A Treatise on Electricity and Magnetism/Volume 1

by James Clerk Maxwell

Clarendon Press Series

 

A Treatise

on

Electricity and Magnetism

Maxwell

 

vol. I.

 

London

Macmillan and Co.

Publishers to the University of

Oxford

Clarendon Press Series

 

A TREATISE

ON

ELECTRICITY AND MAGNETISM

 

BY

JAMES CLERK MAXWELL, M.A.

LLD. EDIN., F.R.SS. LONDON AND EDINBURGH

HONORARY FELLOW OF TRINITY COLLEGE,

AND PROFESSOR OF EXPERIMENTAL PHYSICS

IN THE UNIVERSITY OF CAMBRIDGE

 

VOL. I

 

Oxford

AT THE CLARENDON PRESS

1873

[All rights reserved]

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 phenomena, amber, ἤλεκτρον, 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 phenomena 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, μάγνης, 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 Electromagnetism.

In the following Treatise I propose to describe the 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 phenomena, 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 phenomena among which we are to look for illustrations or explanations of the electromagnetic phenomena.

In describing the phenomena, I shall select those which most clearly illustrate the fundamental ideas of the theory, omitting others, or reserving them till the reader 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 phenomena chiefly with a view to their measurement, describing the methods of measurement, and defining the standards on which they depend.

In the application of mathematics to the calculation of electrical 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.

The internal relations of the different branches of the science which we have to study are more numerous and complex than those of any other science hitherto developed. Its external relations, on the one hand to dynamics, and on the other to heat, light, chemical action, and the constitution of bodies, seem to indicate the special importance of electrical science as an aid to the interpretation of nature.

It appears to me, therefore, that the study of electromagnetism 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 phenomena have also been investigated, and the probability 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 reduction 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 subjects of enquiry, and furnishing us with means of investigation.

It is hardly necessary to enlarge upon the beneficial 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 measurements, 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 experimental opportunities for acquiring it, have been already very great, both in stimulating the energies of advanced 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.

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 memories which are of great importance in electrical science, but they lie concealed in the bulky Transactions of learned societies; they do not form a connected 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 considerably from that of several excellent electrical works, published, most of them, in Germany, and it may appear that scant justice is done to the speculations of several eminent electricians and mathematicians. One reason of this is that before I began the study of electricity I resolved to read no mathematics on the subject till I had first read through Faraday's 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 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^[1], 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 proceeded with the study of Faraday, I perceived 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 mathematicians.

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 coincided, 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 in which we begin with the whole and arrive at the parts by analysis, while the ordinary mathematical methods were founded on the principle of beginning with the parts and building up the whole by synthesis.

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, considered 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 summation of the electrified particles divided each by its distance from a given point. Hence many of the mathematical 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 Faraday.

Great progress has been made in electrical science, chiefly in Germany, by cultivators of the theory of action at a distance. The valuable electrical measurements 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, 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, additional 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 mathematical 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 leaves undecided.

In a philosophical point of view, moreover, it is exceedingly important that two methods should be compared, both of which have succeeded in explaining the principal electromagnetic phenomena, and both of which have attempted to explain the propagation of light is 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 supporters, and will be expounded with a skill worthy of its ingenuity.

I have not attempted an exhaustive account of electrical 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' Reibungselektrizität, Beer's Einleitung in die Elektrostatik, &c.

I have confined myself almost entirely to the mathematical treatment of the subject, but I would recommend the student, after he has learned, experimentally if possible, what are the phenomena to be observed, to read carefully Faraday's Experimental 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 accurately describing scientific operations and their results^[2].

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 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 consecutively. If by anything I have here written I 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 elementary 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.

CONTENTS.

 

PRELIMINARY.

On the Measurement of Quantities.

Art.

