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Full text of "A treatise on electricity and magnetism"

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A TREATISE 



ON 



ELECTRICITY AND MAGNETISM 



MAXWELL 



VOL. I. 



Uonfron 

MACMILLAN AND CO. 




PUBLISHERS TO THE UNIVERSITY OF 



Clarendon press Series 



A TREATISE 



ON 



ELECTRICITY AND MAGNETISM 



BY 



JAMES CLERK MAXWELL, M.A 

LLD. EDIN., P.E.SS. LONDON AND EDINBURGH 

HONORARY FELLOW OF TRINITY COLLEGE, 

AND PROFESSOR OF EXPERIMENTAL PHYSICS 

IN THE UNIVERSITY OF CAMBRIDGE 



VOL. I 



AT THE CLARENDON PRESS 
1873 

[All rights reserved] 



v, I 

Y\ 

; 



4/w>wa 




. Depti 



PREFACE. 

THE fact that certain bodies, after being rubbed, 
appear to attract other bodies, was known to the 
ancients. In modern times, a great variety of other 
phenomena have been observed, and have been found 
to be related to these phenomena of attraction. They 
have been classed under the name of Electric phe 
nomena, amber, ?i\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, /zayi^?, 
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 

812245 



Ti 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 elec- 
tromagnetism 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 electromagnet-ism 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 w r ho 
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 analysis, 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 summa 
tion of the electrified particles divided each by its dis 
tance from a given point. Hence many of the mathe 
matical 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, Riemann, J. and C. Neumann, Lorenz, &c. is 
founded on the theory of action at a distance, but 
depending either directly on the relative velocity of the 
particles, or on the gradual propagation of something, 



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, 
Riess Reibiingselektricitat, 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 ac 
curately describing scientific operations and their re 
sults *. 

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. 



CONTENTS, 



PRELIMINARY. 



ON THE MEASUREMENT OF QUANTITIES. 

Art. Page 

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 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 v on a vector function .. .. 27 

26. Xature of the operation v 2 29 



xvi CONTENTS. 



PART I. 

ELECTROSTATICS. 
CHAPTER I. 

DESCRIPTION OP 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 elec 

trification 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 



CONTENTS. xvi i 

Art. Pae* 

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. 

ELEMENTAKY 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 

82. Lines of electric induction 84 

83. Specific inductive capacity 86 

VOL. I. b 



xviii CONTENTS. 

CHAPTER III. 

SYSTEMS OF CONDUCTORS. 

Art. 

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 po 

tential 89 

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

of these coefficients 90 

88. Reciprocal property of the coefficients 91 

89. A theorem due to Green 92 

90. Relative magnitude of the coefficients of potential 92 

91. And of induction 93 

92. The resultant mechanical force on a conductor expressed in 

terms of the charges of the different conductors of the system 
and the variation of the coefficients of potential 94 

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

coefficients of induction 94 

94. Comparison of electrified systems 96 



CHAPTER IV. 

GENERAL THEOREMS. 

95. Two opposite methods of treating electrical questions .. .. 98 

96. Characteristics of the potential function 99 

97. Conditions under which the volume-integral 

dV dV a 



vanishes 100 

98. Thomson s theorem of the unique minimum of 

1 

^ ( 2 -f o 2 + <? 2 ) dxdydz 103 

99. Application of the theorem to the determination of the dis 

tribution 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 



CONTENTS. XIX 



CHAPTER V. 

MECHANICAL ACTION BETWEEN ELECTRIFIED BODIES. 
Art. Page 

103. Comparison of the force between different electrified systems .. 119 

104. Mechanical action on an element of an electrified surface .. 121 

105. Comparison between theories of direct action and theories of 

stress 122 

106. The kind of stress required to account for the phenomeuou .. 123 

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

science 126 

108. The hypothesis of stress shewn to account for the equilibrium 

of the medium and for the forces acting between electrified 
bodies 128 

109. Statements of Faraday relative to the longitudinal tension and 

lateral pressure of the lines of force 131 

110. Objections to stress in a fluid considered 131 

111. Statement of the theory of electric polarization 132 

CHAPTER VI. 

POINTS AND LINES OF EQUILIBRIUM, 

112. Conditions of a point of equilibrium 135 

113. Number of points of equilibrium 136 

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

line of self-intersection of the equipotential surface .. .. 137 

115. Angles at which an equipotential surface intersects itself .. 138 

116. The equilibrium of an electrified body cannot be stable .. .. 139 

CHAPTER VII. 

FORMS OF EQUIPOTENTIAL SURFACES AND LINES OF FLOW. 

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

cases 142 

118. Two electrified points, ratio 4 : 1. (Fig. I) 143 

119. Two electrified points, ratio 4 : 1. (Fig. II) 144 

120. An electrified point in a uniform field offeree. (Fig. Ill) .. 145 

121. Three electrified points. Two spherical equipotential sur 

faces. (Fig. IV) 145 

122. Faraday s use of the conception of lines of force 146 

123. Method employed in drawing the diagrams 147 

b 2 



XX CONTENTS. 



CHAPTER VIII. 

SIMPLE CASES OF ELECTRIFICATION. 
Art. 1 age 

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 sur 

rounded by cylinders at different potentials 155 



CHAPTER IX. 

SPHERICAL HARMONICS. 

128. Singular points at which the potential becomes infinite .. .. 157 

129. Singular points of different orders defined by their axes .. .. 158 

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

to its axes .................... 160 

131. This expression is perfectly definite and represents the most 

general type of the harmonic of i degrees ........ 162 

132. The zonal, tesseral, and sectorial types .......... 163 

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

of negative degree .................. 165 

134. Application to the theory of electrified spherical surfaces .. 166 

135. The external action of an electrified spherical surface compared 

with that of an imaginary singular point at its centre .. .. 167 

136. Proof that if Y i and Y$ are two surface harmonics of different 

degrees, the surface-integral / / Y i Yj dS = 0, the integration 
being extended over the spherical surface ........ 169 

137. Value of // Y i YjdS where Y L and Yj are surface harmonics 

of the same degree but of different types ........ 169 

138. On conjugate harmonics ................ 170 

139. If Yj is the zonal harmonic and Y i any other type of the 

same degree 



where Y i(j) is the value of Y i at the pole of Y j ...... 171 

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

monics .................... ..172 

141. Surface-integral of the square of a symmetrical harmonic .. 173 



CONTENTS. xxi 

Art. Page 

142. Different methods of treating spherical harmonics 174 

143. On the diagrams of spherical harmonics. (Figs. V, Vf, VII, 

VHI, 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. 

COXFOCAJL 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 co 

ordinates 182 

149. Expression of a, 0, y 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 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 

1 59. 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 



xxii CONTENTS. 

Art. Page 

165. Finite systems of successive images 203 

166. Case of two spherical surfaces intersecting at an angle ^ ..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 po 

tential V 221 

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

point in the continuation of the plane or spherical surface .. 222 

179. The rest of the sphere supposed uniformly electrified .. .. 223 

180. The bowl maintained at potential V and uninfluenced .. .. 223 

181. Induction on the bowl due to a point placed anywhere .. .. 224 



CHAPTER XII. 

CONJUGATE FUNCTIONS IN TWO DIMENSIONS. 

182. Cases in which the quantities are functions of x and y only .. 226 

183. Conjugate functions 227 

184. Conjugate functions may be added or subtracted 228 

185. Conjugate functions of conjugate functions are themselves 

conjugate 229 

186. Transformation of Poisson s equation 231 

187. Additional theorems on conjugate functions 232 

188. Inversion in two dimensions .. 232 

189. Electric images in two dimensions 233 

190. Neumann s transformation of this case 234 

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

by two plane surfaces 236 

192. Ellipses and hyperbolas. (Fig. X) 237 

193. Transformation of this case. (Fig. XI) 238 



CONTENTS. xxiu 



Art. 

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

ducting 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 mea 
surement 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 



xxvi CONTENTS. 

Art. Page 

269. Dissipation of the ions and loss of polarization 321 

270. Limit of polarization 321 

271. Bitter s secondary pile compared with the Leyden jar .. .. 322 

272. Constant voltaic elements. Daniell s cell 325 



CHAPTER VI. 

MATHEMATICAL THEORY OF THE DISTRIBUTION OF ELECTRIC CURRENTS. 

273. Linear conductors 329 

274. Ohm s Law 329 

275. Linear conductors in series 329 

276. Linear conductors in multiple arc 330 

277. Resistance of conductors of uniform section 331 

278. Dimensions of the quantities involved in Ohm s law .. .. 332 

279. Specific resistance and conductivity in electromagnetic measure 333 

280. Linear systems of conductors in general 333 

281. Reciprocal property of any two conductors of the system .. 335 

282. Conjugate conductors .. .. 336 

283. Heat generated in the system 336 

284. The heat is a minimum when the current is distributed ac 

cording 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 

offlow 341 

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

meters .. 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 



CONTENTS. xxvii 

CHAPTER VIII. 

RESISTANCE AND CONDUCTIVITY IX THEEE DIMENSIONS. 

Art. Page 

297. Equations of resistance 345 

298. Equations of conduction 346 

299. Kate 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 sub 
stances 369 

320. If neither of the substances has the rotatory property denoted 

by I 7 the compound conductor is free from it 370 

321. If the substances are isotropic the direction of greatest resist 

ance is normal to the strata 371 

322. Medium containing parallelepipeds of another medium .. .. 371 

323. The rotatory property cannot be introduced by means of con 

ducting channels 372 

324. Construction of an artificial solid having given coefficients of 

longitudinal and transverse conductivity 373 



xxviii CONTENTS. 



CHAPTER X. 

CONDUCTION IN DIELECTRICS. 

Art. Page 

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

charge 374 

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

insulator 375 

327. No residual charge due to simple conduction 376 

328. Theory of a composite accumulator 376 

329. Residual charge and electrical absorption 378 

330. Total discharge 380 

331. Comparison with the conduction of heat 381 

332. Theory of telegraph cables and comparison of the equations 

with those of the conduction of heat 381 

333. Opinion of Ohm on this subject 384 

334. Mechanical illustration of the properties of a dielectric .. .. 385 



CHAPTER XI. 

MEASUREMENT OF THE ELECTRIC RESISTANCE OF CONDUCTORS. 

335. Advantage of using material standards of resistance in electrical 

measurements 388 

336. Different standards which have been used and different systems 

which have been proposed 388 

337. The electromagnetic system of units .. 389 

338. Weber s unit, and the British Association unit or Ohm .. ,. 389 

339. Professed value of the Ohm 10,000,000 metres per second .. 389 

340. Reproduction of standards 390 

341. Forms of resistance coils 391 

342. Coils of great resistance 392 

343. Arrangement of coils in series 392 

344. Arrangement in multiple arc 393 

345. On the comparison of resistances. (1) Ohm s method .. .. 394 

346. (2) By the differential galvanometer 394 

347. (3) By Wheatstone s Bridge 398 

348. Estimation of limits of error in the determination 399 

349. Best arrangement of the conductors to be compared .. .. 400 

350. On the use of Wheatstone s Bridge 402 

351. Thomson s method for small resistances 404 

352. Matthiessen and Hockin s method for small resistances ., .. 406 



CONTENTS. xxix 

Art. Page 

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 Riihlinann 425 



ERRATA. VOL. I. 

Page 26, 1. 3 from bottom, dele As we have made no assumption , &c. 
down to 1. 7 of p. 27, the expression may then be written , and 
substitute as follows : 

Let us now suppose that the curves for which a is constant 
form a series of closed curves, surrounding the point on the surface 
for which a has its minimum value, a , the last curve of the series, 
for which a = a lt coinciding with the original closed curve s. 

Let us also suppose that the curves for which /3 is constant form 
a series of lines drawn from the point at which a = a to the 
closed curve s, the first, /3 , and the last, fa, being identical. 

Integrating (8) by parts, the first term with respect to a and 
the second with respect to /3, the double integrals destroy each 
other. The line integral, 



f^ 1 (X 
/h d fi 



. 

is zero, because the curve a = a is reduced to a point at which 
there is but one value of X and of x. 
The two line integrals, 



* 



destroy each other, because the point (a, /3J is identical with the 
point (a, /3 ). 

The expression (8) is therefore reduced to 






Since the curve a = a l is identical with the closed curve s, we 
may write this expression 
p. 80, in equations (3), (4), (6), (8), (17), (18), (19), (20), (21), (22), for 

R read N. 
p. 82, 1. 3, for Rl read Nl. 

d-V d*V 

p. 83, in equations (28), (29), (30), (31), for -^ read j^* 

in equation (29), insert before the second member. 
p. 105, 1. 2, for Q read 8irQ. 
p. 108, equation (1), for p read //. 

(2), for p read p. 

(3), for a- read (/. 

(4), for a- read <r. 

p. 113, 1. 4, for KR read ^- KR. 

1. 5, for KRRfcosc read --KRRfwse. 

T: 7T 

p. 114, 1. 5, for S l read S. 

p. 124, last line, for e l -\-e l read e l + e 2 . 

p. 125, lines 3 and 4, transpose within and without; 1. 16, for v 

read V ; and 1. 18, for V read v. 
p. 128, lines 11, 10, 8 from bottom, for dx read dz. 
p. 149, 1. 24, for equpotential read equipotential. 



2 ERRATA. VOL. I. 

p. 159, 1. 3, for F read f. 

,, 1. 2 from bottom, for M read M 2 . 
p. 163, 1. 20, for \i- s +i read AJ-^+I. 

p. 164, equation (34), Jor (_iy- J= read (_!)*-*_ 
p. 179, equation (76), for i+l read 27+1. 

X 2 Z 2 X 2 Z 2 

p. 185, equation (24), for ~ ~=l read -^ T^r 2 = 1 - 

p. 186, 1. 5 from bottom, for The surface-density on the elliptic plate 

read The surface-density on either side of the elliptic plate, 
p. 186, equation (30), for 2n read 4ir. 
p. 188, equation (38), for v 2 read 2n 2 . 
p. 196, 1. 27, for e..e read e 1 ..e 2 . 

Ee e 2 a 3 
p. 197, equation (10) should be M = 1 

p. 204, 1. 15 from bottom, dele either, 
p. 215, 1. 4, for \/2k read */2k. 

E 
p. 234, equation (13), for 2JZ read 

p. 335, dele last 14 lines, 
p. 336, 1. 1, dele therefore. 

1. 2, for l the potential at C to exceed that at D by P, read a 

current, (7, from X to Y. 

1. 4, for C to D will cause the potential at A to exceed that at 
B by the same quantity P, read X to Y will cause an equal 
current G from A to B. 
p. 351, 1. 3, for R 2 y? + R 2 v 2 + R 2 w 2 read R^ 



dt> 



1. 5, read + 

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

~db* 
p. 356, equation (12), for read ~ 



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

E E 

p. 366, equation (3), for ~ read 

r i r z 

p. 367, 1. 5, for 2^8 read 2k 2 S. 
p. 368, equation (14), for <// read //. 

p. 397, 1. 1, for ~8 read ~S . 

FJ rj 

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

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



a+y a+ y 



ELECTRICITY AND MAGNETISM. 



ELECTEICITY AND MAGNETISM. 



PRELIMINARY. 

ON THE MEASUREMENT OF QUANTITIES. 

1.] EVERY expression of a Quantity consists of two factors or 
components. One of these is the name of a certain known quan 
tity of the same kind as the quantity to be expressed, which is 
taken as a standard of reference. The other component is the 
number of times the standard is to be taken in order to make up 
the required quantity. The standard quantity is technically called 
the Unit, and the number is called the Numerical Value of the 
quantity. 

There must be as many different units as there are different 
kinds of quantities to be measured, but in all dynamical sciences 
it is possible to define these units in terms of the three funda 
mental units of Length, Time, and Mass. Thus the units of area 
and of volume are defined respectively as the square and the cube 
whose sides are the unit of length. 

Sometimes, however, we find several units of the same kind 
founded on independent considerations. Thus the gallon, or the 
volume of ten pounds of water, is used as a unit of capacity as well 
as the cubic foot. The gallon may be a convenient measure in 
some cases, but it is not a systematic one, since its numerical re 
lation to the cubic foot is not a round integral number. 

2.] In framing a mathematical system we suppose the funda 
mental units of length, time, and mass to be given, and deduce 
all the derivative units from these by the simplest attainable de 
finitions. 

The formulae at which we arrive must be such that a person 

VOL. i. B 






2 PRELIMINARY. [3. 

of any nation, by substituting for the different symbols the nu 
merical value of the quantities as measured by his own national 
units, would arrive at a true result. 

Hence, in all scientific studies it is of the greatest importance 
to employ units belonging to a properly defined system, and to 
know the relations of these units to the fundamental units, so that 
we may be able at once to transform our results from one system to 
another. 

This is most conveniently done by ascertaining the dimensions 
of every unit in terms of the three fundamental units. When a 
given unit varies as the ^th power of one of these units, it is said 
to be of n dimensions as regards that unit. 

For instance, the scientific unit of volume is always the cube 
whose side is the unit of length. If the unit of length varies, 
the unit of volume will vary as its third power, and the unit of 
volume is said to be of three dimensions with respect to the unit of 
length. 

A knowledge of the dimensions of units furnishes a test which 
ought to be applied to the equations resulting from any lengthened 
investigation. The dimensions of every term of such an equa 
tion, with respect to each of the three fundamental units, must 
be the same. If not, the equation is absurd, and contains some 
error, as its interpretation would be different according to the arbi 
trary system of units which we adopt *. 

The Three Fundamental Units. 

3.] (1) Length. The standard of length for scientific purposes 
in this country is one foot, which is the third part of the standard 
yard preserved in the Exchequer Chambers. 

In France, and other countries which have adopted the metric 
system, it is the metre. The metre is theoretically the ten mil 
lionth part of the length of a meridian of the earth measured 
from the pole to the equator ; but practically it is the length of 
a standard preserved in Paris, which was constructed by Borda 
to correspond, when at the temperature of melting ice, with the 
value of the preceding length as measured by Delambre. The metre 
has not been altered to correspond with new and more accurate 
measurements of the earth, but the arc of the meridian is estimated 
in terms of the original metre. 

* The theory of dimensions was first stated by Fourier, Tkeorie de Chaleur, 160. 



5-] THE THREE FUNDAMENTAL UNITS. 3 

In astronomy the mean distance of the earth from the sun is 
sometimes taken as a unit of length. 

In the present state of science the most universal standard of 
length which we could assume would be the wave length in vacuum 
of a particular kind of light, emitted by some widely diffused sub 
stance such as sodium, which has well-defined lines in its spectrum. 
Such a standard would be independent of any changes in the di 
mensions of the earth, and should be adopted by those who expect 
their writings to be more permanent than that body. 

In treating of the dimensions of units we shall call the unit of 
length []. If I is the numerical value of a length, it is under 
stood to be expressed in terms of the concrete unit [_Z/], so that 
the actual length would be fully expressed by I \L~\. 

4.] (2) Time. The standard unit of time in all civilized coun 
tries is deduced from the time of rotation of the earth about its 
axis. The sidereal day, or the true period of rotation of the earth, 
can be ascertained with great exactness by the ordinary observa 
tions of astronomers ; and the mean solar day can be deduced 
from this by our knowledge of the length of the year. 

The unit of time adopted in all physical researches is one second 
of mean solar time. 

In astronomy a year is sometimes used as a unit of time. A 
more universal unit of time might be found by taking the periodic 
time of vibration of the particular kind of light whose wave length 
is the unit of length. 

We shall call the concrete unit of time \_T~\, and the numerical 
measure of time t. 

5.] (3) Mass. The standard unit of mass is in this country the 
avoirdupois pound preserved in the Exchequer Chambers. The 
grain, which is often used as a unit, is defined to be the 7000th 
part of this pound. 

In the metrical system it is the gramme, which is theoretically 
the mass of a cubic centimetre of distilled water at standard tem 
perature and pressure, but practically it is the thousandth part 
of a standard kilogramme preserved in Paris. 

The accuracy with which the masses of bodies can be com 
pared by weighing is far greater than that hitherto attained in 
the measurement of lengths, so that all masses ought, if possible, 
to be compared directly with the standard, and not deduced from 
experiments on water. 

In descriptive astronomy the mass of the sun or that of the 

B 2 



4 PRELIMINARY. [5. 

earth is sometimes taken as a unit, but in the dynamical theory 
of astronomy the unit of mass is deduced from the units of time 
and length, combined with the fact of universal gravitation. The 
astronomical unit of mass is that mass which attracts another 
body placed at the unit of distance so as to produce in that body 
the unit of acceleration. 

In framing a universal system of units we may either deduce 
the unit of mass in this way from those of length and time 
already defined, and this we can do to a rough approximation in 
the present state of science ; or, if we expect * soon to be able to 
determine the mass of a single molecule of a standard substance, 
we may wait for this determination before fixing a universal 
standard of mass. 

We shall denote the concrete unit of mass by the symbol \M ] 
in treating of the dimensions of other units. The unit of mass 
will be taken as one of the three fundamental units. When, as 
in the French system, a particular substance,, water, is taken as 
a standard of density, then the unit of mass is no longer inde 
pendent, but varies as the unit of volume, or as [I/ 3 ]. 

If, as in the astronomical system, the unit of mass is defined 
with respect to its attractive power, the dimensions of [If] are 
[Z 3 ? 7 - 2 ]. 

For the acceleration due to the attraction of a mass m at a 

fflL 

distance r is by the Newtonian Law -j . Suppose this attraction 

to act for a very small time t on a body originally at rest, and to 
cause it to describe a space s, then by the formula of Galileo, 



whence m = 2 -^ . Since r and s are both lengths, and t is a 
t 

time, this equation cannot be true unless the dimensions of m are 
[i/ 3 ^? 7 " 2 ]. The same can be shewn from any astronomical equa 
tion in which the mass of a body appears in some but not in all 
of the terms f. 

* See Prof. J. Loschmidt, Zur Grosse der Luftmolecule, Academy of Vienna,) 
Oct. 12, 1865; G. J. Stoney on The Internal Motions of Gases, Phil. Mag., Aug. 
1868 ; and Sir W. Thomson on The Size of Atoms, Nature, March 31, 1870. 

f If a foot and a second are taken as units, the astronomical unit of mass would 
be about 932,000,000 pounds. 



6.] DERIVED UNITS. 5 

Derived Units. 

6.] The unit of Velocity is that velocity in which unit of length 
is described in unit of time. Its dimensions are [j&T 7 " 1 ]. 

If we adopt the units of length and time derived from the 
vibrations of light, then the unit of velocity is the velocity of 
light. 

The unit of Acceleration is that acceleration in which the velo 
city increases by unity in unit of time. Its dimensions are [I/T~ 2 ]. 

The unit of Density is the density of a substance which contains 
unit of mass in unit of volume. Its dimensions are [J/.Z/~ 3 ]. 

The unit of Momentum is the momentum of unit of mass moving 
with unit of velocity. Its dimensions are [MLT~ l ~]. 

The unit of Force is the force which produces unit of momentum 
in unit of time. Its dimensions are [MLT~ 2 ]. 

This is the absolute unit of force, and this definition of it is 
implied in every equation in Dynamics. Nevertheless, in many 
text books in which these equations are given, a different unit of 
force is adopted, namely, the weight of the national unit of mass ; 
and then, in order to satisfy the equations, the national unit of mass 
is itself abandoned, and an artificial unit is adopted as the dynamical 
unit, equal to the national unit divided by the numerical value of 
the force of gravity at the place. In this way both the unit of force 
and the unit of mass are made to depend on the value of the 
force of gravity, which varies from place to place, so that state 
ments involving these quantities are not complete without a know 
ledge of the force of gravity in the places where these statements 
were found to be true. 

The abolition, for all scientific purposes, of this method of mea 
suring forces is mainly due to the introduction of a general system 
of making observations of magnetic force in countries in which 
the force of gravity is different. All such forces are now measured 
according to the strictly dynamical method deduced from our 
definitions, and the numerical results are the same in whatever 
country the experiments are made. 

The unit of Work is the work done by the unit of force acting 
through the unit of length measured in its own direction. Its 
dimensions are [J/Z 2 T~ 2 ]. 

The Energy of a system, being its capacity of performing work, 
is measured by the work which the system is capable of performing 
by the expenditure of its whole energy. 



6 PRELIMINARY. [7. 

The definitions of other quantities, and of the units to which 
they are referred, will be given when we require them. 

In transforming the values of physical quantities determined in 
terms of one unit, so as to express them in terms of any other unit 
of the same kind, we have only to remember that every expres 
sion for the quantity consists of two factors, the unit and the nu 
merical part which expresses how often the unit is to be taken. 
Hence the numerical part of the expression varies inversely as the 
magnitude of the unit, that is, inversely as the various powers of 
the fundamental units which are indicated by the dimensions of the 
derived unit. 

On Physical Continuity and Discontinuity. 

7.] A quantity is said to vary continuously when, if it passes 
from one value to another, it assumes all the intermediate values. 

We may obtain the conception of continuity from a consideration 
of the continuous existence of a particle of matter in time and space. 
Such a particle cannot pass from one position to another without 
describing a continuous line in space, and the coordinates of its 
position must be continuous functions of the time. 

In the so-called equation of continuity, as given in treatises 
on Hydrodynamics, the fact expressed is that matter cannot appear 
in or disappear from an element of volume without passing in or out 
through the sides of that element. 

A quantity is said to be a continuous function of its variables 
when, if the variables alter continuously, the quantity itself alters 
continuously. 

Thus, if u is a function of x, and if, while x passes continuously 
from # to fl? 1} u passes continuously from n to u lt but when x 
passes from x l to # 2 , u passes from u-f to u 2 , % being different from 
%, then u is said to have a discontinuity in its variation with 
respect to x for the value x = x l} because it passes abruptly from u^ 
to u{ while x passes continuously through # r 

If we consider the differential coefficient of u with respect to x for 
the value x = x^ as the limit of the fraction 



when # 2 and # are both made to approach ^ without limit, then, 
if X Q and x 2 are always on opposite sides of as ly the ultimate value of 
the numerator will be u^u ly and that of the denominator will 
be zero. If u is a quantity physically continuous, the discontinuity 



8.] CONTINUITY AND DISCONTINUITY. 7 

can exist only with respect to the particular variable x. We must 
in this case admit that it has an infinite differential coefficient 
when x X L . If u is not physically continuous, it cannot be dif 
ferentiated at all. 

It is possible in physical questions to get rid of the idea of 
discontinuity without sensibly altering the conditions of the case. 
If # is a very little less than x l} and x. 2 a very little greater than 
.r 15 then U Q will be very nearly equal to u^ and u 2 to u{. We 
may now suppose u to vary in any arbitrary but continuous manner 
from ?/ to u 2 between the limits X Q and x 2 . In many physical 
questions we may begin with a hypothesis of this kind, and then 
investigate the result when the values of # and x 2 are made to 
approach that of ^ and ultimately to reach it. The result will 
in most cases be independent of the arbitrary manner in which we 
have supposed u to vary between the limits. 

Discontinuity of a Fimction of more than One Variable. 

8.] If we suppose the values of all the variables except x to be 
constant, the discontinuity of the function will occur for particular 
values of #, and these will be connected with the values of the 
other variables by an equation which we may write 

$ = < (x, y, z 3 &c.) = 0. 

The discontinuity will occur when <f> = 0. When $ is positive the 
function will have the form F 2 (x } y, z, &c.). When < is negative 
it will have the form F 1 (x, y, z, &c.). There need be no necessary 
relation between the forms F and F 2 . 

To express this discontinuity in a mathematical form, let one of 
the variables, say .r, be expressed as a function of </> and the other 
variables, and let F 1 and F 2 be expressed as functions of <, y, z, &c. 
We may now express the general form of the function by any 
formula which is sensibly equal to F 2 when <p is positive, and to 
F when c/> is negative. Such a formula is the following 

F n< ^ 2 

As long as n is a finite quantity, however great, F will be a 
continuous function, but if we make n infinite F will be equal to 
F 2 when < is positive, and equal to F^ when </> is negative. 

Discontinuity of the Derivatives of a Continuous Function. 
The first derivatives of a continuous function may be discon- 



8 PRELIMINARY. [9. 

tinuous. Let the values of the variables for which the discon 
tinuity of the derivatives occurs be connected by the equation 

< = <(#,y, 2...) = 0, 

and let F L and F 2 be expressed in terms of $ and nl other 
variables, say (y> z . . .). 

Then, when $ is negative, F l is to be taken, and when $ is 
positive F 2 is to be taken, and, since F is itself continuous, when 
</> is zero, F^ = F 2 . 

Hence, when d> is zero, the derivatives - and -~ may be 

d(p dfy 

different, but the derivatives with respect to any of the other 

variables, such as 7^ and =- must be the same. The discon- 
du dy 

t/ */ 

tinuity is therefore confined to the derivative with respect to 0, all 
the other derivatives being- continuous. 



Periodic and Multiple Functions. 

9.] If u is a function of x such that its value is the same for 
a?, x + a t x + na, and all values of x differing by a, u is called a 
periodic function of x, and a is called its period. 

If x is considered as a function of u, then, for a given value of 
U, there must be an infinite series of values of x differing by 
multiples of a. In this case x is called a multiple function of u, 
and a is called its cyclic constant. 

dx 

The differential coefficient - has only a finite number of values 

du 

corresponding to a given value of u. 

On the Relation of Physical Quantities to Directions in Space. 

10.] t ln distinguishing the kinds of physical quantities, it is of 
great importance to know how they are related to the directions 
of those coordinate axes which we usually employ in defining the 
positions of things. The introduction of coordinate axes into geo 
metry by Des Cartes was one of the greatest steps in mathematical 
progress, for it reduced the methods of geometry to calculations 
performed on numerical quantities. The position of a point is made 
to depend on the length of three lines which are always drawn in 
determinate directions, and the line joining two points is in like 
manner considered as the resultant of three lines. 

But for many purposes in physical reasoning, as distinguished 



II.] VECTORS, OR DIRECTED QUANTITIES. 9 

from calculation, it is desirable to avoid explicitly introducing the 
Cartesian coordinates, and to fix the mind at once on a point of 
space instead of its three coordinates, and on the magnitude and 
direction of a force instead of its three components. This mode 
of contemplating geometrical and physical quantities is more prim 
itive and more natural than the other, although the ideas connected 
with it did not receive their full development till Hamilton made 
the next great step in dealing with space, by the invention of his 
Calculus of Quaternions. 

As the methods of Des Cartes are still the most familiar to 
students of science, and as they are really the most useful for 
purposes of calculation, we shall express all our results in the 
Cartesian form. I am convinced, however, that the introduction 
of the ideas, as distinguished from the operations and methods of 
Quaternions, will be of great use to us in the study of all parts 
of our subject, and especially in electrodynamics, where we have to 
deal with a number of physical quantities, the relations of which 
to each other can be expressed far more simply by a few words of 
Hamilton s, than by the ordinary equations. 

11.] One of the most important features of Hamilton s method is 
the division of quantities into Scalars and Vectors. 

A Scalar quantity is capable of being completely defined by a 
single numerical specification. Its numerical value does not in 
any way depend on the directions we assume for the coordinate 
axes. 

A Vector, or Directed quantity, requires for its definition three 
numerical specifications, and these may most simply be understood 
as having reference to the directions of the coordinate axes. 

Scalar quantities do not involve direction. The volume of a 
geometrical figure, the mass and the energy of a material body, 
the hydrostatical pressure at a point in a fluid, and the potential 
at a point in space, are examples of scalar quantities. 

A vector quantity has direction as well as magnitude, and is 
such that a reversal of its direction reverses its sign. The dis 
placement of a point, represented by a straight line drawn from 
its original to its final position, may be taken as the typical 
vector quantity, from which indeed the name of Vector is derived. 

The velocity of a body, its momentum, the force acting on it, 
an electric current, the magnetization of a particle of iron, are 
instances of vector quantities. 

There are physical quantities of another kind which are related 






1 PRELIMINARY. [ I 2. 

to directions in space, but which are not vectors. Stresses and 
strains in solid bodies are examples of these, and the properties 
of bodies considered in the theory of elasticity and in the theory 
of double refraction. Quantities of this class require for their 
definition nine numerical specifications. They are expressed in the 
language of Quaternions by linear and vector functions of a vector. 

The addition of one vector quantity to another of the same kind 
is performed according to the rule given in Statics for the com 
position of forces. In fact, the proof which Poisson gives of the 
parallelogram of forces is applicable to the composition of any 
quantities such that a reversal of their sign is equivalent to turning 
them end for end. 

When we wish to denote a vector quantity by a single symbol, 
and to call attention to the fact that it is a vector, so that we must 
consider its direction as well as its magnitude, we shall denote 
it by a German capital letter, as fy, 33, &c. 

In the calculus of Quaternions, the position of a point in space 
is defined by the vector drawn from a fixed point, called the origin, 
to that point. If at that point of space we have to consider any 
physical quantity whose value depends on the position of the point, 
that quantity is treated as a function of the vector drawn from 
the origin. The function may be itself either scalar or vector. 
The density of a body, its temperature, its hydrostatic pressure, 
the potential at a point, are examples of scalar functions. The 
resultant force at the point, the velocity of a fluid at that point, 
the velocity of rotation of an element of the fluid, and the couple 
producing rotation, are examples of vector functions. 

12.] Physical vector quantities may be divided into two classes, 
in one of which the quantity is defined with reference to a line, 
while in the other the quantity is defined with reference to an 
area. 

For instance, the resultant of an attractive force in any direction 
may be measured by finding the work which it would do on a 
body if the body were moved a short distance in that direction 
and dividing it by that short distance. Here the attractive force 
is defined with reference to a line. 

On the other hand, the flux of heat in any direction at any 
point of a solid body may be defined as the quantity of heat which 
crosses a small area drawn perpendicular to that direction divided 
by that area and by the time. Here the flux is defined with 
reference to an area. 



13.] FORCES AND FLUXES. 11 

There are certain cases in which a quantity may be measured 
with reference to a line as well as with reference to an area. 

Thus, in treating- of the displacements of elastic solids, we may 
direct our attention either to the original and the actual position 
of a particle, in which case the displacement of the particle is 
measured by the line drawn from the first position to the second, 
or we may consider a small area fixed in space, and determine 
what quantity of the solid passes across that area during the dis 
placement. 

In the same way the velocity of a fluid may be investigated 
either with respect to the actual velocity of the individual parti 
cles, or with respect to the quantity of the fluid which flows through 
any fixed area. 

But in these cases we require to know separately the density of 
the body as well as the displacement or velocity, in order to apply 
the first method, and whenever we attempt to form a molecular 
theory we have to use the second method. 

In the case of the flow of electricity we do not know anything 
of its density or its velocity in the conductor, we only know the 
value of what, on the fluid theory, would correspond to the product 
of the density and the velocity. Hence in all such cases we must 
apply the more general method of measurement of the flux across 
an area. 

In electrical science, electromotive force and magnetic force 
belong to the first class, being defined with reference to lines. 
When we wish to indicate this fact, we may refer to them as 
Forces. 

On the other hand, electric and magnetic induction, and electric 
currents, belong to the second class, being defined with reference 
to areas. When we wish to indicate this fact, we shall refer to them 
as Fluxes. 

Each of these forces may be considered as producing, or tending 
to produce, its corresponding flux. Thus, electromotive force pro 
duces electric currents in conductors, and tends to produce them 
in dielectrics. It produces electric induction in dielectrics, and pro 
bably in conductors also. In the same sense, magnetic force pro 
duces magnetic induction. 

13.] In some cases the flux is simply proportional to the force 
and in the same direction, but in other cases we can only affirm 
that the direction and magnitude of the flux are functions of the 
direction and magnitude of the force. 



12 PRELIM1NAKY. [14. 

The case in which the components of the flux are linear functions 
of those of the force is discussed in the chapter on the Equations 
of Conduction, Art. 296. There are in general nine coefficients 
which determine the relation between the force and the flux. In 
certain cases we have reason to believe that six of these coefficients 
form three pairs of equal quantities. In such cases the relation be 
tween the line of direction of the force and the normal plane of the 
flux is of the same kind as that between a diameter of an ellipsoid 
and its conjugate diametral plane. In Quaternion language, the 
one vector is said to be a linear and vector function of the other, and 
when there are three pairs of equal coefficients the function is said 
to be self-conjugate. 

In the case of magnetic induction in iron, the flux, (the mag 
netization of the iron,) is not a linear function of the magnetizing 
force. In all cases, however, the product of the force and the 
flux resolved in its direction, gives a result of scientific import 
ance, and this is always a scalar quantity. 

14.] There are two mathematical operations of frequent occur 
rence which are appropriate to these two classes of vectors, or 
directed quantities. 

In the case of forces, we have to take the integral along a line 
of the product of an element of the line, and the resolved part of 
the force along that element. The result of this operation is 
called the Line-integral of the force. It represents the work 
done on a body carried along the line. In certain cases in which 
the line-integral does not depend on the form of the line, but 
only on the position of its extremities, the line-integral is called 
the Potential. 

In the case of fluxes, we have to take the integral, over a surface, 
of the flux through every element of the surface. The result of 
this operation is called the Surface-integral of the flux. It repre 
sents the quantity which passes through the surface. 

There are certain surfaces across which there is no flux. If 
two of these surfaces intersect, their line of intersection is a line 
of flux. In those cases in which the flux is in the same direction 
as the force, lines of this kind are often called Lines of Force. It 
would be more correct, however, to speak of them in electrostatics 
and magnetics as Lines of Induction, and in electrokinematics as 
Lines of Flow. 

15.] There is another distinction between different kinds of 
directed quantities, which, though very important in a physical 



1 6.] LINE-INTEGRALS. 1 3 

point of view, is not so necessary to be observed for the sake of 
the mathematical methods. This is the distinction between longi 
tudinal and rotational properties. 

The direction and magnitude of a quantity may depend upon 
some action or effect which takes place entirely along a certain 
line, or it may depend upon something of the nature of rota 
tion about that line as an axis. The laws of combination of 
directed quantities are the same whether they are longitudinal or 
rotational, so that there is no difference in the mathematical treat 
ment of the two classes, but there may be physical circumstances 
which indicate to which class we must refer a particular pheno 
menon. Thus, electrolysis consists of the transfer of certain sub 
stances along a line in one direction, and of certain other sub 
stances in the opposite direction, which is evidently a longitudinal 
phenomenon, and there is no evidence of any rotational effect 
about the direction of the force. Hence we infer that the electric 
current which causes or accompanies electrolysis is a longitudinal, 
and not a rotational phenomenon. 

On the other hand, the north and south poles of a magnet do 
not differ as oxygen and hydrogen do, which appear at opposite 
places during electrolysis, so that we have no evidence that mag 
netism is a longitudinal phenomenon, while the effect of magnetism 
in rotating the plane of polarized light distinctly shews that mag 
netism is a rotational phenomenon. 

On Line-integrals. 

16.] The operation of integration of the resolved part of a vector 
quantity along a line is important in physical science generally, 
and should be clearly understood. 

Let X) y, z be the coordinates of a point P on a line whose 
length, measured from a certain point A, is s. These coordinates 
will be functions of a single variable s. 

Let R be the value of the vector quantity at P, and let the 
tangent to the curve at P make with the direction of R the angle e, 
then .72 cose is the resolved part of R along the line, and the 



integral 



C* 
= / 

^o 



R cos e 



is called the line-integral of R along the line s. 
We may write this expression 



14 PRELIMINARY. [l6. 

where X, T, Z are the components of R parallel to #, y, z respect 
ively. 

This quantity is, in general, different for different lines drawn 
between A and P. When, however, within a certain region, the 

quantity 

Xdx+ Ydy + Zdz = -DV, 

that is, is an exact differential within that region, the value of L 
becomes Jj = \^ A \^ P) 

and is the same for any two forms of the path between A and P, 
provided the one form can be changed into the other by con 
tinuous motion without passing out of this region. 

On Potentials. 

The quantity ^ is a scalar function of the position of the point, 
and is therefore independent of the directions of reference. It is 
called the Potential Function, and the vector quantity whose com 
ponents are X, Y, Z is said to have a potential ^, if 

*-- --($ --<> 

When a potential function exists, surfaces for which the po 
tential is constant are called Equipotential surfaces. The direction 
of R at any point of such a surface coincides with the normal to 

dty 

the surface, and if n be a normal at the point P. then R = =- 

dn 

The method of considering the components of a vector as the 
first derivatives of a certain function of the coordinates with re 
spect to these coordinates was invented by Laplace * in his treat 
ment of the theory of attractions. The name of Potential was first 
given to this function by Green f, who made it the basis of his 
treatment of electricity. Green s essay was neglected by mathe 
maticians till 1846, and before that time most of its important 
theorems had been rediscovered by Gauss, Chasles, Sturm, and 
Thomson J. 

In the theory of gravitation the potential is taken with the 
opposite sign to that which is here used, and the resultant force 
in any direction is then measured by the rate of increase of the 

* Mec. Celeste, liv. iii. 

t Essay on the Application of Mathematical Analysis to the Theories of Electricity 
and Magnetism, Nottingham, 1828. Eeprinted in Crelle s Journal, and in Mr. Ferrer s 
edition of Green s Works. 

J Thomson and Tait, Natural Philosophy, 483. 



17.] RELATION BETWEEN FORCE AND POTENTIAL. 15 

potential function in that direction. In electrical and magnetic 
investigations the potential is defined so that the resultant force 
in any direction is measured by the decrease of the potential in 
that direction. This method of using the expression makes it 
correspond in sign with potential energy, which always decreases 
when the bodies are moved in the direction of the forces acting 
on them. 

17.] The geometrical nature of the relation between the poten 
tial and the vector thus derived from it receives great light from 
Hamilton s discovery of the form of the operator by which the vector 
is derived from the potential. 

The resolved part of the vector in any direction is, as we have 
seen, the first derivative of the potential with respect to a co 
ordinate drawn in that direction, the sign being reversed. 

Now if i t /, k are three unit vectors at right angles to each 
other, and if X, Y, Z are the components of the vector 5 resolved 
parallel to these vectors, then 

9 = IX+jY+kZ; (1) 

and by what we have said above, if ^ is the potential, 



If we now write V for the operator, 



(4) 

The symbol of operation V may be interpreted as directing us 
to measure, in each of three rectangular directions, the rate of 
increase of &, and then, considering the quantities thus found as 
vectors, to compound them into one. This is what we are directed 
to do by the expression (3). But we may also consider it as directing 
us first to find out in what direction ^ increases fastest, and then 
to lay off in that direction a vector representing this rate of 
increase. 

M. Lame, in his Traite des Fonctions Inverses, uses the term 
Differential Parameter to express the magnitude of this greatest 
rate of increase, but neither the term itself, nor the mode in which 
Lame uses it, indicates that the quantity referred to has direction 
as well as magnitude. On those rare occasions in which I shall have 
to refer to this relation as a purely geometrical one, I shall call the 
vector g the Slope of the scalar function ^, using the word Slope 



16 PRELIMINARY. [l8. 

to indicate the direction, as well as the magnitude, of the most 
rapid decrease of #. 

18.] There are cases, however, in which the conditions 

dZ dY dX dZ dY dX 

;- = 0, -j = 0, and =- = 0, 

dy dz dz ax dx dy 

which are those of Xdx + Ydy -f Zdz being a complete differential, 
are fulfilled throughout a certain region of space, and yet the line- 
integral from A to P may be different for two lines, each of 
which lies wholly within that region. This may be the case if 
the region is in the form of a ring, and if the two lines from A 
to P pass through opposite segments of the ring. In this case, 
the one path cannot be transformed into the other by continuous 
motion without passing out of the region. 

We are here led to considerations belonging to the Geometry 
of Position, a subject which, though its importance was pointed 
out by Leibnitz and illustrated by Gauss, has been little studied. 
The most complete treatment of this subject has been given by 
J. B. Listing *. 

Let there be p points in space, and let I lines of any form be 
drawn joining these points so that no two lines intersect each 
other, and no point is left isolated. We shall call a figure com 
posed of lines in this way a Diagram. Of these lines, p 1 are 
sufficient to join the p points so as to form a connected system. 
Every new line completes a loop or closed path, or, as we shall 
call it, a Cycle. The number of independent cycles in the diagram 
is therefore K = I jo+1. 

Any closed path drawn along the lines of the diagram is com 
posed of these independent cycles, each being taken any number of 
times and in either direction. 

The existence of cycles is called Cyclosis, and the number of 
cycles in a diagram is called its Cyclomatic number. 

Cyclosis in Surfaces and Regions. 

Surfaces are either complete or bounded. Complete surfaces are 
either infinite or closed. Bounded surfaces are limited by one or 
more closed lines, which may in the limiting cases become finite 
lines or points. 

A finite region of space is bounded by one or more closed 
surfaces. Of these one is the external surface, the others are 

* Der Census RaumlicTier Complexe, Gott. Abh., Bd. x. S. 97 (1861). 



19.] CYCLIC REGIONS. 17 

included in it and exclude each other, and are called internal 
surfaces. 

If the region has one bounding- surface, we may suppose that 
surface to contract inwards without breaking its continuity or 
cutting itself. If the region is one of simple continuity, such as 
a sphere, this process may be continued till it is reduced to a 
point; but if the region is like a ring, the result will be a closed 
curve; and if the region has multiple connexions, the result will 
be a diagram of lines, and the cyclomatic number of the diagram 
will be that of the region. The space outside the region has the 
same cyclomatic number as the region itself. Hence, if the region 
is bounded by internal as well as external surfaces, its cyclomatic 
number is the sum of those due to all the surfaces. 

When a region encloses within itself other regions, it is called a 
Periphractic region. 

The number of internal bounding surfaces of a region is called 
its periphractic number. A closed surface is also periphractic, its 
number being unity. 

The cyclomatic number of a closed surface is twice that of the 
region which it bounds. To find the cyclomatic number of a 
bounded surface, suppose all the boundaries to contract inwards, 
without breaking continuity, till they meet. The surface will then 
be reduced to a point in the case of an acyclic surface, or to a linear 
diagram in the case of cyclic surfaces. The cyclomatic number of 
the diagram is that of the surface. 

19.] THEOREM I. If throughout any acyclic region 

Xdx+Ydy + Zch =-*, 

the value of the line-integral from a point A to a point P taken 
along any path within the region will le the same. 

We shall first shew that the line-integral taken round any closed 
path within the region is zero. 

Suppose the equipotential surfaces drawn. They are all either 
closed surfaces or are bounded entirely by the surface of the region, 
so that a closed line within the region, if it cuts any of the sur 
faces at one part of its path, must cut the same surface in the 
opposite direction at some other part of its path, and the corre 
sponding portions of the line-integral being equal and opposite, 
the total value is zero. 

Hence if AQP and AQ P are two paths from A to P, the line- 
integral for AQ P is the sum of that for AQP and the closed path 

VOL. i. c 



18 PRELIMINARY. [20. 

AQ PQA. But the line-integral of the closed path is zero, there 
fore those of the two paths are equal. 

Hence if the potential is given at any one point of such a 
region, that at any other point is determinate. 

20.] THEOREM II. In a cyclic region in which the equation 

Xdx+Ydy + Zdz = -D* 

is everywhere fulfilled, the line-integral from, A to P, along a 
line drawn within the region, will not in general be determinate 
unless the channel of communication between A and P be specified, 

Let K be the cyclomatic number of the region, then K sections 
of the region may be made by surfaces which we may call Dia 
phragms, so as to close up K of the channels of communication, 
and reduce the region to an acyclic condition without destroying 
its continuity. 

The line-integral from A to any point P taken along a line 
which does not cut any of these diaphragms will be, by the last 
theorem, determinate in value. 

Now let A and P be taken indefinitely near to each other, but 
on opposite sides of a diaphragm, and let K be the line-integral 
from A to P. 

Let A and P be two other points on opposite sides of the same 
diaphragm and indefinitely near to each other, and let K be the 
line-integral from A to F. Then K = K. 

For if we draw AA and PP 7 , nearly coincident, but on opposite 
sides of the diaphragm, the line-integrals aloug these lines will be 
equal. Suppose each equal to Z, then the line-integral of A P* is 
equal to that of A A + AP + PP= -L+K+L = K = that of AP. 

Hence the line-integral round a closed curve which passes through 
one diaphragm of the system in a given direction is a constant 
quantity K. This quantity is called the Cyclic constant corre 
sponding to the given cycle. 

Let any closed curve be drawn within the region, and let it cut 
the diaphragm of the first cycle p times in the positive direction 
and p times in the negative direction, and let p p = % . Then 
the line-integral of the closed curve will be % K r 

Similarly the line-integral of any closed curve will be 



where n K represents the excess of the number of positive passages 
of the curve through the diaphragm of the cycle K over the 
number of negative passages. 



21.] SURFACE-INTEGRALS. 19 

If two curves are such that one of them may be transformed 
into the other by continuous motion without at any time passing* 
through any part of space for which the condition of having a 
potential is not fulfilled,, these two curves are called Reconcileabje 
curves. Curves for which this transformation cannot be effected 
are called Irreconcileable curves *. 

The condition that Xdx+Ydy -\- Zdz is a complete differential 
of some function ^ for all points within a certain region, occurs in 
several physical investigations in which the directed quantity and 
the potential have different physical interpretations. 

In pure kinematics we may suppose X, Y, Z to be the com 
ponents of the displacement of a point of a continuous body whose 
original coordinates are x, y, z, then the condition expresses that 
these displacements constitute a non-rotational strain f. 

If X, Y, Z represent the components of the velocity of a fluid at 
the point x, y, z, then the condition expresses that the motion of the 
fluid is irrotational. 

If X, Y, Z represent the components of the force at the point 
#, i/j z, then the condition expresses that the work done on a 
particle passing from one point to another is the difference of the 
potentials at these points, and the value of this difference is the 
same for all reconcileable paths between the two points. 

On Surface-Integrals. 

21.] Let dS be the element of a surface, and c the angle which 
a normal to the surface drawn towards the positive side of the 
surface makes with the direction of the vector quantity R, then 

R cos tdS is called the surface-integral of It over the surface S. 



ff 



THEOREM III. The surface-integral of the flux through a closed 
surface may be expressed as the volume-integral of its convergence 
taken within the surface. (See Art. 25.) 

Let X, Y, Z be the components of R, and let I, m, n be the 
direction-cosines of the normal to S measured outwards. Then the 
surface-integral of R over S is 

{(R cos e dS = ffxidS + JJYmdS + j j ZndS 

= IJXdydz+jJYdzdx -f- ft Zdxdy, (1) 

* See Sir W. Thomson C 0n Vortex Motion, Trans. R. S. Edin., 1869. 
t See Thomson and Tait s Natural Philosophy, 190 (). 

C 2 



20 PRELIMINARY. [21. 

the values of X, Y, Z being those at a point in the surface, and 
the integrations being extended over the whole surface. 

If the surface is a closed one, then, when y and z are given, 
the coordinate sc must have an even number of values, since a line 
parallel to x must enter and leave the enclosed space an equal 
number of times provided it meets the surface at all. 

Let a point travelling from # = oo to # = +oo first enter 
the space when os = as l9 then leave it when x = a? 2 , and so on; 
and let the values of X at these points be X 1 , X 2 , &c., then 

,- X 3 ) + &c. 4 (li.-.Xi.-,)} <fyfe. (2) 

If Jf is a quantity which is continuous, and has no infinite values 
between and # 2 , then 

dX 



where the integration is extended from the first to the second 
intersection, that is, along the first segment of x which is within 
the closed surface. Taking into account all the segments which lie 
within the closed surface, we find 



the double integration being confined to the closed surface, but 
the triple integration being extended to the whole enclosed space. 
Hence, if X, J, Z are continuous and finite within a closed surface 
$, the total surface-integral of R over that surface will be 

IT*, ffr/dX dY dz \ - 

JJ****-JJJfc + ^ + jg)*** (5) 

the triple integration being extended over the whole space within & 
Let us next suppose that X, Y, Z are not continuous within the 

closed surface, but that at a certain surface F (x, y> z] = the 

values of X } Y y Z alter abruptly from X, Y, Z on the negative side 

of the surface to X , Y , Z on the positive side. 

If this discontinuity occurs, say, between a? t and # 2 , the value 



J x . dx 



where in the expression under the integral sign only the finite 
values of the derivative of X are to be considered. 

In this case therefore the total surface-integral of R over the 
closed surface will be expressed by 



22.] SOLENOIDAL DISTRIBUTION. 21 



+ 



jj(Y -Y)dzdx + ff(Z -Z)dxdy; (7) 

or, if / , m , ft are the direction-cosines of the normal to the surface 
of discontinuity, and dS an element of that surface, 



, (8) 

where the integration of the last term is to be extended over the 
surface of discontinuity. 

If at every point where X, Y, Z are continuous 
dX dY dZ 

^ + ^ + ^ = > (9) 

and at every surface where they are discontinuous 

I X + m T + n Z = I X+ m Y+ n Z, (10) 

then the surface-integral over every closed surface is zero, and the 
distribution of the vector quantity is said to be Solenoidal. 

We shall refer to equation (9) as the General solenoidal con 
dition, and to equation (10) as the Superficial solenoidal condition. 

22.] Let us now consider the case in which at every point 
within the surface 8 the equation 

+ + *?.0 (11) 

dx dy dz 

is fulfilled. We have as a consequence of this the surface-integral 
over the closed surface equal to zero. 

Now let the closed surface S consist of three parts S lt S , and 
S 2 . Let S 1 be a surface of any form bounded by a closed line L r 
Let S Q be formed by drawing lines from every point of L always 
coinciding with the direction of E. If I, m, n are the direction- 
cosines of the normal at any point of the surface $ , we have 

RcoB* = Xl+Ym + Zn = 0. (12) 

Hence this part of the surface contributes nothing towards the 
value of the surface-integral. 

Let # 2 be another surface of any form bounded by the closed 
curve L. 2 in which it meets the surface S . 

Let Q 1} Q , Q 2 be the surface-integrals of the surfaces S IS S ,S 2 , 
and let Q be the surface-integral of the closed surface S. Then 



22 PRELIMINARY. [2,2. 

and we know that Q Q = ; (14) 

therefore Q 2 = - Q l ; (15) 

or, in other words, the surface-integral over the surface $ 2 is equal 
and opposite to that over S L whatever be the form and position 
of 2 , provided that the intermediate surface S is one for which R 
is always tangential. 

If we suppose Z^ a closed curve of small area,, $ will be a 
tubular surface having the property that the surface-integral over 
every complete section of the tube is the same. 

Since the whole space can be divided into tubes of this kind 

provided dX dY dZ 

-T- + -T- + -j- = 0, (16) 

dx du dz 

J 

a distribution of a vector quantity consistent with this equation is 
called a Solenoidal Distribution. 

On Tubes and Lines of Flow. 

If the space is so divided into tubes that the surface-integral 
for every tube is unity, the tubes are called Unit tubes, and the 
surface-integral over any finite surface S bounded by a closed 
curve L is equal to the number of such tubes which pass through 
S in the positive direction, or, what is the same thing, the number 
which pass through the closed curve L. 

Hence the surface-integral of S depends only on the form of 
its boundary L, and not on the form of the surface within its 
boundary. 

On Peripkractic Regions. 

If, throughout the whole region bounded externally by the single 
closed surface S lt the solenoidal condition 
dX dY dZ^_ 
dx dv dz 

t/ 

is fulfilled, then the surface-integral taken over any closed surface 
drawn within this region will be zero, and the surface-integral 
taken over a bounded surface within the region will depend only 
on the form of the closed curve which forms its boundary. 

It is not, however, generally true that the same results follow 
if the region within which the solenoidal condition is fulfilled is 
bounded otherwise than by a single surface. 

For if it is bounded by more than one continuous surface, one of 
these is the external surface and the others are internal surfaces, 



22.] PERIPHRACTIC REGIONS. 23 

and the region S is a periphractic region, having within it other 
regions which it completely encloses. 

If within any of these enclosed regions, S^ the solenoidal con 
dition is not fulfilled, let 



*-// 



R cos e dS l 



be the surface-integral for the surface enclosing this region, and 
let Q 2 , Q 3 , &c. be the corresponding quantities for the other en 
closed regions. 

Then, if a closed surface S is drawn within the region S t the 
value of its surface-integral will be zero only when this surface 
/S" does not include any of the enclosed regions S 19 S 2 , &c. If it 
includes any of these, the surface-integral is the sum of the surface- 
integrals of the different enclosed regions which lie within it. 

For the same reason, the surface-integral taken over a surface 
bounded by a closed curve is the same for such surfaces only bounded 
by the closed curve as are reconcileable with the given surface by 
continuous motion of the surface within the region S. 

When we have to deal with a periphractic region, the first thing 
to be done is to reduce it to an aperiphractic region by drawing 
lines joining the different bounding surfaces. Each of these lines, 
provided it joins surfaces which were not already in continuous 
connexion, reduces the periphractic number by unity, so that the 
whole number of lines to be drawn to remove the periphraxy is 
equal to the periphractic number, or the number of internal sur 
faces. When these lines have been drawn we may assert that if 
the solenoidal condition is fulfilled in the region S, any closed surface 
drawn entirely within S, and not cutting any of the lines, has its 
surface-integral zero. 

In drawing these lines we must remember that any line joining 
surfaces which are already connected does not diminish the peri 
phraxy, but introduces cyclosis. 

The most familiar example of a periphractic region within which 
the solenoidal condition is fulfilled is the region surrounding a mass 
attracting or repelling inversely as the square of the distance. 

In this case we have 

/>* y* g 

X = m > Y= m > Z = m ; 
r 3 r 3 r 3 

where m is the mass supposed to be at the origin of coordinates. 
At any point where r is finite 

dX dY dZ 

T + -j- + -r = 
ax ay dz 



24 PRELIMINARY. [23. 

but at the origin these quantities become infinite. For any closed 
surface not including the origin, the surface-integral is zero. If 
a closed surface includes the origin, its surface-integral is 4?m. 

If, for any reason, we wish to treat the region round m as if it 
were not periphractic, we must draw a line from m to an infinite 
distance, and in taking surface-integrals we must remember to add 
4Trm whenever this line crosses from the negative to the positive 
side of the surface. 



On Right-handed and Left-handed Relations in Space. 

23.] In this treatise the motions of translation along any axis 
and of rotation about that axis, will be assumed to be of the same 
sign when their directions correspond to those of the translation 
and rotation of an ordinary or right-handed screw *. 

For instance, if the actual rotation of the earth from west to east 
is taken positive, the direction of the earth s axis from south to 
north will be taken positive, and if a man walks forward in the 
positive direction, the positive rotation is in the order, head, right- 
hand, feet, left-hand. 

If we place ourselves on the positive side of a surface, the positive 
direction along its bounding curve will be opposite to the motion 
of the hands of a watch with its face towards us. 

This is the right-handed system which is adopted in Thomson 
and Tait s Natural Philosophy, 243. The opposite, or left-handed 
system, is adopted in Hamilton s and Tait s Quaternions. The 
operation of passing from the one system to the other is called, by 
Listing, Perversion. 

The reflexion of an object in a mirror is a perverted image of the 
object. 

"When we use the Cartesian axes of %, y, z, we shall draw them 



* The combined action of the muscles of the arm when we turn the upper side of 
the right-hand outwards, and at the same time thrust the hand forwards, will 
impress the right-handed screw motion on the memory more firmly than any verbal 
definition. A common corkscrew may be used as a material symbol of the same 
relation. 

Professor W. H. Miller has suggested to me that as the tendrils of the vine are 
right-handed screws and those of the hop left-handed, the two systems of relations in 
space might be called those of the vine and the hop respectively. 

The system of the vine, which we adopt, is that of Linnseus, and of screw-makers 
in all civilized countries except Japan. De Candolle was the first who called the 
hop-tendril right-handed, and in this he is followed by Listing, and by most writers 
on the rotatory polarization of light. Screws like the hop-tendril are made for the 
couplings of railway-carriages, and for the fittings of wheels on the left side of ordinary 
carriages, but they are always called left-handed screws by those who use them. 



24.] LINE-INTEGRAL AND SURFACE-INTEGRAL. 25 

so that the ordinary conventions about the cyclic order of the 
symbols lead to a right-handed system of directions in space. Thus, 
if x is drawn eastward and y northward, z must be drawn upward. 

The areas of surfaces will be taken positive when the order of 
integration coincides with the cyclic order of the symbols. Thus, 
the area of a closed curve in the plane of xy may be written either 



\xdy 



or 



the order of integration being x, y in the first expression, and y^ x 
in the second. 

This relation between the two products dx dy and dy d-x may 
be compared with that between the products of two perpendicular 
vectors in the doctrine of Quaternions, the sign of which depends 
on the order of multiplication, and with the reversal of the sign 
of a determinant when the adjoining rows or columns are ex 
changed. 

For similar reasons a volume-integral is to be taken positive when 
the order of integration is in the cyclic order of the variables x t y, z, 
and negative when the cyclic order is reversed. 

We now proceed to prove a theorem which is useful as esta 
blishing a connexion between the surface-integral taken over a 
finite surface and a line-integral taken round its boundary. 

24.] THEOREM IV. A line-integral taken round a closed curve 
may be expressed in terms of a surface-integral taken over a 
surface bounded by the curve. 

Let X, Y } Z be the components of a vector quantity 51 whose line- 
integral is to be taken round a closed curve s. 

Let S be any continuous finite surface bounded entirely by the 
closed curve s, and let f, 77, f be the components of another vector 
quantity 33, related to X, Y, Z by the equations 



_ 
dy dz ~ dz dx ~ dx dy 

Then the surface-integral of 3 taken over the surface S is equal to 
the line-integral of 51 taken round the curve s. It is manifest that 
, YJ, f fulfil of themselves the so lenoidal condition 

d( drj dC 

_z i __ I _j __ __ o. 

dx dy dz 
Let /, m, n be the direction-cosines of the normal to an element 



26 PEELIMINARY. [24. 

of the surface dS t reckoned in the positive direction. Then the 
value of the surface-integral of 33 may be written 



(2) 

In order to form a definite idea of the meaning of the element 
dS, we shall suppose that the values of the coordinates x, y, z for 
every point of the surface are given as functions of two inde 
pendent variables a and p. If ft is constant and a varies, the point 
(>, y, z) will describe a curve on the surface, and if a series of values 
is given to /3, a series of such curves will be traced, all lying on 
the surface S. In the same way, by giving a series of constant 
values to a, a second series of curves may be traced, cutting the 
first series, and dividing the whole surface into elementary portions, 
any one of which may be taken as the element dS. 

The projection of this element on the plane of y, z is, by the 
ordinary formula, 

, 70 ,dy dz dy dz^ _ . * 

IdS = (-T- TS - -A -T-) dP da - (3) 

\a dp dp da 

The expressions for mdS and ndS are obtained from this by sub 
stituting x, y> z in cyclic order. 

The surface-integral which we have to find is 



(4) 

or, substituting the values of , 77, f in terms of X, Y, Z, 
dX dX dY dY dZ dZ 



The part of this which depends on X may be written 
dXsdzdx dz dx dX ,dx dy dx dy 
~ 



, ,. , , . dXdx dx . . . 

adding and subtracting -= 7 - , this becomes 
3 dx da dp 

f C ( dx ,dX dx dX dy dX dz^ 
JJ (dp \da da dy da dz do) 



dx ,dX dx dX dy dX , 

~^" + ~ + ~ P a 



As we have made no assumption as to the form of the functions 
a and /3, we may assume that a is a function of X, or, in other 
words, that the curves for which a is constant are those for which 



25.] HAMILTON S OPERATOR v. 27 

7 1?" 

X is constant. In this case -^- = 0, and the expression becomes 

dp 

by integration with respect to a, 

C CdX dx -, f ^r dx , 

/ / dQda. = X-T-d8; (9) 

JJ da dp J dp 

where the integration is now to be performed round the closed 
curve. Since all the quantities are now expressed in terms of one 
variable & we may make s, the length of the bounding curve, the 
independent variable, and the expression may then be written 

AS* < io > 

where the integration is to be performed round the curve s. We 
may treat in the same way the parts of the surface -integral which 
depend upon T and Z, so that we get finally, 



where the first integral is extended over the surface , and the 
second round the bounding curve s *. 

On the effect of the operator V on a vector function. 
25.] We have seen that the operation denoted by V is that by 
which a vector quantity is deduced from its potential. The same 
operation, however, when applied to a vector function, produces 
results which enter into the two theorems we have just proved 
(III and IV). The extension of this operator to vector displacements, 
and most of its further development, is due to Professor Tait f. 

Let o- be a vector function of p, the vector of a variable point. 
Let us suppose, as usual, that 

p = ix+jy + kz, 
and o- = iX+jY+kZ; 

where X y Y, Z are the components of o- in the directions of the 
axes. 

We have to perform on cr the operation 

. d . d 7 d 
V = i-j- +J-T +b-r 
dx ay dz 

Performing this operation, and remembering the rules for the 

* This theorem was given by Professor Stokes. Smith s Prize Examination, 1854, 
question 8. It is proved in Thomson and Tait s Natural Philosophy, 190 (f). 

t See Proc. R. S. Edin., April 28, 1862. On Green s and other allied Theorems, 
Trans. R. S. Edin., 1869-70, a very valuable paper ; and On some Quaternion 
Integrals, Proc. R. S. Edin., 1870-71. 



28 PRELIMINARY. [25. 

multiplication of i, /, /, we find that V a- consists of two parts, 
one scalar and the other vector. 
The scalar part is 

,dX dY dZ. TTT 

#V <T = (-7- + -j- + -j-}t see Theorem III, 
\dx dy dz 

and the vector part is 

. ( dZ dY. ,,dX d2\ /./^_^\ 
* VJJT ~ ~3i) + <? \dz~~fa> + ^ ~dgJ 
If the relation between X, Y } Z and , r/, f is that given by 
equation (1) of the last theorem, we may write 

F V o- = & + y rj + Jc C See Theorem IV. 

It appears therefore that the functions of X, Y, Z which occur 
in the two theorems are both obtained by the operation V on the 
vector whose components are X, Y, Z. The theorems themselves 
may be written 

jjjSVvds =jfs.vUvds, (III) 

and fsvdp =JJ8.V<FUvd9i (IV) 

where d 9 is an element of a volume, ds of a surface, dp of a curve, 
and Uv a unit- vector in the direction of the normal. 

To understand the meaning 1 of these functions of a vector, let us 
suppose that o- is the value of o- at a point P, and let us examine 
the value of o- o- in the neighbourhood of P. 
If we draw a closed surface round P } then, if the 
I/ surface-integral of o- over this surface is directed 
inwards, S V o- will be positive, and the vector 
p O-O-Q near the point P will be on the whole 

/ X^ directed towards P, as in the figure (1). 

I propose therefore to call the scalar part of 
jv i V (T the convergence of o- at the point P. 

To interpret the vector part of Vo-, let us 

suppose ourselves to be looking in the direction of the vector 

whose components are f, 77, and let us examine 

-* the vector o- o- near the point P. It will appear 

I p . as in the figure (2), this vector being arranged on 

the whole tangentially in the direction opposite to 

the hands of a watch. 

I propose (with great diffidence) to call the vector 
part of V o- the curl, or the version of o- at the point P. 



26.] CONCENTRATION. 29 

At Fig. 3 we have an illustration of curl combined with con 
vergence. 

Let us now consider the meaning of the equation , 

VV a = 0. \ 

This implies that V <r is a scalar, or that the vector X 

o- is the slope of some scalar function $?. These f 

applications of the operator V are due to Professor ^ 

Tait *. A more complete development of the theory 
is given in his paper On Green s and other allied Theorems f/ 
to which I refer the reader for the purely Quaternion investigation 
of the properties of the operator V. 

26.] One of the most remarkable properties of the operator V is 
that when repeated it becomes 



an operator occurring in all parts of Physics, which we may refer to 
as Laplace s Operator. 

This operator is itself essentially scalar. When it acts on a 
scalar function the result is scalar, when it acts on a vector function 
the result is a vector. 

If, with any point P as centre, we draw a small sphere whose 
radius is r, then if q is the value of q at the centre, and q the 
mean value of q for all points within the sphere, 

2o-2 = iV>- 2 v 2 2 ; 

so that the value at the centre exceeds or falls short of the mean 
value according as V 2 q is positive or negative. 

I propose therefore to call V 2 <? the concentration of q at the 
point P, because it indicates the excess of the value of q at that 
point over its mean value in the neighbourhood of the point. 

If q is a scalar function, the method of finding its mean value is 
well known. If it is a vector function, we must find its mean 
value by the rules for integrating vector functions. The result 
of course is a vector. 

* Proceedings R. S. Edin., 1862. t Trans. R. 8. Edin., 1869-70. 



PART I. 

ELECTROSTATICS. 
CHAPTEE I. 

DESCRIPTION OF PHENOMENA. 

Electrification by Friction. 

27.] EXPERIMENT I *". Let a piece of glass and a piece of resin, 
neither of which exhibits any electrical properties, be rubbed to 
gether and left with the rubbed surfaces in contact. They will 
still exhibit no electrical properties. Let them be separated. They 
will now attract each other. 

If a second piece of glass be rubbed with a second piece of 
resin, and if the pieces be then separated and suspended in the 
neighbourhood of the former pieces of glass and resin,, it may be 
observed 

(1) That the two pieces of glass repel each other. 

(2) That each piece of glass attracts each piece of resin. 

(3) That the two pieces of resin repel each other. 

These phenomena of attraction and repulsion are called Elec 
trical phenomena, and the bodies which exhibit them are said to 
be electrified, or to be charged with electricity. 

Bodies may be electrified in many other ways, as well as by 
friction. 

The electrical properties of the two pieces of glass are similar 
to each other but opposite to those of the two pieces of resin, 
the glass attracts what the resin repels and repels what the resin 
attracts. 

* See Sir W. Thomson On the Mathematical Theory of Electricity, Cambridge 
and Dublin Mathematical Journal, March, 1848. 



28.] ELECTRIFICATION. 31 

If a body electrified in any manner whatever behaves as the 
glass does, that is, if it repels the glass and attracts the resin, the 
body is said to be vitreously electrified, and if it attracts the glass 
and repels the resin it is said to be resinously electrified. All 
electrified bodies are found to be either vitreously or resinously 
electrified. 

It is the established practice of men of science to call the vitreous 
electrification positive, and the resinous electrification negative. 
The exactly opposite properties of the two kinds of electrification 
justify us in indicating them by opposite signs, but the applica 
tion of the positive sign to one rather than to the other kind must 
be considered as a matter of arbitrary convention, just as it is a 
matter of convention in mathematical diagrams to reckon positive 
distances towards the right hand. 

No force, either of attraction or of repulsion, can be observed 
between an electrified body and a body not electrified. When, in 
any case, bodies not previously electrified are observed to be acted 
on by an electrified body, it is because they have become electrified 
by induction. 

Electrification by Induction. 

28.] EXPERIMENT II *. Let a hollow vessel of metal be hung 
up by white silk threads, and let a similar thread 
be attached to the lid of the vessel so that the vessel 
may be opened or closed without touching it. 

Let the pieces of glass and resin be similarly sus 
pended and electrified as before. 

Let the vessel be originally unelectrified, then if 
an electrified piece of glass is hung up within it by 
its thread without touching the vessel, and the lid 
closed, the outside of the vessel will be found to 
be vitreously electrified, and it may be shewn that 
the electrification outside of the vessel is exactly the 
same in whatever part of the interior space the glass 
is suspended. 

If the glass is now taken out of the vessel without touching it, 
the electrification of the glass will be the same as before it was 
put in, and that of the vessel will have disappeared. 

This electrification of the vessel, which depends on the glass 

* This, and several experiments which follow, are due to Faraday, On Static 
Electrical Inductive Action, 1 Phil. Mag., 1843, or Exp. Res., vol. ii. p. 279. 




32 ELECTROSTATIC PHENOMENA. [29. 

being within it, and which vanishes when the glass is removed, is 
called Electrification by induction. 

Similar effects would be produced if the glass were suspended 
near the vessel on the outside, but in that case we should find 
an electrification vitreous in one part of the outside of the vessel 
and resinous in another. When the glass is inside the vessel 
the whole of the outside is vitreously and the whole of the inside 
resinously electrified. 

Electrification ly Conduction. 

29.] EXPERIMENT III. Let the metal vessel be electrified by 
induction, as in the last experiment, let a second metallic body 
be suspended by white silk threads near it, and let a metal wire, 
similarly suspended, be brought so as to touch simultaneously the 
electrified vessel and the second body. 

The second body will now be found to be vitreously electrified, 
and the vitreous electrification of the vessel will have diminished. 

The electrical condition has been transferred from the vessel to 
the second body by means of the wire. The wire is called a con 
ductor of electricity, and the second body is said to be electrified 
by conduction. 

Conductors and Insulators. 

EXPERIMENT IV. If a glass rod, a stick of resin or gutta-percha, 
or a white silk thread, had been used instead of the metal wire, no 
transfer of electricity would have taken place. Hence these latter 
substances are called Non-conductors of electricity. Non-conduc 
tors are used in electrical experiments to support electrified bodies 
without carrying off their electricity. They are then called In 
sulators. 

The metals are good conductors ; air, glass, resins, gutta-percha, 
vulcanite, paraffin, &c. are good insulators; but, as we shall see 
afterwards, all substances resist the passage of electricity, and all 
substances allow it to pass, though in exceedingly different degrees. 
This subject will be considered when we come to treat of the 
Motion of electricity. For the present we shall consider only two 
classes of bodies, good conductors, and good insulators. 

In Experiment II an electrified body produced electrification in 
the metal vessel while separated from it by air, a non-conducting 
medium. Such a medium, considered as transmitting these electrical 
effects without conduction, has been called by Faraday a Dielectric 



31.] SUMMATION OF ELECTRIC EFFECTS. 33 

medium, and the action which takes place through it is called 
Induction. 

In Experiment III the electrified vessel produced electrification 
in the second metallic body through the medium of the wire. Let 
us suppose the wire removed, and the electrified piece of glass taken 
out of the vessel without touching it, and removed to a sufficient 
distance. The second body will still exhibit vitreous electrifica 
tion, but the vessel, when the glass is removed, will have resinous 
electrification. If we now bring the wire into contact with both 
bodies, conduction will take place along the wire, and all electri 
fication will disappear from both bodies, shewing that the elec 
trification of the two bodies was equal and opposite. 

30.] EXPERIMENT V. In Experiment II it was shewn that if 
a piece of glass, electrified by rubbing it with resin, is hung up in 
an insulated metal vessel, the electrification observed outside does 
not depend on the position of the glass. If we now introduce the 
piece of resin with which the glass was rubbed into the same vessel, 
without touching it or the vessel, it will be found that there is 
no electrification outside the vessel. From this we conclude that 
the electrification of the resin is exactly equal and opposite to that 
of the glass. By putting in any number of bodies, electrified in 
any way, it may be shewn that the electrification of the outside of 
the vessel is that due to the algebraic sum of all the electrifica 
tions, those being reckoned negative which are resinous. We have 
thus a practical method of adding the electrical effects of several 
bodies without altering the electrification of each. 

31.] EXPERIMENT VI. Let a second insulated metallic vessel, J5, 
be provided, and let the electrified piece of glass be put into the 
first vessel A, and the electrified piece of resin into the second vessel 
B. Let the two vessels be then put in communication by the metal 
wire, as in Experiment III. All signs of electrification will dis 
appear. 

Next, let the wire be removed, and let the pieces of glass and of 
resin be taken out of the vessels without touching them. It will 
be found that A is electrified resinously and B vitreously. 

If now the glass and the vessel A be introduced together into a 
larger insulated vessel C, it will be found that there is no elec 
trification outside C. This shews that the electrification of A is 
exactly equal and opposite to that of the piece of glass, and that 
of B may be shewn in the same way to be equal and opposite to that 
of the piece of resin. 

VOL. I. D 



34 ELECTROSTATIC PHENOMENA. [32. 

We have thus obtained a method of charging a vessel with a 
quantity of electricity exactly equal and opposite to that of an 
electrified body without altering the electrification of the latter, 
and we may in this way charge any number of vessels with exactly 
equal quantities of electricity of either kind, which we may take 
for provisional units. 

32.] EXPERIMENT VII. Let the vessel B, charged with a quan 
tity of positive electricity, which we shall call, for the present, 
unity, be introduced into the larger insulated vessel C without 
touching it. It will produce a positive electrification on the out 
side of C. Now let B be made to touch the inside of C. No change 
of the external electrification will be observed. If B is now taken 
out of C without touching it, and removed to a sufficient distance, 
it will be found that B is completely discharged, and that C has 
become charged with a unit of positive electricity. 

We have thus a method of transferring the charge of B to C. 
Let B be now recharged with a unit of electricity, introduced 
into C already charged, made to touch the inside of C, and re 
moved. It will be found that B is again completely discharged, 
so that the charge of C is doubled. 

If this process is repeated, it will be found that however highly 
C is previously charged, and in whatever way B is charged, when 
B is first entirely enclosed in C, then made to touch C, and finally 
removed without touching C, the charge of B is completely trans 
ferred to C, and B is entirely free from electrification. 

This experiment indicates a method of charging a body with 
any number of units of electricity. We shall find, when we come 
to the mathematical theory of electricity, that the result of this 
experiment affords an accurate test of the truth of the theory. 

33.] Before we proceed to the investigation of the law of 
electrical force, let us enumerate the facts we have already esta 
blished. 

By placing any electrified system inside an insulated hollow con 
ducting vessel, and examining the resultant effect on the outside 
of the vessel, we ascertain the character of the total electrification 
of the system placed inside, without any communication of elec 
tricity between the different bodies of the system. 

The electrification of the outside of the vessel may be tested 
with great delicacy by putting it in communication with an elec 
troscope. 

We may suppose the electroscope to consist of a strip of gold 



34-] ELECTRICITY AS A QUANTITY. 35 

leaf hanging between two bodies charged, one positively, and the 
other negatively. If the gold leaf becomes electrified it will incline 
towards the body whose electrification is opposite to its own. By 
increasing the electrification of the two bodies and the delicacy of 
the suspension, an exceedingly small electrification of the gold leaf 
may be detected. 

When we come to describe electrometers and multipliers we 
shall find that there are still more delicate methods of detecting 
electrification and of testing the accuracy of our theorems, but at 
present we shall suppose the testing to be made by connecting the 
hollow vessel with a gold leaf electroscope. 

This method was used by Faraday in his very admirable de 
monstration of the laws of electrical phenomena *. 

34.] I. The total electrification of a body, or system of bodies, 
remains always the same, except in so far as it receives electrifi 
cation from or gives electrification to other bodies. 

In all electrical experiments the electrification of bodies is found 
to change, but it is always found that this change is due to want 
of perfect insulation, and that as the means of insulation are im 
proved, the loss of electrification becomes less. We may therefore 
assert that the electrification of a body placed in a perfectly in 
sulating medium would remain perfectly constant. 

II. When one body electrifies another by conduction, the total 
electrification of the two bodies remains the same, that is, the one 
loses as much positive or gains as much negative electrification as 
the other gains of positive or loses of negative electrification. 

For if the two bodies are enclosed in the hollow vessel, no change 
of the total electrification is observed. 

III. When electrification is produced by friction, or by any 
other known method, equal quantities of positive and negative elec 
trification are produced. 

For the electrification of the whole system may be tested in 
the hollow vessel, or the process of electrification may be carried 
on within the vessel itself, and however intense the electrification of 
the parts of the system may be, the electrification of the whole, 
as indicated by the gold leaf electroscope, is invariably zero. 

The electrification of a body is therefore a physical quantity 
capable of measurement, and two or more electrifications can be 
combined experimentally with a result of the same kind as when 

* On Static Electrical Inductive Action, Phil. Mag., 1843, or Exp. Res., vol. ii. 
p. 249. 

D 2 



36 ELECTROSTATIC PHENOMENA. [35. 

two quantities are added algebraically. We therefore are entitled 
to use language fitted to deal with electrification as a quantity as 
well as a quality, and to speak of any electrified body as ( charged 
with a certain quantity of positive or negative electricity/ 

35.] While admitting electricity, as we have now done, to the 
rank of a physical quantity, we must not too hastily assume that 
it is, or is not, a substance, or that it is, or is not, a form of 
energy, or that it belongs to any known category of physical 
quantities. All that we have hitherto proved is that it cannot 
be created or annihilated, so that if the total quantity of elec 
tricity within a closed surface is increased or diminished, the in 
crease or diminution must have passed in or out through the closed 
surface. 

This is true of matter, and is expressed by the equation known as 
the Equation of Continuity in Hydrodynamics. 

It is not true of heat, for heat may be increased or diminished 
within a closed surface, without passing in or out through the 
surface, by the transformation of some other form of energy into 
heat, or of heat into some other form of energy. 

It is not true even of energy in general if we admit the imme 
diate action of bodies at a distance. For a body outside the closed 
surface may make an exchange of energy with a body within 
the surface. But if all apparent action at a distance is the 
result of the action between the parts of an intervening medium, 
and if the nature of this action of the parts of the medium is 
clearly understood, then it is conceivable that in all cases of the 
increase or diminution of the energy within a closed surface we 
may be able to trace the passage of the energy in or out through 
that surface. 

There is, however, another reason which warrants us in asserting 
that electricity, as a physical quantity, synonymous with the total 
electrification of a body, is not, like heat, a form of energy. An 
electrified system has a certain amount of energy, and this energy 
can be calculated by multiplying the quantity of electricity in 
each of its parts by another physical quantity, called the Potential 
of that part, and taking half the sum of the products. The quan 
tities Electricity and Potential, when multiplied together, 
produce the quantity Energy. It is impossible, therefore, that 
electricity and energy should be quantities of the same category, for 
electricity is only one of the factors of energy, the other factor 
being Potential. 



36.] THEORIES OF ELECTRICITY. 37 

Energy, which is the product of these factors, may also be con 
sidered as the product of several other pairs of factors, such as 

A Force x A distance through which the force is to act. 

A Mass X Gravitation acting through a certain height. 

A Mass x Half the square of its velocity. 

A Pressure X A volume of fluid introduced into a vessel at 

that pressure. 
A Chemical Affinity x A chemical change, measured by the number 

of electro-chemical equivalents which enter 

into combination. 

If we obtain distinct mechanical ideas of the nature of electric 
potential, we may combine these with the idea of energy to 
determine the physical category in which Electricity is to be 
placed. 

36.] In most theories on the subject, Electricity is treated as 
a substance, but inasmuch as there are two kinds of electrification 
which, being combined, annul each other, and since we cannot 
conceive of two substances annulling each other, a distinction has 
been drawn between Free Electricity and Combined Electricity. 

Theory of Two Fluids. 

In the theory called that of Two Fluids, all bodies, in their 
unelectrified state, are supposed to be charged with equal quan 
tities of positive and negative electricity. These quantities are 
supposed to be so great that no process of electrification has ever 
yet deprived a body of all the electricity of either kind. The pro 
cess of electrification, according to this theory, consists in taking 
a certain quantity P of positive electricity from the body A and 
communicating it to B, or in taking a quantity N of negative 
electricity from B and communicating it to A^ or in some com 
bination of these processes. 

The result will be that A will have P + N units of negative 
electricity over and above its remaining positive electricity, which 
is supposed to be in a state of combination with an equal quantity 
of negative electricity. This quantity P + N is called the Free 
electricity, the rest is called the Combined, Latent, or Fixed elec 
tricity. 

In most expositions of this theory the two electricities are called 
Fluids, because they are capable of being transferred from one 
body to another, and are, within conducting bodies, extremely 



38 ELECTROSTATIC PHENOMENA. [36. 

mobile. The other properties of fluids, such as their inertia, 
weight, and elasticity, are not attributed to them by those who 
have used the theory for merely mathematical purposes ; but the 
use of the word Fluid has been apt to mislead the vulgar, including 1 
many men of science who are not natural philosophers, and who 
have seized on the word Fluid as the only term in the statement 
of the theory which seemed intelligible to them. 

We shall see that the mathematical treatment of the subject has 
been greatly developed by writers who express themselves in terms 
of the Two Fluids theory. . Their results, however, have been 
deduced entirely from data which can be proved by experiment, 
and which must therefore be true, whether we adopt the theory of 
two fluids or not. The experimental verification of the mathe 
matical results therefore is no evidence for or against the peculiar 
doctrines of this theory. 

The introduction of two fluids permits us to consider the negative 
electrification of A and the positive electrification of B as the effect 
of any one of three different processes which would lead to the same 
result. We have already supposed it produced by the transfer of 
P units of positive electricity from A to B, together with the 
transfer of N units of negative electricity from B to A. But if 
P+N units of positive electricity had been transferred from A 
to B, or if P + N units of negative electricity had been transferred 
from B to A, the resulting free electricity on A and on B would 
have been the same as before, but the quantity of combined 
electricity in A would have been less in the second case and greater 
in the third than it was in the first. 

It would appear therefore, according to this theory, that it is 
possible to alter not only the amount of free electricity in a body, 
but the amount of combined electricity. But no phenomena have 
ever been observed in electrified bodies which can be traced to the 
varying amount of their combined electricities. Hence either the 
combined electricities have no observable properties, or the amount 
of the combined electricities is incapable of variation. The first 
-of these alternatives presents no difficulty to the mere mathema 
tician, who attributes no properties to the fluids except those of 
.attraction and repulsion, for in this point of view the two fluids 
simply annul one another, and their combination is a true mathe 
matical zero. But to those who cannot use the word Fluid without 
thinking of a substance it is difficult to conceive that the com 
bination of the two fluids shall have no properties at all, so that 



37-] THEORIES OF ONE AND OP TWO FLUIDS. 39 

the addition of more or less of the combination to a body shall not 
in any way affect it, either by increasing- its mass or its weight, or 
altering some of its other properties. Hence it has been supposed 
by some, that in every process of electrification exactly equal quan 
tities of the two fluids are transferred in opposite directions, so 
that the total quantity of the two fluids in any body taken to 
gether remains always the same. By this new law they contrive 
to save appearances, forgetting that there would have been no need 
of the law except to reconcile the two fluids theory with facts, 
and to prevent it from predicting non-existent phenomena. 

Theory of One Fluid. 

37.] In the theory of One Fluid everything is the same as in 
the theory of Two Fluids except that, instead of supposing the two 
substances equal and opposite in all respects, one of them, gene 
rally the negative one, has been endowed with the properties and 
name of Ordinary Matter, while the other retains the name of The 
Electric Fluid. The particles of the fluid are supposed to repel 
one another according to the law of the inverse square of the 
distance, and to attract those of matter according to the same 
law. Those of matter are supposed to repel each other and attract 
those of electricity. The attraction, however, between units of the 
different substances at unit of distance is supposed to be a very little 
greater than the repulsion between units of the same kind, so that 
a unit of matter combined with a unit of electricity will exert a 
force of attraction on a similar combination at a distance, this 
force, however, being exceedingly small compared with the force 
between two uncombined units. 

This residual force is supposed to account for the attraction of 
gravitation. Unelectrified bodies are supposed to be charged with 
as many units of electricity as they contain of ordinary matter. 
When they contain more electricity or less, they are said to be 
positively or negatively electrified. 

This theory does not, like the Two-Fluid theory, explain too 
much. It requires us, however, to suppose the mass of the electric 
fluid so small that no attainable positive or negative electrification 
has yet perceptibly increased or diminished either the mass or the 
weight of a body, and it has not yet been able to assign sufficient 
reasons why the vitreous rather than the resinous electrification 
should be supposed due to an excess of electricity. 

One objection has sometimes been urged against this theory by 



40 ELECTROSTATIC PHENOMENA. [38. 

men who ought to have reasoned better. It has been said that 
the doctrine that the particles of matter uncombined with elec 
tricity repel one another, is in direct antagonism with the well- 
established fact that every particle of matter attracts every other 
particle throughout the universe. If the theory of One Fluid were 
true we should have the heavenly bodies repelling one another. 

But it is manifest that the heavenly bodies, according to this 
theory, if they consisted of matter uncombined with electricity, 
would be in the highest state of negative electrification, and would 
repel each other. We have no reason to believe that they are in 
such a highly electrified state, or could be maintained in that 
state. The earth and all the bodies whose attraction has been 
observed are rather in an unelectrified state, that is, they contain 
the normal charge of electricity, and the only action between them 
is the residual force lately mentioned. The artificial manner, how 
ever, in which this residual force is introduced is a much more 
valid objection to the theory. 

In the present treatise I propose, at different stages of the in 
vestigation, to test the different theories in the light of additional 
classes of phenomena. For my own part, I look for additional 
light on the nature of electricity from a study of what takes place 
in the space intervening between the electrified bodies. Such is the 
essential character of the mode of investigation pursued by Faraday 
in his Experimental Researches, and as we go on I intend to exhibit 
the results, as developed by Faraday, W. Thomson, &c., in a con 
nected and mathematical form, so that we may perceive what 
phenomena are explained equally well by all the theories, and what 
phenomena indicate the peculiar difficulties of each theory. 

Measurement of the Force between Electrified Bodies. 
38.] Forces may be measured in various ways. For instance, 
one of the bodies may be suspended from one arm of a delicate 
balance, and weights suspended from the other arm, till the body, 
when unelectrified, is in equilibrium. The other body may then 
be placed at a known distance beneath the first, so that the 
attraction or repulsion of the bodies when electrified may increase 
or diminish the apparent weight of the first. The weight which 
must be added to or taken from the other arm, when expressed 
in dynamical measure, will measure the force between the bodies. 
This arrangement was used by Sir W. Snow Harris, and is that 
adopted in Sir W. Thomson s absolute electrometers. See Art. 217. 



39-] MEASUREMENT OF ELECTRIC FORCES. 41 

It is sometimes more convenient to use a torsion-balance in 
which a horizontal arm is suspended by a fine wire or fibre, so as 
to be capable of vibrating about the vertical wire as an axis, and 
the body is attached to one end of the arm and acted on by the 
force in the tangential direction, so as to turn the arm round the 
vertical axis, and so twist the suspension wire through a certain 
angle. The torsional rigidity of the wire is found by observing 
the time of oscillation of the arm, the moment of inertia of the 
arm being otherwise known, and from the angle of torsion and 
the torsional rigidity the force of attraction or repulsion can be 
deduced. The torsion-balance was devised by Michell for the de 
termination of the force of gravitation between small bodies, and 
was used by Cavendish for this purpose. Coulomb, working in 
dependently of these philosophers, reinvented it, and successfully 
applied it to discover the laws of electric and magnetic forces; 
and the torsion-balance has ever since been used in all researches 
where small forces have to be measured. See Art. 215. 

39.] Let us suppose that by either of these methods we can 
measure the force between two electrified bodies. We shall suppose 
the dimensions of the bodies small compared with the distance 
between them, so that the result may not be much altered by 
any inequality of distribution of the electrification on either body, 
and we shall suppose that both bodies are so suspended in air as 
to be at a considerable distance from other bodies on which they 
might induce electrification. 

It is then found that if the bodies are placed at a fixed distance 
and charged respectively with e and e of our provisional units of 
electricity, they will repel each other with a force proportional 
to the product of e and /. If either e or e is negative, that is, 
if one of the charges is vitreous and the other resinous, the force 
will be attractive, but if both e and e are negative the force is again 
repulsive. 

We may suppose the first body, A, charged with m units of 
vitreous and n units of resinous electricity, which may be con 
ceived separately placed within the body, as in Experiment V. 

Let the second body, B, be charged with m units of positive 
and n units of negative electricity. 

Then each of the m positive units in A will repel each of the m 
positive units in B with a certain force, say f, making a total effect 
equal to mm f. 

Since the effect of negative electricity is exactly equal and 



42 ELECTROSTATIC PHENOMENA. [40. 

opposite to that of positive electricity, each of the m positive units 
in A will attract each of the n negative units in B with the same 
force/*, making a total effect equal to mn f. 

Similarly the n negative units in A will attract the m f positive 
units in B with a force nm f, and will repel the n negative units 
in B with a force nn f. 

The total repulsion will therefore be (mm + nn*)f\ and the total 
attraction will he (mn + m n}f. 

The resultant repulsion will be 

(mm -f nn mn nm )f or (m n) (m f n )f. 

Now m n = e is the algebraical value of the charge on A, and 
m n =. e f is that of the charge on B, so that the resultant re 
pulsion may be written eeff, the quantities e and e being always 
understood to be taken with their proper signs. 

Variation of the Force with the Distance. 

40.] Having established the law of force at a fixed distance, 
we may measure the force between bodies charged in a constant 
manner and placed at different distances. It is found by direct 
measurement that the force, whether of attraction or repulsion, 
varies inversely as the square of the distance, so that if f is the 
repulsion between two units at unit distance, the repulsion at dis 
tance r will be/ 1 /*" 2 , and the general expression for the repulsion 
between e units and e units at distance r will be 



Definition of the Electrostatic Unit of Electricity . 

41.] We have hitherto used a wholly arbitrary standard for our 
unit of electricity, namely, the electrification of a certain piece of 
glass as it happened to be electrified at the commencement of our 
experiments. We are now able to select a unit on a definite prin 
ciple, and in order that this unit may belong to a general system 
we define it so that/ 1 may be unity, or in other words 

The electrostatic unit of electricity is that quantity of electricity 
which, when placed at unit of distance from an equal quantity -, repels 
it with unit of force. 

