The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 1 of 39

THE MATHEMATICAL THEORY

OF

ELECTRICITY AND MAGNETISM

CAMBRIDGE

UNIVERSITY PRESS LONDON : Fetter Lane

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1. 1

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THE MATHEMATICAL THEORY

OF

ELECTRICITY AND MAGNETISM

BY

J. H. JEANS, D.Sc, LL.D., F.R.S.

FORMERLY STOKES LECTURER IN APPLIED MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE; SOMETIME PROFESSOR OF APPLIED MATHEMATICS IN PRINCETON UNIVERSITY

FIFTH EDITION

A>

CAMBRIDGE

AT THE UNIVERSITY PRESS 1927

43 13 at

First Edition 1908

Second Edition 1911

Third Edition 1915

iW*A Edition 1920 (reprinted 1923)

i^A ^cfe'ftbra 1925 ( „ 1927)

PRINTED IN GREAT BRITAIN

PREFACE

[TO THE FIRST EDITION]

THERE is a certain well-defined range in Electromagnetic Theory, which every student of physics may be expected to have covered, with more or less of thoroughness, before proceeding to the study of special branches of developments of the subject. The present book is intended to give the mathematical theory of this range of electromagnetism, together with the mathematical analysis required in its treatment.

The range is very approximately that of Maxwell's original Treatise, but the present book is in many respects more elementary than that of Maxwell. Maxwell's Treatise was written for the fully-equipped mathematician: the present book is written more especially for the student, and for the physicist of limited mathematical attainments.

The questions of mathematical analysis which are treated in the text have been inserted in the places where they are first needed for the development of the physical theory, in the belief that, in many cases, the mathematical and physical theories illuminate one another by being studied simultaneously. For example, brief sketches of the theories of spherical, zonal and ellipsoidal harmonics are given in the chapter on Special Problems in Electrostatics, interwoven with the study of harmonic potentials and electrical applications: Stokes' Theorem is similarly given in connection with the magnetic vector- potential, and so on. One result of this arrangement is to destroy, at least in appearance, the balance of the amounts of space allotted to the different parts of the subject. For instance, more than half the book appears to be devoted to Electrostatics, but this space will, perhaps, not seem excessive when it is noticed how many of the pages in the Electrostatic part of the book are devoted to non-electrical subjects in applied mathematics (potential-theory, theory of stress, etc.), or in pure mathematics (Green's Theorem, harmonic analysis, complex variable, Fourier's series, conjugate functions, curvilinear coordinates, etc.).

A number of examples, taken mainly from the usual Cambridge examina- tion papers, are inserted. These may provide problems for the mathematical student, but it is hoped that they may also form a sort of compendium of results for the physicist, shewing what types of problem admit of exact mathematical solution.

It is again a pleasure to record my thanks to the officials of the University Press for their unfailing vigilance and help during the printing of the book.

J. H. JEANS.

Princeton,

December, 1907.

vi Preface

[TO THE SECOND EDITION]

The second Edition will be found to differ only very slightly from the first in all e.\cept the last few chapters. The chapter on Electromagnetic Theory of Light has, however, been largely rewritten and considerably amplified, and two new chapters appear in the present edition, on the Motion of Electrons and on the General Equations of the Electromagnetic Field. These last chapters attempt to give an introduction to the more recent developments of the subject. They do not aim at anything like completeness of treatment, even in the small parts of the subjects with which they deal, but it is hoped they will form a useful introduction to more complete and specialised works and monographs.

J. H. JEANS.

Cambridge,

August, 1911.

[TO THE THIRD EDITION]

In preparing a third Edition I have made only a few changes in the latter chapters, which were necessary to bring the book up to date.

J. H. JEANS.

London,

November, 1914.

[TO THE FOURTH EDITION]

It will be found that the main changes in the fourth Edition consist in a rearrangement of the later chapters and the addition of a wholly new chapter on the Theory of Relativity. It need hardly be said that no attempt is made to give a full account of the Theory; I have tried to present its broad outlines in the simplest possible way, and in striving after simplicity I have intentionally omitted all elaboration and detail. It is hoped that the new chapter will pro- vide a suitable introduction to the Theory of Relativity for the student who approaches the subject for the first time, equipped with such knowledge of general electrical theory as can be gained from the rest of the book.

J. H. JEANS.

Dorking,

December, 1919.

