universe. 4. The body, from which no more heat is allowed to escape, is now sup- posed to be still further compressed until it has regained its original condition PI> v i> V- Work is done on the expanding substance, but appears as heat in the substance, not as external work either positive -or negative^ and the temperature rises, for no heat is allowed to escape. The whole energy, supplied to the working substance from the source is p^v t log -1 ; that wasted is p 2 v 2 log ; that utilised is ^i r i lo g ^ - P2 v z lo g ^f i - 7 ^ of the whole. This can be proved (Verdet, Ther- Pi v i lo S {L modynamiquej p. 127) equal to Q3 ?3LJ of the whole, for ^- = ^L when p l v 1 v 2 Vl the area ABCD is very smalL * But p : v t = E,0 l ; p 2 v 2 = R0 2 , where O t and 2 are the respective temperatures. Hence the proportion of energy utilised is R& - R0 9 6> - 9 1 - 2 or -J ? of the whole. The working substance operating in such a cycle acts as a distributor of energy ; it divides H, the heat-energy supplied- to it from the Source of heat, into two parts : one part, H lf passing to the Condenser, is lost by conduction and radiation ; the remainder, W, is usefully converted into external Work. The heat H is supplied at the higher temperature 0^ ; the quantity of heat Hj is lost to surrounding objects at the lower temperature & 2 > the Efficiency of such an ideal arrangement, the ratio ^Heat utilised _ H - H x _ W _ O l - 2 ' Heat supplied = H ~ H " ~~0^~ Thus, so far as Carnofs cycle is concerned, even though we could find a working ^ubstance and construct a machine which could carry the cycle out in practice, yet there would be a great waste of heat-energy, unavoidable unless >we had a condenser at a temperature of absolute zero. If the temper- ature of the boiler of an ideal engine competent to work out Carnot's cycle wqre 120 C. (393 Abs.), and that of the condenser C. (273 Abs.), the OQO _ 97*? work done^by such an engine could not exceed - , or about 30 '6 per cent of the whole energy supplied as heat. The cycle above considered is reversible ; each step in it can be retraced or could be retraced if we could construct an engine capable of working without waste of energy in noise, friction, excessive conduction and radiation of heat, and the like work being done not by but upon the engine as it is driven backwards. The effect of reversing such a cycle would be that work W being done upon the engine, the quantity H t of heat would be taken from the condenser, and the quantity- H -of heat would be communicated to the source. Any engine must operate through periodic cycles if its action is to be continuous : and in every engine there is an absolutely necessary waste of 2 A 354 HEAT. [CHAP. energy arising from the necessity of restoring the engine to its primitive position in order that its piston may repeat its effective thrusts. Carnot's ideal " perfect " engine is one which, with a working substance capable of returning to its primitive condition, will work out the reversible cycle above described, and thus attain the efficiency above indicated : an engine which wastes no energy otherwise than by restoring the primitive condition of its working substance. The perfection of a perfect engine depends not on the nature of the working substance, but on the reversibility of the cycle which it operates, and the efficiency of such a reversible engine depends only on the temper- atures between which it works. Carnot's Principle, as enounced by himself is the motive power of heat is independent of the material agents employed to realise it ; its quantity is determined solely by the temperatures between which the " transport of Caloric"* is effected. W The efficiency = (t, t - t), where t = X and t =. 6 1 - 2 . = o - ^ (o . The efficiency depends upon t, the difference of temperatures between the source and condenser, and upon -^ (), a function of t which is called Carnot's function, C. ~. T , , difference of temperatures t We have also seen that efficiency = - = temperature of source t Hence C = ; and Carnot's function is the reciprocal of the absolute tem- perature. The efficiency of a Carnot's reversible engine is greater than that of any other engine. If it were possible to devise a more efficient engine it might be employed with the expenditure of a certain amount of heat to drive a reversible engine backwards ; the source and the condenser of the Carnot's engine might be the same as those of 'the better engine : the Carnot's engine would be occupied in restoring to the source the heat taken from it by the better engine ; on the whole, a surplus of work would during each cycle be done by the conjoined mechanism a surplus not accounted for by heat lost by any body a creation of energy. If the better engine were employed in driving a larger Carnot's engine backwards, there might be no surplus, no external work done ; but a greater amount of heat would be conveyed to the source by the reversed Carnot than would be taken from it by the more efficient but smaller engine, and the whole heat of the universe might be, step by step, induced to travel through the condenser into the source of the conjoined mechanism a conclusion evidently absurd. That it is absurd is the aim of the Second Law of Thermodynamics to state : Heat cannot of itself pass from a colder body to a hotter one, nor can it be made so to pass by any inanimate material mechanism : and no mechanism can be driven by a simple cooling of any material object below the temperature of surrounding objects. The word simple, or some equivalent word, is necessary in the above statement of the second law for the following reason : A quantity of com- * An expression implying, as in his day, the material theory of heat. XIIL] SECOND LAW OF THEKMODYNAMICS. 355 pressed gas can do external work, and in so doing cool itself below the temperature of surrounding objects ; but its cooling is not a simple loss of heat-energy ; there is a concurrent change of condition of the gas, a change which cannot be reversed without the expenditure of heat exceeding in amount the heat converted into work by the expanding gas. This being admitted, we may reason backwards and arrive at the ratio B B of efficiency 1 ? in a reversible engine as a direct corollary of the pro- ?! position ; and the statement of that ratio of efficiency in a reversible engine is also known as the Second Law of Thermodynamics. This Protean law assumes another form, apparently different from but essentially identical with both the preceding. Temperature being assumed proportional to the total heat-energy, the total amount of heat-energy H supplied at the higher temperature # x is proportional to O l ; H = c6 t ; JT j- = c. Similarly H x , the heat lost to the condenser at the lower temper- TT TT TT ature 2 , is H 1 = c^ 2 ; 1 = c. Here _i = 0. In general, in the #2 "l ^2 TT /"/Iff most complex reversible cycle, 2 - = 0, or / = (Thomson) ; an ex- u / u pression very convenient for mathematical purposes, but difficult to translate into words. In a perfect, a reversible cycle, the sum of the equivalences of all the transformations is zero (Clausius). In a non-reversible process the sum of the transformations is positive. According to Rankine's mode of expression, substantially identical with the preceding, the second law is : If the absolute temperature of a uniformly-hot substance be divided into any number of equal parts, the effect of each of those parts in causing work to be performed is equal. This implies that the absolute temperature is propor- tional to the total heat-energy, and so merges into the preceding form of the second law. Lastly, Carnot's principle itself is often called the Second Law of Thermo- dynamics. We have already studied the direct transformation of heat into work in the radiometer. In the steam-engine the heat of the steam may be in part converted into work; the piston is bombarded by the particles of the steam, and if the resistance to its onward movement be not excessive, it is thrust forward by the joint impact of the particles which impinge on it, their several components of motion parallel to the piston-rod being effective in this respect. Even under the most favourable circumstances which can be conceived heat cannot be wholly converted into work by any form of continuously -acting mechanism. The efficiency of the ideal perfect engine small though that efficiency be is never approached in practice ; and the efficiency of the human body considered as a machine one-fifth of the total energy supplied 356 HEAT. [CHAP. to it being capable of utilisation is remarkable when we consider the narrow limits within which it operates. Work can thus be wholly converted into heat, but heat can never be wholly converted into work ; whence a universal tendency to the Degradation of Energy into Heat, the lowest of its forms. MEASUREMENT OF HEAT. Temperature we have now seen to be, when measured from an absolute zero a zero of absolute cold (1) proportional to the absolute amount of molecular kinetic energy, and (2) to be the reciprocal of Carnot's function. What is meant by equal degrees of heat ? Why is the differ- ence between C. and 1 C. supposed to be equal to that between 100 C. and 101 C. ? In a perfect gas equal differences of temperature correspond to equal increments of energy. In a diagram containing a system of adiabatic and isothermal lines, the isothermal lines must be so drawn as to cut off equal areas between the adiabatic lines. Absolute zero would correspond to total absence of heat- energy. If we had a perfect gas at command we might measure temperature by its means in either of two ways : (1) We might observe its pressure at constant volume: equal increments of pressure correspond to equal increments of temperature. (2) We might observe its varying volume at constant pressure : the volume is inversely proportional to the pressure, and within narrow ranges equal increments of pressure approximately correspond to equal increments of temperature. The former is the more accurate method. We have no perfect gases to experiment upon : air, etc. are not perfect gases. Yet we may perform either of the above opera- tions on a quantity of air confined in a flask, and thus construct an air thermometer. The former method that of observation of pressure is here doubly preferable to the latter that of observation of expansion because in the former there is no waste of energy in doing internal work, and the increase of pressure is appreciably the same as that of a perfect gas. The indications of an air thermometer used in this way may hence be assumed as an accurate standard of comparison. XIII.] TEMPERATURE. 357 By it we find that for a fall of 1 C. (from 1 to C.) on the mercurial thermometer, the pressure sinks in the ratio of 274 to 273; hence the tem- perature sinks in the same ratio, absolute zero is 273 C., and Carnot's function has the numerical value of for a temperature of C., and of for a temperature of t C. 273 + t Two bodies are said to be at different temperatures when the one has a tendency to lose heat to the other ; to have the same temperature when there is no such tendency : and bodies are at the same temperature when they have the same heat-energy per molecule, not per unit of weight. Differences of temperature may be roughly perceived by the hand ; the sense of temperature can even be cultivated like that of musical pitch so as to arrive at approximate accuracy without reference to a standard of known temperature. Any of the effects of heat may be used for detecting the presence of heat and for constructing a thermoscope. Arbitrary graduation of any thermoscope will enable it to be used as a thermometer. Br6guet's metallic thermometer is a spiral strip composed of three metal strips soldered together by their broad surfaces : the different rates of expansion cause the spiral to roll or unroll ac- cording to the variations of temperature, and thereby to move a pointer. The air thermometer, one of whose forms is shown in Fig. 139, is principally used as a standard of reference. AD is a manometer, in which above D there is a Torricellian vacuum : H is an air chamber, E an auxiliary cistern of mercury. As far as the mercury A is depressed below a certain mark, so far is the level of mercury at B raised by rais- ing the mercury cistern E, closing the stopcock, and effecting a fine adjustment by means of the screw C. The volume of the gas between B and A is thus made constant, and the column of mercury AD measures the pressure on the gas. The same thermometer may, by an adjustment of the height of the column AD, be used as a stant-pressure-and- variable-volume air-thermometer. Fig.139. con- 358 HEAT. [CHAP. The air thermometer presents the disadvantage of being ex- tremely unwieldy ; but it has the advantage that the expansion of the glass produces little effect in causing a difference between the apparent and the real expansion of the air, or in vitiating the adjustment to constant volume. The ordinary mercurial thermometer is a familiar object. Its simplest form would be that of a flask with a long neck. If the neck were open, the mercury would be in danger of accidental loss and of evaporation ; the neck must therefore be closed. If it were simply closed, the air contained in the neck would at high temperatures be compressed ; the bulb would burst ; hence a vacuum must be produced in the upper part of the neck. This vacuum is produced by closing the tube while mercury is boiling within it ; on cooling the mercury contracts and retracts, leaving a space containing only a certain quantity of the vapour of mercury. As to the graduation of the mercurial thermometer, this might be effected by comparison with an air thermometer, a troublesome process, resulting in degrees true but unequal in size ; or by taking advantage (Eenaldini) of the fact that the " freezing point" of water or, better, the melting point of ice and the " boiling point" of water or, better, the temperature of steam at the pressure of 760 mm. Hg. are constant tempera- tures, and may be taken as fixed points ; that the height assumed by the mercurial column at these two temperatures may be marked on the tube ; and that the tube between these two marks may (Newton) be mechanically graduated by equal division into degrees a method certainly con- venient, but only approximately correct. The boiling point of water is estimated by inserting the thermometer in an atmosphere of steam surrounded by a steam-jacket (Fig. 140), intended (Berthelot) to check irregular condensation. The pressure must be the standard, 760 mm. Hg. The "freezing point" must be determined by the position assumed by the mercury when the water which trickles off melting ice flows in a stream over the mercury bulb, the whole being surrounded by a jacket of melting ice. On the Centigrade thermometer (Linnaeus) the " freezing point" and the boiling point are respectively and 100 C. ; on the Fahrenheit scale they are 32 F. and 212 F. ; F. being the lowest temperature attained by Fahrenheit (Phil. Trans. 1724) by means of a mixture of ice, water, and salt or sal-ammoniac. FI HO XIIL] THERMOMETERS. Fahrenheit did not use the boiling point of water as a imagined his zero to be an absolute zero, and then made or i the freezing point of water to stand at one-third between this absolul and the temperature of the human body, which for convenience he called 96. Water has been used as the expanding substance in thermometers ; it is objectionable on account of its point of maximum density. Alcohol is used at very low temperatures, because it is not readily frozen. Mercury, which is very advantageous on account of its low specific heat and its ready response was brought prominently into notice by the astronomer Halley. The sensitiveness of thermometers the power of reveal- ing minute variations of temperature is increased by narrowing the tube or by enlarging the bulb. A large bulb is, however, inconvenient; because it is difficult of insertion in apertures a fault which may be remedied by giving the bulb a cylindrical form ; because it may alter materially the temperature of the object whose temperature is to be ascertained ; because it slowly equalises its temperature with that of the object. A narrow tube is inconvenient because a narrow thread of mercury is difficult to see ; this may be remedied by using a tube of flat elliptical sec- tion, and by enamelling the back of it. The main causes of error in the use of a thermometer are, that the graduation alters, the "zero rises," or a thermometer inserted in melting ice comes in course of time apparently to indicate a temperature somewhat above C. or 32 F., this effect being probably due to a slow yielding of the bulb to atmospheric pressure; and further, that it is not always possible to ensure that the whole of the mercury is at the same temperature. In testing a thermometer it is important to see that the " freezing point " and the " boiling point " are accurately indicated by it, or that it agrees with a thermometer in this respect correct ; and that the bore of the tube is uniform, so that a little detached portion of the thread of mercury may occupy an equal length in all parts of it. For accurate comparison of thermometers they should be immersed together in a cooling fluid rather than in one which is being heated (Fourier) ; the temperature indicated by a thermo- meter in a cooling fluid is always a little higher than that of the fluid. For practical details connected with testing thermometers see Gscheid- len, Physiologische Methodik, p. 76. For observations of the temperature of the skin it is well (Colin) not to cover the bulb with flannels, or to leave the thermometer in such circum- stances for too long a time, for the skin assumes the temperature of the interior ; rather should quickly-acting thermometers be used. Apply a ther- 360 HEAT. [CHAP. mometer quickly, fresh from the pocket or the hand : keep it closely in contact with the skin ; avoid flowing on the bulb ; put a little cupola of paper or cotton over the bulb, but not in contact with it. Special forms of Mercury Thermometers. In the maximum thermometer above the column of mercury a small bubble of air is intro- duced ; above this a little thread of mercury. When the temperature rises, the air is compressed, the thread is pushed upwards ; when the temperature falls back, the thread of mercury does not return. The minimum thermometer is usually a spirit thermometer with a little broad-headed piece of wire loosely fitting in the spirit. It is ad- justed with its head touching the surface of the thermometric liquid. When the liquid contracts, surface tension drags the wire with it ; when the temperature rises, the liquid passes the wire without forcing it upwards : the position of the end of the wire nearest the free surface indicates the lowest level to which the surface had sunk, and therefore the lowest level which had been attained since the last observation. In Metastatic thermometers any part of the mercury may be removed from the column and shaken aside into an apical cavity, or restored in whole or in part to the main thread ; the thermometer, a very delicate one, being thus competent to read to very small fractions of a degree at any part of the scale chosen at will. See Gscheidlen, p. 84. The principle of overflow liquid being caused to expand and overflow, or vapour (iodine, mercury) being boiled out of a heated flask, what remains being weighed when cooled is utilised in the construction of some pyrometers. As for the measurement of temperature by electricity, that will be ex- plained on page 559. Calorimetry or the quantitative measurement of heat. The Calorie (Co) is the amount of heat required to raise the temperature of 1 kilo, of water (or kilos, of any substance x whose specific heat is x) from C. to 1 C. The calorie or small calorie (ca) is the amount of heat similarly required to heat one gramme to the same extent. Convenience rules the choice between these units. 1. The Method of Mixtures. This may be illustrated by a numerical example. How many calories of heat does a gramme of mercury absorb when it is heated from C. to 1 C ? a question identical with an enquiry as to the specific heat of mercury. One gramme of mercury at 100 C. and one of water at 0* C. are mixed: the result is a* uniform temperature of 3'1 9 4 C. The water has gained 3'194 calories; the mercury has lost the same. Mercury on losing 3*194 ca per gramme is cooled through 96-8 C. ; cooling through 1 C. involves a loss of '033 ca. The specific heat of mercury is thus '033, and the amount of heat contained in the mass of mercury mixed with the water was XIII.] CALORIMETRY. 361 (373 x -033) ca, 373 being its absolute temperature and its mass being unity. A modification of this method is that of Fig. 141. A globe filled with mercury : the free surface of the mercury at a ; the screw s, which alters the position of the surface a so as to bring it Fig.141. to the zero of a scale marked on the horizontal tube ab ; the hot substance, introduced into a depression at o, heats the mercury, expands it, and causes the capillary surface a to assume a new position. Otherwise, instead of observing the direct expansion of the fluid heated, its temperature may be taken by a thermo- meter. Dulong's water calorimeter is of this kind: a copper chamber containing a living animal supplied with air by afferent and efferent pipes : round this a water-jacket, the water in which assumes a certain observed temperature. The number of Calories taken up by the water and by the copper or other vessels (con- sidered as equivalent to x times their weight in water, x being their specific heat) is found, and that amount thus measured is the amount of heat given out by the enclosed animal. 2. Latent heat methods. The amount of heat of a hot body may be measured by the amount of ice melted by it this being ascertained roughly (Lavoisier and Laplace) by the amount of water which trickles from ice amid which the hot body is thrust ; or, better (Bunsen), by observing the actual decrease in volume of a mixed mass of ice and water when some of the ice is melted, or by the amount of liquid water, ether, acetic alde- hyde which the heat of a hot body (or living animal Eosenthal, Arch. f. Anat. und PhysioL, 1878) can evaporate. 362 HEAT. [CHAP. TRANSFERENCE OF HEAT. When two masses or parts of the same mass are in contact, the molecular agitation of each is in part communicated to the other: if they be equally hot, each receives as much heat as it gives up : if they be not equally hot, that which has the more molecular energy loses more than it receives, while the other, the colder, gains more heat than it loses. The flow in one direction thus overpowers that in the other, and, on the whole, heat is transferred from the hotter mass to the colder. We regard in general only the difference, and not the common part ; the surplus which flows from the hotter and not the compensated and non-apparent flow from the colder body. This tendency is universal. Heat always tends to pass on the whole from hotter to colder bodies, and if these be in con- tact, the transference is effected by conduction ; whence all bodies possess some degree of conductivity or power of trans- ferring heat through their substance. "When two points in a substance are at temperatures con- stantly differing by 8r, and are at a distance d, a flow of heat is set up between them. The amount of heat which passes from the one point to the other in time t is proportional (1) to the length of time during which the flow proceeds ; (2) to ST, the difference of temperatures ; and (3) it is inversely proportional to d, the distance between the points. f- 5? 4 Otherwise, the flow / oc 5 or/= K . . Here K is a coefficient, the d d coefficient of conductivity. The coefficient of conductivity varies from substance to substance, being greatest in the metals; some substances permit a rapid, some only a slow transfer of heat : compare a horn spoon and a silver one inserted in a hot liquid. If the one point be maintained at the temperature T and the other at the temperature T /} intermediate points have tempera- tures which from point to point sink uniformly with the distance from the hotter point. There is thus set up a condition of Steady Flow of Heat. Across a plate of thickness e whose sides are maintained at an actual and constant difference (T T x ), the flow of heat per xiii. ] CONDUCTIVITY. 363 /"J 1 f \ unit of area will in time t be K . ' ; across area A the e flow will be KA *-( T "" T /> . e If a bar be heated at one extremity, the amount of heat which will arrive at a sectional area a given distance along the bar will depend upon the thickness of the bar and its proportional surface. A thin iron wire may be melted at one end and not have its temperature raised by 1 C. at a distance of 6 feet; so much heat is lost on the way, being spent in warming the sur- rounding air and in keeping up radiation from the surface. In bars of different thickness the distances from the heated extremity at which the same temperature can be kept up by heating the extremity of the bars to the same temperature are to one another as the square roots of the thicknesses ; and in bars of the same thicknesses but of different lengths the flow of heat into the bar varies as the square root of the cube of the length. A hot point in space conceived to be maintained permanently hot will be the centre of a flow of heat symmetrical in all directions. The points in the surrounding space which are at the same temperature may be connected and found to lie on concentric spheres, or spherical Isothermal Surfaces. The heat travels by the shortest path from one surface to another, by T lines of Propagation, or Lines of Flow, at right angles to both ; and there is on the whole no lateral propagation over an isothermal surface. The difference of temperature per unit of distance along the lines of propagation decreases with the distance, being proportional to . The greater the curva- radius 2 ture of a hot body, the greater will be its loss of heat by conduction. Hence an ellipsoidal body maintained at a uniform temperature loses most heat where the curvature is greatest a proposition closely resembling one in the theory of electricity. We must distinguish a Flow of Heat from a Flow of Temperature. The latter depends on the specific heat; and if we compare the passage of heat through two substances similarly heated, we find that even though the one substance have a greater conductivity than the other, yet, if its specific heat be greater in a still greater proportion, a given temperature may take a longer time, travelling in the better conductor, to reach a point at a given distance from the source of heat, than it does in the worse conductor. Where a substance is not physically similar in all directions, as in the case of crystals, the conductivity may be unequal in three directions. Thus, a plate cut out of any crystal belonging to the biaxial system, and covered with a film of wax, will, if heated by a hot wire passed through its centre, so conduct the 364 HEAT. [CHAP. heat that the wax melts not in a uniform circle as in glass or a crystal of the regular system it will do but in an ellipse. Some solids are extremely bad conductors of heat. Down is perhaps the worst of all conductors ; hare's fur, sand, asbestos, are examples of substances within which warm objects may be placed and remain without losing their heat to any material degree for some time. Flannel, cork, etc., appear warm when they are touched by the bare skin, because they carry away by conduction less heat than the air had been removing before these materials had been touched. Wood is in the radial direction a bad con- ductor : this has a certain effect in preserving the tree in life. The actual amount of the loss of heat suffered by a cooling body depends directly on the effective cooling surface : whence the natural tendency in warm weather to lie at full length, in winter to roll the body up into small compass. The conductivity of the skin as a whole is greatly diminished by a layer of fatty tissue. The muscles are exceedingly bad conductors/ . When a hot body is surrounded by one or more concentric jackets with layers of air between them, the loss of heat is remarkably diminished. A single layer of linen diminishes the loss of heat from the human body by about two-thirds ; a double layer effects a much greater economy of heat, and so forth. The practice in cold countries of using double windows pro- ceeds on this principle, and hence also the hygienic advice to multiply the number of light garments in cold weather rather than their weight. The conductivity of liquids is as a rule greater than that of gases, which in the form of true conduction as distinguished from convection is very small. It is impossible to keep the hands in water at 52 C., while it is quite possible, as observed by Banks, to remain for five minutes in air near the boiling point of water. When a hot body is placed in air it sets up a number of Convection currents. Air becomes heated and rises, carrying away the heat of the hot body : colder air takes its place. Newton's law of cooling in a current of air is, that at each instant the amount of heat lost varies as the difference of temperature between the solid and the air. This law seems to be adhered to within narrow limits. In an undisturbed atmosphere the law of cooling is, that the velocity of cooling is proportional to jp'V 1 - 233 , where a is a constant ('45 for air), p the pressure, and t the excess of temperature. The carbonic acid, etc., of the atmosphere are mixed thoroughly and equably, not by diffusion, which would take several hundred thousand years to accomplish the task, but by convection currents. xiii.] CONVECTION. 365 Convection currents as they pass colder or warmer strata of air exchange molecules with them by diffusion ; the temperature of the whole mass thus rapidly becomes uniform. Convection currents may be demonstrated by throwing some coloured powder into cold water and proceeding to heat the liquid over a lamp ; by looking at distant objects through the heated gases which arise from a heated boiler or wall : the rise of smoke itself is an example of solid particles borne upward by convection currents particles which, when the ascending air has become cool, again fall and may aid in producing fogs. Though two bodies be not in contact with one another, they may yet exchange heat across the intervening space, and the hotter body, giving out more heat than it receives, is said to radiate heat to the colder body. This transfer of heat is effected by means of the ether of space, and we shall, in the meantime, defer the consideration of the transfer of heat by radiation until we can take a general view of waves in the Ether. Transport of Heat from place to place may be effected by storing up work-energy in springs which, on being released, set a mechanism at work which evolves heat by friction; or by storing up heat as " latent heat," or raising the temperature of a substance whose specific heat is high. The former method is not effective, because so large a number of units of work correspond to so small an amount of heat ; the latter are exemplified in heating by hot water or by steam. A hot- water bottle contains several Calories of heat, according to its size and its temperature ; these can be liberated by conduction at any desired situation. Steam, when condensed, liberates at the point of condensation 546 calories of heat for every gramme of water condensed, and can still, in the form of hot water, surrender more heat to sur- rounding objects whose temperatures are below 100 C. CHAPTEE XIV. ON SOUND. THE word Sound is used in three different senses : 1. The physiological sensation perceived by means of the ear. 2. The complex harmonic motion of sounding bodies the Fourier motion, the periodic or vibratory motion of elastic masses whose vibration is the physical cause of sound. 3. The disturbances of the air which affect the ear. "Sounds," says Newton (Princip. ii. Prop. L, Prob. xii. SchoL), " since they arise in tremulous bodies, are no other than waves (pulsus) pro- pagated in the air." A sounding body is a vibrating body. Cause a tuning-fork to sound in the usual way by striking it on the knee or drawing a violin-bow across it, or by forcing a steel rod between its prongs and drawing it through the point of the fork. Apply the point of the vibrating tuning-fork to the lips, to the surface of water, to a piece of glass. Bring a vibrating tuning-fork under a light splinter of wood lying upon two points of support ; on contact the light body will be hurled upwards. Cautiously bring a vibrating tuning-fork or bell into contact with a pith-ball suspended by a thread. Pluck one of the strings of a violin, and let it go : look at it as it vibrates : touch it. Look at a harmonium or concertina reed while it is in action. Observe the distinct tremor caused by a large organ pipe while sounding, or even by a large drum. Kelatively deep, grave sounds are produced by slower vibrations; higher, shriller sounds by more rapid vibrations. Take a long strip of iron say a strip 4 feet long ; fix it in a vice ; pull it aside and let it go ; it will oscillate transversely at a rate such that the oscillations can be counted ; remove it, and refix it so that only 2 feet of it are now free to move ; it will now oscillate four times as frequently : 1 foot free sixteen times as frequently as at first ; 6 inches free sixty-four times as frequently, and so on. The oscillations now become so rapid, the number of them in a second (i.e., their frequency) becomes so great, that they can no longer be counted directly ; now we hear a sound; the shorter the vibrating part, the more rapid become the vibrations, the shriller the sound. CHAP, xiv.] NATUKE OF SOUND. 367 The transmission of sound from a vibrating body to the ear involves as a rule the formation of sound-waves in the air. This may be rendered impossible, e.g., where the sounding "body a bell suspended or placed upon wadding within the bell of an air-pump from which the air is exhausted has no contact with the air and no means of transferring its own vibration to the surrounding air ; in such a case the ear perceives no sound con- veyed through the air, even though the bell be struck. But it may be impossible for another reason. Air will not oscillate in waves such as can be propagated to a distance, unless there be some well-marked compression or rarefaction produced at the centre of disturbance. Take as extreme instances of sound produced by well-marked compressions or rarefactions the effect of the discharge of a cannon, which abruptly adds a mass of gas to the already-present atmosphere, and thereby produces great and sudden compression ; or the rarefaction produced by the sudden collapse of a weak boiler when the steam contained in it has cooled down. Thus a vibrating body, before it can act as a sounding body, must produce alternate compressions and rarefac- tions in the air, and these must be well marked. If, however, the vibrating body be so small that at each oscillation the sur- rounding air has time to flow round it, there is at every oscillation a local rearrangement a local flow and reflow of the air, but the air at a little distance is almost wholly unaffected by this. The same result follows if the medium surrounding the vibrating body be rare e.g., hydrogen or rarefied e.g., rarefied air ; then, on account of the small inertia of the medium, it is easily induced to flow round the vibrating body ; in such cases there is but little wave-motion caused at any distance, and thus there is but little sound produced. \ A string stretched between two points of a rigid and massive framework produces surprisingly little sound when caused to vibrate : it does not act upon the air otherwise than by setting up local flow and reflow. If the same string be stretched over bridges upon a sounding-board, the string gives at each oscillation an impulse to the sounding-board which causes it to yield slightly ; and thus the string causes the sounding - board to vibrate. But though the amplitude of its vibration is small, the sounding-board is broad, and the air cannot, by flowing round its edge, evade compression and rarefac- tion ; the air is, accordingly, alternately compressed and rarefied, and thus a system of waves is effectively set up in it. ' Thus the loudness of the sound produced by a string may, by the use of a sounding-board, be multiplied many thousandfold. A similar experiment may be performed with a vibrating tuning-fork suspended in the air by a string, and the same fork vibrating while its shank is pressed against the panel of a door. 368 SOUND. [CHAP. In these cases the energy of vibration of the string or tuning-fork is very much more rapidly dissipated, while the large -surfaced sounding-board is enabled to produce an intenser or louder sound than is produced when the string or the fork vibrates alone ; and the vibration sooner comes to an end. The speaking-trumpet is in part an application of the same principle. Instead of a comparatively small surface, the oral aperture, being the source of sound, the much broader aperture of the trumpet is practically converted into the source, and the broad sound-waves thence issuing are not weakened at their origin by lateral flow. As a general rule it is therefore advisable, when sound is to be heard at a distance, to make the sources of sound of the largest size convenient. Smallness of size may, however, be compensated by quickness of vibration. Thus the chirp of certain insects is produced by such extremely rapid movement^ as many as 12,000 to-and-fro vibrations per second that the #ir is alternately compressed and rarefied on each side of the wings or in the neighbourhood of the stridulating organs without having time to flow round them. Characteristics of Sounds. The Fourier-motions which may produce sounds differ amongst themselves in their (a.) Frequency the number per second of the slowest component-oscillations. An oscillation is a complete qscillation, once to-and-fro. The frequency of a seconds pendulum is ^, because in one second it performs half a com- plete oscillation. In French works we find that a " vibration simple " is half a complete oscillation, a swing over from one side to the other ; and a seconds pendulum is held to effect such " vibrations simples " at the rate of one per second. The reason for the apparently more artificial mode of defining an oscillation here used will be seen on considering the meaning of period in S.H.M. ; a complete oscillation restores the oscillating body to its starting point. (&.) They differ as to their Energy. Proportioned to the energy are the Intensity and the Square of the Amplitude. (c.) They also differ as to the Relative Amplitudes of their Components. Of these three particulars, the first, the frequency, depends on the vibrating body itself, its form, its material, etc., but is very slightly affected by its viscosity; the second depends entirely on external causes ; the third depends partly on the form, the tension, the rigidity, etc., of the vibrating body, partly on the manner in which it is set in motion. By variations in these particulars an infinite variety of Fourier-motions may be produced in vibrating or sounding bodies ; and as a natural consequence we might expect to find, as we do xiv.] CHARACTERISTICS OF SOUNDS. 369 find, an infinite variety of musical sounds actually occurring in nature. Musical sounds may differ from one another in three cor- responding respects, viz. Pitch, Loudness, and Quality or Character. Pitch. The pitch of a clear musical sound depends on the Frequency of the Fundamental Vibration of the sounding body. Suppose a string to vibrate harmonically, and its component vibra- tions to occur 261, 522, 783, 1044, etc. times per second: then that string would have a fundamental vibration whose fre- quency is 261 per second; and a sound of this fundamental frequency is recognised by our musical sense as the note The loudness of a sound increases with the amplitude of oscilla- tion of the vibrating body ; if two strings, otherwise similar and similarly circumstanced, oscillate through ranges of -J- and ^ inch respectively, the latter has twice the amplitude and tends to pro- duce four times as much sound as the former : the loudness or intensity of sound being among sounds of the same pitch propor- tional to the energy of vibration, and therefore to the square of the amplitude. Mark, however, that the relative loudness of different sounds is not to be measured by their physical intensity or the square of the amplitude of the vibrations at their source, for the ear is not necessarily, and is not in fact, equally sensitive to sound of every pitch. Viscosity of a sounding body, while it scarcely affects the pitch, aids in causing the amplitude of the vibration, and therefore the loudness of the sound produced, gradually to dwindle away. As to their Quality or Character, we find among sounds an infinite variety. We can distinguish a sound produced by a violin from one of the same pitch and loudness produced by a clarionet, a flute, or a pianoforte ; we can distinguish the sound of a viola from that of a violin ; one violin from another ; one player from another on the same violin; one person's voice from that of another; the voice of the same person in different moods or states of health. The basis of all this variety lies in the endless differences that may exist between Fourier-motions which, though they agree as to the frequency of their fundamental or slowest component and as to the total energy involved in their movement, do not necessarily coincide in the relative amplitudes of their component harmonic motions. But if, as this theory indicates, an extended series of com- 2 B 370 SOUND. [CHAP. ponent vibrations go to make up the aggregate vibration of a sounding body, ought we not in the sound produced by a sounding body to hear a series of tones corresponding to the series of vibrational components ? If a string produce the note {& _jEE, corresponding to a fundamental vibration whose frequency is 261 per second, ought we not at the same time to hear other sounds corresponding to 522, to 783, to 1044, etc. vibrations per second ? The reply is that we do actually hear such tones ; but we do not attend to them, and for practical purposes we are therefore deaf to them. We are accustomed to interpret a sound produced by a single sounding-body the voice of a person, for example as a single sound; from earliest infancy we unconsciously train ourselves to listen only to the fundamental tone of any single note : and the presence of the other tones of the really- compound sound produced by a single vibrating - body has the apparent result of determining the Character of that tone to which alone we consciously listen. In many cases, when we listen for the higher component sounds, knowing what to listen for, we can hear them, even with the unaided ear: after practice the ear acquires the power of recognising the presence of these Har- monics with great readiness a power which may easily become oppressive to its possessor. The special training which confers this power differs only in degree from that which enables one to discriminate the different notes which make up a chord sounded in harmony ; for to the untrained ear even a chord, if it be well in tune, seems to be a single mass of sound. Noise. If all the keys of a piano within the compass of one or two octaves be simultaneously struck, the result is a confused jangle, a Noise. Here we have the Superposi- tion of Fourier-motions resulting in an apparently irregular disturbance of the air. This may go still further ; the Fourier- motions, which are superposed on one another, may have no relation of frequency and little or no individual persistence. The more markedly this is the case, the less musical will be the sound produced, and the more markedly will it bear the character of noise, f The general hum of a town is made up of sounds and cries, each of which, taken singly, may perhaps not be unmusical ; but because they are not related to one another by any simple numerical ratio of frequency, they together produce the disagreeable effect of a noise. Noises, then, such as the sound of steam escaping from a boiler, wind rushing through trees, the clatter of falling objects, and so forth, may be considered to be xiv.] NOISE. 371 produced by the superposition of a number of distinct musical sounds. Some of these may predominate in intensity and in persistence ; and thus a noise may have a distinguishable pitch. We may recognise differences in pitch between the noises pro- duced by drawing the thumb-nail at various speeds over the cover of a book bound in cloth, by blowing across the mouth of keys or tubes or flasks of various sizes, by letting boards of various sizes fall on a wooden floor, by blowing through glass tubes on which bulbs of various sizes have been blown, and so forth. Even where the original disturbance is in the highest degree irregular, as where bricks are pitched out of a cart, the elasticity of the bricks, small though it be, affects the pitch of the sound pro- duced, for the noise produced by soft porous bricks is graver than that produced by hard glazed-bricks of the same size. If we listen to a continuous noise with the aid of a resonator (p. 383) tuned to some particular tone, we can often recognise the presence of that tone as a component of the noise ; the resonator will, if it be present as a component, sound it forth continuously if it be continuously present; intermittently if it occur at intervals only. Even a single vibrating -body may, when struck, produce a noise. A bell is not, with ease, so cast as to be perfectly uniform; when struck it tends, if not quite uniform, to divide into unequal sectors, each of which pulsates at its own rate ; the physical re- sult is a number of simultaneous vibrations bearing no simple relation to one another, and the physiological result is a mixed sensation, a jangle, a kind of noise. Thus sounds originate in Fourier -motions ; a musical note in a single Fourier-motion ; (a noise in a number of simultaneous Fourier-motions whose fundamental frequencies bear to one an- other no simple numerical relation A and, as we shall afterwards see, the sensation of harmony in a number of simultaneous Fourier -motions whose fundamental frequencies have a simple numerical relation to one another. The simplest possible sound would be one produced by a vibration in which the Fourier-motion was represented by one component ; such a sound would be a pure tone. The pitch of the sound or note produced by a vibrating body is the pitch of the gravest component; the fundamental Tone; and it may be specified in two ways : (1.) Physically, by stating the number of vibrations per second which correspond to that fundamental tone ; 372 SOUND. [CHAP. (2.) Musically, by referring the tone to its place in an arbitrary scale of pitch in conventional use among musicians. To find the frequency of vibration corresponding to any given note : As the note in question let us take, for the sake of example, that produced by an ordinary " A " tuning-fork. A card or strip of metal is placed so as to touch at one end the cogs of a little cog-wheel, while the other end is firmly fixed ; the wheel is rotated slowly each cog makes one click ; more rapidly the clicks blend into a hum; still more rapidly the hum rises in pitch, and the faster the rotation the shriller becomes the sound ; at a certain rate of rotation the sound is neither graver nor shriller than that produced by the tuning-fork ; this rate of rotation is such that the card is struck 435 times per second; 435 impulses per second given to the card, and by the card to the air, produce the sound =22=, "#'=435." Higher sounds are due to more rapid, lower sounds to slower, vibrations than this. This arrangement is known as Savart's Wheel. Another contrivance, devised to the same end, is the Syren. A rotating disc is pierced by holes arranged equidistantly in a circle, whose centre is in the axis of rotation of the disc. A tube brings a current of air to a spot near the disc so situated that in some positions of the disc the air can blow clear through one or other of the holes, while in others the current of air is almost cut off by the disc itself. Eotate the disc; the current of air is alternately cut off by the disc and allowed to blow through it. If there be 87 holes in the circle of holes, and if the disc rotate five times per second, there are then produced 435 puffs of air per second, and the note " a! " is heard : its quality is, however, decidedly inferior, for the principal sound heard is the noise made by the current of air when it strikes the disc. If the current be divided by 87 pipes, so as to blow through the 87 holes simul- taneously, and to be simultaneously cut off from them all, the sound is very much clearer and louder than when there is only a single stream of air blowing through one hole at a time. Instead of 87 pipes issuing from a wind-chest, we may employ a wind- chest capped by a fixed disc containing 87 holes, arranged in a circle like that of the rotating disc : the rotating disc rotates in the immediate vicinity of the fixed one : simultaneously the air rushes through all the apertures of the rotating disc, simul- taneously it is cut off from them all. The number 87 is in prac- tice never used ; some such number as 24 or 48 is chosen. Connected with the rotating disc is some form of mechanism for xiv.] PITCH. 373 recording the number of rotations effected by it in a given time ; the rotating disc is caused to rotate at such a speed as causes the desired sound to be produced : the number of apertures in the disc, multiplied by the number of rotations per second, gives the number of impulses per second imparted to the air, and thus determines the frequency of the tone in question. The syren works under water as well as it does in air. The experiment already described on page 366 also gives roughly the means of finding the frequency of any given note. The thin strip of metal is, by successive trial, carefully with- drawn into the vice, until its free part gives, when set in vibra- tion, a sound of precisely the same pitch as the tone whose frequency is to be determined. Say that this length is 1 inch ; and also that if 3 inches of the strip be free, it executes 2 9 complete oscillations per minute. The number of oscillations varies in- versely as the square of the length ; whence (1 inch) 2 : (30 inches) 2 : 29 :x, or x = 26,100 vibrations per minute, 435 per second. Still another method of determination of the frequency of vibration of sound of a given pitch is graphically to record the actual vibrations of the sounding body. A tuning-fork has a little feather-barb attached by cement to one of its prongs : the extremity of the barb is brought into contact with slightly - smoked paper spread over the surface of a cylinder. The cylin- der is caused to rotate ; the point of the barb draws a straight line on the smoked paper. The fork is caused to vibrate : the barb now describes, on the rotating cylinder, a sinuous line which records the oscillations of the tuning-fork. An independent mechanism can be made to mark the cylinder once every second, and thus the absolute number of oscillations made by the tuning- fork in the course of each second can be counted on the perma- nent record. The same principle may be applied to many forms of vibrating body, such as strips of metal, membranes, etc. Musical Pitch. The arbitrary scale of pitch in common use, and typified by the white keys of a pianoforte, is the following : Thirty-two foot Octave Subcontra Octave. British C n German C M French ut_, fc E M F,, g;; soi_; A n A,, |;; Orations" } 16 ' 3125 18-3515625 20-390625 21-75 24-46875 27-1875 30-5859375. Ratios 16 18 : 20 : 21-3 : 24 : 26-6 : 30. 374 SOUND. Sixteen-foot Octave Contra Octave. [CHAP. g British G, A, H B. c, D, E, F, German c, D, F, G, A, H, French Ut o re o mi fa. so! la. si No. of Vi- ) brations j 32-625 36703125 40-78125 43-5 48'9375 54-375 61-171875. Katios 32 : 36 : 40 .: 42-6 : 48 : 53-3 : 60. Eight-foot Octave Great Octave. D E F G A B British ~C German C D E F G A H French ut, re, mi, fa, sol, la, si, No.ofVi-) ,, , brations f 6525 73-40625 81-5625 87 97-875 108-75 122-34375. Ratios 64 : 72 : 80 : 85-3 : 96 : 106-6 : 120. Four- foot Octave Little Octave. fe "77= = H c d e f g a 11 b Ut 2 re 2 mi 2 fa, so! 2 la, S1 2 No.ofVi.), 30 . 5 brations } 1305 146-8125 163-125 174 195-75 217-5 244-6875. Ratios 128 : 144 : 160 : 170-6 : 192 : 213-3 : 240. Two-foot Octave One-stroked Octave. 7T 3L cT -ts>- c r*y d 1 e 1 f g' a 1 b 1 Ut 3 re 3 mi s fa 3 S0l 3 las' si. No.ofVi-) 261 brations | 26 293-625 326-25 348 391-5 435 * 489-375. Ratios 256 : 288 : 320 : 341-3 : 384 : 426-6 : 480. One-foot Octave Two-stroked Octave. c" d" e" f" g" a" b" Ut 4 re 4 mi 4 fa 4 so! 4 Ia 4 si 4 . of Vi- ations - 522 587-25 652-5 696 783 870 978-75. tios 512 : 576 : 640 : 682-6 : 768 : 853-3 : 960. Six-inch or Three-stroked Octave. Three-inch or Four-stroked Octave. I sis uts res mis fas sols las sis ut res mi fae sol fi Ia 6 sis utr No. of ^ Vibra- V 1044 1174 '5 1305 1392 1566 1740 1957'5 2088 2349 2610 2784 3132 3480 3915 4176. tions. ) . . Ratios 1024 : 1152 : 1280:1365-3:1536:1706-6:1920 : 2048 : 2304 : 2560:2730-6: 3072: 3413'3: 3840: 4096 xiv.] PITCH. 375 The starting-point of this notation is the a! tuning-fork, made to vibrate 435 times per second, or the second string of the violin, made to vibrate in unison with such a fork. Under this system the c" tuning-fork makes 522 complete oscillations per second. This is entirely a matter of convention. The number 435 was chosen by the Academic des Sciences of Paris ; 440 by the German Society of Nature-researchers at Stuttgart in 1834 ; while a pitch a! =42 6 '6 has been highly recommended, on the ground that under such a system the tones /J} C t , C, c, c f t etc., are produced by 16, 32, 64, 128, 256, etc. vibrations per second an arrangement which has the advantage of giving very simple numbers to deal with, but which has, on the other hand, the practical disadvantage of giving a pitch which is too low to please instrumentalists, and the didactic disadvantage of tending to conceal the real arbitrariness of the convention which assigns to the a! or the c" fork the particular number of vibrations chosen in practice. In practice there is, indeed, a great lack of agreement ; instrument- makers are constantly raising the pitch for the sake of increasing the brilliancy of orchestral music, while vocalists are made to suffer. Modern concert pitch has thus risen as high as of = 460 vibrations per second, about 1-^ semitone above what it was in England in the time of Handel (af= 424), while the organ-pitch in England was, in the middle of the eighteenth century, as low as of = 388. If the standard number of vibrations chosen for of be any other than 435, the whole series of numbers given in the table must suffer a proportionate increase or reduction. The accuracy of such a scale depends not upon precision of absolute numbers of vibrations so much as upon correctness of the ratios of the several numbers to one another. The successive tones of the scale of C are related to one an- other, with respect to their frequency, in the following manner: T r n f S 1 t d 1 256 : 288 : 320 : 341-3 : 384 : 426-6 : 480 : 512. 1 : : I 4 . 3 : g : 2. Here C (c'= 256) is a keynote, and upon it we have raised a diatonic major scale, d r m f s 1 t d 1 . Such a scale is found by experience to be satisfying to the ears of the Western nations ; and whatever tone be chosen as the keynote, there can always be sung or played on instruments of the violin or of the trombone class a scale of this kind, in 3*76 SOUND. [CHAP. which the intervals are felt to be pleasing and in tune, in which the intonation is felt to be just, and in which each tone, when it is carefully listened to while the keynote is borne in mind, is felt to have its own peculiar mental effect, this depending on its relative placa-ia the scale^and not onits_absolute vIBfatiojial frequency. Singers who have simg^ucTTtogether, string players whonaTe practised together without pianoforte accompaniment, naturally use the tones of such a scale, without caring or even knowing what the numerical ratio of the frequencies of the various tones of the scale may be. Intervals. We may now identify the various intervals occurring within the diatonic scale Minor second, "semitone " m : f or t : d 1 . . . 15:16 Grave major-second . r:mors:l . . . 9:10 Major second . . d:r, f:s, l:t . . 8: 9 Grave (or Pythagorean) minor-third . . r:f . . . .27:32 Minor third . . m : s or 1 : d 1 . . 5:6 . Major third. . . d:m, f : 1 , s : t . 4:5 Fourth . . d:f, r:s, mTT7 sTdT t:m' 3 : 4 Acute fourth . " ~~. FTr' T~ .~ . "T 20 : 27 Augmented fourth . fTt . . . .32:45 Grave fifth . . . rTf . . . .27:40 Fifth .... dTs, m:t, f:d' . 2:3 Minor sixth . . t|is, m:d', 1 : f ' . 5:8 Major sixth. . . d : 1, r : t, s : m 1 . 3:5 Acute major-sixth . f:r' . . . .16:27 Grave minor-seventh . r : d 1 , s : f ' , t : 1' . 9:16 Minor seventh . . m : r 1 , 1 : s 1 . . . 5:9 Seventh . . . d:t, f \v? . . . 8:15 Octave . . . dTd 1 , r:r', etc. . . 1:2 Musical intervals are equal to one another when the con- stituent notes in each have the same relative frequency. Thus d : 8 : : 1 : -f, and m : t : : J : ^ ; the ratio of 1 to - is equal to that of ^ to y that is, it is 2:3; whence the musical interval between d and s is equal to that between m and t. Transition. Any tone may be chosen as a keynote. Let us choose g 1 = 384 as our keynote, and then compare the tones xiv.] MUSICAL INTERVALS. 377 of the scale of the key of G with those of the scale of C. Retaining the same ratios, the scale of G is d : r : m : f : S : 1 : t : d 1 . 1 : 1 : f : f : f : f : V 5 : 2 - 384 : 432 : 480 : 512 : 576 : 640 : 720 : 768. Comparing the two scales we find : Scale of C(" Key C"). c' d' e' f g a' b' c" d" e" f" g",etc. . . . . 384 : 426-6*: 480: 512: 576: 640: 682-6 : 768. Scale of G("Key G"). 384: 432 -.480:512:576:640: 720:768. Hft /JJh fe The tones agree with the exception of the a's and the /s. The a! of the scale of C and the a f of the scale of G differ from one another in the ratio of 426'6 : 432, or 8.0 : 81. The two tones are perfectly distinct, and an ear that has become accustomed to the pure scale of C is pained, especially in harmony, by the sub- stitution, for the proper a! in that scale, of the slightly sharper c! which belongs to Key G. The difference between the two tones is called a Comma, and they may be respectively written a and 'of. The f" of Key C and the corresponding note in the scale of G differ more widely from one another ; their frequency-ratio is 682'6 : 720, and the interval between them is sometimes called a semitone, ^| ^. In order to play in correct tune music written in Key G as well as in Key C, we would require not only the tones of the Key of C, but also two additional tones in each octave. Every transition from one Key to another "more remote from" the Key of C multiplies the demand for new tones ; and that to an extent twice as great as the current notation, which neglects differences of a comma, would seem to indicate. In the table, pages 378 and 379, are given the tones of the scale of C, together with a number of tones derived from selected keys. The relative, not the absolute, number of vibrations has been shown in each case. If a singer were called upon to produce a note of 324 vibrations per second, the feat would be impossible. This number is, however, 1-265625 x 256; and hence if c' have 256 vibrations per second, the note required is the re of Key D. A tuning-fork c'= 256 is set in vibration ; call the note of the fork do ; sing do, re ; fix the attention on re (d') ; call it 378 SOUND. [CHAP. . r' ' .. Gt ^ W ^ Nil s> II 1 II CO CM CO CO rH CM II CO rH OS co CM Q OS CM OS OS 1 P rH ? w Ml PH rH rH rH ^ rH rH H ^ rt t S S. fe PH O g =**= "rf r^ 1? S ^ ^ 5 5 ^ fc OO O OS OO O t^. rH CO XO Ml CO lO CO XO % 2 CO CO OS OS O CO O CO XO O J3 s CM OS O CM OS CO OS t CO OS t CO CO CO | | CM OS t^ CO CO cc OS CM OS CO CO OS CO CO i "C 5> Jv. XO Ml t">- XO CM XO XO CM xo MI !i rH O O rH O rH O p p p p pop o o - CO rH O XO - xo O t"* CO O CM CO .CO i o CO 00 XO CM hH 'c3 32 O CO OO CO rH N Ml OO -T-H O ^c CO o <^: OO CO P M O J- I rH CO CO ; ; i-H rH ; *>- rH XO O -!>* CO O : : xo CO cc XO oo : oo MI W .0 g, *"ovo co"'"-" 1 iH ' fH .^ CM CO . CO CO 3 -oo i-* ^j TH rH rH i-H ,-H ^ rH rH rH rH rH ^ rH rH rH H g |f CO 00 "^ G^ i} O rH _ ^ CM rH rH OS 00 rH CO CM XO ^ CO CO OS CO s CO CM CO 1 || ^H lO vO ^^ " " ^2 O) " *5 CM CO ^, rH CM XO CM CO t^. CO ' '10 00 CM ^ 01 ' XO CM XO Ml 1 rH rH rH O! PH . H . fe o* ' ' ; ; ; ^ ^ M & | 1 ! : w ti .a" ! ! fe ! : ^ rs I ~ H w .a i : : ^ : .2 ^ d- O IB" ^ ^ ^ J 1 0* o* _, M M S fr 1. rt ;rS ! I o o ^ ' ' 'o "'o o 'S "g *-! O *s : 1 o O tf O -^^ = ^ = p^ . -P p ."p^" "rf W" P" -H t=- fe p- tf F H ^ ^ 3 *H * ... 3 ' s * I 13 ^ . ? . PK 0> a I _o 5 g S '1 . 1 . 1 f 111 g rg 1 1 1 *-! 1 1 -2 S 1 fn fH ^ ' 2 g ! 1 1 1 4 w s o ii p^ ^ -* * A pn J^ * a 1 I - ^ 1 5 XIV.] MUSICAL INTERVALS. 379 "ll Ok rH * (* & a 3> S> S> S> II II II II - g g g t>> CO QO CO rH rH rH rH CM ^ ^ r - O 1O CM O t -*J 8 S g S | >O 00 Ot^OOOCOOOCO i- CO 1OCO1O1O1OCO1O1OCO >O> CM OSCOOSOOSCOO4OSCO >co t>. cococooscococococo 1 10 t*^ 1O CNI 1O "^J< *O CM lO 1O CM ?rH OrHOCSOrHOOi 1 p opppppppp 0049049 0053950 1 ^ ^ iii / *- IQ 1O CO CO rH *^ CM CO i-H O CO CO OS OS CM OS CO CO O . CO CM co : o o : co O 1O * t>- OS SO CO'"O1O1OO>OCOCOCO O COCO*>.-CMCOrHCO CMCM -^-^t^i>-l>-i>-OOOS t> rH..CO CM.1O OS CO CM. CO O CO. CO ~~" " " rH 1 s ** ' O> "^^ OS 1O * X^* CO OO o> o ' * CM CM'CO -^ ^ o"co t^ t~- rHCM CMtNCMCMCMCMCMOICM I O OS o 3 CO 10 3 1 S o o o o o o OO OOOOOOOOO o o o CM ^ CO ^ rH CM !> O OO 1O -t^* 1O O $n CO-CO t~- 1>- O-OO CO- lO CM CO 1O d 1 -* -* *P rH rH ^ CO -tOt>.Ot>.OO'>HOS -j, -A^^-" 2 SOCMCOr-( S l-H CO i-H CO CO -* CM CO CO ' ' 1O t^."lO 1O CO OS'O O CO' CM CMCMCMrH lOiH-^K CM rH N . - ^ ^ ^ ! | . . . ft . . : : : i _ 'o ..'._. i 1 : ! ' _C^ la ' ' ' * ^-^V ^ : ^ w" ^* -^ >i ^ pp< ^^ > r ^"^ C^ ^^ >* ** S g -I* : J* g 3j > ww|>'g.jr. 2 & . 1 1 '2 'g 2 3 " s 1 J 2 3 '2 S o $ -& o -If ?: ^ -^ :f PQ ? M P - ^ J3 V OB r-l ** ^* " ^a 5 ' ' ' "8 1 * ^ 73:5 ^ |.g 1 - 1 * |l| ii - 1 * so SD d ^ i &o o | g I 4 I gn ! i S 3 fc Is 1 o O 380 SOUND. [CHAP. do without changing its pitch ; dwell on it a moment ; then sing some such phrase as do, mi, sol, do, mi, re, do; and the desired note, Y, a note differing by a comma from e', the mi of Key C, has been produced, the sense of tonality and key -relationship having carried the singer into the correct sound. The column headed " logarithmic increments " contains figures which measure the intervals between the successive tones ; for it gives the logarithms of the frequency-ratios between each tone and its predecessor and its successor ; and the most convenient method of comparing ratios is to compare their logarithms. When the logs, are equal the ratios are equal ; when the ratios are equal the intervals are equal. Thus the intervals between C and Cj, Db and N D, V D and ^DJ, D and D| Eb and E, E and E|, F and Ff , 'F and 'F#, 'Gb and N G, Gb and G, G and Gf Ab and A, A and A#, 'A and 'Af, r Bb and N B> Bb and B, B and B#, are all equal, being measured in that column by the logarithm 0177288, which is the log. of the ratio -ff. Again we find a number of lesser intervals whose log. is "005395, and whose ratio is f 1 : these are Qjf and 'QJ, N D and D, \D| and Df, N Eb and Eb, E and % F and 'F, Fjf and 'F|, N Gb and Gb, N G and G, v Ab and Ab, A and 'A, AJ and 'Aft, N Bb and Bb, N B and B, B and 'B, V and c. Between these the interval is a Comma. The scale may be seen to be roughly divisible into 53 steps or divisions ; but these are not equal to one another ; if they were equal the logarithms would at each step acquire an equal incre- ment, for the ratio between each tone and its predecessor would be equal throughout the scale. Roughly and for diagrammatic purposes it is, however, convenient to represent the interval be- tween C and D by 9 steps, while that between D and E is represented by 8 : and the table is so arranged. A thoroughly accurate table of this kind would be one engraved on metal, the intervals between any two tones in column 3 being made directly proportional to the log. increment between them. The intervals marked Pythagorean in the table are thus derived : Start from c and go upwards by successive fifths, c, g, d f , 'of, f e", r b" '; going downwards we arrive at F, ^p /} ^E^, ^At^. The following exercises will perhaps aid the reader : (a) If the violin be tuned JpEgE, to correct fifths, starting with a'; -ry- show that these notes are respectively e", of, *d f , \/. (&) The scale of "Bb major" is obtained by transition from the key of C to that of F, and from that of F to that of Bb. The scale of " G minor" xiv.] MUSICAL INTEKVALS. 381 has g = la, and thus do = 6i?. Show that the respective descending scales are ,d' t 1 s f m r d ) , /. .. M vrr f Bb maior. . -. . 67, a, $r, /, ?, d, c, B? .1 l' s f ' m r d t, 1,. . . The columns in the table, page 378, headed "Equally- tempered Scale," show the nature of the system of equal temperament, which is as nearly as practicable applied to the pianoforte and organ.- The intervals are equal ; the ratio between a tone and its predecessor and successor is in every case the same ; between each pair of tones the logarithmic increment is equal: it is log 2 = ' 30130 = -0250583. The result differs widely from 12 12 pure intonation ; but we are accustomed to it. On a pianoforte equally tempered the fifths are not appreciably out of tune, though they are a little flat : but the thirds, three of which are forced to make an octave instead of extending only from C to B, are too sharp ; and though this be not offensive on the piano- forte, to which indeed their sharpness lends somewhat of brilliancy, yet in slow sustained harmony these sharp thirds are really dis- cordant, as may be well heard on a loud harmonium tuned in the usual manner, and on which thirds alone are played. Loudness. The physical Intensity of a sound depends initi- ally on the square of the amplitude of the vibration of the sounding body ; but the corresponding sensation of loudness depends not only upon peculiarities of sensitiveness of the ear, but also on the amount of physical disturbance of its drum, and if the sound be conducted to the ear by the air, it depends on the intensity of vibration of the air near the ear ; and this varies not only (1) as the square of the amplitude of the original vibra- tion, but also, in the open air, (2) inversely as the square of the distance of the sounding object. To compare the relative loudnesses of two sounds of nearly the same pitch, place the sounding bodies at such distances that they become just inaudible, and no more : say^that the one becomes inaudible at 10, the other at 50, yards : then the loudness of the one at 50 yards' distance is at the ear equal to that of the other at 10 yards : their initial intensities must be as 10 2 : 50 2 , or 1 : 25. If the sound be not propagated in free air, but be confined in a tube, the loudness of sound may diminish at a much less rate, o S L! SOUND. [CHAP. for ultimately the waves become plane-fronted, and move down the tube without any loss of intensity other than what is due to such loss of energy as is brought about by friction against the sides of the tube or by the viscosity of the air itself. Hence sounds can be carried along sewers, speaking-tubes, etc., to great distances without great diminution of loudness. Similarly, if sound be propagated by parallel or convergent waves in the air, as when it issues from a wide aperture, or after reflexion from a curved surface, it may lose little of its intensity, or may even concentrate its intensity on some particular place. The loudness of a sound also depends, if it be conveyed by a gaseous medium, on the density of that medium at the place where the vibration is imparted to it. The denser the medium, the greater its inertia, and the more readily it is compressed against itself : the greater the compression, the greater the amount of energy imparted to the medium, and the louder the sound produced. A body vibrating in racuo produces no sound: in rarefied air or hydrogen, or any other rare or rarefied gas, it produces a comparatively feeble sound ; in carbonic acid it pro- duces a louder sound than in air. A cannon fired on a mountain- top produces little sound ; one fired beneath is heard distinctly and loudly in a balloon even at a great height Concentration of sound-waves renders sounds louder, as in ear-trumpets and in those stethoscopes the auditory extremity of which fits into the ear. Quality of Sound. If a body vibrate so as to produce a sound of the fundamental pitch C = 64, and if all the harmonics be present, the series is the following : 1. 2. 3L 4. 5. 6. 7. & 9. 10. 11. 12. 13 14. 15, etc: 64 128 192 256 320 3S4 44S 512 576 640 704 768 832 896 960. <." * 9 * * / *'- * f f /*+ / '+ If- V, etc. * * * * These are all tones of the scale of C, with the exception of the 7th and its octave the 14th, the llth and the 13th. The 7th and 14th correspond to a very flat B of 112 vibrations, lying between A and 'A2: the llth to a sharp F of 88 vibra- tions, lying between 'F and F; the 13th to a flat A. lying between A? and A Analysis of a sound into its components may be effected by several methods, of which we shall first consider one due to Prof. Mayer. As our example we take the sound produced by a vibrating organ-reed-pipe, a sound which we recognise as peculiar xiv.] ANALYSIS OF SOUND. 383 and characteristic. We are provided with a set of tuning-forks, one of which vibrates in exact unison with the fundamental tone of the organ -pipe, and the rest of which respectively vibrate 2, 3, 4, 5, etc. times as rapidly. As to the organ-pipe, a part of its wall has been replaced by a piece of inelastic thin morocco leather, or some similar substance, which vibrates exactly as the air within the pipe does. To one point of this is attached a bundle of silkworm-cocoon-threads, 40 inches or so in length : each of these is attached to one of the tuning-forks and tightened somewhat The organ -pipe is caused to sound ; the leather vibrates ; the silk fibres are all set in motion, and each alternately tugs and releases its own tuning-fork. If the vibration appropriate to any one of the tuning-forks be present in the original compound vibration, the corresponding fork is set in motion : if it be not present, that fork remains silent : if the vibration be ample," the fork sounds out loudly : if it be not, the sound is feeble. This arrangement analyses the sound into its components, for it can be seen which of the tuning-forks are set in vibration ; and if the organ-pipe cease sounding, the forks go on sounding for some time, and by their joint action produce a compound sound closely resembling the sound of the reed-pipe which had been the means of setting them in vibration. This action is very exact: the slightest difference between the natural rate of any tuning-fork and that of the corresponding organ-pipe-vibration causes the fork to sound with comparative feebleness, or not to sound at all Eesonators are extensively used as a means of analysis of sound. A resonator consists in its most usual form of a bulb, generally of glass, with a large aperture, a, at one side and a small one, &, at the other. The air within such a bulb has a natural period of vibration which depends upon the cubic contents of the resonator and upon the size of the orifices. This period can be found by the piteh of the sound produced on tapping the resonator with a soft substance, or by blowing brief blasts of air across its mouth. If the air convey a system of waves which agree in period, either absolutely or approximately, with the natural free vibration of the air in the resonator, the air in the resonator will absorb the energy of those waves, will be set in motion, and will act as a sounding body. If we be provided with a set of such resonators, the air in one of which freely vibrates in unison with 384 SOUND. [CHAP. an a! tuning-fork, and in the others respectively 2, 3, 4, 5, 6, 7, etc. times as rapidly, then, on listening to an a! organ-reed-pipe, one ear being closed and the other adapted to each resonator in succession (this being done by fitting the nipple, b, of the resonator (Fig. 142) into the ear), we shall, if the proper sound of any of the resonators be contained in the complex sound to which we listen, hear that resonator loudly sing out its proper tone ; while, if it be not present, we shall simply hear the ordinary sound of the pipe through the resonator, without any reinforcement. And further, if we fill our ears with the sound of the tone thus sung out by the resonator, and remember its pitch, we shall, when the note is again sounded out by the organ-pipe, even without a resonator, have no difficulty in hearing the harmonic tone : and by dint of practice we may hear at will, or even independently of will, many if not all of those component harmonic -tones which, by accompanying that fundamental tone to which alone in ordinary circumstances we are accustomed to listen, help to make up the note of the organ-pipe. A very convenient form of resonator may be made of a common tall lamp chimney or a similar piece of tubing. If it be held vertical, as its lower end is immersed in water to various depths, its natural pitch varies : and a tube thus gradually lowered into water is capable of resounding in succession to the different harmonics of a fundamental note, so that the ear, placed near the tube, can recognise their several presences. In Wintrich's resonator an aperture at the side may be closed by the finger. By aid of the same resonator an observer is thus enabled to listen alternately to the grave-pitched muscle -sound of the heart and to the sharper valve-stretching sound. See Gscheidlen, Physiol. MethodiJc. Eesonators may be otherwise employed. If the small aperture "b be stopped with wax, and if the resonator be brought near a sounding body, it will absorb the energy of any vibration corre- sponding to its own proper tone, and may then be removed and listened to: thus each one of a set of resonators may be made to select one tone out of the group of tones present in an ordinary musical sound, and to bring it to the ear to be listened to. A resonator and a sounding body to which it is in response seem to be mutually repelled, in consequence of the stresses set up in the intervening air (Dvorak). Eesonators may further be used to transfer the ene.rgy which they thus take up to relatively-massive bodies such as tuning- forks. A resonator may be made in the form of a thin wooden box with open ends : a tuning-fork precisely in tune with it is XIV.] RESONATORS. 