This unit is called the Electrostatic- unit to distinguish it from 
the Electromagnetic unit, to be afterwards defined. 

We may now write the general law of electrical action in the 
simple form F = ee r~* -, or, 



43-] LAW OF ELECTRIC FORCE. 43 

The repulsion between two small bodies charged respectively with, e and 
e units of electricity is numerically equal to the product of the charges 
divided by the square of the distance. 

Dimensions of the Electrostatic Unit of Quantity. 
42.] If [Q] is the concrete electrostatic unit of quantity itself, 
and e> e the numerical values of particular quantities ; if [Z] is 
the unit of length, and r the numerical value of the distance ; and 
if \F~\ is the unit of force, and F the numerical value of the force, 
then the equation becomes . 



whence [Q] = 



This unit is called the Electrostatic Unit of electricity. Other 
units may be employed for practical purposes, and in other depart 
ments of electrical science, but in the equations of electrostatics 
quantities of electricity are understood to be estimated in electro 
static units, just as in physical astronomy we employ a unit of 
mass which is founded on the phenomena of gravitation, and which 
differs from the units of mass in common use. 

Proof of the I/aw of Electrical Force. 

43.] The experiments of Coulomb with the torsion-balance may 
be considered to have established the law of force with a certain 
approximation to accuracy. Experiments of this kind, however, 
are rendered difficult, and in some degree uncertain, by several 
disturbing causes, which must be carefully traced and corrected for. 

In the first place, the two electrified bodies must be of sensible 
dimensions relative to the distance between them, in order to be 
capable of carrying charges sufficient to produce measurable forces. 
The action of each body will then produce an effect on the dis 
tribution of electricity on the other, so that the charge cannot be 
considered as evenly distributed over the surface, or collected at 
the centre of gravity ; but its effect must be calculated by an 
intricate investigation. This, however, has been done as regards 
two spheres by Poisson in an extremely able manner, and the 
investigation has been greatly simplified by Sir W. Thomson in 
his Theory of Electrical Images. See Arts. 172-174. 

Another difficulty arises from the action of the electricity 
induced on the sides of the case containing the instrument. By 



4:4: ELECTROSTATIC PHENOMENA. [44. 

making the inside of the instrument accurately cylindric, and 
making- its inner surface of metal, this effect can be rendered 
definite and measurable. 

An independent difficulty arises from the imperfect insulation 
of the bodies, on account of which the charge continually de 
creases. Coulomb investigated the law of dissipation, and made 
corrections for it in his experiments. 

The methods of insulating charged conductors, and of measuring 
electrical effects, have been greatly improved since the time of 
Coulomb, particularly by Sir W. Thomson; but the perfect ac 
curacy of Coulomb s law of force is established, not by any direct 
experiments and measurements (which may be used as illustrations 
of the law), but by a mathematical consideration of the pheno 
menon described as Experiment VII, namely, that an electrified 
conductor B, if made to touch the inside of a hollow closed con 
ductor C and then withdrawn without touching C, is perfectly dis 
charged, in whatever manner the outside of C may be electrified. 
By means of delicate electroscopes it is easy to shew that no 
electricity remains on B after the operation, and by the mathe 
matical theory given at Art. 74, this can only be the case if the 
force varies inversely as the square of the distance, for if the law 
had been of any different form B would have been electrified. 

The Electric Field. 

44.] The Electric Field is the portion of space in the neigh 
bourhood of electrified bodies, considered with reference to electric 
phenomena. It may be occupied by air or other bodies, or it 
may be a so-called vacuum, from which we have withdrawn every 
substance which we can act upon with the means at our dis 
posal. 

If an electrified body be placed at any part of the electric field 
it will be acted on by a force which will depend, in general, on 
the shape of the body and on its charge, if the body is so highly 
charged as to produce a sensible disturbance in the previous elec 
trification of the other bodies. 

But if the body is very small and its charge also very small, 
the electrification of the other bodies will not be sensibly disturbed, 
and we may consider the body as indicating by its centre of gravity 
a certain point of the field. The force acting on the body will 
then be proportional to its charge, and will be reversed when the 
charge is reversed. 



46.] ELECTRIC POTENTIAL. 45 

Let e be the charge of the body, and F the force acting on the 
body in a certain direction, then when e is very small F is propor 
tional to e, or F = Re 

where R is a quantity depending on the other bodies in the field. 
If the charge e could be made equal to unity without disturbing 
the electrification of other bodies we should have F = R. 

We shall call R the Resultant electric force at the given point 
of the field. 

Electric Potential. 

45.] If the small body carrying the small charge e be moved 
from the given point to an indefinite distance from the electrified 
bodies, it will experience at each point of its course a force Re, 
where R varies from point to point of the course. Let the whole 
work done on the body by these electrical forces be Ve, then V is 
the potential at the point of the field from which the body started. 
If the charge e could be made equal to unity without disturbing 
the electrification of other bodies, we might define the potential at 
any point as the work done on a body charged with unit of elec 
tricity in moving from that point to an infinite distance. 

A body electrified positively tends to move from places of greater 
positive potential to places of smaller positive, or of negative 
potential, and a body negatively electrified tends to move in the 
opposite direction. 

In a conductor the electrification is distributed exactly as if 
it were free to move in the conductor according to the same law. 
If therefore two parts of a conductor have different potentials, 
positive electricity will move from the part having greater potential 
to the part having less potential as long as that difference con 
tinues. A conductor therefore cannot be in electrical equilibrium 
unless every point in it has the same potential. This potential is 
called the Potential of the Conductor. 

Equipotential Surfaces. 

46.] If a surface described or supposed to be described in the 
electric field is such that the electric potential is the same at every 
point of the surface it is called an Equipotential surface. 

An electrified point constrained to rest upon such a surface will 
have no tendency to move from one part of the surface to another, 
because the potential is the same at every point. An equipotential 
surface is therefore a surface of equilibrium or a level surface. 



43 ELECTROSTATIC PHENOMENA. [46. 

The resultant force at any point of the surface is in the direction 
of the normal to the surface, and the magnitude of the force is such 
that the work done on an electrical unit in passing from the surface 
rtothesmfecePis T-T , 

No two equipotential surfaces having different potentials can 
one another, because the same point cannot have more than 
potential, but one equipotential surface may meet itself, and 
this takes place at all points and lines of equilibrium. 

Gftte surface of a conductor in electrical eonfibrium is necessarily 
an equipotential nmfki,. If the electrification of the conductor is 



the whole surface, then the potrntinl will diminish as 
away from the surface on every side, and the conductor 
will be surrounded by a series of surfaces of lower potential. 

But if (& to ike action of external electrified bodies) some 
legions of Ike conductor are electrified positively and others ne 
gatively, the complete equipotential surface will consist of the 
of the conductor itself together with a system of other 
meeting the surface of the conductor in the lines which 
divide the positive from the negative regions. These lines will 
be lines of equilibrium, so that an electrified point placed on one 
of these lines will experience no force in any direction. 

Whem the surface of a conductor is electrified positively in some 
parts and negatively in others, there most be some other electrified 
body in the field besides itself. For if we allow a positively 
electrified point, starting from a positively electrified part of the 
surface, to vote always in the direction of the resultant force upon 
it, the potential at the point will continually diminish till the point 
reaches either a negatively electrified surface at a potential less than 
that of the first conductor, or uaeiei off to an infinite distance. 
Since the fiiiBBiJal at an infinite distance is zero, the latter case 
can only occur when the potential of the conductor is positive. 

In the same way a negatively electrified point, moving off from 
a negatively electrified pert of the surface, must either reach a posi 
tively electrified surface, or pass off to infinity, and the latter case 
can only happen when the potential of the conductor is negative. 

Therefore, if both positive and negative electrification exists on 
a conductor, there must be some other body in the field whose 
potential has the same sign as that of the conductor but a greater 
numerical value, and if a conductor of any form is alone in the 
field the electrification of every part is of the same sign as the 
potential of the conductor. 



49-] ELECTBIC TENSION. 47 

Line* of Force. 

47.] The line described by a point moving- always in the direc 
tion of the resultant force is called a Line of force. It cuts the 
equipotential surfaces at right angles. The properties of lines of 
force will be more fully explained afterwards, because Faraday has 
expressed many of the laws of electrical action in terms of his 
conception of lines of force drawn in the electric field, and indicating 
both the direction and the magnitude of the force at every point. 

Efectrlc Tension. 

48.] Since the surface of a conductor is an equipotential surface, 
the resultant force is normal to the surface, and it will be shewn 
in Art. 78 that it is proportional to the superficial density of the 
electrification. Hence the electricity on any small area of the 
surface will be acted on by a force tending from the conductor 
and proportional to the product of the resultant force and the 
density, that is, proportional to the square of the resultant force 

This force which acts outwards as a tension on every part of 
the conductor will be called electric Tension. It is measured like 
ordinary mechanical tension, by the force exerted on unit of area. 

The word Tension has been used by electricians in several vague 
senses, and it has been attempted to adopt it in mathematical 
language as a synonym for Potential ; but on examining the cases 
in which the word has been used, I think it will be more con 
sistent with usage and with mechanical analogy to understand 
by tension a pulling force of so many pounds per square inch 
exerted on the surface of a conductor or elsewhere. We shall find 
that the conception of Faraday, that this electric tension exists not 
only at the electrified surface but all along the lines of force, leads 
to a theory of electric action as a phenomenon of stress in a 
medium. 

Bbctrowotitt Force. 

49.] When two conductors at different potentials are connected 
by a thin conducting wire, the tendency of electricity to flow 
along the wire is measured by the difference of the potentials of 
the two bodies. The difference of potentials between two con 
ductors or two points is therefore called the Electromotive force 
between them. 

Electromotive force mav arise from other causes than difference 



48 ELECTROSTATIC PHENOMENA. [50. 

of potential, but these causes are not considered in treating of sta 
tical electricity. We shall consider them when we come to chemical 
actions, motions of magnets, inequalities of temperature, &c. 

Capacity of a Conductor. 

50.] If one conductor is insulated while all the surrounding con 
ductors are kept at the zero potential by being put in commu 
nication with the earth, and if the conductor, when charged with 
a quantity E of electricity, has a potential F, the ratio of E to V 
is called the Capacity of the conductor. If the conductor is com 
pletely enclosed within a conducting vessel without touching it, 
then the charge on the inner conductor will be equal and op 
posite to the charge on the inner surface of the outer conductor, 
and will be equal to the capacity of the inner conductor multiplied 
by the difference of the potentials of the two conductors. 

Electric Accumulators. 

A system consisting of two conductors whose opposed surfaces 
are separated from each other by a thin stratum of an insulating 
medium is called an electric Accumulator. Its capacity is directly 
proportional to the area of the opposed surfaces and inversely pro 
portional to the thickness of the stratum between them. A Leyden 
jar is an accumulator in which glass is the insulating medium. 
Accumulators are sometimes called Condensers, but I prefer to 
restrict the term condenser to an instrument which is used not to 
hold electricity but to increase its superficial density. 

PROPERTIES OF BODIES IN RELATION TO STATICAL ELECTRICITY. 

Resistance to the Passage of Electricity through a Body. 

51.] When a charge of electricity is communicated to any part 
of a mass of metal the electricity is rapidly transferred from places 
of high to places of low potential till the potential of the whole 
mass becomes the same. In the case of pieces of metal used in 
ordinary experiments this process is completed in a time too short 
to be observed, but in the case of very long and thin wires, such 
as those used in telegraphs, the potential does not become uniform 
till after a sensible time, on account of the resistance of the wire 
to the passage of electricity through it. 

The resistance to the passage of electricity is exceedingly dif 
ferent in different substances, as may be seen from the tables at 



51.] ELECTRIC RESISTANCE. 49 

Arts. 362, 366, and 369, which will be explained in treating of 
Electric Currents. 

All the metals are good conductors, though the resistance of 
lead is 12 times that of copper or silver, that of iron 6 times, 
and that of mercury 60 times that of copper. The resistance of all 
metals increases as their temperature rises. 

Selenium in its crystalline state may also he regarded as a con 
ductor, though its resistance is 3.7 x 10 12 times that of a piece 
of copper of the same dimensions. Its resistance increases as the 
temperature rises. Selenium in the amorphous form is a good 
insulator, like sulphur. 

Many liquids conduct electricity by electrolysis. This mode of 
conduction will be considered in Part II. For the present, we may 
regard all liquids containing water and all damp bodies as con 
ductors, far inferior to the metals, but incapable of insulating a 
charge of electricity for a sufficient time to be observed. 

On the other hand, the gases at the atmospheric pressure, whether 
dry or moist, are insulators so nearly perfect when the electric tension 
is small that we have as yet obtained no evidence of electricity passing 
through them by ordinary conduction. The gradual loss of charge 
by electrified bodies may in every case be traced to imperfect insu 
lation in the supports, the electricity either passing through the 
substance of the support or creeping over its surface. Hence, when 
two charged bodies are hung up near each other, they will preserve 
their charges longer if they are electrified in opposite ways, than if 
they are electrified in the same way. For though the electromotive 
force tending to make the electricity pass through the air between 
them is much greater when they are oppositely electrified, no per 
ceptible loss occurs in this way. The actual loss takes place through 
the supports, and the electromotive force through the supports is 
greatest when the bodies are electrified in the same way. The result 
appears anomalous only when we expect the loss to occur by the 
passage of electricity through the air between the bodies. 

Certain kinds of glass when cold are marvelously perfect in 
sulators, and Sir W. Thomson has preserved charges of electricity 
for years in bulbs hermetically sealed. The same glass, however, 
becomes a conductor at a temperature below that of boiling water. 

Gutta-percha, caoutchouc, vulcanite, paraffin, and resins are good 
insulators, the resistance of gutta-percha at 75F. being about 
6 x 10 19 times that of copper. 

Ice, crystals, and solidified electrolytes, are also insulators. 

VOL. I. E 



50 ELECTROSTATIC PHENOMENA. [52. 

Certain liquids, such as naphtha, turpentine, and some oils, are 
insulators, but inferior to most of the solid insulators. 

The resistance of most substances, except the metals, and selenium 
and carbon, seems to diminish as the temperature rises. 

DIELECTRICS. 

Specific Inductive Capacity. 

52.] All bodies whose insulating power is such that when they 
are placed between two conductors at different potentials the elec 
tromotive force acting on them does not immediately distribute 
their electricity so as to reduce the potential to a constant value, are 
called by Faraday Dielectrics. 

Faraday discovered that the capacity of an accumulator depends 
on the nature of the insulating medium between the two conductors, 
as well as on the dimensions and relative position of the conductors 
themselves. By substituting other insulating media for air as the 
dielectric of the accumulator, without altering it in any other 
respect, he found that when air and other gases were employed as 
the insulating medium the capacity of the accumulator remained the 
same, but that when shell-lac, sulphur, glass, &c., were substituted 
for air, the capacity was increased in a ratio which was different 
for each substance. 

The ratio of the capacity of an accumulator formed of any di 
electric medium to the capacity of an accumulator of the same form 
and dimensions filled with air, was named by Faraday the Specific 
Inductive Capacity of the dielectric medium. It is equal to unity 
for air and other gases at all pressures, and probably at all tempe 
ratures, and it is greater than unity for all other liquid or solid 
dielectrics which have been examined. 

If the dielectric is not a good insulator, it is difficult to mea 
sure its inductive capacity, because the accumulator will not hold a 
charge for a sufficient time to allow it to be measured ; but it is 
certain that inductive capacity is a property not confined to good 
insulators, and it is probable that it exists in all bodies. 

Absorption of Electricity. 

53.] It is found that when an accumulator is formed of certain 
dielectrics, the following phenomena occur. 

When the accumulator has been for some time electrified and is 
then suddenly discharged and again insulated, it becomes recharged 



54-] ELECTRIC ABSORPTION. 51 

in the same sense as at first, but to a smaller degree, so that it may 
be discharged again several times in succession, these discharges 
always diminishing. This phenomenon is called that of the Re 
sidual Discharge. 

The instantaneous discharge appears always to be proportional 
to the difference of potentials at the instant of discharge, and the 
ratio of these quantities is the true capacity of the accumulator; 
but if the contact of the discharger is prolonged so as to include 
some of the residual discharge, the apparent capacity of the accu 
mulator, calculated from such a discharge, will be too great. 

The accumulator if charged and left insulated appears to lose its 
charge by conduction, but it is found that the proportionate rate 
of loss is much greater at first than it is afterwards, so that the 
measure of conductivity, if deduced from what takes place at first, 
would be too great. Thus, when the insulation of a submarine 
cable is tested, the insulation appears to improve as the electrifi 
cation continues. 

Thermal phenomena of a kind at first sight analogous take place 
in the case of the conduction of heat when the opposite sides of a 
body are kept at different temperatures. In the case of heat we 
know that they depend on the heat taken in and given out by the 
body itself. Hence, in the case of the electrical phenomena, it 
has been supposed that electricity is absorbed and emitted by the 
parts of the body. We shall see, however, in Art. 329, that the 
phenomena can be explained without the hypothesis of absorption of 
electricity, by supposing the dielectric in some degree heterogeneous. 

That the phenomenon called Electric Absorption is not an 
actual absorption of electricity by the substance may be shewn by 
charging the substance in any manner with electricity while it is 
surrounded by a closed metallic insulated vessel. If, when the 
substance is charged and insulated, the vessel be instantaneously 
discharged and then left insulated, no charge is ever communicated 
to the vessel by the gradual dissipation of the electrification of the 
charged substance within it. 

54.] This fact is expressed by the statement of Faraday that 
it is impossible to charge matter with an absolute and independent 
charge of one kind of electricity *. 

In fact it appears from the result of every experiment which 
has been tried that in whatever way electrical actions may take 

* Exp. Res., vol. i. series xi. f ii. On the Absolute Charge of Matter, and (1244). 

E 2 



52 ELECTROSTATIC PHENOMENA. [55. 

place among a system of bodies surrounded by a metallic vessel, the 
charge on the outside of that vessel is not altered. 

Now if any portion of electricity could be forced into a body 
so as to be absorbed in it, or to become latent, or in any way 
to exist in it, without being connected with an -equal portion of 
the opposite electricity by lines of induction, or if, after having 
being absorbed, it could gradually emerge and return to its or 
dinary mode of action, we should find some change of electrifica 
tion in the surrounding vessel. 

As this is never found to be the case, Faraday concluded that 
it is impossible to communicate an absolute charge to matter, and 
that no portion of matter can by any change of state evolve or 
render latent one kind of electricity or the other. He therefore 
regarded induction as the essential function both in the first 
development and the consequent phenomena of electricity/ His 
* induction is (1298) a polarized state of the particles of the 
dielectric, each particle being positive on one side and negative 
on the other, the positive and the negative electrification of each 
particle being always exactly equal. 

Disruptive Discharge *. 

55.] If the electromotive force acting at any point of a dielectric 
is gradually increased, a limit is at length reached at which there 
is a sudden electrical discharge through the dielectric, generally 
accompanied with light and sound, and with a temporary or per 
manent rupture of the dielectric. 

The intensity of the electromotive force when this takes place 
depends on the nature of the dielectric. It is greater, for instance, 
in dense air than in rare air, and greater in glass than in air, but 
in every case, if the electromotive force be made great enough, 
the dielectric gives way and its insulating power is destroyed, so 
that a current of electricity takes place through it. It is for this 
reason that distributions of electricity for which the electric resultant 
force becomes anywhere infinite cannot exist in nature. 

The Electric Glow. 

Thus, when a conductor having a sharp point is electrified, 
the theory, based on the hypothesis that it retains its charge, 
leads to the conclusion that as we approach the point the super 
ficial density of the electricity increases without limit, so that at 
the point itself the surface-density, and therefore the resultant 

* See Faraday, Exp. Rts., vol. i., series xii. and xiii. 



55-] ELECTRIC GLOW. 53 

electrical force, would be infinite. If the air, or other surrounding 
dielectric, had an invincible insulating power, this result would 
actually occur ; but the fact is, that as soon as the resultant force 
in the neighbourhood of the point has reached a certain limit, the 
insulating power of the air gives way, so that the air close to 
the point becomes a conductor. At a certain distance from the 
point the resultant force is not sufficient to break through the 
insulation of the air, so that the electric current is checked, and 
the electricity accumulates in the air round the point. 

The point is thus surrounded by particles of air charged with 
electricity of the same kind with its own. The effect of this charged 
air round the point is to relieve the air at the point itself from 
part of the enormous electromotive force which it would have ex 
perienced if the conductor alone had been electrified. In fact the 
surface of the electrified body is no longer pointed, because the 
point is enveloped by a rounded mass of electrified air, the surface 
of which, rather than that of the solid conductor, may be regarded 
as the outer electrified surface. 

If this portion of electrified air could be kept still, the elec 
trified body would retain its charge, if not on itself at least in its 
neighbourhood, but the charged particles of air being free to move 
under the action of electrical force, tend to move away from the elec 
trified body because it is charged with the same kind of electricity. 
The charged particles of air therefore tend to move off in the direc 
tion of the lines of force and to approach those surrounding bodies 
which are oppositely electrified. When they are gone, other un 
charged particles take their place round the point, and since these 
cannot shield those next the point itself from the excessive elec 
tric tension, a new discharge takes place, after which the newly 
charged particles move off, and so on as long as the body remains 
electrified. 

In this way the following phenomena are produced : At and 
close to the point there is a steady glow, arising from the con 
stant discharges which are taking place between the point and the 
air very near it. 

The charged particles of air tend to move off in the same general 
direction, and thus produce a current of air from the point, con 
sisting of the charged particles, and probably of others carried along 
by them. By artificially aiding this current we may increase the 
glow, and by checking the formation of the current we may pre 
vent the continuance of the glow. 



54 ELECTROSTATIC PHENOMENA. [56. 

The electric wind in the neighbourhood of the point is sometimes 
very rapid, but it soon loses its velocity, and the air with its charged 
particles is carried about with the general motions of the atmo 
sphere, and constitutes an invisible electric cloud. When the charged 
particles come near to any conducting surface, such as a wall, they 
induce on that surface an electrification opposite to their own, and 
are then attracted towards the wall, but since the electromotive 
force is small they may remain for a long time near the wall 
without being drawn up to the surface and discharged. They 
thus form an electrified atmosphere clinging to conductors, the pre 
sence of which may sometimes be detected by the electrometer. 
The electrical forces, however, acting between charged portions 
of air and other bodies are exceedingly feeble compared with the 
forces which produce winds arising from inequalities of density 
due to differences of temperature, so that it is very improbable 
that any observable part of the motion of ordinary thunder clouds 
arises from electrical causes. 

The passage of electricity from one place to another by the 
motion of charged particles is called Electrical Convection or Con- 
vective Discharge. 

The electrical glow is therefore produced by the constant passage 
of electricity through a small portion of air in which the tension 
is very high, so as to charge the surrounding particles of air which 
are continually swept off by the electric wind, which is an essential 
part of the phenomenon. 

The glow is more easily formed in rare air than in dense air, 
and more easily when the point is positive than when it is negative. 
This and many other differences between positive and negative elec 
trification must be studied by those who desire to discover some 
thing about the nature of electricity. They have not, however, 
been satisfactorily brought to bear upon any existing theory. 

The Electric Brush. 

56.] The electric brush is a phenomenon which may be pro 
duced by electrifying a blunt point or small ball so as to produce 
an electric field in which the tension diminishes, but in a less, rapid 
manner, as we leave the surface. It consists of a succession of 
discharges, ramifying as they diverge from the ball into the air, 
and terminating either by charging portions of air or by reaching 
some other conductor. It is accompanied by a sound, the pitch of 
which depends on the interval between the successive discharges, 
and there is no current of air as in the case of the glow. 



57-] ELECTRIC SPARK. 55 

The Electric Spark. 

57.] When the tension in the space between two conductors is 
considerable all the way between them, as in the case of two balls 
whose distance is not great compared with their radii, the discharge, 
when it occurs, usually takes the form of a spark, by which nearly 
the whole electrification is discharged at once. 

In this case, when any part of the dielectric has given way, 
the parts on either side of it in the direction of the electric force 
are put into a state of greater tension so that they also give way, 
and so the discharge proceeds right through the dielectric, just as 
when a little rent is made in the edge of a piece of paper a tension 
applied to the paper in the direction of the edge causes the paper to 
be torn through, beginning at the rent, but diverging occasionally 
where there are weak places in the paper. The electric spark in 
the same way begins at the point where the electric tension first 
overcomes the insulation of the dielectric, and proceeds from that 
point, in an apparently irregular path, so as to take in other weak 
points, such as particles of dust floating in air. 

0)i the Electric Force required to produce a Spark in Air. 

In the experiments of Sir W. Thomson * the electromotive force 
required to produce a spark across strata of air of various thick 
nesses was measured by means of an electrometer. 

The sparks were made to pass between two surfaces, one of which 
was plane, and the other only sufficiently convex to make the sparks 
occur always at the same place. 

The difference of potential required to cause a spark to pass was 
found to increase with the distance, but in a less rapid ratio, so that 
the electric force at any point between the surfaces, which is the 
quotient of the difference of potential divided by the distance, can 
be raised to a greater value without a discharge when the stratum 
of air is thin. 

When the stratum of air is very thin, say .00254 of a centimetre, 
the resultant force required to produce a spark was found to be 
527.7, in terms of centimetres and grammes. This corresponds to 
an electric tension of 1 1.29 grammes weight per square centimetre. 

When the distance between the surfaces is about a millimetre 
the electric force is about 130, and the electric tension .68 grammes 
weight per square centimetre. It is probable that the value for 

* Proc. K. S., I860 ; or, Reprint, chap. xix. 



56 ELECTROSTATIC PHENOMENA* [58. 

greater distances is not much less than this. The ordinary pressure 
of the atmosphere is about 1032 grammes per square centimetre. 

It is difficult to explain why a thin stratum of air should require 
a greater force to produce a disruptive discharge across it than a 
thicker stratum. Is it possible that the air very near to the sur 
face of dense bodies is condensed, so as to become a better insu 
lator ? or does the potential of an electrified conductor differ from 
that of the air in contact with it by a quantity having a maximum 
value just before discharge, so that the observed difference of 
potential of the conductors is in every case greater than the dif 
ference of potentials on the two sides of the stratum of air by a 
constant quantity equivalent to the addition of about .005 of an 
inch to the thickness of the stratum ? See Art. 370. 

All these phenomena differ considerably in different gases, and in 
the same gas at different densities. Some of the forms of electrical 
discharge through rare gases are exceedingly remarkable. In some 
cases there is a regular alternation of luminous and dark strata, so 
that if the electricity, for example, is passing along a tube contain 
ing a very small quantity of gas, a number of luminous disks will 
be seen arranged transversely at nearly equal intervals along the 
axis of the tube and separated by dark strata. If the strength of 
the current be increased a new disk will start into existence, and 
it and the old disks will arrange themselves in closer order. In 
a tube described by Mr. Gassiot"* the light of each of the disks 
is bluish on the negative and reddish on the positive side, and 
bright red in the central stratum. 

These, and many other phenomena of electrical discharge, are 
exceedingly important, and when they are better understood they 
will probably throw great light on the nature of electricity as well 
as on the nature of gases and of the medium pervading space. At 
present, however, they must be considered as outside the domain of 
the mathematical theory of electricity. 

Electric Phenomena of Tourmaline. 

58.] Certain crystals of tourmaline, and of other minerals, possess 
what may be called Electric Polarity. Suppose a crystal of tour 
maline to be at a uniform temperature, and apparently free from 
electrification on its surface. Let its temperature be now raised, 
the crystal remaining insulated. One end will be found positively 

* Intellectual Observer, March, 1866. 



59-] ELECTRIFICATION OF TOURMALINE. 57 

and the other end negatively electrified. Let the surface be de 
prived of this apparent electrification by means of a flame or other 
wise, then if the crystal be made still hotter, electrification of the 
same kind as before will appear, but if the crystal be cooled the 
end which was positive when the crystal was heated will become 
negative. 

These electrifications are observed at the extremities of the crys- 
tallographic axis. Some crystals are terminated by a six-sided 
pyramid at one end and by a three- sided pyramid at the other. 
In these the end having the six-sided pyramid becomes positive 
when the crystal is heated. 

Sir W. Thomson supposes every portion of these and other hemi- 
hedral crystals to have a definite electric polarity, the intensity 
of which depends on the temperature. When the surface is passed 
through a flame, every part of the surface becomes electrified to 
such an extent as to exactly neutralize, for all external points, 
the effect of the internal polarity. The crystal then has no ex 
ternal electrical action, nor any tendency to change its mode of 
electrification. But if it be heated or cooled the interior polariza 
tion of each particle of the crystal is altered, and can no longer 
be balanced by the superficial electrification, so that there is a 
resultant external action. 

Plan of this Treatise. 

59.] In the following treatise I propose first to explain the ordinary 
theory of electrical action, which considers it as depending only 
on the electrified bodies and on their relative position, without 
taking account of any phenomena which may take place in the 
surrounding media. In this way we shall establish the law of the 
inverse square, the theory of the potential, and the equations of 
Laplace and Poisson. We shall next consider the charges and 
potentials of a system of electrified conductors as connected by 
a system of equations, the coefficients of which may be supposed 
to be determined by experiment in those cases in which our present 
mathematical methods are not applicable, and from these we shall 
determine the mechanical forces acting between the different elec 
trified bodies. 

We shall then investigate certain general theorems by which 
Green, Gauss, and Thomson have indicated the conditions of so 
lution of problems in the distribution of electricity. One result 
of these theorems is, that if Poisson s equation is satisfied by any 



58 ELECTROSTATIC PHENOMENA. [59. 

function, and if at the surface of every conductor the function 
has the value of the potential of that conductor, then the func 
tion expresses the actual potential of the system at every point. We 
also deduce a method of finding problems capable of exact solution. 

In Thomson s theorem, the total energy of the system is ex 
pressed in the form of the integral of a certain quantity extended 
over the whole space between the electrified bodies, and also in 
the form of an integral extended over the electrified surfaces only. 
The equation between these two expressions may be thus inter 
preted physically. We may conceive the relation into which the 
electrified bodies are thrown, either as the result of the state of 
the intervening medium, or as the result of a direct action between 
the electrified bodies at a distance. If we adopt the latter con 
ception, we may determine the law of the action, but we can go 
no further in speculating on its cause. If, on the other hand, 
we adopt the conception of action through a medium, we are led to 
enquire into the nature of that action in each part of the medium. 

It appears from the theorem, that if we are to look for the seat 
of the electric energy in the different parts of the dielectric me 
dium, the amount of energy in any small part must depend on 
the square of the intensity of the resultant electromotive force at 
that place multiplied by a coefficient called the specific inductive 
capacity of the medium. 

It is better, however, in considering the theory of dielectrics 
in the most general point of view, to distinguish between the elec 
tromotive force at any point and the electric polarization of the 
medium at that point, since these directed quantities, though re 
lated to one another, are not, in some solid substances, in the same 
direction. The most general expression for the electric energy of 
the medium per unit of volume is half the product of the electro 
motive force and the electric polarization multiplied by the cosine 
of the angle between their directions. 

In all fluid dielectrics the electromotive force and the electric 
polarization are in the same direction and in a constant ratio. 

If we calculate on this hypothesis the total energy residing 
in the medium, we shall find it equal to the energy due to the 
electrification of the conductors on the hypothesis of direct action 
at a distance. Hence the two hypotheses are mathematically equi 
valent. 

If we now proceed to investigate the mechanical state of the 
medium on the hypothesis that the mechanical action observed 



6O.] STRESS IN DIELECTRICS. 59 

between electrified bodies is exerted through and by means of 
the medium, as in the familiar instances of the action of one body 
on another by means of the tension of a rope or the pressure of 
a rod, we find that the medium must be in a state of mechanical 
stress. 

The nature of this stress is, as Faraday pointed out *, a tension 
along the lines of force combined with an equal pressure in all 
directions at right angles to these lines. The magnitude of these 
stresses is proportional to the energy of the electrification, or, in 
other words, to the square of the resultant electromotive force mul 
tiplied by the specific inductive capacity of the medium. 

This distribution of stress is the only one consistent with the 
observed mechanical action on the electrified bodies, and also with 
the observed equilibrium of the fluid dielectric which surrounds 
them. I have therefore thought it a warrantable step in scientific 
procedure to assume the actual existence of this state of stress, and 
to follow the assumption into its consequences. Finding the phrase 
electric tension used in several vague senses, I have attempted to 
confine it to what I conceive to have been in the mind of some 
of those who have used it, namely, the state of stress in the 
dielectric medium which causes motion of the electrified bodies, 
and leads, when continually augmented, to disruptive discharge. 
Electric tension, in this sense, is a tension of exactly the same 
kind, and measured in the same way, as the tension of a rope, 
and the dielectric medium, which can support a certain tension 
and no more, may be said to have a certain strength in exactly 
the same sense as the rope is said to have a certain strength. 
Thus, for example, Thomson has found that air at the ordinary 
pressure and temperature can support an electric tension of 9600 
grains weight per square foot before a spark passes. 

60.] From the hypothesis that electric action is not a direct 
action between bodies at a distance, but is exerted by means of 
the medium between the bodies, we have deduced that this medium 
must be in a state of stress. We have also ascertained the cha 
racter of the stress, and compared it with the stresses which may 
occur in solid bodies. Along the lines of force there is tension, 
and perpendicular to them there is pressure, the numerical mag 
nitude of these forces being equal, and each proportional to the 
square of the resultant force at the point. Having established 
these results, we are prepared to take another step, and to form 

* Exp. Bes., series xi. 1297. 



60 ELECTROSTATIC PHENOMENA. [60. 

an idea of the nature of the electric polarization of the dielectric 
medium. 

An elementary portion of a body may be said to be polarized 
when it acquires equal and opposite properties on two opposite 
sides. The idea of internal polarity may be studied to the greatest 
advantage as exemplified in permanent magnets, and it will be 
explained at greater length when we come to treat of magnetism. 

The electric polarization of an elementary portion of a dielectric 
is a forced state into which the medium is thrown by the action 
of electromotive force, and which disappears when that force is 
removed. We may conceive it to consist in what we may call 
an electrical displacement, produced by the electromotive force. 
When the electromotive force acts on a conducting medium it 
produces a current through it, but if the medium is a non-con 
ductor or dielectric, the current cannot flow through the medium, 
but the electricity is displaced within the medium in the direction 
of the electromotive force, the extent of this displacement de 
pending on the magnitude of the electromotive force, so that if 
the electromotive force increases or diminishes the electric displace 
ment increases and diminishes in the same ratio. 

The amount of the displacement is measured by the quantity 
of electricity which crosses unit of area, while the displacement 
increases from zero to its actual amount. This, therefore, is the 
measure of the electric polarization. 

The analogy between the action of electromotive force in pro 
ducing electric displacement and of ordinary mechanical force in 
producing the displacement of an elastic body is so obvious that 
I have ventured to call the ratio of the electromotive force to the 
corresponding electric displacement the coefficient of electric elasticity 
of the medium. This coefficient is different in different media, and 
varies inversely as the specific inductive capacity of each medium. 

The variations of electric displacement evidently constitute electric 
currents. These currents, however, can only exist during the 
variation of the displacement, and therefore, since the displace 
ment cannot exceed a certain value without causing disruptive 
discharge, they cannot be continued indefinitely in the same direc 
tion, like the currents through conductors. 

In tourmaline, and other pyro-electric crystals, it is probable that 
a state of electric polarization exists, which depends upon tem 
perature, and does not require an external electromotive force to 
produce it If the interior of a body were in a state of permanent 



60.] ELECTRIC DISPLACEMENT. 61 

electric polarization, the outside would gradually become charged 
in such a manner as to neutralize the action of the internal elec 
trification for all points outside the body. This external superficial 
charge could not be detected by any of the ordinary tests, and 
could not be removed by any of the ordinary methods for dis 
charging superficial electrification. The internal polarization of 
the substance would therefore never be discovered unless by some 
means, such as change of temperature, the amount of the internal 
polarization could be increased or diminished. The external elec 
trification would then be no longer capable of neutralizing the 
external effect of the internal polarization, and an apparent elec 
trification would be observed, as in the case of tourmaline. 

If a charge e is uniformly distributed over the surface of a 
sphere, the resultant force at any point of the medium surrounding 
the sphere is numerically equal to the charge e divided by the square 
of the distance from the centre of the sphere. This resultant force, 
according to our theory, is accompanied by a displacement of elec 
tricity in a direction outwards from the sphere. 

If we now draw a concentric spherical surface of radius r , the whole 
displacement, E, through this surface will be proportional to the 
resultant force multiplied by the area of the spherical surface. But 
the resultant force is directly as the charge e and inversely as the 
square of the radius, while the area of the surface is directly as the 
square of the radius. 

Hence the whole displacement, E, is proportional to the charge e, 
and is independent of the radius. 

To determine the ratio between the charge , and the quantity 
of electricity, E, displaced outwards through the spherical surface, 
let us consider the work done upon the medium in the region 
between two concentric spherical surfaces, while the displacement 
is increased from E to E+bE. If V^ and Y 2 denote the potentials 
at the inner and the outer of these surfaces respectively, the elec 
tromotive force by which the additional displacement is produced 
is V l F 2 , so that the work spent in augmenting the displacement 



If we now make the inner surface coincide with that of the 
electrified sphere, and make the radius of the other infinite, V^ 
becomes F, the potential of the sphere, and V.> becomes zero, so 
that the whole work done in the surrounding medium is TE. 

But by the ordinary theory, the work done in augmenting the 
charge is Fbe, and if this is spent, as we suppose, in augmenting 



62 ELECTEOSTATIC PHENOMENA. [6 1. 

the displacement, bU = be, and since E and e vanish together, 
Ee, or 

The displacement outwards through any spherical surface concentric 
with the sphere is equal to the charge on the sphere. 

To fix our ideas of electric displacement, let us consider an accu 
mulator formed of two conducting plates A and j5, separated by a 
stratum of a dielectric C. Let W be a conducting wire joining 
A and JB, and let us suppose that by the action of an electromotive 
force a quantity Q of positive electricity is transferred along the 
wire from B to A. The positive electrification of A and the 
negative electrification of will produce a certain electromotive 
force acting from A towards in the dielectric stratum, and this 
will produce an electric displacement from A towards JB within the 
dielectric. The amount of this displacement, as measured by the 
quantity of electricity forced across an imaginary section of the 
dielectric dividing it into two strata, will be, according to our 
theory, exactly Q. See Arts. 75, 76, 111. 

It appears, therefore, that at the same time that a quantity 
Q of electricity is being transferred along the wire by the electro 
motive force from B towards A, so as to cross every section of 
the wire, the same quantity of electricity crosses every section 
of the dielectric from A towards B by reason of the electric dis 
placement. 

The reverse motions of electricity will take place during the 
discharge of the accumulator. In the wire the discharge will be 
Q from A to B, and in the dielectric the displacement will subside, 
and a quantity of electricity Q will cross every section from B 
towards A. 

Every case of electrification or discharge may therefore be con 
sidered as a motion in a closed circuit, such that at every section 
of the circuit the same quantity of electricity crosses in the same 
time, and this is me case, not only in the voltaic circuit where 
it has always been recognised, but in those cases in which elec 
tricity has been generally supposed to be accumulated in certain 
places. 

61.] We are thus led to a very remarkable consequence of the 
theory which we are examining, namely, that the motions of elec 
tricity are like those of an incompressible fluid, so that the total 
quantity within an imaginary fixed closed surface remains always 
the same. This result appears at first sight in direct contradiction 
to the fact that we can charge a conductor and then introduce 



62.] THEORY PROPOSED. 63 

it into the closed space, and so alter the quantity of electricity 
within that space. But we must remember that the ordinary theory 
takes no account of the electric displacement in the substance of 
dielectrics which we have been investigating, but confines its 
attention to the electrification at the bounding surfaces of the 
conductors and dielectrics. In the case of the charged conductor 

O 

let us suppose the charge to be positive, then if the surrounding 
dielectric extends on all sides beyond the closed surface there will be 
electric polarization, accompanied with displacement from within 
outwards all over the closed surface, and the surface-integral of 
the displacement taken over the surface will be equal to the charge 
on the conductor within. 

Thus when the charged conductor is introduced into the closed 
space there is immediately a displacement of a quantity of elec 
tricity equal to the charge through the surface from within out 
wards, and the whole quantity within the surface remains the 
same. 

The theory of electric polarization will be discussed at greater 
length in Chapter V, and a mechanical illustration of it will be 
given in Art. 334, but its importance cannot be fully understood 
till we arrive at the study of electromagnetic phenomena. 

62.] The peculiar features of the theory as we have now de 
veloped them are : 

That the energy of electrification resides in the dielectric medium, 
whether that medium be solid, liquid, or gaseous, dense or rare, 
or even deprived of ordinary gross matter, provided it be still 
capable of transmitting electrical action. 

That the energy in any part of the medium is stored up in 
the form of a state of constraint called electric polarization, the 
amount of which depends on the resultant electromotive force at 
the place. ^ 

That electromotive force acting on a diele^iicr produces what 
we have called electric displacement, the relation between the force 
and the displacement being in the most general case of a kind 
to be afterwards investigated in treating of conduction, but in 
the most important cases the force is in the same direction as 
the displacement, and is numerically equal to the displacement 
multiplied by a quantity which we have called the coefficient of 
electric elasticity of the dielectric. 

That the energy per unit of volume of the dielectric arising from 
the electric polarization is half the product of the electromotive 



64 ELECTROSTATIC PHENOMENA. [62. 

force and the electric displacement multiplied, if necessary, by the 
cosine of the angle between their directions. 

That in fluid dielectrics the electric polarization is accompanied 
by a tension in the direction of the lines of force combined with 
an equal pressure in all directions at right angles to the lines 
of force, the amount of the tension or pressure per unit of area 
being numerically equal to the energy per unit of volume at the 
same place. 

That the surfaces of any elementary portion into which we may 
conceive the volume of the dielectric divided must be conceived 
to be electrified, so that the surface- density at any point of the 
surface is equal in magnitude to the displacement through that 
point of the surface reckoned inwards, so that if the displacement 
is in the positive direction, the surface of the element will be elec 
trified negatively on the positive side and positively on the negative 
side. These superficial electrifications will in general destroy one 
another when consecutive elements are considered, except where 
the dielectric has an internal charge, or at the surface of the 
dielectric. 

That whatever electricity may be, and whatever we may under 
stand by the movement of electricity, the phenomenon which we 
have called electric displacement is a movement of electricity in the 
same sense as the transference of a definite quantity of electricity 
through a wire is a movement of electricity, the only difference 
being that in the dielectric there is a force which we have called 
electric elasticity which acts against the electric displacement, and 
forces the electricity back when the electromotive force is removed ; 
whereas in the conducting wire the electric elasticity is continually 
giving way, so that a current of true conduction is set up, and 
the resistance depends, not on the total quantity of electricity dis 
placed from its position of equilibrium, but on the quantity which 
crosses a section of the conductor in a given time. 

That in every case the motion of electricity is subject to the 
same condition as that of an incompressible fluid, namely, that 
at every instant as much must flow out of any given closed space 
as flows into it. 

It follows from this that every electric current must form a 
closed circuit. The importance of this result will be seen when we 
investigate the laws of electro-magnetism. 

Since, as we have seen, the theory of direct action at a distance 
is mathematically identical with that of action by means of a 



62.] METHOD OF THIS WORK. 65 

medium, the actual phenomena may be explained by the one 
theory as well as by the other, provided suitable hypotheses be 
introduced when any difficulty occurs. Thus, Mossotti has deduced 
the mathematical theory of dielectrics from the ordinary theoiy 
of attraction by merely giving an electric instead of a magnetic 
interpretation to the symbols in the investigation by which Poisson 
has deduced the theory of magnetic induction from the theory of 
magnetic fluids. He assumes the existence within the dielectric of 
small conducting elements, capable of having their opposite surfaces 
oppositely electrified by induction,, but not capable of losing or 
gaining electricity on the whole, owing to their being insulated 
from each other by a non-conducting medium. This theory of 
dielectrics is consistent with the laws of electricity, and may be 
actually true. If it is true, the specific inductive capacity of a 
dielectric may be greater, but cannot be less, than that of air or 
vacuum. No instance has yet been found of a dielectric having 
an inductive capacity less than that of air, but if such should 
be discovered, Mossotti s theory must be abandoned, although his 
formulae would all remain exact, and would only require us to alter 
the sign of a coefficient. 

In the theory which I propose to develope, the mathematical 
methods are founded upon the smallest possible amount of hypo 
thesis, and thus equations of the same form are found applicable to 
phenomena which are certainly of quite different natures, as, for 
instance, electric induction through dielectrics ; conduction through 
conductors, and magnetic induction. In all these cases the re 
lation between the force and the effect produced is expressed by 
a set of equations of the same kind, so that when a problem in 
one of these subjects is solved, the problem and its solution may 
be translated into the language of the other subjects and the 
results in their new form will also be true. 



VOL. I. 



CHAPTER II. 

% 

ELEMENTARY MATHEMATICAL THEORY OF STATICAL 
ELECTRICITY. 

Definition of Electricity as a Mathematical Quantity. 

63.] We have seen that the actions of electrified bodies are such 
that the electrification of one body may be equal to that of another, 
or to the sum of the electrifications of two bodies, and that when 
two bodies are equally and oppositely electrified they have no elec 
trical effect on external bodies when placed together within a closed 
insulated conducting vessel. We may express all these results in 
a concise and consistent manner by describing an electrified body as 
charged with a certain quantity of electricity, which we may denote 
by e. When the electrification is positive, that is, according to the 
usual convention, vitreous, e will be a positive quantity. When the 
electrification is negative or resinous, e will be negative, and the 
quantity e may be interpreted either as a negative quantity of 
vitreous electricity or as a positive quantity of resinous electricity. 

The effect of adding together two equal and opposite charges of 
electricity, +e and e, is to produce a state of no electrification 
expressed by zero. We may therefore regard a body not electrified 
as virtually charged with equal and opposite charges of indefinite 
magnitude, and an electrified body as virtually charged with un 
equal quantities of positive and negative electricity, the algebraic 
sum of these charges constituting the observed electrification. It is 
manifest, however, that this way of regarding an electrified body 
is entirely artificial, and may be compared to the conception of the 
velocity of a body as compounded of two or more different velo 
cities, no one of which is the actual velocity of the body. When 
we speak therefore of a body being charged with a quantity e of 
electricity we mean simply that the body is electrified, and that 
the electrification is vitreous or resinous according as e is positive 
or negative. 



64.] ELECTRIC DENSITY. 67 

ON ELECTRIC DENSITY. 

Distribution in Three Dimensions. 

64.] Definition. The electric volume-density at a given point 
in space is the limiting ratio of the quantity of electricity within 
a sphere whose centre is the given point to the volume of the 
sphere, when its radius is diminished without limit. 

We shall denote this ratio by the symbol p, which may be posi 
tive or negative. 

Distribution on a Surface. 

It is a result alike of theory and of experiment, that, in certain 
cases, the electrification of a body is entirely on the surface. The 
density at a point on the surface, if defined according to the method 
given above, would be infinite. We therefore adopt a different 
method for the measurement of surface-density. 

Definition. The electric density at a given point on a surface is 
the limiting ratio of the quantity of electricity within a sphere 
whose centre is the given point to the area of the surface contained 
within the sphere, when its radius is diminished without limit. 

We shall denote the surface-density by the symbol <r. 

Those writers who supposed electricity to be a material fluid 
or a collection of particles, were obliged in this case to suppose 
the electricity distributed on the surface in the form of a stratum 
of a certain thickness 0, its density being p , or that value of p 
which would result from the particles having the closest contact 
of which they are capable. It is manifest that on this theory 

p 6 = a: 

When cr is negative, according to this theory, a certain stratum 
of thickness is left entirely devoid of positive electricity, and 
filled entirely with negative electricity. 

There is, however, no experimental evidence either of the elec 
tric stratum having any thickness, or of electricity being a fluid 
or a collection of particles. We therefore prefer to do without the 
symbol for the thickness of the stratum, and to use a special symbol 
for surface-density. 

Distribution, along a Line. 

It is sometimes convenient to suppose electricity distributed 
on a line, that is, a long narrow body of which we neglect the 



68 ELECTROSTATICS. [65. 

thickness. In this case we may define the line-density at any point 
to be the limiting 1 ratio of the electricity on an element of the 
line to the length of that element when the element is diminished 
without limit. 

If A denotes the line-density, then the whole quantity of elec 
tricity on a curve is e = I A da, where ds is the element of the curve. 

Similarly, if o- is the surface-density, the whole quantity of elec 
tricity on the surface is 



e 



where dS is the element of surface. 

If p is the volume-density at any point of space, then the whole 
electricity within a certain volume is 

e = I I I p dx dy dz, 

where dx dy dz is the element of volume. The limits of integration 
in each case are those of the curve, the surface, or the portion of 
space considered. 

It is manifest that e, A, o- and p are quantities differing in kind, 
each being one dimension in space lower than the preceding, so that 
if a be a line, the quantities e, a\, a 2 a; and a^p will be all of the 
same kind, and if a be the unit of length, and A, o-, p each the 
unit of the different kinds of density, a\, a 2 a; and a 3 p will each 
denote one unit of electricity. 

Definition of the Unit of Electricity. 

65.] Let A and B be two points the distance between which 
is the unit of length. Let two bodies, whose dimensions are small 
compared with the distance AB, be charged with equal quantities 
of positive electricity and placed at A and B respectively, and 
let the charges be such that the force with which they repel each 
other is the unit of force, measured as in Art. 6. Then the charge 
of either body is said to be the unit of electricity. If the charge of 
the body at B were a unit of negative electricity, then, since the 
action between the bodies would be reversed, we should have an 
attraction equal to the unit of force. 

If the charge of A were also negative, and equal to unity, the 
force would be repulsive, and equal to unity. 

Since the action between any two portions of electricity is not 



68.] LAW OF ELECTRIC FORCE. 69 

affected by the presence of other portions, the repulsion between 
e units of electricity at A and / units at B is <?/, the distance 
AE being unity. See Art. 39. 

Law of Force between Electrified Bodies. 

66.] Coulomb shewed by experiment that the force between 
electrified bodies whose dimensions are small compared with the 
distance between them, varies inversely as the square of the dis 
tance. Hence the actual repulsion between two such bodies charged 
with quantities e and e f and placed at a distance r is 

ee f 
f* 

We shall prove in Art. 74 that this law is the only one con 
sistent with the observed fact that a conductor, placed in the inside 
of a closed hollow conductor and in contact with it, is deprived of 
all electrical charge. Our conviction of the accuracy of the law 
of the inverse square of the distance may be considered to rest 
on experiments of this kind, rather than on the direct measure 
ments of Coulomb. 

Resultant Force between Two Bodies. 

67.] In order to find the resultant force between two bodies 
we might divide each of them into its elements of volume, and 
consider the repulsion between the electricity in each of the elements 
of the first body and the electricity in each of the elements of the 
second body. TVe should thus get a system of forces equal in 
number to the product of the numbers of the elements into which 
we have divided each body, and we should have to combine the 
effects of these forces by the rules of Statics. Thus, to find the 
component in the direction of x we should have to find the value 
of the sextuple integral 



rrrrrr P P (*-* ) dxd y a* d^d/M 

JJJJJJ {(x - x Y + (y-,,J + (z-zJ}? 

where #, y, z are the coordinates of a point in the first body at 
which the electrical density is p, and x , y , /, and p are the 
corresponding quantities for the second body, and the integration 
is extended first over the one body and then over the other. 

Resultant Force at a Point, 

68.] In order to simplify the mathematical process, it is con 
venient to consider the action of an electrified body, not on another 



70 ELECTROSTATICS. [68. 

body of any form, but on an indefinitely small body, charged with 
an indefinitely small amount of electricity, and placed at any point 
of the space to which the electrical action extends. By making 
the charge of this body indefinitely small we render insensible its 
disturbing action on the charge of the first body. 

Let e be the charge of this body, and let the force acting on 
it when placed at the point (#, y, z) be Re, and let the direction- 
cosines of the force be I, m, n, then we may call R the resultant 
force at the point (#, y, z). 

In speaking of the resultant electrical force at a point, we do not 
necessarily imply that any force is actually exerted there, but only 
that if an electrified body were placed there it would be acted on 
by a force Re, where e is the charge of the body. 

Definition. The Resultant electrical force at any point is the 
force which would be exerted on a small body charged with the unit 
of positive electricity, if it were placed there without disturbing the 
actual distribution of electricity. 

This force not only tends to move an electrified body, but to 
move the electricity within the body, so that the positive electricity 
tends to move in the direction of R and the negative electricity 
in the opposite direction. Hence the force R is also called the 
Electromotive Force at the point (x, y, z). 

When we wish to express the fact that the resultant force is a 
vector, we shall denote it by the German letter @. If the body 
is a dielectric, then, according to the theory adopted in this 
treatise, the electricity is displaced within it, so that the quantity 
of electricity which is forced in the direction of ( across unit 
of area fixed perpendicular to @ is 

S) = -K<$ 9 

47T 

where S) is the displacement, ( the resultant force, and K the 
specific inductive capacity of the dielectric. For air, K = 1. 

If the body is a conductor, the state of constraint is continually 
giving way, so that a current of conduction is produced and main 
tained as long as the force @ acts on the medium. 

Components of the Resultant Force. 
If X, Y, Z denote the components of R, then 

X=Rl, Y=Rm, Z=Rn-, 
where /, m, n are the direction -cosines of R. 



69.] ELECTROMOTIVE FORCE. 71 

Line-Integral of Electric Force, or Electromotive Force along 
an Arc of a Curve. 

69.] The Electromotive force along a given arc AP of a curve is 
numerically measured by the work which would be done on a unit 
of positive electricity carried along the curve from the beginning, 
A, to P, the end of the arc. 

If s is the length of the arc, measured from A, and if the re 
sultant force R at any point of the curve makes an angle c with 
the tangent drawn in the positive direction, then the work done 
on unit of electricity in moving along the element of the curve 
ds will be 



and the total electromotive force V will be 
F= jfJiiooBc*, 

"0 

the integration being extended from the beginning to the end 
of the arc. 

If we make use of the components of the force R } we find 



. 
o ds ds ds 

If X, Y, and Z are such that Xda+Ydy + Zdz is a complete 
differential of a function of x, y, z, then 



where the integration is performed in any way from the point A 
to the point P, whether along the given curve or along any other 
line between A and P. 

In this case V is a scalar function of the position of a point in 
space, that is, when we know the coordinates of the point, the value 
of V is determinate, and this value is independent of the position 
and direction of the axes of reference. See Art. 16. 

On Functions of the Position of a Point. 

In what follows, when we describe a quantity as a function of 
the position of a point, we mean that for every position of the point 
the function has a determinate value. We do not imply that this 
value can always be expressed by the same formula for all points of 
space, for it may be expressed by one formula on one side of a 
given surface and by another formula on the other side. 



72 ELECTROSTATICS. [70. 

On Potential Functions. 

70.] The quantity Xdx+Ydy+Zdz is an exact differential 
whenever the force arises from attractions or repulsions whose in 
tensity is a function of the distance only from any number of 
points. For if i\ be the distance of one of the points from the point 
(#, y, z\ and if R^ be the repulsion, then 



with similar expressions for Y 1 and Z 19 so that 

X l dx -\-Y 1 d^ + Z dz = R l dr^ ; 

and since R l is a function of r l only, R l dr^ is an exact differential 
of some function of r lt say V-^ 

Similarly for any other force R 2 , acting from a centre at dis 
tance r 2 , 

X 2 dx + Y<idy + Z^ dz = R z dr z = dV^ . 

But X = X 1 -f X 2 + &c. and Y and Z are compounded in the same 
way, therefore 

Xdx+Ydy + Zdz = d7i + dYt + &G. = dV. 

V, the integral of this quantity, under the condition that V =. 
at an infinite distance, is called the Potential Function. 

The use of this function in the theory of attractions was intro 
duced by Laplace in the calculation of the attraction of the earth. 
Green, in his essay On the Application of Mathematical Analysis 
to Electricity/ gave it the name of the Potential Function. Gauss, 
working independently of Green, also used the word Potential. 
Clausius and others have applied the term Potential to the work 
which would be done if two bodies or systems were removed to 
an infinite distance from one another. We shall follow the use of 
the word in recent English works, and avoid ambiguity by adopting 
the following definition due to Sir W. Thomson. 

Definition of Potential. The Potential at a Point is the work 
which would be done on a unit of positive electricity by the elec 
tric forces if it were placed at that point without disturbing the 
electric distribution, and carried from that point to an infinite 
distance. 

71.] Expressions for the Resultant Force and its components in 
terms of the Potential. 

Since the total electromotive force along any arc AB is 



72.] POTENTIAL. 73 

if we put ds for the arc AB we shall have for the force resolved 
in the direction of ds, 

7? dV 

R cos e = -- 7-; 
as 

whence, by assuming ds parallel to each of the axes in succession, 
we get dV dV dV 

A = -- -j- i I = -- -= j Z = -- J 

ax ay dz 



dy dz j 
We shall denote the force itself, whose magnitude is R and whose 
components are X, Y, Z, by the German letter <, as in Arts. 17 
and 68. 

The Potential at all Points within a Conductor is the same. 

72.] A conductor is a body which allows the electricity within 
it to move from one part of the body to any other when acted on 
by electromotive force. When the electricity is in equilibrium 
there can be no electromotive force acting within the conductor. 
Hence R = throughout the whole space occupied by the con 
ductor. From this it follows that 

dV _ dV _ dV _ 

~fa = ^ Tz~ 
and therefore for every point of the conductor 

r= c, 

where C is a constant quantity. 

Potential of a Conductor. 

Since the potential at all points within the substance of the 
conductor is C, the quantity C is called the Potential of the con 
ductor. C may be defined as the work which must be done by 
external agency in order to bring a unit of electricity from an 
infinite distance to the conductor, the distribution of electricity 
being supposed not to be disturbed by the presence of the unit. 

If two conductors have equal potentials, and are connected by 
a wire so fine that the electricity on the wire itself may be neg 
lected, the total electromotive force along the wire will be zero, 
and no electricity will pass from the one conductor to the other. 

If the potentials of the conductors A and B be VA and V^ then 
the electromotive force along any wire joining A and B will be 

r A -r B 



74: ELECTROSTATICS. [73. 

in the direction AB, that is, positive electricity will tend to pass 
from the conductor of higher potential to the other. 

Potential, in electrical science, has the same relation to Elec 
tricity that Pressure, in Hydrostatics, has to Fluid, or that Tem 
perature, in Thermodynamics, has to Heat. Electricity, Fluids, 
and Heat all tend to pass from one place to another, if the Poten 
tial, Pressure, or Temperature is greater in the first place than in 
the second. A fluid is certainly a substance, heat is as certainly 
not a substance, so that though we may find assistance from ana 
logies of this kind in forming clear ideas of formal electrical rela 
tions, we must be careful not to let the one or the other analogy 
suggest to us that electricity is either a substance like water, or 
a state of agitation like heat. 

Potential due to any Electrical System. 

73.] Let there be a single electrified point charged with a quantity 
e of electricity, and let / be the distance of the point # ,/, / from it, 

then 7 = r Edr = f4^ = - 

J r J r ? 2 r 

Let there be any number of electrified points whose coordinates 
are (^y^z^, (# 2 > ^2> ^2) & c - an d tne i r charges e lt e 29 &c., and 
let their distances from the point (# ,./, /) be r l9 r 2 , &c., then the 
potential of the system at x, y\ / will be 



Let the electric density at any point (#, y, z) within an elec 
trified body be p, then the potential due to the body is 



where r= {(x-x }* + (y-y )* +(z-z ) 2 }*, 

the integration being extended throughout the body. 

On the Proof of the Law of the Inverse Square. 
74.] The fact that the force between electrified bodies is inversely 
as the square of the distance may be considered to be established 
by direct experiments with the torsion-balance. The results, how 
ever, which we derive from such experiments must be regarded 
as affected by an error depending on the probable error of each 
experiment, and unless the skill of the operator be very great, 



74.] PROOF OF THE LAW OF FORCE. 75 

the probable error of an experiment with the torsion-balance is 
considerable. As an argument that the attraction is really, and 
not merely as a rough approximation, inversely as the square of the 
distance, Experiment VII (p. 34) is far more conclusive than any 
measurements of electrical forces can be. 

In that experiment a conductor B, charged in any manner, was 
enclosed in a hollow conducting vessel C, which completely sur 
rounded it. C was also electrified in any manner. 

B was then placed in electric communication with C, and was then 
again insulated and removed from C without touching it, and ex 
amined by means of an electroscope. In this way it was shewn 
that a conductor, if made to touch the inside of a conducting vessel 
which completely encloses it, becomes completely discharged, so 
that no trace of electrification can be discovered by the most 
delicate electrometer, however strongly the conductor or the vessel 
has been previously electrified. 

The methods of detecting the electrification of a body are so 
delicate that a millionth part of the original electrification of B 
could be observed if it existed. No experiments involving the direct 
measurement of forces can be brought to such a degree of accuracy. 

It follows from this experiment that a non-electrified body in the 
inside of a hollow conductor is at the same potential as the hollow 
conductor, in whatever way that conductor is charged. For if it 
were not at the same potential, then, if it were put in electric 
connexion with the vessel, either by touching it or by means of 
a wire, electricity would pass from the one body to the other, and 
the conductor, when removed from the vessel, would be found to be 
electrified positively or negatively, which, as we have already stated, 
is not the case. 

Hence the whole space inside a hollow conductor is at the same 
potential as the conductor if no electrified body is placed within it. 
If the law of the inverse square is true, this will be the case what 
ever be the form of the hollow conductor. Our object at present, 
however, is to ascertain from this fact the form of the law of 
attraction. 

For this purpose let us suppose the hollow conductor to be a thin 
spherical shell. Since everything is symmetrical about its centre, 
the shell will be uniformly electrified at every point, and we have 
to enquire what must be the law of attraction of a uniform spherical 
shell, so as to fulfil the condition that the potential at every point 
within it shall be the same. 



76 ELECTROSTATICS. [74. 

Let the force at a distance r from a point at which a quantity e 
of electricity is concentrated be R, where R is some function of r. 
All central forces which are functions of the distance admit of a 

potential, let us write i- for the potential function due to a unit 

of electricity at a distance r. 

Let the radius of the spherical shell be #, and let the surface- 
density be a. Let P be any point within the shell at a distance 
p from the centre. Take the radius through P as the axis of 
spherical coordinates, and let r be the distance from P to an element 
dS of the shell. Then the potential at P is 

r=mm-*8, 



I i 

L L r 



sn 

o 



Now r 2 = a 2 2 ajo cos -f jo 2 , 

r dr = ap sin c?0. 



Hence F= 2 TT <r - 



xy /*a+p 

- / /(r) dr ; 
pJa-p 



and F must be constant for all values of p less than a. 

Multiplying both sides by p and differentiating with respect to p, 



Differentiating again with respect to />, 

=f(a+p)-f(a-p). 
Since a and p are independent, 

f (r) = C, a constant. 
Hence f(r) = Cr+C , 

and the potential function is 



The force at distance r is got by differentiating this expression 
with respect to r, and changing the sign, so that 



/ 

or the force is inversely as the square of the distance, and this 
therefore is the only law of force which satisfies the condition that 
the potential within a uniform spherical shell is constant*. Now 

* See Pratt s Mechanical Philosophy, p. 144. 



76.] ELECTRIC INDUCTION. 77 

this condition is shewn to be fulfilled by the electric forces with 
the most perfect accuracy. Hence the law of electric force is 
verified to a corresponding degree of accuracy. 

Surface-Integral of Electric Induction^ and Electric Displacement 
through a Surface. 

75.] Let R be the resultant force at any point of the surface, 
and e the angle which R makes with the normal drawn towards the 
positive side of the surface, then R cos e is the component of the 
force normal to the surface, and if dS is the element of the surface, 
the electric displacement through dS will be, by Art. 68, 

KR cos e dS. 
4n 

Since we do not at present consider any dielectric except air, K= 1 . 
We may, however, avoid introducing at this stage the theory of 
electric displacement, by calling R cos dS the Induction through 
the element dS. This quantity is well known in mathematical 
physics, but the name of induction is borrowed from Faraday. 
The surface-integral of induction is 



JJ 



R cos e dS, 

and it appears by Art. 21, that if X, J", Z are the components of R, 
and if these quantities are continuous within a region bounded by a 
closed surface S, the induction reckoned from within outwards is 

dX dY 

Tx + ^ + 

the integration being extended through the whole space within the 
surface. 

Induction through a Finite Closed Surface due to a Single Centre 

of Force. 

76.] Let a quantity e of electricity be supposed to be placed at a 
point 0, and let r be the distance of any point P from 0, the force 

Q 

at that point is R = in the direction OP. 

Let a line be drawn from in any direction to an infinite 
distance. If is without the closed surface this line will either 
not cut the surface at all, or it will issue from the surface as many 
times as it enters. If is within the surface the line must first 



78 ELECTROSTATICS. [76. 

issue from the surface, and then it may enter and issue any number 
of times alternately, ending- by issuing from it. 

Let e be the angle between OP and the normal to the surface 
drawn outwards where OP cuts it, then where the line issues from 
the surface cos will be positive, and where it enters cos e will 
be negative. 

Now let a sphere be described with centre and radius unity, 
and let the line OP describe a conical surface of small angular 
aperture about as vertex. 

This cone will cut off a small element da from the surface of the 
sphere, and small elements dS l} dS 2 , &c. from the closed surface at 
the various places where the line OP intersects it. 

Then, since any one of these elements dS intersects the cone at a 
distance r from the vertex and at an obliquity , 

dS = r 2 sec c da> ; 
and, since R = er~ 2 , we shall have 

ficosclS = edto ; 

the positive sign being taken when r issues from the surface, and 
the negative where it enters it. 

If the point is without the closed surface, the positive values 
are equal in number to the negative ones, so that for any direction 
of *> 2R cos e dS = 0, 



and therefore / / R cos c dS = 0, 

the integration being extended over the whole closed surface. 

If the point is within the closed surface the radius vector OP 
first issues from the closed surface, giving a positive value of e da, 
and then has an equal number of entrances and issues, so that in 
this case SRcosedS = e d<*. 

Extending the integration over the whole closed surface, we shall 
include the whole of the spherical surface, the area of which is 47r, 
so that 

I I R cos e dS = e \ I da = 4 -n e. 

Hence we conclude that the total induction outwards through a 
closed surface due to a centre of force e placed at a point is 
zero when is without the surface, and 47te when is within 
the surface. 

Since in air the displacement is equal to the induction divided 



77-] EQUATIONS OF LAPLACE AND POISSON. 79 

by 4-7T, the displacement through a closed surface, reckoned out 
wards, is equal to the electricity within the surface. 

Corollary. It also follows that if the surface is not closed but 
is bounded by a given closed curve, the total induction through 
it is we, where o> is the solid angle subtended by the closed curve 
at 0. This quantity, therefore, depends only on the closed curve, 
and not on the form of the surface of which it is the boundary. 

On the Equations of Laplace and Poisson. 

77.] Since the value of the total induction of a single centre 
of force through a closed surface depends only on whether the 
centre is within the surface or not, and does not depend on its 
position in any other way, if there are a number of such centres 
e l , 2 , &c. within the surface, and /, <?/, &c. without the surface, 
we shall have 

1 I RcosedS = btie; 

where e denotes the algebraical sum of the quantities of elec 
tricity at all the centres of force within the closed surface, that is, 
the total electricity within the surface, resinous electricity being 
reckoned negative. 

If the electricity is so distributed within the surface that the 
density is nowhere infinite, we shall have by Art. 64, 

47T = 4-7T ]l I pdxdydz, 
and by Art. 75, 

t* AY dZ. 



If we take as the closed surface that of the element of volume 
dx dy dz y we shall have, by equating these expressions, 
dX dY dZ 



and if a potential V exists, we find by Art. 7 1 , 



This equation, in the case in which the density is zero, is called 
Laplace s Equation. In its more general form it was first given by 
Poisson. It enables us, when we know the potential at every point, 
to determine the distribution of electricity. 



80 ELECTROSTATICS. [78. 

We shall denote, as at Art. 26, the quantity 



and we may express Poisson s equation in words by saying that 
the electric density multiplied by 4?r is the concentration of the 
potential. Where there is no electrification, the potential . has no 
concentration, and this is the interpretation of Laplace s equation. 

If we suppose that in the superficial and linear distributions of 
electricity the volume-density p remains finite, and that the elec 
tricity exists in the form of a thin stratum or narrow fibre, then, 
by increasing p and diminishing the depth of the stratum or the 
section of the fibre, we may approach the limit of true superficial 
or linear distribution, and the equation being true throughout the 
process will remain true at the limit, if interpreted in accordance 
with the actual circumstances. 

On the Conditions to be fulfilled at an Electrified Surface. 

78.] We shall consider the electrified surface as the limit to 
which an electrified stratum of density p and thickness v approaches 
when p is increased and v diminished without limit, the product pv 
being always finite and equal to a- the surface-density. 

Let the stratum be that included between the surfaces 

F(v t y,z) = F= a (1) 

and F = a + h. (2) 



If we put R 2 = 



dF 
dx 



dF 
dy 



dF 



(3) 



and if /, m, n are the direction-cosines of the normal to the surface, 

jU dF dF dF m 

Hi = -=- > Mm = -= > lin = _ MM 

dx dy dz 

Now let F l be the value of the potential on the negative side 
of the surface F = a, V its value between the surfaces F = a and 
F = a + Ti y and V 2 its value on the positive side of F a + k. 

Also, let pj, p , and p 2 be the values of the density in these three 
portions of space. Then, since the density is everywhere finite, 
the second derivatives of V are everywhere finite, and the first 
derivatives, and also the function itself, are everywhere continuous 
and finite. 

At any point of the surface F = a let a normal be drawn of 



78.] ELECTRIFIED SURFACE. 81 

length y, till it meets the surface F = a + h, then the value of F at 
the extremity of the normal is 

,dF dF dF. 



or a + h = a + vR + &c. /// (6) 

The value of V at the same point is 

fl dV dV dV\ 
^ = ^+*(l-^+m^ + ^) + &c, (7) 

or r.-r^j^+ta. //y (8 ) 

Since the first derivatives of V continue always finite, the second 
side of the equation vanishes when Ji is diminished without limit, 
and therefore if Y 2 and V- denote the values of V on the outside 
and inside of an electrified surface at the point x, y, z, 

7i = r t . 0) 

If x -f dx, y + dy, z + dz be the coordinates of another point on 

the electrified surface, F=a and 7^= T 2 at this point also ; whence 

dF, dF , dF 





^-V^-f&c.; (11) 

tt A/ 

and when dx, dy, dz vanish, we find the conditions 

-7 = um, f (12) 

dy dy 
fjJL-.*%L=Ci 

where C is a quantity to be determined. 

dV 
Next, let us consider the variation of F and -y- along the 

ordinate parallel to x between the surfaces F= a and F = a + h. 

dF cPF 

We have F= a -f -j-dx + \ -r^(dx} 2 + &c., (13) 

dV d7 l d^V L d 3 7 j 2 
Hence, at the second surface, where F=a + k, and V becomes F 2 , 

VOL. I. G 



82 ELECTROSTATICS. [79. 



whence -^- dx + &c. = Cl, (16) 

by the first of equations (12). 
Kf Multiplying by jfl, and remembering that at the second surface 

Rldx=k (17) 



we find - rT h=CJtl*. (18) 

clx 

Similarly &T _ 



(19) 

and ~cT^^ = CRn*- (20) 

Adding (-7-0- + -T-O- + -7-0-) A = CR ; (21) 

\ //7^ r/ JJ CM 2 * 

but -y-o- +.-T-O- + -TO- =477^ and ^ = vR ; (22) 



hence (7 = 4 TT/V = 4 w <r, (23) 

where cr is the surface-density; or, multiplying the equations 
(12) by I, m, n respectively, and adding, 



This equation is called the characteristic equation of V at a surface. 
This equation may also be written 
dV, dF 2 



where r l5 z^ 2 are the normals to the surface drawn towards the 
first and the second medium respectively, and 7^, T 2 the potentials 
at points on these normals. We may also write it 

-S 2 coS 2 + J S 1 cose 1 + 47ro- = ; (26) 

where R^ R z are the resultant forces, and c lt e 2 the angles which 
they make with the normals drawn from the surface on either 
side. 

79.] Let us next determine the total mechanical force acting on 
an element of the electrified surface. 

The general expression for the force parallel to x on an element 
whose volume is dx dy dz, and volume-density p, is 

dX = -=- p dx dy dz. (27) 



80.] FORCE ACTING ON AN ELECTRIFIED SURFACE. 83 



In the present case we have for any point on the normal v 
dV dV, d*~PS 



(28) 



- 
dx dx dxfyiv 

also, if the element of surface is dS, that of the volume of the 
element of the stratum may be written dSdv ; and if X is the whole 
force on a stratum of thickness v, 



. (29) 

Integrating with respect to v, we find 

" (30) 



evnce . = + + c . ; (31) 

( 32) 

When v is diminished and // increased without limit, the product 
p v remaining always constant and equal to o-, the expression for 
the force in the direction of x on the electricity a- dS on the element 

of surface (IS is ^ 701X^1 dT 9 \ 

X=-^4(-^ + -^); (33) 

that is, the force acting on the electrified element o- dS in any given 
direction is the arithmetic mean of the forces acting on equal 
quantities of electricity placed one just inside the surface and the 
other just outside the surface close to the actual position of the 
element, and therefore the resultant mechanical force on the elec 
trified element is equal to the resultant of the forces which would 
act on two portions of electricity, each equal to half that on the 
element, and placed one on each side of the surface and infinitely 
near to it. 

80.] When a conductor is in electrical equilibrium, the whole of the 
electricity is on the surface. 

We have already shewn that throughout the substance of the 
conductor the potential V is constant. Hence y 2 V is zero, and 
therefore by Poisson s equation, p is zero throughout the substance 
of the conductor, and there can be no electricity in the interior 
of the conductor. 

Hence a superficial distribution of electricity is the only possible 
one in the case of conductors in equilibrium. A distribution 
throughout the mass can only exist in equilibrium when the body 
is a non-conductor. 

G 2 



84 ELECTKOSTATICS. [8 1. 

Since the resultant force within a conductor is zero, the resultant 
force just outside the conductor is along the normal and is equal to 
4 TT a; acting outwards from the conductor. 

81.] If we now suppose an elongated body to be electrified, we 
may, by diminishing its lateral dimensions, arrive at the conception 
of an electrified line. 

Let ds be the length of a small portion of the elongated body, 
and let c be its circumference, and o- the superficial density of the 
electricity on its surface ; then, if A. is the electricity per unit of 
length, A. = co-, and the resultant electrical force close to the 
surface will be A 

477 (T = 4 77 
C 

If, while X remains finite, c be diminished indefinitely, the force 
at the surface will be increased indefinitely. Now in every di 
electric there is a limit beyond which the force cannot be increased 
without a disruptive discharge. Hence a distribution of electricity 
in which a finite quantity is placed on a finite portion of a line 
is inconsistent with the conditions existing in nature. 

Even if an insulator could be found such that no discharge could 
be driven through it by an infinite force, it would be impossible 
to charge a linear conductor with a finite quantity of electricity, 
for an infinite electromotive force would be required to bring the 
electricity to the linear conductor. 

In the same way it may be shewn that a point charged with 
a finite quantity of electricity cannot exist in nature. It is con 
venient, however, in certain cases, to speak of electrified lines and 
points, and we may suppose these represented by electrified wires, 
and by small bodies of which the dimensions are negligible com 
pared with the principal distances concerned. 

Since the quantity of electricity on any given portion of a wire 
diminishes indefinitely when the diameter of the wire is indefinitely 
diminished, the distribution of electricity on bodies of considerable 
dimensions will not be sensibly affected by the introduction of very 
fine metallic wires into the field, so as to form electrical connexions 
between these bodies and the earth, an electrical machine, or an 
electrometer. 

On Lines of Force. 

82.] If a line be drawn whose direction at every point of its 
course coincides with that of the resultant force at that point, the 
line is called a Line of Force. 



82.] LINES OF FOKCE. 85 

If lines of force be drawn from every point of a line they will 
form a surface such that the force at any point is parallel to the 
tangent plane at that point. The surface-integral of the force with 
respect to this surface or any part of it will therefore be zero. 

If lines of force are drawn from every point of a closed curve L l 
they will form a tubular surface S . Let the surface S 19 bounded 
by the closed curve L lt be a section of this tube, and let S 2 be any 
other section of the tube. Let Q , Q 19 Q 2 be the surface-integrals 
over S 0) S lt S 2 , then, since the three surfaces completely enclose a 
space in which there is no attracting matter, we have 

Qo+Qi+ Qz = 0. 

But o = > therefore Q 2 = Q lt or the surface-integral over 
the second section is equal and opposite to that over the first : but 
since the directions of the normal are opposite in the two cases, we 
may say that the surface-integrals of the two sections are equal, the 
direction of the line of force being supposed positive in both. 

Such a tube is called a Solenoid*, and such a distribution of 
force is called a Solenoidal distribution. The velocities of an in 
compressible fluid are distributed in this manner. 

If we suppose any surface divided into elementary portions such 
that the surface-integral of each element is unity, and if solenoids 
are drawn through the field of force having these elements for their 
bases, then the surface-integral for any other surface will be re 
presented by the number of solenoids which it cuts. It is in this 
sense that Faraday uses his conception of lines of force to indicate 
not only the direction but the amount of the force at any place in 
the field. 

We have used the phrase Lines of Force because it has been used 
by Faraday and others. In strictness, however, these lines should 
be called Lines of Electric Induction. 

In the ordinary cases the lines of induction indicate the direction 
and magnitude of the resultant electromotive force at every point, 
because the force and the induction are in the same direction and 
in a constant ratio. There are other cases, however, in which it 
is important to remember that these lines indicate the induction, 
and that the force is indicated by the equipotential surfaces, being 
normal to these surfaces and inversely proportional to the distances 
of consecutive surfaces. 

* From ffw\r)i>, a tube. Faraday uses (3271) the term Sphondyloid in the same 
sense. 



86 ELECTROSTATICS. [83. 

On Specific Inductive Capacity. 

83.] In the preceding investigation of surface-integrals I have 
adopted the ordinary conception of direct action at a distance, 
and have not taken into consideration any effects depending on the 
nature of the dielectric medium in which the forces are observed. 

But Faraday has observed that the quantity of electricity 
induced by a given electromotive force on the surface of a conductor 
which bounds a dielectric is not the same for all dielectrics. The 
induced electricity is greater for most solid and liquid dielectrics 
than for air and gases. Hence these bodies are said to have a 
greater specific inductive capacity than air, which is the standard 
medium. 

We may express the theory of Faraday in mathematical language 
by saying that in a dielectric medium the induction across any 
surface is the product of the normal electric force into the coefficient 
of specific inductive capacity of that medium. If we denote this 
coefficient by K, then in every part of the investigation of sur 
face-integrals we must multiply X, I 7 ", and Z by K, so that the 
equation of Poisson will become 

d ^dV d v dV d ^dV 

-=-.K^- + -j-.K-j- + -j-.K^-+4;Trp = 0. 

dx dx dy dy dz dz 

At the surface of separation of two media whose inductive capa 
cities are K and K 2 , and in which the potentials are V and T 2i 
the characteristic equation may be written 



where v is the normal drawn from the first medium to the second, 
and o- is the true surface-density on the surface of separation ; 
that is to say, the quantity of electricity which is actually on the 
surface in the form of a charge, and which can be altered only by 
conveying electricity to or from the spot. This true electrification 
must be distinguished from the apparent electrification (/_, which is 
the electrification as deduced from the electrical forces in the neigh 
bourhood of the surface, using the ordinary characteristic equation 



If a solid dielectric of any form is a perfect insulator, and if 
its surface receives no charge, then the true electrification remains 
zero, whatever be the electrical forces acting on it. 



83.] SPECIFIC INDUCTIVE CAPACITY. 87 



dF 2 K^dVi , 
Hence T, = T 2 -d^> and 

dV 4V(/K 



The surface-density </ is that of the apparent electrification 
produced at the surface of the solid dielectric by induction. It 
disappears entirely when the inducing force is removed, but if 
during the action of the inducing force the apparent electrification 
of the surface is discharged by passing a flame over the surface, 
then, when the inducing force is taken away, there will appear an 
electrification opposite to <r *. 

In a heterogeneous dielectric in which K varies continuously, if 
p be the apparent volume-density, 



Comparing this with the equation above, we find 
dKdV dKdV dKdV 



The true electrification, indicated by p, in the dielectric whose 
variable inductive capacity is denoted by K, will produce the same 
potential at every point as the apparent electrification, indicated by 
p 7 , would produce in a dielectric whose inductive capacity is every 
where equal to unity. 

* See Faraday s Kemarks on Static Induction, Proceedings of the Royal In 
stitution, Feb. 12, 1858. 



CHAPTER III. 



SYSTEMS OP CONDUCTORS. 

On the Superposition of Electrical Systems. 

84.] Let E l be a given electrified system of which the potential 
at a point P is T 1} and let U 2 be another electrified system of which 
the potential at the same point would be F 2 if E l did not exist. 
Then, if E and E z exist together, the potential of the combined 
system will be /^-f F 2 . 

Hence, if V be the potential of an electrified system E, if the 
electrification of every part of E be increased in the ratio of n to 1 , 
the potential of the new system nE will be n V. 

Energy of an Electrified System. 

85.] Let the system be divided into parts, A 19 A 2 , &c. so small 
that the potential in each part may be considered constant through 
out its extent. Let e l , 2 , &c. be the quantities of electricity in 
each of these parts, and let T 19 F" 2 , &c. be their potentials. 

If now e 1 is altered to ne^ e% to ne^ &c., then the potentials will 
become nT lt nV^ &c. 

Let us consider the effect of changing n into n + dn in all these 
expressions. It will be equivalent to charging A 1 with a quantity 
of electricity e l dn, A 2 with e 2 dn, &c. These charges must be sup 
posed to be brought from a distance at which the electrical action 
of the system is insensible. The work done in bringing e 1 dn of 
electricity to A 19 whose potential before the charge is nV 19 and after 
the charge (n + dn) F lf must lie between 

n Fj e 1 dn and (n -f dn) V^ e dn. 

In the limit we may neglect the square of dn, and write the 
expression 



86.] COEFFICIENTS OF POTENTIAL AND OF INDUCTION. 89 

Similarly the work required to increase the charge of A^ is 
~P 2 e 2 ndn, so that the whole work done in increasing the charge 
of the system is 



If we suppose this process repeated an indefinitely great number 
of times, each charge being indefinitely small, till the total effect 
becomes sensible, the work done will be 

2 ( 7e)fn tin = J 2 ( Ve] (n*-n*) ; 

where 2 ( Ve) means the sum of all the products of the potential of 
each element into the quantity of electricity in that element when 
n 1, and n Q is the initial and % the final value of n. 

If we make n = and % = I, we find for the work required to 
charge an unelectrified system so that the electricity is e and the 
potential V in each element, 



General Theory of a System of Conductors. 

86.] Let A i, A 2 , ...A n be any number of conductors of any 
form. Let the charge or total quantity of electricity on each of 
these be E^ E. 2J ... E n3 and let their potentials be T 19 F 2 , ... J n 
respectively. 

Let us suppose the conductors to be all insulated and originally 
free of charge, and at potential zero. 

Now let A 1 be charged with unit of electricity, the other bodies 
being without charge. The effect of this charge on A 1 will be to 
raise the potential of A l tojo n , that of A 2 to p^, and that of A n to 
j} ln , where j u , &c. are quantities depending on the form and rela 
tive position of the conductors. The quantity j n may be called the 
Potential Coefficient of A l on itself, and p l2 may be called the Po 
tential Coefficient of A on A 2 , and so on. 

If the charge upon A is now made E l , then, by the principle of 
superposition, we shall have 



Now let A 1 be discharged, and A 2 charged with unit of electricity, 
and let the potentials of A lt A 2 , ... A n be ^21^22? --Pzn 
potentials due to E. 2 on A^ will be 



Similarly let us denote the potential of A s due to a unit charge 
on A r by j) rs , and let us call^ r5 the Potential Coefficient of A r on A s , 



90 SYSTEMS OF CONDUCTORS. [87. 

then we shall have the following equations determining the po 
tentials in terms of the charges : 



(1) 



We have here n linear equations containing n 2 coefficients of 
potential. 

87.] By solving these equations for E 19 E 2 , &c. we should obtain 
n equations of the form 



(2) 



n n . .. n nn 

The coefficients in these equations may be obtained directly from 
those in the former equations. They may be called Coefficients of 
Induction. 

Of these q n is numerically equal to the quantity of electricity 
on A l when A l is at potential unity and all the other bodies are 
at potential zero. This is called the Capacity of A^ It depends 
on the form and position of all the conductors in the system. 

Of the rest q rs is the charge induced on A r when A s is main 
tained at potential unity and all the other conductors at potential 
zero. This is called the Coefficient of Induction of A 8 on A r . 

The mathematical determination of the coefficients of potential 
and of capacity from the known forms and positions of the con 
ductors is in general difficult. We shall afterwards prove that they 
have always determinate values, and we shall determine their values 
in certain special cases. For the present, however, we may suppose 
them to be determined by actual experiment. 

Dimensions of these Coefficients. 

Since the potential of an electrified point at a distance r is the 
charge of electricity divided by the distance, the ratio of a quantity 
of electricity to a potential may be represented by a line. Hence 
all the coefficients of capacity and induction (q) are of the nature of 
lines, and the coefficients of potential (p) are of the nature of the 
reciprocals of lines. 



88.] RECIPROCAL PROPERTY OF THE COEFFICIENTS. 91 

88.] THEOREM I. The coefficients of A r relative to A 8 are equal to 
those of A 8 relative to A r . 

If E r , the charge on A r , is increased by bfl r , the work spent in 
bringing bfl r from an infinite distance to the conductor A r whose 
potential is V ry is by the definition of potential in Art. 70, 

r r *E r , 

and this expresses the increment of the electric energy caused by 
this increment of charge. 

If the charges of the different conductors are increased by 
&c., the increment of the electric energy of the system will be 



If, therefore, the electric energy Q is expressed as a function 
of the charges lt E. 2 , &c., the potential of any conductor may be 
expressed as the partial differential coefficient of this function with 
respect to the charge on that conductor, or 



Since the potentials are linear functions of the charges, the energy 
must be a quadratic function of the charges. If we put 

CE r E s 

for the term in the expansion of Q which involves the product 
E r E 8 , then, by differentiating with respect to E s , we find the term 
of the expansion of V s which involves E r to be CE r . 

Differentiating with respect to E r , we find the term in the 
expansion of V r which involves E s to be CE S . 

Comparing these results with equations (1), Art. 86, we find 

Prs = C = Psr, 

or, interpreting the symbols p rs and p sr : 

The potential of A 8 due to a unit charge on A r is equal to the 
potential of A r due to a unit charge on A s . 

This reciprocal property of the electrical action of one conductor 
on another was established by Helmholtz and Sir W. Thomson. 

If we suppose the conductors A r and A s to be indefinitely small, 
we have the following reciprocal property of any two points : 

The potential at any point A 8 , due to unit of electricity placed 
at A r in presence of any system of conductors, is a function of the 
positions of A r and A 8 in which the coordinates of A r and of A s 
enter in the same manner, so that the value of the function is 
unchanged if we exchange A r and A f . 



92 SYSTEMS OF CONDUCTOKS. [89. 

This function is known by the name of Green s Function. 

The coefficients of induction q rs and q sr are also equal. This is 
easily seen from the process by which these coefficients are obtained 
from the coefficients of potential. For, in the expression for q rs , 
p rs and p sr enter in the same way as p sr and p rs do in the expression 
for q sr . Hence if all pairs of coefficients p rs and p sr are equal, the 
pairs q rs and q sr are also equal. 



89.] THEOREM II. Let a charge E r be placed on A r) and let all 
the other conductors he at potential zero, and let the charge 
induced on A 8 be n rs E r , then if A r is discharged and insulated, 
and A s brought to potential V 8 , the other conductors being at 
potential zero } then the potential of A r will be + n rs ~P~ 8 . 

For, in the first case, if V r is the potential of A r , we find by 
equations (2), 

E 8 = q rs Y r , and E r = q rr 7 r . 

Hence E 8 = ^E r , and n rs = - ^ 
q rr q rr 

In the second case, we have 



Hence V r = -V 8 = n r j % . 

"rr 

From this follows the important theorem, due to Green : 
If a charge unity, placed on the conductor A Q in presence of 
conductors A 19 A 2 , &c. at potential zero induces charges n lt 
n 2 , &c. in these conductors, then, if A Q is discharged and in 
sulated, and these conductors are maintained at potentials V^ T 2 , 
&c., the potential of A will be 



The quantities (n) are evidently numerical quantities, or ratios. 

The conductor A may be supposed reduced to a point, and 
A 19 A 2 , &c. need not be insulated from each other, but may be 
different elementary portions of the surface of the same conductor. 
We shall see the application of this principle when we investigate 
Green s Functions. 

90.] THEOREM III. The coefficients of potential are all positive, 

but none of the coefficients p r8 is greater thanp rr or p 8S . 
For let a charge unity be communicated to A r , the other con 
ductors being uncharged. A system of equipotential surfaces will 



91.] PROPERTIES OF THE COEFFICIENTS. 93 

be formed. Of these one will be the surface of A ri and its potential 
will be p rr . If A f is placed in a hollow excavated in A r so as to be 
completely enclosed by it, then the potential of A s will also be p^. 

If, however, A g is outside of A r its potential p rs will lie between 
p rr and zero. 

For consider the lines of force issuing from the charged con 
ductor A r . The charge is measured by the excess of the number 
of lines which issue from it over those which terminate in it. 
Hence, if the conductor has no charge, the number of lines which 
enter the conductor must be equal to the number which issue from 
it. The lines which enter the conductor come from places of greater 
potential, and those which issue from it go to places of less poten 
tial. Hence the potential of an uncharged conductor must be 
intermediate between the highest and lowest potentials in the field, 
and therefore the highest and lowest potentials cannot belong to 
any of the uncharged bodies. 

The highest potential must therefore be p rr , that of the charged 
body A r , and the lowest must be that of space at an infinite dis 
tance, which is zero, and all the other potentials such as p ra must 
lie between p rr and zero. 

If A 8 completely surrounds A^ then^ rs = p ri . 

91.] THEOREM IV. None of the coefficients of induction are positive, 
and the sum of all those belonging to a single conductor is not 
numerically greater than the coefficient of capacity of that con 
ductor, which is always positive. 

For let A r be maintained at potential unity while all the other 
conductors are kept at potential zero, then the charge on A r is q^, 
and that on any other conductor A s is q rs . 

The number of lines of force which issue from A r isp rr . Of these 
some terminate in the other conductors, and some may proceed to 
infinity, but no lines of force can pass between any of the other 
conductors or from them to infinity, because they are all at potential 
zero. 

No line of force can issue from any of the other conductors such 
as A s , because no part of the field has a lower potential than A s . 
If A s is completely cut off from A r by the closed surface of one 
of the conductors, then q rs is zero. If A s is not thus cut off, q rs is a 
negative quantity. 

If one of the conductors A t completely surrounds A r , then all 
the lines of force from A r fall on A t and the conductors within it, 



94 SYSTEMS OF CONDUCTORS. [92. 

and the sum of the coefficients of induction of these conductors with 
respect to A r will be equal to q rr with its sign changed. But if 
A r is not completely surrounded by a conductor the arithmetical 
sum of the coefficients of induction q rs , &c. will be less than q rr . 

We have deduced these two theorems independently by means 
of electrical considerations. We may leave it to the mathematical 
student to determine whether one is a mathematical consequence 
of the other. 

Resultant Mechanical Force on any Conductor in terms of the Charges. 
92.] Let 8$ be any mechanical displacement of the conductor, 
and let 4> be the the component of the force tending to produce that 
displacement, then <J>8< is the work done by the force during 
the displacement. If this work is derived from the electrification 
of the system, then if Q is the electric energy of the system, 

= 0, (3) 



Here Q = i (E l 7 1 + E 2 F 2 + &c.) (5) 

If the bodies are insulated, the variation of Q must be such that 
E^ E ZJ &c. remain constant. Substituting therefore for the values 
of the potentials, we have 

Q = 4S r S.(*, &.*), (6) 

where the symbol of summation 2 includes all terms of the form 
within the brackets, and r and s may each have any values from 
1 to n. From this we find 



as the expression for the component of the force which produces 
variation of the generalized coordinate </>. 

Resultant Mechanical Force in terms of the Potentials. 
93.] The expression for <I> in terms of the charges is 

*=-iS r S.(* r fl.f!p, / I../;.. (8) 

where in the summation r and s have each every value in suc 
cession from 1 to n. 