Page

001. The expression of a quantity consists of two factors, the numerical value, and the name of the concrete unit

001

002. Dimensions of derived units

001

003–5. The three fundamental units—Length, Time and Mass

2, 3

006. Derived units

005

007. Physical continuity and discontinuity

006

008. Discontinuity of a function of more than one variable

007

009. Periodic and multiple functions

008

 10. Relation of physical quantities to directions in space

008

 11. Meaning of the words Scalar and Vector

009

 12. Division of physical vectors into two classes, Forces and Fluxes

 10

 13. Relation between corresponding vectors of the two classes

 11

 14. Line-integration appropriate to forces, surface-integration to fluxes

 12

 15. Longitudinal and rotational vectors

 12

 16. Line-integrals and potentials

 13

 17. Hamilton's expression for the relation between a force and its potential

 15

 18. Cyclic regions and geometry of position

 16

 19. The potential in an acyclic region is single valued

 17

 20. System of values of the potential in a cyclic region

 18

 21. Surface-integrals

 19

 22. Surfaces, tubes, and lines of flow

 21

 23. Right-handed and left-handed relations in space

 24

 24. Transformation of a line-integral into a surface-integral

 25

 25. Effect of Hamilton's operation $\nabla$ on a vector function

 27

 26. Nature of the operation $\nabla ^{2}$

 29

PART I.

Electrostatics.

CHAPTER I.

Description of Phenomena.

Art.

Page

 27. Electrification by friction. Electrification is of two kinds, to which the names of Vitreous and Resinous, or Positive and Negative, have been given

 30

 28. Electrification by induction

 31

 29. Electrification by conduction. Conductors and insulators

 32

 30. In electrification by friction the quantity of the positive electrification is equal to that of the negative electrification

 33

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

 33

 32. To discharge a conductor completely into a metallic vessel

 34

 33. Test of electrification by gold-leaf electroscope

 34

 34. Electrification, considered as a measurable quantity, may be called Electricity

 35

 35. Electricity may be treated as a physical quantity

 36

 36. Theory of Two fluids

 37

 37. Theory of One fluid

 39

 38. Measurement of the force between electrified bodies

 40

 39. Relation between this force and the quantities of electricity

 41

 40. Variation of the force with the distance

 42

 41, 42. Definition of the electrostatic unit of electricity.—Its dimensions

 42

 43. Proof of the law of electric force

 43

 44. Electric field

 44

 45. Electric potential

 45

 46. Equipotential surfaces. Example of their use in reasoning about electricity

 45

 47. Lines of force

 47

 48. Electric tension

 47

 49. Electromotive force

 47

 50. Capacity of a conductor

 48

 51. Properties of bodies.—Resistance

 48

 52. Specific Inductive capacity of a dielectric

 50

 53. 'Absorption' of electricity

 50

 54. Impossibility of an absolute charge

 51

 55. Disruptive discharge.—Glow

 52

 56. Brush

 54

 57. Spark

 55

 58. Electrical phenomena of Tourmaline

 56

 59. Plan of the treatise, and sketch of its results

 57

 60. Electric polarization and displacement

 59

 61. The motion of electricity analogous to that of an incompressible fluid

 62

 62. Peculiarities of the theory of this treatise

 62

CHAPTER II.

Elementary Mathematical Theory of Electricity.

 63. Definition of electricity as a mathematical quantity

 66

 64. Volume-density, surface-density, and line-density

 67

 65. Definition of the electrostatic unit of electricity

 68

 66. Law of force between electrified bodies

 69

 67. Resultant force between two bodies

 69

 68. Resultant force at a point

 69

 69. Line-integral of electric force; electromotive force

 71

 70. Electric potential

 72

 71. Resultant force in terms of the potential

 72

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

 73

 73. Potential due to an electrified system

 74

 74. Proof of the law of the inverse square

 74

 75. Surface-integral of electric induction

 77

 76. Introduction through a closed surface due to a single centre of force

 77

 77. Poisson's extension of Laplace's equation

 79

 78. Conditions to be fulfilled at an electrified surface

 80

 79. Resultant force on an electrified surface

 82

 80. The electrification of a conductor is entirely on the surface

 83

 81. A distribution of electricity on lines or points is physically impossible

 84

 82. Lines of electric induction

 84

 83. Specific inductive capacity

 86

CHAPTER III.

