CONTENTS.
PRELIMINARY.
ON THE MEASUREMENT OF QUANTITIES.
Art.
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
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 Ñ on a vector function.
27
26. Nature of the operation Ñ 2.
29
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.
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.
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 apposite methods of treating electrical questions.
98
96. Characteristics of the potential function.
99
97. Conditions under which the volume-integral vanishes.
100
98. Thomson's theorem of the unique minimum of (a2 + b2 + c2) dxdydz.
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.
MECILANICAL ACTION BETWEEN ELECTRIFIED BODIES.
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 shown 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.
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 harmonies 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 Yi and Yj are two surface harmonies of different degrees, the surface-integral Yi Yj dS = 0, the integration being extended over the spherical surface.
169
137. Value of Yi Yj dS where Yi and Yj are surface harmonics of the same degree but of different types.
169
138. On conjugate harmonies.
170
139. If Yj is the zonal harmonic and Yi any other type of the same degree Yi Yj dS =
where Yi(j) is the value of Yi at the pole of Yj.
171
140. Development of a function in terms of spherical surface harmonies.
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 a, b, g 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 p /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. Neuman'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. Faraday'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. Effect 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. Scebeck'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 at the electrodes.
313
262. Conservation 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 state 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 sccondary pile compared with the Leyden jar.
322
272. Constant voltaie 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 are.
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.
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 parallclepipeds 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.
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 are.
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