Art.


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


2. Dimensions of derived units.


35. The three fundamental units  Length, Time and Mass.


6. Derived units.


7. Physical continuity and discontinuity.


8. Discontinuity of a function of more than one variable.


9. Periodic and multiple functions.


10. Relation of physical quantities to directions in space.


11. Meaning of the words Scalar and Vector.


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


13. Relation between corresponding vectors of the two classes.


14. Lineintegration appropriate to forces, surfaceintegration to fluxes.


15. Longitudinal and rotational vectors.


16. Lineintegrals and potentials.


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


18. Cyclic regions and geometry of position.


19. The potential in an acyclic region is single valued.


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


21. Surfaceintegrals.


22. Surfaces, tubes, and lines of flow.


23. Righthanded and lefthanded relations in space.


24. Transformation of a lineintegral into a surfaceintegral.


25. Effect of Hamilton's operation Ñ on a vector function.


26. Nature of the operation Ñ ^{2}.


PART I.


ELECTROSTATICS. 

CHAPTER I.


DESCRIPTION OF PHENOMENA. 

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


28. Electrification by induction.


29. Electrification by conduction. Conductors and insulators.


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


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


32. To discharge a conductor completely into a metallic vessel.


33. Test of electrification by goldleaf electroscope.


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


35. Electricity may be treated as a physical quantity.


36. Theory of Two fluids.


37. Theory of One fluid.


38. Measurement of the force between electrified bodies.


39. Relation between this force and the quantities of electricity.


40. Variation of the force with the distance.


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


43. Proof of the law of electric force.


44. Electric field.


45. Electric potential.


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


47. Lines of force.


48. Electric tension.


49. Electromotive force.


50. Capacity of a conductor.


51. Properties of bodies.  Resistance.


52. Specific Inductive capacity of a dielectric.


53. `Absorption' of electricity.


54. Impossibility of an absolute charge.


55. Disruptive discharge.  Glow.


56. Brush.


57. Spark.


58. Electrical phenomena of Tourmaline.


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


60. Electric polarization and displacement.


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


62. Peculiarities of the theory of this treatise.


CHAPTER II.


ELEMENTARY MATHEMATICAL THEORY OF ELECTRICITY. 

63. Definition of electricity as a mathematical quantity.


64. Volumedensity, surfacedensity, and linedensity.


65. Definition of the electrostatic unit of electricity.


66. Law of force between electrified bodies.


67. Resultant force between two bodies.


68. Resultant force at a point.


69. Lineintegral of electric force; electromotive force.


70. Electric potential.


71. Resultant force in terms of the potential.


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


73. Potential due to an electrified system.


74. Proof of the law of the inverse square.


75. Surfaceintegral of electric induction.


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


77. Poisson's extension of Laplace's equation.


78. Conditions to be fulfilled at an electrified surface.


79. Resultant force on an electrified surface.


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


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


82. Lines of electric induction.


83. Specific inductive capacity.


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 apposite methods of treating electrical questions.


96. Characteristics of the potential function.


97. Conditions under which the volumeintegral vanishes.


98. Thomson's theorem of the unique minimum of (a^{2} + b^{2} + c^{2}) dxdydz.


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.


MECILANICAL 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 shown 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 selfintersection 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 4: 1. (Fig. I).


119. Two electrified points, ratio 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 i degrees.


132. The zonal, tesseral, and sectorial types.


133. Solid harmonies 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 Y_{i} and Y_{j} are two surface harmonies of different degrees, the surfaceintegral Y_{i} Y_{j} dS = 0, the integration being extended over the spherical surface.


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


138. On conjugate harmonies.


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


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


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


141. Surfaceintegral 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 V in terms of ellipsoidal coordinates.


149. Expression of a, b, g 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 z.


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


163. Geometrical theorems about 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 p /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 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 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 x and 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. Neuman'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 guardring.


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 waterdropping 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 guardring. Thomson's Absolute Electrometer


218. Heterostatic method.


219. Selfacting electrometers.  Thomson's Quadrant Electrometer


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 guardring 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. Faraday'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. Effect 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. Scebeck'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 at the electrodes.


262. Conservation 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 state of the ions at the electrodes.


269. Dissipation of the ions and loss of polarization.


270. Limit of polarization.


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


272. Constant voltaie 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 are.


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. Currentsheets and currentfunctions.


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 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. Surfaceconditions.


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 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 parallclepipeds 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 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 are.


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. Guttapercha.


368. Glass.


369. Gases.


370. Experiments of Wiedemann and Rühlmann.