Systems of Conductors.

Art.

Page

 84. On the superposition of electrified systems

 88

 85. Energy of an electrified system

 88

 86. General theory of a system of conductors. Coefficients of potential

 89

 87. Coefficients of induction. Capacity of a conductor. Dimensions of these coefficients

 90

 88. Reciprocal property of the coefficients

 91

 89. A theorem due to Green

 92

 90. Relative magnitude of the coefficients of potential

 92

 91. And of induction

 93

 92. 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 coefficients of potential

 94

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

 94

 94. Comparison of electrified systems

 96

CHAPTER IV.

General Theorems.

 95. Two opposite methods of treating electrical questions

 98

 96. Characteristics of the potential function

 99

 97. Conditions under which the volume-integral

$\iiint {\Big (}u{\frac {dV}{dx}}+v{\frac {dV}{dy}}+w{\frac {dV}{dz}}{\Big )}dx\,dy\,dz$

vanishes

100

 98. Thomson's theorem of the unique minimum of

$\iiint {\frac {1}{K}}(a^{2}+b^{2}+c^{2})\,dx\,dy\,dz$

103

 99. Application of the theorem to the determination of the distribution of electricity

107

100. Green's theorem and its physical interpretation

108

101. Green's functions

113

102. Method of finding limiting values of electrical coefficients

115

CHAPTER V.

Mechanical Action between Electrified Bodies.

Art.

Page

103. Comparison of the force between different electrified systems

119

104. Mechanical action on an element of an electrified surface

121

105. Comparison between theories of direct action and theories of stress

122

106. The kind of stress required to account for the phenomenon

123

107. The hypothesis of stress considered as a step in electrical science

126

108. The hypothesis of stress shewn to account for the equilibrium of the medium and for the forces acting between electrified bodies

128

109. Statements of Faraday relative to the longitudinal tension and lateral pressure of the lines of force

131

110. Objections to stress in a fluid considered

131

111. Statement of the theory of electric polarization

132

CHAPTER VI.

Points and Lines of Equilibrium.

112. Conditions of a point of equilibrium

135

113. Number of points of equilibrium

136

114. At a point or line of equilibrium there is a conical point or a line of self-intersection of the equipotential surface

137

115. Angles at which an equipotential surface intersects itself

138

116. The equilibrium of an electrified body cannot be stable

139

CHAPTER VII.

Forms of Equipotential Surfaces and Lines of Flow.

117. Practical importance of a knowledge of these forms in simple cases

142

118. Two electrified points, ratio $4:1$. (Fig. I)

143

119. Two electrified points, ratio $4:-1$. (Fig. II)

144

120. An electrified point in a uniform field of force. (Fig. III)

145

121. Three electrified points. Two spherical equipotential surfaces. (Fig. IV)

145

122. Faraday's use of the conception of lines of force

146

123. Method employed in drawing the diagrams

147

CHAPTER VIII.

Simple Cases of Electrification.

Art.

Page

124. Two parallel planes

150

125. Two concentric spherical surfaces

152

126. Two coaxal cylindric surfaces

154

127. Longitudinal force on a cylinder, the ends of which are surrounded by cylinders at different potentials

155

CHAPTER IX.

Spherical Harmonics.

128. Singular points at which the potential becomes infinite

157

129. Singular points of different orders defined by their axes

158

130. Expression for the potential due to a singular point referred to its axes

160

131. This expression is perfectly definite and represents the most general type of the harmonic of $i$ degrees

162

132. The zonal, tesseral, and sectorial types

163

133. Solid harmonics of positive degree. Their relation to those of negative degree

165

134. Application to the theory of electrified spherical surfaces

166

135. The external action of an electrified spherical surface compared with that of an imaginary singular point at its centre

167

136. Proof that if $Y_{i}$ and $Y_{j}$ are two surface harmonics of different degrees, the surface-integral $\iint Y_{i}Y_{j}dS=0$, the integration being extended over the spherical surface