385 Gas fitted on its upper surface ; a sound causes the air in the box to vibrate, the air acts upon the box, and the box upon the tuning-fork ; if all be in exact unison, the energy accumulates in the tuning-fork, which comes to vibrate energetically and to produce a loud sound. Again, the oscillation of the air in a resonator may be rendered visible by the following device : A cavity in a block of wood (Fig. 143) is divided into two parts by a membrane, such as thin goldbeater's skin. The one Pig 143 moiety of the cavity is con- nected with the cavity of a resonator: the other is connected with a supply | of coal gas which enters at B and passes out at C on its way to be burned at the jet D. This contrivance is called Koenig's manometric capsule. When a sound is produced outside F containing as one of its component tones the proper tone of the resonator the air in the resonator oscillates in sympathy with that com- ponent, the diaphragm vibrates with it, and the flame at D is rendered alternately higher and lower by the action of the vibrating diaphragm on the stream of gas. The flame obvi- ously alters its character; and the change undergone by it can be studied by looking at it while the head is turned rapidly from side to side, the eyes being kept fixed relatively to the head, or by looking at the flame through an opera-glass, which is rapidly moved across the field of view, or, best of all, by looking not directly at the flame but at its image in a rapidly -rotating mirror : in which cases the flame or its image appears to spread not into a uniform band of light, but into a band with serrated edges, or even into a chain of bead-like separate images. A sufficiently-extensive set of resonators would thus enable us to effect the analysis of sounds of any degree of complexity : but resonators do not furnish us with as delicate a means of investi- gation as the means first described, unless indeed they be each allied with a tuning-fork ; they respond in general with excessive readiness to any tone in the proximity of their natural tone. Synthesis of Sound. Helmholtz has shown that any quality of sound may be built up by the superposed effect upon the ear of simultaneously -sounding tuning-forks of the proper number, pitch, and relative loudness. 2 c 386 SOUND. [CHAP. Complex Sound- Waves. The pitch, the loudness, and the quality of a sound may be studied together by causing sound- waves to impinge directly upon some sensitive body without any intermediate process of selection or nitration. Thus, if instead of a resonator, as in Fig. 143, a cone be adapted to a manometric capsule, and if sound be produced at the mouth of the cone, sound-waves will impinge directly upon the membrane in A. The membrane will go through a complex motion somewhat resembling the original compound- vibration of the sounding body, and the flame will demonstrate this by its variations of height. The image of this oscillating flame will appear in a mirror, if the mirror be made to revolve, as a band of light, serrated by large teeth, whose outline is broken by subsidiary serrations ; the num- ber and size of the greater serrations indicate the frequency and amplitude of the fundamental vibration : those of the subsidiary serrations vary with the number and variety of the subsidiary vibrations. This experiment may be roughly carried out, if there be no revolving mirror at hand, by whirling the gas-flame itself (a rat's-tail jet at the end of a flexible tube) before the eye. It is interesting to carry out this experiment by singing into the open end of the cone ; even among notes of the same pitch sung to the same vowel, the association of different forms of the flame-image with different qualities of tone and different subjective sensations is very striking ; and it is possible for a singer to attain to the production of very pure tone such pure tone having, however, a somewhat hollow quality by finding out for himself how to control the larynx so as to keep the serrations visibly open and simple. If the membrane A have a small mirror attached to it, the mirror will share in the vibrations of the membrane, and, if it be jointed on a hinge, will reflect a beam of light in such fashion as to produce 'a curve upon photographic paper uniformly rolled past the vibrating membrane ; this curve will indicate the frequency, the amplitude, the -complexity, of the vibrations of the mem- brane. Sound-waves, however complex, may again be caused per- manently to record the succession and variation of their own impulses. Le"on Scott's Phonautograph is a conical vessel, closed at its narrower end by a membrane ; to the membrane is attached a writing-point ; the extremity of the writing-point is brought into contact with a smoked revolving-cylinder. As long as there is no sound the writing-point describes a uniform line on the rotating cylinder : when sound-waves enter the cone the mem- brane is set in vibration, and the writing-point now describes an xiv.] PHON AUTOGRAPH. 387 undulating line which varies in its form according to the frequency, the amplitude, and the complexity of the original vibration. It must be observed that membranes thus made use of do not exactly reproduce the original motion at any point of their surface : those components are exaggerated which approximately or exactly coincide in frequency with some normal mode of free vibration of the membrane itself. Edison's Phonograph is a phonautograph whose writing- point is somewhat blunt; and it records the vibrations of its membrane by being driven through variable distances into a sheet of soft tin fixed on a rotating cylinder : it leaves a permanently- deforming mark, a groove of varying depth. If the membrane after having made such a mark be raised from the rotating cylinder, and the cylinder turned back to its initial position ; if the membrane be now readjusted in its former position, or, better, a little nearer the cylinder ; and if the cylinder be again rotated in its former direction, with the same velocity as at first, the depressions in the tin, being of variable depth, cause the blunt writing-point under which they pass to move alternately towards and away from the cylinder ; this compels the membrane to exe- cute vibrations, and in so doing to set up vibrations and sound- waves in the air, which, being received by the ear, produce a sound similar to the original. Not exactly, however : the process is not perfectly reversible. Some consonants are not well reproduced, especially the explosives (b, p, t, d, k, g) and the sibilants (s, z, th) ; and further it is generally found that there has been some ex- aggeration of some of the higher components in the course of transmission through the membrane, the effect of which is to render the sound reproduced one whose quality is somewhat metallic, nasal, or even squeaky and Punchinello-like. LAWS OF VIBRATION OF SOUNDING BODIES. These laws form properly a part of kinetics ; but the means of research into the phenomena of vibration which lies most readily at our disposal is the observation of the pitch of the sound produced by vibrating bodies ; for which reason some part of the consideration of these laws has been deferred to this place. In general any vibration of a vibrating or sounding body is a periodic motion, a Fourier-motion ; though in particular cases we may find that the vibration is not a single Fourier- motion either simple or complex, but may be resolved into a number of such motions, simultaneous and superposed. 388 SOUND. [CHAP. Transverse Vibrations of Strings. If a string be stretched and drawn aside from its mean position, it tends to return to that position. . In Fig. 144 let the string AB subjected to the tension of a weight of T grammes (that is, Tg dynes) be drawn into the position ACB, the particle C being supposed for simplicity's sake to have been initially at 0, the centre of AB : the tension tending to bring back the particle C to O is the component of the total tension resolved in the direction CO : this varies directly as CO that is, the restitution-stress varies directly as the displacement the criterion of harmonic motion: and it can be shown as a consequence of the fact that the string is fixed at A and B, that the string will oscillate in some such manner ,that its aggregate motion can be analysed into a number of simple oscillations whose periods are commensurable : in other words, that the motion of the string is a Fourier-motion. According to the mode of dis- turbance striking, plucking, bowing of the string, or the dura- tion of these operations, there may be an infinite variety in the relative amplitudes of these component simple oscillations. Some of these components may even be altogether absent : where, for example, a string is plucked at its centre, it is not possible that any of those components which have a point of rest at the centre of the string should be present, and the vibration of a string so plucked is one in which all the even components are absent. In general, a vibrating string does not present any component oscillation whose node is at the point of disturbance. Frequency of Oscillation. In general the velocity of propagation of /IT a wave is / ; here we must substitute for K (the restitution-coefficient) the force here acting that is, the tension per unit of sectional area of the string ; this, measured in dynes, is '-j when r is the radius of the string. Hence v = I =\ / = ~ A. / But the length of eacli wave V p V Tr.r 2 p r V TT . p is fixed by the condition of the string : it is bound at each end, and if we confine our attention to the slowest component, the fundamental tone, we see that A, the wave-length, is equal to 2AB or 2Z, twice the length of the string. xiv.] VIBRATIONS OF STRINGS. 389 Thus, since n, the number of complete oscillations per second, is equal to --._ ~ A 2* 21 This determines the frequency of the fundamental vibration : the harmonics will have frequencies 2, 3, 4, etc. times as great Problem. A wire of steel (p = 7' 8) 1 met. long and 1-2 mm. thick is stretched by the weight of 40 kilogrs. and set in transverse vibration : what will be the frequency of its fundamental vibration ? What its pitch ? 1 1 A_L? = JL J_ /I n ~ 21 ' r V TT.O 200 * '06 V 3" 4000 x 981 1416 x 7*8 = 106^ vibrations per second ; ^1 = (when c' = 256). So for a perfectly-flexible string : the effect of rigidity of wire or string is to diminish the number of vibrations, and to cause the motion to assume the character of a number of superposed harmonic motions of incommensur- able period. The vibrations of a violin string differ much from those of a pianoforte string. In the violin the oscillating string sometimes travels in the same direction as the bow, sometimes away from it. When the bow and the string travel in the same direction, the bow drags the string with it, distorts it, pulls it out to an extent greater than that which it would have travelled if allowed freely to vibrate. When the string returns the bow fails to retain it, loses it, and as it is returning bites and catches it again by means of some rough resinous particle at another part of the bow. The friction, and consequently the adhesion, between the string and the bow are relatively greater when both move in the same direction, for at low relative-speeds friction increases. The string is thus distorted and assumes successively a number of forms, of which no one is curved : and the form of the vibrating string at any instant presents an angle between two straight lines, a form differing considerably from the curve of sines. But it is periodic, and it is true Fourier-motion. The mathematical problem is What superposition of commensurate S.H.M.'s will produce a vibration-curve such that for a certain distance the flexures so balance one another as to produce a straight line, and then so aid one another so as to produce an abrupt angle followed by a straight line ? This can be solved, and the result is that the vibration of a bowed string must be composed of a fundamental vibration, of weak components 2d to 390 SOUND. [CHAP. 6th, and of ample higher components. This agrees with the result of resonator- analysis of the sound of a violin. In the sound of a violin the upper harmonics are loud and piercing ; the nearer harmonics are feeble, and the fundamental tone stands apparently ,alone, but rendered penetrating in quality by the high mass of harmonics. Purity of violin tone depends upon perfect periodicity of the peculiar motion of the string ; this is difficult to attain, for a good elastic violin, uniform strings, a uni- form bow, uniformly resined and evenly handled, are necessary : and any stumbling of the bow over the string or irregular move- ment of the string under the bow are revealed by scratchiness of tone. The sharper the angle made at the point of disturbance, the richer the tone in high harmonics. A string plucked with a quill, as. in the old harpsichord, has thus a metallic tinkling quality, and its fundamental tone is feeble. A string struck suddenly at one point has a form differing greatly from that of the curve of sines. Part of the string remains unaffected, while the part struck is distorted. This distortion travels along the string, and results in a periodic motion abound- ing in high components ; the tone produced is tinkling. If the same cord or wire be struck gently by a soft elastic hammer, the blow being deliberate, and, as it were, gradually insisting upon the displacement of the string at the point struck, the disturbance is more evenly spread over the whole string, and the fundamental component-vibration is more prominent, the higher components are relatively more feeble, and the tone is purer. In a pianoforte string struck by an elastic soft-hammer the harmonics up to the sixth are present ; the seventh is obliterated, or nearly so, by the hammer being made to strike the string at a spot one-seventh of its length from the end of the string- that is, at a spot which would have been a node of the seventh component if that component had existed in the compound vibration : and the components beyond the seventh are feebly represented. The Monochord is a box of thin light wood containing air which communicates with the exterior air by lateral apertures. Upon this box rest two bridges (" banjo-bridges ") one near each end. Over the bridges is stretched a wire ; of which one end is firmly fixed to one end of the box, while the other is either passed over 'a pulley and made to support a weight, or else connected with a tuning-peg, which may be turned by a tuning-key, the tension on the stretched string being thus varied. xiv.] VIBRATIONS OF STRINGS. 391 Experiments with the Monochord. For experiments it is better to use a form of monochord in which there are two wires, of which one is tightened by a peg, the other by a sus- pended weight; in the latter the tension on the wire can be directly measured, in the former it must be inferred. 1. Suppose a wire 1/2 mm. thick, whose free vibrating part is 1*2 metres long, to be stretched by the weight of 48 kilogrs., and the pitch of the sound produced to be G = 96 vibrations. What weight ought to be added in order to raise the pitch to d ? The pitch is raised G : d i.e., a fifth : the vibrations are rendered more numerous in the ratio 2:3; the tension must be increased in the ratio 2 2 : 3 2 , or 4 : 9 ; the stretching- weight must be increased from that of 48 to that of 108 kilogrs.; the mass which would have to be added is 60 kilogrammes. 2. Two wires of equal thickness are stretched one by the tuning-peg and the other by the weight of a heavy mass so as to vibrate in unison. The weighted wire is removed and replaced by one of a different thickness stretched by the same weight. A thinner wire gives a higher tone, a thicker one a lower. If in Ex. 1 a wire 1 mm. thick be employed, what will be the pitch of the sound produced ? The frequency varies inversely as the radius: it therefore exceeds that of a wire 1/2 mm. thick in the ratio of 5 : 6 ; the note produced will be B. 3. A brass wire and a steel wire of equal gauge are equally stretched : they are free to vibrate in equal lengths. Brass has a density p = 8*38, steel = 7'8. The brass wire gives a sound lower in pitch than that given by the steel wire: the respective frequencies are in the ratio of ^7-8" to N 7 8'38 or 1 : 1-03663. A catgut string, whose density is small, gives a higher note than a steel wire. 4. In order to vary the free vibrating -length of wire, a movable bridge is arranged under the string. If this be so placed that 60 cm. of the wire are free to vibrate instead of 120 as before, the sound produced will be the Octave ; if 40 (= HP), the Twelfth ; if 30 ( = ^F), the Fifteenth ; if 24 ( = HP), the Seven- teenth and so forth above the fundamental note emitted by the freely-vibrating string of 120 cm. length. The number of vibrations of a string varies inversely as its length. Hence if we wish with a string which sounds C to pro- duce the note D, whose frequency is that of C X I , we must allow the string to vibrate not as a whole, but only in f of its 392 SOUND. [CHAP. length. To produce the scale on one string, the parts of the string which are allowed to vibrate are as follows : d r m f s 1 t d'| eto i f f f f * & iJ The application of this principle is familiar in violin playing. 5. Nodes and Loops can also be shown on the monochord. If the wire, 120 cm. long, be lightly touched at 20 cm. from the end, and if the twenty-centimetre-part of the wire be set in vibra- tion by a bow, the whole wire is found to be in vibration from end to end ; but not as a whole. It divides itself into segments Fig.HS. or vibrating loops, separated by nodes or points of rest. Each segment is 2 cm. long : and the sound given out is that which might be emitted by half-a-dozen separate wires each 2 cm. long that is, it bears to the note emitted by the whole string the same proportion as g" does to C two octaves and a fifth. Similarly for other fractional divisions of the wire or string. Lightly stop- ping the string has the effect of destroying or of checking the formation of all those modes of vibration which have not a node at the point touched ; hence the 6th, 12th, 18th, etc. components of the vibration of the whole string are unchecked, while the other components are rendered impossible. The nodes and loops of a string vibrating in this way are rendered manifest by paper riders placed some at the nodes, some on the loops ; when the string enters into vibration those riders which had been placed on the nodes will retain their place, while those which had been placed on the loops will be jerked off. Nodes and loops on vibrating strings may be illustrated on a large scale as follows : Take an indiarubber tube 1 feet long, filled with sand, or a long spiral of iron or brass wire, and fix one end of this to a wall; hold the other end in the hand. On moving the hand gently the natural period of oscillation of the cord can be easily found. Give with the hand a series of trans- XIV.] VIBRATIONS OF STRINGS. - sy 393 natural osciUareCB&-:/ the Fig.US. v verse impulses, so timed as to aid the tube or spiral will enter as a whole into wide oscillations, give such impulses twice as often : the cord will divide into two segments, pivotting in a striking manner round the central point. Do so three times as often as at first : the cord will divide into three segments or loops, pivotting on two nodes. By increasing the frequency of the movements of the hand, the cord can be made to oscillate in 4, 5, 6, 7, or even 8 or 9 segments, accord- ing to the dexterity of the experimenter. The experiment is a very striking one ; and it may be varied by causing the hand to move in a circle or an ellipse instead of a straight line. Melde's Experiments. A tuning-fork is provided with a little hook on one of its prongs ; to this hook is attached a fine white-silk thread. This thread is passed over a wheel and at- tached to a suspended mass partly immersed in water; the quantity of this mass can be coarsely adjusted by the addition or removal of sand ; its effective weight can be finely adjusted by varying the quantity of water in W (Fig. 146). The fork is set in vibration ; waves appear to travel up and down the thread ; if the string be illuminated by a beam of light in a dark room, the effect is singularly beautiful. As the tension is increased the segments vary : at length the thread vibrates as a whole, and seems to form an opalescent spindle. Its frequency of vibration is half that of the fork ; the thread, when at its limit, is pulled back by the retreating fork into its mean position, but is relaxed and allowed to swing over upon the return of the fork ; whence two oscillations of the fork correspond to one of the thread. If the tension be reduced to -J-, the vibrating part of the string must be shortened to ^- in order to keep time with the fork, or else, if the string be not shortened, the string will divide into two equal seg- ments or loops separated by a node : if the tension be reduced to ^, the string divides into three loops with two nodes ; and so forth. If the tuning-fork be turned round through 90, so as not now to tighten and relax the thread, but to give it a series of transverse impulses, a similar series of phenomena will be ob- served ; but the fundamental vibration is now simply synchronous with that of the tuning-fork 394 SOUND. [CHAP. When the thread is attached to two tuning-forks whose fre- quencies bear an aliquot ratio, the tuning-forks being placed at distances from one another such as to tighten the thread to the required amount, the motion of the thread becomes periodic, and presents a complex of beautiful loops and nodes which are ob- tained with comparative ease. Transverse vibrations of cords may be studied with respect to the motion of each particle by casting a beam of light along a vibrating cord, and looking at a particular bright or brightened spot on the cord. The bright spot appears, when the cord is looked at end-on, to give in quick succession a large variety of such forms as we have already seen to be produced by the com- position of S.H.M.'s. A bright spot on the cord may also be looked at through a microscope whose object-glass is borne upon a vibrating tuning-fork ; the apparent motion of the spot pro- duced by the motion of the object-glass (this being parallel to the length of the cord) is compounded in the eye with its real motion ; the apparent up-and-down motion of the spot, as looked at transversely, is spread out into an open curve, and thus be- comes more intelligible, for the eye can more readily comprehend open curves than simple up-and-down movements. Longitudinal vibrations of a string may be excited by drawing one point of a violin bow along the string : a very shrill tone is produced. / rr The velocity of propagation = / ; the wave-length is twice the length of the string, or A, = 21; the number of fundamental vibrations per v 1 /K second, n =-?- = . / . A 21 V p Problem. A steel wire elasticity 2,520,000,000 g, and density 7'8 of one metre in length, is clamped at the two ends and set in longitudinal vibration. What will be the pitch of the sound produced ? 1 /K 1 1 2,520,000,000 x 981 n =-/ = / _? ? = 3000 vibrations 21 V p 200V 7'8 per second = /"" + . As a rule the longitudinal vibrations of a string or wire are much more frequent than the transverse ones, and thus produce a much shriller sound, and further, they are not so much affected by tension applied to the string, for a variation of tension which would materially modify the frequency of transverse vibration would have little effect upon either the elasticity K, or the density p, upon which the longitudinal vibrations depend. xiv.] VIBRATIONS OF STRINGS. 395 A violin ^''-string gives, when rubbed longitudinally by one point of the bow, a sound in the neighbourhood of ('/ $)"" ; while, when let down so as to sound only e', it gives out a longitudinal vibration-sound not so low as (/ $ )"" > tne longitudinal vibration hardly falls a comma, while the trans- verse falls an octave. All the catgut strings of a violin may be observed to give out nearly the same longitudinal note, for this does not depend on their thickness. From this we see how important it is to use the bow in such a way as to bring out transverse vibrations only, and by no means to wield it so that any component of its motion over the string can excite longitudinal vibrations, resulting, as these do, in shrill discordant tones. By means of the monochord we may learn that a string while vibrating longitudinally divides into loops separated by nodes, just as its does while executing transverse vibrations. Longitudinal vibrations of rods resemble those of strings or wires. A glass rod grasped by its centre and rubbed longitudi- nally by a resined cloth will enter into longitudinal vibration and will produce a shrill sound. A glass tube treated in the same manner may be made to vibrate so vehemently that it shivers into segments. Transverse vibrations of rods obey the rule that if be the thickness of the rod, / its length, K its coefficient of elasticity, and p its density, then 6 /K /K pV -' or n = const x p v -; 71 m The constant varies according to circumstances. If the rod be free or clamped at both ends, it is 1*78 ; if it be free at one end only, the constant is only 0*28. As examples of rods free at both ends and vibrating trans- versely, we may take the common glass or metal harmonicon plates of glass or metal supported by threads at the nodal lines and struck by hammers. As examples of rods clamped at one end and vibrating transversely, we may take reeds such as those of the harmonium or concertina. Their pitch is raised by filing off their substance towards their free ends ; it is lowered by thinning them towards their base. Tuning-forks afford another example of vibrating rods ; they are tuned in the same way. In rods of the same thickness the frequency of vibration varies inversely /rF as the square of the length, as the formula n = c * / indicates ; but if the thickness and the length I vary together, so that different rods have 396 SOUND. [CHAP. the same shape, the frequency depends on the relative length only. Thus a tuning-fork 4 inches long and one 2 inches long, of the same shape, produce notes which differ by an octave : the same rule applies to reeds. A rod does not vibrate, as a whole, in halves, thirds, etc., but the com- ponent vibrations have frequencies 1 :4 : 9 : 16, etc. ; but each component has its own rate of travelling through the solid, and thus the periodic nature of the vibration is disturbed. Such a vibration of a rod is an extreme case of the vibration of a rigid or thick wire. A rod of circular section can vibrate transversely with indifference in all directions. One whose section is oblong can oscillate more widely in a plane at right angles to the broad face than it can in a plane parallel to that face e.g., a vibrating reed, in which the latter oscillation is absolutely insignificant. If a rod of circular section be filed at one side, all component oscillations which tend to bend the rod upon the filed face are retarded ; those which are at right angles to these are unaffected. Thus a rod of steel, grasped by a vice at a particular spot, may be so tuned that when it is set in vibration by a violin bow its point may execute vibrations in directions at right angles to each other, and bearing to each other any predetermined ratio. Take a knitting-needle ; fix it in a vice ; mark on the needle the height at which it stands in the vice ; touch the free tip of the needle with a little gum : scatter a little starch or powdered antimony over the needle tip ; some will adhere. Suppose the ratio desired is 4 : 5 ; refer to Fig. 38. File the rod always towards one aspect until the movement of the tip of the rod, as revealed by the brilliant particles of starch or antimony, comes to present the curve sought. If the filing have been carried too far, a little metal may be removed from the needle at an aspect at right angles to that of the previous operation. If after a needle has been tuned in this way, so that its com- ponent vibrations have been rendered commensurate, it be grasped by the vice at a point a little above or below the original point, the intervals cease to be commensurate, and the curves seen pass through a series of changes exemplified by those of Fig. 41. A tuning-fork or vibrating reed made to write its own vibrations on a rotating cylinder describes a sinuous curve which is almost identical with the curve of sines. This shows that the motion is pendular. Again, if two harmonium reeds have their tips silvered so as to reflect light, and if they be arranged at right angles to one another ; and if a lamp and lens be so placed that a beam of light falls first upon one reed-tip, then upon the other, and finally upon a screen ; then if one reed be set in vibration, the spot of light opens out into a line, while if both vibrate the line opens out into some figure of the order of those shown in Figs. 35-40. This figure retraces itself and retains its form if the reeds be accurately tuned to an aliquot ratio of frequencies, while, on the other hand, if the reeds be not so in tune, the figure undergoes rapid changes changes painful to the eye, as the accompanying beats are to the ear. Rotatory Vibration of Rods. When a rod is clamped by one end in a vice, and a violin bow drawn round it, it may be caused to execute vibrations in which it successively twists and untwists itself round its own axis ; and it is found to do so with a frequency 0'6 x that of the longitudinal vibrations in the same rod. xiv.] VIBRATIONS OF DISCS. 397 Vibration of Discs or Plates. A disc of metal or of glass may be caused to vibrate by means of a violin bow drawn across its edges. The point of support of the disc is necessarily a nodal point ; any number of points may be supported or fixed, and all these must also be nodal points. The disc or plate may, under such arbitrary conditions, adjust itself so as to vibrate according to circumstances with great variety of nodal lines and vibrating segments. A disc of brass or glass may be fixed at its centre to a heavy stand. If the circumference be touched at any point while the whole is set in vibration by a violin bow, the point touched will be a nodal point ; the spot where the violin bow is applied tends to become the centre of a loop ; according to the relative situations of the points held fixed and of the point of application of the disturbing cause, will vary the manner and the pitch of the resultant vibration. Different discs of the same shape and vibrating in similar 9 'K ways have relative vibrational frequencies varying as / ; i * p the same law as holds good in the transverse vibration of bars. The form of the nodal lines may be studied by strewing sand and lyco- podiuni powder upon a vibrating disc or plate : the sand collects on the nodal lines ; the lycopodium, by reason of the disturbance of the air, is blown towards the centre of each vibrating segment. Contiguous sectors are in opposite phases of vibration. If the ear be placed immediately opposite the centre of figure of a circular vibrating-disc, there will be but little sound heard : the receding and the approaching sectors neutralise each other's effects upon the air and upon the ear. If the hand be held above a vibrating disc so as partly to cut off the effect of one of the sectors, the sound heard opposite the centre of the disc is en- hanced ; if two contiguous sectors be thus shaded from hearing, the sound is, as at first, very feeble ; if two sectors not contiguous but vibrating in the same sense be thus covered, the sound pro- duced is much louder. If a Koenig's manometric capsule be provided with a tube which bifurcates, and if the branch tubes each terminate in a cone, one cone may be placed over a vibrating sector, while the other may be moved about over the disc. As it passes round the circumference of the disc, it will be found that the gas-flame of the capsule is alternately much agitated and steady agitated when both cones are over sectors vibrating in the same sense ; steady, or nearly so, when they are over sectors vibrating in opposite phases. 398 SOUND. [CHAP Vibration of Membranes. If a membrane be subject to a tension equal over its whole circumference, such as that of a drum, it vibrates as a whole ; its higher component vibrations are not commensurable with the slower ones, and the higher tones faintly to be heard in the sound of a drum discord with the fundamental tone. If the membrane be not equally stretched in all directions, it will, when set in vibration, so arrange itself as to vibrate feebly along the line of least tension, and strongly in the direction of greatest tension. Thus a square piece of thin indiarubber, clamped by the two opposite edges and stretched, will vibrate at the same rate as an indiarubber cord of the same free length and exposed to the same tension per unit of sectional area ; and it may be idealised as a collection of indiarubber cords, arranged side by side, attached to each other, and vibrating in unison. Vibration of Bells. A bell-shaped body, set in motion by being struck, or by being rubbed with the resined or wetted finger carried round the circumference, enters into vibration simul- taneously radial and tangential. The bell divides into an even number of sectors ; of these one half dilate, while the other half (individually alternating with the former) contract radially. At the same time sectors of the bell, moving tangentially, twist to and fro round the axis of the bell ; alternate sectors are opposed to one another in the direction of their twist ; hence at some of the nodes which separate the sectors there is compression, at others dilatation of the substance of the bell. The loops of the radial motion are the nodes of the tangential motion ; thus where there is least expansion or contraction there is the greatest amount of twist. In the circumference of a vibrating bell there are generally four loops corresponding to each motion. The effect of loading a vibrating body is to lower its pitch ; if the load be distributed uniformly, all the components are lowered ; if it be suspended from points of the vibrating body, those component vibrations, if any there be, which have their nodes at those points remain unaffected. The preceding propositions have related to the vibrations into which a body may enter when it is disturbed and then left to itself. The vibration in such cases is called free vibration, the period of which depends on the nature and the form of the vibrating body itself. If a body capable of vibration be acted upon by a series of impulses ab externo, the result depends upon xiv.] FORCED VIBRATIONS. 399 the period of recurrence of these impulses. These may be so timed as to aid the natural free vibrations of the body, adding energy, and therefore increasing the amplitude at every oscillation ; or they may be so timed as sometimes to aid, sometimes to thwart the natural oscillations, and thus to produce, on the whole, no effect so far as concerns the amplitude of these. In the former case the interval between two successive impulses ab externo is equal to the period of the natural vibration ; while, when this interval differs materially from the period of the free vibration, the amplitude of vibration is not increased, but the energy com- municated in the successive impulses is dissipated in heat. A heavy bell has a natural period of pendular swing, and if a person gently pull the bell-rope, a very slight swing may be obtained, perhaps barely perceptible. If the pull be repeated while the rope is tending to slacken in the ringer's hand, the original small swing is increased, perhaps doubled ; a succession of well-timed pulls causes ultimately a wide oscillation of the bell ; and when the bell has been set fairly ringing the amplitude of the oscillation is kept up, and all loss of energy replaced by a series of well-timed impulses. If, however, the impulses be so timed that they sometimes act in favour of the oscillations of the bell, and are sometimes delivered against a tightening rope, the bell will either not ring at all or will do so very irregularly. If a body capable of freely vibrating with n oscillations per second be placed in material communication with a body actually vibrating n times per second, the former will take up energy and enter into vibration. If a cylinder Fi AB be fixed at the extremity of the rod CD ; if another cylinder EF of like material and dimensions be fixed at the other end of CD ; and if AB be set in longitudinal vibration, ~~c """!> then EF will be set in vibration of equal period. Here it must be observed (1) that expansion of AB occurs at the same time with compression of EF, and vice versd ; and (2) that the centre of mass of the whole system remains unchanged in its position. Two organ-pipes of exactly the same pitch, mounted on the same wind-chest, may fall into a similar opposition of phase, and retain it for long periods of time : as long as they do this, they can together emit but little sound, and the sound produced is rendered louder by silencing one of the pipes. Two similar strings of equal length equally stretched over 400 SOUND. [CHAP. the same solid framework parallel to one another, will so adjust their vibrations as simultaneously to approach or to diverge from one another. A tuning-fork may be regarded as made up of two rods connected with a common basis ; the prongs simultaneously approach and diverge from one another when the fork is in vibration. Two clocks which keep good time together beat, when placed on the same table, synchronously. A tuning-fork can be set in vibration by another tuning-fork of exactly the same pitch vibrating within the same room. The material communication between the two forks is effected by the air ; the sounding-fork causes waves in the air ; these cause well-timed impulses against the second fork ; the effect of these is cumulative, and the second fork takes up the vibration of the first. In the same way a mass of air in a vessel or flask, since it has a natural period of vibration, may, if air- waves dash against the open mouth of the vessel at such intervals as correspond to its natural vibration, be caused by the accumulated effect of these waves to enter into violent oscillation. This principle we have seen made use of in resonators ; and from the analogy of these in- struments all phenomena of this kind may be called phenomena of Resonance ; the law of which is, that any undulation or vibra- tion is taken up its energy is absorbed by any body capable of freely vibrating synchronously with it, free to do so, and exposed to its periodic impulses. If the undulation or vibration which concusses the vibratile body be complex-harmonic, those com- ponents only which correspond to the natural period of the vibratile body are taken up by that body. On this principle resonators are used to detect the higher components of a complex sound. Forced Vibrations. Under certain circumstances a vibratile body may be compelled to surrender its own preference for a particular mode of vibration, and to vibrate with more or less accuracy in an arbitrary manner imposed upon it by external force. Huyghens discovered that two clocks which did not keep time separately, kept time together when placed on the same table ; the more rapid clock forced up the speed of the slower one while it was itself delayed. Two prongs of a tuning-fork, slightly unequal in size, will force one another to agree in their xiv.] FORCED VIBRATIONS. 401 periods of vibration. If the one prong be powerfully pulled to and fro by an external mechanism so predominant that its period cannot be altered by any resistance offered by the fork, the fork as a whole will be forced to vibrate at a rate determined by the exterior mechanism or outside force ; and it will maintain this rate as long as it is compelled to do so, but no longer : this being the case of a tuning-fork controlled by an electromagnetic inter- rupter, which is found to return to its normal rate of vibration as soon as the electric current ceases. The nearer the rate of the forced vibration is to the rate of the free vibration of the fork, the wider is the oscillation. A resonating chamber of air will also on the same principle resound to some extent under the influence of a tuning-fork of pitch slightly differing from its own natural pitch. Here we observe that a steel fork can force the less massive air slightly to alter the period of its vibration ; towards the end of each swing of the particles of the air, they are in very slow motion, and can with comparative ease be constrained to shorten or to prolong the period of their oscillation. Air-waves, on the other hand, cannot force a massive tuning-fork in this way, and in order that a tuning-fork may take up a vibration conveyed to it through air, the vibration of the primary sounding-body must be in exact unison with the free vibration of the tuning-fork acted upon. There is difficulty in constraining a denser substance to take up a vibra- tion communicated to it by a rarer one. The mechanical action of the ear involves, however, a problem of this kind ; air -waves have to be com- municated to the denser sense-organs ; and this is effected by diminishing their amplitude and consequently increasing the force of their impulse a principle now familiar to us. Perhaps the best examples of bodies which can be forced into vibrations of any period are found among membranes ; to some degree of amplitude any period of vibration can be forced upon a membrane, though all stretched membranes as, e.g., the orches- tral drum have natural periods of free vibration which can be modified by varying the tension. If a vibratile body which has a natural period of vibration be concussed by an external Fourier- motion, the resultant forced-motion of the vibratile body will still be Fourier-motion ; but those components of the original motion which nearly coincide with the natural oscillations of the vibra- tile body will be represented in the resultant forced complex- vibration by components proportionately exaggerated a principle we have already seen to affect the working of the Phonograph. 2 D 402 SOUND. [CHAP. If a membrane be of unequal width and be stretched in one direction, a forced vibration imposed upon it will affect only certain parts of its area. In Fig. 148 ABC is a membrane, Fig.HS. triangular in form and exposed to a tension parallel to CB. Consider a single very narrow strip, ef, of this membrane; imagine it to be isolated from the rest of the membrane. Such a strip would have a certain length, thickness, tension, density ; it would therefore, if it entered alone into transversal vibrations, produce a note of a certain definite pitch. Let that note be sounded in the neighbourhood of the membrane ; the membrane will vibrate strongly at ef, the disturbance rapidly shading off into rest on either side of ef; and, further, there will be some disturbance in those parts of the membrane whose length is 2ef, 3cf, and so forth. Each external sound is responded to by a different part of the membrane, which plays the part for the time being of an imperfectly-isolated string. At right angles to the direction of tension there is but little vibration. If the external sound be complex, several such strips are set in motion. Musical Instruments. For the ends of musical art it is necessary that the vibrating body used as the source of sound should be capable, at the will of the performer, of producing several sounds. In the old Kussian horn-bands each player had only one sound at his disposal, and by dint of practice and drill learned to produce his solitary note at the right instant ; but this kind of orchestral music is quite exceptional. We have already seen that all the notes of the scale may be produced on the monochord by varying the length of the free vibrating part of the string. xiv.] MUSICAL INSTRUMENTS. 403 Some stringed instruments the Lyre, the Harp, the Violin, etc., when played pizzicato, the Banjo, the Guitar are played by plucking the strings. In some cases the Lyre, the Harp the number of sounds which can be produced is limited by the number of strings present ; in others the Banjo, the Guitar each string is made to produce a number of sounds which depend upon the number of frets by which the finger of the executant is guided in shortening the string; in instruments of the Violin class there is no mechanical aid to the performer's fingers, and he is left to his own judgment as to the precise amount by which any given string should be shortened in order that it may emit the particular sound of which he has already formed a mental idea. All these instruments are provided with sounding boards which increase the surface by which vibrations are communicated to the air; and when their strings are plucked, the sound pro- duced is of short duration and rich in high harmonics, poor in lower ones. The strings of the Harpsichord were plucked by quills which were actuated by hammers. The sound was poor in quality, being feeble in the fundamental tone and disproportionately strong in the higher harmonics ; and it was feeble in intensity as compared with the pizzicato notes of a violin, because the sounding-board was wanting in flexibility and had little effect on the air. The Pianoforte contains very strong wire tightly stressed ; the total stress on a Broadwood grand pianoforte exceeds 35,000 Ibs. ; that on a Steinway is 72,000 Ibs.; whence the modern pianoforte is, as regards its framework, necessarily a very much more massive instrument than its predecessors. The longer the string corresponding to a given note, and the greater the tension upon it, the more precisely will the harmonic tones be in tune with the fundamental, and the fuller and richer will be the sound produced. The sound of a pianoforte string struck in the usual way is rich in harmonics up to the sixth ; the seventh is pur- posely prevented by the choice of the spot at which the hammer strikes; the eighth and those beyond are feebly represented. The sounds of the higher strings approximate in character to pure tones. The compass of the modern pianoforte is from A /y ( = 27-18*75) in the thirty-two-feet octave to a!'" ( = 3480), or even to c //w ( = 4l76), the sound of an organ-pipe an inch-and-a- half long. In a vibrating pianoforte - string those components disappear whose 404 SOUND. [CHAP. periods are , f, ^, etc. of the period of contact between the hammer and the string. Hence the varying quality of tone obtained from the same string by hammers of different degrees of repair or varying in the hardness or elastic softness of the leather, or by striking the pianoforte keys in different ways. The slower the stroke, the longer the contact, the greater the disappearance of higher harmonics. The Violin, whose strings are tuned to ^g, V, a! , e", has a compass ranging from \7 to about e tm ; the Viola is tuned to V, *g, *d' ', of ; the Violoncello is tuned to N C, ^G-, X a ; the Double-bass is tuned to A /7 , *D 7 , ^Gr 7 ; these instruments give the strings of the orchestra an aggregate compass of from A /; to about e"" t or nearly seven octaves. In the Violin three strings are of catgut ; their pitch depends upon their thickness ; the fourth string is weighted by a spiral of silver wire an arrangement which to a great degree obviates rigidity. The belly of the violin acts as a sounding board. The air in the cavity aids in trie resonance and improves the tone, for it is easily forced to take up the vibrations of the solid parts of the instrument, especially those component vibrations which are already the most prominent ; and this action of the internal air is important, as may be found on covering they holes with tissue paper, for then the tone of the instrument is materially deteriorated. The quality of the tone is found also to depend greatly upon the empirical form of the bridge. Transverse vibrations of reeds are utilised in the Musical Box : reeds of various lengths are struck, vibrate in the free air, and produce sound of little intensity. In the Harmonium and Con- certina the passage of air from a bellows into a resonating air- chamber is obstructed by reeds, which almost completely close the air passage. The pressure accumulates and bends the reed ; the air escapes and the pressure is relieved ; the reed returns and swings past its mean position, again to be driven outwards. The vibration of such reeds contain as harmonics tones bearing to the fundamental the ratios -J-, -f-,-^, etc. Such reeds vibrating in the open air produce very little sound : the air of the resonating cavity acts as intermediary, and communicates the vibration by a broader surface to the external air. In Mr. Baillie Hamilton's String-organ very various qualities of rich and full tone are produced by connecting in various ways a vibrating reed with a string stretched over a sounding-board. In reed organ-pipes a reed vibrates at the bottom of a tube, the cavity of which it alternately opens to and closes against the access of air from the wind-chest : the air in the cavity of the xiv.] MUSICAL INSTRUMENTS. 405 organ-pipe resounds to the vibration of the reed, and does this most loudly when its own natural pitch coincides with the pitch of the reed. The Clarionet, the Oboe, the Bassoon, are instruments of this order ; in them as in the organ-pipe the reed must be cut to a proper size : but each can produce several tones, for the vibration of a reed made of cane is more irregular than that of a metal reed, and as the size of the resonating air-cavity can be altered by manipulating the finger-holes and keys, to each altered size of the resonating cavity corresponds a different note or com- ponent of the motion of the reed, which the tube is capable of selecting, and to which it resounds. In the ordinary Organ-pipe a sheet of air is blown across the embouchure of the pipe, and vibrates like an invisible reed just outside the pipe : the air inside is set in vibration. The pitch of the sound produced by it is not so high as we might be led to infer from the dimensions of the column of air in the pipe, its elasticity, and its density alone : the reason is that the column of air in the pipe is not an isolated mass of air. While vibrating it has at its extremity to thrust aside the surrounding air at each expansion ; the inertia of the surrounding air hampers the vibra- tion, and, as it were, loads and retards the vibrating column. Nodes in an organ-pipe may be demonstrated by Koenig's manometric capsules attached to the sides of the pipe. Near the centre of an open pipe there is a node, a maximum of varia- tion of density, and hence the air at the node is alternately squeezed together and relaxed, the membrane of the manometric capsule placed at the node will vibrate, and the flame will either be extinguished or will present variations of height. If an organ -pipe be strongly blown, the sound will rise slightly : if the force of the current exceed a certain amount, the sound suddenly breaks into the upper octave a phenomenon familiar to flute-players and near the centre of the pipe is now a loop, while there are now two nodes. Hence, in a pipe so over- blown, the manometric capsule fixed at a point near the centre almost ceases to indicate variations of density, and the flame remains almost steady : never absolutely so in practice, for there is not even any one point at the centre of any loop at which all variations of pressure entirely vanish. If a pipe be filled with hydrogen the sound produced is nearly two octaves above that produced by air at the same temperature; for the velocity of the sound-waves is A/ - (p. 33 1); 406 SOUND. [CHAP. K is the same in all gases, and in the case of hydrogen p = x the density of air ; whence the frequency of vibra- tion in hydrogen is to that in air as 1 : \/14*47 = 1 : 3'8021 or : : C : 'I + . If the barometric pressure vary, the pitch is unaffected : for A/ = A/ ; but the density p increases or diminishes P P T) pari passu with the pressure p : thus is constant, the velocity P is constant, and the pitch is constant. If the temperature vary, the ratio of K to p is disturbed. pv oc : pv = whence cc and velocity = \J oc \/ 0. P P P- The pitch varies as the square root of the absolute temperature. A pipe which gives a sound a = 435 at the temperature of 10 C. will vibrate 446 fl 79 times per second at the temperature of 25 C. (As v/283": \/298~ : : 435 : 446-79) : the pitch is thus sharpened in the ratio 1:1*02'7105, or more than two commas; while the reeds are affected to a very much less degree, and thus an organ tuned in a cool room becomes discordantly out of tune with itself when the room becomes warm. Problem. Of what length should an organ-pipe be which sounds C / at 0" C. and 760 mm. bar. pressure ? C, = 64 vibrations. .1* i * ~ v 1 /kU 1 /I -41 x 1033 x 981 A = / = / = 518'2 centimetres. n V p 64 V y^j- A pipe of this length really gives a sound somewhat lower than C, ; the wider the pipe the lower the pitch. A C,-pipe is called a 16-foot pipe, though really somewhat longer. The common Flute, the Fife, the Piccolo, the Flageolette, are modifications of the open organ-pipe : the size of the vibrating column of air is altered in these instruments by opening or closing lateral apertures. In the Flageolette the mouthpiece is so constructed that the stream of air cannot but pass in the direction empirically found to produce the best sound : in the xiv.] MUSICAL INSTRUMENTS. 407 Flute, Fife, and Piccolo the direction of the exciting stream of air, and to some extent the corresponding quality of tone, and even the pitch, are under the control of the player. Brass instruments vary in shape from the Bugle (conical from mouthpiece to bell) to the Trumpet (slowly- widening tube and suddenly -widening bell), with the intermediate forms, the Cornet and the French Horn. The lips of the player are made to vibrate : the cavity of the instrument resounds. The scale of the instrument is confined to harmonics, which are obtained by varying the method and force of blowing and the tension of the lips; but since the size of the cavity of the instrument may be modified, any note of the scale within the compass of the instru- ment may be brought in as a harmonic of some fundamental note. The size of the cavity may be modified by slides, as in Trombones instruments which are capable of giving pure inton- ation ; by pistons which lengthen the tube by predetermined amounts, as in the Cornet and Cornopean, the Euphonium, and Bombardon; by levers which shorten the tube, as in the Ophicleide. In some cases, however, as in the French Horn and Trumpet, the instrument is confined to one note and its harmonics; but its cavity may be altered in size by the addition of additional pieces of tubing or " crooks," whose dimensions are so adjusted as to cause the fundamental tone of the instrument to change by an interval predetermined for each crook; and the note produced may be to some extent modified both in pitch and in quality by the action of the lips, and of the right hand placed in the bell or the open mouth of the instrument. The pitch of these instruments is that of an open pipe. Their tone is rich in quality and full of high harmonics ; and though we are inclined to associate their peculiar quality with metal, it depends only on the form of the internal cavity and of the bell mouth, not on the material of the walls of the instrument, provided that these be rigid. A cornet sound will be produced by a cornet-shaped tube of guttapercha if the tube be so thick as to be rigid. Stopped Organ-Pipes. A longitudinal column of air steadied at one end should vibrate at one half the rate of a column free to expand at both ends ; but a stopped organ-pipe does not give, as this rule would indicate, the octave of the open pipe of the same length, but its seventh, or even its minor seventh. When overblown it breaks into harmonics in the same way as an open pipe does. 408 SOUND. [CHAP. Other Sources of Sound Singing and Sensitive Flames. A tube terminating in a blowpipe -nozzle is connected with the gas supply : the gas is ignited at the nozzle, and a small flame is thus produced. This flame is slowly passed up a wide glass tube, such as an Argand lamp-cylinder. At some particular position in the tube the flame alters its character and begins to sound forth a note somewhat higher than the natural note (in cool air) of the tube itself; or, should it not spontaneously burst into song, it may be induced to do so by singing to it a note whose pitch is nearly that which it will emit when in action. When the action has commenced, the flame is found, as by observation with a rotating mirror, to be alternately extinguished and rekindled ; its image appears to be broken up into separate beads. A flame of hydrogen is prompter in its action than a flame of coal gas ; for it excites more disturbance in the column of cool air surrounding it. When the action has commenced, the vibrations of the flame and of the column of air react on one another. The flame must be of a certain height adapted to each size of tube. If the apparatus be arranged so that the flame stands about one-quarter up the tube, and if the height of the flame be carefully altered by slowly turning the controlling gas-tap, it will be found that at a certain definite height of flame the action will spontaneously commence, doing so with small initial intensity, while, when the gas-flame is lowered to one-half of this effective height, the sound breaks into the octave above. When a jet of gas is allowed to flow vertically upwards under a pressure of about J inch of water, through a fine vertical-jet, above which a sheet of fine wire-gauze is arranged horizontally at a distance of about 2 inches, the gas may be lit on the upper side of the wire gauze, through which the flame will not descend. The distance of the jet from the gauze may be so adjusted that the flame is yellow at the tip, and at the tip only. The flame is now a sensitive flame, and responds by sinking down to the gauze when sound-waves strike it. If it be surrounded by a wide tube it will sing spontaneously. If, while the flame is so surrounded, the gas be turned down until the singing just ceases, the flame becomes extraordinarily sensitive to all very high sounds, such as hissing, the rattling of keys, etc., and it sings loudly as long as these stimuli are maintained. A simple flame issuing from an exceedingly narrow steatite-burner, under a very great pressure of gas ("10 inches of water"), is sensitive to sounds too xiv.] OTHER SOURCES OF SOUND. 409 acute to be perceptible to the human ear; it alters its form under their influence. Trevelyan's Rocker. A mass of lead so shaped as to rest upon two long but very narrow linear feet. Placed upon a hot body, the points of contact of the rocker with the hot body are suddenly expanded by heat ; the rocker is jerked upwards ; before it has fallen back, the heated points have cooled and re- turned to their normal dimensions. The process is repeated. The whole oscillates rapidly and makes a humming noise. Radiophony. When a beam of light or radiant heat falls upon a body capable of absorbing heat, that body becomes warmed and expands. A flash of light produces an instantaneous expansion, which immediately dies away. An intermittent beam produces a succession of expansions and contractions; in other words, the surface of the body vibrates. The amplitude of its movement may, with beams of light of moderate intensity, exceed the ten -millionth part of a centimetre. Lord Kayleigh has shown that this amplitude is sufficient for the production of sound ; and the power of converting the energy of an intermittent beam of light or radiant heat into that of sound has been shown by Prof. Bell, the inventor of the telephone, to belong to all matter, with a few doubtful exceptions. If an intermittent beam be focussed upon a mass of lamp- black, at each flash of light it becomes warm, and the air within it is dilated ; if it be contained in a test tube the open end of which is connected with the ear by an indiarubber tube, as the successive flashes produce successive dilatations and pulses in the air, these pulses are perceived by the ear as sound ; if the lampblack be contained within a resonator, the frequency of whose natural vibration is equal to that of the frequency of suc- cession of the flashes, the resonator emits a loud sound, audible at a distance. PROPAGATION OF SOUND. Propagation of Sound occurs in all elastic media, and is effected by waves of alternate Compression and Karefaction. Propagation of Sound in Solids. Sound being a vibra- tion is propagated along the ground ; we may put our ears to the ground to listen for distant railway trains, distant vehicles, distant firing or marching. It is conducted along wood, as in the ordinary stethoscope ; or in the experiment of closing the teeth upon a long piece of wood, to the other end of which an 410 SOUND. [CHAI>. assistant holds a vibrating tuning-fork ; or of causing a long strip of wood to rest by one end against the panel of a door, the other end being in contact with a vibrating tuning-fork, the vibration in this case being communicated to the panel, which acts as a sounding-board, and itself sounds out loudly. Sound is conducted by wires ; taps on a telegraph wire are audible at a great distance to an ear applied to the wire, provided that there be no interven- ing tunnel or bridge to form a resonating cavity and to absorb the energy of the vibration. In the Wire Telephone the central points of two stretched membranes or boards of thin wood are connected by a long wire, which, if sufficiently heavy and tense, may be suspended from posts, or even stretched upon carpet or bent round corners, and thus serve the purposes of domestic tele- graphy. When sound-waves impinge upon one membrane of the wire telephone, as when it is directly spoken at, the complex motion of the air is transferred to the membrane ; by the mem- brane it is transferred to the wire ; by the wire to the second membrane ; by that membrane to the air, and by this to the ear of the distant listener. A slender apparatus of this kind, consist- ing of two parchment membranes stretched on rings, with an intervening silk thread, is sold as a toy under the name of the lover's telegraph. Propagation in Liquids. Divers while under water hear the sound of waves beating against the shore. A tumbler of water standing on a resonance box, if the handle of a vibrating tuning-fork be dipped in the water, will convey the vibration to the resonance box, the air in which will resound. An inverted bell filled with water and set in vibration will cause the water to assume beautiful wave-forms an experiment which may be performed with an inverted propagating-glass set in vibration by a wetted finger drawn round its edge, or with a capacious wine- glass set in vibration with a violin bow. In the latter case the interest of the experiment is increased by substituting for the water some strongly-alcoholic liquid ; the agitation breaks the surface of the liquid into drops which, by evaporation, lose some alcohol and dance on the surface of the vibrating liquid. An organ-pipe may be blown by water under water; the water vibrates in the place of air. Propagation in Air and other Gases. In general sound travels in air in concentric spherical waves. If it be restricted to tubes, the waves may become plane-fronted. Though these waves are invisible their existence is beyond xiv.] PROPAGATION OF SOUND. 411 doubt, for when they strike any solid object they produce mechanical effects, and the phenomena of sound, so far as these depend upon propagation through the air, obey the laws of wave- motion. The breaking of glass windows by the discharge of artillery, the* destruction of the drum of the ear which has been known to be caused by the explosion of dynamite, the destruction of property by the explosion of a gunpowder magazine, are only exaggerated instances of that conveyance of energy by the air which is asso- ciated with the production of sound. In Edison's Phonomotor a membrane is stretched over a frame ; at its posterior aspect it is connected with a broad hook which rests on the broad margin of a heavy wheel : the margin of this wheel is provided with small roughnesses so shaped that it is easy for the broad hook to slip over them in one direction, but in one direction only ; on its return it is caught in the small teeth and tends to pull the wheel round. The membrane is spoken at ; it trembles ; the hook catches some of the teeth ; on its return it gives an impulse to the wheel ; continuous sound causes the wheel to rotate, and this with considerable speed and power ; and if a little crank be fixed to the rotating wheel, the energy of the human voice may be made to perform obvious mechanical work. Sound-waves in air are amenable to the laws of ordinary tridi- mensional Wave-motion already discussed. In Figs. 43-48 we find curves whose forms depend upon the assumption that when two vibrations concur, the amplitude of the resultant is obtained by addition of the amplitudes of the com- ponents. To a first approximation this is true, but it leads to a curious result. If the amplitudes and periods of two vibrations be equal, the resultant vibration (Fig. 44), having twice the amplitude, will have four times the energy of either ; the motions cannot be so superposed without a draft of energy from else- where. If two equal waves arrive in the same phase at the entrance of the same channel, there is found to be reflexion of a negative wave from the mouth of that channel. Superposition of vibrations is familiar in Acoustics as a cause of Beats. Two tuning-forks or reeds are brought into exact unison ; they emit jointly a smooth sound. Suppose their pitch to be c" = 522. Load one with wax until it vibrates, say, 521 times per second. Once in the course of a second they will aid one another, and their action on the air at any given spot coincides ; once in the course of every second they will thwart 412 SOUND. [CHAP. one another ; their joint effect will be an alternate fading-away and swelling-out of the sound ; but the pitch of that sound will be 521^ vibrations per second. As the loading of the one fork increases, the number of beats increases until they be- come too rapid to be counted ; but before this occurs the two notes have ceased to be blended into one note of average pitch, and the effect is that of a painful discord. The easiest as well as the most accurate way of tuning a fork to a given note is to have at hand a standard fork which makes four vibrations less per second than correspond to that note ; then adjust the fork to be tuned until it makes exactly four beats per second with that artificial standard. Diffraction. Sound-waves have in general the properties of waves whose wave-length is not small in^ comparison with the apertures through which they pass, the surfaces by which they are reflected, or the obstacles round which they flow. A sound- wave coming through a chink would surfer great lateral diffraction, as shown in Fig. 5 6 ; and the effects of obstacles intervening in the path of a wave of sound may not be such as even to produce a sound -shadow. If, however, the apertures, or surfaces, or obstacles in question be very large in comparison with the wave- length, there may be true sound-shadows and limited beams of acoustic disturbance, reminding us of the shadows and beams of light met with in optics. The air-waves produced by the note C (= 64) have a length of (33,200 -^ 64 =) 518*75 cm., or between 1 6 and 1 7 feet ; those produced by the note c frrrf have a length of about an inch and a-half. When a mixture of long waves and ripples strikes an obstacle, the long waves may pass round it, while the obstacle may intercept the ripples, for rela- tively to these it may be wide enough to cast a sound-shadow. Thus the sound of a brass band suddenly changes in quality when the band comes round a corner into sight. It is plain that for acoustic purposes all the auditors at a concert, though they are not absolutely prevented from hearing by not seeing the per- formers, ought to be in full view of the orchestra. It sometimes happens that a person at the same level as a source of sound, and in full optical view of it, hears nothing : the sound-waves, passing through strata of different density, curve upwards, and being very broad present no diffraction, and pass over the head of the observer, who may again come within their range by elevating his position. Reflexion of Sound follows the ordinary laws of the reflexion xiv.] REFLEXION OF SOUND. 413 of waves. Sound can be reflected by a mirror; a high-pitched bell ringing round the corner of a house can be rendered audible by a sufficiently -large mirror placed at a proper angle. If a stretched sheet of tracing-paper be placed at the same angle it will reflect a proportion of the sound and transmit some of it. A dry handkerchief will transmit a considerable amount of sound ; so will even a half-inch layer of felt ; but if wetted these become better reflectors, while they become almost impervious to sound. The reflexive power of flame is nearly the same as that of tracing paper ; and hot air above a gas-flame can reflect sound almost as well as the gas-flame itself. In clear weather the air is rarely uniform ; there are ascending and descending currents of hotter and colder air ; at each surface of each of these sound is partly re- flected, partly transmitted, and so, ere long, it is wholly dissipated. Sound is often heard better in foggy or even in rainy or snowy weather than in clear, for then the air is more uniform. Sound coming through a fog (vesicles or minute drops), or a shower of rain, or a shower of snow, must be to a certain extent lost by repeated reflexion ; but this effect is often balanced by the in- crease of intensity arising from the concentration of the waves in the narrowed channels between the drops or flakes. In many buildings there are whispering galleries or places where a faint whisper uttered at a particular spot is heard at a distant part of the edifice. This phenomenon may arise in two ways : (1) Eeflexion from the vaulted roof, which acts like a con- cave mirror and causes the waves received by it to converge after reflexion upon a particular focus a phenomenon very common in ellipsoidal roofs, a whisper uttered at one focus of the ellipsoid being reflected to the other focus, and distinctly heard there ; a similar phenomenon also occurring in elliptical rooms where the sound is reflected by the walls : or (2) by the sound undergoing successive reflexions, and thus travelling round the walls. Keflexion of sound is familiarly illustrated by the Echo. Sound striking a broad cliff or wall is reflected, the reflected waves sometime travelling with singular absence of diffraction and precision of direction. If a person can utter ten syllables per second, and if he speak loudly at that rate in presence of a cliff or high wall directly opposite him and at a distance of 1660 cm. (55 feet), just as he is commencing the second syllable the reflected sound of the first syllable begins to arrive at his ear, and at the instant when he ceases to speak the sound of the last syllable spoken begins to be heard ; the sound travels to the cliff 414 SOUND. [CHAP. and back during the tenth part of a second. If the cliff were 5320 cm. distant, the speaker would hear two syllables repeated. When sound is re-echoed from cliff to cliff, or to-and-fro between two smooth walls directly opposite to one another, the result may be a multiple echo, which repeats a sound several times. The roll of thunder is partly due to multiple reflexion from cloud to cloud, partly to the varying distance of the points of disturbance, and the successive breaking on the ear of sound-waves produced along a long line. When the distance of the reflecting surface from the source of sound is too small to produce a distinct and separate echo, the echo may be heard merely as a reinforcement of the sound pro- duced ; whence the practice of placing sounding boards behind and above pulpits and orchestras. The action of the ear-trumpet depends, in the first place, upon multiple reflexion ; sound-waves on their arrival are reflected by the bell into the tube ; then they travel, plane-fronted or nearly so, down the tube to the ear, narrowing in breadth and increasing in intensity as they do so. Refraction of Sound. If a lens be constructed of two large sheets of collodion cemented together at their edges and inflated with carbonic acid, sound-waves diverging from a watch placed at one side of this lens may, after passing through it, converge upon a focus on the other side of it; this showing that refraction occurs when the sound-waves enter the denser medium, the carbonic acid. Interference of Sound. If A and B in Fig. 75 represent the position of two tuning-forks kept accurately in unison, being driven by the same interrupted electric-current, the ear, placed successively at of, Z/, c', etc., perceives alternate sound and silence. The same occurs when A and B in that fig. represent apertures in the side of a padded box within which an organ-pipe or bell is caused to produce sound. Velocity of Sound. There is but little information as to the properties of sound-waves travelling in a substance whose elasticity is not the same in all directions. In Scotch fir the longitudinal propagation is more rapid than that across the fibres of the wood in the ratio of 5 : 4 ; hence sound-waves in that wood must be spheroidal. This is ascertained by observing the form of the nodal lines in a vibrating plate of that wood. In practice we meet with spherical waves such as those in the air, and plane-fronted waves such as those which run along xiv.] VELOCITY OF SOUND. 