Now E r = 2j ( %q rt ) where t may have any value from 1 to n, 
so that 



93-] RESULTANT FORCE IN TERMS OF POTENTIALS. 95 

*=-kWS t (W an %f). - (9) 

Now the coefficients of potential are connected with those of 
induction by n equations of the form 

S r (Arfcr)= 1, (10) 

and \n(n\) of the form 

S r Qrfrr) = 0. (11) 

Differentiating with respect to < we get %n(n + 1) equations of 
the form ^ ^ 

M*-|?)+Mfc^)=0, (12) 

where a and 3 may be the same or different. 
Hence, putting a and b equal to r and s, 



(13) 
but 2 g (figure) = V r , so that we may write 

* = 4S,S,(J^fe), (14) 

where r and may have each every value in succession from 1 
to n. This expression gives the resultant force in terms of the 
potentials. 

If each conductor is connected with a battery or other con 
trivance by which its potential is maintained constant during the 
displacement, then this expression is simply 



under the condition that all the potentials are constant. 

The work done in this case during the displacement 8< is 4>6$, 
and the electrical energy of the system of conductors is increased 
by 8Q; hence the energy spent by the batteries during the dis 
placement is 

(16) 



It appears from Art. 92, that the resultant force < is equal to 

~ , under the condition that the charges of the conductors are 

* dQ 

constant. It is also, by Art. 93, equal to y^, under the con 

dition that the potentials of the conductors are constant. If the 
conductors are insulated, they tend to move so that their energy 
is diminished, and the work done by the electrical forces during 
the displacement is equal to the diminution of energy. 

If the conductors are connected with batteries, so that their 



96 SYSTEMS OF CONDUCTORS. [94. 

potentials are maintained constant, they tend to move so that the 
energy of the system is increased, and the work done by the 
electrical forces during the displacement is equal to the increment 
of the energy of the system. The energy spent by the batteries 
is equal to double of either of these quantities, and is spent half 
in mechanical, and half in electrical work. 

On the Comparison of Similar Electrified Systems. 

94.] If two electrified systems are similar in a geometrical sense., 
so that the lengths of corresponding lines in the two systems 
are as L to L , then if the dielectric which separates the conducting 
bodies is the same in both systems, the coefficients of induction 
and of capacity will be in the proportion of L to U . For if we 
consider corresponding portions, A and A , of the two systems, and 
suppose the quantity of electricity on A to be E, and that on A 
to be E , then the potentials V and V at corresponding points 
B and ^, due to this electrification, will be 

E 



But AS is to A l? as L to L , so that we must have 

E:E : .L7:L 7 . 

But if the inductive capacity of the dielectric is different in the 
two systems, being K in the first and K in the second, then if the 
potential at any point of the first system is to that at the cor 
responding point of the second as V to V 9 and if the quantities 
of electricity on corresponding parts are as E to E t we shall have 



By this proportion we may find the relation between the total 
electrification of corresponding parts of two systems, which are 
in the first place geometrically similar, in the second place com 
posed of dielectric media of which the dielectric inductive capacity 
at corresponding points is in the proportion of K to K 9 and in 
the third place so electrified that the potentials of corresponding 
points are as V to V. 

From this it appears that if q be any coefficient of capacity or 
induction in the first system, and c[ the corresponding one in the 

second > q : q : : LK : L K ; 

and if p and p denote corresponding coefficients of potential in 

the two systems, 1 1 



94-] COMPARISON OF SIMILAR SYSTEMS. 97 

If one of the bodies be displaced in the first system, and the 
corresponding body in the second system receive a similar dis 
placement, then these displacements are in the proportion of L 
to If, and if the forces acting on the two bodies are as F to F, 
then the work done in the two systems will be as FL to F L . 

But the total electrical energy is half the sum of the quantities 
of electricity multiplied each by the potential of the electrified 
body, so that in the similar systems, if Q and Q be the total 

electrical energy, 

Q : q f : : E7 : E V , 

and the difference of energy after similar displacements in the two 
systems will be in the same proportion. Hence, since FL is pro 
portional to the electrical work done during the displacement, 

FL :F L ::E7:E Y . 

Combining these proportions, we find that the ratio of the 
resultant force on any body of the first system to that on the 
corresponding body of the second system is 

F : F : : 



or 



L 2 K 



The first of these proportions shews that in similar systems the 
force is proportional to the square of the electromotive force and 
to the inductive capacity of the dielectric, but is independent of the 
actual dimensions of the system. 

Hence two conductors placed in a liquid whose inductive capacity 
is greater than that of air, and electrified to given potentials, will 
attract each other more than if they had been electrified to the 
same potentials in air. 

The second proportion shews that if the quantity of electricity 
on each body is given, the forces are proportional to the squares 
of the electrifications and inversely to the squares of the distances, 
and also inversely to the inductive capacities of the media. 

Hence, if two conductors with given charges are placed in a 
liquid whose inductive capacity is greater than that of air, they 
will attract each other less than if they had been surrounded with 
air and electrified with the same charges of electricity. 



VOL. I. 



CHAPTER IV. 

GENERAL THEOREMS. 

95.] IN the preceding chapter we have calculated the potential 
function and investigated its properties on the hypothesis that 
there is a direct action at a distance between electrified bodies, 
which is the resultant of the direct actions between the various 
electrified parts of the bodies. 

If we call this the direct method of investigation, the inverse 
method will consist in assuming that the potential is a function 
characterised by properties the same as those which we have already 
established, and investigating the form of the function. 

In the direct method the potential is calculated from the dis 
tribution of electricity by a process of integration, and is found 
to satisfy certain partial differential equations. In the inverse 
method the partial differential equations are supposed given, and 
we have to find the potential and the distribution of electricity. 

It is only in problems in which the distribution of electricity 
is given that the direct method can be used. When we have to 
find the distribution on a conductor we must make use of the 
inverse method. 

"We have now to shew that the inverse method leads in every 
case to a determinate result, and to establish certain general 
theorems deduced from Poisson s partial differential equation 



The mathematical ideas expressed by this equation are of a 
different kind from those expressed by the equation 



/+GO r + ao r + x> 
/ / t 
00 J CD J 00 / 



dsf. 



In the differential equation we express that the values of the 
second derivatives of V in the neighbourhood of any point, and 



96.] CHARACTERISTICS OF THE POTENTIAL. 99 

the density at that point are related to each other in a certain 
manner, and no relation is expressed between the value of V at 
that point and the value of p at any point at a sensible distance 
from it. 

In the second expression, on the other hand, the distance between 
the point (x 3 y , z ) at which p exists from the point (#, y, z] at 
which V exists is denoted by r, and is distinctly recognised in the 
expression to be integrated. 

The integral, therefore, is the appropriate mathematical expression 
for a theory of action between particles at a distance, whereas the 
differential equation is the appropriate expression for a theory of 
action exerted between contiguous parts of a medium. 

We have seen that the result of the integration satisfies the 
differential equation. We have now to shew that it is the only 
solution of that equation fulfilling certain conditions. 

We shall in this way not only establish the mathematical equi 
valence of the two expressions, but prepare our minds to pass from 
the theory of direct action at a distance to that of action between 
contiguous parts of a medium. 

Characteristics of the Potential Function. 

96.] The potential function V, considered as derived by integration 
from a known distribution of electricity either in the substance of 
bodies with the volume-density p or on certain surfaces with the 
surface-density a, p and a- being everywhere finite, has been shewn 
to have the following characteristics : 

(1) Fis finite and continuous throughout all space. 

(2) V vanishes at an infinite distance from the electrified system. 

(3) The first derivatives of V are finite throughout all space, and 
continuous except at the electrified surfaces. 

(4) At every point of space, except on the electrified surfaces, the 
equation of Poisson 



is satisfied. We shall refer to this equation as the General 
Characteristic equation. 

At every point where there is no electrification this equation 
becomes the equation of Laplace, 



100 GENERAL THEOREMS. [97. 

(5) At any point of an electrified surface at which the surface- 
density is cr, the first derivative of F, taken with respect to the 
normal to the surface, changes its value abruptly at the surface, 
so that A y f dV 



where v and // are the normals on either side of the surface, and 
V and V are the corresponding- potentials. We shall refer to this 
equation as the Superficial Characteristic equation. 

(G) If V denote the potential at a point whose distance from 
any fixed point in a finite electrical system is r, then the product 
Vr, when r increases indefinitely, is ultimately equal to E, the total 
charge in the finite system. 

97.] Lemma. Let V be any continuous function of x, y y z, and 
let u, v, w be functions of #, y, z, subject to the general solenoidal 
condition du dv dw 

Tx + Ty + -dz = *> 

where these functions are continuous, and to the superficial sole 
noidal condition 

l(u l u 2 } + m(v^-v 2 ]-\-n(w l w^ = 0, (2) 

at any surface at which these functions become discontinuous, 
, m, n being the direction-cosines of the normal to the surface, 
and u lt v lt w l and u 2 , v 2 , w 2 the values of the functions on opposite 
sides of the surface, then the triple integral 

(ff f dV dV dV^ . 

M = I I I (u -j- + v -j- + w -=- ) dx dy dz (3) 

J J J ^ dx dy dz 

vanishes when the integration is extended over a space bounded by 
surfaces at which either V is constant, or 

lu + mv + nw = 0, (4) 

/, m, n, being the direction-cosines of the surface. 

Before proceeding to prove this theorem analytically we may 
observe, that if u, v, w be taken to represent the components of the 
velocity of a homogeneous incompressible fluid of density unity, 
and if V be taken to represent the potential at any point of space 
of forces acting on the fluid, then the general and superficial equa 
tions of continuity ((1) and (2)) indicate that every part of the 
space is, and continues to be, full of the fluid, and equation (4) 
is the condition to be fulfilled at a surface through which the fluid 
does not pass. 

The integral M represents the work done by the fluid against 
the forces acting on it in unit of time. 



97-] LEMMA. 

Now, since the forces which act on the fluid are derived from 
the potential function F } the work which they do is subject to the 
law of conservation of energy, and the work done on the whole 
fluid within a certain space may be found if we know the potential 
at the points where each line of flow enters the space and where 
it issues from it. The excess of the second of these potentials over 
the first, multiplied by the quantity of fluid which is transmitted 
along each line of flow, will give the work done by that portion 
of the fluid, and the sum of all such products will give the whole 
work. 

Now, if the space be bounded by a surface for which V= C } a 
constant quantity, the potential will be the same at the place 
where any line of flow enters the space and where it issues from 
it, so that in this case no work will be done by the forces on the 
fluid within the space, and M = 0. 

Secondly, if the space be bounded in whole or in part by a 
surface satisfying equation (4), no fluid will enter or leave the space 
through this surface, so that no part of the value of M can depend 
on this part of the surface. 

The quantity M is therefore zero for a space bounded externally 
by the closed surface F= C, and it remains zero though any part 
of this space be cut off from the rest by surfaces fulfilling the 
condition (4). 

The analytical expression of the process by which we deduce the 
work done in the interior of the space from that which takes place 
at the bounding surface is contained in the following method of 
integration by parts. 

Taking the first term of the integral M, 



where 2 (u F) = u 7 l n 2 F 2 + U B F 3 ?/ 4 F 4 + &c. ; 

and where %F 15 w 2 F 2 , &c. are the values of u and v at the points 
whose coordinates are (a lt y, z), (x. 2) y, z), &c., sf ly a? 2 , &c. being the 
values of x where the ordinate cuts the bounding surface or surfaces, 
arranged in descending order of magnitude. 

Adding the two other terms of the integral M, we find 

J/ = 



102 GENERAL THEOREMS. [97. 

If l } m, n are the direction-cosines of the normal drawn inwards 
from the bounding surface at any point, and dS an element of that 
surface, then we may write 



the integration of the first term being extended over the bounding 
surface, and that of the second throughout the entire space. 

For all spaces within which u, v, w are continuous, the second 
term vanishes in virtue of equation (1). If for any surface within 
the space u, v, w are discontinuous but subject to equation (2), we 
find for the part of M depending on this surface, 



= / / 
= / / 



t ) dS 



I) 



2 



where the suffixes ^ and 2 , applied to any symbol, indicate to which 
of the two spaces separated by the surface the symbol belongs. 
Now, since V is continuous, we have at every point of the surface, 

F 1= F 2= F; 
we have also dS l = dS 2 = d8; 

but since the normals are drawn in opposite directions, we have 

/! = 1 2 = I, m 1 = m 2 = m, % = n 2 n ; 
so that the total value of M, so far as it depends on the surface of 
discontinuity, is 



The quantity under the integral sign vanishes at every point in 
virtue of the superficial solenoidal condition or characteristic (2). 

Hence, in determining the value of M, we have only to consider 
the surface-integral over the actual bounding surface of the space 
considered, or 

M = F(lu + mv + nw)dS. 
Case 1 . If V is constant over the whole surface and equal to (7, 

(lu + mv + nw] dS. 



= C( 



The part of this expression under the sign of double integration 
represents the surface-integral of the flux whose components are 
u, v, w, and by Art. 2 1 this surface-integral is zero for the closed 
surface in virtue of the general and superficial solenoidal conditions 
(1) and (2). 



98.] THOMSON S THEOREM. 103 

Hence M = for a space bounded by a single equipotential 
surface. 

If the space is bounded externally by the surface V =. C, and 
internally by the surfaces 7 C l} F= C 2 , &c., then the total value 
of M for the space so bounded will be 

Jf-JKi-j^&c., 

where M is the value of the integral for the whole space within the 
surface V = C, and M l , M 2 are the values of the integral for the 
spaces within the internal surfaces. But we have seen that M } 
M!, M 2 , &c. are each of them zero, so that the integral is zero also 
for the periphractic region between the surfaces. 

Case 2. If lu + mv + nw is zero over any part of the bounding 
surface, that part of the surface can contribute nothing to the value 
of 31, because the quantity under the integral sign is everywhere 
zero. Hence M will remain zero if a surface fulfilling this con 
dition is substituted for any part of the bounding surface, provided 
that the remainder of the surface is all at the same potential. 

98.] We are now prepared to prove a theorem which we owe to 
Sir William Thomson *. 

As we shall require this theorem in various parts of our subject, 
I shall put it in a form capable of the necessary modifications. 

Let a, 6, c be any functions of x, y, z (we may call them the 
components of a flux) subject only to the condition 

da db dc 

-J- + -j- + ~T 
dx dy dz 

where p has given values within a certain space. This is the general 
characteristic of a, b, c. 

Let us also suppose that at certain surfaces (S) a, b, and c are 
discontinuous, but satisfy the condition 

I(a l aj + m(6 1 6j + n(c 1 cj + lv<r = 0; (6) 

where I, m, n are the direction-cosines of the normal to the surface, 
a L , 1 1} q the values of a, b, c on the positive side of the surface, and 
a 2 , b 2 , c. 2 those on the negative side, and o- a quantity given for 
every point of the surface. This condition is the superficial charac 
teristic of a, b, c. 

Next, let us suppose that V is a continuous function of #, y, z, 
which either vanishes at infinity or whose value at a certain point 
is given, and let V satisfy the general characteristic equation 

* Cambridge and Dublin Mathematical Journal, February, 1848. 



104 GEKEKAL THEOREMS. [98 

d dV d d7 d dV 



, (7) 

dx dy dy dz dz 
/ / 



and the superficial characteristic at the surfaces 

7/rr dV* dV<>\ /xr dF-, dV^ 

I (JT, -j-i JC *\ + m(K 1 - r i -Z 2 -=-1) 

^ ^ dx 2 dx ^ * dy 2 dy 

.7 \/ 



^T being a quantity which may be positive or zero but not negative, 
given at every point of space. 

Finally, let 8 TT Q represent the triple integral 



8 TT q = (a 2 + 1>* + c 2 ) dx dy dz, (9) 

extended over a space bounded by surfaces, for each of which either 

V = constant, 

or la -\-mb-\-nc = Kl -=- + Km-^- + Kn-^- = q, (10) 

dm dy dz 

where the value of q is given at every point of the surface ; then, if 
a, 6, c be supposed to vary in any manner, subject to the above 
conditions, the value of Q will be a unique minimum, when 

dV dV dV 

a = A-r-> o = JK.-^-> c K-^- 

due dy dz 



Proof. 
If we put for the general values of a, b, c, 



then, by substituting these values in equations (5) and (7), we find 
that u, v, w satisfy the general solenoidal condition 

. du dv dw 

(!) T" + -J- + T = - 
dx dy dz 

\s 

We also find, by equations (6) and (8), that at the surfaces of 
discontinuity the values of %, v lf w and u 29 v 2 , w 2 satisfy the 
superficial solenoidal condition 

(2) I(u l u 2 ) + m(v 1 v 2 ) + n(w 1 ^w 2 ) = 0. 
The quantities u, v, w, therefore, satisfy at every point the sole 
noidal conditions as stated in the preceding lemma. 



98.] UNIQUE MINIMUM OF Q. 105 

We may now express Q in terms of u, v, w and V, 



~TT W = 



The last term of Q may be written 2 Jf, where ^f is the quantity 
considered in the lemma, and which we proved to be zero when the 
space is bounded by surfaces, each of which is either equipotential 
or satisfies the condition of equation (10), which may be written 

(4) lu + mv + nw = 0. 
Q is therefore reduced to the sum of the first and second terms. 

In each of these terms the quantity under the sign of integration 
consists of the sum of three squares, and is therefore essentially 
positive or zero. Hence the result of integration can only be 
positive or zero. 

Let us suppose the function V known, and let us find what values 
of u, v, w will make Q a minimum. 

If we assume that at every point u = 0, v = 0, and w 0, these 
values fulfil the solenoidal conditions, and the second term of Q 
is zero, and Q is then a minimum as regards the variation of 
, v, w. 

For if any of these quantities had at any point values differing 
from zero, the second term of Q would have a positive value, and 
Q would be greater than in the case which we have assumed. 

But if u = 0, v = 0, and w = 0, then 



dx dy dz 

Hence these values of a, 3, c make Q a minimum. 

But the values of #, 6, c, as expressed in equations (12), are 
perfectly general, and include all values of these quantities con 
sistent with the conditions of the theorem. Hence, no other values 
of a, b, c can make Q a minimum. 

Again, Q is a quantity essentially positive, and therefore Q is 
always capable of a minimum value by the variation of a y b, c. 
Hence the values of a, b, c which make Q a minimum must have 
a real existence. It does not follow that our mathematical methods 
are sufficiently powerful to determine them. 

Corollary I. If a, b, c and K are given at every point of space, 
and if we write 



106 GENERAL THEOREMS. [98. 

(12) a = % + *, > = K% + *, c = K d ^ + w , 

with the condition (1) 

du dv dw 

~d^ + d^ + ~d~z^ 

then 7, u, v, w can be found without ambiguity from these four 
equations. 

Corollary II. The general characteristic equation 

d dV d dV Cl 



where Fis a finite quantity of single value whose first derivatives 
are finite and continuous except at the surface S, and at that surface 
fulfil the superficial characteristic 



dy 



can be satisfied by one value of 7, and by one only, in the following 
cases. 

Case 1 . When the equations apply to the space within any closed 
surface at every point of which 7 = C. 

For we have proved that in this case #, b, c have real and unique 
values which determine the first derivatives of 7, and hence, if 
different values of 7 exist, they can only differ by a constant. But 
at the surface 7 is given equal to (7, and therefore 7 is determinate 
throughout the space. 

As a particular case, let us suppose a space within which p = 
bounded by a closed surface at which 7=C. The characteristic 
equations are satisfied by making V C for every point within the 
space, and therefore V C is the only solution of the equations. 

Case 2. When the equations apply to the space within any closed 
surface at every point of which 7 is given. 

For if in this case the characteristic equations could be satisfied 
by two different values of V, say 7 and 7 , put U = 7 7 , then 
subtracting the characteristic equation in 7 from that in 7 t we 
find a characteristic equation in U. At the closed surface 7=0 
because at the surface 7 = V, and within the surface the density 
is zero because p = p . Hence, by Case 1, U= throughout the 
enclosed space, and therefore 7 = 7 throughout this space. 



99-] APPLICATION OF THOMSONS THEOREM. 107 

Case 3. When the equations apply to a space bounded by a 
closed surface consisting of two parts, in one of which V is given at 
every point, and in the other 

r ,dV dV dV 

Kl-j- +Km -=- +Kn -=- = a, 
dx dy dz 

where q is given at every point. 

For if there are two values of F 9 let U represent, as before, their 
difference, then we shall have the equation fulfilled within a closed 
surface consisting of two parts, in one of which U = 0, and in the 

other JU dU dU 

I h m -= h n -j = ; 

ax dy dz 

and since U = satisfies the equation it is the only solution, and 
therefore there is but one value of V possible. 

Note. The function V in this theorem is restricted to one value 
at each point of space. If multiple values are admitted, then, 
if the space considered is a cyclic space, the equations may be 
satisfied by values of V containing terms with multiple values. 
Examples of this will occur in Electromagnetism. 

99.] To apply this theorem to determine the distribution of 
electricity in an electrified system, we must make K = 1 throughout 
the space occupied by air, and K=& throughout the space occupied 
by conductors. If any part of the space is occupied by dielectrics 
whose inductive capacity differs from that of air, we must make K 
in that part of the space equal to the specific inductive capacity. 

The value of F, determined so as to fulfil these conditions, will 
be the only possible value of the potential in the given system. 

Green s Theorem shews that the quantity Q, when it has its 
minimum value corresponding to a given distribution of electricity, 
represents the potential energy of that distribution of electricity. 
See Art. 100, equation (11). 

In the form in which Q is expressed as the result of integration 
over every part of the field, it indicates that the energy due to the 
electrification of the bodies in the field may be considered as the 
result of the summation of a certain quantity which exists in every 
part of the field where electrical force is in action, whether elec 
trification be present or not in that part of the field. 

The mathematical method, therefore, in which Q, the symbol 
of electrical energy, is made an object of study, instead of p, the 
symbol of electricity itself, corresponds to the method of physical 
speculation, in which we look for the seat of electrical action in 



108 GENERAL THEOREMS. [lOO. 

every part of the field, instead of confining our attention to the 
electrified bodies. 

The fact that Q attains a minimum value when the components 
of the electric force are expressed in terms of the first derivatives 
of a potential, shews that, if it were possible for the electric force 
to be distributed in any other manner, a mechanical force would 
be brought into play tending to bring the distribution of force 
into its actual state. The actual state of the electric field is 
therefore a state of stable equilibrium, considered with reference 
to all variations of that state consistent with the actual distribution 
of free electricity. 

Green s Theorem. 

100.] The following remarkable theorem was given by George 
Green in his essay On the Application of Mathematics to Electricity 
and Magnetism. 

I have made use of the coefficient K, introduced by Thomson, to 
give greater generality to the statement, and we shall find as we 
proceed that the theorem may be modified so as to apply to the 
most general constitution of crystallized media. 

We shall suppose that U and V are two functions of #, y, z, 
which, with their first derivatives, are finite and continuous within 
the space bounded by the closed surface S. 

We shall also put for conciseness 

d v dU d ^dU d ^dU 

~r K ~^- + -7~ K ^r + ^-K-r = 4Trp, (1) 

das dx dy dy dz dz 

d V d7 d ^dV d ^dV 

and K + K -=- + K -=4770 , (2) 

dy dx dy dy dz dz ^ 

where K is a real quantity, given for each point of space, which 
may be positive or zero but not negative. The quantities p and 
p correspond to volume-densities in the theory of potentials, but 
in this investigation they are -to be considered simply as ab 
breviations for the functions of U and V to which they are here 
equated. 

In the same way we may put 



^ 

~r +mK + nK-j- =4770-, (3) 

dx dy dz 

and lK^+mK^+nK^*]*&. (4) 

dx dy dz 

where I, m, n are the direction-cosines of the normal drawn inwards 



ioo.] GREEN S THEOREM. 109 

from the surface S. The quantities a- and <r correspond to super 
ficial densities, but at present we must consider them as defined by 
the above equations. 

Green s Theorem is obtained by integrating by parts the ex 
pression 



TUT r ^ , (IA 

4 TT M = IKi-j--^- + -J--J- + - r - r )dxdydz (o) 

JJJ \dx dx dy dy dz dz 

throughout the space within the surface S. 

If we consider -j- as a component of a force whose potential is T 3 
and K as a component of a flux, the expression will give the 

work done by the force on the flux. 

If we apply the method of integration by parts, we find 



or 



(7) 



In precisely the same manner by exchanging V and T, we should 

find rr rrr 

4irM=+ 4:TTcrUdS + 1 1 / 47rpc/ dxdydz. (8) 

The statement of Green s Theorem is that these three expressions 
for M are identical, or that 

M = / 1 v r 7dS+ I II p Vdx dydz = / <rUdS + p Udx dy dz 
JJ JJJ JJ JJJ 

dx dy dy dz dz 

Correction of Green s Theorem for Cyclosis. 

There are cases in which the resultant force at any point of a 
certain region fulfils the ordinary condition of having a potential, 
while the potential itself is a many-valued function of the coor 
dinates. For instance, if 



we find Frrtan- 1 -, a many -valued function of x and y, the 
x 

values of V forming an arithmetical series whose common difference 



110 GENERAL THEOREMS. [lOO. 

is 2 TT, and in order to define which of these is to be taken in 
any particular case we must make some restriction as to the line 
along which we are to integrate the force from the point where 
V = to the required point. 

In this case the region in which the condition of having a 
potential is fulfilled is the cyclic region surrounding the axis of z, 
this axis being a line in which the forces are infinite and therefore 
not itself included in the region. 

The part of the infinite plane of xz for which x is positive may 
be taken as a diaphragm of this cyclic region. If we begin at 
a point close to the positive side of this diaphragm, and integrate 
along a line which is restricted from passing through the diaphragm, 
the line-integral will be restricted to that value of V which is 
positive but less than 2 IT. 

Let us now suppose that the region bounded by the closed surface 
S in Green s Theorem is a cyclic region of any number of cycles, 
and that the function V is a many-valued function having any 
number of cyclic constants. 

dV dV dV 

The quantities -= > -r= > and -= will have definite values at all 
ax dy dz 

points within S, so that the volume-integral 

f \ 

\dx dx dy dy dz dz 

/ t/ 

has a definite value, a and p have also definite values, so that if U 
is a single valued function, the expression 



has also a definite value. 

The expression involving V has no definite value as it stands, 
for Fis a many- valued function, and any expression containing it 
is many-valued unless some rule be given whereby we are directed 
to select one of the many values of V at each point of the region. 

To make the value of V definite in a region of n cycles, we must 
conceive n diaphragms or surfaces, each of which completely shuts 
one of the channels of communication between the parts of the 
cyclic region. Each of these diaphragms reduces the number of 
cycles by unity, and when n of them are drawn the region is still 
a connected region but acyclic, so that we can pass from any one 
point to any other without cutting a surface, but only by recon- 
cileable paths. 



100.] INTERPRETATION OF GREENES THEOREM. Ill 

Let $! be the first of these diaphragms, and let the line-integral 
of the force for a line drawn in the acyclic space from a point 
on the positive side of this surface to the contiguous point on 
the negative side be KJ , then ^ is the first cyclic constant. 

Let the other diaphragms, and their corresponding cyclic con 
stants, be distinguished by suffixes from 1 to n, then, since the 
region is rendered acyclic by these diaphragms, we may apply to 
it the theorem in its original form. 

We thus obtain for the complete expression for the first member 
of the equation 



The addition of these terms to the expression of Green s Theorem, 
in the case of many- valued functions, was first shewn to be necessary 
by Helmholtz*, and was first applied to the theorem by Thomson. 

Physical Interpretation of Green s Theorem. 

The expressions a- (IS and pdxdydz denote the quantities of 
electricity existing on an element of the surface S and in an 
element of volume respectively. We may therefore write for either 
of these quantities the symbol e, denoting a quantity of electricity. 
We shall then express Green s Theorem as follows 



where we have two systems of electrified bodies, e standing in 
succession for e lt e. 2 , &c., any portions of the electrification of the 
first system, and Y denoting the potential at any point due to all 
these portions, while e stands in succession for e^, e. 2 , &c., portions 
of the second system, and V denotes the potential at any point 
due to the second system. 

Hence Ve denotes the product of a quantity of electricity at a 
point belonging to the second system into the potential at that 
point due to the first system, and 2 ( Ye } denotes the sum of all 
such quantities, or in other words, 2 ( Ye } represents that part of 
the energy of the whole electrified system which is due to the 
action of the second system on the first. 

In the same way 2 ( Y e) represents that part of the energy of 

* Ueber Integrate der Hydrodynamischen Gleichungen welche den Wirbelbe- 
wegungen entsprechen, Crelle, 1858. Translated by Tait in Phil. Mag., 1867, ("). 
t On Vortex Motion, Trans. It. 8. Edin., xxv. part i. p. 241 (1868). 



112 GENERAL THEOREMS. [lOO. 

the whole system which is due to the action of the first system on 
the second. 

If we define V as 2 (-), where r is the distance of the quantity e 

of electricity from the given point, then the equality between these 
two values of M may be obtained as follows, without Green s 
Theorem 



This mode of regarding the question belongs to what we have 
called the direct method, in which we begin by considering certain 
portions of electricity, placed at certain points of space, and acting 
on one another in a way depending on the distances between these 
points, no account being taken of any intervening medium, or of 
any action supposed to take place in the intervening space. 

Green s Theorem, on the other hand, belongs essentially to what 
we have called the inverse method. The potential is not supposed 
to arise from the electrification by a process of summation, but 
the electrification is supposed to be deduced from a perfectly 
arbitrary function called the potential by a process of differen 
tiation. 

In the direct method, the equation is a simple extension of the 
law that when any force acts directly between two bodies, action 
and reaction are equal and opposite. 

In the inverse method the two quantities are not proved directly 
to be equal, but each is proved equal to a third quantity, a triple 
integral which we must endeavour to interpret. 

If we write R for the resultant electromotive force due to the 
potential V, and 1 3 m> n for the direction-cosines of R, then, by 

Art. 71, 

dV - D1 dV dV 

-- =-- = EL -- =- = Em. -- 7- = En. 
dx dy dz 

If we also write E for the force due to the second system, and 
I j m , ri for its direction-cosines, 

ar vr AV dv , 

-- ^ = K i . --- $ = K m , -- = = K n ; 
dx dy dz 

and the quantity M may be written 

M = JL jjf(KRK cos e) dx dy dz, (10) 



ioi.] GREEN S FUNCTION. 113 

where cos e = IV -f mm + nri , 

e being the angle between the directions of R and Iff. 

Now if K is what we have called the coefficient of electric 
inductive capacity, thenJOZ will be the electric displacement due 
to the electromotive force R, and the product \KRRf cose will 
represent the work done by the force Rf on account of the dis 
placement caused by the force R, or in other words, the amount 
of intrinsic energy in that part of the field due to the mutual 
action of R and Rf. 

We therefore conclude that the physical interpretation of Green s 
theorem is as follows : 

If the energy which is known to exist in an electrified system 
is due to actions which take place in all parts of the field, and 
not to direct action at a distance between the electrified bodies, 
then that part of the intrinsic energy of any part of the field 
upon which the mutual action of two electrified systems depends 
is KRRf cos e per unit of volume. 

The energy of an electrified system due to its action on itself is, 
by Art, 85, 4S(*F), 

which is by Green s theorem, putting U F, 






and this is the unique minimum value of the integral considered 
in Thomson s theorem. 

Green s Function. 

101.] Let a closed surface S be maintained at potential zero. 
Let P and Q be two points on the positive side of the surface S 
(we may suppose either the inside or the outside positive), and 
let a small body charged with unit of electricity be placed at P; 
the potential at the point Q will consist of two parts, of which one 
is due to the direct action of the electricity on P, while the other 
is due to the action of the electricity induced on S by P. The 
latter part of the potential is called Green s Function, and is 
denoted by G pq . 

This quantity is a function of the positions of the two points 
P and Q, the form of which depends on that of the surface S. It 
has been determined in the case in which 8 is a sphere, and in 
a very few other cases. It denotes the potential at Q due to the 
electricity induced on S by unit of electricity at P. 

VOL. I. I 



114 GENERAL THEOREMS. [lOI. 

The actual potential at any point Q due to the electricity at P 
and on 8 is 



where r pq denotes the distance between P and Q. 

At the surface S and at all points on the negative side of S 9 the 
potential is zero, therefore i 

pa ~~ ~~ \ / 

where the suffix a indicates that a point A on the surface 8 is taken 
instead of Q. 

Let <r pa ? denote the surface-density induced by P at a point A 
of the surface 8, then, since G pq is the potential at Q due to the 
superficial distribution, 



where dS is an element of the surface 8 at A , and the integration 
is to be extended over the whole surface 8. 

But if unit of electricity had been placed at Q, we should have 
had by equation (1), i 



where v qa is the density induced by Q on an element dS at A, and 
^ is the distance between A and A . Substituting this value of 
- in the expression for G pq , we find 



Since this expression is not altered by changing p into g and 
into we find that g _ ^ . , fi v 

** ~ VTqp * (6) 

a result which we have already shewn to be necessary in Art. 88, 
but which we now see to be deducible from the mathematical process 
by which Green s function may be calculated. 

If we assume any distribution of electricity whatever, and place 
in the field a point charged with unit of electricity, and if the 
surface of potential zero completely separates the point from the 
assumed distribution, then if we take this surface for the surface S t 
and the point for P, Green s function, for any point on the same 
side of the surface as P, will be the potential of the assumed dis 
tribution on the other side of the surface. In this way we may 
construct any number of cases in which Green s function can be 



102.] MINIMUM VALUE OF Q. 116 

found for a particular position of P. To find the form of the 
function when the form of the surface is given and the position 
of P is arbitrary, is a problem of far greater difficulty, though, 
as we have proved, it is mathematically possible. 

Let us suppose the problem solved, and that the point P is 
taken within the surface. Then for all external points the potential 
of the superficial distribution is equal and opposite to that of P. 
The superficial distribution is therefore centrobaric *, and its action 
on all external points is the same as that of a unit of negative 
electricity placed at P. 

Method of Approximating to the Values of Coefficients of Capacity, fyc. 

102.] Let a region be completely bounded by a number of 
surfaces , S^ S 2 , &c., and let K be a quantity, positive or zero 
but not negative, given at every point of this region. Let V 
be a function subject to the conditions that its values at the 
surfaces S 19 S 2t &c. are the constant quantities C lt C 2 , &c., and that 
at the surface S Q dV /^ 

^ = 
where v is a normal to the surface . Then the integral 



taken over the whole region, has a unique minimum when V satisfies 

the equation d ^dV d ^dV d ^dV , . 

A -7 \--j-K \- -=- K -=- = (3) 

dx dx dy dy dz dz 

throughout the region, as well as the original conditions. 

We have already shewn that a function V exists which fulfils the 
conditions (1) and (3), and that it is determinate in value. We 
have next to shew that of all functions fulfilling the surface-con 
ditions it makes Q a minimum. 

Let F be the function which satisfies (1) and (3), and let 

F=F +*7 (4) 

be a function which satisfies (1). 

It follows from this that at the surfaces S 19 8 2 , &c. U= 0. 

The value of Q becomes 



Thomson and Tait s Natural Philosophy, 526. 
I 2 



116 GENERAL THEOREMS. [lO2. 

Let us confine our attention to the last of these three groups 
of terms, merely observing that the other groups are essentially 
positive. By Green s theorem 

/7F" fJIT flV //77 HV /7/7v CC HV 

/ttr Q U U ttr Q ClU (tr Q au \ ., , , ///rr/"7^ 

V dx dx dy dy dz dz JJ dv 

_[[fu( K^+K^ + ~K^}d xddz (6) 
J JJ ^da) dx dy dy dz dz 



fff 



dy dy 

the first integral of the second member being extended over the 
surface of the region and the second throughout the enclosed space. 
But on the surfaces S^ S 2> &c. U= 0, so that these contribute 
nothing to the surface-integral. 

Again, on the surface S QJ -~- = 0, so that this surface contributes 

Cv V 

nothing to the integral. Hence the surface-integral is zero. 

The quantity within brackets in the volume-integral also dis 
appears by equation (3), so that the volume-integral is also zero. 
Hence Q is reduced to 



Both these quantities are essentially positive, and therefore the 
minimum value of Q is when 

-^-^-o (8) 

dx dy dz 

or when U is a constant. But at the surfaces S, &c. U = 0. Hence 
U = everywhere, and F gives the unique minimum value of Q. 

Calculation of a Superior Limit of the Coefficients of Capacity. 

The quantity Q in its minimum form can be expressed by means 
of Green s theorem in terms of F 19 F 2 , &c., the potentials of S lt S 2J 
and JE 19 U 2 , &c., the charges of these surfaces, 

q = ^(r 1 M 1 +r t JB t +& .) i (9) 

or, making use of the coefficients of capacity and induction as defined 
in Article 87, 

Q = i(^i 2 ^ii+^ 2 ^2 + &c.)+F 1 r 2 ^ 12 +&c. (10) 

The accurate determination of the coefficients q is in general 
difficult, involving the solution of the general equation of statical 
electricity, but we make use of the theorem we have proved to 
determine a superior limit to the value of any of these coefficients. 



102.] METHOD OF APPEOXIMATION. 117 

To determine a superior limit to the coefficient of capacity q u , 
make V = 1, and V 2J V^ &c. each equal to zero, and then take 
any function V which shall have the value 1 at S 13 and the value 
at the other surfaces. 

From this trial value of V calculate Q by direct integration, 
and let the value thus found be Q . We know that Q is not less 
than the absolute minimum value Q, which in this case is \ q n . 

Hence q u is not greater than 2 (JX. (11) 

If we happen to have chosen the right value of the function 
F, then q n = 2 Q , but if the function we have chosen differs 
slightly from the true form, then, since Q is a minimum, Q will 
still be a close approximation to the true value. 

Superior Limit of the Coefficients of Potential. 

We may also determine a superior limit to the coefficients of 
potential denned in Article 86 by means of the minimum value 
of the quantity Q in Article 98, expressed in terms of a, b, c. 

By Thomson s theorem, if within a certain region bounded by the 
surfaces S , 15 &c. the quantities a, 6, c are subject to the condition 

da clb dc 



and if la + ml + nc = q (1 3) 

be given all over the surface, where I, m, n are the direction-cosines 
of the normal, then the integral 

e = tiff s: (*+**+*)*"& * ( 14 ) 

is an absolute and unique minimum when 

dV ^clV ^dV , . 

a K-^-t l = K^-> c = K-j-> (15) 

dx ay dz 

When the minimum is attained Q is evidently the same quantity 
which we had before. 

If therefore we can find any form for a, b, c which satisfies the 
condition (12) and at the same time makes 

JS. 2 &c. , (16) 

and if Q" be the value of Q calculated by (14) from these values of 
a, b, c, then Q" is not less than 

(17) 



118 GENERAL THEOREMS. [lO2. 

If we take the case in which one of the surfaces, say S 2 , sur 
rounds the rest at an infinite distance, we have the ordinary case 
of conductors in an infinite region ; and if we make E z = U 19 and 
E for all the other surfaces, we have F~ 2 = at infinity, and 

2 Q" 
jp n is not greater than ^- 

In the very important case in which the electrical action is 
entirely between two conducting surfaces S 1 and $ 2 , of which S 2 
completely surrounds ^ and is kept at potential zero, we have 
E l = fi 2 and q n p u = 1. 

Hence in this case we have 

XT 

q u not less than -^- f \ (18) 

and we had before ,. -, 

q n not greater than 2 Q ; (19) 

so that we conclude that the true value of q ll9 the capacity of the 
internal conductor, lies between these values. 

This method of finding superior and inferior limits to the values 
of these coefficients was suggested by a memoir On the Theory 
of Resonance/ by the Hon. J. W. Strutt, Phil. Trans., 1871. See 
Art. 308. 



CHAPTER V. 

MECHANICAL ACTION BETWEEN ELECTRIFIED BODIES. 

103.] Let Y C be any closed equipotential surface, C being 
a particular value of a function T, the form of which we suppose 
known at every point of space. Let the value of V on the outside 
of this surface be T\, and on the inside 7. 2 . Then, by Poisson s 
equation 

(1) 



we can determine the density p at every point on the outside, and 
the density p 2 at every point on the inside of the surface. We shall 
call the whole electrified system thus explored on the outside U 13 
and that on the inside E 2 . The actual value of Y arises from the 
combined action of both these systems. 

Let R be the total resultant force at any point arising from 
the action of E l and K 2 , R is everywhere normal to the equi 
potential surface passing through the point. 

Now let us suppose that on the equipotential surface Y = C 
electricity is distributed so that at any point of the surface at 
which the resultant force due to E and K 2 reckoned outwards 
is Rj the surface-density is a, with the condition 

R = 4 T: o- ; (2) 

and let us call this superficial distribution the electrified surface S, 
then we can prove the following theorem relating to the action of 
this electrified surface. 

If any equipotential surface belonging to a given electrified 
system be coated with electricity, so that at each point the surface- 
density o- = , where R is the resultant force, due to the original 

47T 

electrical system, acting outwards from that point of the surface, 
then the potential due to the electrified surface at any point on 



120 ELECTRIC ATTRACTION. [103. 

the outside of that surface will be equal to the potential at the 
same point due to that part of the original system which was on 
the inside of the surface, and the potential due to the electrified 
surface at any point on the inside added to that due to the part of 
the original system on the outside will be equal to C, the potential 
of the surface. 

For let us alter the original system as follows : 

Let us leave everything the same on the outside of the surface, 
but on the inside let us make T 2 everywhere equal to C, and let us 
do away with the electrified system E z on the inside of the surface, 
and substitute for it a surface-density a at every point of the 
surface S, such that It ^n a. (3) 

Then this new arrangement will satisfy the characteristics of V at 
every point. 

For on the outside of the surface both the distribution of elec 
tricity and the value of V are unaltered, therefore, since V originally 
satisfied Laplace s equation, it will still satisfy it. 

On the inside V is constant and p zero. These values of V and p 
also satisfy the characteristic equations. 

At the surface itself, if V-^ is the potential at any point on the 
outside and V^ that on the inside, then, if I, m, n are the direction- 
cosines of the normal to the surface reckoned outwards, 

,dV, dV, dV, (A . 

I i +m i+^-i = -^ = -471(7 ; (4) 

dx dy dz 

and on the inside the derivatives of V vanish, so that the superficial 
characteristic 

d7 2 . f dV, dV^ f dV, dV^ 
--T^)+^(-7- 1 --r-)+^(-T J -- --7-^ + 4 w<r=0 (5) 
dx dx ^ dy dy ^ dz dz 

J u 

is satisfied at every point of the surface. 

Hence the new distribution of potential, in which it has the 
old value on the outside of the surface and a constant value on 
the inside, is consistent with the new distribution of electricity, 
in which the electricity in the space within the surface is removed 
and a distribution of electricity on the surface is substituted for 
it. Also, since the original value of V-^ vanishes at infinity, the 
new value, which is the same outside the surface, also fulfils this 
condition, and therefore the new value of V is the sole and only 
value of V belonging to the new arrangement of electricity. 



1 04.] EQUIVALENT ELECTRIFIED SURFACE. 121 

On the Mechanical Action and Reaction of the Systems E 1 and E 2 . 

104.] If we now suppose the equipotential surface V C to 
become rigid and capable of sustaining the action of forces, we 
may prove the following theorem. 

If on every element dS of an equipotential surface a force 

R 2 dS be made to act in the direction of the normal reckoned 
Sir 

outwards, where R is the electrical resultant force along the 
normal, then the total statical effect of these forces on the 
surface considered as a rigid shell will be the same as the total 
statical effect of the electrical action of the electrified system E 
outside the shell on the electrified system E 2 inside the shell, the 
parts of the interior system E 2 being supposed rigidly connected 
together. 

We have seen that the action of the electrified surface in the last 
theorem on any external point was equal to that of the internal 
system E 2J and, since action and reaction are equal and opposite, 
the action of any external electrified body on the electrified surface, 
considered as a rigid system, is equal to that on the internal system 
E 2 . Hence the statical action of the external system E- on the 
electrified surface is equal in all respects to the action of E 1 on the 
internal system E. 2 . 

But at any point just outside the electrified surface the resultant 
force is R in a direction normal to the surface, and reckoned positive 
when it acts outwards. The resultant inside the surface is zero, 
therefore, by Art. 79, the resultant force acting on the element 
dS of the electrified surface is \RadS, where cr is the surface- 
density. 

Substituting the value of a- in terms of R from equation (2), and 
denoting by p dS the resultant force on the electricity spread over 
the element dS, we find 



O7T 

This force always acts along the normal and outwards, whether 
R be positive or negative, and may be considered as equal to a 

pressure p= R 2 acting on the surface from within, or to a tension 
STT 

of the same numerical value acting from without. 

* See Sir W. Thomson On the Attractions of Conducting and Non-conducting 
Electrified Bodies, Cambridge Mathematical Journal, May 1843, and Reprint, 
Art. VII, 147. 



122 ELECTKIC ATTRACTION. 

Now R is the resultant due to the combined action of the 
external system E and the electrification of the surface S. Hence 
the effect of the pressure/? on each element of the inside of the surface 
considered as a rigid body is equivalent to this combined action. 

But the actions of the different parts of the surface on each other 
form a system in equilibrium, therefore the effect of the pressure p on 
the rigid shell is equivalent in all respects to the electric attraction 
of EI on the shell, and this, as we have before shewn, is equivalent 
to the electric attraction of E-^ on E 2 considered as a rigid system. 

If we had supposed the pressure p to act on the outside of the 
shell, the resultant effect would have been equal and opposite, that 
is, it would have been statically equivalent to the action of the 
internal system E 2 on the external system E^. 

Let us now take the case of two electrified systems E^ and 
E 2t such that two equipotential surfaces F = C l and F = C 2 , which 
we shall call S l and S 2 respectively, can be described so that E^ is 
exterior to S 19 and S l surrounds S 2 , and E% lies within S 2 . 

Then if R l and R 2 represent the resultant force at any point of 
S l and S 2 respectively, and if we make 



the mechanical action between E l and E 2 is equivalent to that 
between the shells ^ and S 2 , supposing every point of S 1 pressed 
inwards, that is, towards S 2 with a pressure p lt and every point of 
S 2 pressed outwards, that is, towards S 1 with a pressure p 2 . 

105.] According to the theory of action at a distance the action 
between E^ and E 2 is really made up of a system of forces acting in 
straight lines between the electricity in E l and that in H 29 and the 
actual mechanical effect is in complete accordance with this theory. 

There is, however, another point of view from which we may 
examine the action between E l and E 2 . When we see one body 
acting on another at a distance, before we assume that the one 
acts directly on the other we generally inquire whether there is 
any material connexion between the two bodies, and if we find 
strings, or rods, or framework of any kind, capable of accounting 
for the observed action between the bodies, we prefer to explain 
the action by means of the intermediate connexions, rather than 
admit the notion of direct action at a distance. 

Thus when two particles are connected by a straight or curved 
rod, the action between the particles is always along the line joining 
them, but we account for this action by means of a system of 



106.] INTERNAL FORCES. 123 

internal forces in the substance of the rod. The existence of these 
internal forces is deduced entirely from observation of the effect 
of external forces on the rod, and the internal forces themselves 
are generally assumed to be the resultants of forces which act 
between particles of the rod. Thus the observed action between 
two distant particles is, in this instance, removed from the class 
of direct actions at a distance by referring it to the intervention 
of the rod ; the action of the rod is explained by the existence 
of internal forces in its substance ; and the internal forces are 
explained by means of forces assumed to act between the particles 
of which the rod is composed, that is, between bodies at distances 
which though small must be finite. 

The observed action at a considerable distance is therefore ex 
plained by means of a great number of forces acting between 
bodies at very small distances, for which we are as little able to 
account as for the action at any distance however great. 

Nevertheless, the consideration of the phenomenon, as explained 
in this way, leads us to investigate the properties of the rod, and 
to form a theory of elasticity \\hich we should have overlooked 
if we had been satisfied with the explanation by action at a distance, 

106.] Let us now examine the consequence of assuming that the 
action between electrified bodies can be explained by the inter 
mediate action of the medium between them, and let us ascertain 
what properties of the medium will account for the observed action. 

We have first to determine the internal forces in the medium, 
and afterwards to account for them if possible. 

In order to determine the internal forces in any case we proceed 
as follows : 

Let the system M be in equilibrium under the action of the 
system of external forces F. Divide M by an imaginary surface 
into two parts, M^ and M. 2 , and let the systems of external forces 
acting on these parts respectively be F L and F. 2 . Also let the 
internal forces acting on M l in consequence of its connexion with 
M. 2 be called the system /. 

Then, since M l is in equilibrium under the action of F l and /, 
it follows that / is statically equivalent to F l reversed. 

In the case of the electrical action between two electrified systems 
E 1 and E. 2 > we described two closed equipotential surfaces entirely 
surrounding E 2 and cutting it off" from E 13 and we found that the 
application of a certain normal pressure at every point of the inner 
side of the inner surface, and on the outer side of the outer surface, 



124 ELECTRIC ATTRACTION. [lo6. 

would, if these surfaces were each rigid, act on the outer surface 
with a resultant equal to that of the electrical forces on the outer 
system U lt and on the inner surface with a resultant equal to that 
of the electrical forces on the inner system. 

Let us now consider the space between the surfaces, and let us 
suppose that at every point of this space there is a tension in the 

direction of R and equal to R* per unit of area. This tension 

07T 

will act on the two surfaces in the same way as the pressures on 
the other side of the surfaces, and will therefore account for the 
action between E^ and E 2 , so far as it depends on the internal force 
in the space between S and S 2 . 

Let us next investigate the equilibrium of a portion of the shell 
bounded by these surfaces and separated from the rest by a surface 
everywhere perpendicular to the equipotential surfaces. We may 
suppose this surface generated by describing any closed curve on 
8 lf and drawing from every point of this curve lines of force till 
they meet S 2 . 

The figure we have to consider is therefore bounded by the two 
equipotential surfaces 8 l and S 2 , and by a surface through which 
there is no induction, which we may call S . 

Let us first suppose that the area of the closed curve on 8 1 is very 
small, call it dS lt and that C 2 = C l + dT r . 

The portion of space thus bounded may be regarded as an element 
of volume. If v is the normal to the equipotential surface, and 
dS the element of that surface, then the volume of this element 
is ultimately dSdv. 

The induction through dS l is RdS lt and since there is no in 
duction through S , and no free electricity within the space con 
sidered, the induction through the opposite surface dS 2 will be 
equal and opposite, considered with reference to the space within 
the closed surface. 

There will therefore be a quantity of electricity 

%!P.T15*i 
on the first equipotential surface, and a quantity 

* 2 = l^ R * d8 * 

on the second equipotential surface, with the condition 

= 0. 



1O6.] RESULTANT OF ELECTRIC TENSIONS. 125 

Let us next consider the resultant force due to the action of the 
electrified systems on these small electrified surfaces. 

The potential within/ the surface S l is constant and equal to C lt 
and ^without the surface S 2 it is constant and equal to C. 2 . In the 
shell between these surfaces it is continuous from Q to C 2 . 

Hence the resultant force is zero except within the shell. 

The electrified surface of the shell itself will be acted on by forces 
which are the arithmetical means of the forces just within and just 
without the surface, that is, in this case, since the resultant force 
outside is zero, the force acting on the superficial electrification is 
one-half of the resultant force just within the surface. 

Hence, if XdSdv be the total moving force resolved parallel 
to x, due to the electrical action on both the electrified surfaces of 
the element dSdv, 



where the suffixes denote that the derivatives of Vare to be taken 
at dS 1 and dS. 2 respectively. 

Let I, m, n be the direction-cosines of F, the normal to the 
equipotential surface, then making 

dx I dv, dy m dv, and dz = n dv, 



. ( ^ fl 

- ) = (-J-) +(-7-5 +m -= r + n -j ~) dv + &c. ; 
dx z dx -L \ dx 2 dxdy 



and since e 2 = e l , we may write the value of X 

XdSdv = i e, -j- (l +m-j- +n -j-} dv. 
1 dx ^ dx dy dz 

But e, = -- -EdS and (l* +m ( -j- +n-j-) = R; 

v 



dx d dz 

therefore XdSdv = R~ 

Sir dx 

or, if we write 

dr 



v i dp v i dp 7 i d P . 

then *=i^> Y =*fy Z =^> 

or the force in any direction on the element arising from the action 
of the electrified system on the two electrified surfaces of the 
element is equal to half the rate of increase of p in that direction 
multiplied by the volume of the element. 



126 ELECTRIC ATTRACTION. [106. 

This result is the same if we substitute for the forces acting on 
the electrified surfaces an imaginary force whose potential is \p^ 
acting on the whole volume of the element and soliciting it to 
move so as to increase \p. 

If we now return to the case of a figure of finite size, bounded 
by the equipotential surfaces S l and S 2 and by the surface of no 
induction S Q9 we may divide the whole space into elements by a 
series of equipotential surfaces and two series of surfaces of no 
induction. The charges of electricity on those faces of the elements 
which are in contact will be equal and opposite, so that the total 
effect will be that due to the electrical forces acting on the charges 
on the surfaces S l and $ 2 , and by what we have proved this will be 
the same as the action on the whole volume of the figure due to a 
system of forces whose potential is \p. 

But we have already shewn that these electrical forces are 
equivalent to a tension p applied at all points of the surfaces S 1 
and S. 2 . Hence the effect of this tension is to pull the figure in 
the direction in which p increases. The figure therefore cannot be 
in equilibrium unless some other forces act on it. 

Now we know that if a hydrostatic pressure p is applied at every 
point of the surface of any closed figure, the effect is equal to 
that of a system of forces acting on the whole volume of the figure 
and having a potential p. In this case the figure is pushed in 
the direction in which p diminishes. 

We can now arrange matters so that the figure shall be in 
equilibrium. 

At every point of the two equipotential surfaces S l and $ 2 , let 
a tension p be applied, and at every point of the surface of no 
induction $ let a pressure = p be applied. These forces will keep 
the figure in equilibrium. 

For the tension p may be considered as a pressure p combined 
with a tension 2 p. We have then a hydrostatic pressure^? acting 
at every point of the surface, and a tension 2 p acting on ^ and S 2 
only. 

The effect of the tension 2p at every point of Sj_ and S 2 is double 
of that which we have just calculated, that is, it is equal to that 
of forces whose potential is p acting on the whole volume of the 
figure. The effect of the pressure p acting on the whole surface 
is by hydrostatics equal and opposite to that of this system of 
forces, and will keep the figure in equilibrium. 

107.] We have now determined a system of internal forces in 



1 07.] STRESS IN A DIELECTRIC MEDIUM. 127 

the medium which is consistent with the phenomena so far as 
we have examined them. We have found that in order to account 
for the electric attraction between distant bodies without admitting 
direct action, we must assume the existence of a tension p at every 
point of the medium in the direction of the resultant force R at 
that point. In order to account for the equilibrium of the medium 
itself we must further suppose that in every direction perpendicular 
to R there is a pressure p*. 

By establishing the necessity of assuming these internal forces 
in the theory of an electric medium, we have advanced a step in 
that theory which will not be lost though we should fail in 
accounting for these internal forces, or in explaining the mechanism 
by which they can be maintained in air, glass, and other dielectric 
media. 

We have seen that the internal stresses in solid bodies can be 
ascertained with precision, though the theories which account for 
these stresses by means of molecular forces may still be doubtful. 
In the same way we may estimate these internal electrical forces 
before we are able to account for them. 

In order, however, that it may not appear as if we had no 
explanation of these internal forces, we shall shew that on the 
ordinary theory they must exist in a shell bounded by two equipo- 
tential surfaces, and that the attractions and repulsions of the elec 
tricity on the surfaces of the shell are sufficient to account for them. 

Let the first surface S l be electrified so that the surface-density is 



and the second surface S 2 so that the surface-density is 
* 2 = ^j 

then, if we suppose that the value of V is C at every point within 
S 19 and C 2 at every point outside of S. 2 , the value of F between these 
surfaces remaining as before, the characteristic equation of Fwill 
be satisfied everywhere, and V is therefore the true value of the 
potential. 

We have already shewn that the outer and inner surfaces of the 
shell will be pulled towards each other with a force the value of 
which referred to unit of surface is p, or in other words, there is a 
tension p in the substance of the shell in the direction of the lines 
of force. 

* See Faraday, Exp. Res. (1224) and (1297). 



128 ELECTRIC ATTRACTION. [108. 

If we now conceive the shell divided into two segments by a 
surface of no induction, the two parts will experience electrical 
forces the resultants of which will tend to separate the parts with 
a force equivalent to the resultant force due to a pressure p acting 
on every part of the surface of no induction which divides them. 

This illustration is to be taken merely as an explanation of what 
is meant by the tension and pressure, not as a physical theory to 
account for them. 

108.] We have next to consider whether these internal forces 
are capable of accounting for the observed electrical forces in every 
case, as well as in the case where a closed equipotential surface can 
be drawn surrounding one of the electrified systems. 

The statical theory of internal forces has been investigated by 
writers on the theory of elasticity. At present we shall require only 
to investigate the effect of an oblique tension or pressure on an 
element of surface. 

Let p be the value of a tension referred to unit of a surface to 
which it is normal, and let there be no tension or pressure in any 
direction normal to p. Let the direction-cosines of p be I, m, n. 
Let dy dz be an element of surface normal to the axis of x, and let 
the effect of the internal force be to urge the parts on the positive 
side of this element with a force whose components are 

p xx dy dz in the direction of #, 

Pxydydz y, and 

p xz dy dz z. 

From every point of the boundary of the element dy dz let lines 
be drawn parallel to the direction of the tension j9, forming a prism 
whose axis is in the line of tension, and let this prism be cut by a 
plane normal to its axis. 

The area of this section will be I dy dx, and the whole tension 
upon it will be p I dy d^ and since there is no action on the sides 
of the prism, which are normal to jo, the force on the base dy dz 
must be equivalent to the force p I dy dd acting in the direction 
(I, m, n). Hence the component in the direction of #, 
Pxx c ty dz = pi 2 dy dz ; or 

P* = PP- 
Similarly p xy plm, (1) 

p xz = pin. 

If we now combine with this tension two tensions p and p" in 
directions (I , m , n ) and (l" } m" , n") respectively, we shall have 



IO8.] COMPONENTS OF STRESS. 129 



p xy = pirn + p r m + p" I" m", (2) 

p xz = pln+p l n +p"l"n". 

In the case of the electrical tension and pressure the pressures 
are numerically equal to the tension at every point, and are in 
directions at right angles to the tension and to each other. Hence, 
putting p =p"=-p, (3) 



we find j^ = (2^ 2 

(5) 



for the action of the combined tension and pressures. 

Also, since p = - R 2 , where R denotes the resultant force, and 

OTT 

since El = X, Em = Y, En Z, 

p xx = (X*-Y* 



, = />... (6) 

STT 



where X, J", ^ are the components of R, the resultant electromotive 
force. 

The expressions for the component internal forces on surfaces 
normal to y and z may be written down from symmetry. 

To determine the conditions of equilibrium of the element dxdydz. 

This element is bounded by the six planes perpendicular to the 
axes of coordinates passing through the points (a?, y, z) and (x -f dx t 
y + dy, z + dz). 

The force in the direction of x which acts on the first face dy dz 
is p xx dydz, tending to draw the element towards the negative 
side. On the second face dy dz, for which x has the value x -f dx^ 
the tension p xx has the value 

p xx dy dz+ (fa Pxx) <fa dy dz, 

and this tension tends to draw the element in the positive direction. 
If we next consider the two faces dzdx with respect to the 

VOL. I. K 



130 ELECTRIC ATTRACTION. [108. 

tangential forces urging them in the direction of x, we find the 
force on the first face p yx dz dx> and that on the second 

p yx dz dx + (-j- p 



Similarly for the faces dx dy, we find that a force p zx dx dy acts 
on the first face, and 

p KX dx dy -f (_j-p zx ) dx dy dz 

on the second in the direction of x. 

If dxdydz denotes the total effect of all these internal forces 
acting parallel to the axis of x on the six faces of the element, we find 

f dx dy dz = (-j-p xx + -7- p vx + - r p zx ) dx dy dz : 
dx dy dz 

or, denoting by f the internal force, referred to unit of volume, and 
resolved parallel to the axis of x, 

= d_ d_ i_ 

with similar expressions for r; and the component forces in the 
other directions *. 

Differentiating the values of p xx ,p yx , and p zx given in equations 
(6), we find 

But by Art. 77 

,dX dY dZ^ 



Hence ( 

Similarly 77 = p7, (10) 

C=pZ. 

Thus, the resultant of the tensions and pressures which we have 
supposed to act upon the surface of the element is a force whose 
components are the same as those of the force, which, in the 
ordinary theory, is ascribed to the action of electrified bodies on the 
electricity within the element. 

If, therefore, we admit that there is a medium in which there 
is maintained at every point a tension p in the direction of the 

* This investigation may be compared with that of the equation of continuity 
in hydrodynamics, and with others in which the effect on an element of volume 
is deduced from the values of certain quantities at its bounding surface. 



no.] FARADAY S THEORY. 131 

resultant electromotive force R> and such that R 2 = 8717?, combined 
with an equal pressure p in every direction at right angles to the 
resultant 7?, then the mechanical effect of these tensions and 
pressures on any portion of the medium, however bounded, will be 
identical with the mechanical effect of the electrical forces according 
to the ordinary theory of direct action at a distance. 

109.] This distribution of stress is precisely that to which Fara 
day was led in his investigation of induction through dielectrics. 
He sums up in the following words : 

(1297) The direct inductive force, which may be conceived to 
be exerted in lines between the two limiting and charged con 
ducting surfaces, is accompanied by a lateral or transverse force 
equivalent to a dilatation or repulsion of these representative lines 
(1224.); or the attracting force which exists amongst the par 
ticles of the dielectric in the direction of the induction is ac 
companied by a repulsive or a diverging force in the transverse 
direction. 

(1298) Induction appears to consist in a certain polarized state 
of the particles, into which they are thrown by the electrified body 
sustaining the action, the particles assuming positive and negative 
points or parts, which are symmetrically arranged with respect 
to each other and the inducting surfaces or particles. The state 
must be a forced one, for it is originated and sustained only by 
force, and sinks to the normal or quiescent state when that force 
is removed. It can be continued only in insulators by the same 
portion of electricity, because they only can retain this state of the 
particles. 

This is an exact account of the conclusions to which we have 
been conducted by our mathematical investigation. At every point 
of the medium there is a state of stress such that there is tension 
along the lines of force and pressure in all directions at right angles 
to these lines, the numerical magnitude of the pressure being equal 
to that of the tension, and both varying as the square of the 
resultant force at the point. 

The expression electric tension has been used in various senses 
by different writers. I shall always use it to denote the tension 
along the lines of force, which, as we have seen, varies from point 
to point, and is always proportional to the square of the resultant 
force at the point. 

110.] The hypothesis that a state of stress of this kind exists 
in a fluid dielectric, such as air or turpentine, may at first sight 

K 2 



132 ELECTRIC ATTRACTION. [ill. 

appear at variance with the established principle that at any point 
in a fluid the pressures in all directions are equal. But in the 
deduction of this principle from a consideration of the mobility 
and equilibrium of the parts of the fluid it is taken for granted 
that no action such as that which we here suppose to take place 
along- the lines of force exists in the fluid. The state of stress 
which we have been studying is perfectly consistent with the 
mobility and equilibrium of the fluid, for we have seen that, if 
any portion of the fluid is devoid of electric charge, it experi 
ences no resultant force from the stresses on its surface, however 
intense these may be. It is only when a portion of the fluid 
becomes charged, that its equilibrium is disturbed by the stresses 
on its surface, and we know that in this case it actually tends to 
move. Hence the supposed state of stress is not inconsistent with 
the equilibrium of a fluid dielectric. 

The quantity Q, which was investigated in Thomson s theorem, 
Art. 98, may be interpreted as the energy in the medium due to 
the distribution of stress. It appears from that theorem that the 
distribution of stress which satisfies the ordinary conditions also 
makes Q an absolute minimum. Now when the energy is a 
minimum for any configuration, that configuration is one of equi 
librium, and the equilibrium is stable. Hence the dielectric, 
when subjected to the inductive action of electrified bodies, will 
of itself take up a state of stress distributed in the way we have 
described. 

It must be carefully borne in mind that we have made only one 
step in the theory of the action of the medium. We have supposed 
it to be in a state of stress, but we have not in any way accounted 
for this stress, or explained how it is maintained. This step, 
however, seems to me to be an important one, as it explains, by 
the action of the consecutive parts of the medium, phenomena which 
were formerly supposed to be explicable only by direct action at 
a distance. 

111.] I have not been able to make the next step, namely, to 
account by mechanical considerations for these stresses in the 
dielectric. I therefore leave the theory at this point, merely 
stating what are the other parts of the phenomenon of induction 
in dielectrics. 

I. Electric Displacement. When induction takes place in a 
dielectric a phenomenon takes place which is equivalent to a 
displacement of electricity in the direction of the induction. For 



III.] ELECTRIC POLARIZATION. 133 

instance, in a Leyden jar, of which the inner coating is charged 
positively and the outer coating negatively, the displacement in 
the substance of the glass is from within outwards. 

Any increase of this displacement is equivalent, during the time 
of increase, to a current of positive electricity from within outwards, 
and any diminution of the displacement is equivalent to a current 
in the opposite direction. 

The whole quantity of electricity displaced through any area 
of a surface fixed in the dielectric is measured by the quantity which 
we have already investigated (Art. 75) as the surface-integral of 

induction through that area, multiplied by K, where K is the 

specific inductive capacity of the dielectric. 

II. Superficial Electrification of the Particles of the Dielectric. 
Conceive any portion of the dielectric, large or small, to be separated 
(in imagination) from the rest by a closed surface, then we must 
suppose that on every elementary portion of this surface there is 
an electrification measured by the total displacement of electricity 
through that element of surface reckoned Inwards. 

In the case of the Leyden jar of which the inner coating is 
charged positively, any portion of the glass will have its inner 
side charged positively and its outer side negatively. If this 
portion be entirely in the interior of the glass, its superficial elec 
trification will be neutralized by the opposite electrification of the 
parts in contact with it, but if it be in contact with a conducting 
body which is incapable of maintaining in itself the inductive state, 
the superficial electrification will not be neutralized, but will con 
stitute that apparent electrification which is commonly called the 
Electrification of the Conductor. 

The electrification therefore at the bounding surface of a con 
ductor and the surrounding dielectric, which on the old theory 
was called the electrification of the conductor, must be called in the 
theory of induction the superficial electrification of the surrounding 
dielectric. 

According to this theory, all electrification is the residual effect 
of the polarization of the dielectric. This polarization exists 
throughout the interior of the substance, but it is there neutralized 
by the juxtaposition of oppositely electrified parts, so that it is only 
at the surface of the dielectric that the effects of the electrification 
become apparent. 

The theory completely accounts for the theorem of Art. 7 7, that 



134 ELECTRIC ATTRACTION. [ill. 

the total induction through a closed surface is equal to the total 
quantity of electricity within the surface multiplied by 4-n". For 
what we have called the induction through the surface is simply 
the electric displacement multiplied by 47r, and the total displace 
ment outwards is necessarily equal to the total electrification within 
the surface. 

The theory also accounts for the impossibility of communicating 
an absolute charge to matter. For every particle of the dielectric 
is electrified with equal and opposite charges on its opposite sides, 
if it would not be more correct to say that these electrifications are 
only the manifestations of a single phenomenon, which we may call 
Electric Polarization. 

A dielectric medium, when thus polarized, is the seat of electrical 
energy, and the energy in unit of volume of the medium is nu 
merically equal to the electric tension on unit of area, both quan 
tities being equal to half the product of the displacement and the 
resultant electromotive force, or 



, 

where p is the electric tension, & the displacement, < the electro 
motive force, and K the specific inductive capacity. 

If the medium is not a perfect insulator, the state of constraint, 
which we call electric polarization, is continually giving way. The 
medium yields to the electromotive force, the electric stress is 
relaxed, and the potential energy of the state of constraint is con 
verted into heat. The rate at which this decay of the state of 
polarization takes place depends on the nature of the medium. 
In some kinds of glass, days or years may elapse before the polar 
ization sinks to half its original value. In copper, this change 
may occupy less than the billionth of a second. 

We have supposed the medium after being polarized to be simply 
left to itself. In the phenomenon called the electric current the 
constant passage of electricity through the medium tends to restore 
the state of polarization as fast as the conductivity of the medium 
allows it to decay. Thus the external agency which maintains the 
current is always doing work in restoring the polarization of the 
medium, which is continually becoming relaxed, and the potential 
energy of this polarization is continually becoming transformed 
into heat, so that the final result of the energy expended in main 
taining the current is to raise the temperature of the conductor. 



CHAPTER VI. 

ON POINTS AND LINES OF EQUILIBKIUM. 

112.] IF at any point of the electric field the resultant force is 
zero, the point is called a Point of equilibrium. 

If every point on a certain line is a point of equilibrium, the line 
is called a Line of equilibrium. 

The conditions that a point shall be a point of equilibrium are 
that at that point 

d7 dV dV 

- 0, -=- = 0, -j- = 0. 
dx dy dz 

At such a point, therefore, the value of V is a maximum, or 
a minimum, or is stationary, with respect to variations of the 
coordinates. The potential, however, can have a maximum or a 
minimum value only at a point charged with positive or with 
negative electricity, or throughout a finite space bounded by a 
surface electrified positively or negatively. If, therefore, a point 
of equilibrium occurs in an unelectrified part of the field it must 
be a stationary point, and not a maximum or a minimum. 

In fact, the first condition of a maximum or minimum is that 



9 

dx 2 dy* 
must be all negative or all positive, if they have finite values. 

Now, by Laplace s equation, at a point where there is no elec 
trification, the sum of these three quantities is zero, and therefore 
this condition cannot be fulfilled. 

Instead of investigating the analytical conditions for the cases 
in which the components of the force simultaneously vanish, we 
shall give a general proof by means of the equipotential surfaces. 

If at any point, P, there is a true maximum value of F, then, at 
all other points in the immediate neighbourhood of P, the value of 
V is less than at P. Hence P will be surrounded by a series of 



136 POINTS AND LINES OF EQUILIBRIUM. [ 11 3 

closed equipotential surfaces, each outside the one before it, and at 
all points of any one of these surfaces the electrical force will be 
directed outwards. But we have proved, in Art. 76, that the surface- 
integral of the electrical force taken over any closed surface gives 
the total electrification within that surface multiplied by 4 IT. Now, 
in this case the force is everywhere outwards, so that the surface- 
integral is necessarily positive, and therefore there is positive elec 
trification within the surface, and, since we may take the surface as 
near to P as we please, there is positive electrification at the point P. 

In the same way we may prove that if V is a minimum at P, 
then P is negatively electrified. 

Next, let P be a point of equilibrium in a region devoid of elec 
trification, and let us describe a very small closed surface round 
P, then, as we have seen, the potential at this surface cannot be 
everywhere greater or everywhere less than at P. It must there 
fore be greater at some parts of the surface and less at others. 
These portions of the surface are bounded by lines in which the 
potential is equal to that at P. Along lines drawn from P to 
points at which the potential is less than that at P the electrical 
force is from P, and along lines drawn to points of greater po 
tential the force is towards P. Hence the point P is a point of 
stable equilibrium for some displacements, and of unstable equili 
brium for other displacements. 

113.] To determine the number of the points and lines of equi 
librium, let us consider the surface or surfaces for which the 
potential is equal to <?, a given quantity. Let us call the regions 
in which the potential is less than C the negative regions, and 
those in which it is greater than C the positive regions. Let 
T be the lowest, and V^ the highest potential existing in the 
electric field. If we make C = V , the negative region will in 
clude only the electrified point or conductor of lowest potential, 
and this is necessarily electrified negatively. The positive region 
consists of the rest of space, and since it surrounds the negative 
region it is periphractic. See Art. 18. 

If we now increase the value of C the negative region will 
expand, and new negative regions will be formed round negatively 
electrified bodies. For every negative region thus formed the 
surrounding positive region acquires one degree of periphraxy. 

As the different negative regions expand, two or more of them 
may meet in a point or a line. If n + 1 negative regions meet, 
the positive region loses n degrees of periphraxy, and the point 



114.] THEIR NUMBER. 137 

or the line in which they meet is a point or line of equilibrium 
of the nth degree. 

When C becomes equal to F l the positive region is reduced to 
the electrified point or conductor of highest potential, and has 
therefore lost all its periphraxy. Hence, if each point or line of 
equilibrium counts for one, two, or n according to its degree, the 
number so made up by the points or lines now considered will 
be one less than the number of negatively electrified bodies. 

There are other points or lines of equilibrium which occur where 
the positive regions become separated from each other, and the 
negative region acquires periphraxy. The number of these, reck 
oned according to their degrees, is one less than the number of 
positively electrified bodies. 

If we call a point or line of equilibrium positive when it is the 
meeting-place of two or more positive regions, and negative when 
the regions which unite there are negative, then, if there are p 
bodies positively and n bodies negatively electrified, the sum of 
the degrees of the positive points and lines of equilibrium will be 
p 1, and that of the negative ones 1. 

But, besides this definite number of points and lines of equi 
librium arising from the junction of different regions, there may 
be others, of which we can only affirm that their number must be 
even. For if, as the negative region expands, it meets itself, it 
becomes a cyclic region, and it may acquire, by repeatedly meeting 
itself, any number of degrees of cyclosis, each of which corresponds 
to the point or line of equilibrium at which the cyclosis was 
established. As the negative region continues to expand till it 
fills all space, it loses every degree of cyclosis it has acquired, and 
becomes at last acyclic. Hence there is a set of points or lines 
of equilibrium at which cyclosis is lost, and these are equal in 
number of degrees to those at which it is acquired. . 

If the form of the electrified bodies or conductors is arbitrary, 
we can only assert that the number of these additional points or 
lines is even, but if they are electrified points or spherical con 
ductors, the number arising in this way cannot exceed (n l)(n 2), 
where n is the number of bodies. 

114.] The potential close to any point P may be expanded in 
the series 

F= ro+^ + tfa + fcc.; 

where H lt H 2 > &c. are homogeneous functions of #, y, z, whose 
dimensions are 1, 2, &c. respectively. 



138 POINTS AND LINES OF EQUILIBRIUM. [H5- 

Since the first derivatives of V vanish at a point of equilibrium, 
H = 0, if P be a point of equilibrium. 

Let H i be the first function which does not vanish, then close to 
the point P we may neglect all functions of higher degrees as 
compared with H i . 

Now H i 

is the equation of a cone of the degree i, and this cone is the cone 
of closest contact with the equipotential surface at P. 

It appears, therefore, that the equipotential surface passing 
through P has, at that point, a conical point touched by a cone 
of the second or of a higher degree. 

If the point P is not on a line of equilibrium this cone 
does not intersect itself, but consists of i sheets or some smaller 
number. 

If the nodal line intersects itself, then the point P is on a line 
of equilibrium, and the equipotential surface through P cuts itself 
in that line. 

If there are intersections of the nodal line not on opposite points 
of the sphere, then P is at the intersection of three or more lines 
of equilibrium. For the equipotential surface through P must cut 
itself in each line of equilibrium. 

115.] If two sheets of the same equipotential surface intersect, 
they must intersect at right angles. 

For let the tangent to the line of intersection be taken as the 



axis of 0, then -jy = 0. Also let the axis of x be a tangent to 

d 2 7 
one of the sheets, then -=-g = 0. It follows from this, by Laplace s 

dPV 
equation, that -j-^ = 0, or the axis of y is a tangent to the other 

y 

sheet. 

This investigation assumes that H 2 is finite. If H 2 vanishes, let 
the tangent to the line of intersection be taken as the axis of z, and 
lei as = r cos 0, and y = r sin 6, then, since 

d 2 7 



dz 2 " dx 



or 

dr* ^ r dr ^ r< 

the solution of which equation in ascending powers of r is 



1 1 6.] THEIR PROPERTIES. 139 

At a point of equilibrium A l is zero. If the first term that does 
not vanish is that in r*, then 

V F = ^r cos (0 + 0^) + terms in higher powers of r. 
This gives i sheets of the equipotential surface F= F , intersecting 
at angles each equal to - . This theorem was given by Rankine *. 

It is only under certain conditions that a line of equilibrium can 
exist in free space, but there must be a line of equilibrium on the 
surface of a conductor whenever the electrification of the conductor 
is positive in one portion and negative in another. 

In order that a conductor may be oppositely electrified in different 
portions of its surface, there must be in the field some places where 
the potential is higher than that of the body and others where it is 
lower. We must remember that at an infinite distance the potential 
is zero. 

Let us begin with two conductors electrified positively to the 
same potential. There will be a point of equilibrium between the 
two bodies. Let the potential of the first body be gradually raised. 
The point of equilibrium will approach the other body, and as the 
process goes on it will coincide with a point on its surface. If the 
potential of the first body be now increased, the equipotential 
surface round the first body which has the same potential as the 
second body will cut the surface of the second body at right angles 
in a closed curve, which is a line of equilibrium. 

Earnshaw s Theorem. ^ 

116.] An electrified body placed in a field of electric force cannot 

be in stable equilibrium. 

First, let us suppose the electricity of the moveable body (A), and 

also that of the system of surrounding bodies (), to be fixed in 

those bodies. 

Let V be the potential at any point of the moveable body due to 

the action of the surrounding bodies (B)> and let e be the electricity 

on a small portion of the moveable body A surrounding this point. 

Then the potential energy of A with respect to B will be 

M= 2(F*), 

where the summation is to be extended to every electrified portion 

of ^. 

* Summary of the Properties of certain Stream Lines, Phil. Mag., Oct. 1864. 
See also, Thomson and Tait s Natural Philosophy, 780 ; and Rankine ami Stokes, 
in the Proc. R. S., 1867, p. 468 ; also W. R. Smith, Proc. R. S. Edin., 1869-70, p. 79. 



140 POINTS AND LINES OF EQUILIBRIUM. 

Let a, bj c be the coordinates of any electrified part of A with 
respect to axes fixed in A, and parallel to those of x, y, z. Let the 
coordinates of the point fixed in the body through which these axes 
pass be 77, f 

Let us suppose for the present that the body A is constrained to 
move parallel to itself, then the absolute coordinates of the point 
a, b 3 c will be 

x f+, y y + b, z f+ c. 

The potential of the body A with respect to B may now be 
expressed as the sum of a number of terms, in each of which V 
is expressed in terms of <z, b } c and f, 17, and the sum of these 
terms is a function of the quantities #, b, c, which are constant for 
each point of the body, and of 77, f, which vary when the body is 
moved. 

Since Laplace s equation is satisfied by each of these terms it is 
satisfied by their sum, or 



" Jf " ~W 
Now let a small displacement be given to A, so that 
d = ldr t dr] = mdr, d = ndr; 

then -= dr will be the increment of the potential of A with respect 

to the surrounding system B. 

If this be positive, work will have to be done to increase r, and 

there will be a force -= tending to diminish r and to restore A to 

its former position, and for this displacement therefore the equi 
librium will be stable. If, on the other hand, this quantity is 
negative, the force will tend to increase r, and the equilibrium will 
be unstable. 

Now consider a sphere whose centre is the origin and whose 
radius is r, and so small that when the point fixed in the body 
lies within this sphere no part of the moveable body A can coincide 
with any part of the external system B. Then, since within the 
sphere \? 2 M = 0, the surface-integral 



-i-dS= 0, 
dr 

taken over the surface of the sphere. 

Hence, if at any part of the surface of the sphere 7 is positive, 

CIT 

there must be some other part of the surface where it is negative, 



Il6.] EQUILIBRIUM ALWAYS UNSTABLE. 141 

and if the body A be displaced in a direction in which =- is 

negative, it will tend to move from its original position, and its 
equilibrium is therefore necessarily unstable. 

The body therefore is unstable even when constrained to move 
parallel to itself, a fortiori it is unstable when altogether free. 

Now let us suppose that the body A is a conductor. We might 
treat this as a case of equilibrium of a system of bodies, the move- 
able electricity being considered as part of that system, and we 
might argue that as the system is unstable when deprived of so 
many degrees of freedom by the fixture of its electricity, it must 
a fortiori be unstable when this freedom is restored to it. 

But we may consider this case in a more particular way, thus 

First, let the electricity be fixed in A, and let it move through 
the small distance dr. The increment of the potential of A due to 

... . dM , 

this cause is r dr. 
dr 

Next, let the electricity be allowed to move within A into its 
position of equilibrium, which is always stable. During this motion 
the potential will necessarily be diminished by a quantity which we 
may call Gdr. 

Hence the total increment of the potential when the electricity 
is free to move will be 

f dM /A* 

(-dr-- C ^ r > 

and the force tending to bring A back towards its original position 
will be dM 

~dr~" Cj 
where C is always positive. 

Now we have shewn that v is neg-ative for certain direc- 

dr 

tions of r, hence when the electricity is free to move the instability 
in these directions will be increased. 



CHAPTER VII. 

FORMS OF THE EQUIPOTENTIAL SURFACES AND LINES OF 
INDUCTION IN SIMPLE CASES. 

117.] WE have seen that the determination of the distribution 
of electricity on the surface of conductors may be made to depend 
on the solution of Laplace s equation 



^ _ 

~d^ + df + dz* = 
V being a function of x, y^ and z, which is always finite and con 
tinuous, which vanishes at an infinite distance, and which has 
a given constant value at the surface of each conductor. 

It is not in general possible by known mathematical methods 
to solve this equation so as to fulfil arbitrarily given conditions, 
but it is always possible to assign various forms to the function 
V which shall satisfy the equation, and to determine in each case 
the forms of the conducting surfaces, so that the function V shall 
be the true solution. 

It appears, therefore, that what we should naturally call the 
inverse problem of determining the forms of the conductors from 
the potential is more manageable than the direct problem of de 
termining the potential when the form of the conductors is given. 

In fact, every electrical problem of which we know the solution 
has been constructed by an inverse process. It is therefore of 
great importance to the electrician that he should know what 
results have been obtained in this way, since the only method by 
which he can expect to solve a new problem is by reducing it 
to one of the cases in which a similar problem has been con 
structed by the inverse process. 

This historical knowledge of results can be turned to account in 
two ways. If we are required to devise an instrument for making 
electrical measurements with the greatest accuracy, we may select 
those forms for the electrified surfaces which correspond to cases 
of which we know the accurate solution. If, on the other hand, 



Jl8.] USE OF DIAGRAMS. 143 

we are required to estimate what will be the electrification of bodies 
whose forms are given, we may begin with some case in which one 
of the equipotential surfaces takes a form somewhat resembling the 
given form,, and then by a tentative method we may modify the pro 
blem till it more nearly corresponds to the given case. This method 
is evidently very imperfect considered from a mathematical point 
of view, but it is the only one we have, and if we are not allowed 
to choose our conditions, we can make only an approximate cal 
culation of the electrification. It appears, therefore, that what we 
want is a knowledge of the forms of equipotential surfaces and 
lines of induction in as many different cases as we can collect 
together and remember. In certain classes of cases, such as those 
relating to spheres, we may proceed by mathematical methods. In 
other cases we cannot afford to despise the humbler method of 
actually drawing tentative figures on paper, and selecting that 
which appears least unlike the figure we require. 

This latter method I think may be of some use, even in cases in 
which the exact solution has been obtained, for I find that an eye- 
knowledge of the forms of the equipotential surfaces often leads to a 
right selection of a mathematical method of solution. 

I have therefore drawn several diagrams of systems of equipotential 
surfaces and lines of force, so that the student may make himself 
familiar with the forms of the lines. The methods by which such 
diagrams may be drawn will be explained as we go on, as they 
belong to questions of different kinds. 

118.] In the first figure at the end of this volume we have the 
equipotential surfaces surrounding two points electrified with quan 
tities of electricity of the same kind and in the ratio of 20 to 5. 

Here each point is surrounded by a system of equipotential 
surfaces which become more nearly spheres as they become smaller, 
but none of them are accurately spheres. If two of these surfaces, 
one surrounding each sphere, be taken to represent the surfaces 
of two conducting bodies, nearly but not quite spherical, and if 
these bodies be charged with the same kind of electricity, the 
charges being as 4 to 1, then the diagram will represent the 
equipotential surfaces, provided we expunge all those which are 
drawn inside the two bodies. It appears from the diagram that 
the action between the bodies will be the same as that between 
two points having the same charges, these points being not exactly 
in the middle of the axis of each body, but somewhat more remote 
than the middle point from the other body. 



144 EQUIPOTENTIAL SURFACES \_ 11 9 

The same diagram enables us to see what will be the distribution 
of electricity on one of the oval figures, larger at one end than 
the other, which surround both centres. Such a body, if electrified 
with a charge 25 and free from external influence, will have the 
surface-density greatest at the small end, less at the large end, 
and least in a circle somewhat nearer the smaller than the larger end. 

There is one equipotential surface, indicated by a dotted line, 
which consists of two lobes meeting at the conical point P. That 
point is a point of equilibrium, and the surface-density on a body 
of the form of this surface would be zero at this point. 

The lines of force in this case form two distinct systems, divided 
from one another by a surface of the sixth degree, indicated by a 
dotted line, passing through the point of equilibrium, and some 
what resembling one sheet of the hyperboloid of two sheets. 

This diagram may also be taken to represent the lines of force 
and equipotential surfaces belonging to two spheres of gravitating 
matter whose masses are as 4 to 1. 

119.] In the second figure we have again two points whose 
charges are as 4 to 1, but the one positive and the other negative. 
In this case one of the equipotential surfaces, that, namely, corre 
sponding to potential zero, is a sphere. It is marked in the diagram 
by the dotted circle Q. The importance of this spherical surface 
will be seen when we come to the theory of Electrical Images. 

We may see from this diagram that if two round bodies are 
charged with opposite kinds of electricity they will attract each other 
as much as two points having the same charges but placed some 
what nearer together than the middle points of the round bodies. 

Here, again, one of the equipotential surfaces, indicated by a 
dotted line, has two lobes, an inner one surrounding the point whose 
charge is 5 and an outer one surrounding both bodies, the two 
lobes meeting in a conical point P which is a point of equilibrium. 

If the surface of a conductor is of the form of the outer lobe, a 
roundish body having, like an apple, a conical dimple at one end of 
its axis, then, if this conductor be electrified, we shall be able to 
determine the superficial density at any point. That at the bottom 
of the dimple will be zero. 

Surrounding this surface we have others having a rounded 
dimple which flattens and finally disappears in the equipotential 
surface passing through the point marked M. 

The lines of force in this diagram form two systems divided by a 
surface which passes through the point of equilibrium. 



121.] AND LINES OF INDUCTION. 145 

If we consider points on the axis on the further side of the point 
J5, we find that the resultant force diminishes to the double point P, 
where it vanishes. It then changes sign, and reaches a maximum 
at M, after which it continually diminishes. 

This maximum, however,, is only a maximum relatively to other 
points on the axis, for if we draw a surface perpendicular to the 
axis, H is a point of minimum force relatively to neighbouring 
points on that surface. 

120.] Figure III represents the equipotential surfaces and lines 
of force due to an electrified point whose charge is 10 placed at 
A, and surrounded by a field of force, which, before the intro 
duction of the electrified point, was uniform in direction and 
magnitude at every part. In this case, those lines of force which 
belong to A are contained within a surface of revolution which 
has an asymptotic cylinder, having its axis parallel to the un 
disturbed lines of force. 

The equipotential surfaces have each of them an asymptotic 
plane. One of them, indicated by a dotted line, has a conical 
point and a lobe surrounding the point A. Those below this surface 
have one sheet with a depression near the axis. Those above have 
a closed portion surrounding A and a separate sheet with a slight 
depression near the axis. 

If we take one of the surfaces below A as the surface of a con 
ductor, and another a long way below A as the surface of another 
conductor at a different potential, the system of lines and surfaces 
between the two conductors will indicate the distribution of electric 
force. If the lower conductor is very far from A its surface will 
be very nearly plane, so that we have here the solution of the 
distribution of electricity on two surfaces, both of them nearly 
plane and parallel to each other, except that the upper one has 
a protuberance near its middle point, which is more or less pro 
minent according to the particular equipotential line we choose for 
the surface. 

121.] Figure IV represents the equipotential surfaces and lines 
of force due to three electrified points A } B and C, the charge of A 
being 1 5 units of positive electricity, that of B 1 2 units of negative 
electricity, and that of C 20 units of positive electricity. These 
points are placed in one straight line, so that 

AB = 9, BC = 16, AC = 25. 

In this case, the surface for which the potential is unity consists 
of two spheres whose centres are A and C and their radii 15 and 20. 

VOL. i. L 



146 EQUIPOTENTIAL SURFACES 

These spheres intersect in the circle which cuts the plane of the 
paper in I) and I/, so that B is the centre of this circle and its 
radius is 12. This circle is an example of a line of equilibrium, for 
the resultant force vanishes at every point of this line. 

If we suppose the sphere whose centre is A to be a conductor 
with a charge of 3 units of positive electricity, and placed under 
the influence of 20 units of positive electricity at C, the state of 
the case will be represented by the diagram if we leave out all the 
lines within the sphere A. The part of this spherical surface within 
the small circle DD will be negatively electrified by the influence 
of C. All the rest of the sphere will be positively electrified,, and 
the small circle DD itself will be a line of no electrification. 

We may also consider the diagram to represent the electrification 
of the sphere whose centre is C, charged with 8 units of positive 
electricity, and influenced by 1 5 units of positive electricity placed 
at A. 

The diagram may also be taken to represent the case of a con 
ductor whose surface consists of the larger segments of the two 
spheres meeting in Lit, charged with 23 units of positive elec 
tricity. 

We shall return to the consideration of this diagram as an 

o 

illustration of Thomson s Theory of Electrical Images. See Art. 168. 

122.] I am anxious that these diagrams should be studied as 
illustrations of the language of Faraday in speaking of lines of 
force/ the forces of an electrified body, &c. 

In strict mathematical language the word Force is used to signify 
the supposed cause of the tendency which a material body is found 
to have towards alteration in its state of rest or motion. It is 
indifferent whether we speak of this observed tendency or of its 
immediate cause, since the cause is simply inferred from the effect, 
and has no other evidence to support it. 

Since, however, we are ourselves in the practice of directing the 
motion of our own bodies, and of moving other things in this way, 
we have acquired a copious store of remembered sensations relating 
to these actions, and therefore our ideas of force are connected in 
our minds with ideas of conscious power, of exertion, and of fatigue, 
and of overcoming or yielding to pressure. These ideas, which give 
a colouring and vividness to the purely abstract idea of force, do 
not in mathematically trained minds lead to any practical error. 

But in the vulgar language of the time when dynamical science 
was unknown, all the words relating to exertion, such as force, 



123.] AND ^ INES OF INDUCTION. 147 



energy, power, &c., were confounded with each other, though some 
of the schoolmen endeavoured to introduce a greater precision into 
their language. 

The cultivation and popularization of correct dynamical ideas 
since the time of Galileo and Newton has effected an immense 
change in the language and ideas of common life, but it is only 
within recent times, and in consequence of the increasing im 
portance of machinery, that the ideas of force, energy, and power 
have become accurately distinguished from each other. Very few, 
however, even of scientific men, are careful to observe these dis 
tinctions ; hence we often hear of the force of a cannon-ball when 
either its energy or its momentum is meant, and of the force of an 
electrified body when the quantity of its electrification is meant. 

Now the quantity of electricity in a body is measured, according 
to Faraday s ideas, by the number of lines of force, or rather of 
induction, which proceed from it. These lines of force must all 
terminate somewhere, either on bodies in the neighbourhood, or on 
the walls and roof of the room, or on the earth, or on the heavenly 
bodies, and wherever they terminate there is a quantity of elec 
tricity exactly equal and opposite to that on the part of the body 
from which they proceeded. By examining the diagrams this will 
be seen to be the case. There is therefore no contradiction between 
Faraday s views and the mathematical results of the old theor}^, 
but, on the contrary, the idea of lines of force throws great light 
on these results, and seems to afford the means of rising by a con 
tinuous process from the somewhat rigid conceptions of the old 
theory to notions which may be capable of greater expansion, so 
as to provide room for the increase of our knowledge by further 
researches. 

123.] These diagrams are constructed in the following manner : 

First, take the case of a single centre of force, a small electrified 

-p 

body with a charge E. The potential at a distance r is F = ; 

T? 
hence, if we make r= -^ . we shall find r, the radius of the sphere 

for which the potential is F. If we now give to F the values 
1, 2, 3, &c., and draw the corresponding spheres, we shall obtain 
a series of equipotential surfaces, the potentials corresponding to 
which are measured by the natural numbers. The sections of these 
spheres by a plane passing through their common centre will be 
circles, which we may mark with the number denoting the potential 

L 2 



148 EQUIPOTENTIAL SURFACES [ I2 3- 

of each. These are indicated by the dotted circles on the right 
hand of Fig-. 6. 

