A treatise on electricity and magnetism, vol i, j c maxwell

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Clarendon Press Series

Clarendon Press Series

aa TREATISE

ON °

ELECTRICITY AND MAGNETISM

BY

JAMES CLERK MAXWELL, M.A

LL) EVIN,, P.R.83, LONDON AND EDINBURGH HONORARY PELLOW OF TRINITY COLLEGE,

PREFACE

THE fact that certain bodies, after being rubbed,

appear to attract other bodies, was known to the

ancients In modern times, a great variety of other phenomena have been observed, and have been found

to be related to these phenomena of attraction They have been classed under the name of Electric phe-

nomena, amber, #Aextpoy, having been the substance

in which they were first described

Other bodies, particularly the loadstone, and pieces

of iron and steel which have been subjected to certain processes, have also been long known to exhibit phe-

nomena of action at a distance These phenomena, with others related to them, were found to differ from

the electric phenomena, and have been classed under

the name of Magnetic phenomena, the loadstone, uưyune,

being found in the Thessalian Magnesia

These two classes of phenomena have since been

found to be related to each other, and the relations between the various phenomena of both classes, so

far as they are known, constitute the science of Elec-

tromagnetism

In the following Treatise I propose to describe the

vi PREFACE,

most important of these phenomena, to shew how they may be subjected to measurement, and to trace the mathematical connexions of the quantities measured Having thus obtained the data for a mathematical theory of electromagnetism, and having shewn how this theory may be applied to the calculation of phe- nomena, I shall endeavour to place in as clear a light

as I can the relations between the mathematical form

of this theory and that of the fundamental science of Dynamics, in order that we may be in some degree prepared to determine the kind of dynamical pheno- mena among which we are to look for illustrations or explanations of the electromagnetic phenomena

In describing the plienomena, I shall select, those

which most clearly illustrate the fundamental ideas of

the theory, omitting others, or reserving them till the

readcr is more advanced

The most important aspect of any phenomenon from

a mathematical point of view is that of a measurable

quantity I shall therefore consider electrical pheno-

mena chiefly with a view to their measurement, de-

scribing the methods of measurement, and defining the standards on which they depend

In the application of mathematics to the calculation

of clectrical quantities, I shall endeavour in the first place to deduce the most general conclusions from the data at our disposal, and in the next place to apply the results to the simplest cases that can be chosen

I shall avoid, as much as I can, those questions which, though they have elicited the skill of mathematicians,

have not enlarged our knowledge of science.

dynamics, and on the other to heat, light, chemical action, and the constitution of bodies, seem to indicate

the special importance of electrical scicnce as an aid

to the interpretation of nature

It appears to me, therefore, that the study of elec- tromagnetism in all its extent has now become of the first importance as a means of promoting the progress

of science

The mathematical laws of the different classes of

phenomena have been to a great extent satisfactorily

made out

The connexions between the different classes of phe- nomena have also been investigated, and the proba- bility of the rigorous exactness of the experimental laws has been greatly strengthened by a more extended knowledge of their relations to each other

Finally, some progress has been made in the re- duction of electromagnetism to a dynamical science,

by shewing that no electromagnetic phenomenon is contradictory to the supposition that it depends on purely dynamical action

What has been hitherto done, however, has by no

means exhausted the field of electrical research It has rather opened up that field, by pointing out sub- jects of enquiry, and furnishing us with means of investigation

It is hardly necessary to enlarge upon the beneficial

Vili PREFACK,

results of magnetic research on navigation, and the importance of a knowledge of the true direction of the compass, and of the effect of the iron in a ship But the labours of those who have endeavoured to render navigation more secure by means of magnetic observations have at the same time greatly advanced the progress of pure science

Gauss, as a member of the German Magnetic Union, brought his powerful intellect to bear on the theory

of magnetism, and on the methods of observing it, and he not only added greatly to our knowledge of the theory of attractions, but reconstructed the whole

of magnetic science as regards the instruments used, the methods of observation, and the calculation of the

results, so that his memoirs on Terrestrial] Magnetism

may be taken as models of physical research by all those who are engaged in the measurement of any

of the forces in nature,

The important applications of electromagnetism to telegraphy have also reacted on pure science by giving

a commercial value to accurate electrical measure- ments, and by affording to electricians the use of apparatus on a scale which greatly transcends that

of any ordinary laboratory The consequences of this demand for electrical knowledge, and of these experi- mental opportunities for acquiring it, have been already very great, both in stimulating the energies of ad- vanced electricians, and in diffusing among practical: men a degree of accurate knowledge which is likely

to conduce to the general scientific progress of the whole engineering profession.

