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DUBLIN UNIVERSITY PRESS SERIES. 



A HISTORY 



OF THE 



THEORIES OF AETHER AND ELECTRICITY 

FKOM THE AGE OF DESCAKTES TO THE CLOSE OF 
THE NINETEENTH CENTURY. 



BY 



E. T. WH1TTAKER, 

Hon. Sc.D. (DubL}; I.E.S.; Roy at Astronomer of Ireland. 




LONGMANS, GREEN, AND CO., 

39 PATERNOSTER ROW, LONDON, 
NEW YORK, BOMBAY, AND CALCUTTA. 

HODGES, FIGGIS, & CO., LTD., DUBLIN. 
1910. 



MM* 



DUBLIN : 

PRINTED AT UHE UNIVERSITY PRESS, 
BY PONSONBY AND OIBRS. 



THE author desires to record his gratitude to Mr. W. W. 
EOUSE BALL, Fellow of Trinity College, Cambridge, and to 
Professor W. McF. ORR, F.R.S., of the Royal College of Science 
for Ireland ; these friends have read the proof-sheets, and have 
made many helpful suggestions and criticisms. 

Thanks are also 'due to the BOARD OF TRINITY COLLEGE, 
DUBLIN, for the financial assistance which made possible the 
publication of the work. 



236360 



CONTENTS. 

CHAPTEK I. 

y THE THEORY OF THE AETHER IN THE SEVENTEENTH CENTURY. 

Page 

Matter and aether, . . . . . . .1 

The physical writings of Descartes, ..... 2 

Early history of magnetism : Petrus Peregrinus, Gilbert, Descartes, 7 
Fermat attacks Descartes' theory of light : the principle of least 

time, ........ 10 

Hooke's undulat>ry theory : the advance of wave -fronts, . . 11 

Newton overthrows Hooke's theory of colours, . . .15 

Conception of the aether in the writings of Newton, . . 17 

Newton's theories of the periodicity of homogeneous light, and of 

fits of easy transmission, . . ,20 

The velocity of light : Galileo, Roemer, . . . .21 

Huygens' Traite de la lumiere : his theories of the propagation of 

waves, and of crystalline optics, . . .22 

Newton shows that rays obtained by double refraction have sides : 

his objections to the undulatory theory, . . .28 

X 

CHAPTER II. 

ELECTRIC AND MAGNETIC SCIENCE, PRIOR TO THE INTRODUCTION OF 
THE POTENTIALS. 

The electrical researches of Gilbert : the theory of emanations, . 29 
State of physical science in the first half of the eighteenth century, 32 
Gray discovers electric conduction : Desaguliers, . . 37 

The electric fluid, ....... 38 

Du Fay distinguishes vitreous and resinous electricity, . .39 

Xollet's effluent and affluent streams, . . . .40 

The Leyden phial, ..... . . 41 

The one-fluid theory : ideas of Watson and Franklin, . . 42 

Final overthrow by Aepinus of the doctrine of effluvia, . . 48 

Priestley discovers the law of electrostatic force, . . .50 



viii Contents. 

Page 

Cavendish, . ... 51 

Michell discovers the law of magnetic force, . . . .54 

The two-fluid theory : Coulomb, . . . . .56 

Limited mobility of the magnetic fluids, . . .58 

Poisson's mathematical theory of electrostatics, . . .59 

The equivalent surface- and volume-distributions of magnetism : 

Poisson's theory of magnetic induction, . . .64 

Green's Nottingham memoir, . . . . .65 



CHAPTER III. 
GALVANISM, FROM GALVANI TO OHM. 

Sulzer's discovery, ... . .67 

Galvanic phenomena, ....... 68 

Rival hypotheses regarding the galvanic fluid, , . .70 

The voltaic pile, ....... 72 

Nicholson and Carlisle decompose water voltaically, . . 75 

Davy's chemical theory of the pile, ..... 76 

Grothuss' chain, . . . . . . .78 

De La Rive's hypothesis, . . . . . .79 

Berzelius' scheme of electro-chemistry, . . . .80 

Early attempts to discover a connexion between electricity and 

magnetism, . ... 83 

Oersted's experiment : his explanation of it, . . .85 

The law of Biot and Savart, . . . . . .86 

The researches of Ampere on electrodynamics, . . 87 

Seebeck's phenomenon, . . . . . .90 

Davy's researches on conducting power, . . . .94 

Ohm's theory : electroscopic force, . . . . .95 

CHAPTER IV. 
THE LUMINIFEBOUS MEDIUM, FROM BRADLEY TO FRESNEL. 

Bradley discovers aberration, . . . . .99 

John Bernoulli's model of the aether, .... 100 

Maupertuis and the principle of least action, . . . 102 

Views of Euler, Courtivron, Melvill, .... 104 

Young defends the undulatory theory, and explains the colours of 

thin plates, ... ... 105 

Laplace supplies a corpuscular theory of double refraction, . . 109 



Contents. ix 

Page 

Young proposes a dynamical theory of light in crystals, . . 110 

Researches of Malus on polarization, .... Ill 

Recognition of biaxal crystals, ... . 113 

Fresnel successfully explains diffraction, . . . 114 

His theory of the relative motion of aether and matter, . . 115 

Young suggests the transversality of the vibrations of light, . 121 

Fresnel discusses the dynamics of transverse vibrations, . . 123 

Fresnel's theory of the propagation of light in crystals, . . 125 

Hamilton predicts conical refraction, . . . ] 31 

Fresnel's theory of reflexion, ..... 133 




CHAPTER V. 
I ,THE AETHER AS AN ELASTIC SOLID. 

Astronomical objection to the elastic-solid theory : Stokes' 

hypothesis. . . . . . . .137 

Navier and Cauchy discover the equation of vibration of an elastic 

solid, 139 

Poisson distinguishes condensational and distortional waves, . 141 

Cauchy's first and second theories of light iq, crystals, . . 143 

Cauchy's first theory of reflexion, ..... 145 

His second theory of reflexion, ..... 147 

The theory of reflexion of MacCullagh and Neumann, . . 148 

Green discovers the correct conditions at the boundaries, . . 151 

Green's theory of reflexion : objections to it, . . . 152 

MacCullagh introduces a new type of elastic solid, . . . 154 

W. Thomson's model of a rotationally-elastic body, . . 157 

Cauchy's third theory of reflexion : the contractile aether, . . 158 

Later work of W. Thomson and others on the contractile aether, . 159 

Green's first and second theories of light in crystals, . . 161 

Influence of Green, ....... 167 

Researches of Stokes on the relation of the direction of vibration of 

light to its plane of polarization, .... 168 

The hypothesis of aeolotropic inertia, .... 171 

Rotation of the plane of polarization of light by active bodies, . 173 

MacCullagh's theory of natural rotatory power, . . 175 

MacCullagh's and Cauchy's theory of metallic reflexion, . . 177 

Extension of the elastic -solid theory to metals, . . 179 

Lord Rayleigh's objection, .... . 181 

Cauchy's theory of dispersion, . . 182 

Boussinesq's elastic-solid theory, ..... 185 



x Contents. 

CHAPTEE VI. 

FARADAY. 

Page 

Discovery of induced currents : lines of magnetic force, . . 189 

Self-induction, . . . . . . .193 

Identity of frictional and voltaic electricity : Faraday's views on the 

nature of electricity, . . . . . 194 

Electro-chemistry, . . ".. . *. . . . 197 

Controversy between the adherents of the chemical and contact 

hypotheses, . . . . . . 201 

The properties of dielectrics, . . . . . 206 

Theory of dielectric polarization : Faraday, W. Thomson, and 

Mossotti, . . : . . . . .211 

The connexion between magnetism and light, . . . 213 

Airy's theory of magnetic rotatory polarization, . . 214 

Faraday's Thoughts on Ray -Vibrations, . ..''-. . . 217 

Researches of Faraday and Pliicker on diamagnetism, . . 218 

CHAPTER VII. 

THE MATHEMATICAL ELECTRICIANS OF THE MIDDLE OF THE NINETEENTH 

CENTURY. 

F. Neumann's theory of induced currents : the electrodynamic 

potential, . . . . . ; . . 222 

W. Weber's theory of electrons, . . . . .225 

Riemann's law, . . . ... . 231 

v-Proposals to modify the law of gravitation, . .. . . 232 

Weber's theory of paramagnetism and diamagnetism : later theories, 234 

Joule's law : energetics of the voltaic cell, .... 239 

Researches of Helmholtz on electrostatic and electrodynamic energy, 242 
W. Thomson distinguishes the circuital and irrotational magnetic 

vectors, ........ 244 

His theory of magnecrystallic action, .... 245 

His formula for the energy of a magnetic field, . . . 247 

Extension of this formula to the case of fields produced by currents, 249 
Kirchhoff identifies Ohm's electroscopic force with electrostatic 

potential, . . . . . . / 251 

The discharge of a Leyden jar : W. Thomson's theory, . . 253 

The velocity of electricity and the propagation of telegraphic signals, 254 

Clausius' law of force between electric charges : crucial experiments, 261 

Nature of the current, ...... 263 

The thermo-electric researches of Peltier and W. Thomson, 264 



Contents. xi 

CHAPTER VIII. 
MAXWELL. 

Page 

Gauss and Riemann on the propagation of electric actions, . . 268 

Analogies suggested by W. Thomson, .... 269 

Maxwell's hydrodynamical analogy, ..... 271 

The vector potential, ...... 273 

Linear and rotatory interpretations of magnetism, . . . 274 

Maxwell's mechanical model of the electromagnetic field, . . 276 

Electric displacement, ...... 279 

Similarity of electric vibrations to those of light, . . . 281 

Connexion of refractive index and specific inductive capacity, . 283 
Maxwell's memoir of 1864, . . ... .284 

The propagation of electric disturbances in crystals and in metals, . 288 

Anomalous dispersion, ...... 291 

The Max well -Sellmeier theory of dispersion, . . . 292 

Imperfections of the electromagnetic theory of light, . . 295 

The theory of L. Lorenz, ...... 297 

Maxwell's theory of stress in the electric field, . . . 300 

The pressure of radiation, ...... 303 

Maxwell's theory of the magnetic rotation of light, . . . 307 

CHAPTER IX. 

MODELS OF THE AETHER. 

Analogies in which a rotatory character is attributed to magnetism, 310 

Models in which magnetic force is represented as a linear velocity, 311 
Researches of W. Thomson, Bjerknes, and Leahy, on pulsating and 

oscillating bodies, ...... 316 

MacCullagh's quasi-elastic solid as a model of the electric medium, 318 
The Hall effect, . . . . . .320 

Models of Riemann and Fitz Gerald, . . . . 324 

Vortex-atoms, . . . . . . .326 

The vortex-sponge theory of the aether : researches of W. Thomson, 

Fitz Gerald, and Hicks, , . . . . .327 

CHAPTER X. 

THE FOLLOWERS OF MAXWELL. 

Helmholtz and H. A. Lorentz supply an electromagnetic theory of 

reflexion, ....... 337 

Crucial experiments of Helmholtz and Schiller, . . . 338 



xii Contents. 

Page 

Convection -currents : Rowland's experiments, . . . 339 
The moving charged sphere : researches of J. J. Thomson, Fitz Gerald, 

and Heaviside, . . . . . . . 340 

Conduction of rapidly -alternating currents, .... 344 

Fitz Gerald devises the magnetic radiator, .... 345 

Poynting's theorem, ....... 347 

Poynting and J. J. Thomson develop the theory of moving lines of 

force, . . . . . . . 349 

Mechanical momentum in the electromagnetic field, . . 352 

New derivation of Maxwell's equations by Hertz, . . . 353 

Hertz's assumptions and Weber's theory, .... 356 

Experiments of Hertz on electric waves, .... 357 

The memoirs of Hertz and Heaviside on fields in which material 

bodies are in motion, ...... 365 

The current of dielectric convection, ..... 367 

Kerr's magneto-optic phenomenon, . ... 368 

Rowland's theory of magneto-optics, .... 369 

The rotation of the plane of polarization in naturally active bodies, 370 



CHAPTER XI. 

CONDUCTION IN SOLUTIONS AND GASES, FROM FARADAY TO 
J. J. THOMSON. 

The hypothesis of Williamson and Clausius, . . . 372 

Migration of the ions, ...... 373 

The researches of Hittorf and Kohlrausch, .... 374 

Polarization of electrodes, ...... 375 

Electrocapillarity, .... . 376 

Single differences of potential, . . . . . 379 

Helmholtz' theory of concentration-cells, .... 381 

Arrhenius' hypothesis, ... ... 383 

The researches of Nernst, ... . 386 

Earlier investigations of the discharge in rarefied gases, . . 390 

Faraday observes the dark space, ..... 391 

Researches of Pliicker, Hittorf, Goldstein, and Varley, on the 

cathode rays, .... . 393 

Crookes and the fourth state of matter, .... 394 
Objections and alternatives to the charged-particle theory of 

cathode rays, ....... 395 

Giese's and Schuster's ionic theory of conduction in gases, . . 397 

J. J. Thomson measures the velocity of cathode rays, . . 400 



Contents. xiii 

Page 

Discovery of X-rays : hypotheses regarding them, . . 401 

Further researches of J. J. Thomson on cathode rays : the ratio m/e, 404 

Vitreous and resinous electricity, . . . 406 

Determination of the ionic charge by J. J. Thomson, . . 407 

Becquerel's radiation : discovery of radio-active substances, . 408 



CHAPTER XII. 

THE THEORY OF AETHER AND ELECTRONS IN THE CLOSING YEARS 
OF THE NINETEENTH CENTURY. 

Stokes' theory of aethereal motion near moving bodies, . . 411 

Astronomical phenomena in which the velocity of light is involved, 413 

Crucial experiments relating to the optics of moving bodies, . 416 

Lorentz' theory of electrons, ...... 419 

The current of dielectric convection : Rontgen's experiment, . 426 

The electronic theory of dispersion, ..... 428 

Deduction of Fresnel's formula from the theory of electrons, . 430 

Experimental verification of Lorentz' hypothesis, . . . 431 

Fitz Gerald's explanation of Michelson's experiment, . . 432 
Lorentz' treatise of 1895, . . . . . . . 433 

Expression of the potentials in terms of the electronic charges, . 436 

Further experiments on the relative motion of earth and aether, . 437 
Extension of Lorentz' transformation : Larmor discovers its 

connexion with Fitz Gerald's hypothesis of contraction, . 440 
Examination of the supposed primacy of the original variables : 

fixity relative to the aether : the principle of relativity, . 444 

The phenomenon of Zeeman, ..... 449 

Connexion of Zeeman's effect with the magnetic rotation of light, . 452 

The optical properties of metals, ..... 454 

The electronic theory of metals, ..... 456 

Thermionics, ........ 464 

INDEX, . 470 



MEMOKANDUM ON NOTATION. 



VECTORS are denoted by letters in clarendon type, as E. 

The three components of a vector E are denoted by E x , E y , E z ; 
and the magnitude of the vector is denoted by E, so that 



The vector product of two vectors E and H, which is denoted 
by [E . H], is the vector whose components are (E y H z - E^H^ 
E Z H X - E*H Z , EtHy - E y H x }. Its direction is at right angles to the 
direction of E and H, and its magnitude is represented by twice the 
area of the triangle formed by them. 



The scalar product of E and H is E X H X + E y E y + E^. It is 
denoted by (E . H). 

OJ^j (1 jjj O Jjj 

The quantity -f y -I- is denoted by div E. 

The vector whose components are 

J f * t * ^ . y _ *\ 

is denoted by curl E. 

If V denote a scalar quantity, the vector whose components are 
8F 8F 9F\ 
- 5T * ^7' - -5T 1S denoted b 7 g rad ^ 



The symbol V is used to denote the vector operator whose 

898 

components are , , . 
dx dy 82 

Differentiation with respect to the time is frequently indicated by 
a dot placed over the symbol of the variable which is differentiated. 



THEORIES OF AETHER AND ELECTRICITY. 

CHAPTEK I. 

THE THEORY OF THE AETHER IN THE SEVENTEENTH CENTURY. 

THE observation of the heavens, which has been pursued con- 
tinually from the earliest ages, revealed to the ancients the 
regularity of the planetary motions, and gave rise to the 
conception of a universal order. Modern research, building on 
this foundation, has shown how intimate is the connexion 
between the different celestial bodies. They are formed of the 
same kind of matter ; they are similar in origin and history ; 
and across the vast spaces which divide them they hold 
perpetual intercourse. 

Until the seventeenth century the only influence which was 
known to be capable of passing from star to star was that of 
light. Newton added to this the force of gravity ; and it is now 
recognized that the power of communicating across vacuous 
regions is possessed also by the electric and magnetic attractions. 

It is thus erroneous to regard the heavenly bodies as isolated 
in vacant space; around and between them is an incessant 
conveyance and transformation of energy. To the vehicle of this 
activity the name aetlier has been given. 

The aether is the solitary tenant of the universe, save for 
that infinitesimal fraction of space which is occupied by ordinary 
matter. Hence arises a problem which has long engaged 
attention, and is not yet completely solved : What relation 
subsists between the medium which fills the interstellar void 
and the condensations of matter that are scattered throughout 
it? 

B 



$5 l ' r The ^Theory of the -Aether 

The history of this problem may be traced back continuously 
to the earlier half of the seventeenth century. It first emerged 
clearly in that reconstruction of ideas regarding the physical 
universe which was effected by Eene Descartes. 

Descartes was born in 1596, the son of Joachim Descartes, 
Counsellor to the Parliament of Brittany. As a young man he 
followed the profession of arms, and served in the campaigns of 
Maurice of Nassau, and the Emperor ; but his twenty-fourth 
year brought a profound mental crisis, apparently not unlike 
those which have been recorded of many religious leaders ; and 
he resolved to devote himself thenceforward to the study of 
philosophy. 

The age which preceded the birth of Descartes, and that in 
which he lived, were marked by events which greatly altered 
the prevalent conceptions of the world. The discovery of 
America, the circumnavigation of the globe by Drake, the over- 
throw of the Ptolemaic system of astronomy, and the invention 
of the telescope, all helped to loosen the old foundations and to 
make plain the need for a new structure. It was this that 
Descartes set himself to erect. His aim was the most ambitious 
that can be conceived ; it was nothing less than to create from 
the beginning a complete system of human knowledge. 

Of such a system the basis must necessarily be metaphysical ; 
and this part of Descartes' work is that by which he is most 
widely known. But his efforts were also largely devoted to the 
mechanical explanation of nature, which indeed he regarded as 
one of the chief ends of Philosophy.* 

The general character of his writings may be illustrated by 
a comparison with those of his most celebrated contemporary, f 
Bacon clearly defined the end to be sought for, and laid down 
the method by which it was to be attained; then, recognizing 
that to discover all the laws of nature is a task beyond the 

* Of the works M'hich bear on our present subject, the Dioptrique and the 
Me'teores were published at Leyden in 1638, and the Principia Philosophiae at 
Amsterdam in 1644, six years before the death of its author. 

t The principal philosophical works of Bacon were written about eighteen years 
before those of Descartes. 



in the SeventeentJi Century. 3 

powers of one man or one generation, he left to posterity the 
work of filling in the framework which he had designed. 
Descartes, on the other hand, desired to leave as little as possible 
for his successors to do ; his was a theory of the universe, worked 
out as far as possible in every detail. It is, however, impossible 
to derive such a theory inductively unless there are at hand 
sufficient observational data on which to base the induction ; 
and as such data were not available in the age of Descartes, 
he was compelled to deduce phenomena from preconceived 
principles and causes, after the fashion of the older philosophers. 
To the inherent weakness of this method may be traced the 
errors that at last brought his scheme to ruin. 

The contrast between the systems of Bacon and Descartes is 
not unlike that between the Eoman republic and the empire of 
Alexander. In the one case we have a career of aggrandizement 
pursued with patience for centuries ; in the other a growth of 
fungus-like rapidity, a speedy dissolution, and an immense 
influence long exerted by the disunited fragments. The 
grandeur of Descartes' plan, and the boldness of its execution, 
stimulated scientific thought to a degree before unparalleled ; 
and it was largely from its ruins that later philosophers 
constructed those more valid theories which have endured to 
our own time. 

Descartes regarded the world as an immense machine, 
operating by the motion and pressure of matter. " Give me 
matter and motion," he cried, " and I will construct the universe." 
A peculiarity which distinguished his system from that which 
afterwards sprang from its decay was the rejection of all forms 
of action at a distance ; he assumed that force cannot be com- 
municated except by actual pressure or impact. By this 
assumption he was compelled to provide an explicit mechanism 
in order to account for each of the known forces of nature a 
task evidently much more difficult than that which lies before 
those who are willing to admit action at a distance as an 
ultimate property of matter. 

Since the sun interacts with the planets, in sending them 

B 2 



4 The Theory of the Aether 

light and heat and influencing their motions, it followed from 
Descartes' principle that interplanetary space must be a plenum,, 
occupied by matter imperceptible to the touch but capable of 
serving as the vehicle of force and light. This conclusion in 
turn determined the view which he adopted on the all- important 
question of the nature of matter. 

Matter, in the Cartesian philosophy, is characterized not by 
impenetrability, or by any quality recognizable by the senses,, 
but simply by extension ; extension constitutes matter, and 
matter constitutes space. The basis of all things is a primitive,, 
elementary, unique type of matter, boundless in extent and 
infinitely divisible. In the process of evolution of the universe 
three distinct forms of this matter have originated, correspond- 
ing respectively to the luminous matter of the sun, the 
transparent matter of interplanetary space, and the dense, 
opaque matter of the earth. " The first is constituted by what 
has been scraped off the other particles of matter when they 
were rounded ; it moves with so much velocity that when it 
meets other bodies the force of its agitation causes it to be 
broken and divided by them into a heap of small particles that 
are of such a figure as to fill exactly all the holes and small 
interstices which they find around these bodies. The next type 
includes most of the rest of matter ; its particles are spherical, 
and are very small compared with the bodies we see on the 
earth ; but nevertheless they have a finite magnitude, so that 
they can be divided into others yet smaller. There exists in 
addition a third type exemplified by some kinds of matter 
namely, those which, on account of their size and figure, cannot be 
so easily moved as the preceding. I will endeavour to show that 
all the bodies of the visible world are composed of these three 
forms of matter, as of three distinct elements ; in fact, that the sun 
and the fixed stars are formed of the first of these elements, the 
interplanetary spaces of the second, and the earth, with the 
planets and comets, of the third. For, seeing that the sun and 
the fixed stars emit light, the heavens transmit it, and the earth, 
the planets, and the comets reflect it, it appears to me that there 



in the Seventeenth Century. 5 

is ground for using these three qualities of luminosity, trans- 
parence, and opacity, in order to distinguish the three elements 
of the visible world.* 

According to Descartes' theory, the sun is the centre of an 
immense vortex formed of the first or subtlest kind of inatter.f 
The vehicle of light in interplanetary space is matter of the 
second kind or element, composed of a closely packed assemblage 
of globules whose size is intermediate between that of the 
vortex-matter and that of ponderable matter. The globules of 
the second element, and all the matter of the first element, are 
constantly straining away from the centres around which they 
turn, owing to the centrifugal force of the vortices ;J so that the 
globules are pressed in contact with each other, and tend to 
move outwards, although they do not actually so move. It is 
the transmission of this pressure which constitutes light ; the 
action of light therefore extends on all sides round the sun and 
fixed stars, and travels instantaneously to any distance. |j In 
the Dwptrique$ vision is compared to the perception of the 
presence of objects which a blind man obtains by the use of his 
stick ; the transmission of pressure along the stick from the 
object to the hand being analogous to the transmission of 
pressure from a luminous object to the eye by the second kind 
of matter. 

Descartes supposed the " diversities of colour and light " to 
he due to the different ways in which the matter moves.** In 
the Meteores,^ the various colours are connected with different 
rotatory velocities of the globules, the particles winch rotate most 
rapidly giving the sensation of red, the slower ones of yellow, and 
the slowest of green and blue the order of colours being taken 
from the rainbow. The assertion of the dependence of colour 

* Principia, Part iii, 52. 

t It is curious to speculate on the impression which would have been produced 
had the spirality of nehulse heen discovered hefore the overthrow of the Cartesian 
theory of vortices. 

J Ibid., 55-59. Ibid., 63. || Ibid., 64. IT Discours premier. 

** Principia, Part iv, 195. ft Discours Huitieme. 



6 The Theory of the Aether 

on periodic time is a curious foreshadowing of one of the 
great discoveries of Newton. 

The general explanation of light on these principles was 
amplified by a more particular discussion of reflexion and 
refraction. The law of reflexion that the angles of incidence 
and refraction are equal had been known to the Greeks ; but 
the law of refraction that the sines of the angles of incidence 
and refraction are to each other in a ratio depending on the 
media was now published for the first time.* Descartes gave 
it as his own ; but he seems to have been under considerable 
obligations to Willebrord Snell (b. 1591, d. 1626), Professor of 
Mathematics at Leyden, who had discovered it experimentally 
(though not in the form in which Descartes gave it) about 
1621. Snell did not publish his result, but communicated it in 
manuscript to several persons, and Huygens affirms that this 
manuscript had been seen by Descartes. 

Descartes presents the law as a deduction from theory. 
This, however, he is able to do only by the aid of analogy ;. 
when rays meet ponderable bodies, " they are liable to be 
deflected or stopped in the same way as the motion of a ball or 
a stone impinging 011 a body " ; for " it is easy to believe that 
the action or inclination to move, which I have said must be 
taken for light, ought to follow in this the same laws as 
motion."f Thus he replaces light, whose velocity of propagation 
he believes to be always infinite, by a projectile whose velocity 
varies from one medium to another. The law of refraction is 
then proved as follows J : 

Let a ball thrown from A meet at B a cloth CBE, so weak 
that the ball is able to break through it and pass beyond, but 
with its resultant velocity reduced in some definite proportion,, 
say 1 : k. 

Then if BI be a length measured on the refracted ray 
equal to AB, the projectile will take k times as long to 
describe BI as it took to describe AB. But the component 

* Dioptrique, Discount second. t Jbid., Discows premier. 

% Ibid., Discotirs second. 



in the Seventeenth Century. 7 

of velocity parallel to the cloth must be unaffected by the 
impact; and therefore the projection BE of the refracted ray 
must be k times as long as the projection BC of the incident 




I 

ray. So if i and r denote the angles of incidence and refraction, 
we have 

BE BC 



or the sines of the angles of incidence and refraction are in a 
constant ratio ; this is the law of refraction. 

Desiring to include all known phenomena in .his system, 
Descartes devoted some attention to a class of effects which 
were at that time little thought of, but which were destined to 
play a great part in the subsequent development of Physics. 

The ancients were acquainted with the curious properties 
possessed by two minerals, amber (riXtKrpov) and magnetic 
iron ore (77 \iOos Mayv?}r/e). The former, when rubbed, 
attracts light bodies : the latter has the power of attracting 
iron. 

The use of the magnet for the purpose of indicating direc- 
tion at sea does not seem to have been derived from classical 
antiquity ; but it was certainly known in the time of the 
Crusades. Indeed, magnetism was one of the few sciences 
which progressed during the Middle Ages ; for in the thirteenth 
century Petrus Peregrinus,* a native of Maricourt in Picardy, 
made a discovery of fundamental importance. 

Taking a natural magnet or lodestone, which had been 
rounded into a globular form, he laid it on a needle, and marked 

* His Epistola was written in 1269. 



8 The Theory of the Aether 

the line along which the needle set itself. Then laying the 
needle on other parts of the stone, he obtained more lines in 
the same way. When the entire surface of the stone had been 
covered with such lines, their general disposition became evident; 
they formed circles, which girdled the stone in exactly the same 
way as meridians of longitude girdle the earth ; and there were 
two points at opposite ends of the stone through which all the 
circles passed, just as all the meridians pass through the Arctic 
and Antarctic poles of the earth.* Struck by the analogy, 
Peregrinus proposed to call these two points the poles of the 
magnet : and he observed that the way in which magnets set 
themselves and attract each other depends solely on the position 
of their poles, as if these were the seat of the magnetic power. 
Such was the origin of those theories of poles and polarization 
which in later ages have played so great a part in Natural 
Philosophy. 

The observations of Peregrinus were greatly extended not 
long before the tune of Descartes by William Gilberd or Gilbertf 
(6. 1540, d. 1603). Gilbert was born at Colchester: after 
studying at Cambridge, he took up medical practice in London, 
and had the honour of being appointed physician to Queen 
Elizabeth. In 1600 he published a work* on Magnetism and 
Electricity, with which the modern history of both subjects 
begins. 

Of Gilbert's electrical researches we shall speak later : in 
magnetism he made the capital discovery of the reason why 
magnets set in definite orientations with respect to the earth ; 
which is, that the earth is itself a great magnet, having one of 
its poles in high northern and the other in high southern 
latitudes. Thus the property of the compass was seen to be 
included in the general principle, that the north-seeking pole of 

* " Procul dubio oranes lineae hujusmodi in duo puncta concurrent sicut omnes 
orbes meridian! in duo concurrunt polos mundi oppositos." 

t The form in the Colchester records is Gilberd. 

J Gulielmi Gilberti de Magnete, Magneticisque corporibus, et de magno magnete 
tellure : London, 1600. An English translation by P. F. Mottelay was published 
in 1893. 



in the Seventeenth Century. 9 

every magnet attracts the south-seeking pole of every other 
magnet, and repels its north-seeking pole. 

Descartes attempted* to account for magnetic phenomena 
by his theory of vortices. A vortex of fluid matter was 
postulated round each magnet, the matter of the vortex entering 
by one pole and leaving by the other : this matter was supposed 
to act on iron and steel by virtue of a special resistance to its 
motion afforded by the molecules of those substances. 

Crude though the Cartesian system was in this and many 
other features, there is no doubt that by presenting definite 
conceptions of molecular activity, and applying them to so wide 
a range of phenomena, it stimulated the spirit of inquiry, and 
prepared the way for the more accurate theories that came after. 
In its own day it met with great acceptance: the confusion which 
had resulted from the destruction of the old order was now, as 
it seemed, ended by a reconstruction of knowledge in a system 
at once credible and complete. Nor did its influence quickly 
wane ; for even at Cambridge it was studied long after Newton 
had published his theory of gravitation ;f and in the middle of 
the eighteenth century Euler and two of the Bernoullis based 
the explanation of magnetism on the hypothesis of vertices.* 

Descartes' theory of light rapidly displaced the conceptions 
which had held sway in the Middle Ages. The validity 
of his explanation of refraction was, however, called in 
question by his fellow-countryman Pierre de Ferinat (b. 1601, 
d. 1665), and a controversy ensued, which was kept up 
by the Cartesians long after the death of their master. Fermat 

* Principia, Part iv, 133 sqq. 

f Winston has recorded that, having returned to Cambridge after his 
ordination in 1693, he resumed his studies there, " particularly the Mathematicks, 
and the Cartesian Philosophy : which was alone in Vogue with us at that Time. 
But it was not long before I, with immense Pains, but no Assistance, set myself 
with the utmost Zeal to the study of Sir Isaac Newton's M-onderful Discoveries." 
\Vhiston's Memoirs (1749), i, p. 36. 

J Their memoirs shared a prize of the French Academy in 1743, and were 
printed in 1752 in the Heciieil des pieces qui ontremporte les prix de VAcad., tome v. 
Renati Descartes Epistolae, Pars tertia ; Amstelodami, 1683. The Fennat 
correspondence is comprised in letters xxix to XLVI. 



10 The Theory of the Aether 

eventually introduced a new fundamental law, from which he 
proposed to deduce the paths of rays of light. This was the 
celebrated Principle of Least Time, enunciated* in the form, 
" Nature always acts by the shortest course." From it the law 
of reflexion can readily be derived, since the path described by 
light between a point 011 the incident ray and a point on the 
reflected ray is the shortest possible consistent with the con- 
dition of meeting the reflecting surfaces. t In order to obtain the 
law of refraction, Fermat assumed that " the resistance of the 
media is different," and applied his "method of maxima and 
minima " to find the path which would be described in the least 
time from a point of one medium to a point of the other. In 
1661 he arrived at the solution.* "The result of my work," he 
writes, " has been the most extraordinary, the most unforeseen, 
and the happiest, that ever was ; for, after having performed all 
the equations, multiplications, antitheses, and other operations 
of my method, and having finally finished the problem, I have 
found that my principle gives exactly and precisely the same 
proportion for the refractions which Monsieur Descartes has 
established." His surprise was all the greater, as he had 
supposed light to move more slowly in dense than in rare media, 
whereas Descartes had (as will be evident from the demonstration 
given above) been obliged to make the contrary supposition. 
Although Fermat's result was correct, and, indeed, of high 
permanent interest, the principles from which it was derived 
were metaphysical rather than physical in character, and con- 
sequently were of little use for the purpose of framing a 
mechanical explanation of light. Descartes' theory therefore 
held the field until the publication in 1667 of the Micrographics 

* Epist. XLII, written at Toulouse in August, 1657, to Monsieur de la 
Chambre ; reprinted in (Euvres de Fermat (ed. 1891), ii, p. 354. 

t That reflected light follows the shortest path was no new result, for it had 
been affirmed (and attributed to Hero of Alexandria) in the Ke<t>aA.cua rwv OTTTIKUHT 
of Heliodorns of Larissa, a work of which several editions were published in the 
seventeenth, century. 

J Epist. XLIII, written at Toulouse on Jan. 1, 1662 ; reprinted in (Euvres de 
Fermat, ii, p. 457 ; i, pp. 170, 173. 

The imprimatur of Viscount Brouncker, P.R.S., is dated Nov. 23, 1664. 



in the Seventeenth Centnry. 11 

of Eobert Hooke (b. 1635, d. 1703), one of the founders of the 
Eoyal Society, and at one time its Secretary. 

Hooke, who was both an observer and a theorist, made two 
experimental discoveries which concern our present subject ; but 
in both of these, as it appeared, he had been anticipated. The 
first* was the observation of the iridescent colours which are 
seen when light falls on a thin layer of air between two glass 
plates or lenses, or on a thin film of any transparent substance. 
These are generally known as the " colours of thin plates," or 
" Newton's rings " ; they had been previously observed by Boyle.f 
Hooke's second experimental discovery,^ made after the date of 
the Micrographia, was that light in air is not propagated exactly 
in straight lines, but that there is some illumination within the 
geometrical shadow of an opaque body. This observation had 
been published in 1665 in. a posthumous work of Francesco 
Maria Grimaldi (b. 1618, d. 1663), who had given to the phe- 
nomenon the name diffraction. 

Hooke's theoretical investigations on light were of great 
importance, representing as they do the transition from the 
Cartesian system to the fully developed theory of undulations. 
He begins by attacking Descartes' proposition, that light is a 
tendency to motion rather than an actual motion. " There is," 
he observes, 1 1 " no luminous Body but has the parts of it in 
motion more or less " ; and this motion is " exceeding quick." 
Moreover, since some bodies (e.g. the diamond when rubbed or 
heated in the dark) shine for a considerable time without being 
wasted away, it follows that whatever is in motion is not per- 
manently lost to the body, and therefore that the motion must 
be of a to-and-fro or vibratory character. The amplitude of the 
vibrations must be exceedingly small, since some luminous bodies 
(e.g. the diamond again) are very hard, and so cannot yield or 
bend to any sensible extent. 

* Micrographia, p. 47. t Boyle's Works (ed. 1772), i, p. 742. 

% Hooke's Posthumous Works, p. 186. 

Pkysico- Mathesis de lumine, coloribits, et iride. Bologna, 1665 ; book i, prop. i. 
|| Micrographia, p. 55. 



12 The Theory of the Aether 

Concluding, then, that the condition associated with the 
emission of light by a luminous body is a rapid vibratory motion 
of very small amplitude, Hooke next inquires how light travels 
through space. " The next thing we are to consider," he says, 
" is the way or manner of the trajection of this motion through 
the interpos'd pellucid body to the eye : And here it will be 
easily granted 

" First, that it must be a body susceptible and impartible of 
this motion that will deserve the name of a Transparent ; and 
next, that the parts of such a body must be homogeneous, or of 
the same kind. 

" Thirdly, that the constitution and motion of the parts must 
be such that the appulse of the luminous body may be commu- 
nicated or propagated through it to the greatest imaginable 
distance in the least imaginable time, though I see no reason to 
affirm that it must be in an instant. 

" Fourthly, that the motion is propagated every way through 
an Homogeneous medium by direct or straight lines extended every 
way like Eays from the centre of a Sphere. 

" Fifthly, in an Homogeneous medium this motion is propa- 
gated every way with equal velocity, whence necessarily every 
pulse or vibration of the luminous body will generate a Sphere, 
which will continually increase, and grow bigger, just after the 
same manner (though indefinitely swifter) as the waves or rings 
on the surface of the water do swell into bigger and bigger 
circles about a point of it, where by the sinking of a Stone the 
motion was begun, whence it necessarily follows, that all the 
parts of these Spheres undulated through an Homogeneous medium 
cut the Kays at right angles." 

Here we have a fairly definite mechanical conception. It 
resembles that of Descartes in postulating a medium as the 
vehicle of light ; but according to the Cartesian hypothesis the 
disturbance is a statical pressure in this medium, while in 
Hooke's theory it is a rapid vibratory motion of small amplitude. 
In the above extract Hooke introduces, moreover, the idea of 
the wave-swrface, or locus at any instant of a disturbance gene- 



in the Seventeenth Century. 13 

rated originally at a point, and affirms that it is a sphere, 
whose, centre is the point in question, and whose radii are 
the rays of light issuing from the point. 