If there be another centre of force, we may in the same way draw 
the equipotential surfaces belonging to it, and if we now wish to 
find the form of the equipotential surfaces due to both centres 
together, we must remember that if V^ be the potential due to one 
centre, and V 2 that due to the other, the potential due to both will be 
V^ + V^ = V. Hence, since at every intersection of the equipotential 
surfaces belonging to the two series we know both V^ and F 2 , we 
also know the value of V. If therefore we draw a surface which 
passes through all those intersections for which the value of V is 
the same, this surface will coincide with a true equipotential surface 
at all these intersections, and if the original systems of surfaces 
be drawn sufficiently close, the new surface may be drawn with 
any required degree of accuracy. The equipotential surfaces due to 
two points whose charges are equal and opposite are represented by 
the continuous lines on the right hand side of Fig. 6. 

This method may be applied to the drawing of any system of 
equipotential surfaces when the potential is the sum of two po 
tentials, for which we have already drawn the equipotential surfaces. 

The lines of force due to a single centre of force are straight 
lines radiating from that centre. If we wish to indicate by these 
lines the intensity as well as the direction of the force at any point, 
we must draw them so that they mark out on the equipotential 
surfaces portions over which the surface-integral of induction has 
definite values. The best way of doing this is to suppose our 
plane figure to be the section of a figure in space formed by the 
revolution of the plane figure about an axis passing through the 
centre of force. Any straight line radiating from the centre and 
making an angle 6 with the axis will then trace out a cone, 
and the surface-integral of the induction through that part of any 
surface which is cut off by this cone on the side next the positive 
direction of the axis, is 2ir^E(l cos 6). 

If we further suppose this surface to be bounded by its inter 
section with two planes passing through the axis, and inclined at 
the angle whose arc is equal to half the radius, then the induction 
through the surface so bounded is 

E (1 cos 0) = 2 V, say ; 

and = cos" 1 (l 2 -^ f )- 
If we now give to ^ a series of values 1, 2, 3 ... E, we shall find 



VoC. 2. 



1C face, f 148. 



Fig. 6. 




lanes of Ihrce. 



l Surfaces 



Jfet/uxl* of 

Zirtes of Forre <??ta 



urface. 



CZa render: . 



123.] AND LINES OF INDUCTION. 149 

a corresponding series of values of 9, and if E be an integer, the 
number of corresponding lines of force, including the axis, will be 
equal to E. 

We have therefore a method of drawing lines of force so that 
the charge of any centre is indicated by the number of lines which 
converge to it, and the induction through any surface cut off in the 
way described is measured by the number of lines of force which 
pass through it. The dotted straight lines on the left hand side 
of Fig. 6 represent the lines of force due to each of two electrified 
points whose charges are 10 and 10 respectively. 

If there are two centres of force on the axis of the figure we 
may draw the lines of force for each axis corresponding to values 
of ^ and ^ 2 , and then, by drawing lines through the consecutive 
intersections of these lines, for which the value of ^ + ^2 is the 
same, we may find the lines of force due to both centres, and in 
the same way we may combine any two systems of lines of force 
which are symmetrically situated about the same axis. The con 
tinuous curves on the left hand side of Fig. 6 represent the lines 
of force due to the tsvo electrified points acting at once. 

After the equipotential surfaces and lines of force have been 
constructed by this method the accuracy of the drawing may be 
tested by observing whether the two systems of lines are every 
where orthogonal, and whether the distance between consecutive 
eqiipotential surfaces is to the distance between consecutive lines 
of force as half the distance from the axis is to the assumed unit of 
length. 

In the case of any such system of finite dimensions the line of 
force whose index number is ^ has an asymptote which passes 

through the centre of gravity of the system, and is inclined to the 

^/ 
axis at an angle whose cosine is 1 2 -^ , where E is the total 

electrification of the system, provided ^ is less than E. Lines of 
force whose index is greater than E are finite lines. 

The lines of force corresponding to a field of uniform force parallel 
to the axis are lines parallel to the axis, the distances from the 
axis being the square roots of an arithmetical series. 

The theory of equipotential surfaces and lines of force in two 
dimensions will be given when we come to the theory of conjugate 
functions *. 

* See a paper On the Flow of Electricity in Conducting Surfaces, by Prof. W. R. 
Smith, Proc. R. S. Edin., 1869-70, p. 79. 



CHAPTER VIII. 

SIMPLE CASES OP ELECTEIFICATIOtf. 

Two Parallel Planes. 

124.] We shall consider, in the first place, two parallel plane 
conducting surfaces of infinite extent, at a distance c from each 
other, maintained respectively at potentials A and B. 

It is manifest that in this case the potential V will be a function 
of the distance z from the plane A, and will be the same for all 
points of any parallel plane between A and J3, except near the 
boundaries of the electrified surfaces, which by the supposition 
are at an infinitely great distance from the point considered. 

Hence, Laplace s equation becomes reduced to 



__ 

== 

the integral of which is 

7= C, + C 2 z; 

and since when z = 0, V = A, and when z = <?, V = B, 



For all points between the planes, the resultant electrical force 
is normal to the planes, and its magnitude is 



c 

In the substance of the conductors themselves, R = 0. Hence 
the distribution of electricity on the first plane has a surface- 
density <r, where AB 

47TO- = R = -- 

c 

On the other surface, where the potential is jB, the surface- 
density a- will be equal and opposite to <r, and 



1 24.] SIMPLE CASES. PARALLEL PLANES. 151 

Let us next consider a portion of the first surface whose area 
is S, taken so that no part of S is near the boundary of the 
surface. 

The quantity of electricity on this surface is E = S<r, and, by 
Art. 79, the force acting on every unit of electricity is \R, so that 
the whole force acting on the area S, and attracting it towards 
the other plane, is 



Here the attraction is expressed in terms of the area S, the 
difference of potentials of the two surfaces (A B), and the distance 
between them c. The attraction, expressed in terms of the charge 
E } on the area S, is 2 TT 

The electrical energy due to the distribution of electricity on the 
area S, and that on an area S on the surface B denned by projecting 
S on the surface B by a system of lines of force, which in this case 
are normals to the planes, is 
Q= 

- 
2 



- 27r E*c 

- -3- A c, 

= Fc. 

The first of these expressions is the general expression of elec 
trical energy. 

The second gives the energy in terms of the area, the distance, 
and the difference of potentials. 

The third gives it in terms of the resultant force R, and the 
volume Sc included between the areas S and S , and shews that the 
energy in unit of volume isp where 8 nfl = R 2 . 

The attraction between the planes is jo/S> or in other words, there 
is an electrical tension (or negative pressure) equal to p on every 
unit of area. 

The fourth expression gives the energy in terms of the charge. 

The fifth shews that the electrical energy is equal to the work 
which would be done by the electric force if the two surfaces were 
to be brought together, moving parallel to themselves, with their 
electric charges constant. 



152 SIMPLE CASES. 

To express the charge in terms of the difference of potentials, 
we have i 



1 o 

The coefficient = q represents the charge due to a differ 
ence of potentials equal to unity. This coefficient is called the 
Capacity of the surface S, due to its position relatively to the 
opposite surface. 

Let us now suppose that the medium between the two surfaces 
is no longer air but some other dielectric substance whose specific 
inductive capacity is K, then the charge due to a given difference 
of potentials will be K times as great as when the dielectric is air, 
or 



The total energy will be 



_ 2^ 

- gjgJSl C. 

The force between the surfaces will be 

_ KS (B-A)* 



--E* 
~ KS l 

Hence the force between two surfaces kept at given potentials 
varies directly as K, the specific capacity of the dielectric, but the 
force between two surfaces charged with given quantities of elec 
tricity varies inversely as K. 

Two Concentric Spherical Surfaces. 

125.] Let two concentric spherical surfaces of radii a and , of 
which I is the greater, be maintained at potentials A and B 
respectively, then it is manifest that the potential V is a function 
of r the distance from the centre. In this case, Laplace s equation 
becomes d*V 2 dV 

~W + r ~dr = 

The integral of this is 

F=Q+Qr-i; 

and the condition that V A when r = a, and V = B when r = 6, 
gives for the space between the spherical surfaces, 



12 5-] CONCENTRIC SPHERICAL SURFACES. 153 

Aa-Bb A-B 



r= 



d 1) & i I) 
dV A-B 



_ 2 



If (7 15 <r 2 are the surface-densities on the opposed surfaces of a 
solid sphere of radius a, and a spherical hollow of radius b, then 

1 A-B 1 B-A 



If EI and ^2 be the whole charges of electricity on these surfaces, 



Tlie capacity of the enclosed sphere is therefore 7 

If the outer surface of the shell be also spherical and of radius c, 
then, if there are no other conductors in the neighbourhood, the 
charge on the outer surface is 

E 3 = Be. 

Hence the whole charge on the inner sphere is 



and that of the outer 



If we put = oo, we have the case of a sphere in an infinite 
space. The electric capacity of such a sphere is a, or it is nu 
merically equal to its radius. 

The electric tension on the inner sphere per unit of area is 

(A -By 2 



STT a 2 (b-a) 2 

The resultant of this tension over a hemisphere is ira 2 j) = F 
normal to the base of the hemisphere, and if this is balanced by a 
surface tension exerted across the circular boundary of the hemi 
sphere, the tension on unit of length being T, we have 

F= 2iraT. 

b* (A-B) 2 Ef 
Hence 



l - 
8 (b of 8 a 



16iro (b-a 



154 SIMPLE CASES. [126. 

If a spherical soap bubble is electrified to a potential A, then, if 
its radius is a, the charge will be Aa, and the surface-density 
will be I A 

47T a 

The resultant electrical force just outside the surface will be 4770-, 
and inside the bubble it is zero, so that by Art. 79 the electrical 
force on unit of area of the surface will be 27ro- 2 , acting outwards. 
Hence the electrification will diminish the pressure of the air 
within the bubble by 27ro- 2 , or 



But it may be shewn that if T is the tension which the liquid 
film exerts across a line of unit length, then the pressure from 

T 
within required to keep the bubble from collapsing is 2 - . If the 

electrical force is just sufficient to keep the bubble in equilibrium 
when the air within and without is at the same pressure 

A 2 = IGvaT. 



Two Infinite Coaxal Cylindric Surfaces. 

126.] Let the radius of the outer surface of a conducting cylinder 
be , and let the radius of the inner surface of a hollow cylinder, 
having the same axis with the first, be I. Let their potentials 
be A and B respectively. Then, since the potential V is in this 
case a function of r, the distance from the axis, Laplace s equation 
becomes 

d 2 F \_dV_ 

dr 2 + r~fo == 

whence V = Q + C 2 log r. 

Since V = A when r = a, and V = B when r = b, 

A log |- .Slog - 

V = r a - 



If o-j, o- 2 are the surface-densities on the inner and outer 
surfaces, 

A-B B-A 

47701 = - -, 4770*2 = 



I2/.] COAXAL CYLINDERS. 155 

If E l and E 2 are the charges on a portion of the two cylinders of 
length I, measured along the axis, 

^A B 

The capacity of a length I of the interior cylinder is therefore 

!** 

If the space between the cylinders is occupied by a dielectric of 
specific capacity K instead of air, then the capacity of the inner 
cylinder is L IK 

2 - 

The energy of the electrical distribution on the part of the infinite 
cylinder which we have considered is 

lK(ABf 

4 , b 







1 
Ir 


i 




i 







Fig. 5. 

127.] Let there be two hollow cylindric conductors A and B, 
Fig. 5, of indefinite length,, having the axis of x for their common 
axis, one on the positive and the other on the negative side of the 
origin, and separated by a short interval near the origin of co 
ordinates. 

Let a hollow cylinder C of length 2 1 be placed with its middle 
point at a distance x on the positive side of the origin, so as to 
extend into both the hollow cylinders. 

Let the potential of the positive hollow cylinder be A, that of 
the negative one J3, and that of the internal one C, and let us put 
a for the capacity per unit of length of C with respect to A, and 
/3 for the same quantity with respect to B. 

The capacities of the parts of the cylinders near the origin and 
near the ends of the inner cylinder will not be affected by the 
value of x provided a considerable length of the inner cylinder 
enters each of the hollow cylinders. Near the ends of the hollow 



156 SIMPLE CASES. [127. 

cylinders, and near the ends of the inner cylinder, there will be 
distributions of electricity which we are not yet able to calculate, 
but the distribution near the origin will not be altered by the 
motion of the inner cylinder provided neither of its ends comes 
near the origin, and the distributions at the ends of the inner 
cylinder will move with it, so that the only effect of the motion 
will be to increase or diminish the length of those parts of the 
inner cylinder where the distribution is similar to that on an in 
finite cylinder. 

Hence the whole energy of the system will be, so far as it depends 
on x, 

Q= a(l + x)(C-A) 2 + %(3 (l-x] (C-B) 2 + quantities 

independent of x ; 
and the resultant force parallel to the axis of the cylinders will be 



If the cylinders A and B are of equal section, a = (3 } and 
X = a(3-A)(C-lU + #)). 

It appears, therefore, that there is a constant force acting on 
the inner cylinder tending to draw it into that one of the outer 
cylinders from which its potential differs most. 

If C be numerically large and A + B comparatively small, then 
the force is approximately x = a(B A)C; 

so that the difference of the potentials of the two cylinders can be 
measured if we can measure X, and the delicacy of the measurement 
will be increased by raising C, the potential of the inner cylinder. 

This principle in a modified form is adopted in Thomson s 
Quadrant Electrometer, Art. 219. 

The same arrangement of three cylinders may be used as a 
measure of capacity by connecting B and C. If the potential of 
A is zero, and that of B and C is F, then the quantity of electricity 
on A will be ^ = (q l3 + a (l+x)) V\ 

so that by moving C to the right till x becomes os+ f the capacity of 
the cylinder becomes increased by the definite quantity af, where 



.. . 

a and b being the radii of the opposed cylindric surfaces. 



CHAPTER IX. 



SPHERICAL HARMONICS. 

On Singular Points at which the Potential becomes Infinite. 

128.] We have already shewn that the potential due to a 
quantity of electricity e, condensed at a point whose coordinates 
are (a, b, c\ is V--- 

where r is the distance from the point (a, 5, c) to the point (x, y, z), 
and Y is the potential at the point (#, y, z]. 

At the point (a, b, c) the potential and all its derivatives hecome 
infinite, hut at every other point they are finite and continuous, 
and the second derivatives of V satisfy Laplace s equation. 

Hence, the value of F, as given by equation (1), may be the 
actual value of the potential in the space outside a closed surface 
surrounding the point (a, b, c], but we cannot, except for purely 
mathematical purposes, suppose this form of the function to hold 
up to and at the point (#, b, c) itself. For the resultant force close 
to the point would be infinite, a condition which would necessitate 
a discharge through the dielectric surrounding the point, and 
besides this it would require an infinite expenditure of work to 
charge a point with a finite quantity of electricity. 

We shall call a point of this kind an infinite point of degree zero. 
The potential and all its derivatives at such a point are infinite, 
but the product of the potential and the distance from the point 
is ultimately a finite quantity e when the distance is diminished 
without limit. This quantity e is called the charge of the infinite 
point. 

This may be shewn thus. If V be the potential due to other 
electrified bodies, then near the point V is everywhere finite, and 
the whole potential is 



whence Vr = T r+e. 



158 SPHEEICAL HARMONICS. 

When r is indefinitely diminished T f remains finite, so that 
ultimately y r _ t 

129.] There are other kinds of singular points, the properties of 
which we shall now investigate, but, before doing so, we must define 
some expressions which we shall find useful in emancipating our 
ideas from the thraldom of systems of coordinates. 

An axis is any definite direction in space. We may suppose 
it defined in Cartesian coordinates by its three direction-cosines 
I, m, n, or, better still, we may suppose a mark made on the surface 
of a sphere where the radius drawn from the centre in the direction 
of the axis meets the surface. We may call this point the Pole 
of the axis. An axis has therefore one pole only, not two. 

If through any point x. y, z a plane be drawn perpendicular to 
the axis, the perpendicular from the origin on the plane is 

p = Ix + my + nz. (2) 

The operation d d d d 

-jj- = l-=- +m + -=- (3) 

an, ax ay az 

is called Differentiation with respect to an axis h whose direction- 
cosines are I, m, n. 

Different axes are distinguished by different suffixes. 

The cosine of the angle between the vector r and any axis 7^ 
is denoted by A and the vector resolved in the direction of the 
axis by^, where 

A;/ = lifs + mty + niZ =&. (4) 

The cosine of the angle between two axes h t and 7tj is denoted by 
Hi where My = /, lj + % m, + , , . (5) 

From these definitions it is evident that 



*- V 



Now let us suppose that the potential at the point (so, y, z) due 
to a singular point of any degree placed at the origin is 



If such a point be placed at the extremity of the axis h, the 
potential at (x, y, z) will be 

Mf((x-lk), (y-mh), (z-nh)); 



INFINITE POINTS. 159 

and if a point in all respects equal and of opposite sign be placed 
at the origin, the potential due to the pair of points will be 

r=Mf{(x-lX), (y-mh), (z-nh)}-Mf(x,y, z\ 

7 

= Mh -^F(x, y, z) + terms containing h 2 . 

If we now diminish h and increase M without limit, their product 
Mh remaining constant and equal to M , the ultimate value of the 
potential of the pair of points will be 



V> -) satisfies Laplace s equation, then V, which is the 
difference of two functions, each of which separately satisfies the 
equation, must itself satisfy it. 

If we begin with an infinite point of degree zero, for which 

F o = M -> (10) 

we shall get for a point of the first degree 



A point of the first degree may be supposed to consist of two 
points of degree zero, having equal and opposite charges M Q and 
ir o , and placed at the extremities of the axis h. The length 
of the axis is then supposed to diminish and the magnitude of the 
charges to increase, so that their product M^k is always equal to 
Jfj. The ultimate result of this process when the two points 
coincide is a point of the first degree, whose moment is J/ x and 
whose axis is ^. A point of the first degree may therefore be 
called a Double point. 

By placing two equal and opposite points of the first degree at 
the extremities of the second axis h. 2 , and making M^ 2 = M. 2 , we 
get by the same process a point of the second degree whose potential 






160 SPHERICAL HARMONICS. 

We may call a point of the second degree a Quadruple point, 
because it is constructed by making four points approach each 
other. It has two axes, h^ and 7/ 2 , and a moment M 2 . The di 
rections of these two axes and the magnitude of the moment com 
pletely define the nature of the point. 

130.] Let us now consider an infinite point of degree i having 
i axes, each of which is defined by a mark on a sphere or by two 
angular coordinates, and having also its moment M it so that it is 
defined by 2^+1 independent quantities. Its potential is obtained 
by differentiating F with respect to the i axes in succession, so 
that it may be written 



The result of the operation is of the form 



where Y it which is called the Surface Harmonic, is a function of the 
i cosines, A x . . . A^ of the angles between r and the i axes, and of the 
\i(i\) cosines, j* 12 , &c. of the angles between the different axes 
themselves. In what follows we shall suppose the moment Mi unity. 
Every term of Y i consists of products of these cosines of the form 

Ml2 ^34 M2s-l 2s ^2s + l \J 

in which there are s cosines of angles between two axes, and i2s 
cosines of angles between the axes and the radius vector. As each 
axis is introduced by one of the i processes of differentiation, the 
symbol of that axis must occur once and only once among the 
suffixes of these cosines. 

Hence in every such product of cosines all the indices occur 
once, and none is repeated. 

The number of different products of s cosines with double suffixes, 
and i 2s cosines with single suffixes, is 

N= - ~=r-5 (15) 

- 



For if we take any one of the N different terms we can form 
from it 2 s arrangements by altering the order of the suffixes of the 
cosines with double suffixes. From any one of these, again, we 
can form \s_ arrangements by altering the order of these cosines, 
and from any one of these we can form ; i-2s arrangements by 
altering the order of the cosines with single suffixes. Hence, with 
out altering the value of the term we may write it in 2 8 s^ i-2s 



130.] TRIGONOMETRICAL EXPRESSION. 161 

different ways, and if we do so to all the terms, we shall obtain 
the whole permutations of i symbols, the number of which is <j_. 
Let the sum of all terms of this kind be written in th<T ab 

breviated form vf\f-2 > 

^ (^ M ) 

If we wish to express that a particular symbol j occurs among 
the A s only, or among the n s only, we write it as a suffix to the \ 
or the fji. Thus the equation 

2 (A- 2 M ) = 2 (A/- 2 - ft) + 2 (A*- 2 - M /) (16) 

expresses that the whole system of terms may be divided into two 
portions, in one of which the symbol/ occurs among the direction- 
cosines of the radius vector, and in the other among the cosines 
of the angles between the axes. 

Let us now assume that up to a certain value of i 
r, = 4, 2 (A*) + A Ll 2 (A- 2 M !) + &c. 

-M t , 8 2(A - 2 V) + &c. (17) 

This is evidently true when i \ and when i = 2. We shall shew 
that if it is true for i it is true for i + 1 . We may write the series 

r;. = s{4,,2(v- v)}, (is) 

where S indicates a summation in which all values of s not greater 
than \ i are to be taken. 

Multiplying by _i_r~( i+l \ and remembering that p { = r\ i} we 
obtain by (14), for the value of the solid harmonic of negative 
degree, and moment unity, 

V { = \^S{A itS r 2s - 2i - l I,( I j i - 2s ^}. (19) 

Differentiating V i with respect to a new axis whose svmbol is 
y, we should obtain J^ +1 with its sign reversed, 



r 2 - 2i - 1 2 (/- 2 -V/ +1 )}- (20) 
If we wish to obtain the terms containing s cosines with double 
suffixes we must diminish s by unity in the second term, and we find 



)]}. (21) 

If we now make 



1 _ s (22) 

then T i+l = ilS {^ +l _^2.-2 ( i + i)-i 2 ^+1-2.^ (23) 

and this value of J^ +1 is the same as that obtained by changing i 



VOL. i. 



162 SPHERICAL HARMONICS. [ I 3 I 

into i+l in the assumed expression, equation (19), for V { . Hence 
the assumed form of 7J", in equation (19), if true for any value of i, 
is true for the next higher value. 

To find the value of A Ls , put s = in equation (22), and we find 

4 + i.o = ^^ 4.o ; (24) 

-f- i 
and therefore, since A 1 is unity, 

I2t 

(25) 



and from this we obtain, by equation (22), for the general value of 
the coefficient 12-2s 



and finally, the value of the trigonometrical expression for T t is 



This is the most general expression for the spherical surface- 
harmonic of degree i. If i points on a sphere are given, then, if any 
other point P is taken on the sphere, the value of Y i for the point 
P is a function of the i distances of P from the i points, and of the 
\i(i 1) distances of the i points from each other. These i points 
may be called the Poles of the spherical harmonic. Each pole 
may be defined by two angular coordinates, so that the spherical 
harmonic of degree i has 2i independent constants, exclusive of its 
moment, M i9 

131.] The theory of spherical harmonics* was first given by 
Laplace in the third book of his Mecanique Celeste. The harmonics 
themselves are therefore often called Laplace s Coefficients. 

They have generally been expressed in terms of the ordinary 
spherical coordinates and 0, and contain 2i+l arbitrary con 
stants. Gauss appears* to have had the idea of the harmonic 
being determined by the position of its poles, but I have not met 
with any development of this idea. 

In numerical investigations I have often been perplexed on ac 
count of the apparent want of definiteness of the idea of a Laplace s 
Coefficient or spherical harmonic. By conceiving it as derived by 

the successive differentiation of with respect to i axes, and as 
expressed in terms of the positions of its i poles on a sphere, I 

* Gauss. Werlse, bd.v. s. 361. 



132.] SYMMETRICAL SYSTEM. 163 

have made the conception of the general spherical harmonic of any 
integral degree perfectly definite to myself, and I hope also to those 
who may have felt the vagueness of some other forms of the ex 
pression. 

When the poles are given, the value of the harmonic for a given 
point on the sphere is a perfectly definite numerical quantity. 
When the form of the function, however, is given, it is by no 
means so easy to find the poles except for harmonics of the first 
and second degrees and for particular cases of the higher degrees. 

Hence, for many purposes it is desirable to express the harmonic 
as the sum of a number of other harmonics, each of which has its 
axes disposed in a symmetrical manner. 

Symmetrical System. 

132.] The particular forms of harmonics to which it is usual to 
refer all others are deduced from the general harmonic by placing 
i (T of the poles at one point, which we shall call the Positive Pole 
of the sphere, and the remaining a- poles at equal distances round 
one half of the equator. 

In this case A 1? A 2 , ... A,-^ are each of them equal to cos 0, and 
A.f-s+1 ... A^ are of the form sin 9 cos(( /3). We shall write /u for 
cos 6 and v for sin 0. 

Also the value of /*,-/ is unity if j and f are both less than i cr, 
zero when one is greater and the other less than this quantity, 

and cos n - when both are greater. 

When all the poles are concentrated at the pole of the sphere, 
the harmonic becomes a zonal harmonic for which a- = 0. As the 
zonal harmonic is of great importance we shall reserve for it the 
symbol 

We may obtain its value either from the trigonometrical ex 
pression (27), or more directly by differentiation, thus 



n -n- 



It is often convenient to express Q f as a homogeneous function of 
cos and sin 6, which we shall write //, and v respectively, 

M 2 



164 SPHERICAL HARMONICS. [ X 3 2 - 



(30) 



In this expansion the coefficient of /^. is unity, and all the other 
terms involve v. Hence at the pole, where ^=1 and v=0, Q { = 1. 

It is shewn in treatises on Laplace s Coefficients that Q { is the 
coefficient of Ji l in the expansion of (1 2^/ + ^ 2 )~^. 

The other harmonics of the symmetrical system are most con 
veniently obtained by the use of the imaginary coordinates given by 
Thomson and Tait, Natural Philosophy, vol. i. p. 148, 

The operation of differentiating with respect to a axes in suc 
cession, whose directions make angles with each other in the 
plane of the equator, may then be written 

*1 = ^1 + ^1. (32) 

The surface harmonic of degree i and type a is found by 
differentiating - with respect to i axes, cr of which are at equal 

intervals in the plane of the equator, while the remaining i a 
coincide with that of z, multiplying the result by r i+l and dividing 
by _*_. Hence 

ro (+.)& (33) 



Now <T + ?] a = 2 r cr 2; "cos(o-(^ + /3), (35) 



and ^ ^ = (-1)-J=^ ^). (36) 



Hence Y = 2 ^ 

where the factor 2 must be omitted when o- = 0. 

The quantity 3 ." i g a function of 0, the value of which is given 
in Thomson and Tait s Natural Philosophy, vol. i. p. 149. 

It may be derived from Q { by the equation 



_ 

where Q t - is expressed as a function of /x only. 



1 33.] SOLID HARMONICS OF POSITIVE DEGREE. 165 

Performing the differentiations on Q { as given in equation (29), 
we obtain 



We may also express it as a homogeneous function of /* and y, 

ir-- |i ^^r - /1 -~,^{. (40) 

2 2<r r 



In this expression the coefficient of the first term is unity, and 
the others may be written down in order by the application of 
Laplace s equation. 

The following relations will be found useful in Electrodynamics. 
They may be deduced at once from the expansion of Q /i . 

-" = = (41) 



-1" 15" ~T J 

0# &>&W Harmonics of Positive Degree. 

133.] We have hitherto considered the spherical surface harmonic 
Y i as derived from the solid harmonic 



This solid harmonic is a homogeneous function of the coordinates 
of the negative degree (i+1). Its values vanish at an infinite 
distance and become infinite at the origin. 

We shall now shew that to every such function there corresponds 
another which vanishes at the origin and has infinite values at an 
infinite distance, and is the corresponding solid harmonic of positive 
degree i. 

A solid harmonic in general may be defined as a homogeneous 
function of x, y^ and z, which satisfies Laplace s equation 
d 2 F d*7 d*V 
~d^ + ~df + dz* ~~ 
Let H t be a homogeneous function of the degree ^, such that 

H t = l^M^Yi = r 2i+l F { . (43) 

Then = 2i+lr 2 - 



166 SPHERICAL HARMONICS. [ 1 34- 

Hence 



,/ dV, dV: dV^ t 

r^- l (x-^+y^ + z-^) + r 2^i_ l + --^ + -i . 44) 
> dx dy dz V# 2 dy- dz 2J 

t/ 7 

Now, since V i is a homogeneous function of negative degree i+1, 



The first two terms therefore of the right hand member of 
equation (44) destroy each other, and, since ^ satisfies Laplace s 
equation, the third term is zero, so that H i also satisfies Laplace s 
equation, and is therefore a solid harmonic of degree i. 

We shall next shew that the value of H i thus derived from V i is 
of the most general form. 

A homogeneous function of a?, y, z of degree i contains 

i(t+i)(t+2) 

terms. But 



is a homogeneous function of degree ^ 2, and therefore contains 
\i(i 1) terms, and the condition ^ 2 H L = requires that each of 
these must vanish. There are therefore \i(il) equations between 
the coefficients of the \ (i + 1)(^ + 2) terms of the homogeneous 
function, leaving 2^+1 independent constants in the most general 
form of H^ 

But we have seen that J f i has 2^+1 independent constants, 
therefore the value of H t is of the most general form. 

Application of Solid Harmonics to the Theory of Electrified Spheres. 

134.] The function 7J satisfies the condition of vanishing at 
infinity, but does not satisfy the condition of being everywhere 
finite, for it becomes infinite at the origin. 

The function II i satisfies the condition of being finite and con 
tinuous at finite distances from the origin, but does not satisfy the 
condition of vanishing at an infinite distance. 

But if we determine a closed surface from the equation 

^=#0 (46) 

and make H i the potential function within the closed surface and 



1 35.] ELECTRIFIED SPHERICAL SURFACE. 167 

/^ the potential outside it, then by making- the surface-density a- 
satisfy the characteristic equation 



, (47) 

we shall have a distribution of potential which satisfies all the 
conditions. 

It is manifest that if H i and V i are derived from the same value 
of J" i5 the surface H { = 1\ will be a spherical surface, and the 
surface-density will also be derived from the same value of 1^. 

Let a be the radius of the sphere, and let 

(48) 



Then at the surface of the sphere, where r = a, 



dV dH 

and -= --- = = 4770-; 

dr dr 

T) 

or (j + i)__ + 2 V-M = 477(7; 

whence we find ff i and J f i in terms of C, 



We have now obtained an electrified system in which the potential 
is everywhere finite and continuous. This system consists of a 
spherical surface of radius a, electrified so that the surface-density 
is everywhere CY it where C is some constant density and Y i is a 
surface harmonic of degree i. The potential inside this sphere, 
arising- from this electrification, is everywhere ff t , and the potential 
outside the sphere is T\. 

These values of the potential within and without the sphere 
might have been obtained in any given case by direct integration, 
but the labour would have been great and the result applicable only 
to the particular case. 

135.] We shall next consider the action between a spherical 
surface, rigidly electrified according to a spherical harmonic, and 
an external electrified system which we shall call E. 

Let V be the potential at any point due to the system E, and 
Y i that due to the spherical surface whose surface-density is cr. 



168 SPHERICAL HARMONICS. [_ L 35- 

Then, by Green s theorem, the potential energy of E on the 
electrified surface is equal to that of the electrified surface on E, or 

(50) 

where the first integration is to be extended over every element dS 
of the surface of the sphere, and the summation 2 is to be extended 
to every part dE of which the electrified system E is composed. 

But the same potential function V { may be produced by means 
of a combination of 2* electrified points in the manner already 
described. Let us therefore find the potential energy of E on 
such a compound point. 

If M is the charge of a single point of degree zero, then M F 
is the potential energy of V on that point. 

If there are two such points, a positive and a negative one, at 
the positive and negative ends of a line h lt then the potential energy 
of E on the double point will be 



and when M increases and & L diminishes indefinitely, but so that 

1/ ^ = Jl/i, 
the value of the potential energy will be for a point of the first degree 



Similarly for a point of degree i the potential energy with respect 
to E will be 

1 



This is the value of the potential energy of E upon the singular 
point of degree i. That of the singular point on E is ^dU 3 and, 
by Green s theorem, these are equal. Hence, by equation (50), 



[[ &V 



If o- = CT i where C is a constant quantity, then, by equations 

(49) and (14), 

. (51) 



Hence, if V is any potential function whatever which satisfies 
Laplace s equation within the spherical surface of radius a, then the 



I37-] SURFACE-INTEGRAL OF THE PRODUCT OF HARMONICS. 169 

integral of VY i dS, extended over every element dS t of the surface 
of a sphere of radius a, is given by the equation 

^"-rfiirfar^s: < 52 ) 



where the differentiations of V are taken with respect to the axes 
of the harmonic Y it and the value of the differential coefficient is 
that at the centre of the sphere. 

136.] Let us now suppose that V is a solid harmonic of positive 
degree j of the form j 

T=^Y, (53) 

At the spherical surface, r = a, the value of V is the surface har 
monic YJ, and equation (52) becomes 



II YY /<? 

r < T > d = 






where the value of the differential coefficient is that at the centre 
of the sphere. 

When / is numerically different from j, the surface-integral of 
the product Y t Yj vanishes. For, when i is less than j, the result 
of the differentiation in the second member of (54) is a homogeneous 
function of x, y, and z, of degree j i, the value of which at the 
centre of the sphere is zero. If i is equal toj the result is a constant, 
the value of which will be determined in the next article. If the 
differentiation is carried further, the result is zero. Hence the 
surface-integral vanishes when i is greater than j. 

137.] The most important case is that in which the harmonic 
rJYj is differentiated with respect to i new axes in succession, the 
numerical value of J being the same as that of i, but the directions 
of the axes being in general different. The final result in this case 
is a constant quantity, each term being the product of i cosines of 
angles between the different axes taken in pairs. The general 
form of such a product may be written symbolically 



which indicates that there are s cosines of angles between pairs of 
axes of the first system and $ between axes of the second system, 
the remaining i2s cosines being between axes one of v.hich 
belongs to the first and the other to the second system. 

In each product the suffix of every one of the 2i axes occurs 
once, and once only. 



170 SPHEKICAL HARMONICS. 

The number of different products for a given value of # is 

([fji 

N = ( 55 ) 



The final result is easily obtained by the successive differen 
tiation of 

r,F. = . , S {(_ 1V^=L r 2. 2 (y- V)} . 
j j | j U t 2J-* j-s 

Differentiating this i times in succession with respect to the new 
axes, so as to obtain any given combination of the axes in pairs, 
we find that in differentiating r 2s with respect to s of the new axes, 
which are to be combined with other axes of the new system, we 
introduce the numerical factor 2s (2s 2) ... 2, or 2 s \s_. In con 
tinuing the differentiation the j>/s become converted into /x s, but 
no numerical factor is introduced. Hence 



(56) 



Substituting this result in equation (54) we find for the value of 
the surface-integral of the product of two surface harmonics of the 
same degree, taken over the surface of a sphere of radius a, 



JJY i Y i dS = 



This quantity differs from zero only when the two harmonics are 
of the same degree, and even in this case, when the distribution of 
the axes of the one system bears a certain relation to the distribution 
of the axes of the other, this integral vanishes. In this case, the 
two harmonics are said to be conjugate to each other. 

On Conjugate Harmonics. 

138.] If one harmonic is given, the condition that a second 
harmonic of the same degree may be conjugate to it is expressed 
by equating the right hand side of equation (57) to zero. 

If a third harmonic is to be found conjugate to both of these 
there will be two equations which must be satisfied by its 2i 
variables. 

If we go on constructing new harmonics, each of which is con 
jugate to all the former harmonics, the variables will be continually 
more and more restricted, till at last the (2i+ l)th harmonic will 
have all its variables determined by the 2i equations, which must 



1 3 9.] CONJUGATE HARMONICS. 171 

be satisfied in order that it may be conjugate to the 2i preceding 
harmonics. 

Hence a system of 2i+l harmonics of degree i may be con 
struct ed, each of which is conjugate to all the rest. Any other 
harmonic of the same degree may be expressed as the sum of this 
system of conjugate harmonics each multiplied by a coefficient. 

The system described in Art. 132, consisting of 2^+1 har 
monics symmetrical about a single axis, of which the first is zonal, 
the next i 1 pairs tesseral, and the last pair sectorial, is a par 
ticular case of a system of 2i+l harmonics, all of which are 
conjugate to each other. Sir W. Thomson has shewn how to 
express the conditions that 2 i -f 1 perfectly general harmonics, 
each of which, however, is expressed as a linear function of the 
2 / -f 1 harmonics of this symmetrical system, may be conjugate 
to each other. These conditions consist of i(2i+l) linear equa 
tions connecting the (2^+l) 2 coefficients which enter into the 
expressions of the general harmonics in terms of the symmetrical 
harmonics. 

Professor Clifford has also shewn how to form a conjugate system 
of 2+l sectorial harmonics having different poles. 

Both these results were communicated to the British Association 
in 1871. 

139.] If we take for Yj the zonal harmonic Q Jt we obtain a 
remarkable form of equation (57). 

In this case all the axes of the second system coincide with each 
other. 

The cosines of the form //, v will assume the form A. where A. is the 
cosine of the angle between the common axis of Qj and an axis of 
the first system. 

The cosines of the form ^ will all become equal to unity. 

The number of combinations of s symbols, each of which is 
distinguished by two out of i suffixes, no suffix being repeated, is 

N = (58) 



and when each combination is equal to unity this number represents 
the sum of the products of the cosines p^, or 2 (/&,-/). 

The number of permutations of the remaining I 2s symbols of 
the second set of axes taken all together is i-2s. Hence 

2 fr/,./- 2 ) = :-2* 2 A - 2 . (59) 

Equation (57) therefore becomes, when Y j is the zonal harmonic, 



172 SPHERICAL HAKMONICS. 



r, wl (so) 

where J^-) denotes the value of Y i in equation (27) at the common 
pole of all the axes of Qj. 

140.] This result is a very important one in the theory of 
spherical harmonics, as it leads to the determination of the form 
of a series of spherical harmonics, which expresses a function having 
any arbitrarily assigned value at each point of a spherical surface. 

For let F be the value of the function at any given point of the 
sphere, say at the centre of gravity of the element of surface dS, 
and let Q t be the zonal harmonic of degree i whose pole is the point 
P on the sphere, then the surface-integral 



extended over the spherical surface will be a spherical harmonic 
of degree i, because it is the sum of a number of zonal harmonics 
whose poles are the various elements dS, each being multiplied by 
FdS. Hence, if we make 



we may expand F in the form 

F= AJo + A^ + bc. + AiYi, (62) 



or 

1 



471 a* 



. (63) 



This is the celebrated formula of Laplace for the expansion in 
a series of spherical harmonics of any quantity distributed over 
the surface of a sphere. In making use of it we are supposed to 
take a certain point P on the sphere as the pole of the zonal 
harmonic Q { , and to find the surface-integral 



over the whole surface of the sphere. The result of this operation 
when multiplied by 2i+I gives the value of A i Y i at the point P. 
and by making P travel over the surface of the sphere the value of 
A { Y { at any other point may be found. 



SPHERICAL HARMONIC ANALYSIS. 173 

But A{ i is a general surface harmonic of degree ?, and we wish 
to break it up into the sum of a series of multiples of the 2e-f- 1 
conjugate harmonics of that degree. 

Let P i be one of these conjugate harmonics of a particular type, 
and let B i % be the part of A i Y i belonging to this type. 

We must first find r r 

(64) 



which may be done by means of equation (57), making the second 
set of poles the same, each to each, as the first set. 

We may then find the coefficient B i from the equation 

* = -sff FP * & (63) 

For suppose F expanded in terms of spherical harmonics, and let 
BjPj be any term of this expansion. Then, if the degree of Pj is 
different from that of P i3 or if, the degree being the same, Pj is 
conjugate to P i3 the result of the surface-integration is zero. Hence 
the result of the surface-integration is to select the coefficient of the 
harmonic of the same type as P { . 

The most remarkable example of the actual development of a 
function in a series of spherical harmonics is the calculation by 
Gauss of the harmonics of the first four degrees in the expansion 
of the magnetic potential of the earth, as deduced from observations 
in various parts of the world. 

He has determined the twenty-four coefficients of the three 
conjugate harmonics of the first degree, the five of the second, 
seven of the third, and nine of the fourth, all of the symmetrical 
system. The method of calculation is given in his General Theory 
of Terrestrial Magnetism. 

141.] When the harmonic P i belongs to the symmetrical system 
we may determine the surface-integral of its square extended over 
the sphere by the following method. 

The value of i* Y? is, by equations (34) and (36), 



and by equations (33) and (54), 



Performing the difierentiations, we find that the only terms 
which do not disappear are those which contain z i ~ <T . Hence 



174 SPHERICAL HARMONICS. [H 2 - 

(66) 



except when o- = 0, in which case we have, by equation (GO), 



These expressions give the value of the surface-integral of the 
square of any surface harmonic of the symmetrical system. 

We may deduce from this the value of the integral of the square 
of the function 3>), given in Art. 132, 

9 2 2 " ia- (} v \ 2 
n)Va = " - _ - (l ~> . (68) 

Ul 2i+l \i + <r 

This value is identical with that given by Thomson and Tait, and is 
true without exception for the case in which a = 0. 

142.] The spherical harmonics which I have described are those 
of integral degrees. To enter on the consideration of harmonics 
of fractional, irrational, or impossible degrees is beyond my purpose, 
which is to give as clear an idea as I can of what these harmonics 
are. I have done so by referring the harmonic, not to a system 
of polar coordinates of latitude and longitude, or to Cartesian 
coordinates, but to a number of points on the sphere, which I 
have called the Poles of the harmonic. Whatever be the type 
of a harmonic of the degree i, it is always mathematically possible 
to find i points on the sphere which are its poles. The actual 
calculation of the position of these poles would in general involve 
the solution of a system of 2i equations of the degree i. The 
conception of the general harmonic, with its poles placed in any 
manner on the sphere^ is useful rather in fixing our ideas than in 
making calculations. For the latter purpose it is more convenient 
to consider the harmonic as the sum of 2i-\- 1 conjugate harmonics 
of selected types, and the ordinary symmetrical system, in which 
polar coordinates are used, is the most convenient. In this system 
the first type is the zonal harmonic Q { , in which all the axes 
coincide with the axis of polar coordinates. The second type is 
that in which i 1 of the poles of the harmonic coincide at the pole 
of the sphere, and the remaining one is on the equator at the origin 
of longitude. In the third type the remaining pole is at 90 of 
longitude. 

In the same way the type in which i or poles coincide at the 
pole of the sphere, and the remaining a are placed with their axes 



1 43.] FIGURES OF SPHERICAL HARMONICS. 175 

at equal intervals round the equator, is the type 2 <r, if one of the 
poles is at the origin of longitude, or the type 2 a- -f 1 if it is at 
longitude 

143.] It appears from equation (60) that it is always possible 
to express a harmonic as the sum of a system of zonal harmonics 
of the same degree, having their poles distributed over the surface 
of the sphere. The simplification of this system, however, does not 
appear easy. I have however, for the sake of exhibiting to the 
eye some of the features of spherical harmonics, calculated the zonal 
harmonics of the third and fourth degrees, and drawn, by the 
method already described for the addition of functions, the equi- 
potential lines on the sphere for harmonics which are the sums of 
two zonal harmonics. See Figures VI to IX at the end of this 
volume. 

Fig. VI represents the sum of two zonal harmonics of the third 
degree whose axes are inclined 120 in the plane of the paper, and 
the sum is the harmonic of the second type in which a = 1 , the axis 
being perpendicular to the paper. 

In Fig. VII the harmonic is also of the third degree, but the 
axes of the zonal harmonics of which it is the sum are inclined 
90, and the result is not of any type of the symmetrical system. 
One of the nodal lines is a great circle, but the other two which are 
intersected by it are not circles. 

Fig. VIII represents the difference of two zonal harmonics of 
the fourth degree whose axes are at right angles. The result is a 
tesseral harmonic for which i = 4, a- = 2. 

Fig. IX represents the sum of the same zonal harmonics. The 
result gives some notion of one type of the more general har 
monic of the fourth degree. In this type the nodal line on the 
sphere consists of six ovals not intersecting each other. Within 
these ovals the harmonic is positive, and in the sextuply connected 
part of the spherical surface which lies outside the ovals, the har 
monic is negative. 

All these figures are orthogonal projections of the spherical 
surface. 

I have also drawn in Fig. V a plane section through the axis 
of a sphere, to shew the equipotential surfaces and lines of force 
due to a spherical surface electrified according to the values of a 
spherical harmonic of the first degree. 



176 SPHERICAL HARMONICS. [ T 44- 

Within the sphere the equipotential surfaces are equidistant 
planes, and the lines of force are straight lines parallel to the axis, 
their distances from the axis being as the square roots of the 
natural numbers. The lines outside the sphere may be taken as a 
representation of those which would be due to the earth s magnetism 
if it were distributed according to the most simple type. 

144.] It appears from equation (52), by making i = 0, that if 
V satisfies Laplace s equation throughout the space occupied by a 
sphere of radius #, then the integral 

(69) 



where the integral is taken over the surface of the sphere, dS being 
an element of that surface, and F is the value of V at the centre 
of the sphere. This theorem may be thus expressed. 

The value of the potential at the centre of a sphere is the mean 
value of the potential for all points of its surface, provided the 
potential be due to an electrified system, no part of which is within 
the sphere. 

It follows from this that if V satisfies Laplace s equation through 
out a certain continuous region of space, and if, throughout a 
finite portion, however small, of that space, Fis constant, it will 
be constant throughout the whole continuous region. 

If not, let the space throughout which the potential has a 
constant value C be separated by a surface S from the rest of 
the region in which its values differ from C, then it will always 
be possible to find a finite portion of space touching S and out 
side of it in which V is either everywhere greater or everywhere 
less than C. 

Now describe a sphere with its centre within S, and with part 
of its surface outside S, but in a region throughout which the value 
of V is every where greater or everywhere less than C. 

Then the mean value of the potential over the surface of the 
sphere will be greater than its value at the centre in the first case 
and less in the second, and therefore Laplace s equation cannot 
be satisfied throughout the space occupied by the sphere, contrary 
to our hypothesis. It follows from this that if V^=C throughout 
any portion of a connected region, V C throughout the whole 
of the region which can be reached in any way by a body 01 
finite size without passing through electrified matter. (We sup 
pose the body to be of finite size because a region in which V is 
constant may be separated from another region in which it is 



45-] THEOREM OF GAUSS. 177 

variable by an electrified surface, certain points or lines of which 
are not electrified, so that a mere point might pass out of the 
region through one of these points or lines without passing 
through electrified matter.) This remarkable theorem is due to 
Gauss. See Thomson and Tait s Natural Philosophy^ 497. 

It may be shewn in the same way that if throughout any finite 
portion of space the potential has a value which can be expressed 
by a continuous mathematical formula satisfying Laplace s equation, 
the potential will be expressed by the same formula throughout 
every part of space which can be reached without passing through 
electrified matter. 

For if in any part of this space the value of the function is V , 
different from V, that given by the mathematical formula, then, 
since both V and V satisfy Laplace s equation, U= V V does. 
But within a finite portion of the space [7=0, therefore by what 
we have proved U = throughout the whole space, or T = V. 

145.] Let Y { be a spherical harmonic of i degrees and of any 
type. Let any line be taken as the axis of the sphere, and let the 
harmonic be turned into n positions round the axis, the angular 

o 

distance between consecutive positions being -- 

If we take the sum of the n harmonics thus formed the result 
will be a harmonic of i degrees, which is a function of 6 and of the 
sines and cosines of n$. 

If_ n is less than i the result will be compounded of harmonics for 
which s is zero or a multiple of n less than i, but if n is greater 
than / the result is a zonal harmonic. Hence the following theorem : 

Let any point be taken on the general harmonic Y it and let a 
small circle be described with this point for centre and radius 0, 
and let n points be taken at equal distances round this circle, then 
if Q; is the value of the zonal harmonic for an angle 0, and if Y- is 
the value of Y i at the centre of the circle, then the mean of the 
n values of Y i round the circle is equal to Q t Y{ provided n is greater 
than i. 

If n is greater than i -f s, and if the value of the harmonic at 
each point of the circle be multiplied by sin<S( or cos sty where 
s is less than i, and the arithmetical mean of these products be 
A s , then if 3?*^ * s the value of W for the angle 6, the coefficient 
of sin sty or cos 8$ in the expansion of Y t will be 



VOL. I. N 



-178 SPHERICAL HARMONICS. [ 146. 

In this way we may analyse Y i into its component conjugate 
harmonics by means of a finite number of ascertained values at 
selected points on the sphere. 

Application of Spherical Harmonic Analysis to the Determination 
of the Distribution of Electricity on Spherical and nearly Spherical 
Conductors under the Action of known External Electrical Forces. 

146.] We shall suppose that every part of the electrified system 
which acts on the conductor is at a greater distance from the 
centre of the conductor than the most distant part of the conductor 
itself, or, if the conductor is spherical, than the radius of the 
sphere. 

Then the potential of the external system, at points within this 
distance, may be expanded in a series of solid harmonics of positive 
degree y = A ^ + ^ r YI + & c + j . j. ^ (7 0) 

The potential due to the conductor at points outside it may be 
expanded in a series of solid harmonics of the same type, but of 
negative degree 

(71) 



At the surface of the conductor the potential is constant and 
equal, say, to C. Let us first suppose the conductor spherical and 
of radius a. Then putting r = a, we have U+ ~F= C, or, equating 
the coefficients of the different degrees, 

3 Q = a(C-AJ, 

JB 1 =-a^A 19 (72) 

_#. =- + ! ^. 

The total charge of electricity on the conductor is B Q . 

The surface-density at any point of the sphere may be found 
from the equation 



dV dU 

4 770- = -= -- -y- 

dr dr 



i Y i . (73) 



Distribution of Electricity on a nearly Spherical Conductor. 
Let the equation of the surface of the conductor be 

r = a(l+JF), (74) 



146.] NEARLY SPHERICAL CONDUCTOR. 179 

where F is a function of the direction of r, and is a numerical 
quantity the square of which may be neglected. 

Let the potential due to the external electrified system be ex 
pressed, as before, in a series of solid harmonics of positive degree, 
and let the potential U be a series of solid harmonics of negative 
degree. Then the potential at the surface of the conductor is 
obtained by substituting the value of r from equation (74) in these 
series. 

Hence, if C is the value of the potential of the conductor and 
.Z? the charge upon it, 
C= 4, 



..-(j+l)B,arWFYr (75) 

Since F is very small compared with unity, we have first a set 
of equations of the form (72), with the additional equation 

= - Q -F + 3A 1 aFY 1 + 8tc. + (i+l)A i a i FY i 

+ 2(.# / 0-U +1 >7,)-2 ((j+VSja-U+VFYj). (76) 

To solve this equation we must expand F, FY 1 . . . FY i in terms of 
spherical harmonics. If F can be expanded in terms of spherical 
harmonics of degrees lower than Jc } then FY i can be expanded in 
spherical harmonics of degrees lower than i + k. 

Let therefore 

B Q - F- 3A 1 aFY 1 - ...-(2i+l)A i W<= 2 (Bj a-U+DJ}), (77) 

d 

then the coefficients Bj will each of them be small compared with 
the coefficients B Q ... B i on account of the smallness of F, and 
therefore the last term of equation (76), consisting of terms in BjF, 
may be neglected. 

Hence the coefficients of the form Bj may be found by expanding 
equation (76) in spherical harmonics. 

For example, let the body have a charge _Z? , and be acted on by 
no external force. 

Let F be expanded in a series of the form 

F = S 1 Y l + &c. + S t Y lk . (78) 

Then S l Y l + &c. + 8 1g Y t = 2(S J a-V+VY j ), (79) 

N 2 



180 SPHEEICAL HAKMONICS. 

or the potential at any point outside the body is 



(80) 



and if o- is the surface-density at any point 

dU 

4-770- = -- > 

dr 



or 47700- = (l+fl 2 r a +...+ (-1)^7,). (81) 

Hence, if the surface differs from that of a sphere by a thin 
stratum whose depth varies according to the values of a spherical 
harmonic of degree /, the ratio of the difference of the superficial 
densities at any two points to their sum will be k I times the 
ratio of the difference of the radii of the same two points to their 
sum. 



CHAPTER X. 



CONTOCAL QUADRIC SURFACES*. 



147.] Let the general equation of a confocal system be 



~ 2 



where X is a variable parameter, which we shall distinguish by the 
suffix A 1 for the hyperboloids of two sheets, A, 2 for the hyperboloids 
of one sheet, and A 3 for the ellipsoids. The quantities 

0, A 15 b, \. 2 , c, A 3 

are in ascending order of magnitude. The quantity a is introduced 
for the sake of symmetry, but in our results we shall always suppose 
a = 0. 

If we consider the three surfaces whose parameters are A 15 A 2 , A 3 , 
we find, by elimination between their equations, that the value of 
x 2 at their point of intersection satisfies the equation 

X*(6 Z -a*)(C*-a*) = (A 1 2 - 2 )(A 2 2 - 2 )(A 3 2 - 2 ). (2) 

The values of f and z 2 may be found by transposing a, b, c 
symmetrically. 

Differentiating this equation with respect to \ ly we find 

dx Aj / 3 x 

~~T = r~9 - 9 * ^ 

d\ Aj 2 a 2 

If ds^ is the length of the intercept of the curve of intersection of 
A 2 and A 3 cut off between the surfaces A x and Aj + ^A^ then 

-^J 2 jb 2 ^ ~di\* ^ ~di\ 2 __ A 1 2 (A 2 2 -A 1 2 )(A 3 2 -A 1 2 ) 

3^1 = ^XT h ^| h ^r! B w-^w-^xv-^)- 

* This investigation is chiefly borrowed from a very interesting work, Lemons sur 
les Fonctions Inverses des Transcendantes et les Surfaces Isotherme*. Par G. 
Paris, 1857. 



182 CONFOCAL QUADRIC SURFACES. 

The denominator of this fraction is the product of the squares of 
the semi-axes of the surface A x . 
If we put 

7)2 _ \ 2 A 2 7)2 _ A 2 \ 2 <) n A 7) 2 _ \ 2 _ \ 2 /K\ 

**1 A 3 A 2 ^2 A 3 A l J anQ -^3 A 2 A l > \P) 

and if we make a = 0, then 

d _ D 2 D 3 ( 



It is easy to see that Z^ 2 and D 3 are the semi-axes of the central 
section of A x which is conjugate to the diameter passing 1 through 
the given point, and that D 2 is parallel to ds 2 , and D 3 to ds 3 . 

If we also substitute for the three parameters \ lt A 2 , A 3 their 
values in terms of three functions a, (3, y, denned by the equations 

da c . 

-j = , . > A., = when a = 0, 



/ 2 /2 /^f =1 > A 2 = * When = > ( 7 ) 

VA 2 2 b 2 Vc 2 A 2 2 

/? 

A 3 = c when y = ; 



then ^ = -D 2 D 3 da, ds 2 = -D 3 D 1 dp, ds 3 -D^D^ dy. (8) 

C 

148.] Now let V be the potential at any point a, /3, y, then the 
resultant force in the direction of ds is 



__ _ L _ dV c 
1- ds[" Jad Sl ~ "Jal)^!^ 

Since ^, ds 2 , and ^ 3 are at right angles to each other, the 
surface-integral over the element of area ds 2 ds 3 is 

- dV c DD D,D 



Now consider the element of volume intercepted between the 
surfaces a, /3, y, and a + ^a, fi + dfa y + dy. There will be eight 
such elements^ one in each octant of space. 

We have found the surface-integral for the element of surface 
intercepted from the surface a by the surfaces (3 and p + dfi, y and 



I49-] TRANSFORMATION OF POISSON s EQUATION. 183 

The surface-integral for the corresponding element of the surface 
a-f da will be 



da c 

since D^ is independent of a. The surface-integral for the two 
opposite faces of the element of volume, taken with respect to the 
interior of that volume, will be the difference of these quantities, or 



Similarly the surface-integrals for the other two pairs of forces 
will be 



. and 

c dy 2 c 

These six faces enclose an element whose volume is 

727 2 7 2 



and if p is the volume-density within that element, we find by 
Art. 77 that the total surface-integral of the element, together with 
the quantity of electricity within it, multiplied by 4 TT is zero, or, 
dividing by dadfidy, 



which is the form of Poisson s extension of Laplace s equation re 
ferred to ellipsoidal coordinates. 

If p = the fourth term vanishes, and the equation is equivalent 
to that of Laplace. 

For the general discussion of this equation the reader is referred 
to the work of Lame already mentioned. 

149.] To determine the quantities a, 0, y, we may put them in 
the form of ordinary elliptic functions by introducing the auxiliary 
angles 0, $, and \//-, where 

A x = sin0, 

A 2 = V c 2 sin 2 $ + b 2 cos^), (13) 



sm\// 

If we put 5 = h, and F + /2 - 1, we may call k and It the two 
complementary moduli of the confocal system, and we find 



184 CONFOCAL QUADKIC SURFACES. [l 50. 

an elliptic integral of the first kind, which we may write according 
to the usual notation F(kO}. 
In the same way we find 



13 = 



, 2^ 

1 / 2 cos 2 </> 
where FJc is the complete function for modulus k , 



y * 7 o * o " 

V 1 k* sm 2 \lr 

Here a is represented as a function of the angle 0, which is a 
function of the parameter A 15 /3 as a function of </> and thence of A 2 , 
and y as a function of \j/ and thence of A 3 . 

But these angles and parameters may be considered as functions 
of a, (3, y. The properties of such inverse functions, and of those 
connected with them, are explained in the treatise of M. Lame on 
that subject. 

It is easy to see that since the parameters are periodic functions 
of the auxiliary angles, they will be periodic functions of the 
quantities a, /3, y : the periods of Aj and A 3 are 4 F(k) and that of A 2 
is 2 F(Jc ). 

Particular Solutions. 

150.] If V is a linear function of a, (3, or y, the equation is 
satisfied. Hence we may deduce from the equation the distribution 
of electricity on any two confocal surfaces of the same family 
maintained at given potentials, and the potential at any point 
between them. 

The Hyperboloids of Two Sheets. 

When a is constant the corresponding surface is a hyperboloid 
of two sheets. Let us make the sign of a the same as that of x in 
the sheet under consideration. We shall thus be able to study one 
of these sheets at a time. 

Let a x , a 2 be the values of a corresponding to two single sheets, 
whether of different hyperboloids or of the same one, and let F 15 F 2 
be the potentials at which they are maintained. Then, if we make 

-a 2 r i + a(7 7 i Tg) fio\ 
, (18) 



the conditions will be satisfied at the two surfaces and throughout 
the space between them. If we make V constant and. equal to V 
in the space beyond the surface a l5 and constant and equal to F 2 



150.] DISTRIBUTION OF ELECTRICITY. 185 

in the space beyond the surface a 2 , we shall have obtained the 
complete solution of this particular case. 

The resultant force at any point of either sheet is 
R _ _dF_ _dFda 
ds l ~ da ds 

or ^ = r i~ r 2 C . (20) 

If pi be the perpendicular from the centre on the tangent plane 
at any point, and P l the product of the semi-axes of the surface, 
then p l D 2 D. 3 = P 1 . 

Hence we find ^1^2 C P\ ^oi\ 

1 = a ~p~ 

or the force at any point of the surface is proportional to the per 
pendicular from the centre on the tangent plane. 

The surface-density a- may be found from the equation 

4-770- = ^. (22) 

The total quantity of electricity on a segment cut off by a plane 
whose equation is x = a from one sheet of the hyperboloid is 

--iV (23) 



2 a l -a 2 

The quantity on the whole infinite sheet is therefore infinite. 
The limiting forms of the surface are : 

(1) When a = F^ the surface is the part of the plane of xz on 
the positive side of the positive branch of the hyperbola whose 
equation is #2 z z 

To o := 1 \ / 

b 2 c 2 

(2) When a = the surface is the plane of yz. 

(3) When a = F^ the surface is the part of the plane of xz on 
the negative side of the negative branch of the same hyperbola. 

The Hyperloloids of One Sheet. 

By making /3 constant we obtain the equation of the hyperboloid 
of one sheet. The two surfaces which form the boundaries of the 
electric field must therefore belong to two different hyperboloids. 
The investigation will in other respects be the same as for the 
hyperboloids of two sheets, and when the difference of potentials 
is given the density at any point of the surface will be proportional 
to the perpendicular from the centre on the tangent plane, and the 
whole quantity on the infinite sheet will be infinite. 



186 CONFOCAL QUADRIC SURFACES. 

Limiting Forms. 

(1) When /3 = the surface is the part of the plane of xz 
between the two branches of the hyperbola whose equation is 
written above, (24). 

(2) When = F(k ) the surface is the part of the plane of xy 
which is on the outside of the focal ellipse whose equation is 



The Ellipsoids. 

For any given ellipsoid y is constant. If two ellipsoids, y : and y 2 , 
be maintained at potentials V^ and V^ then, for any point y in the 
space between them, we have 

^) (26) 



71 72 
The surface-density at any point is 



where p 3 is the perpendicular from the centre on the tangent plane, 
and P 3 is the product of the semi-axes. 

The whole charge of electricity on either surface is 



a finite quantity. 

When y = F(k) the surface of the ellipsoid is at an infinite 
distance in all directions. 

If we make V 2 = and y 2 = F(k), we find for the quantity of 
electricity on an ellipsoid maintained at potential V in an infinitely 
extended field, V , . 

^ c WTr\ v / 

F(k)*-y 

The limiting form of the ellipsoids occurs when y = 0, in which 
case the surface is the part of the plane of xy within the focal 
ellipse, whose equation is written above. (25). 

The surface-density on the elliptic plate whose equation is (25), and 
whose eccentricity is $, is 

o- = - V __ X , (30) 



/ 

V 



and its charge is _ V 






151.] SURFACES OF REVOLUTION. 187 

Particular Cases. 

151.] If k is diminished till it becomes ultimately zero, the 
system of surfaces becomes transformed in the following manner : 

The real axis and one of the imaginary axes of each of the 
hyperboloids of two sheets are indefinitely diminished, and the 
surface ultimately coincides with two planes intersecting in the 
axis of z. 

The quantity a becomes identical with 6, and the equation of the 
system of meridional planes to which the first system is reduced is 

1-2 ,,,2 

_? ?. = o. (32) 

(sin a) 2 (cos a) 2 

The quantity /3 is reduced to 

^=/ s - = lo ^ tan t (33) 



whence we find 

2 
smd) = -5 - 5> cosc> = 




e* -f e~ 

If we call the exponential quantity \(e^ + er^) the hyperbolic 
cosine of /3, or more concisely the hypocosine of /3, or cos h ft, and if 
we call i (e^ e-P} the hyposine of ft, or sin^ ft, and if by the same 
analogy we call 

the hyposecant of ft, or sec h ft, 



cos h ft 

1 

sin^/3 
sin hf$ 
cost ft 



the hypocosecant of ft, or cosec Ji ft, 
the hypotangent of ft, or tan h ft, 



and COS 1 P the hypoeotang-ent of ft, or cot Ji ft ; 
sm/fc/3 

then A 2 = c sec h ft, and the equation of the system of hyperboloids 
of one sheet is 

= C 2 (35) 

" 



(seek ft) 2 

The quantity y is reduced to \ff, so that A 3 = c cosec y, and the 
equation of the system of ellipsoids is 

O O 9 

* + y + * = C 2. (36) 

(secy) 2 (tany) 2 

Ellipsoids of this kind, which are figures of revolution about their 
conjugate axes, are called Planetary ellipsoids. 



188 CONFOCAL QUADRIC SURFACES. 

The quantity of electricity on a planetary ellipsoid maintained at 
potential V in an infinite field, is 

gastf-JL-, (37) 



where c sec y is the equatorial radius, and c tan y is the polar radius. 
If y = 0, the figure is a circular disk of radius c, and 

V 
a = -- . (38) 

W *,v/e 2 -r a 
Q = c~ (39) 

2 

152.] to0^ Ciw*. Let b c, then = 1 and = 0, 

_ 9$ 
a = log tan - , whence A x = c tan ^ a, (40) 

and the equation of the hyperboloids of revolution of two sheets 
becomes #2 ^2 



(sec/*a) 2 " 

The quantity /3 becomes reduced to <, and each of the hyper 
boloids of one sheet is reduced to a pair of planes intersecting in 
the axis of x whose equation is 



(sin/3) 2 (cos/3) 2 
This is a system of meridional planes in which (3 is the longitude. 

The quantity y becomes log tan -- 5 whence A 3 = c cot k y, 
and the equation of the family of ellipsoids is 

>2 .,2 I 2 



(cosec/5y) 2 ~ 

These ellipsoids, in which the transverse axis is the axis of revo- 
lution, are called Ovary ellipsoids. 

The quantity of electricity on an ovary ellipsoid maintained at a 
potential V in an infinite field is 

Q = c (44) 

If the polar radius is A = c cot h y, and the equatorial radius is 
B = c cosec Ji y, 



- /AK . 

= log -- --- (45) 



1 54.] CYLINDERS AND PARABOLOIDS. 189 

If the equatorial radius is very small compared to the polar radius, 
as in a wire with rounded ends, 



y = log-,, j and Q = ; - A ; 5- ( 46 ) 

& B log A logjfj 

When both b and c become zero, their ratio remaining finite, 
the system of surfaces becomes two systems of confocal cones, and 
a system of spherical surfaces of which the radius is inversely 
proportional to y. 

If the ratio of b to c is zero or unity, the system of surfaces 
becomes one system of meridian planes, one system of right cones 
having a common axis, and a system of concentric spherical surfaces 
of which the radius is inversely proportional to y. This is the 
ordinary system of spherical polar coordinates. 

Cylindric Surfaces. 

153.] When c is infinite the surfaces are cylindric, the generating 
lines being parallel to z. One system of cylinders is elliptic, with 
the equation 

/yi2 ;/ - 

x + __! = 2 (47) 

(cos/fcci) 2 ^ (sin/U) 2 

The other is hyperbolic, with the equation 

-r 2 <?/2 

y _ 2 (48) 

(cos/3) 2 (sin/3) 2 " 
This system is represented in Fig. X, at the end of this volume. 

Confocal Paraboloids. 

154.] If in the general equations we transfer the origin of co 
ordinates to a point on the axis of x distant t from the centre of 
the system, and if we substitute for x, A, d, and c, t + x, 1 + A, l + b, 
and l + c respectively, and then make I increase indefinitely, we 
obtain, in the limit, the equation of a system of paraboloids whose 
foci are at the points x = b and x = c, 



If the variable parameter is A for the first system of elliptic 
paraboloids, JJL for the hyperbolic paraboloids, and v for the second 
system of elliptic paraboloids, we have A, 5, /u, c, v in ascending 
order of magnitude, and 



190 



CONFOCAL QUADRIC SUltFACES. 



[ I 54- 



x = 



c-b 



z * 

~~c=b~ 

\ = \ (b + c) \(c b) cos ha, 



x 



= 2 (c b) sin^- sin - 
v 22 



b) (cos fry cos/3 cos /i a), " 

1 

2 



%(c b)cosh -cos- sin ^ - 
LI LI & 



(50) 



(51) 



(52) 



When 5 = c we have the case of paraboloids of revolution about 
the axis of a?, and x = a (e^e 2 ^, 

y = 2ae a+ y cos/3, (53) 



The surfaces for which /3 is constant are planes through the axis, 
/3 being the angle which such a plane makes with a fixed plane 
through the axis. 

The surfaces for which a is constant are confocal paraboloids. 
When a=0 the paraboloid is reduced to a straight line terminating 
at the origin. 

We may also find the values of a, ft, y in terms of r, 6, and $, 
the spherical polar coordinates referred to the focus as origin,, and 
the axis of the parabolas as axis of the sphere, 



a = log 



cos i 0), 



ft = <P> (54) 

y = log (f* sin 4 &} 

We may compare the case in which the potential is equal to a, 
with the zonal solid harmonic r i Q 4 . Both satisfy Laplace s equa 
tion, and are homogeneous functions of x, y, #, but in the case 
derived from the paraboloid there is a discontinuity at the axis, and 
i has a value not differing by any finite quantity from zero. 

The surface-density on an electrified paraboloid in an infinite 
field (including the case of a straight line infinite in one direction) 
is inversely as the distance from the focus, or, in the case of 
the line, from the extremity of the line. 



CHAPTER XL 

THEORY OF ELECTRIC IMAGES AND ELECTRIC INVERSION. 

155.] We have already shewn that when a conducting sphere 
is under the influence of a known distribution of electricity, the 
distribution of electricity on the surface of the sphere can be 
determined by the method of spherical harmonics. 

For this purpose we require to expand the potential of the in 
fluencing system in a series of solid harmonics of positive degree, 
having the centre of the sphere as origin, and we then find a 
corresponding series of solid harmonics of negative degree, which 
express the potential due to the electrification of the sphere. 

By the use of this very powerful method of analysis, Poisson 
determined the electrification of a sphere under the influence of 
a given electrical system, and he also solved the more difficult 
problem to determine the distribution of electricity on two con 
ducting spheres in presence of each other. These investigations 
have been pursued at great length by Plana and others, who have 
confirmed the accuracy of Poisson. 

In applying this method to the most elementary case of a sphere 
under the influence of a single electrified point, we require to expand 
the potential due to the electrified point in a series of solid har 
monics, and to determine a second series of solid harmonics which 
express the potential, due to the electrification of the sphere, in the 
space outside. 

It does not appear that any of these mathematicians observed 
that this second series expresses the potential due to an imaginary 
electrified point, which has no physical existence as an electrified 
point, but which may be called an electrical image, because the 
action of the surface on external points is the same as that which 
would be produced by the imaginary electrified point if the spherical 
surface were removed. 



192 ELECTKIC IMAGES. [156. 

This discovery seems to have been reserved for Sir W. Thomson, 
who has developed it into a method of great power for the solution 
of electrical problems, and at the same time capable of being pre 
sented in an elementary geometrical form. 

His original investigations, which are contained in the Cambridge 
and Dublin Mathematical Journal, 1848, are expressed in terms of 
the ordinary theory of attraction at a distance, and make no use of 
the method of potentials and of the general theorems of Chapter IV, 
though they were probably discovered by these methods. Instead, 
however, of following the method of the author, I shall make free 
use of the idea of the potential and of equipotential surfaces, when 
ever the investigation can be rendered more intelligible by such 
means. 

Theory of Electric Images. 

156.] Let A and B, Figure 7, represent two points in a uniform 

dielectric medium of infinite extent. 
Let the charges of A and B be e 1 
and e 2 respectively. Let P be any 
point in space whose distances from 
A and B are r^ and r 2 respectively. 
Then the value of the potential at P 
will be TT e \ e 2 

Fig. 7. TI , r * 

The equipotential surfaces due to 

this distribution of electricity are represented in Fig. I (at the end 
of this volume) when e 1 and e 2 are of the same sign, and in Fig. II 
when they are of opposite signs. We have now to consider that 
surface for which V = 0, which is the only spherical surface in 
the system. When e 1 and e 2 are of the same sign, this surface is 
entirely at an infinite distance, but when they are of opposite signs 
there is a plane or spherical surface at a finite distance for which 
the potential is zero. 

The equation of this surface is 

fL + .4 = 0. (2) 

Its centre is at a point C in AB produced, such that 

AC .BC:-. e 2 : e 2 2 , 
and the radius of the sphere is 




The two points A and B are inverse points with respect to this 



15 7-] INVERSE POINTS. 193 

sphere, that is to say, they lie in the same radius, and the radius is 
a mean proportional between their distances from the centre. 

Since this spherical surface is at potential zero, if we suppose 
it constructed of thiii metal and connected with the earth, there 
will be no alteration of the potential at any point either outside or 
inside, but the electrical action everywhere will remain that due to 
the two electrified points A and B. 

If we now keep the metallic shell in connexion with the earth 
and remove the point , the potential within the sphere will become 
everywhere zero, but outside it will remain the same as before. 
For the surface of the sphere still remains at the same potential, 
and no change has been made in the exterior electrification. 

Hence, if an electrified point A be placed outside a spherical 
conductor which is at potential zero, the electrical action at all 
points outside the sphere will be that due to the point A together 
with another point B within the sphere, which we may call the 
electrical image of A. 

In the same way we may shew that if B is a point placed inside 
the spherical shell, the electrical action within the sphere is that 
due to B, together with its image A. 

157.] Definition of an Electrical Image. An electrical image is 
an electrified point or system of points on one side of a surface 
which would produce on the other side of that surface the same 
electrical action which the actual electrification of that surface 
really does produce. 

In Optics a point or s} T stem of points on one side of a mirror 
or lens which if it existed would emit the system of rays which 
actually exists on the other side of the mirror or lens, is called a 
virtual image. 

Electrical images correspond to virtual images in optics in being 
related to the space on the other side of the surface. They do not 
correspond to them in actual position, or in the merely approximate 
character of optical foci. 

There are no real electrical images, that is, imaginary electrified 
points which would produce, in the region on the same side of the 
electrified surface, an effect equivalent to that of the electrified surface. 

For if the potential in any region of space is equal to that due 
to a certain electrification in the same region it must be actually 
produced by that electrification. In fact, the electrification at any 
point may be found from the potential near that point by the 
application of Poisson s equation. 

VOL. I. O 



194 ELECTRIC IMAGES. [l 57 . 

Let a be the radius of the sphere. 

Let/* be the distance of the electrified point A from the centre C. 

Let e be the charge of this point. 

Then the image of the point is at S, on the same radius of the 

o 

sphere at a distance = , and the charge of the image is e-f 
J J 

We have shewn that this image 

will produce the same effect on the 
opposite side of the surface as the 
actual electrification of the surface 
does. We shall next determine the 
surface-density of this electrifica 
tion at any point P of the spherical 
surface, and for this purpose we shall 
make use of the theorem of Coulomb, 

Art. 80, that if R is the resultant force at the surface of a conductor, 

and o- the superficial density, 

R = 477 0-, 

R being measured away from the surface. 

We may consider R as the resultant of two forces, a repulsion 

Z> /Tf 1 

- =_ acting along AP, and an attraction e -, ^-^ acting along PB. 

A.L J JL Jj 

Resolving these forces in the directions of AC and CP, we find 
that the components of the repulsion are 

Pi P ft 

- along AC, and along CP. 




Those of the attraction are 



f P 3 * BP> CP 

o a 
Now BP r AP, and BC = , so that the components of 

J J 
the attraction may be written 

1 f 2 I 

AC > and ~ e 



The components of the attraction and the repulsion in the 
direction of AC are equal and opposite, and therefore the resultant 
force is entirely in the direction of the radius CP. This only 
confirms what we have already proved, that the sphere is an equi- 
potential surface, and therefore a surface to which the resultant 
force is everywhere perpendicular. 



158.] DISTRIBUTION OF ELECTRICITY. 195 

The resultant force measured along CP, the normal to the surface 
in the direction towards the side on which A is placed, is 



a AP 3 

If A is taken inside the sphere f is less than , and we must 
measure R inwards. For this case therefore 

V^A: > 

In all cases we may write 

AD. Ad 1 ,^ 

R = - e ~Cp-AP*> 

where AD, Ad are the segments of any line through A cutting the 
sphere, and their product is to be taken positive in all cases. 

158.] From this it follows, by Coulomb s theorem, Art. 80, 
that the surface-density at P is 

AD. Ad 1 , R . 

*=- e I^CPAP* 

The density of the electricity at any point of the sphere varies 
inversely as the cube of its distance from the point A. 

The effect of this superficial distribution, together with that of 
the point J, is to produce on the same side of the surface as the 
point A a potential equivalent to that due to e at A, and its image 

e -p at B> and on the other side of the surface the potential is 

t/ 

everywhere zero. Hence the effect of the superficial distribution 
by itself is to produce a potential on the side of A equivalent to 

that due to the image e- at B, and on the opposite side a 

J 
potential equal and opposite to that of e at A. 

The whole charge on the surface of the sphere is evidently e - 
since it is equivalent to the image at B. 

We have therefore arrived at the following theorems on the 
action of a distribution of electricity on a spherical surface, the 
surface-density being inversely as the cube of the distance from 
a point A either without or within the sphere. 

Let the density be given by the equation 



. . . 

where C is some constant quantity, then by equation (6) 

. (8) 



196 ELECTRIC IMAGES. 

The action of this superficial distribution on any point separated 
from A by the surface is equal to that of a quantity of electricity 
<?, or 4 TT a C 

AD .Ad 
concentrated at A. 

Its action on any point on the same side of the surface with A is 
equal to that of a quantity of electricity 



fAD.Ad 

concentrated at B the image of A. 

The whole quantity of electricity on the sphere is equal to the 
first of these quantities if A is within the sphere, and to the second 
if A is without the sphere. 

These propositions were established by Sir W. Thomson in his 
original geometrical investigations with reference to the distribution 
of electricity on spherical conductors, to which the student ought 
to refer. 

159.] If a system in which the distribution of electricity is 
known is placed in the neighbourhood of a conducting sphere of 
radius a, which is maintained at potential zero by connexion with 
the earth, then the electrifications due to the several parts of the 
system will be superposed. 

Let A I} A 2 , &c. be the electrified points of the system, f^f^ &c. 
their distances from the centre of the sphere, e 19 e 2 , &c. their 
charges, then the images _Z? ls .Z? 2 , &c. of these points will be in the 

a 2 a 2 
same radii as the points themselves, and at distances -~- > -^- &c. 

/l e/2 

from the centre of the sphere, and their charges will be 

a a 

- , _e?--&c, 

fl /2 

The potential on the outside of the sphere due to the superficial 
electrification will be the same as that which would be produced by 
the system of images 19 H 2 , &c. This system is therefore called 
the electrical image of the system A 1 , A 2 , &c. 

If the sphere instead of being at potential zero is at potential 7 ? , 
we must superpose a distribution of electricity on its outer surface 
having the uniform surface-density 

7 



The effect of this at all points outside the sphere will be equal to 



l6o.] IMAGE OF AN ELECTRIFIED SYSTEM. 197 

that of a quantity Va of electricity placed at its centre, and at 
all points inside the sphere the potential will be simply increased 
by 7. 

The whole charge on the sphere due to an external system of 
influencing points A ly A 2 , &c. is 

E= Fa-e"-e.,^--&e., (9) 

/I /2 

from which either the charge E or the potential V may be cal 
culated when the other is given. 

When the electrified system is within the spherical surface the 
induced charge on the surface is equal and of opposite sign to the 
inducing charge, as we have before proved it to be for every closed 
surface, with respect to points within it. 

160.] The energy due to the mutual action between an elec 
trified point e, at a distance f from the centre of the sphere greater 
than a the radius, and the electrification of the spherical surface 
due to the influence of the electrified point and the charge of the 
sphere, is 



where Fis the potential, and .2? the charge of the sphere. 

The repulsion between the electrified point and the sphere is 
therefore, by Art. 92, 

,V ef . 



Hence the force between the point and the sphere is always an 
attraction in the following cases 

(1) When the sphere is uninsulated. 

(2) When the sphere has no charge. 

(3) When the electrified point is very near the surface. 

In order that the force may be repulsive, the potential of the 

/ 3 
sphere must be positive and greater than e ^ , and the 

charge of the sphere must be of the same sign as e and greater 



At the point of equilibrium the equilibrium is unstable, the force 



198 ELECTRIC IMAGES. [l6l. 

being an attraction when the bodies are nearer and a repulsion 
when they are farther off. 

When the electrified point is within the spherical surface the 
force on the electrified point is always away from the centre of 
the sphere, and is equal to 



The surface-density at the point of the sphere nearest to the 
electrified point where it lies outside the sphere is 



The surface-density at the point of the sphere farthest from the 
electrified point is 



When E, the charge of the sphere, lies between 
W-) nd 



Af+ 

the electrification will be negative next the electrified point and 
positive on the opposite side. There will be a circular line of division 
between the positively and the negatively electrified parts of the 
surface, and this line will be a line of equilibrium. 

If E= e 



the equipotential surface which cuts the sphere in the line of equi 
librium is a sphere whose centre is the electrified point and whose 
radius is \/f 2 a 2 . 

The lines of force and equipotential surfaces belonging to a case 
of this kind are given in Figure IV at the end of this volume. 

Images in an Infinite Plane Conducting Surface. 

161.] If the two electrified points A and B in Art. 156 are 
electrified with equal charges of electricity of opposite signs, the 
surface of zero potential will be the plane, every point of which is 
equidistant from A and B. 



1 62.] 



IMAGES IN AN INFINITE PLANE. 



199 




Hence, if A be an electrified point whose charge is e, and AD 
a perpendicular on the plane, produce AD 
to B so that D = ^, and place at B 
a charge equal to e, then this charge 
at B will be the image of A } and will 
produce at all points on the same side of 
the plane as A, an effect equal to that 
of the actual electrification of the plane. 
For the potential on the side of A due 
to A and B fulfils the conditions that 
V 2 F= everywhere except at A, and 
that V = at the plane, and there is only Fig. 8. 

one form of V which can fulfil these conditions. 

To determine the resultant force at the point P of the plane, we 

M 

observe that it is compounded of two forces each equal to --^ , 

one acting along AP and the other along PB. Hence the resultant 
of these forces is in a direction parallel to AB and equal to 



Hence R, the resultant force measured from the surface towards the 
space in which A lies, is 



and the density at the point P is 



eAD 



On Electrical Inversion. 

162.] The method of electrical images leads directly to a method 
of transformation by which we may derive from any electrical 
problem of which we know the solution any number of other 
problems with their solutions. 

We have seen that the image of a point at a distance r from the 
centre of a sphere of radius R is in the same radius and at a distance 
/ such that r/=R 2 . Hence the image of a system of points, lines, 
or surfaces is obtained from the original system by the method 
known in pure geometry as the method of inversion, and described 
by Chasles, Salmon, and other mathematicians. 



200 ELECT1UC IMAGES. [162. 

If A and B are two points, A and B their images, being the 
A centre of inversion, and R the radius of the 
sphere of inversion, 

OA.OA = R 2 = OB.OB . 
Hence the triangles OAB } OB A are similar, 
and AB : A B : : A : OB f : : OA.OB : R 2 . 




B 



Fi - 9> If a quantity of electricity e be placed at A, 

its potential at B will be e 

= AS 
If e be placed at A its potential at B will be 

V = ~IW 

In the theory of electrical images 

e:e ::OA:R::R: OA . 
Hence V \V :: R:OB, (17) 

or the potential at B due to the electricity at A is to the potential 
at the image of B due to the electrical image of A as R is to OB. 

Since this ratio depends only on OB and not on OA, the potential 
at B due to any system of electrified bodies is to that at B due 
to the image of the system as R is to OB. 

If r be the distance of any point A from the centre, and / that 
of its image A , and if e be the electrification of A, and / that of A } 
also if L, 8, K be linear, superficial, and solid elements at A, and 
If, S , K their images at A , and A, <r, p, A , </, p the corresponding 
line-surface and volume-densities of electricity at the two points, 
V the potential at A due to the original system, and V the potential 
at A due to the inverse system, then 
/ _ L _ R 2 _ / a S _ ^ 4 _ / 4 K _R* _ / 6 
"7 := T " 7 2 ~~ ~~.P 1T = ~7 ~~ ^* If ~~ r r ~" ]* 

_/ JS ^ A^_ ^__-# 

~~e ~" ~r ~~~R T "" ^ "" / 

(/ r 3 .S 3 p r 5 _ 72 5 

(T ^ 3 /^ p .S 5 "~ r 5 

V _ r _ R 

T -Jf == V 

If in the original system a certain surface is that of a conductor, 

* See Thomson and Tait s Natural Philosophy, 515. 



163.] GEOMETRICAL THEOREMS. 201 

and has therefore a constant potential P, then in the transformed 

T) 

system the image of the surface will have a potential P -^ - But 

by placing at 0, the centre of inversion, a quantity of electricity 
equal to PR, the potential of the transformed surface is reduced 
to zero. 

Hence, if we know the distribution of electricity on a conductor 
when insulated in open space and charged to the potential P, we 
can find by inversion the distribution on a conductor whose form is 
the image of the first under the influence of an electrified point with 
a charge PR placed at the centre of inversion, the conductor 
being in connexion with the earth. 

163.] The following geometrical theorems are useful in studying 
cases of inversion. 

Every sphere becomes, when inverted, another sphere, unless 
it passes through the centre of inversion, in which case it becomes 
a plane. 

If the distances of the centres of the spheres from the centre of 
inversion are a and a , and if their radii are a and a , and if we 
define the power of the sphere with respect to the centre of in 
version to be the product of the segments cut off by the sphere 
from a line through the centre of inversion, then the power of the 
first sphere is cr a 2 , and that of the second is a 2 a 2 . We 
have in this case 

a a R 2 a?*-^ 

- = =- 8 s= 3= i (19) 

a a a- a- F 

or the ratio of the distances of the centres of the first and second 
spheres is equal to the ratio of their radii, and to the ratio of the 
power of the sphere of inversion to the power of the first sphere, 
or of the power of the second sphere to the power of the sphere 
of inversion. 

The centre of either sphere corresponds to the inverse point of 
the other with respect to the centre of inversion. 

In the case in which the inverse surfaces are a plane and a 
sphere, the perpendicular from the centre of inversion on the plane 
is to the radius of inversion as this radius is to the diameter of 
the sphere, and the sphere has its centre on this perpendicular and 
passes through the centre of inversion. 

Every circle is inverted into another circle unless it passes 
through the centre of inversion, in which case it becomes a straight 
line. 



202 ELECTRIC IMAGES. [164. 

The angle between two surfaces, or two lines at their intersection, 
is not changed by inversion. 

Every circle which passes through a point, and the image of that 
point with respect to a sphere, cuts the sphere at right angles. 

Hence, any circle which passes through a point and cuts the 
sphere at right angles passes through the image of the point. 

164.] "We may apply the method of inversion to deduce the 
distribution of electricity on an uninsulated sphere under the 
influence of an electrified point from the uniform distribution on 
an insulated sphere not influenced by any other body. 

If the electrified point be at A, take it for the centre of inversion, 
and if A is at a distance f from the centre of the sphere whose 
radius is #, the inverted figure will be a sphere whose radius is a 
and whose centre is distant f\ where 

a f R2 (20} 

== 



The centre of either of these spheres corresponds to the inverse 
point of the other with respect to A, or if C is the centre and B the 
inverse point of the first sphere, C will be the inverse point, and B 
the centre of the second. 

Now let a quantity e of electricity be communicated to the 
second sphere, and let it be uninfluenced by external forces. It 
will become uniformly distributed over the sphere with a surface- 
density ^ 



Its action at any point outside the sphere will be the same as 
that of a charge e placed at B the centre of the sphere. 
At the spherical surface and within it the potential is 

P = 7 (22) 

a constant quantity. 

Now let us invert this system. The centre B becomes in the 
inverted system the inverse point B, and the charge e at B 

AB 
becomes ^ -JT~ at B, and at any point separated from B by the 

surface the potential is that due to this charge at B. 

The potential at any point P on the spherical surface, or on the 
same side as _Z?, is in the inverted system 



AP 



165.] SYSTEMS OF IMAGES. 203 

If we now superpose on this system a charge e at A, where 



the potential on the spherical surface, and at all points on the same 
side as B, will be reduced to zero. At all points on the same side 
as A the potential will be that due to a charge e at A, and a charge 

e ~ at B. 

But e j T =-e a j 7 = -e a r (24) 

as we found before for the charge of the image at B. 

To find the density at any point of the first sphere we have 

<> 



Substituting for the value of a- in terms of the quantities be 
longing to the first sphere, we find the same value as in Art. 158, 

22 (26) 

V / 



On Finite Systems of Successive Images. 

165.] If two conducting planes intersect at an angle which is 
a submultiple of two right angles, there will be a finite system of 
images which will completely determine the electrification. 

For let AOB be a section of the two conducting planes per 
pendicular to their line of inter 
section, and let the angle of 

intersection AOB = -, let P 
n 

be an electrified- point, and let 

PO = r, and POB = 0. Then, 

if we draw a circle with centre 

and radius OP, and find points 

which are the successive images 

of P in the two planes beginning 

with OB, we shall find Q x for the Fig. 10. 

image of P in OB, P 2 for the image of Q l in OA, Q 3 for that of P 2 

in OB, P 3 for that of Q 3 in OA, and Q. 2 for that of P 3 in OB. 

If we had begun with the image of P in AO we should have 
found the same points in the reverse order Q 2 , P 3 , Q 3 , P 2 , Q 19 
provided AOB is a submultiple of two right angles. 




204 ELECTRIC IMAGES. [l66. 

For the alternate images P 15 P 2 , P 3 are ranged round the circle 
at angular intervals equal to 2 AOB, and the intermediate images 
Qi> 625 QB are a ^ intervals of the same magnitude. Hence, if 
2 AOB is a sub multiple of 27r, there will be a finite number of 
images, and none of these will fall within the angle AOB. If, 
however, AOB is not a submultiple of it, it will be impossible to 
represent the actual electrification as the result of a finite series of 
electrified points. 

If AOB = -, there will be n negative images Q lt Q 2) &c., each 

equal and of opposite sign to P, and n 1 positive images P 2 , 
P 3 , &c., each equal to P, and of the same sign. 

The angle between successive images of the same sign is 

fv 

If we consider either of the conducting planes as a plane of sym 
metry, we shall find the positive and negative images placed 
symmetrically with regard to that plane, so that for every positive 
image there is a negative image in the same normal, and at an 
equal distance on the opposite side of the plane. 

If we now invert this system with respect to any point, the two 
planes become two spheres, or a sphere and a plane intersecting 

at an angle - , the influencing point P being within this angle. 

p 

The successive images lie on the circle which passes through P 
and intersects both spheres at right angles. 

To find the position of the images we may either make use of 
the principle that a point and its image are in the same radius 
of the sphere, and draw successive chords of the circle beginning 
at P and passing through the centres of the two spheres al 
ternately. 

To find the charge which must be attributed to each image, take 
any point in the circle of intersection, then the charge of each 
image is proportional to its distance from this point, and its sign 
is positive or negative according as it belongs to the first or the 
second system. 

166.] We have thus found the distribution of the images when 
any space bounded by a conductor consisting of two spherical surfaces 

meeting at an angle - , and kept at potential zero, is influenced by 

an electrified point. 

We may by inversion deduce the case of a conductor consisting 



1 66.] TWO INTERSECTING SPHERES. 205 

o/ two spherical segments meeting at a re-entering angle - , charged 

to potential unity and placed in free space. 

For this purpose we invert the system with respect to P. The 
circle on which the images formerly lay now becomes a straight 
line through the centres of the spheres. 

If the figure (11) represents 
a section through the line of 
centres AB, and if D, J/ are the 
points where the circle of in 
tersection cuts the plane of the 
paper, then, to find the suc 
cessive images, draw DA a 
radius of the first circle, and 
draw DC, DB, &i?., making 

7T 2lT 

angles-, , &c. with DA. Fig. 11. 

n n 

The points C, B, &c. at which they cut the line of centres will 
he the positions of the positive images, and the charge of each 
will be represented by its distances from D. The last of these 
images will be at the centre of the second circle. 

To find the negative images draw DP, DQ, &c., making angles 

- > , &c. with the line of centres. The intersections of these 

lines with the line of centres will give the positions of the negative 
images, and the charge of each will be represented by its distance 
from D. 

The surface-density at any point of either sphere is the sum 
of the surface-densities due to the system of images. For instance, 
the surface-density at any point S of the sphere whose centre is 
A, is 




where A, B, C, &c. are the positive series of images. 

When S is on the circle of intersection the density is zero. 

To find the total charge on each of the spherical segments, we 
may find the surface-integral of the induction through that segment 
due to each of the images. 

The total charge on the segment whose centre is A due to the 
image at A whose charge is DA is 



206 ELECTRIC IMAGES. 



where is the centre of the circle of intersection. 

In the same way the charge on the same segment due to the 
image at B is \ (DB-\- OB), and so on, lines such as OB measured 
from to the left being reckoned negative. 

Hence the total charge on the segment whose centre is A is 
i (DA + DB + DC+ &c.) + i (OA + OB + 0(7+ &c.), 



167.] The method of electrical images may be applied to any 
space bounded by plane or spherical surfaces all of which cut one 
another in angles which are submultiples of two right angles. 

In order that such a system of spherical surfaces may exist, every 
solid angle of the figure must be trihedral, and two of its angles 
must be right angles, and the third either a right angle or a 
submultiple of two right angles. 

Hence the cases in which the number of images is finite are 

(1) A single spherical surface or a plane. 

(2) Two planes, a sphere and a plane, or two spheres intersecting 
at an angle - 

(3) These two surfaces with a third, which may be either plane 
or spherical, cutting both orthogonally. 

(4) These three surfaces with a fourth cutting the first two 
orthogonally and the third at an angle , . Of these four surfaces 

one at least must be spherical. 

We have already examined the first and second cases. In the 
first case we have a single image. In the second case we have 
2^1 images arranged in two series in a circle which passes 
through the influencing point and is orthogonal to both surfaces. 
In the third case we have, besides these images, their images with 
respect to the third surface, that is, 4^1 images in all besides the 
influencing point. 