Preface vii

[TO THE FIFTH EDITION]

In preparing a Fifth Edition I have introduced the changes that seemed to be called for by the now established position of the new theories of relativity and quanta. I have not attempted any detailed account of the theory of quanta but have added a chapter on " The Electrical Structure of Matter " which will introduce the reader to this theory.

It is a pleasure to record my thanks to friends and correspondents who have helped me by making suggestions and pointing out errors and misprints in earlier editions. My thanks are especially due to Dr A. Russell, F.R.S., Dr Harold Jeffreys, F.R.S., Professor E. P. Adams, Dr R. E. Baynes, Mr L. A. Pass and Dr H. L. Curtis.

J. H. JEANS. Dorking,

March, 1925.

CONTENTS

INTRODUCTION

The three divisions of Electromagnetism

PAGE 1

ELECTROSTATICS AND CURRENT ELECTRICITY

CIIAP.

I.

II.

III.

IV.

Physical Principles . ....

The Electrostatic Field of Force

Conductors and Condensers ....

Systems of Conductors

V. Dielectrics and Inductive Capacity , VI. The State of the Medium in the Electrostatic Field VII. General Analytical Theorems .... VIII. Methods for the Solution of Special Problems . IX. Steady Currents in Linear Conductors X. Steady Currents in Continuous Media

5

24

66

88

115

140

156

185

300

341

MAGNETISM

XL Permanent Magnetism XII.

Induced Magnetism

364

408

ELECTROMAGNETISM

XIII. The Magnetic Field produced by Electric Currents

XIV. Induction of Currents in Linear Circuits . XV. Induction of Currents in Continuous Media

XVI. Dynamical Theory of Currents .... XVII. Displacement Currents and Electromagnetic Waves XVIII. The Electromagnetic Theory of Light

XIX. The Motion of Electrons

XX. The Theory of Relativity

XXI. The Electrical Structure of Matter .

Index

425 452 473 485 510 532 559 593 629

647

INTRODUCTION

TEE THREE DIVISIONS OF ELECTROMAGNETISM

1. The fact that a piece of amber, on being rubbed, attracted to itself other small bodies, was known to the Greeks, the discovery of this fact being- attributed to Thales of Miletus (640-548 B.C.). A second fact, namely, that a certain mineral ore (lodestone) possessed the property of attracting iron, is mentioned by Lucretius. These two facts have formed the basis from which the modern science of Electromagnetism has grown. It has been found that the two phenomena are not isolated, but are insignificant units in a vast and intricate series of phenomena. To study, and as far as possible interpret, these phenomena is the province of Electromagnetism. And the mathematical development of the subject must aim at bringing as large a number of the phenomena as possible within the power of exact mathe- matical treatment.

2. The first great branch of the science of Electromagnetism is known as Electrostatics. The second branch is commonly spoken of as Magnetism, but is more accurately described as Magnetostatics. We may say that Electrostatics has been developed from the single property of amber already mentioned, and that Magnetostatics has been developed from the single property of the lodestone. These two branches of Electromagnetism deal solely with states of rest, not with motion or changes of state, and are therefore concerned only with phenomena which can be described as statical. The developments of the two statical branches of Electromagnetism, namely Electrostatics and Magnetostatics, are entirely independent of one another. The science of Electrostatics could have been developed if the properties of the lodestone had never been discovered, and similarly the science of Magnetostatics could have been developed without any knowledge of the properties of amber.

The third branch of Electromagnetism, namely, Electrodynamics, deals with the motion of electricity and magnetism, and it is in the development of this branch that we first find that the two groups of phenomena of electricity and magnetism are related to one another. The relation is

J. 1

2 Introduction

a reciprocal relation: it is found that magnets in motion produce the same effects as electricity at rest, while electricity in motion produces the same effects is magnets at rest. The third division of Electromagnetism, then, connects the two former divisions of Electrostatics and Magnetostatics, and is in a sense symmetrically placed with regard to them. Perhaps we may compare the whole structure of Electromagnetism to an arch made of three stones. The two side stones can be placed in position independently, neither in any way resting on the other, but the third cannot be placed in position until tie two side stones are securely fixed. The third stone rests equally on the two other stones and forms a connection between them.