415 wires. In general the propagation of sound is amenable to the yg~ Where the compression produced by the wave as P it runs has little or no effect upon the temperature of the medium, as in the case of solids and liquids, the velocity of sound is appreciably equal to vK -f- p ; where heat is developed by the compression, as in gases, and the heat so developed has no time to be appreciably conducted or radiated away from the com- pressed part of the wave, the velocity of sound is equal to -T- p, where k is the ratio between the two specific heats. The velocity of sound in Solids may be found by determining the pitch of longitudinal vibrations set up in long thin rods. The length of the rod is half a wave-length, ; the pitch gives n, the number of vibrations per second ; the equation v = n\ gives the velocity of sound in cm. per second. The velocity of sound in Liquids may be determined by direct experiment, as by sounding a bell under water, and by listening at a distant station for the arrival of the sound, the precise instant of the production of which is signalled ; or by comparing the pitch of organ-pipes blown in different liquids. /K The formula v =/ gives for water the velocity of 143,900 cm. V p per second ; the observed value is 148,900 cm. The velocity of sound in Gases as, for example, air has been directly determined by firing cannon at a known distance, and by observing the interval of time which elapses between seeing the flash and hearing the report. The objections to this method are that such violent concussions as those of cannon produce aerial vibrations which can be shown to travel faster than disturbances of less intensity, and that the velocity is not equal in all direc- tions round the cannon or at all distances from it. The length of an organ-pipe producing a note of a given pitch has been taken as a means of measurement. The column of air in the pipe, if it vibrated alone, would be half a wave-length in length ; but the air is not isolated ; the sound actually pro- duced is graver than that corresponding to an isolated column of air, and the velocity so measured is not even approximately correct. On similar principles the length of a pipe closed at one end and subjected at the other to the aerial impulses derived from a 416 SOUND. [CHAP. vibrating tuning-fork this length being so adjusted by a movable piston that the air in the tube resounds its loudest is taken as , and the velocity of sound is again obtained only approximately, however (Fig. 149). Fiff.149. The velocity of sound in air may also be determined by methods based on interference. At A (Fig. 150) sound enters; the waves pass along the two channels B and C to E. B can be lengthened ; it is lengthened until no sound is heard at E ; B is now half a wave-length longer than C. At E may be placed the ear, a resonator, a manometric capsule, or any other indicator of sound-waves. No energy passes down DE when no sound is heard at E : the waves are reflected at D and pass back to A, the B-waves returning along C, the C-waves along B ; they arrive at A in the same phase. Even if they arrive at D in the same phase and pass on to E, there must be reflexion of a negative wave from D along B and C. A glass tube vibrating longitudinally will emit a sound ; the length of the glass is half a wave-length. If the tube contain air, the air will be set in vibration ; but it must divide itself into segments, each half as long as an air-wave corresponding to the same tone. Finely-powdered silica placed in the tube will be distributed by the vibrating air in such a way as to accumulate at the nodes of the aerial vibrations. The number of air-segments thus indicated shows the comparative speed of waves in air and in glass. This is Kundt's method. The theoretical value for the velocity is found from the equa- tion v = \ ~ , where k can be found by thermodynamic experiments (page 331), K and p by direct observation. The best average value for the velocity of sound in air seems to be about 33,200 cm. per second ; the extremes being 330'6 (330-7 Eegnault) and 333-7 metres. The velocity of sound in air is unaffected by variations of pressure, as we have already seen ; it varies as the square root of the absolute temperature ; it is affected by humidity, for damp xiv.] VELOCITY OF SOUND. 417 air is lighter than dry air under the same pressure. It is also affected by wind, which not only retards the passage of sound to windward, but may also distort the waves and cause them to pass upwards. In general, the greater the intensity of sound the greater its velocity : sound therefore continuously slackens in speed of transmission of speed as it becomes fainter. It is still a moot point whether acute sounds are more or less rapidly conveyed through air than grave sounds : they are certainly not conveyed so far, for they are more affected by the viscosity of the air, and thus the higher components of an inharmonious sound are abolished by distance, and the whole effect is softened. The velocity of sound' being known, it becomes possible to measure certain distances by its aid. A lightning flash is seen, practically, instantaneously ; if the sound take 5 sec. to travel, it must have travelled 1660 metres, or about a mile. Again, a stone is dropped down a well ; the sound of the splash is heard in 6 seconds : at what depth is the surface of the water ? The stone falls for x seconds through a space equal to \gy? ; the sound comes up for y seconds through a space equal to 33,200?/ cm. From the two equations x + y = 6, J x 981x 2 = 33,200?/, we find x = 5 '543, and the depth equal to 15,172 cm. Propagation of Sound in gases according to the Kinetic Theory.* Suppose AB to be a cylinder, CD an oscillating piston. The molecules which make up the gas in AB are represented by dots ; for the lg ' nonce we shall suppose these molecules to be arranged in straight files, and we shall consider only one such file. The members of such a file remain in the same straight line, striking and rebounding from one another. We keep in mind the proposition that two perfectly-elastic bodies when they enter into collision rebound from one another with exchanged velocities ; and that other, that a perfectly-elastic body striking a rigid obstacle returns with a velocity equal to the relative velocity with which it struck the obstacle. If the piston be at rest, it does not change the velocities of those particles which strike it, except in their direction. If it be moving towards them as they strike it, they rebound from it with a velocity greater than that with which they approached it ; this increased velocity they communi- cate by exchange to others with which they come in collision : they then return to the approaching piston, and again rebound with increased velocity. These molecules thus act as carriers of energy ; they borrow from the advancing piston a certain amount of energy, which they pass on to the mole- cules beyond ; from molecule to molecule this energy is transmitted and a transitory crowding together of the molecules, commencing at the piston, is propagated through the whole gaseous mass. Conversely, when the piston is in retreat, those molecules which overtake it rebound with a diminished velocity ; they exchange this diminished velocity with those molecules which * See Tolver Preston, Phil. Mag. iii. (1877). 2 E 418 SOUND. [CHAP. they tardily encounter, and which they do not turn back until these have travelled farther than they normally could have done ; a diminution of velocity and an accompanying rarefaction are propagated throughout the gas. If the particles all lay in straight lines, the speed of propagation of sound would be the average speed at which the particles move ; just as the rate of propagation of a message by couriers would be the average rate at which they ride. Nothing would be gained in point of speed by multiplying the number of such couriers if their horses were not susceptible to fatigue ; and so it is a matter of indifference what the number of molecules is by the intervention of which the exchange and transmission of energy are effected, provided always that the collision of the molecules occupies a time inappre- ciable in comparison with the interval spent by them in traversing their free path, and further that their own size is very small in comparison with the average length of that free path. Thus the velocity of sound does not depend on the density or on the pressure of the gas. If the speed be changed, the case is different ; an increased average speed causes an increased velocity of propagation. This may occur within the same gas when the temperature is altered ; the absolute temperature measures the kinetic energy of the molecules ; to the square root of this the mean velocity is proportional ; the rate of sound-propagation is equal to this mean velocity ; whence the rate of sound-propagation must vary as the square root of the absolute temperature. On comparing t\vo gases we find that the mean velocity varies inversely as the square root of the molecular weight, and therefore as the square root of the density ; whence the velocities of sound in two different gases are to one another inversely as the square roots of their respective densities. We cannot, however, affirm that the particles of a gas lie in straight lines or files ; they move on the whole with perfect symnletry with reference to every point. Professor Clerk Maxwell showed that, taking this into account, we ought not to expect the rate of propagation of sound to be equal to the average velocity of the particles, but proportional to it; and that, on the assumption that the particles were small as compared with their mean dis- tances, and that each one was smooth and round, so as riot to be set in rotation by impacts, then the rate of propagation of sound should bear to the mean velocity of the particles the ratio of \/5 : 3, or '745 : 1. Kundt and Warpurg found exactly this ratio in the case of the vapour of mercury. In hydrogen the mean velocity is 184,260 cm. per sec. ; the velocity of sound in hydrogen is, according to the mean of several observations, 126,9 17 '6 cm. per sec. ; the ratio is -6888 : 1, less than that given above. The inference is that the molecules are to some extent set in rotatory as well as in translatory movement. Doppler's Principle. From a sounding body, approaching or approached, sound-waves reach the ear in greater number than when the source of sound and the listener are relatively at rest, and vice versd. To a person standing at a railway station while an express rushes whistling through, the pitch of the whistle, seems suddenly to fall as the engine passes him. Even the puffs of an approaching goods-engine seem appreciably more numerous to the ear than those of a receding one. xiv.] SOUND. 419 THE HUMAN EAR. Aerial waves are communicated to the air in the external auditory meatus. This is short in comparison with the length of the average sound-wave. Its own proper sound is about g"'", and sounds in the neighbourhood of this tone are painfully rein- forced by the resonance of the meatus. The movements of the air in the meatus do not materially differ from those of a single point in the wave-front : the physical problem to be solved in the organ of hearing is one of the same kind as would be presented if the eye were called upon by the inspection of a single point on the surface of a multifariously- rippled sheet of water to discriminate all the component undula- tions of the extended surface. The movements of the air in the meatus are, in consequence of the changes of pressure which they occasion, communicated to the drum of the ear, the membrana tympani. The drum of the ear may receive some vibrations by direct transmission from the bones of the skull. We remark here (1.) The natural note of so small a membrane is very high ; but weighted as it is by the chain of bones of the internal ear, it can take up vibrations of a much less frequency than this note. (2.) The vibration of the membrane is a forced one, and, as regards the amplitude of very high components, does not precisely coincide in character with that of the air. (3.) At the same time the form of the membrane is such that it vibrates more at its edges than at its centre, and the tendency of the membrane to set up vibrations of its own, or to alter those forced upon it, is mitigated. (4.) The membrane is normally under tension : it is pulled inwards by the handle of the malleus; considerable pressures upon it cause very small inward movements, especially since its radial fibres have very slight extensibility. (5.) It is easier for a rarefaction of the air in the meatus to cause an outward movement, which slackens the membrane, than for a condensation to drive the membrane inwards and thus to tighten it a fact of importance in reference to combinational tones, of which hereafter. (6.) When the membrane does move inwards, it pushes 420 SOUND. [CHAP. inwards the handle of the malleus, which is firmly attached to it : but only through a very small distance. This small amplitude of movement, about one-fortieth of that of the air in the meatus, implies that the handle of the malleus is wielded with con- siderable force one step in the increase of the force of the aerial vibrations on their way to the internal ear. (7.) The movements of the drum of the ear are astoundingly small. The greatest displacement seems to be about O'l mm. or 1-2 5 Oth of an inch. A sound produced by an/ $ ( = 181) pipe under an air pressure of 40 mm. of water can be distinctly heard at a distance of 115 metres. Topler and Boltzmann calculated that at such a distance the movements of the air must be reduced to *000,04 mm.; but those of the more massive drum, with its appendages, cannot be more than "000,001 mm. or the twenty- five-millionth part of an inch an oscillation so minute as to be beyond direct microscopic observation. The drum of the ear sets in motion the handle of the Malleus. The malleus is a small bone, somewhat resembling a hammer, with a head and a handle. It is so suspended by ligaments, head upwards, that when its handle is thrust inwards, its head is made to rotate to a limited extent. The head of the malleus is connected by a smooth joint of peculiar form with a second bone, the Incus. The action of the joint is such that when the handle of the malleus is forced inwards, the head, as it rotates, locks in the incus and forces it round ; while, if the handle of the malleus be driven violently outwards, the head, rotating in a reversed direction, does not pull the incus with it, but glides over it, rotating through as much as 5 before the two bones again begin to move as one piece. If air be driven through the Eustachian tube from the mouth-cavity, as it always is during swallowing, it presses against the membrane from within ; if it were not for this peculiar joint there would be a decided risk of the chain of bones malleus, incus, and stapes being torn away from their connection with the internal ear. While the two bones are thus unlocked, as during swallowing, there is an impairment in their power of transmitting vibrations, and there arises a partial deafness, especially for loud sounds. The incus has a process or long projection which, when the handle of the malleus moves inwards, moves inwards also. The point of this is attached to a little stirrup -shaped bone, the Stapes. Motion is thus communicated through malleus, incus, stapes ; but the stapes move only two-thirds as much as the end of the handle of the malleus another step in the increase of xiv.] THE HUMAN EAR. 421 force and diminution of amplitude of the vibration conveyed to the ear by the air. Communication of sound through the bones of the head seems to be effected, for the most part, through the malleus, incus, and stapes. Com- munication of sound to the bones of the head may be facilitated by means of the audiphone. This is a plate of thin vulcanite, which is bent and kept by strings under a certain degree of tension, and the edge of which is placed in contact with the teeth. Even a piece of stiff brown-paper, loosely doubled and grasped by its opposite edges between the teeth, will act as an audiphone and take up waves of sound from the air, and will convey them to the bones of the head. A certain proportion of the sound travels directly to the nervous apparatus embedded in the skull, which is directly shaken ; sound is thus rendered to some extent audible to those whose auditory ossicles fail in their function. The footplate of the stirrup is blended with a membrane occupying a small aperture the fenestra ovalis in the hard mass of the temporal bone. The footplate of the stirrup has itself a form closely resembling that of a footprint of the right foot. If now the reader will place his right foot on a soft carpet and forcibly drive into the carpet the outer edge of that foot, he will see that the inner edge of his foot is tilted upwards ; this describes the motion of the stapes when driven inwards against the membrane of the fenestra ovalis. Beyond this membrane lies the fluid of the internal ear, con- tained in membranous bags, which float in channels hollowed out in the temporal bone. This system of membranous bags contain- ing fluid consists of the vestibule, the cochlea (in front), and the semicircular canals (behind). Here we have to do with the two former. The fluid lying in the vestibule is immediately behind the membrane of the fenestra ovalis ; the vibrations of the stapes are communicated to it. This fluid is in direct communication with the fluid lying in a part of the cochlea called the scala vestibuli. The structure of the cochlea seems at first sight somewhat complex. It is a snail-shell-like object; if unwound and laid Fig.152. flat it might be diagrammatically represented by Fig. 152. S is the stapes, blending with the membrane of the fenestra ovalis ; V 422 SOUND. [CHAP. is the vestibule ; S.V. is the scala vestibuli, a cavity extending to the tip of the cochlea ; there it is continuous with S.T., the scala tympani ; this ends at />., the fenestra rotunda, an aperture closed by a strong membrane. The fluid filling S.V. (the peri- lymph) is continuous with that in S.T., and hence strong pressure on the stapes will cause />., the membrane of the fenestra rotunda, to bulge outwards. The scala tympani and the fenestra rotunda are perhaps a safety arrangement. The vibrations which are important to us are those of the fluid in the scala vestibuli, Between the scala vestibuli and the scala tympani lies a par- tition, incomplete at the apex of the cochlea. This partition is partly of bone, partly of membrane : the purely membranous part is called the membrana basilaris. Transverse section of the cochlea shows us further that we have not only to do with the two scales and the intervening par- tition, but also with a third cavity. Such a section is shown F1&153 diagrammatically in Fig. 153. The scala vestibuli rests only upon the bony part of the partition ; the third cavity, the scala cochlece, S.C., lies mainly above the membranous part, the basilar membrane, m.b. The scala cochlece contains fluid (endo- lymph), and is practically a closed cavity. When the liquid in the scala vestibuli vibrates, the endo- lymph in the scala cochlece is at each impulse forced into similar movement before the liquid in S.V. has had time to pass into S.T. ; and it in its turn acts upon the basilar membrane. The basilar membrane is triangular in form, being widest at the tip of the cochlea. It takes up vibrations of definite pitch, in response to which it vibrates not as a whole but locally ; just as when we sing to a piano with the dampers down, only those strings respond which are in unison with the sound produced by the voice. It responds to vibrations of considerable slowness compared with its natural vibrations, for not only does it lie be- tween two liquids, but it is also somewhat heavily loaded ; upon it are mounted certain osseous structures arranged in two rows, the rods of Corti ; and upon these are arranged the cells of Corti, each of which is connected with a single nerve-fibre. Each of these nerve fibres can only be stimulated to sensation by the vibratile movement of that cell of Corti from which it runs ; and xiv.] THE HUMAN EAR. 423 it is deaf to every sound but that one to which the vibration of the particular underlying part of the basilar membrane corre- sponds. As there are from 16,000 to 20,000 such cells of Corti, and therefore the same number of nerve -fibres which, although they merge into a common strand the auditory nerve and enter the brain side by side, are isolated from one another, we may say that we have not one, but from 16,000 to 20,000 distinct Senses of Hearing, each with its own special Organ of Hearing ; the Ear, as we have described it, being merely a mechanical means for the transmission of vibration from the external world to these sense organs, and its due distribution among them. There is yet some difficulty in seeing how the rods of Corti, 3000 pairs in number, affect differently the half-dozen cells of Corti borne by each pair ; hence some deny that more than 3000 different sounds can be perceived otherwise than by a process of comparison ; even this would enable us to distinguish tones differing by less than the thirtieth part of a semitone. When an external sound does not coincide with any of the above 16,000 or 20,000, the cells most nearly corresponding to it will be disturbed ; one of these cells will vibrate more than its neighbours. The phenomenon is now one of unconscious com- parison of their relative disturbances. Within the middle range of hearing, sounds differing by one vibration in three seconds can be distinguished by some persons ; when the notes chosen are very high in pitch, grave errors in the discrimination of pitch may, on the other hand, be readily committed. When a compound sound is produced, the basilar membrane is set in motion in a number of limited regions at the same time, and the effect is mingled in the brain. The nature of the com- pound sensation, by which we thus recognise the different quali- ties of sound, is a question which passes the bounds of physics. The ear has a certain power of persisting in vibrations once set up in it ; but only in small degree, and to an extent the greater the lower the pitch of the note. If the sound c' be broken up by alternate flashes of sound and silences of equal duration, when these each number 1 30 per second, the sound seems continuous. Each pitch has its own duration of persistence ; and Mayer has pointed out (see Phil. Mag. 1876, vol. ii.) that in a mixture of sounds, alternately admitted to the ear and shut off from it by apertures in a rotating disc, some may be rendered evidently intermittent by the action of the disc, while others may appear continuous, and that thus we have a new means of analysing compound sounds. As to the limits of hearing, sounds may be perceived by the human ear which range from 16 (Preyer), or 34 (Helmholtz), to 424 SOUND. [CHAP. about 32,000 (Despretz), or even 40,000 (Appunn and Preyer) ; there being between different individuals curious differences in the power of perception, especially of high sounds. A small number of abrupt clicks cannot, with ease, blend into a con- tinuous sound ; if they seem to do so, it is generally some high harmonic that is really heard. A very small number of true pendular vibrations seems to produce sound when sufficient surface is acted upon, or the ear otherwise firmly enough set in vibration ; whence the very deep hum of a river -steamer slackening speed, or that sound of contracting muscle 19 vibrations per second which is heard when the forefingers are pressed into the ears and the elbows pressed against the table. The sensible loudness of sounds does not coincide very closely with their physical intensity. This arises partly from modifica- tions in the form of the vibration induced by so complicated a transmission through the auditory apparatus, partly from causes purely physiological. It is curious that, as Mayer has shown, high notes are heard with diffi- culty in the presence of lower ones. Hence sixteen violins in an orchestra produce by no means so great an effect as sixteen violins alone. A lower note tends to drown a higher one, especially if the higher note be thoroughly in tune, or form a correct acoustic interval with the lower. In a single musical sound the fundamental tone drowns the harmonics, even though the latter may, as in a tinkling piano, very greatly exceed it in physical intensity. The sensitiveness of different ears to sound may be compared by measuring the relative distances at which a given sound be- comes inaudible; as the squares of these distances, so are the sensibilities of the listening ears. If one ear can hear a certain sound at 3 feet, the other only at 3 inches, then the duller ear is (36 2 -h3 2 ):=144 times less sensitive than its fellow. Even beyond the ear and within the brain there is some mechanism of which we are still ignorant. Professor Silvanus Thompson has shown that two sounds which beat with one another will, when conveyed separately one to each ear, produce beats which appear to jar the vertex of the brain ; and further, that two sounds of the same pitch and phase, arriving separately in the ears, appear to be heard in the ears, while, if they arrive in opposite phases, the effect is as if the sound were heard not in the ears at all, but within the vertex of the cranium. The former phenomenon may be roughly shown by a tuning-fork held to each ear ; the latter by a pair of telephones, one to each ear, one being provided with a commutator, by which the current in that telephone can be reversed at will. Direction of sound can hardly be determined if the head be held fixed ; we turn the head slightly while listening, and interpret unconsciously the consequent variations of intensity in the two ears. XIV.] SOUND. HARMONY AND DISSONANCE. >- Just as it is disagreeable to the eye to be exposed f6^s^^eW< ing light, so it is painful to the ear to be exposed to audiETe" flickering such as that produced by two sounds which beat when sounded together. The climax of unpleasantness or Discord is reached when the beats amount to about 32 per second. When the beats are still more numerous than this, the two notes which are sounded together become more distinctly separable by the ear, and the beats are less prominent to the sense of hearing. Were this the only cause in operation, the intervals (16:17), which all differ by 32 vibrations per second, would be equally painful to the ear. To some extent it is the case that an interval concordant between high notes is painful in the lower parts of the scale, as may be found on playing a major third in different octaves on a harmonium : but another cause affects the degree of painfulness of a discordant interval. When two contiguous sounds affect the basilar membrane simultaneously, the vibration of the basilar membrane is not limited to those fibres or narrow strips which exactly correspond to the two contiguous sounds heard ; the result is a confused vibration of the membrane in a wider strip, and a sound is heard compounded of all the notes within the region of that strip ; this is interpreted by the ear either where the two component notes are very close as a single note of average pitch, or, where they are further apart, as two separate notes coupled with a painful sensation. Discord is thus due partly to beats, partly to difficulty in identifying pitch. The beats produced by mis-timed consonant-intervals correspond to curious vibrational curves, for which see Bosanquet, Phil. Mag. 1881. When two notes are produced by separate sources of sound, the upper harmonics may possibly clash and beat with one another. This may be specially observed in the roughness of full chords of a brass band within an enclosed space. If, for instance, C and G be sounded together, the aggregate harmonics are the following : of which only g, , g f in a similar way has the primary differential tones Ab, Eb, c, and the secondary differential tones c, l\>, d f , g'\>, b\>, c" ; and the primary chord is thus embedded in a mass of combination tones comprising, among others, the discordant series The consonance of c', e'b, g f is therefore necessarily much harsher than that of c', e f , and . 83 foot-pounds per square foot of the earth's surface per second ; hence, to use Sir William Thomson's phrase, the " mechanical value of a cubic mile of sunlight," the energy of the waves comprised within a cubic mile of ether near the earth, is about 12,050 foot-pounds. Heat-waves and light-waves in ether are not waves of compres- sion and rarefaction, like those of sound in air. The propagation of an ether -wave is effected after a different fashion, somewhat difficult to realise. The analogy of a transverse vibration running along a cord, or of a wave of up-and-down oscillation running over the surface of water or over a thin membrane, must be extended to the ether, with its three dimensions in space. At any point where the movement of the ether is examined, it is found to be-- an oscillation at right angles to the direction in which the wave is being propagated, and therefore parallel to the wave- front. The vibration of the ether is initially of the nature of a forced vibration ; it is probably excited by the oscillation of a part of the ether, which is in some way entangled within, or which en- velops, the vibrating molecule. The molecular vibration which excites the ether-waves is a true vibration of the molecule, not a translational oscillation from place to place. The molecules of ordinary matter must be supposed, in virtue of their small size, to vibrate very rapidly. "We have already stated that the average diameter of molecules is about the -roocfoooo P art ^ a millimetre, and that they may perhaps consist of ether rolling within ether in vibratile vortices. A steel tuning-fork 2 inches (50 mm.) long may, if it be of the proper form, vibrate 480 times a second ; if it were ^^^-/Q mm. long, and of the same shape, it would vibrate 30,000,000000 times per second ; if made not of steel, but of ether, its frequency would be greater in the ratio of the velocity of propagation in steel to that in ether, and would therefore amount to about 1,740,240000,000000 oscillations per second. The vibration of a molecule is more like that of a disc than that of a tuning-fork ; but the rough analogy just mentioned may serve to show that it is, even a priori, probable that some such number may denote the average frequency of the ether-waves, an average modified in the direction of retardation by the formation of heavier molecules through the coalescence of smaller molecules, or perhaps by the reaction of the ether which is set in forced vibration, or modified, on the other hand, in the direction of acceleration by the formation of higher-pitched vibra- tions, which may be, to use the musical analogy, dissonant with one another when the structural arrangement of the molecules is unsymmetrical. The molecule of sodium-vapour acts somewhat like a disc which is slightly un- symmetrical : such a disc would give out two tones very near one another in pitch : and a vibrating sodium-molecule gives rise to two sets of ether- waves which differ only slightly in frequency. Limits of Frequency. We are not acquainted with any xv.] LIMITS OF FREQUENCY. 433 ether -waves except those whose frequencies lie between the limits of about 107,000000,000-000 (Langley), and about 40,000,000000,000000 oscillations per second a range, to use a musical analogy, of about eight-and-a-half octaves ; but of these our eyes are sensitive to scarcely one octave to those, namely, which range between about 392,000000,000000 per second (extreme red of the spectrum), and about 757,000000,000000 per second (extreme violet). There may perhaps be ether- waves more or less rapid than the extreme limits mentioned, but we have no sense by which their existence is made known to us, and, at present, no experimental means of investigating them. Velocity and Wave-length. These waves all travel through the ether/of' space at the same rate, namely, about 30057,400000 cm. (186,680 miles) per second. Ether- waves while traversing the ether present no essential differences, except in respect of their frequencies, and hence also of their wave-lengths ; the latter vary in a vacuum from about 10 ^ 00 cm. to about ^300000 cm -> and those waves to which our unassisted eyes are sensitive, the waves of light, have wave-lengths ranging between T3 ^ 42 cm. and 25165 cm - These wave-lengths are usually ^pecified in terms of "tenth-metres;" a tenth -metre being , , or 0*000000,01 cm. Extreme red and extreme violet have thus in a vacuum the respective wave-lengths of 7667 and 3970 tenth- metres. Ether-waves do not traverse all substances with equal speed : hence their wave-lengths in different substances vary; if any particular kind of radiation have to be spoken of, it may be defined by specifying its wave-length in some specified medium, but it is better to state its numerical frequency. To do the latter implies, however, that we assume and we are apparently justified in assuming that all kinds of light pass through a vacuum that is, through the ether of space with equal speed. Kinds of Radiation. When a succession of waves impinges on a mass of ordinary matter, the effect varies according to the nature and the condition of the body which receives their shock ; if it be an ordinary opaque mass, that mass may be warmed, wave- motion being transformed into heat, and the waves which have impinged upon it are ex post facto called a beam of Radiant 2 F 434 OF ETHER-WAVES. [CHAP. Heat ; if they fall upon the eye, they may produce a sensation of light, and the wave-system is then called a beam of Light : fall- ing upon a sensitised photographic plate, or a living green leaf, it may operate chemical decomposition, and it is then called a beam of Actinic rays. The word " rays " in the last phrase may be understood to mean not imaginary lines at right angles to the wave-front, but kinds of radiation; and hence we speak of Heat rays, of Light rays, of Chemical or Actinic rays ; these names being given to one and the same train of waves according to the effects which it is found competent to produce. But while ether- waves are in course of traversing the ether, there is neither heat, light, nor chemical decomposition ; merely wave-motion, and transference of energy by wave-motion. Hence none of these names can in strictness be applied to a train of waves while these are actually travelling through the ether. Ether-waves which differ in their frequency differ to some extent in their degree of power of producing the motion of heat, the sensation of light, or of doing the work of chemical decomposi- tion. All ether-waves can produce heat, for their energy is con- verted into heat when they fall upon and are absorbed by such a substance as a thick layer of lampblack, which for the most part arrests and extinguishes them. The slowest waves known those whose frequency is less than 392,000000,000,000 per second are too slow either to affect the eye with the sensation of light, or, in the ordinary case, to impart to molecules an agitation brisk enough to shake them to pieces, and thus to operate chemical decomposition. Such slow waves, whose presence can only be recognised after their impact by the conversion of their energy into Heat, are called Dark- Heat- Waves. If they fall upon an ordinary photographic plate they do not operate chemical decomposition ; but if the molecules upon which they impinge be specially heavy and complex, even these slow heat-waves may be found to toss and shake them with briskness sufficient to break them up. The waves may, on the other hand, be so rapid above 757,000000,000000 per second as to produce no visual effect on the eye ; the eye is normally, physiologically blind to them, and is unable to feel their impact; but they may effect chemical decomposition ; their successive impulses may aid the natural free vibrations of the molecule, which thus become increasingly ample : and just as a resonant tumbler into which its own note is steadily sung vibrates, shivers, and breaks into fragments, so a molecule, xv.] KINDS OF RADIATION. 435 quivering under the steady, regular, and continuously well-timed blows of the rapid ether-waves, may yield and break up into its constituent atoms, or into groups of atoms, which constitute simpler molecules. Such rapid waves are called Invisible or Ultra- violet Chemical Rays. The power of operating chemical decomposition possessed by the more rapid waves depends upon their frequency more than upon their intensity. . The slowest waves may thus produce heat, or perhaps chemi- cal decomposition of heavy complex-molecules ; waves of medium rapidity may produce heat, the sensation of light, or chemical effect ; the more rapid ones may produce heat or chemical effect according to the substance upon which they fall. The invisible chemical-rays, though they can operate chemical decomposition, are yet of very feeble physical intensity ; their aggregate kinetic energy is, in the radiations from the sun, as we receive them filtered through our atmosphere, millions of times less than that of the slower red or dark heat-rays : even those rays which are visible are effective not so much in virtue of their intensity, which is but small, as in virtue of the extraordinary sensitiveness of the eye to light that is, to the impact of ether- waves of a certain range of frequency. Colour. Within the limits of visibility -392 billions to 757 billions there is an indefinite variety of integral and fractional numbers, each of which represents the frequency of a particular kind of radiation, a particular kind of light. Physically there are as many kinds of light as there are possible frequencies between the limits mentioned. These kinds of light, each physi- cally characterised by the number of waves which strike the eye during a second, are recognised by the eye as being distinct, not as the result of any conscious process of counting the number of impulses suffered by the eye during a second, which would be absolutely impossible, but in consequence of the distinct and peculiar Sensation attending the reception in the eye of wave- motion of each particular frequency a sensation known in each case as that of a particular Colour. Thus, when we look at a Bunsen burner, the flame of which is caused to emit a dingy-yellow light by contact with common salt, we recognise the sensation as one of yellow light. Colour is a sensation : it is not a material existence ; but the physical basis and cause of the special sensa- tion of yellow light is in this case the* joint simultaneous impact on the eye of two kinds of ether- waves, which have the respective 436 OF ETHER- WAVES. [CHAP. frequencies of 508,905810,000000 and 510,604000,000000 per second, or the respective wave-lengths in air of 5895 and 5889-04 tenth-metres. Either of these trains of waves impinging singly on the eye would produce a sensation of yellow, the slower one giving a yellow very slightly more orange in its tint than the other does. The term " yellow light," which means primarily a certain sensa- tion, means, secondarily, the physical cause of this sensation that is, a train of ether- waves of a particular frequency. Any particular colour is best specified by a statement of the frequency of the single wave-motion, which can produce it when it enters the eye ; the analogy between light of any given Colour and a sound of any given Pitch being obvious. When there fall successively upon the eye trains of light- waves which differ only slightly in their frequency, the respective colour sensations produced by them may resemble one another generically, though not precisely. When, in gradual succession, luminous waves of all possible frequencies are caused to strike the eye, we obtain in successive gradation the sensations of all the colours of the spectrum. The slowest waves which can affect the eye produce a sensation of red, those somewhat more rapid a sen- sation of scarlet ; then in succession we find, as the frequencies increase, that the sensations produced are those of orange -red, reddish -orange, orange, yellow -orange, orange -yellow, yellow, greenish -yellow, yellowish -green, green, bluish -green, greeiiish- blue, blue, blue-violet, violet. Waves of still greater rapidity than those which produce the sensation of violet are practically invisible ; but it must be admitted that they are not per- fectly so. Even beyond the ordinary range of visibility some eyes are affected by ultra-violet ether -waves ; a sensation of la vender -gray colour results : a spectrum is often seen, especially if the dispersion be small, to contain three bright bands of lavender-gray in the ultra-violet region. This light is in intensity about 1-1 200th part of that which shines in the same region of the spectrum when it is rendered visible by fluorescence. The following table, modified from Ogden Kood's Modern Chromatics, gives the frequencies and the wave-lengths in air of the several undulations which correspond to the several leading colours of the spectrum, and to some of the so-called Fraunhofer Lines : XV.] FREQUENCIES AND WAVE-LENGTHS. 437 Frequencies. Wave-lengths in centimetres. Line A 395,000000,000000 00007604 Centre of red . , . ... 00007000 Line B ... 473,300000,000000 00006867 Line C 457,700000,000000 00006562 Centre of orange-red . . . ... 00006208 Centre of orange . . . ... 00005972 Line D 1 508,905810,000000 00005895 Line D 2 . . . . . 510,604000,000000 00005889 Centre of orange-yellow . ... 00005879 Centre of yellow . . . ... 00005808 Centre of green ... 00005271 Line E . , ' . . 570,000000,000000 00005269 Line b . . . . ... ... ... 00005183 Centre of blue-green . . ; ... 00005082 Centre of cyan-blue . ... 00004960 Line F 617,900000,000000 00004861 Centre of blue . . . ... 00004732 Centre of violet-blue . ... 00004383 Line G .... 697,300000,000000 00004307 Centre of puce-violet . . . . ... 00004059 LineHj . ^ . . 756,900000,000000 00003968 Line H 2 763,600000,000000 00003933 When a source of light is receding from the eye, fewer waves per second strike the eye ; the light approximates towards red. Conversely, the light of an approaching luminous object is, as it were, sharpened in pitch. By this application of D dp pier's principle, the speed of relative approach or recession of the earth and many fixed stars has been estimated. That which we call white light is, in the state in which we receive it from such a body as a white-hot bar of iron or, per- haps in its purest form, from the crater of the positive pole of the electric light, a mixture of long and short waves ; waves of all periods are either continuously present, or, if absent for a time, are absent in such feeble proportions or for such short intervals that they are not appreciably missed by the eye. White light of this kind is comparable to an utterly-discordant chaos of sound of every audible pitch ; such a noise would produce no distinct impression of pitch of any kind ; and so white light is uncoloured. If a beam of light of one kind, one wave-length, one colour, homogeneous or monochromatic light, be caused to converge upon and pass through a slit in an opaque screen, it may be rendered parallel by the aid of a lens ; when it falls upon a screen 438 OF ETHER- WAVES. [CHAP. it may cast upon that screen a coloured image of the slit. If the light, on issuing from the lens, instead of being received directly upon a screen, be made to pass through a glass prism, the narrow edge of which is held parallel to the slit, it will be refracted by that prism, and the image of the slit will now be found in a new position on the screen. If a beam of white light be so dealt with, a number of coloured images of the slit will be formed, each in its proper place on the screen, each image over- lapping its neighbour if the slit be wide; there will thus be formed a many-coloured band of light, in which the colours are marshalled in the order of the frequency of their waves, the slowest waves, the red, being least refracted by the glass prism ; the quickest waves, the violet, being most refracted. This is the spectrum : every component of the original white-light is dis- played in the spectrum, each in its distinct place ; and thus the prism furnishes us with a means of analysing light that is, of finding what its components are. But the spectrum extends beyond the visible part of it ; the more rapid invisible rays, being more refrangible than the violet, form an invisible part an ultra-violet region which we detect by the phenomena of fluorescence (p. 454), or by casting the whole spectrum upon a sensitive photographic-plate, upon which we afterwards find a record of a region of the spectrum invisible to the eye ; and the slower dark-heat rays form an invisible part of the spectrum beyond the red, the heat-spectrum or ultra-red region, not visible, but demonstrable by means of any apparatus such as a thermometer or a thermopile (Fig. 2 1 2), which is sensitive to heat. If the prism used be made of quartz, or if the spectrum be produced by reflexion from a diffraction-grating (p. 493), it will be found that the ultra-violet region is, if the light analysed be that of the electric arc, from six to eight times as long as the whole of the visible part of the spectrum ; while, if the prism used be of glass, it absorbs to a remarkable degree these rapid ultra-violet waves. If the light analysed be that of the sun, the ultra-violet part of the spectrum is comparatively very short, on account of absorption by the atmosphere. This effect of the atmosphere is of extreme importance. Sunlight is originally bright blue, and is extremely rich in the more refrangible rays, but nitration through two absorbent atmospheres that of the sun and that of the earth renders it a yellowish- white (Langley). The ultra-violet part of the spectrum is enormously brighter at high altitudes.; Compound Coloured -Light. Let us now cast a beam of xv.] COMPOUND COLOURED LIGHT. 439 sunshine or of electric light, shining through a slit in an opaque screen, upon a piece of greenish-blue glass, and receive upon a white screen the light which passes through this coloured glass : by the aid of a lens we may obtain a greenish-blue image of the slit upon the screen. So far as we have yet learned, such coloured light, whatever be the mechanism of its production, is a single kind of light perhaps due to waves of only a single frequency : whether this be so in the particular case may be tested by inter- posing a prism in the path of the coloured beam of light : if the greenish-blue light, be homogeneous, we shall again have on the screen an image of the slit, altered in position, but not in colour. This is not what we find : a short and imperfect spectrum is pro- duced; the transmitted greenish -blue light is analysed by the prism into green Iight 3 blue light, yellow light, with perhaps some other colours, more or less faintly represented. This phenomenon is very singular. It shows that two widely-differing physical causes are capable of producing exactly the same colour-sensation : the one being, as we have already seen, the impact of ether- waves of a single definite frequency, the other being the joint impact on the retina of a number of wave-Systems, each of which is capable, if it were to act independently, of producing a distinct sensation ; and the colour-sensation which is produced by the joint action of these wave-systems may differ from that which char- acterises any one of them. It is as if a listener to concerted music were to hear the strains of an orchestra compounded into some sort of loud melody of average pitch, he being wholly unable, by his unaided ear, to recognise the really compound nature of the sound heard by him. Then, whether the instruments all played in unison or diverged into pre-calculated harmony, the effect on his ear might remain the same. Further, many such mixtures may produce the same apparently simple sensation ; and accordingly such a phrase as " green light " or " violet light " is perfectly vague, unless it be accompanied by a specification of its physical cause. Complementary Colours. The greenish -blue glass in the instance just alluded to has in whole or in part prevented the transmission of violet light, of red, of orange, and of other kinds of light which are present in white sunlight; the complex of undulations thus denied transmission would, if collectively allowed to impinge on the eye, have produced a single sensation of red light. If this compound red-light had not been obstructed by the coloured glass, the transmitted beam would have been white ; this compound red-light thus obstructed by the greenish glass, and the compound greenish -light transmitted by it, will pass together through a piece of clear glass, and will together produce the sensation of white light. To the eye it is a matter of 440 OF ETHER- WAVES. [CHAP. indifference whether the red or the greenish light be monochro- matic or compound ; monochromatic red-light and monochromatic greenish-blue light, allowed to fall upon the same spot in the eye, will mingle, and, if they be of the proper tint, will produce the compound sensation of white light. These colours, red and greenish-blue, each of the proper tint, are thus Complementary to one another ; together they make up white light. The following pairs of colours are, among others, thus complementary to one another : Red and a very greenish blue, orange and cyan-blue (a rather greenish blue), yellow and ultramarine blue, greenish -yellow and violet, green and "purple," the latter being a colour not in the spectrum, but formed by the superposition of blue and red. The expression "white light" standing alone is thus also wholly vague ; physiologically it means light which produces the sensation of white ; physically it may mean (1) a mixture of all possible light- waves, long and short, in certain proportions, or (2) a mixture of two complementary simple colours, or (3) of a simple colour blended with a compound one of any degree of complexity. The white light of sunlight at sea-level is made up (Vierordt and Rood) by a mixture ( = 1000) of the following coloured lights : Red, 54 ; Orange- red, 140 ; Orange, 80 ; Orange-yellow, 114 ; Yellow, 54 ; Greenish-yellow, 206 ; Yellowish-green, 121 ; Green and blue-green, 184 ; Cyan -blue, 32 ; Blue, 40 ; Ultra-marine and blue- violet, 20 ; Violet, 5. EADIATIONS OF A HOT BODY. The hotter a body the greater the intensity of the aggregate disturbance which it sets up in the ether ; and further, the greater the frequency of the most rapid components of that disturbance. A white-hot iron ball is visible in a dark room ; it emits dark heat-rays, light-rays, and also the rapid ultra-violet rays : it can be seen and photographed, and its warmth can be felt at a distance. If it be intensely hot it may emit so great a proportion of violet and blue light that it appears bluish ; it is " blue-hot." As it cools down, the more rapid vibrations die away ; the ultra-violet waves cease to be formed; the mass becomes some- what less easy to photograph by its own light. Gradually the violet rays cease to be emitted : the light radiated is now apparently tinged with yellow : the apparent colour becomes orange, then red ; a body at a red-heat is difficult to photograph, xv.] EADIATIONS OF A HOT BODY. 441 though it continues perfectly visible in the dark. When its temperature sinks to a point below 525 C., it ceases to radiate light and becomes invisible ; it continues, however, to radiate heat, as may be felt for some time by the cooler hand placed near it. Becquerel asserts that if the brightness of the light given out by a body at 500 C. be equal to 1, at 1000 C. its intensity is 100, and at 1500 it is 10,000. It never ceases to radiate heat; it could not cease to do so unless it were cooled down to absolute zero. Since the molecules of all bodies are in repeated collision with their fellow- molecules, as they rebound at each collision they shiver and they vibrate. They must therefore continuously originate ether- waves waves which, when the temperature of the body is below 525 C., are too slow to affect the eye. Exchange of Radiations. Two bodies placed opposite to one another with intervening ether of which we cannot get rid, and with or without intervening air, may present the two follow- ing cases : 1. Both may be of the same temperature, in which case the one loses by imparting to the ether exactly as much energy as it takes up from those ether-waves which strike it, having been originated by the other hot body ; whence two bodies equally hot exchange their energies by radiation, but do this to an equal extent, and there is thus no change in their relative temperatures. 2. The one may, on the other hand, be hotter than the other. The hotter body sets up a more vehement system of ether- waves than the colder one can; in doing this it expends its energy to a greater extent than the colder one does ; the hotter loses more energy than it gains ; the colder gains more than it loses ; in course of time their energies, and therefore their temperatures, become equal : when the temperatures have become equal, though the two bodies still go on imparting energy to each other, neither profits by the exchange, and their temperatures remain equal. The absolute amount of radiation of energy from a body does not depend on the condition or even on the presence of surround- ing objects, but solely on the condition of the body itself. It is easy to see that the absolute physical brightness of the sun or of a candle is at any moment independent of the presence of illu- minated objects ; it is not, however, at first sight so clear that a fire not only warms a room, but the room also warms the fire ; 442 OF ETHER- WAVES. [CHAP. that the sun warms the earth while the earth to a lesser extent it is true warms the sun ; and that the warming of a colder body by a hotter one depends upon the difference of two similar but unequally- opposed actions. When a lump of ice is placed near an object at the ordinary temperature, that object is cooled ; it loses to the ice more heat than it gets from the ice : the ice apparently radiates cold. When one body is surrounded by another, the body enclosed and the walls of the enclosure come to have the same temperature. A thermometer whose bulb is immersed in a cavity will come to indicate the temperature of the walls of the cavity, whether it be in contact with them or not. This equalisation of temperature by radiation is quite independent of the form of the walls of the F1&154 cavity; a cavity of any form acts in the same way as a spherical cavity would do. In figure 154 the irre- gular hollow body ABC surrounds a body E ; both E and ABC assume after some time a common temper- ature, and remain at an equal tem- perature. The irregular body ABC might be re- placed by the hollow spherical-body FGH, or by the hollow sphere KLM. From this ensue the following propositions. 1. The amount of energy received by a receiving surface the amount of heat received, the brightness of light there varies inversely as the square of the distance from the source of radiation. The advantage of extensive surface possessed by the larger sphere KLM is exactly neutralised by its disadvantage of distance ; its surface is greater, the radiation received by it per unit of area is less, both in the ratio of the squares of the radii, and the total radiation received by it is the same, whatever be its radius. A candle at the distance of 1 foot can illuminate a printed page as brightly as a 2 5 -candle gas-burner at a distance of 5 feet. A bright wall is equally bright at all distances when looked at through a narrow tube. Close at hand it appears brighter, but less of it can be seen ; at a distance it appears dimmer, but more of its surface can be seen ; in all cases the amount of light falling on a given area of the retina is the same. 2. When a plane wave whose area is AB strikes squarely and simul- taneously all parts of a surface whose area is also AB the normals to the wave being also normals to the receiving surface the receiving surface receives a certain number of units of energy per second. In Fig. 155 AB is a hot or bright body radiating ether-waves towards CD ; CD receives e units of energy per second per unit of its area. If the receiving surface be tilted, say into the position DE, the wave- front, striking obliquely, is now able to cover the larger surface DE ; no more XV.] EXCHANGE OF EADIATIONS. 443 of the wave can now reach DE than would previously have reached CD. The same quantity of energy is thus distributed over a larger surface : the , Fig.166. quantity of energy communicated to it per unit of its area is diminished in the ratio , or is equal to (e x cos CDE) units. DE In accordance with this the intensity of sunlight at noon is greater than during the earlier and later portions of the day, when the surface of the earth is presented obliquely to the sun's radiation. 3. Let AB receive energy from CD or DE ; then, whether the surface be the smaller CD vertically facing it, or the larger DE arranged obliquely, is a matter of indifference ; in either case there will be radiated towards AB the same amount of energy. DE therefore radiates towards AB in the direction DB, less energy per unit of its surface than CD does in that direction when equally heated, and that in the ratio of cos CDE : 1. Were this not so, and did a hot surface radiate equally in all directions, then a body placed within a hot enclosure might become hotter than the walls of that enclosure. This principle explains the apparent uniformity of brightness of the sun's disc. Towards the margin of the sun's apparent disc, areas which seem equal to similar areas near the centre are in reality much larger ; but we see them obliquely ; their larger superficial area exactly compensates the effect of their oblique aspect. The law just stated that bodies are always radiating and receiving energy that the amount of radiation depends on the temperature of the radiating body that at constant temper- atures bodies radiate as much energy as they receive is known as Pre vest's Law of Exchanges. From this it follows that good radiators are good absorbents, and conversely, good absorbents are good radiators. If a hot-water vessel be intended to retain its heat for a comparatively long period in the open air, it must be polished externally ; a polished surface, being a good reflector, is a bad absorbent, and is therefore a bad radiator ; while a blackened surface, being a good absorbent, is a good radiator, and heat is with comparative rapidity lost through a coating of lampblack, provided that it be not so thick as to impede conduction of heat to the surface. Prevost's Law is not only true of the aggregate energy gained or lost by a body through radiation ; it is also true, as Balfour- Stewart has pointed out, with regard to each particular form or kind of radiation by means of which energy may be conveyed between neighbouring objects. 444 OF ETHER- WAVES. [CHAP. If a piece of yellow glass be placed within a hot shell of iron, the glass and the iron may both shine by their own light, and the glass may be looked at through a minute aperture in the wall of the hollow shell. Yellow glass absorbs ultramarine light, and a white-hot object looked at through it appears yellow, provided that the glass be colder than the source of the white light ; but when the yellow glass is itself as hot as the source of white light, as it must be in this instance, in which we look through the white-hot glass at the white-hot wall of the iron shell, the glass seems perfectly transparent to the whole white light, a phenomenon which may be interpreted as showing that while the glass only transmits yellow, it itself radiates blue, light ; the aggre- gate radiation, the transmitted yellow and the radiated blue, producing in the eye an aggregate effect of pure white. If the yellow glass be hotter than the source of light behind it, it seems relatively blue. The conclusion is, that as yellow glass absorbs blue light, so when itself heated it radiates blue light. Stokes's Law. A body which absorbs any particular kind of radiation will in general, when heated, become a source of radia- tion of the same kind ; just as a resonator will impart to the air when it vibrates the same kind of sound of which it may rob the air when it is relatively at rest. If a screen of strings tuned, say to the note of a, be arranged between a sounding a organ-pipe and a listener, the latter will hear comparatively little of the sound produced by the pipe ; by resonance the strings have taken up the energy, and have converted part of it into heat. If a mixed sound were produced on the further side of such a screen, the sound of a would not be transmitted to the listener; the rest of the mixed sound would be heard by him. When mixed ether-waves strike a system of molecules of which some are tuned to particular frequencies, those molecules will take up the energy of vibrations of those frequencies : the body will appear to be opaque to the corresponding waves. From the reciprocity of absorption and radiation it follows that if a given substance be divided into portions of which the one, A, is hot, while the other, B, is comparatively cool, radia- tions from A will be absorbed by B ; the cooler portion, B, is opaque to radiations from the hotter portion, A. Thus if car- bonic oxide be burned, its flame contains hot carbonic-acid ; the radiations from such a flame cannot pass through pure carbonic- acid, and are checked in very large proportion by air containing even a very small percentage of that gas. A hydrogen flame contains hot aqueous- vapour ; the heat radiated from this very slow dark heat-waves cannot pass through aqueous vapour : in this way, as Prof. Tyndall has shown, while the sun's light and heat can reach the earth's surface through the humid atmosphere, their effect is to xv.] STOKES'S LAW. 445 warm the earth and cause it to produce slow waves of dark heat ; these resemble in frequency the waves produced by hot aqueous -vapour in a hydrogen flame, and they cannot pass away through the aqueous vapour of the atmosphere. The atmosphere thus acts as a kind of heat-trap, and the surface of the earth is preserved from extremes of cold produced by excessive radiation. But for the atmosphere the earth's temperature would be below - 45 C., even under the vertical rays of a tropical sun (Langley). Burning sodium- vapour emits a particular yellow light ; if looked at through a mass of sodium-vapour, it can hardly be seen ; sodium -vapour absorbs the light given out by hotter sodium-vapour. Even though light of that particular kind do not happen in any particular instance to have been emitted by burning sodium, if the attempt be made to transmit it through sodium- vapour, the sodium- vapour will be found opaque to that kind of light. If an electric lamp produce a beam of light which contains amongst others this particular kind of light, and if a spirit lamp have salt (NaCl) placed in its wick so that it gives out this particular yellow light, this denoting that the spirit-lamp flame contains incandescent sodium- vapour ; and if the electric arc be looked at through the spirit-lamp flame, then the colour of its light would appear, if the eye were sufficiently sensitive, to be altered ; it is bluer ; the sodium-yellow light of the electric arc is absorbed as it passes through the comparatively cool spirit-lamp flame, which, by its own comparatively-feeble, radiation, does not repair the damage done by it, and the light which has passed through the spirit-flame is comparatively (not absolutely) wanting in that particular kind of yellow. The beam may be made, after passing through the sodium-vapour, to traverse a slit and a prism, and thus to form a spectrum on a screen. It will be found, if this be done, that the spectrum is discontinuous ; at the place where the particular yellow light ought to have been found, and would have been found had no spirit-lamp flame intervened, we find a dark line a dark image of the slit, which, if the slit be fine and the focussing accurate, is found to be a double line ; a line not absolutely lightless, but shining with the comparatively- feeble rays of the spirit lamp, and therefore dark in comparison with its environment. If the temperature of the spirit-flame be increased, the dark line brightens up ; if the temperature of the absorber be equal to that of the source, there is no dark line ; if the temperature of the absorber be higher than that of the source, more of the particular light is emitted than is absorbed by it, and the line is relatively bright. The prism, which resolves any com- pound light into differently-coloured linear images of a slit, 446 OF ETHER- WAVES. [CHAP. images which stand side by side so closely as to blend into one another, but any defect or redundancy of brightness in any one or in any group of which can be at once detected, offers a more delicate means of investigation than the eye can afford. In Spectrum Analysis a prism or a diffraction-grating is used to disperse into a spectrum the light which passes through a narrow slit from a luminous body ; by inspection of the spectrum we can at once see what kinds of light are emitted, and what kinds are not emitted, by a luminous body. But the kinds of light emitted by incandescent substances are generally (since they depend on the vibrational frequencies of the molecules of the substances) distinctively characteristic of each chemical element, and, to a certain extent, of each physical state of each degree of temper- ature of the incandescent substance. The spectrum of the limelight is continuous ; that of the sun is 'not. It presents dark lines; among others, the double sodium- line : the presence of this indicates a bright central source of light, a hot region of the sun's atmosphere, containing incan- descent sodium-vapour, the light from which is absorbed by the cooler sodium-vapour in the upper and cooler regions of the same atmosphere. These lines, discovered by Fraunhofer, and named after him, are distinguished by letters ; and the best -marked of the numerous Fraunhofer-lines are known as A, B, and C in the red, D (a double line) on the orange side of yellow, E in the green, F in the blue, G at the beginning and H X and H 2 near the end of the violet. The position of any colour is often roughly specified by stating its proximity to one or other of these Fraunhofer-lines. When the body radiating energy consists of a gas, each mole- cule, as it proceeds in its free path, executes free vibrations, like a vibrating tuning-fork thrown through the air ; and the mass thus vibrating may impress upon the ether only one kind of vibration, or perhaps, if the structure of the vibrating molecule be complex, a large though not an indefinite number of simul- taneous oscillations whose frequencies may or may not be com- mensurable. Thus a white-hot vapour may emit only a few distinct kinds of light, and may produce a line-spectrum a spectrum consisting of a few isolated, linear, diversely -coloured images of the slit. A very rare gas may emit very little heat or light even at such tempera- tures as 1500 C. : gases are bad radiators. The outer shell of a flame is non-luminous. Vapours are nearer their points of liquefaction than gases are, and are better radiators. xv.] SPECTRUM ANALYSIS. 447 When the particles are so close together as to have no free path, or but a small one, they very frequently collide and re- bound, and thus vibrate in an irregular manner; no rate of vibration is long enough absent for the eye to detect its absence. From the radiations of an incandescent solid or liquid, no kind of radiation appears to be absent up to the most rapid which is given out by the incandescent body ; and the spectrum of such a body is continuous, so far as it extends. It is not, however, necessarily equally bright throughout, for didymium and erbium oxides give well-marked bright bands in the spectrum of the light which they emit while incandescent,, If a heated gas or vapour be compressed, the shocks between its molecules become proportionately more numerous : if its tem- perature be increased, the energy of each shock becomes greater ; in either of these cases the vibrations of the molecules tend towards irregularity and complexity ; and there may, in addition to the main free- vibration of the molecules which is well-marked if there be any appreciable free path be a number of additional vibrations of all or of many frequencies : a condition which is indicated by the broadening of the lines in a linear spectrum of a gas into the bands of a band - spectrum. As pressure is relieved the spectrum merges into that of an ordinary incandes- cent gas, or on the other hand, as the pressure is increased, into the continuous spectrum of an incandescent liquid or solid. Continuity between the gaseous or vapourous and the liquid states is thus indicated on an independent ground. Light from incandescent solids or liquids travels from some distance within the surface ; for it is polarised at right angles to the plane of in- cidence ; this shows that it has been refracted on its outward passage through the surface of the incandescent body and into the rarer surrounding medium. Light from incandescent gases is not polarised ; sunlight is not polarised ; hence sunlight is due to incandescent gas or vapour. x Variations in the light emitted by one and the same sub- stance under different conditions, and therefore in the spectrum of that light, serve to indicate molecular changes in the substance which radiates light. Salts have a different mode of vibration, and therefore a different spectrum, from their component elements ; heat or, if heat fail, a discharge of high-tension electricity will break them up into their elements. Even the elements are reduced to comparatively-simple forms of aggregation by high temperatures ; their continuous spectrum breaks up into one of bands ; a still higher temperature, such as that of a high-tension 448 OF ETHER- WAVES. [CHAP. electric spark if other means fail, converts the spectrum into a line spectrum, the line spectra being perhaps due to atoms, the band spectra to molecules. The spectra of the same substance at different temperatures are often remarkably dissimilar. At temperatures beyond our reach, such as those of some of the fixed stars, or the lower levels of the sun's atmosphere the high temperature of which may be inferred from the great amount of Jbhe highly-refrangible rays emitted by them the elements themselves may be broken up and reduced to simpler forms of matter. This lends probability to the belief that the various elements are modifications of one kind of matter a belief some- what strengthened by numerous coincidences between the lines of the spectra of different elements. TRANSMISSION, EEFLEXION, AND ABSORPTION. When ether-waves fall upon a transparent body, they pass through it : they are propagated through the ether which lies between the molecules. When a body is thus pervious to light it is specially said to be transparent; when pervious to dark heat it is said to be diathermanous, no special term being- used to denote transparency to actinic radiation. A body impervious to light is opaque, impervious to dark heat it is adiathermanous. A perfectly -transparent body is invisible. Colourless thin glass, with a dustless, polished, clean surface, approaches this char- acter : objects are seen beyond it, and, as we say, through it : they appear, if the glass be thin, inappreciably distorted. Light may be reflected from the polished surface of glass, and the pre- sence of the glass may thus be rendered manifest to one standing in a particular position ; the sun shining on the windows of a distant house makes the window-glass visible. If glass be roughened at its surface, it presents numerous facets which reflect light so as to make the glass visible in all directions ; and light passing through it is irregularly turned out of its path in all directions ; objects beyond cannot be seen dis- tinctly, though light can pass through the whole mass, and rough- ened glass, though not perfectly transparent, is translucent. When glass is powdered, the powder presents so many facets and reflects so often the light which falls upon it that the whole is practically opaque : it is a powder which reflects in every xv.] TRANSMISSION. 449' direction the light incident upon it in white light a white powder, in red light a red powder. When ether- waves of any kind impinge upon a body impervi- ous to them their progress is arrested ; in part they are reflected or scattered; in part they are absorbed by the impervious body; the ether loses energy, ordinary matter gains it, and the imper- vious body is heated to an extent corresponding with the amount of energy absorbed this heat being first communicated to the superficial layer of the body. If ether- waves impinge upon a body which is transparent and diathermanous, that body is not heated, for the ether- waves pass through it and are not absorbed. Thus, clear mountain-air is not heated by the sunshine which streams through it ; in the shade it may be very cold. Sunshine may stream through clear ice, or even through hoar-frost, without melting it. If there be any particles of dust in the air or in the ice, these, being opaque, will become heated, and the air is then, by conduction, rendered warm, or the ice is melted. Some bodies are impervious to all kinds of radiation ; others, having a power of Selective Absorption, are impervious to some kinds only. Thus radiant heat can pass, while the more rapid light-waves cannot pass, through a thin piece of black vulcanite, or through a strong solution of iodine in bisulphide of carbon : while a solution of alum in water is, on the other hand, transparent to light, hut is adiathermanous, impervious to heat- rays. Lampblack, again, is very transparent to the slowest heat-waves. A soap-bubble film is remarkably adiathermanous, cutting off about half the heat of an incident beam. Glass is transparent and diathermanous, but is somewhat opaque to the ultra-violet rapid ether-waves ; a quartz prism or lens allows a great amount of ultra-violet radiation to pass through it which a glass prism or lens would extinguish. The absorptive power of a substance may not be so extensive as to enable it to absorb and extinguish light-rays or heat-rays of all kinds ; it may arrest some only. A piece of green glass can only allow a certain number of kinds of light to pass through it ; by their joint impact these produce the sensation of green. Sun- light contains other waves than these ; they have been absorbed ; the green glass is opaque to them. These waves would together have produced a sensation of purple-coloured light. If this purple light had alone fallen upon the green glass, it would not have been transmitted ; the glass would have appeared to be opaque. When sunlight is directed first through purple glass and then 2 G 450 OF ETHER- WAVES. [CHAP. through green, the eye perceives blackness : the two pieces of glass are together opaque, though each of them is transparent to its own kind of light. Very dark- red glass and green glass together produce a similar effect of blackness : pale-red glass allows some green light to traverse it, and so, when it is combined with green glass, the result is dark-green light. Nickel nitrate absorbs red and violet, and is therefore green when in solution. Cobalt solution is red. A mixture of strong solutions of the two metals is black : diluted it becomes, however, almost colourless. Copper, when it receives the impact of white light, emits orange light, together with superficially -reflected white light. Electrically -deposited copper, while immersed in a solution of sulphate of copper, which does not allow the transmission of orange light, looks as white as plaster-of-Paris does in the same liquid. The colour of a coloured object, as seen by transmitted light, is produced by subtraction of the light absorbed from the light incident upon the object. The kind of light transmitted may vary with the thickness of the absorbing medium. A solution of chloride of chromium in a thin layer absorbs much yellow, orange, and yellowish-green light ; in a thicker layer it absorbs all but the red and some green and blue ; in a still thicker layer the only colour transmitted is red. Thus a wedge-shaped layer of this solution appears to vary in colour, according to the thickness, from a greenish-blue, through purple, to red. Iodine vapour transmits a blue group and a red group, as also ultra-violet rays ; together these produce an impres- sion of purple : in thicker layers the blue rays alone are trans- mitted, and the vapour appears blue. When a strong solution of blood is interposed in the path of a beam of light, no light but red is transmitted ; dilute the solution gradually, and successively the solution appears more and more yellowish, and of increas- ingly paler hue. The special absorptions of absorbent bodies are most thoroughly studied, not by means of their visible colours, but by the prismatic analysis of the light which passes through them. It is then found that some substances absorb several distinct kinds of light, belonging to different regions of the spectrum. Transparent coloured-objects, through which light is filtered, give dark bands across the spectrum the so-called " Absorption- bands," which indicate what kind of light has been stopped and extinguished by the absorbent object these bands varying in breadth with the degree of concentration of the absorbent solution employed, and varying in position with its nature. xv.] ABSORPTION. 