In the fourth case we first draw through the influencing point 
a circle orthogonal to the first two surfaces, and determine on it 
the positions and magnitudes of the n negative images and the 
n \ positive images. Then through each of these 2n points, 
including the influencing point, we draw a circle orthogonal to 
the third and fourth surfaces, and determine on it two series of 



i68.] 



TWO SPHERES CUTTING ORTHOGONALLY. 



207 



images, n in each series. We shall obtain in this way, besides the 
influencing point, 2nn 1 positive and 2 n n negative images. 
These 4 n n points are the intersections of n circles with n other 
circles, and these circles belong to the two systems of lines of 
curvature of a cy elide. 

If each of these points is charged with the proper quantity of 
electricity, the surface whose potential is zero will consist of n-\-n 
spheres, forming two series of which the successive spheres of the 

first set intersect at angles - , and those of the second set at angles 

, while every sphere of the first set is orthogonal to every sphere 
of the second set. 

Case of Two Spheres cutting Orthogonally. See Fig. IV at the 

end of this volume. 

168.] Let A and B, Fig. 12, be the centres of two spheres cutting 
each other orthogonally in D and 
I/, and let the straight line DJ/ cut 
the line of centres in C. Then C 
is the image of A with respect to 
the sphere B, and also the image 
of B with respect to the sphere 
whose centre is A. If AD = a, 
BD = /3, then AB = \/a 2 + /3 2 , and 
if we place at A } B, C quantities Fig. 12. 

of electricity equal to a, /3, and ^ respectively, then both 

spheres will be equipotential surfaces whose potential is unity. 

We may therefore determine from this system the distribution of 
electricity in the following cases : 

(1) On the conductor PDQD* formed of the larger segments of 
both spheres. Its potential is 1, and its charge is 

a/3 




+ /3 



= AD+BD-CD. 



This quantity therefore measures the capacity of such a figure 
when free from the inductive action of other bodies. 

The density at any point P of the sphere whose centre is A, and 
the density at any point Q of the sphere whose centre is B, are 
respectively 



208 ELECTRIC IMAGES. [l68. 

At the points of intersection, D, J/, the density is zero. 

If one of the spheres is very much larger than the other, the 
density at the vertex of the smaller sphere is ultimately three times 
that at the vertex of the larger sphere. 

(2) The lens P DQ L formed by the two smaller segments of 

the spheres, charged with a quantity of electricity = -. , 

Va a -f-/3 2 

and acted on hy points A and B, charged with quantities a and /3, 
is also at potential unity, and the density at any point is expressed 
by the same formulae. 

(3) The meniscus DPD Q formed by the difference of the 
segments charged with a quantity a, and acted on by points B 

and C, charged respectively with quantities /3 and , is also 

va 2 -f /3 2 
in equilibrium at potential unity. 

(4) The other meniscus QDP Z/ under the action of A and C. 
We may also deduce the distribution of electricity on the following 

internal surfaces. 

The hollow lens PDQD under the influence of the internal 
electrified point C at the centre of the circle DD . 

The hollow meniscus under the influence of a point at the centre 
of the concave surface. 

The hollow formed of the two larger segments of both spheres 
under the influence of the three points A, B, C. 

But, instead of working out the solutions of these cases, we shall 
apply the principle of electrical images to determine the density 
of the electricity induced at the point P of the external surface of 
the conductor PDQI/ by the action of a point at charged with 
unit of electricity. 

Let OA = a, OB = 6, OP = r, BP =p, 
AD = a, BD= (3, AB = Jo* + pP. 

Invert the system with respect to a sphere of radius unity and 
centre 0. 

The two spheres will remain spheres, cutting each other ortho 
gonally, and having their centres in the same radii with A and B. 
If we indicate by accented letters the quantities corresponding to 
the inverted system, 

a I , a /3 

= 



1 68.] TWO SPHERES CUTTING ORTHOGONALLY. 209 

If, in the inverted system, the potential of the surface is unity, 
then the density at the point P f is 



If, in the original system, the density at P is <r, then 



and the potential is - . By placing at a negative charge of 

electricity equal to unity, the potential will become zero over the 
surface, and the density at P will be 






4 TT ar 



This gives the distribution of electricity on one of the spherical 
surfaces due to a charge placed at 0. The distribution on the 
other spherical surface may be found by exchanging a and b, a and 
/3, and putting q or AQ instead of p. 

To find the total charge induced on the conductor by the elec 
trified point at 0, let us examine the inverted system. 

In the inverted system we have a charge a at J , and /3 at ff, 

/ Qfi 

and a negative charge at a point C in the line 

such that AC : CB : : a 2 : /3 2 . 

If OA = a , = V, OC = c } we find 



_ 



Inverting this system the charges become 



, tiff 1 a/3 

and 



c 



Hence the whole charge on the conductor due to a unit of 
negative electricity at is 

a ., _ _ a 



VOL. I. 



210 



ELECTRIC IMAGES. 



[I6 9 . 



Distribution of Electricity on Three Spherical Surfaces ivhich 

Intersect at Right Angles. 

169.] Let the radii of the spheres be a, , y, then 
* CA = 7 2 + a 2 ~ 




Fig. 13. 



Let PQR, Fig. 1 3, be the feet 
of the perpendiculars from ABC 
on the opposite sides of the tri 
angle, and let be the inter 
section of perpendiculars. 

Then P is the image of B in 
the sphere y, and also the image 
of C in the sphere /3. Also is 
the image of P in the sphere o. 

Let charges a } /3, and y be 
placed at A, B, and C. 

Then the charge to be placed 
at P is 



/ V 02 + y 2 



Also ^P = 



sidered as the image of P, is 
o/3y 



, so that the charge at 0, con- 



In the same way we may find the system of images which are 
electrically equivalent to four spherical surfaces at potential unity 
intersecting at right angles. 

If the radius of the fourth sphere is 8, and if we make the charge 
at the centre of this sphere = 8, then the charge at the intersection 
of the line of centres of any two spheres, say a and p, with their 
plane of intersection, is 1 



v 



O~ I r\f) 



The charge at the intersection of the plane of any three centres 
ABC with the perpendicular from D is 



A/i+i+i 



I/O.] FOUR SPHERES CUTTING ORTHOGONALLY. 211 

and the charge at the intersection of the four perpendiculars is 
1 

V"l 1 1 f 
ci 2 /3 2 y 2 8 2 

System of Four Spheres Intersecting at Right Angles under the 
Action of an Electrified Point. 

170.] Let the four spheres be A,B,C,D, and let the electrified point 
be 0. Draw four spheres A It B l} C 1} D^, of which any one, A lt 
passes through and cuts three of the spheres, in this case B, 
C, and D, at right angles. Draw six spheres (ab), (ac), (ad), (be), 
(bd), (cd) } of which each passes through and through the circle 
of intersection of two of the original spheres. 

The three spheres B I} C lt D^ will intersect in another point besides 
0. Let this point be called A , and let ff, C , and I/ be the 
intersections of C l} J) lt A lf of D lt A I} B I} and of A 19 B, C re 
spectively. Any two of these spheres, A l , B 1 , will intersect one of 
the six (cd) in a point (ab ). There will be six such points. 

Any one of the spheres, A lf will intersect three of the six (ab), 
(ac), (ad) in a point a . There will be four such points. Finally, 
the six spheres (ab), (ac), (ad), (cd), (db), (be), will intersect in one 
point S. 

If we now invert the system with respect to a sphere of radius 
R and centre 0, the four spheres A, B, C, D will be inverted into 
spheres, and the other ten spheres will become planes. Of the 
points of intersection the first four A , B f , C , If will become the 
centres of the spheres, and the others will correspond to the other 
eleven points in the preceding article. These fifteen points form 
the image of in the system of four spheres. 

At the point A , which is the image of in the sphere A, we 

must place a charge equal to the image of 0, that is, , where a 

a 

is the radius of the sphere A, and a is the distance of its centre 
from 0. In the same way we must place the proper charges at 
ff, C , I/. 

The charges for each of the other eleven points may be found from 
the expressions in the last article by substituting a , /3 , y , 6 for 
a, /3, y, 8, and multiplying the result for each point by the distance 
of the point from 0, where 



c 2 -y 2 



P 2 



212 



ELE.CTRIC IMAGES. 



[171- 



Two Spheres not Intersecting. 

171.] When a space is bounded by two spherical surfaces which 
do not intersect, the successive images of an influencing point 
within this space form two infinite series, all of which lie beyond 
the spherical surfaces, and therefore fulfil the condition of the 
applicability of the method of electrical images. 

Any two non-intersecting spheres may be inverted into two 
concentric spheres by assuming as the point of inversion either 
of the two common inverse points of the pair of spheres. 

We shall begin, therefore, with the case of two uninsulated 
concentric spherical surfaces, subject to the induction of an elec 
trified point placed between them. 

Let the radius of the first be b, and that of the second be 1 ***, and 
let the distance of the influencing point from the centre be r = be u . 

Then all the successive images will be on the same radius as the 
influencing point. 

Let Q , Fig. 14, be the image of P in the first sphere, P 1 that 
of Q G in the second sphere, Q 1 that of P 1 in the first sphere, and 
so on ; then 

and OP S .OQ S _ 1 = b 2 e 2 , 
also OQ = de~ u , 

f)T> 7 3 yjtt + 2 SJ 

\JJL -I C/C/ * 

OQ 1 = fo-(+a*n, &c. 
Hence OP 8 = h 
OQ S = *< 

If the charge of P is denoted by P, 
Fi S- 14 - then 

P s = Pe>, Q s =-Pe-(+*\ 

Next, let Q/ be the image of P in the second sphere, P/ that of 
!/ in the first, &c., 




OQ, = 



OP 2 = 



P; = Pe~* } Q s = Pe^~ u . 

Of these images all the P s are positive, and all the Q s negative, 
all the P / s and Q s belong to the first sphere, and all the P s and 
/5 s to the second. 



1/2.] 



TWO SPHEEES NOT INTERSECTING. 



213 



The images within the first sphere form a converging series, the 
sum of which is 

-P 



This therefore is the quantity of electricity on the first or interior 
sphere. The images outside the second sphere form a diverging 
series, but the surface-integral of each with respect to the spherical 
surface is zero. The charge of electricity on the exterior spherical 
surface is therefore 

e ~ tt 



V*_l ) 



If we substitute for these expressions their values in terms of 
OA, OB, and OP, we find 

charge H *< 

n OS AP 

n J B = -P__. 

If we suppose the radii of the spheres to become infinite, the case 
becomes that of a point placed between two parallel planes A and H. 
In this case these expressions become 

T)~D 

charge on A = P , 



charge on = P - . 



172.] In order to pass from this case to that of any two spheres 
not intersecting each 
other, we begin by 
finding the two com 
mon inverse points 0, 
(7 through which all 
circles pass that are 
orthogonal to both 
spheres. Then, invert 
ing the system with 
respect to either of 
these points, the spheres 
become concentric, as 

rig. 15. 

in the first case. 

The radius OAPB on which the successive images lie becomes 
an arc of a circle through and (7, and the ratio of (/P to OP is 




214 ELECTRIC IMAGES. [ r 7 2 - 

equal to Ce n where C is a numerical quantity which for simplicity 
we may make equal to unity. 
We therefore put 

... -or . VA _ , as 



Let /3 a = <*, u a = 0. 

Then all the successive images of P will lie on the arc OAPB&. 
The position of the image of P in A is Q where 

(70 

^(oMg^| = 2a-^ 

That of Qo in is P x where 



Similarly 

u(P 6 ) = 

In the same way if the successive images of P in B, A, B, &c. 
are Q , P/, /, &c., 



To find the charge of any image P s we observe that in the 
inverted figure its charge is 



OP 

In the original figure we must multiply this by (XP 8 . Hence the 
charge of P s in the dipolar figure is 



A / P * 

V "OP: 



If we make f = VOP.(/P 3 and call the parameter of the 
point P, then we may write 



or the charge of any image is proportional to its parameter. 

If we make use of the curvilinear coordinates u and v t such that 



x+ V - 

Jc sin hu ksvbv m 

coshucosv ~~ cosku -cosv 9 



172.] TWO SPHERES NOT INTERSECTING. 215 

x 2 + (y fccotv) 2 = k z cosec 2 v, 
(x + k cot huf + f = & cosec h?u, 






\/COS /& W COS V 

Since the charge of each image is proportional to its parameter, 
, and is to be taken positively or negatively according as it is of 
the form P or Q, we find 

P\/coshu cost; 

JT t = > 

V cos ^ (u + 2 5 -BJ) cos v 



PVcoshu cosv 



V cos 7^(2 a w 2*-sr) cos v 



Pvcoshu cos v 
Vcos A(u 2siv) cos v 



PV cos hit cosv 



Vcosk(2fi u + 2s &) cosv 

We have now obtained the positions and charges of the two 
infinite series of images. We have next to determine the total 
charge on the sphere A by finding the sum of all the images within 
it which are of the form Q or P / . We may write this 



COSV 



<*m\ / z, n , 

1 Vcosn(u 231*) cosv 

. _ *== 
PVcoshu cosv 



- //cos ^(2 a u 2 six) cosv 
In the same way the total induced charge on B is 



COS V * t - 



cosv 



COSV 



t -n I i i n ^ 

- Vcos/i(2j3 u + 2s>&) cosv 
* In these expressions we must remember that 



and the other functions of u are derived from these by the same definitions as the 
corresponding trigonometrical functions. 

The method of applying dipolar coordinates to this case was given by Thomson in 
Lioui ille s Journal for 1847. See Thomson s reprint of Electrical Papers, 211, 212. 
In the text I have made use of the investigation of Prof. Betti, Nuovo Cimento, 
vol. xx, for the analytical method, but I have retained the idea of electrical images as 
used by Thomson in his original investigation, Phil. Mag., 1853. 



216 ELECTEIC IMAGES. [ T 73- 

173.] We shall apply these results to the determination of the 
coefficients of capacity and induction of two spheres whose radii are 
a and #, and the distance of whose centres is <?. 

In this case 



sin/$.a= > sin/./3=T 
a b 

Let the sphere A be at potential unity, and the sphere B at 
potential zero. 

Then the successive images of a charge a placed at the centre 
of the sphere A will be those of the actual distribution of electricity. 
All the images will lie on the axis between the poles and the 
centres of the spheres. 

The values of u and v for the centre of the sphere A are 

u 2 a, v = 0. 

Hence we must substitute a or k - j for P. and 2 a for u t and 

sin fia 

v = in the equations, remembering that P itself forms part of the 
charge of A. We thus find for the coefficient of capacity of A 



=o sn aw -a 
for the coefficient of induction of A on B or of B on A 



and for the coefficient of capacity of B 



sn 



To calculate these quantities in terms of a and #, the radii of the 
spheres, and of c the distance between their centres, we make use 
of the following quantities 






TWO ELECTRIFIED SPHERES. 217 

We may now write the hyperbolic sines in terms of p^ q, r ; thus 



2S = OC 
.=1 




Proceeding 1 to the actual calculation we find, either by this 
process or by the direct calculation of the successive images as 
shewn in Sir W. Thomson s paper, which is more convenient for 
the earlier part of the series, 



- 



o 

" 



CLO U/~U~ u, u o 



ab 2 _ _ & 

- f + "* 



174.] We have then the following equations to determine the 
charges E a and E b of the two spheres when electrified to potentials 
7 a and 7 b respectively, 



If We put qaaQbb qab 2 = & = ff > 

and p aa = q bb I/, p ab = q a b D , Pvb <1. &> 
whence Paalhb Pai? = & > 

then the equations to determine the potentials in terms of the 
charges are = 



^^, and^> b6 are the coefficients of potential. 
The total energy of the system is, by Art. 85, 



+ 2 



218 ELECTRIC IMAGES. [!74- 

The repulsion between the spheres is therefore, by Arts. 92, 93, 

F= +2 + 



where c is the distance between the centres of the spheres. 

Of these two expressions for the repulsion, the first, which 
expresses it in terms of the potentials of the spheres and the 
variations of the coefficients of capacity and induction, is the most 
convenient for calculation. 

We have therefore to differentiate the q s with respect to c. 
These quantities are expressed as functions of k, a, /3, and -57, and 
must be differentiated on the supposition that a and b are constant. 
From the equations 

. 7 j . , n sin h a sin h 8 
K = a sin h a = b sm hfi = c : j 

, da sin h a cos hQ 

we find -y- = j. 7 > 

dc k sin li TX 

dp _ cos k a sin h /3 
dc ~ 

d ST 1 



dk cos Ji a cos 



dc sin h -ar 

whence we find 

dq aa cos h a cos h$ q aa s=&(sc a cos h (3) cos h (s TV a) 



dc sin/fc ar k ^*s=o c (sin^(s-s7 a)) ; 

dq ab ___ cos h a cos h ft q ab 

d<tbb _ 



c?<? sin h & k *s=o c (sin h ({3 + *-BT)) 2 

Sir William Thomson has calculated the force between two 
spheres of equal radius separated by any distance less than the 
diameter of one of them. For greater distances it is not necessary 
to use more than two or three of the successive images. 

The series for the differential coefficients of the ^s with respect 
to c are easily obtained by direct differentiation 



1 7 5.] TWO SPHERES IN CONTACT. 219 

dq aa _ 2a 2 bc 2a*b 2 c(2c 2 -2b 2 -a 2 ) 

7/7 : - ^2-Zp2 ~ ~c*-* ~ " 



__ab a 2 b 2 (3 c 2 - a 2 - b 2 ) 
"~~ 



a*b*{(5c 2 -a 2 -b 2 )(c 2 -a 2 -b 2 )-a 2 b 2 } 
C 2 (c 2 -a 2 -b 2 + ab) 2 (c 2 -a 2 -b 2 - ab) 2 



clq bb _ 2ab 2 c 2a 2 b*c(2c 2 -2a 2 -b 2 ) 

~ ~ ~ ~ G 



Distribution of Electricity on Tivo Spheres in Contact. 

175.] If we suppose the two spheres at potential unity and not 
influenced by any other point, then, if we invert the system with 
respect to the point of contact, we shall have two parallel planes, 

distant and 7 from the point of inversion, and electrified by 

the action of a unit of electricity at that point. 

There will be a series of positive images, each equal to unity, at 

distances s( (- j\ from the origin, where s may have any integer 

value from oo to -foo. 

There will also be a series of negative images each equal to 1, 
the distances of which from the origin, reckoned in the direction of 

1 A K 
a, are - +s(- + 7)- 
a \a b 

When this system is inverted back again into the form of the 
two spheres in contact, we have a corresponding series of negative 
images, the distances of which from the point of contact are of the 

form - , where s is positive for the sphere A and negative 
/I 1\ 

*(-a + d 

for the sphere B. The charge of each image, when the potential 
of the spheres is unity, is numerically equal to its distance from the 
point of contact, and is always negative. 

There will also be a series of positive images whose distances 
from the point of contact measured in the direction of the centre 

of a, are of the form - - 



When s is zero, or a positive integer, the image is in the sphere A. 
When s is a negative integer the image is in the sphere S. 



220 ELECTEIC IMAGES. [ 1 75> 

The charge of each image is measured by its distance from the 
origin and is always positive. 

The total charge of the sphere A is therefore 

=oo i ab 

= 



Each of these series is infinite, but if we combine them in the form 



-6*=i s (a + b)(s(a+b}-d) 

the series becomes converging. 

In the same way we find for the charge of the sphere B } 

^*=oo ab ab 

tt h = 7, . 



The values of E a and U b are not^ so far as I know, expressible 
in terms of known functions. Their difference, however,, is easily 
expressed, for 



i:ab 

cot 



a + b a-\-b 
When the spheres are equal the charge of each for potential unity 



s 



= log e 2 = 1.0986^. 

When the sphere A is very small compared with the sphere B 
the charge on A is 

E* = j %I^ approximately; 

^ a 2 
or E a = -- r 

The charge on B is nearly the same as if A were removed, or 

E b = b. 

The mean density on each sphere is found by dividing the charge 
by the surface. In this way we get 



1 77.] SPHERICAL BOWL. 221 



E 



Hence, if a very small sphere is made to touch a very large one, 
the mean density on the small sphere is equal to that on the large 

o 

sphere multiplied by , or 1.644936. 

Application of Electrical Inversion to the case of a Spherical Bowl. 
176.] One of the most remarkable illustrations of the power of 
Sir W. Thomson s method of Electrical Images is furnished by his 
investigation of the distribution of electricity on a portion of a 
spherical surface bounded by a small circle. The results of this 
investigation, without proof, were communicated to M. Liouville 
and published in his Journal in 1847. The complete investigation 
is given in the reprint of Thomson s Electrical Papers, Article XV. 
I am not aware that a solution of the problem of the distribution 
of electricity on a finite portion of any curved surface has been 
given by any other mathematician. 

As I wish to explain the method rather than to verify the 
calculation, I shall not enter at length into either the geometry 
or the integration, but refer my readers to Thomson s work. 

Distribution of Electricity on an Ellipsoid. 

177.] It is shewn by a well-known method* that the attraction 
of a shell bounded by two similar and similarly situated and 
concentric ellipsoids is such that there is no resultant attraction 
on any point within the shell. If we suppose the thickness of 
the shell to diminish indefinitely while its density increases, we 
ultimately arrive at the conception of a surface-density varying 
as the perpendicular from the centre on the tangent plane, and 
since the resultant attraction of this superficial distribution on any 
point within the ellipsoid is zero, electricity, if so distributed on 
the surface,, will be in equilibrium. 

Hence, the surface-density at any point of an ellipsoid undis 
turbed by external influence varies as the distance of the tangent 
plane from the centre. 

* Thomson and Tait s Natural Philosophy, 520, or Art. 150 of this book. 



222 ELECTRIC IMAGES. 

Distribution of Electricity on a Disk. 

By making two of the axes of the ellipsoid equal, and making 
the third vanish, we arrive at the case of a circular disk, and at an 
expression for the surface-density at any point P of such a disk 
when electrified to the potential V and left undisturbed by external 
influence. If <r be the surface- density on one side of the disk, 
and if KPL be a chord drawn through the point P, then 

V 



Application of the Principle of Electric Inversion. 

178.] Take any point Q as the centre of inversion, and let R 
be the radius of the sphere of inversion. Then the plane of the 
disk becomes a spherical surface passing through Q, and the disk 
itself becomes a portion of the spherical surface bounded by a circle. 
We shall call this portion of the surface the bowl. 

If S is the disk electrified to potential V and free from external 
influence, then its electrical image S will be a spherical segment at 
potential zero, and electrified by the influence of a quantity Y R of 
electricity placed at Q. 

We have therefore by the process of inversion obtained the 
solution of the problem of the distribution of electricity on a 
bowl or a plane disk when under the influence of an electrified 
point in the surface of the sphere or plane produced. 

Influence of an Electrified Point placed on the unoccupied part of the 

Spherical Surface. 

The form of the solution, as deduced by the principles already 
given and by the geometry of inversion, is as follows : 

If C is the central point or pole of the spherical bowl S, and 
if a is the distance from C to any point in the edge of the segment, 
then, if a quantity q of electricity is placed at a point Q in the 
surface of the sphere produced, and if the bowl S is maintained 
at potential zero, the density a- at any point P of the bowl will be 



~ 27T 2 QP 2 A/ a 2 -OP 2 

CQ, CP, and QP being the straight lines joining the points, <?, Q, 
and P. 

It is remarkable that this expression is independent of the radius 
of the spherical surface of which the bowl is a part. It is therefore 
applicable without alteration to the case of a plane disk. 



l8o.] SPHERICAL BOWL. 223 

Influence of any Number of Electrified Points. 

Now let us consider the sphere as divided into two parts, one of 
which, the spherical segment on which we have determined the 
electric distribution, we shall call the bowl, and the other the 
remainder, or unoccupied part of the sphere on which the in 
fluencing- point Q is placed. 

If any number of influencing points are placed on the remainder 
of the sphere, the electricity induced by these on any point of the 
bowl may be obtained by the summation of the densities induced 
by each separately. 

179.] Let the whole of the remaining surface of the sphere 
be uniformly electrified, the surface-density being p, then the 
density at any point of the bowl may be obtained by ordinary 
integration over the surface thus electrified. 

We shall thus obtain the solution of the case in which the bowl 
is at potential zero, and electrified by the influence of the remaining 
portion of the spherical surface rigidly electrified with density p. 

Now let the whole system be insulated and placed within a 
sphere of diameter f, and let this sphere be uniformly and rigidly 
electrified so that its surface-density is //. 

There will be no resultant force within this sphere, and therefore 
the distribution of electricity on the bowl will be unaltered, but 
the potential of all points within the sphere will be increased by 
a quantity V where 



Hence the potential at every point of the bowl will now be V. 

Now let us suppose that this sphere is concentric with the sphere 
of which the bowl forms a part, and that its radius exceeds that 
of the latter sphere by an infinitely small quantity. 

We have now the case of the bowl maintained at potential V and 
influenced by the remainder of the sphere rigidly electrified with 
superficial density p -f p . 

180.] We have now only to suppose p-j-p = 0, and we get the 
case of the bowl maintained at potential V and free from external 
influence. 

If <T is the density on either surface of the bowl at a given point 
when the bowl is at potential zero, and is influenced by the rest 
of the sphere electrified to density p, then, when the bowl is main 
tained at potential F, we must increase the density on the outside 
of the bowl by p , the density on the supposed enveloping sphere. 



224 



ELECTRIC IMAGES. 



[181. 



The result of this investigation is that if f is the diameter of 
the sphere, a the chord of the radius of the bowl, and r the chord 
of the distance of P from the pole of the bowl, then the surface- 
density <r on the inside of the bowl is 



27r 2 / 

and the surface-density on the outside of the bowl at the same 
point is V 



+ 



27T/ 



In the calculation of this result no operation is employed more 
abstruse than ordinary integration over part of a spherical surface. 
To complete the theory of the electrification of a spherical bowl 
we only require the geometry of the inversion of spherical surfaces. 

181.] Let it be required to find the surface-density induced at 
any point of the bowl by a quantity q of electricity placed at a 
point Q, not now in the spherical surface produced. 

Invert the bowl with respect to Q, the radius of the sphere of 
inversion being R. The bowl 8 will be inverted into its image $", 
and the point P will have P for its image. We have now to 
determine the density </ at P when the bowl 8 is maintained at 
potential V , such that q = V R, and is not influenced by any 
external force. 

The density o- at the point P of the original bowl is then 



<J = 



QP 1 



this bowl being at potential zero, and influenced by a quantity q of 

electricity placed at Q. 

The result of this process is as follows : 

Let the figure represent a section 
through the centre, 0, of the sphere, 
the pole, C } of the bowl, and the in 
fluencing point Q. D is a point 
which corresponds in the inverted 
figure to the unoccupied pole of the 
rim of the bowl, and may be found 
by the following construction. 

Draw through Q the chords EQE 
and FQF , then if we suppose the 
radius of the sphere of inversion to 
be a mean proportional between the 

segments into which a chord is divided at Q, E F will be the 




Fig. 16. 



1 8 1.] SPHERICAL BOWL. 225 



image of EF. Bisect the arc F CE in I/, so that F &=!/&, and 
draw J/QD to meet the sphere in D. D is the point required. 
Also through 0, the centre of the sphere, and Q draw HOQH. 
meeting the sphere in H and H . Then if P be any point in the 
bowl, the surface-density at P on the side which is separated from 
Q by the completed spherical surface, induced by a quantity q of 
electricity at Q, will be 



~ 






where a denotes the chord drawn from <?, the pole of the bowl, 
to the rim of the bowl. 

On the side next to Q the surface-density is 

q QH.QH 
f 27T 2 HH .PQ* 



VOL. I. 



CHAPTER XII. 



THEORY OF CONJUGATE FUNCTIONS IN TWO DIMENSIONS. 



182.] THE number of independent cases in which the problem 
of electrical equilibrium has been solved is very small. The method 
of spherical harmonics has been employed for spherical conductors, 
and the methods of electrical images and of inversion are still more 
powerful in the cases to which they can be applied. The case of 
surfaces of the second degree is the only one, as far as I know, 
in which both the equipotential surfaces and the lines of force are 
known when the lines of force are not plane curves. 

But there is an important class of problems in the theory of 
electrical equilibrium, and in that of the conduction of currents, 
in which we have to consider space of two dimensions only. 

For instance, if throughout the part of the electric field under 
consideration, and for a considerable distance beyond it, the surfaces 
of all the conductors are generated by the motion of straight lines 
parallel to the axis of z, and if the part of the field where this 
ceases to be the case is so far from the part considered that the 
electrical action of the distant part on the field may be neglected, 
then the electricity will be uniformly distributed along each gene 
rating line, and if we consider a part of the field bounded by two 
planes perpendicular to the axis of z and at distance unity, the 
potential and the distribution of electricity will be functions of x 
and y only. 

If p dx dy denotes the quantity of electricity in an element whose 
base is dx dy and height unity, and a- ds the quantity on an element 
of area whose base is the linear element ds and height unity, then 
the equation of Poisson may be written 



183.] PROBLEMS IN TWO DIMENSIONS. 227 

When there is no free electricity, this is reduced to the equation 
of Laplace, 



The general problem of electric equilibrium may be stated as 
follows : 

A continuous space of two dimensions, bounded by closed curves 
C 19 C 2 , &c. being given, to find the form of a function, F, such that 
at these boundaries its value may be F 15 F 2 , &c. respectively, being 
constant for each boundary, and that within this space V may be 
everywhere finite, continuous, and single valued, and may satisfy 
Laplace s equation. 

I am not aware that any perfectly general solution of even this 
question has been given, but the method of transformation given in 
Art. 190 is applicable to this case, and is much more powerful than 
any known method applicable to tbree dimensions. 

The method depends on the properties of conjugate functions of 
two variables. 

Definition of Conjugate Functions. 

183.] Two quantities a and /3 are said to be conjugate functions 
of x and y^ if a + V T /3 is a function of x -f \/ 1 y. 
It follows from this definition that 

do. d(3 da d8 

= and + = ; w 



dx 2 dy* ~ dz 2 , dy* 

Hence both functions satisfy Laplace s equation. Also 

dadft__dadft_d^ 2 Jo* _~dfi 2 ^ 2 _ 2 
dx dy dy dx ~ dx\ + dy\ ~" dx h ~fy\ ~~ 

If x and y are rectangular coordinates, and if ds^ is the intercept 
of the curve (/3 = constant) between the curves a and a -f da, and 
ds. 2 the intercept of a between the curves /3 and /3 -f ^/3, then 
ds,_d^_ 1 
da ~ dp " R 
and the curves intersect at right angles. 

If we suppose the potential F = F -f,a, where k is some con 
stant, then V will satisfy Laplace s equation, and the curves (a) will 
be equipotential curves. The curves (/3) will be lines of force, and 



228 CONJUGATE FUNCTIONS. [184. 

the surface-integral of a surface whose projection on the plane of 
xy is the curve AB will be k(pBpA\ where $ A and (3 B are the 
values of (3 at the extremities of the curve. 

If a series of curves corresponding 1 to values of a in arithmetical 
progression is drawn on the plane, and another series corresponding 
to a series of values of /3 having the same common difference, then 
the two series of curves will everywhere intersect at right angles, 
and, if the common difference is small enough, the elements into 
which the plane is divided will be ultimately little squares, whose 
sides, in different parts of the field, are in different directions and of 
different magnitude, being inversely proportional to R. 

If two or more of the equipotential lines (a) are closed curves 
enclosing a continuous space between them, we may take these for 
the surfaces of conductors at potentials (V Q + ka^, (F" 4-^a 2 ), &c. 
respectively. The quantity of electricity upon any one of these 

Jc 
between the lines of force ^ and /3 2 will be Oa &) 

The number of equipotential lines between two conductors will 
therefore indicate their difference of potential, and the number of 
lines of force which emerge from a conductor will indicate the 
quantity of electricity upon it. 

We must next state some of the most important theorems 
relating to conjugate functions, and in proving them we may use 
either the equations (1), containing the differential coefficients, or 
the original definition, which makes use of imaginary symbols. 

184.] THEOREM I. If x and y are conjugate functions with respect 
to x and y, and if x" and y" are also conjugate functions with 
respect to x and y, then the functions x + x" and y +y" will 
le conjugate functions with respect to x 



dx _dy dx" _dy" . 

~7~ - ~T~ 5 <*-U.U ~ - = - - , 

dx ay dx dy 



therefore 



dx dy 



A1 dx dy dx" dy" 

Also = _^L an d - = - J ; 

dy dx dy dx 



therefore 

dy dx 

or x -fa?" and /+/ are conjugate with respect to x and y. 



185.] GRAPHIC METHOD. 229 

Graphic Representation of a Function which is the Sum of Two 
Given Functions. 

Let a function (a) of x and y be graphically represented by a 
series of curves in the plane of xy, each of these curves corre 
sponding to a value of a which belongs to a series of such values 
increasing by a common difference, b. 

Let any other function, (3, of x and y be represented in the same 
way by a series of curves corresponding to a series of values of j3 
having the same common difference as those of a. 

Then to represent the function a + ft in the same way, we must 
draw a series of curves through the intersections of the two former 
series from the intersection of the curves (a) and (/3) to that of the 
curves (a +6) and (/3 8), then through the intersection of (a 4- 2 b) 
and (/3 2 b), and so on. At each of these points the function will 
have the same value, namely a + ft. The next curve must be drawn 
through the points of intersection of a and /3 + 5, of a + b and ft, 
of a + 2 b and /3 8, and so on. The function belonging to this 
curve will be a + /3 + 8. 

In this way, when the series of curves (a) and the series (/3) are 
drawn, the series (a + (3) may be constructed. These three series of 
curves may be drawn on separate pieces of transparent paper, and 
when the first and second have been properly superposed, the third 
may be drawn. 

The combination of conjugate functions by addition in this way 
enables us to draw figures of many interesting cases with very 
little trouble when we know how to draw the simpler cases of 
which they are compounded. We have, however, a far more 
powerful method of transformation of solutions, depending on the 
following theorem. 

185.] THEOREM II. If x" and y" are conjugate functions with 
respect to the variables x and y , and if x and y are conjugate 
functions with respect to x and y> then x" and y" will be con 
jugate functions with respect to x and y. 

dx" dx" dx dx" dy 

For T- = -7-7- -5- + -^-7- -y- > 

dx dx dx dy dx 

df dy dy" dx 
dy dy dx dy 

*r_. 

dy 



230 CONJUGATE FUNCTIONS. 

dx" dx" dx dx" dy 

and = - r - + -j -f- , 
dy dx dy dy dy 

dy" dy dy" dx 
dy dx dx dx 



dx 

and these are the conditions that x" and y" should be conjugate 
functions of x and y. 

This may also be shewn from the original definition of conjugate 
functions. For #"-f\/^iy is a function of x f + V 1 y , and 
x + \/ iy is a function of # + \/ ly. Hence, x" + \/ \y" 
is a function of # + \/ \y. 

In the same way we may shew that if af and y are conjugate 
functions of x and y y then x and y are conjugate functions of x 
and y. 

This theorem may be interpreted graphically as follows : 

Let x , y be taken as rectangular coordinates, and let the curves 
corresponding to values of x" and of y" taken in regular arithmetical 
series be drawn on paper. A double system of curves will thus be 
drawn cutting the paper into little squares. Let the paper be also 
ruled with horizontal and vertical lines at equal intervals, and let 
these lines be marked with the corresponding values of x and y . 

Next, let another piece of paper be taken in which x and y are 
made rectangular coordinates and a double system of curves of, y 
is drawn, each curve being marked with the corresponding value 
of x or y . This system of curvilinear coordinates will correspond, 
point for point, to the rectilinear system of coordinates x , y on the 
first piece of paper. 

Hence, if we take any number of points on the curve x" on the 
first paper, and note the values of x and y at these points, and 
mark the corresponding points on the second paper, we shall find 
a number of points on the transformed curve x" . If we do the 
same for all the curves x" , y" on the first paper, we shall obtain on 
the second paper a double series of curves of , y" of a different form, 
but having the same property of cutting the paper into little 
squares. 



1 86.] 



THEOREMS. 



231 



186.] THEOREM III. If V is any function of V and y , and if of 

and if are conjugate functions of oo and y, then 



integration being between the same limits. 

dV dVdx dYdif 

-j- = -j-> -J- + -T-* -j- 
a? a* 00 dy dx 



^_ 
dx z ~ dx * 



dx dy 
" h dx d dx dx + d 2 dx 



and 



dx 



dy\ dx dy dy dy 



Adding the last two equations, and remembering the conditions 
of conjugate functions (1), we find 



_ 

~ dx 72 

_ ,d 2 Y 

~ W 2 



2 "7~?2 

dx ; 



Hence 



dy 
dx dy dy dx 






If V is a potential, then, by Poisson s equation 



and we may write the result 

ffpdxdy=ffp <lx dy , 

or the quantity of electricity in corresponding portions of two 
systems is the same if the coordinates of one system are conjugate 
functions of those of the other. 



232 CONJUGATE FUNCTIONS. [187. 

Additional Theorems on Conjugate Functions. 

187.] THEOREM IV. If x^ and y l3 and also # 2 and y^ are con 
jugate functions of x and y, then, if 

X=x lL x z - M2 , and r= ^^ + 3^, 
X and T will be conjugate functions of x and y. 

For X+ y^Tr = + V 



THEOREM V. If </> be a solution of the equation 

^ ^0 

dx* + df 

and e = 



will be conjugate functions of x and y. 

For R and are conjugate functions of 7^ and -~- , and these 

, r , . J r & , dx dy 

are conjugate functions of x and y. 



EXAMPLE I. Inversion. 

188.] As an example of the general method of transformation 
let us take the case of inversion in two dimensions. 

If is a fixed point in a plane, and OA a fixed direction, and 
if r = OP = ae? y and = AOP, and if x, y are the rectangular 
coordinates of P with respect to 0, 



p and Q are conjugate functions of x and y. 

If // = np and = nO, p and ^ will be conjugate functions of p 
and 0. In the case in which n = 1 we have 

/ = , and B =-e, (6) 

which is the case of ordinary inversion combined with turning the 
figure 180 round OA. 

Inversion in Two Dimensions. 
In this case if r and / represent the distances of corresponding 



i8 9 -] 



ELECTRIC IMAGES IN TWO DIMENSIONS. 



233 



points from 0, e and / the total electrification of a body, 8 and & 
superficial elements, V and V solid elements, a- and </ surface- 
densities, p and p volume densities, $ and <f> corresponding po 
tentials, 



a* / 2 



~s 



EL 
~r 



(7) 



EXAMPLE II. Electric Images in Two Dimensions. 

189.] Let A be the centre of a circle of radius AQ = b, and let 
E be a charge at A 3 then the potential 
at any point P is 

b Q. 

and if the circle is a section of a hollow 
conducting cylinder, the surface-density 
E 




at any point Q is -7 



Fig. 17. 



Invert the system with respect to a point 0, making 

AO = mb, and a 2 = (m 2 -l)b 2 ; 
then we have a charge at A equal to that at A } where AA = 

The density at Q is 

E W-ZIf 
~2ri> A Q 2 
and the potential at any point P f within the circle is 

< = = 2 E (log b log AP), 

= 2^ (log 0P -log^ P -log). (9) 

This is equivalent to a combination of a charge ^ at ^ , and a 

charge J at 0, which is the image of A , with respect to the 

circle. The imaginary charge at is equal and opposite to that 

If the point P is defined by its polar coordinates referred to the 
centre of the circle, and if we put 

p = log r log b, and p = log AA log 5, 
then AP = be?. AA ^be?*, A0 = be-p* , (10) 



234 CONJUGATE FUNCTIONS. [190. 

and the potential at the point (p, 6} is 



E log (e 2 Po 2 eK e? cos + e 2 ?) -f 2 Ep . (11) 
This is the potential at the point (p, 0) due to a charge E, placed 

at the point (p , 0), with the condition that when p = 0, <p = 0. 
In this case p and are the conjugate functions in equations (5) : 

p is the logarithm of the ratio of the radius vector of a point to 

the radius of the circle, and 6 is an angle. 

The centre is the only singular point in this system of coordinates, 

and the line-integral of / -=- ds round a closed curve is zero or 2 TT, 

J U/S 

according as the closed curve excludes or includes the centre. 

EXAMPLE III. Neumann s Transformation of this Case *. 

190.] Now let a and ft be any conjugate functions of x and y, 
such that the curves (a) are equipotential curves, and the curves 
(ft) are lines of force due to a system consisting of a charge of half 
a unit at the origin, and an electrified system disposed in any 
manner at a certain distance from the origin. 

. Let us suppose that the curve for which the potential is a is 
a closed curve, such that no part of the electrified system except the 
half-unit at the origin lies within this curve. 

Then all the curves (a) between this curve and the origin will be 
closed curves surrounding the origin, and all the curves (ft) will 
meet in the origin, and will cut the curves (a) orthogonally. 

The coordinates of any point within the curve (a ) will be determ 
ined by the values of a and ft at that point, and if the point travels 
round one of the curves a in the positive direction, the value of ft 
will increase by 2 TT for each complete circuit. 

If we now suppose the curve (a ) to be the section of the inner 
surface of a hollow cylinder of any form maintained at potential 
zero under the influence of a charge of linear density E on a line of 
which the origin is the projection, then we may leave the external 
electrified system out of consideration, and we have for the potential 
at any point (a) within the curve 

<*> = 2^(a-a ), (12) 

and for the quantity of electricity on any part of the curve a 
between the points corresponding to ft 1 and ft z , 

q = 2J0G8.-/9,). (13) 

See Crelie s Journal, 1861. 



190.] NEUMANN S TRANSFORMATION. 235 

If in this way, or in any other, we have determined the dis 
tribution of potential for the case of a given curve of section when 
the charge is placed at a given point taken as origin, we may pass 
to the case in which the charge is placed at any other point by an 
application of the general method of transformation. 

Let the values of a and /3 for the point at which the charge is 
placed be a x and ft, then substituting in equation (11) a ct for p, 
and ft ft for 6, we find for the potential at any point whose co 
ordinates are a and ft 
$ = Elog (l-2e a -icos((3-l3 1 ) + e 2 ( a - a J) 

-.E log (l-2^ +a i- 2a ocos(/3-^ 1 ) + ^ 2 ( a+a i- 2a o)) + 2^(a 1 -a ). (14) 

This expression for the potential becomes zero when a=a , and is 
finite and continuous within the curve a except at the point a t ft , 
at which point the first term becomes infinite, and in its immediate 
neighbourhood is ultimately equal to 2E log/, where / is the 
distance from that point. 

We have therefore obtained the means of deducing the solution 
of Green s problem for a charge at any point within a closed curve 
when the solution for a charge at any other point is known. 

The charge induced upon an element of the curve a between the 
points /3 and /3 + dp by a charge E placed at the point a : ft is 



_ _ - _ J B (l ~ 

27T l-2^ a l- a o)cOS(/3-ft) + * 2 ( a l-o) 

From this expression we may find the potential at any point 
a i A within the closed curve, when the value of the potential at 
every point of the closed curve is given as a function of j3, and 
there is no electrification within the closed curve. 

For, by Theorem II of Chap. Ill, the part of the potential at 
a x ft, due to the maintenance of the portion dj3 of the closed curve 
at the potential F, is n V, where n is the charge induced on df$ by 
unit of electrification at c^ft. Hence, if V is the potential at a 
point on the closed curve defined as a function of ft and $ the 
potential at the point a, ft within the closed curve, there being no 
electrification within the curve, 

~ (16) 

- 



2e(i- a <i>cos(/3 j 



236 CONJUGATE FUNCTIONS. [ I 9 i - 

EXAMPLE IV. Distribution of Electricity near an Edge of a 
Conductor formed by Two Plane Faces. 

191.] In the case of an infinite plane face of a conductor charged 
with electricity to the surface -density o- , we find for the potential 
at a distance y from the plane 



where C is the value of the potential of the conductor itself. 

Assume a straight line in the plane as a polar axis, and transform 
into polar coordinates, and we find for the potential 

V= C 4 77 o- # P sin 0, 

and for the quantity of electricity on a parallelogram of breadth 
unity, and length ae? measured from the axis 

E = (T^ae?. 

Now let us make p = np and = n6 , then, since p and are 
conjugate to p and 0, the equations 

V = (74 77 o- ae n ? sin nO 
and E = v^ae 1 "? 

express a possible distribution of electricity and of potential. 

If we write r for ae? y r will be the distance from the axis, and 
6 the angle, and we shall have 

y.n 

V = C 4TTO-Q n , sin n 6, 
a 



V will be equal to C whenever nd = 77 or a multiple of 77. 

Let the edge be a salient angle of the conductor, the inclination 
of the faces being a, then the angle of the dielectric is 2 IT a, so 
that when 6 = 2 TT a the point is in the other face of the conductor. 
We must therefore make 

n(2it a) = 77, 
Then r=C-4ir<r a(-\~ "sin ^ 

W 277 



The surface-density <r at any distance r from the edge is 

a TT 

dE -ir ,r**-* 



1Q2.] ELLIPSES AND HYPERBOLAS. 237 

When the angle is a salient one a is less than 77, and the surface- 
density varies according- to some inverse power of the distance 
from the edge, so that at the edge itself the density becomes 
infinite, although the whole charge reckoned from the edge to any 
finite distance from it is always finite. 

Thus, when a = the edge is infinitely sharp, like the edge of a 
mathematical plane. In this case the density varies inversely as 
the square root of the distance from the edge. 

When a = the edge is like that of an equilateral prism, and the 

o 

density varies inversely as the ^ power of the distance. 

When a = - the edge is a right angle, and the density is in- 
2 

versely as the cube root of the distance. 

o _ 

When a = - the edge is like that of a regular hexagonal prism, 

o 

and the density is inversely as the fourth root of the distance. 

When a = 77 the edge is obliterated, and the density is constant. 

When a = -f TT the edge is like that in the inside of the hexagonal 
prism, and the density is directly as the square root of the distance 
from the edge. 

When a=f 77 the edge is a re-entrant right angle, and the density 
is directly as the distance from the edge. 

When a=-f 77 the edge is a re-entrant angle of 60, and the 
density is directly as the square of the distance from the edge. 

In reality, in all cases in which the density becomes infinite at 
any point, there is a discharge of electricity into the dielectric at 
that point, as is explained in Art. 55. 

EXAMPLE V. Ellipses and Hyperbolas. Fig. X. 
192.] We have seen that if 

#j_ = e* cos ty, y^ = e* sin x//-, (1) 

x and y will be conjugate functions of $ and ^. 

Also, if x 2 = er* cos \j/, y% = e~* sin \^, (2) 

x. 2 and y 2 will be conjugate functions. Hence, if 
2 a? = ^ + ^2= (* + <?-*) cos VT, 2y = ft + #j = (<?*-*-*) sin ijr, (3) 

x and y will also be conjugate functions of $ and \f/. 

In this case the points for which $ is constant lie in the ellipse 
whose axes are & -f er* and e* er*. 



238 CONJUGATE FUNCTIONS. 

The points for which \l/ is constant lie in the hyperbola whose 
axes are 2 cos \j/ and 2 sin \//. 

On the axis of x, between x = I and x = + I, 

<f) = 0, \js = cos- 1 x. (4) 

On the axis of #, beyond these limits on either side, we have 

x> 1, $ = 0, < = log (# + y^T), (5) 



Hence,, if <p is the potential function, and \j/ the function of flow, 
we have the case of electricity flowing- from the negative to the 
positive side of the axis of x through the space between the points 
1 and + 1 , the parts of the axis beyond these limits being 
impervious to electricity. 

Since, in this case, the axis of y is a line of flow, we may suppose 
it also impervious to electricity. 

We may also consider the ellipses to be sections of the equi- 
potential surfaces due to an indefinitely long flat conductor of 
breadth 2, charged with half a unit of electricity per unit of length. 

If we make \ff the potential function, and < the function of flow, 
the case becomes that of an infinite plane from which a strip of 
breadth 2 has been cut away and the plane on one side charged to 
potential IT while the other remains at zero. 

These cases may be considered as particular cases of the quadric 
surfaces treated of in Chapter X. The forms of the curves are 
given in Fig. X. 

EXAMPLE VI. Fig. XI. 
193.] Let us next consider oc and y f as functions of x and y, where 

^!/ 

x b log V# 2 -f y 2 , y = I tan- 1 - ( 6 ) 

af and y will be also conjugate functions of $ and \/r. 

The curves resulting from the transformation of Fig. X with 
respect to these new coordinates are given in Fig. XI. 

If x and y are rectangular coordinates, then the properties of the 
axis of x in the first figure will belong to a series of lines parallel 
to x in the second figure for which y bn ir, where n is any 
integer. 

The positive values of x f on these lines will correspond to values 
of x greater than unity, for which, as we have already seen, 

__ / /~2^ \ 

\l? = rnr, $ log(>+ ^/ X 2 1) = log U & + V e b I/. (7) 



195 ] EDGE OF AN ELECTRIFIED PLATE. 239 

The negative values of x on the same lines will correspond to 
values of x less than unity, for which, as we have seen, 


<f> = 0, \l/ = cos~ l x = cos- 1 e b . (8) 

The properties of the axis of y in the first figure will belong to 
a series of lines in the second figure parallel to # , for which 

/ = $*( + ). (9) 

The value of \ff along these lines is \j/ = 77 (n r -f J-) for all points 
both positive and negative, and 

= log (y + vV + = lo U* + V e b ~ + I/. (10) 

194.] If we consider <p as the potential function, and \/r as the 
function of flow, we may consider the case to be that of an in 
definitely long strip of metal of breadth it d with a non-conducting 
division extending from the origin indefinitely in the positive 
direction, and thus dividing the positive part of the strip into two 
separate channels. We may suppose this division to be a narrow 
slit in the sheet of metal. 

If a current of electricity is made to flow along one of these 
divisions and back again along the other, the entrance and exit of 
the current being at an indefinite distance on the positive side of 
the origin, the distribution of potential and of current will be given 
by the functions $ and \/r respectively. 

If, on the other hand, we make i/r the potential, and $ the 
function of flow, then the case will be that of a current in the 
general direction of y, flowing through a sheet in which a number 
of non-conducting divisions are placed parallel to x, extending from 
the axis of y to an indefinite distance in the negative direction. 

195.] We may also apply the results to two important cases in 
statical electricity. 

(1) Let a conductor in the form of a plane sheet, bounded by a 
straight edge but otherwise unlimited, be placed in the plane of xz 
on the positive side of the origin, and let two infinite conducting 
planes be placed parallel to it and at distances \-nb on either side. 
Then, if \js is the potential function, its value is for the middle 
conductor and J TT for the two planes. 

Let us consider the quantity of electricity on a part of the middle 
conductor, extending to a distance 1 in the direction of z, and from 
the origin to x = a. 

The electricity on the part of this strip extending from x l to x. 2 

is -<- 



240 CONJUGATE FUNCTIONS. [196. 

Hence from the origin to x = a the amount is 

E= log(e b + v e b l) - (11) 

47T 

If a is large compared with &, this becomes 



Hence the quantity of electricity on the plane hounded by the 
straight edge is greater than it would have been if the electricity 
had been uniformly distributed over it with the same density that 
it has at a distance from the boundary, and it is equal to the 
quantity of electricity having the same uniform surface-density, 
but extending to a breadth equal to b log e 2 beyond the actual 
boundary of the plate. 

This imaginary uniform distribution is indicated by the dotted 
straight lines in Fig. XI. The vertical lines represent lines of 
force, and the horizontal lines equipotential surfaces, on the hypo 
thesis that the density is uniform over both planes, produced to 
infinity in all directions. 

196.] Electrical condensers are sometimes formed of a plate 
placed midway between two parallel plates extending considerably 
beyond the intermediate one on all sides. If the radius of curvature 
of the boundary of the intermediate plate is great compared with 
the distance between the plates, we may treat the boundary as 
approximately a straight line, and calculate the capacity of the 
condenser by supposing the intermediate plate to have its area 
extended by a strip of uniform breadth round its boundary, and 
assuming the surface-density on the extended plate the same as 
it is in the parts not near the boundary. 

Thus, if 8 be the actual area of the plate, L its circumference, 
and B the distance between the large plates, we have 

b = -B, (13) 

TT 

and the breadth of the additional strip is 

= ^.-B, (14) 

IT 

so that the extended area is 

8 = 8 + 3L-}o2. (15) 



7T 



196.] DENSITY NEAR THE EDGE. 241 

The capacity of the middle plate is 



Correction for the Thickness of the Plate. 

Since the middle plate is generally of a thickness which cannot 
be neglected in comparison with the distance between the plates, 
we may obtain a better representation of the facts of the case by 
supposing the section of the intermediate plate to correspond with 
the curve -fy = \j/ . 

The plate will be of nearly uniform thickness, (3 = 2#\//, at a 
distance from the boundary, but will be rounded near the edge. 

The position of the actual edge of the plate is found by putting 
/ = 0, whence x f I log cos ^ . (17) 

The value of (p at this edge is 0, and at a point for which x = a 
it is a + I log e 2 

~~b~ 
Hence the quantity of electricity on the plate is the same as 

if a strip of breadth JR. , wv n ft > 

a = log e (2cos^) 

had been added to the plate, the density being assumed to be every 
where the same as it is at a distance from the boundary. 

Density near the Edge. 

The surface-density at any point of the plate is 

* 

1 d$ _ 1 <? T 

4 TT dx f ~ 477$ /~2*T~ 
V* 6 ~_l 

1 / _!. _ 4 ^ \ 

=aiVl^i & + |^ 6 -&C.A (19) 



The quantity within brackets rapidly approaches unity as x 
increases, so that at a distance from the boundary equal to n times 
the breadth of the strip a, the actual density is greater than the 

normal density by about 2n+1 of the normal density. 

In like manner we may calculate the density on the infinite planes 



(20) 



When # = 0, the density is 2~^ of the normal density. 
VOL. i. R 



242 CONJUGATE FUNCTIONS. 

At n times the breadth of the strip on the positive side, the 
density is less than the normal density by about +1 

At n times the breadth of the strip on the negative side, the 
density is about of the normal density. 

These results indicate the degree of accuracy to be expected in 
applying this method to plates of limited extent, or in which 
irregularities may exist not very far from the boundary. The same 
distribution would exist in the case of an infinite series of similar 
plates at equal distances, the potentials of these plates being 
alternately + V and V. In this case we must take the distance 
between the plates equal to B. 

197.] (2) The second case we shall consider is that of an infinite 
series of planes parallel to x z at distances = lib, and all cut off by 
the plane of yz> so that they extend only on the negative side of this 
plane. If we make (/> the potential function, we may regard these 
planes as conductors at potential zero. 

Let us consider the curves for which ^> is constant. 

When y = nirb, that is, in the prolongation of each of the planes, 
we have j = 6 log i ^ + e -^ ^ 

when y = (n+\}bir, that is, in the intermediate positions 

^=5 log i(* --*). (22) 

Hence, when < is large, the curve for which is constant is 
an undulating line whose mean distance from the axis of y is . 
approximately 



a = bfr-log.!), (23) 

and the amplitude of the undulations on either side of this line is 



When <p is large this becomes be~ 2 ^ } so that the curve approaches 
to the form of a straight line parallel to the axis of y at a distance 
a from ab on the positive side. 

If we suppose a plane for which of = a, kept at a constant 
potential while the system of parallel planes is kept at a different 
potential, then, since 6$ = a + b log e 2, the surface-density of 
the electricity induced on the plane is equal to that which would 
have been induced on it by a plane parallel to itself at a potential 
equal to that of the series of planes, but at a distance greater 
than that of the edges of the planes by b log 2. 



1 99.] A GROOVED SURFACE. 243 

If B is the distance between two of the planes of the series, 
B = TT b, so that the additional distance is 

a =5^. (25) 



198.] Let us next consider the space included between two of 
the equipotential surfaces, one of which consists of a series of parallel 
waves, while the other corresponds to a large value of $, and may 
be considered as approximately plane. 

If D is the depth of these undulations from the crest to the trough 

of each wave, then we find for the corresponding value of <, 

D 

^. (26) 



The value of of at the crest of the wave is 

6 log i(e* + (?-*). (27) 

Hence, if A is the distance from the crests of the waves to the 
opposite plane, the capacity of the system composed of the plane 
surface and the undulated surface is the same as that of two planes 
at a distance A + a where 

= v lo s ^- (28) 

7T 7T - 

l+e B 

199.] If a single groove of this form be made in a conductor 
having the rest of its surface plane, and if the other conductor is 
a plane surface at a distance A, the capacity of the one conductor 
with respect to the other will be diminished. The amount of this 

diminution will be less than the -th part of the diminution due 

n 

to n such grooves side by side, for in the latter case the average 
electrical force between the conductors will be less than in the 
former case, so that the induction on the surface of each groove will 
be diminished on account of the neighbouring grooves. 

If L is the length, B the breadth, and D the depth of the groove, 
the capacity of a portion of the opposite plane whose area is S will be 

8 LB a 



4 TT A 4iK A.A + a 
If A is large compared with B or a , the correction becomes 



(29) 



R 2 



244: CONJUGATE FUNCTIONS. [2OO. 

and for a slit of infinite depth, putting D = oo, the correction is 



To find the surface-density on the series of parallel plates we 
must find o- = -f- f when d> = 0. We find 



V e 2& - 1 

The average density on the plane plate at distance A from the 
edges of the series of plates is o- = 7 . Hence, at a distance from 
the edge of one of the plates equal to na the surface- density is 
of this average density. 



200.] Let us next attempt to deduce from these results the 
distribution of electricity in the figure formed by rotating the 
plane of the figure about the axis y =R. In this case, Poisson s 
equation will assume the form 

dV 

(88) 



Let us assume F^=<, the function given in Art. 193, and determine 
the value of p from this equation. We know that the first two 
terms disappear, and therefore 

" = -i^?7^- (34) 

If we suppose that, in addition to the surface-density already 
investigated, there is a distribution of electricity in space according 
to the law just stated, the distribution of potential will be repre 
sented by the curves in Fig. XI. 

Now from this figure it is manifest that -=-7 is generally very 

" 
small except near the boundaries of the plates, so that the new 

distribution may be approximately represented by what actually 
exists, namely a certain superficial distribution near the edges of 
the plates. 

If therefore we integrate / I pdaicty between the limits ,/=0 and 
y =-5, and from # = oo to # = +oo, we shall find the whole 

2i 

additional charge on one side of the plates due to the curvature. 



201.] THEORY OF THOMSONS GUARD-RING. 245 

. d(fr d\j/ 
Since - = ^-, > 
ay dx 



Integrating with respect to y , we find 



This is the total quantity of electricity which we must suppose 
distributed in space near the positive side of one of the cylindric 
plates per unit of circumference. Since it is only close to the edge 
of the plate that the density is sensible, we may suppose it all 
condensed on the surface of the plate without altering sensibly its 
action on the opposed plane surface, and in calculating the attraction 
between that surface and the cylindric surface we may suppose this 
electricity to belong to the cylindric surface. 

The superficial charge on the positive surface of the plate per 
unit of length would have been J, if there had been no curvature. 

T) 

Hence this charge must be multiplied by the factor (l +i~j) 
to get the total charge on the positive side. 

In the case of a disk of radius R placed midway between two 
infinite parallel plates at a distance , we find for the capacity 
of the disk T>,, } 

* + S**!*S + l3. (38) 

.> 77 

Theory of Thomson s Guard-ring. 

201.] In some of Sir W. Thomson s electrometers, a large plane 
surface is kept at one potential, and at a distance a from this surface 
is placed a plane disk of radius R surrounded by a large plane plate 
called a Guard-ring with a circular aperture of radius R concentric 
with the disk. This disk and plate are kept at potential zero. 

The interval between the disk and the guard-plate may be 
regarded as a circular groove of infinite depth, and of breadth 
KR, which we denote by B. 



246 CONJUGATE FUNCTIONS. \_2O2. 

The charge on the disk due to unit potential of the large disk, 

7?2 

supposing the density uniform, would be -- 

4 ^L 

The charge on one side of a straight groove of breadth B and 
length It = 27T.Z2, and of infinite depth, would be 

RB 



But since the groove is not straight, but has a radius of curvature 

j) 
R, this must be multiplied by the factor (l + i -~-) 

The whole charge on the disk is therefore 



8A SA 

The value of a cannot be greater than 



R>*-R* g 





, =0.22 B nearly. -, 

If B is small compared with either A or R this expression will 
give a sufficiently good approximation to the charge on the disk 
due to unity of difference of potential. The ratio of A to R 
may have any value, but the radii of the large disk and of the 
guard-ring must exceed R by several multiples of A. 

EXAMPLE VII. Fig. XII. 

202.] Helmholtz, in his memoir on discontinuous fluid motion *, 
has pointed out the application of several formulae in which the 
coordinates are expressed as functions of the potential and its 
conjugate function. 

One of these may be applied to the case of an electrified plate 
of finite size placed parallel to an infinite plane surface connected 
with the earth. 

Since x^ = A < and y^ A ty, 

and also # 2 = A e* cos \//- and y^ = A e^ sin \j/, 

are conjugate functions of < and \fr, the functions formed by adding 
#! to # 2 and^ to y^ will be also conjugate. Hence, if 
x = A$ + Ae$ cos \js, 
y = A \lr-\- A e$ sin if/, 

* Konigl. AkacL der Wissenschaften, zu Berlin, April 23, 1868. 



202.] TWO EQUAL DISKS. 247 

then x and y will be conjugate with respect to $ and \\r, and < and 
\j/ will be conjugate with respect to x and y. 

Now let x and y be rectangular coordinates, and let k-ty be the 
potential, then (/> will be conjugate to k^, Jc being any constant. 

Let us put \lf = TT, then y = ^TT, # = J (< **). 

If < varies from oo to 0, and then from to +00, x varies 
from oo to A and from A to oo. Hence the equipotential 
surface for which k^\f TT is a plane parallel to # at a distance b = it A 
from the origin, and extending from oo to x = A. 

Let us consider a portion of this plane, extending from 
x = (A -f a) to x = A and from z = to z = c, 
let us suppose its distance from the plane of xz to be y b = ATT, 
and its potential to be V = k ^ = Jc TT. 

The charge of electricity on any portion of this part of the plane 
is found by ascertaining the values of $ at its extremities. 

If these are fa and fa, the quantity of electricity is 

ck(fa-fa). 
We have therefore to determine <f> from the equation 



will have a negative value fa and a positive value fa at the edge 
of the plane, where x = A, <p = 0. 

Hence the charge on the negative side is cJcfa y and that on 
the positive side is ckfa. 

If we suppose that a is large compared with A, 



If we neglect the exponential terms* in fa we shall find that the 
charge on the negative surface exceeds that which it would have 
if the superficial density had been uniform and equal to that at a 
distance from the boundary, by a quantity equal to the charge on a 

strip of breadth A - with the uniform superficial density. 
The total capacity of the part of the plane considered is 



24:8 CONJUGATE FUNCTIONS. [203. 

The total charge is CF, and the attraction towards the infinite 
plane is 

A 



A 



a 
g-r 

A 

The equipotential lines and lines of force are given in Fig. XII. 

EXAMPLE VIII. Theory of a Grating of Parallel Wires. Fig. XIII. 

203.] In many electrical instruments a wire grating is used to 
prevent certain parts of the apparatus from being electrified by 
induction. We know that if a conductor be entirely surrounded 
by a metallic vessel at the same potential with itself, no electricity 
can be induced on the surface of the conductor by any electrified 
body outside the vessel. The conductor, however, when completely 
surrounded by metal, cannot be seen, and therefore, in certain cases, 
an aperture is left which is covered with a grating of fine wire. 
Let us investigate the effect of this grating in diminishing the 
effect of electrical induction. We shall suppose the grating to 
consist of a series of parallel wires in one plane and at equal 
intervals, the diameter of the wires being small compared with the 
distance between them, while the nearest portions of the electrified 
bodies on the one side and of the protected conductor on the other 
are at distances from the plane of the screen, which are considerable 
compared with the distance between consecutive wires. 

204.] The potential at a distance / from the axis of a straight 
wire of infinite length charged with a quantity of electricity A. per 
unit of length is F = - 2 A log / + <?. ( 1 ) 

We may express this in terms of polar coordinates referred to an 
axis whose distance from the wire is unity, in which case we must 
make / 2 = 1 -f 2 r cos + r 2 , (2) 

and if we suppose that the axis of reference is also charged with 
the linear density A , we find 

V A. log (1 2r cos + r 2 ) 2 A log r + C. (3) 

If we now make 



- /yl v 

r = e a , = - - > (4) 



205-] INDUCTION THROUGH A GRATING. 249 

then, by the theory of conjugate functions, 



/ ^JL z-r t^\ 

= Alo \l-2e a cos- - + e a / 



. 
log \l-2e a cos- - + e a / 2 A loge a +C, (5) 

where x and y are rectangular coordinates, will be the value of the 
potential due to an infinite series of fine wires parallel to z in the 
plane of yz, and passing through points in the axis of x for which 
# is a multiple of a. 

Each of these wires is charged with a linear density A. 

The term involving A indicates an electrification, producing a 

constant force -- - in the direction of y. 
a J 

The forms of the equipotential surfaces and lines of force when 
A = are given in Fig. XIII. The equipotential surfaces near the 
wires are nearly cylinders, so that we may consider the solution 
approximately true, even when the wires are cylinders of a dia 
meter which is finite but small compared with the distance between 
them. 

The equipotential surfaces at a distance from the wires become 
more and more nearly planes parallel to that of the grating. 

If in the equation we make y = 6 lt a quantity large compared 
with a, we find approximately, 

r x = - i (A + A ) j, C nearly. (6) 

Cl 

If we next make y = 1 2 where b. 2 is a negative quantity large 
compared with a, we find approximately, 

F. 2 = - i- 2 (A - A ) + C nearly. (7) 



If c is the radius of the wires of the grating, c being small 
compared with a, we may find the potential of the grating itself 
by supposing that the surface of the wire coincides with the equi 
potential surface which cuts the plane of yz at a distance c from the 
axis of z. To find the potential of the grating we therefore put 
x = c, and y 0, whence 

205.] We have now obtained expressions representing the elec 
trical state of a system consisting of a grating of wires whose 
diameter is small compared with the distance between them, and 
two plane conducting surfaces, one on each side of the grating, 
and at distances which are great compared with the distance 
between the wires. 



250 CONJUGATE FUNCTIONS. 



The surface-density o-j on the first plane is got from the equa- 
n(6) 4 i= J = _il (X4 .V). (9) 

That on the second plane <r 2 from the equation (7) 



db% a 

If we now write a , / - , vc^ 



and eliminate A and X from the equations (6), (7), (8), (9), (10), 
we find 

, + *,+ ?*ll* = Fil + 2^?- r,- F^, (12) 



r, i + 2 --FJ. (is) 



, 

a / a a 



When the wires are infinitely thin, a becomes infinite, and the 
terms in which it is the denominator disappear, so that the case 
is reduced to that of two parallel planes without a grating in 
terposed. 

If the grating is in metallic communication with one of the 
planes, say the first, V=. V^ , and the right-hand side of the equation 
for o-j becomes V^ F z . Hence the density ^ induced on the first 
plane when the grating is interposed is to that which would have 

been induced on it if the grating were removed, the second plane 

2 A 
being maintained at the same potential, as 1 to 1 H - 



We should have found the same value for the effect of the grating 
in diminishing the electrical influence of the first surface on the 
second, if we had supposed the grating connected with the second 
surface. This is evident since # x and d 2 enter into the expression 
in the same way. It is also a direct result of the theorem of 
Art. 88. 

The induction of the one electrified plane on the other through 
the grating is the same as if the grating were removed, and the 
distance between the planes increased from 3 X -|- b 2 to 



If the two planes are kept at potential zero, and the grating 
electrified to a given potential, the quantity of electricity on the 
grating will be to that which would be induced on a plane of equal 
area placed in the same position as 

2 d l 6 2 is to 2 6 l & 2 + a (6 1 



206.] METHOD OF APPROXIMATION. 251 

This investigation is approximate only when b^ and b. 2 are large 
compared with #, and when a is large compared with c. The 
quantity a is a line which may be of any magnitude. It becomes 
infinite when c is indefinitely diminished. 

If we suppose c =. \a there will be no apertures between the 
wires of the grating, and therefore there will be no induction 
through it. We ought therefore to have for this case a = 0. The 
formula (11), however, gives in this case 



which is evidently erroneous, as the induction can never be altered 
in sign by means of the grating. It is easy, however, to proceed 
to a higher degree of approximation in the case of a grating of 
cylindrical wires. I shall merely indicate the steps of this process. 

Method of Approximation. 

206.] Since the wires are cylindrical, and since the distribution 
of electricity on each is symmetrical with respect to the diameter 
parallel to y y the proper expansion of the potential is of the form 

F= tf logr + 2C^cosi0, (14) 

where r is the distance from the axis of one of the wires, and the 
angle between r and y, and, since the wire is a conductor, when 
r is made equal to the radius V must be constant, and therefore 
the coefficient of each of the multiple cosines of 6 must vanish. 

For the sake of conciseness let us assume new coordinates , 17, &c. 
such that 

a^Z-nx, arj = 27ry, ap = 2itr, /3 = 2 TT , &c. (15) 
and let F ft = log (e^+0 + r-fo+W 2 cos). (16) 

Then if we make 

() 



by giving proper values to the coefficients A we may express any 
potential which is a function of 77 and cos f, and does not become 
infinite except when 77 + /3 = and cos = 1 . 

When (3 = the expansion of F in terms of p and is 

F = 2logp + -zp 2 cos26- T r zp*cos46 + &c. (18) 

For finite values of ft the expansion of F is 
1 4- e~P 



. (19) 



252 CONJUGATE FUNCTIONS. [206. 

In the case of the grating with two conducting planes whose 
equations are 77 = /^ and rj = j3 2 , that of the plane of the grating 
being rj = 0, there will be two infinite series of images of the 
grating. The first series will consist of the grating itself together 
with an infinite series of images on both sides, equal and similarly 
electrified. The axes of these imaginary cylinders lie in planes 
whose equations are of the form 

n = 2*(A+/3a), (20) 

n being an integer. 

The second series will consist of an infinite series of images for 
which the coefficients A G , A^ A^ &c. are equal and opposite to the 
same quantities in the grating itself, while A 19 A^ &c. are equal 
and of the same sign. The axes of these images are in planes whose 
equations are of the form 

rj = 2/3 2 + 2*(/3 1 + /3 2 ), (21) 

m being an integer. 

The potential due to any finite series of such images will depend 
on whether the number of images is odd or even. Hence the 
potential due to an infinite series is indeterminate, but if we add to 
it the function Brj+C, the conditions of the problem will be sufficient 
to determine the electrical distribution. 

We may first determine V^ and F" 2 , the potentials of the two 
conducting planes, in terms of the coefficients A , A 1} &c., and of 
JB and C. We must then determine o-j and o- 2 , the surface- density 
at any point of these planes. The mean values of a- L and cr 2 are 
given by the equations 

4770-!=^-^, 47T<T 2 = J Q + J?. (22) 

We must then expand the potentials due to the grating itself 
and to all the images in terms of p and cosines of multiples of Q y 
adding to the result BpcosB+C. 

The terms independent of 6 then give V the potential of the 
grating, and the coefficient of the cosine of each multiple of 
equated to zero gives an equation between the indeterminate co 
efficients. 

In this way as many equations may be found as are sufficient 
to eliminate all these coefficients and to leave two equations to 
determine o- x and o- 2 in terms of T 19 F" 2 , and F". 

These equations will be of the form 



F 2 -F = 47rcr 1 (a + y) + 4wa 2 (* 2 + o-y). (23) 



206.] METHOD OF APPROXIMATION. 253 

The quantity of electricity induced on one of the planes protected 
by the grating, the other plane being at a given difference of 
potential, will be the same as if the plates had been at a distance 

instead of 6-, +6*. 

a-f y 

The values of a and y are approximately as follows, 

a ( , a 5 
a= 2^r 0g 2^-3 



, 6 -iii / _A-h __! 

(24) 



CHAPTER XIII. 



ELECTROSTATIC INSTRUMENTS. 



On Electrostatic Instruments. 

THE instruments which we have to consider at present may be 
divided into the following classes : 

(1) Electrical machines for the production and augmentation of 
electrification. 

(2) Multipliers, for increasing electrification in a known ratio. 

(3) Electrometers, for the measurement of electric potentials and 
charges. 

(4) Accumulators, for holding large electrical charges. 

Electrical Machines. 

207.] In the common electrical machine a plate or cylinder of 
glass is made to revolve so as to rub against a surface of leather, 
on which is spread an amalgam of zinc and mercury. The surface 
of the glass becomes electrified positively and that of the rubber 
negatively. As the electrified surface of the glass moves away 
from the negative electrification of the rubber it acquires a high 
positive potential. It then comes opposite to a set of sharp metal 
points in connexion with the conductor of the machine. The posi 
tive electrification of the glass induces a negative electrification 
of the points, which is the more intense the sharper the points 
and the nearer they are to the glass. 

When the machine works properly there is a discharge through 
the air between the glass and the points, the glass loses part of 
its positive charge, which is transferred to the points and so to 
the insulated prime conductor of the machine, and to any other 
body with which it is in electric communication. 

The portion of the glass which is advancing towards the rubber 
has thus a smaller positive charge than that which is leaving it 
at the same time, so that the rubber, and the conductors in com 
munication with it, become negatively electrified. 



208.] ELECTROPHORUS. 255 

The highly positive surface of the glass where it leaves the 
rubber is more attracted by the negative charge of the rubber than 
the partially discharged surface wnich is advancing towards the 
rubber. The electrical forces therefore act as a resistance to the force 
employed in turning the machine. The work done in turning the 
machine is therefore greater than that spent in overcoming ordinary 
friction and other resistances, and the excess is employed in pro 
ducing a state of electrification whose energy is equivalent to this 
excess. 

The work done in overcoming friction is at once converted into 
heat in the bodies rubbed together. The electrical energy may 
be also converted either into mechanical energy or into heat. 

If the machine does not store up mechanical energy, all the 
energy will be converted into heat, and the only difference between 
the heat due to friction and that due to electrical action is that the 
former is generated at the rubbing surfaces while the latter may be 
generated in conductors at a distance *. 

We have seen that the electrical charge on the surface of the 
glass is attracted by the rubber. If this attraction were sufficiently 
intense there would be a discharge between the glass and the 
rubber, instead of between the glass and the collecting points. To 
prevent this, flaps of silk are attached to the rubber. These become 
negatively electrified and adhere to the glass, and so diminish the 
potential near the rubber. 

The potential therefore increases more gradually as the glass 
moves away from the rubber, and therefore at any one point there 
is less attraction of the charge on the glass towards the rubber, and 
consequently less danger of direct discharge to the rubber. 

In some electrical machines the moving part is of ebonite instead 
of glass, and the rubbers of wool or fur. The rubber is then elec 
trified positively and the prime conductor negatively. 

The Electrophorus of Volta. 

208.] The electrophorus consists of a plate of resin or of ebonite 
backed with metal, and a plate of metal of the same size. An 
insulating handle can be screwed to the back of either of these 
plates. The ebonite plate has a metal pin which connects the metal 

* It is probable that in many cases where dynamical energy is converted into heat 
by friction, part of the energy may be first transformed into electrical energy and 
then converted into heat as the electrical energy is spent in maintaining currents of 
short circuit close to the rubbing surfaces. See SirW. Thomson. On the Electro- 
dynamic Qualities of Metals. Phil. Trans., 1856, p. 650. 



256 ELECTROSTATIC INSTRUMENTS. [209. 

plate with the metal back of the ebonite plate when the two plates 
are in contact. 

The ebonite plate is electrified negatively by rubbing it with 
wool or cat s skin. The metal plate is then brought near the 
ebonite by means of the insulating handle. No direct discharge 
passes between the ebonite and the metal plate, but the potential 
of the metal plate is rendered negative by induction, so that when 
it comes within a certain distance of the metal pin a spark passes, 
and if the metal plate be now carried to a distance it is found 
to have a positive charge which may be communicated to a con 
ductor. The metal at the back of the ebonite plate is found to 
have a negative charge equal and opposite to the charge of the metal 
plate. 

In using the instrument to charge a condenser or accumulator 
one of the plates is laid on a conductor in communication with 
the earth, and the other is first laid on it, then removed and applied 
to the electrode of the condenser, then laid on the fixed plate and 
the process repeated. If the ebonite plate is fixed the condenser will 
be charged positively. If the metal plate is fixed the condenser will 
be charged negatively. 

The work done by the hand in separating the plates is always 
greater than the work done by the electrical attraction during the 
approach of the plates, so that the operation of charging the con 
denser involves the expenditure of work. Part of this work is 
accounted for by the energy of the charged condenser, part is spent 
in producing the noise and heat of the sparks, and the rest in 
overcoming other resistances to the motion. 

On Machines producing Electrification by Mechanical Work. 

209.] In the ordinary frictional electrical machine the work done 
in overcoming friction is far greater than that done in increasing 
the electrification. Hence any arrangement by which the elec 
trification may be produced entirely by mechanical work against 
the electrical forces is of scientific importance if not of practical 
value. The first machine of this kind seems to have been Nicholson s 
Revolving Doubler, described in the Philosophical Transactions for 
1788 as ( an instrument which by the turning of a Winch produces 
the two states of Electricity without friction or communication with 
the Earth/ 

210.] It was by means of the revolving doubler that Volta 
succeeded in developing from the electrification of the pile an 



210.] THE REVOLTING DOUBLER. 257 

electrification capable of affecting- his electrometer. Instruments 
on the same principle have been invented independently by Mr. 
C. F. Varley*, and SirW, Thomson. 

These instruments consist essentially of insulated conductors of 
various forms, some fixed and others moveable. The moveable 
conductors are called Carriers, and the fixed ones may be called 
Inductors, Receivers, and Regenerators. The inductors and receivers 
are so formed that when the carriers arrive at certain points in 
their revolution they are almost completely surrounded by a con 
ducting body. As the inductors and receivers cannot completely 
surround the carrier and at the same time allow it to move freely 
in and out without a complicated arrangement of moveable pieces, 
the instrument is not theoretically perfect without a pair of re 
generators, which store up the small amount of electricity which 
the carriers retain when they emerge from the receivers. 

For the present, however, we may suppose the inductors and 
receivers to surround the carrier completely when it is within them, 
in which case the theory is much simplified. 

We shall suppose the machine to consist of two inductors A and 
C, and of two receivers B and D, with two carriers F and G. 

Suppose the inductor A to be positively electrified so that its 
potential is A, and that the carrier F is within it and is at potential 
F. Then, if Q is the coefficient of induction (taken positive) between 
A and F 3 the quantity of electricity on the carrier will be Q (FA}. 

If the carrier, while within the inductor, is put in connexion with 
the earth, then F = 0, and the charge on the carrier will be QA, 
a negative quantity. Let the carrier be carried round till it is 
within the receiver B, and let it then come in contact with a spring 
so as to be in electrical connexion with B. It will then, as was 
shewn in Art. 32, become completely discharged, and will com 
municate its whole negative charge to the receiver B. 

The carrier will next enter the inductor (7, which we shall suppose 
charged negatively. While within C it is put in connexion with 
the earth and thus acquires a positive charge, which it carries off 
and communicates to the receiver D, and so on. 

In this way, if the potentials of the inductors remain always 
constant, the receivers B and D receive successive charges, which 
are the same for every revolution of the carrier, and thus every 
revolution produces an equal increment of electricity in the re 
ceivers. 

* Specification of Patent, Jan. 27, 1860, No. 206. _-^ 

VOL. I. S 



258 ELECTROSTATIC INSTRUMENTS. [2IO. 

But by putting the inductor A in communication with the re 
ceiver D, and the inductor C with the receiver _Z?, the potentials 
of the inductors will be continually increased, and the quantity 
of electricity communicated to the receivers in each revolution will 
continually increase. 

For instance, let the potential of A and be U, and that of 
and C, F, and when the carrier is within A let the charge on A 
and C be #, and that on the carrier z, then, since the potential 
of the carrier is zero, being in contact with earth, its charge is 
z= QU. The carrier enters B with this charge and communicates 
it to B. If the capacity of B and C is B, their potential will be 

changed from V to V ~~ U. 
B 

If the other carrier has at the same time carried a charge QF 
from C to I), it will change the potential of A and from U to 

Q 

U - V 3 if Q is the coefficient of induction between the carrier 

and C, and A the capacity of A and D. If, therefore, U n and F n 
be the potentials of the two inductors after n half revolutions, and 
U n+1 and F n+1 after n+1 half revolutions, 

77 77 ^ V 

u n+l u n ~J "n 

7. + i = r.- 1 ry n . 

If we write jt? 2 = - and 2 = -^ , we find 
-o A 



Hence 



It appears from these equations that the quantity pU+qT con 
tinually diminishes, so that whatever be the initial state of elec 
trification the receivers are ultimately oppositely electrified, so that 
the potentials of A and B are in the ratio of p to q. 

On the other hand, the quantity pUqV continually increases, 
so that, however little pU may exceed or fall short of q Tat first, 
the difference will be increased in a geometrical ratio in each 



211.] THE RECIPROCAL ELECTROPHORUS. 259 

revolution till the electromotive forces become so great that the 
insulation of the apparatus is overcome. 

Instruments of this kind may be used for various purposes. 

For producing* a copious supply of electricity at a high potential, 
as is done by means of Mr. Varley s large machine. 

For adjusting the charge of a condenser, as in the case of 
Thomson s electrometer, the charge of which can be increased or 
diminished by a few turns of a veiy small machine of this kind, 
which is then called a Replenishes 

For multiplying small differences of potential. The inductors 
may be charged at first to an exceedingly small potential, as, for 
instance, that due to a thermo-electric pair, then, by turning the 
machine, the difference of potentials may be continually multiplied 
till it becomes capable of measurement by an ordinary electrometer. 
By determining by experiment the ratio of increase of this difference 
due to each turn of the machine, the original electromotive force 
with which the inductors were charged may be deduced from the 
number of turns and the final electrification. 

In most of these instruments the carriers are made to revolve 
about an axis and to come into the proper positions with respect 
to the inductors by turning an axle. The connexions are made by 
means of springs so placed that the carriers come in contact with 
them at the proper instants. 

211.] Sir AY. Thomson*, however, has constructed a machine for 
multiplying electrical charges in which the carriers are drops of 
water falling out of the inside of an inductor into an insulated 
receiver. The receiver is thus continually supplied with electricity 
of opposite sign to that of the inductor. If the inductor is electrified 
positively, the receiver will receive a continually increasing charge 
of negative electricity. 

The water is made to escape from the receiver by means of a 
funnel, the nozzle of which is almost surrounded by the metal of 
the receiver. The drops falling from this nozzle are therefore 
nearly free from electrification. Another inductor and receiver of 
the same construction are arranged so that the inductor of the 
one system is in connexion with the receiver of the other. The 
rate of increase of charge of the receivers is thus no longer constant, 
but increases in a geometrical progression with the time, the 
charges of the two receivers being of opposite signs. This increase 
goes on till the falling drops are so diverted from their course by 
* Proc. E. S., June 20, 1867. 
S 3 



260 



ELECTROSTATIC INSTRUMENTS. 



[212. 



the electrical action that they fall outside of the receiver or even 
strike the inductor. 

In this instrument the energy of the electrification is drawn 
from that of the falling drops. 

212.] Several other electrical machines have been constructed 
in which the principle of electric induction is employed. Of these 
the most remarkable is that of Holtz, in which the carrier is a glass 
plate varnished with gum-lac and the inductors are pieces of 
pasteboard. Sparks are prevented from passing between the parts 
of the apparatus by means of two glass plates, one on each side 
of the revolving carrier plate. This machine is found to be very 
effective, and not to be much affected by the state of the atmo 
sphere. The principle is the same as in the revolving doubler and 
the instruments developed out of the same idea, but as the carrier 
is an insulating plate and the inductors are imperfect conductors, 
the complete explanation of the action is. more difficult than in 
the case where the carriers are good conductors of known form 
and are charged and discharged at definite points. 

213.] In the electrical machines already described sparks occur 

whenever the carrier comes in 
contact with a conductor at a 
different potential from its 
own. 

Now we have shewn that 
whenever this occurs there is 
a loss of energy, and therefore 
the whole work employed in 
turning the machine is not con 
verted into electrification in an 
available form, but part is spent 
in producing the heat and noise 
of electric sparks. 
I have therefore thought it desirable to shew how an electrical 
machine may be constructed which is not subject to this loss of 
efficiency. I do not propose it as a useful form of machine, but 
as an example of the method by which the contrivance called in 
heat-engines a regenerator may be applied to an electrical machine 
to prevent loss of work. 

In the figure let A, B, C, A , 1?, C f represent hollow fixed 
conductors, so arranged that the carrier P passes in succession 
within each of them. Of these A, A and B, & nearly surround the 




Fig. 17. 



2 1 3.] MACHINE WITHOUT SPARKS. 261 

carrier when it is at the middle point of its passage, but C, C do not 
cover it so much. 

We shall suppose A, J3, C to be connected with a Leyden jar 
of great capacity at potential F, and A , , C to be connected with 
another jar at potential F . 

P is one of the carriers moving in a circle from A to C", &c., 
and touching in its course certain springs, of which a and a are 
connected with A and A respectively, and e, e are connected with 
the earth. 

Let us suppose that when the carrier P is in the middle of A 
the coefficient of induction between P and A is A. The capacity 
of P in this position is greater than A, since it is not completely 
surrounded by the receiver A. Let it be A + a. 

Then if the potential of P is U t and that of A, F, the charge 
on P will be (A + a)U-A7. 

Now let P be in contact with the spring a when in the middle 
of the receiver A, then the potential of P is F, the same as that 
of A, and its charge is therefore a V. 

If P now leaves the spring a it carries with it the charge a V. 
As P leaves A its potential diminishes, and it diminishes still more 
when it comes within the influence of C , which is negatively 
electrified. 

If when P comes within C its coefficient of induction on C is 
C", and its capacity is C + c , then, if U is the potential of P 
the charge on P is 



If C V -aV, 

then at this point U the potential of P will be reduced to zero. 

Let P at this point come in contact with the spring e which is 
connected with the earth. Since the potential of P is equal to that 
of the spring there will be no spark at contact. 

This conductor C , by which the carrier is enabled to be connected 
to earth without a spark, answers to the contrivance called a 
regenerator in heat-engines. We shall therefore call it a Re 
generator. 

Now let P move on, still in contact with the earth-spring /, till 
it comes into the middle of the inductor .5, the potential of which 
is F. If B is the coefficient of induction between P and B at 
this point, then, since U = the charge on P will be BV. 

When P moves away from the earth-spring it carries this charge 
with it. As it moves out of the positive inductor B towards the 



262 ELECTROSTATIC INSTRUMENTS. [214. 

negative receiver A its potential will be increasingly negative. At 
the middle of A , if it retained its charge, its potential would be 



A + af 

and if B7 \$ greater than a V its numerical value will be greater 
than that of V . Hence there is some point before P reaches the 
middle of A where its potential is V. At this point let it come 
in contact with the negative receiver-spring a . There will be no 
spark since the two bodies are at the same potential. Let P move 
on to the middle of A } still in contact with the spring, and therefore 
at the same potential with A. During this motion it communicates 
a negative charge to A. At the middle of A it leaves the spring 
and carries away a charge a V towards the positive regenerator 
C, where its potential is reduced to zero and it touches the earth- 
spring e. It then slides along the earth-spring into the negative 
inductor J? , during which motion it acquires a positive charge B V 
which it finally communicates to the positive receiver A, and the 
cycle of operations is repeated. 

During this cycle the positive receiver has lost a charge a V and 
gained a charge B f V . Hence the total gain of positive electricity 
is B V -aV. 

Similarly the total gain of negative electricity is BVa V. 

By making the inductors so as to be as close to the surface of 
the carrier as is consistent with insulation, B and B may be made 
large, and by making the receivers so as nearly to surround the 
carrier when it is within them, a and a may be made very small, 
and then the charges of both the Leyden jars will be increased in 
every revolution. 

The conditions to be fulfilled by the regenerators are 
C V = aV, and CF= a V. 

Since a and a f are small the regenerators do not require to be 
either large or very close to the carriers. 

On Electrometers and Electroscopes. 

214.] An electrometer is an instrument by means of which 
electrical charges or electrical potentials may be measured. In 
struments by means of which the existence of electric charges or 
of differences of potential may be indicated, but which are not 
capable of affording numerical measures, are called Electroscopes. 

An electroscope if sufficiently sensible may be used in electrical 
measurements, provided we can make the measurement depend on 



2 1 5.] COULOMB S TORSION BALANCE. 263 

the absence of electrification. For instance, if we have two charged 
bodies A and we may use the method described in Chapter I to 
determine which body has the greater charge. Let the body A 
be carried by an insulating support into the interior of an insulated 
closed vessel C. Let C be connected to earth and again insulated. 
There will then be no external electrification on C. Now let A 
be removed, and B introduced into the interior of C, and the elec 
trification of C tested by an electroscope. If the charge of B is 
equal to that of A there will be no electrification, but if it is greater 
or less there will be electrification of the same kind as that of B, or 
the opposite kind. 

Methods of this kind, in which the thing to be observed is the 
non-existence of some phenomenon, are called null or zero methods. 
They require only an instrument capable of detecting the existence 
of the phenomenon. 

In another class of instruments for the registration of phe 
nomena the instrument may be depended upon to give always the 
same indication for the same value of the quantity to be registered, 
but the readings of the scale of the instrument are not proportional 
to the values of the quantity, and the relation between these 
readings and the corresponding value is unknown, except that the 
one is some continuous function of the other. Several electrometers 
depending on the mutual repulsion of parts of the instrument 
which are similarly electrified are of this class. The use of such 
instruments is to register phenomena, not to measure them. Instead 
of the true values of the quantity to be measured, a series of 
numbers is obtained, which may be used afterwards to determine 
these values when the scale of the instrument has been properly 
investigated and tabulated. 

In a still higher class of instruments the scale readings are 
proportional to the quantity to be measured, so that all that is 
required for the complete measurement of the quantity is a know 
ledge of the coefficient by which the scale readings must be 
multiplied to obtain the true value of the quantity. 

Instruments so constructed that they contain within themselves 
the means of independently determining the true values of quan 
tities are called Absolute Instruments. 



Coulomb s Torsion Balance. 
215.] A great number of the experiments by which Coulomb 



264 ELECTROSTATIC INSTRUMENTS. [215. 

established the fundamental laws of electricity were made by mea 
suring the force between two small spheres charged with electricity, 
one of which was fixed while the other was held in equilibrium by 
two forces, the electrical action between the spheres, and the 
torsional elasticity of a glass fibre or metal wire. See Art. 38. 

The balance of torsion consists of a horizontal arm of gum-lac, 
suspended by a fine wire or glass fibre, and carrying at one end a 
little sphere of elder pith, smoothly gilt. The suspension wire is 
fastened above to the vertical axis of an arm which can be moved 
round a horizontal graduated circle, so as to twist the upper end 
of the wire about its own axis any number of degrees. 

The whole of this apparatus is enclosed in a case. Another little 
sphere is so mounted on an insulating stem that it can be charged 
and introduced into the case through a hole, and brought so that 
its centre coincides with a definite point in the horizontal circle 
described by the suspended sphere. The position of the suspended 
sphere is ascertained by means of a graduated circle engraved on 
the cylindrical glass case of the instrument. 

Now suppose both spheres charged, and the suspended sphere 
in equilibrium in a known position such that the torsion-arm makes 
an angle with the radius through the centre of the fixed sphere. 
The distance of the centres is then 2 a sin \ 0, where a is the radius 
of the torsion-arm, and if F is the force between the spheres the 
moment of this force about the axis of torsion is Fa cos i 0. 

Let both spheres be completely discharged, and let the torsion- 
arm now be in equilibrium at an angle $ with the radius through 
the fixed sphere. 

Then the angle through which the electrical force twisted the 
torsion-arm must have been </>, and if M is the moment of 
the torsional elasticity of the fibre, we shall have the equation 
Fa cos ^0 = M(0-4>). 

Hence, if we can ascertain M, we can determine F } the actual 
force between the spheres at the distance 2 a sin \ 0. 

To find My the moment of torsion, let / be the moment of inertia 
of the torsion-arm, and T the time of a double vibration of the arm 
under the action of the torsional elasticity, then 

M= 



In all electrometers it is of the greatest importance to know 
what force we are measuring. The force acting on the suspended 



2 1 5.] INFLUENCE OF THE CASE. 265 

sphere is due partly to the direct action of the fixed sphere, but 
partly also to the electrification, if any, of the sides of the case. 

If the case is made of glass it is impossible to determine the 
electrification of its surface otherwise than by very difficult mea 
surements at every point. If, however, either the case is made 
of metal, or if a metallic case which almost completely encloses the 
apparatus is placed as a screen between the spheres and the glass 
case, the electrification of the inside of the metal screen will depend 
entirely on that of the spheres, and the electrification of the glass 
case will have no influence on the spheres. In this way we may 
avoid any indefiniteness due to the action of the case. 