PREFACE, ix

There are several treatises in which electrical and magnetic phenomena are described in a popular way

These, however, are not what is wanted by those who

have been brought face to face with quantities to be

measured, and whose minds do not rest satisfied with

lecture-room experiments

There is also a considerable mass of mathematical memoirs which are of great importance in electrical science, but they lie concealed in the bulky Trans- actions of learned societies; they do not form a con- nected system; they are of very unequal merit, and

they are for the most part beyond the comprehension

of any but professed mathematicians

I have therefore thought that a treatise would be useful which should have for its principal object to take up the whole subject in a methodical manner, and which should also indicate how each part of the subject is brought within the reach of methods of verification by actual measurement

The general complexion of the treatise differs con- siderably from that of several excellent clectrical works, published, most of them, in Germany, and it may appear that scant justice is done to the specu- lations of several eminent electricians and mathema- ticians, One reason of this is that before I began the study of electricity I resolved to read no mathe- matics on the subject till I had first read through

Earadays Experimental Researches on Electricity I

was aware that there was supposed to be a difference between Faraday’s way of conceiving phenomena and

that of the mathematicians, so that neither he nor

x PREFACE

they were satisfied with each other’s language I had also the conviction that this discrepancy did not arise from either party being wrong I was first convinced

of this by Sir William Thomson *, to whose advice and assistance, as well as to his published papers, I owe most of what I have learned on the subject

As I procecded with the study of Faraday, I per- ceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited

in the conventional form of mathematical symbols, I] also found that these methods were capable of being expressed in the ordinary mathematical forms, and thus compared with those of the professed mathcma- ticians,

For instance, Faraday, in his mind’s eye, saw lines

of force traversing all space where the mathematicians saw centres of force attracting at a distance : Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on the electric fluids,

When I had translated what I considered to be Faraday’s ideas into a mathematical form, I found that in general the results of the two methods coin-

cided, so that the same phenomena were accounted

for, and the same laws of action deduced by both methods, but that Faraday’s methods resembled those

* I take this opportunity of acknowledging my obligations to Sir

W Thomson and to Professor Tait for many valuable suggestions made during the printing of this work

I also found that several of the most fertile methods

of research discovered by the mathematicians could be expressed much better in terms of ideas derived from Faraday than in their original form

The whole theory, for instance, of the potential, con-

sidered as a quantity which satisfies a certain partial differential equation, belongs essentially to the method which I have called that of Faraday According to the other method, the potential, if it is to be considered

at all, must be regarded as the result of a summa-

tion of the electrified particles divided each by its dis- tance from a given point Hence many of the mathe- matical discoveries of Laplace, Poisson, Green and Gauss find their proper place in this treatise, and their appropriate expression in terms of conceptions mainly derived from Haraday

Great progress has been made in electrical science, chiefly in Germany, by cultivators of the theory of action at a distance The valuable electrical measure- ments of W Weber are interpreted by him according

to this theory, and the electromagnetic speculation which was originated by Gauss, and carried on by

Weber, Riemann, J and C Neumann, Lorenz, &c is

founded on the theory of action at a distance, but depending either directly on the relative velocity of the particles, or on the gradual propagation of something,

x1 PREFACE

whether potential or force, from the one particle to the other The great success which these eminent

men have attained in the application of mathematics

to electrical phenomena gives, as is natural, addi- tional weight to their theoretical speculations, so that those who, as students of electricity, turn to them as the greatest authorities in mathematical electricity, would probably imbibe, along with their mathematica] methods, their physical hypotheses,

These physical hypotheses, however, are entirely alien from the way of looking at things which I adopt, and one object which I have in view is that some of those who wish to study electricity may, by reading this treatise, come to see that there is another way of treating the subject, which is no less fitted to

explain the phenomena, and which, though in some

parts it may appear less definite, corresponds, as I think, more faithfully with our actual knowledge, both

in what it affirms and in what it Icaves undecided

In a philosophical point of view, moreover, it is exceedingly important that two methods should be compared, both of which have suecceded in explaining the principal clectromagnetic phenomena, and both of which have attempted to explain the propagation of light as an electromagnetic phenomenon, and have actually calculated its velocity, while at the same time the fundamental conceptions of what actually takes place, as well as most of the secondary conceptions of the quantities concerned, are radically different

I have therefore taken the part of an advocate rather than that of a judge, and have rather exemplified one

method than attempted to give an impartial description

of both I have no doubt that the method which I

have called the German one will also find its sup-

porters, and will be expounded with a skill worthy

of its ingenuity

I have not attempted an exhaustive account of elec-

trical phenomena, experiments, and apparatus The

student who desires to read all that is known on these

subjects will find great assistance from the Traité

d’Electricité of Professor A de la Rive, and from several

German treatises, such as Wiedemann’s Galvanismus,

Riess’ Reibungselektricitdét, Beer’s Hinleitung in die Elek-

trostatzk, &c

I have confined myself almost entirely to the ma-

thematical treatment of the subject, but I would

recommend the student, after he has learned, expcri-

mentally if possible, what are the phenomena to be

observed, to read carefully Faraday’s Laperimental

Researches in Electricity He will there find a strictly

contemporary historical account of some of the greatest

electrical discoveries and investigations, carried on in

an order and succession which could hardly have been

improved if the results had been known from the

first, and expressed in the language of a man who

devoted much of his attention to the methods of ac-

curately describing scientific operations and their re-

sults *

It is of great advantage to the student of any

subject to read the original memoirs on that subject,

for science is always most completely assimilated when

* Life and Letters of laratay, vol i, p 395

XIV PREFACE

it is in the nascent state, and in the case of Faraday’s Researches this is comparatively easy, as they are

published in a separate form, and may be read con-

secutively If by anything I have here written 1 may assist any student in understanding Faraday’s modes of thought and expression, I shall regard it as the accomplishment of one of my principal aims—to communicate to others the same delight which I have found myself in reading Faraday’s Researches

The description of the phenomena, and the elc- mentary parts of the theory of each subject, will be found in the earlier chapters of each of the four Parts into which this treatise is divided The student will find in these chapters enough to give him an elementary acquaintance with the whole science