451 When a strong solution of blood is interposed in the path of a beam of light which is on its way to form a spectrum on a screen, all the spectrum, with the exception of the red part of it, disappears. As the liquid is diluted the spectrum lengthens out : orange, yellows, greens, blues, are successively added ; but there always remain two relatively-dark absorption-bands in the spectrum, in the yellow and in the green, between the Fraunhofer lines known as D and E. If the blood be treated with sulphide of ammonium, it will be reduced ; its oxyhaemoglobin will become reduced hemoglobin ; the chemical constitu- tion changes, and with it the absorbent power ; the absorption-band is now a single band placed between the two preceding. On the evidence of absorption-bands Capt. Abney has brought to light the existence of traces of benzene vapour between the earth and the sun, and Prof. Langley has shown that there are very peculiar gaps in the heat-spectrum which are probably due to absorption by the upper regions of the solar atmosphere. The kind of light absorbed by a body may also vary with its molecular constitution. It is supposed by Helmholtz that each absorption depends on the presence of a particular kind of molecule, differing from the simple chemical mole- cule. Chlorine has many absorption-bands in its spectrum, and it must either arrange itself in many kinds of molecules, or else its ordinary atoms, considered as vibrating bodies, must be extremely complex. By changes in the absorption-bands we may learn that sub- stances change their molecular constitution when heated. Iodine vapour gives an extensive absorption ; when highly heated, the absorption-spectrum becomes reduced to a few bands ; when the vapour is still more highly heated, some of the absorption-bands disappear, and one of them is replaced by a group of fine lines. In a red solution of cobalt the chloride, for example when heat is applied to it, the salt enters into a different state of hydration ; its molecular structure is changed ; the solution becomes blue. Sulphur- vapour changes its absorption-spectrum when its density changes at 1000 C. ; N 2 4 , when it becomes N0 2 , changes its spectrum, though it does not do so when it becomes a liquid ; iodine, on the other hand, when dissolved in carbon disulphide, has the same absorption-spectrum as when it is in the state of vapour. If there be no molecular difference between a substance in- candescent and the same substance .absorptive of light from a hotter object, a condition probably realised in the case of didy- niium, erbium, and terbium compounds, the incandescence- and the absorption-spectra will be mutually complementary ; the one presenting bright lines where the other presents dark The Colour of a coloured object seen by reflected light is 452 OF ETHER- WAVES. [CHAP. also generally due to absorption. An object seen by reflected sunlight does not appear to be coloured in any degree unless there have been absorption of some of the components of the incident white-light, and the colour of a coloured object is com- plementary to the colour which would have been produced by these absorbed components had they jointly impinged on the eye. Some of the light incident on a piece of coloured glass is re- flected at its surface ; there is no absorption ; if the incident light be white, the light reflected is also white. If a piece of green glass be laid upon black paper, and if it be looked at in such a direction that daylight is not directly reflected from it into the eye, it will be nearly invisible, and will be devoid of colour ; it will appear black. If coloured glass be ground to powder, the powder is white ; white light is reflected at every facet, while the light reflected from the lower surfaces of the fragments, and again issuing into the air, has nowhere traversed a layer of sufficient thickness to cause the extinction of all the absorbable components of the incident sunlight. The finer the powder, the whiter it is ; the coarser it is, the more marked is its colour. If the upper surface of a sheet of green glass be ground, it will appear almost white ; if the ground surface be looked at through the glass, it will appear green, for the light issuing from the glass is white light, which has undergone a certain amount of absorption. If the green powder be immersed in water or oil, there is less reflexion at the several facets ; there is deeper penetration of the light into the mass. and consequently more absorption ; the colour appears to deepen. Hence the value of oil as a medium in painting. A solution of chloride of copper placed in a deep black- walled vessel will not appear to have any colour ; it will seem black ; it reflects no light except from its surface. If powdered chalk be mixed with it, light is now reflected from the white particles of chalk, and passes out in every direction, through every part of the surface ; so much of the reflected light is absorbed that it appears green when it reaches the eye, the milky mass appears green. In a similar way a piece of malachite is penetrated by light to a very small depth ; internal reflexion occurs ; absorption of all the outpassing light takes place, with the exception of certain kinds, which jointly appear green ; the malachite is green. A piece of polished gold reflects white light at its surface; it also reflects interiorly, and from within the substance of the gold at a very small depth there is reflected in xv.] ABSORPTION. 453 all directions a quantity of light which, by absorption before leaving the surface, has become of an orange colour. If the layer of gold be very thin, that part of the light which would be absorbed by a thicker layer may, in part, pass through and issue into transparent media before it is wholly con- verted into heat. A thin piece of gold leaf thus appears trans- parent and allows a greenish -blue kind of light to pass through it, which, if the leaf be rendered very thin by the action upon it of a solution of cyanide of potassium, may become violet, for both green and violet light then find their way through. The object-glass of an astronomical telescope may be covered with a thin layer of silver, which will reflect the heat and some of the light, allowing a pleasant greenish-light to pass. When a beam of light enters the eye after undergoing repeated reflexion from gold to gold, it is of a deep-orange colour ; this is the true colour of gold. As we ordinarily see gold, the orange light coming from its deeper particles is mixed with much white light irregularly reflected from its sur- face. The true colour of copperas scarlet, of silver a yellowish -bronze colour, of brass a rich golden-red. By reason of such repeated reflexion, a deep metal -vase, equally polished within and without, appears to be of a much richer colour internally than it is externally, and velvets appear of a richer colour than silks, for light undergoes repeated reflexions between the vertical fibres which constitute the outer aspect of the former. When there is little opportunity for reflexion from the inner particles of a body, as where light falls exceedingly obliquely upon a gold mirror from a white object and is reflected into the eye, the image of the white object in the polished gold-mirror appears not gold-coloured, but white. Some metals can be rendered transparent, not by being reduced to thin films, but by being reduced to the liquid state : potassium and sodium can be dissolved in anhydrous liquid-ammonia ; the solution is blue, and the true colour of these metals is therefore a copper colour. If the incident light be already coloured, it may be that the whole of it is absorbed. An object, blue or red in daylight, if illuminated by a sodium-flame, may absorb all the light that falls upon it ; if it do so, it appears black ; a bunch of flowers, looked at in such a light, where it is not yellow, appears black ; it must either reflect some or none of the light which falls upon it. A piece of red cloth illuminated by the red regions of the spectrum glows with a bright red; when moved into other regions it becomes black, for it absorbs the incident light. The blue colour of opalescent bodies which in general present a multitude of reflecting particles embedded in a uniform matrix, and of which we may take as a type the sky-blue liquid obtained by adding to water a very small proportion of milk, is not primarily due to absorption. The principle is an established one, that where there is most refraction of light there is the greatest proportion of reflected light. A beam of mixed light 454 OF ETHER- WAVES. [CHAP. falls upon a colourless transparent-body : all the rays are both refracted and reflected ; the blue and violet are the more sharply refracted, and a greater proportion of them is reflected than of the less-refrangible rays. Even after one reflexion the image of an object in a mirror is bluer than the object itself. After multiple reflexion light may become distinctly blue. Multi- plicity of reflexion is favoured by smallness of the individual particles. The light which is not reflected is wholly, or in part, absorbed : the sun, looked at through a thin layer of dilute milk, appears yellow ; through a thicker layer, orange or red ; through a still thicker layer, it cannot be seen. Similar phenomena are presented by water into which a little very dilute alcoholic-solution of resin or mastic has been dropped with stirring, by salt water into which a few drops of a very dilute solution of nitrate of silver have been stirred, by a thin haze, by smoke ; all these appear blue by reflected, yellow or red by transmitted, light. Even the Sky itself is a haze of this kind, the air being rendered visible against the dark background of black space by sunlight reflected from its fine suspended-dust ; while the light transmitted is always more or less yellowish, and, in the afternoon and evening, when sunlight comes to us through a greater thickness of the more dusty layers, verges towards orange or even red. When the particles of a haze increase in size they jointly offer a greater resistance to the entry of light into the fog : light is reflected more promptly, and the reflected light presents a large proportion of white light. This phenomenon is familiar to the smoker ; the thick clouds of smoke produced by vigorous smoking are obviously different from the thin fine blue columns which ascend from a cigar laid aside for a moment. Those rays which are absorbed in the greatest proportion by any substance are reflected by it in the least; when a beam of sunshine falls on a green leaf, the actinic rays are absorbed and spent in doing chemical work ; the light reflected from such a leaf is feeble in actinic rays, and foliage is conse- quently riot easy to photograph. Light which is absorbed is generally converted into Heat; this may presently be radiated away; shorter, quicker light -waves strike the body; longer, slower waves of dark heat leave it. FLUORESCENCE, PHOSPHORESCENCE, AND CALORESCENCE. Fluorescence and Phosphorescence. The molecular dis- turbances of the interior particles of a body impinged upon by light may, however, give rise to other waves which are not so slow as to be invisible ; the ether- waves absorbed may thus give rise to Light. In this case the body may not only reflect light, but it may also seem to emit light from within; it is fluor- escent. The particles down to a very small depth, being set in agitation, originate a new set of ether- waves, which are propagated from each particle in every direction. xv.] FLUORESCENCE 455 The phenomena of Fluorescence may be shown by a solution of sesculin, which may be very simply prepared by stirring some horse-chestnut twigs in water ; a beam of light is caused to pass through this solution, and then for some distance within the solution the liquid seems self-luminous and shines in a dark room with an opalescent shimmer along the track of the beam of light. This effect is partly due to the impact of the light rays, but is prin- cipally due to the rapid invisible ultra-violet waves. If a piece of paper be wetted with a solution of sesculin, and if this paper be then used as a screen on which the image of a slit is thrown through a quartz prism, the ultra- violet part of the spectrum is rendered visible ; a compound blue light radiates from the paper over an area six or eight times as long as the ordinary visible coloured-spectrum ; the light refracted by a prism may, with the same effect, fall on the walls of a glass vessel containing the fluorescent solution. Quinine chloride or disulphate, on paper or in solution, gives a blue light that blue which is seen about the edge of the upper surface of a solution of quinine in a phial ; petroleum or shale oil a green ; turmeric solution in alcohol, or much better in castor oil, a green ; uranium compounds, especially uranium glass, a green light ; chlorophyll in solution, or lying undissolved in the cells of leaves, a red ; an alcoholic solution of soot or one of datura stramonium, a greenish blue. Among fluorescent substances we find also such compounds as eosin(tetrabromofluorescein),fluorescein (resorcin-phthalein), anthracene, fluor-spar (especially chlorophane, which, when heated by conduc- tion or by radiant heat, shines with an emerald-green light), many sulphides, especially those of barium and calcium, and, to a slight degree, the cornea and the crystalline lens, and the rods and cones of the retina. Very frequently a body goes on vibrating for some time after ether- waves have ceased to strike it ; this is familiar when the waves given out by it are Heat-waves. Sometimes, however, the body thus vibrating produces Light, and such a body Balmain's luminous paint, for example which goes on visibly shining or nuorescing for some time after ether- waves have ceased to im- pinge upon it, is said to be phosphorescent. Among such bodies we find barium and calcium sulphides, diamonds, chlorophane, dry paper, silk, sugar, teeth, the alkalies and alkaline earths and their salts in general, and compounds of uranium. These substances may be placed in a Geissler tube in a dark room ; an electric curr. ent passes ; the solids commence to fluoresce in the light produced by the discharge, but the observer's eyes are kept shut ; the current is stopped, and the eyes are at once opened to look at the tubes ; the solids are seen shining in the dark room. For substances the duration of whose phosphorescence is very small Becquerel's Phosphoroscope may be employed. In rapid succession a phos- phorescent body is exposed to bright light and brought against a dark back- ground before the eye of an observer situated in darkness. Most objects are found by this means to be to some extent phosphorescent. The compound nature of the light produced by fluorescence or by phos- phorescence can be ascertained by means of a slit and a prism. It is a very singular fact that the red rays of the spectrum and the invisible heat-rays have the effect of accelerating the exhaustion of a phos- 456 OF ETHER-WAVES. [CHAP. phorescing body. If a body, phosphorescing after exposure to white light, or better, to violet and ultra-violet rays, have a spectrum instantaneously thrown upon it, the body thereafter phosphoresces more brightly in the area occupied by the ultra-red part of the spectrum ; if the exposure to the spectral image be relatively prolonged, the phosphorescence becomes exhausted in those regions on which heat-rays had fallen, and now the Fraunhofer dark- lines in the invisible part of the spectrum are rendered manifest by local phosphorescences of those parts of the screen which have not been affected by the impact of heat-waves (Becquerel). A similar action of these rays has been long known : they often reverse the chemical action of the actinic rays. As a rule a fluorescent or phosphorescent body emits for a longer or shorter time, on exposure to light, or, specially, on exposure to actinic rays, the same kind of light which, when light falls upon it, it absorbs ; and thus, in some instances, the light emitted by fluorescent and phosphorescent bodies presents bright bands where the absorption -spectrum of the same sub- stance presents dark bands ; but the whole series of phenomena of fluorescence is one full of anomalies ; we do not fully know the laws of the molecular groupings of different substances, simple and compound, their necessary modes of vibrations, or their rela- tions to the ether. A mixed beam of sunlight which has passed through a fluor- escent solution cannot affect another solution of the same kind ; fluorescent solutions rapidly absorb those rays which are the effective cause of their luminosity. We sometimes find transformation of slower waves into more rapid ones. When a solution of naphthaline-red has been shone upon by a beam of deep -red light, it emits by fluorescence an orange-yellow light. Chlorophyll presents an analogous pheno- menon ; it fluoresces with a red light, even though it be shone upon by a slower red-light. In the case of chlorophane, the impact of slow radiant -heat -waves is competent to set up an emerald-green light. Calorescence. When a beam of light is filtered through a solution of iodine in bisulphide of carbon, so that dark heat-rays can alone pass through, these heat-rays may be brought to a focus by a lens, and absorbed by a piece of platinum placed at the focus ; this will become luminous and give rise to ether-waves of all kinds ; if its light be examined by a prism it will be found to give a continuous spectrum. This phenomenon is called by Tyndall the calorescence of heat-rays. XV.] ETHER-WAVES. 457 SOURCES OF ETHER- WAVES. Vibrations of Molecules. Light, Heat, and Chemical Eadia- tion being primarily due to the vibration of particles of ordinary Matter in the midst of Ether, the energy of ether -waves is derived from the kinetic energy of vibrating particles ; and what- ever increases the Kinetic Energy of these vibrating particles increases their vibratory movement, and gives rise to increased radiation. When by any action a given amount of energy is liberated in or communicated to a system of material particles, the rapidity of their resultant vibration, and therefore that of the ether-waves set up by them, depends on the rapidity with which that action occurs. When energy is slowly imparted to or liberated among them, the vibrations of the particles may remain relatively slow, and radiant heat may alone be the result ; while if the particles be suddenly set in violent commotion, their vibra- tion will be complex and irregular, the particles will become incandescent, and they will at once originate not only heat, but also light or even actinic waves. When a flash of lightning or an electric spark rushes through the air it jars the particles of air, and renders the air incandescent and luminous ; and it even originates actinic waves, for an electric spark can be photographed as well as seen. When the electric discharge through the air is continuous or rapidly intermittent, its light is, to the eye, apparently continuous, and we have the Electric Light. When a flint and steel are struck together the concussion agitates the molecules of these particles, of steel which are knocked off, and a luminous spark is produced ; so also when a bullet strikes a target there is a flash of light. Within a gas-flame molecules of a hydro- carbon are robbed of part of their hydrogen by a process of destructive dis- tillation ; the residues are heavy, almost purely-carbonaceous molecules, and these, in virtue of the energy supplied by the combustion of the hydrogen, become strongly agitated and incandescent, oscillating within the gas-flame, and therein act as sources of light until the current takes them into the zone of perfect combustion in the outer region of the flame ; there they become completely oxidised into gaseous carbonic-acid, and thereupon lose in great part their radiative power. The brightness of a gas-flame is favoured by external pressure, or by a relatively-small internal pressure and velocity of outflow, by the long continuance of carbon particles or other solid particles (which in a candle-flame cast a shadow in sunlight) within the flame in which they are incandescent, and by heating the gas before it enters the flame. When a crystal is cleft it often emits a flash of light; work is done in splitting the crystal : the energy of part of this work appears as that of ether-waves. 458 OF ETHER- WAVES. [CHAP. When salts suddenly crystallise out of a liquid menstruum it not unfrequently happens that the formation of crystals is attended with a flash of light ; the salt leaves the water and coheres with particles of its own substance ; the agitation attend- ing this process causes ether-waves to be set up. Even the application of moderate heat, falling far short of such a temperature as might produce incandescence, may cause a body to become luminous, as in the case of the fluorescence of fluor-spar and the diamond, which shine when heat is imparted to them by conduction. We have already seen that light may result from the impact of ether-waves upon a body. Chemical union is often attended with both heat and light : as when we drop copper filings or powdered antimony into chlo- rine gas, or in the ordinary phenomena of combustion. Even slow combustion, such as that of eremacausis or decay may cause light, as in the luminosity of decaying wood ; or the green luminosity visible on the surface of some fish when in a state of incipient decay ; or the slow oxidation of a piece of phosphorus in the air at ordinary temperatures, or of sulphur, or of the metal arsenic at higher temperatures. Even during the life of organisms they may become luminous either abnormally, as when the skin of the human body evolves phosphuretted hydrogen ; or normally, as in the glow-worm, in the noctiluca, in medusoids, and in many other invertebrate animals : light being in these cases produced at the expense of the animal heat which might otherwise have been evolved. Vibrations communicated to the Ether. In all these cases the origin of the light plainly is in the agitation of ordinary matter, but there is a certain deficiency of knowledge in respect of the next step in the transference of energy. How is any ether- wave set up in the Ether by the motion of any particle of ordinary matter within it ? A full answer to this question would involve a full knowledge of the constitution of the Ether, and of the relation of the Ether to the particles of ordinary matter which are imbedded in it a question still under discussion. Some hold that the Ether is entirely independent of ordinary matter, being unaffected in density by its presence ; others hold that it is of various densities in various substances, these densities being in different transparent substances inversely proportional to the squares of the velocities of light within them. Some hold that it is so independent of ordinary matter that a moving solid xv.] . ORIGIN OF ETHER- WAVES. 459 body moves freely through ether like an ideal net through ideally- frictionless water ; in which case it would be difficult to under- stand how a vibrating molecule could set up vibrations in it. If this were so, the most rapidly-moving solid transparent object would allow the transmission of light through the ether which permeates it, as if it were itself at rest. The contrary view seems probable ; a ray of light is said to be retarded a little by being made to pass up a running stream of water ; the effect, quite per- ceptible in the case of water circulating at the comparatively- slow rate of two metres per second, is, however, imperceptible in a current of air. A beam of light was found by Fizeau to be retarded when made to pass through a rotating cylinder of glass in such a direction that the rotation of the glass tended to carry back the light while in the act of passing through it. The consequence of such an adhesion between the ether and the matter embedded in it is, that the earth must to some extent drag the ether with it as it rolls through space. The whole subject is as yet one of the most recondite in physics. PROPAGATION OF WAVES THROUGH THE ETHER. At present it is usual in discussing the propagation of ether- waves to assume the wave to have been effectually set up ; the wave-motion is studied as it diverges from a small wave-front formed in the immediate neighbourhood of the vibrating mole- cule ; and in discussing the transmission of ether- waves of different wave-lengths through different transparent bodies, we shall have to take for granted that the interaction of the Ether and the ordinary Matter an action which cannot be very great, for, if it were, Transparence would be impossible is such as, in different media, unequally to retard ether -waves of different wave-lengths. This retarding effect depends somehow upon the nature of the transparent body; and this holds good not only with regard to light in general as where a diamond is found to transmit light much more slowly than water does but also with reference to each particular kind of light. Each transparent substance has its own rate of transmission for ether-waves of each particular frequency; and this is found for each case only by experiment. A denser substance may sometimes trans- mit ether-waves more rapidly than a rarer one does : light passes more rapidly through water, for example, than through alcohol or oil of turpentine. A substance through which light travels more slowly is said, however, to be optically denser. 460 OF ETHER- WAVES. [CHAP. On the assumption that the density of the ether is different in different substances, it would follow that all wave-lengths must be diminished or increased in equal proportions, that all kinds of waves must be equally retarded or accelerated, and all kinds of light, heat, or chemical rays there- fore equally refracted, on passing from one medium into another a conclu- sion contradicted by the simplest experiment with a prism. Cauchy, on the arbitrary assumption that the ether consisted of separate particles of an average size extremely minute as compared with the average distance between them, found that the amount of retardation was affected by the frequency of undulation, and that thus prismatic dispersion became explicable ; an assump- tion which more modern writers unwilling to admit that ether, which is not found to be capable of having waves of compression and rarefaction set up in it, and whose parts yet preserve or tend to preserve their mean posi- tions, can be a fluid composed of separate molecules have converted by interpretation into the following, namely, that there is some kind of discon- tinuity in the relations between the ether and the ordinary matter which it permeates ; a discontinuity which is held to show that while ether may be considered to be a homogeneous jelly-like solid, which can yield to powerful stresses after the manner of a fluid, the matter, apparently homogeneous, which is embedded in it, is not truly homogeneous throughout. The Ether resembles a very weak solution of gelatine : to relatively-great momenta it acts as a fluid, and it closes up behind moving particles ; to small stresses it acts as a solid, and it suffers tangential strain under the influence of a tangential stress. Ether-vibration transverse. When any part of the ether is displaced by a vibrating molecule, the displaced portion always tends to return to its normal position : in doing so it sets up waves. These are waves of transverse vibration like those of an elastic string or rod plucked laterally. According to some the ether is absolutely incompressible, and it is impossible to form waves of compression in it ; according to others waves of compression are at first formed, but very rapidly die out. The latter view assimilates the motion of the ether to that of an ordinary elastic-solid in which both longitudinal and tangential displacements occur and waves both transverse and compressional are produced. According to Clerk Maxwell's view the ether is a homogeneous body, a non-conductor of electricity : periodic electric-stresses applied to this produce waves which travel at the rate of about 300,000000 metres per second ; these waves are waves of transverse vibration, and there is no vibration longitudinal or normal to the wave-front. These waves, due to electric dis- placement, are quite competent to explain the ordinary phenomena of light, and this theory explains on mathematical grounds that absence of the normal or compressional vibration which is a source of great perplexity in all the mechanical theories of light. According to this view, each particle of a body through which light is shining is in rapid succession exposed to alternately- opposite electric stresses : at each half-vibration it becomes oppositely elec- trified ; but the ordinary effects of electricity are not generally observed when light shines through or on a body, for the electrification produced by any one half- vibration simply reverses the effect of that produced by the previous half-vibration. xv.] VELOCITY OF ETHER- WAVES. 461 The Velocity of propagation of ether-waves through the Ether of space is found by two astronomical methods. 1. Jupiter's Satellites. These pass out of sight behind the mass of Jupiter and again reappear : when the earth is nearest to Jupiter the eclipses and reappearances appear to take place 8j minutes earlier, when the earth has wheeled round to the opposite side of its orbit, and is at its furthest from Jupiter 8j minutes later than they would have appeared if the earth had been at the centre of its orbit. The suddenly commencing or ceasing light takes 16J minutes to cross the earth's orbit, a distance of 299,270,000000- metres : it therefore travels 302,300,000 metres per second. 2. Aberration. No star is seen in its true place : every star seems to describe a little ellipse in the heavens, and seems to travel round the ellipse once a year. The reason is, that as the earth wheels onward in its orbit, bearing the observing telescope with it, rays of light coming from distant stars on their way down the telescope tend, short though the telescope tube be, to verge towards the hinder side of that tube : for which reason, in order to see the star in the centre of the field, the eyepiece must be tilted appreci- ably backwards in a direction opposed to that of the earth's orbital motion : the telescope, when the star is seen in the centre of its field, is therefore directed not towards the true position of the star, but towards a point in advance of it. In the course of a year, therefore, as the earth bowls round its elliptical orbit, the successive points to which it is necessary to direct the telescope are found to have been situated on the circumference of an ellipse. The size of this ellipse indicates the amount of tilting of the telescope : from this can be inferred the proportion between the length of the telescope and the distance traversed by the ocular during the time spent by the ether- waves in passing down the telescope tube ; the speed of the waves of light can be calculated from these data, and is found to be 299,300,000 metres per second. Do waves of different frequencies travel at the same or at different rates ? If their rates were different, then a suddenly- appearing satellite of Jupiter, or a suddenly-brightening variable star, would be first rendered visible by that light which first arrives at and enters the eye, and it might consequently appear violet or blue ; and when it disappears it would continue for the longest time visible by that component of light which is slowest in travelling, and therefore might appear red before vanishing ; or again, aberration of light would necessarily have the effect of giving us an image of each star drawn out into a spectrum. Nothing of the kind is observed ; all kinds of ether- waves must therefore travel through the ether of space at the same rate. Terrestrial experiments for ascertaining the velocity of light are based upon one of two principles. 1. Fizeau's principle. A ray of light is rendered intermittent by flashing between the teetji of a rotating cogwheel. It travels to a distant mirror ; each flash is there reflected along its former path. Before a flash can again reach the cogwheel, the cogwheel may have rotated so far that one of 462 OF ETHER- WAVES. [CHAP. its cogs now obstructs the returning ray ; if a sufficiently-increased speed be imparted to the cogwheel, the light is allowed again to pass between the teeth of the wheel through a neighbouring notch, which has now come to occupy the position at first occupied by that notch through which the light had flashed on its onward journey. Given, then, that the light has travelled to a certain distance and back, and that in the meantime the cogwheel has been rotated through a certain angle, it is, in principle, easy to find the speed of propagation of the light. Fizeau found this to be 314 million metres : Cornu, by similar experiments, obtained the value 300,400,000 metres. 2. Foucault's principle. A beam of light starts from a source S ; it strikes a mirror M, and is reflected to a distant mirror R, on which it is focussed by a lens : it is there reflected and retraces its journey : it is again reflected from M and returns to S. If, however, the mirror M have, in the meantime, been rotated perceptibly before the beam of light has had time to return from the distant R, the light can no longer be reflected from M towards the original point S ; it illuminates some other point T. The dis- tance between S and T can be measured ; the amount of rotation of the mirror M in the time taken by the light to go from M to R and back can be inferred from this ; the amount of rotation of the mirror M can be read off on a speed -indicator attached to the rotating apparatus : the distance traversed by light in one second can be ascertained by calculation from these data. There is no need to use instantaneous flashes of light from S ; the steady beam from S reflected from the rotating mirror M only encounters the small fixed mirror R for an instant once in the course of each revolution, and is thus rendered practically instantaneous. By this means, with a mirror rotating 1000 times in a second, Foucault demonstrated that light takes a measurable time to pass through a dis- tance of 7 or 8 yards. Lord Rayleigh has shown that these different methods cannot be expected to give the same results, for it is not precisely the same thing which is observed in all these cases. In some (aberration method) the speed of single waves is observed ; in others (Fizeau, Jupiter's satellites) the speed of propagation of a certain peculiarity, intermittence ; in others (Foucault) these are blended. As a mean result it may be stated that the velocity of ether- waves in a vacuum that is, in the ether of space is 300,574,000 metres per second. From this it follows that K, the rigidity of the ether, and />, its density in vacuo, are definite in amount, and bear to one another the relation K = p x (30057,400,000) 2 ; for?; = { \/K -^ /oj- centimetres. The mean velocity in air is less than that in vacuo in the ratio of 1 to 1-000294. It is generally believed that light of all colours travels with -equal velocities through air, though some doubt has been cast on this result by the recent experiments of Forbes and Young, who find that blue light travels more rapidly in air than red light does, in the ratio of 1018 to 1000. By a modification of Foucault's method, above described, the relative .speeds of light in two different transparent media, or in the same medium -at different temperatures or under different pressures, may be compared. xv.] VELOCITY OF ETHER-WAVES. 463 The light between M and R has to traverse a space in which a certain thick- ness of the medium whose retarding power is to be examined may be laid in the path of the beam : the beam may be exposed by having to pass through this medium to a retardation which is rendered manifest and measurable by an alteration of the position of the image T. The Physical Intensity of light at a place is measured by the energy transmitted through that place in a second of time ; for light of constant colour this intensity is also proportional to its brightness as perceived by the eye. Hence there are two methods of measuring the intensity of a beam of light : 1. Calorimetrical: allow the beam to fall upon a thermopile, and estimate the intensity of the light by the amount of heat into which it is converted upon absorption ; the beam in this case having undergone a preliminary sifting through alum -water, which absorbs the heat-rays*. 2. Photometrical: two sources of light are placed at such distances from an illuminated body that they appear to produce the same effect, such as equal shadows, or equal illumination of the two sides of a disc ; but this method is only accurate when the two lights to be compared are of exactly the same colour. The intensity of actinic radiation may be estimated by observing the depth of tint produced in a piece of photographic paper exposed for a given time. The total intensity of radiation may be measured calorimetrically. MODE OF PROPAGATION POLARISATION. Waves of light under which term we shall for the moment include all forms of ether-waves have the peculiarities of propa- gation characterising waves whose wave-length is generally small in comparison with the breadth of their wave-front. They do not usually diverge laterally from the directions mapped out by the normals to their wave -fronts; or, as it is commonly expressed, Light travels in straight lines; they can only so diverge when they are made to pass through apertures or round obstacles not very much greater in breadth than their own wave-length. The light from a single luminous point is propagated in spherical waves ; that from such an extended object as a candle-flame in waves which, at some distance from the source, are approximately spherical. If light from a wide source be made to pass through a narrow tube, or successively to traverse equal apertures in two opaque screens, at such a distance from the source that the wave passing through the second screen has a plane front (see Fig. 57), then on the further side of the second screen there may be an unwidening or parallel beam of light. Such a parallel beam of light, as it traverses space, may be compared to a vibrating string of ether, isolated in the ether, 464 OF ETHER-WAVES. [CHAP. vibrating independently, and practically unaffected by the ether situated laterally with respect to it. Such an isolated cord of ether may enter into vibrations of different kinds, analogous to the vibrations of strings. 1. It may transversely vibrate simply up-and-down, or from side to side, or in any other single direction, its vibrations are restricted to one plane ; it is then called a beam of Plane-Polar- ised Light, By a convention as to which there is some dispute, a plane cutting the beam in its whole length, but at right angles to the plane in which the ether vibrates, is called the Plane of Polar- isation, The vibration of the ether is thus effected at right angles to the plane of polarisation. 2. Its vibration may be resoluble into simultaneous trans verse - vibrations in two planes at right angles to one another. (.) These may be of equal period, and the vibration in one plane may be J period behind or in front of that in the other ; looked at endwise, any part of the ether in such a beam would necessarily be seen if it could be rendered visible like a bright point on a vibrating string to execute small circular vibrations. Such a beam of light is said to be Circularly Polarised. Looked at from one side the vibration would apparently progress like a screw. Fig.156. R A L A common corkscrew is a right-handed spiral. A simple experiment with a string, one end of which is fixed to a wall, while the other is held in the hand, will show that, in order to impress upon the string the right-handed spiral form, we must rotate the free end in a direction opposed to that of the hands of a watch. Such is the movement of a so-called right -handed circularly -polarised beam of light. When the rotation is in the opposite sense the circularly-polarised ray is left-handed. Fig. 156 shows the direction of propagation and of rotation, and the forms assumed by the vibrating ether in a right (R) and in a left-handed (L) ray respectively. (&.) The circle may. by a difference of phase other than ^ period, be converted into an ellipse. A beam, the ether in which rotates in an ellipse, is called a beam of Elliptically-Polarised light ; | this again may be right or left-handed. (c.) The periods of vibration in the two planes may not be equal, but may be commensurable. A beam of light of this kind would present movements which, looked at xv.] POLARISATION. 465 end-on, would present, for each portion of the vibrating ether, figures like those of 35-40, etc. (d.) The periods of vibration in the two planes may not be commensurable or even constant, and further, the vibration in each plane may be variable in its amplitude. In such cases the vibration would rapidly run through a great variety of figures, circles, ellipses, figures of eight, and non-reentrant complex har- monic-curves of every kind. This is the condition of a beam of Common Light. No single plane has any advantage. If for a moment the amplitude in any particular plane preponderate, this is but momentary : and since the most irregular transverse-vibra- tion can be resolved into a vibration up-and-down, and one side- to-side, or may be resolved in any other two planes arbitrarily chosen at right angles to each other, a beam of common light may be held to be the result of the superposition of two simultaneous irregular trans verse -vibrations, each plane -polarised, each pos- sessed of half the energy of the whole vibration, and both pro- pagated with the same velocity through the ether. The doctrines of composition and resolution of harmonic motion are applicable to each small portion of the ether within such a transversely -vibrating beam, just as they are to trans- versely-vibrating strings. A beam of common light encountering an object which is selectively transparent to vibrations in one plane, but opaque to vibrations in a plane at right angles to this, will have the latter vibrations extinguished ; it will lose half its energy ; the beam to which the object is transparent the transmitted beam will have half the energy of the original beam ; and all its vibrations being executed in one plane, it will be a beam of Plane-polarised Light. A body which acts in this way on a beam of common light is called a Polariser. A beam of plane-polarised light falling on a polariser will, should the plane in which its vibrations are executed happen to coincide with the plane of those vibrations to which the polariser is transparent, be found to pass through it freely : if the plane of vibration be, on the other hand, a plane at right angles to the plane of the freely-transmitted vibrations, no vibration can get through, no light is transmitted, and to light polarised in such a plane the selectively-transparent polariser proves perfectly opaque. If the condition be intermediate that is, if the plane of the actual vibration and the plane of free transmission through the polariser be neither coincident nor at right angles to one another 2 H 466 OF ETHER- WAVES. [CHAP. then the actual vibrations of the plane-polarised light must be resolved into two component plane-polarised vibrations at right angles to one another ; these, looked at end-on, carry out the principle of Fig. 42 ; and of these components the one in the plane of free transmission, the other at right angles to that plane the former is transmitted, while the latter is extinguished by absorption, its energy becoming converted into heat. Wlien ordinary light has had a certain proportion of its vibrations in a given plane quenched, while in the plane at right angles to this they -are not quenched at all, or not quenched in equal proportion, it is in a state of partial polarisation, and is called Partially-Polarised Light. A plane-polarised beam of light may not only be resolved into two at right angles to one another and coincident in phase, but also (see Figs. 46 and 47) into two circularly -polarised beams, the one left-handed, the other right-handed. If it be supposed that a transparent body or a region of space is so peculiarly constituted or stressed that a left-handed circularly- polarised beam travels more rapidly through it than a right- handed one can, then, on passing a plane-polarised beam of light through such a region, the left-handed circular component emerges with its phase less advanced than the right-handed one ; but the plane-polarised light equivalent to the synthesis of two such circularly-polarised beams can no longer be due to a vibra- tion in the original plane ; the plane has been turned round a longitudinal axis in the centre "of the beam ; and the further a plane-polarised beam travels through such a body or region of space, the greater, in a direct ratio, will be the rotation of the plane of polarisation of that beam a result observed in many cases, and to be described under the head of the so-called Rotatory Polarisation. When the left-handed component is relatively accelerated in its transmis- sion, or retarded in its phase of emergence, the plane is rotated to the left that is, to an observer stationed at the source of light the plane of polarisation of the receding beam is seen to rotate in a direction opposed to that of the hands of a watch : the more-rapidly travelling left-handed component is at any point less advanced in phase than the more-slowly travelling right-handed component at the same point ; the contrary-to-clock rotation of the right- handed ray prevails over the less advanced clockwise rotation of the left-handed ray. KEFLEXION AND REFRACTION. When a ray of light travelling in a rarer medium strikes the surface of an optically denser transparent medium, some light is xv.] REFLEXION AND REFRACTION. 467 reflected, some refracted ; and if there had been neither absorption nor scattering, the energy of the reflected ray, together with that of the refracted ray, would have been equal to that of the incident ray. The incident ray and the reflected ray are in one plane : this is called the plane of incidence. The plane of incidence is at right angles to the reflecting surface at the point of incidence and reflexion. The vibration of the incident light may be either at right angles to the plane of incidence, i.e., parallel to the reflecting sur- face, or it may be in that plane, in which case the vibrating ether will not brush, but will strike the reflecting surface : or it may be in any intermediate direction or sequence of directions. Fresnel, in investigating this subject, made the following assumptions, viz. (1) that of the conservation of vires vivce,* or, as we would now say, the Conservation of Energy ; (2) that the movement of the incident ray merges continuously into that of the refracted ray ; (3) that while it does so continu- ously, it does so very rapidly at the surface of separation ; and (4) a very arbitrary assumption that differences of velocity of ether-waves in different substances are due to differences of density of the ether, whose elasticity remains unaffected. From these postulates he showed (1.) If the incident beam be a plane -polarised beam, vibrating parallel to the reflecting surface, the- refracted and the re- flected are also plane-polarised beams whose vibrations are parallel to the original direction. (2.) The frequency of vibration is unaffected by reflexion and refraction : the colours of the incident, reflected, and refracted rays are the same. (3.) The angle of incidence is equal to the angle of reflexion. (4.) The sine of the angle of refraction bears to the sine of the angle of incidence a constant ratio, n t the "Index of Ke frac- tion ;" the numerical value of n depends upon the nature of the two media. Sin i = n sin r f (5.) The amplitudes of the three rays are related to one another in the following way: The angle of incidence is i ; that of refraction is r ; the intensity of the incident ray = /. Then v, the amplitude of the reflected ray, is equal to f- 5m . -r-MJ while u, the amplitude of the refracted ray, is equal to (2 sin r cos t\ _, ). Jbrom the former formula we learn sin (i + r) J mv 1 * Vis viva = 2mv* ; energy = S -5- 468 OF ETHER- WAVES. [OHAP. (a) That the greater the angle of incidence i, the greater is v, the ampli- tude of the reflected ray. (&) That when the incident ray is so nearly parallel to the surface of the glass as simply to graze it, the reflected ray is equal to the incident one and the refracted ray is nil. (c) That when i is greater than r, v is negative ; while when i is less than r, v is positive ; or in words, when a ray strikes the surface of a denser medium, the reflected ray is a direct continuation of the incident ray, changed in direction ; while when light travels in a denser medium, half a wave-length is lost on reflexion at the sur- face of a rarer medium. This conclusion is independent of any such hypothesis as to particles as that by which we have already illustrated the same propositions on pages 112 and 118. (6.) The respective intensities of a reflected, a refracted, and the incident ray are in the ratios of sin 2 (i r) f sin 2i sin 2r sin 2 (i + r) sin 2 (i + r) (7.) When light travelling in a denser medium strikes the surface of separation between the denser and a rarer medium at such an angle i that sin i is greater than n, both reflexion and refraction are possible ; but if the angle of incidence be such, that sin i = n, then sin r = 1, and the ray refracted into the rarer medium grazes the reflecting surface, for r = 90; and any light falling still more obliquely upon the surface will be Totally Reflected, the reflected ray possessing the whole energy of the original incident -ray. In the 'last case sin r = , which is n greater than 1; and r, the angle of refraction, is an impossible angle ; there is therefore no refraction. In Fig. 157 the ray AO is partly reflected to A" and partly FI refracted to A'; the ray BO is partly reflected to B" and partly refracted to B', the refracted por- tion grazing the refracting sur- face ; the ray CO is wholly re- flected to C", and is not refracted at all into the rarer medium. As examples of Total Beflex- ion we may take a tumbler of water held above the head; it will give a clear mirror -image of the objects on the table below it ; a bubble of air in water, or a test tube containing air immersed in water, will, when looked at XV.] REFLEXION AND REFRACTION. 469 Fig.158. at a certain angle, appear to have as bright a mirror-surface as that of mercury. In Fig. 158 light entering a total -reflexion prism at M is totally reflected at 0, and travels towards R, which it reaches with- out refraction. (8.) The energy of a ray totally re- flected is equal to that of the incident ray; there is, however, a slight retarda- tion of its phase. These laws all apply to vibrations executed at right angles to the plane of incidence, and were deduced by Fresnel from the fundamental hypotheses already mentioned. Let us now turn to the reflexion and the refraction of plane- polarised light whose vibrations are at right angles to these, and are thus executed in the plane of incidence. In the refracted and reflected rays the vibrations will still be in the plane of incidence, but they cannot, after encountering the refracting surface, remain parallel to their original direction. The consequence deduced by Fresnel from this is, that while the ordinary laws of refraction and reflexion are obeyed by such plane-polarised light so far as directions are concerned, the relative intensities of the incident, the reflected and the refracted light, are not the same as they were in the preceding case, but are now respectively in the ratio tan 2 (i ?) ^ sin 2 i sin 2 r ' tan 2 (t + r) ' sin 2 (i + r) cos' 2 (i - r) incident : reflected : refracted. The intensity of the reflected light is, in a particular case, nil ; that is, when {tan 2 (i - r) + tan 2 (i + r) = 0, or when (i + r) = 90. In Fig. 159 the ray AO, whose vibration is in the plane of Fig. 169. incidence, falls at such an angle i that it is refracted along OA'; if it had been reflected at all it would have been reflected along OA 7/ ; at one par- ticular angle of incidence, AON, the refracted ray OA' and the reflected ray OA" (if there .had been such a ray) would have been at right angles to one another. When the angle of in- cidence and the angle of refraction together make up a right angle, there is no reflected ray ; no vibration effected in the plane of incidence is reflected at all, and a plane -polarised ray of this kind falling at the appropriate angle of incidence, however bright 470 OF ETHER- WAVES. [CHAP. it may be, will fail to be reflected from a mirror. This is a case of Total Refraction ; the whole of the energy of the incident ray is in the refracted ray. When the incident light grazes the refract- ing surface, the reflected beam also grazes it, and there is no refraction. When light whose vibration is in the plane of incidence is totally reflected, it undergoes a slight retardation of phase, less than that observed in the case of light whose vibration is at right angles to the plane of incidence. From these results it is easy to pass to the case of light whose vibration may be considered to be the result of the composition of vibrations parallel to the plane of incidence with others at right angles to that plane. Such light may be plane-polarised, vibrating in some plane neither that of incidence nor one at right angles to it ; it may be circularly- or elliptically-polarised light, or it may be common light. In all these cases each component is reflected and refracted according to its own laws. In this way the reflected and refracted rays may come to differ in character from one another, and from the original ray. As an extreme case, let a beam of common light fall upon a piece of glass at such an angle that i 4- r = 90 (Fig. 159) ; of that part of the incident beam which is due to vibrations executed at right angles to the Plane of Incidence there is reflected a certain proportion; of that part of the incident ray which is due to vibrations parallel to this plane there is reflected none. The light reflected from glass at such an angle has its vibrations thus restricted to a plane at right angles to the plane of incidence ; it is plane-polarised light. A piece of glass held in the course of a beam of common light at the^proper angle may thus be used as a simple means of obtaining a beam of light Plane-Polarised by Eeflexion. The precise angle of incidence i, for which i + r = 90, is called the Angle of Complete Polarisation. It depends upon n, the refractive index of the refractive substance, and the angle i is such that tan i = n : for sin i = n sin r = n cos i. Such are the consequences deduced by Fresnel from the hypo- thesis that the Ether is condensed around the particles of ordinary matter while its elasticity remains unaffected. Neumann and MacCullagh, from the contrary hypothesis that the density of the Ether is the same in all substances, while its elasticity is different in different substances deduced a set of conclusions precisely similar to those above given, with this remarkable difference, that the pro- perties attributed by Fresnel to plane-polarised ether- waves whose oscillations are effected at right angles to the plane of incidence are by the latter writers xv.] PLANE OF POLARISATION. 471 found to be associated with plane-polarised light whose vibrations are parallel to that plane, and vice versd. The fundamental postulates of the two theories are closely associated with these consequences. We must now turn to a point of terminology. When a beam falls upon a mirror at the angle of complete polarisation, the re- flected ray, if there be any, is plane-polarised ; and it is said to be polarised in the plane of incidence. According to Fresnel's view, the vibrations of this beam are supposed to be executed in a direction at right angles to the plane in which it is said to be polarised. The question between the followers of Fresnel and Cauchy on the one hand, and those of Neumann and MacCullagh on the other, may thus be stated : Are the vibrations of plane-polarised light executed at right angles or parallel to the plane in which the light is said to be polarised a plane which by convention is called the plane of polarisation ? This question is still sub judice. Though it seemed until recently that Fresnel's view had definitely prevailed, and that the vibration of polarised light is at right angles to its plane of polarisation, yet in recent times the contrary view has met with increasing favour, especially since the electromagnetic theory of the propaga- tion of light has been promulgated ; this theory, due to Clerk Maxwell, demands that the vibrations of a plane-polarised ray be in the plane of polar- isation, not at right angles to it. A further consideration, which lends probability to this view, is derived from certain mathematical difficulties which arise from the adhiission of Fresnel's second hypothesis, that the move- ment of the ether in the second medium is continuous with that in the first ; for this hypothesis is found to lead directly to the conclusion that the density of the ether in the two media must be the same, and is therefore one which is incompatible with his fourth hypothesis. When ordinary monochromatic-light is reflected at any angle other than that of complete polarisation, the reflected and the refracted beam must both be partially polarised, and each will be polarised to an equal extent, though in contrary senses. In the reflected beam, light polarised in the plane of incidence preponderates until the incidence is a grazing one : in the refracted ray, light polarised at right angles to that plane preponderates to an exactly equal extent, so far as the energy of the vibration is concerned. When mixed light, such as white light, falls upon a refracting surface, then since n, the index of refraction, is different for each kind of light, the proportions of each coloured light present in the reflected and the refracted rays respectively are different ; white light, when reflected from a refracting surface, always becomes bluer, the refracted light redder ; and we have seen this to account for the blue colour of haze. The intensity of a reflected ray is represented by {sin 2 (i - r) + sin 2 (i + r)}. If we pass to a ray of greater refrangibility we alter r to r, and our intensity 472 OF ETHER- WAVES. [CIIAP. becomes {sin 2 (i r)-r-sin 2 (i + r)}, which is always greater than the former. Where the actual refraction is greater, the . corresponding angle of refraction is less ; wherefore in this case f is a smaller angle than r. Near the incidence of total reflexion some colours may be totally reflected, others in part refracted ; near the incidence of total refraction or complete polarisation an analogous result is obtained. The slower waves of heat have a lower refractive index, and must therefore strike a refracting surface at an angle somewhat more vertical or nearer the normal than those of light, in order to become completely polarised. We thus learn that refraction and reflexion may materially modify the character of light which strikes on a refracting surface. If, however, we attend only to the Direction of the respective rays, and not to their states of polarisation, or to their colour, etc., we may study the effect of mirrors or lenses in modifying the direction of an incident beam of light, whether this be plane- fronted, convergent or divergent. "We shall first consider the case of monochromatic light. Mirrors. Plane mirrors reflect light in such a way that the reflected waves are, as regards their direction, precisely such as might have come from an object or source of light situated behind the reflecting surface, and at a distance behind it equal to the distance between the object and the mirror. This is illustrated in Fig. 61, and it is a matter of familiar knowledge in the use of looking-glasses, and in the appearance presented by the inverted image of objects on the shore when these are seen reflected on a surface of smooth water, the images then seen being apparently at the same horizontal distance from the eye as the objects them- selves ; while the image of a slightly-clouded sky, as reflected in very smooth turbid water, appears extremely deep. The apparent inversion of an image in a mirror is a natural result of the fact that the image of eac k p i n k j s apparently situated be- hind the mirror. Fig. 160 explains this result. The reader may construct a diagram to show how it is that a mirror about half a man's height, and placed opposite the upper half of his body, will give him a full-length image of himself. The use of mirrors fixed at an angle of 45, in order to cast the light of the sky into a room, or in order, when fixed outside xv.] MIRRORS. 473 a window, to enable a person within a room to see the passengers in the street outside, is sufficiently intelligible. In medicine the same principle is utilised in the Laryngoscope. Light falling from a lamp upon a concave mirror is cast upon a small plane-mirror, held by means of a long handle at an angle of 45 within the pharynx of the patient ; the light is reflected and passes down towards the larynx, which is illuminated and becomes a source of light ; rays returning from this, passing upwards, strike the small pharyngeal-mirror, and are diverted by it so that they traverse, horizontally, the cavity of the mouth, and pass through a small aperture in the centre of the concave mirror into the eye of the observer, who is thus enabled to see the larynx and windpipe. The brightness and distinctness of an image depend upon the polish of the mirror, and on its not scattering the light which falls upon it. A smooth clean mirror is, itself, almost invisible. A piece of polished platinum reflects light as well when it is white-hot as when it is cold. The image of an object in a mirror can never be brighter than the object itself, however smooth the mirror be. Hence, if a candle be held between two parallel mirrors, the long series of images produced by multiple reflexion grows fainter as the images seem to grow more distant. When we have multiple reflexions of light between two polished plates, if the plates be parallel and the incidence oblique, the reflexions are more numerous the nearer the plates are to one another. If the two plates be in- clined to one another at an angle of 60, the images of a point lying between them, the image of which is multiplied by repeated reflexion, are so situated that the first, second, .... sixth form together a symmetrical kaleido- scope-image of the point, a group of images ranged round a central axis, while the seventh and further images coincide with their predecessors. Simi- larly for such angles as 90, 45, 30, 24, and other aliquot parts of 360. When a mirror is rotated a beam of light reflected from it is deflected through an angle equal to A pA ' twice that of the rotation of the mirror itself. In Fig. 161 S is a source of light ; AB a mirror ; SM an incident ray ; MR a reflected ray. If the mirror be turned into the position A'B', the reflected beam is now MR' : the reflected ray has swept through the angle RMR', which is equal to twice the angle AMA'. We have already (p. 116) considered some cases of the re- flexion of waves at parabolic and elliptical mirrors, and on seg- < <7v / f- 7^ JJ M 474 OF ETHER- WAVES. [CHAP. ments of spheres. Parabolic mirrors are used when it is desired to bring the plane-fronted light of a distant star accurately to a focus, or to produce a parallel beam of light : in the latter case the source of light is placed at the focus of the paraboloid. Spherical mirrors may be considered, if we restrict our atteation to those rays which fall very near the centre of the mirror, to have an approximate focus at the tip of their " caustic by re- flexion" (Fig. 64). When the beam is broad the rays do not converge accurately to a focus, and the image of a point is a circle, brightest towards its centre. This is the Spherical Aberration of a mirror, which renders its definition, especially of somewhat broad objects, very bad. In consequence of this it is often necessary to cut off the lateral rays by a diaphragm, which increases the clearness of definition, though it diminishes the brightness of the image. When light coming from a source at a distance f is re- flected by a concave spherical-mirror, it is reflected back to an approximate focus at a distance /'. The distance of the source /, the distance of the approximate focus f ', and r, the radius of the mirror, are connected by the law 11 2 7 f = r~ When the source is infinitely distant, -7 = 0, and /' = J r ; the case of Fig. 64, page 116. The focus to which the light converges in this case is called the Principal Focus of the mirror. If the course of the light be reversed, and / = J r t f = j an( l the light is reflected to a focus at an infinite distance ; it is approximately parallel-rayed or plane-fronted. _ When the source is at a definite distance, beyond the centre of the sphere, the focus is between the principal focus and the geometrical centre of the sphere ; conversely, when a source of light is between the principal focus and the centre of the sphere, the reflected light converges upon a point at a defi- nite point beyond the centre. Light radiating from the centre of a spherical mirror, after reflexion, again converges upon the same point. When the source is between the principal T focus and the mirror, i.e., when / is less than , /' is negative, and the reflected rays seem to diverge from a point on the other side of the mirror, and the Image of the point is imaginary or Virtual. Pairs of points at the respective distances / and /', as defined by this formula, are called Conjugate Foci. Conjugate foci are found in one of the four following relations : (a.) Coincident, both being at the geometrical centre of the mirror. (6.) One between the principal focus and the centre ; the other beyond the centre. XV.] MIRRORS. 475 Fig.162. (c.) One at the principal focus ; the other at an infinite distance beyond the centre. (d.) One between the principal focus and the mirror itself ; the other, a virtual focus, apparently behind the mirror. If the source of light be not in the axis of the mirror, the light is not brought to a focus in the axis, but at some point situated laterally. In Fig. 162 the rays proceeding from A con- verge upon A', and an eye situated in the direction of B looking to- wards the mirror will receive rays which appear to diverge from a Real image at the point A', a point which the waves really traverse. If -the source of light be an extended object, AZ (Fig. 163), there will be formed an inverted real image, A'Z', an image produced by an actual divergence of the rays from a series of points situated at 71 'A'. In Fig. 163 the object AZ produces a diminished and inverted Fig.163. real -image at Z'A'; or, conversely, an object at Z'A' will present an inverted and enlarged real-image to the eye placed beyond AZ, an image which appears nearer than the object itself. Suppose that the object AZ is the face of a person looking at a concave mirror, it is plain that the eyes in that face are eyes situated beyond Z'A' : a real image of the face is seen, diminished and inverted, and apparently situated at Z'A', between the observer and the mirror. As the observer ap- proaches the mirror, the image approaches him : it appears to grow larger. When his eye is at the geometrical centre of the mirror, the image is blurred : when the eye is placed between the centre and the principal focus, the rays not having either really or apparently crossed one another, the image is still blurred. When the eye passes between the principal focus and the surface of the mirror, the rays reflected seem to come from a virtual image behind the mirror, which is erect, and is larger the nearer the eye is to the mirror. The student will have little difficulty in drawing the diagrams appropri- ate to each case, and in verifying the results even by means of such a simple concave-mirror as the inside of a watch-case, or of a common large spoon. 476 OF ETHER-WAVES. [CHAP. Fig.164. Convex Spherical -Mirrors. In Fig. 164 we see that a beam of light, plane-fronted or parallel -rayed, and therefore _ coming, apparently or really, from an infinite distance, is so reflected that it appears, after reflexion, to diverge from the principal focus of the principal mirror, this point being the tip of its caustic; while a beam diverging from a point A ap- pears, after reflexion, to diverge similarly from a point A' situated nearer the mirror than the principal focus. The formula is When mixed coloured-light falls upon a mirror, all the re- flected rays, whatever be their wave-lengths or relative intensities, are reflected in the same directions. When a mirror is flexible and has a variable curvature, as the form of the mirror is made to vary irregularly, intermittently, or in accordance with some harmonic law of simple or complex variation, so the intensity of light at any point near the focus varies irregularly, intermittently, or harmonically. Fig.165. Refraction of Light. Light-rays are bent when they reach the surface of separation between an optically rarer and an optically denser medium. In Fig. 165 a body situated at S seems to be situated approximately at S'. In this way, if a coin be placed in a '*''/' - basin, and the eye placed in such a s/- - position as just not to see it, water poured into the basin will bring the coin into view. The sun is seen before he has astronomically risen, and continues to be seen after the true sunset. A stick placed in water appears bent, for the image of each point of its surface appears raised within the water by an amount propor- tioned to its depth beneath the surface of the water ; each point of the image appears indistinct, being brought only to an approxi- mate focus, and the image of the whole is blue on one side, red on the other, for, as shown in Fig. 165, the different colours of xv.] PRISMS. 4*77 white light travelling from any point S are unequally refracted into the air. When light passes through a number of parallel-sided transparent plates of different densities, the total angular-deviation produced is the same as if the last of them had alone stood in the course of the transmitted light. Prisms Monochromatic Light. If we confine our atten- tion to a single ray of light A Fi impinging upon a prism, we may trace the course of the corresponding wave-front, as in Fig. 166. The light travel- ling from S strikes at the angle of incidence i; it is refracted at the angle r; s'mi = n sin r (L) It strikes the second face at the angle r' ; it leaves the glass at the angle i' ; sin i f = n sin r' (ii.) Further, the angles r + r r can be proved together equal to the angle A* ; r + / = A (iii.) Lastly, if. the angle between the incident ray SD and the deviated ray ES be V, then i + i' = V + A (iv.) f From these four equations, which in- volve n, together with the six angles, i, i f , r, r', A, V, we may, if we know any three of these angles (say A, i t and V), determine, for the particular monochromatic light employed, n, the Index of Eefraction of the material of the prism ; this numerical quantity n expresses, as compared with unity, the relative slowness of ether-waves in the prism as compared with that in the medium surrounding the prism. Thus, if a particular glass prism have the index 1*5 in air for a mono- chromatic yellow-light, that light travels 1'5 times as fast in air as it does in the glass out of which the prism has been cut. To find the Kefractive Index of a Liquid, the amount of re- fraction must be directly observed. Two telescopes arranged radially on a vertical circle or alidade ; light traverses the one, and is rendered parallel by it ; it then impinges on the level sur- * The normals to the faces drawn at the points of entrance and exit of the ray cross one another at X ; the quadrilateral AX has its internal angles equal to four right angles ; two of these (at D and E) are right angles : the remaining two (A and X) are therefore equal to two right angles. In the triangle DEX, the angles r + r + X are again equal to 180 ; whence r + / = A. t The angles r + r = A ; r + r + V = V + A ; V = OED + ODE ; .: r + r' + OED -TODE = V + A ; but r + ODE = XDO = i ; and r' + OED = XEO = H ; . 