3. In the present book these three divisions will be developed in the ■order in which they have been mentioned, namely Electrostatics, Magneto- statics, Electromagnetism. The earlier chapters will give an explanation of the physical ideas adopted by Maxwell in his Treatise on Electricity and Magnetism side by side with a purely mathematical theory. Maxwell's treat- ment of Electrical Science was differentiated from that of other writers by his insistence on Faraday's conception of electric and magnetic energy as residing in the medium. According to this view, the forces acting on electrified or magnetised bodies did not form the whole system of forces in action, but served only to reveal the presence of a vastly more intricate system of forces, which acted throughout the ether by which the material bodies were supposed to be surrounded. It was only through the presence of matter that the sup- posed system of forces became perceptible to human observation, so that it was necessary to try to reconstruct the whole system of forces from no data except those given by the resultant effect of the forces on matter, where matter was present. As might be expected, these data proved insufficient to give full and definite knowledge of the system of ethereal forces; it was found that a great number of systems of ethereal forces could be constructed, each of which would produce the same effects on matter as are observed. Of these systems, however, a single one seemed so very much more probable than any of the others, that it was unhesitatingly adopted both by Maxwell and by Faraday.

As soon as the step had been taken of attributing the mechanical forces acting on matter to a system of forces acting throughout the whole ether, a further physical development was made not only possible but also necessary. A stress in the ether might be supposed to represent either an electric or a magnetic force, but could not be both. Faraday supposed a stress in the ether to be identical with electrostatic force. There was no longer any possibility, in this scheme of the universe, of regarding magnetostatic forces as evidence of simple stresses in the ether.

The three divisions of Electromagnetism 3

It has, however, been said that magnetostatic forces are found to be produced by the motion of electric charges. Now if electric charges at rest produce simple stresses in the ether, the motion of electric charges must obviously be accompanied by changes in the stresses in the ether. It accord- ingly became possible to identify magnetostatic force with change in the system of stresses in the ether. This interpretation of magnetic force formed an essential part of Maxwell's theory. Comparing the ether to an elastic material medium, we may say that the electric forces were interpreted as the statical pressures and strains which accompanied the compression, dilatation or displacement of the medium, while magnetic forces were interpreted as the pressures and strains in the medium caused by its motion and momentum. Thus electrostatic energy was regarded as the potential energy of the medium, while magnetic energy was regarded as its kinetic energy. Maxwell shewed that the whole series of known electrostatic and magnetostatic phenomena might be consistently interpreted as phenomena produced by the stresses and motion of a medium, this motion being in conformity with the laws of dynamics. This hypothesis is examined in the earlier chapters of the book, although, as will be seen later, recent developments call for at least a drastic modification, and more probably for the complete abandonment of the whole hypothesis.

4. The observational fact that magnetostatic forces were produced by the motion of electric charges inevitably raised the question of the interpretation of general magnetic phenomena in electrical terms. A solution of the problem suggested by Ampere and Weber needs but little modification to represent the answer to which modern investigations have led. Recent experimental researches shew that all matter must be supposed to consist solely of electrically charged particles, and it seems highly probable that all magnetic phenomena can be explained by the motion of these charges. If the motion of the charges is governed by a regularity of a certain kind, the body as a whole will shew magnetic properties. If this regularity does not obtain, the magnetic forces produced by the motions of the individual charges will on the whole neutralise one another, and the body will appear to be non-magnetic. On this view the electricity and magnetism which at first sight appeared to exist independently in the universe, are resolved into electricity alone — electricity and magnetism become electricity at rest and electricity in motion.

This discovery of the ultimate identity of electricity and magnetism is by no means the last word of the science of Electromagnetism. As far back as the time of Maxwell and Faraday, it was recognised that the forces at work in chemical phenomena must be regarded largely, if not entirely, as electrical forces. Later, Maxwell shewed light to be an electromagnetic phenomenon, so that the whole science of Optics became a branch of Electromagnetism.

1—2

4 Introduction

Gradually the conviction grew that all physical forces, with the possible exception of Gravitation, would prove to be ultimately of Electromagnetic origin, ;o that by the end of the nineteenth century most scientists believed that the science of Electromagnetism would advance along the road opened out by Maxwell until the whole physical universe had been explained in the terms of electromagnetic theory. Recently this belief has experienced two very severe checks.