The remaining chapters of each Part are occupied with the higher parts of the theory, the processes of

numerical calculation, and the instruments and methods

of experimental research

The relations between electromagnetic phenomena and those of radiation, the theory of molecular electric currents, and the results of speculation on the nature

of action at a distance, are treated of in the last four

chapters of the second volume,

Feb, 1, 1873,

ON THE MEASUREMENT OF QUANTITIES,

1 The expression of a quantity consists of two factors, the nu-

merical value, and the name of the concrete unit

Dimensions of derived units

The three fandamental units—Length, Time and “Mase

"Derived units ¬

Phy sical continuity and discontinuity

Discontinuity of a function of more than one variable

Periodic and multiple functions vee

Relation of physical quantities to directions in space

Meaning of the words Scalar and Vector Division of physical vectors into two classes, Forces ‘and Fluxes Relation between corresponding vectors of the two classes

Line-integration appropriate to forces, surface-integration to

Longitudinal and rotational vectors

Line-integrals and potentials — sẻ ¬

Hamilton’s expression for the relation between a force and its

Cyclic regions and geometry of position

The potential in an acyclic region is single valued

System of values of the potential in a cyclic region

Surface-integrals

Surfaces, tubes, and lines of flow

Right-handed and left-handed relations in space

Transformation of a line-integral into a surface-integral

Effect of Hamilton’s operation y on a vector function

NÑature of the operation g`

Electrification by conduction, Conductors and insulators

- In electrification by friction the quantity of the positive clee- trification is equal to that of the negative electrification

To charge a vessel with o quantity of electricity equal and opposite to that of an excited body seas

To discharge a conductor completely into a metallic vessel

Test of electrification by gold-leaf clectroscope ‹ Electrification, considered as 1 measurable quantity, may be called Electricity

tees Electricity may be treated ag a physical quantity

Theory of Two fluids

Measurement of the force between electrified bodies

Relation between this force aud the quantities of clectricity

Variation of the force with the distance ¬

42 Definition of the electrostatic unit of electricity — Its dimensions

tea Proof of the law of electric force

Electric field

Equipotential surfteos, Example of their use in reasoning about electricity

Impossibility of an absolute charge

Disruptive discharge —Glow

Brush

Electrical phenomena of Tourmaline “

Plan of the treatise, and sketch of its results

Electric polarization and displacement “

The motion of electricity nhalogone to that ofa an incompr ssgible

fluid

32 Peculiarities of the theory of this treatise

CHAPTER 11

ELEMENTARY MATHEMATICAL THEORY OF ELECTRICITY

Definition of electricity as a mathematical quantity

sf Volume-density, surface-density, and line-density

5 Definition of the clectrostatic unit of clectricity tos

Law of foree between clectrified bodies

37 Resultant force between two bodies

38 Resultant force ut a point

Tine-integral of electric force ; eleetromotive force

Eloctric potential

Resultant force in terms of the potential

72 The potential of all points of a conductor is the same

73 Potential due to an clectrified system

74 Proof of the Jaw of the inverse square a

75 Surface-integral of clectric induction °

76 Introduetion through a closed surface duc to q single centre

77 Poisson's extension of Laplace's equation teas

78 Conditions to be fulfilled at an electrificd surface

Resultant force on an electrified surface

The electrification of a conductor is entirely on the wurface

_ A distribution of electricity on lincs or points is physically

Lines of electric induction - «wee

3 Specific inductive capacity

Qn the superposition of electrified systems te ae BB

General theory of a system of conductors, Coefficients of po- tential

Coefficients of induction Capacity of a conductor, Dimensions

of these coeflivients th HH HH cv vu v00 Reciproenl property of the coeficients ha ve ĐỊ

A theorem duetoGren 0 Khó ke cà 92 Relative magnitude of the coeflicients of potential 1 6 gg

The resultant mechanical force on a conductor expressed in terms of the charges of the different conductors of the system and the variation of the coeflicients of potential 9g