478 OF ETHER- WAVES. [CHAP. face of liquid in a central vessel, penetrates it, leaves the vessel, passing normally through the glass bottom of the vessel, and enters the second telescope, which it traverses. The relative angles made by the two telescopes with a horizontal bar indicate the angles of incidence and of refraction, which, being known, the index n is known. The Kefractive Index of a Gas is found by processes which depend on the amount of retardation suffered by light in a long column of that gas, as will be seen under " Interference," and as has already been explained under " Velocity of Light " (the rotating- mirror method, page 462). If instead of a prism filled, say, with water in air, we use a prism filled with air submerged in water, the deviation of a ray travelling in the water will be equal, but of opposite sense to that of a ray travelling in air and refracted by the water-prism. The refractive index of organic substances is found to have a close rela- tion to their chemical constitution. If a substance contain carbon, hydrogen, and oxygen, its formula being C a H 6 O e , and if its density be /o, it is found that the numerical quantity ( x molecular weight ) is constant, even \ P ) though the density be varied by changes of temperature, and it is called the Molecular Refractive Power of the substance. This refractive power may be predicted from the chemical formula by finding the value of | 5a+ 1-36 + 3c j.. Isomeric substances have thus the same refractive power. If in the last figure (166) light start from S', it will retrace the line S'EDS ; and the same result will follow if a mirror at S' turn back the light coming. from S: on its direction being reversed, light will retrace its path a proposition applicable to waves in general. Minimum Deviation. It is only when the prism is turned into such a position that i, the angle of incidence, becomes equal to the angle i f y that light diverging from a source S (Fig. 166) can converge upon a single focus lying towards S' ; and this is the position in which the incident ray of light is, on the whole, least deviated by the prism. If mixed coloured -light be passed through a prism, each colour has its own index of refraction, n, its own path through the prism ; the whole light is broken up by Dispersion into a bundle of monochromatic rays, each travelling in a separate direction, and diverging the more widely from one another the further they travel ; and, received on a screen, these form a Spectrum. Each kind of light is differently deviated by the prism ; and for each xv.] PRISMS. kind of light the minimum deviation of the prism is different ; image of a slit looked at through a prism, or cast upon a screen, sharply in focus in all parts of the spectrum at once ; only one colour accurately in focus at any one time ; and to put any particular colour of spectrum into focus, the prism must be rotated one way and another in the beam of light until that position is found for it which corresponds to the greatest approach of the particular colour towards the red end of the spectrum : this is for that colour the position of minimum deviation and of most accu- rate definition. The Rainbow is produced by reflexion and refraction of sunlight in the drops of water which make up falling rain. Parallel sunlight falls from behind the spectator ; in each drop the light is dispersively re- fracted, and then reflected from the further face of the drop; it travels back through the drop and emerges in a state of chromatic dispersion. Drops which for the momenk are situated at a certain angular height send violet light into the eye of the observer, but the red light from them misses his eye, for it strikes too low. Drops at a greater angular height send red light into the observer's eye, but violet light proceeding from them is, as it were, aimed too high, and does not enter his eye. Drops in intermediate positions send intermediate coloured-light into his eye. Drops above or below a certain range of angular height do not send light into his eye at all. The whole phenomenon is symmetrical round an axis containing the sun and passing through the observer's eye, and the bow is, according to the height of the sun in the heavens, a greater or lesser portion of a circle whose parts are equidistant from that axis. The rainbow as seen by the one eye is not formed by the same water-drops as the rainbow seen by the other eye. Repeated reflexions and refractions in raindrops frequently give rise to secondary rainbows, which, under experimental conditions, have been observed to the number of eighteen. The halo seen around the sun when it shines through a frozen cloud is due to refraction of sunlight through the crystals. Conceive a circle of prisms round the sun arranged in such a position as to send a maximum of sunlight into the eye : a circular spectrum would be seen, red externally ; among the particles of ice in the cloud some must be in the favourable posi- tion of these prisms. Recomposition of White Light. If a bundle of coloured rays, emergent from a prism, be pi 16? received on a second prism, similar to the first but reversed in position, these monochromatic rays are again recombined, and again form, on emergence, a beam of the original mixed coloured-light, parallel to its original direction ; on the whole there is neither dispersion nor angular deviation. Deviation without Dispersion. If white light fall upon a flint-glass prism of such an angle as to make on a screen, at a distance of, say, 10 feet, a spectrum whose length between two 480 OF ETHER-WAVES. [CHAP. definite colours or Fraunhofer lines is, say, 3 inches ; and if another prism of crown glass which is able to produce a spectrum of an exactly-equal length between the same definite colours or lines be so placed as to reverse the dispersive action of the former prism, .according to the principles of the last paragraph, the light leaving the prism is approximately white. It is not, however, parallel to its original course ; for when we pass from prisms of one substance to prisms of another, we find a phenomenon known as the Irrationality of Dispersion; we find that the amount of mean deviation and the amount of dispersion pro- duced by a given prism are independent of one another ; and hence, to reverse dispersion is not necessarily to reverse devi- ation, if this be effected by a prism of a second, a different sub- stance. A flint-glass prism and a crown-glass prism thus combined may produce deviation without producing any dispersion, and the emergent light is approximately white ; and thus, for two kinds of light a flint-glass prism may be achromatised by. a second prism of crown glass. The reconi- position of the colour is not perfect, except for the two colours (or lines) chosen ; it would have been perfect were the spectrum of crown glass pre- cisely similar to that of flint glass in respect to the proportionate lengths of the coloured areas in it ; but it is not so ; in the crown-glass spectrum the orange and yellow are proportionately more refracted, and are spread over a proportionately greater area than they are in the flint-glass spectrum, and the blue and violet less so ; the former are for accurate recomposition too much, the latter too little refracted by the achromatising crown-glass ; the issuing beam, white at the centre, is yellowish on one side, bluish on the other. Three prisms may be combined so as to blend three colours in the emergent ray ; and so forth. Eeflexion at the surface of a mirror may be said to furnish an 1 example of deviation without dispersion. Sometimes a reflecting prism (Fig. 158) is preferred to a mirror, especially in lantern projection-apparatus. Light enters normally at one face of the prism, is totally reflected at the second, and passes normally through the third. The image produced by such a prism is inverted, and if the incident beam be parallel there is no refrac- tion, and therefore no chromatic dispersion, while the loss of light is very small. Dispersion without Deviation. If crown-glass prisms and flint-glass prisms be alternated they can be made to produce dis- persion while the issuing rays are parallel to the original direction of the entering light. This principle is applied in the Direct-vision spectro- scope, which simply consists of a train of such prisms, to which xv.] DISPERSION. 481 the light is admitted by a slit at A (Fig. 168), and from which the light issuing at B is caused to pass through a lens so adjusted Fig.168. as to give the eye a clear imagaKof the slit for each colour ; and the eye accordingly receives the impression of a continuous spec- trum situated at the focus of the lens. Abnormal Dispersion. The amount of deviation of each kind of coloured light can be directly measured when a spectrum is formed ; so can the angle of the prism and the angle of incidence ; thus for each transparent substance, and for each wave-frequency of incident ether- waves, the index of refraction may be calculated and recorded. The spectra produced by similar prisms of different substances differ not only in their absolute lengths, but also in the proportionate length of each colour, and even in their arrangements. A hollow glass prism filled with iodine vapour refracts red light most, and violet least ; it gives a spectrum the order of succession in which is ultra-violet, violet, blue, red, the re- verse of the ordinary; the intermediate parts of the spectrum are lost by absorption. A weak alcoholic-solution of fuchsin in a hollow glass prism refracts the blue and violet less than it refracts the yellow and red ; the spectrum thus presents the following order : Fraunhofer lines, F to H i.e. green, blue, violet then A to D, red, orange and yellow, not quite up to the E line : the green from E, nearly as far as F (E inclusive), being absent on account of absorption. Aniline violet, aniline blue, indigo carmine, give a green -blue -orange spectrum. A concentrated solution of cyanin in a hollow glass prism gives a spectrum consisting of first, green-blue, then a dark band, then red, and traces of orange. In general all bodies which, like many of the aniline dyes or crystals of permanganate of potash, act upon some kinds of light as metallic reflectors, and present, when in the solid state, superficial colours differing from the body-colours, are found to give, when their solution occupies a hollow glass prism, an analogous refraction-spectrum not to be confounded with the ordinary absorption-spectrum in which the order of the colours is not that of the spectrum as produced by a prism of solid glass. On the redward side of an absorption-band the index of refraction is found to increase so rapidly as to throw the part of the spectrum near the redward side of the absorption -band over towards the violet ; and conversely, the part of the spectrum situated near the violet-ward side of an absorption-band is thrown back towards the red. The effect may be simply to render the absorption- band narrower than it would have been had there been no effect of this kind, or to cause overlapping of different parts of the spectra ; or, as in the case of fuchsin and cyanin, actually to throw over the redward and the violet-ward parts of the spectrum into each other's places, and to make red light more refrangible than violet. These phenomena, experimentally systematised by Kundt, have been discussed by Helmholtz on the supposition that the absorbed light being in tune with the molecules of the body, these molecules 2 I 482 OF ETHER- WAVES. [CHAP. Fig.169. are set in motion which is modified by intramolecular friction so as to react on the transmitted light. The question is still obscure. The heat region of the spectrum is very much shortened or compressed by the use of glass prisms. Lenses. Lenses are of two main kinds : a. Thin -edged, thick in the centre; either convex on both sides, plano-convex, or convexo-concave, with a shallow concavity; a parallel-rayed beam of light falling upon one of these lenses is made to converge after transmission upon a Real Focus at the opposite side of the lens. I. Thick -edged, thinnest in the centre; bi-concave, plano- concave, or concavo-convex, with a deep concavity : a plane- fronted beam of light after transmission diverges so as to seem to come from a Vir- tual Focus on the same side of the lens as the source of light itself. Fig. 169 shows these dif- ferent forms and their action on a parallel beam of light travelling from left to right. Every lens has a prin- cipal focus ; this is the point to which a parallel beam of rays is caused to converge, or from which it is apparently caused to con- verge, as the case may be. The distance of this principal focus from the lens is called /, the Focal Length of the lens. If r and r f be the distances of the centres of curvature of the right and left faces of the lens respectively, both distances being measured from the centre of the lens positively that is, towards the right on a horizontal line ; and if a parallel beam come from the left to a focal point on the right, where n is the index of refraction of the lens in air. This formula applies without alteration to concavo-convex lenses, the convexity of which is turned towards the left, and both of whose centres of curvature are therefore situated in a positive direction. In a bi-convex lens r is negative, and j = (n- 1) I - - - -, J ; in a bi-concave one / is negative, and v * ( 1) ( i 7 J ; in a convexo-plane lens (convexity to xv.] LENSES. 483 the left) r = oo , and = (n 1) ( _ J ; in a concave -plane lens (con- cavity to the right) r' = oo , and - = (n 1) - . To find the focal length of a thin-edged lens, find by experi- ment the distance at which it will make a clear image of the sun upon a screen : light coming from the sun is practically plane- fronted, and is caused to converge upon the principal focus. Lenses, like mirrors, have Conjugate Foci at distances p and p t ; rays coming from an object placed at one focus are caused to produce an image, real or virtual, at the other focus. Convergent or thin-edged lenses have their principal focal distance and their conjugate foci (or distances of object and image) related according to the formula I^I__I. s ' v f A beam comes from an infinite distance ; p = oo ; then p / = f; light converges upon the principal focus, as we have seen. Conversely, if a beam come from a source of light at the principal focus, p = /;- . . p t = oo ; the image is real, but at an infinite distance behind the lens. If the source of light be at a distance less than infinity, but greater than twice the focal length, the image is real, and is on the other side of the lens, at a point between the principal focus and a point twice the focal length from the lens. Conversely, if the object be at a distance greater than /, but less than 2/, the image is real, at a distance greater than 2/. If the object be at a distance 2/, the image is also at a distance 2/, on the other side of the lens. The object and the image are then equal in size. Hence another method of finding the focal length of a thin-edged lens. Adjust an object, a lens, and a screen, so that the image on the screen is equal in size to the object : the screen and the object are now both situated at distances equal to 2/ from the centre of the lens ; half the distance of either of them from the centre of the lens is the focal length. If the object be between the lens and its principal focus, the rays are not made sufficiently convergent to cross at any place ; they seem to come from a virtual image behind the principal focal-point, further from the lens than the object, and therefore behind it As to the inversion or erectness of the image produced by a thin-edged lens: an object at (Fig. 170) produces a smaller Fig.170. inverted image at I, and an eye placed beyond I that is, at a 484 OF ETHEE-WAVES. [CHAP. Fig.171. sufficient distance from the lens will perceive the image of a distant object, inverted, smaller than the object, and apparently situated between him and the lens; the contrary being the general impression. If, however, the object be brought so near the lens as to lie at a distance from the lens less than the focal length, then to an eye situated at any distance on the other side of the lens a virtual image will be apparent, erect, magnified, and more distant than the ob- ject. Hence these lenses are commonly used as magnifying glasses (Fig. 171). Divergent or thick-edged lenses. The formula for these is III P ' ' ~Pi 7" When p = oo, p t /; a virtual image at the principal focus. When p = f, p, = oo ; a real image at an infinite distance. When p is of any value > 0, < oo , p t is less than /. The focus of a thick-edged lens is most conveniently found by coupling it with a thin-edged one, found by trial among a sufficiently extensive series, so that together they shall produce no change in the apparent size of an object seen through them. As to inversion and size of image, Fig. 172 shows that the image is erect, diminished, virtual, and nearer the lens than the object itself; and, since there is no subsequent cross- ing of the rays beyond the lens, there is no inversion of the image. Lenses of this kind may be used as diminish- ing glasses. For some purposes flexible lenses may be used in which the curvature may be slightly varied. Cusco's opthalmoscopic lens consists of two pieces of thin'microscopic cover-glass fixed in a frame : water fills the cavity between them ; by forcing more or less water into the cavity the curvature may be varied. Even when monochromatic light is transmitted through lenses, the focussing can never be exact if their surfaces be spherical ; each point of an extended object forms a slightly-blurred image ; this effect can be reduced somewhat by the use of diaphragms, Fig.172. xv.] SPHERICAL AND CHROMATIC ABERRATION. 485 which cut off the external parts of broad beams of light, and thus diminish the spherical aberration of the lens (as in the pupil of the eye), but it can only be brought to a minimum by modify- ing the curvature of the lenses used. Some lenses are so curved as to bring all the points of an extendedlmage into the same plane, and thus to produce a flat field ; others to bring points differing in distance to foci which differ very little from one another, and thus to secure penetration. The calculation of the various curved- surfaces necessary for these ends often involves considerable mathematical skill. When mixed coloured-light is passed through a thin-edged lens, violet light is most refracted, and comes to a focus sooner than the red rays do ; beyond the red focus is the heat-focus ; between the violet focus and the lens is the region of the photo- graphic focus. Makers of photographic lenses have shown much skill in making the photographic and the visual focus coincide ; for special photographic work, such as Kutherford's lunar photography, lenses have had to be constructed whose curvature is calculated with reference to the focus of the highly-re- frangible actinic rays alone ; and, while nothing can be distinctly seen through such lenses, photographs of extraordinary clearness have been taken by their aid. If a beam of white light be passed through a single con- vergent-lens, a screen placed at the violet focus will give an image with a red border the red rays not having yet converged ; if it be placed a little further off, at the red focus, the image is now surrounded by a violet border, for the violet rays are already divergent. Consequently no clear definition can be obtained by the use of such simple lenses, and it is necessary to render them Achromatic. A biconvex lens of flint glass, more convergent than is necessary, is coupled with a biconcave lens of crown glass of proper curvature ; the latter destroys the dispersion, by bringing two colours to the same focus, without wholly doing away with the deviation ; the couplet acts on the whole as a single lens, producing a somewhat smaller refraction than either of the lenses. This arrangement may be seen in the object-glass of any common telescope. For still further accuracy, three, four, or even a greater number of lenses may be combined, by which three, four, or more colours are brought to the same focus ; as in the achromatic objectives of microscopes. The most complex system of lenses may be reduced, so far as its action upon light is concerned, to a single lens, which gives images of the same size and position as 486 OF ETHER- WAVES. [CHAP. the system in question. Thus an achromatic microscope-objective may be treated in diagrams as a single lens. The Eye may be ideally reduced for many purposes to a single lens com- posed of aqueous or vitreous humour, having its back coincident with the retina, and its anterior aspect a spherical surface of 5 '1248 mm. radius, situated at its most anterior point 2 '3448 mm. behind the actual anterior surface of the cornea. Such a lens would refract incident light, and bring images of distant points to a focus upon the retina in the same way as the actual eye does. Eadiant Heat may be shown to be reflected and refracted like Light by concentrating rays of dark heat upon a thermopile by means of a lens or a mirror, or by refracting them by means of a prism into a new path, in the course of which the thermopile must somewhere be placed before it will indicate the impact of Heat-waves. INTERFERENCE. Ether-waves are capable of Interference. Two systems of equal waves arriving at the same point in opposite phases will produce at that point no effect either of light or of heat or of photographic action : at that point the ether will be at rest ; and thus light added to light may produce darkness. In Fig. 75 the two points A and B are centres of wave-motion, and at the points &', d! 3 f, on the screen MN", there is no disturbance, while at inter- vening points, of, c f , e f , the amplitude of disturbance is doubled. Interference of waves thus affects the distribution of energy in a system of undulation, and such a screen produces a system of negative reflected waves from a!, c', e f , etc. Let us now consider a monochromatic beam of plane-polarised light. Such a beam may be divided in two parts by reflexion from a silvered or platinum mirror bent in the middle at an angle very nearly equal to 180, or else by refraction through a biprism, whose angle is very nearly 180. The last case is shown in Fig. 173. S is a source of light; the light from it is transmitted through a polariser N" : it is now a polarised beam. The rays are received by a convergent lens, which makes them 1 XV.] INTERFERENCE. 487 converge upon S". In its course it is passed through a piece of glass R, coloured red with suboxide of copper : it is now to a rough approximation monochromatic. It is then passed through the biprism P, which refracts it in such a way that it seems to come from two equal and equally-distant foci at S w and S ;/// . The light may then be received, either on a screen, or directly in the observer's eye placed in the onward path of the beam. A series of dark and bright fringes will be seen corresponding to the alternate fringes of rest and disturbance of figure 75. The two beams apparently travelling from S w and S /;// are polarised in the same plane, and any irregularity of amplitude characterising the one is participated in by the other. Hence they are in a position to interfere fully and regularly with one another. If, on the other hand, they had been polarised in planes at right angles to one another, they could not have extinguished one another at any point. When common light is used, it may be at once filtered through a piece of red glass and then passed through a convergent lens. To procure monochromatic light it is better to project a spectrum upon a screen in which there is a slit, and then, behind the screen, to make use of that part of the spectrum whose light falls upon and traverses the slit. It is very easy to procure a bright spot which may represent a simple luminous point by making a small hole in a metal screen, and in this insert- ing a drop of glycerine. This acts as a powerfully-convergent lens, and if sunlight be concentrated upon it there will appear on the dark side of the Fig.m. screen an intensely bright little spot of light which may be used as a source of light for many experiments ; with such a source of light Fresnel dis- covered the laws of diffraction. More elaborately, the same result may be better attained by means of the electric light made to converge by an achro- matic lens of exceedingly short focus, a high-power microscopic objective. When monochromatic common light, proceeding from a lumin- ous point, is passed through a biprism, its vibrations in each of two planes, at right angles to one another, produce the effects of inter- ference independently of one another, but produce their respective fringes and bands in coincident positions on the screen. When 488 OF ETHER- WAVES. [CHAP. mixed coloured-light or white light is treated in this way, the red fringes do not coincide with the violet fringes ; the violet fringes are more numerous than the red fringes, and are closer together. This will be understood from Fig. 7 5 ; if the wave-length be in- creased, the points a f , l r , c f , d f , e f , must become further distant from one another. A violet fringe is seen near the axial line of the beam ; it is overlapped by a blue, the blue by a green, and so on : each coloured fringe produced by the interference of white light presents a complete spectrum. The number of such spectra is limited ; at a little distance from the axial line of the beam the fringes overlap one another so as to produce what appears to the eye to be simply white light, but the spectrum of which shows a series of alternately dark and light bands : all the colours being equally encroached upon by dark bands, the result seems white. A bent mirror used instead of a biprism produces, by reflexion of white light upon a screen, alternate fringes of white light and darkness. Measurement of -wave-length. If S'"S"" be the apparent position of the two images or apparent sources of light, which must be monochromatic ; N the position of the central fringe, illuminated by the joint action of S"'a,nd S"" ; the angle S'"NS"" = 2i ; N' the position of, say, the fourth bright fringe ; S'"N' is shorter than S""N' by four wave-lengths ; this difference is very nearly equal to \ NN' x S'"S"" -f- AN } = NN' x 2 tan i. Fig.175. f/ N The angle 2t can be measured with a theodolite ; the distance NN' can be measured with a micrometer ; the value of the four wave-lengths, and therefore of one wave-length, can be determined from these data. Fig. 75 shows that the line of propagation of these fringes in space is hyperbolic ; the foci of these hyperbolae being the two apparent sources. If the light from one of the sources be retarded by being made to pass through a layer of a substance in which light travels more slowly than in air, the whole of the fringes will be shifted somewhat towards the side on which the retardation takes place. From the amount of this shifting may be calculated the amount of retardation ; and by means of this the relative velocities of light in (and therefore the refractive indices of) such things as hot air, cold air, hydrogen gas, normal glass, compressed glass, compressed liquids, and so forth, may be estimated. Colours of thin films. Thin films of transparent substances, such as oil upon the surface of water, iron oxide upon the sur- XV.] COLOURS OF THIN FILMS. 489 face of tempered steel, oxides deposited upon metals by the galvanic battery, soap bubbles, glass blown out to an extreme tenuity or exfoliating under the influence of slow decomposition, present curious colours when shone upon by a comparatively bright light. Such films may be rendered permanent ; a solution of bitumen and a little caoutchouc in a mixture of benzene and oil of naphtha, dropped into water, forms films which solidify and may be caused to adhere to a sheet of paper. In Fig. 176 light from the source S is incident upon a thin transparent-film AB of uniform thickness. A part of the light is at once reflected to R from the first surface of the film. Another part is refracted to R' after havin underone *3 one reflexion at the second surface. If ' the path of the beam in the film be an even* number of half wave-lengths, the beam travelling to R' is opposed in phase to that travelling to R, and an eye placed at RR' (these points being supposed very close together) will receive no impres- sion of light ; or, rather, it will receive but a feeble impression, for the ray to R' cannot be quite equal in intensity to that travelling to R. Again, an eye placed at TT' will perceive but a feeble impression of light; not absolute darkness, for the ray to T is considerably more intense than that to T', and is not completely neutralised by it. Let the film be of variable thickness; a film of air between a glass plate and a biprism, or between a convex lens and a plate of glass, varies in thickness with the distance from the centre ; in the former case the thickness of the film of air varies as the distance, in the latter, approximately as the square of the distance from the central point. Monochromatic light re- flected from such a system presents the appearance of alternately dark and bright bands or circles bright where the directly-re- flected light and the light reflected from the second surface of * This seems strange ; we might have expected a retardation of an odd number of half wave-lengths to produce a difference in phase of half a period ; but it will be remembered that the beam reflected at one of the surfaces of the film that sur- face, namely, which separates an optically denser from a rarer medium suffers a loss of half a wave-length, which is independent of the thickness of the film. 490 OF ETHEK-WAVES. [CHAP. the film are similar in phase dark where they are opposed. In the case of a lens pressed against a plate they are known as Newton's rings. The less the curvature of the lens the greater the distance between two consecutive rings. If mixed coloured or white light be employed, the dark and bright rings of the several components cannot coincide, and the result is a series of circular spectra, in each of which the violet circle is the narrow- est. These spectra overlap one another at a little distance from the centre, and blend into what appears to the eye to be white light. A series of dark rings or fringes may be obtained by rubbing a film of soap on black glass, drying it, and breathing gently upon one point of this through a glass tube ; this, done in the sunshine, gives rise to bright colours. It is not possible actually to obtain monochromatic light ; even that emitted by incandescent sodium- vapour, in which some five hundred rings can be seen, is not quite monochromatic. The centre of Newton's rings is dark if there be approximate contact ; perfect contact there never can be, for a dustless surface it is impossible to obtain ; even when there is no appreciable thickness of film traversed, the fact that one ray is reflected from the upper, and the other from the lower surface, the one at the bounding surface of an optically-denser, the other at the surface of an optically-rarer medium, causes the one to lose, while the other does not lose, half a wave-length on reflexion ; they thus become opposed in phase, and the centre is dark. If, however, both reflexions be made to take place from the surface of an optically-denser medium, as in Young's experiment, in which light travelling through a lens of crown glass was reflected first from the upper surface of a film of oil of sassafras, lying between that lens and a plate of flint glass, sassafras being intermediate in its refractive power between crown and flint glass, there is no such relative retardation, and the centre of the system of rings is bright. The iridescence of mother of pearl and of objects with a finely-grooved or striated surface, such as butterfly's scales, is an effect of interference. Sunlight falls upon their surface ; some of this is reflected from the ridges, some from the grooves, and in this way a difference of path is set up among the reflected rays, which causes differences of phase among them, and, in the case of some of them, opposition of phase and extinction. When the incidence of the reflected light is very oblique, the ridges alone may reflect, the differences of phase and of path produced will be very small ; there will be little iridescence and very considerable reflexion. The propagation of light " in straight lines " within the same isotropic medium is itself a result of interference. From it is derived the power of making a geometrical Shadow. In Fig. 178 XV.] SHADOWS. 491 a real focus at F acts as a source of light. It casts a sharply- defined shadow of an object upon a screen. If the source of light be an extended one, not a mere point, the shadow consists of two regions, a central umbra and a marginal penumbra. Fig.180. In Fig. 179 the sun, S, shines upon the earth, E; the earth being smaller than the sun, there is formed a cone of darkness behind the earth ; if the moon travel wholly or partially into this cone of shadow, it will be wholly or partly unillumined, and we have a total or a partial eclipse of the moon. But outside this shadow there is a penumbral region, in which a body, or any point of a body, will be in " half -shadow," not fully illuminated, because able only to see a portion of the illuminating body. When light radiating from an extended object passes through a small aperture, the waves arriving at the aperture from the object traverse the aper- ture, and there cross each other; they then diverge, and a screen placed on the opposite side of the aperture receives an in- verted imaga of the object, whose size varies with the distance of the screen, as in the well-known Camera Obscura. An aperture of no appreciable breadth would, at whatever distance the screen might be placed, give a perfect image in the natural colours, an image of which no part would be out of focus ; one of ^V-inch diameter will give on a screen at 40 inches distance an., image which, though wanting in brightness, is as perfectly defined an image as any possible lens placed at the aperture can produce : one of -^inch will produce the same definition at a distance of 250 inches ; or, in general (Rayleigh), if A be the wave-length, r the semi-diameter of the aperture, f the least distance of good defini- tion, / = (2r 2 -f- A,). When the screen is nearer than this, each point of the 492 OF ETHER- WAVES. [CHAP. object makes an image on the screen, which has the same shape as the aperture, and the superposition of these makes a blurred image. Light thus travels in straight lines, and is incapable of passing round corners under ordinary circumstances, and as examined by our ordinary senses. A closer examination of the subject shows, however, that light does to a certain degree pass even round corners. The pheno- mena of Diffraction in which this is observed are explicable on the ordinary principles of interference. Let S (Fig. 77) be the source of light; waves diverge from this as a centre. These waves impinge upon a screen AB. Fig. 77 shows that beyond the screen AB there is a series of fringes within the geometrical shadow ; that even in the part directly in view of the source of light there are bands of relative darkness ; that the central point of the shadow may be nearly as brightly illuminated as if there had been no screen AB ; that the broader the object AB, the narrower will be the fringes; that the forms in space of the regions of approximate darkness are hyperboloids ; while if the source of light be removed to an infinite distance, the hyperbolic lines of relative rest in the illuminated region are practically reduced to straight lines, but sweep past the obstacle without touching it. When the obstacle is circular a minute circle of tinfoil pasted on a piece of clear glass the shadow cast upon a screen, or received in the eye directly or by the aid of a lens or telescope focussed on the obstacle, is seen to be surrounded by a series of dark and bright rings ; or, if the light from S be mixed, coloured, or white light, by a series of spectra ; while the shadow is also modified by a series of such bands or spectra, and its centre is- bright. A similar construction for a little circular aperture in an opaque screen at AB will show that the bright spot produced on a screen beyond AB will have fringes blurring the sharpness of its edges, and that at certain distances of the second screen from AB the centre of the bright spot will be dark. When the obstacle or chink is linear and parallel-sided, the fringes or spectra are parallel to one another ; when it is not so they assume a curved form ; when it is angular the fringes may assume a great variety of remarkable and beautiful forms. The phenomenon of diffraction can be roughly observed by looking at a distant gas-flame, edge on, with the half-closed eyes ; the sun shining on the eye-lashes will also produce a similar effect ; the morning sun, shining on twigs of trees situated between xv.] DIFFRACTION. 493 the sun and the eye, causes the shadows of some of them to become bright in the centre, and a curious silvery appearance results. The image of any point seen through a telescope or microscope has its clearness of definition interfered with by the diffraction of rays of light round the edges of the diaphragm, or round the edges of the lens. This effect is generally insignificant in terres- trial telescopes ; it is very noticeable in astronomical telescopes, where the source of light, a distant star, ought to appear reduced to a point, but is often apparently enlarged into a perceptible disc, or surrounded by rings; and in the microscope it sets a limit to the powers attainable, for high powers involve small lenses and small apertures, and these bring diffraction in their train. If a very large number of lines, some parallel, and others parallel to each other and at right angles to the former, be ruled upon glass, light issuing from a slit or from the image of a slit will, if transmitted through this so-called Diffraction-grating, and received upon a screen or in the eye, be found to be resolved into a central bright image of the slit, on each side of which is a dark space, and then a series of successive spectra, separated by dark spaces ; these spectra have their violet ends turned towards the central bright image. By multiplying the number of lines in the Diffraction -grating, as in Prof. Eowland's new gratings, which have 43,000 lines to the inch, the spectra may be rendered almost perfectly pure, so that Fraunhofer's lines may be easily seen in them. A microscopical preparation of muscular tissue will often be found to act as a more or less efficient diffraction-grating ; the striations of the muscular fibres take the place of the grooves engraved on the glass. A metallic surface in which grooves have been cut will produce by reflexion a diffraction-spectrum if the image of a slit be reflected from it. The value of diffraction-spectra is that the deviation in the successive spectra depends directly upon the wave-length. If any kind of light have, in air, the wave-length A. centimetres, and if n be the average number of lines per centimetre engraved on the grating ; and if 8 be the angular deviation of any particular coloured light (or, better, of any particular Fraunhofer line), then sin 8 is equal to nX. for the first spectrum, to 2nA for the second spectrum, and so forth ; and since n and d can be measured, A. can be accurately found. The Twinkling of Stars is another effect of interference : light, coming to the eye from a star so distant as to be practically 494 OF ETHER- WAVES. [CHAP. a single luminous point, arrives in rays which have traversed slightly unequal distances in an irregularly-refracting atmosphere, and thus enter the eye in irregularly-unequal phases. Now one colour is extinguished, now another ; the eye perceives coloured light complementary to that momentarily lost. No two persons can, as a rule, see any star twinkling in precisely the same manner. The planets twinkle only at their edges^: their discs present many points or sources of light, whose scintillations, on the whole, mask one another. If a planet and a twinkling star say Jupiter and Sirius be severally looked at through an opera-glass which is rapidly whirled across the field of view, the image of the planet will appear to be drawn out into a continuous streak, while that of the star will be broken up into a chain of unequally- bright and differently-coloured spots of light. The colours of light from a bright point twinkling through a dusty haze, or shining through a piece of glass covered with lycopodium ; the halo which surrounds the moon as it shines through an atmosphere charged with particles of condensed aqueous vapour ; the coloured rings seen when particles float in the vitreous humour of the eye, these are all different diffractive effects of interference ; and the greater the size of the particles which produce them, the greater the breadth of the coloured rings. The interference of Actinic Eays may be shown by photography; of Dark Heat, by passing a delicate Thermopile or Tasimeter (p. 565) or a Bolometer (Langley's Thermic Balance, p. 634) through an invisible diffraction-fringe system of dark heat-waves, obtained by treating rays of dark heat with a bent mirror or a biprism, under which circumstances the instrument employed will alter- nately indicate and cease to indicate the impact of heat-waves. DOUBLE EEFRACTION. If a transparent medium have the same properties in all directions it is homogeneous, or, optically, isotropic. A wave of mechanical disturbance starting from a single point of disturb- ance in it will be spherical. The properties of the ether-waves within transparent substances are, in some fashion, correlated with the molecular structure of the substance, and thus the ether- waves propagated from centres within homogeneous or isotropic sub- stances are themselves also spherical. ' Substances in which the propagation of light is in spherical waves are either amorphous, or else belong to the cubical system of crystals, the system in which the three crystallographic axes of the crystal are equal. XV.] DOUBLE REFRACTION. Fiff.181. In some crystalline substances one of the cry axes differs from the other two ; the crystal is then in reference to this axis only, and is said to be mechanical disturbance is propagated in such a crystal in the* form of an ellipsoid. A slice cut out of such a crystal in such a way that its sides are parallel to this principal axis, is said to have been cut parallel to the Principal Section of the crystal. The propagation of an ether-wave in a uniaxial crystal is peculiar. Fig. 181 shows an equal - sided rhombohedron cut out s^~\~^\ of a crystal of Iceland f \ spar by splitting it along | .c its natural cleavage- \ / planes ; its axis AB joins the opposite ob- tuse - angles. Let a point C on this axis be a centre of optical disturbance. Then two concentric sets of waves are produced; the one spherical, the other ellipsoidal ; one of the axes of the ellipsoid coincides with the axis of the 'crystal, and is equal for a given interval of time to the diameter of the sphere developed in an equal time ; the other two axes, which, to avoid circumlocution, we shall here call the extraordinary axes, are equal to one another, and are either longer or shorter than the former, accord- ing to the nature of the crystal. The next question is Which part of a general disturbance at C is propagated in the spherical, and which in the ellipsoidal, wave ? It may roughly be stated that just as we have seen beams of polarised light differently affected by simple reflexion and refrac- tion according to the plane of their polarisation, so in double refraction the behaviour of a beam of light depends upon its state of polarisation. On referring to Fig. 59 we find that the construction there given for the course of a refracted plane-fronted wave may be reduced to the following construction (due to Huyghens) for a single ray refracted at the surface of an ordinary isotropic medium. AB is an incident ray travelling through the medium M ; CD a circular arc, drawn from centre B, with radius proportionate to the velocity of light in the medium M. Continue the arc CD into the second medium M' ; produce AB until it cuts that arc 496 OF ETHEK-WAVES. [CHAP. *!.( in E ; from E draw a tangent line (or plane) cutting the refracting Fig.iss. surface in T. From B as centre draw a semicircular arc in the second medium M', with a radius proportionate to the velocity of light in M'.- From T draw a tangent to this arc; the tangent touches the arc at B'; join BB'. BB' is the course of the re- fracted ray (Fig. 182). A series of somewhat similar constructions will enable us to study a certain number of cases of double refraction. Suppose a block to be cut out of a doubly-refracting crystal in such a way that one of its cut surfaces is parallel to the axis ; and suppose an incident beam to fall upon that surface in a direction at right angles to the axis. Fig. 183 shows that if GH represent such a block, and if the incident beam be in the plane of the paper, the axis is in such a case looked at end- ' on ; and then we find that the incident ray is divided into two parts, which travel at different rates, the slower one, BO, in the central sphere, the more rapid one BE, in the outer ellip- soid, which, looked at in this aspect, has a circular section ; the former, BO, the Ordinary Ray (which obeys the ordinary law sin i = n sin T), more refracted than the latter, BE, the Extra- ordinary Ray. Both these rays are in this case in the plane of the paper, like the original incident-ray. The relative radii of the two circles may be found from the respective amounts of refraction of 'the two rays at this kind of incidence. For the light emitted by sodium-vapour, the ordinary index and the extraordinary index of Iceland spar are respectively 1*65850 and 1*48635 ; the reciprocals of these numbers represent the relative velocities of the ordinary and the extraordinary rays in Iceland spar as compared with that of light in air, this being reckoned as unity. In such crystals as those of Iceland spar the ordinary ray is more retarded than the extra- ordinary. Let us now turn the block of spar round so that its axis is XV.] DOUBLE REFRACTION. 497 brought into the plane of the jgaper that is, into the same plane with the incident light; the incident light now travels in a principal section of the crystal. One of the extraordinary axes of the ellipsoid, being at right angles to the axis of the crystal, is at right angles to the refract- ing surface ; its semi-axis, BF in the sectional figure (Fig. 184), bears to the radius of the circle the ratio of T65850 to T48635 if the light used be that emitted by sodium-vapour. Fig.185. r\ If the block of spar be cut by a plane at right angles to the principal sections, but not parallel to the axis, we obtain the result shown in Fig. 185. The incident light is in the plane of the paper; the axis of the crystal is also in the plane of the paper. When the surface which receives the incident beam has been cut at right angles to the axis, and the light falls upon it normally (that is, at right angles to the surface or parallel to the axis), there is no double refraction ; the ordinary and the extra- ordinary rays coincide. A parallel -sided slice of Iceland spar cut in any other direction than at right angles to the axis will divide an incident ray into an ordinary and an extraordinary ray, as in Fig. 186 (except in the case in which one of the rays is so refracted as to become parallel with the axis, in which case the other ray coincides with it) ; and thus an observer at A will see two images of a spot at B : while if he turn the slice round, the extraordinary image will rotate round the ordinary one. This 'can be readily observed with an ordinary crystal of Iceland spar. Light striking on a plate or a common crystal of Iceland spar is thus split into two rays, and a single point or a page of 2 K 498 OF ETHER- WAVES. [CHAP. print looked at through, such a^crystal gives a double image. Conversely, a pair of points, C, lfc^ looked at by an observer at A A, will havP their images blended, and by find- ing for various distances between C and D the E angle ABE, at which these points appear to f blend, the two refractive indices may be found : the rays CB, DB, and BA being caused to lie all in a principal section of the crystal. I When the incident ray is oblique to the principal section, the extra- ordinary ray is no longer in the same plane with the incident and the ordinary refracted ray, but is deflected to one or the other side : the tangent plane to the ellipsoid does not touch it in the plane of incidence. - The above figures are drawn for crystals such as Iceland spar, beryl, emerald, mica, ruby, sapphire, tourmaline, the ordinary index of refraction of which is greater than the extraordinary, and in which the ordinary ray travels more slowly than the extraordinary, and lies between the extraordinary ray and the axis ; such crystals are called Negative Crystals. In others, such as ice, quartz, boracite, the extraordinary ray lies between the ordinary ray and the axis : such crystals are called Positive Crystals. In the latter, the extraordinary axes of the ellipsoid are shorter than the diameter of the sphere, which thus encloses the ellipsoid : the extraordinary index of refraction is in them greater than the ordinary index. The two rays, the ordinary and the extraordinary, are found to be polarised in planes almost exactly at right angles to one another. The ordinary ray is polarised in a plane contain- ing both the incident ray and the crystalline axis. If the incidence be that of Fig. 184, the incident ray, the reflected ray, and both refracted rays are in the same plane, the plane of the paper, and the axis is parallel to that plane; the ordinary ray is said to be polarised in that plane ; light polarised in such a plane of incidence passes through the spar as an ordinary ray. The definition of the plane of polarisation .is a terminological convention ; and the question as to the relation of the direction of the vibrations of the ordinary ray, thus said to be polarised in that plane, to the plane itself, we have already seen to be a question still under discussion. The extraordinary ray, when the whole three rays thus travel in a principal section of the crystal, is found to be polarised in a plane exactly at right angles with the plane of polarisation of the ordinary ray. xv.] DOUBLE EEFRACTION. 499 The second face of the block of crystal may be so cut that it receives the ordinary M^ the extraordinary rays at such an angle as to transmit the 'onte, Uit totally to reflect the other. In Nicol's prism a long raomb of Iceland spar is cut in this way, and the portions are so cemented by Canada balsam that when common light enters the Nicol it is divided into two rays, of which one, the Ordinary, is totally reflected when it meets the cemented surf ace, while the Extraordinary ray is transmitted and emerges (the faces of the prism having been, in order to permit this, c ^BP own * the proper angle), in a direction parallel to thatSsPlhe incident ray. The whole arrangement is thus capable of acting as a polariser ; and if polarised light be sent through it in one rotational position, the Mcol will transmit it freely; while if the Nicol be rotated through 90 in either direction, on either side of the most favourable position, it will transmit none of it. It can thus serve as a means not only of producing polarised light, but also of detecting polarised light, and of finding in what plane it is polarised ; and when it does this duty it is called an analyser. In tourmaline there is double refraction ; but one of the rays, the ordinary, is absorbed, and the extraordinary alone passes through. Thus a thin plate of tourmaline acts as a polariser of common light incident upon it, and another plate rotating in front of it may act as an analyser, a convenient arrangement, were it not that tourmaline is always dark in colour, and absorbs much of the light incident upon it. For this reason Nicol's prisms are commonly used as sources of polarised light. Crystals of sulphate of iodo-quinine act like tourmaline, but are useless because they are small and brittle. We may here recall the different modes of obtaining a beam of plane- polarised light. 1. Reflexion of ordinary light from glass at the angle of complete polarisation. 2. Transmission through a pile of glass plates with parallel sides ; the angle of incidence being the angle of complete polarisation, or an angle approximating to it. ^Separation of the ordinary from the extraordinary ray produced by ^ double refraction ; this being done (a) by tourmaline, which extinguishes the ordinary ray ; (6) by a Nicol or a Foucault prism, which turns aside the ordinary ray. Some crystals, such as topaz and arragonite, have two axes, and are called Biaxial Crystals : in these the wave-surface is very complex, and they have three indices of refraction. 500 OF ETHER- WAVES. [CHAP. In general, in these crystals no ray obeys the ordinary law of refraction that sin i = n sin r ; but that ray which does so most nearly in general, and which does so perfectly when the incidence is in^Si&of the principal sections, is called the ordinary ray ; while the other of ^he jjro rays, into which a ray of incident light is divided on non-axial incidence, is called the extraordinary ray. In such crystals the positions of the optic axes, which have no neces- sary relation to the crystallographic axes, are variable ; they vary with the temperature of the crystal, and with the kind of light employed ; and in some cases a crystal is found to be binaxial for one, uniaxial for another kind of light ; Glauberite (native sulphate of soda and lime), for example, being binaxial to red, uniaxial to violet light. When light has passed through a crystal f fteland spar and been divided into an ordinary and an extraordinary ray, if it be caused to fall upon a second crystal whose faces are parallel to those of the first, the two rays pass through, suffering no further division ; the ordinary ray emerging from the first crystal is still the ordinary ray in the second crystal, which acts like a mere pro- longation of the first. If the second crystal be turned 90 round a longitudinal axis parallel to the line AB in Fig. 188, there is still no division of the rays ; but the ordinary ray on emergence from the first crystal is an extraordinary ray relative to the second crystal, and is refracted as such in that crystal ; and the converse applies to the extraordinary ray emerging from the first crystal. If the second crystal occupy any rotational position intermediate between these, each ray incident on it is decomposed into an ordinary and an extraordinary ray. There are thus, in the ordinary case, four images of a bright point seen through a pair of crystals arranged end to end, at a distance from one another, and these images blend into two when the crystals are, by rota- tion, placed parallel or at right angles to one another. Interposed Lamina. When a polariser and an analyser of any kind are arranged at right angles, so that a plane-fronted beam incident on the system is wholly cut off or deflected Jpr it, an eye placed beyond the analyser can perceive no light ;^but if a thin film of mica, or other double-refracting substance, uniaxial or biaxial, of uniform thickness, be caused to intervene between the polariser and the analyser, the field may become filled with light, coloured or white, according to the position of the interposed film. XV.] DOUBLE REFRACTION. 501 In Fig. 189 the line AB represents a plane vertical to the paper, and cutting the paper in AB : we call this the vertical plane, or the plane AB. Then let us by any convenient means produce a beam of plane -polarised monochromatic light, polarised in the plane AB, and let suppose this beam to be travelling away Fig.189 us seen end-on, travelling away from the observer's eye. Interpose a thin plate of some birefringent substance in the path of the beam : let the axis of this lie in the plane CD. The beam AB is broken up by the interposed plate into two : one in which the plane of polarisation is parallel to CD, one in which it is at right angles to that plane. The former is transmitted through the interposed plate as an ordinary ray, the latter as an extraordinary. The lines, Oa, O/, Oc, indicate respectively the relative ampli- tudes of the incident polarised-beam, of the extraordinary and of the ordinary transmitted beams. The interposed plate may be so thin that although the in- cident beam is divided into two transmitted beams, these have not perceptibly separated from one another, and on emergence are not only parallel, but are also practically coincident. In a wide -fronted wave-system this coincidence may be held to be absolute except at the edges of the beam. Though the two beams coincide in direction, their undulations do not coincide in phase ; in positive crystals the extraordinary, in negative crystals the ordinary, ray is more retarded than its companion. Let us suppose that the more retarded ray has lost one wave-length : then the result of superposition of the two emergent rays will be a plane-polarised beam similar to that which had originally fallen upon the interposed plate ; if half a wave-length ( = \ be lost, the result will again be an equal plane-polarised beam, polarised in the plane EE' (Fig. 189). If, again, CD coincide with AB that is, if the principal plane of the inter- posed crystalline plate be parallel to the plane of polarisation of the incident light there is no extraordinary beam, O/; and the light, having been trans- mitted through the interposed film as an ordinary ray, emerges plane-polar- ised, as it entered. If, again, CD be at right angles to AB, the incident beam is wholly transmitted as an extraordinary ray, and emerges polarised in the original plane. Let us now suppose that CD is inclined to AB at an angle of 45 : if one of the rays be retarded by some even multiple of , the result is plane-polar- ised light, either polarised in the original plane (when the retardation may be measured in whole wave-lengths), or in one at right angles to it (when the retardation is some odd number of half wave-lengths), for EE' is at right angles to AB when CD makes 45 with it. Again, if the retardation be some odd multiple of , the extraordinary and ordinary rays are compounded into 502 OF ETHER- WAVES. [CHAP. Fig-. 190. a circularly-polarised ray of light ; and if the retardation be of any value other than some multiple of a quarter wave-length, the result is an elliptic- ally-polarised beam, the ellipse being, according to the amount of retarda- tion, some one of those indefinitely numerous ellipses which may be de- scribed within the rectangle EaE'fr (Fig. 190). In the general case, AB (Fig. 190) being the plane of the incident beam, CD the principal section of the interposed plate, the angle AOC having any value, and Oa, Oc, and Og being respectively the relative amplitudes of the incident ray polarised in the plane of AB, and of the ordinary and extraordinary rays emergent from the interposed plate ; compounded, their result is an Elliptically-Polarised beam, of which the limits are : (a) A plane-polarised beam, whose plane of polarisation is AB and whose amplitude is represented by Oa. (1) When CD coincides with AB. (2) When CD is at right angles to AB. (3) When the relative retardation of cd smdfg is 0, or an even multiple f X of-. (6) An equal plane-polarised beam whose plane of polarisation passes through EE' ; the angle AOE' being equal to twice AOC : this is the result when the relative retardation is an odd number of half wave-lengths. (c) A circularly-polarised beam when the angle AOC is one of 45, and the relative retardation is some odd multiple of (1) Right-handed (rotation contrary to hands of a watch) when the component polarised in Of loses (together with any number of whole wave-lengths) one quarter wave-length or gains three quarters relatively to Oc. (2) Left-handed when it relatively gains one quarter or loses three. Elliptically-polarised light is produced in every other relative position of CD. This is left-handed if the relative retardation of the ordinary ray trans- mitted through CD lie between and or between nX and n\ + ; right- 2 2 handed if its relative retardation lie between _ and A, or between n\ + 2 2 and (n + 1) A. If the plane CD lie so that the angle AOC lies to the left of AB (the observer being, as hitherto, supposed to be stationed near the source of light), these conditions of left and right-handedness respectively are reversed. A plate of refracting substance of such a thickness that when it is inter- posed in the path of a beam of plane-polarised light of a particular colour, XV.] DOUBLE REFRACTION. 503 with its principal section at an angle of + 45 to the plane of polarisation, it converts that plane-polarised light into circularly-polarised light, is called a quarter-undulation plate. Quarter-undulation plates are of two kinds : (a) Where the thick- ness is just such as to cause a relative retardation of quarter wave-lengths ; (&) Where the plates are thicker, but are opposed in their action. In Fig. 191 two plates cut out of a doubly - refracting crystal are shown fitted together ; the one is cut so that its axis is parallel to the plane of the paper ; the other has its axis at right angles to the paper. Incident light arrives already polar- ised ; it is divided by double refraction into two rays, an ordinary and an extraordinary ; then, since the second plate has its axis at right angles to the axis of the first, the ordinary ray of the first plate is refracted in this as an extraordinary ray, while the extraordinary ray of the former passes through as an ordinary ray. On emergence both rays are parallel and practically coincident ; and the amount of relative retardation is equal to that produced by a thin plate equal in thickness to the difference between the thicknesses of the two plates. When light, plane-polarised, is totally reflected from glass, it is found to be elliptically polarised, unless it had been originally polarised in the plane of incidence, or in a plane at right angles to this. Reflexion from metals presents this peculiarity at all angles of incidence. The vibratory movement actually extends beyond the surface of the glass into the rarer medium be- yond, as may be proved on bringing a second piece of glass close to the totally reflecting surface, when interference-colours will be seen. As a result of this, a difference of phases is set up between the two components (polarised in and at right angles to the plane of incidence) into which the incident light may be resolved. A similar result occurs in metallic reflexion, for some of the light penetrates to a slight depth below the reflecting surface. A wave can- not have its direction abruptly changed : and during the gradual change of its direction, its phase becomes altered to a slight extent : and this effect differs in amount according to the direction of vibration of the incident waves. The change of phase in light reflected from gold or silver is such as to indicate (Jamin and Quincke) that light travels faster in those metals than in air. When the angle of incidence is such that the difference of phase set up is , two such total reflexions would convert a plane- F1 192 polarised ray into a circularly-polarised one. If a rhomb of glass be cut in such a form that a ray of light may pass normally through one surface, strike a second surface at the appropriate angle of incidence, and be there totally reflected, strike the third surface at an equal angle, and pass out normally through a fourth surface, a ray so travel- ling through it will, on emergence, be found to be circularly polarised. Such a rhomb is known as a Fresnel's Rhomb, and acts as a quarter- undulation plate for every kind of light, while a film of mica, real or virtual, can only act as such towards light of one kind. If plane -polarised light pass successively through two similar quarter- 504 OF ETHER- WAVES. [CHAP. Fig.198. undulation plates, similarly placed, the emergent light is plane -polarised in a plane at right angles to the original plane of polarisation ; whereas, if the two quarter-undulation plates be opposed in their action, the light is restored by the second to its original plane-polarised state. A second quarter-undula- tion plate of known action affords us a means of distinguishing right from left-handed elliptically or circularly-polarised light. The two vibrations which, make up the circular or elliptic vibration of the ether in a circularly or elliptically-polarised beam of light are not in a condition to interfere with one another on account of their difference of phase, because they are executed in planes at right angles to one another. If a beam circularly or elliptically polarised by an interposed lamina be received upon a birefringent prism, it is split into two parts, one an ordinary ray, the other an extraordinary ray, and each of these is plane-polar- ised. In Fig. 193 AB is the plane of original polarisation, CD a principal section of the interposed lamina, EE' a principal section of the analysing crystal. Then a plane -polarised ray whose am- plitude is represented in magni- tude by the line Oa, and whose plane of polarisation is AB, is resolved into two, Oc and O/, which are upon emergence com- pounded into a plane or an elliptically or circularly-polarised beam, according to their relative retardations. When this strikes the analyser, its components Oc and Of are themselves resolved each into a pair of components parallel and at right angles to EE'; these are respectively Oe' and Ok from Oc, and Oe and Og from O/. In the plane EE' we have therefore two vibrations, Oef and Oe; in the plane at right angles to EE' we have the vibrations Og and Oh. But Oef and Oe differ in phase ; so do Off and Ok. These are therefore in a condition for interference. The ordinary ray, passing through the analyser, is made up of the mutually-interfering components Oe f and Oe, and the extraordinary of Og and Ok; the effect of interference is to cause a distribution of energy such that the ordinary ray gains or loses as much energy as the extraordinary loses or gains, and thus the energies of the ordinary and the extraordinary rays are, taken together, equal to the energy of the incident plane- xv.] DOUBLE REFRACTION. 505 polarised ray. The amount of relative retardation caused by the interposition of the doubly -refracting plate, when measured in wave-lengths, depends upon the particular kind of light employed. Hence when the original plane-polarised light is a white light, each colour obeys its own law ; each colour, if strong in the ordi- nary, is weak in the extraordinary ray, and vice versd ; thus the extraordinary ray and the ordinary are coloured, and their colours are complementary. The following are the limiting cases : 1. There is no extraordinary ray when (a) AB, CD, and EE' (Fig. 193) coincide. (6) AB and EE' coincide, and CD is at right angles to them. 2. There is no ordinary image when (a) AB and CD coincide, and EE' is at right angles to them. (6) CD and EE' coincide, and AB is at right angles to them. 3. The two images are equal for every colour, and are therefore white (a) When AB and CD coincide, and the angle AOE' = 45. (6) When AB and CD are at right angles, and the angle AOE' = 45. (c) When CD and EE' coincide, both being at an angle of 45 with AB. (y the eye itself, while the biconcave lens employed makes up the pair of lenses. The ophthalmoscope may also be used in such a way as to give an in- verted image, not so much magnified as in the preceding case, but more extensive in its field, brighter, and more easy of attainment. A beam of light reflected from the mirror converges upon and passes through a focus ; it then diverges on its way towards the eye, but encounters a thin-edged lens which causes it rapidly to converge into and then to pass through a focus within the eye, and, after traversing this focus, to illuminate a wide area of the fundus of the eye. Light starting backwards from a large area of the illuminated fundus crosses within the eye and leaves it with a wide diver- gence, but is collected by the biconcave lens before mentioned, and passes through the aperture in the mirror into the eye of the observer, who then perceives an inverted and magnified image of the fundus of the eye, an image which may be still further enlarged by means of a second convergent-lens placed behind the aperture of the mirror. VISUAL PERCEPTION. The retina is not a uniform surface, but is made up of elements whose average distance from one another, in the yellow spot, is about "005 mm. Distant points whose angular distance is such that their images on the retina are less than '005 mm. from one another, seem to blend into one, and thus two stars, whose angular distance is less than 70", appear 'to the eye as a single star. The stimulation of nerves is associated with chemical work in the nerve-ends ; and this with absorption. In this respect it is interesting to find that the retina, which is particularly sensitive to yellow and green light, absorbs green and yellow light, and in white light appears purple. It has been pointed out that the blindness of the eye to heat-waves and actinic waves is of advan- tage : for the energy of heat-radiation is relatively so great that everything would appear intensely bright, and our ordinary vision of objects would be impossible if the rays of dark heat were visible ; while if the ultra-violet rays were visible, the image of every point would be shrouded in a haze due to chromatic aberration. xv.] VISUAL PERCEPTION. 515 It is now the received view that each element of the eye which is broad enough to perceive white light consists of three ultimate elements, each of which is capable of perceiving one of three physiologically -primary colours. All colour -perceptions, infinite as these may be in intensity and in hue, are due to the simultaneous excitation of the three sets of nerve-ends by stimuli of given absolute or relative amount. As to what the physio- logically-primary colours are, opinion is still somewhat divided : red, green, and violet (Young and Helmholtz) ; vermilion, emerald green, and artificial ultramarine blue (Maxwell) ; red, green, and blue (Fick). Any three such colours may be made the basis of a systematic classification of colours. When orange light, mono- chromatic or compound, affects the eye, the nerve-ends sensitive to red are affected; those sensitive to green are simultaneously affected, but less so ; while those sensitive to violet are very feebly affected. When the red and the green nerve-ends (as we may call them) are equally affected, the resultant impression is one of yellow ; red light and green light together make yellow light. This may be shown by several methods of mixture of colours. 1. A source of light : a prism : a screen upon which a spectrum is formed : two slits in the screen, so placed as to admit the passage of two selected colours of the spectrum : achromatic lenses behind the screen converge the two coloured beams towards a common crossing point : a screen there placed indicates the mixed colour. 2. A V-shaped slit in a screen (Helmholtz) ; a prism behind this : the two spectra produced overlap each other and produce a very extensive series of combination-colours. 3. By Maxwell's discs : a disc of red and one of green-painted card- board : each disc slit down to the centre, and cut out at the centre so as to be fitted upon a rotating top : the one disc being slipped through the other, the relative proportions of red and green in view can be modified at will : the whole is rotated at such a rate that the successive impressions of red and green enter the eye at least from twenty-five to fifty times per second : each local impression of red in the retina is still vivid while that of green has already commenced, and vice versa ; the colours blend in the eye, and various shades of orange-red, orange, yellow, or yellowish-green are produced, according to the relative proportion of the colours blended. Red and yellow make orange ; yellow and green, yellowish green ; green and blue, a bluish green ; or in general, colours near one another in the spectrum an average or intermediate sensation. Eed and green in different proportions may produce all the colours of the spectrum between red and green : green and violet all the colours between green and the violet colour employed. 516 OF ETHER- WAVES. [CHAP. When blue and yellow lights are mixed the impression pro- duced on the eye is that of a white light ; blue and yellow are complementary colours. This is contrary to the general impres- sion that yellow and blue make green : when yellow and blue pigments are mixed, the yellow and the blue lights reflected from the mixture destroy one another, forming white light ; and the residual green, never absent from the purest blue or yellow pig- ment-reflected light, is perceived, somewhat wanting in brightness, and diluted by the white light produced by the complementary colours. The phenomena of double refraction enable us to produce an indefinite number of pairs of complementary colours. Some peculiarities of perception of colour are readily ex- plicable on this theory of three kinds of nerve-ends, which is due to Young. A spectrum formed by light travelling from a waning source is found to modify its tints as the light fades ; the orange- red seems to become more purely red, the yellow-green more purely green, and so on ; at length the faint spectrum is approxi- mately restricted to red, green, and violet, or violet-blue ; of each triplet of nerve-ends, one is feebly stimulated by a given colour, the other two are inappreciably so, though if one be stimulated the others can never remain wholly unaffected. On the other hand, if a coloured light be rendered exceedingly bright, the other nerve-ends participate in the excitement : very bright red seems somewhat orange ; violet very easily passes over into whiteness when its brilliancy is excessive. A black colour is due to the absence of stimulation of any of the nerve- ends ; and between bright white and black there is a gradation of weak whites which are called grays. Fatigue of the retina causes it to become insensible to a colour long looked at : when white light is then looked at, it appears of a hue complementary to the colour the sense for which has been temporarily exhausted. When some of the nerve-ends of the retina are stimulated, the stimula- tion spreads to some degree : a very narrow white-hot wire appears, especially from a little distance, to be much wider than it really is ; this phenomenon being named Irradiation. In consequence of this the crescent moon appears larger than that part of the moon which is illuminated by light reflected from the earth ; and a candle- or gas -flame appears continuous, though its incandescent particles are by no means in contact with one another. Perception of Form. The two eyes receive images of different form ; these are blended by a mental operation into a compound image, which education has taught us to associate with xv.] VISUAL PERCEPTION. 