Hooke's next effort was to produce a mechanical theory of 
refraction, to replace that given by Descartes. " Because," he 
says, "all transparent mediums are not Homogeneous to one 
another, therefore we will next examine how this pulse or motion 
will be propagated through differingly transparent mediums. 
And here, according to the most acute and excellent Philosopher 
Des Cartes, I suppose the sine of the angle of inclination in the 
first medium to be to the sine of refraction in the second, as the 
density of the first to the density of the second. By density, I 
mean not the density in respect of gravity (with which the 
refractions or transparency of mediums hold no proportion), but 
in respect only to the trajeetion of the Kays of light, in which 
respect they only differ in this, that the one propagates the 
pulse more easily and weakly, the other more slowly, but 
more strongly. But as for the pulses themselves, they will 
by the refraction acquire another property, which we shall now 
endeavour to explicate. 

"We will suppose, therefore, in the first Figure, ACFD to be 




a physical Kay, or ABC and DEFto be two mathematical Kays r 
trajected from a very remote point of a luminous body through 



14 The Theory of the Aether 

an Homogeneous transparent medium LL, and DA, EB, FC, to be 
small portions of the orbicular impulses which must therefore 
cut the Rays at right angles : these Rays meeting with the plain 
surface NO of a medium that yields an easier transitus to the 
propagation of light, and falling obliquely on it, they will in the 
medium MM be refracted towards the perpendicular of the 
surface. And because this medium is more easily trajected than 
the former by a third, therefore the point of the orbicular 
pulse FG will be moved to If four spaces in the same time that 
F, the other end of it, is moved to three spaces, therefore the 
whole refracted pulse to H shall be oblique to the refracted Rays 
GHK and /." 

Although this is not in all respects successful, it represents 
a decided advance on the treatment of the same problem by 
Descartes, which rested on a mere analogy. Hooke tries to 
determine what happens to the wave-front when it meets 
the interface between two media ; and for this end he intro- 
duces the correct principle that the side of the wave-front 
which first meets the interface will go forward in the second 
medium with the velocity proper to that medium, while the 
other side of the wave-front which is still in the first medium 
is still moving with the old velocity : so that the wave-front 
will be deflected in the transition from one medium to the 
other. 

This deflection of the wave-front was supposed by Hooke to 
be the origin of the prismatic colours. He regarded natural or 
white light as the simplest type of disturbance, being consti- 
tuted by a simple and uniform pulse at right angles to the 
direction of propagation, and inferred that colour is generated 
by the distortion to which this disturbance is subjected in the 
process of refraction. "The Ray,"* he says, " is dispersed, split, and 
opened by its Refraction at the Superficies of a second medium, 
and from a line is opened into a diverging Superficies, and 
so obliquated, whereby the appearances of Colours are produced." 

* Hooke, Posthnmo/is Works, p. 82. 



in the Seventeenth Century. \ 5 

" Colour/' he says in another place,* " is nothing but the 
disturbance of light by the communication of the pulse to other 
transparent mediums, that is by the refraction thereof." His 
precise hypothesis regarding the different colours wasf "that 
Blue is an impression on the Retina of an oblique and confus'd 
pulse of light, whose weakest part precedes, and whose 
strongest follows. And, that red is an impression on the Retina 
of an oblique and confus'd pulse of light, whose strongest part 
precedes, and whose weakest follows." 

Hooke's theory of colour was completely overthrown, within 
a few years of its publication, by one of the earliest discoveries 
of Isaac Xewton (b. 1642, d. 1727). Newton, who was elected 
a Fellow of Trinity College, Cambridge, in 1667, had in the 
beginning of 1666 obtained a triangular prism, " to try- 
therewith the celebrated Phaenomena of Colours." For this 
purpose, " having darkened my chamber, and made a small hole 
in my window-shuts, to let in a convenient quantity of the 
Sun's light, I placed my Prisme at his entrance, that it might 
be thereby refracted to the opposite wall. It was at first a 
very pleasing divertisement, to view the vivid and intense 
colours produced thereby ; but after a while applying myself to 
consider them more circumspectly, I became surprised to see 
them in an oblong form, which, according to the received laws 
of Refraction, I expected should have been circular" The 
length of the coloured spectrum was in fact about five times as 
great as its breadth. 

This puzzling fact he set himself to study ; and after more 
experiments the true explanation was discovered namely, 
that ordinary white light is really a mixture of rays of every 
variety of colour, and that the elongation of the spectrum is 
due to the differences in the refractive power of the glass for 
these different rays. 

" Amidst these thoughts," he tells us,+ " I was forced from 

*To the Royal Society, February 15, 1671-2. 

t Micrographia, p. 64. 

J Phil. Trans., Xo. 80, February 19, 1671-2. 



16 The Theory of the Aether 

Cambridge by the intervening Plague " ; this was in 1666, and 
his memoir on the subject was not presented to the Koyal 
Society until five years later. In it he propounds a theory of 
colour directly opposed to that of Hooke. " Colours," he says, 
"are not Qualifications of light derived from Refractions, or 
Reflections of natural Bodies (as 'tis generally believed), but 
Original and connate properties, which in divers Rays are divers. 
Some Rays are disposed to exhibit a red colour and no other : 
some a yellow and no other, some a green and no other, and so 
of the rest. Nor are there only Rays proper and particular to 
the more eminent colours, but even to all their intermediate 
gradations. 

" To the same degree of Refrangibility ever belongs the 
same colour, and to the same colour ever belongs the same 
degree of Refrangibility." 

" The species of colour, and degree of Refrangibility proper 
to any particular sort of Rays, is not mutable by Refraction, nor 
by Reflection from natural bodies, nor by, any other cause, that 
I could yet observe. When any one sort of Rays hath been 
well parted from those of other kinds, it hath afterwards 
obstinately retained its colour, notwithstanding my utmost 
endeavours to change it." 

The publication of the new theory gave rise to an acute 
controversy. As might have been expected, Hooke was foremost 
among the opponents, and led the attack with some degree of 
asperity. When it is remembered that at this time Newton 
was at the outset of his career, while Hooke was an older man, 
with an established reputation, such harshness appears par- 
ticularly ungenerous; and it is likely that the unpleasant 
consequences which followed the announcement of his first 
great discovery had much to do with the reluctance which 
Newton ever afterwards showed to publish his results to the 
world. 

In the course of the discussion Newton found occasion to 
explain more fully the views which he entertained regarding 
the nature of light. Hooke charged him with holding the 



in the Seventeenth Century. 17 

doctrine that light is a material substance. Now Newton had, as 
a matter of fact, a great dislike of the more imaginative kind of 
hypotheses ; he altogether renounced the attempt to construct 
the universe from its foundations after the fashion of Descartes, 
and aspired to nothing more than a formulation of the laws 
which directly govern the actual phenomena. His theory of 
gravitation, for example, is strictly an expression of the results 
of observation, and involves no hypothesis as to the cause of the 
attraction which subsists between ponderable bodies ; and his 
own desire in regard to optics was to present a theory free from 
speculation as to the hidden mechanism of light. Accordingly, 
in reply to Hooke's criticism, he protested* that his views on 
colour were in no way bound up with any particular conception 
of the ultimate nature of optical processes. 

Xewton was, however, unable to carry out his plan of 
connecting together the phenomena of light into a coherent 
and reasoned whole without having recourse to hypotheses. The 
hypothesis of Hooke, that light consists in vibrations of an 
aether, he rejected for reasons which at that time were perfectly 
cogent, and which indeed were not successfully refuted for over 
a century. One of these was the incompetence of the wave- 
theory to account for the rectilinear propagation of light, and 
another was its inability to embrace the facts discovered, as 
we shall presently see, by Huygens, and first interpreted 
correctly by Newton himself of polarization. On the whole, 
he seems to have favoured a scheme of which the following may 
be taken as a summaryf : 

All space is permeated by an elastic medium or aether, which 
is capable of propagating vibrations in the same way as the 

*Phil. Trans, vii, 1672, p. 5086. 

t Cf. Newton's memoir in Phil. Trans, vii, 1672 ; his memoir presented to the 
Royal Society in December, 1675, which is printed in Birch, iii, p. 247; his 
Opticks, especially Queries 18, 19, 20, 21, 23, 29; the Scholium at the end of 
the Principia ; and a letter to Boyle, written in February, 1678-9, which is printed 
in Horsley's Newtoni Opera, p. 385. 

In the Principia, Book I., section xiv, the analogy between rays of light and 
streams of corpuscles is indicated ; but Newton does not commit himself to any 
theory of light based on this. 

C 



18 The Theory of the Aether 

air propagates the vibrations of sound, but with far greater 
velocity. 

This aether pervades the pores of all material bodies, and 
is the cause of their cohesion ; its density varies from one body 
to another, being greatest in the free interplanetary spaces. It 
is not necessarily a single uniform substance : but just as air 
contains aqueous vapour, so the aether may contain various 
" aethereal spirits," adapted to produce the phenomena of 
electricity, magnetism, and gravitation. 

The vibrations of the aether cannot, for the reasons already 
mentioned, be supposed in themselves to constitute light. 
Light is therefore taken to be " something of a different kind, 
propagated from lucid bodies. They, that will, may suppose 
it an aggregate of various peripatetic qualities. Others may 
suppose it multitudes of unimaginable small and swift 
corpuscles of various sizes, springing from shining bodies 
at great distances one after another; but yet without any 
sensible interval of time, and continually urged forward by a 
principle of motion, which in the beginning accelerates them, 
till the resistance of the aethereal medium equals the force of 
that principle, much after the manner that bodies let fall in 
water are accelerated till the resistance of the water equals the 
force of gravity. But they, that like not this, may suppose 
light any other corporeal emanation, or any impulse or motion 
of any other medium or aethereal spirit diffused through the 
main body of aether, or what else they can imagine proper for 
this purpose. To avoid dispute, and make this hypothesis 
general, let every man here take his fancy ; only whatever 
light be, I suppose it consists of rays differing from one another 
in contingent circumstances, as bigness, form, or vigour."* 

In any case, light and aether are capable of mutual inter- 
action; aether is in fact the intermediary between light and 
ponderable matter. When a ray of light meets a stratum of 
aether denser or rarer than that through which it has lately 
been passing, it is, in general, deflected from its rectilinear 
* Royal Society, Dec. 9, 1675. 



in the Seventeenth Century. 19 

course ; and differences of density of the aether between one 
material medium and another account on these principles for 
the reflexion and refraction of light. The condensation or 
rarefaction of the aether due to a material body extends to 
some little distance from the surface of the body, so that the 
inflexion due to it is really continuous, and not abrupt; and 
this further explains diffraction, which Newton took to be 
" only a new kind of refraction, caused, perhaps, by the 
external aethers beginning to grow rarer a little before it 
came at the opake body, than it was in free spaces." 

Although the regular vibrations of Newton's aether were not 
supposed to constitute light, its irregular turbulence seems to 
have represented fairly closely his conception of heat. He 
supposed that when light is absorbed by a material body, 
vibrations are set up in the aether, and are recognizable as 
the heat which is always generated in such cases. The 
conduction of heat from hot bodies to contiguous cold ones he 
conceived to be effected by vibrations of the aether propagated 
between them ; and he supposed that it is the violent agitation 
of aethereal motions which excites incandescent substances to 
emit light. 

Assuming with Newton that light is not actually con- 
stituted by the vibrations of an aether, even though such 
vibrations may exist in close connexion with it, the most 
definite and easily conceived supposition is that rays of light 
are streams of corpuscles emitted by luminous bodies. Although 
this was not the hypothesis of Descartes himself, it was so 
thoroughly akin to his general scheme that the scientific men 
of Newton's generation, who were for the most part deeply 
imbued with the Cartesian philosophy, instinctively selected 
it from the wide choice of hypotheses which Newton had offered 
them ; and by later writers it was generally associated with 
Newton's name. A curious argument in its favour was drawn 
from a phenomenon which had then been known for nearly half 
a century : Vincenzo Cascariolo, a shoemaker of Bologna, had 
discovered, about 1630, that a substance, which afterwards 

C 2 



20 The Theory of the Aether 

received the name of Bologna stone or Bologna phosphorus, has- 
the property of shining in the dark after it has been exposed 
for some time to sunlight ; and the storage of light which 
seemed to be here involved was more easily explicable on the 
corpuscular theory than on any other. The evidence in 
this quarter, however, pointed the other way when it was 
found that phosphorescent substances do not necessarily emit 
the same kind of light as that which was used to stimulate 
them. 

In accordance with his earliest discovery, Newton considered 
colour to be an inherent characteristic of light, and inferred 
that it must be associated with some definite quality of the 
corpuscles or aether-vibrations. The corpuscles corresponding 
to different colours would, he remarked, like sonorous bodies of 
different pitch, excite vibrations of different types in the 
aether ; and " if by any means those [aether- vibrations] of 
unequal bignesses be separated from one another, the largest 
beget a Sensation of a Red colour, the least or shortest of a 
deep Violet, and the intermediate ones, of intermediate colours ; 
much after the manner that bodies, according to their several 
sizes, shapes, and motions, excite vibrations in the Air of various 
bignesses, which, according to those bignesses, make several 
Tones in Sound."* 

This sentence is the first enunciation of the great principle 
that homogeneous light is essentially periodic in its nature, and 
that differences of period correspond to differences of colour. 
The analogy with Sound is obvious ; and it may be remarked 
in passing that Newton's theory of periodic vibrations in an 
elastic medium, which he developed! in connexion with the 
explanation of Sound, would alone entitle him to a place among 
those who have exercised the greatest influence on the theory 
of light, even if he had made no direct contribution to the 
latter subject. 

* Phil. Trans, vii (1672), p. 5088. 

t Newton's Prmcipia, Book ii., Props, xliii.-l. 



in the Seventeenth Century. 21 

Newton devoted considerable attention to the colours of 
thin, plates, and determined the empirical laws of the 
phenomena with great accuracy. In order to explain them, he 
supposed that " every ray of light, in its passage through any 
refracting surface, is put into a certain transient constitution or 
state, which, in the progress of the ray, returns at equal 
intervals, and disposes the ray, at every return, to be easily 
transmitted through the next refracting surface, and, between 
the returns, to be easily reflected by it."* The interval 
between two consecutive dispositions to easy transmission, or 
" length of fit," he supposed to depend on the colour, being 
greatest for red light and least for violet. If then a ray of 
homogeneous light falls on a thin plate, its fortunes as regards 
transmission and reflexion at the two surfaces will depend on 
the relation which the length of fit bears to the thickness of 
the plate ; and on this basis he built up a theory of the colours 
of thin plates. It is evident that Newton's "length of fit" 
corresponds in some measure to the quantity which in the 
undulatory theory is called the wave-length of the light ; but 
the suppositions of easy transmission and reflexion were soon 
found inadequate to explain all Newton's experimental results 
.at least without making other and more complicated additional 
assumptions. 

At the time of the publication of Hooke's Micrographia, and 
Newton's theory of colours, it was not known whether light 
is propagated instantaneously or not. An attempt to settle 
the question experimentally had been made many years 
previously by Galileo,f who had stationed two men with 
lanterns at a considerable distance from each other ; one of 
them was directed to observe when the other uncovered his 
light, and exhibit his own the moment he perceived it. But 
the interval of time required by the light for its journey was 
too small to be perceived in this way ; and the discovery was 

* Optic ks, Book ii., Prop. 12. 

t Discorri e dimostrazioiti matemaliche, p. 43 of the Elzevir edition of 1638. 



22 The Theory of the Aether 

ultimately made by an astronomer. It was observed in 1675 
by Olof Roemer* (b. 1644, d. 1710) that the eclipses of the first 
satellites of Jupiter were apparently affected by an unknown 
disturbing cause ; the time of the occurrence of the phenomenon 
was retarded when the earth and Jupiter, in the course of their 
orbital motions, happened to be most remote from each other, 
and accelerated in the contrary case. Eoemer explained this 
by supposing that light requires a finite time for its pro- 
pagation from the satellite to the earth ; and by observations of 
eclipses, he calculated the interval required for its passage from 
the sun to the earth (the light-equation, as it is called) to be 
11 minutes, f 

Shortly after Roemer's discovery, the wave-theory of light 
was greatly improved and extended by Christiaan Huygens 
(b. 1629, d. 1695). Huygens, who at the time was living in 
Paris, communicated his results in 1678 to Cassini, Eoemer, 
De la Hire, and the other physicists of the French Academy, 
and prepared a manuscript of considerable length on the subject. 
This he proposed to translate into Latin, and to publish in that 
language together with a treatise on the Optics of Telescopes ; 
but the work of translation making little progress, after a delay 
of twelve years, he decided to print the work on wave-theory 
in its original form. In 1690 it appeared at Ley den, J under 
the title Traite de la lumiere ou sont expliquees les causes de ce 
qui luy arrive dans la reflexion et dans la refraction. Et parti- 



*Mem. de 1'Acad. x. (1666-1699), p. 575. 

t It was soon recognized that Roemer's value was too large ; and the 
astronomers of the succeeding half-century reduced it to 7 minutes. Delambre, 
by an investigation whose details appear to have been completely destroyed, 
published in 1817 the value 493 -2 s , from a discussion of eclipses of Jupiter's 
satellites during the previous 150 years. Glasenapp, in an inaugural dissertation 
published in 1875, discussed the eclipses of the first satellite between 1848 and 
1870, and derived, by different assumptions, values between 496 s and 501 s , the 
most probable value being 500-8 8 . Sampson, in 1909, derived 498'64 S from his 
own readings of the Harvard Observations, and 498'79 S from the Harvard readings, 
with probable errors of about + 0'02". The inequalities of Jupiter's surface give 
rise to some difficulty in exact determinations. 

% Huygens had by this time returned to Holland. 



in the Seventeenth Century. 23 

culierement dans Vetrange refraction du cristal d'Islande. Par 
C.ff.D.Z* 

The truth of Hooke's hypothesis, that light is essentially a 
form of motion, seemed to Huygens to be proved ]}y the effects 
observed with burning-glasses ; for in the combustion induced at 
the focus of the glass, the molecules of bodies are dissociated ; 
which, as he remarked, must be taken as a certain sign of motion, 
if, in conformity to the Cartesian philosophy, we seek the cause 
of all natural phenomena in purely mechanical actions. 

The question then arises as to whether the motion is that 
of a medium, as is supposed in Hooke's theory, or whether it 
may be compared rather to that of a flight of arrows, as in the 
corpuscular theory. Huygens decided that the former alter- 
native is the only tenable one, since beams of light proceeding 
in directions inclined to each other do not interfere with each 
other in any way. 

Moreover, it had previously been shown by Torricelli that 
light is transmitted as readily through a vacuum as through 
air ; and from this Huygens inferred that the medium or aether 
in which the propagation takes place must penetrate all matter, 
and be present even in all so-called vacua. 

The process of wave-propagation he discussed by aid of a 
principle which was nowf introduced for the first time, and has 
since been generally known by his name. It may be stated 
thus : Consider a wave-front,* or locus of disturbance, as it 
exists at a definite instant t : then each surface-element of the 
wave-front may be regarded as the source of a secondary wave, 
which in a homogeneous isotropic medium will be propagated 
outwards from the surface-element in the form of a sphere 
whose radius at any subsequent instant t is proportional to 
(t-t ) ; and the wave-front which represents the whole distur- 

* i.e. Cbristiaan Huygens de Zuylichem. The custom of indicating names by 
initials was not unusual in that age. 

t Traite de la lum., p. 17. 

I It maybe remarked that Huygens' " waves " are really what modern writers, 
following Hooke, call " pulses "; Huygens never considered true wave-trains 
having the property of periodicity. 



24 The Theory of the Aether 

bance at the instant t is simply the envelope of the secondary 
waves which arise from the various surface elements of the 
original wave-front.* The introduction of this principle enabled 
Huygens to succeed where Hooke and other contemporary 
wave-theoristsf had failed, in achieving the explanation of 
refraction and reflexion. His method was to combine his own 
principle with Hooke's device of following separately the fortunes 
of the right-hand and left-hand sides of a wave-front when it 
reaches the interface between two media. The actual explana- 
tion for the case of reflexion is as follows : 

Let AB represent the interface at which reflexion takes 
place, AHC the incident wave-front at an instant , GMB the 
position which the wave-front would occupy at a later instant t 
if the propagation were not interrupted by reflexion. Then by 




"G 

Huygens' principle the secondary wave from A is at the instant 
t a sphere ENS of radius equal to AG : the disturbance from H t 
after meeting the interface at K, will generate a secondary 
wave TV oi radius equal to KM, and similarly the secondary 
wave corresponding to any other element of the original wave- 

* The justification for this was given long afterwards by Fresnel, Annales de 
chimie, xxi. 

t e.g. Ignace Gaston Pardies and Pierre Ango, the latter of whom published 
a work on Optics at Paris'in 1682. 



in the Seventeenth Century. 25 

front can be found. It is obvious that the envelope of these 
secondary waves, which constitutes the final wave-front, will be 
a plane BN, which will be inclined to AB at the same angle as 
AC. This gives the law of reflexion. 

The law of refraction is established by similar reasoning, 
on the supposition that the velocity of light depends on the 
medium in which it is propagated. Since a ray which passes 
from air to glass is bent inwards towards the normal, it may be 
inferred that light travels more slowly in glass than in air. 

Huygens offered a physical explanation of the variation in 
velocity of light from one medium to another, by supposing 
that transparent bodies consist of hard particles which interact 
with the aethereal matter, modifying its elasticity. The 
opacity of metals he explained by an extension of the same 
idea, supposing that some of the particles of metals are hard 
(these account for reflexion) and the rest soft : the latter destroy 
the luminous motion by damping it. 

The second half of the Theorie de la lumiere is concerned with 
a phenomenon which had been discovered a few years pre- 
viously by a Danish philosopher, Erasmus Bartholin (b. 1625, 
d. 1698). A sailor had brought from Iceland to Copenhagen a 
number of beautiful crystals which he had collected in the Bay 
of Eoerford. Bartholin, into whose hands they passed, noticed* 
that any small object viewed through one of these crystals 
appeared double, and found the immediate cause of this in the 
fact that a ray of light entering the crystal gave rise in general 
to two refracted rays. One of these rays was subject to the 
ordinary law of refraction, while the other, which was called 
the extraordinary ray, obeyed a different law, which Bartholin 
did not succeed in determining. 

The matter had arrived at this stage when it was taken up 
by Huygens. Since in his conception each ray of light corresponds 
to the propagation of a wave-front, the two rays in Iceland 
spar must correspond to two different wave-fronts propagated 

* Ejcperimenta cristatti Islandici disdiaclastici : 1669. 



26 The Theory of the Aether 

simultaneously. In this idea he found no difficulty ; as he says : 
" It is certain that a space occupied by more than one kind of 
matter may permit the propagation of several kinds of waves, 
different in velocity; for this actually happens in air mixed 
with aethereal matter, where sound-waves and light- waves are 
propagated together." 

Accordingly he supposed that a light-disturbance generated 
at any spot within a crystal of Iceland spar spreads out in the 
form of a wave-surface, composed of a sphere and a spheroid 
having the origin of disturbance as centre. The spherical wave- 
front corresponds to the ordinary ray, and the spheroid to the 
extraordinary ray ; and the direction in which the extraordinary 
ray is refracted may be determined by a geometrical construc- 
tion, in which the spheroid takes the place which in the 
ordinary construction is taken by the sphere. 

Thus, let the plane of the figure be at right angles to the 
intersection of the wave-front with the surface of the crystal ; 
let AB represent the trace of the incident wave-front ; and 
suppose that in unit time the disturbance from B reaches the 
interface at T. In this unit-interval of time the disturbance 
from A will have spread out within the crystal into a sphere 
and spheroid : so the wave-front corresponding to the 





ordinary ray will be the tangent-plane to the sphere through 
the line whose trace is T, while the wave-front corresponding 
to the extraordinary ray will be the tangent-plane to the 
spheroid through the same line. The points of contact N 



in the Seventeenth Century. 27 

and M will determine the directions AN and A M of the two- 
refracted rays* within the crystal. 

Huygens did not in the Thtoi-ie de la lumiere attempt a detailed 
physical explanation of the spheroidal wave, but communicated 
one later in a letter to Papin,f written in December, 1690. " As 
to the kinds of matter contained in Iceland crystal," he says, 
" I suppose one composed of small spheroids, and another which 
occupies the interspaces around these spheroids, and which serves 
to bind them together. Besides these, there is the matter of 
aether permeating all the crystal, both between and within the 
parcels of the two kinds of matter just mentioned ; for I suppose 
both the little spheroids, and the matter which occupies the 
intervals around them, to be composed of small fixed particles, 
amongst which are diffused in perpetual motion the still finer 
particles of the aether. There is now no reason why the 
ordinary ray in the crystal should not be due to waves propa- 
gated in this aethereal matter. To account for the extraordinary 
refraction, I conceive another kind of waves, which have for 
vehicle both the aethereal matter and the two other kinds of 
matter constituting the crystal. Of these latter, I suppose that 
the matter of the small spheroids transmits the waves a little 
more quickly than the aethereal matter, while that around the 
spheroids transmits these waves a little more slowly than the 
same aethereal matter. . . . These same waves, when they travel 
in the direction of the breadth of the spheroids, meet with 
more of the matter of the spheroids, or at least pass with less 
obstruction, and so are propagated a little more quickly in this 
sense than in the other ; thus the light-disturbance is propagated 
as a spheroidal sheet." 

Huygens made another disco veryj of capital importance when 

* The word ray in the wave-theory is always applied to the line which goes 
from the centre of a wave (i.e. the origin of the disturbnnce) to a point on its 
surface, whatever may be the inclination of this line to the surface-element on 
which it abuts; for this line has the optical properties of the "rays" of the 
emission theory. 

t Huygens' (Envres, ed. 1905, x., p. 177. 

+ T/ieorie de la lumiere, p. 89. 



28 Theory of the Aether in the Seventeenth Century. 

experimenting with the Iceland crystal. He observed that the 
two rays which are obtained by the double refraction of a single 
ray afterwards behave in a way different from ordinary light 
which has not experienced double refraction ; and in particular, 
if one of these rays is incident on a second crystal of Iceland 
spar, it gives rise in some circumstances to two, and in others 
to only one, refracted ray. The behaviour of the ray at this 
second refraction can be altered by simply rotating the second 
crystal about the direction of the ray as axis ; the ray under- 
going the ordinary or extraordinary refraction according as the 
principal section of the crystal is in a certain direction or in the 
direction at right angles to this. 

The first stage in the explanation of Huygens' observation 
was reached by Newton, who in 1717 showed* that a ray 
obtained by double refraction differs from a ray of ordinary 
light in the same way that a long rod whose cross-section is a 
rectangle differs from a long rod whose cross-section is a circle : 
in other words, the properties of a ray of ordinary light are the 
same with respect to all directions at right angles to its direction 
of propagation, whereas a ray obtained by double refraction 
must be supposed to have sides, or properties related to special 
directions at right angles to its own direction. The refraction 
of such a ray at the surface of a crystal depends on the relation 
of its sides to the principal plane of the crystal. 

That a ray of light should possess such properties seemed to 
Newton f an insuperable objection to the hypothesis which 
regarded waves of light as analogous to waves of sound. On 
this point he was in the right : his objections are perfectly 
valid against the wave-theory as it was understood by his 
contemporaries J, although not against the theory which was put 
forward a century later by Young and Fresnel. 

* The second edition of Newton's Opticks, Query 26. t Opticks, Query 28. 

J In which the oscillations are performed in the direction in which the wave 
advances. 

In which the oscillations are performed in a direction at right angles to that 
in which the wave advances. 



29 ) 



CHAPTEE II. 

ELECTRIC AND MAGNETIC SCIENCE PRIOR TO THE INTRODUCTION 
OF THE POTENTIALS. 

THE magnetic discoveries of Peregrinus and Gilbert, and the 
vortex-hypothesis by which Descartes had attempted to explain 
them,* had raised magnetism to the rank of a separate science 
by the middle of the seventeenth century. The kindred science 
of electricity was at that time in a less developed state ; but it 
had been considerably advanced by Gilbert, whose researches in 
this direction will now be noticed. 

For two thousand years the attractive power of amber had 
been regarded as a virtue peculiar to that substance, or possessed 
by at most one or two others. Gilbert provedf this view to be 
mistaken, showing that the same effects are induced by friction 
in quite a large class of bodies ; among which he mentioned 
glass, sulphur, sealing-wax, and various precious stones. 

A force which was manifested by so many different kinds of 
matter seemed to need a name of its own; and accordingly 
Gilbert gave to it the name electric, which it has ever since 
retained. 

Between the magnetic and electric forces Gilbert remarked 
many distinctions. The lodestone requires no stimulus of friction 
such as is needed to stir glass and sulphur into activity. 
The lodestone attracts only magnetizable substances, whereas 
electrified bodies attract everything. The magnetic attraction 
between two bodies is not affected by interposing a sheet of 
paper, or a linen cloth, or by immersing the bodies in water j 
whereas the electric attraction is readily destroyed by screens. 
Lastly, the magnetic force tends to arrange bodies in definite 

*Cf. pp. 7-9. t De Magnete, lib. ii., cap. 2. 



30 Electric and Magnetic Science 

orientations ; while the electric force merely tends to heap them 
together in shapeless clusters. 

These facts appeared to Gilbert to indicate that electric 
phenomena are due to something of a material nature, which 
under the influence of friction is liberated from the glass or 
amber in which under ordinary circumstances it is imprisoned. 
In support of this view he adduced evidence from other quarters. 
Being a physician, he was well acquainted with the doctrine 
that the human body contains various humours or kinds of 
moisture phlegm, blood, choler, and melancholy, which, as 
they predominated, were supposed to determine the temper of 
mind; and when he observed that electrifiable bodies were 
almost all hard and transparent, and therefore (according to the 
ideas of that time) formed by the consolidation of watery liquids, 
he concluded that the common menstruum of these liquids must 
be a particular kind of humour, to the possession of which the 
electrical properties of bodies were to be referred. Friction 
might be supposed to warm or otherwise excite or liberate the 
humour, which would then issue from the body as an effluvium 
and form an atmosphere around it. The effluvium must, he 
remarked, be very attenuated, for its emission cannot be detected 
by the senses. 

The existence of an atmosphere of effluvia round every 
electrified body might indeed have been inferred, according to 
Gilbert's ideas, from the single fact of electric attraction. For 
he believed that matter cannot act where it is not ; and hence 
if a body acts on all surrounding objects without appearing to 
touch them, something must have proceeded out of it unseen. 

The whole phenomenon appeared to him to be analogous to 
the attraction which is exercised by the earth on falling bodies. 
For in the latter case he conceived of the atmospheric air as the 
effluvium by which the earth draws all things downwards to 
itself. 

Gilbert's theory of electrical emanations commended itself 
generally to such of the natural philosophers of the seventeenth 
century as were interested in the subject ; among whom were 



prior to the Introduction of the Potentials. 31 

numbered Niccolo Cabeo (b. 1585, d. 1650), an Italian Jesuit 
who was. perhaps the first to observe that electrified bodies repel 
as well as attract ; the English royalist exile, Sir Kenelm 
Digby (b. 1603, d. 1665); and the celebrated Robert Boyle 
(b. 1627, d. 1691). There were, however, some differences of 
opinion as to the manner in which the effluvia acted on the small 
bodies and set them in motion towards the excited electric; 
Gilbert himself had supposed the emanations to have an inherent 
tendency to reunion with the parent body ; Digby likened their 
return to the condensation of a vapour by cooling ; and other 
writers pictured the effluvia as forming vortices round the 
attracted bodies in the Cartesian fashion. 

There is a well-known allusion to Gilbert's hypothesis in 
Newton's Opticks.* 

" Let him also tell me, how an electrick body can by friction 
emit an exhalation so rare and subtle,t and yet so potent, as by 
its emission to cause no sensible diminution of the weight of the 
electrick body, and to be expanded through a sphere, whose 
diameter is above two feet, and yet to be able to agitate and 
carry up leaf copper, or leaf gold, at a distance of above a foot 
from the electrick body ? " 

It is, perhaps, somewhat surprising that the Newtonian 
doctrine of gravitation should not have proved a severe blow to 
the emanation theory of electricity ; but Gilbert's doctrine was 
now so firmly established as to be unshaken by the overthrow 
of the analogy by which it had been originally justified. It was, 
however, modified in one particular about the beginning of the 
eighteenth century. In order to account for the fact that 
electrics are not perceptibly wasted away by excitement, the 
earlier writers had supposed all the emanations to return 
ultimately to the body which had emitted them ; but the 
corpuscular theory of light accustomed philosophers to the 
idea of emissions so subtle as to cause no perceptible loss ; and 

* Query 22. 

t " Subtlety," says Johnson, " which in its original import means exility of 
particles, is taken in its metaphorical meaning for nicety of distinction." 



32 Electric and Magnetic Science 

after the time of Newton the doctrine of the return of the- 
electric effluvia gradually lost credit. 

Newton died in 1727. Of the expositions of his philosophy 
which were published in his lifetime by his followers, one at 
least deserves to be noticed for the sake of the insight which 
it affords into the state of opinion regarding light, heat, and 
electricity in the first half of the eighteenth century. This was 
the Physices elementa matlwmatica experimentis confirmata of 
Wilhelm Jacob s'Gravesande (b. 1688, d. 1742), published at 
Ley den in 1720. The Latin edition was afterwards reprinted 
several times, and was, moreover, translated into French and 
English : it seems to have exercised a considerable and, on the 
whole, well-deserved influence on contemporary thought. 

s'Gravesande supposed light to consist in the projection of 
corpuscles from luminous bodies to [the eye ; the motion being 
very swift, as is shown by astronomical observations. Since 
many bodies, e.g. the metals, become luminous when they- -are 
heated, he inferred that every substance possesses a natural 
store of corpuscles, which are expelled when it is heated to 
incandescence ; conversely, corpuscles may become united to a 
material body ; as happens, for instance, when the body is exposed 
to the rays of a fire. Moreover, since the heat thus acquired is 
readily conducted throughout the substance of the body, he 
concluded that corpuscles can penetrate all substances, however 
hard and dense they be. 

Let us here recall the ideas then current regarding the 
nature of material bodies. From the time of Boyle (1626-1691) 
it had been recognized generally that substances perceptible to 
the senses may be either elements or compounds or mixtures ; 
the compounds being chemical individuals, distinct from mere 
mixtures of elements. But the substances at that time accepted 
as elements were very different from those which are now known 
by the name. Air and the calces* of the metals figured in the 
list, while almost all the chemical elements now recognized were 



prior to the Introduction oj the Potentials. 33 

omitted from it ; some of them, such as oxygen and hydrogen, 
because they were as yet undiscovered, and others, such as the 
metals, because they were believed to be compounds. 

Among the chemical elements, it became customary after 
the time of Newton to include light-corpuscles.* That some- 
thing which is confessedly imponderable should ever have been 
admitted into this class may at first sight seem surprising. But 
it must be remembered that questions of ponderability counted 
for very little with the philosophers of the period. Three- 
quarters of the eighteenth century had passed before Lavoisier 
enunciated the fundamental doctrine that the total weight of 
the substances concerned in a chemical reaction is the same 
after the reaction as before it. As soon as this principle came 
to be universally applied, light parted company from the true 
elements in the scheme of chemistry. 

We must now consider the views which were held at this 
time regarding the nature of heat. These are of interest for our 
present purpose, on account of the analogies which were set up 
between heat and electricity. 

The various conceptions which have been entertained 
concerning heat fall into one or other of two classes, according as 
heat is represented as a mere condition producible in bodies, or 
as a distinct species of matter. The former view, which is that 
universally held at the present day, was advocated by the great 
philosophers of the seventeenth century. Bacon maintained it in 
the Novum Organum : " Calor," he wrote, " est niotus expansivus, 
cohibitus, et nitens per partes minores."f Boyle+ affirmed that 
the " Nature of Heat " consists in " a various, vehement, and 
intestine commotion of the Parts among themselves." Hooke 
declared that " Heat is a property of a body arising from the 
motion or agitation of its parts." And Newton|| asked : " Do not 

* Newton himself (Oplicks, p. 349) suspected that light-corpuscles and 
ponderable matter might be transmuted into each other : much later, Boscovich 
(Theoria, pp. 215, 217) regarded the matter of light as a principle or element in 
the constitution of natural bodies. 

t Nov. Org., Lib. n., Aphor. xx. J Mechanical Production of Heat and Cold. 

Micrographia, p. 37. || Opticks. 

D 



34 Electric and Magnetic Science 

all fixed Bodies, when heated beyond a certain Degree, emit 
light and shine ; and is not this Emission performed by the 
vibrating Motion of their Parts ? " and, moreover, suggested the 
converse of this, namely, that when light is absorbed by a 
material body, vibrations are set up which are perceived by the 
senses as heat. 

The doctrine that heat is a material substance was main- 
tained in Newton's lifetime by a certain school of chemists. The 
most conspicuous member of the school was Wilhelm Homberg 
(b. 1652, d. 1715) of Paris, who* identified heat and light with the 
sulphureous principle, which he supposed to be one of the primary 
ingredients of all bodies, and to be present even in the inter- 
planetary spaces. Between this view and that of Newton it 
might at first seem as if nothing but sharp opposition was to be 
expected, j- But a few years later the professed exponents of the 
Principia and the Opticks began to develop their system under 
the evident influence of Homberg's writings. This evolution 
may easily be traced in s'Gravesande, whose starting-point is 
the admittedly Newtonian idea that heat bears to light a 
relation similar to that which a state of turmoil bears to regular 
rectilinear motion ; whence, conceiving light as a projection of 
corpuscles, he infers that in a hot body the material particles 
and the light-corpusclesj are in a state of agitation, which 
becomes more violent as the body is more intensely heated. 

s'Gravesande thus holds a position between the two opposite 
camps. On the one hand he interprets heat as a mode of 
motion ; but on the other he associates it with the presence of 
a particular kind of matter, which he further identifies with the 
matter of light. After this the materialistic hypothesis made 

* Mem. del'Acad., 1705, p. 88. 

t Though it reminds us of a curious conjecture ofNewtoa'i: "Is not the 
strength and vigour of the action between light and sulphureous bodies one reason 
M-liy sulphureous bodies take fire more readily and burn more vehemently than 
other bodies do? " 

J I have thought it best to translate s'Gravesande's ignis by " light-corpuscles." 
This is, I think, fully justified by such of his statements as Quando ignis per 
lineas rectas oculos nostros intrat, ex motu gttein fibris in fundo oculi cont/tninicai 
ideam luminis excitat. 



prior to the Introduction of the Potentials. 35 

rapid progress. It was frankly advocated by another member 
of the Dutch school, Hermann Boerhaave* (6. 1668, d. 1738), 
Professor in the University of Leyden, whose treatise on 
chemistry was translated into English in 1727. 