If, as Maxwell believed, the ultimate seat of electromagnetic and optical phenomena is the ether, it ought to be possible to find out something about the ether by electromagnetic and optical means. It ought, for instance, at least to be possible to determine the velocity with which we move through the ether. A series of experiments devised to this end have one and all failed to disclose this velocity. To every experimental enquiry, Nature seems to give the answer either that there is no ether or that natural phenomena go on exactly as if there were no ether. If this view is finally established, and at present there seems only a very meagre chance of any alternative, Maxwell's theory of the electromagnetic ether must necessarily fall out of science; it will have served its purpose as a scaffolding which will have enabled the structure of electromagnetic theory to have been built in perfect form, but it will not be part of that structure. Nevertheless the time for finally deciding how much of Maxwell's theory is scaffolding and how much is part of the essential structure has hardly yet come, so that in the present book we shall first develop the theory along the general lines initiated by Maxwell, and then shall devote a chapter to the development of a more modern theory and to a discussion of how far the existence of an ether is essential to electromagnetic theory.

The second check to Maxwell's theory has originated from the study of radiation and the ultimate electrical structure of matter; phenomena of primary importance have been found not to be reconcileable with Maxwell's original theory. In a sense the new facts hardly cut at the roots of the theory; they must rather be thought of as restricting the spread of the branches. There is no question that the electrical phenomena of everyday life, thunder- storms, telephones and dynamos, are all governed by Maxwell's laws; it is only when we pass to the phenomena arising from the most intimate electrical structure of matter that Maxwell's laws appear to be inadequate. Our final chapter will contain an explanation of the failure of Maxwell's Electrodynamics to deal with these problems, and a very brief introduction to the new theory which has taken its place.

CHAPTER I

PHYSICAL PRINCIPLES

The Fundamental Conceptions of Electrostatics

I. State of Electrification of a Body.

5. We proceed to a discussion of the fundamental conceptions which form the basis of Electrostatics. The first of these is that of a state of electrification of a body. When a piece of amber has been rubbed so that it attracts small bodies to itself, we say that it is in a state of electrification — or, more shortly, that it is electrified.

Other bodies besides amber possess the power of attracting small bodies after being rubbed, and are therefore susceptible of electrification. Indeed it is found that all bodies possess this property, although it is less easily recognised in the case of most bodies, than in the case of amber. For instance a brass rod with a glass handle, if rubbed on a piece of silk or cloth, will shew the power to a marked degree. The electrification here resides in the brass ; as will be explained immediately, the interposition of glass or some similar substance between the brass and the hand is necessary in order that the brass may retain its power for a sufficient time to enable us to observe it. If we hold the instrument by the brass rod and rub the glass handle we find that the same power is acquired by the glass.

II. Conductors and Insulators.

6. Let us now suppose that we hold the electrified brass rod in one hand by its glass handle, and that we touch it with the other hand. We find that after touching it its power of attracting small bodies will have completely disappeared. If we immerse it in a stream of water or pass it through a flame we find the same result. If on the other hand we touch it with a piece of silk or a rod of glass, or stand it in a current of air, we find that its power of attracting small bodies remains unimpaired, at any rate for a time. It appears therefore that the human body, a flame or water

6 Electrostatics — Physical Principles [ch. i

have the power of destroying the electrification of the brass rod when placed in contact with it, while silk and glass and air do not possess this property. It is for this reason that in handling the electrified brass rod, the substance in direct contact with the brass has been supposed to be glass and not the hand.

In this way we arrive at the idea of dividing all substances into two classes according as they do or do not remove the electrification when touch- ing the electrified body. The class which remove the electrification are called ( onductors, for as we shall see later, they conduct the electrification away from the electrified body rather than destroy it altogether; the class which allow the electrified body to retain its electrification are called non- conductors or insulators. The classification of bodies into conductors and insulators appears to have been first discovered by Stephen Gray (1696- 1736).

At the same time it must be explained that the difference between insulators and conductors is one of degree only. If our electrified brass rod were left standing for a week in contact only with the air surrounding it and the glass of its handle, we should find it hard to detect traces of electrifica- tion after this time — the electrification would have been conducted away by the air and the glass. So also if we had been able to immerse the rod in a flame for a billionth of a second only, we might have found that it retained considerable traces of electrification. It is therefore more logical to speak of good conductors and bad conductors than to speak of conductors and insula- tors. Nevertheless the difference between a good and a bad conductor is so enormous, that for our present purpose we need hardly take into account the feeble conducting power of a bad conductor, and may without serious incon- sistency, speak of a bad conductor as an insulator. There is, of course, nothing to prevent us imagining an ideal substance which has no conducting power at all. It will often simplify the argument to imagine such a substance, although we cannot realise it in nature.