517 the distance of the several parts of the object. This is applied in the Stereoscope : two pictures of images taken from different photographic standpoints are formed, one in each eye, and the effect is that of outstanding relief. This may be exaggerated with singular effect where the photographs are taken from stand- points situated at a mutual distance of several feet : mountain scenery is thus brought into perspective. The same exaggerated effect may be observed when a landscape is looked at through a pair of telescopes, parallel but at several inches' distance from one another, the light traversing each being brought into the corre- sponding eye by an arrangement of reflecting prisms. The images in the two eyes may often differ in brightness : when this is the case, there is a struggle between the two fields of view, which causes the impression known to us as that of Lustre ; this effect being especially well marked in the case of metals. CHAPTEE XVI. ELECTRICITY AND MAGNETISM. ELECTRICITY and Magnetism are not forms of Energy ; neither are they forms of Matter. They may perhaps be provisionally defined as properties or Conditions of Matter; but whether this Matter be the ordinary matter, or whether it be, on the other hand, that all-prevading Ether by which ordinary matter is everywhere surrounded, is a question which has been under discussion, and which may now be fairly held to be settled in favour of the latter view. At first sight it would appear that the electricity of an electrified body is a condition of that body itself. When a small piece of resin and a small piece of glass are rubbed together, it is found that after they are pulled asunder, the resin and the glass are in such a condition that they attract one another with a definite and measurable force ; and that this force varies inversely as the square of the distance between them. This attraction across an intervening space is by some held to be due to a so-called Mutual Action at a Distance; but when the bodies are pulled away from one another, work is done upon them which will be restored when they are allowed to approach one another, and it seems probable that this work has been done not upon two isolated bodies mutually acting at a distance, but upon a system which consists of the two bodies together with the Ether between them, which has been stressed by their separation ; the tendency of the two bodies to approach one another is the elastic tendency of the Ether to recover its original condition ; and pheno- mena of electric attraction and repulsion may be explained as phenomena of Ether-stress. Two masses of resin rubbed on glass are found to repel one another ; two masses of glass which have been rubbed with resin also repel one another; in other words, two masses in a similar electric condition generally repel one another. CHAP, xvi.] GENERAL PROPOSITIONS. 519 According to the nature, the size, the dryness, of the pieces of material exposed to mutual friction, and according to some other circumstances, it is found that after friction and separation the force of mutual attraction or repulsion of two electrified bodies varies. One body may thus be more or less highly electrified than another ; it is said to possess or be charged with a greater or a less quantity of electricity. When two electrified bodies, at a mutual distance of one centi- metre, attract or repel one another with a force which balances one dyne, they are each said to be charged with a quantity equal to one Absolute or C.G.S. Electrostatic Unit of Electricity. If one of these bodies, charged with a unit of electricity, be brought to an exact centimetre's distance from a body charged with an unknown quantity of electricity, the force between the two electrified bodies may be measured directly ; and if it be equal to n dynes, the body tested is shown to bear a charge of n units of electricity. Further, if a body bearing m units be brought to the same distance from a body charged with n units, the force between them will be equal to m x n = mn dynes. A piece of glass, after being rubbed with resin, is said to bear a charge of vitreous electricity ; the resin, on the other hand, is said to be charged with resinous electricity. If any body become electrified in any way, it must become either vitreously or resin- ously electrified. Similarly-electrified bodies repel one another ; dissimilarly- electrified bodies attract one another; these statements being, when the bodies are very near one another, subject to an excep- tion hereafter to appear (p. 535). When a jet of water issues from a metallic nozzle connected with an electric machine, the particles of the issuing stream, being similarly electrified, repel one another, and the jet is broken up into spray. When the nozzle has a capillary orifice, the surface tension at the aperture is overcome by the electric self-repulsion, and the liquid rapidly issues as if its viscosity were greatly diminished. If a body charged with resinous electricity and one equally charged with vitreous be brought into contact, the charges of both apparently disappear and the bodies resume a neutral state. Vitreous and resinous electricities are thus found to bear to one another the same relation as positive and negative quantities in algebra, and by a purely arbitrary convention charges of vitreous electricity are said to be positive, and resinous negative. The above statements are comprised within the statement that if / be the 520 ELECTRICITY AND MAGNETISM. [CHAP. force of repulsion between two charges of electricity, these charges being m units in one body and m' units in another, and d the distance between them, f=Jc ^- ; and if our units of quantity be so chosen that k = 1, / = 2 If m and m be both positive or both negative, the product mm' is positive, and the stress is expansive or repulsive ; while, if one of the charges be resinous and the other vitreous, mm' is negative, and the stress is such that the bodies appear to attract one another. When an electrified body presents a charge of m units uni- formly distributed over a superficial area of s sq. cm., its charge wi per sq. cm. is - = : and since the difference of potentials of any \r r J rr two equipotential surfaces is numerically equal to the work done in transfer- ring a unit-charge from the surface of lower to the surface of higher potential, it follows that the work done by or on a unit-charge, on its moving or being moved from distance r to distance r' from a charge Q, is \ T t r ' TT If a body charged with electricity be not free itself to move along the lines of force, we find this most remarkable phenomenon that in a field of force, the points of which corresponding to xvi.] CURRENT. 525 the extremities of the body are at different potentials, the electrical condition of the body tends to travel : one aspect of the charged body that, namely, which looks towards the direction in which the charged body would itself travel if it were free to do so tends to become more strongly charged or to acquire a greater density, while the other aspect tends to become less strongly, or, it may be, even oppositely charged. This redistribu- tion of the electric charge, if it take place, has the effect of equalising the potential throughout the body placed within the field of force. The new distribution once assumed is permanent so long as the field of force which immediately surrounds the body, and which tends to determine a difference of potential between its opposite aspects, remains unchanged ; but while that distribution is being assumed we have a brief Current of electricity. A difference of potential, in whatever way it may be set up within a body, produces a tendency to prompt equalis- ation of potential throughout that body, and thus to the estab- lishment of a momentary current of Electricity ; a permanent difference of potential, in whatever way kept up, tends to produce a continuous current. A lightning conductor is the seat of a continuous current so long as the earth at its base and the air at its apex are at different potentials. Difference of potential is analogous to difference of level or Head of Water in hydraulics ; and when it determines a flow of electricity, it is often called electromotive force or E.M.F. a term which might with advantage be abandoned, and instead of which we shall use the phrase electromotive difference of potential or E.M.D.P. In place of this phrase the reader who has any reason for doing so will easily read the words electromotive force, the objection to which is simply that a difference of potential, like a difference of water-level, is not itself a force, and does not even completely specify the force which determines an electric flow: to determine this the form and the dimensions of the electric conductor must be known, as well as the difference of electric potential between its extremities, just as the dimensions of a water-pipe must be known, as well as the available head of water, before we can calculate the local falls of pressure and the forces producing flow. If two bodies be at different potentials, when they are con- nected by a metallic wire, the charge over them will be readjusted 526 ELECTRICITY AND MAGNETISM. [CHAP. by a momentary current through the wire, and they will come to the same potential. Two bodies are said to be at the same potential when elec- tricity has no tendency to travel from one to the other, even though they be brought into communication by a metallic wire. The earth itself is arbitrarily assumed to be at zero potential : and bodies in such a condition that, when they are placed in contact or in metallic communication with the earth, their electric condition is unaltered, have a potential whose value is equal to this arbitrary zero. The arbitrary or conventional potential or, briefly, The Potential of a point in an electric field of force is, numerically, the number of ergs of work necessary to bring a unit of electricity up to the point in question from a region of nominal zero-potential i.e., from the surface of the earth. Between a positively-charged body within a room and the negatively-charged wall of the room there must lie, in the inter- vening field of force, one equipotential surface which has a Zero Potential, its potential being the same as that of the earth outside the room. Within this equipotential closed surface there is a region of Positive Potential; exterior to it there is a region of Negative Potential. The potential of the inner region is greatest at the surface of the electrified body ; the potential in the negative region is most negative on the surface of the walls. The potential cannot be a maximum or a minimum at any point within a field of force if that point be not upon the surface of one of the conductors whose surfaces bound the field. Conductors and Non-conductors. In the familiar case of a lightning-conductor we see a marked distinction between the conductive copper in which a continuous current of electricity can flow, and the air or an unprotected building which can only be traversed by a disruptive discharge. A conductor is, when a charge is borne by it and retained by it in equilibrium, a sub- stance throughout the whole volume and over the whole surface of which the potential is uniform ; while if inequalities of poten- tial were set up within it, the conducting material of a perfect conductor would offer no resistance to the readjustment of poten- tial by means of a current. A perfect non-conductor or dielectric would, on the other hand, be a substance the different parts of which may, after an electric disturbance, remain, without any process of readjustment and for an indefinite period of time, xvi.] CONDUCTORS AND NON-CONDUCTORS. 527 at potentials differing to any extent. There are no bodies which are absolute non-conductors ; all conduct electricity more or less slowly. There are no bodies which are perfect conductors : all offer more or less resistance to the flow of electricity. Bodies which conduct extremely badly are called Non-conductors or insulators : bodies which offer comparatively small resistance to the passage of electricity through them are in practice called Conductors. When a charged body is placed upon an insulator, such as ebonite, guttapercha, indiarubber, dry glass, sealing-wax, it is said to be insulated ; its potential cannot become equal to that of the earth for a long period of time; it is said to retain its charge for a long period. Dry cold air is one of the best insulators : but even .within dry cold air bodies charged with electricity gradually lose their charge ; a partial vacuum is a good conductor ; a good vacuum is again a good insulator. Ice insulates, water is a bad conductor ; obsidians and lavas insulate when hot ; glass when dry is an insulator, but when very hot is a conductor. A body charged and supported upon a dry-glass stem within a vacuum or a very dry cold atmos- phere will retain its charge for a very long period. The phenomena of electricity present themselves within a conductor only while a current is actually passing through it ; for then only are there any differences of potential within the con- ductor. At other times that is, when there is no current, but a. more or less permanent condition of Statical Equilibrium of the charge electrical phenomena are restricted to the Field of Force that is, to the non-conductor or Dielectric between two com- plementary charges; for within non-conductors alone, not in conductors, can any electrical stress or difference of potential, per- manently or for any length of time, be maintained. If the air had been as good a conductor as copper we would probably never have known anything about electricity, for our attention would never have been directed to any electrical phenomena. Phenomena of electricity in a state of equilibrium, associated with more or less permanent differences of potential and evinced within a dielectric, are said to be Electrostatic ; those evinced in a conductor during adjustment of the electric potential by the passage of a current are said to be Electrokinetic. If electrostatic phenomena be due to stresses in the ether, electrokinetic are due to movements of the same ; and a moment- ary current of electricity in a copper wire is a throb due to release from stress of that part of the all-pervading ether which pervades 528 ELECTRICITY AND MAGNETISM. [CHAP. that wire a throb which is propagated not without influence, as we shall see, upon the ether external to the wire. " Free " and " Bound " Charges. A distinction is frequently /^made between a free and a bound charge of electricity. The former is understood to be a charge borne by an insulated body, and independent of surrounding objects, while the latter is such a charge as is held in position by the presence and attraction of a \charge of the opposite character upon a neighbouring body. In truth, however, all charges are bound charges ; the complementary distribution must be somewhere ; the field of force may be great or small, but it must have its limits. It may be small, as when a little electrified body is suspended within a metal flask ; it may be great, like the field of force between a thundercloud and the earth : in the former case the complementary charge is distributed over the inner surface of the flask ; in the latter it travels about, and is at its densest upon the surface of the earth beneath the travelling thundercloud, or else upon adjacent clouds. Even when an electrified body is placed at an extremely great distance from all surrounding objects, it cannot be held to have a free charge, for its charge is bound by the complementary distribution upon the far-distant objects ; and a particle isolated in otherwise vacuous infinite space, if such a thing were possible, could not be charged with electricity at all, for the complementary charge could, in such a case, have no locus. If the ether be stretched or compressed, it must be stretched or compressed between at least two points, which may be near or far from one another. Bearing this in mind, however, it is undoubt- edly convenient in many respects to permit ourselves the use of such expressions as " a body freely charged with m units of + elec- tricity," and in so doing to omit, provisionally, all consideration of the complementary charge, which is supposed sufficiently distant. Division of Charge. When a conductor charged with elec- tricity is brought into contact with another at a different potential, the electric potentials of the two conductors become equalised ; and if the two bodies be of the same form, size, temperature, and chemical nature, and if they be symmetrically arranged, they will, after separation, each bear a charge equal to one-half of the alge- braical sum of the original charges of the two bodies. This change of distribution involves a readjustment of the lines of force and of the equipotential surfaces throughout the surrounding dielectric, and an alteration of the distribution of the complementary charge over its opposite boundary. XVI.] DIVISION OF CHARGE. 529 When the bodies are unequal in size, etc., or are unsymmetric- ally arranged, the division of the charge between them is not equal. Two similar but unequal spheres are found, after being brought into communication by a long thin wire, which is then removed, to bear charges proportioned to their radii. Capacity. When a conductor has a charge of electricity im- parted to it, the potential of its surface and of its whole volume is raised, positively or negatively (i.e., lowered), as the case may be. When an insulated body requires a charge of C units of electricity to be imparted to it in order to raise its potential by one unit that is, from zero to unity, or from V to V + 1 it is said to have a Capacity of C units. When a body whose capacity is C has its potential raised by the amount V, the Charge of elec- tricity imparted to the conductor is equal to VC units. When a series of conductors whose capacities are C y , C /y , C /y/ , etc., and which are charged to potentials V,, V //} V /y/ , etc., so that their several charges are respectively C,V,, C^V,,, C y// V //y etc. are connected by wire, the potential thereupon assumed by the whole system is equal to The whole charge = ,0, + V,,^, + V,,,C,,, . . . = 2YG The whole capacity C, + C,, + C,,, . . . ~~ 2C. ' The capacity of a conductor is the same whether it be solid or hollow : the merest film of gold leaf supported on a wooden ball has as great a capacity as a solid metallic sphere. Electrostatic stress can only exist within the field of force, the dielectric, which is limited by the surface of the conductor ; beneath this surface it is a matter of indifference what the metallic thickness may be, since within a conductor there can be no permanent difference of potential, no permanent electrostatic stress. The Work spent in charging a conductor is equal to half the product of Q, the Charge imparted to it, into V, the Eise of Poten- tial produced in it. To bring a quantity, Q, of electricity from a place of zero potential to a place of constant potential, V, involves the expendi- ture of QV units of work, by our definition of Poten- tial. To bring a charge Q by successive instalments into a region whose poten- tial is at first zero, but steadily rises as the suc- cessive instalments arrive, o Quantity work must be done which is equal to half the product of the final rise of potential into the whole 2 M Fig.198. 530 ELECTRICITY AND MAGNETISM. [CHAP. charge brought up. In Fig. 198 the small rectangles represent the work done in bringing each successive instalment up to the corresponding poten- tial ; these rectangles increase, and their sum, which represents the total work done in bringing up the whole charge OQ to the final potential OP, is represented by the triangle OQP, whose area = JOQ x OP = J whole quantity x final rise of potential = J QV. The work done in charging a conductor is equal to the elec- trical energy stored up in that conductor ; and since this is equal to -^QV, we see that the energy of an electrified body depends not only upon the quantity of electricity borne by that body, but also upon the potential; just as the potential energy of a mill- pond depends not only upon the quantity of water contained in it, but also upon the average elevation of that water above surround- ing objects. For which reason a mere quantity of Electricity is not a quantity of Energy, and Electricity is not a form of Energy. The energy of a charged conductor is measured by JQV; but this is equal (since Q = CV, where C is the capacity of the conductor) to i-CV 2 or to The energy of a system of connected conductors is equal to JV 2 . 20, or to iQ 2 -^ 20, where 20 is the aggregate capacity of the whole system. Suppose now that two conductors, of which the one is charged to potential V, while the other is at zero potential, and of which the respective capacities are C, and C //5 are placed in metallic communication ; on contact they form a joint conductor whose capacity is (C, + C,,). The energy of the single Q2 Q2 charged conductor was J . ^-; that of both taken together is i - , a smaller quantity. There is therefore an apparent Loss of Energy equal to Q 2 Q 2 Q 2 C C * c; - * C^TC,, = 2c; / .(c / + c // ) or 07/0: of tlie energy of the original charge. If C, = C,,, half the energy of the charged conductor is apparently lost by partial discharge. Wherever there is a readjustment of electricity in the form of a running-down of electricity from a place of high potential to a place of low potential, there is a loss of energy of electrification, just as when a full pond is allowed partly to discharge itself into an empty one, the average level of the whole is lowered, and the energy of position partly disappears, to reappear in the form of heat. In general, where electrified conductors are connected by metallic wires, if there be a current, the potential energy of the sys- tem sinks to a minimum ; heat and if a spark pass light, sound, and mechanical effect being produced. Where the components of an electrified and insulated system are allowed to approach or to recede from one another in obedience to the electric forces, the energy of xvi.] INDUCTION. 531 electrification becomes in part converted into mechanical work, and therefore falls in amount ; while if they be pulled asunder or made to approach against the electric forces, the mechanical work done upon the insulated system from without is converted into the energy of a higher electrification. Induction. When an electrified body, or a number of such bodies, is placed within a hollow metallic -shell (Fig. 199) with which there is no communication except through non-conductors, the shell becomes charged by Induction across the intervening dielectric. If the bodies placed within the shell be positively charged, the inner surface of the shell becomes negatively, the outer surface positively, electrified. The opposite charge thus induced on the inner surface of the shell, the similar charge induced upon its outer surface, and the original inducing charge on the internally-suspended system, are all equal in amount, if the shell completely or even if it very largely surround the elec- trified body suspended within it. Thus the positive and the negative charges called into existence by Induction are together algebraically equal to zero. The distribution of the induced charge on the interior surface of a completely-surrounding shell is such that on external points it produces an effect equal and opposite to that of the interior insulated charge ; the two inner charges therefore produce together no effect upon external bodies, and the induced charge on the outer surface is the only charge which can affect particles situated in the outer air. The two interior charges are bound to each other, for they are of opposite character, and there is a field of force between them; the outer charge is said to be free. The distribution of electricity over the inner and outer surfaces of the shell is, if the shell be spherical, governed by the Fig.200. law that the superficial density o- at any point E is (CE 2 ~ CM 2 ) Q = ~ -- ~ ' F j where C is the geometrical centre of the shell, and M the point' at which the charge Q is situated. Such being the law of distribution, the inner charge Q at M, and the opposite charge - Q on the surface (the inner surface of the shell), produce together no effect on surrounding particles. If the outer surface of the shell be connected with the surface of the earth, the shell and the earth become one extended con- 532 ELECTRICITY AND MAGNETISM. [CHAP. ductor, and the positive charge on the outer surface of the shell is repelled to the most distant region of the earth's surface ; it now blends with and neutralises the negative charge previously borne by the earth in consequence of the original positive electri- fication of the inducing body. As a result of this we have, within the shell, a purely local field of force restricted to the space between the internally -suspended body and the interior surface of the shell, and giving rise to no phenomena outside that cavity. If, on the other hand, the insulated body within be made to touch the enveloping shell, the internal field of force will be destroyed ; but the outer induced charge will remain, distributed over the outer surface of the shell. Two oppositely-charged conducting surfaces thus. separated by a dielectric or limited field of force constitute an Accumulator, or, as it is often termed, a Condenser. The capacity of an accumulator is greater than that of either or both of the con- ducting surfaces of which it is made up. Suppose an accumulator to be made up of an inner conducting sphere (solid or hollow) of radius R, and, concentric with the sphere, an outer shell whose inner spherical surface has a radius r. Let the inner electrified sphere bear a charge Q ; this charge acts as if it were concentrated at the centre and Q produces a potential, V = -jr , which is the same throughout and over the sur- face of the inner sphere. Similarly, the potential at any point within the outer shell, due to the charge Q upon its inner surface is - The outer surface of the shell may or may not have a free charge. In any case, the potential of the joint system is the excess of its internal over its external potential. If there be no charge on the external surface, the external poten- tial is zero. The total internal-potential is then simply the sum of the several potentials + ^- and This sum is equal to Q . The Capacity of si r Rr the accumulator is Quantity -h Potential = Q -r ! Q f r J j- = E =: ; this is greater than the capacities of either of the two component spher- ical-surfaces, these capacities being respectively R and r; for capacity charge Q Q = potential = Q "^ R =: R f r the 8 P here > and Q ^ ^- = r for the shell. The smaller the difference between r and R that is, the thinner the dielectric between the two metallic surfaces the greater is the capacity of the accumulator, and correspondingly, the less will be the potential to which a given charge will raise it. The nature of the dielectric between the plates of the ac- cumulator is not a matter of indifference. It is found and this xvi.] SPECIFIC INDUCTIVE CAPACITY. 533 proves that in the phenomena of electric attraction the dielectric plays an important part that the capacity of an accumulator varies with the nature of the interposed dielectric, and is propor- tional to a constant special to each substance, and called the Specific Inductive Capacity of that substance. A thickness of 3 '2 mm. of sulphur is thus equivalent as a dielectric to a thickness of 1 mm. of air, and the sp. ind. cap. of air being taken as a standard and equal to unity, that of sulphur is 3 '2. Sometimes the sp. ind. cap. of a vacuum, which differs very little from that of air, is taken as unity, in which case the dielectric is the ether itself. All gases have the same inductive capacity whatever their temperature or density. If their pressure be raised, their sp. ind. cap. is also increased proportionately; and conversely, when a gas is employed as a dielectric, charging the plates diminishes its pressure. If a given charge will raise an accumulator in which air is the dielectric to a potential V it will only raise a similar accumulator whose dielectric has V sp. ind. cap. = Jc to the potential r- ; and since in the special case of a con- ducting material used in the place of a dielectric the difference between the inner and outer coats is zero, the sp. ind. cap. of a conductor may be con- sidered infinite, for = 00 The sp. ind. cap. of a dielectric diminishes with the time, and is therefore difficult to measure directly ; and when an accumulator is discharged by- metallic communication set up between its two coatings, its charge does not at once completely vanish, but the condition of the dielectric is apparently very similar to that of a body which, being imperfectly elastic, recovers slowly and irregularly its primitive form and condition after deformation ; and it is curious that the same means shaking, jarring, etc. which facili- tate the return of such a body to its normal condition after a strain, facilitate the prompt and complete discharge of an accumulator whose two coatings are kept in metallic connection. Quartz employed as a dielectric has one-ninth the residual capacity of glass ; Iceland spar seems to have no residual capacity at all, and permits prompt discharge. The dielectric of an accumulator may become double-refracting under the influence of Electric Stress, which tends to dilate it: its optical axis is parallel to the lines of force. Solids slowly, liquids instantly, acquire or lose this condition of strain. The capacity of a spherical accumulator in which the dielectric has a sp. ind. cap. = k is k -^ If the two radii be very nearly equal, this may Jc be taken as equal to r 2 , where t is the thickness of the dielectric ; and this is equal to . surface . This last is the general formula for the capacity t 4<7T of a Ley den jar. 534 ELECTKICITY AND MAGNETISM. [CHAP. The form of accumulator known as a Ley den jar usually consists of a glass vessel lined internally and externally with tinfoiL The inner coating communicates by wire with a smooth metallic knob projecting externally and insulated from the outer coating. By contact between the knob and a charged conductor the inner coating is charged. By induction through the glass there is produced an electrical separation in the external tinfoil. The external surface of this is temporarily connected with the earth. Thereafter there remains a field of force in the glass between the two tinfoil coatings. This may be discharged by establishing a metallic communication between the two coat- ings, the outer tinfoil being first touched, then the inner. A Ley den jar when charged dilates somewhat, and as it expands its capacity increases, and the potential to which a given charge can raise it falls. When discharged it makes a dull sound, and the glass glows at the edges of the tinfoil, while the internal air becomes warm. A submarine telegraph cable is in effect a very long Leyden jar. The copper core is the inner coating ; the guttapercha or other insulator repre- sents the glass ; the outer coating of tinfoil is represented by the protecting iron wire or by the bounding surface of the sea- water. When a charge of electricity is passed into a deep-sea cable, the cable takes some time to become fully charged : it then bears an electrostatic charge upon the surface of its copper core. An ordinary aerial telegraph-wire is, again, but to a less marked degree, a Leyden jar. The inner coating is the surface of the wire itself ; the dielectric is the air ; the outer coating is the surface of the earth. The electrostatic capacity of an aerial wire is small in comparison with that of a submarine cable ; but it is not insignificant. Batteries of Leyden Jars. When a Leyden jar has its inner coating placed in simultaneous metallic communication with the inner coats of a series of uninsulated jars, the whole becomes in effect one great Leyden jar of increased surface, and the jars are said to form a battery connected in Surface. The charge of the first jar being then distributed throughout an enlarged conductor brings it to a reduced potential, and energy is lost in the production of sparks when the battery is charged by the first jar. A series or battery of Leyden jars is said to be charged in Cascade when the outer coat of one jar is connected by metal with the inner coat of the next, and so forth, while a charge is imparted to the inner coating of the first. The difference of potential between the inner coating of the first jar and the outer coating of the last is distributed between the jars of the battery, and thus the risk is diminished of any of the jars being destroyed through an excessive difference of potential in any one jar causing a spark to pass and perforate the glass. The charge of the whole system is only equal to that V V of a single jar, and the difference of potential in each of n jars is - ", where V, and V o are the potentials of the first and the last coatings ; whence the energy of the whole ( = charge x potential-difference) is the nib. part of the energy of a single jar loaded with the same charge as the battery. xvi.] INDUCTION. 535 If the conductors surrounding an inducing charge do not completely enclose it, the charges induced upon them are each numerically less than the inducing charge, and the sum of those of each kind is also numerically less than that charge. The effect of induction is seen when an electrified body such as a glass rod rubbed with a dry silk -handkerchief is brought into the neighbourhood of light bodies suspended or floating in the air. Over each of these bodies there is a separa- tion of electricities ; the aspect nearer to the inducing body is charged with electricity of the opposite kind, and is attracted ; the farther aspect is charged with electricity of the same kind, and is repelled, but to a less extent, because it is more distant ; its charge not being " bound " is, besides, more readily dispersed into the surrounding air. On the whole, these light bodies are attracted. If they come in contact with the inducing body, they acquire a part of its charge, and are thereupon repelled. As another effect of induction we find that while two similarly-charged bodies at the same potential within the same field will always attract one another, yet if they be not at precisely the same potential, the one of higher potential will, by its presence, alter the distribution of electricity over the other, the weaker, in such a sense that the weaker one may even become oppositely charged over the nearer aspect, and the attraction of the more highly-charged body for this side of the weaker may prevail over its repul- sion of the further side ; and 011 the whole, two such bodies will, if they be placed at a sufficiently small distance and if no spark pass between them, attract one another. In a certain intermediate position there will be unstable equilibrium, and at all greater distances there will be repulsion. When a conducting body is brought into the neighbourhood of a system of insulated and charged conductors, the energy of that system falls, for the interposed body causes by its presence a redistribution of the charge of the system ; and such a redistri- bution of the charge causes a fall of the potential and of the energy of the system. If the body introduced be a dielectric, the effect produced is similar but smaller. Electric Screens. A conducting sphere surrounding an in- sulated electrified body and connected with the earth will, as we have seen, shelter an external particle from the inductive action of the enclosed electrified body; and conversely it will shelter the internal electrified body from the distribution-disturbing anu potential-lowering influence of the outside particle. A screen of perforated tinfoil or a cage of wire gauze has nearly an equal effect : such screens are used to protect delicate instruments from the inductive effect of external electrified bodies. 536 ELECTRICITY AND MAGNETISM. [CHAP. The place of an enveloping sphere may be taken by a plate of metal connected with the earth. If the diameter of this be infinite, or practically, if it be very great as compared with the distance between the electrified and the protected particle, the screening action will be perfect. In Fig. 201 A is an insulated body positively charged by a galvanic battery or a frictional electric - machine ; D is a large metallic screen ; B is a metallic body con- nected by a wire with the earth; this wire passes round the magnetic needle of a Galvanometer, G ; the screen D is suddenly removed : there is a sudden separation of electricities in B: a positive charge escapes round the galvanometer and deflects its needle by an instantaneous twitch. The phenomena of electricity in equilibrium are very similar in their mathematical aspect to those of the steady flow of Heat ; equipotential sur- faces represent isothermal surfaces ; lines of force represent lines of flow of heat ; specific inductive capacity takes the place of conductivity. Again, the calculation of the variation of the force throughout a dielectric field resembles very closely that of the distribution of the flow in a steadily- flowing mass of fluid ; just as the stream lines in a field of liquid-flow may be held to exert lateral pressure upon one another, so do the lines of force in an electric field laterally repel one another, as is specially manifest at the surface of a conductor, where the elements of charge repel one another ; in each of the tubes of force or tubes of flow the product of the force or of the flow into the cross-sectional area is constant (Law of Continuity in Hydro- dynamics) : and the energy per unit of volume in a field of force or of flow is at each point numerically equal to the electrostatic or hydrostatic pressure per unit of area at that point. In Faraday and Maxwell's theory of Ether-stress, the flow of charge across an electrified surface is insisted on : this flow occurs whenever a separation of electricities takes place. When a thin insulated sheet of tinfoil is exposed to the inductive influence of a charged conductor, there is a separation of positive from negative charge across the conductor influenced, and on each side there is a charge induced whose density is o-. The quantity of electri- city thus induced to flow in either direction is, over any given area A, equal T|1 to (T . A : this quantity is equal to A, since F = ITTCT. If any other di- 4?r electric than air intervene between the inducing charge and the conductor acted upon, the quantity of fl o w, the so-called Electric Displacement, F is k x A. A conductor offers no permanent resistance to the displace- ment of an electric charge through it, and as long as there is maintained between the extremities of a conductor a permanent difference of potential, so xvi.] ELECTRIC STRESS. 537 long will the electric displacement produced be continuously relieved by an electric flow or Current; but in a non-conductor the extremities may remain under a permanent difference of potential, for the Electric Displacement^ the flow set up in the first instant of exposure to electric stress, is arrested by a certain Electric Elasticity of the Dielectric, which, being repre- i j i. .L-L f i' electric stress force acting across each unit of area sented by the fraction - = - s - electnc strain quantity of flow across each unit of area = F -7- f /c . \ 3L, is inversely proportional to the specific inductive capacity of the dielectric. Dimensions of Electrostatic Measures. The Absolute or C.G.S. unit of Quantity of electricity is that quantity which, placed at a certain distance from a similar and equal charge, repels it with a certain force. The dimensions of a force are [MLT~ 2 ] (see p. 54) ; force between two electric [Q] x [Ql quantities, r^ 2 k = F = [MLT~ 2 ] ; whence the dimensions of a quantity are[Q] = [MiLIT- 1 ]. Density o- ; quantity of electricity on a given area: its dimensions Difference of Potential, E: quantity .of work required to move a quantity of electricity from one point to another : its dimensions are Work done = j- ML 2 T _ 2 -j _=_ [MILiT-i] = [M*L*T-n L Quantity moved Electric Force, F: the Force at any point in a field is the force acting upon a unit quantity of electricity placed there : its dimensions are - [MLIT - 1] + [L] - [ML - *T - ij Capacity, C : the Quantity necessary to produce a certain rise of potential : its dimensions are - - - - = [M1L2T" 1 ] -f- Difference of potential [M*LiT] = [L], a Length simply. The relative capacities of conductors of similar form are simply proportional to their diameters. Specific Inductive Capacity, k: the ratio of two quantities of electricity developed under similar circumstances : it is therefore simply a Number : its dimensions are accordingly 0. OBSERVATION OF DIFFERENCES OF POTENTIAL. Observation of Differences of Potential is effected by means of instruments called Electroscopes and Electrometers; the former indicate the nature, the latter mea&ure the amount, of differences of potential. Gold-Leaf Electroscope. A glass flask with a vulcanite stopper : through the stopper passes a metal rod surmounted by a metallic sphere or 538. ELECTRICITY AND MAGNETISM. [CHAP. plate and terminated below by a pair of freely-suspended strips of gold leaf. If the metallic part of the electroscope be charged by contact with an electrified body, the gold leaves, becoming similarly charged, repel one another, and diverge, slightly if the charge be feeble, widely if it be great. The electroscope may also be temporarily charged by induction: a + electrified body brought into the neighbourhood of the sphere or plate causes that sphere or plate to become negatively, while the more distant gold leaves within the flask are positively, charged. If, while the electroscope is electrified by induction, its upper extremity be momentarily touched by the experi- menter, the gold leaves collapse, for their charge escapes to the earth : the plate or sphere, however, retains its charge, and when the inducing body is removed, the opposite charge borne by the sphere or plate becomes free to distribute itself over all the metal of the electroscope, and the leaves again diverge, for the instrument is now permanently charged. If the electroscope be permanently charged, the approach of a body similarly charged will cause a further divergence of the leaves : the approach of a body oppositely charged will cause the leaves to repel each other with less force : whence the nature of the electrification of a given charged-body can be ascertained. The deficiencies of the electroscope are : that its indications are quali- tative, not accurately quantitative ; and that the glass does not thoroughly screen the gold leaf from the direct inductive action of external charged- bodies. In order to obviate the latter defect, the inner surface of the flask is sometimes lined with perforated tinfoil, or the whole is surrounded by a cage of wire gauze. The gold-leaf electroscope is a development of earlier instruments, in which straws, plain balls of elder-pith, or gilt pith-balls, were employed. In the discharging electroscope the gold leaves when they diverge come in contact with two metallic uprights which communicate with the earth : they are thus discharged and collapse, again to be charged : the number of oscillations of the gold leaves affords a rough measure of the quantity of electricity borne by a conductor which is discharged to earth through such an electroscope. Peltier's Electroscope. A vertical brass ring, insulated ; attached to its inner circumference, at the lowest point, a vertical pointed-rod ; on the pointed rod is poised either a magnetic needle or else a metallic rod whose direction is determined by a small magnetic needle attached to it. The whole is turned round a vertical axis until the ring and the poised metallic rod lie in the same plane. If the ring be charged, the charge is shared with the poised metallic mass, and the ring and the poised mass repel one another ; the latter swings round until the force of electrical repulsion is balanced by the tendency of the magnet to point to the magnetic north and south. This instrument may, by imparting to it a series of successive known charges, be so adjusted as to act as a electrometer. Bohnenberger's Electroscope. Two vertical dry piles (p. 553), the one with its + pole, the other with its pole upper- most; between these oppositely-charged poles there is a field of force within which a strip of gold leaf is suspended. If un- charged this strip hangs vertically ; if charged it is repelled by one pole and attracted towards the other. XVI.] BOHNENBERGER'S ELECTROSCOPE. 539 Fig.202. Instead of two piles the two extremities of one and the same dry pile may be used to make such an electro- scope. In Fig. 202 AB is a dry pile whose poles are connected with the metallic plates C andD, between which there is thus formed a field of force in which the gold ^ A leaf E is suspended. On the same principle the Quadrant Electrometer of Sir William Thomson is based. In Fig. 203 the two opposite Fig.203. quadrants A and D are connected with one another by wire, but are insulated from B and C. A and D are thus at the same potential, while B and C are also at the same potential, a potential which may differ from that of A and D. A and D may be brought to the potential of the earth by means of a wire con- nected with gas or water pipes ; B and C may be brought to the potential of any given object by connecting them with it by means of a wire. The quadrants A, D and B, C are thus at different potentials, and a metallic needle an aluminium needle of a flat dumb-bell shape will, if it be suspended symmetrically over the quadrants, and if it be kept charged by constant connec- tion with one coating of a Leyden jar, which may be replenished when necessary, indicate, by the amount of torsion which it can 540 ELECTRICITY AND MAGNETISM. [CHAP. impose on a system of suspension consisting- of two parallel threads, the nature and approximately the amount of the dif- ference of potential between the two pairs of quadrants, and therefore between the earth and the object whose Potential is to be measured. If the quadrants be made hollow, and the needle suspended within them, the arrangement is better adapted for electrometric purposes. The amount of deflection of the suspended needle may be ob- served by connecting with it a very light mirror, upon which a very narrow beam of light shines ; as the needle is deflected the beam of light reflected from the mirror is deflected through an angle twice as great as that of the deflection of the mirror ; and the beam of light, if received upon a distant scale, thus acts as a weightless pointer. * Upon the scale the deflection of the spot of light may be read off ; that deflection is proportional to the tangent of twice the angle of deflection of the mirror : for small angles it is nearly proportional directly to twice the angle (Fig. 161). Coulomb's Torsion Balance. A long vertical slender hard wire or silk fibre, AB, Fig. 204, by which there is sus- pended in a horizontal position a thin counterpoised rod of glass or shellac, CD, which bears at one of its extremities a little gilt sphere D. In one position of the suspending wire the gilt sphere D comes into contact with a sphere-ended metal rod EF : this rod projects through the walls of the glass case in which the whole is encaged, and is therefore insulated. This metal rod terminates externally in a sphere E, Fig.204. which may be charged by contact with an electrified body, such as a proof-plane. A proof-plane is a small metallic disc provided with an insulating glass or ebonite handle. It is used by laying the disc upon the surface of an electrified body : when the disc is withdrawn it bears with it a charge propor- tional to the charge previously borne by that part of the surface of the electrified body with which it had been placed in contact : it is then made to touch the sphere E of the torsion balance. EF being charged, ^ E the two spheres, F and D, when they come in contact, * become charged with electricity of the same kind and repel one another : they do this until there is equili- brium between the electric repulsion and the torsion of the suspending wire AB. The proof-plane may be used directly in the place of EF ; and instead of a proof plane a proof-sphere may be used when the curvature of the body whose charge is to be examined is but small. Different charges may be compared by comparing the amounts of torsion XVI.] COULOMB'S TOKSIOX BALANCE. 541 necessary to bring the two mutually-repellent bodies, D and F, to equal dis- tances. A preliminary charge is given to the ball D ; a charge Q of the same kind is imparted to F, or brought in by a proof-plane or proof -sphere. Let the repulsion between Q and the charge on D be such that the suspended horizontal fibre makes an angle FBD of 10 with that position in which D is in contact with F, while the upper end A is twisted in the contrary direction so as, as it were, to tend to force F and D together through an angle of 410 ; the total torsion of the wire AB is 420. Now remove the charge Q and substitute a charge Q' ; the index at A indicates 95 of rota- tion there when D is in its former position : the total torsion of the wire is now 105. The charges Q and Q' are proportional to the torsions which their repulsions balance ; and Q : Q' : : 4 : 1. The Absolute Difference of Potential between two bodies may be ascertained by measuring the attraction between two metallic plates which are respectively connected by metallic wires with the two bodies in question. In Fig. 205 AB is a galvanic battery, the ex- tremities of which are permanently at different potentials : it is desired to find the difference be- tween these potentials. Connect A and B with the plates C and D. The field of force between C and D is uniform at its centre. D is fixed. But E, the central part of C, is movable. The attrac- tion between E and D may be measured by observing the dis- tortion of a spring which tends to pull E upwards while the electrical 'attraction tends to pull E downwards, this observation being made when the distance of D is so adjusted that the lower surface of E is flush with that of C. It is sometimes found ad- vantageous in the use of instruments of this kind to connect D alternately with B and with the earth : the spring tends to become differently distorted in the two cases, and in order to render its distortion equal in both cases the distance of D must be varied. The amount of approximation or retraction of D may be measured by a micrometer-screw. The spring which keeps up E against the attraction of D may be replaced by transforming E into one pan of a delicate balance, of which the other pan may be loaded with known weights. The potential of E is V E ; that of D is V D : the difference of potential to be measurecf is V E V D : the force in the field of force between E and D V V is , where t is the thickness of the dielectric ; it is also uniformly 542 ELECTRICITY AND MAGNETISM. [CHAP. equal to 47ro-. The total attraction between the two plates E and D is equal to the sum of the attractions of the plate D for every little element of the surface of E. The attraction of a large plate of superficial density o- for a unit of quantity placed opposite its centre is 27nr (p. 168, 8) ; for the whole surface of E, a surface of area S and superficial density or, it is STTO- x So- = V V V V 2irSo- 2 . But since 47rcr = as above, o- = ; and there- t 4-rrt fore the Attraction = 27rS-> 1 O O OO O ~0~ o 10 8 CO o o o o o o o o o o o I-H 10 O O O i-H CO O> CO O O < 564 ELECTRICITY AND MAGNETISM. [CHAP. The numbers in the above table have (with the exception of those for the last four substances) been calculated from the data of the authorities named, on the assumption that a rod of mercury 1 metre long and 1 sq. Q C O mm. in section has a resistance of 90()> oooooo,000000 C ' G ' S ' electrostatic units = -953 Ohm; or, in other words, that the Ohm, the British Association or practical unit of resistance (the resistance of a certain standard wire, which is indirectly intended to represent the 900,000,000000th part of a C.G.S. electrostatic unit of resistance), is the resistance of a column of mercury 1 sq. mm. in section and 1 '0493 18 metre in length. If, in the report of the Committee of the Paris Electrical Congress of 1881, appointed to consider this matter, it be declared that any other value of the Ohm (as measured in lengths of mercury-column) is the true one, then the absolute values in the first two numerical columns of the above table will have been found to be erroneous, and will all have to be proportionately corrected ; but the relative values in the third column will remain. In the meantime, Lord Kayleigh and Mrs. Sidgwick find that the value of the British Asso- ciation Standard Ohm is 0*98651 times its intended value ; and Messrs. Glazebrook, Dodds, and Sargant assign to it the value 0*986439 times the theoretical Ohm ; while Joule assigned to it the value 0-9870. From Lord Rayleigh and Mrs. Sidgwick's measurement it follows that the resistance of a metre of mercury 1 sq. mm. in section is not 0*953 but 0*9413 Ohms. If the table be read without the multiplier or divisor, i? 2 , it then expresses the specific resistances and conductivities in another system the Electro- magnetic system of absolute or C.G.S. units, from which the Ohm and the Volt are primarily derived, the Ohm being 10 9 electromagnetic units of resistance, and the Volt 10 8 electromagnetic units of potential - difference. This system depends upon the laws of Magnetism, afterwards to be explained. The conductivity of metals decreases, that of some bad con- ductors or insulators increases, with their temperatures : a heated wire or 'dynamo-electric machine increases the resistance in the circuit of which it forms a part. Very roughly, the resistance of a metallic conductor is proportional to its absolute temperature. Alloys are in general worse conductors than the arithmetical con- sideration of their percentage composition and the conductivities of their component metals would lead us to expect. There is a broad resemblance between the conductivities of metals for electricity and for heat : the best conductors of the one are in general the best conductors of the other ; and in both cases alloys offer a relatively-high resistance. The series are, however, not identical. Variable Conductivity. Conductivity varies not only with varying temperature, but also with varying tension, torsion, or pressure. In powders or porous material, such as metal filings, platinum sponge, charcoal, it in- creases with the pressure ; and if the pressure vary within small Kmits, the variations of conductivity follow and are proportional to the variations of pressure. This is the principle of the Microphone. In such materials Heat xvi.] CONDUCTIVITY. 565 raises the internal pressure and therefore the contact, and this modifies the amount of resistance and the heat produced within the conductor : this last itself affects the conductivity, as in the Tasimeter, which detects changes in temperature by the variation of a current passing through a rod of carbon fixed between metallic supports, and exposed to varying temperatures. Selenium varies in conductivity with its state of aggregation, its temperature, the length of time during which a current has been passing through it ; and crystalline selenium, when acted upon by light (especially the yellow and the red), and to a less extent when acted upon by dark rays, increases in conductivity : in the case of very bright sunlight this increase being some- times even tenfold. Light of variable intensity produces corresponding and rapidly-responding variations in the conductivity of the crystalline selenium upon which it may fall a fact utilised in the construction of the Photophone. Metals, unlike selenium, become worse conductors as the temperature rises ; but Siemens asserts that at 210 C. selenium changes its character and comes to act like a metal. Reduced resistance and reduced length of a Conductor. This may be explained by a few numerical examples. We sup- pose the unit of resistance to be the Ohm, as above defined, the resistance of freshly-distilled mercury in a column of 1 sq. mm. section and 1*049318 metres in length. 1. What length of soft copper -wire of 1 sq. mm. sectional area will give a resistance equal to one Ohm? 1*049318 x 61*70 = 64-7429106 metres. The figure 61*70 is taken from the table of conductivities above. The Eesistance of 64*7429106 metres of copper is thus equal to that of 1*049318 metres of mercury: the Eeduced Length of 64*7429106 metres of copper is 1*049318 metres of mercury. 2. What will be the resistance of a column of mercury 100 metres long and 1 sq. cm. in section ? It will be equal to that of a column of mercury 1 sq. mm. in section x 1 metre in length multiplied by I = 100, and divided by s = 10 2 . It is therefore *953 Ohm. Its reduced length is 1 metre of standard mercury- column. 3. What will be the absolute resistance, and what the resist- ance in Ohms, of 1 metres of platinum wire whose diameter is millimetre ? Its sectional area s = irr 2 = TT \-g7jJ sq. cm. =: sc l* cm * ^ l en gth ^ = 100,000 cm. ; its specific resistance r is 14752*04 -f- v* ; the ^Resistance of the wire is ^_ 100000x 14752-04 6400 _ 3,005,320,000000 S V 2 7T V* C.G.S. electrostatic units, or 3005*32 Ohms. 566 ELECTRICITY AND MAGNETISM. [CHAP. The intensity of a current is measured by a Galvanometer (pp. 620, 633), round the magnetic needle of which the current is passed : in the Tangent Galvanometer the tangent of the angle of deflection of the needle is proportional to the intensity of the current. The intensity of a current is equal throughout all parts of a circuit in which there is a steady flow. A magnetic needle is equally deflected when "brought into the neighbourhood of any part of the circuit, whether the circuit be locally composed of solid, of liquid, or of heated or rarefied gas. The practical unit of intensity is the intensity of that current which is produced in a conductor whose total resistance is 1 Ohm ( = AAAAAA y uu,uuu, uuuuuu C.G.S. electrostatic unit), when there is kept up between its extremities a potential-difference which constantly amounts to one Volt, or ? C.G.S. electrostatic unit. 1 E 1 Volt 300 C.G.S.E.S. unit SinCe I = R = FOhn^ ' C.G.S.E.S.unit 900,000,000000 the practical unit of intensity, the Ampere, is equal to 3000,000000 C.G.S. electrostatic units of intensity. In a current whose intensity is one Ampere, the practical unit of quantity, the Coulomb, passes any given section during each second : the Coulomb is thus equal to 3000,000000 C.G.S. electrostatic units of quantity. Electrical engineers have adopted the Ohm, the Volt, etc., as means of practical measurement. The Ohm and the Volt in electrical workshops are not abstract calculations, but standard wires and standard batteries (or multiples or fractions of these), by comparison with which the resistance or the E.M.D.P., the so-called electromotive force, of any given combination of materials may be relatively measured. Dimensions of Electrostatic Measure. Intensity a quantity passing per second : .. -gj = [MHJT-i] - [T] - [MJLIT-* p . . T, [El Resistance: R = !=J = [MiLiT- 1 ] -^ [MJLiT- 2 ] = [TL- 1 ]. L 1 ] T? ["TH FT 21 Specific Resistance: r = -7- = kJ f^-4 = [T]. I [Lj [LJ Specific Conductivity : c = = ryp; The above dimensions are based on the assumption that the Quantity of electricity in a current is the same thing as the Quantity of an electro- static charge : they are therefore called Dimensions in Electrostatic Measure. P) xvi.] FALL OF POTENTIAL IN A CURRENT. 567 Fall of Potential in a Homogeneous Conductor of uniform thickness. During the maintenance of a steady current one end of a homogeneous conductor is at a higher, the other at a lower potential, and between these points the fall is gradual, so that intermediate points are at in- termediate potentials. Fig. 215 shows that if the length of the conductor be repre- c sented by CZ, the end C con- nected with the positive pole of the battery is at a potential which differs by (P)P' from the potential of the end Z, connected with the negative pole. The Fall is steady, and depends (1) upon the difference of potential between the ends of the conductor, and (2) upon the length of the conductor; it is measured by the Slope of the line PP 7 , the amount of fall of potential per unit of length. It is difficult with short pieces of wire to observe the different potentials at different parts of the conductor ; but when long wires are used, or when the current is made to pass through a column of water, electroscopes may then be attached to different parts of the wire or of the column of water, and will, if the battery and conductor be not insulated, show that the conductor is, near C, at a positive potential, that towards the mid-point this diminishes, that the midpoint is a point of zero potential (the potential of the earth), and that as we approach Z we find the potential increasingly negative. The same diagram represents for a uniform conductor, such as a wire, the variations in the density of the superficial charge borne by the wire, and in virtue of which the potential falls : the potential within the wire varies as its superficial density, and the equipotential surfaces within the wire are equidistant and approxi- mately plane. In conductors of less simple form the relation between the superficial charge and the internal equipotential surfaces is more complicated than this. Resistance in a Heterogeneous Conductor. When a con- ductor is made up of a succession of conductors which, on account of their differing materials or conditions or thicknesses, present different resistances to the current, it may become necessary to consider each conductor as reduced to an equivalent length of a standard conductor, such as a column of mercury one sq. mm. in cross-section. For example : a current passes successively along (1) a metre of mercury 1 sq. mm. in section; (2) 10 metres of 568 ELECTRICITY AND MAGNETISM. [CHAP. mercury 1 sq. cm. in section ; (3) 1 mm. of pure water 1 cm. in section; (4) 6 5 '4 6 metres of soft copper- wire 4 sq. mm. in cross- section : what is the total resistance of this combination ? We must reduce all to a common term, to reduced lengths of our standard mercury- column. Then above (1) is equivalent to a metre of such a column, (2) is equivalent to -^ metre, (3) to 140,845 metres, and (4) to ^ metre of such a mercury column; and the whole resistance is that of 140,846*35 metres of the standard conductor. In a galvanic circuit we have to consider two sets of resistances : those internal to the cells, the internal resistance, R 4 ; those in the conducting p media, the external resistance, R e . Then Ohm's Law is I = p p Let n cells be arranged, copper to copper, zinc to zinc ; the E.M.D.P. of the combination is the same as that of one cell, and = E Volts ; the internal resistance (the combination being virtually one cell of ?i-fold surface) is Hi Ohms ; the external resistance is unaltered. The intensity is therefore E ttE If the internal resistance be extremely small in comparison with the Tfl Tjl external, R, may vanish from this expression ; then I = -=r- = ~- Amperes, and there is found to be little advantage in the use of many cells ; but if the -pi external resistance be extremely small, the intensity becomes =- and the H t arrangement in surface (or " in quantity ") is the best If n cells be arranged in file, copper to zinc, the E.M.D.P. is riE Volts ; the internal resistance is nil, Ohms ; and the external, as before, R e Ohms. TJie intensity is now nR , R Amperes. This arrangement of cells in series (or " in tension ") is the best when the internal resistance is extremely small in comparison with the external ; for then, R, vanishing, the intensity is nE -p- Amperes ; while if the external resistance, on the other hand, be exceed- E F ingly small in comparison with the internal, the intensity is -=- = =r- , ns\ t K t ~F which differs but little from , the intensity of the current produced by one cell. * + K For extremely great external resistances, then, arrange in series ; for extremely small external resistances, arrange in surface. When neither the internal resistance nor the external can be considered as vanishingly small the one in comparison with the other, the best arrange- ment is to unite cells, ab in number, into a series of 6 each : in each series of 6, the b cells are placed side by side, copper to copper, zinc to zinc ; then a such series are arranged in file, the copper terminal of each series being con- xvi.] RESISTANCE OF CELLS. 569 nected with the zinc of the next. In this way we virtually make up a large cells, each of 6-fold surface. In each of these virtual large cells the E.M.D.P. is E Volts ; the resist- ance is -rth of r Ohms, the resistance of a single celL Now couple a such large cells ; the E.M.D.P. of the combination is aE Volts ; the internal resistance of the whole, R i? is equal to a x T Ohms ; the intensity of the current produced is aE aE E r ~ a?r ~ ar R. a T + R, - + R + - b n n a Amperes, where n = db. The denominator of the last fraction is the least possible, and the value of the intensity consequently the greatest, when - = -r . When the intensity is greatest, R, is thus equal to y , or the external resistance is equal to R f , the internal. If the external resistance be equal to abr, and still more if it be greater than abr, the problem of the most advantageous arrangement of the cells in rank and file becomes an insoluble one, and the cells must be arranged in series. Problem. Sixty Grove cells, in each of which the resistance is *6 Ohm, are at disposal : a resistance of 10 kilometres of soft copper- wire of 4 mm. diameter is to be encountered ; what is the best arrangement of the cells ? The ex- ternal resistance, R e , is that of 1,000,000 cm. of copper wire of cross-section 125664 sq. cm. and relative conductivity 6170 : this is equal to 12-29228 Ohms. Now in the equation R e = -^-r = -^-r, R e = 12*29228, r = '6, n = 60 ; whence a = 31 '8. The nearest feasible number corresponding to this value of a is 30 ; and the best arrangement is the division of the 60 cells into 30 virtual double-surface cells. If the external resistance be that of one kilometre of such wire, a being found equal to 11 '08, the best arrangement is 12 series set in tension, and each containing 5 cells joined in surface. The Fall of Potential in a Heterogeneous Conductor. If we draw a diagram, setting out on a base-line and using as abscissae the reduced lengths of the several successive conductors which make up a heterogeneous conductor, and if for a moment we let drop from view the local differences of potential set up by contact of different materials, then the line of potentials slopes uniformly down from one end of the heterogeneous conductor to the other end, and from such a diagram we may find the total fall of potential within each component conductor. Fig. 216 very diagrammatically represents the fall of potential in the composite conductor specified above. AA' is the potential at the junction 5VO ELECTRICITY AND MAGNETISM. [CHAP, of the slender and the thicker column of mercury, BB r that at the one surface, CC' that at the other surface of the water, 00' and DD' the terminal potentials. Mercury If now we follow this up with another diagram in which the real lengths of the conductors are supposed to be represented, we find a remarkable appearance presented by it. The potential-line, which indicates the successive falls of potential, is represented by the line O'A'B'C'D' in Fig. 217. The fall of potential is ex- Pig.217. C' D ceedingly rapid in the bad conductors, for bad conductors keep up a great difference of potential within their substance ; and the whole fall of potential is distributed among the component con- ductors, to each according to its Reduced Resistance. If there be local differences of potential within the conductor, these must be added to or subtracted from the total fall of potential for which the conductor has to provide. Let AB be a conductor, of which one half consists of copper wire, the other half of zinc wire, of an equal thickness, and let its extremities be kept at potentials which differ by 2 AX. In Fig. 218 AC is the Fig.218. '- F reduced length of the copper wire, and CB ( = ^ AC ) \ I / * ) / the XVI.] FALL OF POTENTIAL IN A CURRENT. 571 reduced length of the zinc wire. Between the copper and the zinc there is a rise of potential represented by DE, which makes the slope of the line of potentials steeper throughout the con- ductor. To BX' add X'F, which is equal to DE : connect X and F by a dotted line. Of this the portion XD represents the fall of potential within the copper : the sudden rise of potential at D brings the line of potentials up to E, whence it is continued parallel to XF, along the course EX', arriving at the terminal potential X 7 . From this diagram another may be constructed in which, instead of the reduced lengths AC and CB, the correspond- ing true lengths may be represented and the corresponding true slope of the potential-line found for each. If we take as another example of a composite conductor the whole circuit of a single one-fluid galvanic cell, with its connecting wire, we find that the corresponding diagram presents, when the conductors are represented by their reduced lengths, a form such as that indicated in Fig. 219. The zinc is negative, the copper positive, relatively to the fluid; and at the N x\v Fig.2i9. junction of copper and zinc the potential takes a sudden rise, for the zinc is positive at the point of contact with the copper. "We might again reduce this dia- gram to another in which the reduced lengths of the different parts of the circuit would be replaced by their true lengths, and the true slope of the potential-line found for each. Flow through large Conductors. If a conductor be very wide in comparison with the wires leading to and from it, the current widens out, and no part of the conductor is free from equipotential surfaces and lines of flow. If it be practically infinite the resistance offered by it depends on the radius of the wires or plates connecting it with the battery, and on the specific resistance of the conductor itself : not on the distance traversed by the current in the wide conductor. Some hold this to explain the fact that the earth itself can be used in 572 ELECTRICITY AND MAGNETISM. [CHAP. telegraphy in place of a return wire ; others hold that the earth in that case does not directly act as a conductor, but as a means of bringing the two plates embedded in it, and connected with the extremities of the single wire em- ployed, to zero potential. In Fig. 220 the battery B is connected with two Fig.220. galvanometers, G and G', by a long telegraphic wire interrupted at A : at K there is a key by which contact may be made or broken. CD is a wire, the continuity of which may be broken by another key. When C and D are connected, and connection made at K, both galvanometers are deflected. If, however, the connection CD be broken, and connection be suddenly made at K, the galvanometer G is alone deflected : the earth between E and E' does not simply replace the wire CD between C and D. On the other hand it is beyond doubt that currents do run in the earth's crust. A telephone, part of whose circuit runs in the straight line joining two telegraph stations, will pick up signals from the earth-currents. Fig.22i. Derived Currents. When a current in its course finds the conductor to divide and reunite, it divides into portions which run along the several paths open to it. In Fig. 221 the current arriving at A divides into two moieties ; if the two paths be equal in their re- sistance, these moieties will be equal. If the resistances be not equal, the current passing along each branch will be inversely proportional to the re- sistance, for the difference of potential between the extremities is the same for every branch, and in each branch the product of the intensity into the resistance is equal to the difference of potential. The double path acts like a single conductor whose resistance is equal to 1 J_ , 2_> where r' and r" are the resistances of the two branches. The r' + 7' conducting power of the double path (the reciprocal of its resistance) is the sum of the conducting powers of the two branches ; these are respectively xvi. 1 DERIVED CURRENTS. 5*73 and ; their sum is f + , J , and the resistance of the double path Kirchhoff's Laws. I. Where a steady current branches, the quantity of electricity arriving by the single wire is equal to the quantity leaving the junction by the branches. The algebraical sum of the intensities of the currents passing towards^Sfpassing from the junction is equal to zero ; 21 = 0. II. In a metallic circuit comprising within it a source of permanent difference of potential E, the products of the intensity of the current within each part of the circuit into the correspond- ing resistance are together equal to E ; 2 (Ir) = E. In a metallic circuit in which there is no source of permanent difference of potential, E= 0, and 2 (lr) = 0. This law applies to each several mesh of a wire network as well as to a single metallic loop, and it holds good even when an extraneous current is passed through the loop. Wheatstone's Bridge. In Fig. 222 there is represented an arrange- ment of conductors known by this name. The respective resistances, intensities, and directions of the current are indicated in that figure. Kirch- hoff's Laws give us the rela- tions between these. Law I. shows that at A, I (the inten- sity in the wire CuA) = Ij 4- 1 4 (1); that at B,: (2) ; and that at C, Fig.222. I 5 + I 2 + I 3 = I (3). Law II. shows that within the loop CZnCuABC, in which there is a source of difference of potential E, the E.M.D.P. of the battery, E = IR + 1^ + I 2 r 2 (4) ; while within the loop ABD there is no galvanic cell, E = and I 1 r 1 I 5 r 5 - I 4 r 4 = (5) ; and similarly within the loop BCD, I 2 r 2 - I 3 r 3 + I 5 r. = (6). From these equations the value of I 5 may be shown to be equal to zero, when r x : r 2 : : r 4 : r 3 , or as regards resistances, AB : BC : : AD : DC. To compare the resistances of two conductors, one is placed between A and B, the other between B and C ; then the end of the wire BD is moved along AC until the galvanometer G gives no deflection. At that moment the resistance of the conductor in AB is to the resistance of the conductor in BC as the length of AD is to the length of DC. A scale under the wire AC enables this last ratio to be read off. If one of the conductors compared say that in AB be a wire of known resistance, a Standard Resistance -Coil, the 574 ELECTRICITY AND MAGNETISM. [CHAP. Fig.223. resistance of the other may be absolutely measured. If the resistance in AB be exchanged for a tenfold resistance, the value of the resistance in BC will seem to be numerically diminished to a tenth, and thus resistances ten times as great can be measured by being placed between B and C. In this way the range of the instrument can be increased. When the above ratio obtains between the several resistances, the currents will remain unchanged whether BD remain open or closed, or be closed intermittently by a key. Measurement of the E.M.D.P. of a Galvanic Cell or Battery while in action. One method out of many may be selected as an illustration. Take that of Wiedemann. Two batteries or cells, the one a standard ; connect in tension : the joint E.M.D.P. is E, + E 7/ ; the total resistance is r! + r t " + R a ; E + E the intensity of the current produced is I = - " Now turn one r t + r/ + K, of the batteries round, and connect so that the two now oppose one another ; the joint E.M.D.P. is E, E 7/ : the total resistance is as before: the intensity -E - E is I' = , ' ,/ "-p- Hence E, : E,, : : I - I' : I + 1'. E, the E.M.D.P. of the r i ~T r i ~T K> e standard cell, is known ; I and I', the intensities, can be observed : whence E' can be calculated. The Internal Resistance of a Battery may be measured (by Mance's method) by making it one of the four resistances within a Wheatstone's bridge (Fig. 223); one other of the resistances, say AB, being rendered adjustable either by making AB consist of standard re- sistance-coils or by the use of a*Rheostat or Rheo- chord, by which variable quantities of wire or mer- cury, or fluids of various kinds equivalent to so many Ohms resistance, may be introduced into AB. A galvanometer is placed in AGO ; a key in BD. The adjustable resistance in AB is varied in amount until the deflection of the galvanometer becomes un- affected by making or breaking contact in BD. The relation RAB : RBC : : AD : DC again holds good. The Intensity of a Current is easily measured in Amperes when the number of Ohms resistance and the number of Volts which generate it are known. It may also be inferred from some of the effects of the current. The Energy of a Current. In a current of intensity I, a quantity, I, of electricity passes during each second from a place where the potential is V, to a place where it is V o ; but V y V o , the fall in its potential, is E, the electromotive difference of poten- tial within the circuit. This fall is constant, for the electromotive difference of potential is kept up : the Energy of the current per second is therefore I x (V 7 VJ = IE ; or, in the course of that xvi.] ENERGY OF A CURRENT. 