Somewhat later it was found that the heating effects of the 
rays from incandescent bodies may be separated from their 
luminous effects by passing the rays through a plate of glass, 
which transmits the light, but absorbs the heat. After this 
discovery it was no longer possible to identify the matter of heat 
with the corpuscles of light ; and the former was consequently 
accepted as a distinct element, under the name of caloric.^ In 
the latter part of the eighteenth and early part of the nineteenth 
centuries} caloric was generally conceived as occupying the 
interstices between the particles of ponderable matter an idea 
which fitted in well with the observation that bodies commonly 
expand when they are absorbing heat, but which was less com- 
petent to explain the fact that water expands when freezing. 
The latter difficulty was overcome by supposing the union 
between a body and the caloric absorbed in the process of 
melting to be of a chemical nature; so that the consequent 
changes in volume would be beyond the possibility of prediction. 

As we have already remarked, the imponderability of heat 
did not appear to the philosophers of the eighteenth century to 
be a sufficient reason for excluding it from the list of chemical 
elements ; and in any case there was considerable doubt as to 
whether caloric was ponderable or not. Some experimenters 
believed that bodies were heavier when cold than when hot; 
others that they were heavier when hot than when cold. The 
century was far advanced before Lavoisier and Eumford finally 

* Boerhaave followed Homberg in supposing the matter of heat to be present ia 
all so-called vacuous spaces. 

t Scheele in 1777 supposed caloric to be a compound of oxygen and phlogiston, 
and light to be oxygen combined with a greater proportion of phlogiston. 

J In suite of the experiments of Benjamin Thompson, Count Eumford (b. 1753, 
.d. 1814), in the closing years of the eighteenth century. These should have 
-sufficed to re-establish the older conception of heat. 

This had been known since the time of Boyle. 

D 2 



36 Electric and Magnetic Science 

proved that the temperature of a body is without sensible 
influence on its weight. 

Perhaps nothing in the history of natural philosophy is more 
amazing than the vicissitudes of the theory of heat. The true 
hypothesis, after having met with general acceptance throughout 
a century, and having been approved by a succession of illus- 
trious men, was deliberately abandoned by their successors 
in favour of a conception utterly false, and, in some of its 
developments, grotesque and absurd. 

We must now return to s'Gravesande's book. The pheno- 
mena of combustion he explained by assuming that when a body 
is sufficiently heated the light-corpuscles interact with the 
material particles, some constituents being in consequence sepa- 
rated and carried away with the corpuscles as flame and smoke. 
This view harmonizes with the theory of calcination which had 
been developed by Becher and his pupil Stahl at the end of the- 
seventeenth century, according to which the metals were sup- 
posed to be composed of their calces and an element phlogiston. 
The process of combustion, by which a metal is changed into its- 
calx, was interpreted as a decomposition, in which the phlogiston 
separated from the metal and escaped into the atmosphere ; 
while the conversion of the calx into the metal was regarded as 
a union with phlogiston.* 

s'Gravesande attributed electric effects to vibrations induced 
in effluvia, which he supposed to be permanently attached to 
such bodies as amber. " Glass," he asserted, " contains in it, and 
has about its surface, a certain atmosphere, which is excited by 
Friction and put into a vibratory motion ; for it attracts and 

* The correct idea of combustion had been advanced by Hooke. "The disso- 
lution of inflammable bodies," he asserts in the Micrographia, " is performed by a 
substance inherent in and mixed with the air, that is like, if not the very same 
with, that which is fixed in saltpetre." But this statement met with little favour 
at the time, and the doctrine of the compound nature of metals survived in full 
vigour until the discovery of oxygen by Priestley and Scheele in 1771-5. In 1775 
Lavoisier reaffirmed Hooke's principle that a metallic calx is not the metal minus 
phlogiston, but the metal plus oxygen; and this idea, which carried with it the 
recognition of the elementary nature of metals, was generally accepted by the end' 
of the eighteenth century. 



prior to the Introduction of the Potentials. 37 

repels light Bodies. The smallest parts of the glass are agitated 
by the Attrition, and by reason of their elasticity, their motion is 
vibratory, which is communicated to the Atmosphere above- 
mentioned : and therefore that Atmosphere exerts its action the 
further, the greater agitation the Parts of the Glass receive when 
a greater attrition is given to the glass." 

The English translator of s'Gravesande's work was himself 
destined to play a considerable part in the history of electrical 
science. Jean Theophile Desaguliers (b. 1683, d. 1744) was an 
Englishman only by adoption. His father had been a Huguenot 
pastor, who, escaping from France after the revocation of the Edict 
of Nantes, brought away the boy from La Kochelle, concealed, it is 
said, in a tub. The young Desaguliers was afterwards ordained, 
and became chaplain to that Duke of Chandos who was so 
ungratefully ridiculed by Pope. In this situation he formed 
friendships with some of the natural philosophers of the capital, 
and amongst others with Stephen Gray, an experimenter of 
whom little is known* beyond the fact that he was a pensioner 
of the Charterhouse. 

In 1729 Gray communicated, as he says,f " to Dr. Desaguliers 
and some other Gentlemen " a discovery he had lately made, 
" showing that the Electrick Vertue of a Glass Tube may be 
conveyed to any other Bodies so as to give them the same 
Property of attracting and repelling light Bodies as the Tube 
does, when excited by rubbing : and that this attractive Vertue 
might be carried to Bodies that were many Feet distant from 
the Tube." 

This was a result of the greatest importance, for previous 
workers had known of no other way of producing the attractive 
emanations than by rubbing the body concerned.* It was found 

* Those M*ho are interested in the literary history of the eighteenth century will 
recall the controversy as to whether the verses on the death of Stephen Gray were 
written hy Anna "Williams, whose name they bore, or by her patron Johnson. 

| Phil. Trans, xxxvii (1731), pp. 18, 227, 285, 397. 

j Otto von Guericke (b. 1602, d. 1686) bad, as a matter of fact, observed the 
conduction of electricity along a linen thread ; but this experiment does not seem 
to have been followed up. Cf. Experimenta novamagdeburgica, 1672. 



38 Electric and Magnetic Science 

o 

that only a limited class of substances, among which the metals 
were conspicuous, had the capacity of acting as channels for the 
transport of the electric power ; to these Desaguliers, who. con- 
tinued the experiments after Gray's death in 1736, gavfc^ the 
name non-electrics or conductors. 

After Gray's discovery it was no longer possible to believe 
that the electric effluvia are inseparably connected with the 
bodies from which they are evoked by rubbing ; and it became 
necessary to admit that these emanations have an independent 
existence, and can be transferred from one body to another. 
Accordingly we find them recognized, under the name of the 
electric fluidft as one of the substances of which the world is 
constituted. The imponderability of this fluid did not, for the 
reasons already mentioned, prevent its admission by the side of 
light and caloric into the list of chemical elements. 

The question was actively debated as to whether the electric 
fluid was an element sui generis, or, as some suspected, was 
another manifestation of that principle whose operation is seen 
in the phenomena of heat. Those who held the latter view 
urged that the electric fluid and heat can both be induced by 
friction, can both induce combustion, and can both be transferred 
from one body to another by mere contact ; and, moreover, that 
the best conductors of heat are also in general the best con- 
ductors of electricity. On the other hand it was contended that 
the electrification of a body does not cause any appreciable rise 
in its temperature; and an experiment of Stephen Gray's 
brought to light a yet more striking difference. Gray,J in 1729,. 
made two oaken cubes, one solid and the other hollow, and 
showed that when electrified in the same way they produced 
exactly similar effects ; whence he concluded that it was only 
the surfaces which had taken part in the phenomena. Thus 
while heat is disseminated throughout the substance of a body, 
the electric fluid resides at or near its surface. In the middle of 

* Phil. Trans, xli. (1739), pp. 186, 193, 200, 209: Dissertation concerning 
Electricity, 1742. 

t The Cartesians defined a fluid to be a body whose minute parts are in a 
continual agitation. J Phil. Trans, xxxvii., p. 35. 



prior to the Introduction of the Potentials. 39 

the eighteenth century it was generally compared to an envelop- 
ing atmosphere. " The electricity which a non-electric of great 
length (for example, a hempen string 800 or 900 feet long) 
receives, runs from one end to the other in a sphere of electrical 
Effluvia" says Desaguliers in 1740 ^and a report of the French 
Academy in 1733 says :f " Around an electrified body there is 
formed a vortex of exceedingly fine matter in a state of agitation,, 
which urges towards the body such light substances as lie 
within its sphere of activity. The existence of this vortex is 
more than a mere conjecture ; for when an electrified body i& 
brought close to the face it causes a sensation like that of 
encountering a cobweb. "J 

The report from which this is quoted was prepared in 
connexion with the discoveries of Charles-Francois du Fay 
(b. 1698, d. 1739), superintendent of gardens to the King of 
France. Du Fay accounted for the behaviour of gold leaf when 
brought near to an electrified glass tube by supposing that at 
first the vortex of the tube envelopes the gold-leaf, and so attracts 
it towards the tube. But when contact occurs, the gold-leaf 
acquires the electric virtue, and so becomes surrounded by a 
vortex of its own. The two vortices, striving to extend in 
contrary senses, repel each other, and the vortex of the tube, 
being the stronger, drives away that of the gold-leaf. " It is 
then certain/' says du Fay,H " that bodies which have become 
electric by contact are repelled by those which have rendered 
them electric ; but are they repelled likewise by other electrified 
bodies of all kinds ? And do electrified bodies differ from each 
other in no respect save their intensity of electrification ? An 
examination of this matter has led me to a discovery which I 
should never have foreseen, and of which I believe no one 
hitherto has had the least idea." 

* Phil. Trans, xli., p. 636. t Hist, de 1'Acad., 1733, p. 6. 

t This observation had been made first by Hawksbee at the beginning of the 
century. 

Mem. de 1'Acad. des Sciences, 1733, pp. 23, 73, 233, 457 ; 1734, pp. 341, 
503; 1737, p. 86 ; Phil. Trans, xxxviii. (1734), p. 258. 

|| Mem. de 1'Acad., 1733, p. 464. 



40 Electric and Magnetic Science 

He found, in fact, that when gold-leaf which had been 
electrified by contact with excited glass was brought near to an 
excited piece of copal,* an attraction was manifested between 
them. " I had expected," he writes, " quite the opposite effect, 
since, according to my reasoning, the copal and gold-leaf, which 
were both electrified, should have repelled each other." 
Proceeding with his experiments he found that the gold-leaf, 
when electrified and repelled by glass, was attracted by all 
electrified resinous substances, and that when repelled by the 
latter it was attracted by the glass. " We see, then," he continues, 
" that there are two electricities of a totally different nature 
namely, that of transparent solids, such as glass, crystal, &c., 
and that of bituminous or resinous bodies, such as amber, copal, 
sealing-wax, &c. Each of them repels bodies which have 
contracted an electricity of the same nature as its own, and 
attracts those whose electricity is of the contrary nature. We 
see even that bodies which are not themselves electrics can 
acquire either of these electricities, and that then their effects 
are similar to those of the bodies which have communicated it 
to them." 

To the two kinds of electricity whose existence was thus 
demonstrated, du Fay gave the names vitreous and resinous, by 
which they have ever since been known. 

An interest in electrical experiments seems to have spread 
from du Fay to other members of the Court circle of Louis XV ; 
and from 1745 onwards the Memoirs of the Academy contain a 
series of papers on the subject by the Abbe Jean-Antoine Nollet 
{&. 1700, d. 1770), afterwards preceptor in natural philosophy 
to the Koyal Family. Nollet attributed electric phenomena to 
the movement in opposite directions of two currents of a fluid, 
" very subtle and inflammable," which he supposed to be present 
in all bodies under all circumstances.f When an electric is 
excited by friction, part of this fluid escapes from its pores, 
forming an effluent stream; and this loss is repaired by an 

* A hard transparent resin, used in the preparation of varnish. 
t Cf. Nollet' s lieeherchet, 1749, p. 245. 



prior to the Introduction of the Potentials. 41 

dtfiucnt stream of the same fluid entering the body from outside. 
Light bodies in the vicinity, being caught in one or other of 
these streams, are attracted or repelled from the excited electric. 

Nollet's theory was in great vogue for some time ; but six or 
seven years after its first publication, its author came across a 
work purporting to be a French translation of a book printed 
originally in England, describing experiments said to have been 
made at Philadelphia, in America, by one Benjamin Franklin. 
"He could not at first believe," as Franklin tells us in his 
AutobiograpJvy, " that such a work came from America, and said 
it must have been fabricated by his enemies at Paris to decry 
his system. Afterwards, having been assured that there really 
existed such a person as Franklin at Philadelphia, which he had 
doubted, he wrote and published a volume of letters, chiefly 
addressed to me, defending his theory, and denying the verity 
of my experiments, and of the positions deduced from them." 

We must now trace the events which led up to the discovery 
which so perturbed Nollet. 

In 1745 Pieter van Musschenbroek (6. 1692, d. 1761), 
Professor at Leyden, attempted to find a method of preserving 
electric charges from the decay which was observed when the 
charged bodies were surrounded by air. With this purpose he 
tried the effect of surrounding a charged mass of water by an 
envelope of some non-conductor, e.g., glass. In one of his 
experiments, a phial of water was suspended from a gun- 
barrel by a wire let down a few inches into the water through 
the cork; and the gun-barrel, suspended on silk lines, was 
applied so near an excited glass globe that some metallic fringes 
inserted into the gun-barrel touched the globe in motion. 
Under these circumstances a friend named Cimaeus, who 
happened to grasp the phial with one hand, and touch the gun- 
barrel with the other, received a violent shock ; and it became 
evident that a method of accumulating or intensifying the 
electric power had been discovered.* 

* The discovery was made independently in the same year by Ewald Georg 
von Kleist, Dean of Kumrain. 



42 Electric and Magnetic Science 

o 

Shortly after the discovery of the Leyden phial, as it was 
named by Nollet, had become known in England, a London 
apothecary named William Watson (6. 1715, d. 1787)* noticed 
that when the experiment is performed in this fashion the 
observer feels the shock " in no other parts of his body but his 
arms and breast " ; whence he inferred that in the act of 
discharge there is a transference of something which takes the 
shortest or best- conducting path between the gun-barrel and 
the phial. This idea of transference seemed to him to bear 
some similarity to Nollet's doctrine of afflux and efflux; and 
there can indeed be little doubt that the Abbe's hypothesis, 
though totally false in itself, furnished some of the ideas from 
which Watson, with the guidance of experiment, constructed 
a correct theory. In a memoiirt)read to the Eoyal Society 
in October, 1746, he propounded the doctrine that electrical 
actions are due to the presence of an " electrical aether/' which 
in the charging or discharging of a Leyden jar is transferred, but 
is not created or destroyed. The excitation of an electric, 
according to this view, consists not in the evoking of anything 
from within the electric itself without compensation, but in the 
accumulation of a surplus of electrical aether by the electric at 
the expense of some other body, whose stock is accordingly 
depleted. All bodies were supposed to possess a certain natural 
store, which could be drawn upon for this purpose. 

" I have shewn," wrote Watson, " that electricity is the 
effect of a very subtil and elastic fluid, occupying all bodies in 
contact with the terraqueous globe ; and that every-where, in 
its natural state, it is of the same degree of density ; and that 
glass and other bodies, which we denominate electrics per se y . 
have the power, by certain known operations, of taking this fluid 
from one body, and conveying it to another, in a quantity 
sufficient to be obvious to all our senses; and that, under 

* Watson afterwards rose to eminence in the medical profession, and was 
knighted. 

t Phil. Trans, xliv., p. 718. It may here he noted that it was Watson who 
improved the phial by coating it nearly to the top, both inside and outside, with 
tinfoil. 



prior to the Introduction of the Potentials. 43 

certain circumstances, it was possible to render the electricity in 
some bodies more rare than it naturally is, and, by communi- 
cating this to other bodies, to give them an additional quantity, 
and make their electricity more dense." 

In the same year in which Watson's theory was proposed, a 
certain Dr. Spence, who had lately arrived in America from 
Scotland, was showing in Boston some electrical experiments. 
Among his audience was a man who already at forty years of 
age was recognized as one of the leading citizens of the English 
colonies in America, Benjamin Franklin of Philadelphia (b. 1706, 
d. 1790). Spence's experiments " were," writes Franklin,* 
" imperfectly performed, as he was not very expert ; but, being 
on a subject quite new to me, they equally surprised and 
pleased me." Soon after this, the "Library Company" of 
Philadelphia (an institution founded by Franklin himself) 
received from Mr. Peter Collinson of London a present of a glass 
tube, with some account of its use. In a letter written to 
Collinson on July llth, 1747,f Franklin described experiments 
made with this tube, and certain deductions which he had 
drawn from them. 

If one person A, standing on wax so that electricity cannot 
pass from him to the ground, rubs the tube, and if another 
person B, likewise standing on wax, passes his knuckle along 
near the glass so as to receive its electricity, then both A and B 
will be capable of giving a spark to a third person C standing 
on the floor; that is, they will be electrified. If, however, A 
and B touch each other, either during or after the rubbing, they 
will not be electrified. 

This observation suggested to Franklin the same hypothesis 
that (unknown to him) had been propounded a few months 
previously by Watson : namely, that electricity is an element 
present in a certain proportion in all matter in its normal 
condition ; so that, before the rubbing, each of the persons A, 
B, and C has an equal share. The effect of the rubbing is to 

* Franklin's Autobiography. 

t Franklin's New Experiments and Observations on Electricity, letter ii. 



44 Electric and Magnetic Science 

transfer some of A's electricity to the glass, whence it is 
transferred to B. Thus A has a deficiency and B a superfluity 
of electricity ; and if either of them approaches C, who has the 
normal amount, the distribution will be equalized by a spark. 
If, however, A and B are in contact, electricity flows between 
them so as to re-establish the original equality, and neither is 
then electrified with reference to C. 

Thus electricity is not created by rubbing the glass, but 
only transferred to the glass from the rubber, so that the 
rubber loses exactly as much as the glass gains ; the, total 
quantity of electricity in any insulated system is invariable. This 
assertion is usually known as the principle of conservation of 
electric charge. 

The condition of A and B in the experiment can evidently 
be expressed by plus and minus signs : A having a deficiency 
- e and B a superfluity + e of electricity. Franklin, at the 
commencement of his own experiments, was not acquainted 
with du Fay's discoveries ; but it is evident that the electric 
fluid of Franklin is identical with the vitreous electricity of 
du Fay, and that du Fay's resinous electricity is, in Franklin's 
theory, merely the deficiency of a stock of vitreous electricity 
supposed to be possessed naturally by all ponderable bodies. 
In Franklin's theory we are spared the necessity for admitting 
that two quasi-material bodies can by their union annihilate each 
other, as vitreous and resinous electricity were supposed to do. 

Some curiosity will naturally be felt as to the considerations 
which induced Franklin to attribute the positive character to 
vitreous rather than to resinous electricity. They seem to have 
been founded on a comparison of the brush discharges from 
conductors charged with the two electricities; when the 
electricity was resinous, the discharge was observed to spread 
over the surface of the opposite conductor " as if it flowed from 
it." Again, if a Ley den jar whose inner coating is electrified 
vitreously is discharged silently by a conductor, of whose pointed 
ends one is near the knob and the other near the outer coating, 
the point which is near the knob is seen in the dark to be illumi- 



prior to the Introduction of the Potentials. 45 

nated with a star or globule, while the point which is near the 
outer coating is illuminated with a pencil of rays; which 
suggested to Franklin that the electric fluid, going from the 
inside to the outside of the jar, enters at the former point and 
issues from the latter. And yet again, in some cases the flame 
of a wax taper is blown away from a brass ball which is 
discharging vitreous electricity, and towards one which is 
discharging resinous electricity. But Franklin remarks that 
the interpretation of these observations is somewhat conjectural, 
and that whether vitreous or resinous electricity is the actual 
electric fluid is not certainly known. 

Regarding the physical nature of electricity, Franklin held 
much the same ideas as his contemporaries ; he pictured it as 
an elastic* fluid, consisting of " particles extremely subtile, since 
it can permeate common matter, even the densest metals, with 
such ease and freedom as not to receive any perceptible 
resistance." He departed, however, to some extent from the 
conceptions of his predecessors, who were accustomed to ascribe 
all electrical repulsions to the diffusion of effluvia from the 
excited electric to the body acted on ; so that the tickling 
sensation which is experienced when a charged body is brought 
near to the human face was attributed to a direct action of the 
effluvia on the skin. This doctrine, which, as we shall see, 
practically ended with Franklin, bears a suggestive resemblance 
to that which nearly a century later was introduced by 
Faraday ; both explained electrical phenomena without intro- 
ducing action at a distance, by supposing that something which 
forms an essential part of the electrified system is present at 
the spot where any electric action takes place ; but in the older 
theory this something was identified with the electric fluid 
itself, while in the modern view it is identified with a state of 
stress in the aether. In the interval between the fall of one 
school and the rise of the other, the theory of action at a 
distance was dominant. 

The germs of the last-mentioned theory may be found in 

*i.c., repulsive of its own particles. 



46 Electric and Magnetic Science 

Franklin's own writings. It originated in connexion with the 
explanation of the Ley den jar, a matter which is discussed 
in his third letter to Collinson, of date September 1st, 1747. 
In charging the jar, he says, a quantity of electricity is taken 
away from one side of the glass, by means of the coating 
in contact with it, and an equal quantity is communi- 
cated to the other side, by means of the other coating. The 
glass itself he supposes to be impermeable to the electric 
fluid, so that the deficiency on the one side can permanently 
coexist with the redundancy on the other, so long as the two 
sides are not connected with each other ; but when a con- 
nexion is set up, the distribution of fluid is equalized through 
the body of the experimenter, who receives a shock. 

Compelled by this theory of the jar to regard glass as 
impenetrable to electric effluvia, Franklin was nevertheless well 
aware* that the interposition of a glass plate between an 
electrified body and the objects of its attraction does not shield 
the latter from the attractive influence. He was thus driven to 
supposef that the surface of the glass which is nearest the 
excited body is directly affected, and is able to exert an 
influence through the glass on the opposite surface ; the latter 
surface, which thus receives a kind of secondary or derived 
excitement, is responsible for the electric effects beyond it. 

This idea harmonized admirably with the phenomena of 
the jar ; for it was now possible to hold that the excess of 
electricity on the inner face exercises a repellent action through 
the substance of the glass, and so causes a deficiency on the 
outer faces by driving away the electricity from it.J 

Franklin had thus arrived at what was really a theory of 
action at a distance between the particles of the electric fluid ; 
and this he was able to support by other experiments. " Thus," 
he writes, " the stream of a fountain, naturally dense and con- 
tinual, when electrified, will separate and spread in the form of 
a brush, every drop endeavouring to recede from every other 

* New Experiments, 1750, 28. t Hid., 1750, 34. 

J Ibid., 1750, 32. Letter v. 



prior to the Introduction of the Potentials. 47 

drop.' In order to account for the attraction between 
oppositely charged bodies, in one of which there is an excess of 
electricity as compared with ordinary matter, and in the other 
an excess of ordinary matter as compared with electricity, he 
assumed that " though the particles of electrical matter do repel 
each other, they are strongly attracted by all other matter " ; so 
that " common matter is as a kind of spunge to the electrical 
fluid." 

These repellent and attractive powers he assigned only to 
the actual (vitreous) electric fluid; and when later on the 
mutual repidsion of resinously electrified bodies became known 
to him,* it caused him considerable perplexity.f As we shall see, 
the difficulty was eventually removed by.Aepinus. 

In spite of his belief in the power of electricity to act at a 
distance, Franklin did not abandon the doctrine of effluvia. 
"The form of the electrical atmosphere," he says,} "is that of the 
body it surrounds. This shape may be rendered visible in a still 
air, by raising a smoke from dry rosin dropt into a hot tea- 
spoon under the electrified body, which will be attracted, and 
spread itself equally on all sides, covering and concealing the 
body, And this form it takes, because it is attracted by all 
parts of the surface of the body, though it cannot enter the 
substance already replete. Without this attraction, it would 
not remain round the body, but dissipate in the air." He 
observed, however, that electrical effluvia do not seem to 
affect, or be affected by, the air ; since it is possible to breathe 
freely in the neighbourhood of electrified bodies ; and moreover 
a current of dry air does not destroy electric attractions and 
repulsions. 

Kegarding the suspected identity of electricity with the 
matter of heat, as to which Nollet had taken the affirmative 
position, Franklin expressed no opinion. " Common fire," he 

* He refers to it in his Paper read to the Royal Society, December 18, 1755. 
t Cf. letters xxxvii and xxxviii, dated 1761 and 1762. 
1 New Experiment* , 1750, 15. 
Letter vii, 1751. 



48 Electric and Magnetic Science 

writes,* " is in all bodies, more or less, as well as electrical fire. 
Perhaps they may be different modifications of the same 
element ; or they may be different elements. The latter is by 
some suspected. If they are different things, yet they may and 
do subsist together in the same body." 

Franklin's work did not at first receive from European 
philosophers the attention which it deserved ; although Watson 
generously endeavoured to make the colonial writer's merits 
known,f and inserted some of Franklin's letters in one of his own 
papers communicated to the Eoyal Society. But an account of 
Franklin's discoveries, which had been printed in England, 
happened to fall into the hands of the naturalist Buffon, who was 
so much impressed that he secured the issue of a French transla- 
tion of the work ; and it was this publication which, as we have 
seen, gave such offence to Nollet. The success of a plan proposed 
by Franklin for drawing lightning from the clouds soon engaged 
public attention everywhere; and in a short time the triumph 
of the one-fluid theory of electricity, as the hypothesis of 
Watson and Franklin is generally called, was complete. Collet, 
who was obdurate, "lived to see himself the last of his sect, 
except Monsieur B of Paris, his eleve and immediate 
disciple." J 

The theory of effluvia was finally overthrown, and replaced 
by that of action at a distance, by the labours of one of 
Franklin's continental followers, Francis Ulrich Theodore 
Aepinus (&. 1724, d. 1802). The doctrine that glass is 
impermeable to electricity, which had formed the basis of 
Franklin's theory of the Ley den phial, was generalized by Aepinus|| 
and his co-worker Johann Karl Wilcke (5. 1732, d. 1796) 
into the law that all non-conductors are impermeable to the 

* Letter v. 

Cx_- tPhil. Trans, xlvii, p. 202. Watson agreed with Nollet in rejecting Franklin's 
J theory of the impermeability of glass. 
J Franklin's Autobiography. 

This philosopher's surname had been hellenized from its original form Hoeck 
to alveivos by one of his ancestors, a distinguished theologian. 

|| F. V. T. Aepinus Tentamen Thcoriae Elcctricitatis et Magnetismi : 
St. Petersburg, 1759. 



prior to the Introduction of the Potentials. 49 

electric fluid. That this applies even to air they proved by 
constructing a machine analogous to the Leyden jar, in which, 
however, air took the place of glass as the medium between 
two oppositely charged surfaces. The success of this experi- 
ment led Aepinus to deny altogether the existence of electric 
effluvia surrounding charged bodies :* a position which he 
regarded as strengthened by Franklin's observation, that the 
electric field in the neighbourhood of an excited body is not 
destroyed when the adjacent air is blown away. The electric 
fluid must therefore be supposed not to extend beyond the 
excited bodies themselves. The experiment of Gray, to which 
we have already referred, showed that it does not penetrate 
far into their substance; and thus it became necessary to 
suppose that the electric fluid, in its state of rest, is con- 
fined to thin layers on the surfaces of the excited bodies. 
This being granted, the attractions and repulsions observed 
between the bodies compel us to believe that electricity acts 
at a distance across the intervening air. 

Since two vitreously charged bodies repel each other, the 
force between two particles of the electric fluid must (on 
Franklin's one-fluid theory, which Aepinus adopted) be 
repulsive : and since there is 'an attraction between oppositely 
charged bodies, the force between electricity and ordinary 
matter must be attractive. These assumptions had been made, 
as we have seen, by Franklin; but in order to account for 
the repulsion between two resinously charged bodies, Aepinus 
introduced a new supposition namely, that the particles 
of ordinary matter repel each other. This, at first, startled 
his contemporaries; but, as he pointed out, the "unelectrified" 
matter with which we are acquainted is really matter saturated 
with its natural quantity of the electric fluid, and the forces 
due to the matter and fluid balance each other ; or perhaps, 
as he suggested, a slight want of equality between these 
forces might give, as a residual, the force of gravitation. 

Assuming that the attractive and repellent forces increase as " 

* This was also maint.iined about the same time by Giacomo Battista Beet-aria 
of Turin (b. 1716, d. 1781;. 

E 



50 Electric and Magnetic Science 



<v 



the distance between the acting charges decreases, Aepinus 
applied his theory to explain a phenomenon which had been 
more or less indefinitely observed by many previous writers, and 
specially studied a short time previously by John Canton* 
(&. 1718, d. 1772) and by Wilckef namely, that if a conductor 
is brought into the neighbourhood of an excited body without 
actually touching it, the remoter portion of the conductor 
acquires an electric charge of the same kind as that of the 
excited body, while the nearer portion acquires a charge of the 
opposite kind. This effect, which is known as the induction of 
electric charges, had been explained by Canton himself and by 
Franklin} in terms of the theory of electric effluvia. Aepinus 
showed that it followed naturally from the theory of action at a 
distance, by taking into account the mobility of the electric fluid 
in conductors ; and by discussing different cases, so far as was 
possible with the means at his command, he laid the foundations 
of the mathematical theory of electrostatics. 

Aepinus did not succeed in determining the law according to 
which the force between two electric charges varies with the 
distance between them ; and the honour of having first accom- 
plished this belongs to Joseph Priestley (b. 1733, d. 1804), the 
discoverer of oxygen. Priestley, who was a friend of Franklin's, 
had been informed by the latter that he had found cork balls to 
be wholly unaffected by the electricity of a metal cup within 
which they were held ; and Franklin desired Priestley to repeat 
and ascertain the fact. Accordingly, on December 21st, 1766, 
Priestley instituted experiments, which showed that, when a 
hollow metallic vessel is electrified, there is no charge on the inner 
surface (except near the opening), and no electric force in the air 
inside. From this he at once drew the correct conclusion, which 
was published in 1767. " May we not infer," he says, "from 

*Phil. Trans, xlviii (1753), p. 350. 

t Disputatio physica experimentalis de electricitatibus contrariis : Rostock, 1757. 

J In liis paper read to the Royal Society on Dec. 18th, 1755. 

J. Priestley, The History and Present State of Electricity, with Original 
Experiments ; London, 1767: page 732. That electrical attraction follows the 
law of the inverse square had been suspected -by Daniel Bernoulli in 1760: Cf. 
Sochi's Experiments, Ada Helvetica, iv, p. 214. 



prior to the Introduction of the Potentials. 51 

this experiment that the attraction of electricity is subject to 
the same laws with that of gravitation, and is therefore according 
to the squares of the distances ; since it is easily demonstrated 
that were the earth in the form of a shell, a body in the inside 
of it would not be attracted to one side more than another ? " 

This brilliant inference seems to have been insufficiently 
studied by the scientific men of the day ; and, indeed, its author 
appears to have hesitated to claim for it the authority of a com- 
plete and rigorous proof. Accordingly we find that the question 
of the law of force was not regarded as finally settled for eighteen 
years afterwards.* 

By Franklin's law of the conservation of electric charge, and 
Priestley's law of attraction between charged bodies, electricity 
was raised to the position of an exact science. It is impossible 
to mention the names of these two friends in such a connexion 
without reflecting on the curious parallelism of their lives. In 
both men there was the same combination of intellectual bold- 
ness and power with moral earnestness and public spirit. Both 
.of them carried on a long and tenacious struggle with the reac- 
tionary influences which dominated the English Government in 
.the reign of George III ; and both at last, when overpowered in 
the conflict, reluctantly exchanged their native flag for that of 
the United States of America. The names of both have been 
held in honour by later generations, not more for their 
scientific discoveries than for their services to the cause of 
religious, intellectual, and political freedom. 

The most celebrated electrician of Priestley's contemporaries 
in London was the Hon. Henry Cavendish (b. 1731, d. 1810), 
whose interest in the subject was indeed hereditary, for his 
father, Lord Charles Cavendish, had assisted in Watson's experi- 
ments of 1747.f In 1771 Cavendish} presented to the Koyal 
Society an " Attempt to explain some of the principal phenomena 
of Electricity, by means of an elastic fluid." The hypothesis j 

* In 1769 Dr. John Robison (b. 1739, d. 1805), of Edinburgh, endeavoured to 
determine the law of force by direct experiment, and found it to be tbat of the 
inverse 2'06 th power of the distance. 

t Phil. Trans, xlv, p. 67 (1750). J Phil. Trans. Ixi, p. 584 (1771). 

E 2 



52 Electric and Magnetic Science 

adopted is that of the one-fluid theory, in much the same form 
as that of Aepinus. It was, as he tells us, discovered indepen- 
dently, although he became acquainted with Aepinus' work 
before the publication of his own paper. 

In this memoir Cavendish makes no assumption regarding 
the law of force between electric charges, except that it is 
" inversely as some less power of the distance than the cube " ; 
but he evidently inclines to believe in the law of the inverse 
square. Indeed, he shows it to be " likely, that if the electric 
attraction or repulsion is inversely as the square of the distance, 
almost all the redundant fluid in the body will be lodged close 
to the surface, and there pressed close together, and the rest of 
the body will be saturated"; which approximates closely to the 
discovery made four years previously by Priestley. Cavendish 
did, as a matter of fact, rediscover the inverse square law shortly 
afterwards; but, indifferent to fame, he neglected to communicate 
to others this and much other work of importance. The value of 
his researches was not realized until the middle of the nineteenth 
century, when William Thomson (Lord Kelvin) found in Caven- 
dish's manuscripts the correct value for the ratio of the electric 
charges carried by a circular disk and a sphere of the same radius 
which had been placed in metallic connexion. Thomson urged 
that the papers should be published ; which came to pass* in 
1879, a hundred years from the date of the great discoveries 
which they enshrined. It was then seen that Cavendish had 
anticipated his successors in several of the ideas which will 
presently be discussed amongst others, those of electrostatic 
capacity and specific inductive capacity. 

In the published memoir of 1771 Cavendish worked out the 
consequences of his fundamental hypothesis more completely 
than Aepinus ; and, in fact, virtually introduced the notion of 
electric potential, though, in the absence of any definite assump- 
tion as to the law of force, it was impossible to develop this idea 
to any great extent. 

* The Electrical Researches of the Hon. Henry Cavendish, edited by J. Clerk 
Maxwell, 1879. 



prior to the Introduction of the Potentials. 53 

One of the investigations with which Cavendish occupied 
himself was a comparison between the conducting powers of 
different materials for electrostatic discharges. The question 
had been first raised by Beccaria, who had shown* in 1753 that 
when the circuit through which a discharge is passed contains 
tubes of water, the shock is more powerful when the cross-section 
of the tubes is increased. Cavendish went into the matter 
much more thoroughly, and was able, in a memoir presented to 
the Eoyal Society in 1775,f to say : " It appears from some 
experiments, of which I propose shortly to lay an account before 
this Society, that iron wire conducts about 400 million times 
better than rain or distilled water that is, the electricity meets 
with no more resistance in passing through a piece of iron wire 
400,000,000 inches long than through a column of water of the 
same diameter only one inch long. Sea- water, or a solution of 
one part of salt in 30 of water, conducts 100 times, or a saturated 
solution of sea-salt about 720 times, better than rain-water." 

The promised account of the experiments was published in 
the volume edited in 1879. It appears from it that the method 
of testing by which Cavendish obtained these, results was 
simply that of physiological sensation; but the figures given 
in the comparison of iron and sea- water are remarkably exact. 

While the theory of electricity was being established on a sure 
foundation by the great investigators of the eighteenth century, 
a no less remarkable development was taking place in the 
kindred science of magnetism, to which our attention must now 
be directed. 

The law of attraction between magnets was investigated at 
an earlier date than the corresponding law for electrically 
charged bodies. Newton, in the Principia says : " The power of 
gravity is of a different nature from the power of magnetism. 
For the magnetic attraction is not as the matter attracted. 
Some bodies are attracted more by the magnet, others less ; most 
bodies not at all. The power of magnetism, in one and the same 

* G. B. Beccaria, DdV ehttridsmo artificiale e natural*, Turin. 1753, p. 113. 
+ Phil. Trans. Ixvi (1776), p. 196. % Book iii, Prop, vi, cor. 5. 



54 Electric and Magnetic Science 

body, may be increased and diminished ; and is sometimes far 
stronger, for the quantity of matter, than the power of gravity ; 
and in receding from the magnet, decreases not in the duplicate, 
but almost in the triplicate proportion of the distance, as nearly 
as I could judge from some rude observations." 

The edition of ihePrincipia which was published in 1742 by 
Thomas Le Seur and Francis Jacquier contains a note on this 
corollary, in which the correct result is obtained that the 
directive couple exercised on one magnet by another is 
proportional to the inverse cube of the distance. 

The first discoverer of the law of force between magnetic 1 
\ poles was John Michell (b. 1724, d. 1793), at that time a young 
Fellow of Queen's College, Cambridge,* who in 1750 published 
A Treatise of Artificial Magnets ; in ivhich is shown an easy 
and expeditious method of making them superior to the lest 
natural ones. In this he states the principles of magnetic 
theory as followsf : 

" Wherever any Magnetism, is found, whether in the Magnet 
itself, or any piece of Iron, etc., excited by the Magnet, there are 
always found two Poles, which are generally called North and 
South ; and the North Pole of one Magnet always attracts the 
South Pole, and repels the North Pole of another: and wee versa" 
This is of course adopted from Gilbert. 