93 The same in terms of the potentials, and the variation of the cocflicients of induction

44 Comparison of cleetritied 8ÿBEINB ¿2 cố vo vo cu 06

CHAPTER IV, GENERAL THEOREMS

95 Two opposite methods of treating clectrical questions 98

96 Characteristics of the potential function 19

97, Conditions under which the volume-integral

100 Green’s theorem and its physical interpretation = 108

102, Method of finding limiting values of electrical cocfheicnts 11ã

CONTENTS, XIX

CHAPTER V

MECHANICAL ACTION BETWEEN ELECTRIFIED KODIES,

103, Comparison of the force between different electrified systems 119

104 Mechanical action on an clement of an electrified surface 121

105 Comparison between theories of direct action and theories of

106 The kind of stress requir ed to account for the phenomenon 128

107, The hypothesis of stress considered as a step ín clectrical

108, The hypothesis of stross sewn to account for the equilibriam

of the medium and for the forees acting between electrified

109, Statements of Faraday relative { to the | longitndinal tension and

lateral pressure of the lines of foree 6 131

110 Objections to stress ina fluid considered 0.00 6 38)

111 Statement of the theory of electric polarization cà 139

¡ 112 Conditions of a point of equilibrium tae ee BS

113 Number of points of equilibrium 0.0 woe 186

i 114 Ata point or line of equilibrium there is a conical point or a

: line of self-intersection of the equipotential surface 4 137

115 Angles at which an equipotential surface intersects itself 138

116 Phe equilibrium of an clectrified body cannot be stable 139

CHAPTER VII

FORMS OF EQUJPOTENTIAL SURFACFS AND LINES OF FLOW,

117 Practical importance of 2 Knowledge of these forms in simple

(uses ` ¬—¬ s s «142

118 Two clectrified points, ratio 4:1 (Big TQ) 148

119 Two electrified points, ratio 4: 1 (Fig Il) oo «144

120, An electrified point in a uniform field of foree, (Wig TTI) 2 145

121 Three clectrified points Two spherical equipetential sur-

122 Faraday’s use of the conecption of lines at foree we,

123 Method employed in drawing the diagrams „147

b2

SEMPLE CASES OF ELECTRIFICATION,

Two parallel planes

Two concentric spherical surfaces

Two coaxal cylindric surfaces

Longitudinal foree on weylinder, the ends of which are sup-

rounded by cylinders at different potentials

CHAPTER IX, SPHERICAL ILARMONICS, Singular points at which the potential becomes infinite

Singular points of diflerent orders defined hy their axes ,

Expression for the potential due to a sinưular point referred to its axes

This expression is perfectly definite and represents the most general type of the harmonic of 7 degrees

Solid harmonics of positive degree ‘Their relation to those of negative degree ,

Application to the theory of electrified spherical surfaces

The external action of an clectrified spherical surface compared

with that of an imaginary singular point at its centre

Proof that if P, and Ƒ; are two surface harmonies of different

degrees, the surface-integral II Y, Yd = 0, the intesration

being extended over the spherical surface

- Value of | | VY; dS where Y, and ?; are surface harmonies

of the same degree but of different types

If ¥; is the zonal harmonie and ¥; any other type of the same degree

ae

I YjdN¥a 22" yp 270 -+ ] dna ¡2!

Whtre 222; is the value oŸ 7; ñE the pole of Y; N

142 Different methods of trenting spherical harmonics 171

143 On the diagrams of spherical harmonics, (Figs, V, VỊ, VI,

144 If the potential is constant throughout 4 my finite portion of space it is so throughout the whole region contimuious with it within which Laplace’s equation is satisfied « 176

145 To analyse a spherical harmonic into a system of conjugate harmonics by means of a finite number of measurements at selected points of the sphere wo 177

146 Application to spherical and nearly spherical conduetors 178

CHAPTER X

CONFOCAL SURFACES OF THE SECOND DEGREE

147, The lines of intersection of two Systems and their intereepts

148 The churacteristic equation of Vin terms of ellipsoidal co-

149, Expression of a, 8, y in ten ms of elliptic functions ` 183

150 Particular solutions of electrical distribution on the confocal surfaces and their limiting forms tà Ö 184

151 Continuous transformation into a figure of rev olution abont

152 ‘Transformation into a flere of revolution about the axis of « 188

153 Transformation into a system of cones and spheres 189

CHAPTER XI

THEORY OF ELECTRIC IMAGES,

156 When two points are oppositely and unequally elects fi, the surface for which the potential is zero is a sphere 192

158 Distribution of leetricity 0 on the sur face of the sphere 195

159 Image of any given distribution of electricity 196

160 Resultant foree between an electrified point and sphere 197

161 Images in an infinite plane pondncling surface 198

163 Geomctrical theorems about i inversion 201

164 Application of the :aethod to the problem of Art 1i 58 202

166 Case of two spherical surfaces intersecting at an angle 5 204

167, Enumeration of the cases in which the number of images is

168 Case of two spheres intersecting orthogonally 207

169, Case of three spheres interseeting orthogonally 210

170 Case of four spheres intersecting orthogonally eee BLT

171 Infinite series of images Case of two concentric spheres 212

213

172 Any two spheres not intersecting each other

173 Culculation of the cveflicients of capacity and induction 216

174 Calculation of the charges of the spheres, and of the force

175, Distribution of electricity on two spheres in contact Proof

176, Thomson's investigation of an clectrified spherical bowl 221

177 Distribution on an ellipsoid, and on a circular disk at po-

178 Induction on an uwninsulated disk or bowl by an electrified point in the continuation of the plane or spherical surface 222

