A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 1 of 28

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A TREATISE

ON

ELECTRICITY AND MAGNETISM

MAXWELL

TOL. I.

Honfcon HENRY FROWDE

OXFORD UNIVERSITY PRESS "WAREHOUSE 7 PATERNOSTER ROW

Clarrnion iress Merles

A TREATISE

ELECTRICITY AND MAGNETISM

BY

JAMES CLERK MAXWELL, M.A.

LL.D. EDIN., D.C.L., F.R.SS. LONDON AND EDINBURGH

HONORARY FELLOW OF TRINITY COLLEGE, , AND PROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE

VOL. I

SECOND EDITION

AT THE CLARENDON PRESS

1881 [ All rights reserved ]

A5/KONOMY

ASTRONOMY JJBRAKY ,

PREFACE TO THE FIRST EDITION,

1HE 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 phe- nomena, amber, %\eKTpov, 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 phe- nomena 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, nayvw, 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 Elec- tromagnetism.

In the following Treatise I propose to describe the

M677187

vi PREFACE.

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 phe- nomena, 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 pheno- mena 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 pheno- mena chiefly with a view to their measurement, de- scribing 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.

PREFACE. vii

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 electro- magnetism 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 phe- nomena have also been investigated, and the proba- bility 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 re- duction 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 sub- jects of enquiry, and furnishing us with means of investigation.

It is hardly necessary to enlarge upon the beneficial

viii PREFACE.

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 measure- ments, 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 experi- mental opportunities for acquiring it, have been already very great, both in stimulating the energies of ad- vanced 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.

PREFACE. ix

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 memoirs which are of great importance in electrical science, but they lie concealed in the bulky Trans- actions of learned societies ; they do not form a con- nected 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 con- siderably from that of several excellent electrical works, published, most of them, in Germany, and it may appear that scant justice is done to the specu- lations of several eminent electricians and mathema- ticians. One reason of this is that before I began the study of electricity I resolved to read no mathe- matics 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

x PREFACE.

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 *, 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 per- ceived 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 mathema- ticians.

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 coin- cided, 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

• 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.

PREFACE. xi

in which we begin with the whole and arrive at the parts by anlaysis, while the ordinary mathematical methods were founded on the principle of beginning with the parts and building up the whole by syn- thesis.

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, con- sidered 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 measure- ments 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, Eiemann, 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,

xii PREFACE.

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, addi- tional 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 as 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

PREFACE. xiii

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 sup- porters, and will be expounded with a skill worthy of its ingenuity.

I have not attempted an exhaustive account of elec- trical phenomena, experiments, and apparatus. The student who desires to read all that is known on these subjects will find great assistance from the Traite d' Electricite of Professor A. de la Rive, and from several German treatises, such as Wiedemann's Galvanismus, Eiess' Reilungselektricitat, Beer's Einleitung in die Elek- trostatik, &c.

I have confined myself almost entirely to the ma- thematical treatment of the subject, but I would recommend the student, after he has learned, experi- mentally 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 *.

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

• Life and Letters of Faraday, vol. i. p. 395.

xiv PREFACE,

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 con- secutively. 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 ele- mentary 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.

PREFACE TO THE SECOND EDITION.

WHEN I was asked to read the proof-sheets of the second edition of the Electricity and Magnetism the work of printing had already reached the ninth chapter, the greater part of which had been revised by the author.

Those who are familiar with the first edition will see from a comparison with the present how extensive were the changes intended by Professor Maxwell both in the substance and in the treatment of the subject, and how much this edition has suffered from his premature death. The first nine chapters were in some cases entirely re- written, much new matter being added and the former contents rearranged and simplified.

From the ninth chapter onwards the present edition is little more than a reprint. The only liberties I have taken have been in the insertion here and there of a step in the mathematical reasoning where it seemed to be an advantage to the reader, and of a few foot-notes on parts of the subject which my own experience or that of pupils attending my classes shewed to require further elucidation. These footnotes are in square brackets.

There were two parts of the subject in the treatment

xvi PREFACE.

of which it was known to me that the Professor con- templated considerable changes : viz. the mathematical theory of the conduction of electricity in a network of wires, and the determination of coefficients of induction in coils of wire. In these subjects I have not found myself in a position to add, from the Professor's notes, anything substantial to the work as it stood in the former edition, with the exception of a numerical table, printed in vol. ii, pp. 317-319. This table will be found very useful in calculating coefficients of induction in circular coils of wire.