"Each Pole attracts or repels exactly equally, at equal 
distances, in every direction." This, it may be observed, over- 
throws the theory of vortices, with which it is irreconcilable. 
" The Magnetical Attraction and Eepulsion are exactly equal to 
each other." This, obvious though it may seem to us, was really 
a most important advance, for, as he remarks, " Most people, who 

* Michell had taken his degree only two years previously. Later in life he was 
on terms of friendship with Priestley, Cavendish, and William Herschel ; it was 
he who taught Herschel the art of grinding mirrors for telescopes. The plan of 
determining the density of the earth, which was carried out by Cavendish in 1798, 
and is generally known as the " Cavendish Experiment," was due to Michell. 
Michell was the first inventor of the torsion-balance ; he also made many valuable 
contributions to Astronomy. In 1767 he became Rector of Thornhill, Yorks, 
and lived there until his death. 

t Loc. cit., p. 17. 



-^ 

prior to the Introduction of the Potentials. 55 

have mention'd any thing relating to this property of the Magnet, 
have agreed, not only that the Attraction and Repulsion of 
Magnets are not equal to each other, but that also, they do not 
observe the same rule of increase and decrease." 

" The Attraction and Eepulsion of Magnets decreases, as the 
Squares of the distances from the respective poles increase." 
This great discovery, which is the basis of the mathematical 
theory of Magnetism, was deduced partly from his own observa- 
tions, and partly from those of previous investigators (e.g. 
Dr. Brook Taylor and P. Muschenbroek), who, as he observes, 
had made accurate experiments, but had failed to take into 
account all the considerations necessary for a sound theoretical 
discussion of them. 

After Michell the law of the inverse square was maintained 
by Tobias Mayer* of Gottingen (&. 1723, d. 1762), better known 
as the author of Lunar Tables which were long in use ; and by 
the celebrated mathematician, Johann Heinrich Lambertf (b. 
1728, d. 1777). 

The promulgation of the one-fluid theory of electricity, in 
the middle of the eighteenth century, naturally led to attempts 
to construct a similar theory of magnetism ; this was effected in 
1759 by AepinusJ, who supposed the "poles "to be places at 
which a magnetic fluid was present in amount exceeding or 
falling short of the normal quantity. The permanence of 
magnets was accounted for by supposing the fluid to be entangled 
in their pores, so as to be with difficulty displaced. The particles 
of the fluid were assumed to repel each other, and to attract the 
particles of iron and steel ; but, as Aepinus saw, in order to satis- 
factorily explain magnetic phenomena it was necessary to assume 
also a mutual repulsion among the material particles of the 
magnet. 

Subsequently two imponderable magnetic fluids, to which 

* Noticed in Gottinger Gelehrter Anzeiger, 1760 : cf. Aepinus, Nov. Comm. 
Acad. Petrop., 1768, and Mayer's Opera Inedita, herausg. von G. C. Lichtenberg. 
\-Histoirede V Acad. de Berlin, 1766, pp. 22, 49. 
% In the Tentamen, to which reference has already been made. 



56 Electric and Magnetic Science 

the names boreal and austral were assigned, were postulated by 
the Hollander Anton Brugmans (5. 1732, d. 1789) and by 
Wilcke. These fluids were supposed to have properties of 
mutual attraction and repulsion similar to those possessed by 
vitreous and resinous electricity. 

The writer who next claims our attention for his services 
both to magnetism and to electricity is the French physicist, 
Charles Augustin Coulomb* (ft. 1736, d. 1806). By aid of the 
torsion-balance, which was independently invented by Michell 
and himself, he verified in 1785 Priestley's fundamental law 
that the repulsive force between two small globes charged with 
the same kind of electricity is in the inverse ratio of the square 
of the distance of their centres. In the second memoir he 
extended this law to the attraction of opposite electricities. 

Coulomb did not accept the one-fluid theory of Franklin, 
Aepinus, and Cavendish, but preferred a rival hypothesis which 
had been proposed in 1759 by Kobert Symmer.f " My notion," 
said Symmer, " is that the operations of electricity do not depend 
upon one single positive power, according to the opinion generally 
received; but upon two distinct, positive, and active powers, 
which, by contrasting, and, as it were, counteracting each other, 
produce the various phenomena of electricity ; and that, when a 
body is said to be positively electrified, it is not simply that it is 
possessed of a larger share of electric matter than in a natural 
state ; nor, when it is said to be negatively electrified, of a less ; 
but that, in the former case, it is possessed of a larger portion 
of one of those active powers, and in the latter, of a larger 
portion of the other ; while a body in its natural state remains 
unelectrified, from an equal ballance of those two powers within 
it." 

Coulomb developed this idea : " Whatever be the cause of 
electricity," he says,J " we can explain all the phenomena by 

* Coulomb's First, Second, and Third Memoirs appear in Memoires de 1'Acad., 
1785 ; the Fourth in 1786, the Fifth in 1787, the Sixth in 1788, and the Seventh 
in 1789. 

t Phil. Trim*, li (1759), p. 371. j Sixth Memoir, p. 561. 



prior to the Introduction of the Potentials. 57 

supposing that there are two electric fluids, the parts of the 
same fluid repelling each other according to the inverse square 
of the distance, and attracting the parts of the other fluid 
according to the same inverse square law." " The supposition ^ 
of two fluids," he adds, " is moreover in accord with all those 7 
discoveries of modern chemists and physicists, which have made 
known to us various pairs of gases whose elasticity is destroyed 
by their admixture in certain proportions an effect which could 
not take place without something equivalent to a repulsion 
between the parts of the same gas, which is the cause of its 
elasticity, and an attraction between the parts of different 
gases, which accounts for the loss of elasticity on combination." J 

According, then, to the two-fluid theory, the " natural fluid " 
contained in all matter can be decomposed, under the influence 
of an electric field, into equal quantities of vitreous and 
resinous electricity, which, if the matter be conducting, can then 
fly to the surface of the body. The abeyance of the characteristic 
properties of the opposite electricities when in combination was f 
sometimes further compared to the neutrality manifested by . 
the compound of an acid and an alkali. 

The publication of Coulomb's views led to some controversy 
between the partisans of the one-fluid and two-fluid theories ; the 
latter was soon generally adopted in France, but was stoutly 
opposed in Holland by Van Marum and in Italy by Volta. 
The chief difference between the rival hypotheses is that, in the ^ 
two-fluid theory, both the electric fluids are movable within the 
substance of a solid conductor ; while in the one-fluid theory the 
actual electric fluid is mobile, but the particles of the conductor 
are fixed. The dispute could therefore be settled only by a deter- 
mination of the actual motion of electricity in discharges ; and 
this was beyond the reach of experiment. 

In his Fourth Memoir Coulomb showed that electricity in 
equilibrium is confined to the surface of conductors, and does 
not penetrate to their interior substance ; and in the Sixth 
Memoir* he virtually establishes the result that the electric 

* Page 677. 



58 Electric and Magnetic Science 

force near a conductor is proportional to the surface-density of 
electrification. 

Since the overthrow of the doctrine of electric effluvia by 
Aepinus, the aim of electricians had been to establish their 
science upon the foundation of a law of action at a distance, 
resembling that which had led to such triumphs in Celestial 
Mechanics. When the law first stated by Priestley was at 
length decisively established by Coulomb, its simplicity and 
beauty gave rise to a general feeling of complete trust in it as 
the best attainable conception of electrostatic phenomena. 
The result was that attention was almost exclusively focused 
on action-at-a-distance theories, until the time, long afterwards,, 
when Faraday led natural philosophers back to the right' 
path. 

Coulomb rendered great services to magnetic theory. It was 
he who in 1777, by simple mechanical reasoning, completed 
the overthrow of the hypothesis of vortices.* He also, in the 
second of the Memoirs already quoted,f confirmed Michell's 
law, according to which the particles of the magnetic fluids 
attract or repel each other with forces proportional to the 
inverse square of the distance. Coulomb, however, went beyond 
this, and endeavoured to account for the fact that the two 
magnetic fluids, unlike the two electric fluids, cannot be 
obtained separately; for when a magnet is broken into 
two pieces, one containing its north and the other its south 
pole, it is found that each piece is an independent magnet 
possessing two poles of its own, so that it is impossible 
to obtain a north or south pole in a state of isolation. 
Coulomb explained this by supposing^ that the mag- 
netic fluids are permanently imprisoned within the molecules 
of magnetic bodies, so as to be incapable of crossing from 
one molecule to the next ; each molecule therefore under all 
circumstances contains as much of the boreal as of the 

* Mem. presences par divers Savans, ix (1780), p. 165. 

t Mem de 1'Acad., 1785, p. 593. Gauss finally established the law by a 
much more refined method. 

J In his Seventh Memoir, Mem, de 1'Acad., 1789, p. 488. 



prior to the Introduction of the Potentials. 59 

austral fluid, and magnetization consists simply in a separation 
of the two fluids to opposite ends of each molecule. Such 
a hypothesis evidently accounts for the impossibility of 
separating the two fluids to opposite ends of a body of finite 
size. The same idea, here introduced for the first time, has 
since been applied with success in other departments of 
electrical philosophy. 

In spite of the advances which have been recounted, 
the mathematical development of electric and magnetic theory 
was scarcely begun at the close of the eighteenth century ; and 
many erroneous notions were still widely entertained. In a 
Eeport* which was presented to the French Academy in 1800, 
it was assumed that the mutual repulsion of the particles of 
electricity on the surface of a body is balanced by the 
resistance of the surrounding air; and for long afterwards 
the electric force outside a charged conductor was confused 
with a supposed additional pressure in the atmosphere. 

Electrostatical theory was, however, suddenly advanced to 
quite a mature state of development by Simeon Denis Poisson 
(b. 1781, d. 1840), in a memoir which was read to the French 
Academy in 1812.f As the opening sentences show, he accepted 
the conceptions of the two-fluid theory. 

" The theory of electricity which is most generally accepted," 
he says, " is that which attributes the phenomena to two 
different fluids, which are contained in all material bodies. 
It is supposed that molecules of the same fluid repel each 
other and attract the molecules of the other fluid ; these 
forces of attraction and repulsion obey the law of the inverse 
square of the distance ; and at the same distance the attractive 
power is equal to the repellent power; whence it follows 
that, when all the parts of a body contain equal quantities 
of the two fluids, the latter do not exert any influence on 
the fluids contained in neighbouring bodies, and consequently 
no electrical effects are discernible. This equal and uniform 

* On Yolla's discoveries. 

t Mem. de Plnstitut, 1811, Part i., p. 1, Part ii., p. 163. 



60 Electric and Magnetic Science 

distribution of the two fluids is called the natural state ; when this 
state is disturbed in any body, the body is said to be electrified, 
and the various phenomena of electricity begin to take place. 

"Material bodies do not all behave in the same way with 
respect to the electric fluid : some, such as the metals, do 
not appear to exert any influence on it, but permit it to 
move about freely in their substance ; for this reason they 
are called conductors. Others, on the contrary very dry air, 
for example oppose the passage of the electric fluid in their 
interior, so that they can prevent the fluid accumulated in 
conductors from being dissipated throughout space." 

When an excess of one of the electric fluids is communi- 
cated to a metallic body, this charge distributes itself over the 
surface of the body, forming a layer whose thickness at any 
point depends on the shape of the surface. The resultant force 
due to the repulsion of all the particles of this surface-layer 
must vanish at any point in the interior of the conductor, since 
otherwise the natural state existing there would be disturbed ; 
and Poisson showed that by aid of this principle it is possible 
in certain cases to determine the distribution of electricity in 
the surface-layer. For example, a well-known proposition of 
the theory of Attractions asserts that a hollow shell whose 
bounding surfaces are two similar and similarly situated 
ellipsoids exercises 110 attractive force at any point within the 
interior hollow; and it may thence be inferred that, if an 
electrified metallic conductor has the form of an ellipsoid, the 
charge will be distributed on it proportionally to the normal 
distance from the surface to an adjacent similar and similarly 
situated ellipsoid. 

Poisson went on to show that this result was by no means all 

that might with advantage be borrowed from the theory of 

I Attractions. Lagrange, in a memoir on the motion of gravitating 

bodies, had shown* that the components of the attractive force 

* Mem. de Berlin, 1777. The theorem was afterwards published, and ascribed 
to Laplace, in a memoir by Legendre on the Attractions of Spheroids, which will 
be found in the Mem. par divers Snvanx, published in 178o. 



prior to the Introduction of the Potentials. 61 

at any point can be simply expressed as the derivates of the 
function which is obtained by adding together the masses of all 
the particles of an attracting system, each divided by its 
distance from the point; and Laplace had shown* that this 
function V satisfies the equation 



in space free from attracting matter. Poisson himself showed 
later, in 1813,f that when the point (z, y, z) is within the 
substance of the attracting body, this equation of Laplace must 
be replaced by 

W VV VV 

^ + w~~v r: p> 

where p denotes the density of the attracting matter at the 
point. In the present memoir Poisson called attention to the 
utility of this function F in electrical investigations, remarking 
that its value over the surface of any conductor must be 
constant. 

The known formulae for the attractions of spheroids show 
that when a charged conductor is spheroidal, the repellent force 
acting on a small charged body immediately outside it will be 
directed at right angles to the surface of the spheroid, and will 
be proportional to the thickness of the surface-layer of electricity 
at this place. Poisson suspected that this theorem might be 
true for conductors not having the spheroidal form a result 
which, as we have seen, had been already virtually given by 
Coulomb ; and Laplace suggested to Poisson the following 
proof, applicable to the general case. The force at a point 
immediately outside the conductor can be divided into a 
part s due to the part of the charged surface immediately 
adjacent to the point, and a part S due to the rest of 
the surface. At a point close to this, but just inside the con- 
ductor, the force j^jpll still act; but the forces will evidently 



* Mem. de 1'Acad., 1782 (published in 1785), p. 113. 
t Bull, de la Soc. Philomathique. iii. (1813,, p. 388. 



62 Electric and Magnetic Science 

be reversed in direction. Since the resultant force at the latter 
point vanishes, we must have S=s ; so the resultant force at the 
exterior point is 2s. But s is proportional to the charge per 
unit area of the surface, as is seen by considering the case of 
an infinite plate ; which establishes the theorem. 

When several conductors are in presence of each other, the 
distribution of electricity on their surfaces may be determined 
by the principle, which Poisson took as the basis of his work, 
that at any point in the interior of any one of the conductors, 
the resultant force due to all the surf ace -layers must be zero. 
He discussed, in particular, one of the classical problems of 
electrostatics namely, that of determining the surface-density 
on two charged conducting spheres placed at any distance from 
each other. The solution depends on Double Gamma Functions 
in the general case ; when the two spheres are in contact, it 
depends on ordinary Gamma Functions. Poisson gave a solution 
in terms of definite integrals, which is equivalent to that in 
terms of Gamma Functions ; and after reducing his results to 
numbers, compared them with Coulomb's experiments. 
f The rapidity with which in a single memoir Poisson passed 
from the barest elements of the subject to such recondite 
problems as those just mentioned may well excite admiration. 
His success is, no doubt, partly explained by the high state of 
development to which analysis had been advanced by the great 
mathematicians of the eighteenth century ; but even after 
allowance has been made for what is due to his predecessors, 
Poisson' s investigation must be accounted a splendid memorial 
u of his genius. 

Some years later Poisson turned his attention to magnetism ; 
and, in a masterly paper* presented to the French Academy in 
1824, gave a remarkably complete theory of the subject. 

His starting-point is Coulomb's doctrine of two imponderable 
magnetic fluids, arising from the decomposition of a neutral 
fluid, and confined in their movements to the individual elements 

* Mem. <le 1'Acad., v, p. 247. 



prior to the Introduction of the Potentials. 63 

of the magnetic body, so as to be incapable of passing from one 
element to the next 

Suppose that an amount m of the positive magnetic fluid is 
located at a point (x y, z) ; the components of the magnetic 
intensity, or force exerted on unit magnetic pole, at a point 
(, f, ) will evidently be 

-m-f-X -m~(-\ -m-(-) 

where r denotes ((? - xf + (n - ?/) 2 + (Z - z) 2 j*. Hence if we 
consider next a magnetic element in which equal quantities of 
the two magnetic fluids are displaced from each other parallel 
to_ the ic-axis, the components of the magnetic intensity at 
(g, i|, 2) will be the negative derivates, with respect to ij, 
respectively, of the function 



where the quantity A, which does not involve (f, j, ), may be 
called the magnetic moment of the element : it may be measured 
by the couple required to maintain the element in equilibrium 
at a definite angular distance from the magnetic meridian. 

If the displacement of the two fluids from each other in the 
element is not parallel to the axis of x t it is easily seen that the 
expression corresponding to the last is 



where the vector (A, B, C) now denotes the magnetic moment 
of the element. 

Thus the magnetic intensity at an -external point (, 77, ) 
due to any magnetic body has the components 



; - 017 
where 



ex oy 
integrated throughout the substance of the magnetic body, and 



64 Electric and Magnetic Science 

where the vector (A, B, C) or I represents the magnetic moment 
per unit- volume, or, as it is generally called, the magnetization. 
The function Fwas afterwards named by Green the magnetic 
potential. 

Poisson, by integrating by parts the preceding expression for 
the magnetic potential, obtained it in the form 

F = [[(I . dS). \ - fjp div I dx dy dz* 

the first integral being taken over the surface $ of the magnetic 
body, and the second integral being taken throughout its volume. 
This formula shows that the magnetic intensity produced by the 
body in external space is the same as would be produced by a 
fictitious distribution of magnetic fluid, consisting of a layer 
over its surface, of surface-charge (I .- dS) per element dS y 
together with a volume-distribution of density - div I through- 
out its substance. These fictitious magnetizations are generally 
known as Poisson's equivalent surface- and volume-distributions 
of magnetism. 

Poisson, moreover, perceived that at a point in a very small 
cavity excavated within the magnetic body, the magnetic 
potential has a limiting value which is independent of the shape 
of the cavity as the dimensions of the cavity tend to zero ; but 
that this is not true of the magnetic intensity, which in such a 
small cavity depends on the shape of the cavity. Taking the 
cavity to be spherical, he showed that the magnetic intensity 
within it is 

grad F 4 ^-7rl,f 
where I denotes the magnetization at the place. 

* If the components of a vector a are denoted by (a x , a y , a z ), the quantity 
drbjc + a y b y -f- a t k z is called the scalar product of two vectors a and b, and is denoted 
by (a . b). 

The quantity ^ ' + ^ + ^ is called the divergence of the vector a, and is 

fix dy 02 

denoted by div a. 

t The vector whose components are - , - ?, - - is denoted by grad V. 

C dy dz J 



prior to the Introduction of the Potentials. 65 

This memoir also contains a discussion of the magnetism 
temporarily induced in soft iron and other magnetizable metals 
by the approach of a permanent magnet. Poisson accounted for 
the properties of temporary magnets by assuming that they 
contain embedded in their substance a great number of small 
spheres, which are perfect conductors for the magnetic fluids ; so 
that the resultant magnetic intensity in the interior of one of 
these small spheres must be zero. He showed that such a sphere, 
when placed in a field of magnetic intensity F,* must acquire a 

magnetic moment of amount -.- F x the volume of the sphere, 

in order to counteract within the sphere the force F. Thus if 
k p denote the total volume of these spheres contained within a 
unit volume of the temporary magnet, the magnetization will be 
I, where 4-TrI = k p F, 

and F denotes the magnetic intensity within a spherical cavity 
excavated in the body. This is Poisson s laiv of induced magnetism. 

It is known that some substances acquire a greater degree 
of temporary magnetization than others when placed in the 
same circumstances : Poisson accounted for this by supposing that 
the quantity k p varies from one substance to another. But the 
experimental data show that for soft iron k p must have a value 
very near unity, which would obviously be impossible if k p is to 
mean the ratio of the volume of spheres contained within a 
region to the total volume of the region.f The physical inter- 
pretation assigned by Poisson to his formulae must therefore be 
rejected, although the formulae themselves retain their value. 

Poisson's electrical and magiietical investigations were 
generalized and extended in 1828 by George Green* (b. 1793, 
d. 1841). Green's treatment is based on the properties of the 
function already used by Lagrange, Laplace, and Poisson, which 

* In the present work, vectors will generally be distinguished by heavy type. 

t This objection was advanced by Maxwell in 430 of his Treatise. An attempt 
to overcome it was made by Betti : cf. p. 377 of his Lessons on the Potential. 

J A.n essay on the application of mathematical analysis to the theories of electricity 
and magnetism, Nottingham, 1828 : reprinted in The Mathematical Papers of the late 
George Green, p. 1. 

F 



66 Electric and Magnetic Science. 

represents the sum of all the electric or magnetic charges in the 
field, divided by their respective distances from some given point : 
to this function Green gave the name potential, by which it has 
always since been known.* 

Near the beginning of the memoir is established the 
celebrated formula connecting surface and volume integrals, 
which is now generally called G-reeris Theorem, and of which 
Poisson's result on the equivalent surface- and volume-distribu- 
tions of magnetization is a particular application. By using 
this theorem to investigate the properties of the potential, 
Green arrived at many results of remarkable beauty and 
interest. We need only mention, as an example of the power 
of his method, the following : Suppose that there is a hollow 
conducting shell, bounded by two closed surfaces, and that a 
number of electrified bodies are placed, some within and some 
without it ; and let the inner surface and interior bodies be 
called the interior system, and the outer surface and exterior 
botlies be called the exterior system. Then all the electrical 
phenomena of the interior system, relative to attractions, 
repulsions, and densities, will be the same as if there were no 
exterior system, and the inner surface were a perfect conductor, 
put in communication with the earth ; and all those of the 
exterior system will be the same as if the interior system did not 
exist, and the outer surface were a perfect conductor, containing 
a quantity of electricity equal to the whole of that originally 
contained in the shell itself and in all the interior bodies. 

It will be evident that electrostatics had by this time 
attained a state of development in which further progress could 
be hoped for only in the mathematical superstructure, unless 
experiment should unexpectedly bring to light phenomena of 
an entirely new character. This will therefore be a convenient 
place to pause and consider the rise of another branch of 
electrical philosophy. 

* Euler in 1744 (De melhodis inveniendi . . .) had spoken of the vis potentialis 
what would now be called the potential energy possessed by an elastic body 
when bent. 



CHAPTEE III. 

GALVANISM, FROM GALVANI TO OHM. 

UNTIL the last decade of the eighteenth century, electricians 
were occupied solely with statical electricity. Their attention 
was then turned in a different direction. 

In a work entitled Recherches sur Vorigine des sentiments 
agreables et cUsagr cables, which was published* in 1752, 
Johann Georg Sulzer (b. 1720, d. 1779) had mentioned that, if 
two pieces of metal, the one of lead and the other of silver, be 
joined together in such a manner that their edges touch, and if 
they be placed on the tongue, a taste is perceived " similar to 
that of vitriol of iron," although neither of these metals applied 
separately gives any trace of such a taste. " It is not probable," 
he says, " that this contact of the two metals causes a solution 
of either of them, liberating particles which might affect the 
tongue : and we must therefore conclude that the contact sets 
up a vibration in their particles, which, by affecting the nerves 
of the tongue, produces the taste in question." 

This observation was not suspected to have any connexion 
with electrical phenomena, and it played no part in the incep- 
tion of the next discovery, which indeed was suggested by a 
mere accident. 

Luigi Galvani, born at Bologna in 1737, occupied from 1775 
onwards a chair of Anatomy in his native city. For many years 
before the event which made him famous he had been studying 
the susceptibility of -the nerves to irritation ; and, having been <- 
formerly a pupil of Beccaria, he was also interested in electrical 
experiments. One day in the latter part of the year 1780 he ' 
had, as he tells us,f " dissected and prepared a frog, and laid it 
on a table, on which, at some distance from the frog, was an 
electric machine. It happened by chance that one of my 

* Mem. de 1'Acad. de Berlin, 1752, p. 356. 

t Aloysii Galvani, De Viribus E 'lee trie itatis in Motu Mnsculari : Commentarii 
Bononiensi, vii (1791), p. 363. 

F 2 



68 Galvanism, from Galvani to Ohm. 

assistants touched the inner crural nerve of the frog with the 
point of a scalpel ; whereupon at once the muscles of the limbs 
were violently convulsed. 

" Another of those who used to help me in electrical experi- 
ments thought he had noticed that at this instant a spark was 
drawn from the conductor of the machine. I myself was at the 
time occupied with a totally different matter; but when he 
drew my attention to this, I greatly desired to try it for myself,. 
and discover its hidden principle. So I, too, touched one or 
other of the crural nerves with the point of the scalpel, at the 
same time that one of those present drew a spark ; and the same 
phenomenon was repeated exactly as before."* 

After this, Galvani conceived the idea of trying whether the 
electricity of thunderstorms would induce muscular contractions 
equally well with the electricity of the machine. Having 
successfully experimented with lightning, he " wished," as he 
writes,! " to try the effect of atmospheric electricity in calm 
weather. My reason for this was an observation I had made,, 
that frogs which had been suitably prepared for these experi- 
ments and fastened, by brass hooks in the spinal marrow, to 
the iron lattice round a certain hanging-garden at my house,, 
exhibited convulsions not only during thunderstorms, but 
sometimes even when the sky was quite serene. I suspected 
these effects to be due to the changes which take place during 
the day in the electric state of the atmosphere ; and so, with 
some degree of confidence, I performed experiments to test the 
point; and at different hours for many days I watched frogs 
which I had disposed for the purpose ; but could not detect any 
motion in their muscles. At length, weary of waiting in vain, 
I pressed the brass hooks, which were driven into the spinal 
marrow, against the iron lattice, in order to see whether 
contractions could be excited by varying the incidental circum- 

* According to a story which has often been repeated, but which rests on no 
sufficient evidence, the frog was one of a number which had been procured for th& 
Signora Galvani, who, being in poor health, had been recommended to take a soup, 
made of these animals as a restorative. f Loc. cit., p. 377. 



Galvanism, from Galvani to Ohm. 69 

stances of the experiment. I observed contractions tolerably 
often, but they did not seem to bear any relation to the changes 
in the electrical state of the atmosphere. 

" However, at this time, when as yet I had not tried the 
experiment except in the open air, I came very near to adopt- 
ing a theory that the contractions are due to atmospheric 
electricity, which, having slowly entered the animal and accu- 
mulated in it, is suddenly discharged when the hook comes in 
contact with the iron lattice. For it is easy in experimenting 
to deceive ourselves, and to imagine we see the things we wish 
to see. 

" But I took the animal into a closed room, and placed it on 
an iron- plate ; and when I pressed the hook which was fixed 
in the spinal marrow against the plate, behold ! the same 
spasmodic contractions as before. I tried other metals at 
different hours on various days, in several places, and always 
with the same result, except that the contractions were more 
violent with some metals than with others. After this I tried 
various bodies which are not conductors of electricity, such as 
glass, gums, resins, stones, and dry wood ; but nothing happened. 
This was somewhat surprising, and led me to suspect that 
electricity is inherent in the animal itself. This suspicion was 
strengthened by the observation that a kind of circuit of subtle 
nervous fluid (resembling the electric circuit which is manifested 
in the Leyclen jar experiment) is completed from the nerves to 
the muscles when the contractions are produced. 

" For, while I with one hand held the prepared frog by the 
hook fixed in its spinal marrow, so that it stood with its feet 
on a silver box, and with the other hand touched the lid of 
the box, or its sides, with any metallic body, I was surprised 
to see the frog become strongly convulsed every time that I 
applied this artifice."* 

Galvani thus ascertained that the limbs of the frog are con- 
vulsed whenever a connexion is made between the nerves and 
muscles by a metallic arc, generally formed of more than one 

*This observation was made in 1786. 



70 Galvanism > from Galvani to Ohm. 

kind of metal ; and he advanced the hypothesis that the convul- 
sions are caused by the transport of a peculiar fluid from the 

' nerves to the muscles, the arc acting as a conductor. To this 
fluid the names Galvanism and .Animal Electricity were soon 
generally applied. Galvani himself considered it to be the same 
as the ordinary electric fluid, and, indeed, regarded the entire 
phenomenon as similar to the discharge of a Leyden jar. 

*' The publication of Gralvani's views soon engaged the attention 
of the learned world, and gave rise to an animated controversy 
between those who supported Galvani's own view, those who 
believed galvanism to be a fluid distinct from ordinary electricity, 
and a third school who altogether refused to attribute the effects 
to a supposed fluid contained in the nervous system. The leader 
of the last-named party was Alessandro Volta (b. 1745, d. 1827), 
Professor of Natural Philosophy in the University of Pavia, who 
in 1792 put forward the view* that the stimulus in Galvani's 
experiment is derived essentially from the connexion of two 
different metals by a moist body. "The metals used in the 

* experiments, being applied to the moist bodies of animals, can by 
themselves, and of their proper virtue, excite and dislodge the 
electric fluid from its state of rest ; so that the organs of the 

* animal act only passively." At first he inclined to combine this 
theory of metallic stimulus with a certain degree of belief in 
such a fluid as Galvani had supposed; but after the end of 17!. '3 
he denied the existence of animal electricity altogether. 

From this standpoint Volta continued his experiments and 
worked out his theory. The following quotation from a lettert 
which he wrote later to Gren, the editor of the Neucs Journal //. 
Physik, sets forth his view in a more developed form : 

"The contact of different conductors, particularly the metallic, 
including pyrites and other minerals, as well as charcoal, which 
I call dry conductors, or of the first class, with moist conductors, 
or conductors of the second class, agitates or disturbs the electric 

f fluid, or gives it a certain impulse. Do not ask in what manner : 
it is enough that it is a principle, and a general principle. This 

*Phil. Trans., 1793, pp. 10, 27. tPhil. Mag. iv (1799), pp. 59, 163, 306. 



Galvanism , from Galvani to Okm. 71 

impulse, whether produced by attraction or any other force, is 
different or unlike, both in regard to the different metals and to 
the different moist conductors ; so that the direction, or at least 
the power, with which the electric fluid is impelled or excited, is 
different when the conductor A is applied to the conductor B, or 
to another C. In a perfect circle of conductors, where either 
one of the second class is placed between two different from each 
other of the first class, or, contrariwise, one of the first class is 
placed between two of the second class different from each other, 
an electric stream is occasioned by the predominating force either 
to the right or to the left a circulation of this fluid, which ceases 
only when the circle is broken, and which is renewed when the 
circle is again rendered complete." 

Another philosopher who, like Volta, denied the existence of 
a fluid peculiar to animals, but who took a somewhat different 
view of the origin of the phenomenon, was Giovanni Fabroni, of 
Florence (b. 1752, d. 1822), who,* having placed two plates of 
different metals in water, observed that one of them was partially 
oxidized when they were put in contact ; from which he rightly 
concluded that some chemical action is inseparably connected 
with galvanic effects. 

The feeble intensity of the phenomena of galvanism, which 
compared poorly with the striking displays obtained in electro- 
statics, was responsible for some falling off of interest in them 
towards the end of the eighteenth century ; and the last years 
of their illustrious discoverer were clouded by misfortune. Being 
attached to the old order which was overthrown by the armies 
of the French Ke volution, he refused in 1798 to take the oath of 
allegiance to the newly constituted Cisalpine Eepublic, and was 
deposed from his professorial chair. A profound melancholy, 
which had been induced by domestic bereavement, was aggra- 
vated by poverty and disgrace ; and, unable to survive the loss 
of all he held dear, he died broken-hearted before the end of 
the year.f 

* Phil. Journal, 4to, iii. 308 ; iv. 120 ; Journal de Physique, vi. 348. 
t A decree of reinstatement had been granted, but had not come into operation 
at the time of Galvani's death. 



< 



72 Galvanism, Jrom Galvani to O/it/i. 

Scarcely more than a year after the death of Galvani, the 
new science suddenly regained ' the eager attention of philo- 
sophers. This renewal of interest was due to the discovery by 
Volta, in the early spring of 1800, of a means of greatly increasing 
the intensity of the effects. Hitherto all attempts to magnify 
the action by enlarging or multiplying the apparatus had ended 
in failure. If a long chain of different metals was used instead 
of only two, the convulsions of the frog were no more violent. 
But Volta now showed* that if any number of couples, each 
consisting of a zinc disk and a copper disk in contact, were taken, 
and if each couple was separated from the next by a disk of moist- 
ened pasteboard (so that the order was copper, zinc, pasteboard, 
copper, zinc, pasteboard, &c.), the effect of the pile thus formed 
was much greater than that of any galvanic apparatus previously 
introduced. When the highest and lowest disks were simul- 
taneously touched by the fingers, a distinct shock was felt ; and 
this could be repeated again and again, the pile apparently 
possessing within itself an indefinite power of recuperation. It 
thus resembled a Leyden jar endowed with a power of automati- 
cally re-establishing its state of tension after each explosion; 
with, in fact, " an inexhaustible charge, a perpetual action or 
impulsion on the electric fluid." 

Volta unhesitatingly pronounced the phenomena of the pile 
to be in their nature electrical. The circumstances of Galvani's 
original discovery had prepared the minds of philosophers for 
this belief, which was powerfully supported by the similarity of 
the physiological effects of the pile to those of the Leyden jar, 
and by the observation that the galvanic influence was conducted 
only by those bodies e.g. the metals which were already 
known to be good conductors of static electricity. But Volta 
now supplied a still more convincing proof. Taking a disk of 
copper and one of zinc, 'he held each by an insulating handle 
and applied them to each other for an instant. After the disks 
had been separated, they were brought into contact with a deli- 

* I'hil. Trans., 1800, p. 403. 



Galvanism, from Galvani to Ohm. 73 

oate electroscope, which indicated by the divergence of its straws 
that the disks were now electrified the zinc had, in fact, acquired 
a positive and the copper a negative electric charge.* Thus the 
mere contact of two different metals, such as those employed in / 
the pile, was shown to be sufficient for the production of effects ' 
undoubtedly electrical in character. 

On the basis of this result Volta in the same year (1800) 
put forward a definite theory of the action of the pile. Suppose 
first that a disk of zinc is laid on a disk of copper, which in turn 
rests on an insulating support. The experiment just described 
shows that the electric fluid will be driven from the copper to 
the zinc. We may then, according to Volta, represent the state 
or " tension " of the copper by the number - J, and that of the 
zinc by the number + J, the difference being arbitrarily taken as 
unity, and the sum being (on account of the insulation) zero. It 
will be seen that Volta's idea of " tension " was a mingling of 
two ideas, which in modern electric theory are clearly distin- 
guished from each other namely, electric charge and electric 
potential. 

Now let a disk of moistened pasteboard be laid on the zinc, 
and a disk of copper on this again. Since the uppermost 
copper is not in contact with the zinc, the contact-action does 
not take place between them ; but since the moist pasteboard is 
a conductor, the copper will receive a charge from the zinc. 
Thus the states will now be represented by - f for the lower 
copper, + J for the zinc, and + \ for the upper copper, giving a 
zero sum as before. 

If, now, another zinc disk is placed on the top, the states 
will be represented by - 1 for the lower copper, for the lower 
zinc and upper copper, and + 1 for the upper zinc. 

In this way it is evident that the difference between the 
numbers indicating the tensions of the uppermost and lowest 

* Abraham Bennet (b. 1750, d. 1799) had previously shown (Xew Experiments 
in Electricity, 1789, pp. 86-102) that many bodies, when separated after contact, f 
are oppositely electrified ; he conceived that different bodies have different attrac- 
tions or capacities for electricity. 



74 Galvanism , from Galvani to O/im. 

disks in the pile will always be equal to the number of pairs of 
metallic disks contained in it. If the pile is insulated, the 
sum of the numbers indicating the states of all the disks must 
be zero; but if the lowest disk is connected to earth, the 
tension of this disk will be zero, and the numbers indicating the 
states of all the other disks will be increased by the same 
amount, their mutual differences remaining unchanged. 

The pile as a whole is thus similar to a Leyden jar ; 
when the experimenter touches the uppermost and lowest 
disks, he receives the shock of its discharge, the intensity being 
proportional to the number of disks. 

The moist layers played no part in Volta's theory beyond 
j. that of conductors.* It was soon found that when the moisture 
is acidified, the pile is more efficient; but this was attributed 
solely to the superior conducting power of acids. 

Yolta fully understood and explained the impossibility of 
constructing a pile from disks of metal alone, without making 
use of moist substances. As he showed in 1801, if disks of 
various metals are placed in contact in any order, the extreme 
metals will be in the same state as if they touched each other 
directly without the intervention of the others ; so that the 
whole is equivalent merely to a single pair. When the metals 
are arranged in the order silver, copper, iron, tin, lead, zinc, 
each of them becomes positive with respect to that which 
precedes it, and negative with respect to that which follows it ; 
but the moving force from the silver to the zinc is equal to the 
sum of the moving forces of the metals comprehended between 
them in the series. 

When a connexion was maintained for some time between 
the extreme disks of a pile by the human body, sensations 
were experienced which seemed to indicate a continuous activity 
in the entire system. Yolta inferred that the electric current 
persists during the whole time that communication by con- 

* Volta had inclined, in his earlier experiments on galvanism, to locate the seat 
of power at the interfaces of the metals with the rnoist conductors. Cf. his letter 
to Gren, Phil. Mag. iv (1799), p. 62. 



Galvanism, from Gaivani to Ohm. 75 

ductors exists all round the circuit, and that the current is 
suspended only when this communication is interrupted. 
" This endless circulation or perpetual motion of the electric 
fluid," he says, "may seem paradoxical, and may prove 
inexplicable ; but it is none the less real, and we can, so to 
speak, touch and handle it." 