169

137. Value of $\iint Y_{i}Y_{j}dS$ where $Y_{i}$ and $Y_{j}$ are surface harmonics of the same degree but of different types

169

138. On conjugate harmonics

170

139. If $Y_{j}$ is the zonal harmonic and $Y_{i}$ any other type of the same degree

$\iint Y_{i}Y_{j}dS={\frac {4\pi a^{2}}{2i+1}}Y_{i(j)}$

where $Y_{i(j)}$ is the value of $Y_{i}$ at the pole of $Y_{j}$

171

140. Development of a function in terms of spherical surface harmonics

172

141. Surface-integral of the square of a symmetrical harmonic

173

142. Different methods of treating spherical harmonics

174

143. On the diagrams of spherical harmonics. (Figs. V, VI, VII, VIII, IX)

175

144. If the potential is constant throughout any finite portion of space it is so throughout the whole region continuous 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

177

146. Application to spherical and nearly spherical conductors

178

CHAPTER X.

Confocal Surfaces of the Second Degree.

147. The lines of intersection of two systems and their intercepts by the third system

181

148. The characteristic equation of $V$ in terms of ellipsoidal coordinates

182

149. Expression of $\alpha$, $\beta$, $\gamma$ in terms of elliptic functions

183

150. Particular solutions of electrical distribution on the confocal surfaces and their limiting forms

184

151. Continuous transformation into a figure of revolution about the axis of $z$

187

152. Transformation into a figure of revolution about the axis of $x$

188

153. Transformation into a system of cones and spheres

189

154. Confocal paraboloids

189

CHAPTER XI.

Theory of Electric Images.

155. Thomson's method of electric images

191

156. When two points are oppositely and unequally electrified, the surface for which the potential is zero is a sphere

192

157. Electric images

193

158. Distribution of electricity on the surface of the sphere

195

159. Image of any given distribution of electricity

196

160. Resultant force between an electrified point and sphere

197

161. Images in an infinite plane conducting surface

198

162. Electric inversion

199

163. Geometrical theorems about inversion

201

164. Application of the method to the problem of Art. 158

202

165. Finite systems of successive images

203

166. Case of two spherical surfaces intersecting at an angle ${\tfrac {\pi }{n}}$

204

167. Enumeration of the cases in which the number of images is finite

206

168. Case of two spheres intersecting orthogonally

207

169. Case of three spheres intersecting orthogonally

210

170. Case of four spheres intersecting orthogonally

211

171. Infinite series of images. Case of two concentric spheres

212

172. Any two spheres not intersecting each other

213

173. Calculation of the coefficients of capacity and induction

216

174. Calculation of the charges of the spheres, and of the force between them

217

175. Distribution of electricity on two spheres in contact. Proof sphere

219

176. Thomson's investigation of an electrified spherical bowl

221

177. Distribution on an ellipsoid, and on a circular disk at potential $V$

221

178. Induction on an uninsulated 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 $V$ and uninfluenced

223

181. Induction on the bowl due to a point placed anywhere

224

CHAPTER XII.

Conjugate Functions in Two Dimensions.

182. Cases in which the quantities are functions of $x$ and $y$ only

226

183. Conjugate functions

227

184. Conjugate functions may be added or subtracted

228

185. Conjugate functions of conjugate functions are themselves conjugate

229

186. Transformation of Poisson's equation

231

187. Additional theorems on conjugate functions

232

188. Inversion in two dimensions

232

189. Electric images in two dimensions

233

190. Neumann's transformation of this case

234

191. Distribution of electricity near the edge of a conductor formed by two plane surfaces

236

192. Ellipses and hyperbolas. (Fig. X)

237

193. Transformation of this case. (Fig. XI)

238

194. Application to two cases of the flow of electricity in a conducting sheet

239

195. Application to two cases of electrical induction

239

196. Capacity of a condenser consisting of a circular disk between two infinite planes

240

197. Case of a series of equidistant planes cut off by a plane at right angles to them

242

198. Case of a furrowed surface

243

199. Case of a single straight groove

243

200. Modification of the results when the groove is circular

244

201. Application to Sir W. Thomson's guard-ring

245

202. Case of two parallel plates cut off by a perpendicular plane. (Fig. XII)

246

203. Case of a grating of parallel wires. (Fig. XIII)

248

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

248

205. The grating used as a shield to protect a body from electrical influence

249

206. Method of approximation applied to the case of the grating

251

CHAPTER XIII.