It may be mentioned here that of all substances the metals are by very much the best conductors. Next come solutions of salts and acids, and lastly as very bad conductors (and therefore as good insulators) come oils, waxes, silk, glass and such substances as sealing wax, shellac, indiarubber. Gases under ordinary conditions are good insulators. Indeed it is worth noticing that if this had not been so, we should probably never have become acquainted with electric phenomena at all, for all electricity would be carried away by conduction through the air as soon as it was generated. Flames, however, conduct well, and, for reasons which will be explained later, all gases become good conductors when in the presence of radium or of so-called radio-active substances. Distilled water is an almost perfect insulator, but any other sample of water will contain impurities which generally cause it to conduct

6, 7] The Fundamental Conceptions of Electrostatics 7

tolerably well, and hence a wet body is generally a bad insulator. So also an electrified body suspended in air loses its electrification much more rapidly in damp weather than in dry, owing to conduction by water-particles in the air.

When the body is in contact with insulators only, it is said to be "insulated." The insulation is said to be good when the electrified body retains its electrification for a long interval of time, and is said to be poor when the electrification disappears rapidly. Good insulation will enable a body to retain most of its electrification for some days, while with poor insula- tion the electrification will last only for a few minutes or seconds.

III. Quantity of Electricity.

7. We pass next to the conception of a definite quantity of electricity, this quantity measuring the degree of electrification of the body with which it is associated. It is found that the quantity of electricity associated with any body remains constant except in so far as it is conducted away by con- ductors. To illustrate, and to some extent to prove this law, we may use an instrument known as the gold-leaf electroscope. This consists of a glass vessel, through the top of which a metal rod is passed, supporting at its lower end two gold-leaves which under normal conditions hang flat side by side, touching one another throughout their length. When an electrified body touches or is brought near to the brass rod, the two gold-leaves are seen to separate, for reasons which will become clear later (§ 21), so that the instru- ment can be used to examine whether or not a body is electrified.

Let us fix a metal vessel on the top of the brass rod, the vessel being closed but having a lid through which bodies can be in- serted. The lid must be supplied with an insulating handle for its manipulation. Suppose that we have electrified some piece of matter — to make the picture definite, suppose that we have electrified a small brass rod by rubbing it on silk — and let us suspend this body inside the vessel by an insulating thread in such a manner that it does not touch the sides of the vessel. Let us close the lid of the vessel, so that the vessel entirely surrounds the electrified body, and note the amount of separation of the gold-leaves of the electro- scope. Let us try the experiment any number of times, placing the electrified body in different positions inside the closed vessel, taking care only that it does not come into contact with the sides of the vessel or with any other conductors. We shall find that in every case the separation of the gold-leaves is exactly the same.

Via. l.

8 Electrostatics — Physical Principles [ch. i

In this way then, we get the idea of a definite quantity of electrification associated with the brass rod, this quantity being independent of the position of the r >d inside the closed vessel of the electroscope. We find, further, that the divergence of the gold-leaves is not only independent of the position of the rod inside the vessel, but is independent of any changes of state which the rod may have experienced between successive insertions in the vessel, provided only that it has not been touched by conducting bodies. We might for instance heat the rod, or, if it was sufficiently thin, we might bend it into a different shape, and on replacing it inside the vessel we should find that it produced exactly the same deviation of the gold-leaves as befoie. We may, then, regard the electrical properties of the rod as being due to a quantity of electricity associated with the rod, this quantity remaining permanently the same, except in so far as the original charge is lessened by contact with conductors, or increased by a fresh supply.

8. We can regard the electroscope as giving an indication of the magni- tude of a quantity of electricity, two charges being equal when they produce the same divergence of the leaves of the electroscope.

In the same way we can regard a spring-balance as giving an indication of the magnitude of a weight, two weights being equal when they produce the same extension of the spring.

The question of the actual quantitative measurement of a quantity of electricity as a multiple of a specified unit has not yet been touched. We can, however, easily devise means for the exact quantitative measurement of electricity in terms of a unit. We can charge a brass rod to any degree we please, and agree that the charge on this rod is to be taken to be the standard unit charge. By rubbing a number of rods until each produces exactly the same divergence of the electroscope as the standard charge, we can prepare a number of unit charges, and we can now say that a charge is equal to n units, if it produces the same deviation of the electroscope as would be produced by n units all inserted in the vessel of the electroscope at once. This method of measuring an electric charge is of course not one that any rational being would apply in practice, but the object of the present explanation is to elucidate the fundamental principles, and not to give an account of practical methods.