Yolta announced his discovery in a letter to Sir Joseph 
Banks, dated from Como, March 20th, 1800. Sir Joseph, who 
was then President of the Eoyal Society, communicated the 
news to William Nicholson (b. 1753, d. .1815), founder of the 
Journal which is generally known by his name, and his 
friend Anthony Carlisle (b. 1768, d. 1840), afterwards a 
distinguished surgeon. On the 30th of the following month, 
Nicholson and Carlisle set up the first pile made in England. In 
repeating Volta's experiments, having made the contact more 
secure at the upper plate of the pile by placing a drop of water 
there, they noticed* a disengagement of gas round the con- 
ducting wire at this point ; whereupon they followed up the 
matter by introducing a tube of water, into which the wires 
from the terminals of the pile were plunged. Bubbles of an 
inflammable gas were liberated at one wire, while the other 
wire became oxidised ; when platinum wires were used, oxygen 
and hydrogen were evolved in a free state, one at each wire. 
This effect, which was nothing less than the electric decom- 
position of water into its constituent gases, was obtained on 
May 2nd, 1800.f 

Although it had long been known that frictional electricity 
is capable of inducing chemical action,* the discovery of 
Nicholson and Carlisle was of the first magnitude. It was at 
once extended by William Cruickshank, of Woolwich (b. 1745, 



i's Journal (4to), iv, 179 (1800) ; Phil. Mag. vii, 337 (1800). 

t It was obtained independently four months later l>y J. "W. Hitter. 

J Beccaria (Lettere deW elettricismo, Bologna, 1758, p. 282) had reduced mercury 
and other metals from their oxides by discharges ot fractional electricity ; and 
Priestley had obtained an inflammable gas from certain organic liquids in the 
same way. Cavendish in 1781 had established the constitution of water by 
electrically exploding hydrogen and oxygen. 



76 Galvanism > from Galvani to Ohm. 

d. 1800), who* showed that solutions of metallic salts are also 
decomposed by the current; and William Hyde Wollaston 
(ft. 1766, d. 1828) seized on it as a testf of the identity of the 
electric currents of Volta with those obtained by the discharge 
of f rictional electricity. He found that water could be decom- 

vy posed by currents of either type, and inferred that all differences 
between them could be explained by supposing that voltaic 
electricity as commonly obtained is " less intense, but produced 
in much, larger quantity." Later in the same year (1801), 
Martin van Mar um (ft. 1750, d. 1837) and Christian Heinrich 
Pfaff (ft. 1773, d. 1852) arrived at the same conclusion by 
carrying out on a large scale} Volta's plan of using the pile to 

V charge batteries of Leyden jars. 

The discovery of Nicholson and Carlisle made a great 
impression on the mind of Humphry Davy (ft. 1778, d. 1829), a 
young Cornishman who about this time was appointed Professor 
of Chemistry at the E-oyal Institution in London. Davy at once 
began to experiment vvitli Voltaic piles, and in November, 1800, 
showed that they give no current when the water between the 

y pairs of plates is pure, and that their power of action is " in 
great measure proportional to the power of the conducting 
fluid substance between the double plates to oxydate the 
zinc." This result, as he immediately perceived, did not 
harmonize well with Volta's views on the source of electricity 
in the pile, but was, on the other hand, in agreement with 
, Eabroni's idea that galvanic effects are always accompanied by 
chemical action. After a series of experiments he definitely 

1 concluded that " the galvanic pile of Volta acts only when the 
conducting substance between the plates is capable of oxydating 
the zinc ; and that, in proportion as a greater quantity of 
oxygen enters into combination with the zinc in a given time, 
so in proportion is the power of the pile to decompose water 
and to give the shock greater. It seems therefore reasonable 

* Nicholson's Journal (4to), iv (1800), pp. 187,245: Phil. Mag., vii (1800), 
p. 337. 

t Phil. Mag., 1801, p. 427. J Phil. Mag., xii (1802), p. 161. 

Nicholson's Journal (4to), iv (1800) ; Davy's Works, ii, p. 155. 



Galvanism, from Galvani (o Ohm. 77 

to conclude, though with our present quantity of facts we are 
unable to explain the exact mode of operation, that the </ 
oxydatioii of the zinc in the pile, and the chemical changes 
connected with it, are somehow the cause of the electrical effects ^ 
it produces." This principle of oxidation guided Davy in 
designing many new types of pile, with elements chosen from 
the whole range of the known metals. 

Davy's chemical theory of the pile was supported by 
Wollaston* and by Nicholson,f the latter of whom urged that 
the existence of piles in which only one metal is used (with more 
than one kind of fluid) is fatal to any theory which places the 
seat of the activity in the contact of dissimilar metals. 

Davy afterwards proposed J a theory of the voltaic pile 
which combines ideas drawn from both the "contact" and 
" chemical " explanations. Ho supposed that before the circuit 
is closed, the copper and zinc disks in each contiguous pair 
assume opposite electrostatic states, in consequence of inherent 
"electrical energies" possessed by the metals; and when a > 
communication is made between the extreme disks by a wire, 
the opposite electricities annihilate each other, as in the dis- 
charge of a Leyden jar. If the liquid (which Davy compared 
to the glass of a Leyden jar) were incapable of decomposition, 
the current would cease after this discharge. But the liquid in 
the pile is composed of two elements which have inherent 
attractions for electrified metallic surfaces : hence arises 
chemical action, which removes from the disks the outermost 
layers of molecules, whose energy is exhausted, and exposes 
new metallic surfaces. The electrical energies of the copper and 
zinc are consequently again exerted, and the process of electro- 
motion continues. Thus the contact of metals is the cause 
which disturbs the equilibrium, while the chemical changes 
continually restore the conditions under which the contact 
energy can be exerted. 

In this and other memoirs Davy asserted that chemical 

*Phil. Trans., 1801, p. 427. t Nicholson'* Journal, i (1802), p. 142. 

; Phil. Trans., 1807, p. 1. 



78 Galvanism, from Galvani to Ohm. 

J affinity is essentially of an electrical nature. " Chemical and 
electrical attractions," he declared,* "are produced by the 
same cause, acting in one case on particles, in the other on 
masses, of matter; and the same property, under different 
modifications, is the cause of all the phenomena exhibited by 
different voltaic combinations." 

The further elucidation of this matter came chiefly from 

- researches on electro-chemical decomposition, which we must 
now consider. 

A phenomenon which had greatly surprised Nicholson and 
Carlisle in their early experiments was the appearance of 
the products of galvanic decomposition at places remote from 
each other. The first attempt to account for this was made in 
1806 by Theodor von Grothussf (b. 1785, d. 1822) and by Davy,} 
who advanced a theory that the terminals at which water is 
decomposed have attractive and repellent powers ; that the pole 
whence resinous electricity issues has the property of attracting 
hydrogen and the metals, and of repelling oxygen and acid 
substances, while the positive terminal has the power of attract- 
ing oxygen and repelling hydrogen ; and that these forces are 
sufficiently energetic to destroy or suspend the usual operation 
of chemical affinity in the water-molecules nearest the 
terminals. The force due to each terminal was supposed to 
diminish with the distance from the terminal. When the 
molecule nearest one of the terminals has been decomposed by 
the attractive and repellent forces of the terminal, one of its 
constituents is liberated there, while the other constituent, by 
virtue of electrical forces (the oxygen and hydrogen being in 
opposite electrical states), attacks the next molecule, which 
is then decomposed. The surplus constituent from this attacks 
the next molecule, and so on. Thus a chain of decompositions 
and recompositions was supposed to be set up among the 
molecules intervening between the terminals. 

* Phil. Trans., 1826, p. 383. f Ann. de Cliim., Iviii (1806), p. 54. 

t Bukerian lecture for 1806, Phil. Trans., 1807, p. 1. A theory similar to that 
of Grothuss and Davy was communicated by Peter Mark Eoget (b. 1779, d. 1869) 
in 1807 to the Philosophical Society of Manchester : cf. Roget's Galvanism, 106. 



Galvanism^ from ^Galvani to Ohm. 79 

The hypothesis of Grothuss and Davy was attacked in 1825 
by Aiiguste De La Kive* (6. 1801, d. 1873) of Geneva, on the 
ground of its failure to explain what happens when different 
liquids are placed in series in the circuit. If, for example, a 
solution of zinc sulphate is placed in one compartment, and 
water in another, and if the positive pole is placed in the 
solution of zinc sulphate, and the negative pole in the water, 
De La Rive found that oxide of zinc is developed round the 
latter; although decomposition and recomposition of zinc 
sulphate could not take place in the water, which contained 
none of it. Accordingly, he supposed the constituents of the 
decomposed liquid to be bodily transported across the liquids, 
in close union with the moving electricity. In the electrolysis 
of water, one current of electrified hydrogen was supposed to 
leave the positive pole, and become decomposed into hydrogen 
and electricity at the negative pole, the hydrogen being 
there liberated as a gas. Another current in the same way 
carried electrified oxygen from the negative to the positive 
pole. In this scheme the chain of successive decompositions 
imagined by Grothuss does not take place, the only molecules 
decomposed being those adjacent to the poles. 

The appearance of the products of decomposition at the 
separate poles could be explained either in Grothuss' fashion 
by assuming dissociations throughout the mass of liquid, or 
in De La Rive's by supposing particular dissociated atoms 
to travel considerable distances. Perhaps a preconceived 
idea of economy in Nature deterred the workers of that time 
from accepting the two assumptions together, when either of 
them separately would meet the case. Yet it is to this apparent 
redundancy that later researches have pointed as the truth. 
Nature is what she is, and not what we would make her. 

De La Rive was one of the most thoroughgoing opponents 
of Volta's contact theory of the pile ; even in the case when 
two metals are in contact in air only, without the intervention 

* Annales de Cnimie, xxviii, 190. 



80 Galvanism, from Galvani to Ohm. 

of any liquid, he attributed the electric effect wholly to the 
chemical affinity of the air for the metals. 

During the long interval between the publication of the rival 
hypotheses of Grothuss and De La Bive, little real progress 
was made with the special problems of the cell ; but mean- 
while electric theory was developing in other directions. One 
of these, to which our attention will first be turned, was the 
electro-chemical theory of the celebrated Swedish chemist, 
Jons Jacob Berzelius (b. 1779, d. 1848). 

Berzelius founded his theory,* which had been in one or two 
of its features anticipated by Davy,f on inferences drawn from 
Volta's contact effects. " Two bodies," he remarked, " which 
have affinity for each other, and which have been brought into 
mutual contact, are found upon separation to be in opposite 
electrical states. That which has the greatest affinity for 
oxygen usually becomes positively electrified, and the other 
negatively." 

This seemed to him to indicate that chemical affinity arises 
from the play of electric forces, which in turn spring from 
electric charges within the atoms of matter. To be precise, 
he supposed each atom to possess two poles, which are the 
seat of opposite electrifications, and whose electrostatic field is 
the cause of chemical affinity. 

By aid of this conception Berzelius drew a simple and vivid 
picture of chemical combination. Two atoms, which are about 
to unite, dispose themselves so that the positive pole of one 
touches the negative pole of the other ; the electricities of these 
two poles then discharge each other, giving rise to the heat and 
light which are observed to accompany the act of combination.! 
The disappearance of these leaves the compound molecule with 
the two remaining poles ; and it cannot be dissociated into its 
constituent atoms again until some means is found of restoring 
to the vanished poles their charges. Such a means is afforded 

* Memoirs of the Acad. of Stockholm, 1812 ; Nicholson's Journal of Nat. Phil., 
xxxiv (1813), 142, 153, 240, 319; xxxv, 38, 118, 159. 

t Pnil. Trans., 1807. J This idea was Davy's. 



Galvanism, from Gaivani to Ohm. 81 

by the action of the galvanic pile in electrolysis : the opposite 
electricities of the current invade the molecules of the 
electrolyte, and restore the atoms to their original state of 
polarization. 

If, as Berzelius taught, all chemical compounds are formed 
by the mutual neutralization of pairs of atoms, it is evident / 
that they must have a binary character. Thus he conceived a 
salt to be compounded of an acid and an oxide, and each of 
these to be compounded of two other constituents. Moreover, 
in any compound the electropositive member would be replace- 
able only by another electropositive member, and the electro- 
negative member only by another member also electronegative ; 
so that the substitution of, e.g., chlorine for hydrogen in a 
compound would be impossible an inference which was 
overthrown by subsequent discoveries in chemistry. 

Berzelius succeeded in bringing the most curiously diverse 
facts within the scope of his theory. Thus " the combination 1 
of polarized atoms requires a motion to turn the opposite 
poles to each other; and to this circumstance is owing the 
facility with which combination takes place when one of the 
two bodies is in the liquid state, or when both are in that 
state ; and the extreme difficulty, or nearly impossibility, of 
effecting an union between bodies, both of which are solid. 
And again, since each polarized particle must have an electric 
atmosphere, and as this atmosphere is the predisposing cause of 
combination, as we have seen, it follows, that the particles 
cannot act but at certain distances, proportioned to the 
intensity of their polarity ; and hence it is that bodies, which 
have affinity for each other, always combine nearly on the 
instant when mixed in the liquid state, but less easily in the 
gaseous state, and the union ceases to be possible under a 
certain degree of dilatation of the gases ; as we know by the 
experiments of Grothuss, that a mixture of oxygen and 
hydrogen in due proportions, when rarefied to a certain 
degree, cannot be set on fire at any temperature whatever." j 
And again : " Many bodies require an elevation of temperature to 

G 



82 Galvanism, from Galvani to Ohm. 

enable them to act upon each other. It appears, therefore, 
that heat possesses the property of augmenting the polarity of 
these bodies." 

Berzelius accounted for Volta's electromotive series by 
assuming the electrification at one pole of an atom to be some- 
what more or somewhat less than what would be required to 
neutralize the charge at the other pole. Thus each atom would 
possess a certain net or residual charge, which might be of 
either sign ; and the order of the elements in Volta's series 
could be interpreted simply as the order in which they would 
stand when ranged according to the magnitude of this residual 
charge. As we shall see, this conception was afterwards 
overthrown by Faraday. 

Berzelius permitted himself to publish some speculations on 
the nature of heat and electricity, which bring vividly before 
us the outlook of an able thinker in the first quarter of the 
nineteenth century. The great question, he says, is whether 
v the electricities and caloric are matter or merely phenomena. 
If the title of matter is to be granted only to such things as 
are ponderable, then these problematic entities are certainly 
not matter ; but thus to narrow the application of the term is, 
he believes, a mistake; and he inclines to the opinion that 
caloric is truly matter, possessing chemical affinities without 
obeying the law of gravitation, and that light and all radiations 
consist in modes of propagating such matter. This conclusion 
makes it easier to decide regarding electricity. " From 
the relation which exists between caloric and the electricities," 
he remarks, "it is clear that what may be true with regard 
to the materiality of one of them must also be true with 
regard to that of the other. There are, however, a quantity 
of phenomena produced by electricity which do not admit of 
explanation without admitting at the same time that electricity 
is matter. Electricity, for instance, very often detaches 
everything which covers the surface of those bodies which 
conduct it. It, indeed, passes through conductors without 
leaving any trace of its passage ; but it penetrates non-con- 



Galvanism i from Galvani to Ohm. 83 

ductors which oppose its course, and makes a perforation 
precisely of the same description as would have been made 
by something which had need of place for its passage. We 
often observe this when electric jars are broken by an over- 
charge, or when the electric shock is passed through a number 
of cards, etc. We may therefore, at least with some proba- 
bility, imagine caloric and the electricities to be matter, 
destitute of gravitation, but possessing affinity to gravitating 
bodies. When they are not confined by these affinities, they 
tend to place themselves in equilibrium in the universe. The ^ 
suns destroy at every moment this equilibrium, and they send 
the re-united electricities in the form of luminous rays towards 
the planetary bodies, upon the surface of which the rays, being 
arrested, manifest themselves as caloric ; and this last in its 
turn, during the time required to replace it in equilibrium in 
the universe, supports the chemical activity of organic and 
inorganic nature." 

It was scarcely to be expected that anything so speculative 
as Berzelius' electric conception of chemical combination 
would be confirmed in all particulars by subsequent discovery ; 
and, as a matter of fact, it did not as a coherent theory survive 
the lifetime of its author. But some of its ideas have 
persisted, and among them the conviction which lies at its 
foundation, that chemical affinities are, in the last resort, of * 
electrical origin. 

While the attention of chemists was for long directed to 
the theory of Berzelius, the interest of electricians was 
diverted from it by a discovery of the first magnitude in a 
different region. 

That a relation of some land subsists between electricity 
and magnetism had been suspected by the philosophers of the 
eighteenth century. The suspicion was based in part on some 
curious effects produced by lightning, of a kind which may be 
illustrated by a paper published in the Philosophical Transactions 
in 1735.* A tradesman of Wakefield, we are told, "having put 

*Phil. Trans, xxxix (1735), p. 74. 
G 2 



84 Galvanism, from Galvani to Ohm. 

up a great number of knives and forks in a large box, and 
having placed the box in the corner of a large room, there 
happen'd in July, 1731, a sudden storm of thunder, lightning, 
etc., by which the corner of the room was damaged, the Box 
split, and a good many knives and forks melted, the sheaths 
being untouched. The owner emptying the box upon a Counter 
where some Nails lay, the Persons who took up the knives, that 
lay upon the Nails, observed that the knives took up the Nails." 

Lightning thus came to be credited with the power of 
magnetizing steel ; and it was doubtless this which led Franklin* 
in 1751 to attempt to magnetize a sewing-needle by means of 
the discharge of Leyden jars. The attempt was indeed success- 
ful ; but, as Van Marum afterwards showed, it was doubtful 
whether the magnetism was due directly to the current. 

More experiments followed. f In 1805 Jean Nicholas Pierre 
Hachette (b. 1769, d. 1834) and Charles Bernard Desormes 
(b. 1777, d. 1862) attempted to determine whether an insulated 
voltaic pile, freely suspended, is oriented by terrestrial mag- 
netism ; bat without positive result. In 1807 Hans Christian 
Oersted (&. 1777, d. 1851), Professor of Natural Philosophy in 
Copenhagen, announced his intention of examining the action 
of electricity on the magnetic needle ; but it was not for some 
years that his hopes were realized. If one of his pupils is to be 
believed,* he was " a man of genius, but a very unhappy experi- 
menter ; he could not manipulate instruments. He must 
always have an assistant, or one of his auditors who had easy 
hands, .to arrange the experiment." 

During a course of lectures which he delivered in the winter 
of 1819-20 on " Electricity, Galvanism, and Magnetism," the 
idea occurred to him that the changes observed with the 
compass-needle during a thunderstorm might give the clue to 
the effect of which he was in search ; and this led him to think 
that the experiment should be tried with the galvanic circuit 

* Letter vi from Franklin to Collinson. f In 1774 the Electoral Academy 

of Bavaria proposed the question, " Is there a real and physical analogy between 
electric and magnetic forces ? " as the subject of a prize. 

1 Cf. a letter from Hansteen inserted inBence Jones' Life of Faraday y ii, p. 395. 



Galvanism, from Galvani to Ohm. 85 

closed instead of open, and to inquire whether any effect is 
produced on a magnetic needle when an electric current is 
passed through a neighbouring wire. At first he placed the 
wire at right angles to the needle, but observed no result. 
After the end of a lecture in which this negative experiment 
had been shown, the idea occurred to him to place the wire 
parallel to the needle : on trying it, a pronounced deflexion was 
observed, and the relation between magnetism and the electric 
current was discovered. After confirmatory experiments with 
more powerful apparatus, the public announcement was made 
in July, 1820 * 

Oersted did not determine the quantitative laws of the 
;action, but contented himself with a statement of the qualita- 
tive effect and some remarks on its cause, which recall the 
magnetic speculations of Descartes : indeed, Oersted's concep- 
tions may be regarded as linking those of the Cartesian school 
to those which were introduced subsequently by Faraday. " To 
the effect which takes place in the conductor and in the sur- 
rounding space," he wrote, " we shall give the name of the 
-conflict of electricity? " The electric conflict acts only on the 
magnetic particles of matter. All non-magnetic bodies appear 
penetrable by the electric conflict, while magnetic bodies, or 
rather their magnetic particles, resist the passage of this conflict 
Hence they can be moved by the impetus of the contending 
powers. 

" It is sufficiently evident from the preceding facts that the 
electric conflict is not confined to the conductor, but dispersed 
pretty widely in the circumjacent space. 

" From the preceding facts we may likewise collect, that this 
conflict performs circles ; for without this condition, it seems 
impossible that the one part of the uniting wire, when placed 
below the magnetic pole, should drive it toward the east, and 
when placed above it toward the west; for it is the nature of a 

* Schweigger's Journal fur Chemie und Physik, zxix (1820), p. 275 ; Thomson's 
Annals of Philosophy, xvi (1820), p. 273; Ostwald's Klattiter der 
' Wi.ssenseha.ften, Nr. 63. 



86 Galvanism, from Galvani to Ohm. 

circle that the motions in opposite parts should have an opposite 1 
direction." 

Oersted's discovery was described at the meeting of the 
French Academy on September llth, 1820, by an academician 
(Arago) who had just returned from abroad. Several investi- 
gators in France repeated and extended his experiments ; and 
the first precise analysis of the effect was published by two of 
these, Jean-Baptiste Biot (b. 1774, d. 1862) and Felix Savart 
(b. 1791, d. 1841), who, at a meeting of the Academy of Sciences 
on October 30th, 1820, announced* that the action experienced 
by a pole of austral or boreal magnetism, when placed at any 
distance from a straight wire carrying a voltaic current, may be 

4 thus expressed : " Draw from the pole a perpiendicular to the 
wire ; the force on the pole is at right angles to this line and ta 
the wire, and its intensity is proportional to the reciprocal of 
the distance." This result was soon further analysed, the 
attractive force being divided into constituents, each of which 
was supposed to be due to some particular element of the 
current ; in its new form the law may be stated thus : the- 
magnetic force due to an element ds of a circuit, in which a 
current i is flowing, at a point whose vector distance from ds is r,, 
is (in suitable units) 

i ids 

|ds,r|t or curl .+ 
r 3 J r 

It was now recognized that a magnetic field may be produced 
as readily by an electric current as by a magnet ; and, as Arago 
soon showed, this, like any other magnetic field, is capable of 

* Annales de Chimie, xv (1820), p. 222 ; Journal de Phys., xli, p. 51. 

f If a and b denote two vectors, the vector whose components are (a y b z a z b y ^ 
a z b* a*b z , a x b y a y b x ) is called the vector product of a and b, and is denoted by 
[a, b]. Its direction is at right angles to those of a and b, and its magnitude is 
represented by twice the area of the triangle formed by them. 

+ If a denotes any vector, the vector whose components are ^- z - -^, - - ^-* r 

3% 9 a * -, 

z-l - - is denoted by curl a. 

fo ty 

Annales de Chimie, xv (1820), p. 93. 



Galvanism, from Galvani to Ohm. 87 

inducing magnetization in iron. The question naturally sug- 
gested itself as to whether the similarity of properties between 
currents and magnets extended still further, e.g. whether 
conductors carrying currents would, like magnets, experience 
ponderomotive forces when placed in a magnetic field, and 
whether such conductors would consequently, like magnets, 
exert ponderomotive forces on each other. 

The first step towards answering these inquiries was taken 
by Oersted* himself. " As," he said, " a body cannot put 
another in motion without being moved in its turn, when it 
possesses the requisite mobility, it is easy to foresee that the 
galvanic arc must be moved by the magnet " ; and this he 
verified experimentally. 

The next step came from Andre Marie Ampere (b. 1775, 
d. 1836), who at the meeting of the Academy on September 18th, 
exactly a week after the news of Oersted's first discovery had 
arrived, showed that two parallel wires carrying currents 
attract each other if the currents are in the same direction, 
and repel each other if the currents are in opposite directions. 
During the next three years Ampere continued to prosecute 
the researches thus inaugurated, and in 1825 published his 
collected results in one of the most celebrated memoirsf in the 
history of natural philosophy. 

Ampere introduces his work by proclaiming himself a 
follower of that school which explained all physical phenomena 
in terms of equal and oppositely directed forces between pairs 
of particles ; and he renounces the attempt to seek more 
speculative, though possibly more fundamental, explanations 
in terms of the motions of ultimate fluids and aethers. Never- 
theless, he indicates two conceptions of this latter character, on 
which such explanations might be founded. 

In the firstj he suggests that the ponderomotive forces 

* Schweigger's Journal fur Chem. u. Phys., xxix (1820), p. 364 ; Thomson's 
Annals of Philosophy, xvi (1820), p. 375. t Mem. de 1'Acad., vi, p. 175. 

% facueil tF observations electro- dynamiques, p. 215 ; and the memoir just cited, 
pp. 285, 370. 



88 Galvanism, from Galvani to Ohm. 

between circuits carrying electric currents may be due to " the 
reaction of the elastic fluid which extends throughout all 
space, whose vibrations produce the phenomena of light," and 
which is " put in motion by electric currents." This fluid or 
aether can, he says, " be no other than that which results from 
the combination of the two electricities/' 

In the second conception,* Ampere suggests that the 
interspaces between the metallic molecules of a wire which 
carries a current may be occupied by a fluid composed of the 
two electricities, not in the proportions which form the neutral 
fluid, but with an excess of that one of them which is opposite 
to the electricity peculiar to the molecules of the metal, and 
which consequently masks this latter electricity. In this inter- 
molecular fluid the opposite electricities are continually being 
dissociated and recombined ; a dissociation of the fluid within 
one inter-molecular interval having taken place, the positive 
electricity thus produced unites with the negative electricity 
of the interval next to it in the direction of the current, while 
the negative electricity of the first interval unites with the 
positive electricity of the next interval in the other direction. 
Such interchanges, according to this hypothesis, constitute the 
electric current. 

Ampere's memoir is, however, but little occupied with the 
more speculative side of the subject. His first aim was to 
investigate thoroughly by experiment the ponderomotive forces 
on electric currents. 

" When," he remarks, " M. Oersted discovered the action 
which a current exercises on a magnet, one might certainly have 
suspected the existence of a mutual action between two circuits 
carrying currents ; but this was not a necessary consequence ; 
for a bar of soft iron also acts on a magnetized needle, although 
there is no mutual action between two bars of soft iron." 

Ampere, therefore, submitted the matter to the test of the 
laboratory, and discovered that circuits carrying electric 
currents exert ponderomotive forces on each other, and that 

* Recucil d' observations electro-dunamiques, pp. 297, 300, 371. 



Galvanism, from Gaivani to Ohm. 89 

ponderomotive forces are exerted on such currents by magnets. 
To the science which deals with the mutual action of currents 
he gave the name electro-dynamics ;* and he showed that the 
action obeys the following laws : 

(1) The effect of a current is reversed when the direction of 
the current is reversed. 

(2) The effect of a current flowing in a circuit twisted into 
small sinuosities is the same as if the circuit were smoothed out. 

(3) The force exerted by a closed, circuit on an element of 
another circuit is at right angles to the latter. 

(4) The force between two elements of circuits is unaffected 
when all linear dimensions are increased proportionately, the 
current-strengths remaining unaltered. 

From these data, together with his assumption that the force 
between two elements of circuits acts along the line joining them, 
Ampere obtained an expression of this force : the deduction may 
be made in the following way : 

Let ds, ds' be the elements, r the line joining them, and i, i' 
the current-strengths. From (2) we see that the effect of ds on 
ds' is the vector sum of the effects of dx, dy, dz on ds', where 
these are the three components of ds: so the required force 
must be of the form 

r x a scalar quantity which is linear and homogeneous in ds ; 
and it must similarly be linear and homogeneous in ds' ; so 
using (1), we see that the force must be of the form 

F = ill | (ds . ds') 4> (r) + (ds . r) (ds'. r) i/, (r)} , 
where < and i// denote undetermined functions of r. 

From (4) it follows that when ds, ds', r are all multiplied by 
the same number, F is unaffected : this shows that 

4>(r) = - and f (r) = - , 

where A and B denote constants. Thus we have 

, M(ds.ds') (ds.r)(ds'. r)) 

F = n r \ + - - ; 

( r 3 r 6 ) 

*. Loc. cit., p. 298. 



90 Galvanism , from Galvani to Ohm. 

Now, by (3), the resolved part of F along ds' must vanish when 
integrated round the circuit s, i.e. it must be a complete 
differential when dr is taken to be equal to - ds. That is to- 
say, 

^(ds.ds')(r.ds') (ds . r) (ds'. r) 2 

/o-3 .f\> 

must be a complete differential ; or 



must be a complete differential ; and therefore 

7 A B iA 

d '^ = -- 5 ( dS ' r )> 

3 ^ B J 

or ~2^" dr = r* dT ' 

or B = - I A. 

Thus finally we have 

F = Constant x ii'i || (ds . ds') - - 5 (ds . r)(ds'. r) 

This is Ampere's formula : the multiplicative constant depends 
of course on the units chosen, and may be taken to be - 1. 

The weakness of Ampere's work evidently lies in the 
assumption that the force is directed along the line joining the- 
two elements : for in the analogous case of the action between 
two magnetic molecules, we know that the force is not directed 
along the line joining the molecules. It is therefore of interest 
to find the form of F when this restriction is removed. 

For this purpose we observe that we can add to the expression 
already found for F any term of the form 

0(r) . (ds . r) . ds', 
where 0(r) denotes any arbitrary function of r ; for since 



this term vanishes when integrated round the circuit s ; and it 



Galvanism, from Galvani to Ohm. 91 

contains ds and ds' linearly and homogeneously, as it should. 
We can also add any terms of the form 

rf{r..(ds'.r). x (r)|, 

where \(r] denotes any arbitrary function of r, and d denotes 
differentiation along the arc s, keeping ds' fixed (so that 
dr = - ds) ; this differential may be written 

- ds . (ds'. r) . x (r) - r x (r) (ds'. ds) - * x '( r ) r (ds . r) (ds'. r). 

In order that the law of Action and Eeaction may not be 
violated, we must combine this with the former additional term 
so as to obtain an expression symmetrical in ds and ds' : and 
hence we see finally that the general value of F is given by the 
equation 

F = -n'rj j |(ds.ds')-J(ds.r)(ds.r)j 

+ x (-; (ds' . r) ds + x( r ) (ds . r) . ds' + x (r) (ds . ds')r 

+ i x '(r)(ds.r)(ds'.r)r. 
The simplest form of this expression is obtained by taking 



when we obtain 

/ 
F = - {(ds . r) . ds' + (ds'. r)ds - (ds . ds')r} . 

The comparatively simple expression in brackets is the 
vector part of the quaternion product of the three vectors 
ds, r, ds'.* 

From any of these values of F we can find the ponderomotive 
force exerted by the whole circuit s on the element ds' : it is, in 
fact, from the last expression, 

u'f 1 



[ ? - 3 ((ds'.r).ds-(ds.ds>}, 



* The simpler form of F given in the text is, if the term in da' be omitted, the 
form given by Grassmann, Ann. d. Phys. Ixiv (1845), p. 1. For further work on 
this subject cf. Tait, Proc. R. S. Edin. viii (1873), p. 220, and Korteweg, Journal 
fiir Math, xc (1881), p. 45. 



92 Galvanism, from Galvani to Ohm. 






or i [ds'. B], 

where B = 



Now this value of B is precisely the value found by Biot and 
Savart* for the" magnetic intensity at ds' due to the- current i in 
the circuit s. Thus we see that the ponderomotive force on a 
current-element ds' in a magnetic field B is i' [ds'. B]. 

Ampere developed to a considerable extent the theory 
of the equivalence of magnets with circuits carrying currents; 
and showed that an electric current is equivalent, in its 
magnetic effects, to a distribution of magnetism on any 
surface terminated by the circuit, the axes of the magnetic 
molecules being everywhere normal to this surface :f such a 
magnetized surface is called a mayiwtic shell. He preferred, 
I however, to regard the current rather than the magnetic fluid 
as the fundamental entity, and considered magnetism to be 
really an electrical phenomenon : each magnetic molecule owes 
its properties, according to this view, to the presence within it 
/ of a small closed circuit in which an electric current is 
perpetually flowing. 

The impression produced by Ampere's memoir was great 
and lasting. Writing half a century afterwards, Maxwell 
speaks of it as " one of the most brilliant achievements in 
science." " The whole," he says, " theory and experiment, 
seems as if it had leaped, full-grown and full-armed, from the 
brain of the ' Newton of electricity/ It is perfect in form and 
unassailable in accuracy ; and it is summed up in a formula 
from which all the phenomena may be deduced, and which 
must always remain the cardinal formula of electrodynamics." 

Not long after the discovery by Oersted of the connexion 
between galvanism and magnetism, a connexion was discovered 
between galvanism and heat.| In 1822 Thomas Johann Seebeck 

* See ante, p. 86. t Loc. cit., p. 367. 



Galvanism, from ^Galvani to Ohm. 93 

(b. 1770, d. 1831), of Berlin discovered* that an electric current 
can be set up in a circuit of metals, without the interposition 
of any liquid, merely by disturbing the equilibrium of 
temperature. Let a ring be formed of copper and bismuth 
soldered together at the two extremities; to establish a 
current it is only necessary to heat the ring at one of these 
junctions. To this new class of circuits the name thermo- 
electric was given. 

It was found that the metals can be arranged as a 
thermo-electric series, in the order of their power of generating 
currents when thus paired, and that this order is quite different 
from Volta's order of electromotive potency. Indeed antimony 
and bismuth, which are near each other in the latter series, are 
at opposite extremities of the former. 

The currents generated by thermo-electric means are 
generally feeble : and the mention of this fact brings us to 
the question, which was about this time engaging attention, 
of the efficacy of different voltaic arrangements. 

Comparisons of a rough kind had been instituted soon after 
the discovery of the pile. The French chemists Antoine 
FranQois de Fourcroy (b. 1755, d. 1809), Louis Mcolas 
Yauquelin (b. 1763, d. 1829), and Louis Jacques Thenard 
(b. 1777, d. 1857) foundf in 1801, on varying the size of the 
metallic disks constituting the pile, that the sensations 
produced on the human frame were unaffected so long as the 
number of disks remained the same; but that the power 'of 
burning finely drawn wire was altered; and that the latter 
power was proportional to the total surface of the disks 
employed, whether this were distributed among a small number 
of large disks, or a large number of small ones. This was 

* Abhandl. d. Berlin Akad. 1822-3 ; Ann. d. Phys. Ixxiii (1823), pp. 115, 
430 ; vi (1826), pp. 1, 133, 253. 

Volta had previously noticed that a silver plate whose ends were at different 
temperatures appeared to act like a voltaic cell. 

Further experiments were performed by James Gumming (. 1777, d. 1861), 
Professor of Chemistry at Cambridge, Trans. Camb. Phil. Soc. ii (1823), p. 47, 
and by Antoine Cesar Becquerel (b. 1788, d. 1878), Annales de Chimie, xxxi 
(1826), p. 371. t Ann. de Chimie, xxxix (1801), p. 103. 



94 Galvanism, from Galvani to Ohm. 

explained by supposing that small plates give a small quantity 
of the electric fluid with a high velocity, while large plates 
give a larger quantity with no greater velocity. Shocks, 
which were supposed to depend on the velocity of the fluid 
alone, would therefore not be intensified by increasing the size 
of the plates. 

The effect of varying the conductors which connect the 
terminals of the pile was also studied. Nicolas Grautherot 
(b. 1753, d. 1803) observed* that water contained in tubes which 
have a narrow opening does not conduct voltaic currents so 
well as when the opening is more considerable. This experi- 
ment is evidently very similar to that which Beccaria had 
performed half a century previously! with electrostatic 
discharges. 

As we have already seen, Cavendish investigated very 
-completely the power of metals to conduct electrostatic 
discharges; their power of conducting voltaic currents was 
now examined by Davy.J His method was to connect the 
terminals of a voltaic battery by a path containing water 
(which it decomposed), and also by an alternative path 
consisting of the metallic wire under examination. When the 
length of the wire was less than a certain quantity, the water 
ceased to be decomposed ; Davy measured the lengths and 
weights of wires of different materials and cross-sections under 
these limiting circumstances ; and, by comparing them, showed 
that the conducting power of a wire formed of any one metal 
is inversely proportional to its length and directly proportional 
to its sectional area, but independent of the shape of the cross- 
section.! The latter fact, as he remarked, showed that voltaic 
currents pass through the substance of the conductor and not 
along its surface. 

Davy, in the same memoir, compared the conductivities of 
various metals, and studied the effect of temperature : he found 

* Annales de China., xxxix (1801), p. 203. t See p. 53. 

% Phil. Trans., 1821, p. 433. His results were confirmed afterwards by 
Becquerel, Annales de Chiiuie, xxxii (1825), p. 423. 
6 These results had been known to Cavendish. 



Galvanism , from Galvani to Ohm. 95 

that the conductivity varied with the temperature, being 
" lower in some inverse ratio as the temperature was higher." 

He also observed that the same magnetic power is exhibited 
by every part of the same circuit, even though it be formed 
of wires of different conducting powers pieced into a chain, 
so that " the magnetism seems directly as the quantity of 
electricity which they transmit." 

The current which flows in a given voltaic circuit evidently 
depends not only on the conductors which form the circuit, 
but also on the driving-power of the battery. In order to form 
a complete theory of voltaic circuits, it was therefore necessary 
to extend Davy's laws by taking the driving-power into 
account. This advance was effected in 1826 by Georg Simom 
Ohm* (b. 1787, d. 1854). 

Ohm had already carried out a considerable amount of 
experimental work on the subject, and had, e.g., discovered that 
if a number of voltaic cells are placed in series in a circuit, the 
current is proportional to their number if the external 
resistance is very large, but is independent of their number if 
the external resistance is small. He now essayed the task 
of combining all the known results into a consistent theory. 