179 The rest of the sphere supposed uniformly electrified 223

180, The bowl maintained at potential Vand uninfluenced 223

181, Induction on the bowl due to nx point placed anywhere 224

CHAPTER XII

CONJUGATE FUNCTIONS IN TWO DIMENSIONS,

182 Cases in which the quantities are functions of and yonly 226

184 Conjugate functions may be added or subtracted 0 998

185 Conjugate functions of conjugate functions are themselves

186 Transformation of Poisson's "Ôn .aAaAIUỤỪ:

187 Additional theorems on conjugate functions 6 4 239

188 [Inversion iu two dimensions 0.00 232

189, Electric imayes in two dimensions 0 Q 233

190, Newmann’s transformation of thiseuse 0 oe 284

191, Distribution of electricity near the edye of a conductor formed

193 Transformation of this case (Pi XD _ éiáa::

195 Application to two cases of electrical induction 239

196 Capacity of a condenser consisting of a circular disk hetween

197, Case of a serics of equidistant planes cụt of by au plane at right

199 Caso of a single straight groove vee 243

200 Modifiention of the results when the groove is circular 211

201 Application to Six W, Thomson3 guard-ring , 215

202 Case of two parallel plates cut off by a perpendicular plane

203 Cuse of a grating of par allel wires (lig XTIT) 248

204 Case of a single electrified wire transformed into that of the

209, Production of electrification by mechanical wor ve: —Nicholson's

210 Principle of Varley’s and Thomson's clectr ical machines vẻ 256

213 Theory of regencrators applied to cleetr ieal machines 260

214 On clectrometcrs und clectroscopes Indicating instruments and null metnods Difference between registration and mea-

215, Couloml’s Torsion Balance for measuring charges 268

216 Electrometers for measuring potentials, Snow Harris's and

217, Principle of the guar a-ring Thomson’ s Absolute Electr ometer 267

218, Heterostatic method 269

219 Self-acting elcetromietors, —Thomson’s Quadrant Electrometer 27!

220 Measurement of the electric potential of a small body 274

221 Measurement of the potential at a point in the air 275

ợnay§š§ẽ>ằ mm ä<

Art

Page

222, Measurement of the potential of a conductor without touching it 276

223 Measurement of the superficial density of electrification The

226 On electric accumulators, The Leyden jar 281

227 Accumulators of measurable capacity teas oe BBD

228 The guard-ring accumulator Vu vỤ c S 283

229 Comparison of the capacities of accumulators 285

PART IL

ELECTROKINEMATICS,

CHAPTER J, THE ELECTRIC CURRENT,

230 Current produced when conductors are discharged 0, 4, 288

231 Transforenee ofeleetrifieation sò ò 288

232 Description of the voltaic battery 00, 0 4 289

233, Electromotive foree HS SỐ 290

234, Production of a steady current eae ea, 290

235, Properties of the current 6 uc 291

237 Explanation of terms connected with electrolysis 999

238 Different modes of passaye of the earrent ` ` BOQ

239 Magnetic action of the current 0 oe 293

240, The Galvanometer ¬ HH HE HH BOY

CHAPTER II

CONDUCTION AND RESISTANCE,

242 Generation of heat by the current Joule’sLaw 296

243 Analogy between the conduction of electricity and that of heat 297

244 Differences between the two classes of phenomena 997

245 Euraday' doctrine of the impossibility of an absolute charge 298

CONTENTS

CHAPTER II]

ELECTROMOTIVE FORCE BETWEEN BODIES IN CONTACT

Art,

216 Voltrs law af the eontnet foree between different metals at the

248 Thomson's voltaic current in which gravity performs the part

249 Pelticr’s phenomenon Deduction of the thermoelectric clec- tromotive force at a junction ¬

250 Secheck’s discovery of thermoelcctric currents

251 Magnus’s law of « circuit of one metal

252, Cumming’s discovery of thermoelectric inversions

253 Thomson's deductions from these facts, and discovery of the reversible thermal effects of clectric currents in copper and

254, Tait’s law of the electromotive for co of a ther moeleetrie pair

CHAPTER IV

ELECTROLYSIS, Earaday's law of electrochemical equivalents

, Clansius’s theory of molecular agitation

Electrolytic polarization «60 ee

‹ Lest of an electrolyte by polarization

Difficulties in the theory of clectrolysis

» Molecular charges

Secondary actions observed at the eleetr odes

Conservation of energy in clectrolysis “

Measurement of chemical affinity as an electr omotive foree

264 Difficulties of applying Ohm’s law to clectrolytes

265 Ohm’s law nevertheless applicable

266 The effect of polarization distinguished from that of resistance

267 Polarization duc to the presence of the ions at the electrodes

268 Relation between the electromotive force of polar ization and the state of the ions at the clectrodes

271, Litters secondary pile compared with the Leyden ja jar 322

272, Constant voltaic clencnts —Daniell’s cell ¬ 325

CHAPTER VI, MATHEMATICAL THEORY OF THE DISTRIBUTION OF ELECTRIC CURRENTS,

276 Linear conductors in multiple are ,, 330

277 Resistance of conductors of uniform section 331

278, Dimensions of the quantities involved in Ohn’s law 332

279, Specific resistance and conductivity in electromagnetic measure 333

280, Lincar systems of conductors in general , 333

281, Reciprocal property of any two conduetors of the sy stem 335

281, The heat is a minimum when the current is s distribnted uŒ-

CHAPTER VII

CONDUCTION IN THREE DIMENSIONS,

286 Composition and resolution of electri ic currents 338

287 Determination of the quantity which flows through any surface 339

289 Relation belween any three c systems of surfaces of low 310

991, Expression for the components of the flow i in terms of surfitees

of flow te ee eas có oo» 34]

292, Simplification of this expression by a proper choice of para-

293 Unit tubes of flow used as a complete method of determining

29-1, Current-sheety and current- fanetions 342

295 Equation of ‘continuity’ _ 342

296, Quantity of electricity which flows through a given surface 3-14

Rate of gencration of heat

Equation of continuity in a homogencous medium

Theory of the coefficient 7 It probably does not exist

Generalized form of Thomson's theorem

Strutt’s method applied to a wire of variable section, —Lower limit of the value of the resistance