In a work so original, and containing so many details of new results, it was impossible but that there should be a few errors in the first edition. I trust that in the present edition most of these will be found to have been corrected. I have the greater confidence in ex- pressing this hope as, in reading some of the proofs, I have had the assistance of various friends conversant with the work, among whom I may mention particularly my brother Professor Charles Niven, and Mr. J. J. Thom- son, Fellow of Trinity College, Cambridge.

W. D. NIVEN.

TRINITY COLLEGE, CAMBEIDGE, Oct. i, 1881.

CONTENTS,

PRELIMINARY.

ON THE MEASUKEMENT OF QUANTITIES.

Art.

1. The expression of a quantity consists of two factors, the nu-

merical 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 13

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 V on a vector function . . . . 28

26. Nature of the operation V2 ,..,. 29

VOL. I. b

CONTENTS.

PAET I

ELECTKOSTATICS. CHAPTER I.

DESCRIPTION OF PHENOMENA. Art.

27. Electrification by friction. Electrification is of two kinds, to

which the names of Vitreous and Resinous, or Positive and

Negative, have been given 31

28. Electrification by induction 32

29. Electrification by conduction. Conductors and insulators . . 33

30. In electrification by friction the quantity of the positive elec-

trification is equal to that of the negative electrification . . 34

31. To charge a vessel with a quantity of electricity equal and

opposite to that of an excited body 34

32. To discharge a conductor completely into a metallic vessel . . 35

33. Test of electrification- by gold-leaf electroscope 35

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

called Electricity . . 36

35. Electricity may be treated as a physical quantity 37

36. Theory of Two fluids 38

37. Theory of One fluid 40

38. Measurement of the force between electrified bodies 41

39. Relation between this force and the quantities of electricity . . 42

40. Variation of the force with the distance 43

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

dimensions . . 43, 44

43. Proof of the law of electric force 44

44. Electric field 45

45. Electromotive force and potential 46

46. Equipotential surfaces. Example of their use in reasoning

about electricity 47

47. Lines of force 48

48. Electric tension 49

49. Electromotive force 49

50. Capacity of a conductor. Electric Accumulators 49

51. Properties of bodies. — Resistance 50

52. Specific Inductive capacity of a dielectric 52

53. c Absorption' of electricity 53

CONTENTS. xix

Art. Page

54. Impossibility of an absolute charge 54

55. Disruptive discharge. — Glow 54

56. Brush 57

57. Spark 57

58. Electrical phenomena of Tourmaline 58

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

60. Electric polarization and displacement 61

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

fluid 64

62. Peculiarities of the theory of this treatise 65

CHAPTER II.

ELEMENTARY MATHEMATICAL THEOEY OF ELECTRICITY.

63. Definition of electricity as a mathematical quantity 68

64. Volume -density, surface-density, and line-density 68

65. Definition of the electrostatic unit of electricity 70

66. Law of force between electrified bodies 70

67. Resultant force between two bodies 71

68. Resultant intensity at a point 71

69. Line-integral of electric intensity ; electromotive force . . . . 72

70. Electric potential 73

71. Resultant intensity in terms of the potential 74

72. The potential of all points of a conductor is the same . . . . 75

73. Potential due to an electrified system 76

74 a. Proof of the law of the inverse square. Cavendish's experiments 76

74 6. Cavendish's experiments repeated in a modified form . . . . 77

74 c, d, e. Theory of the experiments 79-81

75. Surface-integral of electric induction 82

76. Induction through a closed surface due to a single centre of

force • 83

77. Poisson's extension of Laplace's equation 84

78 a, 6, c. Conditions to be fulfilled at an electrified surface . . 85-88

79. Resultant force on an electrified surface 88

80. The electrification of a conductor is entirely on the surface . . 90

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

impossible • • •• 91

82. Lines of electric induction 92

83 a. Specific inductive capacity 94

836. Apparent distribution of electricity 94

XX CONTENTS.

CHAPTER III.

ON ELECTRICAL WORK AND ENERGY IN A SYSTEM OF CONDUCTORS.

Art. Page 84. On the superposition of electrified systems. Expression for the

energy of a system of conductors 96

85 a. Change of the energy in passing from one state to another . . 97

856. Relations between the potentials and the charges 98

857. Theorems of reciprocity 98

858. Theory of a system of conductors. Coefficients of potential. Ca-

pacity. Coefficients of induction 100

88. Dimensions of the coefficients 103

89 a. Necessary relations among the coefficients of potential . . . . 103

89 6. Relations derived from physical considerations 104

89 c. Relations among coefficients of capacity and induction . . . . 105

89 d. Approximation to capacity of one conductor 105

89 e. The coefficients of potential changed by a second conductor . . 106

90 a. Approximate determination of the coefficients of capacity and

induction of two conductors 107

90 6. Similar determination for two condensers 107

91. Relative magnitudes of coefficients of potential 109

92. And of induction 110

93 a. Mechanical force on a conductor expressed in terms of the

charges of the different conductors of the system 110

93 6. Theorem in quadratic functions Ill

93 c. "Work done by the electric forces during the displacement of a

system when the potentials are maintained constant . . . . Ill 94. Comparison of electrified systems 112

CHAPTER IV.