Electrostatic Instruments.

207. The frictional electrical machine

254

208. The electrophorus of Volta

255

209. Production of electrification by mechanical work.—Nicholson's Revolving Doubler

256

210. Principle of Varley's and Thomson's electrical machines

256

211. Thomson's water-dropping machine

259

212. Holtz's electrical machine

260

213. Theory of regenerators applied to electrical machines

260

214. On electrometers and electroscopes. Indicating instruments and null methods. Difference between registration and measurement

262

215. Coulomb's Torsion Balance for measuring charges

263

216. Electrometers for measuring potentials. Snow Harris's and Thomson's

266

217. Principle of the guard-ring. Thomson's Absolute Electrometer

267

218. Heterostatic method

269

219. Self-acting electrometers.—Thomson's Quadrant Electrometer

271

220. Measurement of the electric potential of a small body

274

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

275

222. Measurement of the potential of a conductor without touching it

276

223. Measurement of the superficial density of electrification. The proof plane

277

224. A hemisphere used as a test

278

225. A circular disk

279

226. On electric accumulators. The Leyden jar

281

227. Accumulators of measurable capacity

282

228. The guard-ring accumulator

283

229. Comparison of the capacities of accumulators

285

PART II.

Electrokinematics.

CHAPTER I.

The Electric Current.

230. Current produced when conductors are discharged

288

231. Transference of electrification

288

232. Description of the voltaic battery

289

233. Electromotive force

290

234. Production of a steady current

290

235. Properties of the current

291

236. Electrolytic action

291

237. Explanation of terms connected with electrolysis

292

238. Different modes of passage of the current

292

239. Magnetic action of the current

293

240. The Galvanometer

294

CHAPTER II.

Conduction And Resistance.

241. Ohm's Law

295

242. Generation of heat by the current. Joule's Law

296

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

297

244. Differences between the two classes of phenomena

297

245. Farady's doctrine of the impossibility of an absolute charge

298

CHAPTER III.

Electromotive force between bodies in contact.

246. Volta's law of the contact force between different metals at the same temperature

299

247. Effects of electrolytes

300

248. Thomson's voltaic current in which gravity performs the part of chemical action

300

249. Peltier's phenomenon. Deduction of the thermoelectric electromotive force at a junction

300

250. Seebeck's discovery of thermoelectric currents

302

251. Magnus's law of a circuit of one metal

302

252. Cumming's discovery of thermoelectric inversions

304

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

304

254. Tait's law of the electromotive force of a thermoelectric pair

305

CHAPTER IV.

Electrolysis.

255. Faraday's law of electrochemical equivalents

307

256. Clausius's theory of molecular agitation

309

257. Electrolytic polarization

309

258. Test of an electrolyte by polarization

310

259. Difficulties in the theory of electrolysis

310

260. Molecular charges

311

261. Secondary actions observed in the electrodes

313

262. Conversation of energy in electrolysis

315

263. Measurement of chemical affinity as an electromotive force

316

CHAPTER V.

Electrolytic polarization.

264. Difficulties of applying Ohm's law to electrolytes

318

265. Ohm's law nevertheless applicable

318

266. The effect of polarization distinguished from that of resistance

318

267. Polarization due to the presence of the ions at the electrodes. The ions not in a free state

319

268. Relation between the electromotive force of polarization and the states of the ions at the electrodes

320

269. Dissipation of the ions and loss of polarization

321

270. Limit of polarization

321

271. Ritter's secondary pile compared with the Leyden jar

322

272. Constant voltaic elements.—Daniell's cell

325

CHAPTER VI.