To illustrate this by an example in which we can calculate all 
the effects, let us suppose that the case is a sphere of radius b, 
that the centre of motion of the torsion-arm coincides with the 
centre of the sphere and that its radius is a ; that the charges on 
the two spheres are E 1 and E Z3 and that the angle between their 
positions is 6; that the fixed sphere is at a distance a^ from the 
centre, and that r is the distance between the two small spheres. 

Neglecting for the present the effect of induction on the dis 
tribution of electricity on the small spheres, the force between 
them will be a repulsion 



and the moment of this force round a vertical axis through the 

centre will be 

EE l aa 1 sin 
-73 

The image of E l due to the spherical surface of the case is a point 

b 2 b 

in the same radius at a distance with a charge E l , and the 

a i a i 

moment of the attraction between E and this image about the axis 

of suspension is 

a sin Q 

x^L 



If b, the radius of the spherical case, is large compared with a 



266 ELECTEOSTATIC INSTRUMENTS. \_2l6. 

and a 13 the distances of the spheres from the centre, we may neglect 
the second and third terms of the factor in the denominator. The 
whole moment tending to turn the torsion-arm may then be written 



Electrometers for the Measurement of Potentials. 

216.] In all electrometers the moveable part is a body charged 
with electricity, and its potential is different from that of certain 
of the fixed parts round it. When, as in Coulomb s method, an 
insulated body having a certain charge is used, it is the charge 
which is the direct object of measurement. We may, however, 
connect the balls of Coulomb s electrometer, by means of fine wires, 
with different conductors. The charges of the balls will then 
depend on the values of the potentials of these conductors and on 
the potential of the case of the instrument. The charge on each 
ball will be approximately equal to its radius multiplied by the 
excess of its potential over that of the case of the instrument, 
provided the radii of the balls are small compared with their 
distances from each other and from the sides or opening of the 
case. 

Coulomb s form of apparatus, however, is not well adapted for 
measurements of this kind, owing to the smallness of the force 
between spheres at the proper distances when the difference of po 
tentials is small. A more convenient form is that of the Attracted 
Disk Electrometer. The first electrometers on this principle were 
constructed by Sir W. Snow Harris*. They have since been 
brought to great perfection, both in theory and construction, by 
SirW. Thomson f. 

When two disks at different potentials are brought face to face 
with a small interval between them there will be a nearly uniform 
electrification on the opposite faces and very little electrification 
on the backs of the disks, provided there are no other conductors 
or electrified bodies in the neighbourhood. The charge on the 
positive disk will be approximately proportional to its area, and to 
the difference of potentials of the disks, and inversely as the distance 
between them. Hence, by making the areas of the disks large 

* Phil Trans. 1834. 

t See an excellent report on Electrometers by Sir W. Thomson. Report of the 
British Association, Dundee, 1867. 



PRINCIPLE OF THE GUARD-RING. 



267 



and the distance between them small, a small difference of potential 
may give rise to a measurable force of attraction. 

The mathematical theory of the distribution of electricity over 
two disks thus arranged is given at Art. 202, but since it is im 
possible to make the case of the apparatus so large that we may 
suppose the disks insulated in an infinite space, the indications of 
the instrument in this form are not easily interpreted numerically. 

217.] The addition of the guard-ring to the attracted disk is one 
of the chief improvements which Sir W. Thomson has made on the 
apparatus. 

Instead of suspending the whole of one of the disks and determ 
ining the force acting upon it, a central portion of the disk is 
separated from the rest to form the attracted disk, and the outer 
ring forming the remainder of the disk is fixed. In this way the 
force is measured only on that part of the disk where it is most 
regular, and the want of uniformity of the electrification near the 



COUNTERPOISE 



LENS 




Fig. 18. 

edge is of no importance, as it occurs on the guard-ring and not 
on the suspended part of the disk. 

Besides this, by connecting the guard-ring with a metal case 
surrounding the back of the attracted disk and all its suspending 
apparatus, the electrification of the back of the disk is rendered 



268 ELECTROSTATIC INSTRUMENTS. [2 I/. 

impossible, for it is part of the inner surface of a closed hollow 
conductor all at the same potential. 

Thomson s Absolute Electrometer therefore consists essentially 
of two parallel plates at different potentials, one of which is made 
so that a certain area, no part of which is near the edge of the 
plate, is moveable under the action of electric force. To fix our 
ideas we may suppose the attracted disk and guard-ring uppermost. 
The fixed disk is horizontal, and is mounted on an insulating stem 
which has a measurable vertical motion given to it by means of 
a micrometer screw. The guard-ring is at least as large as the 
fixed disk ; its lower surface is truly plane and parallel to the fixed 
disk. A delicate balance is erected on the guard-ring to which 
is suspended a light moveable disk which almost fills the circular 
aperture in the guard-ring without rubbing against its sides. The 
lower surface of the suspended disk must be truly plane, and we 
must have the means of knowing when its plane coincides with that 
of the lower surface of the guard-ring, so as to form a single plane 
interrupted only by the narrow interval between the disk and its 
guard-ring. 

For this purpose the lower disk is screwed up till it is in contact 
with the guard-ring, and the suspended disk is allowed to rest 
upon the lower disk, so that its lower surface is in the same plane 
as that of the guard-ring. Its position with respect to the guard- 
ring is then ascertained by means of a system of fiducial marks. 
Sir W. Thomson generally uses for this purpose a black hair 
attached to the moveable part. This hair moves up or down just 
in front of two black dots on a white enamelled ground and is 
viewed along with these dots by means of a piano convex lens with 
the plane side next the eye. If the hair as seen through the lens 
appears straight and bisects the interval between the black dots 
it is said to be in its sighted position, and indicates that the sus 
pended disk with which it moves is in its proper position as regards 
height. The horizontality of the suspended disk may be tested by 
comparing the reflexion of part of any object from its upper surface 
with that of the remainder of the same object from the upper 
surface of the guard-ring. 

The balance is then arranged so that when a known weight is 
placed on the centre of the suspended disk it is in equilibrium 
in its sighted position, the whole apparatus being freed from 
electrification by putting every part in metallic communication. 
A metal case is placed over the guard-ring so as to enclose the 



2 1 8.] THOMSON S ABSOLUTE ELECTROMETER. 269 

balance and suspended disk, sufficient apertures being left to see 
the fiducial marks. 

The guard-ring-, case, and suspended disk are all in metallic 
communication with each other, but are insulated from the other 
parts of the apparatus. 

Now let it be required to measure the difference of potentials 
of two conductors. The conductors are put in communication with 
the upper and lower disks respectively by means of wires, the 
weight is taken off the suspended disk, and the lower disk is 
moved up by means of the micrometer screw till the electrical 
attraction brings the suspended disk down to its sighted position. 
We then know that the attraction between the disks is equal to 
the weight which brought the disk to its sighted position. 

If W be the numerical value of the weight, and g the force of 
gravity, the force is Wg, and if A is the area of the suspended 
disk, D the distance between the disks, and T the difference of the 

potentials of the disks, 

F 2 A 



. T-, 



If the suspended disk is circular, of radius R, and if the radius of 
the aperture of the guard-ring is J? , then 

* and r=4 



218.] Since there is always some uncertainty in determining the 
micrometer reading corresponding to D = 0, and since any error 

* Let us denote the radius of the suspended disk by P, and that of the aperture 
of the guard-ring by R , then the breadth of the annular interval between the 
disk and the ring will be B = R -R. 

If the distance between the suspended disk and the large fixed disk is D, and 
the difference of potentials between these disks is F, then, by the investigation in 
Art. 201, the quantity of electricity on the suspended disk will be 



I 8D 8D D + aJ 

where a = B 1 -^^-, or o = 0. 220635 (R -E). 

If the surface of the guard-ring is not exactly in the plane of the surface of 
the suspended disk, let us suppose that the distance between the fixed disk and 
the guard-ring is not D but D + z D , then it appears from the investigation in 
Art. 225 that there will be an additional charge of electricity near the edge of 
the disk on account of its height z above the general surface of the guard-ring. 
The whole charge in this case is therefore 



270 ELECTROSTATIC INSTRUMENTS. [218. 

in the position of the suspended disk is most important when D 
is small, Sir W. Thomson prefers to make all his measurements 
depend on differences of the electromotive force F. Thus, if V and 
V are two potentials, and D and If the corresponding distances, 

F-r = (D-ff) 



A 

For instance, in order to measure the electromotive force of a 
galvanic battery, two electrometers are used. 

By means of a condenser, kept charged if necessary by a re- 
plenisher, the lower disk of the principal electrometer is maintained 
at a constant potential. This is tested by connecting the lower 
disk of the principal electrometer with the lower disk of a secondary 
electrometer, the suspended disk of which is connected with the 
earth. The distance between the disks of the secondary elec 
trometer and the force required to bring the suspended disk to 
its sighted position being constant, if we raise the potential of the 
condenser till the secondary electrometer is in its sighted position, 
we know that the potential of the lower disk of the principal 
electrometer exceeds that of the earth by a constant quantity which 
we may call V. 

If we now connect the positive electrode of the battery to earth, 
and connect the suspended disk of the principal electrometer to the 
negative electrode, the difference of potentials between the disks 
will be V+v, if v is the electromotive force of the battery. Let 
D be the reading of the micrometer in this case, and let J/ be the 
reading when the suspended disk is connected with earth, then 



In this way a small electromotive force v may be measured 
by the electrometer with the disks at conveniently measurable 
distances. When the distance is too small a small change of 
absolute distance makes a great change in the force, since the 

and in the expression for the attraction we must substitute for A, the area of the 
disk, the corrected quantity 



where R = radius of suspended disk, 

R = radius of aperture in the guard-ring, 
D = distance between fixed and suspended disks, 
D = distance between fixed disk and guard-ring, 
a = 0.220635 (K -E). 

When a is small compared with D we may neglect the second term, and when 

D is small we may neglect the last term. 



2 1 9.] GAUGE ELECTROMETER. 271 

force varies inversely as the square of the distance, so that any 
error in the absolute distance introduces a large error in the result 
unless the distance is large compared with the limits of error of 
the micrometer screw. 

The effect of small irregularities of form in the surfaces of the 
disks and of the interval between them diminish according to the 
inverse cube and higher inverse powers of the distance, and what 
ever be the form of a corrugated surface, the eminences of which 
just reach a plane surface, the electrical effect at any distance 
which is considerable compared to the breadth of the corrugations, 
is the same as that of a plane at a certain small distance behind 
the plane of the tops of the eminences. See Arts. 197, 198. 

By means of the auxiliary electrification, tested by the auxiliary 
electrometer, a proper interval between the disks is secured. 

The auxiliary electrometer may be of a simpler construction, in 
which there is no provision for the determination of the force 
of attraction in absolute measure, since all that is wanted is to 
secure a constant electrification. Such an electrometer may be 
called a gauge electrometer. 

This method of using an auxiliary electrification besides the elec 
trification to be measured is called the Heterostatic method of 
electrometry, in opposition to the Idiostatic method in which the 
whole effect is produced by the electrification to be measured. 

In several forms of the attracted disk electrometer, the attracted 
disk is placed at one end of an arm which is supported by being 
attached to a platinum wire passing through its centre of gravity 
and kept stretched by means of a spring. The other end of the 
arm carries the hair which is brought to a sighted position by 
altering the distance between the disks, and so adjusting the force 
of the electric attraction to a constant value. In these electro 
meters this force is not in general determined in absolute measure, 
but is known to be constant, provided the torsional elasticity of 
the platinum wire does not change. 

The whole apparatus is placed in a Leyden jar, of which the inner 
surface is charged and connected with the attracted disk and 
guard-ring. The other disk is worked by a micrometer screw and 
is connected first with the earth and then with the conductor whose 
potential is to be measured. The difference of readings multiplied 
by a constant to be determined for each electrometer gives the 
potential required. 

219.] The electrometers already described are not self-acting, 



272 ELECTROSTATIC INSTRUMENTS. [ 2I 9 

but require for each observation an adjustment of a micrometer 
screw, or some other movement which must be made by the 
observer. They are therefore not fitted to act as self- registering in 
struments, which must of themselves move into, the proper position. 
This condition is fulfilled by Thomson s Quadrant Electrometer. 

The electrical principle on which this instrument is founded may 
be thus explained : 

A and B are two fixed conductors which may be at the same 
or at different potentials. C is a moveable conductor at a high 
potential, which is so placed that part of it is opposite to the 
surface of A and part opposite to that of J5, and that the proportions 
of these parts are altered as C moves. 

For this purpose it is most convenient to make C moveable about 
an axis, and make the opposed surfaces of A^ of B, and of C portions 
of surfaces of revolution about the same axis. 

In this way the distance between the surface of C and the 
opposed surfaces of A or of IB remains always the same, and the 
motion of C in the positive direction simply increases the area 
opposed to B and diminishes the area opposed to A. 

If the potentials of A and B are equal there will be no force 
urging C from A to B, but if the potential of C differs from that 
of B more than from that of A, then C will tend to move so as 
to increase the area of its surface opposed to B. 

By a suitable arrangement of the apparatus this force may be 
made nearly constant for different positions of C within certain 
limits, so that if C is suspended by a torsion fibre, its Deflexions 
will be nearly proportional to the difference of potentials between 
A and B multiplied by the difference of the potential of C from 
the mean of those of A and B. 

C is maintained at a high potential by means of a condenser 
provided with a replenisher and tested by a gauge electrometer, 
and A and B are connected with the two conductors the difference 
of whose potentials is to be measured. The higher the potential 
of C the more sensitive is the instrument. This electrification of 
C, being independent of the electrification to be measured, places 
this electrometer in the heterostatic class. 

We may apply to this electrometer the general theory of systems 
of conductors given in Arts. 93, 127. 

Let A, B, C denote the potentials of the three conductors re 
spectively. Let a, 5, c be their respective capacities, p the coefficient 
of induction between B and C, q that between C and A, and r that 



2I 9 .] 



QUADRANT ELECTROMETER. 



273 



between A and B. All these coefficients will in general vary with. 
the position of C y and if C is so arranged that the extremities of A 
and B are not near those of C as long as the motion of C is confined 
within certain limits, we may ascertain the form of these coefficients. 
If represents the deflexion of C from A towards B, then the part 
of the surface of A opposed to C will diminish as increases. 
Hence if A is kept at potential 1 while B and C"are kept at potential 
0, the charge on A will be a = a aO, where a and a are 
constants, and a is the capacity of A. 

If A and B are symmetrical, the capacity of B is I 5 -f a Q. 

The capacity of C is not altered by the motion, for the only 
effect of the motion is to bring a different part of C opposite to the 
interval between A and B. Hence c = <? . 

The quantity of electricity induced on C when B is raised to 
potential unity is p =. p^ aQ. 

The coefficient of induction between A and C is q = q Q -\-aO. 

The coefficient of induction between A and B is not altered by 
the motion of C, but remains r = r . 

Hence the electrical energy of the system is 

Q = \A*a+%B*b + \C*c + BCp + CAq + ABr, 
and if is the moment of the force tending to increase 6, 



= -~ , A, B, C being supposed constant, 
du 

. da . ~ db -, n dc 



da 



or 



= a(A-B] (C- 



In the present form of Thomson s Quadrant Electrometer the 
conductors A and B are in the form of 
a cylindrical box completely divided 
into four quadrants, separately insu 
lated, but joined by wires so that two 
opposite quadrants are connected with 
A and the two others with B. 

The conductor C is suspended so as 
to be capable of turning about a 
vertical axis, and may consist of two 
opposite flat quadrantal arcs supported 
by their radii at their extremities. 
In the position of equilibrium these quadrants should be partly 

VOL. I. T 




Fig. 19. 



274 ELECTROSTATIC INSTRUMENTS. [220. 

within A and partly within .5, and the supporting radii should 
be near the middle of the quadrants of the hollow base, so that 
the divisions of the box and the extremities and supports of C 
may be as far from each other as possible. 

The conductor C is kept permanently at a high potential by 
being connected with the inner coating of the Ley den jar which 
forms the case of the instrument. and A are connected, the first 
with the earth, and the other with the body whose potential is to be 
measured. 

If the potential of this body is zero, and if the instrument be 
in adjustment, there ought to be no force tending to make C move, 
but if the potential of A is of the same sign as that of C, then 
C will tend to move from A to with a nearly uniform force, and 
the suspension apparatus will be twisted till an equal force is 
called into play and produces equilibrium. For deflexions within 
certain limits the deflexions of C will be proportional to the 
product (A-B)(C-\(A + B)}. 

By increasing the potential of C the sensibility of the instrument 
may be increased, and for small values of \ (A 4- B) the force will be 
nearly proportional to (AB] C. 



On the Measurement of Electric Potential. 

220.] In order to determine large differences of potential in ab 
solute measure we may employ the attracted disk electrometer, and 
compare the attraction with the effect of a weight. If at the same 
time we measure the difference of potential of the same conductors 
by means of the quadrant electrometer, we shall ascertain the 
absolute value of certain readings of the scale of the quadrant 
electrometer, and in this way we may deduce the value of the scale 
readings of the quadrant electrometer in terms of the potential 
of the suspended part, and the moment of torsion of the suspension 
apparatus. 

To ascertain the potential of a charged conductor of finite size 
we may connect the conductor with one electrode of the electro 
meter, while the other is connected to earth or to a body of 
constant potential. The electrometer reading will give the potential 
of the conductor after the division of its electricity between it 
and the part of the electrometer with which it is put in contact. 
If K denote the capacity of the conductor, and K that of this part 



221.] MEASUREMENT OF POTENTIAL. 275 

of the electrometer, and if V, V denote the potentials of these 
bodies before making 1 contact, then their common potential after 
making contact will be 

= _ KF+KT 

K+K 
Hence the original potential of the conductor was 



If the conductor is not large compared with the electrometer, 
K will be comparable with K, and unless we can ascertain the 
values of K and K the second term of the expression will have 
a doubtful value. But if we can make the potential of the electrode 
of the electrometer very nearly equal to that of the body before 
making contact, then the uncertainty of the values of K and K 
will be of little consequence. 

If we know the value of the potential of the body approximately, 
we may charge the electrode by means of a replenished or other 
wise to this approximate potential, and the next experiment will 
give a closer approximation. In this way we may measure the 
potential of a conductor whose capacity is small compared with 
that of the electrometer. 

To Measure the Potential at any Point in the Air. 

221.] First Method. Place a sphere, whose radius is small com 
pared with the distance of electrified conductors, with its centre 
at the given point. Connect it by means of a fine wire with the 
earth, then insulate it, and carry it to an electrometer and ascertain 
the total charge on the sphere. 

Then, if V be the potential at the given point, and a the 
radius of the sphere, the charge on the sphere will be P r a=Q^ 
and if V be the potential of the sphere as measured by an elec 
trometer when placed in a room whose walls are connected with 
the earth, then Q _ y, 

whence V+ V = 0, 

or the potential of the air at the point where the centre of the 
sphere was placed is equal but of opposite sign to the potential of 
the sphere after being connected to earth, then insulated, and 
brought into a room. 

This method has been employed by M. Delmann of Creuznach in 

T 2 



276 ELECTROSTATIC INSTRUMENTS. 

measuring the potential at a certain height above the earth s 
surface. 

Second Method. We have supposed the sphere placed at the 
given point and first connected to earth, and then insulated, and 
carried into a space surrounded with conducting matter at potential 
zero. 

Now let us suppose a fine insulated wire carried from the elec 
trode of the electrometer to the place where the potential is to 
be measured. Let the sphere be first discharged completely. This 
may be done by putting it into the inside of a vessel of the same 
metal which nearly surrounds it and making it touch the vessel. 
Now let the sphere thus discharged be carried to the end of the 
wire and made to touch it. Since the sphere is not electrified it 
will be at the potential of the air at the place. If the electrode 
wire is at the same potential it will not be affected by the contact, 
but if the electrode is at a different potential it will by contact 
with the sphere be made nearer to that of the air than it was 
before. By a succession of such operations, the sphere being 
alternately discharged and made to touch the electrode, the poten 
tial of the electrode of the electrometer will continually approach 
that of the air at the given point. 

222.] To measure the potential of a conductor without touching 
it, we may measure the potential of the air at any point in the 
neighbourhood of the conductor, and calculate that of the conductor 
from the result. If there be a hollow nearly surrounded by the 
conductor, then the potential at any point of the air in this hollow 
witt be very nearly that of the conductor. 

In this way it has been ascertained by Sir W. Thomson that if 
two hollow conductors, one of copper and the other of zinc, are 
in metallic contact, then the potential of the air in the hollow 
surrounded by zinc is positive with reference to that of the air 
in the hollow surrounded by copper. 

Third Method. If by any means we can cause a succession of 
-small bodies to detach themselves from the end of the electrode, 
the potential of the electrode will approximate to that of the sur 
rounding air. This may be done by causing shot, filings, sand, or 
water to drop out of a funnel or pipe connected with the electrode. 
The point at which the potential is measured is that at which 
the stream ceases to be continuous and breaks into separate parts 
or drops. 

Another convenient method is to fasten a slow match to the 



223.] THEORY OF THE PROOF PLANE. 277 

electrode. The potential is very soon made equal to that of the 
air at the burning end of the match. Even a fine metallic point 
is sufficient to create a discharge by means of the particles of the 
air when the difference of potentials is considerable, but if we 
wish to reduce this difference to zero, we must use one of the 
methods stated above. 

If we only wish to ascertain the sign of the difference of the 
potentials at two places, and not its numerical value, we may cause 
drops or filings to be discharged at one of the places from a nozzle 
connected with the other place, and catch the drops or filings 
in an insulated vessel. Each drop as it falls is charged with a 
certain amount of electricity, and it is completely discharged into 
the vessel. The charge of the vessel therefore is continually ac 
cumulating, and after a sufficient number of drops have fallen, the 
charge of the vessel may be tested by the roughest methods. The 
sign of the charge is positive if the potential of the nozzle is positive 
relatively to that of the surrounding air. 



MEASUREMENT OF SURFACE-DENSITY OF ELECTRIFICATION. 

Theory of the Proof Plane. 

223.] In testing the results of the mathematical theory of the 
distribution of electricity on the surface of conductors, it is necessary 
to be able to measure the surface-density at different points of 
the conductor. For this purpose Coulomb employed a small disk 
of gilt paper fastened to an insulating stem of gum-lac. He ap 
plied this disk to various points of the conductor by placing it 
so as to coincide as nearly as possible with the surface of the 
conductor. He then removed it by means of the insulating stem, 
and measured the charge of the disk by means of his electrometer. 

Since the surface of the disk, when applied to the conductor, 
nearly coincided with that of the conductor, he concluded that 
the surface-density on the outer surface of the disk was nearly 
equal to that on the surface of the conductor at that place, and that 
the charge on the disk when removed was nearly equal to that 
on an area of the surface of the conductor equal to that of one side 
of the disk. This disk, when employed in this way, is called 
Coulomb s Proof Plane. 

As objections have been raised to Coulombs use of the proof 
plane, I shall make some remarks on the theory of the experiment. 



278 ELECTROSTATIC INSTRUMENTS. [224. 

The experiment consists in bringing- a small conducting body 
into contact with the surface of the conductor at the point where 
the density is to be measured, and then removing the body and 
determining its charge. 

"We have first to shew that the charge on the small body when 
in contact with the conductor is proportional to the surface- 
density which existed at the point of contact before the small body 
was placed there. 

We shall suppose that all the dimensions of the small body, and 
especially its dimension in the direction of the normal at the point 
of contact, are small compared with either of the radii of curvature 
of the conductor at the point of contact. Hence the variation of 
the resultant force due to the conductor supposed rigidly electrified 
within the space occupied by the small body may be neglected, 
and we may treat the surface of the conductor near the small body 
as a plane surface. 

Now the charge which the small body will take by contact with 
a plane surface will be proportional to the resultant force normal 
to the surface, that is, to the surface-density. We shall ascertain 
the amount of the charge for particular forms of the body. 

We have next to shew that when the small body is removed no 
spark will pass between it and the conductor, so that it will carry 
its charge with it. This is evident, because when the bodies are 
in contact their potentials are the same, and therefore the density 
on the parts nearest to the point of contact is extremely small. 
When the small body is removed to a very short distance from 
the conductor, which we shall suppose to be electrified positively, 
then the electrification at the point nearest to the small body is 
no longer zero but positive, but, since the charge of the small body 
is positive, the positive electrification close to the small body will 
be less than at other neighbouring points of the surface. Now 
the passage of a spark depends in general on the magnitude of the 
resultant force, and this on the surface-density. Hence, since we 
suppose that the conductor is not so highly electrified as to be 
discharging electricity from the other parts of its surface, it will 
not discharge a spark to the small body from a part of its surface 
which we have shewn to have a smaller surface-density. 

224.] We shall now consider various forms of the small body. 

Suppose it to be a small hemisphere applied to the conductor so 
as to touch it at the centre of its flat side. 

Let the conductor be a large sphere, and let us modify the form 



225.] THE PROOF PLANE. 279 

of the hemisphere so that its surface is a little more than a hemi 
sphere, and meets the surface of the sphere at right angles. Then 
we have a case of which we have already obtained the exact solution. 
See Art. 168. 

If A and B be the centres of the two spheres cutting each other 
at right angles, DD a diameter of the circle of intersection, and C 
the centre of that circle, then if V is the potential of a conductor 
whose outer surface coincides with that of the two spheres, the 
quantity of electricity on the exposed surface of the sphere A is 



and that on the exposed surface of the sphere B is 

4 7 (AD + BD + BC - CD - AC\ 
the total charge being the sum of these, or 



If a and /3 are the radii of the spheres, then, when a is large 
compared with , the charge on B is to that on A in the ratio of 

!5o+i+i $+*>* 

Now let & be the uniform surface-density on A when B is re 
moved, then the charge on A is 

4 TT a 2 <r, 
and therefore the charge on B is 

37r/3 2 o-(l +i^ +&C.), 
v 3 a 

or, when B is very small compared with a, the charge on the 
hemisphere B is equal to three times that due to a surface-density a- 
extending over an area equal to that of the circular base of the 
hemisphere. 

It appears from Art. 175 that if a small sphere is made to touch 
an electrified body, and is then removed to a distance from it, the 
mean surface-density on the sphere is to the surface-density of the 
body at the point of contact as 7r 2 is to 6, or as 1.645 to 1. 

225.] The most convenient form for the proof plane is that of 
a circular disk. We shall therefore shew how the charge on a 
circular disk laid on an electrified surface is to be measured. 

For this purpose we shall construct a value of the potential 
function so that one of the equipotential surfaces resembles a circular 
flattened protuberance whose general form is somewhat like that of 
a disk lying on a plane. 



280 ELECTROSTATIC INSTRUMENTS. [ 22 5- 

Let o- be the surface-density of a plane, which we shall suppose 
to be that of xy. 

The potential due to this electrification will be 
y =477 0-2. 

Now let two disks of radius a be rigidly electrified with surface- 
densities (/ and +</. Let the first of these be placed on the plane 
of xy with its centre at the origin, and the second parallel to it at 
the very small distance c. 

Then it may be shewn, as we shall see in the theory of mag 
netism, that the potential of the two disks at any point is a) a- c, 
where &> is the solid angle subtended by the edge of either disk at 
the point. Hence the potential of the whole system will be 
V = 4 TT <T Z-\-<&<T c. 

The forms of the equipotential surfaces and lines of induction 
are given on the left-hand side of Fig. XX, at the end of Vol. II. 

Let us trace the form of the surface for which V = 0. This 
surface is indicated by the dotted line. 

Putting the distance of any point from the axis of z = r, then, 
when r is much less than , and z is small, 



= 2 77 27T- +&C. 

a 



Hence, for values of r considerably less than a, the equation of 
the zero equipotential surface is 

(* 
= 4 7T (TZ+2 77 <r tf 27T (/ --- (- &C. j 



CT C 

or z n = 



- 
a, 

Hence this equipotential surface near the axis is nearly flat. 

Outside the disk, where r is greater than a, co is zero when z is 
zero, so that the plane of xy is part of the equipotential surface. 

To find where these two parts of the surface meet, let us find at 

dV 
what point of this plane -^- = 0. 

When r is very nearly equal to a 

dV 2</c 

-7- = 4 TT oH -- 
dz ra, 

Hence, when 

dV </c 



The equipotential surface V = is therefore composed of a disk- 



226.] ACCUMULATORS. 281 

like figure of radius r , and nearly uniform thickness z , and of the 
part of the infinite plane of xy which lies beyond this figure. 

The surface-integral over the whole disk gives the charge of 
electricity on it. It may be found, as in the theory of a circular 
current in Part IV, to be 

Q = 4 TT a </ c {log 2}-(-7ro-r 2 . 

r a 

The charge on an equal area of the plane surface is TT a- r 2 , hence 
the charge on the disk exceeds that on an equal area of the plane 

in the ratio of z , Birr . 

1 4- 8 log to unity, 

T Z 

where z is the thickness and r the radius of the disk, z being sup 
posed small compared with r. 

On Electric Accumulators and the Measurement of Capacity. 

226.] An Accumulator or Condenser is an apparatus consisting 
of two conducting surfaces separated by an insulating dielectric 
medium. 

A Ley den jar is an accumulator in which an inside coating of 
tinfoil is separated from the outside coating by the glass of which 
the jar is made. The original Leyden phial was a glass vessel 
containing- water which was separated by the glass from the hand 
which held it. 

The outer surface of any insulated conductor may be considered 
as one of the surfaces of an accumulator, the other being the earth 
or the walls of the room in which it is placed, and the intervening 
air being the dielectric medium. 

The capacity of an accumulator is measured by the quantity of 
electricity with which the inner surface must be charged to make 
the difference between the potentials of the surfaces unity. 

Since every electrical potential is the sum of a number of parts 
found by dividing each electrical element by its distance from a 
point, the ratio of a quantity of electricity to a potential must 
have the dimensions of a line. Hence electrostatic capacity is a 
linear quantity, or we may measure it in feet or metres without 
ambiguity. 

In electrical researches accumulators are used for two principal 
purposes, for receiving and retaining large quantities of electricity 
in as small a compass as possible, and for measuring definite quan 
tities of electricity by means of the potential to which they raise 
the accumulator. 



282 ELECTROSTATIC INSTRUMENTS. [ 22 7- 

For the retention of electrical charges nothing has been devised 
more perfect than the Leyden jar. The principal part of the loss 
arises from the electricity creeping along the damp uncoated surface 
of the glass from the one coating to the other. This may be checked 
in a great degree by artificially drying the air within the jar, and 
by varnishing the surface of the glass where it is exposed to the 
atmosphere. In Sir W. Thomsons electroscopes there is a very 
small percentage of loss from day to day, and I believe that none 
of this loss can be traced to direct conduction either through air 
or through glass when the glass is good, but that it arises chiefly 
from superficial conduction along the various insulating stems and 
glass surfaces of the instrument. 

In fact, the same electrician has communicated a charge to 
sulphuric acid in a large bulb with a long neck, and has then her 
metically sealed the neck by fusing it, so that the charge was com 
pletely surrounded by glass, and after some years the charge was 
found still to be retained. 

It is only, however, when cold, that glass insulates in this 
way, for the charge escapes at once if the glass is heated to 
a temperature below 100C. 

When it is desired to obtain great capacity in small compass, 
accumulators in which the dielectric is sheet caoutchouc, mica, or 
paper impregnated with paraffin are convenient. 

227.] For accumulators of the second class, intended for the 
measurement of quantities of electricity, all solid dielectrics must be 
employed with great caution on account of the property which they 
possess called Electric Absorption. 

The only safe dielectric for such accumulators is air, which has 
this inconvenience, that if any dust or dirt gets into the narrow 
space between the opposed surfaces, which ought to be occupied only 
by air, it not only alters the thickness of the stratum of air, but 
may establish a connexion between the opposed surfaces, in which 
case the accumulator will not hold a charge. 

To determine in absolute measure, that is to say in feet or metres, 
the capacity of an accumulator, we must either first ascertain its 
form and size, and then solve the problem of the distribution of 
electricity on its opposed surfaces, or we must compare its capacity 
with that of another accumulator, for which this problem has been 
solved. 

As the problem is a very difficult one, it is best to begin with an 
accumulator constructed of a form for which the solution is known. 



228.] MEASUREMENT OF CAPACITY. 283 

Thus the capacity of an insulated sphere in an unlimited space is 
known to be measured by the radius of the sphere. 

A sphere suspended in a room was actually used by MM. Kohl- 
ran sch and Weber, as an absolute standard with which they com 
pared the capacity of other accumulators. 

The capacity, however, of a sphere of moderate size is so small 
when compared with the capacities of the accumulators in common 
use that the sphere is not a convenient standard measure. 

Its capacity might be greatly increased by surrounding the 
sphere with a hollow concentric spherical surface of somewhat 
greater radius. The capacity of the inner surface is then a fourth 
proportional to the thickness of the stratum of air and the radii of 
the two surfaces. 

Sir W. Thomson has employed this arrangement as a standard of 
capacity, but the difficulties of working the surfaces truly spherical, 
of making them truly concentric, and of measuring their distance 
and their radii with sufficient accuracy, are considerable. 

We are therefore led to prefer for an absolute measure of capacity 
a form in which the opposed surfaces are parallel planes. 

The accuracy of the surface of the planes can be easily tested, 
and their distance can be measured by a micrometer screw, and 
may be made capable of continuous variation, which is a most 
important property of a measuring instrument. 

The only difficulty remaining arises from the fact that the planes 
must necessarily be bounded, and that the distribution of electricity 
near the boundaries of the planes has not been rigidly calculated. 
It is true that if we make them equal circular disks, whose radius 
is large compared with the distance between them, we may treat 
the edges of the disks as if they were straight lines, and calculate 
the distribution of electricity by the method due to Helmholtz, and 
described at Art. 202. But it will be noticed that in this case 
part of the electricity is distributed on the back of each disk, and 
that in the calculation it has been supposed that there are no 
conductors in the neighbourhood, which is not and cannot be the 
case in a small instrument. 

228.] We therefore prefer the following arrangement, due to 
Sir W. Thomson, which we may call the Guard-ring arrangement, 
by means of which the quantity of electricity on an insulated disk 
may be exactly determined in terms of its potential. 



284: 



ELECTROSTATIC INSTRUMENTS. 



[228. 






1 M 1 1 


U B / 

ft <*<y 

n * 


G 


A 


G 


B 



Fig. 20. 



The Guard-ring Accumulator. 

Bb is a cylindrical vessel of conducting material of which the 
outer surface of the upper face is accurately plane. This upper 

surface consists of two parts, 
a disk A, and a broad ring 
BB surrounding the disk, 
separated from it by a very 
small interval all round, just 
sufficient to prevent sparks 
passing. The upper surface 
of the disk is accurately in 
the same plane with that of 
the guard-ring. The disk is 
supported by pillars of insulating material GG. C is a metal disk, 
the under surface of which is accurately plane and parallel to BB. 
The disk C is considerably larger than A. Its distance from A 
is adjusted and measured by means of a micrometer screw, which 
is not given in the figure. 

This accumulator is used as a measuring instrument as follows : 
Suppose C to be at potential zero, and the disk A and vessel Bb 
both at potential V. Then there will be no electrification on the 
back of the disk because the vessel is nearly closed and is all at the 
same potential. There will be very little electrification on the 
edges of the disk because BB is at the same potential with the 
disk. On the face of the disk the electrification will be nearly 
uniform, and therefore the whole charge on the disk will be almost 
exactly represented by its area multiplied by the surface-density on 
a plane, as given at Art. 124. 

In fact, we learn from the investigation at Art. 201 that the 
charge on the disk is 



( 8A SA A+a) 

where R is the radius of the disk, R that of the hole in the guard- 
ring, A the distance between A and (7, and a a quantity which 

cannot exceed R -R l ^2-* 



If the interval between the disk and the guard-ring is small 
compared with the distance between A and C, the second term will 
be very small, and the charge on the disk will be nearly 



8A 



229.] COMPARISON OF CAPACITIES. 285 

Now let the vessel Bb be put in connexion with the earth. The 
charge on the disk A will no longer be uniformly distributed, but it 
will remain the same in quantity, and if we now discharge A we 
shall obtain a quantity of electricity, the value of which we know 
in terms of V, the original difference of potentials and the measur 
able quantities E, R and A. 

On the Comparison of the Capacity of Accumulators. 

229.] The form of accumulator which is best fitted to have its 
capacity determined in absolute measure from the form and dimen 
sions of its parts is not generally the most suitable for electrical 
experiments. It is desirable that the measures of capacity in actual 
use should be accumulators having only two conducting surfaces, one 
of which is as nearly as possible surrounded by the other. The 
guard-ring accumulator, on the other hand, has three independent 
conducting portions which must be charged and discharged in a 
certain order. Hence it is desirable to be able to compare the 
capacities of two accumulators by an electrical process, so as to test 
accumulators which may afterwards serve as secondary standards. 

I shall first shew how to test the equality of the capacity of two 
guard-ring accumulators. 

Let A be the disk, B the guard-ring with the rest of the con 
ducting vessel attached to it, and C the large disk of one of these 
accumulators, and let A , B , and C be the corresponding parts of 
the other. 

If either of these accumulators is of the more simple kind, having 
only two conductors, we have only to suppress B or B , and to 
suppose A to be the inner and C the outer conducting surface. C 
in this case being understood to surround A. 

Let the following connexions be made. 

Let B be kept always connected with C", and J? with C, that is, 
let each guard-ring be connected with the large disk of the other 
condenser. 

(1) Let A be connected with B and C and with /, the electrode 
of a Ley den jar, and let A be connected with B f and C and with 
the earth. 

(2) Let A, B, and C be insulated from /. 

(3) Let A be insulated from B and C", and A from B f and C . 

(4) Let B and C be connected with B* and C and with the 
earth. 

(5) Let A be connected with A . 



286 ELECTROSTATIC INSTRUMENTS. [229. 

(6) Let A and A be connected with an electroscope E. 
We may express these connexions as follows : 

(1) Q = C=B = A | A==C =J. 

(2) = C=&=A | A==C \J. 

(3) = C=& | A | A\=C . 

(4) = C=ff\A | -4 |-B=C"=0. 

(5) = (7=^|^ = A\=C =0. 

(6) = C=ff | ^= J0= ^ | =<? = 0. 

Here the sign of equality expresses electrical connexion, and the 
vertical stroke expresses insulation. 

In ( 1 ) the two accumulators are charged oppositely, so that A is 
positive and A negative, the charges on A and A being uniformly 
distributed on the upper surface opposed to the large disk of each 
accumulator. 

In (2) the jar is removed, and in (3) the charges on A and A are 
insulated. 

In (4) the guard-rings are connected with the large disks, so that 
the charges on A and A , though unaltered in magnitude, are now 
distributed over their whole surface. 

In (5) A is connected with A . If the charges are equal and of 
opposite signs, the electrification will be entirely destroyed, and 
in (6) this is tested by means of the electroscope JE. 

The electroscope E will indicate positive or negative electrification 
according as A or A has the greater capacity. 

By means of a key of proper construction, the whole of these 
operations can be performed in due succession in a very small 
fraction of a second, and the capacities adjusted till no electri 
fication can be detected by the electroscope, and in this way the 
capacity of an accumulator may be adjusted to be equal to that of 
any other, or to the sum of the capacities of several accumulators, 
so that a system of accumulators may be formed, each of which has 
its capacity determined in absolute measure, i. e. in feet or in metres, 
while at the same time it is of the construction most suitable for 
electrical experiments. 

This method of comparison will probably be found useful in 
determining the specific capacity for electrostatic induction of 
different dielectrics in the form of plates or disks. If a disk of 
the dielectric is interposed between A and C, the disk being con 
siderably larger than A, then the capacity of the accumulator will 



229.] SPECIFIC INDUCTIVE CAPACITY. 287 

be altered and made equal to that of the same accumulator when A 
and C are nearer together. If the accumulator with the dielectric 
plate, and with A and C at distance x, is of the same capacity as 
the same accumulator without the dielectric, and with A and C at 
distance x , then, if a is the thickness of the plate, and K its specific 
dielectric inductive capacity referred to air as a standard, 



x 

The combination of three cylinders, described in Art. 127, has 
been employed by Sir TV. Thomson as an accumulator whose capa 
city may be increased or diminished by measurable quantities. 

The experiments of MM. Gibson and Barclay with this ap 
paratus are described in the Proceedings of the Royal Society, Feb. 2, 
1871, and Phil. Trans., 1871, p. 573. They found the specific in 
ductive capacity of paraffin to be 1.975, that of air being unity, 



PART II. 

ELECTED KINEMATICS. 



CHAPTEE I, 

THE ELECTRIC CURRENT. 

230.] WE have seen, in Art. 45, that when a conductor is in 
electrical equilibrium the potential at every point of the conductor 
must be the same. 

If two conductors A and B are charged with electricity so that 
the potential of A is higher than that of B, then, if they are put 
in communication by means of a metallic wire C touching both of 
them, part of the charge of A will be transferred to B, and the 
potentials of A and B will become in a very short time equalized. 

231.] During this process certain phenomena are observed in 
the wire C, which are called the phenomena of the electric conflict 
or current. 

The first of these phenomena is the transference of positive 
electrification from A to B and of negative electrification from B 
to A. This transference may be also effected in a slower manner 
by bringing a small insulated body into contact with A and B 
alternately. By this process, which we may call electrical con 
vection, successive small portions of the electrification of each body 
are transferred to the other. In either case a certain quantity of 
electricity, or of the state of electrification, passes from one place 
to another along a certain path in the space between the bodies. 

Whatever therefore may be our opinion of the nature of elec 
tricity, we must admit that the process which we have described 
constitutes a current of electricity. This current may be described 



232.] THE VOLTAIC BATTERY. 289 

as a current of positive electricity from A to .5, or a current of 
negative electricity from B to A, or as a combination of these two 
currents. 

According to Fechner s and Weber s theory it is a combination 
of a current of positive electricity with an exactly equal current 
of negative electricity in the opposite direction through the same 
substance. It is necessary to remember this exceedingly artificial 
hypothesis regarding the constitution of the current in order to 
understand the statement of some of Weber s most valuable ex 
perimental results. 

If, as in Art. 36, we suppose P units of positive electricity 
transferred from A to B , and N units of negative electricity trans 
ferred from B to A in unit of time, then, according to Weber s 
theory, P=N, and P or N is to be taken as the numerical measure 
of the current. 

We, on the contrary, make no assumption as to the relation 
between P and N, but attend only to the result of the current, 
namely, the transference of P + A r of positive electrification from A 
to j5, and we shall consider P-f N the true measure of the current. 
The current, therefore, which Weber would call 1 we shall call 2. 

On Steady Currents. 

232.] In the case of the current between two insulated con 
ductors at different potentials the operation is soon brought to 
an end by the equalization of the potentials of the two bodies, 
and the current is therefore essentially a Transient current. 

But there are methods by which the difference of potentials of 
the conductors may be maintained constant, in which case the 
current will continue to flow with uniform strength as a Steady 
Current. 

The Voltaic Battery. 

The most convenient method of producing a steady current is by 
means of the Voltaic Battery. 

For the sake of distinctness we shall describe Daniell s Constant 
Battery : 

A solution of sulphate of zinc is placed in a cell of porous earth 
enware, and this cell is placed in a vessel containing a saturated 
solution of sulphate of copper. A piece of zinc is dipped into the 
sulphate of zinc, and a piece of copper is dipped into the sulphate 
of copper. Wires are soldered to the zinc and to the copper above 

VOL. i. u 



290 THE ELECTRIC CURRENT. [ 2 33* 

the surface of the liquid. This combination is called a cell or 
element of Daniell s battery. See Art. 272. 

233.] If the cell is insulated by being placed on a non-con 
ducting stand, and if the wire connected with the copper is put 
in contact with an insulated conductor A, and the wire connected 
with the zinc is put in contact with J3, another insulated conductor 
of the same metal as A, then it may be shewn by means of a delicate 
electrometer that the potential of A exceeds that of B by a certain 
quantity. This difference of potentials is called the Electromotive 
Force of the Daniell s Cell. 

If A and B are now disconnected from the cell and put in 
communication by means of a wire, a transient current passes 
through the wire from A to J3, and the potentials of A and B 
become equal. A and B may then be charged again by the cell, 
and the process repeated as long as the cell will work. But if 
A and B be connected by means of the wire C, and at the same 
time connected with the battery as before, then the cell will main 
tain a constant current through C, and also a constant difference 
of potentials between A and B. This difference will not, as we 
shall see, be equal to the whole electromotive force of the cell, for 
part of this force is spent in maintaining the current through the 
cell itself. 

A number of cells placed in series so that the zinc of the first 
cell is connected by metal with the copper of the second, and 
so on, is called a Voltaic Battery. The electromotive force of 
such a battery is the sum of the electromotive forces of the cells 
of which it is composed. If the battery is insulated it may be 
charged with electricity as a whole, but the potential of the copper 
end will always exceed that of the zinc end by the electromotive 
force of the battery, whatever the absolute value of either of these 
potentials may be. The cells of the battery may be of very various 
construction, containing different chemical substances and different 
metals, provided they are such that chemical action does not go 
on when no current passes. 

234.] Let us now consider a voltaic battery with its ends insulated 
from each other. The copper end will be positively or vitreously 
electrified, and the zinc end will be negatively or resinously electrified. 

Let the two ends of the battery be now connected by means 
of a wire. An electric current will commence, and will in a very 
short time attain a constant value. It is then said to be a Steady 
Current. 



236.] ELECTKOLYSIS. 291 

Properties of the Current. 

235.] The current forms a closed circuit in the direction from 
copper to zinc through the wires, and from zinc to copper through 
the solutions. 

If the circuit be broken by cutting any of the wires which 
connect the copper of one cell with the zinc of the next in order, the 
current will be stopped, and the potential of the end of the wire 
in connexion with the copper will be found to exceed that of the 
end of the wire in connexion with the zinc by a constant quantity, 
namely, the total electromotive force of the circuit. 

Electrolytic Action of the Current. 

236.] As long as the circuit is broken no chemical action goes 
on in the cells, but as soon as the circuit is completed, zinc is 
dissolved from the zinc in each of the Daniell s cells, and copper is 
deposited on the copper. 

The quantity of sulphate of zinc increases, and the quantity of 
sulphate of copper diminishes unless more is constantly supplied. 

The quantity of zinc dissolved and also that of copper deposited is 
the same in each of the Daniell s cells throughout the circuit, what 
ever the size of the plates of the cell, and if any of the cells be of a 
different construction, the amount of chemical action in it bears 
a constant proportion to the action in the Daniell s cell. For 
instance, if one of the cells consists of two platinum plates dipped 
into sulphuric acid diluted with water, oxygen will be given off 
at the surface of the plate where the current enters the liquid, 
namely, the plate in metallic connexion with the copper of Daniell s 
cell, and hydrogen at the surface of the plate where the current 
leaves the liquid, namely, the plate connected with the zinc of 
Daniell s cell. 

The volume of the hydrogen is exactly twice the volume of the 
oxygen given off in the same time, and the weight of the oxygen is 
exactly eight times the weight of the hydrogen. 

In every cell of the circuit the weight of each substance dissolved, 
deposited, or decomposed is equal to a certain quantity called the 
electrochemical equivalent of that substance, multiplied by the 
strength of the current and by the time during which it has 
been flowing. 

For the experiments which established this principle, see the 
seventh and eighth series of Faraday s Experimental Researches; 

T: 2 



292 THE ELECTRIC CURRENT. [237. 

and for an investigation of the apparent exceptions to the rule, see 
Miller s Chemical Physics and Wiedemann s Galvanismus. 

237.] Substances which are decomposed in this way are called 
Electrolytes. The process is called Electrolysis. The places where 
the current enters and leaves the electrolyte are called Electrodes. 
Of these the electrode by which the current enters is called the 
Anode, and that by which it leaves the electrolyte is called the 
Cathode. The components into which the electrolyte is resolved 
are called Ions : that which appears at the anode is called the 
Anion, and that which appears at the cathode is called the Cation. 

Of these terms, which were, I believe, invented by Faraday with 
the help of Dr. Whewell, the first three, namely, electrode, elec 
trolysis, and electrolyte have been generally adopted, and the mode 
of conduction of the current in which this kind of decomposition 
and transfer of the components takes place is called Electrolytic 
Conduction. 

If a homogeneous electrolyte is placed in a tube of variable 
section, and if the electrodes are placed at the ends of this tube, 
it is found that when the current passes, the anion appears at 
the anode and the cation at the cathode, the quantities of these 
ions being electrochemically equivalent, and such as to be together 
equivalent to a certain quantity of the electrolyte. In the other 
parts of the tube, whether the section be large or small, uniform 
or varying, the composition of the electrolyte remains unaltered. 
Hence the amount of electrolysis which takes place across every 
section of the tube is the same. Where the section is small the 
action must therefore be more intense than where the section is 
large, but the total amount of each ion which crosses any complete 
section of the electrolyte in a given time is the same for all sections. 

The strength of the current may therefore be measured by the 
amount of electrolysis in a given time. An instrument by which 
the quantity of the electrolytic products can be readily measured 
is called a Voltameter. 

The strength of the current, as thus measured, is the same 
at every part of the circuit, and the total quantity of the elec 
trolytic products in the voltameter after any given time is pro 
portional to the amount of electricity which passes any section in 
the same time. 

238.] If we introduce a voltameter at one part of the circuit 
of a voltaic battery, and break the circuit at another part, we may 
suppose the measurement of the current to be conducted thus. 



239-] MAGNETIC ACTION. 293 

Let the ends of the broken circuit be A and H, and let A be the 
anode and B the cathode. Let an insulated ball be made to touch 
A and B alternately, it will carry from A to B a certain measurable 
quantity of electricity at each journey. This quantity may be 
measured by an electrometer, or it may be calculated by mul 
tiplying the electromotive force of the circuit by the electrostatic 
capacity of the ball. Electricity is thus carried from A to B on the 
insulated ball by a process which may be called Convection. At 
the same time electrolysis goes on in the voltameter and in the 
c^lls of the battery, and the amount of electrolysis in each cell may 
be compared with the amount of electricity carried across by the 
insulated ball. The quantity of a substance which is electrolysed 
by one unit of electricity is called an Electrochemical equivalent 
of that substance. 

This experiment would be an extremely tedious and troublesome 
one if conducted in this way with a ball of ordinary magnitude 
and a manageable battery, for an enormous number of journeys 
would have to be made before an appreciable quantity of the electro 
lyte was decomposed. The experiment must therefore be considered 
as a mere illustration, the actual measurements of electrochemical 
equivalents being conducted in a different way. But the experi 
ment may be considered as an illustration of the process of elec 
trolysis itself, for if we regard electrolytic conduction as a species 
of convection in which an electrochemical equivalent of the anion 
travels with negative electricity in the direction of the anode, while 
an equivalent of the cation travels with positive electricity in 
the direction of the cathode, the whole amount of transfer of elec 
tricity being one unit, we shall have an idea of the process of 
electrolysis, which, so far as I know, is not inconsistent with known 
facts, though, on account of our ignorance of the nature of electricity 
and of chemical compounds, it may be a very imperfect repre 
sentation of what really takes place. 

Magnetic Action of the Current. 

239.] Oersted discovered that a magnet placed near a straight 
electric current tends to place itself at right angles to the plane 
passing through the magnet and the current. See Art. 475. 

If a man were to place his body in the line of the current so 
that the current from copper through the wire to zinc should flow 
from his head to his feet, and if he were to direct his face towards 
the centre of the magnet, then that end of the magnet which tends 



294 THE ELECTEIC CUR11ENT. [240. 

to point to the north would, when the current flows, tend to point 
towards the man s right hand. 

The nature and laws of this electromagnetic action will be dis 
cussed when we come to the fourth part of this treatise. What 
we are concerned with at present is the fact that the electric 
current has a magnetic action which is exerted outside the current, 
and by which its existence can be ascertained and its intensity 
measured without breaking the circuit or introducing anything into 
the current itself. 

The amount of the magnetic action has been ascertained to be 
strictly proportional to the strength of the current as measured 
by the products of electrolysis in the voltameter, and to be quite 
independent of the nature of the conductor in which the current 
is flowing, whether it be a metal or an electrolyte. 

240.] An instrument which indicates the strength of an electric 
current by its magnetic effects is called a Galvanometer. 

Galvanometers in general consist of one or more coils of silk- 
covered wire within which a magnet is suspended with its axis 
horizontal. When a current is passed through the wire the magnet 
tends to set itself with its axis perpendicular to the plane of the 
coils. If we suppose the plane of the coils to be placed parallel 
to the plane of the earth s equator, and the current to flow round 
the coil from east to west in the direction of the apparent motion 
of the sun, then the magnet within will tend to set itself with 
its magnetization in the same direction as that of the earth con 
sidered as a great magnet, the north pole of the earth being similar 
to that end of the compass needle which points south. 

The galvanometer is the most convenient instrument for mea 
suring the strength of electric currents. We shall therefore assume 
the possibility of constructing such an instrument in studying the 
laws of these currents, reserving the discussion of the principles of 
the instrument for our fourth part. When therefore we say that 
an electric current is of a certain strength we suppose that the 
measurement is effected by the galvanometer. 



CHAPTER II. 



CONDUCTION AND KESISTANCE. 

241.] IF by means of an electrometer we determine the electric 
potential at different points of a circuit in which a constant electric 
current is maintained, we shall find that in any portion of the 
circuit consisting of a single metal of uniform temperature through 
out, the potential at any point exceeds that at any other point 
farther on in the direction of the current by a quantity depending 
on the strength of the current and on the nature and dimensions 
of the intervening portion of the circuit. The difference of the 
potentials at the extremities of this portion of the circuit is called 
the External electromotive force acting on it. If the portion of 
the circuit under consideration is not homogeneous, but contains 
transitions from one substance to another, from metals to elec 
trolytes, or from hotter to colder parts, there may be, besides the 
external electromotive force, Internal electromotive forces which 
must be taken into account. 

The relations between Electromotive Force, Current, and Resist 
ance were first investigated by Dr. G. S. Ohm, in a work published 
in 1827, entitled Die Galvanische Kette Mathematisch Bearbeitet, 
translated in Taylor s Scientific Memoirs. The result of these in 
vestigations in the case of homogeneous conductors is commonly 
called Ohm s Law. 

Ohm s Laio. 

The electromotive force acting between the extremities of any part 
of a circuit is the product of the strength of the current and the 
Resistance of that part of the circuit. 

Here a new term is introduced, the Resistance of a conductor, 
which is defined to be the ratio of the electromotive force to 
the strength of the current which it produces. The introduction 



296 CONDUCTION AND RESISTANCE. [242. 

of this term would have been of no scientific value unless Ohm 
had shewn,, as he did experimentally, that it corresponds to a real 
physical quantity, that is, that it has a definite value which is 
altered only when the nature of the conductor is altered. 

In the first place, then, the resistance of a conductor is inde 
pendent of the strength of the current flowing through it. 

In the second place the resistance is independent of the electric 
potential at which the conductor is maintained, and of the density 
of the distribution of electricity on the surface of the conductor. 

It depends entirely on the nature of the material of which the 
conductor is composed, the state of aggregation of its parts, and its 
temperature. 

The resistance of a conductor may be measured to within one 
ten thousandth or even one hundred thousandth part of its value, 
and so many conductors have been tested that our assurance of the 
truth of Ohm s Law is now very high, In the sixth chapter we 
shall trace its applications and consequences. 

Generation of Heat by the Current. 

242.] We have seen that when an electromotive force causes 
a current to flow through a conductor, electricity is transferred 
from a place of higher to a place of lower potential. If the transfer 
had been made by convection, that is, by carrying successive 
charges on a ball from the one place to the other, work would have 
been done by the electrical forces on the ball, and this might have 
been turned to account. It is actually turned to account in a 
partial manner in those dry pile circuits where the electrodes have 
the form of bells, and the carrier ball is made to swing like a 
pendulum between the two bells and strike them alternately. In 
this way the electrical action is made to keep up the swinging 
of the pendulum and to propagate the sound of the bells to a 
distance. In the case of the conducting wire we have the same 
transfer of electricity from a place of high to a place of low potential 
without any external work being done. The principle of the Con 
servation of Energy therefore leads us to look for internal work in 
the conductor. In an electrolyte this internal work consists partly 
of the separation of its components. In other conductors it is 
entirely converted into heat. 

The energy converted into heat is in this case the product of 
the electromotive force into the quantity of electricity which passes. 
But the electromotive force is the product of the current into the 



244-] COMPARISON WITH PHENOMENA OF HEAT. 297 

resistance, and the quantity of electricity is the product of the 
current into the time. Hence the quantity of heat multiplied by 
the mechanical equivalent of unit of heat is equal to the square of 
the strength of the current multiplied into the resistance and into 
the time. 

The heat developed by electric currents in overcoming 1 the re 
sistance of conductors has been determined by Dr. Joule, who 
first established that the heat produced in a given time is pro 
portional to the square of the current, and afterwards by careful 
absolute measurements of all the quantities concerned, verified the 
Cation JH=C*Rt, 

where / is Joule s dynamical equivalent of heat, H the number of 
units of heat, C the strength of the current, R the resistance of the 
conductor, and t the time during which the current flows. These 
relations between electromotive force, work, and heat, were first fully 
explained by Sir W. Thomson in a paper on the application of the 
principle of mechanical effect to the measurement of electromotive 
forces *. 

243.] The analogy between the theory of the conduction of 
electricity and that of the conduction of heat is at first sight almost 
complete. If we take two systems geometrically similar, and such 
that the conductivity for heat at any part of the first is proportional 
to the conductivity for electricity at the corresponding part of the 
second, and if we also make the temperature at any part of the 
first proportional to the electric potential at the corresponding point 
of the second, then the flow of heat across any area of the first 
will be proportional to the flow of electricity across the corre 
sponding area of the second. 

Thus, in the illustration we have given, in which flow of elec 
tricity corresponds to flow of heat, and electric potential to tem 
perature, electricity tends to flow from places of high to places 
of low potential, exactly as heat tends to flow from places of high 
to places of low temperature. 

244.] The theory of potential and that of temperature may 
therefore be made to illustrate one another ; there is, however, one 
remarkable difference between the phenomena of electricity and 
those of heat. 

Suspend a conducting body within a closed conducting vessel by 
a silk thread, and charge the vessel with electricity. The potential 

* PhU. Mag., Dec. 1851. 



298 CONDUCTION AND RESISTANCE. [ 2 45- 

of the vessel and of all within it will be instantly raised, but 
however long and however powerfully the vessel be electrified, and 
whether the body within be allowed to come in contact with the 
vessel or not, no signs of electrification will appear within the 
vessel, nor will the body within shew any electrical effect when 
taken out. 

But if the vessel is raised to a high temperature, the body 
within will rise to the same temperature, but only after a con 
siderable time, and if it is then taken out it will be found hot, 
and will remain so till it has continued to emit heat for some time. 

The difference between the phenomena consists in the fact that 
bodies are capable of absorbing and emitting heat, whereas they 
have no corresponding property with respect to electricity. A body 
cannot be made hot without a certain amount of heat being 
supplied to it, depending on the mass and specific heat of the body, 
but the electric potential of a body may be raised to any extent 
in the way already described without communicatiDg any electricity 
to the body. 

245.] Again, suppose a body first heated and then placed inside 
the closed vessel. The outside of the vessel will be at first at the 
temperature of surrounding bodies, but it will soon get hot, and 
will remain hot till the heat of the interior body has escaped. 

It is impossible to perform a corresponding electrical experiment. 
It is impossible so to electrify a body, and so to place it in a 
hollow vessel, that the outside of the vessel shall at first shew no 
signs of electrification but shall afterwards become electrified. It 
was for some phenomenon of this kind that Faraday sought in 
vain under the name of an absolute charge of electricity. 

Heat may be hidden in the interior of a body so as to have no 
external action, but it is impossible to isolate a quantity of elec 
tricity so as to prevent it from being constantly in inductive 
relation with an equal quantity of electricity of the opposite kind. 

There is nothing therefore among electric phenomena which 
corresponds to the capacity of a body for heat. This follows at 
once from the doctrine which is asserted in this treatise, that 
electricity obeys the same condition of continuity as an incom 
pressible fluid. It is therefore impossible to give a bodily charge 
of electricity to any substance by forcing an additional quantity of 
electricity into it. See Arts. 61, 111, 329, 334. 



CHAPTER III. 



ELECTROMOTIVE FORCE BETWEEN BODIES IN CONTACT. 

The Potentials of Different Substances in Contact. 

246.] IF we define the potential of a hollow conducting vessel 
as the potential of the air inside the vessel, we may ascertain this 
potential by means of an electrometer as described in Part I, 
Art, 222. 

If we now take two hollow vessels of different metals, say copper 
and zinc, and put them in metallic contact with each other, and 
then test the potential of the air inside each vessel, the potential 
of the air inside the zinc vessel will be positive as compared with 
that inside the copper vessel. The difference of potentials depends 
on the nature of the surface of the insides of the vessels, being 
greatest when the zinc is bright and when the copper is coated 
with oxide. 

It appears from this that when two different metals are in 
contact there is in general an electromotive force acting from the 
one to the other, so as to make the potential of the one exceed 
that of the other by a certain quantity. This is Volta s theory of 
Contact Electricity. 

If we take a certain metal, say copper, as the standard, then 
if the potential of iron in contact with copper at the zero potential 
is /, and that of zinc in contact with copper at zero is Z, then 
the potential of zinc in contact with iron at zero will be ZI. 

It appears from this result, which is true of any three metals, 
that the differences of potential of any two metals at the same 
temperature in contact is equal to the difference of their potentials 
when in contact with a third metal, so that if a circuit be formed 
of any number of metals at the same temperature there will be 
electrical equilibrium as soon as they have acquired their proper 
potentials, and there will be no current kept up in the circuit. 



300 CONTACT FOKCE. [ 2 47- 

247.] If, however, the circuit consist of two metals and an elec 
trolyte, the electrolyte, according to Volta s theory, tends to reduce 
the potentials of the metals in contact with it to equality, so that 
the electromotive force at the metallic junction is no longer balanced, 
and a continuous current is kept up. The energy of this current 
is supplied by the chemical action which takes place between the 
electrolyte and the metals. 

248.] The electric effect may. however, be produced without 
chemical action if by any other means we can produce an equali 
zation of the potentials of two metals in contact. Thus, in an 
experiment due to Sir W. Thomson*, a copper funnel is placed in 
contact with a vertical zinc cylinder, so that when copper filings 
are allowed to pass through the funnel, they separate from each 
other and from the funnel near the middle of the zinc cylinder, 
and then fall into an insulated receiver placed below. The receiver 
is then found to be charged negatively, and the charge increases 
as the filings continue to pour into it. At the same time the zinc 
cylinder with the copper funnel in it becomes charged more and 
more positively. 

If now the zinc cylinder were connected with the receiver by a 
wire, there would be a positive current in the wire from the cylinder 
to the receiver. The stream of copper filings, each filing charged 
negatively by induction, constitutes a negative current from the 
funnel to the receiver, or, in other words, a positive current from 
the receiver to the copper funnel. The positive current, therefore, 
passes through the air (by the filings) from zinc to copper, and 
through the metallic junction from copper to zinc, just as in the 
ordinary voltaic arrangement, but in this case the force which keeps 
up the current is not chemical action but gravity, which causes the 
filings to fall, in spite of the electrical attraction between the 
positively charged funnel and the negatively charged filings. 

249.] A remarkable confirmation of the theory of contact elec 
tricity is supplied by the discovery of Peltier, that, when a current 
of electricity crosses the junction of two metals, the junction is 
heated when the current is in one direction, and cooled when it 
is in the other direction. It must be remembered that a current 
in its passage through a metal always produces heat, because it 
meets with resistance, so that the cooling effect on the whole 
conductor must always be less than the heating effect. We must 
therefore distinguish between the generation of heat in each metal, 
North British Revieiv, 1864, p. 353 ; and Proc. E. S., June 20, 1867. 



249-] PELTIER S PHENOMENON. 301 

due to ordinary resistance, and the generation or absorption of heat 
at the junction of two metals. We shall call the first the frictional 
generation of heat by the current, and, as we have seen, it is 
proportional to the square of the current, and is the same whether 
the current be in the positive or the negative direction. The second 
we may call the Peltier effect, which changes its sign with that 
of the current. 

The total heat generated in a portion of a compound conductor 
consisting of two metals may be expressed by 



H = c*t- 

where H is the quantity of heat, / the mechanical equivalent of 
unit of heat, R the resistance of the conductor, C the current, and 
t the time ; IT being the coefficient of the Peltier effect, that is, 
the heat absorbed at the junction due to the passage of unit of 
current for unit of time. 

Now the heat generated is mechanically equivalent to the work 
done against electrical forces in the conductor, that is, it is equal 
to the product of the current into the electromotive force producing 
it. Hence, if E is the external electromotive force which causes 
the current to flow through the conductor, 

JH= CUt = RC* t-JU Ct, 
whence E = RC-JU. 

It appears from this equation that the external electromotive 
force required to drive the current through the compound conductor 
is less than that due to its resistance alone by the electromotive 
force /fl. Hence /n represents the electromotive contact force 
at the junction acting in the positive direction. 

This application, due to Sir W. Thomson *, of the dynamical 
theory of heat to the determination of a local electromotive force 
is of great scientific importance, since the ordinary method of 
connecting two points of the compound conductor with the elec 
trodes of a galvanometer or electroscope by wires would be useless, 
owing to the contact forces at the junctions of the wires with 
the materials of the compound conductor. In the thermal method, 
on the other hand, we know that the only source of energy is the 
current of electricity, and that no work is done by the current 
in a certain portion of the circuit except in heating that portion 
of the conductor. If, therefore, we can measure the amount of the 
* Proc. R. S. Edin., Dec. 15, 1851 ; and Trans. R. 8. Edin., 1854. 



302 CONTACT FORCE. [250. 

current and the amount of heat produced or absorbed, we can 
determine the electromotive force required to urge the current 
through that portion of the conductor, and this measurement is 
entirely independent of the effect of contact forces in other parts of 
the circuit. 

The electromotive force at the junction of two metals, as de 
termined by this method, does not account for Volta s electromotive 
force as described in Art. 246. The latter is in general far greater 
than that of this Article, and is sometimes of opposite sign. Hence 
the assumption that the potential of a metal is to be measured by 
that of the air in contact with it must be erroneous, and the greater 
part of Volta s electromotive force must be sought for, not at the 
junction of the two metals, but at one or both of the surfaces which 
separate the metals from the air or other medium which forms the 
third element of the circuit. 

250.] The discovery by Seebeck of thermoelectric currents in 
circuits of different metals with their junctions at different tem 
peratures, shews that these contact forces do not always balance 
each other in a complete circuit. It is manifest, however, that 
in a complete circuit of different metals at uniform temperature the 
contact forces must balance each other. For if this were not the 
case there would be a current formed in the circuit, and this current 
might be employed to work a machine or to generate heat in the 
circuit, that is, to do work, while at the same time there is no 
expenditure of energy, as the circuit is all at the same temperature, 
and no chemical or other change takes place. Hence, if the Peltier 
effect at the junction of two metals a and d be represented by H a& 
when the current flows from a to b, then for a circuit of two metals 
at the same temperature we must have 



and for a circuit of three metals a, 6, c, we must have 

n bc +n ca +n a& = o. 

It follows from this equation that the three Peltier effects are not 
independent, but that one of them can be deduced from the other 
two. For instance, if we suppose c to be a standard metal, and 
if we write P a = /n ac and P b JU bc , then 

JU ab = P a -P b . 

The quantity P a is a function of the temperature, and depends on 
the nature of the metal a. 

251.] It has also been shewn by Magnus that if a circuit is 



251.] THERMOELECTRIC PHENOMENA. 303 

formed of a single metal no current will be formed in it, however 
the section of the conductor and the temperature may vary in 
different parts. 

Since in this case there is conduction of heat and consequent 
dissipation of energy, we cannot, as in the former case, consider this 
result as self-evident. The electromotive force, for instance, between 
two portions of a circuit might have depended on whether the 
current was passing from a thick portion of the conductor to a thin 
one, or the reverse, as well as on its passing rapidly or slowly from a 
hot portion to a cold one, or the reverse, and this would have made 
a current possible in an unequally heated circuit of one metal. 

Hence, by the same reasoning as in the case of Peltier s phe 
nomenon, we find that if the passage of a current through a 
conductor of one metal produces any thermal effect which is re 
versed when the current is reversed, this can only take place when 
the current flows from places of high to places of low temperature, 
or the reverse, and if the heat generated in a conductor of one 
metal in flowing from a place where the temperature is a? to a 
place where it is ?/, is H, then 

JH= RCH-S xy Ct, 

and the electromotive force tending to maintain the current will 
be S xv . 

If x, y, z be the temperatures at three points of a homogeneous 
circuit, we must have 

Svz + S zx + S xy = 0, 

according to the result of Magnus. Hence, if we suppose z to be 
the zero temperature, and if we put 

Q X = S X , and Q y = S yz , 
we find S xy =Q x -Q v , 

where Q x is a function of the temperature x, the form of the 
function depending on the nature of the metal. 

If we now consider a circuit of two metals a and b in which 
the temperature is x where the current passes from a to 5, and 
y where it passes from 6 to a, the electromotive force will be 

F = P ax -P bx + Q bx ~ Q by + P by -P av + Qay- Qat, 

where P ax signifies the value of P for the metal a at the tempera 
ture #, or 



Since in unequally heated circuits of different metals there are in 



304 CONTACT FORCE. [ 2 52. 

general thermoelectric currents, it follows that P and Q are in 
genera] different for the same metal and same temperature. 

252.] The existence of the quantity Q was first demonstrated by 
Sir W. Thomson, in the memoir we have referred to, as a deduction 
from the phenomenon of thermoelectric inversion discovered by 
Gumming *, who found that the order of certain metals in the ther 
moelectric scale is different at high and at low temperatures, so that 
for a certain temperature two metals may be neutral to each other. 
Thus, in a circuit of copper and iron if one junction be kept at the 
ordinary temperature while the temperature of the other is raised, 
a current sets from copper to iron through the hot junction, and 
the electromotive force continues to increase till the hot junction 
has reached a temperature T, which, according to Thomson, is 
about 284C. When the temperature of the hot junction is raised 
still further the electromotive force is reduced, and at last, if the 
temperature be raised high enough, the current is reversed. The 
reversal of the current may be obtained more easily by raising the 
temperature of the colder junction. If the temperature of both 
junctions is above T the current sets from iron to copper through 
the hotter junction, that is, in the reverse direction to that ob 
served when both junctions are below T. 

Hence, if one of the junctions is at the neutral temperature T 
and the other is either hotter or colder, the current will set from 
copper to iron through the junction at the neutral temperature. 

253.] From this fact Thomson reasoned as follows : 

Suppose the other junction at a temperature lower than T. 
The current may be made to work an engine or to generate heat in 
a wire, and this expenditure of energy must be kept up by the 
transformation of heat into electric energy, that is to say, heat 
must disappear somewhere in the circuit. Now at the tempera 
ture T iron and copper are neutral to each other, so that no 
reversible thermal effect is produced at the hot junction, and at 
the cold junction there is, by Peltier s principle, an evolution of 
heat. Hence the only place where the heat can disappear is in the 
copper or iron portions of the circuit, so that either a current in 
iron from hot to cold must cool the iron, or a current in copper 
from cold to hot must cool the copper, or both these effects may 
take place. By an elaborate series of ingenious experiments Thom 
son succeeded in detecting the reversible thermal action of the 
current in passing between parts of different temperatures, and 
* Cambridge Transactions, 1823. 



254-] EXPERIMENTS OF TAIT. 305 

he found that the current produced opposite effects in copper and 
in iron *. 

When a stream of a material fluid passes along a tube from 
a hot part to a cold part it heats the tube, and when it passes 
from cold to hot it cools the tube, and these effects depend on 
the specific capacity for heat of the fluid. If we supposed elec 
tricity, whether positive or negative, to be a material fluid, we 
might measure its specific heat by the thermal effect on an un 
equally heated conductor. Now Thomson s experiments shew that 
positive electricity in copper and negative electricity in iron carry 
heat with them from hot to cold. Hence, if we supposed either 
positive or negative electricity to be a fluid, capable of being 
heated and cooled, and of communicating heat to other bodies, we 
should find the supposition contradicted by iron f6r positive elec 
tricity and by copper for negative electricity, so that we should 
have to abandon both hypotheses. 

This scientific prediction of the reversible effect of an electric 
current upon an unequally heated conductor of one metal is another 
instructive example of the application of the theory of Conservation 
of Energy to indicate new directions of scientific research. Thomson 
has also applied the Second Law of Thermodynamics to indicate 
relations between the quantities which we have denoted by P 
and Q, and has investigated the possible thermoelectric properties 
of bodies whose structure is different in different directions. He 
has also investigated experimentally the conditions under which 
these properties are developed by pressure, magnetization, &c. 

254.] Professor Taitf has recently investigated the electro 
motive force of thermoelectric circuits of different metals, having 
their junctions at different temperatures. He finds that the elec 
tromotive force of a circuit may be expressed veiy accurately by 
the formula 

E= (4-4) ft-* A +4)1 

where ^ is the absolute temperature of the hot junction, t 2 that 
of the cold junction, and t Q the temperature at which the two metals 
are neutral to each other. The factor a is a coefficient depending 
on the nature of the two metals composing the circuit. This law 
has been verified through considerable ranges of temperature by 
Professor Tait and his students, and he hopes to make the thermo 
electric circuit available as a thermometric instrument in his 

* On the Electrodynamic Qualities of Metals. Phil Tram., 1856. 
t Proc. R. S. Edin., Session 18/0-71, p. 308, also Dec. 18, 1871. 
VOL. I. X 



306 CONTACT FORCE. [254. 

experiments on the conduction of heat, and in other cases in which 
the mercurial thermometer is not convenient or has not a sufficient 
range. 

According to Tait s theory, the quantity which Thomson calls 
the specific heat of electricity is proportional to the absolute tem 
perature in each pure metal, though its magnitude and even its 
sign vary in different metals. From this he has deduced by ther- 
modynamic principles the following results. Let h a t, 7c b t, k c t 
be the specific heats of electricity in three metals a, b, c, and let 
T bc , T ca , T ab be the temperatures at which pairs of these metals are 
neutral to each other, then the equations 

k b }T ab = 0, 



express the relation of the neutral temperatures, the value of the 
Peltier effect, and the electromotive force of a thermoelectric circuit. 



CHAPTER IV. 



ELECTROLYSIS. 

Electrolytic Conduction. 

255.] I HAVE already stated that when an electric current in 
any part of its circuit passes through certain compound substances 
called Electrolytes, the passage of the current is accompanied by 
a certain chemical process called Electrolysis, in which the substance 
is resolved into two components called Ions, of which one, called 
the Anion, or the electronegative component, appears at the Anode, 
or place where the current enters the electrolyte, and the other, 
called the Cation, appears at the Cathode, or the place where the 
current leaves the electrolyte. 

The complete investigation of Electrolysis belongs quite as much 
to Chemistry as to Electricity. We shall consider it from an 
electrical point of view, without discussing its application to the 
theory of the constitution of chemical compounds. 

Of all electrical phenomena electrolysis appears the most likely 
to furnish us with a real insight into the true nature of the electric 
current, because we find currents of ordinary matter and currents 
of electricity forming essential parts of the same phenomenon. 

It is probably for this very reason that, in the present imperfectly 
formed state of our ideas about electricity, the theories of electro 
lysis are so unsatisfactory. 

The fundamental law of electrolysis, which was established by 
Faraday, and confirmed by the experiments of Beetz, Hittorf, and 
others down to the present time, is as follows : 

The number of electrochemical equivalents of an electrolyte which 
are decomposed by the passage of an electric current during a given 
time is equal to the number of units of electricity which are trans 
ferred by the current in the same time. 

The electrochemical equivalent of a substance is that quantity 

X 2, 



308 ELECTROLYSIS. [255. 

of the substance which is electrolysed by a unit current passing 
through the substance for a unit of time, or, in other words, by the 
passage of a unit of electricity. When the unit of electricity is 
denned in absolute measure the absolute value of the electro 
chemical equivalent of each substance can be determined in grains 
or in grammes. 

The electrochemical equivalents of different substances are pro 
portional to their ordinary chemical equivalents. The ordinary 
chemical equivalents, however, are the mere numerical ratios in 
which the substances combine, whereas the electrochemical equi 
valents are quantities of matter of a determinate magnitude, de 
pending on the definition of the unit of electricity. 

Every electrolyte consists of two components, which, during the 
electrolysis, appear where the current enters and leaves the elec 
trolyte, and nowhere else. Hence, if we conceive a surface described 
within the substance of the electrolyte, the amount of electrolysis 
which takes place through this surface, as measured by the elec 
trochemical equivalents of the components transferred across it 
in opposite directions, will be proportional to the total electric 
current through the surface. 

The actual transfer of the ions through the substance of the 
electrolyte in opposite directions is therefore part of the phenomenon 
of the conduction of an electric current through an electrolyte. At 
every point of the electrolyte through which an electric current 
is passing there are also two opposite material currents of the anion 
and the cation, which have the same lines of flow with the electric 
current, and are proportional to it in magnitude. 

It is therefore extremely natural to suppose that the currents of 
the ions are convection currents of electricity, and, in particular, 
that every molecule of the cation is charged with a certain fixed 
quantity of positive electricity, which is the same for the molecules 
of all cations, and that every molecule of the anion is charged with 
an equal quantity of negative electricity. 

The opposite motion of the ions through the electrolyte would 
then be a complete physical representation of the electric current. 
We may compare this motion of the ions with the motion of gases 
and liquids through each other during the process of diffusion, 
there being this difference between the two processes, that, in 
diffusion, the different substances are only mixed together and the 
mixture is not homogeneous, whereas in electrolysis they are chemi 
cally combined and the electrolyte is homogeneous. In diffusion 



257-] THEORY OP CLAUSIUS. 309 

the determining cause of the motion of a substance in a given 
direction is a diminution of the quantity of that substance per 
unit of volume in that direction, whereas in electrolysis the motion 
of each ion is due to the electromotive force acting on the charged 
molecules. 

256.] Clausius *, who has bestowed much study on the theory 
of the molecular agitation of bodies, supposes that the molecules 
of all bodies are in a state of constant agitation, but that in solid 
bodies each molecule never passes beyond a certain distance from 
its original position, whereas in fluids a molecule, after moving 
a certain distance from its original position, is just as likely to 
move still farther from it as to move back again. Hence the 
molecules of a fluid apparently at rest are continually changing 
their positions, and passing irregularly from one part of the fluid 
to another. In a compound fluid he supposes that not only the 
compound molecules travel about in this way, but that, in the 
collisions which occur between the compound molecules, the mole 
cules of which they are composed are often separated and change 
partners, so that the same individual atom is at one time associated 
with one atom of the opposite kind, and at another time with another. 
This process Clausius supposes to go on in the liquid at all times, but 
when an electromotive force acts on the liquid the motions of the 
molecules, which before were indifferently in all directions, are now 
influenced by the electromotive force, so that the positively charged 
molecules have a greater tendency towards the cathode than towards 
the anode, and the negatively charged molecules have a greater 
tendency to move in the opposite direction. Hence the molecules 
of the cation will during their intervals of freedom struggle towards 
the cathode, but will continually be checked in their course by 
pairing for a time with molecules of the anion, which are also 
struggling through the crowd, but in the opposite direction. 

257.] This theory of Clausius enables us to understand how it is, 
that whereas the actual decomposition of an electrolyte requires an 
electromotive force of finite magnitude, the conduction of the 
current in the electrolyte obeys the law of Ohm, so that every 
electromotive force within the electrolyte, even the feeblest, produces 
a current of proportionate magnitude. 

According to the theory of Clausius, the decomposition and 
recomposition of the electrolyte is continually going on even when 
there is no current, and the veiy feeblest electromotive force is 
* Fogg. Ann. bd. ci. s. 338 (1857). 



310 ELECTROLYSIS. [25$. 

sufficient to give this process a certain degree of direction, and so 
to produce the currents of the ions and the electric current, which 
is part of the same phenomenon. Within the electrolyte, however, 
the ions are never set free in finite quantity, and it is this liberation 
of the ions which requires a finite electromotive force. At the 
electrodes the ions accumulate, for the successive portions of the 
ions, as they arrive at the electrodes, instead of finding molecules of 
the opposite ion ready to combine with them, are forced into com 
pany with molecules of their own kind, with which they cannot 
combine. The electromotive force required to produce this effect 
is of finite magnitude, and forms an opposing electromotive force 
which produces a reversed current when other electromotive forces 
are removed. When this reversed electromotive force, owing to 
the accumulation of the ions at the electrode, is observed, the 
electrodes are said to be Polarized. 

258.] , One of the best methods of determining whether a body 
is or is not an electrolyte is to place it between platinum electrodes 
and to pass a current through it for some time, and then, dis 
engaging the electrodes from the voltaic battery, and connecting 
them with a galvanometer, to observe whether a reverse current, 
due to polarization of the electrodes, passes through the galvano 
meter. Such a current, being due to accumulation of different 
substances on the two electrodes, is a proof that the substance has 
been electrolytically decomposed by the original current from the 
battery. This method can often be applied where it is difficult, 
by direct chemical methods, to detect the presence of the products 
of decomposition at the electrodes. See Art. 271. 

259.] So far as we have gone the theory of electrolysis appears 
very satisfactory. It explains the electric current, the nature of 
which we do not understand, by means of the currents of the 
material components of the electrolyte, the motion of which, 
though not visible to the eye, is easily demonstrated. It gives a 
clear explanation, as Faraday has shewn, why an electrolyte which 
conducts in the liquid state is a non-conductor when solidified, for 
unless the molecules can pass from one part to another no elec 
trolytic conduction can take place, so that the substance must 
be in a liquid state, either by fusion or by solution, in order to be 
a conductor. 

But if we go on, and assume that the molecules of the ions 
within the electrolyte are actually charged with certain definite 
quantities of electricity, positive and negative, so that the elec- 



260.] MOLECULAR CHARGE. 311 

trolytic current is simply a current of convection, we find that this 
tempting hypothesis leads us into very difficult ground. 

In the first place, we must assume that in every electrolyte each 
molecule of the cation, as it is liberated at the cathode, commu 
nicates to the cathode a charge of positive electricity, the amount 
of which is the same for every molecule, not only of that cation 
but of all other cations. In the same way each molecule of the 
anion when liberated, communicates to the anode a charge of 
negative electricity, the numerical magnitude of which is the same 
as that of the positive charge due to a molecule of a cation, but 
with sign reversed. 

If, instead of a single molecule, we consider an assemblage of 
molecules, constituting an electrochemical equivalent of the ion, 
then the total charge of all the molecules is, as we have seen, one 
unit of electricity, positive or negative. 

260.] We do not as yet know how many molecules there are 
in an electrochemical equivalent of any substance, but the molecular 
theory of chemistry, which is corroborated by many physical con 
siderations, supposes that the number of molecules in an elec 
trochemical equivalent is the same for all substances. We may 
therefore, in molecular speculations, assume that the number of 
molecules in an electrochemical equivalent is N, a number unknown 
at present, but which we may hereafter find means to determine *. 

Each molecule, therefore, on being liberated from the state of 

combination, parts with a charge whose magnitude is -~ , and is 

positive for the cation and negative for the anion. This definite 
quantity of electricity we shall call the molecular charge. If it 
were known it would be the most natural unit of electricity. 

Hitherto we have only increased the precision of our ideas by 
exercising our imagination in tracing the electrification of molecules 
and the discharge of that electrification. 

The liberation of the ions and the passage of positive electricity 
from the anode and into the cathode are simultaneous facts. The 
ions, when liberated, are not charged with electricity, hence, when 
they are in combination, they have the molecular charges as above 
described. 

The electrification of a molecule, however, though easily spoken 
of, is not so easily conceived. 

We know that if two metals are brought into contact at any 
* See note to Art. 5. 



312 ELECTROLYSIS. [260. 

point, the rest of their surfaces will be electrified, and if the metals 
are in the form of two plates separated by a narrow interval of air, 
the charge on each plate may become of considerable magnitude. 
Something like this may be supposed to occur when the two 
components of an electrolyte are in combination. Each pair of 
molecules may be supposed to touch at one point, and to have the 
rest of their surface charged with electricity due to the electro 
motive force of contact. 

But to explain the phenomenon, we ought to shew why the 
charge thus produced on each molecule is of a fixed amount, and 
why, when a molecule of chlorine is combined with a molecule of 
zinc, the molecular charges are the same as when a molecule of 
chlorine is combined with a molecule of copper, although the elec 
tromotive force between chlorine and zinc is much greater than 
that between chlorine and copper. If the charging of the molecules 
is the effect of the electromotive force of contact, why should 
electromotive forces of different intensities produce exactly equal 
charges ? 

Suppose, however, that we leap over this difficulty by simply 
asserting the fact of the constant value of the molecular charge, 
and that we call this constant molecular charge, for convenience in 
description, one molecule of electricity. 

This phrase, gross as it is, and out of harmony with the rest of 
this treatise, will enable us at least to state clearly what is known 
about electrolysis, and to appreciate the outstanding difficulties. 

Every electrolyte must be considered as a binary compound of 
its anion and its cation. The anion or the cation or both may be 
compound bodies, so that a molecule of the anion or the cation 
may be formed by a number of molecules of simple bodies. A 
molecule of the anion and a molecule of the cation combined to 
gether form one molecule of the electrolyte. 

In order to act as an anion in an electrolyte, the molecule which 
so acts must be charged with what we have called one molecule 
of negative electricity, and in order to act as a cation the molecule 
must be charged with one molecule of positive electricity. 

These charges are connected with the molecules only when they 
are combined as anion and cation in the electrolyte. 

When the molecules are electrolysed, they part with their charges 
to the electrodes, and appear as unelectrified bodies when set free 
from combination. 

If the same molecule is capable of acting as a cation in one 



26 1.] SECONDARY PRODUCTS OF ELECTROLYSIS. 313 

electrolyte and as an anion in another, and also of entering into 
compound bodies which are not electrolytes, then we must suppose 
that it receives a positive charge of electricity when it acts as a 
cation, a negative charge when it acts as an anion, and that it 
is without charge when it is not in an electrolyte. 

Iodine, for instance, acts as an anion in the iodides of the metals 
and in hydriodic acid, but is said to act as a cation in the bromide 
of iodine. 

This theory of molecular charges may serve as a method by 
which we may remember a good many facts about electrolysis. 
It is extremely improbable that when we come to understand the 
true nature of electrolysis we shall retain in any form the theory of 
molecular charges, for then we shall have obtained a secure basis 
on which to form a true theory of electric currents, and so become 
independent of these provisional theories. 

261.] One of the most important steps in our knowledge of 
electrolysis has been the recognition of the secondary chemical 
processes which arise from the evolution of the ions at the elec 
trodes. 

In many cases the substances which are found at the electrodes 
are not the actual ions of the electrolysis, but the products of the 
action of these ions on the electrolyte. 

Thus, when a solution of sulphate of soda is electrolysed by a 
current which also passes through dilute sulphuric acid, equal 
quantities of oxygen are given off at the anodes, and equal quan 
tities of hydrogen at the cathodes, both in the sulphate of soda 
and in the dilute acid. 

But if the electrolysis is conducted in suitable vessels, such as 
U-shaped tubes or vessels with a porous diaphragm, so that the 
substance surrounding each electrode can be examined separately, 
it is found that at the anode of the sulphate of soda there is an 
equivalent of sulphuric acid as well as an equivalent of oxygen, 
and at the cathode there is an equivalent of soda as well as two 
equivalents of hydrogen. 

It would at first sight seem as if, according to the old theory 
of the constitution of salts, the sulphate of soda were electrolysed 
into its constituents sulphuric acid and soda, while the water of the 
solution is electrolysed at the same time into oxygen and hydrogen. 
But this explanation would involve the admission that the same 
current which passing through dilute sulphuric acid electrolyses 
one equivalent of water, when it passes through solution of sulphate 



314 ELECTROLYSIS. [261. 

of soda electrolyses one equivalent of the salt as well as one equi 
valent of the water, and this would be contrary to the law of 
electrochemical equivalents. 

But if we suppose that the components of sulphate of soda are 
not SO 3 and NaO but SO 4 and Na, not sulphuric acid and soda 
but sulphion and sodium then the sulphion travels to the anode 
and is set free, but being unable to exist in a free state it breaks 
up into sulphuric acid and oxygen, one equivalent of each. At 
the same time the sodium is set free at the cathode, and there 
decomposes the water of the solution, forming one equivalent of 
soda and two of hydrogen. 

In the dilute sulphuric acid the gases collected at the electrodes 
are the constituents of water, namely one volume of oxygen and 
two volumes of hydrogen. There is also an increase of sulphuric 
acid at the anode, but its amount is not equal to an equivalent. 

It is doubtful whether pure water is an electrolyte or not. The 
greater the purity of the water., the greater the resistance to elec 
trolytic conduction. The minutest traces of foreign matter are 
sufficient to produce a great diminution of the electrical resistance 
of water. The electric resistance of water as determined by different 
observers has values so different that we cannot consider it as a 
determined quantity. The purer the water the greater its resistance, 
and if we could obtain really pure water it is doubtful whether it 
would conduct at all. 

As long as water was considered an electrolyte, and was, indeed, 
taken as the type of electrolytes, there was a strong reason for 
maintaining that it is a binary compound, and that two volumes 
of hydrogen are chemically equivalent to one volume of oxygen. 
If, however, we admit that water is not an electrolyte, we are free 
to suppose that equal volumes of oxygen and of* hydrogen are 
chemically equivalent. 

The dynamical theory of gases leads us to suppose that in perfect 
gases equal volumes always contain an equal number of molecules, 
and that the principal part of the specific heat, that, namely, which 
depends on the motion of agitation of the molecules among each 
other, is the same for equal numbers of molecules of all gases. 
Hence we are led to prefer a chemical system in which equal 
volumes of oxygen and of hydrogen are regarded as equivalent, 
and in which water is regarded as a compound of two equivalents 
of hydrogen and one of oxygen, and therefore probably not capable 
of direct electrolysis. 



262.] DYNAMICAL THEORY. 315 

While electrolysis fully establishes the close relationship between 
electrical phenomena and those of chemical combination, the fact 
that every chemical compound is not an electrolyte shews that 
chemical combination is a process of a higher order of complexity 
than any purely electrical phenomenon. Thus the combinations of 
the metals with each other, though they are good conductors, and 
their components stand at different points of the scale of electri 
fication by contact, are not, even when in a fluid state,, decomposed 
by the current. Most of the combinations of the substances which 
act as anions are not conductors, and therefore are not electrolytes. 
Besides these we have many compounds, containing the same com 
ponents as electrolytes, but not in equivalent proportions, and these 
are also non-conductors, and therefore not electrolytes. 

On the Conservation of Energy in Electrolysis. 

262.] Consider any voltaic circuit consisting partly of a battery, 
partly of a wire, and partly of an electrolytic cell. 

During the passage of unit of electricity through any section of 
the circuit, one electrochemical equivalent of each of the substances 
in the cells, whether voltaic or electrolytic, is electrolysed. 

The amount of mechanical energy equivalent to any given 
chemical process can be ascertained by converting the whole energy 
due to the process into heat, and then expressing the heat in 
dynamical measure by multiplying the number of thermal units by 
Joule s mechanical equivalent of heat. 

Where this direct method is not applicable, if we can estimate 
the heat given out by the substances taken first in the state before 
the process and then in the state after the process during their 
reduction to a final state, which is the same in both cases, then the 
thermal equivalent of the process is the difference of the two quan 
tities of heat. 

In the case in which the chemical action maintains a voltaic 
circuit, Joule found that the heat developed in the voltaic cells is 
less than that due to the chemical process within the cell, and that 
the remainder of the heat is developed in the connecting wire, or, 
when there is an electromagnetic engine in the circuit, part of the 
heat may be accounted for by the mechanical work of the engine. 

For instance, if the electrodes of the voltaic cell are first con 
nected by a short thick wire, and afterwards by a long thin wire, 
the heat developed in the cell for each grain of zinc dissolved is 
greater in the first case than the second, but the heat developed 



316 ELECTROLYSIS. 

in the wire is greater in the second case than in the first. The 
sum of the heat developed in the cell and in the wire for each grain 
of zinc dissolved is the same in both cases. This has been esta 
blished by Joule by direct experiment. 

The ratio of the heat generated in the cell to that generated 
in the wire is that of the resistance of the cell to that of the wire, 
so that if the wire were made of sufficient resistance nearly the 
whole of the heat would be generated in the wire, and if it were 
made of sufficient conducting power nearly the whole of the heat 
woula be generated in the cell. 

Let the wire be made so as to have great resistance, then the 
heat generated in it is equal in dynamical measure to the product 
of the quantity of electricity which is transmitted, multiplied by 
the electromotive force under which it is made to pass through 
the wire. 