9. Positive and Negative Electricity. Let us suppose that we insert in the vessel of the electroscope the piece of silk on which one of the brass rods has been supposed to have been rubbed in order to produce its unit charge. We shall find that the silk produces a divergence of the leaves of the electroscope, and further that this divergence is exactly equal to that which is produced by inserting the brass rod alone into the vessel of the electroscope. If, however, we insert the brass rod and the silk together into the electroscope, no deviation of the leaves can be detected.

7-11] The Fundamental Conceptions of Electrostatics 9

Again, let us suppose that we charge a brass rod A with a charge which the divergence of the leaves shews to be n units. Let us rub a second brass rod B with a piece of silk G until it has a charge, as indicated by the electro- scope, of m units, m being smaller than n. If we insert the two brass rods together, the electroscope will, as already explained, give a divergence corre- sponding to n + m units. If, however, we insert the rod A and the silk G together, the deviation will be found to correspond to n — m units.

In this way it is found that a charge of electricity must be supposed to have sign as well as magnitude. As a matter of convention, we agree to speak of the m units of charge on the silk as m positive units, or more briefly as a charge + m, while we speak of the charge on the brass as m negative units, or a charge — w.

10. Generation of Electricity. It is found to be a general law that, on rubbing two bodies which are initially uncharged, equal quantities of positive and negative electricity are produced on the two bodies, so that the total charge generated, measured algebraically, is nil.

We have seen that the electroscope does not determine the sign of the charge placed inside the closed vessel, but only its magnitude. We can, however, determine both the sign and magnitude by two observations. Let us first insert the charged body alone into the vessel. Then if the divergence of the leaves corresponds to m units, we know that the. charge is either + m or — m, and if we now insert the body in company with another charged body, of which the charge is known to be + n, then the charge we are attempting to measure will be + m or — m according as the divergence of the leaves indicates n + m or n ~ m units. With more elaborate instruments to be described later (electrometers) it is possible to determine both the magnitude and sign of a charge by one observation.

11. If we had rubbed a rod of glass, instead of one of brass, on the silk, we should have found that the silk had a negative charge, and the glass of course an equal 'positive charge. It therefore appears that the sign of the charge produced on a body by friction depends not only on the nature of the body itself, but also on the nature of the body with which it has been rubbed.

The following is found to be a general law : If rubbing a substance A on a second substance B charges A positively and B negatively, and if rubbing the substance B on a third substance G charges B positively and G negatively, then rubbing the substance A on the substance G will charge A positively and G negatively.

It is therefore possible to arrange any number of substances in a list such that a substance is charged with positive or negative electricity when rubbed

10 Electrostatics — Physical Principles [on. I

with a second substance, according as the first substance stands above or below the second substance on the list. The following is a list of this kind, which 'ncludes some of the most important substances :

Cat's skin, Glass, Ivory, Silk, Rock crystal, The Hand, Wood, Sulphur, Flannel, Cotton, Shellac, Caoutchouc, Resins, Guttapercha, Metals, Guncotton.

A substance is said to be electropositive or electronegative to a second substance according as it stands above or below it on a list of this kind. Thus of any pair of substances one is always electropositive to the other, the other being electronegative to the first. Two substances, although chemically the same, must be regarded as distinct for the purposes of a list such as the above, if their physical conditions are different ; for instance, it is found that a hot body must be placed lower on the list than a cold body of the same chemical composition.

IV. Attraction and Repulsion of Electric Charges.

12. A small ball of pith, or some similarly light substance, coated with gold-leaf and suspended by an insulating thread, forms a convenient instru- ment for investigating the forces, if any, which are brought into play by the presence of electric charges. Let us electrify a pith ball of this kind positively and suspend it from a fixed point. We shall find that when we bring a second small body charged with positive electricity near to this first body the two bodies tend to repel one another, whereas if we bring a negatively charged body near to it, the two bodies tend to attract one another. From this and similar experiments it is found that two small bodies charged with electricity of the same sign repel one another, and that two small bodies charged with electricity of different signs attract one another.

This law can be well illustrated by tying together a few light silk threads by their ends, so that they form a tassel, and allowing the threads to hang vertically. If we now stroke the threads with the hand, or brush them with a brush of any kind, the threads all become positively electrified, and there- fore repel one another. They consequently no longer hang vertically but spread themselves out into a cone. A similar phenomenon can often be noticed on brushing the hair in dry weather. The hairs become positively electrified and so tend to stand out from the head.