Mathematical Theory of the Distribution of Electric Currents.

273. Linear conductors

329

274. Ohm's Law

329

275. Linear conductors in series

329

276. Linear conductors in multiple arc

330

277. Resistance of conductors of uniform section

331

278. Dimensions of the quantities involved in Ohm's law

332

279. Specific resistance and conductivity in electromagnetic measure

333

280. Linear systems of conductors in general

333

281. Reciprocal property of any two conductors of the system

335

282. Conjugate conductors

336

283. Heat generated in the system

336

284. The heat is a minimum when the current is distributed according to Ohm's law

337

CHAPTER VII.

Conduction in Three Dimensions.

285. Notation

338

286. Composition and resolution of electric currents

338

287. Determination of the quantity which flows through any surface

339

288. Equation of a surface of flow

340

289. Relation between any three systems of surfaces of flow

340

290. Tubes of flow

340

291. Expression for the components of the flow in terms of surfaces of flow

341

292. Simplification of this expression by a proper choice of parameters

341

293. Unit tubes of flow used as a complete method of determining the current

341

294. Current-sheets and current-functions

342

295. Equation of 'continuity'

342

296. Quantity of electricity which flows through a given surface

344

CHAPTER VIII.

Resistance and Conductivity in Three Dimensions.

Art.

Page

297. Equations of resistance

345

298. Equations of conduction

346

299. Rate of generation of heat

346

300. Conditions of stability

347

301. Equation of continuity in a homogeneous medium

348

302. Solution of the equation

348

303. Theory of the coefficient $T$. It probably does not exist

349

304. Generalized form of Thomson's theorem

350

305. Proof without symbols

351

306. Strutt's method applied to a wire of variable section.—Lower limit of the value of the resistance

353

307. Higher limit

356

308. Lower limit for the correction for the ends of the wire

358

309. Higher limit

358

CHAPTER IX.

Conduction Through Heterogeneous Media.

310. Surface-conditions

360

311. Spherical surface

362

312. Spherical shell

363

313. Spherical shell placed in a field of uniform flow

364

314. Medium in which small spheres are uniformly disseminated

365

315. Images in a plane surface

366

316. Method of inversion not applicable in three dimensions

367

317. Case of conduction through a stratum bounded by parallel planes

367

318. Infinite series of images. Application to magnetic induction

368

319. On stratified conductors. Coefficients of conductivity of a conductor consisting of alternate strata of two different substances

369

320. If neither of the substances has the rotatory property denoted by $T$ the compound conductor is free from it

370

321. If the substances are isotropic the direction of greatest resistance is normal to the strata

371

322. Medium containing parallelepipeds of another medium

371

323. The rotatory property cannot be introduced by means of conducting channels

372

324. Construction of an artificial solid having given coefficients of longitudinal and transverse conductivity

373

CHAPTER X.

Conduction in Dielectrics.

Art.

Page

325. In a strictly homogeneous medium there can be no internal charge

374

326. Theory of a condenser in which the dielectric is not a perfect insulator

375

327. No residual charge due to simple conduction

376

328. Theory of a composite accumulator

376

329. Residual charge and electrical absorption

378

330. Total discharge

380

331. Comparison with the conduction of heat

381

332. Theory of telegraph cables and comparison of the equations with those of the conduction of heat

381

333. Opinion of Ohm on this subject

384

334. Mechanical illustration of the properties of a dielectric

385

CHAPTER XI.

Measurement of the Electric Resistance of Conductors.