Spherical shell placed in a field of uniform flow

Medium in which small spheres arc uniformly disseminated

Images in a plane surface ¬ “

Method of inversion not applicable in three dimensions

Case of conduction pưongh a stratum bounded by parallel

Infinite series of images Application to magnetic induction ,

On stratified conductors Cocflicients of conductivity of a conductor consisting of alternate strata of two different sub-

If neither of the substances has the rotator y proper ty denoted

by 7 the compound conductor is free from it

Tf the substances ure isotropic the direction of greatest resist- ance is normal to the strata

Medium containing parallelepipeds of mother medium

The rotatory property cannot be introduced by means of con-

Construction of an artificial solid having given coeflicients of longitudinal and transverse conductivity

360

362

363 36:1

ae

CHAPTER X, CONDUCTION IN DIELECTRICS,

325 Th a strictly homogeneous medium there can be no internal

326 Theory of a condenser in w hieh the dieleeie § is not a perfect

327 No residual charge due to ‘simple conduction wee 376

428 Theory of a composite nccummlator 0 837

329 Residual charge and electrical absorption Hs ca cà 78

331 Comparison with the condnetion of heat Hà ơn 181

332 Theory of telegraph cables and comparison of the cy uations

with those of the conduction of heat 0.0 8Ị

333 Opinion of Olm on this subject ¬ ¬ BBA

334, Mechanical illustration of the properties of a dieleetrie c “ 8ã

CHAPTER XI, MEASUREMENT OF TITE ELECTRIC RESISTANCE OF CONDUCTORS

335, Advautage of using material standards of resistance in electrical

336 Different standards which have been used and different systems

337 The clectromagnetie system of units os .ò — 889

338 Weber's unit, and the British Association unit or Ohm ue B89

339 Professed value of the Ohm 10,000,000 metres per second = 389

341 Forms of resistanee coils 0 vẻ 391

345 On the comparison of resistances, (1) Ohm's method 394

$46 (2) Dy the differential galvanometer tae ee BO

‘47 (3) By Whentstune’s Bridge = tee ae B98

348, Estimation of limits of error in the deter mination toe 00

349, Best arrangement of the conductors to be compared 4 400

350 Ou the use of Wheatstone’s Bridge 402

351 Thomson’s method for small YOSISEHNCCR 4 wae AOE

352 Matthiessen and Hoekin’s method for small resistanees “ „ 106

Art

353, Comparison of great resistances by the clectrometer

354 By accumulation in a condenser

356 Thomson's method for the resistance of a galvanumcter

357, Mance’s method of determining the resistance of a battery

358, Comparison of electromotive forces

CHAPTER XII

ELECYRIC RESISTANCE OF SUBSTANCES

359 Metals, electrolytes, and dielectrics

ELECTRICITY AND MAGNETISM.

ERRATA VOL I

Page 26, 1 3 from bottom, dele ‘As we have made no assumption’, &e down to 1,7 of p 27, ‘the expression may then be written’, and substitute as follows :—

Let us now suppose that the curves for which a is constant

form a series of closed curves, surrounding the point on the surface

for which a has its minimum value, đạ, the last curve of the series, for which a = @,, coinciding with the original closed curve s Let us also suppose that the curves for which 9 is constant form

a serics of lines drawn from the point at which a= a, to the closed curve s, the first, 3,, and the last, Øj, being identical,

Integrating (8) by parts, the first term with respect to a and the second with respect to 8, the double integrals destroy each other The line integral,

p 83, in equations ; ; » (31), for dae Peed Fy

» 1n cquation (29), tnseré — before the second member

p 105, 1.2, for Q read 87Q

p- 108, equation (1), for p read pi

” » (2), for p’ read p,

” ” (3), for o read a”,

” ” (4), for a” read o

p 113, 1.4, for KR read _Rk

» 1.5, for KRR cose read A KR 4T RF’ cose

.114, 1.5, for 6; read #,

124, last line, for e,t+e, read ete,

125, lines 3 and 4, transpose within and without; 1 16, for v

read V; and 1.18, for V read v,

128, lines 11, 10, 8 from bottom, for da read dz,

149, 1 24, for eyupotential read equipotential

PPP

: - Sosy Nin |2» \i— wo

164, equation (34), for (—1 ipo read (~1)-* Pei fz

185, equation (24), fo eos! read B 2 ce 4 pol

186, 1.5 from bottom, for ‘The surfuce-density on the elliptic plate’ reed ‘The surface-density on cither side of the elliptic plate

186, equation (30), for 2a read 47,

188, equation (38), for m read 27%

196, 1.27, for e e read e, e,

197, equation (10) should be A= Bf 4,

ÖÔ„ ⁄ OF ( fia’)

204, 1.15 from bottom, dele cither

215, 1.4, for J 2k read Jl ok,

234, equation (13), for 22 read on

335, dele last 14 lines

336, 1.1, dele therefore

1, 2, for ‘the potential at © to exceed that at D by BP,’ read a current, C, from V to Y

1.4, for *C to J will cause the potential at A to exceed that ut

4 by the same quantity 2 read X to Y will cause an equal current € from aA to B,

351, 1.3, for RPO+RIAv + Rew? read RK, w+ Rev + Rw,

» L 5, eaử + 2/// (ur +0 ủy + qs) dudydz

355, last line, for §” read „9

365, in equations (12), (15), (1G), for A read Ar

366, cyuation (3), for =2 peal ~2 ry TQ :