For this purpose he adopted the idea of comparing the flow 
of electricity in a current to the flow of heat along a wire, the 
theory of which had been familiar to all physicists since the 
publication of Fourier's Theorie analytique de la chcdeur in 
1822. " I have proceeded," he says, " from the supposition that 
the communication of the electricity from one particle takes 
place directly only to the one next to it, so that no immediate 
transition from that particle to any other situate at a greater 
distance occurs. The magnitude of the flow between two 
adjacent particles, under otherwise exactly similar circum- 
stances, I have assumed to be proportional to the difference of 

*Ann. d. Phys. vi (1826), p. 459 ; vii, pp. 45,117; Die Galvanische Eette 
mathematisch bearbeitet : Berlin, 1827 ; translated in Taylor's Scientific Memoirs, 
ii (1841), p. 401. Cf. also subsequent papers by Ohm in Kastner's Archiv fur 
d. ges. Naturkhre, and Schweigger's Jahrbuch. 



96 Galvanism, from Galvani to Ohm. 

the electric forces existing in the two particles ; just as, in the 
theory of heat, the flow of caloric between two particles is 
regarded as proportional to the difference of their temperatures."' 
'*' The comparison between the flow of electricity and the flow 
of heat suggested the propriety of introducing a quantity 
whose behaviour in electrical problems should resemble that of 
temperature in the theory of heat. The differences in the 
values of such a quantity at two points of a circuit would 
provide what was so much needed, namely, a measure of the 
"driving-power" acting on the electricity between these 
points. To carry out this idea, Ohm recurred to Volta's theory 
of the electrostatic condition of the open pile. It was cus- 
tomary to measure the " tension " of a pile by connecting one 
terminal to earth and testing the other terminal by an 
electroscope. Accordingly Ohm says : " In order to investigate 
the changes which occur in the electric condition of a body A 
in a perfectly definite manner, the body is each time brought, 
under similar circumstances, into relation with a second 
moveable body of invariable electrical condition, called the 
electroscope ; and the force with which the electroscope is 
repelled or attracted by the body is determined. This force is 
termed the electroscopic force of the body A" 

" The same body A may also serve to determine the electro- 
scopic force in various parts of the same body. For this 
purpose take the body A of very small dimensions, so that 
when we bring it into contact with the part to be tested of any 
third body, it may from its smallness be regarded as a substitute 
for this part : then its electroscopic force, measured in the way 
described, will, when it happens to be different at the various 
places, make known the relative differences with regard to 
electricity between these places." 

Ohm assumed, as was customary at that period, that when 
two metals are placed in contact, " they constantly maintain at 
the point of contact the same difference between their electro- 
scopic forces." He accordingly supposed that each voltaic cell 
possesses a definite tension, or discontinuity of electroscopic 



Galvanism, from Galvani to Ohm. 97 

force, which is to be regarded as its contribution to the driving- 
force of any circuit in which it may be placed. This assumption 
confers a definite meaning on his use of the term " electroscopic 
force " ; the force in question is identical with the electrostatic 
potential. But Ohm and his contemporaries did not correctly 
understand the relation of galvanic conceptions to the j 
electrostatic functions of Poisson. The electroscopic force 
in the open pile was generally identified with the thickness 
of the electrical stratum at the place tested ; while Ohm, 
recognizing that electric currents are not confined to the 
surface of the conductors, but penetrate their substance, 
seems to have thought of the electroscopic force at a place in 
a circuit as being proportional to the volume-density of 
electricity there an idea in which he was confirmed by the 
relation which, in an analogous case, exists between the 
temperature of a body and the volume-density of heat 
supposed to be contained in it. 

Denoting, then, by S the current which flows in a wire of 
conductivity y, when the difference of the electroscopic forces at 
the terminals is E, Ohm writes 

S = yE. 

From this formula it is easy to deduce the laws already given 
by Davy. Thus, if the area of the cross-section of a wire 
is A y we can by placing n such wires side by side construct 
a wire of cross-section nA. If the quantity E is the same 
for each, equal currents will flow in the wires ; and therefore 
the current in the compound wire will be ?i times that in 
the single wire ; so when the quantity E is unchanged, the 
current is proportional to the cross-section; that is, the 
conductivity of a wire is directly proportional to its cross-section, 
which is one of Davy's laws. 

In spite of the confusion which was attached to the idea of 
electroscopic force, and which was not dispelled for some years, 
the publication of Ohm's memoir marked a great advance 
in electrical philosophy. It was now clearly understood that 
the current flowing in any conductor depends only on the 

H 



98 Galvanism^ from Galvani to Ohm. 

conductivity inherent in the conductor and on another variable 
which bears to electricity the same relation that temperature 
bears to heat ; and, moreover, it was realized that this latter 
variable is the link connecting the theory of currents with 
the older theory of electrostatics. These principles were a 
sufficient foundation for future progress; and much of the 
work which was published in the second quarter of the century 
was no more than the natural development of- the principles 
laid down by Ohm.* 

It is painful to relate that the discoverer had long to wait 
before the merits of his great achievement were officially 
recognized. Twenty- two years after the publication of the 
memoir on the galvanic circuit, he was promoted to a university 
professorship ; this he held for the five years which remained 
until his death in 1854. 

* Ohm's theory was confirmed experimentally by several investigators, among 
whom may be mentioned Gustav Theodor Feehner(i. 1801, d. 1887) (Maassbestim- 
mungen iiber die Galvanische Kette, Leipzig, 1831), and Charles Wheatstone 
(b. 1802, d. 1875) (Phil. Trans,, 1843, p. 303). 



CHAPTER IV. 

THE LUMINIFEROUS MEDIUM, FROM BRADLEY TO FRESNEL. 

ALTHOUGH Newton, as we have seen, refrained from committing 
himself to any doctrine regarding the ultimate nature of light, 
the writers of the next generation interpreted his criticism of 
the wave-theory as equivalent to an acceptance of the 
corpuscular hypothesis. As it happened, the chief optical 
discovery of this period tended to support the latter theory, 
by which it was first and most readily explained. In 1728 
James Bradley (b. 1692, d. 1762), at that time Savilian 
Professor of Astronomy at Oxford, sent to the Astronomer 
Royal (Halley) an " Account of a new discovered motion of the 
Fix'd Stars."* In observing the star y in the head of the 
Dragon, he had found that during the winter of 1725-6 the 
transit across the meridian was continually more southerly, 
while during the following summer its original position was 
restored by a motion northwards. Such an effect could not be 
explained as a result of parallax ; and eventually Bradley 
guessed it to be due to the gradual propagation of light.f 

Thus, let CA denote a ray of light, falling on the line BA ; 
and suppose that the eye of the observer is travelling ^ 
along BA, with a velocity which is to the velocity 
of light as BA is to CA. Then the corpuscle of 
light, by which the object is discernible to the eye 
at A, would have been at C when the eye was at 
B. The tube of a telescope must therefore be pointed 
in the direction BC, in order to receive the rays 
from an object whose light is really propagated in 
the direction CA. The angle BCA measures the 
difference between the real and apparent positions 

of the object ; and it is evident from the figure that the sine of 

Phil. Trans, xxxv (1728), p. 637. 

t Roemer, in a letter to Huygens of date 30th Dec., 1677, mentions a suspected 
displacement of the apparent position of a star, due to the motion of the earth at 
right angles to the line of sight. Cf . Correspondance de Huygens, viii, p. 53. 

H 2 



100 The Lumini/erous Medium, 

this angle is to the sine of the visible inclination of the object 
to the line in which the eye is moving, as the velocity of the eye 
is to the velocity of light. Observations such as Bradley's will 
therefore enable us to deduce the ratio of the mean orbital 
velocity of the earth to the velocity of light, or, as it is called,. 
the constant of 'aberration ; from its value Bradley calculated that 
light is propagated from the sun to the earth in 8 minutes 
12 seconds, which, as he remarked, "is as it were a Mean 
betwixt what had at different times been determined from the 
eclipses of Jupiter's satellites."* 

With the exception of Bradley's discovery, which was 
primarily astronomical rather than optical, the eighteenth 
century was decidedly barren, as regards both the experimental 
and the theoretical investigation of light ; in curious contrast 
to the brilliance of its record in respect of electrical researches. 
But some attention must be given to a suggestive study f of the 
aether, for which the younger John Bernoulli (b. 1710, d. 1790) 
was in 1736 awarded the prize of the French Academy. His 
ideas seem to have been originally suggested by an attempt}; 

*Struve in 1845 found for the constant of aberration the value 20"'445, which 
lie afterwards corrected to 20"'463. This was superseded in 1883 by the value 
20"-492, determined by M. Nyren. The observations of both Struve and Nyren 
were made with the transit in the prime vertical. The method now generally 
used depends on the measurement of differences of meridian zenith distances 
(Talcott's method, as applied by F. Kiistner, Beobachtungs-Ergebnisse der kon. 
Stern warte zu Berlin, Heft 3, 1888) ; the value at present favoured for the 
constant of aberration is 20"-523. Cf. Chandler, Ast. Journal, xxiii, pp. 1, 12 
(1903). 

The collective translatory motion of the solar system gives rise to aberrational: 
terms in the apparent places of the fixed stars ; but the principal term of this 
character does not vary with the time, and consequently is equivalent to a 
permanent constant displacement. The second-order terms (i.e. those which 
involve the ordinary constant of aberration multiplied by the sun's velocity) 
might be measurable quantities in the case of stars near the Pole ; and the same is 
true of the variations in the first-order terms (i.e. those which involve the sun's 
velocity not multiplied by the constant of aberration) due to the circumstance 
that the star's apparent R. A. and Declination, which occur in these terms, are 
not constant, but are affected by Precession, Nutation, and Aberration. Cf. 
Seeliger, Ast. Nach., cix., p. 273 (1884). 

t Printed in 1752, in the Recueil des pieces qui ont remportes les prix de V Acad.y. 
tome iii. J Acta eniditorum, MDCCI, p. 19. 



from Bradky to Fresnel. 101 

which his father, the elder John Bernoulli (b. 1667, d. 1748), 
had made in 1701 to connect the law of refraction with the 
mechanical principle of the composition of forces. If two 
opposed forces whose ratio is ju maintain in equilibrium a 
particle which is free to move only in a given plane, it follows 
from the triangle of forces that the directions of the forces must 
obey the relation 

sin i = fj. sin r, 

where i and r denote the angles made by these directions with 
the normals to the plane. This is the same equation as that 
which expresses the law of refraction, and the elder Bernoulli 
conjectured that a theory of light might be based on it ; but 
he gave no satisfactory physical reason for the existence of 
forces along the incident and refracted rays. This defect his 
son now proceeded to remove. 

All space, according to the younger Bernoulli, is permeated 
by a fluid aether, containing an immense number of excessively 
small whirlpools. The elasticity which the aether appears to 
possess, and in virtue of which it is able to transmit vibrations, 
is really due to the presence of these whirlpools ; for, owing to 
-centrifugal force, each whirlpool is continually striving to 
dilate, and so presses against the neighbouring whirlpools. It 
will be seen that Bernoulli is a thorough Cartesian in spirit ; 
not only does he reject action at a distance, but he insists that 
even the elasticity of his aether shall be explicable in terms of 
matter and motion. 

This aggregate of small vortices, or " fine-grained turbulent 
motion," as it came to be called a century and a half later,* is 
interspersed with solid corpuscles, whose dimensions are small 
-compared with their distances apart. These are pushed about 
by the whirlpools whenever the aether is disturbed, but never 
travel far from their original positions. 

A source of light communicates to its surroundings a 
disturbance which condenses the nearest whirlpools ; these by 

* Cf . Lord Kelvin's vortex-sponge aether, described later in this work. 



102 The Luminiferous Medium i 

their condensation displace the contiguous corpuscles from their 
equilibrium position ; and these in turn produce condensations 
in the whirlpools next beyond them, so that vibrations are 
propagated in every direction from the luminous point. It is 
curious that Bernoulli speaks of these vibrations as longitudinal, 
and actually contrasts them with those of a stretched cord, 
which, " when it is slightly displaced from its rectilinear form, 
and then let go, performs transverse vibrations in a direction at 
right angles to the direction of the cord." When it is 
remembered that the objection to longitudinal vibrations, on 
the score of polarization, had already been clearly stated by 
Newton, and that Bernoulli's aether closely resembles that 
which Maxwell invented in 1861-2 for the express purpose of 
securing transversality of vibration, one feels that perhaps no 
man ever so narrowly missed a great discovery. 

Bernoulli explained refraction by combining these ideas 
with those of his father. Within the pores of ponderable 
bodies the whirlpools are compressed, so the centrifugal force 
must vary in intensity from one medium to another. Thus a 
corpuscle situated in the interface between two media is acted 
on by a greater elastic force from one medium than from the 
other; and by applying the triangle of forces to find the- 
conditions of its equilibrium, the law of Snell and Descartes 

r may be obtained. 

Not long after this, the echoes of the old controversy 

' between Descartes and Fermat about the law of refraction 
were awakened* by Pierre Louis Moreau deMaupertuis (b. 1698,, 
d. 1759). 

It will be remembered that according to Descartes the 
velocity of light is greatest in dense media, while according to- 
Fermat the propagation is swiftest in free aether. The argu- 
ments of the corpuscular theory convinced Maupertuis that on 
this particular point Descartes was in the right ; but never- 
theless he wished to retain for science the beautiful method by 
which Fermat had derived his result. This he now proposed 

*Mem. de 1'Acad., 1744, p. 417. 



from Bradley to Fresnel. 103 

to do by modifying Fermat's principle so as to make it agree 
with the corpuscular theory; instead of assuming that light 
follows the quickest path, he supposed that " the path described 
is that by which the quantity of action is the least " ; and this 
action he defined to be proportional to the sum of the spaces 
described, each multiplied by the velocity with which it is 
traversed. Thus instead of Fermat's expression 



dt or 



tds 

} v 



(where t denotes time, v velocity, and ds an element of the path) 
Maupertuis introduced 

/v ds 

as the quantity which is to assume its minimum value when the 
path of integration is the actual path of the light. Since 
Maupertuis' v, which denotes the velocity according to the 
corpuscular theory, is proportional to the reciprocal of Fermat's 
v, which denotes the velocity according to the wave- theory, the 
two expressions are really equivalent, and lead to the same law 
of refraction. Maupertuis' memoir is, however, of great 
interest from the point of view of dynamics ; for his suggestion 
was subsequently developed by himself and by Euler and 
Lagrange into a general principle which covers the whole 
range of Nature, so far as Nature is a dynamical system. 

The natural philosophers of the eighteenth century for the 
most part, like Maupertuis, accepted the corpuscular hypothesis ; 
'but the wave-theory was not without defenders. Franklin* 
declared for it ; and the celebrated mathematician Leonhard 
Euler (b. 1707, d. 1783) ranged himself on the same side. In a 
work entitled Nova Theoria Lucis et Colorum, published! while 
he was living under the patronage of Frederic the Great at 
Berlin, he insisted strongly on the resemblance between light 
and sound ; " light is in the aether the same thing as sound in 
air/' Accepting Newton's doctrine that colour depends on 

* Letter xxiii, written in 1752. 

tL. Euleri Opuscula varii argumenti, Berlin, 1746, p. 169. 



104 The Luminiferous Medium , 

wave-length, he in this memoir supposed the frequency greatest 
for red light, and least for violet ; but a few years later* he 
adopted the opposite opinion. 

The chief novelty of Euler's writings on light is his 
explanation of the manner in which material bodies appear 
coloured when viewed by white light ; and, in particular, of the 
way in which the colours of thin plates are produced. He 
denied that such colours are due to a more copious reflexion of 
light of certain particular periods, and supposed that they 
represent vibrations generated within the body itself under the 
stimulus of the incident light. A coloured surface, according 
to this hypothesis, contains large numbers of elastic molecules, 
which, when agitated, emit light of period depending only 
on their own structure. The colours of thin plates Euler 
explained in the same way ; the elastic response and free period 
of the plate at any place would, he conceived, depend on its 
thickness at that place ; and in this way the dependence of the 
colour on the thickness was accounted for, the phenomena as 
a whole being analogous to well-known effects observed in 
experiments on sound. 

An attempt to improve the corpuscular theory in another 
direction was made in 1752 by the Marquis de Courtivron,f and 
independently in the following year by T. Melville These 
writers suggested, as an explanation of the different refran- 
gibility of different colours, that " the differently colour'd rays 
are projected with different velocities from the luminous body : 
the red with the greatest, violet with the least, and the inter- 
mediate colours with intermediate degrees of velocity." On 
this supposition, as its authors pointed out, the amount of 
aberration would be different for every different colour ; and 
the satellites of Jupiter would change colour, from white through 
green to violet, through an interval of more than half a minute 
before their immersion into the planet's shadow ; while at 
emersion the contrary succession of colours should be observed, 

* Mem. del' Acad.de Berlin, 1752, p. 262. t Courtivron's Traite cfoptique, 1752. 
JPhil. Trans, xlviii (1753), p. 262. 



from Bradley to FremeL 105 

beginning with red and ending in white. The testimony of 
practical astronomers was soon given that such appearances are 
not observed ; and the hypothesis was accordingly abandoned. 

The fortunes of the wave-theory began to brighten at the 
end of the century, when a new champion arose. Thomas 
Young, born at Milverton in Somersetshire in 1773, and 
trained to the practice of medicine, began to write on optical 
theory in 1799. In his first paper* he remarked that, according' 1 
to the corpuscular theory, the velocity of emission of a 
corpuscle must be the same in all cases, whether the projecting 
force be that of the feeble spark produced by the friction of two Q 
pebbles, or the intense heat of the sun itself a thing almost 
incredible. This difficulty does not exist in the undulatory 
theory, since all disturbances are known to be transmitted^ 
through an elastic fluid with the same velocity. The reluctance 
which some philosophers felt to filling all space with an elastic 
fluid he met with an argument which strangely foreshadows 
the electric theory of light : " That a medium resembling in 
many properties that which has been denominated ether does 
really exist, is undeniably proved by the phenomena of 
electricity. The rapid transmission of the electrical shock 
shows that the electric medium is possessed of an elasticity as 
great as is necessary to be supposed for the propagation of light. 
Whether the electric ether is to be considered the same with 
the luminous ether, if such a fluid exists, may perhaps at some 
future time be discovered by experiment : hitherto I have not 
been able to observe that the refractive power of a fluid 
undergoes any change by electricity." 

Young then proceeds to show the superior power of the^ 
wave-theory to explain reflexion and refraction. In the 
corpuscular theory it is difficult to see why part of the light 
should be reflected and another part of the same beam reflected ; 
but in the undulatory theory there is no trouble, as is shown 
by analogy with the partial reflexion of sound from a cloud or _, 
denser stratum of air : " Nothing more is necessary than to 

* Phil. Tni., 1800, p. 106. 



106 The JLuminiferous Medium, 

suppose all refracting media to retain, by their attraction, a, 
greater or less quantity of the luminous ether, so as to make its- 
density greater than that which it possesses in a vacuum, 
without increasing its elasticity." This is precisely the 
hypothesis adopted later by Fresnel and Green. 

In 1801 Young made a discovery of the first magnitude* 
when attempting to explain Newton's rings on the principles of 
the wave-theory. Eejecting Euler's hypothesis of induced 
vibrations, he assumed that the colours observed all exist in 
the incident light, and showed that they could be derived from 
it by a process which was now for the first time recognized 
in optical science. 

The idea of this process was not altogether new, for it had 
been used by Newton in his theory of the tides. " It may 
happen," he wrote, f " that the tide may be propagated from the 
ocean through different channels towards the same port, and 
may pass in less time through some channels than through 
others, in which case the same generating tide, being thus 
divided into two or more succeeding one another, may produce 
by composition new types of tide." Newton applied this- 
principle to explain the anomalous tides at Batsha in Tonkin, 
which had previously been described by Halley.J 

Young's own illustration of the principle is evidently 
suggested by Newton's. " Suppose," he says, " a number of 
equal waves of water to move upon the surface of a stagnant 
lake, with a certain constant velocity, and to enter a narrow 
channel leading out of the lake ; suppose then another similar 
cause to have excited another equal series of waves, which 
arrive at the same channel, with the same velocity, and at the 
same time with the first. Neither series of waves will destroy 
the other, but their effects will be combined ; if they enter the 
channel in such a manner that the elevations of one series 
coincide with those of the other, they must together produce a 
series of greater joint elevations ; but if the elevations of one 

* Phil. Trans., 1802, pp. 12, 387. t Principia, Book in, Prop. 24. 

% Phil. Trans, xiv (1684), p. 681. Young's Works, i, p. 202. 



from Bradley to Fremel. 107 

series are so situated as to correspond to the depressions of the 
other, they must exactly fill up those depressions, and the 
surface of the water must remain smooth. Now I maintain 
that similar effects take place whenever two portions of light 
are thus mixed ; and this I call the general law of the interference 
of light." 

Thus, " whenever two portions of the same light arrive to the 
eye by different routes, either exactly or very nearly in the same 
direction, the light becomes most intense when the difference of 
the routes is any multiple of a certain length, and least intense 
in the intermediate state of the interfering portions ; and this 
length is different for light of different colours." 

Young's explanation of the colours of thin plates as seen by 
reflexion was, then, that the incident light gives rise to two 
beams which reach the eye : one of these beams has been 
reflected at the first surface of the plate, and the other at the 
second surface ; and these two beams produce the colours by 
their interference. 

One difficulty encountered in reconciling this theory with 
observation arose from the fact that the central spot in Newton's 
rings (where the thickness of the thin Him of air is zero) is 
black and not white, as it would be if the interfering beams were 
similar to each other in all respects. To account for this Young ' 
showed, by analogy with the impact of elastic bodies, that when -> 
light is reflected at the surface of a denser medium, its phase 
is retarded by half an undulation : so that the interfering 
beams at the centre of Newton's rings destroy each other. The 
correctness of this assumption he verified by substituting essence 
of sassafras (whose refractive index is intermediate between those 
of crown and flint glass) for air in the space between the lenses ; 
as he anticipated, the centre of the ring-system was now white. 

Newton had long before observed that the rings are smaller ~* 
when the medium producing them is optically more dense. 
Interpreted by Young's theory, this definitely proved that the 
wave-length of light is shorter in dense media, and therefore ^ 
that its velocity is less. 



108 The Lumini/erous Medium, 

The publication of Young's papers occasioned a fierce attack 
on him in the Edinburgh Review, from the pen of Henry 
Brougham, afterwards Lord Chancellor of England. Young 
replied in a pamphlet, of which it is said* that only a single 
copy was sold ; and there can be no doubt that Brougham for 
the time being achieved his object of discrediting the wave- 
theory, f 

Young now turned his attention to the fringes of shadows. 
In the corpuscular explanation of these, it was supposed that 
the attractive forces which operate in refraction extend their 
influence to some distance from the surfaces of bodies, and 
inflect such rays as pass close by. If this were the case, the 
amount of inflexion should obviously depend on the strength of 
the attractive forces, and consequently on the refractive indices 
of the bodies a proposition which had been refuted by the 
experiments of s'Gravesande. The cause of diffraction effects 
was thus wholly unknown, until Young, in the Bakerian lecture 
for 1803,J showed that the principle of interference is concerned 
in their formation ; for when a hair is placed in the cone of rays 
diverging from a luminous point, the internal fringes (i.e. those 
within the geometrical shadow) disappear when the light passing 
on one side of the hair is intercepted. His conjecture as to the 
origin of the interfering rays was not so fortunate ; for he attri- 
buted the fringes outside the geometrical shadow to interference 
between the direct rays and rays reflected at the diffracting 
edge ; and supposed the internal fringes of the shadow of a 
narrow object to be due to the interference of rays inflected by 
the two edges of the object. 

The success of so many developments of the wave-theory 
led Young to inquire more closely into its capacity for solving 
the chief outstanding problem of optics that of the behaviour 
of light in crystals. The beautiful construction for the extra- 

* Peacock's Life of Young. 

t" Strange fellow," wrote Macaulay, when half a century afterwards he 
found himself sitting beside Brougham in the House of Lords, " his powers 
gone : his spite immortal." 

I Phil. Trans., 1804; Young's Works, i, p. 179. 



from Bradley ( to FresneL 109 

ordinary ray given by Huygens had lain neglected for a century ; 
and the degree of accuracy with which it represented the 
observations was unknown. At Young's suggestion Wollaston* 
investigated the matter experimentally, and showed that the 
agreement between his own measurements and Huygens' rule 
was remarkably close. " I think," he wrote, " the result must be 
admitted to be highly favourable to the Huygeniaii theory ; 
and, although the existence of two refractions at the same time, 
in the same substance, be not well accounted for, and still less 
their interchange with each other, when a ray of light is made 
to pass through a second piece of spar situated transversely to 
the first, yet the oblique refraction, when considered alone, seems 
nearly as well explained as any other optical phenomenon." 

Meanwhile the advocates of the corpuscular theory were not 
idle ; and in the next few years a succession of discoveries on 
their part, both theoretical and experimental, seemed likely to 
imperil the good position to which Young had advanced the 
rival hypothesis. 

The first of these was a dynamical explanation of the 
refraction of the extraordinary ray in crystals, which was 
published in 1808 by Laplace.f His method is an extension of 
that by which Maupertuis had accounted for the refraction of 
the ordinary ray, and which since Maupertuis' day had been so 
developed that it was now possible to apply it to problems of 
all degrees of complexity. Laplace assumes that the crystalline 
medium acts on the light-corpuscles of the extraordinary ray so 
as to modify their velocity, in a ratio which depends on the 
inclination of the extraordinary ray to the axis of the crystal : 
so that, in fact, the difference of the squares of the velocities of 
the ordinary and extraordinary rays is proportional to the 
square of the sine of the angle which the latter ray makes with 
the axis. The principle of least action then leads to a law of 
refraction identical with that found by Huygens' construction 

* Phil. Trans., 1802, p. 381. 

tMem. de PInst., 1809, p. 300: Journal de Physique, Jan., 1809; Mem. de 
la Soc. d'Arcueil, ii. 



110 The Luminiferous Medium, 

with the spheroid ; just as Maupertuis' investigation led to a 
law of refraction for the ordinary ray identical with that found 
by Huygens' construction with the sphere. 

The law of refraction for the extraordinary ray may also be 
deduced from Fermat's principle of least time, provided that the 
velocity is taken inversely proportional to that assumed in the 
principle of least action ; and the velocity appropriate to 
Fermat's principle agrees with that found by Huygens, being, in 
iact, proportional to the radius of the spheroid. These results 
are obvious extensions of those already obtained for ordinary 
refraction. 

Laplace's theory was promptly attacked by Young,* who 
pointed out the improbability of such a system of forces as 
would be required to impress the requisite change of velocity on 
the light-corpuscles. If the aim of controversial matter is to 
convince the contemporary world, Young's paper must be 
counted unsuccessful ; but it permanently enriched science by 
proposing a dynamical foundation for double refraction on the 
principles of the wave-theory. " A solution," he says, " might 
be deduced upon the Huygenian principles, from the simplest 
possible supposition, that of a medium more easily compressible 
in one direction than in any direction perpendicular to it, as if it 
consisted of an infinite number of parallel plates connected by 
a substance somewhat less elastic. Such a structure of the 
elementary atoms of the crystal may be understood by compar- 
ing them to a block of wood or of mica. Mr. Chladni found that 
the mere obliquity of the fibres of a rod of Scotch fir reduced 
the velocity with which it transmitted sound in the proportion 
of 4 to 5. It is therefore obvious that a block of such wood 
-must transmit every impulse in spheroidal that is, oval 
undulations ; and it may also be demonstrated, as we shall 
show at the conclusion of this article, that the spheroid will be 
truly elliptical when the body consists either of plane and 
parallel strata, or of equidistant fibres, supposing both to be 
^extremely thin, and to be connected by a less highly elastic 

* Quarterly Eeview, Nov., 1809 ; Young's Works, i, p. 220. 



from Bradley to FremeL 111 

substance ; the spheroid being in the former case oblate and in 
the latter oblong." Young then proceeds to a formal proof 
that "an impulse is propagated through every perpendicular 
section of a lamellar elastic substance in the form of an elliptic 
undulation." This must be regarded as the beginning of 
the dynamical theory of light in crystals. It was confirmed 
in a striking way not long afterwards by Brewster,* who found 
that compression in one direction causes an isotropic transparent 
solid to become doubly-refracting. 

Meanwhile, in January, 1808, the French Academy had 
proposed as the subject for the physical prize in 1810, " To 
furnish a mathematical theory of double refraction, and to 
confirm it by experiment." Among those who resolved to 
compete was Etienne Louis Malus (b. 1775, d. 1812), a colonel 
of engineers who had seen service with Napoleon's expedition 
to Egypt. While conducting experiments towards the end of 
1808 in a house in the Kue des Enfers in Paris, Malus happened 
to analyse with a rhomb of Iceland spar the light of the setting 
sun reflected from the window of the Luxembourg, and was 
surprised to notice that the two images were of very different 
intensities. Following up this observation, he found that light 
which had been reflected from glass acquires thereby a modifi- 
cation similar to that which Huygens had noticed in rays 
which have experienced double refraction, and which Newton 
had explained by supposing rays of light to have " sides." This 
discovery appeared so important that without waiting for the 
prize competition he communicated it to the Academy in 
December, 1808, and published it in the following month.f 
" I have found," he said, " that this singular disposition, 
which has hitherto been regarded as one of the peculiar effects 
of double refraction, can be completely impressed on the 
luminous molecules by all transparent solids and liquids." 
" For example, light reflected by the surface of water at an 

* Phil. Trans., 1815, p. 60. 

tNouveau Bulletin des Sciences, par la Soc. Philomatique. i (1809), p. 266; 
Memoires de la Soc. d'Arcueil, ii (1809). 



112 The Luminiferous Medium, 

angle of 5245' has all the characteristics of one of the beams 
produced by the double refraction of Iceland spar, whose 
principal section is parallel to the plane which passes through 
the incident ray and the reflected ray. If we receive this 
reflected ray on any doubly- refracting crystal, whose principal 
section is parallel to the plane of reflexion, it will not be divided 
into two beams as a ray of ordinary light would be, but will be 
refracted according to the ordinary law." 

After this Malus found that light which has been refracted 
at the surface of any transparent substance likewise possesses 
in some degree this property, to which he gave the name 
polarization. The memoir* which he finally submitted to the 
Academy, and which contains a rich store of experimental and 
analytical work on double refraction, obtained the prize in 1810 ; 
its immediate effect as regards the rival theories of the ultimate 
nature of light was to encourage the adherents of the corpuscular 
doctrine ; for it brought into greater prominence the phenomena 
of polarization, of which the wave-theorists, still misled by the 
analogy of light with sound, were unable to give any account. 

The successful discoverer was elected to the Academy of 
Sciences, and became a member of the celebrated club of Arcueil.f 
But his health, which had been undermined by the Egyptian 
campaign, now broke down completely : and he died, at the age 
of thirty-six, in the following year. 

The polarization of a reflected ray is in general incomplete 
i.e. the ray displays only imperfectly the properties of light 
which has been polarized by double refraction ; but for one 
particular angle of incidence, which depends on the reflecting 
body, the polarization of the reflected ray is complete. Malus 
measured with considerable accuracy the polarizing angles for 
glass and water, and attempted to connect them with the other 
optical constants of these substances, the refractive indices and 
dispersive powers, but without success. The matter was 

* Mem. presentes a 1'Inst. par divers Savans, ii (1811), p. 303. 

t So called from the village near Paris where Laplace and Berthollet had 
their country-houses, and where the meetings took place. The club consisted of 
a dozen of the most celebrated scientific men in France. 



from Bradley to Fresnel. 113 

afterwards taken up by David Brewster (b. 1781, d. 1868), who 
in 1815* showed that there is complete polarization by reflexion 
when the reflected and refracted rays satisfy the condition of 
being at right angles to each other. 

Almost at the same time Brewster made another discovery 
which profoundly affected the theory of double refraction. It 
had till then been believed that double refraction is always 
of the type occurring in Iceland spar, to which Huygens' 
construction is applicable. Brewster now found this belief to be 
erroneous, and showed that in a large class of crystals there are 
two axes, instead of one, along which there is no double 
refraction. Such crystals are called Uaxal, the simpler type to 
which Iceland spar belongs being called uniaxal. 

The wave-theory at this time was still encumbered with 
difficulties. Diffraction was not satisfactorily explained ; for 
polarization no explanation of any kind was forthcoming ; the 
Huygenian construction appeared to require two different 
luminiferous media within doubly refracting bodies ; and the 
universality of that construction had been impugned by 
Brewster's discovery of biaxal crystals. 

The upholders of the emission theory, emboldened by the 
success of Laplace's theory of double refraction, thought the 
time ripe for their final triumph ; and as a step to this, in 
March, 1817, they proposed Diffraction as the subject of the 
Academy's prize for 1818. Their expectation was disappointed ; 
and the successful memoir afforded the first of a series of 
reverses by which, in the short space of seven years, the 
corpuscular theory was completely overthrown. 

The author was Augustin Fresnel (b. 1788, d. 1827), the 
son of an architect, and himself a civil engineer in the 
Government service in Normandy. During the brief dominance 
of Napoleon after his escape from Elba in 1815, Fresnel fell into 
trouble for having enlisted in the small army which attempted 
to bar the exile's return ; and it was during a period of enforced 
idleness following on his arrest that he commenced to study 

Phil. Trans., 1815, p. 125. 
I 



114 The Lumini/erous Medium, 

diffraction. In his earliest memoir* he propounded a theory 
similar to that of Young, which was spoiled like Young's 
theory by the assumption that the fringes depend on light 
reflected by the diffracting edge. Observing, however, that the 
blunt and sharp edges of a knife produce exactly the same 
fringes, he became dissatisfied with this attempt, and on July 
15th, 1816, presented to the Academy a supplement to his 
paper,f in which, for the first time, diffraction-effects are 
referred to their true cause namely, the mutual interference 
of the secondary waves emitted by those portions of the original 
wave-front which have not been obstructed by the diffracting 
screen. Fresnel's method of calculation utilized the principles 
of both Huygens and Young ; he summed the effects due to 
different portions of the same primary wave-front, with due 
regard to the differences of phase engendered in propagation. 

The sketch presented to the Academy in 1816 was during 
the next two years developed into an exhaustive memoir, J 
which was submitted for the Academy's prize. 

It so happened that the earliest memoir, which had been 
presented to the Academy in the autumn of 1815, had been 
referred to a Commission of which the reporter was Francois 
Arago (&. 1786, d. 1853) ; Arago was so much impressed that 
he sought the friendship of the author, of whom he was later a 
strenuous champion. 

A champion was indeed needed when the larger memoir was 
submitted ; for Laplace, Poisson, and Biot, who constituted a 
majority of the Commission to which it was referred, were all 
zealous supporters of the corpuscular theory. During the 
examination, however, Fresnel was vindicated in a somewhat 
curious way. He had calculated in the memoir the diffraction- 
patterns of a straight edge, of a narrow opaque body bounded 
by parallel sides, and of a narrow opening bounded by parallel 
edges, and had shown that the results agreed excellently with 

* Annales de Chimie (2), i (1816), p 239 ; (Euvres, i, p. 89. 

t (Euvres, i, p. 129. 

I Mem. de 1'Acad., v (1826), p. 339 ; (Euvres, i, p. 247. 



from Bradley to FresneL 115 

his experimental measures. Poisson, when reading the manu- 
script, happened to notice that the analysis could be extended 
to other cases, and in particular that it would indicate the 
existence of a bright spot at the centre of the shadow of a 
circular screen. He suggested to Fresnel that this and some 
further consequences should be tested experimentally ; this was 
done, and the results were found to confirm the new theory. 
The concordance of observation and calculation was so admirable 
in all cases where a comparison was possible that the prize was 
awarded to Fresnel without further hesitation. 

In the same year in which the memoir on diffraction was 
submitted, Fresnel published an investigation* of the influence 
of the earth's motion on light. We have already seen that 
aberration was explained by its discoverer in terms of the 
corpuscular theory ; and it was Young who first showedf how 
it may be explained on the wave-hypothesis. " Upon con- 
sidering the phenomena of the aberration of the stars," he 
wrote, " I am disposed to believe that the luminiferous aether 
pervades the substance of all material bodies with little or no 
resistance, as freely perhaps as the wind passes through a 
grove of trees." In fact, if we suppose the aether surrounding 
the earth to be at rest and unaffected by the earth's motion, 
the light- waves will not partake of the motion of the telescope , 
which we may suppose directed to the true place of the star, 
and the image of the star will therefore be displaced from the 
central spider-line at the focus by a distance equal to that 
which the earth describes while the light is travelling through 
the telescope. This agrees with what is actually observed. 

But a host of further questions now suggest themselves. 
Suppose, for instance, that a slab of glass with a plane face is 
carried along by the motion of the earth, and it is desired to 
adjust it so that a ray of light coming from a certain star 
shall not be bent when it enters the glass : must the 
.surface be placed at right angles to the true direction of the 

* Annales de Chimie, ix, p. 57 (1818) ; CEnvres, ii, p. 627. 
t Phil. Trans., 1804, p. 12; Young's Works, i, p. 188. 
I 2 



116 The L uminiferous Medium, 

star as freed from aberration, or to its apparent direction as 
affected by aberration ? The question whether rays coming 
from the stars are refracted differently from rays origi- 
nating in terrestrial sources had been raised originally by 
Michell* ; and Kobison and Wilsonf had asserted that the focal 
length of an achromatic telescope should be increased when it 
is directed to a star towards which the earth is moving, owing 
to the change in the relative velocity of light. AragoJ sub- 
mitted the matter to the test of experiment, and concluded that 
the light coming from any star behaves in all cases of reflexion 
and refraction precisely as it would if the star were situated in 
the place which it appears to occupy in consequence of aber- 
ration, and the earth were at rest ; so that the apparent 
refraction in a moving prism is equal to the absolute refraction 
in a fixed prism. 