263.] Now during the time in which an electrochemical equi 
valent of the substance in the cell undergoes the chemical process 
which gives rise to the current, one unit of electricity passes 
through the wire. Hence, the heat developed by the passage 
of one unit of electricity is in this case measured by the electro 
motive force. But this heat is that which one electrochemical 
equivalent of the substance generates, whether in the cell or in the 
wire, while undergoing the given chemical process. 

Hence the following important theorem, first proved by Thomson 
(Phil. Mag. Dec. 1851): 

The electromotive force of an electrochemical apparatus is in 
absolute measure equal to the mechanical equivalent of the chemical 
action on one electrochemical equivalent of the substance/ 

The thermal equivalents of many chemical actions have been 
determined by Andrews, Hess, Favre and Silbermann, &c., and from 
these their mechanical equivalents can be deduced by multiplication 
by the mechanical equivalent of heat. 

This theorem not only enables us to calculate from purely thermal 
data the electromotive force of different voltaic arrangements, and 
the electromotive force required to effect electrolysis in different 
cases, but affords the means of actually measuring chemical affinity. 

It has long been known that chemical affinity, or the tendency 
which exists towards the going on of a certain chemical change, 
is stronger in some cases than in others, but no proper measure 
of this tendency could be made till it was shewn that this tendency 
in certain cases is exactly equivalent to a certain electromotive 



263-] CALCULATION OF ELECTROMOTIVE FORCE. 317 

force, and can therefore be measured according to the very same 
principles used in the measurement of electromotive forces. 

Chemical affinity being therefore, in certain cases, reduced to 
the form of a measurable quantity, the whole theory of chemical 
processes, of the rate at which they go on, of the displacement of 
one substance by another, &c., becomes much more intelligible than 
when chemical affinity was regarded as a quality sui generis, and 
irreducible to numerical measurement. 

When the volume of the products of electrolysis is greater than 
that of the electrolyte, work is done during the electrolysis in 
overcoming the pressure. If the volume of an electrochemical 
equivalent of the electrolyte is increased by a volume v when 
electrolysed under a pressure p, then the work done during the 
passage of a unit of electricity in overcoming pressure is v p, and 
the electromotive force required for electrolysis must include a 
part equal to vp t which is spent in performing this mechanical 
work. 

If the products of electrolysis are gases which, like oxygen and 
hydrogen, are much rarer than the electrolyte, and fulfil Boyle s 
law very exactly, vjo will be very nearly constant for the same 
temperature, and the electromotive force required for electrolysis 
will not depend in any sensible degree on the pressure. Hetfce it 
has been found impossible to check the electrolytic decomposition 
of dilute sulphuric acid by confining the decomposed gases in a 
small space. 

When the products of electrolysis are liquid or solid the quantity 
vp will increase as the pressure increases, so that if v is positive 
an increase of pressure will increase the electromotive force required 
for electrolysis. 

In the same way, any other kind of work done during electro 
lysis will have an effect on the value of the electromotive force, 
as, for instance, if a vertical current passes between two zinc 
electrodes in a solution of sulphate of zinc a greater electromotive 
force will be required when the current in the solution flows 
upwards than when it flows downwards, for, in the first case, it 
carries zinc from the lower to the upper electrode, and in the 
second from the upper to the lower. The electromotive force 
required for this purpose is less than the millionth part of that 
of a Daniell s cell per foot. 



CHAPTER V. 



ELECTROLYTIC POLARIZATION. 



264.] WHEN an electric current is passed through an electrolyte 
bounded by metal electrodes, the accumulation of the ions at the 
electrodes produces the phenomenon called Polarization, which 
consists in an electromotive force acting in the opposite direction 
to the current, and producing an apparent increase of the resistance. 

When a continuous current is employed, the resistance appears 
to increase rapidly from the commencement of the current, and 
at last reaches a value nearly constant. If the form of the vessel 
in which the electrolyte is contained is changed, the resistance is 
altered in the same way as a similar change of form of a metallic 
conductor would alter its resistance, but an additional apparent 
resistance, depending on the nature of the electrodes, has always 
to be added to the true resistance of the electrolyte. 

265.] These phenomena have led some to suppose that there is 
a finite electromotive force required for a current to pass through 
an electrolyte. It has been shewn, however, by the researches of 
Lenz, Neumann, Beetz, Wiedemann*, Paalzowf, and recently by 
those of MM. F. Kohlrausch and W. A. NippoldtJ, that the con 
duction in the electrolyte itself obeys Ohm s Law with the same 
precision as in metallic conductors, and that the apparent resistance 
at the bounding surface of the electrolyte and the electrodes is 
entirely due to polarization. 

266.] The phenomenon called polarization manifests itself in 
the case of a continuous current by a diminution in the current, 
indicating a force opposed to the current. Resistance is also per 
ceived as a force opposed to the current, but we can distinguish 

* Galvanismus, bd. i. t Berlin Monatsbericht, July, 1868. 

Pogg, Ann. bd. cxxxviii. s. 286 (October, 1869). 



267.] DISTINGUISHED FROM RESISTANCE. 319 

between the two phenomena by instantaneously removing or re 
versing the electromotive force. 

The resisting force is always opposite in direction to the current, 
and the external electromotive force required to overcome it is 
proportional to the strength of the current, and changes its direc 
tion when the direction of the current is changed. If the external 
electromotive force becomes zero the current simply stops. 

The electromotive force due to polarization, on the other hand, 
is in a fixed direction, opposed to the current which produced it. 
If the electromotive force which produced the current is removed, 
the polarization produces a current in the opposite direction. 

The difference between the two phenomena may be compared 
with the difference between forcing a current of water through 
a long capillary tube, and forcing water through a tube of moderate 
length up into a cistern. In the first case if we remove the pressure 
which produces the flow the current will simply stop. In the 
second case, if we remove the pressure the water will begin to flow 
down again from the cistern. 

To make the mechanical illustration more complete, we have only 
to suppose that the cistern is of moderate depth, so that when a 
certain amount of water is raised into it, it begins to overflow. 
This will represent the fact that the total electromotive force due 
to polarization has a maximum limit. 

267.] The cause of polarization appears to be the existence at 
the electrodes of the products of the electrolytic decomposition of 
the fluid between them. The surfaces of the electrodes are thus 
rendered electrically different, and an electromotive force between 
them is called into action, the direction of which is opposite to that 
of the current which caused the polarization. 

The ions, which by their presence at the electrodes produce the 
phenomena of polarization, are not in a perfectly free state, but 
are in a condition in which they adhere to the surface of the 
electrodes with considerable force. 

The electromotive force due to polarization depends upon the 
density with which the electrode is covered with the ion, but it 
is not proportional to this density, for the electromotive force does 
not increase so rapidly as this density. 

This deposit of the ion is constantly tending to become free, 
and either to diffuse into the liquid, to escape as a gas, or to be 
precipitated as a solid. 

The rate of this dissipation of the polarization is exceedingly 



320 ELECTROLYTIC POLARIZATION. [268. 

small for slight degrees of polarization, and exceedingly rapid near 
the limiting value of polarization. 

268.] We have seen, Art. 262, that the electromotive force acting 
in any electrolytic process is numerically equal to the mechanical 
equivalent of the result of that process on one electrochemical 
equivalent of the substance. If the process involves a diminution 
of the intrinsic energy of the substances which take part in it, 
as in the voltaic cell, then the electromotive force is in the direction 
of the current. If the process involves an increase of the intrinsic 
energy of the substances, as in the case of the electrolytic cell, 
the electromotive force is in the direction opposite to that of the 
current, and this electromotive force is called polarization. 

In the case of a steady current in which electrolysis goes on 
continuously, and the ions are separated in a free state at the 
electrodes, we have only by a suitable process to measure the 
intrinsic energy of the separated ions, and compare it with that 
of the electrolyte in order to calculate the electromotive force 
required for the electrolysis. This will give the maximum polari 
zation. 

But during the first instants of the process of electrolysis the 
ions when deposited at the electrodes are not in a free state, and 
their intrinsic energy is less than their energy in a free state, 
though greater than their energy when combined in the electrolyte. 
In fact, the ion in contact with the electrode is in a state which 
when the deposit is very thin may be compared with that of 
chemical combination with the electrode, but as the deposit in 
creases in density, the succeeding portions are no longer so in 
timately combined with the electrode, but simply adhere to it, and 
at last the deposit, if gaseous, escapes in bubbles, if liquid, diffuses 
through the electrolyte, and if solid, forms a precipitate. 

In studying polarization we have therefore to consider 

(1) The superficial density of the deposit, which we may call 
o-. This quantity <r represents the number of electrochemical 
equivalents of the ion deposited on unit of area. Since each 
electrochemical equivalent deposited corresponds to one unit of 
electricity transmitted by the current, we may consider o- as re 
presenting either a surface-density of matter or a surface-density of 
electricity. 

(2) The electromotive force of polarization, which we may call p. 
This quantity p is the difference between the electric potentials 
of the two electrodes when the current through the electrolyte 



2/0.] DISSIPATION OF THE DEPOSIT. 321 

is so feeble that the proper resistance of the electrolyte makes no 
sensible difference between these potentials. 

The electromotive force p at any instant is numerically equal 
to the mechanical equivalent of the electrolytic process going on at 
that instant which corresponds to one electrochemical equivalent of 
the electrolyte. This electrolytic process, it must be remembered, 
consists in the deposit of the ions on the electrodes, and the state 
in which they are deposited depends on the actual state of the 
surface of the electrodes, which may be modified by previous 
deposits. 

Hence the electromotive force at any instant depends on the 
previous history of the electrode. It is, speaking very roughly, 
a function of cr, the density of the deposit, such that p when 
a = 0, but p approaches a limiting value much sooner than a- does. 
The statement, however, that p is a function of cr cannot be 
considered accurate. It would be more correct to say that p is 
a function of the chemical state of the superficial layer of the 
deposit, and that this state depends on the density of the deposit 
according to some law involving the time. 

269.] (3) The third thing we must take into account is the 
dissipation of the polarization. The polarization when left to itself 
diminishes at a rate depending partly on the intensity of the 
polarization or the density of the deposit, and partly on the nature 
of the surrounding medium, and the chemical, mechanical, or thermal 
action to which the surface of the electrode is exposed. 

If we determine a time T such that at the rate at which 
the deposit is dissipated, the whole deposit would be removed in 
a time T> we may call T the modulus of the time of dissipation. 
When the density of the deposit is very small, T is very large, 
and may be reckoned by days or months. When the density of 
the deposit approaches its limiting value T diminishes very rapidly, 
and is probably a minute fraction of a second. In fact, the rate 
of dissipation increases so rapidly that when the strength of the 
current is maintained constant, the separated gas, instead of con 
tributing to increase the density of the deposit, escapes in bubbles 
as fast as it is formed. 

270.] There is therefore a great difference between the state of 
polarization of the electrodes of an electrolytic cell when the polari 
zation is feeble, and when it is at its maximum value. For instance, 
if a number of electrolytic cells of dilute sulphuric acid with 
platinum electrodes are arranged in series, and if a small electro- 

VOL. i. y 



322 ELECTROLYTIC POLARIZATION. [ 2 7 J - 

motive force, such as that of one Daniell s cell, be made to act 
on the circuit, the electromotive force will produce a current of 
exceedingly short duration, for after a very short time the elec 
tromotive force arising from the polarization of the cell will balance 
that of the Daniell s cell. 

The dissipation will be very small in the case of so feeble a state 
of polarization, and it will take place by a very slow absorption 
of the gases and diffusion through the liquid. The rate of this 
dissipation is indicated by the exceedingly feeble current which 
still continues to flow without any visible separation of gases. 

If we neglect this dissipation for the short time during which 
the state of polarization is set up, and if we call Q the total 
quantity of electricity which is transmitted by the current during 
this time, then if A is the area of one of the electrodes, and a- 
the density of the deposit, supposed uniform, 

e = ^(r, 

If we now disconnect the electrodes of the electrolytic apparatus 
from the Daniell s cell, and connect them with a galvanometer 
capable of measuring the whole discharge through it, a quantity 
of electricity nearly equal to Q will be discharged as the polari 
zation disappears. 

271.] Hence we may compare the action of this apparatus, which 
is a form of Ritter s Secondary Pile, with that of a Leyden jar. 

Both the secondary pile and the Leyden jar are capable of being 
charged with a certain amount of electricity, and of being after 
wards discharged. During the discharge a quantity of electricity 
nearly equal to the charge passes in the opposite direction. The 
difference between the charge and the discharge arises partly from 
dissipation, a process which in the case of small charges is very 
slow, but which, when the charge exceeds a certain limit, becomes 
exceedingly rapid. Another part of the difference between the charge 
and the discharge arises from the fact that after the electrodes 
have been connected for a time sufficient to produce an apparently 
complete discharge, so that the current has completely disappeared, 
if we separate the electrodes for a time, and afterwards connect 
them, we obtain a second discharge in the same direction as the 
original discharge. This is called the residual discharge, and is a 
phenomenon of the Leyden jar as well as of the secondary pile. 

The secondary pile may therefore be compared in several respects 
to a Leyden jar. There are, however 3 certain important differences. 
The charge of a Leyden jar is very exactly proportional to the 



271.] COMPARISON WITH LEYDEN JAR. 323 

electromotive force of the charge, that is, to the difference of 
potentials of the two surfaces, and the charge corresponding to unit 
of electromotive force is called the capacity of the jar, a constant 
quantity. The corresponding quantity, which may be called the 
capacity of the secondary pile, increases when the electromotive 
force increases. 

The capacity of the jar depends on the area of the opposed 
surfaces, on the distance between them, and on the nature of the 
substance between them, but not on the nature of the metallic 
surfaces themselves. The capacity of the secondary pile depends 
on the area of the surfaces of the electrodes, but not on the distance 
between them, and it depends on the nature of the surface of the 
electrodes, as well as on that of the fluid between them. The 
maximum difference of the potentials of the electrodes in each 
element of a secondary pile is very small compared with the maxi 
mum difference of the potentials of those of a charged Leyden jar, 
so that in order to obtain much electromotive force a pile of many 
elements must be used. 

On the other hand, the superficial density of the charge in the 
secondary pile is immensely greater than the utmost superficial 
density of the charge which can be accumulated on the surfaces 
of a Leyden jar, insomuch that Mr. C. F. Varley *, in describing 
the construction of a condenser of great capacity, recommends a 
series of gold or platinum plates immersed in dilute acid as prefer 
able in point of cheapness to induction plates of tinfoil separated 
by insulating material. 

The form in which the energy of a Leyden jar is stored up 
is the state of constraint of the dielectric between the conducting 
surfaces, a state which I have already described under the name 
of electric polarization, pointing out those phenomena attending 
this state which are at present known, and indicating the im 
perfect state of our knowledge of what really takes place. See 
Arts. 62, 111. 

The form in which the energy of the secondary pile is stored 
up is the chemical condition of the material stratum at the surface 
of the electrodes, consisting of the ions of the electrolyte and the 
substance of the electrodes in a relation varying from chemical 
combination to superficial condensation, mechanical adherence, or 
simple juxtaposition. 

The seat of this energy is close to the surfaces of the electrodes, 

* Specification of C. F. Varley, Electric Telegraphs, &c., Jan. 1860. 

Y 2 



324 ELECTROLYTIC POLARIZATION. 

and not throughout the substance of the electrolyte, and the form 
in which it exists may be called electrolytic polarization. 

After studying- the secondary pile in connexion with the Leyden 
jar, the student should again compare the voltaic battery with 
some form of the electrical machine, such as that described in 
Art. 211. 

Mr. Varley has lately * found that the capacity of one square 
inch is from 175 to 542 microfarads and upwards for platinum 
plates in dilute sulphuric acid, and that the capacity increases with 
the electromotive force, being about 175 for 0.02 of a Daniell s 
cell, and 542 for 1.6 Daniell s cells. 

But the comparison between the Leyden jar and the secondary 
pile may be carried still farther, as in the following experiment, 
due to Bufff. It is only when the glass of the jar is cold that 
it is capable of retaining a charge. At a temperature below 100C 
the glass becomes a conductor. If a test-tube containing mercury 
is placed in a vessel of mercury, and if a pair of electrodes are 
connected, one with the inner and the other with the outer portion 
of mercury, the arrangement constitutes a Leyden jar which will 
hold a charge at ordinary temperatures. If the electrodes are con 
nected with those of a voltaic battery, no current will pass as long 
as the glass is cold, but if the apparatus is gradually heated a 
current will begin to pass, and will increase rapidly in intensity as 
the temperature rises, though the glass remains apparently as hard 
as ever. 

This current is manifestly electrolytic, for if the electrodes are 
disconnected from the battery, and connected with a galvanometer, 
a considerable reverse current passes, due to polarization of the 
surfaces of the glass. 

If, while the battery is in action the apparatus is cooled, the 
current is stopped by the cold glass as before, but the polarization 
of the surfaces remains. The mercury may be removed, the surfaces 
may be washed with nitric acid and with water, and fresh mercury 
introduced. If the apparatus is then heated, the current of polar 
ization appears as soon as the glass is sufficiently warm to conduct it. 

We may therefore regard glass at 100C, though apparently a 
solid body, as an electrolyte, and there is considerable reason 
to believe that in most instances in which a dielectric has a 
slight degree of conductivity the conduction is electrolytic. The 

* Proc. R. 8., Jan. 12, 1871. 

t Annalen der Chemie und Pharmacie, bd. xc. 257 (1854). 



272.] CONSTANT VOLTAIC ELEMENTS. 325 

existence of polarization may be regarded as conclusive evidence of 
electrolysis, and if the conductivity of a substance increases as the 
temperature rises, we have good grounds for suspecting that it is 
electrolytic. 

On Constant Voltaic Elements. 

272.] When a series of experiments is made with a voltaic 
battery in which polarization occurs, the polarization diminishes 
during the time that the current is not flowing, so that when 
it begins to flow again the current is stronger than after it has 
flowed for some time. If, on the other hand, the resistance of the 
circuit is diminished by allowing the current to flow through a 
short shunt, then, when the current is again made to flow through 
the ordinary circuit, it is at first weaker than its normal strength 
on account of the great polarization produced by the use of the 
short circuit. 

To get rid of these irregularities in the current, which are 
exceedingly troublesome in experiments involving exact measure 
ments, it is necessary to get rid of the polarization, or at least 
to reduce it as much as possible. 

It does not appear that there is much polarization at the surface 
of the zinc plate when immersed in a solution of sulphate of zinc 
or in dilute sulphuric acid. The principal seat of polarization is 
at the surface of the negative metal. When the fluid in which 
the negative metal is immersed is dilute sulphuric acid, it is seen 
to become covered with bubbles of hydrogen gas, arising from the 
electrolytic decomposition of the fluid. Of course these bubbles, 
by preventing the fluid from touching the metal, diminish the 
surface of contact and increase the resistance of the circuit. But 
besides the visible bubbles it is certain that there is a thin coating 
of hydrogen, probably not in a free state, adhering to the metal, 
and as we have seen that this coating is able to produce an elec 
tromotive force in the reverse direction, it must necessarily diminish 
the electromotive force of the battery. 

Various plans have been adopted to get rid of this coating of 
hydrogen. It may be diminished to some extent by mechanical 
means, such as stirring the liquid, or rubbing the surface of the 
negative plate. In Smee s battery the negative plates are vertical, 
and covered with finely divided platinum from which the bubbles of 
hydrogen easily escape, and in their ascent produce a current of 
liquid which helps to brush off other bubbles as they are formed. 

A far more efficacious method, however, is to employ chemical 



326 ELECTROLYTIC POLARIZATION. 

means. These are of two kinds. In the batteries of Grove and 
Bunsen the negative plate is immersed in a fluid rich in oxygen, 
and the hydrogen, instead of forming a coating on the plate, 
combines with this substance. In Grove s battery the plate is 
of platinum immersed in strong nitric acid. In Bunsen s first 
battery it is of carbon in the same acid. Chromic acid is also used 
for the same purpose, and has the advantage of being free from the 
acid fumes produced by the reduction of nitric acid. 

A different mode of getting rid of the hydrogen is by using 
copper as the negative metal, and covering the surface with a coat 
of oxide. This, however, rapidly disappears when it is used as 
the negative electrode. To renew it Joule has proposed to make 
the copper plates in the form of disks, half immersed in the liquid, 
and to rotate them slowly, so that the air may act on the parts 
exposed to it in turn. 

The other method is by using as the liquid an electrolyte, the 
cation of which is a metal highly negative to zinc. 

In Daniell s battery a copper plate is immersed in a saturated 
solution of sulphate of copper. When the current flows through 
the solution from the zinc to the copper no hydrogen appears on 
the copper plate, but copper is deposited on it. When the solution 
is saturated, and the current is not too strong, the copper appears 
to act as a true cation, the anion S O 4 travelling towards the zinc. 

When these conditions are not fulfilled hydrogen is evolved at 
the cathode, but immediately acts on the solution, throwing down 
copper, and uniting with S O 4 to form oil of vitriol. When this 
is the case, the sulphate of copper next the copper plate is replaced 
by oil of vitriol, the liquid becomes colourless, arid polarization by 
hydrogen gas again takes place. The copper deposited in this way 
is of a looser and more friable structure than that deposited by true 
electrolysis. 

To ensure that the liquid in contact with the copper shall be 
saturated with sulphate of copper, crystals of this substance must 
be placed in the liquid close to the copper, so that when the solution 
is made weak by the deposition of the copper, more of the crystals 
may be dissolved. 

We have seen that it is necessary that the liquid next the copper 
should be saturated with sulphate of copper. It is still more 
necessary that the liquid in which the zinc is immersed should be 
free from sulphate of copper. If any of this salt makes its way 
to the surface of the zinc it is reduced, and copper is deposited 



272.] 



THOMSON S FORM OF DANIELL S CELL. 



327 



on the zinc. The zinc, copper, and fluid then form a little circuit 
in which rapid electrolytic action goes on, and the zinc is eaten 
away by an action which contributes nothing to the useful effect 
of the battery. 

To prevent this, the zinc is immersed either in dilute sulphuric 
acid or in a solution of sulphate of zinc, and to prevent the solution 
of sulphate of copper from mixing with this liquid, the two liquids 
are separated by a division consisting of bladder or porous earthen 
ware, which allows electrolysis to take place through it, but 
effectually prevents mixture of the fluids by visible currents. 

In some batteries sawdust is used to prevent currents. The 
experiments of Graham, however, shew that the process of diffusion 
goes on nearly as rapidly when two liquids are separated by a 
division of this kind as when they are in direct contact, provided 
there are no visible currents, and it is probable that if a septum 
is employed which diminishes the diffusion, it will increase in 
exactly the same ratio the resistance of the element, because elec 
trolytic conduction is a process the mathematical laws of which 
have the same form as those of diffusion, and whatever interferes 
with one must interfere equally with the other. The only differ 
ence is that diffusion is always going on, while the current flows 
only when the battery is in action. 

In all forms of DanielFs battery the final result is that the 
sulphate of copper finds its way to the zinc and spoils the battery. 
To retard this result indefinitely, Sir W. Thomson * has constructed 
Darnell s battery in the following form. 



SIPHON- 




ELECTRGDES 



LEVEL Cf SIPHON 

ZWSO+ 1 Cu SO* 
COPPER 



Fig. 21. 

In each cell the copper plate is placed horizontally at the bottom 
* Proc. R. 8., Jan. 19, 1871. 



328 ELECTROLYTIC POLARIZATION. [272. 

and a saturated solution of sulphate of zinc is poured over it. The 
zinc is in the form of a grating and is placed horizontally near the 
surface of the solution. A glass tube is placed vertically in the 
solution with its lower end just above the surface of the copper 
plate. Crystals of sulphate of copper are dropped down this tube, 
and, dissolving in the liquid, form a solution of greater density 
than that of sulphate of zinc alone, so that it cannot get to the 
zinc except by diffusion. To retard this process of diffusion, a 
siphon, consisting of a glass tube stuffed with cotton wick, is 
placed with one extremity midway between the zinc and copper, 
and the other in a vessel outside the cell, so that the liquid is 
very slowly drawn off near the middle of its depth. To supply 
its place, water, or a weak solution of sulphate of zinc, is added 
above when required. In this way the greater part of the sulphate 
of copper rising through the liquid by diffusion is drawn off by the 
siphon before it reaches the zinc, and the zinc is surrounded by 
liquid nearly free from sulphate of copper, and having a very slow 
downward motion in the cell, which still further retards the upward 
motion of the sulphate of copper. During the action of the battery 
copper is deposited on the copper plate, and SO 4 travels slowly 
through the liquid to the zinc with which it combines, forming 
sulphate of zinc. Thus the liquid at the bottom becomes less dense 
by the deposition of the copper, and the liquid at the top becomes 
more dense by the addition of the zinc. To prevent this action 
from changing the order of density of the strata, and so producing 
instability and visible currents in the vessel, care must be taken to 
keep the tube well supplied with crystals of sulphate of copper, 
and to feed the cell above with a solution of sulphate of zinc suffi 
ciently dilute to be lighter than any other stratum of the liquid 
in the cell. 

Daniel! s battery is by no means the most powerful in common 
use. The electromotive force of Grove s cell is 192,000,000, of 
DanielPs 107,900,000 and that of Bunsen s 188,000,000. 

The resistance of Daniell s cell is in general greater than that of 
Grove s or Bunsen s of the same size. 

These defects, however, are more than counterbalanced in all 
cases where exact measurements are required, by the fact that 
Daniell s cell exceeds every other known arrangement in constancy 
of electromotive force. It has also the advantage of continuing 
in working order for a long time, and of emitting no gas. 



CHAPTER VI. 



LINEAR ELECTRIC CURRENTS. 

Chi Systems of Linear Conductors. 

273.] ANY conductor may be treated as a linear conductor if it 
is arranged so that the current must always pass in the same manner 
between two portions of its surface which are called its electrodes. 
For instance, a mass of metal of any form the surface of which is 
entirely covered with insulating material except at two places, at 
which the exposed surface of the conductor is in metallic contact 
with electrodes formed of a perfectly conducting material, may be 
treated as a linear conductor. For if the current be made to enter 
at one of these electrodes and escape at the other the lines of flow 
will be determinate, and the relation between electromotive force, 
current and resistance will be expressed by Ohm s Law, for the 
current in every part of the mass will be a linear function of E. 
But if there be more possible electrodes than two, the conductor 
may have more than one independent current through it, and these 
may not be conjugate to each other. See Art. 282. 



Law. 

274.] Let E be the electromotive force in a linear conductor 
from the electrode A l to the electrode A. 2 . (See Art. 69.) Let 
C be the strength of the electric current along the conductor, that 
is to say, let C units of electricity pass across every section in 
the direction A 1 A. 2 in unit of time, and let R be the resistance of 
the conductor, then the expression of Ohm s Law is 

E = CE. (1) 

Linear Conductors arranged in Series. 

275.] Let A lt A 2 be the electrodes of the first conductor and let 
the second conductor be placed with one of its electrodes in contact 



330 LINEAR ELECTRIC CURRENTS. [276. 

with A 2t so that the second conductor has for its electrodes A 2 , A 3 . 
The electrodes of the third conductor may be denoted by A 3 
and A 4 . 

Let the electromotive force along each of these conductors be 
denoted by JS 12 , E^ E M , and so on for the other conductors. 

Let the resistance of the conductors be 

Bl2> ^23 > ^34 > & C - 

Then, since the conductors are arranged in series so that the same 
current C flows through each, we have by Ohm s Law, 

E 12 = CR 12 , EM = CR^, EU = CR^. (2) 

If E is the resultant electromotive force, and R the resultant 
resistance of the system, we must have by Ohm s Law, 

E = CR. (3) 

NOW ^=^12 + ^3+ ^34 (4) 

the sum of the separate electromotive forces, 
= C (R 12 -f 7^ 23 + 7 34 ) by equations (2). 
Comparing this result with (3), we find 

R = RU + RK + RU> (5) 

Or, the resistance of a series of conductors is the sum of the resistances 
of the conductors taken separately. 

Potential at any Point of the Series. 

Let A and C be the electrodes of the series, B a point between 
them, a, c, and b the potentials of these points respectively. Let 
R l be the resistance of the part from A to JB, R 2 that of the part 
from B to C, and R that of the whole from A to C, then, since 

ab = R 1 C, l^c R^C, and ac RC, 
the potential at B is 



which determines the potential at B when those at A and C are 
given. 

Resistance of a Multiple Conductor. 

276.] Let a number of conductors ABZ, ACZ, ADZ be arranged 
side by side with their extremities in contact with the same two 
points A and Z. They are then said to be arranged in multiple 
arc. 

Let the resistances of these conductors be R I} R 2 , R% respect- 



2/7-] SPECIFIC RESISTANCE AND CONDUCTIVITY. 331 

ively, and the currents C lt C 2 , (7 3 , and let the resistance of the 
multiple conductor be R, and the total current C. Then, since the 
potentials at A and Z are the same for all the conductors, they have 
the same difference, which we may call E. We then have 

E = C^ 2?! = (7 2 -^2 == ^3 ^3 = dl* 

but C=C l +C 2 +C 9 , 

1111 ,-v 

=^ + ^ + ^ 3 - 

Or, M reciprocal of the resistance of a multiple conductor is the sum 
of the reciprocals of the component conductors. 

If we call the reciprocal of the resistance of a conductor the 
conductivity of the conductor, then we may say that the con 
ductivity of a multiple conductor is the sum of the conductivities of 
the component conductors. 

Current in any Branch of a Multiple Conductor. 

From the equations of the preceding article, it appears that if 
(\ is the current in any branch of the multiple conductor, and 
R l the resistance of that branch, 

4 = C, ; ... ; (8) 

where C is the total current, and R is the resistance of the multiple 
conductor as previously determined. 

Longitudinal Resistance of Conductors of Uniform Section. 

277.] Let the resistance of a cube of a given material to a current 
parallel to one of its edges be p, the side of the cube being unit of 
length, p is called the specific resistance of that material for unit 
of volume. 

Consider next a prismatic conductor of the same material whose 
length is /, and whose section is unity. This is equivalent to I 
cubes arranged in series. The resistance of the conductor is there 
fore I p. 

Finally, consider a conductor of length I and uniform section s. 
This is equivalent to s conductors similar to the last arranged in 
multiple arc. The resistance of this conductor is therefore 



When we know the resistance of a uniform wire we can determine 



332 LINEAR ELECTRIC CURRENTS. [^78. 

the specific resistance of the material of which it is made if we can 
measure its length and its section. 

The sectional area of small wires is most accurately determined 
by calculation from the length, weight, and specific gravity of the 
specimen. The determination of the specific gravity is sometimes 
inconvenient, and in such cases the resistance of a wire of unit 
length and unit mass is used as the specific resistance per unit of 
weight. 

If r is this resistance, I the length, and m the mass of a wire, then 

*.*?. 

m 

On the Dimensions of the Quantities involved in these Equations. 

278.] The resistance of a conductor is the ratio of the electro 
motive force acting on it to the current produced. The conduct 
ivity of the conductor is the reciprocal of this quantity, or in 
other words, the ratio of the current to the electromotive force 
producing it. 

Now we know that in the electrostatic system of measurement 
the ratio of a quantity of electricity to the potential of the con 
ductor on which it is spread is the capacity of the conductor, and 
is measured by a line. If the conductor is a sphere placed in an 
unlimited field, this line is the radius of the sphere. The ratio 
of a quantity of electricity to an electromotive force is therefore a 
line, but the ratio of a quantity of electricity to a current is the 
time during which the current flows to transmit that quantity. 
Hence the ratio of a current to an electromotive force is that of a 
line to a time, or in other words, it is a velocity. 

The fact that the conductivity of a conductor is expressed in the 
electrostatic system of measurement by a velocity may be verified 
by supposing a sphere of radius r charged to potential V, and then 
connected with the earth by the given conductor. Let the sphere 
contract, so that as the electricity escapes through the conductor 
the potential of the sphere is always kept equal to V. Then the 
charge on the sphere is r V at any instant, and the current is 

-j, :(rV\ but, since V is constant, the current is -IT ^i and the 

electromotive force through the conductor is V. 

The conductivity of the conductor is the ratio of the current to 

the electromotive force, or -=- , that is, the velocity with which the 



28O.] SYSTEM OF LINEAR CONDUCTORS. 333 

radius of the sphere must diminish in order to maintain the potential 
constant when the charge is allowed to pass to earth through the 
conductor. 

In the electrostatic system, therefore, the conductivity of a con 
ductor is a velocity, and of the dimensions [LT 1 ]. 

The resistance of the conductor is therefore of the dimensions 

\L-*T\. 

The specific resistance per unit of volume is of the dimension of 
\T~\j and the specific conductivity per unit of volume is of the 
dimension of [2 7 " 1 ]- 

The numerical magnitude of these coefficients depends only on 
the unit of time, which is the same in different countries. 

The specific resistance per unit of weight is of the dimensions 



279.] We shall afterwards find that in the electromagnetic 
system of measurement the resistance of a conductor is expressed 
by a velocity, so that in this system the dimensions of the resist 
ance of a conductor are [ZI 7 " 1 ]. 

The conductivity of the conductor is of course the reciprocal of 
this. 

The specific resistance per unit of volume in this system is of the 
dimensions \_L 2 T~ l } ) and the specific resistance per unit of weight 
is of the dimensions \L~ 1 T~ 1 M]. 

On Linear Systems of Conductors in general. 

280.] The most general case of a linear system is that of n 
points , AH A. 2 , ... A n , connected together in pairs by \n(n 1) 
linear conductors. Let the conductivity (or reciprocal of the re 
sistance) of that conductor which connects any pair of points, say 
A p and A q , be called K pq , and let the current from A p to A q be C pq . 
Let j^ and P q be the electric potentials at the points A p and A q 
respectively, and let the internal electromotive force, if there be 
any, along the conductor from A p to A q be E pq . 

The current from A p to A q is, by Ohm s Law, 

C M = K M (P P -P,+E M ). (1) 

Among these quantities we have the following sets of relations : 

The conductivity of a conductor is the same in either direction, 
or " K M = K qp . (2) 

The electromotive force and the current are directed quantities , 
so that E pt =-E qp , and C n =-C v . (3) 



334 LINEAR ELECTRIC CURRENTS. [280. 

Let P lt P 2 ,...P n be the potentials at A lt A 2 , ... A n respectively, 
and let Qi> Q 2 , ... Q n be the quantities of electricity which enter 
the system in unit of time at each of these points respectively. 
These are necessarily subject to the condition of continuity 

Qi+Q*.-. +<?= 0, (4) 

since electricity can neither be indefinitely accumulated nor pro 
duced within the system. 

The condition of continuity at any point A p is 

Q P = C pl +C p2 + &c. + C pn . (5) 

Substituting the values of the currents in terms of equation 
(1), this becomes 

Q p = (K fl +K ft + to i .+K r JP f -(K A P 1 + K rt P t + tos.+K,.PJ 

+ (K pq E pl + & C .+K p ^ fn ). (G) 

The symbol K pp does not occur in this equation. Let us therefore 
give it the value 

= - (*,i + K + &c. + *,) ; (7) 

that is, let K pp be a quantity equal and opposite to the sum of 
all the conductivities of the conductors which meet in A p . We 
may then write the condition of continuity for the point A p , 



fl pn -Q p . (8) 

By substituting 1, 2, &c. n for p in this equation we shall obtain 
n equations of the same kind from which to determine the n 
potentials P lf P 2 , &c., P n . 

Since, however, there is a necessary condition, (4), connecting the 
values of Q, there will be only nl independent equations. These 
will be sufficient to determine the differences of the potentials of th e 
points, but not to determine the absolute potential of any. This, 
however, is not required to calculate the currents in the system. 

If we denote by D the determinant 



D = 



(9) 



and by D pq , the minor of K m , we find for the value of P p 
(P p -P n )D= (K 1 ,S 12 +&o.-Q l )J) pl + (K 21 S. 21 + & G .-Q 2 )D l>2 

+ (K ql Z ql +& . + K qn E qn - Q t ) D m + &c. (10) 
In the same way the excess of the potential of any other point, 



28 1.] SYSTEM OF LINEAR CONDUCTORS. 335 

say A q , over that of A n may be determined. We may then de 
termine the current between A p and A q from equation (1), and so 
solve the problem completely. 

281.] We shall now demonstrate a reciprocal property of any 
two conductors of the system, answering to the reciprocal property 
we have already demonstrated for statical electricity in Art. 88. 

The coefficient of Q q in the expression for P p is ~ . That of Q p 
in the expression for P q is ^ 

Now D pq differs from D qp only by the substitution of the symbols 
such as K qp for K pq . But, by equation (2), these two symbols are 
equal, since the conductivity of a conductor is the same both ways. 
Hence D pq = D qp . (11) 

It follows from this that the part of the potential at A p arising 
from the introduction of a unit current at A q is equal to the part of 
the potential at A q arising from the introduction of a unit current 
at A p . 

We may deduce from this a proposition of a more practical form. 

Let A, , C, D be any four points of the system, and let the 
effect of a current Q, made to enter the system at A and leave it 
at B, be to make the potential at C exceed that at D by P. Then, 
if an equal current Q be made to enter the system at C and leave 
it at D, the potential at A will exceed that at B by the same 
quantity P. 

We may also establish a property of a similar kind relating to 
the effect of the internal electromotive force E rs) acting along the 
conductor which joins the points A r and A s in producing an ex 
ternal electromotive force on the conductor from A p to A q , that is 
to say, a difference of potentials P p P q . For since 

%* = *> 

the part of the value of P p which depends on this electromotive 

force is 1 

-p(D pr -D ps )E rs , 

and the part of the value of P q is 

~(D qr -D qa }E rt . 
Therefore the coefficient of E rs in the value of P p P q is 

^-{D p ^D q ,-D p ,-D qr }. (12) 

This is identical with the coefficient of E pq in the value of P r P,. 



336 LINEAR ELECTRIC CURRENTS. [282. 

If therefore an electromotive force E be introduced, acting in the 
conductor from A to B, and if this causes the potential at C to 
exceed that at D by P, then the same electromotive force E intro 
duced into the conductor from C to D will cause the potential at A 
to exceed that at J5 by the same quantity P. 

The electromotive force E may be that of a voltaic battery intro 
duced between the points named, care being taken that the resist 
ance of the conductor is the same before and after the introduction 
of the battery. 

282.] If Dfr+ D v -D pt - Dqr = o, (13) 

the conductor A p A q is said to be conjugate to A r A 8 , and we have 
seen that this relation is reciprocal. 

An electromotive force in one of two conjugate conductors pro 
duces no electromotive force or current along the other. We shall 
find the practical application of this principle in the case of the 
electric bridge. 

The theory of conjugate conductors has been investigated by 
Kirchhoff, who has stated the conditions of a linear system in the 
following manner, in which the consideration of the potential is 
avoided. 

(1) (Condition of continuity. ) At any point of the system the 
sum of all the currents which flow towards that point is zero. 

(2) In any complete circuit formed by the conductors the sum 
of the electromotive forces taken round the circuit is equal to the 
sum of the products of the current in each conductor multiplied by 
the resistance of that conductor. 

We obtain this result by adding equations of the form (1) for the 
complete circuit, when the potentials necessarily disappear. 

Heat Generated in the System. 

283.] The mechanical equivalent of the quantity of heat gene 
rated in a conductor whose resistance is R by a current C in unit of 
time is, by Art. 242, JH = C 2 . (14) 

We have therefore to determine the sum of such quantities as 
RC 2 for all the conductors of the system. 

For the conductor from A p to A q the conductivity is K pq , and the 
resistance 2t pq , where K ^ R ^ = L (15) 

The current in this conductor is, according to Ohm s Law, 

C Pq = K pq (P v -P q ). (16) 



284.] GENERATION OF HEAT. 337 

We shall suppose, however, that the value of the current is not 
that given by Ohm s Law, but X pq , where 



To determine the heat generated in the system we have to find 
the sum of all the quantities of the form 



or JH=2{X rt C* M + 2S rt C r ,Y rt + X M T*n}. (18) 

Giving C pq its value, and remembering the relation between K pq 
and fl pq , this becomes 

2(P p -P q )(C pq + 2Y fq )+K fq Y* M . (19) 

Now since both C and X must satisfy the condition of continuity 
at we have > (20) 



q p = x pl + XM+&C.+XW (21) 

therefore = Y pl + Y P2 + &c. + Y pn . (22) 

Adding together therefore all the terms of (19), we find 

2(fi JP) = 2 P, ft, + S J^IV (23) 

Now since R is always positive and Y 2 is essentially positive, the 
last term of this equation must be essentially positive. Hence the 
first term is a minimum when Y is zero in every conductor, that is, 
when the current in every conductor is that given by Ohm s Law. 

Hence the following theorem : 

284.] In any system of conductors in which there are no internal 
electromotive forces the heat generated by currents distributed in 
accordance with Ohm s Law is less than if the currents had been 
distributed in any other manner consistent with the actual con 
ditions of supply and outflow of the current. 

The heat actually generated when Ohm s Law is fulfilled is 
mechanically equivalent to 2 P p Q q , that is, to the sum of the 
products of the quantities of electricity supplied at the different 
external electrodes, each multiplied by the potential at which it is 
supplied. 



VOL. i. 



CHAPTER VII. 

CONDUCTION IN THREE DIMENSIONS. 

Notation of Electric Currents. 

285.] AT any point let an element of area dS be taken normal 
to the axis of #, and let Q units of electricity pass across this area 
from the negative to the positive side in unit of time, then, if 

J^ becomes ultimately equal to u when dSis indefinitely diminished, 

u is said to be the Component of the electric current in the direction 
of x at the given point. 

In the same way we may determine v and w, the components of 
the current in the directions of y and z respectively. 

286.] To determine the component of the current in any other 
direction OR through the given point 0. 

Let /, m, n be the direction-cosines of OR, then cutting off from 
the axes of &, y, z portions equal to 

r r r 

-j) > and - 
i m n 

respectively at A, and (7, the triangle ABC 
will be normal to OR. 

The area of this triangle ABC will be 




and by diminishing r this area may be diminished without limit. 

The quantity of electricity which leaves the tetrahedron ABCO 
by the triangle ABC must be equal to that which enters it through 
the three triangles OBC, OCA, and OAB. 

o 

The area of the triangle OBC is , and the component of 



287.] COMPONENT AND RESULTANT CURRENTS. 339 

the current normal to its plane is n, so that the quantity which 

enters through this triangle is \ r 2 

mn 

The quantities which enter through the triangles OCA and OAB 

respectively are w 

t ?* ^ > and f r* -^ 
nl Im 

If y is the component of the velocity in the direction OR, then 
the quantity which leaves the tetrahedron through ABC is 



Since this is equal to the quantity which enters through the three 
other triangles, 

i r y = i r 2$ u , v , w I. 
2 Imn 2 \ mn nl Im J 

,,. , . , 2 Imn 
multiplying by ^ > we get 

y lu + mv + nw. (1) 

If we put n 2 + v 2 + w 2 = F 2 , 

and make / , m , n such that 

n = IT, v = m r, and w = nT ; 

then ysr^ + w^ + MwO- ( 2 ) 

Hence, if we define the resultant current as a vector whose 

magnitude is T, and whose direction-cosines are / , m 9 n , and if 

y denotes the current resolved in a direction making an angle 

with that of the resultant current, then 

y = T cos 6 ; (3) 

shewing that the law of resolution of currents is the same as that 

of velocities, forces, and all other vectors. 

287.] To determine the condition that a given surface may 

be a surface of flow. 

Let F(x,y,z) = \ (4) 

be the equation of a family of surfaces any one of which is given by 

making X constant, then, if we make 



T~ 

dx 



~J~ ~J~ 

dy\ dz\ 



,-x 

^ 



the direction-cosines of the normal, reckoned in the direction in 
which X increases, are 

-.-r^A. TIT^A TIT^A. / A x 

l = N--, m^N--i n = N-- (6) 



Z 2 



340 



CONDUCTION IN THREE DIMENSIONS. 



[288. 



Hence, if y is the component of the current normal to the surface, 



-*{ 



dX dX 

u -=- 4- v -j- 

dx dy 



w 



dX\ 
dz] 



(7) 



If y = there will be no current through the surface, and the 
surface may be called a Surface of Flow, because the lines of motion 
are in the surface. 

288.] The equation of a surface of flow is therefore 

^A dX dX , ftX 

u - +v-j- +w -j- = 0. (8) 

ax ay dz 

If this equation is true for all values of A, all the surfaces of the 
family will be surfaces of flow. 

289.] Let there be another family of surfaces, whose parameter 
is A , then, if these are also surfaces of flow, we shall have 
dX dA. dX 



-j- + 
dx 



f- w 



= 
dz 



(9) 



If there is a third family of surfaces of flow, whose parameter 
is A", then dK ,, 



u 



dx 



M 



7 



\-w 



dz 



= 0. 



(10) 



Eliminating between these three equations, u, v, and w disappear 
together, and we find 



dX 
dx 
dX 
dx 
dX" 



dX 



dX 



dX" 



= 



or 



(11) 



(12) 



dX 
~dz 
dX 

dz 
dX" 
v dy dz 

X" = </> (A, A ) ; 
that is, X" is some function of A and X . 

290.] Now consider the four surfaces whose parameters are A, 
A 4- 8 A, X , and A + 5 A . These four surfaces enclose a quadrilateral 
tube, which we may call the tube 8A.8A . Since this tube is 
bounded by surfaces across which there is no flow, we may call 
it a Tube of Flow. If we take any two sections across the tube, 
the quantity which enters the tube at one section must be equal 
to the quantity which leaves it at the other, and since this quantity 
is therefore the same for every section of the tube, let us call it 
I/bX.bX where L is a function of A and A , the parameters which 
determine the particular tube. 



293-] TUBES OF FLOW. 341 

291.] If bS denotes the section of a tube of flow by a plane 

normal to a?, we have by the theory of the change of the inde 
pendent variables, 

*x *x *<^ 7A ^ /A ^^\ ns\ 

oA.oA. = oM r--)i (I*) 

\y dz dz ay 

and by the definition of the components of the current 

\ . (14) 



(15) 



. f dK d\ d\dX^ ~\ 

Hence u = L(-r 7 ~r -j- ) 

v dy dz dz dy * 

,.-..,, T /d\ d\. d\ d\\ 

similarly v = L ( -= = = 7- ) 3 

v #2 aa? fite dz 

-j- /(t A.wA. ^ZA^/A\ 

\flnf. flu flu flnr>. / 



V# 6^ dy dx 

292.] It is always possible when one of the functions A or X is 
known, to determine the other so that L may be equal to unity. 
For instance, let us take the plane of yz, and draw upon it a series 
of equidistant lines parallel to y, to represent the sections of the 
family A. by this plane. In other words, let the function X be 
determined by the condition that when x = A 7 = z. If we then 
make L = 1, and therefore (when x = 0) 



X = 

then in the plane (x = 0) the amount of electricity which passes 

through any portion will be 

/ r 

(16) 



Having determined the nature of the sections of the surfaces of 
flow by the plane of yz y the form of the surfaces elsewhere is 
determined by the conditions (8) and (9). The two functions A 
and A r thus determined are sufficient to determine the current at 
every point by equations (15), unity being substituted for L. 

On Lines of Flow. 

293.] Let a series of values of A and of A be chosen, the suc 
cessive differences in each series being unity. The two series of 
surfaces denned by these values will divide space into a system 
of quadrilateral tubes through each of which there will be a unit 
current. By assuming the unit sufficiently small, the details of 
the current may be expressed by these tubes with any desired 
amount of minuteness. Then if any surface be drawn cutting the 



342 CONDUCTION IN THREE DIMENSIONS. [ 2 94- 

system of tubes, the quantity of the current which passes through 
this surface will be expressed by the number of tubes which cut it, 
since each tube carries unity of current. 

The actual intersections of the surfaces may be called Lines of 
Flow. When the unit is taken sufficiently small, the number of 
lines of flow which cut a surface is approximately equal to the 
number of tubes of flow which cut it, so that we may consider 
the lines of flow as expressing- not only the direction of the current 
but its strength^ since each line of flow through a given section 
corresponds to a unit current. 

On Current-Sheets and Current-Functions. 

294.] A stratum of a conductor contained between two con 
secutive surfaces of flow of one system, say that of A , is called 
a Current-Sheet. The tubes of flow within this sheet are deter 
mined by the function A. If A^ and A P denote the values of A at 
the points A and P respectively, then the current from right to 
left across any line drawn on the sheet from A to P is A P A^. 
If AP be an element, ds, of a curve drawn on the sheet, the current 
which crosses this element from right to left is 

d\ , 

-j-ds. 
ds 

This function A, from which the distribution of the current in 
the sheet can be completely determined, is called the Current- 
Function. 

Any thin sheet of metal or conducting matter bounded on both 
sides by air or some other non-conducting medium may be treated 
as a current-sheet, in which the distribution of the current may 
be expressed by means of a current-function. See Art. 647. 



lion of Continuity. 

295.] If we differentiate the three equations (15) with respect to 
x, y, z respectively, remembering that L is a function of A and A , 
we find du dv dw 



The corresponding equation in Hydrodynamics is called the 
Equation of Continuity. The continuity which it expresses is 
the continuity of existence, that is, the fact that a material sub 
stance cannot leave one part of space and arrive at another, without 
going through the space between. It cannot simply vanish in the 



295-] EQUATION OF CONTINUITY. 343 

one place and appear in the other, but it must travel along a con 
tinuous path, so that if a closed surface be drawn, including the 
one place and excluding the other, a material substance in passing 
from the one place to the other must go through the closed surface. 
The most general form of the equation in hydrodynamics is 

d(pu) d(pv) d(pw) dp , } 

~fa~ ~Ji ~W ^Tt- 

where p signifies the ratio of the quantity of the substance to the 
volume it occupies, that volume being in this case the differential 
element of volume, and (pu} } (pv), and (pw) signify the ratio of the 
quantity of the substance which crosses an element of area in unit 
of time to that area, these areas being normal to the axes of #, ^, and 
z respectively. Thus understood, the equation is applicable to any 
material substance, solid or fluid, whether the motion be continuous 
or discontinuous, provided the existence of the parts of that sub 
stance is continuous. If anything, though not a substance, is 
subject to the condition of continuous existence in time and space, 
the equation will express this condition. In other parts of Physical 
Science, as, for instance, in the theory of electric and magnetic 
quantities, equations of a similar form occur. We shall call such 
equations equations of continuity to indicate their form, though 
we may not attribute to these quantities the properties of matter, 
or even continuous existence in time and space. 

The equation (17), which we have arrived at in the case of 
electric currents, is identical with (18) if we make p = 1, that is, 
if we suppose the substance homogeneous and incompressible. The 
equation, in the case of fluids, may also be established by either 
of the modes of proof given in treatises on Hydrodynamics. In 
one of these we trace the course and the deformation of a certain 
element of the fluid as it moves along. In the other, we fix our 
attention on an element of space, and take account of all that 
enters or leaves it. The former of these methods cannot be applied 
to electric currents, as we do not know the velocity with which the 
electricity passes through the body, or even whether it moves in 
the positive or the negative direction of the current. All that we 
know is the algebraical value of the quantity which crosses unit 
of area in unit of time, a quantity corresponding to (pu) in the 
equation (18). We have no means of ascertaining the value of 
either of the factors p or u, and therefore we cannot follow a par 
ticular portion of electricity in its course through the body. The 
other method of investigation, in which we consider what passes 



344 CONDUCTION IN THREE DIMENSIONS. [296. 

through the walls of an element of volume, is applicable to electric 
currents, and is perhaps preferable in point of form to that which 
we have given, but as it may be found in any treatise on Hydro 
dynamics we need not repeat it here. 

Quantity of Electricity which passes through a given Surface. 

296.] Let F be the resultant current at any point of the surface. 
Let dS be an element of the surface, and let e be the angle between 
F and the normal to the surface, then the total current through 

the surface will be r r 

J T cos e dS, 

the integration being extended over the surface. 

As in Art. 21, we may transform this integral into the form 



in the case of any closed surface, the limits of the triple integration 
being those included by the surface. This is the expression for 
the total efflux from the closed surface. Since in all cases of steady 
currents this must be zero whatever the limits of the integration, 
the quantity under the integral sign must vanish, and we obtain 
in this way the equation of continuity (17). 



CHAPTER VIII. 



RESISTANCE AND CONDUCTIVITY IN THREE DIMENSIONS. 

On tke most General Relations betiveen Current and Electro 
motive Force. 

297.] LET the components of the current at any point be u, v, w. 

Let the components of the electromotive force be X, Y, Z. 

The electromotive force at any point is the resultant force oil 
a unit of positive electricity placed at that point. It may arise 
(1) from electrostatic action, in which case if V is the potential, 

AV ar dr. m 

X= ~te Yz ~Ty -Tz 

or (2) from electromagnetic induction, the laws of which we shall 
afterwards examine ; or (3) from thermoelectric or electrochemical 
action at the point itself, tending to produce a current in a given 
direction. 

We shall in general suppose that X, F, Z represent the com 
ponents of the actual electromotive force at the point, whatever 
be the origin of the force, but we shall occasionally examine the 
result of supposing it entirely due to variation of potential. 

By Ohm s Law the current is proportional to the electromotive 
force. Hence X, Y, Z must be linear functions of u, v, w. We 
may therefore assume as the equations of Resistance, 
X= ^u+Q 3 v + P 2 w^ 

Y = P^ + ^v + Q^^ (2) 

Z 



We may call the coefficients R the coefficients of longitudinal 
resistance in the directions of the axes of coordinates. 

The coefficients P and Q may be called the coefficients of trans 
verse resistance. They indicate the electromotive force in one 
direction required to produce a current in a different direction. 



346 RESISTANCE AND CONDUCTIVITY. [298. 

If we were at liberty to assume that a solid body may be treated 
as a system of linear conductors, then, from the reciprocal property 
(Art. 281) of any two conductors of a linear system, we might shew 
that the electromotive force along z required to produce a unit current 
parallel to y must be equal to the electromotive force along y re 
quired to produce a unit current parallel to z. This would shew 
that P 1 = Q ly and similarly we should find P 2 = Q 2 , and P 3 = Q 3 . 
When these conditions are satisfied the system of coefficients is said 
to be Symmetrical. When they are not satisfied it is called a 
Skew system. 

We have great reason to believe that in every actual case the 
system is symmetrical, but we shall examine some of the con 
sequences of admitting the possibility of a skew system. 

298.] The quantities u, v, w may be expressed as linear functions 
of X, Y, Z by a system of equations, which we may call Equations 
of Conductivity, 

u r 1 X + # 3 Y+ q 2 Z, 

v =toX+r t Y+ Pl Z, (3) 

w = X 



we may call the coefficients r the coefficients of Longitudinal con 
ductivity, andjfl and q those of Transverse conductivity. 

The coefficients of resistance are inverse to those of conductivity. 
This relation may be defined as follows : 

Let [PQR] be the determinant of the coefficients of resistance, 
and [pqr] that of the coefficients of conductivity, then 

P 1 P 2 P,+ Q 1 Q 2 Q 3 +R 1 R 2 R B -P 1 Q 1 R 1 -P 2 Q 2 R 2 -P 3 Q 3 E 3) (4) 

[pqr] = PiPA + qiq<iqz + r l r 2 rs-p l q l r l -p 2 q 2 r z -p 3 q 3 r^ (5) 
[PQR] [pqr] = 1, (6) 

\_PQK\p, = (P 2 P 3 - i A), [pqr] P 1 = (ptPz-q, r,l (7) 

&c. &c. 

The other equations may be formed by altering the symbols 
P, Q, R, p, q, r, and the suffixes 1, 2, 3 in cyclical order. 

Rate of Generation of Heat. 

299.] To find the work done by the current in unit of time 
in overcoming resistance, and so generating heat, we multiply the. 
components of the current by the corresponding components of the 
electromotive force. We thus obtain the following expressions for 
W, the quantity of work expended in unit of time : 



3OO.] COEFFICIENTS OF CONDUCTIVITY. 347 



= S& 2 + R. 2 v 2 +R 3 w 2 + (Pi + <3i) vw + (P 2 + Qa) ww + (P 3 



uv 



By a proper choice of axes, either of the two latter equations may 
be deprived of the terms involving the products of u, v, w or of 
X, Yj Z. The system of axes, however, which reduces W to the form 



is not in general the same as that which reduces it to the form 



It is only when the coefficients P 15 P 2 , P 3 are equal respectively 
to Q 19 Q 2 , <2 3 that the two systems of axes coincide. 
If with Thomson * we write 



and p -s + t, 

then we have 



= S-Tn 

= s t; ) 



and [PQR] r, = R. 2 R 3 -S^+T^, x 

-f^^-^^, (13) 



If therefore we cause S lf S 2 , S 3 to disappear, ^ will not also dis 
appear unless the coefficients T are zero. 

Condition of Stability. 

300.] Since the equilibrium of electricity is stable, the work 
spent in maintaining the current must always be positive. The 
conditions that W must be positive are that the three coefficients 
R 19 R 2 , RV and the three expressions 



must all be positive. 

There are similar conditions for the coefficients of conductivity. 



* Tram. R. S. Edin., 1853-4, p. 165. 



348 RESISTANCE AND CONDUCTIVITY. [3O1. 

Equation of Continuity in a Homogeneous Medium. 

301.] If we express the components of the electromotive force 
as the derivatives of the potential F, the equation of continuity 

du dv div 

-T- + ~r + -r = 

ax ay az 
becomes in a homogeneous medium 

d 2 F d 2 F d 2 7 d 2 7 d 2 F d 2 7 



<i -T-o- 2-r-5- -T5- < iT T 2^~j- 3^~r 
1 das 2 2 dy 2 dz 2 1 dy dz L dzdx das ay 

If the medium is not homogeneous there will be terms arising 
from the variation of the coefficients of conductivity in passing 
from one point to another. 

This equation corresponds to Laplace s equation in an isotropic 
medium. 

302.] If we put 

\rs\ = T! r 2 r 3 + 2 s l s 2 ^-^ s^ -r 2 s 2 2 -r 3 5 3 2 , (17) 

and [AS] = A 1 A 2 A 3 +2 1 3 2 3 3 -A 1 $ l *-A 2 2 2 -A 3 B 3 2 , (18) 

where \rs] A t = r 2 r 3 

(19) 

and so on, the system A, B will be inverse to the system /, s, and 
if we make 

A l x 2 ^A 2 y 2 ^-A^z^ + 2B l yz + 2B z zx-}-2B z xy = \AS\ p 2 , (20) 
we shall find that 

F=^i (21) 

4 77 p 

is a solution of the equation. 

In the case in which the coefficients T are zero, the coefficients A 
and B become identical with R and 8. When T exists this is not 
the case. 

In the case therefore of electricity flowing out from a centre in an 
infinite homogeneous, but not isotropic, medium, the equipotential 
surfaces are ellipsoids, for each of which p is constant. The axes of 
these ellipsoids are in the directions of the principal axes of con 
ductivity, and these do not coincide with the principal axes of 
resistance unless the system is symmetrical. 

By a transformation of this equation we may take for the axes 
of x, y, z the principal axes of conductivity. The coefficients of the 
forms * and B will then be reduced to zero, and each coefficient 



303.] SKEW SYSTEM. 349 

of the form A will be the reciprocal of the corresponding 1 coeffi 
cient of the form r. The expression for p will be 

^ + ^ + *! = ^!_. (22) 



303.] The theory of the complete system of equations of resist 
ance and of conductivity is that of linear functions of three vari 
ables, and it is exemplified in the theory of Strains *, and in other 
parts of physics. The most appropriate method of treating it is 
that by which Hamilton and Tait treat a linear and vector function 
of a vector. We shall not, however, expressly introduce Quaternion 
notation. 

The coefficients T ly T 2 , T 3 may be regarded as the rectangular 
components of a vector T, the absolute magnitude and direction 
of which are fixed in the body, and independent of the direction of 
the axes of reference. The same is true of ^, 2 , # 3 , which are the 
components of another vector t. 

The vectors T and t do not in general coincide in direction. 

Let us now take the axis of z so as to coincide with the vector 
T, and transform the equations of resistance accordingly. They 
will then have the form 

X = XiU + SsV+Ssio-Tv, 

Y = SsU+R^v + S^ + Tu, (23) 

Z = Su + S 



It appears from these equations that we may consider the elec 
tromotive force as the resultant of two forces, one of them depending 
only on the coefficients R and S, and the other depending on T alone. 
The part depending on E and S is related to the current in the 
same way that the perpendicular on the tangent plane of an 
ellipsoid is related to the radius vector. The other part, depending 
on T, is equal to the product of T into the resolved part of the 
current perpendicular to the axis of T, and its direction is per 
pendicular to T and to the current, being always in the direction in 
which the resolved part of the current would lie if turned 90 in 
the positive direction round T. 

Considering the current and T as vectors, the part of the 
electromotive force due to T is the vector part of the product, 
T x current. 

The coefficient T may be called the Rotatory coefficient. We 

* See Thomson and Tail s Natural Philosophy. 154. 



350 RESISTANCE AND CONDUCTIVITY. [304. 

have reason to believe that it does not exist in any known sub 
stance. It should be found, if anywhere, in magnets, which have 
a polarization in one direction, probably due to a rotational phe 
nomenon in the substance. 

304.] Let us next consider the general characteristic equation 
ofF, 

d dV dV dV d , dV dV dV 



d dv 



where the coefficients of conductivity p, q, r may have any positive 
values, continuous or discontinuous, at any point of space, and V 
vanishes at infinity. 

Also, let #, 6, c be three functions of x, y, z satisfying the condition 
da db dc 



dV dV dV 

and let a r, -= -f p~ -=- -f 9 - 

dx A dy a dz 

__ dF dF dF 
dx 2 dy dz 

dV dV dV 



(26) 



Finally, let the triple-integral 

be extended over spaces bounded as in the enunciation of Art. 97, 
where the coefficients P, Q, R are the coefficients of resistance. 

Then W will have a unique minimum value when a, b, c are such 
that u } v, w are each everywhere zero, and the characteristic equation 
(24) will therefore, as shewn in Art. 98, have one and only one 
solution. 

In this case W represents the mechanical equivalent of the heat 
generated by the current in the system in unit of time, and we have 
to prove that there is one way, and one only, of making this heat 
a minimum, and that the distribution of currents (a be) in that case 
is that which arises from the solution of the characteristic equation 
of the potential V. 

The quantity W may be written in terms of equations (25) and (26), 



305.] EXTENSION OF THOMSON S THEOREM. 351 

? clTf d7* 
\^Ty\^T Z 
VdV dV dV 






Since + + ! = o, (29) 

dk ay <z 

the third term of W vanishes within the limits. 

The second term, being the rate of conversion of electrical energy 
into heat, is also essentially positive. Its minimum value is zero, 
and this is attained only when u, v, and w are everywhere zero. 

The value of W is in this case reduced to the first term, and is 
then a minimum and a unique minimum. 

305.] As this proposition is of great importance in the theory of 
electricity, it may be useful to present the following proof of the 
most general case in a form free from analytical operations. 

Let us consider the propagation of electricity through a conductor 
of any form, homogeneous* or heterogeneous. 

Then we know that 

(1) If we draw a line along the path and in the direction of 
the electric current, the line must pass from places of high potential 
to places of low potential. 

(2) If the potential at every point of the system be altered in 
a given uniform ratio, the currents will be altered in the same ratio, 
according to Ohm s Law. 

(3) If a certain distribution of potential gives rise to a certain 
distribution of currents, and a second distribution of potential gives 
rise to a second distribution of currents, then a third distribution in 
which the potential is the sum or difference of those in the first 
and second will give rise to a third distribution of currents, such 
that the total current passing through a given finite surface in the 
third case is the sum or difference of the currents passing through 
it in the first and second cases. For, by Ohm s Law, the additional 
current due to an alteration of potentials is independent of the 
original current due to the original distribution of potentials. 

(4) If the potential is constant over the whole of a closed surface, 



352 RESISTANCE AND CONDUCTIVITY. [3O5- 

and if there are no electrodes or intrinsic electromotive forces 
within it, then there will be no currents within the closed surface, 
and the potential at any point within it will be equal to that at the 
surface. 

If there are currents within the closed surface they must either 
be closed curves, or they must begin and end either within the 
closed surface or at the surface itself. 

But since the current must pass from places of high to places of 
low potential, it cannot flow in a closed curve. 

Since there are no electrodes within the surface the current 
cannot begin or end within the closed surface, and since the 
potential at all points of the surface is the same, there can be 
no current along lines passing from one point of the surface to 
another. 

Hence there are no currents within the surface, and therefore 
there can be no difference of potential, as such a difference would 
produce currents, and therefore the potential within the closed 
surface is everywhere the same as at the surface. 

(5) If there is no electric current through any part of a closed 
surface, and no electrodes or intrinsic electromotive forces within 
the surface, there will be no currents within the surface, and the 
potential will be uniform. 

We have seen that the currents cannot form closed curves, or 
begin or terminate within the surface, and since by the hypothesis 
they do not pass through the surface, there can be no currents, and 
therefore the potential is constant. 

(6) If the potential is uniform over part of a closed surface, and 
if there is no current through the remainder of the surface, the 
potential within the surface will be uniform for the same reasons. 

(7) If over part of the surface of a body the potential of every 
point is known, and if over the rest of the surface of the body the 
current passing through the surface at each point is known, then 
only one distribution of potentials at points within the body can 
exist. 

For if there were two different values of the potential at any 
point within the body, let these be F L in the first case and F 2 in 
the second case, and let us imagine a third case in which the 
potential of every point of the body is the excess of potential in the 
first case over that in the second. Then on that part of the surface 
for which the potential is known the potential in the third case will 
be zero, and on that part of the surface through which the currents 



306.] RESISTANCE OF A WIRE OF VARIABLE SECTION. 353 

are known the currents in the third case will be zero, so that by 
(6) the potential everywhere within the surface will be zero, or 
there is no excess of V^ over F 9 , or the reverse. Hence there is 
only one possible distribution of potentials. This proposition is 
true whether the solid be bounded by one closed surface or by 
several. 

On the Approximate Calculation of the Resistance of a Conductor 
of a given Form. 

306.] The conductor here considered has its surface divided into 
three portions. Over one of these portions the potential is main 
tained at a constant value. Over a second portion the potential has 
a constant value different from the first. The whole of the remainder 
of the surface is impervious to electricity. "We may suppose the 
conditions of the first and second portions to be fulfilled by applying 
to the conductor two electrodes of perfectly conducting material, 
and that of the remainder of the surface by coating it with per 
fectly non-conducting material. 

Under these circumstances the current in every part of the 
conductor is simply proportional to the difference between the 
potentials of the electrodes. Calling this difference the electro 
motive force, the total current from the one electrode to the other 
is the product of the electromotive force by the conductivity of the 
conductor as a whole, and the resistance of the conductor is the 
reciprocal of the conductivity. 

It is only when a conductor is approximately in the circumstances 
above defined that it can be said to have a definite resistance, or 
conductivity as a whole. A resistance coil, consisting of a thin 
wire terminating in large masses of copper, approximately satisfies 
these conditions, for the potential in the massive electrodes is nearly 
constant, and any differences of potential in different points of the 
same electrode may be neglected in comparison with the difference 
of the potentials of the two electrodes. 

A very useful method of calculating the resistance of such con 
ductors has been given, so far as I know, for the first time, by 
the Hon. J. W. Strutt, in a paper on the Theory of Resonance*. 

It is founded on the following considerations. 

If the specific resistance of any portion of the conductor be 
changed, that of the remainder being unchanged, the resistance of 

* Phil. Trans., 1871, p. 77. See Art. 102. 
VOL. I. A a 



354 RESISTANCE AND CONDUCTIVITY. [306. 

the whole conductor will be increased if that of the portion is 
increased, and diminished if that of the portion be diminished. 

This principle may be regarded as self-evident, but it may easily 
be shewn that the value of the expression for the resistance of a 
system of conductors between two points selected as electrodes, 
increases as the resistance of each member of the system in 
creases. 

It follows from this that if a surface of any form be described 
in the substance of the conductor, and if we further suppose this 
surface to be an infinitely thin sheet of a perfectly conducting 
substance, the resistance of the conductor as a whole will be 
diminished unless the surface is one of the equipotential surfaces 
in the natural state of the conductor, in which case no effect will 
be produced by making it a perfect conductor, as it is already in 
electrical equilibrium. 

If therefore we draw within the conductor a series of surfaces, 
the first of which coincides with the first electrode, and the last 
with the second, while the intermediate surfaces are bounded by 
the non-conducting surface and do not intersect each other, and 
if we suppose each of these surfaces to be an infinitely thin sheet 
of perfectly conducting matter, we shall have obtained a system 
the resistance of which is certainly not greater than that of the 
original conductor, and is equal to it only when the surfaces we 
have chosen are the natural equipotential surfaces. 

To calculate the resistance of the artificial system is an operation 
of much less difficulty than the original problem. For the resist 
ance of the whole is the sum of the resistances of all the strata 
contained between the consecutive surfaces, and the resistance of 
each stratum can be found thus : 

Let dS be an element of the surface of the stratum, v the thick 
ness of the stratum perpendicular to the element, p the specific 
resistance, E the difference of potential of the perfectly conducting 
surfaces, and dC the current through dS, then 

dC=EdS, (1) 

pv 

and the whole current through the stratum is 

L4 (2) 

pv 

the integration being extended over the whole stratum bounded by 
the non-conducting surface of the conductor. 



306.] RESISTANCE OP A WIRE OF VARIABLE SECTION. 355 
Hence the conductivity of the stratum is 

" ; " I =//>* " " <> 

and the resistance of the stratum is the reciprocal of this quantity. 

If the stratum be that bounded by the two surfaces for which 
the function F has the values F and F+ dF respectively, then 

(5) 



and the resistance of the stratum is 

dF 



1 

P 

To find the resistance of the whole artificial conductor, we have 
only to integrate with respect to F, and we find 




The resistance R of the conductor in its natural state is greater 
than the value thus obtained, unless all the surfaces we have chosen 
are the natural equipotential surfaces. Also, since the true value 
of R is the absolute maximum of the values of R^ which can thus 
be obtained, a small deviation of the chosen surfaces from the true 
equipotential surfaces will produce an error of R which is com 
paratively small. 

This method of determining a lower limit of the value of the 
resistance is evidently perfectly general, and may be applied to 
conductors of any form, even when p, the specific resistance, varies 
in any manner within the conductor. 

The most familiar example is the ordinary method of determining 
the resistance of a straight wire of variable section. In this case 
the surfaces chosen are planes perpendicular to the axis of the 
wire, the strata have parallel faces, and the resistance of a stratum 
of section S and thickness ds is 

**!=* .,..... ( ) 

and that of the whole wire of length s is 



where S is the transverse section and is a function of s. 

A a 2 



356 RESISTANCE AND CONDUCTIVITY. [37- 

This method in the case of wires whose section varies slowly 
with the length gives a result very near the truth, but it is really 
only a lower limit, for the true resistance is always greater than 
this, except in the case where the section is perfectly uniform. 

307.] To find the higher limit of the resistance, let us suppose 
a surface drawn in the conductor to be rendered impermeable to 
electricity. The effect of this must be to increase the resistance of 
the conductor unless the surface is one of the natural surfaces of 
flow. By means of two systems of surfaces we can form a set of 
tubes which will completely regulate the flow, and the effect, if 
there is any, of this system of impermeable surfaces must be to 
increase the resistance above its natural value. 

The resistance of each of the tubes may be calculated by the 
method already given for a fine wire, and the resistance of the 
whole conductor is the reciprocal of the sum of the reciprocals of 
the resistances of all the tubes. The resistance thus found is greater 
than the natural resistance, except when the tubes follow the 
natural lines of flow. 

In the case already considered, where the conductor is in the 
form of an elongated solid of revolution, let us measure as along the 
axis, and let the radius of the section at any point be 6. Let one 
set of impermeable surfaces be the planes through the axis for each 
of which $ is constant, and let the other set be surfaces of revolution 
for which ^ _ ^ 2j ( 9 ) 

where ty is a numerical quantity between and 1 . 

Let us consider a portion of one of the tubes bounded by the 
surfaces <p and $ + ^$, \j/ and \l/ + d\js, x and x+dx. 

The section of the tube taken perpendicular to the axis is 

ydyd$ = \Wdty d$. (10) 

If 6 be the angle which the tube makes with the axis 



*. - - (11) 

The true length of the element of the tube is dx sec 0, and its 
true section is * i <ty ty cos 0, 

so that its resistance is 



T , A dx a 

Let A=, and 



307.] HIGHER AND LOWER LIMITS. 357 

the integration being extended over the whole length, x, of the 
conductor, then the resistance of the tube d\\r d$ is 



. 

and its conductivity is 



To find the conductivity of the whole conductor, which is the 
sum of the conductivities of the separate tubes, we must integrate 
this expression between $ = and $ = 277, and between \j/ = 
and \fr = I . The result is 



which may be less, but cannot be greater, than the true con 
ductivity of the conductor. 

When -y- is always a small quantity -j will also be small, and we 
may expand the expression for the conductivity, thus 

F=l(-4+t|-4>"> 

The first term of this expression, , is that which we should 

A. 

have found by the former method as the superior limit of the con 
ductivity. Hence the true conductivity is less than the first term 
but greater than the whole series. The superior value of the 
resistance is the reciprocal of this, or 



If, besides supposing the flow to be guided by the surfaces < and 
\ff, we had assumed that the flow through each tube is proportional 
to d\j/ d$, we should have obtained as the value of the resistance 
under this additional constraint 

(17) 

which is evidently greater than the former value, as it ought to be, 
on account of the additional constraint. In Mr. Strutt s paper this 
is the supposition made, and the superior limit of the resistance 
there given has the value (17), which is a little greater than that 
which we have obtained in (16). 



358 KESISTANCE AND CONDUCTIVITY. [38. 

308.] We shall now apply the same method to find the correction 
which must be applied to the length of a cylindrical conductor of 
radius a when it? extremity is placed in metallic contact with a 
massive electrode, which we may suppose of a different metal. 

For the lower limit of the resistance we shall suppose that an 
infinitely thin disk of perfectly conducting matter is placed between 
the end of the cylinder and the massive electrode, so as to bring 
the end of the cylinder to one and the same potential throughout. 
The potential within the cylinder will then be a function of its 
length only, and if we suppose the surface of the electrode where 
the cylinder meets it to be approximately plane, and all its dimen 
sions to be large compared with the diameter of the cylinder, the 
distribution of potential will be that due to a conductor in the form 
of a disk placed in an infinite medium. See Arts. 152, 177. 

If E is the difference of the potential of the disk from that of 
the distant parts of the electrode, C the^current issuing from the 
surface of the disk into the electrode, and /o the specific resistance 
of the electrode, p Q. a R (18) 

Hence, if the length of the wire from a given point to the 
electrode is L, and its specific resistance p, the resistance from that 
point to any point of the electrode not near the junction is 



TT a 4 $ 
and this may be written 

JZ = JL(Z+^), (19) 

na* p 4 

where the second term within brackets is a quantity which must 
be added to the length of the cylinder or wire in calculating its 
resistance, and this is certainly too small a correction. 

To understand the nature of the outstanding error we may 
observe, that whereas we have supposed the flow in the wire up 
to the disk to be uniform throughout the section, the flow from 
the disk to the electrode is not uniform, but is at any point in 
versely proportional to the minimum chord through that point. In 
the actual case the flow through the disk will not be uniform, 
but it will not vary so much from point to point as in this supposed 
case. The potential of the [disk in the actual case will not be 
uniform, but will diminish from the middle to the edge. 

309.] We shall next determine a quantity greater than the true 
resistance by constraining the flow through the disk to be uniform 



309.] CORRECTION FOR THE ENDS OF THE WIRE. 359 

at every point. We may suppose electromotive forces introduced 
for this purpose acting perpendicular to the surface of the disk. 

The resistance within the wire will be the same as before, but 
in the electrode the rate of generation of heat will be the surface- 
integral of the product of the flow into the potential. The rate of 

ri 

flow at any point is - 2 , and the potential is the same as that of 
an electrified surface whose surface-density is o-, where 



. 

p being the specific resistance. 

We have therefore to determine the potential energy of the 
electrification of the disk with the uniform surface-density o-. 

The potential at the edge of a disk of uniform density a- is easily 
found to be 4cr. The work done in adding a strip of breadth 
da at the circumference of the disk is 2 -naa-da . lav, and the 
whole potential energy of the disk is the integral of this, 

or P= a* a*. (21) 

o 

In the case of electrical conduction the rate at which work is 
done in the electrode whose resistance is R f is 

C*R=^P, (22) 

P 
whence, by (20) and (21), 



and the correction to be added to the length of the cylinder is 

P 8 

7 Si* 

this correction being greater than the true value. The true cor- 

f 

rection to be added to the length is therefore an, where n is a 

o P 

number lying- between - and , or between 0.785 and 0.849. 
4 3?r 

Mr. Strutt, by a second approximation, has reduced the superior 
limit of n to 0.8282. 



\\ 



CHAPTER IX. 

CONDUCTION THROUGH HETEROGENEOUS MEDIA. 

On the Conditions to be Fulfilled at the Surface of Separation 
between Two Conducting Media. 

310.] THERE are two conditions which the distribution of currents 
must fulfil in general, the condition that the potential must be 
continuous, and the condition of continuity of the electric currents. 

At the surface of separation between two media the first of these 
conditions requires that the potentials at two points on opposite 
sides of the surface, but infinitely near each other, shall be equal. 
The potentials are here understood to be measured by an elec 
trometer put in connexion with the given point by means of an 
electrode of a given metal. If the potentials are measured by the 
method described in Arts. 222,, 246, where the electrode terminates 
in a cavity of the conductor filled with air, then the potentials at 
contiguous points of different metals measured in this way will 
differ by a quantity depending on the temperature and on the 
nature of the two metals. 

The other condition at the surface is that the current through 
any element of the surface is the same when measured in either 
medium. 

Thus, if Fj and F 2 are the potentials in the two media, then at 
any point in the surface of separation 

7i = r,, a) 

and if n lf v lt w and u 2 , v 2) w 2 are the components of currents in the 
two media, and I, m, n the direction-cosines of the normal to the 
surface of separation, 

% I + #! m -f W-L n = u. 2 l-\-v 2 m + w%n. (2) 

In the most general case the components n, v, w are linear 



310.] SURFACE-CONDITIONS. 361 

functions of the derivatives of F t the forms of which are given in 
the equations 



u = 



v = q 3 X+r 2 Y+p l Z,> (3) 

w = p 2 X+ q 1 Y+ r 3 Z,) 

where X, Y, Z are the derivatives of V with respect to x, y, z 
respectively. 

Let us take the case of the surface which separates a medium 
having these coefficients of conduction from an isotropic medium 
having a coefficient of conduction equal to r. 

Let X , Y } Z be the values of X, Y, Z in the isotropic medium, 
then we have at the surface 

r=r, (4) 

or Xdx+Ydy + Zdz = X dx+Y dy + Z dz, (5) 

when Idx + mdy + ndz = 0. (6) 

This condition gives 

X =X+47TO-^ 7 = 7+4770-02, Z = Z+lll<Tn, (?) 

where a- is the surface-density. 

We have also in the isotropic medium 

u =rX , v =rY , w =rZ , (8) 

and at the boundary the condition of flow is 

u l-\-tfm + w n r= ul + vm + wn, (9) 

or r(lX+mY+nZ+ir<T} 
= l(r 1 X+psY+ c h Z)+m(^X+ r 2 Y+^Z) + Q 2 X+ q 1 Y+ r 3 Z), (10) 

whence 

477 or r = (l(ri r) + mq 3 + np^ X+ (Ip 3 + m(r 2 r) + nq^Y 

+ (lq 2 + mj) 1 + n(r2-r))Z. (11) 

The quantity cr represents the sui-face-density of the charge 
on the surface of separation. In crystallized and organized sub 
stances it depends on the direction of the surface as well as on 
the force perpendicular to it. In isotropic substances the coeffi 
cients^; and q are zero, and the coefficients r are all equal, so that 

4770-= (-^f- l)(lX+m7+nZ), (12) 

where r x is the conductivity of the substance, r that of the external 
medium, and I, m, n the direction-cosines of the normal drawn 
towards the medium whose conductivity is r. 

When both media are isotropic the conditions may be greatly 



362 CONDUCTION IN HETEROGENEOUS MEDIA. [31 1. 

simplified, for if k is the specific resistance per unit of volume, then 

1 dV 1 dV l dV 

u= r -j-> #=____, w , (13) 

k dx k dy k dz 

and if v is the normal drawn at any point of the surface of separation 
from the first medium towards the second, the conduction of con 
tinuity is 1 dV^ 1 dF 2 

&! dv ~ k 2 dv 

If 0j and 6 2 are the angles which the lines of flow in the first and 
second media respectively make with the normal to the surface 
of separation, then the tangents to these lines of flow are in the 
same plane with the normal and on opposite sides of it, and 

#! tan 6 1 = k 2 tan 2 . (15) 

This may be called the law of refraction of lines of flow. 

311.] As an example of the conditions which must be fulfilled 
when electricity crosses the surface of separation of two media, 
let us suppose the surface spherical and of radius #, the specific 
resistance being ^ within and Jc% without the surface. 

Let the potential, both within and without the surface, be ex 
panded in solid harmonics, and let the part which depends on 
the surface harmonic 8 i be 



t (2) 

within and without the sphere respectively. 

At the surface of separation where r = a we must have 

F 1= r 2 and J-^ f^. (3) 

^ dr k 2 dr 

From these conditions we get the equations 






(*) 



J. At 

These equations are sufficient, when we know two of the four 
quantities A 19 A 2 , B^ B 2 , to deduce the other two. 

Let us suppose A^ and B l known, then we find the following 
expressions for A 2 and B, 



(5) 



= 



312.] SPHERICAL SHELL. 363 

In this way we can find the conditions which each term of the 
harmonic expansion of the potential must satisfy for any number of 
strata bounded by concentric spherical surfaces. 

312.] Let us suppose the radius of the first spherical surface 
to be # 15 and let there be a second spherical surface of radius a 2 
greater than a lf beyond which the specific resistance is 3 . If there 
are no sources or sinks of electricity within these spheres there 
will be no infinite values of T, and we shall have B l = 0. 

We then find for A 3 and .Z? 3 , the coefficients for the outer medium, 



*u.:-| 

(6) 



1) 2 = [&(+ 



The value of the potential in the outer medium depends partly 
on the external sources of electricity, which produce currents in 
dependently of the existence of the sphere of heterogeneous matter 
within, and partly on the disturbance caused by the introduction of 
the heterogeneous sphere. 

The first part must depend on solid harmonics of positive degrees 
only, because it cannot have infinite values within the sphere. The 
second part must depend on harmonics of negative degrees, because 
it must vanish at an infinite distance from the centre of the sphere. 

Hence the potential due to the external electromotive forces must 
be expanded in a series of solid harmonics of positive degree. Let 
A 3 be the coefficient of one these, of the form 

44* 

Then we can find J. 19 the corresponding coefficient for the inner 
sphere by equation (6), and from this deduce A%, -Z? 2 > an ^ ^3- Of 
these _Z? 3 represents the effect on the potential in the outer medium 
due to the introduction of the heterogeneous spheres. 

Let us now suppose 3 = k , so that the case is that of a hollow 
shell for which k = k 2 , separating an inner from an outer portion of 
the same medium for which k=Jc l . 

If we put 

1 

(2i^\^k l k^i(i + i}(k 2 -k l Y(\-^ 



364: CONDUCTION IN HETEROGENEOUS MEDIA. [3*3- 

then A 1 = ^^ 2 (2i+l) 2 CA 3 , 



The difference between A 3 the undisturbed coefficient, and A 1 its 
value in the hollow within the spherical shell, is 

A 3 -A 1 = (k 2 ~k^ i(i+i)(i- (^) 2 * hl ) CA 3 . (8) 

Since this quantity is always positive whatever be the values 
of k and & 2 , it follows that, whether the spherical shell conducts 
better or worse than the rest of the medium, the electrical action 
within the shell is less than it would otherwise be. If the shell 
is a better conductor than the rest of the medium it tends to 
equalize the potential all round the inner sphere. If it is a worse 
conductor, it tends to prevent the electrical currents from reaching 
the inner sphere at all. 

The case of a solid sphere may be deduced from this by making 
#! = 0, or it may be worked out independently. 

313.] The most important term in the harmonic expansion is 
that in which i = 1, for which 



c= 



= 3& 2 (2 h 



(9) 



The case of a solid sphere of resistance k 2 may be deduced from 
this by making a 1 = 0. We then have 



(10) 



It is easy to shew from the general expressions that the value 
of B 9 in the case of a hollow sphere having a nucleus of resistance 
u surrounded by a shell of resistance k 2 , is the same as that of 
a uniform solid sphere of the radius of the outer surface, and of 
resistance K, where 




314.] MEDIUM CONTAINING SMALL SPHERES. 365 

314.] If there are n spheres of radius a^ and resistance A lt placed 
in a medium whose resistance is k 2) at such distances from each 
other that their effects in disturbing the course of the current 
may be taken as independent of each other, then if these spheres 
are all contained within a sphere of radius # 2 , the potential at a 

great distance from the centre of this sphere will be of the form 

I 

where the value of B is 

B = AzA a* A. (13) 

The ratio of the volume of the n small spheres to that of the 
sphere which contains them is 

na-, 3 



The value of the potential at a great distance from the sphere 
may therefore be written 



Now if the whole sphere of radius a. 2 had been made of a material 
of specific resistance K, we should have had 



That the one expression should be equivalent to the other, 

2*! + 
= 



This, therefore, is the specific resistance of a compound medium 
consisting of a substance of specific resistance 2 , in which are 
disseminated small spheres of specific resistance & lt the ratio of the 
volume of all the small spheres to that of the whole being p. In 
order that the action of these spheres may not produce effects 
depending on their interference, their radii must be small compared 
with their distances, and therefore p must be a small fraction. 

This result may be obtained in other ways, but that here given 
involves only the repetition of the result already obtained for a 
single sphere. 

When the distance between the spheres is not great compared 

% _ fc 

with their radii, and when ~ - j- is considerable, then other 

2 #! -f- # 2 

terms enter into the result, which we shall not now consider. 
In consequence of these terms certain systems of arrangement of 



366 CONDUCTION IN HETEROGENEOUS MEDIA. 

the spheres cause the resistance of the compound medium to be 
different in different directions. 

Application of the Principle of Images. 

315.] Let us take as an example the case of two media separated 
by a plane surface, and let us suppose that there is a source 8 
of electricity at a distance a from the plane surface in the first 
medium, the quantity of electricity flowing from the source in unit 
of time being S. 

If the first medium had been infinitely extended the current 
at any point P would have been in the direction SP, and the 

potential at P would have been - - where E = - and ^ = SP. 

In the actual case the conditions may be satisfied by taking 
a point 7, the image of S in the second medium, such that 7$ 
is normal to the plane of separation and is bisected by it. Let r 2 
be the distance of any point from 7, then at the surface of separation 

r-, = r 



25 



_ 

~dv~ ~~dv 

Let the potential 7^ at any point in the first medium be that 
due to a quantity of electricity E placed at S, together with an 
imaginary quantity E 2 at 7, and let the potential F 2 at any point 
of the second medium be that due to an imaginary quantity E l at 
8, then if 

(3) 



the superficial condition T = F 2 gives 

and the condition 

1 rlV. 1 dV- 

(5) 




"i 

f\ 7_ 7 T 

whence E l = 

The potential in the first medium is therefore the same as would 
be produced in air by a charge E placed at S, and a charge E^ 
at 7 on the electrostatic theory, and the potential in the second 
medium is the same as that which would be produced in air by 
a charge E l at S. 



317.] STRATUM WITH PARALLEL SIDES. 367 

The current at any point of the first medium is the same as would 

k.k 
have been produced by the source S together with a source y -^ S 

placed at J if the first medium had been infinite, and the current 
at any point of the second medium is the same as would have been 

2 k S 
produced by a source -r-, ,-r placed at S if the second medium had 

(A + #2) 
been infinite. 

We have thus a complete theory of electrical images in the case 
of two media separated by a plane boundary. Whatever be the 
nature of the electromotive forces in the first medium, the potential 
they produce in the first medium may be found by combining their 
direct effect with the effect of their image. 

If we suppose the second medium a perfect conductor, then 
2 = 0, and the image at / is equal and opposite to the course 
at S. This is the case of electric images, as in Thomson s theory 
in electrostatics. 

If we suppose the second medium a perfect insulator, then 
& 2 = oc, and the image at / is equal to the source at S and of the 
same sign. This is the case of images in hydrokinetics when the 
fluid is bounded by a rigid plane surface. 

316.] The method of inversion, which is of so much use in 
electrostatics when the bounding surface is supposed to be that 
of a perfect conductor, is not applicable to the more general case 
of the surface separating two conductors of unequal electric resist 
ance. The method of inversion in two dimensions is, however, 
applicable, as well as the more general method of transformation in 
two dimensions given in Art. 190 *. 

Conduction through a Plate separating Two Media. 

317.] Let us next consider the effect of a plate of thickness AB of 
a medium whose resist 
ance is 2 , and separating ^ 
two media whose resist 
ances are ^ and / 3 , in ~ J~~ J~ 
altering the potential due 
to a source S in the first 
medium. 

The potential will be Fi s- 23 - 

* See Kirchhoff, Pogg. Ann. Ixiv. 497, and Ixvii. 344 ; Quincke, Pogg. xcvii. 382 ; 
and Smith, Proc. R. S. Edin., 1869-70, p. 79. 



- 



368 CONDUCTION IN HETEROGENEOUS MEDIA. [318. 

equal to that due to a system of charges placed in air at certain 
points along the normal to the plate through S. 

Make 

AI=SA, BI^SB, AJi=IiA 9 BI^J^B, AJ 2 = I 2 A, &c. ; 
then we have two series of points at distances from each other equal 
to twice the thickness of the plate. 

318.] The potential in the first medium at any point P is equal to 



that at a point P* in the second 

J^_ 

PI + 



_ 

P S + PI + p 7 / + ~FI + 



and that at a point P" in the third 



where /, / , &c. represent the imaginary charges placed at the 
points 7, &c., and the accents denote that the potential is to be 
taken within the plate. 

Then, by the last Article, for the surface through A we have, 

Z. _ 7. o 7. 

/= ^_^1 E > = J^2 K 

^2 + ^1 * a + *i 

For the surface through B we find 



Similarly for the surface through A again, 

j-f _ ^l~^2 jr T_ 2 &1 j 
1 ~~ Je _U Jf l l ~ Tf JLb 

K \ "r *i K \ *<~ K 2 

and for the surface through B, 

If If <>Jc 



If we make _ &i&2 - / _ 

*l "f" *8 

we find for the potential in the first medium, 

r 



&c - 

-. (15) 



319.] STRATIFIED CONDUCTORS. 369 

For the potential in the third medium we find 



If the first medium is the same as the third, then k = 3 and 
p = //, and the potential on the other side of the plate will be 



If the plate is a very much better conductor than the rest of the 
medium, p is very nearly equal to 1 . If the plate is a nearly perfect 
insulator, p is nearly equal to 1, and if the plate differs little in 
conducting power from the rest of the medium, p is a small quantity 
positive or negative. 

The theory of this case was first stated by Green in his ( Theory 
of Magnetic Induction (Essay, p. 65). His result, however, is 
correct only when p is nearly equal to 1 *. The quantity g which 
he uses is connected with p by the equations 

2p _ #! #a 3ff ._^i~4 m 

9 ~ 3-p ~~ 1 +2/ 2 > P ~~ 2+ff~~ k-L + Jc^ 

p 
If we put p = - , we shall have a solution of the problem of 

1 + 27TK 

the magnetic induction excited by a magnetic pole in an infinite 
plate whose coefficient of magnetization is K. 

On Stratified Conductors. 

319.] Let a conductor be composed of alternate strata of thick 
ness c and <f of two substances whose coefficients of conductivity 
are different. Required the coefficients of resistance and conduc 
tivity of the compound conductor. 

Let the plane of the strata be normal to Z. Let every symbol 
relating to the strata of the second kind be accented, and let 
every symbol relating to the compound conductor be marked with 
a bar thus, X. Then 

X=X=X , (c+c }u = cu + c u, 
Y=Y = Y , (c + c )v = cv + c v ; 
(c -\-c )~Z = cZ+ c Z , w = w w . 

We must first determine , u , #, ?/, Z and Z in terms of 
X, 7 and w from the equations of resistance, Art. 297, or those 

* See Sir W. Thomson s Note on Induced Magnetism in a Plate, Canib. and 
Dub. Math. Journ., Nov. 1845, or Reprint, art. ix. 156. 

VOL. I. B b 



370 CONDUCTION IN HETEROGENEOUS MEDIA. [320. 

of conductivity, Art. 298. If we put D for the determinant of the 
coefficients of resistance, we find 

ur^D = R 2 X- Q 3 Y+wq 2 D, 
v r 3 D = R! Y- P 3 X + wft I), 



Similar equations with the symbols accented give the values 
of u, v and /. Having found u, v and ~w in terms of X, F and ^ 
we may write down the equations of conductivity of the stratified 

conductor. If we make h = and //= . we find 





h + h 1*= 



= 

__ 
_ 



( Pl - 



c + c f 



320.] If neither of the two substances of which the strata are 
formed has the rotatory property of Art. 303, the value of any 
P or p will be equal to that of its corresponding Q or q. From 
this it follows that in the stratified conductor also 

or there is no rotatory property developed by stratification, unless 
it exists in the materials. 

