# A Treatise on Electricity and Magnetism/Contents

Editorial remarks:

• Chapter VI of Part II, "Linear Electric Currents" is referenced as "Mathematical Theory of the Distribution of Electric Currents" here.
• Art. 273 of that chapter, "On Systems of Linear Conductors" is referenced as "Linear Conductors" here.

CONTENTS.

ON THE MEASUREMENT OF QUANTITIES.

 Art. Page 1. The expression of a quantity consists of two factors, the numerical value, and the name of the concrete unit 1 2. Dimensions of derived units 1 .mw-parser-output .nowrap,.mw-parser-output .nowrap a:before,.mw-parser-output .nowrap .selflink:before{white-space:nowrap}3–5. The three fundamental units—Length, Time and Mass 2, 3 6. Derived units 5 7. Physical continuity and discontinuity 6 8. Discontinuity of a function of more than one variable 7 9. Periodic and multiple functions 8 10. Relation of physical quantities to directions in space 8 11. Meaning of the words Scalar and Vector 9 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 ${\displaystyle \nabla }$ on a vector function 27 26. Nature of the operation ${\displaystyle \nabla ^{2}}$ 29

PART I.

ELECTROSTATICS.

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.

Art. Page
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.

84. On the superposition of electrified systems

85. Energy of an electrified system

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

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

88. Reciprocal property of the coefficients

89. A theorem due to Green

90. Relative magnitude of the coefficients of potential

91. And of induction

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

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

94. Comparison of electrified systems

Chapter IV.

General Theorems.

95. Two opposite methods of treating electrical questions

96. Characteristics of the potential function

97. Conditions under which the volume-integral

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

98. Thomson's theorem of the unique minimum of

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

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

100. Green's theorem and its physical interpretation

101. Green's functions

102. Method of finding limiting values of electrical coefficients

Chapter V.

Mechanical Action between Electrified Bodies.

103. Comparison of the force between different electrified systems

104. Mechanical action on an element of an electrified surface

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

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

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

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

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

110. Objections to stress in a fluid considered

111. Statement of the theory of electric polarization

Chapter VI.

Points and Lines of Equilibrium.

112. Conditions of a point of equilibrium

113. Number of points of equilibrium

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

115. Angles at which an equipotential surface intersects itself

116. The equilibrium of an electrified body cannot be stable

Chapter VII.

Forms of Equipotential Surfaces and Lines of Flow.

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

118. Two electrified points, ratio ${\displaystyle 4:1}$. (Fig. I)

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

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

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

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

123. Method employed in drawing the diagrams

Chapter VIII.

Simple Cases of Electrification.

124. Two parallel planes

125. Two concentric spherical surfaces

126. Two coaxal cylindric surfaces

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

Chapter IX.

Spherical Harmonics.

128. Singular points at which the potential becomes infinite

129. Singular points of different orders defined by their axes

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

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

132. The zonal, tesseral, and sectorial types

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

134. Application to the theory of electrified spherical surfaces

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

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

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

138. On conjugate harmonics

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

${\displaystyle \iint Y_{i}Y_{j}dS={\frac {4\pi a^{2}}{2i+1}}Y_{i(j)}}$
where ${\displaystyle Y_{i(j)}}$ is the value of ${\displaystyle Y_{i}}$ at the pole of ${\displaystyle Y_{j}}$

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

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

142. Different methods of treating spherical harmonics

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

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

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

146. Application to spherical and nearly spherical conductors

Chapter X.

Confocal Surfaces of the Second Degree.

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

148. The characteristic equation of ${\displaystyle V}$ in terms of ellipsoidal coordinates

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

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

151. Continuous transformation into a figure of revolution about the axis of ${\displaystyle z}$

152. Transformation into a figure of revolution about the axis of ${\displaystyle x}$

153. Transformation into a system of cones and spheres

154. Confocal paraboloids

Chapter XI.

Theory of Electric Images.

155. Thomson's method of electric images

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

157. Electric images

158. Distribution of electricity on the surface of the sphere

159. Image of any given distribution of electricity

160. Resultant force between an electrified point and sphere

161. Images in an infinite plane conducting surface

162. Electric inversion

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

165. Finite systems of successive images

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

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

168. Case of two spheres intersecting orthogonally

169. Case of three spheres intersecting orthogonally

170. Case of four spheres intersecting orthogonally

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

172. Any two spheres not intersecting each other

173. Calculation of the coefficients of capacity and induction

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

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

176. Thomson's investigation of an electrified spherical bowl

177. Distribution on an ellipsoid, and on a circular disk at potential ${\displaystyle V}$

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

179. The rest of the sphere supposed uniformly electrified

180. The bowl maintained at potential ${\displaystyle V}$ and uninfluenced

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

Chapter XII.

Conjugate Functions in Two Dimensions.

182. Cases in which the quantities are functions of ${\displaystyle x}$ and ${\displaystyle y}$ only

183. Conjugate functions

184. Conjugate functions may be added or subtracted

185. Conjugate functions of conjugate functions are themselves conjugate

186. Transformation of Poisson's equation

187. Additional theorems on conjugate functions

188. Inversion in two dimensions

189. Electric images in two dimensions

190. Neumann's transformation of this case

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

192. Ellipses and hyperbolas. (Fig. X)

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

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

195. Application to two cases of electrical induction

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

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

198. Case of a furrowed surface

199. Case of a single straight groove

200. Modification of the results when the groove is circular

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

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

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

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

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

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

Chapter XIII.