575 period of time during which a quantity Q passes, the .total energy is equal to QE. E Since by Ohm's law I = -p, we find that the Energy E 2 IE = ; and that it is also equal to I 2 K per second. The energy of a current of one Ampere-intensity under an electromotive potential -difference of one Volt is equal, fsince Energy = EI,)to (^ x 3000,000000) = 10,000000 C.G.S. units or Ergs per second. This Unit of Energy is sometimes called an Ampere- Volt or a Watt =10 megergs. This amount of energy per second is nearly equal to T ^ Horse-power, or e Cheval-vapeur. * Static Charge of a Conductor. If an insulated galvanic cell be symmetrically connected with two long wires, also insu- lated, the two wires become charged each with a uniform static charge, positive or negative, as the case may be. The potential is uniform on each side of the galvanic celL If the copper of a cell be abruptly connected with the earth, the copper becomes reduced to zero potential, and the zinc side of the cell, together with any wire which may be attached to it, has its potential doubled ; if, on the other hand, the zinc be suddenly connected with the earth, the positive charge borne by the copper, and by any wire connected with it, is doubled. In the former case there is an instantaneous current from the nega- tive wire through the cell to the earth : in the latter case there is an instantaneous current from the earth to the insulated posi- tive-wire. When an open circuit is abruptly closed for an instant, an instantaneous current is produced in the wire ; this current is not felt simultaneously over the whole circuit. In the case (a) of Fig. 224 =iE ^^ c> the galvanometers B and Cl9 B' twitch first when the _^_ interrupted circuit is ^E^ /^\ ^ momentarily completed; ^L+ < -' in case (b) of that figure, under similar circumstances, the galvano- meters A and A' twitch first. When a current is actually passing along a conductor, the conductor bears a certain Superficial Charge, varying in density 576 ELECTRICITY AND MAGNETISM. [CHAP. from point to point of its surface : if such a conductor be sud- denly isolated by both ends from the circuit, it retains this charge ; and not until a wire has had this requisite charge imparted to it can a Steady Current flow along the wire. Between the instant at which the current begins to flow and that at which it becomes steady, there is a period of adjustment, the Variable Period. When, as in Fig. 225, a battery of which one pole is con- nected to earth has its other pole suddenly brought into com- munication with a long wire whose other extremity is connected Fig.225. Ear with the earth, the time which elapses before the current through the wire becomes steady is found to vary as the square of the length of that wire. This has been proved by experiments on aerial telegraphic lines, and on extremely bad conductors, such as cotton threads, whose conductivity is due almost entirely to the slight film of hygroscopic moisture which covers their surface. The time spent in acquiring at any point of the wire a certain fixed in- tensity of current depends (1) inversely upon the effective difference of potential set up by the galvanic cell ; (2) directly upon the specific resistance of the conductor ; (3) inversely upon its cross-section ; (4) directly upon the square of its length ; and (5) directly upon its electrostatic capacity. The time which elapses before a certain proportion of the ultimate in- tensity is attained varies directly as the specific resistance of the wire, and very roughly as the square of the distance of the point from the cell, and also directly as the capacity and inversely as the cross-section : it is also, in practice, not independent of the effective difference of potential set up by the galvanic cell employed, or of the state of insulation of the wire. A lightning discharge through a lightning conductor is so brief that the laws of steady flow do not hold good : it is of advantage, in order to diminish the risk of lateral divergence, to render the current more uniform, or, in other words, to retard it ; for this purpose the capacity of the conductor should be increased, and therefore its surface ; and lightning conductors should be broad flat plates of metal rather than compact rods. The effect of self-induction of the current also aids in bringing about this result ; currents running parallel and in the same direction retard one another. In a uniform wire, OL, between whose extremities the difference of potential is maintained equal to OP (Fig. 226), XVI.] VARIABLE PERIOD. 577 Fig.226. Fig.227. the ultimate Line of Potentials is PL; the same line indicates the uniformly -varying Density of the Superficial Charge from point to point ; and when such a distribu- tion has once been pro- duced over the surface of the conductor, Ohm's law is obeyed ; but at various instants during the preliminary vari- able period, the distribution of potentials over the wire is such as is indicated by the curved lines, 1, 2, 3, etc., sketched in Fig. 226. When the extremity of a long wire is momentarily charged by contact with a charged conductor or with one pole of a battery, its home end suddenly acquires a high potential, which is im- mediately thereupon reduced by communication of the charge acquired by the extremity of the wire to the rest of the wire. In Fig. 227 the end p of the conductor OL is suddenly raised to the potential OP. A point such as A is found, as it were, to leap up to a high potential and then to descend. A wave of sudden increase of potential thus travels along the conductor, but falls off progres- sively, both in abruptness and in height, the further it travels. At the distant end, for a short interval after the circuit has been actually completed, no effect is perceived ; the current then begins to become sensible : and it then, if the contact be kept up at the home end, appears to increase in intensity after the manner indicated by the Arrival- Curve represented in Fig. 228. A current, even though it be constantly maintained at the home end, would take an infinite time to acquire its maximum value at the distant end of such a conductor as an Atlantic cable, if that 2 P Fig.228. Time 578 ELECTRICITY AND MAGNETISM. [CHAP. conductor had, when the current commenced to traverse it, been uncharged ; it would, however, require only about 108 seconds to attain -f^ of its maximum value, and about the fifth-part of a second to attain T -J^ of its maximum value. The apparent velocity of transmission of signals is thus seen to be entirely an affair of the delicacy of the instruments which detect the current on arrival at the distant end, and is perfectly distinct from the velocity of propagation of an electromagnetic disturbance. When the current suddenly stops after having acquired a steady flow, its cessation at the distant end presents a similar deliberation. When a wire is momentarily connected with a charged body and then connected with the earth, or "put to earth," the arrival-curve at its distant end is a curve due to the superposition of two arrival-curves ; the first of these is the arrival-curve, resembling that of Fig. 228, due to the contact with the charged body ; the second is curved in the opposite sense, and is due to the sudden discharge of the conductor. The dotted curve of Fig. 228 is the result of the superposition of two such opposed arrival-curves. This curve indicates that there is an abrupt and brief variation of potential at the end of the wire distant from the galvanic cell. More effective still than this in producing an abrupt and brief current is the process of following up each positive charge immediately after putting to earth, with a negative one, after which the wire is again put to earth. The disadvantage of this is that the potential, while it abruptly ceases to be positive, sinks at once to a negative condition, as in Fig. Fig.229. 229 ; for which reason it is X, customary so to arrange the "- .^ mechanism at the signalling station that each apparently simple making of contact is in reality a complex operation, in which an odd number of cur- rents of opposite kinds are sent in rapid succession into the wire, the wire being, after each, put to earth ; each of these currents being briefer than its predecessor, and correcting it. The arrival-curve for such a combination indicates an abrupt rise of potential, an abrupt fall, and then a slightly wavy line, which at no point diverges to any material extent from the base-line OL. Even these methods are increased in effect, the arrival-curve being ren- dered still more abrupt, by the use of Accumulators or Condensers, as they are usually called. Each condenser is composed of a large number of plates of tinfoil separated by waxed paper and paraffin : the alternate plates are in metallic communication with one another. One series of alternate plates in each condenser is in communication with the cable ; the other set is in com- munication with the galvanic battery or with the galvanometer G (Fig. 230). Any sudden variation in the potential of the landward plates of the home condenser is immediately followed by an equally-sudden flow of electricity XVI.] SUBMARINE CABLE. 579 into or from the cableward plates of that condenser : this flow takes place either from or into the cable itself ; this disturbance is propagated along the cable ; the potential of the cableward plates of the condenser at the receiving Fig.230. station is affected ; by induction the distribution of electricity in the land- ward plates of that condenser is affected, and a current passes through the galvanometer either from the condenser to the earth or in the reverse direc- tion. On connection of the home condenser with the positive pole of the battery employed, a positive current, runs through the distant galvanometer G to the earth ; and on putting the home condenser to earth a reverse cur- rent passes through G, which may be corrected as before. This cannot be done on land lines on account of defective insulation. Even if the key be kept permanently pressed down at the transmitting station the current passing through G is but momentary, for both condensers quickly assume a condition of electrostatic equilibrium. EFFECTS OF A CURRENT. Production of Heat. If a circuit be completed and allowed, as it were, to run to waste, no external work being done by it, heat is developed within the cell and in the conducting wire. The Heat produced represents the total Energy of the current, and E 2 is equal, like that energy, to I 2 E (Joule's Law), or to -p- , where E is the total resistance of the circuit, and I the intensity of the current actually passing ; this is the heat produced in each second. Problem. A uniform copper-wire whose cross-section is 4 sq. mm., and whose length is 104-9318 metres, connects the poles of a cell whose effective difference of potential is one Volt, and whose internal resistance is 4 Ohms. How much heat will be developed during one minute ? E is one Volt = ^J^ C.G.S. electrostatic units of E.M.D.P. The total /104"Q318 1 \ resistance, K, is 4 Ohms internal + ( - x ^ x ^-^r )0hms ex- ternal = 4-405 Ohms = \1-049318 * 61 90,Q00 4 000, 5 OOOOOO aG - S - ***"*** "* its - The E 2 1 440-5 heat = Energy = _ = _,. 90)000000;000000 = 2,270 >1 48 ergs per second = 136,208,880 ergs per minute = about ^j- Calorie per minute. 580 ELECTRICITY AND MAGNETISM. [CHAP. When a current is made to pass through a heterogeneous conductor composed of different metals, between which there is developed a constant difference of potential, that energy which is wholly converted into heat when no work is done by the current is divided into two parts. Of these one part obeys Joule's Law, and is equal to I 2 R, the product of the total resistance into the square of the actual intensity : the other, which may be positive or negative, goes to produce Peltier's effect, which is the following : Consider a junction of metals, A and B, such that when this junction is made the hot junction of a thermo-electric circuit a current passes through it from A to B : let a current be made to run ab externo through that junction in the same direction, A to B ; that junction will under such circumstances be cooled, while if, on the other hand, the current be made to flow from B to A, the junction will be heated. In Fig. 231 OL is a conductor composed partly of zinc, partly of copper ; a current is made to flow through the conductor from to L, between the Fig.23i. potentials V, and V IV . At the junction J there is a sudden fall of poten- tial (V y/ - VJ. In J the intensity - ' Fal1 f P otential - V > ~ V ' the intensity is V,,,-Vn R.. Resistance of OJ In both it is equal : hence R. in JL R/ ,+ R where E OL is the total fall of potential, Ej the fall at J, and R the total resistance. Hence E OL = RI + Ej. The Energy of the current is E OL x I = RI 2 + EjI. The first part of this expression, RI 2 , represents heat distri- buted over the whole conductor ; the second part, Ejl, represents heat locally developed at J, and proportional to the fall of potential there. If the current be made to pass from copper to zinc there will be a rise, a negative fall : the heat developed at the junction will be a negative quantity, and the junction will be cooled. In a thermo-electric circuit of copper and iron, 'the copper, which is nomi- nally positive to the iron, because the current flows from the copper to the iron across the hot junction, is in reality negative to it. Fig. 232 explains this : the dotted line indicates the fall of potential when the circuit is closed ; the broken horizontal line represents the potentials assumed when-fche circuit xvi.] PELTIER'S EFFECT. 581 is open. The current passes down a fall of potential at A, and therefore tends to heat A, the cool junction ; at B, Fig.232. the hot junction, it passes through a rise A of potential and tends to cool B. The > current therefore tends to heat the cool and cool the hot junction. This is 1^ Peltier's Effect I -. When a current is passed db ext-ern* through iron, copper, iron successively, it again tends to heat the iron-copper and to cool the copper-iron junction. The main current is weakened by the reverse thermo-electric current produced. Thomson's Effect. The same thing may occur even within a single metal. Hot iron is negative (or thermo-electrically positive) to colder iron ; a current passing within a mass of iron from a hotter region to a colder travels against progressively rising potentials and cools the iron ; passing from cold to hot iron, it heats the iron especially in the hotter region. These effects are reversed in copper, for hot copper is positive (or thermo-electric- ally negative) to colder copper. The Thomson effects are opposed to the Peltier, and explain Reversal. When a circuit is composed of various conductors which suc- cessively offer different resistances to the current, the heat produced is distributed among them, to each according to its resistance. Numerical Example : A circuit consisting of one cell whose E.M.D.P. is 1*8 Volts, and whose internal resistance is 0*7237 Ohms, and of an external conductor composed of 6*170 metres of soft copper- wire 4 sq. mm. in cross- section in which is interpolated a piece of platinum wire ^ mm. in diar. and 4 cm. in length, will have a total resistance amounting to Battery 0*7237 Ohm, copper wire ^ Ohm, and platinum wire (equivalent to a mercury column ( x - ) metres long and '007854 sq. mm. in section) '7513 \100 6'46/ Ohm; or on the whole 1*500 Ohms, or 15 C.G.S. electrostatic 9,000000,000000 units. We assume that a steady current can be set up and maintained for a second within such a circuit, and further, that radiation and conduction of heat may be set aside. Of the total heat produced is developed in 025 , '7513 . the battery, in the copper wire, and in the small piece ot platinum wire. The total heat produced in a second is 5i = / ( l '*\*. 15 ] R (^ V3oo/ ' 9,000000,000000 j C.G.S. units or ergs; this is 2 1,600000 ergs or (21,600000 ^- 41,573,025,475) Calories. The heat evolved in the battery '4825 of the whole would be, if we suppose the battery to contain 1 kilogramme of material of a mean specific heat of 0'8, sufficient to raise its temperature by about "000313 C. in a second ; that evolved in the copper wire (whose weight is about 217 grammes and sp. heat = 0*095) by about *0004 C. ; while that liberated in the platinum wire (whose weight is about 0*0276 grins., and whose sp. heat = 0*0325) would be competent to raise it in a second to the temperature of 290 C. 582 ELECTRICITY AXD MAGNETISM. [CHAP. Production of Light. When one part of a circuit presents a relatively-great resistance, the greater part of the heat developed within the circuit is concentrated within that part. When the local resistance is due to a thin platinum wire or a thin filament of carbon or of carbonised paper or vegetable fibre, that bad conductor is so far heated as to emit a considerable amount of light. This is illustrated by the various forms of incandescent lamps. Those in which the carbon filament is arranged within a vacuum give out, according to the type of lamp, the number in circuit, and the intensity of the current employed, a light equal to that of from 10 to 21 candles each; the fall of potential in each lamp is from 50 to 91 Volts, the resistance is from 27w to 137w (i.e., Ohms) each, and the intensity of the current neces- sary to keep up the light is from 0'70 Amp. to T76 Amp.; a light equal to that of from 56 to 115 candles being produced for each horse-power spent on the dynamo-electric machines which generate the electric current. If the carbon filament be so constructed as to present the form of a hollow tube, of relatively-great surface and small actual cross-section, the luminous efficiency of the lamp is greatly increased (Bernstein, Curto). When too strong a current is driven through such a lamp, the superficial particles of the heated carbon are scattered throughout the vacuum ; the carbon is volatilised and condensed on the wall of the lamp: so with platinum and iridium heated above 1*700 C. When a strong electric current is driven through a carbon rod to a thicker piece of carbon, the thin rod becomes heated ; when this takes place in air the carbon burns away rapidly; but if the rod rest loosely by one end upon the thicker mass, the contact is always maintained, and the light is fairly steady so long as any carbon remains. When the interposed resistance is that of a certain thickness of air, the current will not pass unless the interval be so small or the difference of potential on its two sides be so great that a spark can fly across it. When this is the case the current is established across the interval. If the poles be of carbon, their extremities become intensely hot and wear away by oxidation in the air. The intervening air becomes so good a conductor when intensely heated that its resistance alone will not explain the , resistance offered by the voltaic arc to the transmission of the current. This is due to a kind of thermo-electric effect. The positive pole is hotter (4000 C.) than the negative (3000 -3500 C.); it becomes thermo-electrically negative ; this tends to produce a reverse current. A phenomenon similar to this is observed between the various parts of a flame. xvi.] PRODUCTION OF LIGHT. 583 The positive pole, being the hotter, wears away about twice as fast as the negative. The problem of electric lighting is to keep the arc in the same place, the carbons being allowed to approach one another as far as, and only as far as, is necessary in order to make up for their wear. -In Jablochkoff s and Jamin's candles the two carbons are rods, parallel to one another and of equal length ; the arc passes between their apices. If the current passed in one direction only, one carbon would wear away faster than the other; the carbons would thus cease to be of the same length. The currents used must rapidly alternate in their direction ; both carbons are thus equally worn away and the length of the arc is constant ; but all such alter- nating currents make much noise in the lamp. The fall of potential in such a lamp is from 42 to 77 Volts; the intensity of the current producing the light from 3' 5 to 10 Amperes, and the candles per horse-power 125 to 313. In arc-lamps two carbon points are placed opposite to one another, and it is the part of a special regulatory mechanism to keep the carbons at a constant distance apart. Such regulating mechanisms depend for their action (1) upon variations of inten- sity of the current traversing the lamp, or (2) upon variations in the differences of pqtential between the two ends of the arc, or (3) upon departures from a predetermined relation between this difference of potential and the intensity of the current, or (4) upon variations in the amount of heat developed in the arc. The resistance of the voltaic arc is about 1 Jo> ; the fall of potential is from 32 to 58 Volts ; the light produced is from 280 to 800 candles per horse-power, according to the arrangement of the circuit. The heat developed in the arc has been utilised by Messrs. Siemens and Huntington, who produce the electric arc within the interior of a crucible, and by its means fuse very refractory metals with considerable expedition. When the electric arc is produced between carbons in vacuo a beautiful glow is obtained, the negative pole being surrounded by a blue aureole, and the positive by a stratified pale-blue light. The carbon evaporates, the vessel becomes filled with a blue vapour which darkens to indigo, and this condenses and renders the whole opaque. If a very little vapour of bisulphide of carbon be introduced into the vacuum, the light becomes unsupportably bright, and of an extremely brilliant green. Its spectrum presents channelled regions in the red, yellow, green, and violet, which look like duplicates of one another reproduced in different colours (Jamin). Geissler's Vacuum Tubes. When a discharge of high E.D.P., as from an electric frictional-machine, or an " induction coil," or a battery of 400 Groves, is sent through a mass of rarefied gas con- 584 ELECTRICITY AND MAGNETISM. [CHAP. tained within a so-called vacuum tube, that gas glows with a bright light, characteristic, as regards its spectrum, of the gas exposed to this operation. The positive pole is surrounded by a bright glow, the negative by a set of strise. If the vacuum be very good and the tube containing the rarefied gas be somewhat narrow at its middle, the glow breaks up into striae, which flow and flicker if the current which produces them be in the slightest degree variable. The discharge through a vacuum is shown to be disruptive by the fact that the fall of potential in the vacuum tube is inde- pendent of the E.D.P. of the circuit; and the strise seem to indicate that it is pulsatile, not continuous. The approach of external conductors repels the internal glow and causes its deflection ; and the glow is deflected by a magnet in the same way as a wire bearing a current. In very high vacua the discharges from the two poles of a vacuum tube appear to be independent of one another : each pole discharges itself without, as it were, feeling the condition of the opposite pole of the tube. Even where both are positive they may discharge towards one another into the same space. Electrification of Radiant Matter. When the rarefaction of a gas is extreme (one-millionth) its matter becomes radiant. The movement of its molecules may be guided and rendered manifest by electrification. In a Geissler's tube, the matter filling which is radiant, the molecules which come in contact with the negative pole are at once repelled from it in lines at right angles to its surface. Energy is imparted to these mole- cules by the electrified negative-pole, and where these molecules strike each other or other molecules they produce an internal glow ; where they strike glass (or diamond or ruby) they produce light and cause phosphorescence ; they also produce heat, so that when they are directed from a concave negative-pole upon a piece of platinum, the energy conveyed by them brings that piece of metal to its melting point; and when they strike a movable body they produce obvious mechanical effect (Crookes). Two streams of molecules proceeding from a forked negative-pole repel one another like two similarly-electrified gold leaves, and the negatively- electrified particles which constitute such a stream are attracted and deflected from their course by a magnet. Chemical Effect of Electrification. When two metal tubes, of which the one is positively, the other negatively charged, are arranged so as to form a double tube with an annular channel, xvi.] CHEMICAL EFFECTS. 585 and when gases are passed through this channel, then chemical effects are sometimes produced, as where oxygen subjected to this treatment becomes in part converted into ozone. Chemical Effects of a Current Electrolysis. In most cases if a liquid permit the passage of a current through it, it is decomposed by the conductor ; in other words, most liquids which possess conductivity are electrolytes. A few liquids, such as alcohol and ether, though not absolutely non-conductive, are not decomposed by the passage of a current. Let us take as an example the effect of a current upon a solution of hydrochloric acid in water. In such a solution insert two platinum plates, the one, the positive-electrode, connected by wire with the copper plate of a sufficient battery ; the other, the negative-electrode, with the zinc or negative plate. Each mole- cule of hydrochloric acid, HC1, consists of one atom of hydrogen, H, and one of chlorine, Cl ; the former is electropositive, the latter negative. A molecule of HC1 (the chemical symbol being used as an abbreviation of the name) near the negative-electrode is torn asunder; its electropositive hydrogen is attracted by the negative-electrode and liberated on its surface ; the electronega- tive chlorine, by one means or another, finds its way to the positive-electrode, which by a secondary reaction it attacks and dissolves with the formation of PtCl 4 . It is possible that the free chlorine-atom may travel directly to the positive-electrode, for when liberated it is in effect a charged particle within a field of force; but it seems more probable that a chlorine-atom is liberated at the surface of the positive-electrode at the same time as a hydrogen- atom is liberated on the surface of the negative-electrode, and that the remaining atoms and the intervening molecules re- adjust their mutual relations according to a method the ?!UllllL2iL mechanism of which is in all probability that indicated in Fig. 233. In that figure (a) a) represents a line of molecules D arranged end to end along a line of force ; (&) the effect of the decomposition of a molecule r; (H ci) (H ci) ('H ci) ( ci) H, ci, (H~~ct? :'H~cY; H.. ci I'CI~H" ci H (ci hi), 01 * " (H'CII CH~~cVs ('H ci) 9} "' ~ ' at each end of the line ; (c) the __ mode of readjustment by what may be called a wholesale exchange of partners on the part of all the atoms all along the line, and the formation of a new set of 586 ELECTKICITY AND MAGNETISM. [CHAP. molecules out of the materials of the old, this being a process which there is chemical evidence for believing to be always going on irregularly, but which is here directed by the condition of the field of force between the electrodes ; and (d) the reversal of the direction of all the molecules thus formed, every molecule being turned round in the field of force, so that its negative chlorine comes to face the positive-electrode. In course of time all the chlorine will be found to have travelled to the positive- electrode, and all the hydrogen of the hydrochloric acid to the negative-electrode. In such a molecule as that of copper sulphate, CuS0 4 , the copper plays the part of the hydrogen of the previous example, while the part of the atom of chlorine is taken up by the atom-group or salt- radicle S0 4 . The group SO^ could not undertake a trajectory through a fluid ; it would break up into S0 3 and ; whence the necessity for the apparently complicated explanation of the last paragraph for the simpler but analogous case of hydrochloric acid. The copper liberated at the negative- electrode forms a film upon that electrode ; the S0 4 liberated at the positive- electrode reacts secondarily upon the water present ; S0 4 -|- H 2 = H 2 S0 4 -h ; the positive-electrode is surrounded by sulphuric acid, and oxygen is liberated on its surface. If a solution of sulphate of potash be decomposed, the molecule K 2 S0 4 is divided into K 2 and S0 4 ; the S0 4 causes the evolution of oxygen as a secondary product at the positive-electrode ; the potassium, by the reaction K 2 + 2H 2 = H 2 + 2KHO, causes the evolution of hydrogen at the negative-electrode. Here the water seems to have been decomposed ; the apparent decomposition of water is, however, a secondary result of the decomposition of the K 2 S0 4 . Pure water can only with extreme difficulty be decom- posed ; it is an extremely bad conductor, and perhaps if perfectly pure it would, like alcohol, not be found to be an electrolyte at all, and possibly not even a conductor. When metallic salts are dissolved in it, so that its conductivity is improved, it becomes an electrolyte a property not confined to water, but possessed also by such substances as oil and bisulphide of carbon, which in the pure state are non-conductors. The secondary reactions met with in electrolysis depend upon the time which is allowed for them, and are therefore favoured by currents of small intensity. If copper chloride be electrolysed between copper electrodes, the one, the negative-electrode, is thickened by a deposit of copper, xvi.] ELECTROLYSIS. 587 while the other is worn away, being attacked by the chlorine ; and the intervening solution of copper chloride is, if the decomposing current be feeble, maintained in its state of saturation; but if the decomposition be very rapid the solution of the copper electrode does not keep pace with the evolution of chlorine upon it, and the solution becomes weaker in copper and acid in its reaction. The secondary reactions observed are sometimes very peculiar. When hydrochloric acid is decomposed the chlorine evolved at the positive-electrode attacks the water present and liberates oxygen, which in its turn attacks some of the hydrochloric acid formed and produces chloric and perchloric acids. When Na 2 C0 3 is decomposed Na appears at the negative pole (pro- ducing a secondary evolution of hydrogen) and CNa0 3 at the positive pole ; this last group reacts upon water and forms oxygen and NaHC0 3 ; 2CNa0 3 + H 2 = O + 2HNaC0 3 . When NaHCO 3 is decomposed it produces Na and CHO 3 , and then 2CH0 3 = 2C0 2 + H 2 O + 0. When formic acid (H.COOH) is electrolysed it breaks up into H and COOH ; then 2COOH + H 2 O = 2 H.COOH + 0, or formic acid and oxygen; but the oxygen reacts upon some of the formic acid present, and then H.COOH + = H 2 O + CO^ When fused caustic-potash is electrolysed, K appears at the negative, HO at the positive ; this coalesces into H 2 O 2 , and breaks up into H 2 and ; but if the action be slow the K acts upon some of the water, and hydrogen is evolved. The nascent hydrogen evolved at the negative pole will attack aldehydes, forming alcohols, and thus certain ill-tasted rough alcohols may be greatly improved. Faraday's Laws of Electrolysis. First Law. The quantity of an electrolyte decomposed in a given time varies directly as the intensity of the current. A current whose intensity is n Amperes will decompose 0*000,097371 grammes of water per second. A Voltameter is an instrument in which a current is made to pass through acidulated water between platinum electrodes ; the acidulated water is in part decomposed, and the hydrogen and oxygen resulting from the secondary decomposition are collected in a graduated tube; the amount of the mixed gases can be read off, and the weight of the water decomposed readily ascertained ; a simple proportion gives the intensity of the current actually passing through the voltameter. Instead of measuring the quantity of gas evolved, we may weigh the quantity of copper deposited from a solution of copper sulphate; a current of one Ampere intensity deposits -000330676 grammes of copper per second. Instead of sending the whole current through the voltameter, we may send a known fractional part of it by arranging two paths for the current ; one the main wire, resistance r; the other a branch wire entering the volta- meter, and a wire again connecting the voltameter with the main wire ; the resistance in this side track is R ; the intensities in the two paths will be inversely as r and R. If we make R = 39r, and if I be the measured inten- sity of the current in this greater resistance, the current in the main wire will have an intensity of 391. 588 ELECTRICITY AND MAGNETISM. [CHAP. Faraday's Second Law. The law of electro-chemical equi- valence may be divided into the following propositions, of which the third may be regarded as a paraphrase of the law itself : 1. The gramme-equivalent of a metal is that quantity which will replace one gramme of hydrogen. For example: in comparing HC1 (H = 1, Cl = 35'5) with AgCl (Ag = 108, Cl = 35-5), we find Ag ( = 108) to be equivalent to H ( = 1) ; the gramme - equivalent of silver is 108 grammes. In comparing CuS0 4 with H 2 S0 4 we find Cu ( = 63'5) equivalent to H 2 ( = 2) ; the gramme-equivalent of copper is 31*75 grammes. 2. The gramme-equivalent of a salt-radicle or halogen is that quantity which will combine with one gramme of hydrogen. In HC1, 35*5 grammes of chlorine unite with 1 gramme of hydrogen ; the gramme-equivalent of chlorine is 35*5 grammes. In HN0 3 (H = 1, N0 3 =62), 62 grammes of N0 3 unite with 1 grm. of H ; the gramme -equivalent of NO 3 is 62. In H 2 S0 4 (H 2 =2, S0 4 =96) the gramme-equivalent of S0 4 is 48 grammes. In H 3 P0 4 (H 3 = 3, P0 4 = 95) the gramme-equivalent of P0 4 is 31f. 3. When a current whose intensity is, after the current has become steady, equal to n Amperes passes through a solution of a salt, '00 0,0 10,4 15% gramme-equivalents of the salt-radicle or halogen are liberated at the positive pole, and a corresponding quantity of the metal at the negative, during each second. A current is sent simultaneously through a solution of cupric chloride and a solution of cuprous chloride, and continues to pass through both solu- tions for five minutes ; its intensity is 18 Amperes ; compare the amounts of copper deposited on the negative-electrodes in the two solutions. In each the amount of the halogen the chlorine liberated is (-000010415 x 18 Amp. x 300 sec.) gramme -equivalents, or ('000010415 x 18 x 300 x 35*5) grammes. In Cud 2 every 71 parts of chlorine are combined with 6 3 '5 of copper ; the copper deposited from the CuCl 2 is therefore j (-000010415 x 18 x 300 x 35 -5) x ~5 I grammes. In Cu 2 Cl 2 every 71 parts of chlorine are combined with 127 of copper ; the copper deposited from the Cu 2 Cl 2 solution is therefore | (-000010415 x 18 x 300 x 35-5) x 1 grammes, double the quantity deposited by the same current from a solution of cupric chloride. In decomposing an electrolyte Work is done by the current ; its Energy is spent in producing Chemical Decomposition. The energy absorbed in the decomposition of say 9 grammes of water is equal to the heat liberated by chemical action and change of physical state when 1 gramme of hydrogen and 8 of oxygen are exploded together and condensed into water. xvi. ] ELECTROLYSIS. 589 Work done on decomposition, if such work be other^han chemical, causes divergences from Faraday's second law. Such work is done when the pro- ducts of decomposition of a liquid become gaseous, or when the metal of an electrolyte is lifted up towards the negative electrode ; while the energy of the current is increased when these circumstances are reversed. The relation of the energy liberated by the chemical action of the battery to the heat produced in the battery and the energy spent in doing electrolytic work is represented by the equation Battery-energy = Heat evolved in battery + Electrolytic work done. The last term cannot be greater than the first ; if it were, the heat developed in the battery would be a negative quantity, and the battery would go on indefinitely cooling itself as well as all surrounding objects an impossibility. In a Daniell cell the replacement of one grm.-equivt. of copper ( = 31*75 grins.) by one grm.-equivt. of zinc (= 32 '5 grms.) is attended with the evolu- tion of 24 '2 Calories of heat; in a Grove the energy evolved is 47 Ca for every grm.-equivt. of Zn dissolved. When 1 grm. H and 8 grm. O unite to form one grm.-equivt. of water, 34'5Ca of heat are evolved. A single Grove cell can therefore decompose water ; a single Daniell cannot. Two such cells are, however, able to effect decomposition ; the energy supplied by two cells arranged in tension is double that supplied by one, even though the resistances be so adjusted that the current produced is of the same intensity : for Energy = El, and if E be doubled the energy is doubled. A single cell will, however, decompose water if the positive-electrode be of a substance such as copper, which will combine with oxygen, and thereby liberate energy sufficient to make up the deficiencv in that supplied by the celL The process of electrolysis is turned to practical account in the arts of electroplating, etc. ; the article to be covered with a metallic film is made the negative -electrode in a suitable solution of the metal The positive- electrode is often itself made of the metal to be deposited from the solution ; as the metal is deposited from the solution upon the negative-electrode, the positive-electrode is attacked and its substance dissolved in the solution, which is thus kept saturated if the action be not too rapid. When a mixed solution of acetates of lead and copper (Nobili), or a solution of litharge in caustic potash (Becquerel), is electrolysed between two electrodes, of which the negative is a sharp-pointed platinum wire or steel needle, while the positive is a large plate of silver, german-silver, silvered copper (spangle metal), or even thin sheet-iron, the current being one of relatively-great E.D.P. (15 to 20 very small Bunsen cells mounted in file), there is formed on the positive-electrode a series of rings concentric with the point of the negative-electrode, Nobili's rings, produced by deposition of Pb0 2 . If the positive -electrode be complex and consist of two or more points, or if currents be made to run through some of these points towards the plate, while in others the direction is reversed, the rings are modified into representations, in iridescent hues, of complete systems of equipotential surfaces (Guebhard). 590 ELECTRICITY AND MAGNETISM. [CHAP. Polarisation of Electrodes. When platinum electrodes have been used in the decomposition of water, the negative one is found to contain a certain quantity of hydrogen not only adherent to its surface, but also occluded within its substance ; while the positive one carries similarly a certain quantity of oxygen. These oxygen and hydrogen films tend to produce a reverse current, which weakens the main current. When the main current ceases the reverse current continues for some time and dies away gradually. The occluded hydrogen is very gradually reduced by the oxygen brought to it by the reverse current, and a corresponding quantity of hydrogen is liberated on the positive pole, where it is occluded by the electrode or dissolved by the water. Thus, though a current passes, no products visibly appear. In Grove's gas-battery a number of such electrodes' are arranged so as to give a current, or which amounts to the same thing a circuit is arranged in which a current passes through : (1) water ; (2) a plate of platinum standing partly in water, partly in hydrogen gas ; (3) conducting wire ; (4) a plate of platinum standing partly in an atmosphere of oxygen, partly in the water. If electrodes of palladium be used in the decomposition of water, the negative one absorbs as much as 936 vols. of hydrogen, with which it forms an alloy greater in size than the original electrode. The electrodes in such a case are said to be polarised, and their polarisation within a given substance is the best test as to whether that substance is really an electrolyte. Warm glass is thus found to be actually an electrolyte. Secondary Cells and Batteries. When acidulated water is decomposed between electrodes of lead, the negative electrode remains bright, but the positive becomes covered with a film of Pb0 2 . When the current ceases, if the positive and negative electrodes be connected by a conducting wire, a reverse current a polarisation-current, or Secondary Current passes; the film of Pb0 2 is negative, the lead positive. The reverse action appears to be (Gladstone and Tribe) in the main the following : Pb(X H 2 S0 4 H 2 S0 4 Pb = PbO H H 2 S0 4 PbSO, which becomes PbS0 4 | H 2 | H 2 | PbS0 4 . Sulphate of lead is formed on both plates, while Pb0 2 is formed on the posi- tive plate by secondary reactions. An arrangement of this kind, into which a current of electri- city can be passed and a reverse or secondary current obtained at xvi.] SECOND AKY CELLS. 591 will, is a Secondary Cell; and secondary cells may be grouped into Secondary Batteries. Planters original form of secondary cell consisted of two large lead electrodes, separated by a sheet of felt, rolled up into a spiral and immersed in 1 J Q dilute sulphuric acid. Faure improved this by covering both electrodes with a layer of red-lead of about 10 kilogs. per sq. metre, held on by layers of felt and parchment between the opposed plates. When a current has been passed into a so-called Faure-accumulator for some time, the red-lead on the surface of the negative electrode is converted into spongy lead, while that on the positive electrode is oxidised to Pb0 2 . This improvement greatly abridges the tedious pro- cess formerly necessary for charging Plante's accumulators. In the Faure-Sellon-Volckmar accumulators there is no felt ; the plates of lead are pierced or cast with holes, into which there is compressed a quantity of red-lead, of reduced lead, or of a salt of lead. One of these, weighing 140 kgr., and composed of 43 plates, gave (Hospitalier) a current of 120 Amp. for 6 hours. The efficiency of these batteries is in part due to the fact that their porous metal or oxide is in close contact with the lead plate, and is, on account of its porosity, able to retain large quantities of the oxygen or hydrogen which is evolved when an external current is passed through the accumulator. When the secondary current has passed- for some time, both films become mainly converted into similar compounds of lead, and the apparatus is ready for a renewed charge. When a secondary battery is charged by two or three Grove cells and disconnected from them, it will be rapidly discharged if connection be established between its poles by means of a thick wire. From fifty to sixty per cent, or, in the newer forms, a still greater proportion of the energy actually sunk in it can be recovered in the form of the energy of the secondary current. When the cells of a secondary battery, charged side by side, are disconnected from the source, and then connected in series and discharged, the electric current produced is one of high tension or great fall of potential. The E.D.P. of a single cell is about 2 '2 5 Volts, and hence a hundred such cells, first arranged in surface and charged by prolonged connection with a few cells, can be made for a short time to pass a current of E.D.P. = 225 Volts ; such a current pro- duces much heat, together with vibration and crumpling of the conducting wire. The internal resistance of these cells is very small, being only 0'006 Ohm in a single cell whose surface is about 300 sq. cm. 592 ELECTRICITY AND MAGNETISM. [CHAP. A secondary cell of about 18 Ibs. weight, the plates in which are each about 2 sq. feet in area, will, when charged by three Leclanches, keep at a white heat during 5 to 10 minutes a platinum wire of -^ inch. diar. and 8 inches long. A pile of this weight kept constantly connected with three or four Leclanche elements (which require little attention beyond that of keeping them moist) is a very convenient means of heating such a thing as a gal- vaiiocautery wire, which must be raised to a high temperature for a short time. Electrical Storage of Energy. When Energy is stored up in bent steel springs, about 3924 megergs per kilogramme can be stored up i.e., 40 kilogrs. can be lifted through 1 metre by the elasticity of a spring weighing one kilogramme. When it is stored up in compressed air, 1 kilogr. of air com- pressed to one-sixth contains 2,250,000 megergs, of which about 450,000 can be recovered in the form of work. When it is stored in secondary batteries, about 500,000 megergs per kilo, of secondary battery are stored up, of which from 250,000 to 330,000 may be recovered if the batteries be used within a day or two after charging. The great fault of these accumulators in their present form is their want of durability. Equalisation of a Current. A current passed through one plate of an Electrostatic Accumulator will be apparently absorbed when the current is increased, and will be given out equably when the current falls off; such an accumulator is therefore com- petent to play the part taken by a flywheel in the mechanical transmission of power. Similarly, secondary cells may be made to serve as regulators of a current if they be fitted up in the course of the conductor in the manner indicated in Fig. 234. The direction of the secondary z- c+ current is indicated by the dotted lines and arrows connected with the secondary cell ; it opposes the main current in C + DCZ , but aids it in DEC. xvi.] ELECTRIC CURRENTS. 593 THE DYNAMICAL PROPERTIES OF A CURRENT. The phenomena of a steady electric-current are not confined to the conducting wire, for the space surrounding the wire bear- ing a current is found to be in a peculiar condition a condition which can be explained as due to displacement of the ether or other dielectric or medium filling that space, and one which it seems impossible physically to account for on any other satis- factory basis. The condition of the space immediately surrounding a current may be explored by means of iron filings. If a conducting wire be passed vertically through a hole in a piece of card or of silver- paper adjusted to a horizontal position, and if iron filings be then sprinkled upon the card, and if the card be gently tapped down- wards so that the filings may leap into positions spontaneously assumed by them, they will be found to range themselves in con- centric circles round the current, while each filing becomes, for the time being, a little magnet. The space round the current is therefore an Electromagnetic Field of Force, permeated by concentric circular Lines of Force and by Equipotential Surfaces, which are at right angles to these. The equipotential surfaces all have the line of the current for their common edge or boundary. If the current be straight the equipotential surfaces are planes ; and if they were visible, and if the current could be looked at end-on, so as to appear a mere point, these surfaces would seem to radiate from it like equidistant radii from a centre. The Lines of Force mark the direction in which an ordinary magnet, such as a small compass-needle, when placed within the field, tends to place itself. The one end of the magnet is driven in one direction, the other end equally in the opposite direction along these lines of force : the magnet is acted upon by a couple, which acts upon the two extremities or poles like the hands on the handles of a copying press one pole being pushed or repelled, the other being pulled or attracted, until the magnet lies along a line of force. The effective components of the couple gradually diminish as this position is being assumed, and the couple ulti- mately ceases to produce further rotation ; and further, since one pole is attracted as much as the other is repelled, the magnet as a whole undergoes in such a field no movement of translation. The direction in which a current tends to throw the positive or north-seeking pole of a magnet placed in its neighbourhood 2 Q 594 ELECTRICITY AND MAGNETISM. [CHAP. is shown by Fig. 235. This direction is called the Positive Direction of the lines of force. The current in the figure passes P vertically upwards ; the Positive pole is thrown to Fig-.sss the Left Hand of the Current. This expression, left hand of the current, is obtained by supposing the current to be replaced by a person whose head is at B and feet at A, and who turns so as con- stantly to keep the magnet -pole in full view. The relation between the direction of the current and the positive direction of the lines of force is always the same as that between the propulsion of the point and the twist of the hand in the ordinary use of a European corkscrew. Conversely, action and reaction being opposite, stationary positive magnet-poles tend to throw movable upward currents to their left. For negative magnetic-poles the directions given are reversed. Further, Linear Currents act upon other Linear Currents; but they do not throw them to the right or left ; they attract or repel them directly. In Fig. 236 AB is a current ; round it there are a number Fig . 236 of lines of force. Let an- BR Pn other current, parallel and ''"" """"~i m the same direction, be ----- ---... brought to CD : then, be- tween AB and CD the lines of force of the two currents are opposed, but beyond CD they concur in their direc- tion. The result is that the medium beyond CD is in a state of greater con- straint than that between AB and CD ; so is that to the left side of AB ; and the two con- ductors are impelled towards one another. If the current in AB be opposed to that in CD, the directions of the lines of force coincide between AB and CD, but are opposed beyond AB or CD, and the stress is such as to drive AB and CD asunder. If the currents in AB and CD be at right angles, approaching one another (Fig. 237), the current in AB has lines of force whose direction is upwards on the left side of AB, downward on its right. The current CD has lines which would depress a positive C- 5D A c XVI.] DYNAMICAL PROPERTIES OF A CURRENT. 595 A' A& <&>' . C Down o Q 1 1 1 1 Down o. D il 1 1 t pole placed in the upper, and raise a positive pole placed in the lower part of the figure. The concurrence of lines of force is in the upper part of the dia- y gram (Fig. 237) : if the current AB be fixed, the conductor CD is repelled in a direction con- trary to that of the course of Up the current in A. If the cur- rent in CD pass from D to- wards C, the conductor CD tends to move in the same direction as the current in AB. These results were first summarised in Ampere's formula, by which the attraction between very small elements of two linear circuits was stated to vary II' as I I'-- (2 sin 6 sin &. cos = ; cos IV (o = 1 ; F cc I I'-jp (2 sin sin & - cos 6 cos B'}. 1. Let both currents be parallel to AB ; = 0, 0'= 0; sin = sin 0' W = ; cos = cos & 1 ; F oc - 1 1' -p- . Two elements of current, end-on to one another, and running in the same direction, repel one another. 2. Let both be at right angles to AB ; = 6' = 90 ; cos 6 = ; sin 6 = IV 1; Fcc2H' 3-. Two elements of current, parallel and abreast of one another, running in the same direction, attract one another. If they be opposed in direction, the product 1 1' is negative ; then F is negative, and the mutual action is one of repulsion. 3. Let one be at right angles to AB, the other parallel to it ; 9 = 90 ; & = 0. Cos 6 = = sin & ; cos & = 1 = sin 9. F = 0. There is no appreciable action between two extremely small elements one of which points end-on and at right angles to the centre of the other. 4. Let one be at right angles to AB, the other at any other angle, say IV 45. 6 = 90, & = 45. Fee v x 2 H'-^ Two currents diverging from a point attract one another ; if one of the currents be reversed, they repel 596 ELECTRICITY AND MAGNETISM. [CHAP. one another; if both be reversed, so that both converge upon an angle, attract one another. 5. Two elements, B and A, parallel in their direction ; the resultant force is one of attraction between B and A. A similar element at A' attracts B towards A'. Of these two attractions the resultant is towards 0. Pairs of such elements symmetrically ranged to an infinite distance on either side of Fig.239. make up an infinite conductor whose attraction for B is on the aggregate F cc YyB x length of element B. The attraction of an infinite current for an element of current running parallel with it is directed along a line at right angles to both currents, and is inversely proportional to the distance between them. 6. The element B passes towards ; A and A' are equal elements, sym- metrically arranged, of an infinite current along AA' : A attracts B : A' equally repels B ; the resultant is in a line parallel to AA', and tends to drive the current in the direction A' A. Resultant Fig.240. A O The formulae of Grassmann, Clausius, Helmholtz, and others, we here pass over. If a wire bearing a straight current be bent into a closed circuit or loop, its equipotential surfaces are modified into a series of bowl-shaped surfaces which still have the wire, the contour of the circuit, for their common edge or boundary. A circular cur- rent would have equipotential surfaces whose general form, looked at in section, is indicated by the undotted curves of Tig. 241, in which the lines of force are shown at right angles to the equipo- tential surfaces. To one side of the circuit the potential is posi- tive, to the other it is negative. The positive side of the circuit is such that an observer standing girdled by the current, with his head in the positive region, would see the current pass round him from his right towards his left hand. If another circular circuit be brought near, and if the direc- XVI.] FIELD OF FORCE ROUND CLOSED CIRCUIT. \\-* 59-7. tion of the current within it be the same as that i^tft^ the Fig.241. former, the lines of force of the two circuits coalesce, and the two circuits attract one another. The resultant system of surfaces Fig.242. Equipotential Surfaces only. Equipotential Surfaces & Lines of Force. and lines takes the form indicated in Fig. 242. The field of 598 ELECTEICITY AND MAGNETISM. [CHAI>. force between the two circuits is approximately uniform. The lines of force are all closed curves, but some of them, those which pass up the centre of the region between the circuits, take a relatively-wide sweep into space, and seem to radiate from or converge upon the external face of either circuit. A large number of such circular circuits arranged so as to have a common axis, and thus to form, as it were, the outline of a cylinder, would form a so-called Solenoidal system. Such a system would have lines of force radiating from each extremity, taking a more or less ample sweep into space, returning into the Fig 243 PP os ite extremity and passing up the axial region of the cylinder; each line of force being thus a closed curve. The external electromagnetic field of such a solenoid system would be identical with that produced by a system consisting of an attracting disc at the one, and a repelling disc at the other extremity of the solenoid ; and such a solenoid would by one of its extremities attract the north and repel the south pole of a compass-needle ; while by the other it would attract the north and repel the south pole. Such a solenoid would, so far as its external action is concerned, act like a bar-magnet, and Ampere's theory of Magnetism is, that magnets and solenoid systems of currents are fundamentally identical. A solenoid may be roughly realised by winding a wire into a narrow spiral and bringing the two extremities back to the same point. The error introduced in each turn of the spiral by its departure from a perfect ring- form is roughly compensated by the return of the wire (Fig. 243). This identity of action of Magnets and of Solenoidal Current- systems being premised, we now proceed to give a rapid summary of the main phenomena of Magnetism. MAGNETISM. Some bodies a piece of loadstone, a compass-needle, a wire spiral through which a current is passing tend, when suspended by their centre of gravity, to lay themselves in a definite direction, and to place a definite line within them, their Magnetic Axis, in a definite direction, which, roughly speaking, lies north and south. The same bodies have the power of attracting iron. Such bodies are called Magnets. Magnets may be divided into Permanent (loadstone, hard steel xvi. ] MAGNETISM. 599 magnets) or Temporary (a solenoid current, or an electromagnet, a bar of soft iron whose magnetic properties are induced by the presence of an electric current circulating round it, but which endure, in soft iron, no longer than the persistence of that current) ; or again, into Natural (loadstone) and Artificial. The constituent particles of a magnet are themselves magnets. A permanent magnet may be cut into an indefinite number of fragments, each of which will be a little magnet, the original magnetic axis in which will continue to point to the magnetic north and south. When a steel bar is converted into a permanent, or a soft-iron bar into a temporary magnet, some operation must be effected, not upon the mass as a whole, but upon its constituent molecules. The magnetic axis of a bar-magnet or compass-needle coincides more or less closely, but hardly ever with perfect accuracy, with its geometrical axis of figure. The extremities of the magnetic axis are the Poles of the magnet. One mode of expressing the mechanical action of magnets is to feign a distribution of imaginary magnetic matter at the Poles ; positive at the one pole, equal and negative at the other ; the attractions and repulsions observed are exercised mainly to and from these poles. Another method is to feign a distribution of magnetic matter partly over the surface, partly within the substance of the magnet (Poisson), or over the surface only (Gauss and Green). Positive and negative magnetic distribution may be feigned to be either lieapings-up of positive matter towards or at positive poles, and of negative matter towards or at negative ; or else to be distribution in excess and defect respectively (or inversely) of one and the same all-pervading magnetic fluid. These modes of representation are convenient for calculation and exposi- tion merely. A long thin bar so magnetised that all its molecules would, considered as magnets, be absolutely equal, would have its poles at its ends. Such a theoretical bar-magnet is called a Solenoidal Magnet. In practice the action of bar-magnets is the same as that of a theoretical Solenoid whose poles are at a somewhat less distance from one another than the extremities of the bar : for which reason the Poles of a bar-magnet are often said to be within its substance, at a short distance from its ends. The North-seeking pole of a magnet is its Positive pole ; the other, its south pole, is its negative pola In different magnets, unlike poles attract one another ; like poles repel one another. The force between two poles varies inversely as the square of the distance between them. It also varies directly with 600 ELECTRICITY AND MAGNETISM. [CHAP. the strength of each pole, or the quantity m of magnetic matter mm' supposed to be concentrated at each. It is therefore F = ^- , and is repulsive when the poles are similar, both positive or both negative. Two poles are said to be of Equal Strength when they can replace one another in their action upon external magnetic poles. Two poles are said to be each of Unit Strength if they be equal and have between them a repulsion-force equal to one dyne when their mutual distance is one centimetre. Clausius and Siemens suggest that such a pole should be called a Weber. A unit pole placed at 1 cm. from a similar pole of strength m units (or Webers) will be repelled with a force of m dynes. The opposite poles of a magnet are of strengths opposite but numerically equal; these are + m and m; their sum is zero; the sum of the magnetisms of every magnet is always zero. It is not possible to isolate a single magnetic pole or to produce any numerical difference of strength between the two poles of a magnet. Eound a magnet there is a Magnetic Field of Force per- meated by Magnetic Lines of Force and Magnetic Equi- potential Surfaces. An isolated positive-pole (if such a thing were possible) placed in the neighbourhood of the positive extremity of a bar-magnet would be repelled and would travel to the negative extremity, not by the shortest path, but by following the wide sweep of any line of force on which it might happen to lie. The Direction of a magnetic Line of Force is the direction in which a positive pole is driven, or, conversely, a negative pole pulled upon. A magnet within a magnetic field is acted upon by a Couple : its positive pole is driven, its negative pole drawn, in the direction of the Lines of Force passing through them : the axis of the magnet tends to coincide as nearly as possible with a line of force passing through its centre. If this line of force be curved, the axis of the magnet is set tangentially to it. The Condition of a Magnetic Field at a point is determined (1) by the Direction of the Line of Force passing through the point, and (2) by the local Intensity or Strength of the field i.e., by the amount of force with which a unit-pole there situated is repelled or attracted. XVL] MAGNETISM. 601 The unit intensity of field would be that produced by a Weber at 1 cm. distance. Sir William Thomson suggests that this unit of intensity should be called a Gauss. When the local intensity is F, a magnet whose length is I, and whose poles have the respective values ra and m, is acted on by a couple : the force acting on the positive pole is mF, and its moment is (raF x %l) ; the moment of that acting on the negative pole is ( m x F x %l) = JwF: the moment of the couple is thus mFl. At a spot where the intensity F = 1 that is, within a unit magnetic field this moment is equal to ml, the numerical Strength of either pole multiplied by the Length of the magnet. This moment is called the Magnetic Moment of the magnet. If a magnet of length I be broken into n fragments, each of length , the / 1\ U magnetic moment of each fragment is m x ( j ; the sum of the magnetic moments of all the n fragments is n X [ m . J = ml, the magnetic moment of the original magnet, and each fragment possesses poles of strength m, equal to those of the original magnet. When two equal magnets are arranged thus N S, N S, the extreme poles are effective, the intermediate ones mask one another ; when work is done upon them in separating them, the original condition is restored and all the poles are again manifest. When n such magnets are connected in this way, all the poles except the extremes mask one another. A uniform bar-magnet I cm. long and a sq. cm. in cross-section, and therefore having a volume of la cub. cm., may be considered as a collection of la magnets, each 1 cm. in length and 1 sq. cm. in section, and therefore each of unit volume. The magnetic moment of the whole is equal to that of la such magnets : the magnetic moment of each of these unit- volume magnets is that of the entire magnet, ml, divided by la, the volume, and would therefore be equal to 77? -. This is the so-called Intensity of Magnetisation of the a bar-magnet. 602 ELECTRICITY AND MAGNETISM. [CHAP. If the intensity of magnetisation of a bar-magnet were equal throughout, its poles would be situated exactly at its extremities. We generally find, however, that magnets present abnormalities in this respect, and that they may even have secondary poles, produced by local inequalities in the intensity and by consequent deficient compensation of the internal poles. Such complex distributions can, within a bar, be generally represented by the superposition of a number of solenoids of different lengths. A steel sphere will be magnetised uniformly if placed for 245 some time within a uniform magnetic field, and it then has a moment equal to that of a small axial magnet NS, Fig. 245, and it \ tends to lay its axis NS along that line of | force which passes through its centre. / The Earth considered as a magnet is not / uniformly magnetised; its intensity of magnet- isation is not equal throughout ; it does not act upon bar-magnets placed near its surface exactly as a distant bar-magnet would do, for the law of its action is not even approximately expressible by any formula less complicated than one which contains at least twenty-four coefficients (Gauss). To find the Magnetic Moment of a Magnet. We must combine the magnetic moment M of the magnet with H, the horizontal component of the intensity of the earth's magnetic force at the place of observation. By one process we can find the value of MH ; by another we can find that of ; from these data we can find not only M, the magnetic moment (for H = M 2 ^ , but also H, for MH -7- ~ = H 2 . / H 1. To find MH : Suspend a very long magnet by one thread attached to its centre ; load it so that it may swing horizontally round a vertical axis : observe the time of its oscillation under the earth's attraction for one pole and repulsion of the other. The time of oscillation is T = TT / - , where I is v HM the moment of inertia (page 144). In the simple -pendulum formula T = 2?r / ; but in the simple pendulum I = M 2 : and w, the weight of r~-r the mass M, is M QOQ 000 Watts. 10 + r 10 + r If the amount of onflow of a fluid in a pipe were found to vary directly as the motive difference of pressure, any amount of energy might be transferred from one place to another by the smallest flow of water, for any water allowed to flow out of the pipe might be made to escape with any assignable velocity ; xvi.] TRANSMISSION OF ENERGY. 627 provided always that the tube were strong enough at all points to sustain at all intermediate points the necessary pressure. p When the total resistance is the internal resistance B^ I = ; when it is E (R< + R e ), I / = '- . These two distinct sets of circumstances are linked together by (1) The criterion of maximum utility, R i + , = R 4 ; and (2) 58 The energy imparted to the dynamos is a constant, = El = E / I / = N ergs. From these equations we find I = A / - , andE =R, , T = ( R. ) , , , ' V 100 R' " ' \58 ) ( /58 N\ /100t>w i \ A / =. =-] = A / -rrr *V = vR< , JS. We must now choose numencal \V 100 R 4 / V 58 values for R, + e , the total resistance internal and external, and N, the amount of energy imparted to the system. We shall use C.G.S. Electrostatic units. Let the resistance be that of 2000 kilometres of copper wire of 1 sq. cm. iii cross-section, and that of x dynamos and x electromotors. The dynamos are each supposed to generate an E.D.P. of 1000 Volts, or 3^ C.G.S. Electro- static units. Their number must be (E, -=- 3 J). Let the joint resistance of each dynamo and motor be 10 Ohms, or C.G.S. Electrostatic unit of resistance. The resistance of 90000,000000 the (E -f- 3i) pairs of machines will be ) (E 4- 3i) x I \> ' 90000,000000)' n or '- C.G.S. Electrostatic units. 300,000,000000 The wire (4000 kilometres) will offer a resistance of about 648 Ohms, or 216 Electrostatic C.G.S. units. 300,000,000000 The total resistance R, , = ^ - i. E + 216 \ C.G.S. E.S. 300,000,000000 I ' / units. Let N be the energy of the Falls of Niagara, measured also in C.G.S. units or ergs. About 100,000000,000000 grammes of water fall per hour through a height of about 4830 cm. The potential energy lost by the water is about 132,000,000000,000000 ergs per second = N. The equation E y = vR f+e N is now E / = / - I E y + 216 1 x 132,000,000000,000000, a quad- V 7 300,000,000000 ( ratic ; whence E 7 = 440,020 C.G.S.E.S. units or 132,006,000 Volts. If iron telegraphic-wire 4 mm. in diar. were used, its resistance would be (the resistance of iron being fff that of copper) 31680 Ohms ; the total resistance would be - - - [ ^ + 352 1 andE = 450,320C.G.S.E.S. 10000,000000 I 30 ) units, or 135,096,000 Volts. No practicable insulation could be set up, adequate to sustain permanently so great a stress, which corresponds in the last case to a striking distance of about half-a-mile in air ; and no dynamo-electric machines could be ranged in file to the number necessary, for the insulation of their coils would break down.- 628 ELECTRICITY AND MAGNETISM. [CHAP. It is practicable, however, with ordinary telegraphic wires insulated in the ordinary way, and with a 1 6-horse-power dynamo, to drive a 6 -horse- power electromotor at a distance of 30 miles. The wire must also, "by possessing sufficient thickness, offer so little resist- ance that it is not so far heated as to deteriorate in conductivity. THE NATURE OF THE MAGNETIC FIELD. If a plane-polarised beam of light be sent through a magnetic field occupied by a transparent medium, its plane will, by the retardation or acceleration in phase of one of its circular compon- ents, be rotated. The sense of this rotation depends upon the direction of the lines of force and upon the nature of the medium ; its amount depends upon the thickness and the nature of the medium and upon the intensity of the field (resolved in the direc- Fig.253. tion of the ray). In diamagnetic substances, and in flint glass and a few magnetic substances, the direction of rotation of the plane of polar- isation is that shown by Fig. 253, in which AB represents a line of force, and the arrowed circle represents the direction of rotation of the plane. Whether the ray travel in the direction AB or in the direction BA, the absolute rotation im- parted to it on its transmission through the magnetic field is the same; whence, if it be reflected from a mirror and sent back through the field, the rotation of its plane will be doubled. In most magnetic media the direction above-mentioned is reversed. The rotation is better marked in solutions of nickel than of iron, and in uniaxial crystals it is most marked along the axis. Hall's Experiment. A thin cross of gold leaf on a piece of glass. A current from a battery passes through two arms of the cross, but does not affect a galvanometer con- nected with the other two arms. "When the cross is made to face the lines of force of a strong magnetic field, a constant current is indicated by the galvanometer. The current of the pile is, as it were, pushed into the galvanometer circuit. Kerr's Experiment. Light reflected from the polished face of a magnet undergoes rotation of the plane of its polarisation : when reflected from the north pole it is rotated from left to right. Fig.254. xvi.] NATUKE OF THE MAGNETIC FIELD. 629 Maxwell's Theory. These phenomena can he explained and deduced from one another by supposing that there is some kind of rotation round the lines of force ; and this rotation may be regarded, with extreme probability, as rotation within extremely small whirlpools, whose common axes are arranged in the direction of the lines of force. This condition in the magnetic field is not assumed instant- aneously. There is a period of adjustment, and during this period the phenomena of the Variable State and of Induction appear : these are due to the forces called into play when the velocity of the whirlpools or vortices is changing (Clerk Maxwell). During these changes the mechanism connecting the vortices, whatever be the nature of that mechanism, is exposed to Stress which is equivalent to Electromotive Difference of Potential ; if that mechanism yield elastically there is Electric Displacement, and the return to the normal condition gives rise to an Electric Current. Maxwell's Theory of Light. The same kind of stress has been shown by Clerk Maxwell to be perfectly competent to explain the phenomena of Light. The condition of the medium occupying the path of a beam of light is equivalent to that which the same region would assume if exposed to the induction of rapidly-alternating currents. In a non-conductor there is, under the action of such stresses, an electric displacement, and energy is stored up in this displace- ment; but in a conductor any disturbance is propagated as an electric current. If Light be an electromagnetic phenomenon two consequences follow. The first of these is : Since in non- conductors the displacement produces a restitution-force which varies as the displacement a criterion of vibratory movement propagated with a definite velocity ; but in conductors no such force is manifested, and the energy of electric disturbance must soon be converted into heat: That light -vibrations are not possible in conductors, and conductors should be always opaque, while non-conductors ought, if homogeneous, to be trans- parent. With few exceptions this is the rule. The second consequence is that v, the velocity of propagation of an electro- magnetic disturbance in a non-conductor, ought to be the same as that of light. This constant, v, is otherwise proved to be the same as the ratio of the electrostatic to the electromagnetic unit of electrical intensity or quantity presently to be explained ; and this ratio is experimentally found to be such as to give v 630 ELECTRICITY AND MAGNETISM. [CHAP. the mean value of 30,000,000000 cm. per sec., which coincides sufficiently with the velocity of light. Measurement of v. The ratios between the electromagnetic and electrostatic units may be determined by several methods, of which two may be taken as examples. Weber's method. Charge a Ley den jar with a known quantity of electricity, Q ; discharge the jar through a wire, which passes round a gal- vanometer-needle. The quantity of Electricity passing through the galvano- meter may be measured in terms of the deviation undergone by the needle in consequence of being thrown by the instantaneous current. This gives the quantity, q, in electromagnetic measure. These separate measurements of the numerical values, Q and q, of the one quantity of electricity give the ratio between the electrostatic and the electromagnetic unit. Sir William Thomson's method. The two ends of a wire of great resistance, R, are kept at a constant potential-difference, E ; a constant current runs through the wire ; this current is found to have an intensity i C.G.S. electromagnetic units ; the difference of potential, electromagnetically measured, is (by Ohm's law) e = iR. E and e bear to one another the relation of 1 : v ; whence v may be found numerically. If an electrostatically-charged body could be whirled round a magnetic needle at the rate of 30,000,000000 cm. per second, it ought to act upon it in the same way as a circulating electric- current. At very high speeds, such as are physically within our reach, the same effect should be observed in small degree, and Prof. Rowland of Baltimore has succeeded in making it manifest. Electromagnetic Measure. We must distinguish between Units and Measures. A given distance is numerically greater when measured in inches than when measured in miles. A current of given intensity, I in electrostatic units, must be repre- sented by smaller numbers when electromagnetic units are used : a current of I = 60000,000000 is a current of i 2 ; but the C.G.S. electromagnetic unit of intensity is 30,000,000000 times as great as the C.G.S. electrostatic unit. Dimensions. Intensity, i. A current of electricity whose intensity is electrostatically represented by I will be represented electromagnetically by only ( J units ; whence a current of I C.G.S. electrostatic units' inten- sity, which has dimensions M4LIT" 2 , will have in electromagnetic measure dimensions [MiLiT- 2 ] -=- [LT- 1 ] = [M1UT- 1 ]. Quantity, q, = Intensity x Time = [MiUT- 1 ] x [T] = [MiLi]. Potential, or Difference of Potential, e, = work done -j- quantity of electricity upon which work is done = [ML 2 T- 2 ] -*- [MiLt] = [MHJT- 2 ]. Electrical Force, the force acting on electromagnetic unit of quantity. Its dimensions are force -h quantity = [MLT~ 2 ] -i- [M*L*] = [M*LiT~ 2 ]. Resistance = difference of potential -f- current -intensity = [MiL*T~ 2 ] -f- [MiLiT- 1 ] = [LT- 1 ]. xvi. ] ELECTROMAGNETIC MEASURE. 