Fresnel now set out to provide a theory capable of explaining 
Arago's result. To this end he adopted Young's suggestion, 
that the refractive powers of transparent bodies depend on the 
concentration of aether within them ; and made it more precise 
by assuming that the aethereal density in any body is pro- 
portional to the square of the refractive index. Thus, if c 
denote the velocity of light in vacuo, and if c, denote its 
velocity in a given material body at rest, so that /u = c/o { is the 
refractive index, then the densities p and p l of the aether in 
interplanetary space and in the body respectively will be 
connected by the relation 

pi = n*P- 

Fresnel further assumed that, when a body is in motion, part 
of the aether within it is carried along namely, that part which 
constitutes tne excess of its density over the density of aether 
in vacuo ; while the rest of the aether within the space occupied 
by the body is stationary. Thus the density of aether carried 

* Phil. Trans., 1784, p. 35. 

t Trans. E. S. Edin., i, Hist., p. 30. 

J Biot, Astron. Phys., 3rd ed., v, p. 364. The accuracy of Arago's 
experiment can scarcely have been such as to demonstrate absolutely his 
result. 

ft, 



from Bradley to FresneL 117 

along is (pi - p) or (^ - l)/o, while a quantity of aether of 
density p remains at rest. The velocity with which the centre 
of gravity of the aether within the body moves forward in the 
direction of propagation is therefore 



where w denotes the component of the velocity of the body in 
this direction. This is to be added to the velocity of propaga- 
tion of the light- waves within the body ; so that in the moving 
body the absolute velocity of light is 



Many years afterwards Stokes* put the same supposition in 
a slightly different form. Suppose the whole of the aether 
within the body to move together, the aether entering the body 
in front, and being immediately condensed, and issuing from it 
behind, where it is immediately rarefied. On this assumption a 
mass pw of aether must pass in unit time across a plane of area 
unity, drawn anywhere within the body in a direction at right 
angles to the body's motion; and therefore the aether within 
the body has a drift- velocity - wp/p l relative to the body : so 
the velocity of light relative to the body will be Ci - wplp\, and 
the absolute velocity of light in the moving body will be 

v* 






v* k 

or ci + ^i w, as before. 

P 

This formula was experimentally confirmed in 1851 by 
H. Fizeau,f who measured the displacement of interference- 
fringes formed by light which had passed through a column of 
moving water. 

* Phil. Mag. xxviii (1846) p. 76. 

t Annales de Chimie, Ivii (1859), p. 385. Also by A. A. Michelson and 
E. W. Morley, Am. Journ. Science, xxxi (1886), p. 377. 



118 The Luminiferous Medium, 

The same result may easily be deduced from an experiment 
performed by Hoek.* In this a beam of light was divided into 
two portions, one of which was made to pass 
through a tube of water AB and was then reflected 
at a mirror C, the light being afterwards allowed to 
return to A without passing through the water : 
while the other portion of the bifurcated beam was 
made to describe the same path in the reverse 
order, i.e. passing through the water on its return 

,. journey from C instead of on the outward journey, 

On causing the two portions of the beam to inter- 
fere, Hoek found that no difference of phase was produced 
between them when the apparatus was oriented in the direction 
of the terrestrial motion. 

Let w denote the velocity of the earth, supposed to be 
directed from the tube towards the mirror. Let c/n denote the 
velocity of light in the water at rest, and C/A* + <l> the velocity 
of light in the water when moving. Let I denote the length of 
the tube. The magnitude of the distance BC does not affect 
the experiment, so we may suppose it zero. 

The time taken by the first portion of the beam to perform 
its journey is evidently 

If i 



C/fi + ^ W C + W ' 

while the time for the second portion of the beam is 
I I 

+ . 

C - W C/fJL - + W 

The equality of these expressions gives at once, when terms 
of higher orders than the first in w/c are neglected, 

= Ou 2 - 1) w\^\ 
which is FresneFs expression.! 

* Archives Neerl. iii, 180 (1868). 

t Fresnel's law may also be deduced from the principle that the amount of light 
transmitted by a slab of transparent matter must be the same whether the slab is at 
rest or in motion : otherwise the equilibrium of exchanges of radiation would be 
tiated. Cf. Larmor, Phil. Trans, clxxxv (1893), p, 775. 



from Bradley to Fresnel. 



119 



On the basis of this formula, Fresnel proceeded to solve 
the problem of refraction in moving bodies. Suppose that a 
prism A Q (7 B is carried along by the earth's motion in vacuo, its 
face A(, C being at right angles to the direction of motion ; and 




that light from a star is incident normally on this face. The 
rays experience no refraction at incidence ; and we have only to 
consider the effect produced by the second surface A<>I> . Sup- 
pose that during an interval T of time the prism travels from 
the position A Q C Bo to the position A Ci B^ while the luminous 
disturbance at C travels to h and the luminous disturbance at 
A travels to D, so that B v D is the emergent wave-front. 
Then we have 



-1 



10 



A D 



TC, 



If we write CiA\B\ = i, and denote the total deviation 
of the wave-front by 81, we have 

AiD = AJ) - AA Q cos Si = TC - rw cos 81, 



TIC, 



120 The Lumniiferous Medium, 

and therefore (neglecting second-order terms in w/c] 

A 

sin A^B^D c - w cos 81 _ c w w ^ 

- : ~ ~ "f- *" COo Ol 

sin ^ Ci c t c Ci 

Ct-W- 

Denoting by 8 the value of 81 when w is zero, we have 

sin (i -8) c 
sin i d 

Subtracting this equation from the preceding, we have 

8 -Si _ w 
sin c 

Now the telescope by which the emergent wave-front B\ D 
is received is itself being carried forward by the earth's motion; 
and we must therefore apply the usual correction for aberration 
in order to find the apparent direction of the emergent ray. But 
this correction is w sin 8/c, and precisely counteracts the effect 
which has been calculated as due to the motion of the prism. 
So finally we see that the motion of the earth has no first-order 
influence on the refraction of light from the stars. 

Fresnel inferred from his formula that if observations were 
made with a telescope filled with water, the aberration would be 
unaffected by the presence of the water a result which was 
verified by Airy* in 1871. He showed, moreover, that the 
apparent positions of terrestrial objects, carried along with the 
observer, are not displaced by the earth's motion ; that experi- 
ments in refraction and interference are not influenced by any 
motion which is common to the source, apparatus, and observer ; 
and that light travels between given points of a moving material 
system by the path of least time. These predictions have also been 
confirmed by observation: Kespighif in 1861, and Hoek+ in 1868, 
experimenting with a telescope filled with water and a terrestrial 
source of light, found that no effect was produced on the 
phenomena of reflexion and refraction by altering the orienta- 

* Proc. Roy. Soc., xx, p. 35. t Mem. Accad. Sci. Bologna, ii, p. 279. 

I Ast. Nach., Ixxiii, p. 193. 



from Bradtey to Fresnel. 121 

tion of the apparatus relative to the direction of the earth's 
motion. E. Mascart* in 1872 discussed experimentally the 
question of the effect of motion of the source or recipient of 
light in all its bearings, and showed that the light of the sun and 
that derived from artificial sources are alike incapable of revealing 
by diffraction-phenomena the translatory motion of the earth. 

The greatest problem now confronting the investigators of 
light was to reconcile the facts of polarization with the principles 
of the wave-theory. Young had long been pondering over this, 
but had hitherto been baffled by it. In 1816 he received a 
visit from Arago, who told him of a new experimental result 
which he and Fresnel had lately obtained! namely, that two 
pencils of light, polarized in planes at right angles, do not 
interfere with each other under circumstances in which ordinary 
light shows interference-phenomena, but always give by their 
reunion the same intensity of light, whatever be their difference 
of path. 

Arago had not long left him when Young, reflecting on the 
new experiment, discovered the long-sought key to the mystery : 
it consisted in the very alternative which Bernoulli had rejected 
eighty years before, of supposing that the vibrations of light are 
executed at right angles to the direction of propagation. 

Young's ideas were first embodied in a letter to Arago,J 
dated Jan. 12, 1817. "I have been reflecting," he wrote, " on the 
possibility of giving an imperfect explanation of the affection 
of light which constitutes polarization, without departing from 
the genuine doctrine of undulations. It is a principle in this 
theory, that all undulations are simply propagated through 
homogeneous mediums in concentric spherical surfaces like the 

Ann. de 1'Ecole Noemale, (2) i, p. 157. 

t It was not published until 1819, in Annales de Chimie, x ; Fresnel's (Euvres, 
i, p. 509. By means of this result, Fresnel was able to give a complete explana- 
tion of a class of phenomena which Arago had discovered in 1811, viz. that when 
polarized light is transmitted through thin plates of sulphate of lime or mica, and 
afterwards analysed by a prism of Iceland spar, beautiful complementary colours 
are displayed. Young had shown that these effects are due essentially to inter- 
ference, hut had not made clear the part played by polarization. 

J Young's JTorks, i., p. 380. 



122 The Lumini/erous Medium, 

undulations of sound, consisting simply in the direct and retro- 
grade motions of the particles in the direction of the radius,, 
with their concomitant condensation and rarefactions. And 
yet it is possible to explain in this theory a transverse vibration,, 
propagated also in the direction of the radius, and with equal 
velocity, the motions of the particles being in a certain constant 
direction with respect to that radius ; and this is a polarization" 

In an article on " Chromatics," which was written in 
September of the same year* for the supplement to the 
Encyclopaedia Britannica, he says :f " If we assume as a mathe- 
matical postulate, on the undulating theory, without attempting 
to demonstrate its physical foundation, that a transverse motion 
may be propagated in a direct line, we may derive from this 
assumption a tolerable illustration of the subdivision of polarized 
light by reflexion in an oblique plane," by " supposing the polar 
motion to be resolved " into two constituents, which fare 
differently at reflexion. 

In a further letter to Arago, dated April 29th, 1818, Young 
recurred to the subject of transverse vibrations, comparing light 
to the undulations of a cord agitated by one of its extremities.^ 
This letter was shown by Arago to Fresnel, who at once saw 
that it presented the true explanation of the non-interference 
of beams polarized in perpendicular planes, and that the latter 
effect could even be made the basis of a proof of the correctness 
of Young's hypothesis : for if the vibration of each beam be 
supposed resolved into three components, one along the ray and 
the other two at right angles to it, it is obvious from the Arago- 
Fresnel experiment that the components in the direction of the 
ray must vanish : in other words, that the vibrations which 
constitute light are executed in the wave-front. 

It must be remembered that the theory of the propagation 
of waves in an elastic solid was as yet unknown, and light was 

* Peacock's Life of Young, p. 391. t Young's Works, i., p. 279. 

JThis analogy had been given by Hooke in a communication to the Royal 
Society on Feb. 15, 1671-2. But there seems no reason to suppose that Hook e- 
appreciated the point now advanced by Young. 



from Bradley to FresneL 123 

still always interpreted by the analogy with the vibrations of 
sound in air, for which the direction of vibration is the same as 
that of propagation. It was therefore necessary to give some 
justification for the new departure. With wonderful insight 
Fresnel indicated* the precise direction in which the theory of 
vibrations in ponderable bodies needed to be extended in order 
to allow of waves similar to those of light : " the geometers," he 
wrote, " who have discussed the vibrations of elastic fluids hitherto 
have taken account of no accelerating forces except those arising 
from the difference of condensation or dilatation between conse- 
cutive layers." He pointed out that if we also suppose the 
medium to possess a rigidity, or power of resisting distortion, such 
as is manifested by all actual solid bodies, it will be capable of 
transverse vibration. The absence of longitudinal waves in the 
aether he accounted for by supposing that the forces which oppose 
condensation are far more powerful than those which oppose 
distortion, and that the velocity with which condensations are 
propagated is so great compared with the speed of the oscillations 
of light, that a practical equilibrium of pressure is maintained 
perpetually. 

The nature of ordinary non-polarized light was next discussed. 
" If then," Fresnel wrote,f " the polarization of a ray of light 
consists in this, that all its vibrations are executed in the same 
direction, it results from any hypothesis on the generation of 
light-waves, that a ray emanating from a single centre of dis- 
turbance will always be polarized in a definite plane at any 
instant. But an instant afterwards, the direction of the motion 
changes, and with it the plane of polarization ; and these 
variations follow each other as quickly as the perturbations of 
the vibrations of the luminous particle : so that even if we could 

*Annales de Chiinie, xvii (1821), p. 180; (Eiwres, i, p. 629. Young had 
already drawn attention to this point. " It is difficult," he says in his Lectures on 
Natural Philosophy, ed. 1807, vol. i, p. 138, "to compare the lateral adhesion, or 
the force which resists the detrusion of the parts of a solid, with any form of direct 
cohesion. This force constitutes the rigidity or hardness of a solid body, and is 
wholly absent from liquids." 

t Loc. cit, p. 185. 



124 The Luminiferous Medium, 

isolate the light of this particular particle from that of other 
luminous particles, we should doubtless not recognize in it any 
appearance of polarization. If we consider now the effect pro- 
duced by the union of all the waves which emanate from the 
different points of a luminous body, we see that at each instant, 
at a definite point of the aether, the general resultant of all the 
motions which commingle there will have a determinate 
direction, but this direction will vary from one instant to the 
next. So direct light can be considered as the union, or more 
exactly as the rapid succession, of systems of waves polarized in 
all directions. According to this way of looking at the matter, 
the act of polarization consists not in creating these transverse 
motions, but in decomposing them in two invariable directions, 
and separating the components from each other ; for . then, in 
each of them, the oscillatory motions take place always in the 
same plane." 

He then proceeded to consider the relation of the direction of 
vibration to the plane of polarization. " Apply these ideas to 
double refraction, and regard a uniaxal crystal as an elastic 
medium in which the accelerating force which results from 
the displacement of a row of molecules perpendicular to the 
axis, relative to contiguous rows, is the same all round the 
axis ; while the displacements parallel to the axis produce 
accelerating forces of a different intensity, stronger if the 
crystal is "repulsive," and weaker if it is "attractive." The 
distinctive character of the rays which are ordinarily refracted 
being that of propagating themselves with the same velocity 
in all directions, we must admit that their oscillatory motions 
are executed at right angles to the plane drawn through these 
rays and the axis of the crystal; for then the displacements 
which they occasion, always taking place along directions 
perpendicular to the axis, will, by hypothesis, always give rise 
to the same accelerating forces. But, with the conventional 
meaning which is attached to the expression 'plane, of polarization, 
the plane of polarization of the ordinary rays is the plane 
through the axis : thus, in a pencil of polarized light, the 



from Bradley, to Fresnel. 125 

oscillatory motion is executed at right angles to the plane of 
polarization" 

This result afforded Fresnel a foothold in dealing with the 
problem which occupied the rest of his life : henceforth his aim 
was to base the theory of light on the dynamical properties of 
the luminiferous medium. 

The first topic which he attacked from this point of view 
was the propagation of light in crystalline bodies. Since 
Brewster's discovery that many crystals do not conform to the 
type to which Huygens' construction is applicable, the wave 
theory had to some extent lost credit in this region. Fresnel, 
now, by what was perhaps the most brilliant of all his efforts,* 
not only reconquered the lost territory, but added a new domain 
to science. 

He had, as he tells us himself, never believed the doctrine 
that in crystals there are two different luminiferous media, 
one to transmit the ordinary, and the other the extraordinary 
waves. The alternative to which he inclined was that the two 
velocities of propagation were really the two roots of a quadratic 
equation, derivable in some way from the theory of a single 
aether. Could this equation be obtained, he was confident of 
finding the explanation, not only of double refraction, but also 
of the polarization by which it is always accompanied. 

The first step was to take the case of uniaxal crystals, 
which had been discussed by Huygens, and to see whether 
Huygens' sphere and spheroid could be replaced by, or made to 
depend on, a single surface.f 

Now a wave propagated in any direction through a uniaxal 

*His first memoir on Double Refraction was presented to the Academy on 
Nov. 19th, 1821, but has not been published except in his collected works: 
(Eitvres, ii, p. 261. It was followed by other papers in 1822; and the results were 
finally collected in a memoir which was printed in 1827, Mem. de VAcad. vii, 
p. 45, (Euvres, ii, p. 479. 

t In attempting to reconstruct Fresnel's course of thought at this period, the 
present writer has derived much help from the Life prefixed to the (Euvres de 
Fresnel. Both Fresnel and Young were singularly fortunate in their biographers : 
Peacock's Life of Young, and this notice of Fresnel, which was the last work of 
Verdet, are excellent reading. 



126 The Luminiferous Medium, 

crystal can be resolved into two plane-polarized components ; 
one of these, the " ordinary ray," is polarized in the principal 
section, and has a velocity v l9 which may be represented by the 
radius of Huygens' sphere say, 

Vi = &; 

while the other, the " extraordinary ray," is polarized in a plane 
.at right angles to the principal section, and has a wave- velocity v 9 , 
which may be represented by the perpendicular drawn from the 
centre of Huygens' spheroid on the tangent-plane parallel to the 
plane of the wave. If the spheroid be represented by the 
equation 

if + z'" x* 

+ ^ = 1 - 

and if (I, m, n) denote the direction-cosines of the normal to the 
plane of the wave, we have therefore 

v,~ = a*(m* + n*) + ?> 2 / 2 . 

But the quantities 1/Vi and l/t? 8 , as given by these equations, 
are easily seen to be the lengths of the semi-axes of the ellipse 
in which the spheroid 

6 2 (?/ 3 4- z~) + arx- = 1 

is intersected by the plane 

Ix + my + nz = ; 

.and thus the construction in terms of Huygens' sphere and 
spheroid can be replaced by one which depends only on a single 
surface, namely the spheroid 



Having achieved this reduction, Fresnel guessed that the 
<?ase of biaxal crystals could be covered by substituting for the 
latter spheroid an ellipsoid with three unequal axes say, 



x z if z* 
_++_ = 



If I/Vi and l/^ denote the lengths of the semi-axes of the 
.ellipse in which this ellipsoid is intersected by the plane 

Ix 4 my + nz - 0, 



from Bradley to Fresnel. 127 

it is well known that #1 and v z are the roots of the equation in v 



; --0; 



1 . 1 ,1 

tf tf v- 

ti 2 3 

and accordingly Fresnel conjectured that the roots of this 
equation represent the velocities, in a biaxal crystal, of the two 
plane-polarized waves whose normals are in the direction 
(I, m, n). 

Having thus arrived at his result by reasoning of a purely 
geometrical character, he now devised a dynamical scheme to 
suit it. 

The vibrating medium within a crystal he supposed to be 
ultimately constituted of particles subjected to mutual forces ; 
and on this assumption he showed that the elastic force of 
restitution when the system is disturbed must depend linearly 
on the displacement. In this first proposition a difference is 
apparent between Fresnel's and a true elastic-solid theory ; for 
in actual elastic solids the forces of restitution depend not on 
the absolute displacement, but on the strains, i.e., the relative 
displacements. 

In any crystal there will exist three directions at right 
angles to each other, for which the force of restitution acts in 
the same line as the displacement : the directions which possess 
this property are named axes of elasticity. Let these be taken 
as axes, and suppose that the elastic forces of restitution for 
unit displacements in these three directions are 1/5], l/c 2 , l/s 
respectively. That the elasticity should vary with the direction 
of the molecular displacement seemed to Fresnel to suggest that 
the molecules of the material body either take part in the 
luminous vibration, or at any rate influence in some way the 
elasticity of the aether. 

A unit displacement in any arbitrary* direction (a, )3, 7) can 
be resolved into component displacements (cos a, cos /3, cos 7) 
parallel to the axes, and each of these produces its own effect 



128 The Luminiferous Medium ^ 

independently ; so the components of the force of restitution are 

COS a COS )3 COS y 
l ft 3 

This resultant force has not in general the same direction 
as the displacement which produced it ; but it may always 
he decomposed into two other forces, one parallel and the other 
perpendicular to the direction of the displacement ; and the 
former of these is evidently 



The surface 



COS 2 a COS 2 )3 COS 2 7 

I {_ I _ 

fl 2 3 



X 2 V* 



i 2 3 

will therefore have the property that the square of its radius 
vector in any direction is proportional to the component in that 
direction of the elastic force due to a unit displacement in that 
direction : it is called the surface of elasticity. 

Consider now a displacement along one of the axes of the 
section on which the surface of elasticity is intersected by the 
plane of the wave. It is easily seen that in this case the com- 
ponent of the elastic force at right angles to the displacement 
acts along the normal to the wave-front; and Fresnel assumes 
that it will be without influence on the propagation of the 
vibrations, on the ground of his fundamental hypothesis that the 
vibrations of light are performed solely in the wave-front. This 
step is evidently open to criticism ; for in a dynamical theory 
everything should be deduced from the laws of motion without 
special assumptions. But granting his contention, it follows 
that such a displacement will retain its direction, and will be 
propagated as a plane-polarized wave with a definite velocity. 

Now, in order that a stretched cord may vibrate with 
unchanged period, when its tension is varied, its length must be 
increased proportionally to the square root of its tension ; and 
similarly the wave-length of a luminous vibration of given period 
is proportional to the square root of the elastic force (per unit 



from Bradley to Fresnel. 129 

displacement), which urges the molecules of the medium parallel 
to the wave-front. Hence the velocity of propagation of a 
wave, measured at right angles to its front, is proportional to 
the square root of the component, along the direction of dis- 
placement, of the elastic force per unit displacement ; and the 
velocity of propagation of such a plane-polarized wave as we 
have considered is proportional to the radius vector of the 
surface of elasticity in the direction of displacement. 

Moreover, any displacement in the given wave-front can be 
resolved into two, which are respectively parallel to the two 
axes of the diametral section of the surface of elasticity by a 
plane parallel to this wave-front ; and it follows from what has 
been said that each of these component displacements will be 
propagated as an independent plane-polarized wave, the velocities 
of propagation being proportional to the axes of the section,* 
and therefore inversely proportional to the axes of the section of 
the inverse surface of this with respect to the origin, which is 
the ellipsoid 

* + + *-i. 

i 2 3 

But this is precisely the result to which, as we have seen, 
Fresnel had been led by purely geometrical considerations ; and 
thus his geometrical conjecture could now be regarded as 
substantiated by a study of the dynamics of the medium. 

It is easy to determine the wave-surface or locus at any 
instant say, t = 1 of a disturbance originated at some previous 
instant say, = at some particular point say, the origin. For 
this wave-surface will evidently be the envelope of plane waves 
emitted from the origin at the instant t = that is, it will be 
the envelope of planes 

Ix + my + nz - v = 0, 

where the constants /, m, n, v are connected by the identical 
equation I 2 + m* + n z = 1, 

* It is evident from this that the optic axes, or lines of single wave-velocity, 
along which there is no double refraction, will be perpendicular to the two 
circular sections of the surface of elasticity. 

K 



130 The Lumimferous Medium, 

and by the relation previously found namely, 
/ 2 m 2 n~ 



1 



By the usual procedure for determining envelopes, it may be 
shown that the locus in question is the surface of the fourth 
degree 

x z _ _f _fl_ _ n 



which is called Fresnel's wave-surface* It is a two-sheeted surface, 
as must evidently be the case from physical considerations. In 
uniaxal crystals, for which * 2 and c 3 are equal, it degenerates into 
the sphere 

r 2 = l/ e> , 
and the spheroid 

^ + fl (tf + Z 2 ) = 1. 

It is to these two surfaces that tangent-planes are drawn in 
the construction given by Huygens for the ordinary and 
extraordinary refracted rays in Iceland spar. As Fresnel 
observed, exactly the same construction applies to biaxal 
crystals, when the two sheets of the wave-surface are substi- 
tuted for Huygens' sphere and spheroid. 

" The theory which I have adopted," says Fresnel at the end 
of this memorable paper, " and the simple constructions which 
I have deduced from it, have this remarkable character, that 
all the unknown quantities are determined together by the 
solution of the problem. We find at the same time the 
velocities of the ordinary ray and of the extraordinary ray, and 
their planes of polarization. Physicists who have studied 
attentively the laws of nature will feel that such simplicity and 

* Another construction for the wave-surface is the following, which is due to 
MacCullagh, Coll. Works, p. 1. Let the ellipsoid 

*i x ~ + 6 2^" ~*~ *3~~ = * 

be intersected hy a plane through its centre, and on the perpendicular to that plane 
take lengths equal to the semi-axes of the section. The locus of these extremities 
is the wave-surface. 



from Bradley to FresneL 131 

such close relations between the different elements of the 
phenomenon are conclusive in favour of the hypothesis on 
which they are based." 

The question as to the correctness of Fresnel's construction 
was discussed for many years afterwards. A striking conse- 
quence of it was pointed out in 1832 by William Kowan 
Hamilton (b. 1805, d. 1865), Royal Astronomer of Ireland, who 
remarked* that the surface defined by Fresnel's equation has 
four conical points, at each of which there is an infinite number 
of tangent planes ; consequently, a single ray, proceeding from 
a point within the crystal in the direction of one of these 
points, must be divided on emergence into an infinite number of 
rays, constituting a conical surface. Hamilton also showed 
that there are four planes, each of which touches the wave- 
surface in an infinite number of points, constituting a circle of 
contact : so that a corresponding ray incident externally should 
be divided within the crystal into an infinite number of refracted 
rays, again constituting a conical surface. 

These singular and unexpected consequences of the theory 
were shortly afterwards verified experimentally by Humphrey 
Lloyd,f and helped greatly to confirm belief in Fresnel's theory. 
It should, however, be observed that conical refraction only 
shows his form of the wave- surf ace to be correct in its general 
features, and is no test of its accuracy in all details. But it 
was shown experimentally by Stokes in 1872J Glazebrook in 
1879, and Hastings in 1887,1 1 that the construction of Huygens 
and Fresnel is certainly correct to a very high degree of 
approximation; and Fresnel's final formulae have since been 
regarded as unassailable. The dynamical substructure on 
which he based them is, as we have seen, open to objection ; 

* Trans. Roy. Irish Acad., xvii (1833), p. 1. 

t Trans. Roy. Irish Acad., xvii (1833), p. 145. Strictly speaking, the bright 
oone which is usually observed arises from rays adjacent to the singular ray : 
the latter can, however, be observed, its enfeeblement by dispersion into the 
conical form causing it to appear dark. 

I Proc. R. S., xx, p. 443. 

Phil. Trans., clxxi, p. 421. 

|| Am. Jour. Sci. (3), xxxv, p. 60. 

K 2 



132 The Luminiferous Medium, 

but, as Stokes observed*: "If we reflect on the state of the 
subject as Fresnel found it, and as he left it, the wonder is, not 
that he failed to give a rigorous dynamical theory, but that a 
single mind was capable of effecting so much." 

In a second supplement to his first memoir on Double 
Eefraction, presented to the Academy on November 26th, 1821,-]- 
Fresnel indicated the lines on which his theory might be 
extended so as to take account of dispersion. " The molecular 
groups, or the particles of bodies," he wrote, " may be separated 
by intervals which, though small, are certainly not altogether 
insensible relatively to the length of a wave." Such a coarse- 
grainedness of the medium would, as he foresaw, introduce into 
the equations terms by which dispersion might be explained ; 
indeed, the theory of dispersion which was afterwards given by 
Cauchy was actually based on this principle. It seems likely 
that, towards the close of his life, Fresnel was contemplating a 
great memoir on dispersion^ which was never completed. 

Fresnel had reason at first to be pleased with the reception of 
his work on the optics of crystals : for in August, 1822, Laplace 
spoke highly of it in public ; and when at the end of the year a 
seat in the Academy became vacant, he was encouraged to hope 
that the choice would fall on him. In this he was disappointed. . 
Meanwhile his researches were steadily continued ; and in 
January, 1823, the very month of his rejection, he presented to- 
the Academy a theory in which reflexion and refraction] | are 
referred to the dynamical properties of the luminiferous media. 

*Brit. Assoc. Rep., 1862, p. 254. 

t (Euvres, ii, p. 438. 

J Cf. the biography in (Euvres de Fresnel, i, p. xcvi. 

Writing to Young in the spring of 1823, he says : " Tous ces memoires, 
que dernierement j'ai pre'sentes coup sur coup a 1'Academie des Sciences, ne 
m'en ont pas cependant otivert la porte. C'est M. Dulong qui a ete nomine 
pour remplir la place vacante dans la section de physique. . . Vous voyez, 
Monsieur, que la theorie des ondulations ne m'a point porte honheur : mais cela 
ne m'en degoute pas : et je me console de ce malheur en m* occupant d'optique 
avec une nouvelle ardeur." 

|| The MSS- was for some time believed to be lost, but was ultimately found 
among the papers of Fourier, and printed in Mem. de 1'Acad. xi (1832), p. 393 : 
(Euvres, i, p. 767. 



from Bradley to FresneL 133 

As in his previous investigations, he assumes that the 
vibrations which constitute light are executed at right angles 
to the plane of polarization. He adopts Young's principle, that 
reflexion and refraction are due to differences in the inertia of 
the aether in different material bodies, and supposes (as in 
his memoir on Aberration) that the inertia is proportional to 
the inverse square of the velocity of propagation of light in 
the medium. The conditions which he proposes to satisfy at the 
interface between two media are that the displacements of the 
adjacent molecules, resolved parallel to this interface, shall be 
equal in the two media ; and that the energy of the reflected 
and refracted waves together shall be equal to that of the 
incident wave. 

On these assumptions the intensity of the reflected and 
refracted light may be obtained in the following way : 

Consider first the case in which the incident light is 
polarized in the plane of incidence, so that the displacement is 
at right angles to the plane of incidence ; let the amplitude 
of the displacement at a given point of the interface be / 
for the incident ray, g for the reflected ray, and h for the 
refracted ray. 

The quantities of energy propagated per second across unit 
cross-section of the incident, reflected, and refracted beams are 
proportional respectively to 



where c b c 2 , denote the velocities of light, and p l} p z the densities 
of aether, in the two media ; and the cross-sections of the beams 
which meet the interface in unit area are 

cos i, cos i, cos r 

respectively. The principle of conservation of energy therefore 
gives 

c,p! cos i ./ 2 = c,/o! cos i . g z + c 2 /o 2 cos r . h~. 

The equation of continuity of displacement at the interface is 

/ + 9 = h. 



134 The Luminiferous Medium, 

Eliminating li between these two equations, and using the 
formulae 

sin 2 T Co 2 pi 

sin 2 i C* p 2 ' 
we obtain the equation 

Z. _ sm (^ ~ r ) 
g sin (i + r) 

Thus when the light is polarized in the plane of reflexion, the 
amplitude of the reflected wave is 

Q-l -T\ (ft ^ -0*\ 

- \-. r x the amplitude of the incident vibration. 

sin pj + r) 

Fresnel shows in a similar way that when the light is 
polarized at right angles to the plane of reflexion, the ratio of 
the amplitudes of the reflected and incident waves is 

tan (i - r) 
tan (i + r) 

These formulae are generally known as Fresnel' s sine-law and 
FresneTs tangent-law respectively. They had, however, been 
discovered experimentally by Brewster some years previously. 
When the incidence is perpendicular, so that i and r are very 
small, the ratio of the amplitudes becomes 

Limit , 

^ + r 

or 



where ju 2 and //i denote the refractive indices of the media. 
This formula had been given previously by Young* and Poisson,f 
on the supposition that the elasticity of the aether is of the 
same kind as that of air in sound. 

When i + r = 90, tan (i + r) becomes infinite : and thus 
a theoretical explanation is obtained for Brewster 's law, that if 
the incidence is such as to make the reflected and refracted rays- 

* Article Chromatics, Encycl. Britt. Suppl. t Mem. Inst. ii. (1817). 



fro vi Bradley to Fresnel. 135 

perpendicular to each other, the reflected light will be wholly 
polarized in the plane of reflexion. 

Fre&nel's investigation can scarcely be called a dynamical 
theory in the strict sense, as the qualities of the medium are 
not defined. His method was to work backwards from the 
known properties of light, in the hope of arriving at a mechanism 
to which they could be attributed ; he succeeded in accounting 
for the phenomena in terms of a few simple principles, but was 
not able to specify an aether which would in turn account for 
these principles. The " displacement " of Fresnel could not be 
a displacement in an elastic solid of the usual type, since its 
normal component is not continuous across the interface between 
two media.* 

The theory of ordinary reflexion was completed by a dis- 
cussion of the case in which light is reflected totally. This had 
formed the subject of some of Fresnel's experimental researches 
several years before; and in two papersf presented to the 
Academy in November, 1817, and January, 1818, he had shown 
that light polarized in any plane inclined to the plane of reflexion 
is partly "depolarized" by total reflexion, and that this is 
due to differences of phase which are introduced between the 
components polarized in and perpendicular to the plane of 
reflexion. " When the reflexion is total," he said, " rays 
polarized in the plane of reflexion are reflected nearer the 
surface of the glass than those polarized at right angles to the 
same plane, so that there is a difference in the paths described." 
This change of phase he now deduced from the formulae 
already obtained for ordinary reflexion. Considering light 
polarized in the plane of reflexion, the ratio of the amplitudes of 
the reflected and incident light is, as we have seen, 

sin (i - r) 

sin (i + r) ' 
when the sine of the angle of incidence is greater than /i 2 /jui, 

* Fresnel's theory of reflexion can, however, he reconciled with the electro- 
magnetic theory of light, by identifying his "displacement" with the electric 
force. f (Euvres de Fresnel, i., pp. 441, 487. 



136 The Luminiferous Medium. 

so that total reflexion takes place, this ratio may be written in 
the form 

where 6 denotes a real quantity defined by the equation 



tan 



cos ^ 



Fresnel interpreted this expression to mean that the 
amplitude of the reflected light is equal to that of the incident, 
but that the two waves differ in phase by an amount 0. The 
case of light polarized at right angles to the plane of reflexion 
may be treated in the same way, and the resulting formulae are 
completely confirmed by experiment. 

A few months after the memoir on reflexion had been 
presented, Fresnel was elected to a seat in the Academy ; and 
during the rest of his short life honours came to him both from 
France and abroad. In 1827 the Royal Society awarded him 
the Rumford medal ; but Arago, to whom Young had confided 
the mission of conveying the medal, found him dying ; and 
eight days afterwards he breathed his last. 

By the genius of Young and Fresnel the wave-theory of 
light was established in a position which has since remained 
unquestioned ; and it seemed almost a work of supererogation 
when, in 1850, Foucault* and Fizeau,f carrying out a plan long 
before imagined by Arago, directly measured the velocity of 
light in air and in water, and found that on the question so 
long debated between the rival schools the adherents of the 
undulatory theory had been in the right. 

* Comptes Rendus, xxx (1850), p. 551. t Ibid., p. 562. 



( 137 ) 



CHAPTER V. 

THE AETHER AS AN ELASTIC SOLID. 

WHEN Young and Fresnel put forward the view that the 
vibrations of light are performed at right angles to its direction 
of propagation, they at the same time pointed out that this 
peculiarity might be explained by making a new hypothesis 
regarding the nature of the luminiferous medium ; namely, that 
it possesses the power of resisting attempts to distort its shape. 
It is by the possession of such a power that solid bodies are 
distinguished from fluids, which offer no resistance to distortion; 
the idea of Young and Fresnel may therefore be expressed by 
the simple statement that the aether behaves as an elastic solid. 
After the death of Fresnel this conception was developed in a 
brilliant series of memoirs to which our attention must now be 
directed. 

The elastic-solid theory meets with one obvious difficulty at 
the outset. If the aether has the qualities of a solid, how is it that 
the planets in their orbital motions are able to journey through 
it at immense speeds without encountering any perceptible 
resistance ? This objection was first satisfactorily answered by 
Sir George Gabriel Stokes* (b. 1819, d. 1903), who remarked 
that such substances as pitch and shoemaker's wax, though so 
rigid as to be capable of elastic vibration, are yet sufficiently 
plastic to permit other bodies to pass slowly through them. 
The aether, he suggested, may have this combination of qualities 
in an extreme degree, behaving like an elastic solid for vibrations 
so rapid as those of light, but yielding like a fluid to the much 
slower progressive motions of the planets. 

Stokes's explanation harmonizes in a curious way with 
Fresnel's hypothesis that the velocity of longitudinal waves in 

* Trans. Camb. Phil. Soc., viii, p. 287 (1845). 



138 The Aether as an Elastic Solid. 

the aether is indefinitely great compared with that of the 
transverse waves ; for it is found by experiment with actual 
substances that the ratio of the velocity of propagation of 
longitudinal waves to that of transverse waves increases 
rapidly as the medium becomes softer and more plastic. 

In attempting to set forth a parallel between light and the 
vibrations of an elastic substance, the investigator is compelled 
more than once to make a choice between alternatives. He 
may, for instance, suppose that the vibrations of the aether are 
executed either parallel to the plane of polarization of the light 
or at right angles to it ; and he may suppose that the different 
refractive powers of different media are due either to differences 
in the inertia of the aether within the media, or to differences 
in its power of resisting distortion, or to both these causes 
combined. There are, moreover, several distinct methods for 
avoiding the difficulties caused by the presence of longitudinal 
vibrations ; and as, alas ! we shall see, a further source of 
diversity is to be found in that liability to error from which no 
man is free. It is therefore not surprising that the list of 
elastic-solid theories is a long one. 

At the time when the transversality of light was dis- 
covered, no general method had been developed for investi- 
gating mathematically the properties of elastic bodies; but 
under the stimulus of Fresnel's discoveries, some of the best 
intellects of the age were attracted to the subject. The volume 
of Memoirs of the Academy which contains Fresnel's theory of 
crystal-optics contains also a memoir by Claud Louis Marie 
Henri Navier* (&. 1785, d. 1836), at that time Professor of 
Mechanics in Paris, in which the correct equations of vibratory 
motion for a particular type of elastic solid were for the first 
time given. ISTavier supposed the medium to be ultimately 
constituted of an immense number of particles, which act on 
each other with forces directed along the lines joining them, and 
depending on their distances apart ; and showed that if e denote 

* Mem. de 1'Acad. vii, p. 375. The memoir was presented in 1821, and 
published in 1827. 



The Aether as an Elastic Solid. 139 

the (vector) displacement of the particle whose undisturbed 
position is (x, y, z], and if p denote the density of the medium, 
the equation of motion is 

p = - 3n grad div e - n curl curl e, 

ot 

where n denotes a constant which measures the rigidity, or 
power of resisting distortion, of the medium. All such elastic 
properties of the body as the velocity of propagation of waves 
in it must evidently depend on the ratio n/p. 