GENERAL THEOREMS.

95 a, b. Two opposite methods of treating electrical questions 115, 116

96 a. Green's Theorem 118

96 6. When one of the functions is many valued 120

96 c. When the region is multiply connected 120

96 d. When one of the functions becomes infinite in the region . . 121

97 a, b. Applications of Green's method 123,124

98. Green's Function 125

99 a. Energy of a system expressed as a volume integral . . . . 126

CONTENTS.

xxi

Page

99 b. Proof of unique solution for the potential when its value is

given at every point of a closed surface 127

100 a-e. Thomson's Theorem 129-132

101 a-h. Expression for the energy when the dielectric constants

are different in different directions. Extension of Green's Theorem to a heterogeneous medium 133-137

102 a. Method of finding limiting values of electrical coefficients . . 138 102 b. Approximation to the solution of problems of the distribution

of electricity on conductors at given potentials 140

102 c. Application to the case of a condenser with slightly curved

plates 142

CHAPTER V.

MECHANICAL ACTION BETWEEN TWO ELECTBICAL SYSTEMS.

103. Expression for the force at any point of the medium in terms

of the potentials arising from the presence of the two systems 144

104. In terms of the potential arising from both systems . . . . 145

105. Nature of the stress in the medium which would produce the

same force 146

106. Further determination of the type of stress ; . . 148

107. Modification of the expressions at the surface of a conductor. . 149

108. Discussion of the integral of Art. 104 expressing the force

when taken over all space 151

109. Statements of Faraday relative to the longitudinal tension and

lateral pressure of the lines of force 153

110. Objections to stress in a fluid considered 153

111. Statement of the theory of electric polarization 154

CHAPTER VI.

POINTS AND LINES OF EQUILIBKIUM.

112. Conditions for a point of equilibrium 157

113. Number of points of equilibrium 158

114. At a point or line of equilibrium there is a conical point or a

line of self-intersection of the equipotential surface . . . . 159

115. Angles at which an equipotential surface intersects itself . . 160

116. The equilibrium of an electrified body cannot be stable . . . . 161

xxii CONTENTS.

CHAPTER VII.

FOKMS OF EQUIPOTENTIAL SURFACES AND LINES OF FLOW.

Art. Page

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

cases 164

118. Two electrified points, ratio 4:1. (Fig. I) 165

119. Two electrified points, ratio 4 : — 1. (Fig. II) 166

120. An electrified point in a uniform field of force (Fig. Ill) . . 167

121. Three electrified points. Two spherical equipotential sur-

faces. (Fig. IV) 167

122. Faraday's use of the conception of lines of force 168

123. Method employed in drawing the diagrams 169

CHAPTER VIII.

SIMPLE CASES OF ELECTRIFICATION.

124. Two parallel planes . . 172

125. Two concentric spherical surfaces 174

126. Two coaxal cylindric surfaces 176

127. Longitudinal force on a cylinder, the ends of which are sur-

rounded by cylinders at different potentials 177

CHAPTER IX.

SPHERICAL HARMONICS.

128. Heine, Todhunter, Ferrers 179

129 a. Singular points 179

1296. Definition of an axis 180

129 c. Construction of points of different orders 181

129 d. Potential of such points. Surface harmonics Yn .. .. 182

130 a. Solid harmonics. ffn = rnYn 182

130 b. There are 2n+l independent constants in a solid harmonic

of the n*h order 183

131 a. Potential due to a spherical shell 184

1316. Expressed in harmonics 184

131 c. Mutual potential of shell and external system 185

132. Value of ffTmTnds 186

133. Trigonometrical expressions for Tn 187

134. Value of f/YmYnds, when m = n 189

135 a. Special case when Tm is a zonal harmonic 190

1356. Laplace's expansion of a surface harmonic 190

1357. Conjugate harmonics 192

CONTENTS. xxiii

Art. Page

137. Standard harmonics of any order 192

138. Zonal harmonics 193

139. Laplace's coefficient or Biaxal harmonic 194

140 a. Tesseral harmonics. Their trigonometrical expansion .. 194

140 b. Notations used by various authors 197

140 c. Forms of the tesseral and sectorial harmonics 197

141. Surface integral of the square of a tesseral harmonic .. . . 198 142 a. Determination of a given tesseral harmonic in the expansion

of a function 199

142 b. The same in terms of differential coefficients of the function. . 199

143. Figures of various harmonics 200

144 a. Spherical conductor in a given field of force 201

144 b. Spherical conductor in a field for which Green's function is

known 201-

145 a. Distribution of electricity on a nearly spherical conductor . . 204 145 b. When acted on by external electrical force .. .. .. .. 206