321.] If we now suppose that there is no rotatory property, and 
also that the axes of x, y and z are the principal axes, then the 
p and q coefficients vanish, and 

7* 2 + 6V/ C + C 

7 ) 

C + C 




322.] STRATIFIED CONDUCTORS, 371 

If we begin with both substances isotropic, but of different 
conductivities, then the result of stratification will be to make 
the resistance greatest in the direction of a normal to the strata, 
and the resistance in all directions in the plane of the strata will 
be equal. 

322.] Take an isotropic substance of conductivity r, cut it into 
exceedingly thin slices of thickness a, and place them alternately 
with slices of a substance whose conductivity is s, and thickness 
ka. 

Let these slices be normal to x. Then cut this compound con 
ductor into thicker slices, of thickness Z>, normal to y> and alternate 
these with slices whose conductivity is s and thickness Jc. 2 b. 

Lastly, cut the new conductor into still thicker slices, of thick 
ness c, normal to ^, and alternate them with slices whose con 
ductivity is s and thickness 3 c. 

The result of the three operations will be to cut the substance 
whose conductivity is r into rectangular parallelepipeds whose 
dimensions are , b and c 9 where b is exceedingly small compared 
with c } and a is exceedingly small compared with b, and to embed 
these parallelepipeds in the substance whose conductivity is s, so 
that they are separated from each other l\a in the direction of x> 
2 b in that of y } and 3 c in that of z. The conductivities of the 
conductor so formed in the directions of x, y and z are 



_ 



3 

The accuracy of this investigation depends upon the three 
dimensions of .the parallelepipeds being of different orders of mag 
nitude, so that we may neglect the conditions to be fulfilled at 
their edges and angles. If we make k l} k 2 and 3 each unity, then 

3r+5s 



If r 0, that is, if the medium of which the parallelepipeds 
are made is a perfect insulator, then 



= f 



B b 2 



372 CONDUCTION IN HETEKOGENEOUS MEDIA. [3 2 3- 

If r = oo, that is, if the parallelepipeds are perfect conductors, 
r i = i*> r z = %*> r 3 = 2s. 

In every case, provided ^ = & 2 = 3 , it may be shewn that 
r l9 r 2 an( l r s are i n ascending order of magnitude, so that the 
greatest conductivity is in the direction of the longest dimensions 
of the parallelepipeds, and the greatest resistance in the direction 
of their shortest dimensions. 

323.] In a rectangular parallelepiped of a conducting solid, let 
there be a conducting channel made from one angle to the opposite, 
the channel being a wire covered with insulating material, and 
let the lateral dimensions of the channel be so small that the 
conductivity of the solid is not affected except on account of the 
current conveyed along the wire. 

Let the dimensions of the parallelepiped in the directions of the 
coordinate axes be <z, 6, <?, and let the conductivity of the channel, 
extending from the origin to the point (adc), be abcK. 

The electromotive force acting between the extremities of the 
channel is aX+bY+cZ, 

and if C be the current along the channel 

C = Kalc(aX+bY+cZ). 

The current across the face be of the parallelepiped is dcu, and 
this is made up of that due to the conductivity of the solid and 
of that due to the conductivity of the channel, or 

bcu = bc 
or a = (r l 

In the same way we may find the values of v and w. The 
coefficients of conductivity as altered by the effect of the channel 
will be 



In these expressions, the additions to the values of p lt &c., due 
to the effect of the channel, are equal to the additions to the values 
of q lt &c. Hence the values of p-^ and q cannot be rendered 
unequal by the introduction of linear channels into every element 
of volume of the solid, and therefore the rotatory property of 
Art. 303, if it does not exist previously in a solid, cannot be 
introduced by such means. 



324-] COMPOSITE CONDUCTOR. 373 

324.] To construct a framework of linear conductors which shall 
have any given coefficients of conductivity forming a symmetrical 
system. 

Let the space be divided into equal small 
cubes, of which let the figure represent one. 
Let the coordinates of the points 0, L y M, N, 



ntials 


be as 
x 


A 

follows : 
y z Potential. 


\L/ 




i/\ 




N M 

















Fig. 24. 




L 





1 


1 


V + Y+Z, 






M 


1 





1 


Q+Z+X, 






N 


1 


1 





0+X+Y. 







Let these four points be connected by six conductors, 

OL, OH, ON, 3IN, NL, LM, 
of which the conductivities are respectively 

A, JS, C, P, Q 9 R. 
The electromotive forces along these conductors will be 
Y+Z, Z+X, X+Y, Y-Z, Z-X, X-Y, 
and the currents 

A(Y+Z), 3 (Z+X), C(X+Y), P(Y-Z), Q(Z-X), R(X-Y). 
Of these currents, those which convey electricity in the positive 
direction of x are those along LM, LN, OH and ON, and the 
quantity conveyed is 

= (B 
Similarly 
v = (C-R}X 

w = (3-Q)X +(A-P)Y 

whence we find by comparison with the equations of conduction, 
Art. 298, 

4: A = r 2 + r s -r 1 + 2 d p 1 , 4P = 

= r B + r 1 -r 2 + 2p 2 , 4 Q = 

4 G = 



CHAPTER X. 



CONDUCTION IN DIELECTRICS. 

325.] WE have seen that when electromotive force acts on a 
dielectric medium it produces in it a state which we have called 
electric polarization, and which we have described as consisting* 
of electric displacement within the medium in a direction which, 
in isotropic media,, coincides with that of the electromotive force, 
combined with a superficial charge on every element of volume 
into which we may suppose the dielectric divided, which is negative 
on the side towards which the force acts, and positive on the side 
from which it acts. 

When electromotive force acts on a conducting medium it also 
produces what is called an electric current. 

Now dielectric media, with very few, if any, exceptions, are 
also more or less imperfect conductors, and many media which are 
not good insulators exhibit phenomena of dielectric induction. 
Hence we are led to study the state of a medium in which induction 
and conduction are going on at the same time. 

-For simplicity we shall suppose the medium isotropic at every 
point, but not necessarily homogeneous at different points. In this 
case, the equation of Poisson becomes, by Art. 83, 
d (v dV^ d f^dV^ d , v dY^ 

?^*^<*^*C^)*-*^ a) 

where K is the specific inductive capacity. 

The * equation of continuity of electric currents becomes 

i<iiS\:L- (l*I\ d ( idr. d p _ 

dx V ifoJ + dy V dy> + dz V fa) ~ Tt (} 

where r is the specific resistance referred to unit of volume. 

When K or r is discontinuous, these equations must be trans 
formed into those appropriate to surfaces of discontinuity. 



326.] THEORY OF A CONDENSER. 375 

In a strictly homogeneous medium r and K are both constant, so 
that we find 

d*V d*V d*V P dp , ox 

- 7 - T +-7^+- r =-47r-^=r : , (3) 

dx 2 dj/ 2 dz 2 K at 

-*Z t 
whence p = Ce Kr ; (4) 

Kr -L 

or, if we put T= , p Ce ?. (5) 

This result shews that under the action of any external electric 
forces on a homogeneous medium, the interior of which is originally 
charged in any manner with electricity, the internal charges will 
die away at a rate which does not depend on the external forces, 
so that at length there will be no charge of electricity within 
the medium, after which no external forces can either produce or 
maintain a charge in any internal portion of the medium, pro 
vided the relation between electromotive force, electric polarization 
and conduction remains the same. When disruptive discharge 
occurs these relations cease to be true, and internal charge may 
be produced. 

On Conduction through a Condenser. 

326.] Let C be the capacity of a condenser, R its resistance, and 
E the electromotive force which acts on it, that is, the difference of 
potentials of the surfaces of the metallic electrodes. 

Then the quantity of electricity on the side from which the 
electromotive force acts will be CE, and the current through the 
substance of the condenser in the direction of the electromotive 

E 

force will be -^> 
H 

If the electrification is supposed to be produced by an electro 
motive force E acting in a circuit of which the condenser forms 

part, and if -^ represents the current in that circuit, then 

9-!+"- 

Let a battery of electromotive force E Q and resistance i\ be 
introduced into this circuit, then 

, 

Hence, at any time t lt 

(8) 




376 CONDUCTION IN DIELECTRICS. [327. 

Next, let the circuit r be broken for a time t 2 , 

_^_ 
E(=E^=E^e T Z w here T 2 = CR. (9) 

Finally, let the surfaces of the condenser be connected by means 
of a wire whose resistance is r 3 for a time t z , 

E(=E 3 ) = E 2 e-% where T, = |^A. (10) 

If Qs is the total discharge through this wire in the time 3 , 



In this way we may find the discharge through a wire which 
is made to connect the surfaces of a condenser after being charged 
for a time t lt and then insulated for a time t 2 . If the time of 
charging is sufficient, as it generally is, to develope the whole 
charge, and if the time of discharge is sufficient for a complete 
discharge, the discharge is 

-*- 



3.27.] In a condenser of this kind, first charged in any way, next 
discharged through a wire of small resistance, and then insulated, 
no new electrification will appear. In most actual condensers, 
however, we find that after discharge and insulation a new charge 
is gradually developed, of the same kind as the original charge, 
but inferior in intensity. This is called the residual charge. To 
account for it we must admit that the constitution of the dielectric 
medium is different from that which we have just described. We 
shall find, however, that a medium formed of a conglomeration of 
small pieces of different simple media would possess this property. 

Theory of a Composite Dielectric. 

328.] We shall suppose, for the sake of simplicity, that the 
dielectric consists of a number of plane strata of different materials 
and of area unity, and that the electric forces act in the direction 
of the normal to the strata. 

Let a l9 #2> &c. be the thicknesses of the different strata. 

X lt X 2 , &c. the resultant electrical force within each stratum. 

fli,p2> & c ^ ne current due to conduction through each stratum. 

fi>fz> & c - ^ ne electric displacement. 

u lt ^ 2 , &c. the total current, due partly to conduction and partly 
to variation of displacement. 



328.] STRATIFIED DIELECTRIC. 377 

r 1} r. 2 , &c. the specific resistance referred to unit of volume. 

K 1} K 2 , &c. the specific inductive capacity. 

15 2 , &c. the reciprocal of the specific inductive capacity. 

E the electromotive force due to a voltaic battery, placed in 
the part of the circuit leading from the last stratum towards the 
first, which we shall suppose good conductors. 

Q the total quantity of electricity which has passed through this 
part of the circuit up to the time t. 

E Q the resistance of the battery with its connecting wires. 

o-^ the surface-density of electricity on the surface which separates 
the first and second strata. 

Then in the first stratum we have, by Ohm s Law, 



By the theory of electrical displacement, 

*,= 4V1. (2) 

By the definition of the total current, 



_ 

with similar equations for the other strata, in each of which the 
quantities have the suffix belonging to that stratum. 

To determine the surface-density on any stratum, we have an 
equation of the form ^ f f 9 / 4 ) 

and to determine its variation we have 

f/0- 19 ,r\ 

-=*-* 

By differentiating (4) with respect to z5, and equating the result 
to (5), we obtain 

o 

- = ,sa 7> (6) 



or, by taking account of (3), 

u^ = u 2 = &c. = u. (7) 

That is, the total current u is the same in all the strata, and is 
equal to the current through the wire and battery. 
We have also, in virtue of equations (1) and (2), 
1 . 1 dX, 

u = ^^ + j^^ 

from which we may find X l by the inverse operation on u, 



di 



378 CONDUCTION IN DIELECTRICS. [329. 

The total electromotive force E is 

E = a 1 X 1 + a 2 X 2 + &Lc. ) (10) 



an equation between E, the external electromotive force, and u, the 
external current. 

If the ratio of r to k is the same in all the strata, the equation 
reduces itself to 

j (12) 



which is the case we have already examined, and in which, as we 
found, no phenomenon of residual charge can take place. 

If there are n substances having different ratios of r to k, the 
general equation (11), when cleared of inverse operations, will be 
a linear differential equation, of the nth order with respect to E 
and of the (n l)th order with respect to u, t being the independent 
variable. 

From the form of the equation it is evident that the order of 
the different strata is indifferent, so that if there are several strata 
of the same substance we may suppose them united into one 
without altering the phenomena. 

329.] Let us now suppose that at first fi,f 2) &c. are all zero, 
and that an electromotive force E is suddenly made to act, and let 
us find its instantaneous effect. 

Integrating (8) with respect to t, we find 



q = udt = TXi dt + -j- X 1 + const. (13) 

Now, since X x is always in this case finite, / X dt, must be in 

sensible when t is insensible, and therefore, since X is originally 
zero, the instantaneous effect will be 

X l = 47i^Q. (14) 
Hence, by equation (10), 

E= 47r( 1 tf 1 + / 2 tf 2 + &c.), (15) 

and if C be the electric capacity of the system as measured in this 
instantaneous way, 

__ Q __ _ 1 (16) 



E 4w( 1 1 + 2 a + &c.) 



329.] ELECTRIC ABSORPTION/ 379 

This is the same result that we should have obtained if we had 
neglected the conductivity of the strata. 

Let us next suppose that the electromotive force E is continued 
uniform for an indefinitely long time, or till a uniform current of 
conduction equal top is established through the system. 

We have then X 1 = i\p, and therefore 

E = (y 1 fl 1 + /2 2 + &c.) J p. (17) 

If R be the total resistance of the system, 

" P ~ 
In this state we have by (2), 



so that ^(L.- __),, ... (19 ) 

If we now suddenly connect the extreme strata by means of a 
conductor of small resistance, E will be suddenly changed from its 
original value E to zero, and a quantity Q of electricity will pass 
through the conductor. 

To determine Q we observe that if Xf be the new value of X l , 
then by (13), j-/= X 1 + 4 77 ^ Q. (20) 

Hence, by (10), putting E = 0, 

= ^ X l + &c. + 4 77 (a 1 k\ + a z k. 2 + &c.) Q, (21) 

or = ^ + -^ Q. (22) 

Hence Q = C?^ where (7 is the capacity, as given by equation 
(16). The instantaneous discharge is therefore equal to the in 
stantaneous charge. 

Let us next suppose the connexion broken immediately after this 
discharge. We shall then have u = 0, so that by equation (8), 



Xi = X e i , (23) 

where X is the initial value after the discharge. 
Hence, at any time t, 



The value of E at any time is therefore 



380 CONDUCTION IN DIELECTKICS. [33- 

and the instantaneous discharge after any time t is EC. This is 
called the residual discharge. 

If the ratio of r to k is the same for all the strata, the value of E 
will be reduced to zero. If, however, this ratio is not the same, let 
the terms be arranged according to the values of this ratio in 
descending order of magnitude. 

The sum of all the coefficients is evidently zero, so that when 
t = 0, E = 0. The coefficients are also in descending order of 
magnitude, and so are the exponential terms when t is positive. 
Hence, when t is positive, E will be positive, so that the residual 
discharge is always of the same sign as the primary discharge. 

When t is indefinitely great all the terms disappear unless any 
of the strata are perfect insulators, in which case r is infinite for 
that stratum/ and R is infinite for the whole system, and the final 
value of E is not zero but 

E = ^ (l-47ra 1 ^ 1 (7). (25) 

Hence, when some, but not all, of the strata are perfect insulators, 
a residual discharge may be permanently preserved in the system. 

330.] "We shall next determine the total discharge through a wire 
of resistance R Q kept permanently in connexion with the extreme 
strata of the system, supposing the system first charged by means 
of a long-continued application of the electromotive force E. 

At any instant we have 

E= a 1 r l p l + azr 2 p 2 + &c. + JR w = 0, (26) 



and also, by (3), u=^ L + . (27) 

Hence (R + R ) * = i *i ^ + V 2 %& +&c. (28) 

Integrating with respect to t in order to find Q, we get 

(R + JR )Q = ! r, (// -/J + a 2 r 2 (/ 2 -/ 2 ) + &c., (29) 

where f^ is the initial, and/ 1 / the final value ofj^. 

In this case // = 0, and /, = E, ( - ?) 



Hence (R + BJ Q = + + &<s. -3 > CX, (30) 



where the summation is extended to all quantities of this form 
belonging to every pair of strata. 



331.] RESIDUAL DISCHARGE. 381 

It appears from this that Q is always negative, that is to say, in 
the opposite direction to that of the current employed in charging 1 
the system. 

This investigation shews that a dielectric composed of strata of 
different kinds may exhibit the phenomena known as electric 
absorption and residual discharge, although none of the substances 
of which it is made exhibit these phenomena when alone. An 
investigation of the cases in which the materials are arranged 
otherwise than in strata would lead to similar results, though 
the calculations would be more complicated, so that we may 
conclude that the phenomena of electric absorption may be ex 
pected in the case of substances composed of parts of different 
kinds, even though these individual parts should be microscopically 
small. 

It by no means follows that every substance which exhibits this 
phenomenon is so composed, for it may indicate a new kind of 
electric polarization of which a homogeneous substance may be 
capable, and this in some cases may perhaps resemble electro 
chemical polarization much more than dielectric polarization. 

The object of the investigation is merely to point out the true 
mathematical character of the so-called electric absorption, and to 
shew how fundamentally it differs from the phenomena of heat 
which seem at first sight analogous. 

331.] If we take a thick plate of any substance and heat it 
on one side, so as to produce a flow of heat through it, and if 
we then suddenly cool the heated side to the same temperature 
as the other, and leave the plate to itself, the heated side of the 
plate will again become hotter than the other by conduction from 
within. 

Now an electrical phenomenon exactly analogous to this can 
be produced, and actually occurs in telegraph cables, but its mathe 
matical laws, though exactly agreeing with those of heat, differ 
entirely from those of the stratified condenser. 

In the case of heat there is true absorption of the heat into 
the substance with the result of making it hot. To produce a truly 
analogous phenomenon in electricity is impossible, but we may 
imitate it in the following way in the form of a lecture-room 
experiment. 

Let A lt A 29 &c. be the inner conducting surfaces of a series of 
condensers, of which H Q , lt H. 2 , &c. are the outer surfaces. 

Let A 19 A 2 , &c. be connected in series by connexions of resist- 



382 



CONDUCTION IN DIELECTRICS. 



[33 r - 



ance R, and let a current be passed along this series from left to 
right. 

Let us first suppose the plates B Q , R lf 2 , each insulated and 
free from charge. Then the total quantity of electricity on each of 
the plates B must remain zero, and since the electricity on the 
plates A is in each case equal and opposite to that of the opposed 

A 



Fig. 25. 

surface they will not be electrified, and no alteration of the current 
will be observed. 

But let the plates B be all connected together, or let each be 
connected with the earth. Then, since the potential of A l is 
positive, while that of the plates B is zero, A l will be positively 
electrified and B 1 negatively. 

If PU P 2) &c. are the potentials of the plates A lt A 2 , &c., and C 
the capacity of each, and if we suppose that a quantity of electricity 
equal to Q passes through the wire on the left, Q l through the 
connexion R^ and so on, then the quantity which exists on the 
plate A l is Q Q 1 , and we have 



Similarly Qi Q: 

and so on. 

But by Ohm s Law we have 



If we suppose the values of C the same for each plate, and those 
of R the same for each wire, we shall have a series of equations of 
the form 



332.] THEORY OP ELECTRIC CABLES. 383 



If there are n quantities of electricity to be determined, and if 
either the total electromotive force, or some other equivalent con 
ditions be given, the differential equation for determining any one 
of them will be linear and of the nth order. 

By an apparatus arranged in this way, Mr. Varley succeeded in 
imitating the electrical action of a cable 12,000 miles long. 

When an electromotive force is made to act along the wire on 
the left hand, the electricity which flows into the system is at first 
principally occupied in charging the different condensers beginning 
with A I} and only a very small fraction of the current appears 
at the right hand till a considerable time has elapsed. If galvano 
meters be placed in circuit at S 19 jR. 2 , &c. they will be affected 
by the current one after another, the interval between the times of 
equal indications being greater as we proceed to the right. 

332.] In the case of a telegraph cable the conducting wire is 
separated from conductors outside by a cylindrical sheath of gutta- 
percha, or other insulating material. Each portion of the cable 
thus becomes a condenser, the outer surface of which is always at 
potential zero. Hence, in a given portion of the cable, the quantity 
of free electricity at the surface of the conducting wire is equal 
to the product of the potential into the capacity of the portion of 
the cable considered as a condenser. 

If a 1} a 2 are the outer and inner radii of the insulating sheath, 
and if K is its specific dielectric capacity, the capacity of unit of 
length of the cable is, by Art. 126, 

* = --. CD 



Let v be the potential at any point of the wire, which we may 
consider as the same at every part of the same section. 

Let Q be the total quantity of electricity which has passed 
through that section since the beginning of the current. Then the 
quantity which at the time t exists between sections at x and at 
r, is n $n 



and this is, by what we have said, equal to cvbx. 



384 CONDUCTION IN DIELECTRICS. [333- 

Hence cv=-^. (2) 

clx 

Again, the electromotive force at any section is --, and by 

Ohm s Law, ^ ^Q 

__ = -J|, (3) 

dx dt 

where k is the resistance of unit of length of the conductor, and 

-~^ is the strength of the current. Eliminating Q between (2) and 
dt 

(3), we find , dv d 2 v ,.^ 

C/C ~j~ = "7 n (*) 

dt d& 

This is the partial differential equation which must be solved 
in order to obtain the potential at any instant at any point of the 
cable. It is identical with that which Fourier gives to determine 
the temperature at any point of a stratum through which heat 
is flowing in a direction normal to the stratum. In the case of 
heat c represents the capacity of unit of volume, or what Fourier 
calls CD, and k represents the reciprocal of the conductivity. 

If the sheath is not a perfect insulator, and if k is the resist 
ance of unit of length of the sheath to conduction through it in a 
radial direction, then if p is the specific resistance of the insulating 

material, r 

*i=2 Pl log e f. (5) 

2 

The equation (2) will no longer be true, for the electricity is 
expended not only in charging the wire to the extent represented 

v 
by cv, but in escaping at a rate represented by -y- . Hence the rate 

of expenditure of electricity will be 

dv_ 1_ , 

dt + 



whence, by comparison with (3), we get 
,dv 



f . 

-^ 

and this is the equation of conduction of heat in a rod or ring 
as given by Fourier*. 

333.] If we had supposed that a body when raised to a high 
potential becomes electrified throughout its substance as if elec 
tricity were compressed into it, we should have arrived at equa 
tions of this very form. It is remarkable that Ohm himself, 

* Theorie de la Chaleur, art. 105- 



334-] 



HYDROSTATICAL ILLUSTRATION. 



385 



misled by the analogy between electricity and heat, entertained 
an opinion of this kind, and was thus, by means of an erroneous 
opinion, led to employ the equations of Fourier to express the 
true laws of conduction of electricity through a long wire, long 
before the real reason of the appropriateness of these equations had 
been suspected. 

Mechanical Illustration of the Properties of a Dielectric. 

334.] Five tubes of equal sectional area A, B, C, D and P are 
arranged in circuit as in the figure. 
A, B, C and D are vertical and equal, 
and P is horizontal. 

The lower halves of A } B, C, D 
are filled with mercury, their upper 
halves and the horizontal tube P are 
filled with water. 

A tube with a stopcock Q con 
nects the lower part of A and B 
with that of C and D, and a piston 
P is made to slide in the horizontal 
tube. 

Let us begin by supposing that 
the level of the mercury in the four 
tubes is the same, and that it is 
indicated by A Q , B Q , (7 , D Q) that 
the piston is at P , and that the 
stopcock Q is shut. 

Now let the piston be moved from P to P l} a distance a. Then, 
since the sections of all the tubes are equal, the level of the mercury 
in A and C will rise a distance a, or to A and C lt and the mercury 
in B and D will sink an equal distance a, or to B^ and D 1 . 

The difference of pressure on the two sides of the piston will 
be represented by 4#. 

This arrangement may serve to represent the state of a dielectric 
acted on by an electromotive force 4 a. 

The excess of water in the tube D may be taken to represent 
a positive charge of electricity on one side of the dielectric, and the 
excess of mercury in the tube A may represent the negative charge 
on the other side. The excess of pressure in the tube P on the 
side of the piston next D will then represent the excess of potential 
on the positive side of the dielectric. 

VOL. i. c c 



s i^\ 

f p p p X 

/ ! f \ 


-A - 




(: 


. - -^ 


^ 

- c - 






-i 








* 






-A - 




-B - 










2 




8 










~ A 0- 




~ B 0~ 




v 




- - 










- C a- 




*.- 






-a - 








-D - 






/ 








i 

t 




Q 

Fig. 26. 



386 CONDUCTION IN DIELECTEICS. [334- 

If the piston is free to move it will move back to P and be 
in equilibrium there. This represents the complete discharge of 
the dielectric. 

During the discharge there is a reversed motion of the liquids 
throughout the whole tube, and this represents that change of 
electric displacement which we have supposed to take place in a 
dielectric. 

I have supposed every part of the system of tubes filled with 
incompressible liquids, in order to represent the property of all 
electric displacement that there is no real accumulation of elec 
tricity at any place. 

Let us now consider the effect of opening the stopcock Q while 
the piston P is at P l . 

The level of A L and D l will remain unchanged, but that of and 
C will become the same, and will coincide with B Q and C . 

The opening of the stopcock Q corresponds to the existence of 
a part of the dielectric which has a slight conducting power, but 
which does not extend through the whole dielectric so as to form 
an open channel. 

The charges on the opposite sides of the dielectric remain in 
sulated, but their difference of potential diminishes. 

In fact, the difference of pressure on the two sides of the piston 
sinks from 4# to 2 a during the passage of the fluid through Q. 

If we now shut the stopcock Q and allow the piston P to move 
freely, it will come to equilibrium at a point P 2 , and the discharge 
will be apparently only half of the charge. 

The level of the mercury in A and B will be ^a above its 
original level, and the level in the tubes C and D will be \a 
below its original level. This is indicated by the levels A 29 -Z? 2 , 

c 2 , A- 

If the piston is now fixed and the stopcock opened, mercury will 
flow from B to C till the level in the two tubes is again at B Q and 
C . There will then be a difference of pressure = a on the two 
sides of the piston P. If the stopcock is then closed and the piston 
P left free to move, it will again come to equilibrium at a point P 3 , 
half way between P 2 and P . This corresponds to the residual 
charge which is observed when a charged dielectric is first dis 
charged and then left to itself. It gradually recovers part of its 
charge, and if this is again discharged a third charge is formed, the 
successive charges diminishing in quantity. In the case of the 
illustrative experiment each charge is half of the preceding, and the 



334-] HYDROSTATICAL ILLUSTRATION. 387 

discharges, which are J, ^, &c. of the original charge, form a series 
whose sum is equal to the original charge. 

If, instead of opening and closing the stopcock, we had allowed it 
to remain nearly, but not quite, closed during the whole experiment, 
we should have had a case resembling that of the electrification of a 
dielectric which is a perfect insulator and yet exhibits the phe 
nomenon called * electric absorption/ 

To represent the case in which there is true conduction through 
the dielectric we must either make the piston leaky, or we must 
establish a communication between the top of the tube A and the 
top of the tube D. 

In this way we may construct a mechanical illustration of the 
properties of a dielectric of any kind, in which the two electricities 
are represented by two real fluids, and the electric potential is 
represented by fluid pressure. Charge and discharge are repre 
sented by the motion of the piston P, and electromotive force by 
the resultant force on the piston. 



c c 2 



CHAPTER XL 

THE MEASUREMENT OF ELECTRIC RESISTANCE. 

335.] IN the present state of electrical science, the determination 
of the electric resistance of a conductor may be considered as the 
cardinal operation in electricity, in the same sense that the deter 
mination of weight is the cardinal operation in chemistry. 

The reason of this is that the determination in absolute measure 
of other electrical magnitudes, such as quantities of electricity, 
electromotive forces, currents, &c., requires in each case a com 
plicated series of operations, involving- generally observations of 
time, measurements of distances, and determinations of moments 
of inertia, and these operations, or at least some of them, must 
be repeated for every new determination, because it is impossible 
to preserve a unit of electricity, or of electromotive force, or of 
current, in an unchangeable state, so as to be available for direct 
comparison. 

But when the electric resistance of a properly shaped conductor 
of a properly chosen material has been once determined, it is found 
that it always remains the same for the same temperature, so that 
the conductor may be used as a standard of resistance, with which 
that of other conductors can be compared, and the comparison of 
two resistances is an operation which admits of extreme accuracy. 

When the unit of electrical resistance has been fixed on, material 
copies of this unit, in the form of Resistance Coils, are prepared 
for the use of electricians, so that in every part of the world 
electrical resistances may be expressed in terms of the same unit. 
These unit resistance coils are at present the only examples of 
material electric standards which can be preserved, copied, and used 
for the purpose of measurement. Measures of electrical capacity, 
which are also of great importance, are still defective, on account 
of the disturbing influence of electric absorption. 

336.] The unit of resistance may be an entirely arbitrary one, 
as in the case of Jacobins Etalon, which was a certain copper 
wire of 22.4932 grammes weight, 7.61975 metres length, and 0.667 



339-] STANDARDS OF RESISTANCE. 389 

millimetres diameter. Copies of this have been made by Leyser of 
Leipsig, and are to be found in different places. 

According- to another method the unit may be defined as the 
resistance of a portion of a definite substance of definite dimensions. 
Thus, Siemens unit is defined as the resistance of a column of 
mercury of one metre long-, and one square millimetre section, at 
the temperature 0C. 

337.] Finally, the unit may be defined with reference to the 
electrostatic or the electromagnetic system of units. In practice 
the electromagnetic system is used in all telegraphic operations, 
and therefore the only systematic units actually in use are those 
of this system. 

In the electromagnetic system, as we shall shew at the proper 
place, a resistance is a quantity homogeneous with a velocity, and 
may therefore be expressed as a velocity. See Art. 628. 

338.] The first actual measurements on this system were made 
by Weber, who employed as his unit one millimetre per second. 
Sir W. Thomson afterwards used one foot per second as a unit, 
but a large number of electricians have now agreed to use the 
unit of the British Association, which professes to represent a 
resistance which, expressed as a velocity, is ten millions of metres 
per second. The magnitude of this unit is more convenient than 
that of Weber s unit, which is too small. It is sometimes referred 
to as the B.A. unit, but in order to connect it with the name of 
the discoverer of the laws of resistance, it is called the Ohm. 

339.] To recollect its value in absolute measure it is useful 
to know that ten millions of metres is professedly the distance 
from the pole to the equator, measured along the meridian of Paris. 
A body, therefore, which in one second travels along a meridian 
from the pole to the equator would have a velocity which, on the 
electromagnetic system, is professedly represented by an Ohm. 

I say professedly, because, if more accurate researches should 
prove that the Ohm, as constructed from the British Association s 
material standards, is not really represented by this velocity, elec 
tricians would not alter their standards, but would apply a cor 
rection. In the same way the metre is professedly one ten-millionth 
of a certain quadrantal arc, but though this is found not to be 
exactly true, the length of the metre has not been altered, but the 
dimensions of the earth are expressed by a less simple number. 

According to the system of the British Association, the absolute 
value of the unit is originally chosen so as to represent as nearly 



390 



MEASUREMENT OF RESISTANCE. 



[340. 



as possible a quantity derived from the electromagnetic absolute 
system. 

340.] When a material unit representing this abstract quantity 
has been made, other standards are constructed by copying this unit, 
a process capable of extreme accuracy of much greater accuracy 
than, for instance, the copying of foot-rules from a standard foot. 

These copies, made of the most permanent materials, are dis 
tributed over all parts of the world, so that it is not likely that 
any difficulty will be found in obtaining copies of them if the 
original standards should be lost. 

But such units as that of Siemens can without very great 
labour be reconstructed with considerable accuracy, so that as the 
relation of the Ohm to Siemens unit is known, the Ohm can be 
reproduced even without having a standard to copy, though the 
labour is much greater and the accuracy much less than by the 
method of copying. 

Finally, the Ohm may be reproduced 
by the electromagnetic method by which 
it was originally determined. This method, 
which is considerably more laborious than 
the determination of a foot from the seconds 
pendulum, is probably inferior in accuracy 
to that last mentioned. On the other hand, 
the determination of the electromagnetic 
unit in terms of the Ohm with an amount 
of accuracy corresponding to the progress 
of electrical science, is a most important 
physical research and well worthy of being 
repeated. 

The actual resistance coils constructed 
to represent the Ohm were made of an 
alloy of two parts of silver and one of pla 
tinum in the form of wires from .5 milli 
metres to .8 millimetres diameter, and from 
one to two metres in length. These wires 
were soldered to stout copper electrodes. 
The wire itself was covered with two layers 




Fig. 27. 



of silk, imbedded in solid paraffin, and enclosed in a thin brass 
case, so that it can be easily brought to a temperature at which 
its resistance is accurately one Ohm. This temperature is marked 
on the insulating support of the coil. (See Fig. 27.) 



34 1 -] RESISTANCE COILS. 391 

0)i the Forms of Resistance Coils. 

341.] A Resistance Coil is a conductor capable of being 1 easily 
placed in the voltaic circuit, so as to introduce into the circuit 
a known resistance. 

The electrodes or ends of the coil must be such that no appre 
ciable error may arise from the mode of making the connexions. 
For resistances of considerable magnitude it is sufficient that the 
electrodes should be made of stout copper wire or rod well amal 
gamated with mercury at the ends, and that the ends should be 
made to press on flat amalgamated copper surfaces placed in mercury 
cups. 

For very great resistances it is sufficient that the electrodes 
should be thick pieces of brass, and that the connexions should 
be made by inserting a wedge of brass or copper into the interval 
between them. This method is found very convenient. 

The resistance coil itself consists of a wire well covered with 
silk, the ends of which are soldered permanently to the elec 
trodes. 

The coil must be so arranged that its temperature may be easily 
observed. For this purpose the wire is coiled on a tube and 
covered with another tube, so that it may be placed in a vessel 
of water, and that the water may have access to the inside and the 
outside of the coil. 

To avoid the electromagnetic effects of the current in the coil 
the wire is first doubled back on itself and then coiled on the tube, 
so that at every part of the coil there are equal and opposite 
currents in the adjacent parts of the wire. 

When it is desired to keep two coils at the same temperature the 
wires are sometimes placed side by side and coiled up together. 
This method is especially useful when it is more important to 
secure equality of resistance than to know the absolute value of 
the resistance, as in the case of the equal arms of Wheatstone s 
Bridge, (Art. 347). 

AVhen measurements of resistance were first attempted, a resist 
ance coil, consisting of an uncovered wire coiled in a spiral groove 
round a cylinder of insulating material, was much used. It was 
called a Rheostat. The accuracy with which it was found possible 
to compare resistances was soon found to be inconsistent with the 
use of any instrument in which the contacts are not more perfect 
than can be obtained in the rheostat. The rheostat, however, is 



392 



MEASUREMENT OF RESISTANCE. 



[342. 



still used for adjusting the resistance where accurate measurement is 
not required. 

Resistance coils are generally made of those metals whose resist 
ance is greatest and which vary least with temperature. German 
silver fulfils these conditions very well, but some specimens are 
found to change their properties during the lapse of years. Hence 
for standard coils, several pure metals, and also an alloy of platinum 
and silver, have been employed, and the relative resistance of these 
during several years has been found constant up to the limits of 
modern accuracy. 

342.] For very great resistances, such as several millions of 
Ohms, the wire must be either very long or very thin, and the 
construction of the coil is expensive and difficult. Hence tellurium 
and selenium have been proposed as materials for constructing 
standards of great resistance. A very ingenious and easy method 
of construction has been lately proposed by Phillips *. On a piece 
of ebonite or ground glass a fine pencil-line is drawn. The ends 
of this filament of plumbago are connected to metallic electrodes, 
and the whole is then covered with insulating varnish. If it 
should be found that the resistance of such a pencil-line remains 
constant, this will be the best method of obtaining a resistance of 
several millions of Ohms. 

343.] There are various arrangements by which resistance coils 
may be easily introduced into a circuit. 

For instance, a series of coils of which the resistances are 1,2, 
4, 8, 16, &c., arranged according to the powers of 2, may be placed 
in a box in series. 




G4 32 



/& 



Fig. 28. 



The electrodes consist of stout brass plates, so arranged on the 
outside of the box that by inserting a brass plug or wedge between 



Phil Mag., July, 1870. 



344-1 



RESISTANCE BOXES. 



393 



two of them as a shunt, the resistance of the corresponding coil 
may be put out of the circuit. This arrangement was introduced 
by Siemens. 

Each interval between the electrodes is marked with the resist 
ance of the corresponding coil, so that if we wish to make the 
resistance box equal to 107 we express 107 in the binary scale as 
64 + 32 + 8+2 + 1 or 1101011. We then take the plugs out 
of the holes corresponding to 64, 32, 8, 2 and 1, and leave the 
plugs in 16 and 4. 

This method, founded on the binary scale, is that in which the 
smallest number of separate coils is needed, and it is also that 
which can be most readily tested. For if we have another coil 
equal to 1 we can test the equality of 1 and l , then that of 1 + 1 
and 2, then that of 1 + ! -{- 2 and 4, and so on. 

The only disadvantage of the arrangement is that it requires 
a familiarity with the binary scale of notation, which is not 
generally possessed by those accustomed to express every number 
in the decimal scale. 

344.] A box of resistance coils may be arranged in a different 
way for the purpose of mea 
suring conductivities instead of 
resistances. 

The coils are placed so that 
one end of each is connected 
with a long thick piece of 
metal which forms one elec 
trode of the box, and the other 




Fig. 29. 



end is connected with a stout piece of brass plate as in the former 
case. 

The other electrode of the box is a long brass plate, such that 
by inserting brass plugs between it and the electrodes of the coils 
it may be connected to the first electrode through any given set of 
coils. The conductivity of the box is then the sum of the con 
ductivities of the coils. 

In the figure, in which the resistances of the coils are 1,2,4, &c., 
and the plugs are inserted at 2 and 8, the conductivity of the 
box is J+-J = f, and the resistance of the box is therefore f 
or 1.6. 

This method of combining resistance coils for the measurement 
of fractional resistances was introduced by Sir W. Thomson under 
the name of the method of multiple arcs. See Art. 276. 



394 



MEASUREMENT OF EESISTANCE. 



[345- 



On the Comparison of Resistances. 

345.] If E is the electromotive force of a battery, and R the 
resistance of the battery and its connexions, including the galvan 
ometer used in measuring the current, and if the strength of the 
current is / when the battery connexions are closed, and I 19 I 2 
when additional resistances r l3 i\ 2 are introduced into the circuit, 
then, by Ohm s Law, 

E=IR = Ii (R + rJ = / 2 (R + r 2 ). 

Eliminating E, the electromotive force of the battery, and R 
the resistance of the battery and its connexions, we get Ohm s 
formula _ 



This method requires a measurement of the ratios of /, /j and 7 2 , 
and this implies a galvanometer graduated for absolute mea 
surements. 

If the resistances ^ and r 2 are equal, then / x and 7 2 are equal, 
and we can test the equality of currents by a galvanometer which 
is not capable of determining their ratios. 

But this is rather to be taken as an example of a faulty method 
than as a practical method of determining resistance. The electro 
motive force E cannot be maintained rigorously constant, and the 
internal resistance of the battery is also exceedingly variable, so 
that any methods in which these are assumed to be even for a short 
time constant are not to be depended on. 

346.] The comparison of resistances can be made with extreme 

c 




accuracy by either of two methods, in which the result is in 
dependent of variations of R and E. 



346.] COMPARISON OF RESISTANCES. 395 

The first of these methods depends on the use of the differential 
galvanometer, an instrument in which there are two coils, the 
currents in which are independent of each other, so that when 
the currents are made to flow in opposite directions they act in 
opposite directions on the needle, and when the ratio of these 
currents is that of m to n they have no resultant effect on the 
galvanometer needle. 

Let I l} T 2 be the currents through the two coils of the galvan 
ometer, then the deflexion of the needle may be written 



Now let the battery current / be divided between the coils of 
the galvanometer, and let resistances A and B be introduced into 
the first and second coils respectively. Let the remainder of the 
resistance of their coils and their connexions be a and (3 respect 
ively, and let the resistance of the battery and its connexions 
between C and D be ;, and its electromotive force E. 

Then we find, by Ohm s Law, for the difference of potentials 
between C and D, 

C-D = /jM + a) = L(B + f) = E-Ir, 
and since /j + 7 2 /, 

, I=E 



where D = (A + a)(+p)+t(A+d+JB+p). 

The deflexion of the galvanometer needle is therefore 

5= ~{m(B + p)-u(A+a)}, 

and if there is no observable deflexion, then we know that the 
quantity enclosed in brackets cannot differ from zero by more than 
a certain small quantity, depending on the power of the battery, 
the suitableness of the arrangement, the delicacy of the galvan 
ometer, and the accuracy of the observer. 

Suppose that B has been adjusted so that there is no apparent 
deflexion. 

Now let another conductor A be substituted for A, and let 
A be adjusted till there is no apparent deflexion. Then evidently 
to a first approximation A = A. 

To ascertain the degree of accuracy of this estimate, let the 
altered quantities in the second observation be accented, then 



396 MEASUREMENT OF EESISTANCE. [346- 



m 



Hence n (A -A) = - 5 - 5 . 

/^ ^ 

If 8 and , instead of being- both apparently zero, had been only 
observed to be equal, then, unless we also could assert that E = E , 
the right-hand side of the equation might not be zero. In fact, 
the method would be a mere modification of that already described. 

The merit of the method consists in the fact that the thing 
observed is the absence of any deflexion, or in other words, the 
method is a Null method, one in which the non-existence of a force 
is asserted from an observation in which the force, if it had been 
different from zero by more than a certain small amount, would 
have produced an observable effect. 

Null methods are of great value where they can be employed, but 
they can only be employed where we can cause two equal and 
opposite quantities of the same kind to enter into the experiment 
together. 

In the case before us both 8 and b are quantities too small to be 
observed, and therefore any change in the value of E will not affect 
the accuracy of the result. 

The actual degree of accuracy of this method might be ascer 
tained by taking a number of observations in each of which A 
is separately adjusted, and comparing the result of each observation 
with the mean of the whole series. 

But by putting A out of adjustment by a known quantity, as, 
for instance, by inserting at A or at B an additional resistance 
equal to a hundredth part of A or of B, and then observing 
the resulting deviation of the galvanometer needle, we can estimate 
the number of degrees corresponding to an error of one per cent. 
To find the actual degree of precision we must estimate the smallest 
deflexion which could not escape observation, and compare it with 
the deflexion due to an error of one per cent. 

* If the comparison is to be made between A and B, and if the 
positions of A and B are exchanged, then the second equation 
becomes 



* This investigation is taken from Weber s treatise on Galvanometry. Gottingen 
Transactions, x. p. 65. 



346.] DIFFERENTIAL GALVANOMETER. 397 





D jy 

whence (m + n) (BA) = -- 5 -=-, I . 

j j 

If m and , ^ and B, a and are approximately equal, then 
B-A = 



Here 8 8 may be taken to be the smallest observable deflexion 
of the galvanometer. 

If the galvanometer wire be made longer and thinner, retaining 
the same total mass, then n will vary as the length of the wire 
and a as the square of the length. Hence there will be a minimum 

(A + a) 
value of 




a = 



If we suppose r, the battery resistance, small compared with A, 
this gives a = i ^ . 

or, M resistance of each coil of the galvanometer should be one-third 
of the resistance to be measured. 

We then find o ^2 

7? / /S * \ 

> A = 77 (o o ;. 

9 nE v 

If we allow the current to flow through one only of the coils 
of the galvanometer, and if the deflexion thereby produced is A 
(supposing the deflexion strictly proportional to the deflecting 

force), then 

mE 3 nE . _ , 1 , 

A = - - = - T if r and a = - A. 
A+a+r A 3 

B-A 2 5-5 r 
Hence -^ = - 

In the differential galvanometer two currents are made to 
produce equal and opposite effects on the suspended needle. The 
force with which either current acts on the needle depends not 
only on the strength of the current, but on the position of the 
windings of the wire with respect to the needle. Hence, unless 
the coil is very carefully wound, the ratio of m to n may change 
when the position of the needle is changed, and therefore it is 
necessary to determine this ratio by proper methods during each 



398 MEASUREMENT OF RESISTANCE. [347- 

course of experiments if any alteration of the position of the needle 
is suspected. 

The other null method, in which Wheatst one s Bridge is used, 
requires only an ordinary galvanometer, and the observed zero 
deflexion of the needle is due, not to the opposing action of two 
currents, but to the non-existence of a current in the wire. Hence 
we have not merely a null deflexion, but a null current as the 
phenomenon observed, and no errors can arise from want of 
regularity or change of any kind in the coils of the galvanometer. 
The galvanometer is only required to be sensitive enough to detect 
the existence and direction of a current, without in any way 
determining its value or comparing its value with that of another 
current. 

347.] Wheatstone s Bridge consists essentially of six conductors 
connecting four points. An electromotive 
force E is made to act between two of the 
points by means of a voltaic battery in 
troduced between B and C. The current 
between the other two points and A is 
measured by a galvanometer. 

Under certain circumstances this current 
becomes zero. The conductors BC and OA 
are then said to be conjugate to each other, 
which implies a certain relation between the resistances of the 
other four conductors, and this relation is made use of in measuring 
resistances. 

If the current in OA is zero, the potential at must be equal 
to that at A. Now when we know the potentials at B and C we 
can determine those at and A by the rule given at Art. 274, 
provided there is no current in OA, 

n _By+C(3 A _ 

/3 + y 

whence the condition is fin _ c 

where b, c> (3, y are the resistances in CA, AB, BO and OC re 
spectively. 

To determine the degree of accuracy attainable by this method 
we must ascertain the strength of the current in OA when this 
condition is not fulfilled exactly. 

Let A, B, C and be the four points. Let the currents along 
BC, CA and AB be x, y and z, and the resistances of these 




348.] 



WHEATSTONE S BRIDGE. 



399 



conductors a, I and c. Let the currents along OA, OB and OC be 
f, j], and the resistances a, ft and y. Let an electromotive force 
E act along BC. Required the current along OA. 

Let the potentials at the points A, B, C and be denoted 
by the symbols A, B, C and 0. The equations of conduction are 

ax=B-C+E, a = 0-A, 

fy=C-A, prj^O-B, 

cz = A-B, y( = 0-C; 

with the equations of continuity 

+jr-jf= 0, 
Y] + Z X = 0, 
C+x-y = 0. 

By considering the system as made up of three circuits OBC, 
OCA and OAB in which the currents are x, y, z respectively, and 
applying Kirchhoff s rule to each cycle, we eliminate the values 
of the potentials 0, A, B, C, and the currents f, r/_, and obtain the 
following equations for x, y and 0, 



yx 



Hence, if we put 



-ay 



a + /3 + y 


y 


-/3 


-y 


5 + y + a 


a 


- 


a 


e + a + /3 



we find 



XT 

= (5/3 cy), 



and 



a? = 



Tl 





348.] The value of D may be expressed in the symmetrical form, 



or, since we suppose the battery in the conductor a and the 
galvanometer in a, we may put B the battery resistance for a and 
G the galvanometer resistance for a. We then find 

D = 



If the electromotive force E were made to act along OA, the 
resistance of OA being still a, and if the galvanometer were placed 



400 MEASUREMENT OF RESISTANCE. [349- 



in BC) the resistance of BC being still a, then the value of D 
would remain the same, and the current in BC due to the electro 
motive force E acting along- OA would be equal to the current 
in OA due to the electromotive force E acting in BC. 

But if we simply disconnect the battery and the galvanometer, 
and without altering their respective resistances connect the battery 
to and A and the galvanometer to B and C 9 then in the value of 
D we must exchange the values of B and G. If D be the value 
of D after this exchange, we find 



Let us suppose that the resistance of the galvanometer is greater 
than that of the battery. 

Let us also suppose that in its original position the galvanometer 
connects the junction of the two conductors of least resistance /3, y 
with the junction of the two conductors of greatest resistance b } c 3 
or, in other words, we shall suppose that if the quantities />, <?, y, (3 
are arranged in order of magnitude, b and c stand together, and 
y and /3 stand together. Hence the quantities b ft and c y are 
of the same sign, so that their product is positive, and therefore 
D D is of the same sign as B G. 

If therefore the galvanometer is made to connect the junction of 
the two greatest resistances with that of the two least, and if 
the galvanometer resistance is greater than that of the battery, 
then the value of 2) will be less, and the value of the deflexion 
of the galvanometer greater, than if the connexions are exchanged. 

The rule therefore for obtaining the greatest galvanometer de 
flexion in a given system is as follows : 

Of the two resistances, that of the battery and that of the 
galvanometer, connect the greater resistance so as to join the two 
greatest to the two least of the four other resistances. 

349.] We shall suppose that we have to determine the ratio of 
the resistances of the conductors AB and AC, and that this is to be 
done by finding a point on the conductor J30C, such that when 
the points A and are connected by a wire, in the course of which 
a galvanometer is inserted, no sensible deflexion of the galvano 
meter needle occurs when the battery is made to act between B 
and C. 

The conductor BOC may be supposed to be a wire of uniform 
resistance divided into equal parts, so that the ratio of the resist 
ances of BO and OC may be read off at once. 



349-] WHEATSTONE S BRIDGE. 401 

Instead of the whole conductor being- a uniform wire, we may 
make the part near of such a wire, and the parts on each side 
may be coils of any form, the resistance of which is accurately 
known. 

We shall now use a different notation instead of the symmetrical 
notation with which we commenced. 

Let the whole resistance of BAG be R. 

Let c = mE and b = (lm) R. 

Let the whole resistance of BOC be S. 

Let /3 = nS and y = (I n] S. 

The value of n is read off directly, and that of m is deduced from 
it when there is no sensible deviation of the galvanometer. 

Let the resistance of the batteiy and its connexions be B, and 
that of the galvanometer and its connexions G. 

We find as before 
D = G{BR + BS+RS}+m(\-m)R*(B + S} + 

+ (**+* 2m*)RS, 
and if f is the current in the galvanometer wire 

t 
C = 



In order to obtain the most accurate results we must make the 
deviation of the needle as great as possible compared with the value 
of (n mj. This may be done by properly choosing the dimensions 
of the galvanometer and the standard resistance wire. 

It will be shewn, when we come to Galvanometry, Art. 716, 
that when the form of a galvanometer wire is changed while 
its mass remains constant, the deviation of the needle for unit 
current is proportional to the length, but the resistance increases 
as the square of the length. Hence the maximum deflexion is 
shewn to occur when the resistance of the galvanometer wire is 
equal to the constant resistance of the rest of the circuit. 

In the present case, if 8 is the deviation, 

5 = CV 

where C is some constant, and G is the galvanometer resistance 
which varies as the square of the length of the wire. Hence we 
find that in the value of D, when 8 is a maximum, the part 
involving G must be made equal to the rest of the expression. 

If we also put m = , as is the case if we have made a correct 
observation, we find the best value of G to be 
G = n(\n)(R + 8). 

VOL. I. D d 



402 



MEASUREMENT OF RESISTANCE. 



[350. 



This result is easily obtained by considering the resistance from 
A to through the system, remembering that BC, being conjugate 
to A0 y has no effect on this resistance. 

In the same way we should find that if the total area of the 
acting surfaces of the battery is given, the most advantageous 
arrangement of the battery is when 

*=1OT 

Finally, we shall determine the value of S such that a given 
change in the value of n may produce the greatest galvanometer 
deflexion. By differentiating the expression for we find 



g* - 
~ 



If we have a great many determinations of resistance to make 
in which the actual resistance has nearly the same value, then it 
may be worth while to prepare a galvanometer and a battery for 
this purpose. In this case we find that the best arrangement is 

S=R, B=\R, G = 2n(l-n)R, 
and if n = i G= \R. 

On the Use of Wheatstone s Bridge. 

350.] We have already explained the general theory of Wheat- 
stone s Bridge, we shall now consider some of its applications. 




_LJ -i : I ; I. M I I I I I 

Fig. 32. 



The comparison which can be effected with the greatest exact 
ness is that of two equal resistances. 



35o.] USE OF WHEATSTONE S BRIDGE. 403 

Let us suppose that (3 is a standard resistance coil, and that 
we wish to adjust y to be equal in resistance to /3. 

Two other coils, b and c, are prepared which are equal or nearly 
equal to each other, and the four coils are placed with their electrodes 
in mercury cups so that the current of the battery is divided 
between two branches, one consisting of /3 and y and the other 
of b and c. The coils b and c are connected by a wire P7?, as 
uniform in its resistance as possible, and furnished with a scale 
of equal parts. 

The galvanometer wire connects the junction of /3 and y with 
a point Q of the wire PR, and the point of contact at Q is made 
to vary till on closing first the battery circuit and then the 
galvanometer circuit, no deflexion of the galvanometer needle is 
observed. 

The coils /3 and y are then made to change places, and a new 
position is found for Q. If this new position is the same as the 
old one, then we know that the exchange of y3 and y has produced 
no change in the proportions of the resistances, and therefore y 
is rightly adjusted. If Q has to be moved, the direction and 
amount of the change will indicate the nature and amount of the 
alteration of the length of the wire of y, which will make its 
resistance equal to that of (3. 

If the resistances of the coils b and c, each including part of the 
wire PJR up to its zero reading, are equal to that of b and c 
divisions of the wire respectively, then, if x is the scale reading 
of Q in the first case, and y that in the second, 
c + z __ _/3_ c+y __ y_ 

b x~ y by ~~ /3 

whence z! = 1 

/3- 

Since b y is nearly equal to c + x, and both are great with 
respect to x or y, we may write this 



and 



When y is adjusted as well as we can, we substitute for b and c 
other coils of (say) ten times greater resistance. 

The remaining difference between /3 and y will now produce 
a ten times greater difference in the position of Q than with the 

D d 2 



404 



MEASUREMENT OF RESISTANCE. 



original coils b and c, and in this way we can continually increase 
the accuracy of the comparison. 

The adjustment by means of the wire with sliding contact piece 
is more quickly made than by means of a resistance box, and it is 
capable of continuous variation. 

The battery must never be introduced instead of the galvano 
meter into the wire with a sliding contact, for the passage of a 
powerful current at the point of contact would injure the surface 
of the wire. Hence this arrangement is adapted for the case in 
which the resistance of the galvanometer is greater than that of the 
battery. 

On the Measurement of Small Resistances. 

351.] When a short and thick conductor is introduced into a 
circuit its resistance is so small compared with the resistance 
occasioned by unavoidable faults in the connexions, such as want 
of contact or imperfect soldering, that no correct value of the 
^ resistance can be deduced from experi 

ments made in the way described above. 
The object of such experiments is 
generally to determine the specific re 
sistance of the substance, and it is re 
sorted to in cases when the substance 
cannot be obtained in the form of a 
long thin wire, or when the resistance 
to transverse as well as to longitudinal 
conduction has to be measured. 




C 



H 


Q 




f* 




ft 




^ y-J 





V 

Fig. 34. 



Sir W. Thomson,* has described a method applicable to such 
cases, which we may take as an example of a system of nine 
conductors. 

* Proc. K. 8., June 6, 1861. 



35i.] THOMSON S METHOD FOR SMALL RESISTANCES. 405 

The most important part of the method consists in measuring 
the resistance, not of the whole length of the conductor, but of 
the part between two marks on the conductor at some little dis 
tance from its ends. 

The resistance which we wish to measure is that experienced 
by a current whose intensity is uniform in any section of the 
conductor, and which flows in a direction parallel to its axis. 
Now close to the extremities, when the current is introduced 
by means of electrodes, either soldered, amalgamated, or simply 
pressed to the ends of the conductor, there is generally a want of 
uniformity in the distribution of the current in the conductor. 
At a short distance from the extremities the current becomes 
sensibly uniform. The student may examine for himself the 
investigation and the diagrams of Art. 193, where a current is 
introduced into a strip of metal with parallel sides through one 
of the sides, but soon becomes itself parallel to the sides. 

The resistance of the conductors between certain marks S, S 
and TT is to be compared. 

The conductors are placed in series, and with connexions as 
perfectly conducting as possible, in a battery circuit of small resist 
ance. A wire SVT is made to touch the conductors at S and T, 
and S V T is another wire touching them at S and T. 

The galvanometer wire connects the points Fand V of these wires. 

The wires SVT and S V T are of resistance so great that the 
resistance due to imperfect connexion at S, T, S or T may be 
neglected in comparison with the resistance of the wire, and F, V 
are taken so that the resistance in the branches of either wire 
leading to the two conductors are nearly in the ratio of the resist 
ances of the two conductors. 

Calling H and .Fthe resistances of the conductors SS and TT. 
A and C those of the branches SF and FT. 

P and R those of the branches S V and V T . 
Q that of the connecting piece S T . 
,, B that of the battery and its connexions. 
G that of the galvanometer and its connexions. 

The symmetry of the system may be understood from the 
skeleton diagram. Fig. 33. 

The condition that B the battery and G the galvanometer may 
be conjugate conductors is, in this case, 

I. *L (1L A Q 

~C " A + \C A 



406 



MEASUREMENT OF RESISTANCE. 



[352. 



Now the resistance of the connector Q is as small as we can 
make it. If it were zero this equation would be reduced to 

F^_ H 
C^-~A 

and the ratio of the resistances of the conductors to be compared 
would be that of C to A, as in Wheatstone s Bridge in the ordinary 
form. 

In the present case the value of Q is small compared with P 
or with R, so that if we assume the points V, V so that the ratio 
of R to C is nearly equal to that of P to A, the last term of the 
equation will vanish, and we shall have 

FiHn C-.A. 

The success of this method depends in some degree on the per 
fection of the contact between the wires and the tested conductors 
at SS , T and T. In the following method, employed by Messrs. 
Matthiessen and Hockin *, this condition is dispensed with. 



! B 




Fig. 35. 

352.] The conductors to be tested are arranged in the manner 
already described, with the connexions as well made as possible, 
and it is required to compare the resistance between the marks 
SS on the first conductor with the resistance between the marks 
T Ton the second. 

Two conducting points or sharp edges are fixed in a piece of 
insulating material so that the distance between them can be 
accurately measured. This apparatus is laid on the conductor to 
be tested, and the points of contact with the conductor are then 
at a known distance SS . Each of these contact pieces is connected 

* Laboratory. Matthiessen and Hockin on Alloys. 



352.] MATTHIESSEN AND HOOKIES METHOD. 407 

with a mercury cup, into which one electrode of the galvanometer 
may be plunged. 

The rest of the apparatus is arranged, as in Wheatstone s Bridge, 
with resistance coils or boxes A and (7, and a wire PE with a 
sliding contact piece Q, to which the other electrode of the galva 
nometer is connected. 

Now let the galvanometer be connected to S and Q, and let 
A l and C l be so arranged, and the position of Q so determined, that 
there is no current in the galvanometer wire. 

Then we know that ^ A -\-PQ 
W = cl+QR 
where XS, PQ, &c. stand for the resistances in these conductors. 

From this we get 

XS A 



Now let the electrode of the galvanometer be connected to S t 
and let resistance be transferred from C to A (by carrying resistance 
coils from one side to the other) till electric equilibrium of the 
galvanometer wire can be obtained by placing Q at some point 
of the wire, say Q 2 . Let the values of C and A be now <? 2 and A 2 , 
and let A 2 +C. 2 + PR = A^ + C^ + PR = R. 

Then we have, as before, 

XS _A 2 +PQ 2 
XT R 

SS Ay-Ai+QiQt 
Whence jy = - -g- 

In the same way, placing the apparatus on the second conductor 
at TT and again transferring resistance, we get, when the electrode 

is in T , 

XT 



XT R 

and when it is in T, 

XT 



XT R 
Whence 



AI K 

We can now deduce the ratio of the resistances SS and T T, for 

SS* _A 2 -A 1 +Q 1 Q 2 
TT A,-A 3 +Q,Q, 

* 



408 MEASUREMENT OF RESISTANCE. [353- 

When great accuracy is not required we may dispense with the 
resistance coils A and C, and we then find 
SS _ Q, Q 2 

?"T~ Q 3 Q* 

The readings of the position of Q on a wire of a metre in length 
cannot be depended on to less than a tenth of a millimetre, and the 
resistance of the wire may vary considerably in different parts 
owing to inequality of temperature, friction, &c. Hence, when 
great accuracy is required, coils of considerable resistance are intro 
duced at A and (?, and the ratios of the resistances of these coils 
can be determined more accurately than the ratio of the resistances 
of the parts into which the wire is divided at Q. 

It will be observed that in this method the accuracy of the 
determination depends in no degree on the perfection of the con 
tacts at SS or TT . 

This method may be called the differential method of using 
Wheatstone s Bridge, since it depends on the comparison of ob 
servations separately made. 

An essential condition of accuracy in this method is that the 
resistance of the connexions should continue the same during the 
course of the four observations required to complete the determ 
ination. Hence the series of observations ought always to be 
repeated in order to detect any change in the resistances. 

On the Comparison of Great Resistances. 

353.] When the resistances to be measured are very great, the 
comparison of the potentials at different points of the system may 
be made by means of a delicate electrometer, such as the Quadrant 
Electrometer described in Art. 219. 

If the conductors whose resistance is to be measured are placed 
in series, and the same current passed through them by means of a 
battery of great electromotive force, the difference of the potentials 
at the extremities of each conductor will be proportional to the 
resistance of that conductor. Hence, by connecting the electrodes 
of the electrometer with the extremities, first of one conductor 
and then of the other, the ratio of their resistances may be de 
termined. 

This is the most direct method of determining resistances. It 
involves the use of an electrometer whose readings may be depended 
on, and we must also have some guarantee that the current remains 
constant during the experiment. 



355-] GEEAT RESISTANCES. 409 

Four conductors of great resistance may also be arranged as in 
Wheatstone s Bridge, and the bridge itself may consist of the 
electrodes of an electrometer instead of those of a galvanometer. 
The advantage of this method is that no permanent current is 
required to produce the deviation of the electrometer, whereas the 
galvanometer cannot be deflected unless a current passes through 
the wire. 

354.] When the resistance of a conductor is so great that the 
current which can be sent through it by any available electromotive 
force is too small to be directly measured by a galvanometer, a 
condenser may be used in order to accumulate the electricity for 
a certain time, and then, by discharging the condenser through a 
galvanometer, the quantity accumulated may be estimated. This 
is Messrs. Bright and Clark s method of testing the joints of 
submarine cables. 

355.] But the simplest method of measuring the resistance of 
such a conductor is to charge a condenser of great capacity and to 
connect its two surfaces with the electrodes of an electrometer 
and also with the extremities of the conductor. If E is the dif 
ference of potentials as shewn by the electrometer, S the capacity 
of the condenser, and Q the charge on either surface, R the resist 
ance of the conductor and x the current in it, then, by the theory 
of condensers, n ?& 

By Ohm s Law, E = Ex, 

and by the definition of a current, 



Hence 



and Q=Q Q e~ y 

where Q is the charge at first when t = . 

t 

Similarly E = E e~** 

where E is the original reading of the electrometer, and E the 
same after a time t. From this we find 

-R- 

~S{logA-log e #} 

which gives R in absolute measure. In this expression a knowledge 
of the value of the unit of the electrometer scale is not required. 



410 



MEASUREMENT OF RESISTANCE. 



[356- 



If S, the capacity of the condenser, is given in electrostatic 
measure as a certain number of metres, then R is also given in 
electrostatic measure as the reciprocal of a velocity. 

If S is given in electromagnetic measure its dimensions are 

%>2 

- - , and R is a velocity. 
Jj 

Since the condenser itself is not a perfect insulator it is necessary 
to make two experiments. In the first we determine the resistance 
of the condenser itself, 2t , and in the second, that of the condenser 
when the conductor is made to connect its surfaces. Let this be Rf. 
Then the resistance, R, of the conductor is given by the equation 

1 1 1 

R R RQ 

This method has been employed by MM. Siemens. 

Thomsons * Method for the Determination of the Resistance of 

the Galvanometer. 

356.] An arrangement similar to Wheatstone s Bridge has been 
employed with advantage by Sir W. Thomson in determining the 




Fig. 36. 

resistance of the galvanometer when in actual use. It was sug 
gested to Sir W. Thomson by Mance s Method. See Art. 357. 

Let the battery be placed, as before, between and C in the 
figure of Article 347, but let the galvanometer be placed in CA 
instead of in OA. If bfi cy is zero, then the conductor OA is 
conjugate to J3C, and, as there is no current produced in A by the 
battery in JBC, the strength of the current in any other conductor 
* Proc. R. S,, Jan. 19, 1871. 



357-] MANCE S METHOD. 411 

is independent of the resistance in OA. Hence, if the galvano 
meter is placed in CA its deflexion will remain the same whether 
the resistance of OA is small or great. We therefore observe 
whether the deflexion of the galvanometer remains the same when 
and A are joined by a conductor of small resistance, as when 
this connexion is broken, and if, by properly adjusting the re 
sistances of the conductors, we obtain this result, we know that 
the resistance of the galvanometer is 



. 

where c, y, and /3 are resistance coils of known resistance. 

It will be observed that though this is not a null method, in the 
sense of there being no current in the galvanometer, it is so in 
the sense of the fact observed being the negative one, that the 
deflexion of the galvanometer is not changed when a certain con 
tact is made. An observation of this kind is of greater value 
than an observation of the equality of two different deflexions of 
the same galvanometer, for in the latter case there is time for 
alteration in the strength of the battery or the sensitiveness of 
the galvanometer, whereas when the deflexion remains constant, 
in spite of certain changes which we can repeat at pleasure, we are 
sure that the current is quite independent of these changes. 

The determination of the resistance of the coil of a galvanometer 
can easily be effected in the ordinary way of using Wheatstone s 
Bridge by placing another galvanometer in OA. By the method 
now described the galvanometer itself is employed to measure its 
own resistance. 

nance s* Method of determining the Resistance of the Battery. 

357.] The measurement of the resistance of a battery when in 
action is of a much higher order of difficulty, since the resistance 
of the battery is found to change considerably for some time after 
the strength of the current through it is changed. In many of the 
methods commonly used to measure the resistance of a battery such 
alterations of the strength of the current through it occur in the 
course of the operations, and therefore the results are rendered 
doubtful. 

In Mance s method, which is free from this objection, the battery 
is placed in BC and the galvanometer in CA. The connexion 
between and B is then alternately made and broken. 

* Proc. R. S., Jan. 19, 1871. 



412 MEASUREMENT OF RESISTANCE. [357- 

If the deflexion of the galvanometer remains unaltered, we know 
that OB is conjugate to CA, whence cy = a a, and a, the resistance 
of the battery, is obtained in terms of known resistances c, y, a. 

When the condition cy = a a is fulfilled, then the current through 
the galvanometer is 

Ea 



and this is independent of the resistance (3 between and B. To 
test the sensibility of the method let us suppose that the condition 
cy = aa is nearly, but not accurately, fulfilled, and that y is the 




Fig. 37. 

current through the galvanometer when and B are connected 
by a conductor of no sensible resistance, and y the current when 
and B are completely disconnected. 

To find these values we must make /3 equal to and to oo in the 
general formula for y, and compare the results. 

In this way we find 

y*-y\ _ <* cyaa 



where y an( i y\ are supposed to be so nearly equal that we may, 
when their difference is not in question, put either of them equal 
to y, the value of the current when the adjustment is perfect. 

The resistance, c, of the conductor AB should be equal to a, 
that of the battery, a and y, should be equal and as small as 
possible, and b should be equal to a + y. 

Since a galvanometer is most sensitive when its deflexion is 
small, we should bring the needle nearly to zero by means of fixed 
magnets before making contact between and B. 

In this method of measuring the resistance of the battery, the 
current in the battery is not in any way interfered with during the 
operation, so that we may ascertain its resistance for any given 



358.] 



COMPARISON OF ELECTROMOTIVE FORCES. 



413 



strength of current, so as to determine how the strength of current 
effects the resistance. 

If y is the current in the galvanometer, the actual current 
through the battery is # with the key down and x : with the 
key up, where 

/ b \ f I ac x 

*o = y( 1 + -r-) *i = y( l +- + -T 
> a + y y vifl 1 

the resistance of the battery is 

cy 

a = - , 
a 

and the electromotive force of the battery is 



The method of Art. 356 for finding the resistance of the galva 
nometer differs from this only in making and breaking contact 
between and A instead of between and .Z?, and by exchanging 
a and 3 we obtain for this case 



On the Comparison of Electromotive Forces. 

358.] The following method of comparing the electromotive forces 
of voltaic and thermoelectric arrangements, when no current passes 
through them, requires only a set of resistance coils and a constant 
battery. 

Let the electromotive force E of the battery be greater than that 
of either of the electromotors to be compared, then, if a sufficient 




iHHHI- 



E 

Fig. 38. 

resistance, S 19 be interposed between the points A lt S l of the 
primary circuit E B l A l E, the electromotive force from A to A 1 



414 



MEASUREMENT OF RESISTANCE. 



[358. 



may be made equal to that of the electromotor E^. If the elec 
trodes of this electromotor are now connected with the points 
A 19 B no current will flow through the electromotor. By placing 
a galvanometer G l in the circuit of the electromotor E^ 9 and 
adjusting the resistance between A l and 12 l9 till the galvanometer 
G } indicates no current, we obtain the equation 



where R l is the resistance between A l and S 19 and C is the strength 
of the current in the primary circuit. 

In the same way, by taking a second electromotor E 2 and placing 
its electrodes at A. 2 and JB 2 , so that no current is indicated by the 

galvanometer G 2 , 

E 2 = Z2 2 <7, 

where 7? 2 is the resistance between A. 2 and B 2 . If the observations 
of the galvanometers G l and G 2 are simultaneous, the value of C, 
the current in the primary circuit, is the same in both equations, 

and we find 

E, : ^ : : K l : S t . 

In this way the electromotive force of two electromotors may be 
compared. The absolute electromotive force of an electromotor 
may be measured either electrostatically by means of the electro 
meter, or electromagnetically by means of an absolute galvano 
meter. 

This method, in which, at the time of the comparison, there 
is no current through either of the electromotors, is a modification 
of Poggendorff s method, and is due to Mr. Latimer Clark, who 
has deduced the following values of electromotive forces : 



Daniell I. Amalgamated Zinc HS0 4 + 4 aq. 



II. 
III. 

Bunsen I. 
II. 

Grove 



HS0 4 +12aq. 
HS0 4 + 12aq. 



HS0 4 + 4aq. 



Concentrated 




V It 


solution of 




o s. 


CuSO 4 


Copper 


= 1.079 


CuS0 4 


Copper 


= 0.978 


CuN0 6 


Copper 


= 1.00 


HN0 6 


Carbon 


= 1.964 


sp. g. 1.38 


Carbon 


= 1.888 


HN0 6 


Platinum 


= 1.956 



A Volt is an electromotive force equal to 100,000,000 units of the centimetre-gramme- 
second system. 



CHAPTER XII. 



ON THE ELECTRIC RESISTANCE OF SUBSTANCES. 

359.] THERE are three classes in which we may place different 
substances in relation to the passage of electricity through them. 

The first class contains all the metals and their alloys, some 
sulphurets, and other compounds containing metals, to which we 
must add carbon in the form of gas-coke, and selenium in the 
crystalline form. 

In all these substances conduction takes place without any 
decomposition, or alteration of the chemical nature of the substance, 
either in its interior or where the current enters and leaves the 
body. In all of them the resistance increases as the temperature 
rises. 

The second class consists of substances which are called electro 
lytes, because the current is associated with a decomposition of 
the substance into two components which appear at the electrodes. 
As a rule a substance is an electrolyte only when in the liquid 
form, though certain colloid substances, such as glass at 100C, 
which are apparently solid, are electrolytes. It would appear from 
the experiments of Sir B. C. Brodie that certain gases are capable 
of electrolysis by a powerful electromotive force. 

In all substances which conduct by electrolysis the resistance 
diminishes as the temperature rises. 

The third class consists of substances the resistance of which is 
so great that it is only by the most refined methods that the 
passage of electricity through them can be detected. These are 
called Dielectrics. To this class belong a considerable number 
of solid bodies, many of which are electrolytes when melted, some 
liquids, such as turpentine, naphtha, melted paraffin, &c., and all 
gases and vapours. Carbon in the form of diamond, and selenium 
in the amorphous form, belong to this class. 

The resistance of this class of bodies is enormous compared with 
that of the metals. It diminishes as the temperature rises. It 



416 RESISTANCE. [360. 

is difficult, on account of the great resistance of these substances, 
to determine whether the feeble current which we can force through 
them is or is not associated with electrolysis. 

On the Electric Resistance of Metals. 

360.] There is no part of electrical research in which more 
numerous or more accurate experiments have been made than in 
the determination of the resistance of metals. It is of the utmost 
importance in the electric telegraph that the metal of which the 
wires are made should have the smallest attainable resistance. 
Measurements of resistance must therefore be made before selecting 
the materials. When any fault occurs in the line, its position is 
at once ascertained by measurements of resistance, and these mea 
surements, in which so many persons are now employed, require 
the use of resistance coils, made of metal the electrical properties 
of which have been carefully tested. 

The electrical properties of metals and their alloys have been 
studied with great care by MM. Matthiessen, Vogt, and Hockin, 
and by MM. Siemens, who have done so much to introduce exact 
electrical measurements into practical work. 

It appears from the researches of Dr. Matthiessen, that the effect 
of temperature on the resistance is nearly the same for a considerable 
number of the pure metals, the resistance at 100C being to that 
at 0C in the ratio of 1.414 to 1, or of 1 to 70.7. For pure iron 
the ratio is 1.645, and for pure thallium 1.458. 

The resistance of metals has been observed by Dr. C.W. Siemens*" 
through a much wider range of temperature, extending from the 
freezing point to 350C, and in certain cases to 1000C. He finds 
that the resistance increases as the temperature rises, but that the 
rate of increase diminishes as the temperature rises. The formula, 
which he finds to agree very closely both with the resistances 
observed at low temperatures by Dr. Matthiessen and with his 
own observations through a range of 1000C, is 



where T is the absolute temperature reckoned from 273C, and 

a, /3, y are constants. Thus, for 

Platinum ...... r = 0.0393697^+ 0.002164077 7 0.2413, 

Copper ......... r = 0.0265777^+0. 0031443^0.22751, 

Iron ............ r= 0. 0725457^ + 0.0038 1337 7 1.23971. 

* Proc. R. S., April 27, 1871. 



361.] OF METALS. 417 

From data of this kind the temperature of a furnace may be 
determined by means of an observation of the resistance of a 
platinum wire placed in the furnace. 

Dr. Matthiessen found that when two metals are combined to 
form an alloy, the resistance of the alloy is in most cases greater 
than that calculated from the resistance of the component metals 
and their proportions. In the case of alloys of gold and silver, the 
resistance of the alloy is greater than that of either pure gold or 
pure silver, and, within certain limiting proportions of the con 
stituents, it varies very little with a slight alteration of the pro 
portions. For this reason Dr. Matthiessen recommended an alloy 
of two parts by weight of gold and one of silver as a material 
for reproducing the unit of resistance. 

The effect of change of temperature on electric resistance is 
generally less in alloys than in pure metals. 

Hence ordinary resistance coils are made of German silver, on 
account of its great resistance and its small variation with tem 
perature. 

An alloy of silver and platinum is also used for standard coils. 

361.] The electric resistance of some metals changes when the 
metal is annealed ; and until a wire has been tested by being 
repeatedly raised to a high temperature without permanently 
altering its resistance, it cannot be relied on as a measure of 
resistance. Some wires alter in resistance in course of time without 
having been exposed to changes of temperature. Hence it is 
important to ascertain the specific resistance of mercury, a metal 
which being fluid has always the same molecular structure, and 
which can be easily purified by distillation and treatment with 
nitric acid. Great care has been bestowed in determining the 
resistance of this metal by W. and C. F. Siemens, who introduced 
it as a standard. Their researches have been supplemented by 
those of Matthiessen and Hockin. 

The specific resistance of mercury was deduced from the observed 
resistance of a tube of length / containing a weight 10 of mercury, 
in the following manner. 

No glass tube is of exactly equal bore throughout, but if a small 
quantity of mercury is introduced into the tube and occupies a 
length A. of the tube, the middle point of which is distant x from 
one end of the tube, then the area s of the section near this point 

C 

will be s = , where C is some constant. 
A 

VOL. I. E 6 



418 



RESISTANCE. 



[362. 



= p I s dx = 



The weight of mercury which fills the whole tube is 

> 

where n is the number of points, at equal distances along- the 
tube, where A has been measured, and p is the mass of unit of 
volume. 

The resistance of the whole tube is 



( 
J 



_ / 7^ Trli- 

I UtJ, ~rT l/V ~~ 

J * 6 

where r is the specific resistance per unit of volume. 

Hence wR = 

wR 
and 



gives the specific resistance of unit of volume. 

To find the resistance of unit of length and unit of mass we must 
multiply this by the density. 

It appears from the experiments of Matthiessen and Hockin that 
the resistance of a uniform column of mercury of one metre in 
length, and weighing one gramme at 0C, is 13.071 Ohms, whence 
it follows that if the specific gravity of mercury is 13.595, the 
resistance of a column of one metre in length and one square 
millimetre in section is 0.96146 Ohms. 

362.] In the following table R is the resistance in Ohms of a 
column one metre long and one gramme weight at 0C, and r is 
the resistance in centimetres per second of a cube of one centi 
metre, according to the experiments of Matthiessen *. 



Silver 


Specific 
gravity 

10.50 


hard drawn 





R 

.1689 


r 

1609 


Percentage 
increment of 
resistance for 
1C at 20C. 

0.377 


Copper ..... 
Gold 


. 8.95 
19 27 


hard drawn 
hard drawn 




o 


.1469 
.4150 


1642 
2154 


0.388 
0.365 


Lead . ... 


11.391 


pressed 


2 


.257 


19847 


0.387 


Mercury , 


13.595 


liquid 1 


3 


.071 


96146 


0.072 


Gold 2, Silver 1 
Selenium at 100 


..15.218 

C 


hard or annealed 
Crystalline form 


1 


.668 


10988 
6xl0 13 


0.065 
1.00 



Phil. Mag., May, 1865. 



364.] OF ELECTROLYTES. 419 



On the Electric Resistance of Electrolytes. 

363.] The measurement of the electric resistance of electrolytes 
is rendered difficult on account of the polarization of the electrodes, 
which causes the observed difference of potentials of the metallic 
electrodes to be greater than the electromotive force which actually 
produces the current. 

This difficulty can be overcome in various ways. In certain 
cases we can get rid of polarization by using electrodes of proper 
material, as, for instance, zinc electrodes in a solution of sulphate 
of zinc. By making the surface of the electrodes very large com 
pared with the section of the part of the electrolyte whose resist 
ance is to be measured, and by using only currents of short duration 
in opposite directions alternately, we can make the measurements 
before any considerable intensity of polarization has been excited 
by the passage of the current. 

Finally, by making two different experiments, in one of which 
the path of the current through the electrolyte is much longer than 
in the other, and so adjusting the electromotive force that the 
actual current, and the time during which it flows, are nearly the 
same in each case, we can eliminate the effect of polarization 
altogether. 

364.] In the experiments of Dr. Paalzow * the electrodes were 
in the form of large disks placed in separate flat vessels filled with 
the electrolyte, and the connexion was made by means of a long 
siphon filled with the electrolyte and dipping into both vessels. 
Two such siphons of different lengths were used. 

The observed resistances of the electrolyte in these siphons 
being 7^ and Z? 2 , the siphons were next filled with mercury, and 
their resistances when filled with mercury were found to be R^ 
and S 2 . 

The ratio of the resistance of the electrolyte to that of a mass 
of mercury at 0C of the same form was then found from the 

formula r> r> 

M^ ti. 2 

P T> / T> f 

jj -M 2 

To deduce from the values of p the resistance of a centimetre in 
length having a section of a square centimetre, we must multiply 
them by the value of r for mercury at 0C. See Art. 361. 

* Berlin MonatsbericU, July, 1868. 
E e 2 



420 RESISTANCE. 

The results given by Paalzow are as follow : 

Mixtures of Sulphuric Acid and Water. 



nr. 

with 



Resistance compared 
mercury. 

H 2 SO 4 ____ 15C 96950 

H 2 SO 4 + 14H 2 .... 19C 14157 

H 2 SO 4 + 13H 2 O .... 22C 13310 

H 2 SO 4 + 499 H 2 O ____ 22C 184773 

Sulphate of Zinc and Wat&)\ 

ZnS0 4 + 23H 2 O .... 23C 194400 

ZnS0 4 + 24H 2 ____ 23C 191000 

ZnSO 4 +105H 2 O .... 23C 354000 

Sulphate of Copper and Water. 

CuSO 4 + 45H 2 O .... 22C 202410 

CuSO 4 +105H 2 O .... 22C 339341 

Sulphate of Magnesium and Water. 

MgS0 4 + 34H 2 O .... 22C 199180 

MgS0 4 +107H 2 .... 22C 324600 

Hydrochloric Acid and Water. 

HC1 + 15H 2 .... 23C 13626 

HC1 + 500H 2 O ____ 23C 86679 

365.] MM. F. Kohlrausch and W. A. Nippoldt* have de 
termined the resistance of mixtures of sulphuric acid and water. 
They used alternating 1 magneto-electric currents, the electromotive 
force of which varied from ^ to T \- of that of a Grove s cell, and 
by means of a thermoelectric copper-iron pair they reduced the 
electromotive force to 4 a ^ of that of a Grove s cell. They found 
that Ohm s law was applicable to this electrolyte throughout the 
range of these electromotive forces. 

The resistance is a minimum in a mixture containing about one- 
third of sulphuric acid. 

The resistance of electrolytes diminishes as the temperature 
increases. The percentage increment of conductivity for a rise of 
1C is given in the following table. 

* Pogg., Ann. cxxxviii, p. 286, Oct. 1869. 



;66.] 



OF ELECTROLYTES. 



421 



Resistance of Mixtures of Sulphuric Acid and Water at 22C in terms 
of Mercury at 0C. MM. Kohlrauscli and Nippoldt. 



Specific gravity 
at 185 



0.9985 

1.00 

1.0504 

1.0989 

1.1431 

1.2045 

1.2631 

1.3163 

1.3547 

1.3994 

1.4482 

1.5026 



Percentage 
of H 2 SO, 



0.0 
0.2 
8.3 
14.2 
20.2 
28.0 
35.2 
41.5 
46.0 
50.4 
55.2 
60.3 



Resistance 
at 22 7 C 



746300 
465100 
34530 
18946 
14990 
13133 
13132 
14286 
15762 
17726 
20796 
25574 



Percentage 

increment of 

conductivity 

for 1C. 

0.47 

0.47 

0.653 

0.646 

0.799 

1.317 

1.259 

.410 

.674 

.582 

.417 

.794 



On the Electrical Resistance of Dielectrics. 

366.] A great number of determinations of the resistance of 
gutta-percha, and other materials used as insulating- media, in the 
manufacture of telegraphic cables, have been, made in order to 
ascertain the value of these materials as insulators. 

The tests are generally applied to the material after it has been 
used to cover the conducting wire, the wire being used as one 
electrode, and the water of a tank, in which the cable is plunged, 
as the other. Thus the current is made to pass through a cylin 
drical coating of the insulator of great area and small thickness. 

It is found that when the electromotive force begins to act, the 
current, as indicated by the galvanometer, is by no means constant. 
The first effect is of course a transient current of considerable 
intensity, the total quantity of electricity being that required to 
charge the surfaces of the insulator with the superficial distribution 
of electricity corresponding to the electromotive force. This first 
current therefore is a measure not of the conductivity, but of the 
capacity of the insulating layer. 

But even after this current has been allowed to subside the 
residual current is not constant, and does not indicate the true 
conductivity of the substance. It is found that the current con 
tinues to decrease for at least half an hour, so that a determination 



422 RESISTANCE. [366. 

of the resistance deduced from the current will give a greater value 
if a certain time is allowed to elapse than if taken immediately after 
applying the battery. 

Thus, with Hooper s insulating material the apparent resistance 
at the end of ten minutes was four times, and at the end of 
nineteen hours twenty-three times that observed at the end of 
one minute. When the direction of the electromotive force is 
reversed, the resistance falls as low or lower than at first and then 
gradually rises. 

These phenomena seem to be due to a condition of the gutta- 
percha, which, for want of a better name, we may call polarization, 
and which we may compare on the one hand with that of a series 
of Leyden jars charged by cascade, and, on the other, with Ritter s 
secondary pile, Art. 271. 

If a number of Leyden jars of great capacity are connected in 
series by means of conductors of great resistance (such as wet 
cotton threads in the experiments of M. Gaugain), then an electro 
motive force acting on the series will produce a current, as indicated 
by a galvanometer, which will gradually diminish till the jars are 
fully charged. 

The apparent resistance of such a series will increase, and if the 
dielectric of the jars is a perfect insulator it will increase without 
limit. If the electromotive force be removed and connexion made 
between the ends of the series, a reverse current will be observed, 
the total quantity of which, in the case of perfect insulation, will be 
the same as that of the direct current. Similar effects are observed 
in the case of the secondary pile, with the difference that the final 
insulation is not so good, and that the capacity per unit of surface 
is immensely greater. 

In the case of the cable covered with gutta-percha, &c., it is found 
that after applying the battery for half an hour, and then con 
necting the wire with the external electrode, a reverse current takes 
place, which goes on for some time, and gradually reduces the 
system to its original state. 

These phenomena are of the same kind with those indicated 
by the residual discharge of the Leyden jar, except that the 
amount of the polarization is much greater in gutta-percha, &c. 
than in glass. 

This state of polarization seems to be a directed property of the 
material, which requires for its production not only electromotive 
force, but the passage, by displacement or otherwise, of a con- 



368.] OF DIELECTRICS. 423 

siderable quantity of electricity, and this passage requires a con 
siderable time. When the polarized state has been set up, there 
is an internal electromotive force acting 1 in the substance in the 
reverse direction, which will continue till it has either produced 
a reversed current equal in total quantity to the first, or till the 
state of polarization has quietly subsided by means of true con 
duction through the substance. 

The whole theory of what has been called residual discharge, 
absorption of electricity, electrification, or polarization,, deserves 
a careful investigation, and will probably lead to important dis 
coveries relating to the internal structure of bodies. 

367.] The resistance of the greater number of dielectrics di 
minishes as the temperature rises. 

Thus the resistance of gutta-percha is about twenty times as great 
at 0C as at 24C. Messrs. Bright and Clark have found that the 
following formula gives results agreeing with their experiments. 
If r is* the resistance of gutta-percha at temperature T centigrade, 
then the resistance at temperature T+ 1 will be 

PC = r x 0.8878 , 
the number varies between 0.8878 and 0.9. 

Mr. Hockin has verified the curious fact that it is not until some 
hours after the gutta-percha has taken its temperature that the 
resistance reaches its corresponding value. 

The effect of temperature on the resistance of india-rubber is not 
so great as on that of gutta-percha. 

The resistance of gutta-percha increases considerably on the 
application of pressure. 

The resistance, in Ohms, of a cubic metre of various specimens of 
gutta-percha used in different cables is as follows *. 

Name of Cable. 

Red Sea .267 x 10 12 to .362 x 10 12 

Malta- Alexandria 1 .23 x 1 12 

Persian Gulf 1.80 x 10 12 

Second Atlantic 3.42 x 10 12 

Hooper s Persian Gulf Core... 7 4. 7 x 10 12 
Gutta-percha at 2 4C 3.53 x 10 12 

368.] The following table, calculated from the experiments of 

* Jenkin s Cantor Lectures. 



424 KESISTANCE. [369. 

M. Buff, described in Art. 271, shews the resistance of a cubic 
metre of glass in Ohms at different temperatures. 

Temperature. Resistance. 

200C 227000 

250 13900 

300 1480 

350 1035 

400 735 

369.] Mr. C. F. Varley * has recently investigated the conditions 
of the current through rarefied gases, and finds that the electro 
motive force E is equal to a constant E Q together with a part 
depending on the current according to Ohm s Law, thus 



For instance, the electromotive force required to cause the 
current to begin in a certain tube was that of 323 Dani ell s cells, 
but an electromotive force of 304 cells was just sufficient to 
maintain the current. The intensity of the current, as measured 
by the galvanometer, was proportional to the number of cells above 
304. Thus for 305 cells the deflexion was 2, for 306 it was 4, 
for 307 it was 6, and so on up to 380, or 304 + 76 for which the 
deflexion was 150, or 76 x 1.97. 

From these experiments it appears that there is a kind of 
polarization of the electrodes, the electromotive force of which 
is equal to that of 304 DanielFs cells, and that up to this electro 
motive force the battery is occupied in establishing this state of 
polarization. When the maximum polarization is established, the 
excess of electromotive force above that of 304 cells is devoted to 
maintaining the current according to Ohm s Law. 

The law of the current in a rarefied gas is therefore very similar 
to the law of the current through an electrolyte in which we have 
to take account of the polarization of the electrodes. 

In connexion with this subject we should study Thomson s results, 
described in Art. 57, in which the electromotive force required 
to produce a spark in air was found to be proportional not to the 
distance, but to the distance together with a constant quantity. 
The electromotive force corresponding to this constant quantity 
may be regarded as the intensity of polarization of the electrodes. 

370.] MM. Wiedemann and Ruhlmann have recently f investi- 

* Proc. E. S., Jan. 12, 1871. 

f Serichte der Konigl. Sachs. Gesellschafl, Oct. 20, 1871. 



370.] OF DIELECTRICS. 425 

gated the passage of electricity through gases. The electric current 
was produced by Holtz s machine, and the discharge took place 
between spherical electrodes within a metallic vessel containing 
rarefied gas. The discharge was in general discontinuous, and the 
interval of time between successive discharges was measured by 
means of a mirror revolving along with the axis of Holtz s machine. 
The images of the series of discharges were observed by means of 
a heliometer with a divided object-glass, which was adjusted till 
one image of each discharge coincided with the other image of 
the next discharge. By this method very consistent results were 
obtained. It was found that the quantity of electricity in each 
discharge is independent of the strength of the current and of 
the material of the electrodes, and that it depends on the nature 
and density of the gas, and on the distance and form of the 
electrodes. 

These researches confirm the statement of Faraday* that the 
electric tension (see Art. 48) required to cause a disruptive discharge 
to begin at the electrified surface of a conductor is a little less 
when the electrification is negative than when it is positive, but 
that when a discharge does take place, much more electricity passes 
at each discharge when it begins at a positive surface. They also 
tend to support the hypothesis stated in Art. 57, that the stratum 
of gas condensed on the surface of the electrode plays an important 
part in the phenomenon, and they indicate that this condensation 
is greatest at the positive electrode. 

* Exp. Res., 1501. 



VOL. i. 



F1L 



T 



VOL . I . 



Vol.1. 



FIG. I . 
Art. 118 




of F 



orce 



Surfaces . 



A = ZO . B = 5 . f, . font of fyuiti&ruim . 



ofite darmdcn fress . 






riG. n. 

Art . 119 




Lines of Force arid 



<z Surfaces . 



A ^2O. B=-5 F, Point 
Q, Spherical wtrfaoe of Zero pntcnti 
Af, Ibi,nt of Mnucimujns force along the 
The dotted, fane is the Lirte of force Y = O.I 



AP = 2 All 



For -i/i&. Delegates yflk Clarmdcn Tress 






FIG Hi 

Art. 120 




Lines 



A = 



.for ikeDeliyafes oftfe Clarendon, P 



fee J%ajr*0H,s JttecirL 



Vol.L 



FIG. iv. 

Art. 121. 




Lines of Force and -Ay 



-.. . 






f force a.nf^ jyuijyole/it .<i Surfaces fsi- a, 
vfci spherical Surface m w^/^yi t/te siy>erficia/ density 
is ft harmonic o/ f/ie first 



.For tkeDelegates of ike, ClarmdvTiFress. 



FIG. vi 

Art. J43 




Spherical Harmante of the fAmi degree. 



= 3 . 



jFbr i/ieDelepaes oflfe Clarendon fress 






FIG. VR 

Art. 143 




of the fJii7-<L deqree. 
3 . 



fbrlk&Delepafes of the Clarmdvn-fress . 



I Maxwell s Il*ctricit)r. Vo2 f 



FJG. VHI 
Art. 143 




Spkeristil Harmonic of the /ourttv degree, 
i = 4, s = 2 . 



FIG. JX 
Art. 143 




Spherical Jfarmvnic of the fourth, d&jree. 



; 



.. Vol.F. 



x. 

Arc. 192. 




E72ip,se.9 



(Zarerufaifress 



Ma 



- VolL 



FIG XL. 
Art. 193. 




of* Jforce- 



the, 



&/ * 




Limes of* J*orce between two Plates . 



Maucwetl s Jllectncity. . Vcl. 2. 



FIG-, xm 

Art 203 




r 






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