367, 1.5, for 24,8 read 2h,8

368, equution (14), for J read 1

+“

» 404, at the end of Art 350 insert ag follows :—

When y, the resistance to be measured, a, the resistance of the battery, and a, the resistauce of the galvanometer, are given, the best valucs of the other resistances have been shewn by Mr Oliver Heaviside (PAdl Afay., Feb 1873) to be

ELECTRICITY AND MAGNETISM

PRELIMINARY

ON THE MEASUREMENT OF QUANTITIES

1.] Every expression of a Quantity consists of two factors or

components One of these is the name of a certain known quan-

tity of the same kind as the quantity to be expressed, which is taken as a standard of reference The other component is the number of times the standard is to be taken in order to make up the required quantity The standard quantity is technically called the Unit, and the number is called the Numerical Value of the quantity

There must he as many different units as there are different kinds of quantities to be measured, but in all dynamical sciences

it is possible to define these units in terms of the threc funda- mental units of Length, Time, and Mass Thus the units of area and of volume are defined respectively as the square and the cube whose sides are the unit of length

Sometimes, however, we find several units of the same kind founded on independent considerations, Thus the gallon, or the volume of ten pounds of water, is used as a unit of capacity as well

as the cubic foot The gallon may be a convenient measure in some cases, but it is not a systematic one, since its numerical re- lation to the cubic foot is not a round integral number

2.) In framing a mathematical system we suppose the funda- mental] units of length, time, and mass to be given, and deduce all the derivative units from these by the simplest attainable de- finitions

The formulae at which we arrive must be such that a person

B

2 PRELIMINARY [3

of any nation, by substituting for the different symbols the nu- merical value of the quantities as measured by his own national

units, would arrive ata trne result,

lence, in all scientific studies it is of the greatest importance

to employ units belonging to a properly defined system, and to know the relations of these units to the fundamental units, so that

we may be able at once to transform our results from one system to another,

This is most conveniently done by ascertaining the dimensions

of every unit in terms of the three fundamental units When a given unit varies as the 2th power of one of these units, it is said

to be of 2 dimensions as regards that unit

For instance, the scientifie unit of volume is always the cube whose side is the unit of length If the unit of length varies,

the unit of volume will vary as its third power, and the unit of volume is said to be of three dimensions with respect to the unit of length

A knowledge of the dimensions of units furnishes a test which

ought to be applied to the equations resulting from any lengthened

investigation The dimensions of every term of such an equa- tion, with respect to each of the three fundamental units, must

be the same If not, the equation is absurd, and contains some

error, as its interpretation would be different according to the arbi- trary system of units which we adopt *

The Three Fundamental Units

3.] (1) Length, The standard of length for scientific purposes

in this country is one foot, which is the third part of the standard yard preserved in the Exchequer Chambers

In France, and other countries which have adopted the metric system, it is the métre The métre is theoretically the ten mil-

lionth part of the length of a meridian of the earth measured from the pole to the equator; but practically it is the length of

a standard preserved in Paris, which was constructed by Borda

to correspond, when at the temperature of melting ice, with the

value of the preceding length as measured by Delambre The matre has not been altered to correspond with new and more accurate

measurements of the earth, but the are of the meridian is estimated

in terms of the original matre,

* The theory of dimensions was first stated by Fourier, Théorie de Chaleur, § 160,

In astronomy the mean distance of the earth from the sun is

sometimes taken as a unit of length

In the present state of science the most universal standard of

length which we could assume would be the wave length in vacuum

ofa particular kind of light, emitted hy some widely diffused sub-

stance such as sodium, which has well-defined lines in its spectrum,

Such a standard would be independent of any changes in the di-

mensions of the earth, and should be adopted by those who expect

their writings to be more permanent than that body

In treating of the dimensions of units we shall call the unit of

length [Z] If Z is the numerical value of a length, it is under-

stood to be expressed in terms of the concrete unit [Z], so that

the actual length would be fully expressed by ¢[L}

4.) (2) Time The standard unit of time in all civilized coun-

tries is deduced from the time of rotation of the earth about its

axis The sidereal day, or the true period of rotation of the earth ;

can be ascertained with great exactness by the ordinary observa-

tions of astronomers; and the mean solar day can be deduced

from this by our knowledge of the length of the year

The unit of time adopted in all physical researches is one second

of mean solar time

In astronomy a year is sometimes used as a unit of time A

more universal unit of time might be found by taking the periodic

time of vibration of the particular kind of light whose wave length

is the unit of length,

We shall call the concrete unit of time [7], and the numerical

measure of time 4

5.] (3) dfuss, The standard unit of mass is in this country the

avoirdunols pound preserved in the Exchequer Chambers The

grain, which is often used as a unit, is defincd to be the 7000th

part of this pound

In the metrical system it is the gramme, which is theoretically

the mass of a cubic centimétre of distilled water at standard tem-

perature and pressure, but practically it is the thousandth part

of a standard kilogramme preserved in Paris

The accuracy with which the masses of bodies can be com-

pared by weighing is far greater than that hitherto attained in

the measurement of lengths, so that all masses ought, if possible,

to be compared directly with the standard, and not deduced from

experiments on water

In descriptive astronomy the mass of the sun or that of the

B2

4 PRELIMINARY, [5- earth is sometimes taken as a unit, but in the dynamical theory

of astronomy the unit of mass is deduced from the units of time and length, combined with the fact of universal gravitation, The astronomical unit of mass is that mass which attracts another

body placed at the unit of distance so as to produce in that hody

the unit of acceleration

In framing a universal system of units we miy either deduce the unit of mass in this way from those of length and time already defined, and this we can do to a rough approximation in the present state of science ; or, if we expect * soon to be able to determine the mass of a single molecule of a standard substance,

we may wait for this determination before fixing a universal standard of mass

We shall denote the conerete unit of mass hy the symbol [4/]

in treating of the dimensions of other units The unit of mass will be taken as one of the three fundamental units When, as

in the French system, a particular substance, water, is taken ag

a standard of’ density, then the unit of mass js no longer inde- pendent, but varies as the unit of volume, or as [2]