335. Advantage of using material standards of resistance in electrical measurements

388

336. Different standards which have been used and different systems which have been proposed

388

337. The electromagnetic system of units

389

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

389

339. Professed value of the Ohm $10,000,000$ metres per second

389

340. Reproduction of standards

390

341. Forms of resistance coils

391

342. Coils of great resistance

392

343. Arrangement of coils in series

392

344. Arrangement in multiple arc

393

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

394

346. (2) By the differential galvanometer

394

347. (3) By Wheatstone's Bridge

398

348. Estimation of limits of error in the determination

399

349. Best arrangement of the conductors to be compared

400

350. On the use of Wheatstone's Bridge

402

351. Thomson's method for small resistances

404

352. Matthiessen and Hockin's method for small resistances

406

353. Comparison of great resistances by the electrometer

408

354. By accumulation in a condenser

409

355. Direct electrostatic method

409

356. Thomson's method for the resistance of a galvanometer

410

357. Mance's method of determining the resistance of a battery

411

358. Comparison of electromotive forces

413

CHAPTER XII.

Electric Resistance of Substances.

359. Metals, electrolytes, and dielectrics

415

360. Resistance of metals

416

361. Resistance of mercury

417

362. Table of resistance of metals

418

363. Resistance of electrolytes

419

364. Experiments of Paalzow

419

365. Experiments of Kohlrausch and Nippoldt

420

366. Resistance of dielectrics

421

367. Gutta-percha

423

368. Glass

423

369. Gases

424

370. Experiments of Wiedemann and Rühlmann

425

PLATES. (not in original TOC)

Page 26, l. 3 from bottom, dele 'As we have made no assumption', &c. down to l. 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, $a_{0}$, the last curve of the series, for which $a=a_{1}$, coinciding with the original closed curve s.

Let us also suppose that the curves for which $\beta$ is constant form a series of lines drawn from the point at which $a=a_{0}$ to the closed curve s, the first $\beta _{0}$, and the last, $\beta _{1}$, being identical

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

$\int _{\beta _{0}}^{\beta _{1}}(X{\frac {dx}{d\beta }})_{a=a_{0}}\ d\beta$,

is zero, because the curve $a=a_{0}$, is reduced to a point at which there is but one value of $X$ and of $x$.

The two line integrals,

$-\int _{a_{0}}^{a_{1}}(X{\frac {dx}{da}})\ _{\mathrm {\beta =\beta _{1}} }\ da$ + $\int _{a_{0}}^{a_{1}}(X{\frac {dx}{da}})\ _{\mathrm {\beta =\beta _{0}} }\ da$,

destroy each other, because the point $(a,\beta _{1})$ is identical with the point $(a,\beta _{0})$.

The expressions (8) is therefore reduced to

$\int _{\beta _{0}}^{\beta _{1}}(X{\frac {dx}{d\beta }})\ _{\mathrm {a=a_{1}} }\ d\beta$

(9)

Since the curve $a=a_{1}$ is identical with the closed curve s, we may write this expression

p. 80, in equations (3), (4), (6), (e), (17), (18), (19), (20), (21), (22), for $R$ read $N1$.

p. 82, l. 3, for $Rl$ read $N1$.

p. 83, in equations (28), (29), (30). (31), for $({\frac {d^{2}V_{1}}{dx^{3}}})$ read $({\frac {d^{2}V'}{dxdv}})$

p. 83,„ in equation (29), insert - before the second member.

p. 105, 1. 2, for $Q$ read $8\pi Q$.

p. 108, equation (1), for $\rho$ read $\rho '$.

p. 108,„ equation„ (2), for $\rho '$ read $\rho$.

p. 108,„ equation„ (3), for $\sigma$, read $\sigma '$.

p. 108,„ equation„ (4), for $\sigma '$ read $\sigma$.

p. 113, l. 4, for $KR$ read ${\frac {1}{4\pi }}KR$.

p. 113,„ l. 5, for $KRR'cos\epsilon$ read ${\frac {1}{4\pi }}KRR'cos\epsilon$.

p. 111, I. 5, for $S_{1}$ read $S$.

p. 124, last line, for $e_{1}+e_{1}$ read $e_{1}+e_{2}$.

p. 125, lines 3 and 4, transpose within and without; l. 16, for $v$ read $V$; and l. 18, for $V$ read $v$.

p. 128, lines 11, 10, 8 from bottom for $dx$ read $dz$.

p. 149, l. 24, for equpotential read equipotential. p. 159, l. 3, for $F$ read $f$.