631 Capacity is quantity of electricity stored upper unit potential-difference produced by it ; its dimensions are [MiLi] ^- [M1L?T~ 2 ] = [L- 1 *! 2 ]. Specific Conductivity: the intensity of current passing across unit area under the action of unit force. Its dimensions are current-intensity -=- (force x area) = [MiLIT- 1 ] -~ [(M4L4T- 2 x L 2 )] =[L~ 2 T]. Specific resistance, the reciprocal of the specific conductivity : [L 2 T -1 ]. From these dimensions we find that that which is measured electrostatically as a current of intensity I, is electromagnetic- ally a current of intensity ( 1 = i. Similarly, by comparison with the electrostatic measures, we find that electrostatic quantity, Q, is numerically expressible as ^ J electromagnetic units ; po- tential -difference, E in electrostatic measure, as (Ev) in electro- magnetic ; resistance, E electrostatic units, as (Kv 2 ) electromagnetic /C\ units ; capacity, C, as I ^J electromagnetic units ; specific con- ductivity and resistance, equal to c and r electrostatic units, are respectively equal to (cv 2 ) and ( 2) electromagnetic units. Some of the units of the C.G.S. Electromagnetic System are inconveniently large or small. It is therefore the practice not to use the C.G.S. electromagnetic units of electrical quantity, in- tensity, resistance, etc., but to build up an electromagnetic system based on new units of length, I, and of mass, ra. These are respectively 1000,000000 cm. and the 100,000,000000th part of a gramme. The unit of current - intensity is then = [( L\ Ill T J "10 v/100000,000000 M*I> 1 X ( x/1000,000000 = C.G.S. Electromagnetic Unit. This new unit of intensity is the Ampere. In the same way we find the unit of resistance, the Ohm, = 1 9 C.G.S. Electromagnetic Units. The Megohm = 1 million Ohms ; the Microhm one-millionth Ohm. The unit of difference of potential, the Volt,= 10 8 C.G.S. Electromagnetic Units ; the Mega volt = 1 million Volts ; the Microvolt one-millionth Yolt. The unit of capacity, r l T?,= 1000.000000 L = 1000,000000 C ' aa Electromagnetic Umt = 1 Farad. The Farad = 10~ 9 X one C.G.S. Electromagnetic Unit of Capacity; but the latter unit is equal to the electrostatic 632 ELECTEICITY AND MAGNETISM. [CHAP. unit x v 2 , or to 9 x 10 20 Electrostatic Units ; the Farad is there- fore equal to KT 9 x 9 x 10 20 = 9 x 10 11 Electrostatic Units of Capacity. The electrostatic capacity of a sphere is equal to its radius, because the potential over the surface and throughout the Q volume of a sphere of radius E charged with quantity Q is ^> and the capacity, which is equal to charge -f- potential, is there- Q fore equal to Q -4- ~- = K ; a Farad is therefore the capacity of a sphere of 9 x 10 11 cm. radius, and for convenience the standard in use is the Microfarad, the millionth of a farad. .p ,. . TT ., Number of C.G.S. Electro- Number of C. G. S. Electro- static Units. magnetic Units. Intensity Ampere . 3000,000000 Quantity Coulomb . 3000,000000 Potential Volt 100,000000 Resistance-Ohm . 900>00>000000 1000,000000 Capacity-Farad . 900,000,000000 1 00,000000 In electric lighting a certain unit is frequently made use of as a conven- tional basis for estimating the sum due by the consumer. This unit repre- sents 1000 Ampere-Volt-Hours, and is equivalent to the Energy conveyed by a current of one Ampere intensity, passing down a fall of potential of one Volt, and sustained for 1000 hours. This amount of Energy = 1 Ampere- Volt or Watt x 3,600,000 sec., and is therefore equal to {10,000000 x 3,600000} = 36,000000,000000 Ergs or 2,654,340 foot-pounds, or about 865 Calories ; and the commercial unit of current is a current of any intensity continued until this quantity of energy has been transmitted through the consumer's apparatus. The basis of the Electromagnetic System of Measurement is the identity of effect between a closed current of electromagnetic intensity i, and a magnetic shell of the same contour, whose magnetic strength is numerically the same, being also equal to i. If we suppose a wire bearing a current and one cm. in length to be bent into a circular arc whose radius is one cm., and if we suppose a unit magnetic pole to be placed at the centre of the circle of which the circular arc forms a part ; and if we further suppose that the force exerted by the current upon the unit magnet-pole is equal to one dyne ; then the current is one whose intensity is in electromagnetic measure equal to unity. In such a case i = 1 ; and this may be taken as a definition of the electromagnetic unit of U intensity. The general formula is / = m -% , where / is the force exerted xvi.] ELECTROMAGNETIC MEASURE. 633 upon a magnet-pole placed at the centre of such an arc, ra the strength of that pole, i the intensity of the current, I the length and r the radius of the circular arc into which the wire is bent. The potential differs in this case from the force exerted in having r instead of r 2 as its divisor (comp. p. 170). When I = 2 TIT, a whole cir- cumference, it is easy to show that the potential is 2-jri ' m, the same as the mutual potential of a magnetic shell of strength i and a magnet-pole m placed very near its surface. The Intensity of a Current, electromagnetically measured, and the Magnetic Strength of a shell must accordingly have the same dimensions. The Magnetic Strength of a magnetic shell is (p. 604) numerically equal to the product of the Magnetic Quantity per unit of area into the Thickness of the shell ; its dimensions must hence be those of Magnetic Quantity, divided by an Area, and multiplied by a linear Thickness ; but the dimensions of Magnetic Quantity must (since the imaginary magnetic matter obeys laws resembling those of attracting or repelling electric matter, similarly imaginary) be like those of electric quantity, the dimensions of Magnet-Strength are accordingly [L 2 ] x [L] = [M^LfrT- 1 ]; the Electromagnetic Unit of Intensity has the same dimensions, and therefore differs from the electrostatic unit, whose dimensions are [M*L ? T~ 2 ], by the term LT' 1 which represents a Velocity : the numerical value of this velocity must be found by experiment, which shows it to be 30,000,000000 cm. The electromagnetic units of measurement have all been derived in theory from the electromagnetic measurement of inten- sity of a current: the electromagnetic measurement of a current is therefore a fundamental measurement. It is effected by the use of Galvanometers and Electrodynamometers. A current sent through a Tangent-galvanometer for a known very short period r produces a deflection 8 of the needle, which takes place in obedience to the momentary impulse imparted by the current ; the intensity of the T current is i cc . 2 sin iS, where T is the natural period of the needle's 7TT oscillation in the terrestrial magnetic field. The constant which is necessary in order to convert the above proportionality into an equation is the Reduction Factor of the galvanometer, and is equal to the ratio between the intensity of a steady deflecting current passing through the galvanometer (as a numerator) and the tangent of the angle of deflection of the needle pro- duced under its influence (as a denominator). In the former, the elementary forms of which have been described, it is well to produce as uniform a field of force as pos- sible. This is effected by arranging a number of coils so as to surround the field, not wholly but in outline. For Sensitiveness 634 ELECTRICITY AND MAGNETISM. [CHAP, xvi.] each winding of the wire is made to come as near the magnet as is practicable. In the latter the current is passed through a coil which is sus- pended within a strong and uniform magnetic field, such as that produced by powerful electromagnets actuated by a second current, or again by fixed coils through which a second current is passing. The deflection of the suspended coil depends upon the strength of the current passing through it, and also upon the strength of the magnetic field surrounding it. If the same current traverse both the fixed and the suspended coils, the rotating couple is pro- portional to ft, and therefore to the energy of the current. If the movable coil be free to slip up and down the axis of a fixed coil in which a current is passing, the inner coil may be sucked in or repelled with a force which may be balanced and measured by known weights : or if the current in the suspended coil be variable, the tension tending to draw it in to the fixed coil may be made to act against a spring, and graphically to record its own variations upon a uniformly-moving piece of paper. The principle of the differential galvanometer may be here applied, as in Prof. Langley's Thermic Balance. The suspended coil is composed of two separate wires wound together, but insulated from one another : a single current is divided into two equal moieties which run in opposite directions through the two wires of the coil ; there is no effect. The least variation in. one of these moieties, as when the conductivity of its path is affected by the local application of heat, causes imperfect compensation, and practically a small uncompensated current passes : however feeble this may be, it can be rendered manifest and measurable by increasing the strength of the magnetic field within which the double coil is suspended. The part of the divided circuit to which heat may be locally applied may be an exceedingly thin strip of platinum. This may be moved up and down say in the dark region of the spectrum. In some places it is heated, in others dark lines it is not. By thus groping in the dark it discovers the dark lines of the heat spectrum. An instrument of this kind is sensitive to differences of temperature of yyj^ F. It will be kept in mind that all our statements as to positive and negative quantities and currents are based upon the purely arbitrary convention that vitreous electricity is positive, and resinous electricity negative. This convention happens to har- monise with that which regards the north-seeking end of a magnet as its positive pole. We do not really know, however, in what direction an electric current travels, or whether it may not be as there seems some reason for believing that it is the sum of two opposed currents, propagated simultaneously from the two poles of a battery. BIBLIOGKAPHY. THE following representative list may perhaps be found helpful to the student - in the course of his further reading : History of Physics. Poggendorff s " Geschichte der Physik. " HeUer's " Geschichte der Physik" (in course of publication). Rosenberger's "Geschichte der Physik" (in course of publication). Mach, "Die Mechanik in ihrer Ent- wickelung " (Leipzig, Brockhaus, 1883). Tables of Constants. Clarke's "Constants of Nature" (in course of publication by the Smithsonian Institute of Washington, D.C., U.S.A.). Everett's "Units and Physical Constants." Landolt und Bornstein, " Physikalisch-chemische TabeUen " (Berlin : Springer, 1883). Text Books. Jarnin et Bouty, " Cours de Physique" (new edition, in course of publi- cation). Violle, "Cours de Physique' 1 (voL L, July 1883). Verdet, (Euvres. Fernet, "Cours de Physique. " Reis, "Lehrbuchder Physik." PouiUet, " Physique." Wiillner. Ganot's " Physics." Deschanel. MiUer, " Chemi- cal Physics." KirchhoflF, " Vorlesungen iiber d. mathem. Physik." Fick, "Die medicinische Physik." Wundt, "La Physique medicale" (French translation by Monoyer). Grehant, " Manuel de Physique medicale." Desplats et Gariel, "Nouveaux Elements de Physique medicale." Hoh, " Die Physik in der Medicin." Hermann, " Physiologic," 5 vols. Works of Reference. Watt's "Dictionary of Chemistry," with supplements. The physical articles in the "Encyclopaedia Britannica," 9th edition. "Jahres- berichte der Fortschritte der Physik " (yearly classified abstracts published by the Berlin Physical Society, generally about five years in arrears, but other- wise of extreme value). Tolhausen, "technologisches Worterbuch, deutschfranz6sisch - englisch." Wershoven, " technologisches Worterbuch, deutsclf-franzbsisch und deutsch- englisch." Periodicals. Current Summaries and News. " Nature. " ' ' The Electrical Review. " "The Electrician " of London. Ditto of New York. "La Lumiere Elec- trique." "Centralblatt fur Elektrotechnik." "Elektrotechnische Zeit- schrift." "Journal of the Chemical Society" (for Chemical Physics). Silliman's "American Journal of Science and Arts." Original Papers. " Philosophical Transactions of the Royal Society of London." "Comptes rendus de 1* Academic des Sciences." Poggendorff's " Annalen der Physik und d. Chimie." Wiedemann's "Annalen." "Annales de Chimie et de Physique." " London, Edinburgh, and Dublin Philosophical Magazine." "Journal of the Physical Society of London. " "Proceedings of the Royal Society of London." " Proceedings and Transactions of the Royal Society of Edinburgh." (See Royal Society Catalogue of Scientific Papers.) 636 BIBLIOGRAPHY. Dynamics, etc. Sir Isaac Newton, " Philosophise Naturalis Principia Mathematical Thomson and Tait, "Treatise on Natural Philosophy," vol. i., 2 parts, new edition. Clerk Maxwell, "Matter and Motion." Clifford, "Elements of Dynamic" (unfinished). Minchin, "Uniplanar Kinematics." Ball, "Me- chanics." Goodeve, "Mechanics." Airy, " Undulatory Theory of Optics." -Airy on "Tides and Waves." Weber, " Wellenlehre. " Wernicke, ' ' Mechanik. ' ' Rankine, ' ' Applied Mechanics. ' ' Goodeve, ' ' Mechanism. " Twisden, "Practical Mechanics." Chambers, "Astronomy." Tait and Steele, "Dynamics of a Particle." Somoff, " Theoretische Mechanik." Lagrange, "Mecanique Analytique." Laplace, " Mecanique Celeste." Poisson, "Mecanique." Poinsot, "Rotation." Cayley, "Reports on Theoretical Dynamics, British Association Reports, 1846, 1857, 1862." Properties of Matter. Watson, " Kinetic Theory of Gases." Graham, " Chemical and Physical Researches. " Frarikland, "Experimental Researches. " J. F. Daniell, "Chemical Philosophy." Wurtz, "The Atomic Theory." Captain Abney's two "Treatises on Photography" (molecular chemistry). Klein, "Elasticitiit." Dynamics of Fluids. Plateau, " Statique des Liquides." Besant, " Hydrodynamics." Lamb, "Motion of Fluids. "Stanley on the "Motion of Fluids" (many new experimental facts). Stokes, "Mathematical and Physical Papers." Helmholtz, " Wissenschaftliche Abhandlungen. " Hicks, " Br. Assoc. Rep. . Hydrodynamics, 1881, 1882." Sound. Stone, "Lessons on Sound." Sedley Taylor, "Sound and Music." Blaserna, "Theory of Sound." Tyndall, "Sound." Donkin, "Acoustics," part 1 (unfinished). Chladni, "Acoustique." Herschel, Encyl. Metrop., " Sound." Helmholtz, "The Sensations of Tone," Ellis's English translation, with additions. Gavarret, "La Phonation et 1' Audition. " Lord Rayleigh, " Theory of Sound." Koenig, " Quelques Experiences de 1' Acoustique, " 1882. Melde's "Akustik." Heat. Clerk Maxwell's "Theory of Heat. " Baynes's " Thermodynamics. " Dixon on " Heat." Balfour Stewart on ' ' Heat. "Count P. de Saint-Robert, ' ' Prin- cipes de Thermodynamique. " Briot, " Theorie mecanique de la Chaleur." Clausius on " Heat. " Berthelot, "Mecanique Chimique." Thomsen, " Thermochimische Untersuchungen. " Perry on "Steam." Him "Dela Chaleur." Rankine's Papers. Sir William Thomson's Papers. Fourier's "Analytical Theory of Heat." Tait's " Thermodynamics " and " Heat." Light and Radiant Heat. Prevost, "Du Calorique rayonnant." (Euvres de Fresnel, de Cauchy, d'Arago. Weber, "Geschichte der Optik." Beer, " Hohere Optik." Glazebrook, " Physical Optics. "Lloyd's " Undulatory Theory of Light." Becquerel, "LaLumiere: ses causes et ses effets." Moigno, "Repertoire d'Optique." Baden Powell, "View of the Undulatory Theory as applied to the Dispersion of Light." Landholt, "The Polariscope. " Lommel, "Optics and Light," Lockyer, " Spectrum Analysis. " Macmunn, "The Spectroscope in Medicine." Rosenberg, "The Use of the Spectroscope in its application to Scientific Medicine" (New York: Putnam). Tidy, "Optics applied to Medicine" (Cantor Lectures, 1872-3). Leconte, "Sight" (New York: Appleton). Rood, "Modern Chromatics." Helmholtz, " Physiologische Optik" (French translation by Javal and Klein). Stokes's "Collected Papers. " Electricity and Magnetism. Clerk Maxwell, "Elementary Treatise. " Clerk Maxwell, "Treatise," in 2 vols. Mascart, " Traite d'Electricite Statique." BIBLIOGRAPHY. 637 Mascart and Joubert, " Legons sur 1'Electricite " (2 vols. ; one vol. the theo- retical published, and translated by Atkinson). Wiedemann's "Die Lehre von der Elektricitat," 4 vols. (two vols. published ; intended to replace Wiedemann's ' ' Galvanismus ; " a most admirable storehouse of experi- mental information). Gordon, "A Physical Treatise on Electricity and Magnetism," 2 vols. Gariel, "Traite pratique d'Electricite." Fleeming Jenkin, "Electricity and Magnetism. " -Silvanus .Thompson. "Elementary Lessons in Electricity and Magnetism." Noad and Preece, " Student's Text Book." Ferguson's "Electricity," Blyth's edition. Frederick Guthrie. "Magnetism and Electricity." A. C. Becquerel, "Traite d'Electricite." Riess, "die Lehre von der Reibungselektricitat." Neumann, "Theorie der Magnetismus. " Beer's " Einleitung in der Elektrostatik." Airy:. " Magnet- ism." De La Rive, "Electricity." Humphrey Lloyd, "Magnetism." Lament, " Erdmagnetismus. " Frohlich, ^Magnetismus und Elektricitat." Plant e, "Recherches sur 1'Electricite." Gumming, "Theory of Elec- tricity." Serpieri, "II Potenziale Elettrico nell' insegnamento elementare della Elettrostatica." Faraday, " Researches in Electricity. " Sir William Thomson, Reprinted Papers on " Electrostatics and Magnetism," and, in part, his other reprinted papers. Clausius, "Mechanische Behandlung der Elektricitat." Ampere, "Theorie des Phenomenes electrodynamiques. " Gauss, " Gesammelte Werke." Helmholtz, " Wissenschaftliche Abhandlungen, " Bd. I., 1882; Bd. II., 1883. DuMoncel, " Les Applications de 1'Electricite, " 5 vols. 1878. Hospitalier, " Les Applications de 1'Electricite. " British Patent Office Catalogues of Specifi- cations. Paget Hicks, "Electric Transmission of Power." Prescott, "Elec- tricity and the Electric Telegraph. " Prescott, "The Telephone, the Electric Light, and other recent inventions." "Engineering" Summary of Electrical Patents. Schellen, " Die magneto- und dyuamoelectrischen Maschinen. " De Cew, same title (Hartleben's " elektrotechnische Bibliothek"). Ternant, " Telegraphic " (popular). Du Moncel, " The Telephone ;" "L'Eclairage elec- trique" (popular). Schwartze, " Telephon, Mikrophon und Radiophon " (Hartleben's Bibliothek). Ferrini, "Technologic der Elektricitat und des Magnetismus. " Zetzsche, ' ' Handbuch der elektrischen Telegraphic " (4 vols. ; not yet completed). Siemens und Halske in Berlin, " Kataloge." Gra'tz, "Die Elektricitat und ihre Anwendungen" (Stuttgart, 1883). Gore's "Elec- trometallurgy. " Weber, " Elektrodynamische Massbestimmungen. " Latimer Clark, "Electric Measurements." Wilke, "Die Elektrischen Mass- und Pracisions- Instrumente" (Hartleben's Bibliothek). Day, "Exercises in Electric and Magnetic Measurement." Culley's "Handbook of Telegraphy." Schwend- ler's " Instructions for Testing Telegraphic Lines." Kempe, "Handbook of Electrical Testing." Hoskier, "Guide to Electrical Testing." Preece and Sivewright, " Telegraphy. "Latimer Clark and Sabine, "Electrical Tables and Formulae." Hospitalier, "Formulaire de 1'Electricien." Zech, "Elec- trisches Formelbuch " (Hartleben's Bibliothek). Laboratory Work. Balfour Stewart and Gee, "Lessons in Practical Physics" (in the press). Weinhold, " Experimental Physics." Weinhold, ' ' Physikalische Demonstrationen.' ' Rechnagel, ' ' Experimen. Physik. ' ' Pickering, ' ' Phy- sical Manipulation." Kohlrausch, "Physical Measurements." Mousson, "Physik auf Grundlage der Erfahrung." Shelley's "Workshop Appliances." Marey, " La Methode graphique." Airy, " Errors of Observation." Aime, " Cours de Manipulation de Physique " (Paris : Witz, 1883). INDEX. The darker figures serve as a guide to the more explanatory references. ABERRATION of Light, 461 Aberration, Chromatic (non-coincidence be- tween the foci for different colours, etc.), 485, 514 ; Spherical, of lenses, 485 ; of mirrors, 474 Abnormal Dispersion, 481 Abscissa, 11 Absolute Potential, 523 Absolute Temperature, 223, 328 Absolute Units ; see Units Absolute Zero, 327, 328, 356, 441 Absorbents, 443, 449 Absorption-bands, 450 Absorption of Actinic rays, 438, 449, 454 ; of Energy during change of State, 320 ; of Ether- waves by the atmosphere, 438, 449 ; of Gases by solids, 291, by liquids 293 ; of Heat, 449 ; of Light, 450 Acceleration, 16, 63 ; mean , 16 ; in Circular motion, 72, 142, 180 ; in a Curved path, 74 ; due to Gravity, 19, 180 ; The Parallelogram, etc., of s, 62 ; Positive or negative, 63 ; in S.H.M., 76 Accelerated Motion, 63, 146 : compounded with uniform, 65 ; under gravity, 179 Accommodation of the Eye, 513 Accumulator, 532, 578, 592 Achromatism, 480 ; achromatic Lenses, 485 Actinic Rays, 433, 454, 456 Action = Force, 17 Action and Reaction, 6 Adhesion, 31, 228, 229 Adiabatism, 322 ; adiabatic Lines, 352 Adiathermancy, 448 Affinity, Capillary, 250 ; Chemical, 46, 228, 292, 321, 546, 548 Air, density of, 289 ; air dissolved in water, 293 ; air-gun, 38 ; air-pump, 290 ; mer- cury air-pump, 306 ; Sprengel's, 291 ; air- thermometer, 357 Ajutages, 270 Alcoholometer, 196 Alternating Currents, 617 Altitude, 12 ; s indicated by the Baro- meter, 311 Amalgamated Zinc, 320, 547, 549 Ampere, 317, 566, 631 Ampere-currents, 611, 616, 617 ; their direction, 611 Ampere's Fonnula, 595 ; his Theory of Magnetism, 598, 611 Ampere-volt ( = Watt) 35, 575, 632 Amplitude in S.H.M., 76; of Vibration, 368 Analyser, 499 Analysis of a complex Harmonic Motion (Fourier's theorem), 96 ; of a Sound, 382 ; of a Vowel, 429 ; Chemical analysis, 191 Aneroid, 310 Aneurism, 257 Angle of Capillarity, 245 ; of Incidence, 113, 119, 467 ; Limiting , 160 ; of complete Polarisation, 470 ; of Re- flexion, 113, 467 ; of Refraction, 120, 467 ; of Repose, 160 Angular Velocity, 69; inS.H.M., 77 Aorta, 280, 288 Aperture, Wave traversing an, 109, 110, 123, 130-131, 132 Approximate Foci, 122, 474 Aqueous Vapour in the Air, 348, 444 Arc-lamps, 583 Archimedes's Principle, 261, 297 Areometer : Fahrenheit's, 197 ; Nicholson's, 198 Armature, 616 Arrival-curve, 577 Arterial tension, 287 Aspiration, 302, 303 Astatic, 610 Atheroma, 277, 288 Atmospheric Electricity, 559 Atmospheric Pressure, 203, 259, 300-313 ; Standard, 312 Atoms, 211, 218 ; chemical, 211-217 ; physical, 27-225 ; size and nature of, 218,219 Atomic Heat, 329 Atomicity, 217 Atomistic Chemical Formulae, 214 Attraction, 166 ; in particular cases, 167 ; conventionally negative, 168. Capillary , 245 ; Electric, 518, 535 ; Magnetic, 599 Attwood's machine, 29, 181 Audiphone, 421 Auditory nerve, 423 1 Availability of Energy, 44 640 INDEX. Avvogadro, 214 ; Avvogadro's Law (chemi- cal), 215 ; (physical), 221, 222 Axes : Crystalline axes, 494, 495, 499 ; Magnetic Axis, 598; of Reference, 60, 62, 66; of Rotation, 69; of Spontaneous Rotation, 146 Azinmth, 12 BABINET'S Compensator, 506 Balance, 29, 153 ; Spring Balance, 31 ; Langley's Thermic Balance, 634 Band spectrum, 447, 448 Banjo, 403 Bar-magnet, 599, 601, 602 Barometer, 306 ; aneroid , 310 ; ver- nier, 22 Barometric column, 304 ; Pressure, see Atmospheric pr. Basilar membrane, 422, 425 Basis of support, 184 Bassoon, 405 Battery, Galvanic, 547; Leyden, 534; Thermo-electric, 559 Beats, 92, 411, 425 Becquerel's Thermo-electric Pile, 559 Bell-crank, 158 ; Oscillation and Swinging of a bell, 398, 399 ; Vibration of a bell, 366, 398 Belting, 162 Benzene-vapour in space, 225, 451 Biaxial crystals, 499 Biology, 1 Biprism, 486 Biquartz, 510 Black, 516 Blackburn's pendulum, 187 Bladder, 257, 296 Blue colour of opalescent bodies, 453, 471 Blue-hot, 440 Boiling, 344 ; boiling-point, 210, 251, 358; b.-pt. at different pressures, 345 Bolometer (Thermic Balance), 634 Bombardon, 407 Bound charge, 528 Bourdon's steam gauge, 263 Bow (and arrow), 38 ; of Violin, 389, 395 Boyle's Law, 204, 222, 225, 289, 290, 301, 333, 351 ; in the Kinetic Theory, 222 ; in case of Vapours, 347 Breaking weight, 142, 232 British Units of Measurement (Foot, Pound, Second), 14 Brittle, 233 Bubbles in boiling, 344 ; soap electrified, 522 Bugle, 407 Bullet crushed, 138 Bumping, 345 CALLIPERS, 24 Calorescence, 456 Caloric, 315 Calorie, 317, 360 ; calorie, 360 Calorimeters, 361 Calorimetry, 360, 463 Camera obscura, 491 Camphor, 243, 510 Caoutchouc, contraction on heating, 324, 338 Capacity, 529, 537, 555 ; of a sphere, 632 Capillarity, 227, 245, 260 ; angle of, 245 Capillary Blood vessels, 281 ; Tubes, Flow through, 281 Capstan, 155 Carbon monoxide, 210, 212, 292 Carbonic anhydride, 207, 212, 226, 290, 292, (solid) 327, 335, 346 Carnot's Cycle, 351 ; C. 's ideal Engine, 343, 354; C.'s Function, 354; C.'s Principle, 354 Cascade (Leyden jars), 534 Cast iron, 227, 292 Cathetometer, 22 Causality, Law of, 3 Cause and effect, 3 Caustic by Reflexion, 117, 474 ; by Refrac- tion, 122, 123 (see Spherical Aberration) Cavendish's experiment, 179 Cells, Galvanic, 547 ; various forms, 549- 554, 590 ; cells coupled in surface, in tension, 548, 568 ; arrangement of, 569 ; E.M.D.P. of measured, 574; internal Resistance measured, 574 Centigrade thermometer, 358 Centimetre, 11 Centre of Figure, 67, 136, 144, 182, 235, 399 ; of Gravity, 68, 182 ; of Oscillation, 145 ; of Percussion, 145 " Centrifugal Force," 141 C.G.S. (Centigramme -Gramme -Second) or Absolute System of measurement of physi- cal quantities, a system based on the three fundamental units named, 519 Change of Elasticity (Heat), 349 Change of Motion, 6 Change of State, 209, 319-322 Change of Velocity, 16, 62 Character of Sound, 369 Charge, 46, 519, 527 ; Division of, 528 ; "free" and "bound," 528; Static charge of Conductor, 575 Charles's Law, 223, 225 Chemical Action as a source of electrical energy, 546 ; Affinity, 46, 228, 292, 321, 546, 548 ; Combination, 209 ; -- Effect of electric current, 585-592 ; of electrification, 584 ; Equivalence, 212; Equivalents, 46, 213 ; Formulae, 213; - Rays, 434 ; Work, 323 Chemistry, 1, 191, 193 Cheval-vapeur, 35, 575 Chimney roaring, 430 Chirp of insects, 368 Chlorophyll, 43, 454, 511 Chromatic Aberration, 485, 514 Circle of Reference in S.H.M., 75, 76 Circuit: Open, 545; Closed, 546; Gal- vanic, 547 ; Positive side of, 596 Circular Motion, 52, 72, 142 ; = 2 S.H.M.'s, 81 INDEX. 641 Circular Polarisation, 465, 502 ; detected, 505 Clamond's Thermo-electric Pile, 559 Clarionet, 405 Clepsydra, 28 Clerk Maxwell ; see Maxwell Clock, Energy in, 38 ; Pendulum, 28, 186, 338 ; Wheelwork in, 28 Closed Circuit, 546 Cobbler's Wax, 200 Cochlea, 421 Coefficient of Compressibility, 232; of Conductivity, 362 ; of Diffusibility, 250 ; of Elasticity, 235, 286, 287 ; of Elastic Restitution, 235 ; of Expansion, 337 ; of Extensibility, 231 ; of Induced Mag- netisation, 607 ; of Magnetic Induction, 607, 611 ; of Resistance to extension, 233, to shear 234, to torsion 234 ; of Restitution, in impact 140, in elasti- city 235; of Rigidity, 200, 234; of Solubility, 248 ; of Transpiration, 295 ; of Viscosity, 201, 274, 282 Coercitive Force, 606, 608, 610 Cohesion, 228 ; solids, 229 ; liquids, 226, 241, 308 Cohesion-figures, 248 Colour, 435, 436 ; by transmitted light, 450; by reflected light, 452 ; mixture of, 515 ; produced by interposed double-re- fracting lamina, 505 Coloured Light : simple, 436 ; compound, 438 Colours of Thin Films, 488 Colours of Metals, 453 Collodion films, 243 Colloids, 250 Combinational Tones, 426 Combining Proportions, 213 Combustion-equivalent, 320 Combustion, Heat of, 43, 318, 321 Comma, 377, 380 Commensurable, 88, 96 Common Light, 465, 505 Communicating Vases, 260 Compensation- Pendulum, 338 Compensator in Saccharimeter, 510 ; Babi- net's Compensator, 506 Complementary Colours, 439, 506, 515, 516 Complementary Distribution of Electricity, 520 Complex Harmonic Motion ; see Fourier- motion Complex Sound-waves, 386 Components of a Force, 134 ; of a Fourier- motion, 368 ; of a Velocity, 54 Component Tones of a Note, 370, 382 Composition of Forces, 134 Composition of Rotations, 69 ; of Trans- versal Vibrations, 100 ; of Uniform with Accelerated Motion, 65 ; of two Velo- cities, 54 ; of more than two, 59 Composition of S.H.M.'s, 80-96 /ompound Pendulum, 188 2 Compressibility, 232 ; of gases, 289 Concertina, 366, 395, 404 Condensation of gases, 205 ; condensation- temperature, 348 Condenser, 520, 532, 578 ; in submarine telegraphy, 578 Condenser and Source in heat-engine, 353 Conduction of Heat, 362 ; in gases, 223 Conductivity, Electrical, 561, 564; for Heat, 362, 536, 564 ; Coefficient of, 362 ; Variable , 564 Conductor of Frictional Electric Machine, 545 Conductors of Electricity, 526, 536 ; Cur- rent in homogeneous , 567 ; in hetero- geneous , 569 ; in wide , 571 ; bad of Heat, 364 ; Static Charge of, 534, 575 Conical Pendulum, 74 Conservation of Electricity name given by Lippmann to the doctrine of p. 521, lines 9-11 Conservation of Energy, 7, 40, 45, 203, 467 et passim. " Conservation of Force," 45, 135, 203 Conservation of Vires vivce, 467 Conservative System, 37, 167, 236 Constancy of Nature, 2 Contact of Metals (electricity), 543, 545 ; of Non-conductors, 544 Contact-breaker, 615 Contact in Friction, 161 Continuity between Liquids and Gases, 207, 226, 447; between Liquids and Solids, 210 Continuity, Law of, 266, 298, 536 Continuous Spectrum, 447 Convection of Heat, 364 ; convection-cur- rents, 364 Conventions : as to Attraction - and Re- pulsion + , 168; Standard Pitch, 372; as to Gravitation-Potential, 171 ; North- seeking pole of magnet Positive, 599, 634; as to Plane of Polarisation, 464 ; Vitreous electricity Positive, 519, 634 Convergent Lenses, 482, 483 Convergent Rays or Beam, 110 Cooling, during evaporation, 346 ; in still air, 364 ; Newton's law of, 364 Copper electronegative to zinc, positive in the battery, 543, 547 Cornet, 407 ; cornet of guttapercha, 407 Cornopean, 407 Corti, 422 Coulomb, 566, 632 Coulomb's Torsion -Balance, 540 Counterpoising of Forces, 31 Couple, 150 ; examples of, 151 ; Moment of, 151, 601, magnetic, 593 Crank, 82 Crest and Trough, 98 Critical State of Matter, 206 ; Temperature, 207, 211, 226 Crookes's Layer, 325, 327 Crooks of Trumpet, 407 T 642 INDEX. Crossed (Nicol's) .prisms, 506 Crushing, 232 Cryophorus, 345 Crystalline form, 225 Crystalloids, 250 Crystals, Axis of, 495 ; Biaxial, 499 ; Con- duction of heat in, 363 ; Expansion in, 337 ; Positive and negative doubly-re- fracting, 498 Currents, 525, 537, 561-598, 629 ; Direc- tion of, 547 ; Mutual action of Currents, 594 ; Current Sheet, 611 Curvature, 73 ; Radius of, 72, 73, 141 Curve of Sines, 78 Curved path, 51 ; Acceleration in, 74 ; Velocity in, 53 Curved Wave -Front: Reflexion of, 113 ; Refraction of, 120 Cycle, 351 ; Carnot's, 351 Cyclical order, 57, 60 DALTON'S Atomic Theory, 213 ; D.'s Law of gaseous pressures, 223, 225 Damping, 265, 274, 615, 621 Daniell's Cell : its various forms, 551 Dark-Heat-Waves, 434 Dark Lines in Spectrum, 445 ; in Heat- Spectrum, 633 Dead-beat, 621 Dead points of a Crank, 82 Decay, 320, 458 Declination, 604 Decomposition, 209 ; of Water by electro- lysis, 583 ; by a Grove cell, not by *- Daniell, 589. See Dissociation. Degradation of Energy, 356 Degrees of Freedom of a Particle, 66 ; of a Rigid Body, 69 Denser medium, Wave entering, 117 ; Wave leaving, 118 Densimeter, 197 Densities of gases, 216, 222, 289, 321 Density, measurement of, 195-198, 260, 261 ; measurement of Vapour - Density, 347 Density of Air, 289 ; of the Earth, 179 ; of Ether, 208 ; of Hydrogen, 198, 289 ; of Matter,..!^ of Water, 194, 198 ; maxi- mum D. of Water, 321 Density, Electrical superficial, 520, 537; Optical, 459 Derived Currents, 572 Dennis, 255 Deviation : minimum, 478 ; D. without dispersion, 479, 480 ; dispersion without D., 480 Dew, 349 ; dewpoint, 348 Dextro-rotatory, 509 Dialysis, 254 Diamagnetic, 606, 608 Diathermancy, 448 Diatonic scale, 375 Dicrotic pulse, 287 Dielectric, 526, 529, 532 Difference of Potential, 170, 523, 536, 537, 554; Electromotive D. P., 525, 561, 629; measured, 541 ; observed, 536-542 ; produced, 542-561 Differential Galvanometer, 624, 633 Differential Tones, 426 Diffraction, fig. 77 ; of Sound-waves, p. 412 ; of Light, etc., 492 Diffraction-grating, 438, 493 Diffusion of Gases, 220, 223, 294 ; of Li- quids, 220, 250 ; through Membranes, 296 Dimensions of Physical Quantities, 53, 537, 566, 630 Dimensions of Space, 9 Direct- vision Spectroscope, 480 Direction, 51 ; of Current, 547 ; of Electro- magnetic Lines of Force 594, of Mag- netic 600 ; of Sound, 425 Discord, 425 Discs, vibration of, 129, 397, 432 Dispersion, 218, 460, 478 ; abnormal, 481; Irrationality of, 480 ; Deviation without , 479, 480 ; without Deviation, 480 Disruptive Tension, 522 Dissipation of Energy ; see Availability (44) and Degradation (356) of E. Dissipation of Sound in air, 413, 417 Dissociation, 216, 220, 319, 330 Dissonance, 425 Distance, Action at a, 519 Divergent Lenses, 482, 484 ; Rays, 110 Dividing Engine, 24 Divisibility of Matter, 193, 211, 218 Doppler's Principle, 418, 437 Double-Bass, 404 Double Refraction, 202, 494 et seq., 533 Double - refracting Lamina, 500, 506; Power, detection of, 507 ; Double-refrac- tion Dynamometer, 508 Drum, 366, 398, 401 ; of Ear (membr. tympani), 419, 427 Dry bulb, 349 Dry pile. 553 Ductility, 232 Dulong and Petit's Law, 329 Duplex Telegraphy, 624 Dynamo-electric Machine ("dynamo"), 317, 559, 619 Dynamometer (Force), 32; (Energy), 47 ; Double-refraction , 508 Dyne, 18 EAR, the Human, 401, 419-424; the Ear- trumpet, 382, 414 Earth : Crust of, 202 ; Density, 179 ; Mass, 179 ; as a Magnet, 602, 611 ; Positive Pole the Southern, 611 ; its Potential zero, 526 ; Putting to , 578 ; Radii of, 181 ; Rotation of, 181 ; in Telegraphy, 572 : Variations of g over, 20, 180 Ebullition, 344 Echo, 413 Ecliptic, 74 Eddies, 273, 279 Effect, Cause and, 3 INDEX. 643 Efficiency of Heat-engine, 353 ; of Carnot's ideal engine, 354 Effusion of Gases, 294 Elastic Bodies, their Impact, 111, 140 ; E. intermediary, 50, 239 ; Tubes, Flow through, 285-288 ; E. Toughness, 236 Elasticity in Solids, 227, 233-239 ; in Gases, 203 ; Perfect and Imperfect , 236, 237; Limits of, 236 ; Fatigue of, 237 ; physi- ological examples, 238 ; mechanical ad- vantages, 50, 238. Electric of Dielec- tric, 537 Electric Attraction, 518, 535 ; Charge, 519, free 528, bound 528 ; Density, 520 ; Displacement of dielectric, 536 ; Elasticity of dielectric, 537 ; Energy, 42, 43 ; Force, 522, 537 ; Furnace, 583 ; Light, 437, 438, 582 ; Machines (frictional), 545 ; Matter (imaginary), 521 ; Quantity, 519 ; Screen, 535 ; Stress, 533 ; Tension, 522 Electricity not a form of Energy, 518, 530 Electrocapillarity, 554 ; Electrodynamome- ter, 633 ; Electrokinetic, 208, 527 ; Elec- trolysis, 43, 291, 585 ; Electrolyte, 547, 548, 550, 585, 590 (glass) ; Electromagnet, 32, 609 ; Electrometer, 218, 539 ; Elec- tromotor, 625 ; Electronegative, 543, 547 ; Electrophorus, 560 ; Electroplating, 585 ; Electropositive, 543, 547 ; Electro- scopes, 520, 537 ; Electrostatic, 208, 527; Electrostatic Units, see Units. Electrodes, 585 ; Polarisation of, 590 Electromagnetic Field of Force, 593, 597, 604 ; Induction, 611 ; Interrupter for tuning-forks, 237, 401, 622; Measure, 630, 632 ; Units, 564, 629, 632 ; Unit of Heat (= Watt), 317, 575 Element, Chemical, 191, 193, 447 ; Gal- vanic, 547 ; Geometrical, 52 Elliptical Motion = 2 S.H.M.'s, 83 Elliptically-polarised Light, 464, 502 ; de- tected, 505 Elongation in S.H.M., 76 E.M.D.P., Electromotive Difference of Po- tential ( = E. M. F. , Electromotive Force), 525 etc., 561, 629; measured, 547 Eudosmotic Equivalent, 253 Energy, 2, 35 ; Potential Energy, 36 ; Kin- etic E., 39. Availability of E., 44; Conservation of, 7, 40, 45, 467 ct passim; Degradation of, 356 ; Graphic representa- tion of , (areas], 46, 47-51, 350; Measure- ment of, 47; Storage of, 593 ; Transform- ations of, 41 ; Transport of, 593 nergy absorbed or evolved during Change of State, 211, 320 ; of Electrified Body, 530 ; of an Electric Current, 43, 574; in Impact, 140, 141; Intrinsic , 220, 319 ; of Jet, 271 ; of Molecules, 315 ; E. of Niagara Falls, 626; in case of Repulsion, 168, 169 ; of Rotation of a Particle, 143 ; of a Mass 144; of S.H.M., 133 : of Sound-waves, 429 ; of Sunlight, 432';' in Superposed \ Waves, 411 ; of Wave -motion, 133, 288, 368 Engine doing and not doing Work, 43 Entropy, 44 Epoch in S.H.M., 76 Equal Temperament, 381' Equalisation of a Current, 592 Equation to a Curve, 66 Equilibrium, 166 ; Stable, unstable, and neutral, 185 ; E, of Liquids, 255 Equipotential Surfaces, 172-177, 2B8, 522, 536, 593, 597, 600, 604 Equivalence of Forces, 3 Equivalence, chemical, 212 ; equivalents, 46, 213 Erg, 34 ; Ergten, 35 Essential Properties of Matter, 190 Ether, the Luminiferous, 207 ; its Density, 208, 460, 470 ; its Elasticity, 208, 460, 470 ; Transverse Vibrations of, 432, 460 Ether-Stress, 208, 518, 536 Ether- Waves, 315, 431, 458 ; their Length measured, 488 ; their Pressure, 512 Euphonium, 407 Evaporation, 210, 344, 554 ; Cooling dur- ing, 346 ; of Ice, 347 ; Latent Heat of , 346 Evolution or absorption of Energy during Change of State, 320 Exchange of Radiations, 441 Expansion of Gases. Indefinite, 202, 291, 301 ; explained, 221 Expansion by Heat, Coefficient of, 337 ; Measurement of, 339-342 ; examples of , 338 ; in hollow bodies, 338 Experimentation, 4 Exploration of Air - Potential, 559 ; of Fluid Pressure, 264 Extensibility, 230 ; Coefficient of, 231 Extension of Matter in Three Dimensions, 192 Exterior Work done by Heat, 323 Extraordinary Ray, 496, 498 Eye, 486, 512 FAHRENHEIT Thermometer, 359 ; F. Zero, 344, 358 Fall of Potential, 567, 569 Fall under Gravitation, 179 Farad, 632 Faraday's Laws of Electrolysis, 587 Faure's Accumulators, 591 Federmanometer, 264 Fennat's Law, 125 Ferromagnetic, 606 Field of Force, 176; electric, 522, 527, 529; electromagnetic, 593, 597, 604, 613 ; magnetic, 600 Fife, 406 Films, Colours of, 488 Filtration, 302 Fish, air bladder of, 289 ; luminosity of, 458 Fixity of Proj jrtion in chemical com- pounds, 212 INDEX. riple, 461 06 57, 516, 582 ; as a Reflector, ig and Sensitive, 408 13 >rs, 476 ; Lenses, 484 :ical Charge, 536 299, 301 362, 363, 536 : Is, 265-288 ; Forces producing in suspended Loops, 267, rough rigid Tubes, 275-281; lary Tubes, 281 ; in Elastic -288 yerature, 363 436, 438, 454 tion, 53 145 of Mirror ( distance of priii- | 474 ; of Lens, 482 Approximate Focus, 122, 474 ; ror, 474 ; principal , 474 ; 474 ; _ of Lens, 482 ; real, lal, 482 ; principal, 482 ; con- (3 ; Heat Focus and Photo- cus, 485 ; Focus of Eye, 486 ieflexion of Light from, 454 ; mgh, 413 34, 35 ; foot-poundal, 35 , 41, 105, 148 ; said to Act ies 18, 34, to Do Work 34, r e Work done against it 34 ; .gal ," 141-143 ; "Conserva- ," 45, 135 ; Field of , 176, 529, electromagnetic 593, 597, magnetic 600 ; producing ! ; Graphic representation of " ; Lines of , 175, 522, elec- ic 593, 597, direction of 594; 600, their direction 600 ; , 599 ; Measurement of , ; Parallelipipedon of s, 134 ; ram of s, 134 ; Polygon of ; .Force a Bate of Change of n, 18, 221 ; Resultant Electric 537 ; Triangle of s, 136 ; h-, 175, 522, 536 fctions, 398-402, 432 ;ter, 192 ; Perception of , 516 rinciple, 462 ion (a periodic motion com- f commensurate S.H.M.'s), 96 Theorem), 366, 368, 371, 387, eorem, 96 ; applied to vibrating !7 .ines, 436, 446, / 493 528 on, 388-398, 4^2 :ture, 209 ; ?oint, 358 French Horn, 407 Frequency, 105, 366, 368, 388, 432 ; of Ether-waves, 433, 437, 467 Fresnel, 467, 469, 471, 487 ; Fresnel's Ehomb, 503 Friction, 159, 316, 350, 389 ; Coefficient of Statical , 159 ; Coefficient of Kinetical , 161 ; in special cases, 162-165 ; between air and a falling body, 179, 299; within fluids, 273 ; Wheels, 165 Frictional Electric machines, 545 Fringes, 132, 133 Frothing, 247 Fusing point affected by Pressure, 211, 343 Fusion, 324, 343 GALILEO'S Principle, 4 ; his Doublet, 513 Gallon, 13 Galvanic Cell, 317 ; Galvanic Battery, Cell, Circuit, Element, Pile, 547 Galvanometer, 566, 620, 633 ; Differential , 624, 634 Gas, 202, 206 ; Radiations of, 446 Gas-battery, 590 Gate on its hinges, 153, 190 Gaugain's Curves, 557, 558 Gauges, 24 ; Steam Gauge, 264 Gauss, 601 Geissler's Vacuum-tubes, 583 General properties of Matter, 193 Glacier-flow, 338, 344 Gramme, 13 ; gramme-atom, gramme-mole- cule, 329 ; gramme -equivalent, 588 j Graphic Formulae (chem.), 217 I Graphic representation of Energy, 46, 35 ; of Force, 46 Gravitation, 17, 37, 219 ; Law of , 178 ; Potential, 171, 174 ; Universal , 180 Gravity, Acceleration g due to, 20 ; Variations of g, 20, 180 ; Measurement of g, 29, 181 ; Centre of , 68, 182 ; effect of on Flow, 267 Grays, 516 Grove's Cell, 552 Guebhard, 589 Guitar, 403 Gun, 6 Gunpowder, 38 Gyration, Radius of, 144 HAEMODROMOGRAPH, 284 Haemodromometre, 284 Haemotachy metre, 284 Hall's experiment, 628 Halo, 479 Hammock, 138 Hardness, 230 Harmonic Curve, 78 ; Engine, 429 ; Telegraphy, 624 Harmonic Motion, 74 ; Acceleration in, proportional f,o Displacement, 76, 388 . Amplitude in, 76, oc v 107 ; Angulai Velocity in, 77 ; Circle of Reference in 76; Composition of, 80-96; H.M.' INDEX. 645 compounded by Blackburn's pendulum, 187, by vibrating reeds 396, by vibrat- ing strings 388 ; due to Elasticity, 237 ; Elongation in , 76 ; Energy in, vary- ing as (amplitude) 2 , 133 ; Isochronous, 77 ; Pendulum -movement, 80 ; Period in H.M., 76 ; Phase in, 76 ; Projection of, 77 ; Resolution of, 89, 94, 96 Harmonicon, 395 Harmonics, 370, 382-4, 425, 427 Harmonium, 366, 381, 395, 404, 425, 427 Harmony, 425, 427 Harp, 403 Harpsichord, 390, 403 Haze, 454, 471 Head of water, 268, 2/2 ; of a gas, 294, 298 Heart, Valves of, 310, 384 ; Work done by, 285 Heat, 41, 221, ^3Mr^tomic , 329 ; Conduction of, 362, in gases 223 ; produced by Electric Current, 579; Flow of , 362 ; Focus, 484 ; Material theory of, _315-r~Moleeular , 329 ; Radiant , 42, 433 ; , Rays, 434 ; Specific , 316, 328; -Spectrum, 438, 451, 482, 633 ; Transference of, 362 Heavy, 190 ; Heavy Liquids, 257 V 2 Height to which a body rises = ; equa- tion iii., p. 146, when v = and a = - g Henry's Law, 293 Heterogeneous Conductors, 569 Hollow body, Expansion of, 338 Homogeneous Conductors, 567 Homogeneous Light, 437 Hooke's Law, 228, 235, 236, 426 Horizontal Component of Terrestrial Mag- netic Force, 602, 605 Horn-bauds, Russian, 402 Horsepower, 35, 575 Hugher, 608 Humidity, 349, 416 Hydraulic Press, 18, 256, 297 ; Ram, 137, 278 ; Tourniquet, 272 Hydrogen, a possible basis of elements, 191 ; a metal, 291 ; alloy with palla- dium, 291 ; Condensation of, 206 ; Dens- ity of, 198, 289 ; Heat evolved on Combustion of, 318, 321, 329 ; Mole- cular Velocity of, 223, 418 ; Self-repul- sion of, 226, 336 ; Sound-waves in, 367 ; in Organ-pipe, 405 Hydrometer, 196 Hydrostatic Paradox, 259 ; Pendulum, 284, 300 ; Pressure, 256, 536 ; Stress, 199, 229, 267 Hygrometer, 349 Hypertrophy of Heart, 277, 288 Hypotheses, 8 ICE, Fusion of 324, affected by pressure 343; Latent Heat of 314, 324; Viscos- \ ity of, 344 I Iceland Spar, 495, 496, 497 ; as a Dielec- tric, 533 Image in Mirror, 472 ; real, 474 ; virtual, 474, 475 ; of Lens, real 484, virtual 484 Imaginary Electric Matter, 521 Imaginary Magnetic Matter, 600 Imbibition, 250 Impact, 139-141 Impenetrability of Matter, 192 Imperfect Elasticity, 236, 237 Incandescent Lamps, 352 Incidence, Angle of, 113, 119 ; Plane of , 467, 471 Inclination (magnetic), 605 Inclined Plane, 155 ; FaU down, 156 ; Pull up an , 163 Incus, 420 Indiarubber 237, 324, 338, 511 Indestructibility of Energy, 7 ; of Mat- ter, 7, 192 Index of Refraction, 120, 467, 471, 478, 488 Indicator Diagram, 48, 350 Induced Magnetisation, Coefficient of, 607 Induction : -Coil, 583, 615 ; Electric , 531 ; inductive capacity, xx, 533, 536, 537; Electromagnetic, 611-628, ^9, Magnetic , 606 ; lines of magnetic , 607 ; coefficient of , 607, 611 ; general problem of magnetic ,610 Inertia, 5, 136; examples, 136; Moment of, 144, 602 Infinite ( oo ), greater than any assignable number or quantity, 52 Insulation, 522, 527 Intensity of Current, 561, 566 ; measured, 574 ; of wave-motion at a Focus, 110 ; Magnetic , 600 ; of magnetisation, 601, 607 ; of Radiation, 463 ; of Sound, 368 Intensity of S.H.M. (oc Energy or (Ampli- tude)-), 368 Interference of Waves, 129; of Ether- waves, 486 et seq. ; of Sound-waves, 414, 416 Intermolecular, 323 Internal Work, 322, 332, 333-6 ; in Air, etc., 335 Intervals (musical), 376 Intramolecular, 320, 323 Intrinsic Energy, 42, 220, 319 Iodine, 449, 450, 451 Iridescence, 490 Irradiation, 516 Irrationality of Dispersion, 480 Isochronous Oscillations of Elastic body, 237 ; of Pendulum, 28, 187 ; S. H. M. 's, 77 Isothermal Lines, 352 ; Surfaces, 363, 536 Isotropic, 106, 494 JELLY, a Solid, 200 ; the Ether like a , 209 Jet, 66, 246, 268, 270, 'tfl Joule, 41, 316; his Equivalent, 41, 317; INDEX. :periments on Gases, 334-6 ; his f the Heat produced by a Current, , Satellites, 74, 461 onation, 376, 428 SCOPE, 473 rperiment, 628. nging, 430 ime, 12; Kilogramme-metre, 35 ics, 51 ; Kinetics, 134 Energy, 39 ; Friction, 161 ; T of Gases, 218, 219-223, ap- :o Radiometer 324, to Spheroidal 126, to Velocity of Sound 417 !*s Laws, 573 8 Manometric Capsule, 385, 397, 405 i DTATORY, 509 sk, 409, 449 cope, 473 138, 386, 428, 429 [eat, 209, 314, 324, 327 ; . iansion, 333, 336 ; - methods mmetry, 361 Clark's Cell, 552 Continuity, 266, 298, 536 ; Law of c Attraction and Repulsion, 519, Law of Gravitation, 178 ; Law of ".tic Attraction and Repulsion, 599, Laws of Motion. 5 .Nature, 1, 5 :,e, 550 tdof the Current, 594 ded Polarised, 464, 466 Measurement of, 21 ; Units of , 11 [82 ; achromatic, 485 ; convergent, 83 ; divergent, 482, 484 ; flexible, ! aw, 613 water, 260 ; Potential analogous 52-154 jar, 533, 534, 539 <& 208, 433 ; production of, 457 ; I electric lamps, 582 ; monochro- 437, 487, 490 ; .polarised, 464 ; |>n or natural, 4657 505 TMaxwell's ', 208, 460, 629 Jjective), 190 ig - conductor, 576 ; flash, dis- of, 417 Angle (in statical friction), 160 f Elasticity, 228, 237 ; of Hearing, r aves, 97 ; Reflexion of, 110 o Pressure (Indicator-Diagram), 49 tiabatic, 352 ; Isothermal, 352 now, 266, 536; of Heat, 363 Force, 175, 268 ; Electric, 522, Clectronagnetic, 593, 597, direc- 594, c it across 614, 615 ; Mag- 500, direction of 600 Lines of Magnetic Induction, 607 Lines of Propagation of Heat, 363 Line-Spectrum, 446, 448 Liquefaction, 209, 321 Liquid, 226, 255-288 Lissajous' Figures; e.g., figs. 35-41 Litre, 12 Loading a Vibrating body, 398 Locomotive Pulse, 279 Logarithmic Decrement (viscosity), 275 Logarithmic Increment (mus. pitch), 378, 380 Longitudinal Vibration, 99, 103, 394 ; Compression and Displacement in, 104 Loops and Nodes, 127 Loops, suspended, liquid in, 259, 267, 308 Loss of Energy of Electrification, 530 Loss of half a Wave-length, 118, 468, 489 Loudness, 367, 368, 369, 381, 424 Luminiferous Ether, 207 (see Ether). Lungs, 294, 300, 302, 304 Lustre, 517 Lyre, 403 MACHINES, Simple, 152 Magdeburg Hemispheres, 302 Magnet, 598 ; Solenoidal, 599 ; how made, 609 Magnetic Axis, 598 ; Declination, 604 ; Dip, 605 ; Equator, 605 ; Ether- vortices, 208, 629 ; Field, 600, 604, 619 ; its nature, 628 ; Force, 599 ; Inclination, 605 ; Induction, 606 ; coefft. of magn. ind., 607, 608 ; general problem of magnetic induction, 610 ; Intensity, 600 ; Meridian, 603, 605 ; Moment, 601, 602 ; of shell, 603 ; - North, 604 ; Parallels, 605 ; Pole (terrestrial), 605 ; (of magnet), 599 ; Potential, 603 ; Rotatory Polarisation, I 628 ; Screen, 610 ; Shell. 603, 604 ; strength of 604, potential of 632 ; - Strength, 600 ; Storms, 606 ; - Tick, 609 Magnetisation : Induced , Coefft. of, 607, intensity of 601 ; Residual , 606 ; Weber's Theory of, 608 Magnetism, 32, 598-611 ; nature of, 598, 611 ; terrestrial, 604-606 Magneto-electric Machines, 559, 616 Malleability, 232 Malleus, 419, 420 Manometer, 262, 289, 297 ; manometre metallique inscripteur, 264 ; m. compen- sateur, 265 Marie-Davy's Cell, 552 Marine Engine, 82 Mass, 12, 190, 219 ; Measurement of, 29 Massive, 190 Matter, 2, 7, 190 ; its Constitution : chemi- cal views, 211 ; physical views, 217 ; Essential properties of , 190 ; extension of, 192 ; iirrjenetrability of, 192 ; in- destructibility -rf, 192 ; General proper- ties of: inertia, o, 136, 193 ; weight, 12, INDEX. 647 19, 178, 193; divisibility, 19&, 211, 218 ; porosity, 194 ; density, 194 ; Per- ception of Matter, 190, 219 Matter, Imaginary, Electric 521, Mag- netic 600 Maxwell on Coercitive Force, 610 ; on the Conservation of Energy in Physiology, 45; 's "Demon," 45; Discs, 515; Theory of Light, 208, 460, 629 ; Theory of the Magnetic Field, 629 Measurement, 1, 21 ; of Adhesion, 31 ; Calorimetric , 360 ; of Density, 340 ; of E.M.D.P., 574; of Expansion, 340 ; of Flow of liquids, 282 ; of Flow of gases, 299 ; of Force, 29-33 ; of g, the acceleration of Gravity, 30, 181 ; of Intensity of current, 574, 633 ; of Length, 21 ; of Liquid Pressure, 261- 265 ; of Mass, 29 ; of Magnetic Mo- ment, 602, 605 ; of II, the Atmos- pheric Pressure, 310 ; of Pressure in a Stream, 282 ; of Refractive Indices, 477, 488 ; of Resistance of a Battery, 574 ; of Resistance to electric Current, 573 ; of Soap-film Tension, 31 , 244 ; of Surface, 26 ; of Temperature, 356 ; of Time, 28 ; of v, the velocity of propagation of an electromagnetic disturbance, 630 ; of Va- pour-density, 347; of Velocity of Liquid Stream, 282 ; of Velocity of Wave- motion in a rod, 128 ; of Volume, 27 ; of Wave-Lengths in Ether, 488, 493 ; of Weight (spring-balance), 31 Mechanical Equivalent of Heat, 317 ; of Sound, 429 Mechanical Powers, 152-159 ; with friction, 163 Medium, 37 Megalerg, 34 Megavolt, 631 Megohm, 631 Melde's Experiments, 393 Melting point of Ice, 343 Membrana Basilaris, 422, 425 ; m. Tym- pani, 419, 427 Membrane : Diffusion through, 252, 296 ; Vibration of, free 129, 398, forced 401 Mercury, Expansion of, 340 ; Thermo- meter, 358 ; Vapour, 216, 418 Meridian, 9 ; Magnetic , 603, 605 Metallic Reflexion, 503 Metals, Colours of, 453 ; Contact of, 543 Meteoric Dust, 43 ; Meteorites, 43 Method of Mixtures, 360 Method of Oscillations, 33 Metre, 10 Mica, 506 Microfarad, 632 Microhm, 631 Microphone, 565, 623 Microscope, 513 Microvolt, 631 Minimum Deviation, 478 Mirrors, 472 ; concave, 474 ; convex, 476; flexible, ''476; rotated, 473 Miscibility of Liquids, 249 Mixture of Colours, 515 Mobile Liquid, 200 Modulus of Superficial Tension, 244 Molecules, 192, 214, 215, 218 ; Nature of, 219 ; NSnTBer of, 224 ; Free Path of, 223, 224 ; Size of, 218, 223 ; Velocity of , 222 ; in Vibration, 225, 317, 432, 457 Molecular Asymmetry, 217, 432, 510 ; Dis- tance of Action, 246 ; Forces, 225- 228 ; Heat, 329 ; Kinetic Energy, 324 ; Refractive Power, 478 Moment : of a Couple, 151 ; of a Force, 146, 149; of Inertia, 144, 602; Magnetic , 601 ; measurement, 602 ; Principle of M.'s, 149, 152 Momentum, 6, 17, 139, of a System 139 ; in Impact, 139-141 Monochord, 390, 402 Monochromatic Light, 437, 486, 490 Morse-code, 621 Motion, 14 ; in a Circle, 52, 72 ; in Curved paths, 53 ; Laws of , 5 ; The Perpetual , 7, 173, 320, 343 ; parallel to or across Equipotential Surfaces, 173, 175 Multiple Proportions, Laws of, 212 Muscle : Breaking- weight of, 232 ; Diffrac- tion - grating, 493 ; Extensibility of , 231 ; Mechanical disadvantage, 153 ; - -Sound, 384, 424 Musical Box, 404 ; Intervals, 376 ; Nota- tion, 377 ; Pitch, 373 NEGATIVE Crystals, 498 Negative Electric Charge, 519 (resinous) Negative Pole of Magnet (south seeking), 599 Negative Pressure in Thorax, 304 Nerve-ends, 422, 514 Neumann and MacCullagh, 470 Neutral Line, 283 Neutral Point (thermo-elect.), 556 Newton's Laws of Motion, 5, 136, 142; his Law of Cooling, 364 ; Law of Uni- versal Gravitation, 180 ; his Law of v = \/K + p, 238, 331 ; Newton's Rings, 490 Niagara Falls, 316 ; Energy of, 627 Nicol's Prism, 499 Nobili's Rings, 589 Nodal Lines, 397 ; Nodal Points, 101. 126 Nodes and Loops, 126 ; in a Membrane, 129 ; on Monochord, 393 ; in an Organ-pipe, 405 ; on a vibrating String, 392, 395 Noe's Thermo-electric Pile, 559 Noise, 370 Non-commensurable, 88, 94 Non-conductors, 526 ; Contact of, 544 Non-conservative System, 38 Normal to Wave-front, 106 ; Propagation along , 106, 123, 130 Note, 371 OBOE, 405 Ohm, 317, 564, 631 648 INDEX. Ohm's Law, 562, 568, 577 Oil in painting, 452 Opacity, 448 Opalescence, 453 Open Circuit, 545 Opera glass, 513 Ophicleide, 407 Ophthalmoscope, 513 Opposition of Phase, 399 Optical Density, 459 ; 0. measurement of Length, 26 Ordinary Kay, 496, 498 Ordinate, 11 Organ : -pipe, 366, 399 ; open, 405, 410, 415 ; stopped, 407 ; reed-pipe, 383, 404 Oscillation, 33, 368 ; as a means of mea- suring Force, 33 ; Centre of , 145 ; Frequency, 388 ; s of a mercury- column, 137 ; of Molecules, 315 Oscillatory Movements of Systems of Par- ticles, 97-133 Osculating Circle, 73 Osmosis, 252 Ounce, 13 Outflow, 268, 272 ; from elastic tubes, 288 Overturning, 183-185 Oxygen, condensation of, 205 Ozone, condensation of, 206 ; formation of, 585 PACINOTTI'S Inductor, 617 Palladium, 291, 590 Parabolic Mirrors, 115, 474 ; Path, 65 ; Surface of rotating liquid, 261 Parallel Beam, 110 Parallelipipedon of Accelerations, 62 ; of Forces, 134 ; of Velocities, 61 Parallelogram of Accelerations, 62 ; of Forces, 134; experimental proof, 135; of Velocities, 54 Partially-polarised, 466, 471 ; detected, 505 Partials (i.e., Harmonics, q.v.), 427 Pascal's Principle, 256, 297 Peltier's Effect, 580, 581 Pendulum, 28, 30, 138, 182, 185 ; Black- burn's, 187 ; Compensation , 338 ; Com- pound , 188; Conical, 74; oscilla- tions Isochronous, 28, 187 ; movement Harmonic, 79, 186 ; Length of Simple , 187, of Compound 189 ; Simple, 182, 185 ; Work done in moving a simple ,187 Penumbra, 491 Perception of Colour, 514 ; of Form, 516 ; of Matter, 190, 219 Percussion, Centre of, 145 Perfect Conductors and Non-conductors. 526, 527' Perfect Elasticity, 236 Perfect Engine, Carnot's, 343, 354 Perfect gas : defined, 331 ; Internal Work in, 333 ; No physical Gas Perfect, 335 Perfectly Conducting Molecules, 611 * Period of S.H.M., 76, 368; of Wave- motion, 105 Periodic Curve, 96 ; Function, 76 Perpetual Motion, The, 7, 173, 320, 343 Phase in S.H.M., 76 Phonautograph, 386 Phonograph, 387, 401 Phonomotor, 411 Phosphorescence, 454 Phosphoroscope, 455 Phosphorus, 249 ; vapour, 216 Photographic Focus, 485 Photographing Vibrations, 386 Photometry, 463 Photophone, 565, 623 Physiology, 1 Pianoforte, 370, 374, 381, 390, 403 Piccolo, 406 Piezometer-tube, 262, 275, 298 Pigments, 516 Pile, Galvanic, 547 ; Thermo-electric, 486, 558, 559 Pint, 13 Pipette, 306 Piston, 34, 49, 203, 211, 355 Pitch of a Screw, 156 Pitch of a Sound, 369, 371 ; musical specification, 373 ; physical, 371 ; of vibrating String, 388, 394 ; Variations of pitch in an Organ-pipe, 406 ; varia- tions of with Temperature, 406 Pitot's Tubes, 284, 300 Plane in Space, Pendulum swinging in fixed, 138 ; Wheel rotating, 138 Plane of Incidence, 467, 471 Plane of Polarisation, 464, 467, 471, 498 Plane-polarised, 464, 499, .501 ; detected, 505 ; by Reflexion, 471 Plane Wave-front, 108 ; reflected, 112 Plant6, 591 Plasticity, 233 Plates, Vibration of, 397 Pneumothorax, 300 Poiseuille's Law, 275, 281 Polarisation, 463, 549; Plane, Elliptical, Circular, 464; Partial, Rotatory, 466; Angle of Complete , 470 ; Plane of , 464, 467, 471, 498 Polarisation of Electrodes, 590; P. -Current, 590 Polarised Light, 103,464; to detect, 505, 506 Polariser, 465, 499 Pole : Magnetic of Earth, 611 ; of Magnet, 599, 611 ; Secondary, 601 Polygon of Accelerations, 62 ; of Forces, 136 ; of Velocities, 60, 62 ; gauche poly gone, 62 Porosity, 194 Positive Crystals, 498 ; - - Direction of electromagnetic Lines of Force, 594, of magnetic 600 ; Electrical Charge (vitreous), 519 ; Pole of the Earth (Antarctic), 611 ; Pole of a Magnet (North-seeking), 599 ; Side of a Cir- * cuit, 596 INDEX. 649 Potential, 46, 170, 268, 522, 540 ; Analogy of Sea-Level, 171, of water-level 525 ; Absolute , 523 ; Absolute Zero of , 171; Arbitrary, 171; Continuity through Zero-value, 171; -Difference, 523, 536, 537 ; measured, 541 ; produced, 542-561 ; Electromotive Difference of , 525, 561 ; Electrical , 523 ; Gravita- tion , 171, 174; Line, 567, 570, 574, 577; Magnetic , 603; of Magnetic Shell, 604, 633 ; Positive and Negative , 170 ; The , 526, 542 ; Zero , 171 Potential Energy, 36-39 ; in case of Repul- sion, 168, 169, 170 Pound, 12 ; Poundal, 18, 19 Powdered Transparent Substance, 448 Practical Electrical Units, 549, 564, 566, 631 Pressure : Atmospheric, 203, 259, 300-313 ; of Ether-waves, 512; in a (ias = coefft. of elast, 349 ; in Gases, 203, 297 ; by Kinetic Theory, 221 ; pressure and volume in gases, 204 ; produced by Heated Solids, 336 ; diminished in Fluid in Motion, 299 ; -Head, 276, 298 ; Line, 277, 298 ; in Liquids, 256, 536 ; in Heavy Liquids, 257 ; measure- ment, 261-265 ; Saturation , 346 ; in Streams, 275-282 Prevost's Law, 443 Principal Focus of Mirror, 474 ; of Lens, 482 Principal Section of a Crystal, 495 Principle of Moments, 149, 152 Prism, 438, 445, 477 ; Nicol's, 499 Projectile, Path of, 179 Projection, 77; of S.H.M., 78 Proof-plane, 540 ; Proof-sphere, 540 Propagation of Sound, 409-418 ; of Waves along Normals, 106, 123, 130 Properties of Matter : Contingent, 194 ; Essential, 190 ; General, 193 Proportion, Fixity of (chem.), 212; Mul- tiple Proportions, 212 Ptolemy's Law, 125 Pulleys, 157 Pulse-wave, 286 Pump, 306, 309 Putting to Earth, 578 Pythagorean Intervals, 380 QUADRANT Electrometer, 539 Quality of Sound, 369, 382 Quantity : Cells arranged in, 568 ; Electric, 519, 537 ; Unit of, (C.G.S. Electrostatic) 519, (C.G.S. Electromagnetic) 632, (Practical) 566, 632 ; of Matter = Mass, 12; of Motion ( = Momentum), 6, Quarter-undulation plate, 503, 505 Quartz as a Dielectric, 533 ; Prisms and Lenses, 438, 449 ; in Rotary Polarisa- tion, 508 RADIANT Heat, 42, 208, 314, 365, 433, 486 Radiant Matter, 207, 584 Radiation, 365, 431 ; from a hot body, 440 ; from Gas and Vapour, 446, 447 ; from Liquids and Solids, 447 ; Exchange of Radiations, 441 Radii of Earth, 181 Radiometer, 324 Radiophony, 409 Radius of Curvature, 73, 141 ; of Gyration, 144 Railway Wheels, 163 Rainbow, 479 Raindrop, 227, 479 ; Friction on, 164 Rarefied air, Sound in, 367 Rate of Change of Momentum = Force, 18, 221 Ray, 110, 123; Kinds of Radiation, 434; Ordinary and Extraordinary, 496, 504 Reaction, 6 Real image, 475, 484 Reciprocal or Reciprocating Motion, 81, 82 Recoil, 272, 298 Reduced Length, 565, 569 ; Resistance, 565, 570 Reduplication, Principle of (in pulleys), 158 Reeds, vibrating, 395, 396, 404 Reference to Axes, 60, 62 Reflexion, 110-117 ; Angle of, 1 3, 467 ; of Ether- waves, 466 ; Metallic , 503 ; of Sound-waves, 411, 412 ; Total , 468 Refraction, 119-122 ; Angle of , 120, 467; of Ether-waves, 466, 476; Index of , 120, 467, 471, 478, 488 ; of gas 478, of liquid 477, of solid 477 ; of Sound-waves, 414 ; Total -470 Refractive Power, Molecular, 478 Regelation, 344 Relaxation, Time of, 202 Relay, 622 Repose, Angle of, 160 Repulsion conventionally Positive, 168 ; Direction of, 524 ; Potential Energy in case of, 168 ; Work done by, 168 Residual Magnetisation, 606, 620 Resinous (negative), 519 Resistance to Electric Current, 561, mea- sured 673 ; to Deformation, Shear, Torsion, 233, 234; to Flow, 273, 276 ; to Traction, 163 Resolution of Forces, 134 ; of S.H.M.'s, 89, 94, 96 ; of Velocities, 57 ; of Vibrations, longitudinal 103, transversal 100 Resonance, 400 Resonators, 371, 383, 384, 385, 429 Restitution-pressure, 228, 235; Coefft. of R., 140, 235 Resultant Harmonic Motion, 95 ; Motion, 54 ; _ Force, 134 Retina, 514 650 INDEX. Reversal in Thermo-electricity, 557, 581 Reversed Action, 625 Reversibility of Carnot's ideal Engine, 353 Rheochord, Rheostat, 574 Right-handed Polarised, 464, 466 Rigid Body, 67 ; degrees of freedom of, 69 ; Solid, 199 Rigidity, 200, 201, 233 ; Coefficient of, 200, 234 ; affecting Vibrations, 389 Rods, vibrations of, 128, 395-6 Rolling Friction, 165 Rope and post, Friction between, 164 ; Wetted rope, 250 Rotating Mirror, 473 Rotation, 68 ; Composition of s, 69 ; of the Earth, 143, .181 ; Energy of of a particle, 143, of a mass 144 ; Force causing constant in direction, 146 ; Instantaneous Axis of , 69 ; of a Liquid, 261 ; of Molecules, 315, 418 ; - of Plane of Polarisation, 217 ; of Plane of S.H.M., 94 ; Axis of Spontaneous , 146 ; of a Wedge, 164 Rotatory Polarisation, 466, 508, magnetic 628 ; Power, 509 ; Vibration, 396 Rowland, 493, 630 Rupert's Drops, 227 SACCHARIMETER, 509; Soleil's, 510' Sand-blast, 230 Saponiue, '247 Sassafras, 490 Saturated Solution, 249 ; Vapour, 205, sp. heat of 332 Saturation-pressure, 346 Savart's Wheel, 372 Scale (musical), 375 Screen, Electric, 535 ; Magnetic, 610 ; Effect of S. on Waves, 131, 132 Screw, 23, 156, 163 ; male and female, 24 Sea-level, Analogy in Potential, 171 Sealing-wax a Fluid, 200 Second, 9 Secondary Batteries, Cells, 590-2 ; S. Cur- rent, (Induction) 613, (Polarisation) 590-592 ; S. Poles, 602 Selective Absorption, 449 Selenite, 506 Selenium, 511, 565 Self-induction : Currents, 576, 615 ; Mag- netic, 607 Self- Repulsion, 522 ; of Hydrogen, 226, 336 Semitone, true (|f ), 376 ; so-called (iff), 377 Sensitive Flames, 408 Separation of Electricities, 536, 542 Series, Cells coupled in, 548 Sextant -Vernier, 22 Shadow, 491 S.H.M., Simple Harmonic Motion, 74. See Harmonic Motion. Shear, 71 Shell, Magnetic, 603 ; potential of, 604, 633 Sidereal Time, 9 Siemens's Governor, 143 Simple Machines, 152 Simple Pendulum, 182, 185 Simple Rotation, 69 Simple Translation, 67 Simultaneous Causes, 4 ; Currents, 623 ; Forces, 134 ; Motions, 4, 54 Sine Galvanometer, 620 Sines, Curve of, 781 Singing Flames, 408 Siphon, 308 Sky, 454 Sliding, 160, 185 Slope of Potential-Line, 567, 570, 571 Soap-film, 31, 218, 242, 294, 449, 489 Sodium-flame, 435, 445, 453 Soft Iron, 606 Softness, 230 Soft Solid, 200 Solar Time, 9 Soleil's Saccharimeter, 510 Solenoids, 598 ; Solenoidal Magnet, 599 Solid, 199, 229-240 ; rigid, 199 ; soft, 200 Solution in Liquids : of solids, 248, 322 ; of gases, 292, 321 ; coefficient of solu- bility, 248 Solution of Gases in solids, 291 ; of Solids in gases, 207 Solution : saturated, 243 ; supersaturated, 249 Sonorescence, 512 Sound, 42, 366 ; Analysis of, 382 ; Propa- gation of, 381-2, 409-418 ; Velocity in steel, 238 ; Waves in air, 367, 386, 410, 417, 623 Sounding-board, 367 Source and Condenser, 353 Space, 9 ; Dimensions of, 9 ; traversed under uniform Acceleration, 63 Spark, 521, 522, 535 Speaking-trumpet, 368 Specific Conductivity, 562 ; - - Density, 194 ; measurement of, 195-198, 260, 261 ; Gravity, 194 ; Sp. gr. bulbs, 198 ; Heat,^316, 328, 360 ; at const, vol., 330, 333 ; at const, pressure, 330, 333 ; ratio of spec, heats, 331 ; In- ductive Capacity, xx, 533, 536, 537 ; - Resistance, 562 Spectrum, 438, 478 ; abnormal, 481 ; Dif- fraction , 493 Spectrum- Analysis, 191, 446 Spherical Aberration of lens, 485 ; of mirror, 474 . _ Form, 227 ; Magnet, 602 ; - Mirrors, 116, 474 Spheroidal State, 326, 554 Spherorneter, 25 Sphygmoscope, 264 Spirometer, 299 Sprengel, 291 Spring-balance, 31 Standards : See Units ; Resistance-coil, 564, 573 Stapes, 420 Stars, Twinkling of, 493 States of Matter, 199 INDEX. 651 Static Charge, f>75 Statical Friction, 159 Stationary Vibrations, 126 Steady Flow of Heat, 362 Steam, Sp. Heat of, 332 ; Jet issuing into air, 334 ; Steam-engine, 316, 355 Stereoscope, 517 Stethoscope, 382, 409 Stokes's Law, 444 Stopping Component Vibrations, 127 Storage of Energy, 592 Straight Path, 5, 51 Strain, 70 ; homogeneous, 71 ; Strain-ellip- soid, 71 Stream-Lines, 266, 536 ; Streams of Liquid, 265 ; Work done in keeping up, 285 ; of Gas, 297 Strength of Magnet, 600 ; of magnetic Field, 600 ; of magnetic Shell, 604, 632 ; of Materials, 227 ; of Structures, 239 Stress, 37, 166 ; Measurement of, 41 ; , Electric, 2<)8, 518, 533, 536, 629 ; Hydro- static, 199, 229, 267 Striking-Distance, 521 String-Organ, 404 Strings, Vibration of, 388-394, 400 ; when bowed, 389 ; plucked, 388, 390 ; struck, 390. Stroboscopic Disc, 269 Stromuhr, 283 Submarine Cable, 534, 577, 578 ; Tele- graphy, 534, 578, 622 Suction, 270, 303, 304 Sulphurous acid, 290, 326, 346 Summational Tones, 427 Sun's Atmosphere, 446, 448 ; Energy, 43 ; Sunlight, 43, 431, 438, 440, 443, 449 ; Sunset, 476 ; Sun's Disc, 443 Superficial Charge of Conductor, 534, 575 ; Film, 226 ; viscosity of, 247 Surf, 248 Surface ; Adhesion, 273 ; Attraction of a gas, 292 ; Cells coupled in , 548 ; Equipotential s, 172-177, 268, 522, 536, 593, 597, 600, 604 ; of Falling bodies, 179 ; Friction, 273 ; in Gases no free , 202 ; Leyden jars coupled in, 534 ; in Liquids, 226, 241, 261 ; Measurement of , 26 ; Tension, 226, 241, 269. 555, measurement of 244 Synthesis of Sound, 385 ; of Vowel-sound, 429 Syren, 372 TANGENT, 51 ; Tangent Galvanometer, 566, 620, 632 ; Screw, 23 Tangential Velocity, 53, 141 Tasimeter, 565 Teinte de passage, 510 Telegraphic Code (Morse), 621 ; Relays, 622 Telegraphy, 572, 620 ; Deep-sea, 534, 577, 578, 622 : Duplex, 624 ; Harmonic, 624 Telegraph Wire, 534, 626 Telephone, 424, 614, 622; Wire , 410 Telescope, 513 Temperament, Equal, 381 Temperature, 221, 323, 327, 356; Absolute . 223, 327, 328, 354, 356 ; of Con- densation of Vapour, 348 ; Critical , 207, 211, 226 ; Measurement of , 356 Tenacity of a Liquid Stream, 226, 308 Tension : Electric Surface , 522 ; Cells coupled in , 548, 568 : of String in Circular Motion, 142 ; Surface , 241 Tenth-metre, 433, 436 Terrestrial Magnetism, 604-606 Theory, 8 Thermal Capacities, 333 Thermic Balance, 634 Thermodynamics : First Law, 317 ; Second Law, 354 Thermo-electricity, 555 Thermo-electric Circuit, 555, 580 ; Dia- gram, 556 ; effect in Electric Arc, 582 ; Pile, 486, 558, 559 ; Power, 555 ; Thermometer, 559 Thermolysis, 216, 220, 319, 330 Thermometers : Air, 356, 357 ; Mercury, 358 ! Thermopile. See Thermo-electric Pile Thermoscope, 357 Thomson's (Prof. James) Proposition as to Melting-point of Ice, 343 Thomson's (Sir Win.) Gravity Electric Ma- chine, 561 ; his Measurement of v, 630 ; his Quadrant Electrometer, 539 ; his Thermo-electric Effect, 581 Thoracic Duct, 270 Tide, 6, 180, 193 ; Tide -calculating Machine, 96 Time, 9 ; Measurement of, 28 ; Sidereal, 9 ; Solar, 9 ; Unit of, 9 ; Time of Relaxation in Canada Balsam, 202 Tone, 371 Torricelli's Law, 268, 294, 298 ; Vacuum, 305, 308, 312 Torsibility, 234 ; Torsion, 33 Total Reflexion, 468 Total Refraction, 470 Toughness, 233, 237; Elastic, 236 Tourmaline, 499, 554 Traction, 163 Transference of Heat, 362 Transformations of Energy ; 41 et passim Transition in Music, 376 Translation of Molecules, 315; of a Particle, 67 ; of a Rigid body, 69 Translucency, 448 Transmissibility of Fluid Pressures, 256, 297 Transmission of Energy, 162, 204 ; 620 (electrical) Transmutation of Elements, 191, 193 Transparence, 448, 459 Transpiration of Gases, 295 Transport of Heat, 365 Transversal Vibrations, 99-103, 388-394 : of Ether, 432, 460 652 INDEX. Travelling of Electric Condition, 525 ; of Equipotential Surfaces, 177 ; of Wave- Form, 97 Trevelyan's Eocker, 409 Triangle of Accelerations, 62 ; of Forces, 136 ; of Velocities, 56 Trombone, 375, 407 Trough and Crest, 98 Trumpet, 407 Tubes of Flow, 536 ; of Force, 175, 523, 536 Tuning-Fork : 29, 42, 237, 366, 368, 373, 375, 384, 395, 396, 400, 401, 410, 412 ; 428, 430 ; Electromagnetic Interrupter for , 237, 401, 622 Twinkling of Stars, 493 ULTRAGASEOUS matter, 207, 584 Ultra-red Kays, 438 Ultra-violet Kays, 435, 436, 438 Umbra, 491 Uniaxial, 495 Uniform Field, 523 Units of Acceleration, 16 ; Angle, 69 ; Area, 12 ; Conductivity, 562 ; Density, 194 ; Electrical: Electrostatic (C.G.S. Electrostatic or C. G. S. E. S. ) : Capacity, 529, 537 ; Conductivity, specific, 562 ; Density, 520 ; Difference of Potential, 523, 524, 537 ; Force, 537 ; Inductive capacity, 533, 537 ; Intensity, 562 ; Quantity, 519 ; Resistance, 562, specific 562. Electromagnetic: 631 ; Mag- netic Field, intensity of, 601 ; magnetic Force, 600; magnetic Pole, 600. Prac- tical : Capacity (Farad), 632 ; Difference of Potential (Volt). 549, 632 ; Intensity (Ampere), 566, 632 ; Quantity (Coulomb), 566, 632 ; Resistance (Ohm), 564. Energy, 35; Expansion (e = 1 when length is doubled), 233 ; Force, 18 ; Heat, 519 ; Length, 10 ; Mass, 12 ; Space, 10 ; Time, 9 ; Velocity, 14 ; Volume 11 ; Work, 34 Universal Gravitation, 180 Universe, Electricity in the, = 0, 521 ; Energy in the, a constant, 40 Unsaturated Vapour, 205 Utility of Electromotor, 625, 626 U-tube, 260 VACUUM, 210 ; Discharge through a , 584 ; as an Insulator, 527 ; Torricelli's , 305, 312 ; -Tubes, 583 Valves, 309 ; of Heart, 310, 384 Vapour (1) in presence of liquid, ready to condense, 205 ; (2) saturated or unsatur- ated, 205 ; (3) a gas condensible by pressure alone, 206 Vapour-density, Measurement of, 347 Variable Period, 576, 629 Variations in Earth's Magnetic Force, 605 ; of Gravity, 20, 180 Velocity, 14, 52, 53; Absolute, 15; Aver- age under uniform Acceleration, 63. Mean, 15, 63 ; Relative, 15 ; Uniform, 14, 52 ; Variable, 15, 52. Angular , 69 ; Change of , 62 ; in Curved paths, 53 ; Resolution of into Components, 57; Tangential, 53, 141 Velocities : Composition of two , 54 ; of more than two, 59 ; Parallelipipedou of , 61 ; Parallelogram of, 54 ; Polygon of, 60, 62 ; gauclie polygone of , 62 ; Triangle of , 56 Velocity of Ether- Waves, 433, 461, 462; of Propagation of an Electromagnetic Disturbance, 578, 629, 630 ; measure- ment, 630, 632 ; Head, 276, 298 ; of Outflow, 268, 272, 295; of Particles of a Gas, 222 ; of Sound. ^ 41^417 : of a Stream. 282 ; of Transmission of Telegraph Signals, 578 ; of Wave-Motion in a gas, 331 ; in an elastic solid, 238 ; in a liquid, 415 Vena Contracta, 269 Vernier, 21 Vibrations: free , 387-398; forced, 398-402, 432 ; of Bells, 398 ; of Cords, 127, 367, transverse 90-103, 388-394, longitudinal 103, 394 ; of Discs, 397, 432 ; the result of Elasticity, 237 ; ' of the Ether, 432, 458, 460 ; of Mem- branes, 129, 398 ; of Molecules, 225, 317, 432, 457 ; Photographed, 386 ; of Rods, 127, 395, 396; Stationary, 126; Stop- ping component s, 127 " Vibrations Simples," 368 Viola, 404 Violin, 366, 369, 375, 380, 389, 390, 392, 403, 404 Violoncello, 404 Virtual Image, 474, 483, 484 ; Focus, 482 Viscosity, 200, 201 ; of Gases, 223 ; of Ice, 344 ; of Liquids, 226, 274, 282 ; in liquid streams, 267, 273; of elastic Solids, 237 ; of a Sounding-body, 369 ; Superficial in liquids, 247 ; in gaseous streams, 295i 298 Vis viva, 467 Vitreous (positive), 519 Voice, 428 Volatilisation, 210 ; of C, Pt, Ir, 582 ; of Snow, 347 Volt, 549, 631 Voltameter, 587 Volta's Pile, 549, 553 Voltaic, see Galvanic Volume, 192 ; Elasticity of, 203 ; Measure- ment of, 28 ; and Pressure in Gases, 204 Vortex-atom, Vortex-ring, 219 Vortices, Magnetic, 208 Vowel, 428 WALKING, 6, 158, 163, 184, 187 Water, Conductivity of, 563, 586 ; Density of, 194, 198, maximum 321 ; Electro- lysis of, 585 INDEX. 653 Watt, 35, 575 Wave : in Air, see Sound ; in Ether (chap. xv.) ; originated, 458 ; propagated, 459 et seq. ; measured, 488, 493. Form, Travelling of, 97, 104 ; velo- city of propagation of, 98, 105 Front, 104, 106; Direction of, 108; Normal to, 106. Length, 97, 105 ; Crest and Trough, 98 ; Loss of half w.-l., 118, 468, 489 ; of Ether- Waves, 218, 433, 437 ; measure- ment of, 488, 493 ; relation of to Fringes, 133. Motion : Energy of, 133, 288 ; Velo- city of in elastic gas 331, liquid 415, solid 238 Waves, Bidimensional, 106 ; Concentric, 108 ; Flat, 108, distorted 109 ; Linear, 97 ; Tridimeusional, 11/7 Effect of Screen on, 131, 132 ; Fre- quency of, 105 ; Interference of, 129 ; propagated along Normals, 106, 123, 130 ; Reflected, 110-117 ; waves reflected in elastic tubes, 288 ; Refracted, 119 ; Traversing an Aperture, 109, 110, 123, 130-131, 132 Weber, 600 ; Weber's Measurement of v, 630 ; his Theory of induced Magnetisa- tion, 608 Wedge, 157 ; Rotation of, 164 Weight, 12, 19, 178, 193, 297 Welding, 229 Well, depth of, 417 Wheatstone's Bridge, 573 Wheel and Axle, 154 Wheels of Railway Train, 163 ; Friction 165 Wheelwork, 155 ; in a clock, 28 Whispering galleries, 413 White Light, 437, 440 ; decomposed, 478 ; recorapouuded, 479 Whitworth's Measuring Machine, 23, 25 Winch, 155 Wind, 109, 313, 417 Wire Telephone, 409 Work, 34 ; unit of, 34 Work done, 7, 34 ; mean rate of doing , 35 ; Work done against Attraction, 170 ; in Charging a conductor, 529 ; by Ether- waves, 512 ; by Expanding sub- stance, 48, 323, 350 ; in producing Exten- sion, 234 ; in Electrolysis, 588 ; in moving across Equipotential Surfaces, 173 ; by or against Force, 34 ; by the Heart, 285 ; by Heat, 323 ; during Over- turning, 184 ; in moving a Pendulum, 187 ; by Repulsion, 168 ; in producing Shear, 234 ; in keeping up a Stream of liquid, 284 ; in producing Twist, 234 Worm wheel, 23 "1 ~ J-NG'S experiment, 490 ZERO Potential, 526 Zero Temperature : Absolute, 327, 328, 356 ; Centigrade, 358 ; Fahrenheit, 344, 359 Zero the Reciprocal of Infinity ( = o) 169; of Thermometers, 344, 358; ris- ing, 359 Zinc, amalgamated, 320, 547, 549 ; Electro- positive to copper, Negative in the Bat- tery, 543, 547 Printed by R. &. 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