If, as in the astronomical system, the unit of mass is defined with respect to its attractive power, the dimensions of [A] are #2 7¬*] For the acecleration due to the attraction of a mass m at a

distance 7 is by the Newtonian Law a Suppose this attraction

to act for a very small time ¢ on a body originally at rest, and to

cause it to describe a space s, then by the formula of Galileo,

tion in which the mass of a body appears in some but not in all

of the terms f,

* See Prof J Loschmidt, ‘Zur Gréose der Luftinolecule,’ Academy of Vienna

Oct, 12, 1865; GJ Stoney on ‘The Internal Motions of Cases.’ Phil, Mag., Aug

1868; and Sir W Thomson on ‘ The Size of A toms,’ Nudare, March 31, 1870,

+ If a foot and a second are taken as units, the astronomical unit of mass would

be about 932,000,000 pounds.

6 ] DERIVED UNITS 5

Derived Units,

6.) The unit of Velocity is that velocity in which unit of length

is described in unit of time Its dimensions are [Z7'~"],

Tf we adopt the units of length and time derived from the

vibrations of light, then the unit of velocity is the velocity of

light

The unit of Acceleration is that acceleration in which the velo-

city increase’ by unity in unit of time Its dimensions are [2 7" *]

The unit of Density is the density of a substance which contains

unit of mass in unit of volume Its dimensions are [47Z~*]

The unit of Momentum is the momentum of unit of mass moving

with unit of velocity Its dimensions are [J7Z7'~"]

The unit of Force is the foree which produces unit of momentum

in unit of time Its dimensions are [AZ L7'~*]

This is the absolute unit of force, and this definition of it is

implied in every equation in Dynamics Nevertheless, in many

text books in which these equations are given, a different unit of

force is adopted, namely, the weight of the national unit of mass ;

and then, in order to satisfy the equations, the national unit of mass

is itself abandoned, and an artificial unit is adopted as the dynamical

unit, equal to the national unit divided by the numerical value of

the force of wravity at the place, In this way both the unit of force

and the unit of mass are made to depend on the value of the

force of gravity, which varies from place to place, so that state-

ments involving these quantities are not complete without a know-

ledge of the force of gravity in the places where these statements

were found to be true

The abolition, for all scientific purposes, of this method of mea-

suring forces is mainly due to the introduction of a general system

of making observations of magnetic force in countries in which

the force of gravity is different All such forees are now measured

according to the strictly dynamical method deduced from our

delinitions, and the numerical results are the same in whatever

country the experiments are made

The unit of Work is the work done by the unit of force acting

through the unit of length measured in its own direction Its

dimensions are [Ä/7*7-*]

The Energy of a system, being its capacity of performing work,

is measured by the work which the system is capable of performing

by the expenditure of its whole energy

6 PRELIMINARY, L7

The definitions of other quantities, and of the units to which

they are referred, will be given when we require them,

In transforming the values of physical quantities determined in terms of one unit, so as to express them in terms of any other unit

of the same kind, we have only to remember that every expres-

sion for the quantity consists of two factors, the unit and the nu- merical part which expresses how often the unit is to be taken

Hence the numerical part of the expression varies inversely as the magnitude of the unit, that is, inversely as the various powers of the fundamental units which are indieated by the dimensions of the derived unit,

Ou Physical Continuity and Discontinuity, 7.) A quantity is said to vary continuously when, if it passes from one value to another, it assumes all the intermediate values,

We may obtain the conception of continuity from a consideration

of the continnons existence of a particle of matter in time and space, Such a particle cannot pass from one position to another without deseribing a continuous line in space, and the coordinates of its position must be continuous functions of the time

In the so-called ‘ equation of continuity,’ as given in treatises

on Hydrodynamies, the fact expressed is that matter cannot appear

in or disappear from an element of volume without passing in or out

through the sides of that element

A quantity is said to be a continuous function of its variables

when, if the variables alter continuously, the quantity itself alters

continuously,

Thus, if «is a funetion ofa, and if, while a passes continuously

from a to 2,, « passes continuously from a, to a, but when ø

passes from 2, to x,, % passes from 2,’ to w„, my being different from

#,, then « is said to have a discontinuity in its variation with

respect to @ for the value 2 = 2,, because it passes abruptly from x,

to 2,’ while 2 passes continuousl y through 2

Tf we consider the differential coefficient of œ with respect to x for

the value # = 2, as the limit of the fraction

U,,— Uo

#,— 8u

When x, and «, are both made to approach 2, without limit, then,

if a and a, are always on opposite sides of 2,, the ultimate value of

the numerator will be w/—m, and that of the denominator will

be zero If wisa quantity physically continuous, the discontinuity

?