145 c. When enclosed in a nearly spherical and nearly concentric

vessel 207

146. Equilibrium of electricity on two spherical conductors . . . . 208

CHAPTER X.

CONFOCAL SURFACES OF THE SECOND DEGREE.

147. The lines of intersection of two systems and their intercepts

by the third system 215

148. The characteristic equation of V in terms of ellipsoidal co-

ordinates 216

149. Expression of a, {3, y in terms of elliptic functions 217

150. Particular solutions of electrical distribution on the confocal

surfaces and their limiting forms 218

151. Continuous transformation into a figure of revolution about

the axis of z 221

152. Transformation into a figure of revolution about the axis of x. . 222

153. Transformation into a system of cones and spheres 223

154. Confocal paraboloids 223

CHAPTER XI.

THEORY OF ELECTRIC IMAGES.

155. Thomson's method of electric images 226

156. When two points are oppositely and unequally electrified, the

surface for which the potential is zero is a sphere . . . . 227

Xxiv CONTENTS.

Art. Page

157. Electric images 228

158. Distribution of electricity on the surface of the sphere . . . . 230

159. Image of any given distribution of electricity 231

160. Resultant force between an electrified point and sphere . . . . 232

161. Images in an infinite plane conducting surface 234

162. Electric inversion 235

163. Geometrical theorems about inversion 236

164. Application of the method to the problem of Art. 158 . . . . 237

165. Finite systems of successive images 238

166. Case of two spherical surfaces intersecting at an angle - . . 240

n

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

finite 241

168. Case of two spheres intersecting orthogonally 242

169. Case of three spheres intersecting orthogonally 245

170. Case of four spheres intersecting orthogonally 246

171. Infinite series of images. Case of two concentric spheres . . 247

172. Any two spheres not intersecting each other 249

173. Calculation of the coefficients of capacity and induction . . . . 251

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

between them 253

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

sphere 255

176. Thomson's investigation of an electrified spherical bowl. . . . 257

177. Distribution on an ellipsoid, and on a circular disk at po-

tential V 257

178. Induction on an uninsulated disk or bowl by an electrified

point in the continuation of the plane or spherical surface . . 258

179. The rest of the sphere supposed uniformly electrified . . . . 259

180. The bowl maintained at potential V and uninfluenced . . . . 259

181. Induction on the bowl due to a point placed anywhere 260

CHAPTER XII.

CONJUGATE FUNCTIONS IN TWO DIMENSIONS.

182. Cases in which the quantities are functions of x and y only . . 262

183. Conjugate functions 263

184. Conjugate functions may be added or subtracted 264

185. Conjugate functions of conjugate functions are themselves

conjugate 265

186. Transformation of Poisson's equation 267

187. Additional theorems on conjugate functions 268

CONTENTS. XXV

Art. Pa*e

188. Inversion in two dimensions 268

189. Electric images in two dimensions 269

190. Neumann's transformation of this case 270

191. Distribution of electricity near the edge of a conductor formed

by two plane surfaces .. „ 272

192. Ellipses and hyperbolas. (Fig. X) 273

193. Transformation of this case. (Fig. XI) 274

194. Application to two cases of the flow of electricity in a con-

ducting sheet 276

195. Application to two cases of electrical induction 276

196. Capacity of a condenser consisting of a circular disk between

two infinite planes 277

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

angles to them 279

198. Case of a furrowed surface 280

199. Case of a single straight groove 281

200. Modification of the results when the groove is circular . . . . 281

201. Application to Sir W. Thomson's guard-ring 284

202. Case of two parallel plates cut off by a perpendicular plane.

(Fig. XII) 285

203. Case of a grating of parallel wires. (Fig. XIII) 286

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

grating 287

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

influence 288

206. Method of approximation applied to the case of the grating . . 289

CHAPTER XIII.

ELECTEOSTATIC INSTRUMENTS.

207. The frictional electrical machine .". . . 292

208. The electrophorus of Volta 293

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

Revolving Doubler 294

210. Principle of Varley's and Thomson's electrical machines. . . . 294