Electrostatic Instruments.

207. The frictional electrical machine

208. The electrophorus of Volta

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

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

211. Thomson's water-dropping machine

212. Holtz's electrical machine

213. Theory of regenerators applied to electrical machines

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

215. Coulomb's Torsion Balance for measuring charges

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

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

218. Heterostatic method

220. Measurement of the electric potential of a small body

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

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

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

224. A hemisphere used as a test

225. A circular disk

226. On electric accumulators. The Leyden jar

227. Accumulators of measurable capacity

228. The guard-ring accumulator

229. Comparison of the capacities of accumulators

Part II.

Electrokinematics.

Chapter I.

The Electric Current.

230. Current produced when conductors are discharged

231. Transference of electrification

232. Description of the voltaic battery

233. Electromotive force

234. Production of a steady current

235. Properties of the current

236. Electrolytic action

237. Explanation of terms connected with electrolysis

238. Different modes of passage of the current

239. Magnetic action of the current

240. The Galvanometer

Chapter II.

Conduction And Resistance.

241. Ohm's Law

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

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

244. Differences between the two classes of phenomena

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

Chapter III.

Electromotive force between bodies in contact.

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

247. Effects of electrolytes

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

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

250. Seebeck's discovery of thermoelectric currents.

251. Magnus's law of a 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 electric currents in copper and in iron

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

Chapter IV.

Electrolysis

255. Faraday's law of electrochemical equivalents

256. Clausius's theory of molecular agitation

257. Electrolytic polarization

258. Test of an electrolyte by polarization

259. Difficulties in the theory of electrolysis

260. Molecular charges

261. Secondary actions observed in the electrodes

262. Conversation of energy in electrolysis

263. Measurement of chemical affinity as an electromotive force

Chapter V.

Electrolytic polarization

264. Difficulties of applying Ohm's law to electrolytes.

265. Ohm's law nevertheless applicable

266. The effect of polarization distinguished from that of resistance

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

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

269. Dissipation of the ions and loss of polarization

270. Limit of polarization

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

272. Constant voltaic elements.—Daniell's cell

Chapter VI.

Mathematical Theory of the Distribution of Electric Currents.

273. Linear conductors

274. Ohm's Law

275. Linear conductors in series

276. Linear conductors in multiple arc

277. Resistance of conductors of uniform section

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

279. Specific resistance and conductivity in electromagnetic measure

280. Linear systems of conductors in general

281. Reciprocal property of any two conductors of the system

282. Conjugate conductors

283. Heat generated in the system

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

Chapter VII.

Conduction in Three Dimensions.

285. Notation

286. Composition and resolution of electric currents

287. Determination of the quantity which flows through any surface

288. Equation of a surface of flow

289. Relation between any three systems of surfaces of flow

290. Tubes of flow

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

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

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

294. Current-sheets and current-functions

295. Equation of 'continuity'

296. Quantity of electricity which flows through a given surface

Chapter VIII.

Resistance and Conductivity in Three Dimensions.

297. Equations of resistance

298. Equations of conduction

299. Rate of generation of heat

300. Conditions of stability

301. Equation of continuity in a homogeneous medium

302. Solution of the equation

303. Theory of the coefficient ${\displaystyle T}$. It probably does not exist

304. Generalized form of Thomson's theorem

305. Proof without symbols

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

307. Higher limit

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

309. Higher limit

Chapter IX.

Conduction Through Heterogeneous Media.

310. Surface-conditions

311. Spherical surface

312. Spherical shell

313. Spherical shell placed in a field of uniform flow

314. Medium in which small spheres are uniformly disseminated

315. Images in a plane surface

316. Method of inversion not applicable in three dimensions

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

318. Infinite series of images. Application to magnetic induction

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

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

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

322. Medium containing parallelepipeds of another medium

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

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

Chapter X.

Conduction in Dielectrics.

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

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

327. No residual charge due to simple conduction

328. Theory of a composite accumulator

329. Residual charge and electrical absorption

330. Total discharge

331. Comparison with the conduction of heat

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

333. Opinion of Ohm on this subject

334. Mechanical illustration of the properties of a dielectric

Chapter XI.

Measurement of the Electric Resistance of Conductors.

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

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

337. The electromagnetic system of units

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

339. Professed value of the Ohm ${\displaystyle 10,000,000}$ metres per second

340. Reproduction of standards

341. Forms of resistance coils

342. Coils of great resistance

343. Arrangement of coils in series

344. Arrangement in multiple arc

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

346. (2) By the differential galvanometer

347. (3) By Wheatstone's Bridge

348. Estimation of limits of error in the determination

349. Best arrangement of the conductors to be compared

350. On the use of Wheatstone's Bridge

351. Thomson's method for small resistances

352. Matthiessen and Hockin's method for small resistances

353. Comparison of great resistances by the electrometer

354. By accumulation in a condenser

355. Direct electrostatic method

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

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

358. Comparison of electromotive forces

Chapter XII.

Electric Resistance of Substances.

359. Metals, electrolytes, and dielectrics

360. Resistance of metals

361. Resistance of mercury

362. Table of resistance of metals

363. Resistance of electrolytes

364. Experiments of Paalzow

365. Experiments of Kohlrausch and Nippoldt

366. Resistance of dielectrics

367. Gutta-percha

368. Glass

369. Gases

370. Experiments of Wiedemann and Rühlmann