p. 159,„ l. 2 from bottom, for $M$ read $M_{2}$.

p. 163, l. 20, for $\lambda _{i-s+1}$ read $\lambda _{i-\sigma +1}$.

p. 164, equation (34), for $(-1)^{i-s}{\frac {\left|{\underline {2s}}\right.}{2^{2}s\left|{\underline {i}}\right.\left|{\underline {s}}\right.}}$ read $(-1)^{i-\sigma }{\frac {\left|{\underline {2\sigma }}\right.}{2^{2}\sigma \left|{\underline {i}}\right.\left|{\underline {\sigma }}\right.}}$

p. 179, equation (76), for $i+l$ read $2i+1$.

p. 185, equation (24), for ${\frac {x^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1$ read ${\frac {x^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}-b^{2}}}=1$.

p. 186, l. 5 from bottom, for 'The surface-density on the elliptic plate' read The surface-density on either side of the elliptic plate.

p. 186, equation (30), for $2\pi$ read $4\pi$.

p. 188, equation (38), for $\pi ^{2}$ read $2\pi ^{2}$.

p. 196, l. 27, for $e..e$ read $e_{1}..e_{2}$.

p. 197, equation (10) should be $M={\frac {E\,e}{f}}-{\frac {1}{2}}{\frac {e^{2}a^{3}}{f^{2}(f^{2}-a^{2})}}$.

p. 204, l. 15 from bottom, dele either.

p. 215, l. 4, for ${\sqrt {2k}}$ read ${\sqrt {2}}k$.

p. 234, equation (13), for $2E$ read ${\frac {E}{2\pi }}$.

p. 335, dele last 14 lines.

p. 336, l. 1, dele therefore.

p. 336,„ l. 2, for 'the potential at $C$ to exceed that at $D$ by $P$,' read a current, $C$, from $X$ to $Y$.

p. 336,„ l. 4, for '$C$ to $D$ will cause the potential at $A$ to exceed that at $B$ by the same quantity $P$,' read $X$ to $Y$ will cause an equal current $C$ from $A$ to $B$.

p. 351, l. 3, for $R_{1}^{2}u^{2}+R_{2}^{2}v^{2}+R_{3}^{2}w^{2}$ read $R_{1}u^{2}+R_{2}v^{2}+R_{3}w^{2}$.

p. 351,„ l. 5, read $+2\iiint (u{\frac {dV}{dx}}+v{\frac {dV}{dy}}+w{\frac {dV}{dz}})\,dx\,dy\,dz$.

p. 355, last line, for $S'$ read $S$.

p. 356, equation (12), for ${\overline {\left.{\frac {db}{d}}\right|}}^{2}$ read ${\overline {\left.{\frac {db}{dx}}\right|}}^{2}$.

p. 365, in equations (12), (15), (16), for $A$ read $Ar$.

p. 366, equation (3), for ${\frac {E_{2}}{r_{1}}}$ read ${\frac {E_{2}}{r_{2}}}$.

p. 367, l. 5, for $2k_{1}S$ read $2k_{2}S$.

p. 368, equation (14), for $J_{2}^{'}$ read $I_{2}^{'}$.

p. 397, l. 1, for ${\frac {D'}{E}}\delta '$ read ${\frac {D'}{E'}}\delta '$.

p. 404, at the end of Art. 350 insert as follows:—

When $\gamma$, the resistance to be measured, $a$, the resistance of the battery, and $\alpha$, the resistance of the galvanometer, are given, the best values of the other resistances have been shewn by Mr. Oliver Heaviside (Phil. Mag., Feb. 1873) to be

$c={\sqrt {a\alpha }},\quad b={\sqrt {a\gamma {\frac {\alpha +\gamma }{a+\gamma }}}},\quad \beta ={\sqrt {\alpha \gamma {\frac {a+\gamma }{\alpha +\gamma }}}}$.

 

---

1. 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.
2. Life and Letters of Faraday, vol. i, p. 395.