Among the referees of one of Navier's papers was Augustine 
Louis Cauchy (b. 1789, d. 1857), one of the greatest analysts of 
the nineteenth century,* who, becoming interested in the 
question, published in 1828f a discussion of it from an entirely 
different point of view. Instead of assuming, as Navier had 
done, that the medium is an aggregate of point-centres of force, 
and thus involving himself in doubtful molecular hypotheses, 
he devised a method of directly studying the elastic properties 
of matter in bulk, and by its means showed that the vibrations 
of an isotropic solid are determined by the equation 

8 2 e ( 1 4 \ 
p = - [fc + -n\ grad div e - n curl curl e ; 



here n denotes, as before, the constant of rigidity; and the 
constant &, which is called the modulus of compression^. denotes 
the ratio of a pressure to the cubical compression produced by 
it. Cauchy's equation evidently differs from Navier's in that 

* Hamilton's opinion, written in 1833, is worth repeating : " The principal 
theories of algebraical analysis (under which I include Calculi) require to he 
entirely remodelled ; and Cauchy has done much already for this great object. 
Poisson also has done much ; but he does not seem to me to have nearly so logical a 
mind as Cauchy, great as his talents and clearness are ; and both are in my 
judgment very far inferior to Fourier, whom I place at the head of the French 
School of Mathematical Philosophy, even above Lagrange and Laplace, though I 
rank their talents above those of Cauchy and Poisson." (Life of Sir W. It. 
Hamilton, ii, p. 58.) 

t Cauchy, Exercices de Mathematiques iii, p. 160 (1828). 

J This notation was introduced at a later period, but is used here in order to 
avoid subsequent changes. 



140 The Aether as an Elastic Solid. 

two constants, k and n, appear instead of one. The reason for 
this is that a body constituted from point-centres of force in 
Navier's fashion has its moduli of rigidity and compression 
connected by the relation* 



Actual bodies do not necessarily obey this condition; e.g. 

for india-rubber, k is much larger than - n ;f and there seems to 

o 

be no reason why we should impose it on the aether. 

In the same year PoissonJ succeeded in solving the diffe- 
rential equation which had thus been shown to determine the 
wave-motions possible in an elastic solid. The solution, which 
is both simple and elegant, may be derived as follows : Let the 
displacement vector e be resolved into two components, of 
which one c is circuital, or satisfies the condition 

div c = 0, 
while the other b is irrotational, or satisfies the condition 

curl b = 0. 
The equation takes the form 

+ 5 V b = ' 

o Tlj 

* In order to construct a body whose elastic properties are not limited by this 
equation, William John Macquorn Rankine (b. 1820, d. 1872) considered a con- 
tinuous fluid in which a number of point-centres of force are situated : the fluid is 
supposed to be partially condensed round these centres, the elastic atmosphere of 
each nucleus being retained round it by attraction. An additional volume-elasticity 
due to the fluid is thus acquired ; and no relation between k and n is now necessary. 
Cf. Rankine's Miscellaneous Scientific Papers, pp. 81 sqq. 

Sir "William Thomson (Lord Kelvin), in 1889, formed a solid not obeying 
Navier's condition by using pairs of dissimilar atoms. Cf. Thomson's Papers, 
iii, p. 395. Cf. also Baltimore Lectures, pp. 123 sqq. 

t It may, however, be objected that india-rubber and other bodies which 
fail to fulfil Navier's relation are not true solids. On this historic controversy, 
cf. Todhunter and Pearson's History of Elasticity, i, p. 496. 

J Mem. de 1'Acad., viii (1828), p. 623. Poisson takes the equation in the 
restricted form given by Navier ; but this does not affect the question of wave- 
propagation. 



The Aether as an Elastic Solid. 141 

The terms which involve b and those which involve c must 
be separately zero, since they represent respectively the irrota- 
tional and the circuital parts of the equation. Thus, c satisfies. 
the pair of equations 

02- 

p T-J- = ?iV 2 c, div c = ; 

vt 

while b is to be determined from 



dt 
A particular solution of the equations for c is easily seen to be 

c x = A sin A (2 - t /-), 



/-), c y = B sinXfz - t /-), c z = 0, 
\PJ V \PJ 



which represents a transverse plane wave propagated with 
velocity ^/(n/p). It can be shown that the general solution of 
the differential equations for c is formed of such waves as this, 
travelling in all directions, superposed on each other 
A particular solution of the equations for b is 



-t E 

V 



p 
which represents a longitudinal wave propagated with velocity 



the general solution of the differential equation for b is formed 
by the superposition of such waves as this, travelling in all 
directions. 

Poisson thus discovered that the waves in an elastic solid 
are of two kinds : those in c are transverse, and are propagated 
with velocity (n/p)b ; while those in b are longitudinal, and are 
propagated with velocity {(k+ $n)/p}%. The latter are* waves 
of dilatation and condensation, like sound-waves ; in the c-waves, 
on the other hand, the medium is not dilated or condensed, but 

* Cf . Stokes, "On the Dynamical Problem of Diffraction," Camb. Phil. 
Trans., ix (1849). 



142 The Aether as an Elastic Solid. 

only distorted in a manner consistent with the preservation of a 
constant density.* 

The researches which have been mentioned hitherto have 
all been concerned with isotropic bodies. Cauchy in 1828f 
extended the equations to the case of crystalline substances. 
This, however, he accomplished only by reverting to Navier's 
plan of conceiving an elastic body as a cluster of particles which 
attract each other with forces depending on their distances apart ; 
the aelotropy he accounted for by supposing the particles to be 
packed more closely in some directions than in others. 

The general equations thus obtained for the vibrations of an 
elastic solid contain twenty-one constants ; six of these depend 
on the initial stress, so that if the body is initially without 
stress, only fifteen constants are involved. If, retaining the 
initial stress, the medium is supposed to be symmetrical with 
respect to three mutually orthogonal planes, the twenty-one 
constants reduce to nine, and the equations which determine 
the vibrations may be written in the form* 



dx\ 2x ty 9 dz 
and two similar equations. The three constants G, H, I re- 
present the stresses across planes parallel to the coordinate 
planes in the undisturbed state of the aether. 

* It may easily be shown that any disturbance, in either isotropic or crystalline 
media, for which the direction of vibration of the molecules lies in the wave-front 
or surface of constant phase, must satisfy the equation 

div 6 = 0, 

where e denotes the displacement ; if, on the other hand, the direction of vibration 
of the molecules is perpendicular to the wave -front, the disturbance must satisfy 
the equation 

curl e = 0. 
These results were proved by M. O'Brien, Trans. Camb. Phil. Soc., 1842. 

t Exercices de Math., iii (1828), p. 188. 

J These are substantially equations (68) on page 208 of the third volume of 
the Exercices. 

$ G, H, I are tensions when they are positive, and pressures when they are 
negative. 



The Aether as an Elastic Solid. 1 43 

On the basis of these equations, Cauchy worked out a 
theory of light, of which an instalment relating to crystal-optics 
was presented to the Academy in 1830.* Its characteristic 
features will now be sketched. 

By substitution in the equations last given, it is found that 
when the wave-front of the vibration is parallel to the plane 
of yz, the velocity of propagation must be (h + G)% if the vibration 
takes place parallel to the axis of y, and (g+ G)$ if it takes place 
parallel to the axis of z. Similarly when the wave-front is 
parallel to the plane of zx, the velocity must be (h + H)% if the 
vibration is parallel to the axis of x, and (/+ H)^ if it is parallel 
fo the axis of z\ and when the wave-front is parallel to the 
plane of xy, the velocity must be (g + /)* if the vibration is parallel 
to the axis of x, and (/ + /)* if it is parallel to the axis of y. 

Now it is known from experiment that the velocity of a 
ray polarized parallel to one of the planes in question is the 
same, whether its direction of propagation is along one or the 
other of the axes in that plane: so, if we assume that the 
vibrations which constitute light are executed parallel to the 
plane of polarization, we must have 

/+#=/+/, ff + I = g+G, k + H=h+G; 
or, G = H=L 

This is the assumption made in the memoir of 1830 : the theory 
based on it is generally known as Cauchy' s First Theory ;( the 
equilibrium pressures G, H, /, being all equal, are taken to be zero. 

Tf, on the other hand, we make the alternative assumption 
that the vibrations of the aether are executed at right angles to 
the plane of polarization, we must have 



* Mem. de 1'Acad., x, p. 293. 

In the previous year (Mem. de 1'Acad., ix, p. 114) Cauchy had stated that the 
equations of elasticity lead in the case of uniaxal crystals to a wave-surface of 
which two sheets are a sphere and spheroid as in Huygens' theory. 

f The equations and results of Cauchy's First Theory of crystal-optics were 
independently obtained shortly afterwards hy Franz Ernst Neumann (b. 1798, 
d. 1895) : cf. Ann. d. Phys. xxv (1832), p. 418, reprinted as No. 76 of Ostwald's 
Klassiker der exakten Wissenschaften, with notes by A. Wangerin. 



144 The Aether as an Elastic Solid. 

the theory based on this supposition is known as Caucliy's 
Second Theory : it was published in 1836.* 

In both theories, Cauchy imposes the condition that the 
section of two of the sheets of the wave-surface made by any 
one of the coordinate planes is to be formed of a circle and an 
ellipse, as in Fresnel's theory ; this yields the three conditions 

3c = f(b + c +/) ; 3ca = g(c + a + g) ; Sab = h(a + b + Ji). 

Thus in the first theory we have these together with the 

equations 

= 0, H=Q, 1=0, 

which express the condition that the undisturbed state of the 
aether is unstressed ; and the aethereal vibrations are executed 
parallel to the plane of polarization. In the second theory we 
have the three first equations, together with 
f-Q-h-I-g-H; 

and the plane of polarization is interpreted to be the plane at 
right angles to the direction of vibration of the aether. 

Either of Cauchy's theories accounts tolerably well for the 
phenomena of crystal-optics; but the wave-surface (or rather 
the two sheets of it which correspond to nearly transverse 
waves) is not exactly Fresnel's. In both theories the existence 
of a third wave, formed of nearly longitudinal vibrations, is a 
formidable difficulty. Cauchy himself anticipated that the 
existence of these vibrations would ultimately be demonstrated 
by experiment, and in one placef conjectured that they might 
be of a calorific nature. A further objection to Cauchy's 
theories is that the relations between the constants do not 
appear to admit of any simple physical interpretation, being 
evidently assumed for the sole purpose of forcing the formulae 
into some degree of conformity with the results of experiment. 
And further difficulties will appear when we proceed subse- 
quently to compare the properties which are assigned to the 
aether in crystal- op tics with those which must be postulated in 
order to account for reflexion and refraction. 

* Comptes Rendus, ii (1836), p. 341 : Mem. de 1'Acad. xviii (1839), p. 153. 
f Mem. de 1'Acad. xviii, p. 161. 



The Aether as an Elastic Solid. 145 

To the latter problem Cauchy soon addressed himself, his 
investigations being in fact published* in the same year (1830) 
as the first of his theories of crystal-optics. 

At the outset of any work on refraction, it is necessary to 
assign a cause for the existence of refractive indices, i.e. for the 
variation in the velocity of light from one body to another. 
Huygens, as we have seen, suggested that transparent bodies 
consist of hard particles which interact with the aethereal matter, 
modifying its elasticity- Cauchy in his earlier papersf followed 
this lead more or less closely, assuming that the density p of the 
aether is the same in all media, but that its rigidity n varies 
from one medium to another. 

Let the axis of x be taken at right angles to the surface of 
separation of the media, and the axis of z parallel to the inter- 
section of this interface with the incident wave-front; and 
suppose, first, that the incident vibration is executed at right 
angles to the plane of incidence, so that it may be represented 
.by 

e~ = /( - x cos i -y sin i + rL t \ 

where i denotes the angle of incidence ; the reflected wave may 
be represented by 



e z = FX cos i - y sin i + t 
V \/ 

and the refracted wave by 

e z = fi I x cos r y sin r + KLt\ 

where r denotes the angle of refraction, and n' the rigidity of 
the second medium. 

To obtain the conditions satisfied at the reflecting surface, 
Cauchy assumed (without assigning reasons) that the x- and 
^/-components of the stress across the #y-plane are equal in 

* Bull, des Sciences Math. xiv. (1830), p. 6. 

t As will appear, his views on this subject subsequently changed. 

L 



146 The Aether as an Elastic Solid. 

the media on either side the interface. This implies in the 
present case that the quantities 

tie* de z 

n and n 

dx ty 

are to be continuous across the interface : so we have 

n cos i'. (/' - 1") = n' cos r . /', ; n sin i.(f' + F) = n' sin r . f\. 

Eliminating /' we have 

F' _ sin (r - i) 
f sin (r + i) 

Now this is Fresnel's sine-law for the ratio of the intensity 
of the reflected ray to that of the incident ray ; and it is known 
that the light to which it applies is that which is polarized 
parallel to the plane of incidence. Thus Cauchy was driven 
to the conclusion that, in order to satisfy the known facts 
of reflexion and refraction, the vibrations of the aether must be 
supposed executed at right angles to the plane of polarization 
of the light. 

The case of a vibration performed in the plane of incidence 
he discussed in the same way. It was found that Fresnel's 
tangent-law could be obtained by assuming that e x and the 
normal pressure across the interface have equal values in the 
two contiguous media. 

The theory thus advanced was encumbered with many diffi- 
culties. In the first place, the identification of the plane of 
polarization with the plane at right angles to the direction of 
vibration was contrary to the only theory of crystal-optics which 
Cauchy had as yet published. In the second place, no reasons 
were given for the choice of the conditions at the interface. 
Cauchy's motive in selecting these particular conditions was 
evidently to secure the fulfilment of Fresnel's sine-law and 
tangent-law; but the results are inconsistent with the true 
boundary-conditions, which were given later by Green. 

It is probable that the results of the theory of reflexion had 
much to do with the decision, which Cauchy now made,* to 

*Comptes Rendus, ii. (1836), p. 341. 



The Aether as an Elastic Solid. 147 

reject the first theory of crystal-optics in favour of the second. 
After 1836 he consistently adhered to the view that the vibra- 
tions of the aether are performed at right angles to the plane of 
polarization. In that year he made another attempt to frame a 
satisfactory theory of reflexion,* based on the assumption just 
mentioned, and on the following boundary-conditions: At 
the interface between two media curl e is to be continuous, and 
(taking the axis of x normal to the interface) de x /dx is also to 
be continuous. 

Again we find no very satisfactory reasons assigned for the 
choice of the boundary- conditions ; and_as the continuity of e 
itself across the interface is not included amongst the conditions 
cHosen, they are obviously open to criticism ; but they lead to 
Fresnel's sine- and tangent-equations, which correctly express 
the actual behaviour of light. f Cauchy remarks that in order to 
justify them it is necessary to abandon the assumption of his 
earlier theory, that the density of the aether is the same in all 
material bodies. 

It may be remarked that neither in this nor in Cauchy's 
earlier theory of reflexion is any trouble caused by the appear- 
ance of longitudinal waves when a transverse wave is reflected, 
for the simple reason that he assumes the boundary-conditions to 
be only four in number ; and these can all be satisfied without 
the necessity for introducing any but transverse vibrations. 

These features bring out the weakness of Cauchy's method of 
attacking the problem. His object was to derive the properties 
of light from a theory of the vibrations of elastic solids. At the 
outset he had already in his possession the differential equations 
of motion of the solid, which were to be his starting-point, and 
the equations of Fresnel, which were to be his goal. It only 

* Comptes Rendus, ii. (1836), p. 341 : " Meraoire sur la dispersion delalumiere " 
(Nouveaux exercices de Math., 1836), p. 203. 

t These boundary -conditions of Cauchy's are, as a matter of fact, satisfied by 
the electric force in the electro-magnetic theory of light. The continuity of 
<;url e is equivalent to the continuity of the magnetic vector across the interface, 
and the continuity of (tex/dx leads to the same equation as the continuity of 
the component of electric force in the direction of the intersection of the 
interface with the plane of incidence. 

L 2 



] 48 The Aether as an Elastic Solid. 

remained to supply the boundary-conditions at an interface, 
which are required in the discussion of reflexion, and the 
relations between the elastic constants of the solid, which are 
required in the optics of crystals. Cauchy seems to have con- 
sidered the question from the purely analytical point of view. 
Given certain differential equations, what supplementary con- 
ditions must be adjoined to them in order to produce a given 
analytical result ? The problem when stated in this form 
admits of more than one solution ; and hence it is not surprising 
that within the space of ten years the great French mathe- 
matician produced two distinct theories of crystal-optics and 
three distinct theories of reflexion,* almost all yielding correct 
or nearly correct final formulae, and yet mostly irreconcilable 
with each other, and involving incorrect boundary-conditions 
and improbable relations between elastic constants. 

Cauchy's theories, then, resemble Fresnel's in postulating 
types of elastic solid which do not exist, and for whose 
assumed properties no dynamical justification is offered. The 
same objection applies, though in a less degree, to the original 
form of a theory of reflexion and refraction which was, 
discovered about this timef almost simultaneously by James 
MacCullagh (6. 1809, d. 1847), of Trinity College, Dublin, 
and Franz Neumann (b. 1798, d. 1895), of Konigsberg. To 
these authors is due the merit of having extended the laws 
of reflexion to crystalline media; but the principles of the 
theory were originally derived in connexion with the simpler 
ease of isotropic media, to which our attention will for the 
present be confined. 

* One yet remains to be mentioned. 

f The outlines of the theory were published by MacCullagh in Brit. Assoc. Rep. 
1835 ; and his results were given in Phil. Mag. x (Jan., 1837), and in Proc. 
Royal Irish Acad. xviii. (Jan., 1837). Neumann's memoir was presented to the 
Berlin Academy towards the end of 1835, and published in 1837 in Abh. Berl. 
Ak. aus dem Jahre 1835, Math. Klasse, p. 1. So far as publication is concerned, 
the priority would seem to belong to MacCullagh; but there are reasons for 
believing that the priority of discovery really rests with Neumann, who had 
arrived at his equations a year before they were communicated to the Berlin 
Academy. 



The Aether as an Elastic Solid. 149 

MacCullagh and Neumann felt that the great objection 
to FresnePs theory of reflexion was its failure to provide for 
the continuity of the normal component of displacement at the 
interface between two media ; it is obvious that a discontinuity 
in this component could not exist in any true elastic-solid 
theory, since it would imply that the two media do not remain 
in contact. Accordingly, they made it a fundamental con- 
dition that all three components of the displacement must be 
continuous at the interface, and found that the sine-law and 
tangent-law can be reconciled with this condition only by 
supposing that the aether- vibrations are parallel to the plane of 
polarization : which supposition they accordingly adopted. In 
place of the remaining three true boundary-conditions, however, 
they used only a single equation, derived by assuming that 
transverse incident waves give rise only to transverse reflected 
and refracted waves, and that the conservation of energy holds 
for these i.e. that the masses of aether put in motion, 
multiplied by the squares of the amplitudes of vibration, are 
the same before and after incidence. This is, of course, the 
same device as had been used previously by Presnel; it 
must, however, be remarked that the principle is unsound as 
applied to an ordinary elastic solid; for in such a body the 
refracted and reflected energy would in part be carried away 
by longitudinal waves. 

In order to obtain the sine and tangent laws, MacCullagh 
and Neumann found it necessary to assume that the inertia 
of the luminiferous medium is everywhere the same, and 
that the differences in behaviour of this medium in different 
substances are due to differences in its elasticity. The two 
laws may then be deduced in much the same way as in the 
previous investigations of Fresnel and Cauchy. 

Although to insist on continuity of displacement at the 
interface was a decided advance, the theory of MacCullagh and 
Neumann scarcely showed as yet much superiority over the 
quasi-mechanical theories of their predecessors. Indeed, 
MacCullagh himself expressly disavowed any claim to regard 



150 The Aether as an Elastic Solid. 

his theory, in the form to which it had then been brought, as a 
final explanation of the properties of light. " If we are asked," 
he wrote, " what reasons can be assigned for the hypotheses on 
which the preceding theory is founded, we are far from being 
able to give a satisfactory answer. We are obliged to confess 
that, with the exception of the law of vis viva, the hypotheses 
are nothing more than fortunate conjectures. These conjectures 
are very probably right, since they have led to elegant laws 
which are fully borne out by experiments ; but this is all we 
can assert respecting them. We cannot attempt to deduce 
them from first principles ; because, in the theory of light, 
such principles are still to be sought for. It is certain, indeed, 
that light is produced by undulations, propagated, with 
transversal vibrations, through a highly elastic aether ; but the 
constitution of this aether, and the laws of its connexion (if it 
has any connexion) with the particles of bodies, are utterly 
unknown/' 

The needful reformation of the elastic-solid theory of 
reflexion was effected by Green, in a paper* read to the 
Cambridge Philosophical Society in December, 1837. Green, 
though inferior to Cauchy as an analyst, was his superior in 
physical insight ; instead of designing boundary-equations for 
the express purpose of yielding Fresnel's sine and tangent 
formulae, he set to work to determine the conditions which are 
actually satisfied at the interfaces of real elastic solids. 

These he obtained by means of general dynamical principles. 
In an isotropic medium which is strained, the potential energy 
per unit volume due to the state of stress is 

4 \te x de y 



+ (~- + ^} -4r-*~-4~~-4 



where e denotes the displacement, and k and n denote the two 

* Trans. Camb. Phil. Soc., 1838 ; Green's Math. Papers, p. 245. 



The Aether as an, Elastic Solid. 151 

elastic constants already introduced; by substituting this value 
of <f> in the general variational equation 



III' \w &t + 1* ** + TF*-| ****** = - 



* (where p denotes the density), the equation of motion may be 
deduced. 

But this method does more than merely furnish the equation 
of motion 



or, 



/ 4 \ 

pe = - ( k + - n ) grad div e - n curl curl e ; 
\ / 

pe = -lk + -n\ grad div e + nV z e, 



which had already been obtained by Cauchy ; for it also yields 
the boundary-conditions which must be satisfied at the interface 
between two elastic media in contact ; these are, as might be 
guessed by physical intuition, that the three components of the 
displacement* and the three components of stress across the 
interface are to be equal in the two media. If the axis of x 
be taken normal to the interface, the latter three quantities 
are 



, 2 \ de x fie z 3e x \ fde v de y 

--TI dive+ 27i , w (-Ji + ), and n (^ + - 

3 ) dx \to fa J \ty dx 



The correct boundary-conditions being thus obtained, it was 
a simple matter to discuss the reflexion and refraction of an 
incident wave by the procedure of Fresnel and Cauchy. The 
result found by Green was that if the vibration of the aethereal 
molecules is executed at right angles to the plane of incidence, 
the intensity of the reflected light obeys Fresnel's sine-law, pro- 
vided the rigidity n is assumed to be the same for all media, 
but the inertia p to vary from one medium to another. Since 
the sine-law is known to be true for light polarized in the plane 
of incidence, Green's conclusion confirmed the hypotheses of 

* These first three conditions are of course not dynamical but geometrical. 



152 The Aether as an Elastic Solid. 

Fresnel, that the vibrations are executed at right angles to the 
plane of polarization, and that the optical differences between 
media are due to the different densities of aether within them. 

It now remained for Green to discuss the case in which the 
incident light is polarized at right angles to the plane of inci- 
dence, so that the motion of the aethereal particles is parallel to 
the intersection of the plane of incidence with the front of the 
wave. In this case it is impossible to satisfy all the six 
boundary-conditions without assuming that longitudinal vibra- 
tions are generated by the act of reflexion. Taking the plane 
of incidence to be the plane of yz, and the interface to be the 
plane of xy, the incident wave may be represented by the 
equations 

6 = A + lz 



where, if i denote the angle of incidence, we have 

I = . / cos i t m = - / sin i. 
\n Mn 

There will be a transverse reflected wave, 



and a transverse refracted wave, 

y); e z = - C f(t + 1& + my), 

where, since the velocity of transverse waves in the second 
medium is v/W/oz, we can determine ^ from the equation 

^f^.&j 

n 

there will also be a longitudinal reflected wave, 

8 9 

e y = D -f(t -\z + my); e z = D -f(t - \z + my), 



The Aeiher as an Elastic Solid. 153 

where A is determined by the equation 



and a longitudinal refracted wave, 

7\ 7\ 

e y = JE - /(* + Aiz + my) ; e z = E - f(t 
where AI is determined by 



Substituting these values for the displacement in the boundary- 
conditions which have been already formulated, we obtain the 
equations which determine the intensities of the reflected and 
refracted waves ; in particular, it appears that the amplitude of 
the reflected transverse wave is given by the equation 

A- E _ ljj>i m? (pi - p 2 ) 2 
A + B Ip 2 I pz (\p z + A!/?I) 

Now if the elastic constants of the media are such that the 
velocities of propagation of the longitudinal waves are of the 
same order of magnitude as those of the transverse waves, the 
direction-cosines of the longitudinal reflected and refracted rays 
will in general have real values, and these rays will carry away 
some of the energy which is brought to the interface by the 
incident wavev-G^een avoided this difficulty by adopting Fresnel's 
suggestion that the resistance of the aether to compression may V\ 
be very large in comparison with the resistance to distortion, \\ 
as is actually the case with such substances as jelly and 
caoutchouc : in this case the longitudinal waves are degraded in 
much the same way as the transverse refracted ray is degraded 
when there is total reflexion, and so do not carry away energy. 
Making this supposition, so that k\ and & 2 are very large, the 
quantities A and A : have the values m </ - 1, and we have 

A- B li pi m ( PI - p 2 f 



A + B I 2 



154 The Aether as an Elastic Solid. 

Thus if BjA denote the modulus of /A, we have 



p\ 



if>l 

This expression represents the ratio of the intensity of the 
transverse reflected wave to that of the incident wave. It 
does not agree with Fresnel's tangent- formula : and both on this 
account and also because (as we shall see) this theory of reflexion 
does not harmonize well with the elastic-solid theory of crystal- 
optics, it must be concluded that the vibrations of a Greenian 
solid do not furnish an exact parallel to the vibrations which 
constitute light. 

The success of Green's investigation from the standpoint of 
dynamics, set off by its failure in the details last mentioned, 
stimulated MacCullagh to fresh exertions. At length he succeeded 
in placing his own theory, which had all along been free from 
reproach so far as agreement with optical experiments was 
concerned, on a sound dynamical basis ; thereby effecting that 
reconciliation of the theories of Light and Dynamics which had 
been the dream of every physicist since the days of Descartes. 

The central feature of MacCullagh's investigation,* which 
was presented to the Eoyal Irish Academy in 1839, is the intro- 
duction of a new type of elastic solid. He had, in fact, concluded 
from Green's results that it was impossible to explain optical 
phenomena satisfactorily by comparing the aether to an elastic 
solid of the ordinary type, which resists compression and 
distortion ; and he saw that the only hope of the situation was 
to devise a medium which should be as strictly conformable to- 
dynamical laws as Green's elastic solid, and yet should have 
its properties specially designed to fulfil the requirements of 
the theory of light. Such a medium he now described. 

If as before we denote by e the vector displacement of a 
point of the medium from its equilibrium position, it is well 

* Trans. Roy. Irish Acad. xxi. : MacCullagh's Coll. Works, p. 145. 



The Aether as an Elastic Solid. 155 

known that the vector curl e denotes twice the rotation of the 
part of the solid in the neighbourhood of the point (x, y, z) from 
its equilibrium orientation. In an ordinary elastic solid, the 
potential energy of strain depends only on the change of size 
and shape of the volume- elements ; on their compression and 
distortion, in fact. For MacCullagh's new medium, on the 
other hand, the potential energy depends only on the rotation 
of the volume-elements. 

Since the medium is not supposed to be in a state of stress 
in its undisturbed condition, the potential energy per unit 
volume must be a quadratic function of the derivates of e ; so 
that in an isotropic medium this quantity <f> must be formed 
from the only invariant which depends solely on the rotation 
and is quadratic in the derivates, that is from (curl e) 2 ; thus 
we may write 

to, 

*~ 

The equation of motion is now to be determined, as in the 
case of Green's aether, from the variational equation 



the result is 



p z = - fi curl curl e. 



It is evident from this equation that if div e is initially 
zero it will always be zero: we shall suppose this to be the 
case, so that no longitudinal waves exist at any time in the 
medium. One of the greatest difficulties which beset elastic- 
solid theories is thus completely removed. 

The equation of motion may now be written 



156 The Aether as an Elastic Solid. 

which shows that transverse waves are propagated with velocity 



From the variational equation we may also determine the 
boundary-conditions which must be satisfied at the interface 
between two media ; these are, that the three components of e 
are to be continuous across the interface, and that the two 
components of p curl e parallel to the interface are also to be 
continuous across it. One of these five conditions, namely, the 
continuity of the normal component of e, is really dependent on 
the other four ; for if we take the axis of x normal to the 
interface, the equation of motion gives 

p a"? = ~ 3 ^ ( curl e) * + l* curl e) " / ' . | 

and as the quantities p, (n curl e) 2 , and (/n curl e) y are continuous 
across the interface, the continuity of c> 2 e x /dt z follows. Thus the 
only independent boundary-conditions in MacCullagh's theory 
are the continuity of the tangential components of e and of 
fj curl e.* It is easily seen that these are equivalent to the 
boundary-conditions used in MacCullagh's earlier paper, namely, 
the equation of vis viva and the continuity of the three 
components of e : and thus the " rotationally elastic " aether of 
this memoir furnishes a dynamical foundation for the memoir 
of 1837. 

The extension to crystalline media is made by assuming the 
potential energy per unit volume to have, when referred to the 
principal axes, the form 



\dz dx J \c>x ty J 



where A, B, C denote three constants which determine the 
optical behaviour of the medium : it is readily seen that the 
wave-surface is Fresnel's, and that the plane of polarization 

* MacCullagh's equations may readily be interpreted in the electro -magnetic 
theory of light : e corresponds to the magnetic force, p curl e to the electric force, 
and curl e to the electric displacement. 






The Aether as an Elastic Solid. 157 

contains the displacement, and is at right angles to the 
rotation. 

MacCullagh's work was regarded with doubt by his own 
and the succeeding generation of mathematical physicists, and 
can scarcely be said to have been properly appreciated until 
FitzGerald drew attention to it forty years afterwards. But 
there can be no doubt that MacCullagh really solved the 
problem of devising a medium whose vibrations, calculated in 
accordance with the correct laws of dynamics, should have the 
same properties as the vibrations of light. 

The hesitation which was felt in accepting the rotationally 
elastic aether arose mainly from the want of any readily 
conceived example of a body endowed with such a property. 
This difficulty was removed in 1889 by Sir William Thomson 
(Lord Kelvin), who designed mechanical models possessed of 
rotational elasticity. Suppose, for example,* that a structure is 
formed of spheres, each sphere being in the centre of the 
tetrahedron formed by its four nearest neighbours. Let each 
sphere be joined to these four neighbours by rigid bars, which 
have spherical caps at their ends so as to slide freely on the 
spheres. Such a structure would, for small deformations, behave 
like an incompressible perfect fluid. Now attach to each bar a 
pair of gyroscopically-mounted flywheels, rotating with equal 
and opposite angular velocities, and having their axes in the line 
of the bar : a bar thus equipped will require a couple to hold 
it at rest in any position inclined to its original position, and 
the structure as a whole will possess that kind of quasi- 
elasticity which was first imagined by MacCullagh. 

This particular representation is not perfect, since a system 
of forces would be required to hold the model in equilibrium if 
it were irrotationally distorted. Lord Kelvin subsequently 
invented another structure free from this defect. t 

* Comptes Eendus, Sept. 16, 1889 : Kelvin's Math, and Phys. Papers, iii, 
p. 466. 

tProc. Roy. Soc. Edinb., Mar. 17, 1890: Kelvin's Math, and Phys. Papers, 
iii, p. 468. 



158 The Aether as an Elastic Solid. 

The work of Green proved a stimulus not only to 
MacCullagh but to Cauchy, who now (1839) published yet 
a third theory of reflexion.* This appears to have owed its 
origin to a remark of Green's, f that the longitudinal wave 
might be avoided in either of two ways namely, by supposing 
its velocity to be indefinitely great or indefinitely small. Green 
curtly dismissed the latter alternative and adopted the former, 
on the ground that the equilibrium of the medium would be 
unstable if its compressibility were negative (as it must be if 
the velocity of longitudinal waves is to vanish). Cauchy, without 
attempting to meet Green's objection, took up the study of a 
medium whose elastic constants are connected by the equation 

k + n = 0, 

so that the longitudinal vibrations have zero velocity; and showed 
that if the aethereal vibrations are supposed to be executed at 
right angles to the plane of polarization, and if the rigidity 
of the aether is assumed to be the same in all media, a ray 
which is reflected will obey the sine-law and tangent-law of 
Fresnel. The boundary-conditions which he adopted in order to 
obtain this result were the continuity of the displacement e and 
-of its derivate 3e/9#, where the axis of x is taken at right 
angles to the interface.* These are not the true boundary-con- 
ditions for general elastic solids ; but in the particular case now 
under discussion, where the rigidity is the same in the two 
media, they yield the same equations as the conditions correctly 
given by Green. 

The aether of Cauchy's third theory of reflexion is well 
worthy of some further study. It is generally known as the y 
.contractile or labile^ aether, the names being due to William 

* Comptes Rendus, ix, p. 676 (25 NOT., 1839), and p. 726 (2 Dec., 1839). 

t Green's Math. Papers, p. 246. 

J Comptes Eendus, x, p. 347 (March 2, 1840) : xxvii, p. 621 (1848) ; sxviii, p. 25 
(1849). Mem. de 1'Acad., xxii (1848), pp. 17, 29. 

Labile or neutral is a term used of such equilibrium as that of a rigid body oil 
-a perfectly smooth horizontal plane. 



The Aether as an Elastic Solid. 159 

Thomson (Lord Kelvin), who discussed it long afterwards.* It 
may be defined as an elastic medium of (negative) com- 
pressibility such as to make the velocity of the longitudinal 
wave zero : this implies that no work is required to be done 
in order to give the medium any small irrotational disturbance. 
An example is furnished by homogeneous foam free from air 
and held from collapse by adhesion to a containing vessel 

Cauchy, as we have seen, did not attempt to refute Green's 
objection that such a medium would be unstable ; but, as 
Thomson remarked, every possible infinitesimal motion of the 
medium is, in the elementary dynamics of the subject, proved 
to be resolvable into coexistent wave-motions. If, then, the 
velocity of propagation for each of the two kinds of wave-motion 
is real, the equilibrium must be stable, provided the medium 
either extends through boundless space or has a fixed containing 
vessel as its boundary. 

When the rigidity of the luminiferous medium is supposed 
to have the same value in all bodies, the conditions to be satisfied 
at an interface reduce to the continuity of the displacement e, 
of the tangential components of curl e, and of the scalar 
quantity (k + ^n) div e across the interface. 

Now we have seen that when a transverse wave is incident 
on an interface, it gives rise in general to reflected and refracted 
waves of both the transverse ajid the longitudinal species. In 
the case of the contractile aether, for which the velocity of 
propagation of the longitudinal waves is very small, the ordinary 
construction for refracted waves shows that the directions of 
propagation of the reflected and refracted longitudinal waves 
will be almost normal to the interface. The longitudinal 
waves will therefore contribute only to the component of 
displacement normal to the interface, not to the tangential 
components : in other words, the only tangential components of 
displacement at the interface are those due to the three trans- 
verse waves the incident, reflected, and refracted. Moreover, 
the longitudinal waves do not contribute at all to curl e ; and, 

* Phil. Mag. xxvi (1888), p. 414. 



160 The Aether as an Elastic Solid. 

therefore, in the contractile aether, the conditions that the 
tangential components of e and of n curl e shall be continuous 
across an interface are satisfied by the distortional part of the 
disturbance taken alone. The condition that the component 
of e normal to the interface is to be continuous is not satisfied 
by the distortional part of the disturbance taken alone, but is 
satisfied when the distortional and congressional parts are taken 
together. 

The energy carried away by the longitudinal waves is 
infinitesimal, as might be expected, since no work is required in 
order to generate an irrotational displacement. Hence, with 
this aether, the behaviour of the transverse waves at an 
interface may be specified without considering the irrotational 
part of the disturbance at all, by the conditions that the 
conservation of energy is to hold and that the tangential 
components of e and of n curl e are to be continuous. But if 
we identify these transverse waves with light, assuming that 
the displacement e is at right angles to the plane of polarization 
of the light, and assuming moreover that the rigidity n is the 
same in all media* (the differences between media depending on 
differences in the inertia p), we have exactly the assumptions 
of Fresnel's theory of light : whence it follows that transverse 
waves in the labile aether must obey in reflexion the sine-law 
and tangent-law of Fresnel. 

The great advantage of the labile aether is that it overcomes 
the difficulty about securing continuity of the normal com- 
ponent of displacement at an interface between two media : 
the light-waves taken alone do not satisfy this condition of 
continuity ; but the total disturbance consisting of light- waves 
and irrotational disturbance taken together does satisfy it ; 
and this is ensured without allowing the irrotational disturbance 
to carry off any of the energy. f 

* This condition is in any case necessary for stability, as was shown by 
R. T. Glazebrook : cf. Thomson, Phil. Mag. xxvi, p. 500. 

f The labile-aether theory of light may be compared with the electro-magnetic 
theory, by interpreting the displacement e as the electric force, and pe as the 
electric displacement. 



The Aether as an Elastic Solid. 161 

William Thomson (Lord Kelvin, b. 1824, d. 1908), who 
devoted much attention to the labile aether, was at one time 
led to doubt the validity of this explanation of light* ; for when 
investigating the radiation of energy from a vibrating rigid 
globe embedded in an infinite elastic-solid aether, he found that 
in some cases the irrotational waves would carry away a 
considerable part of the energy if the aether were of the labile 
type. This difficulty, however, was removed by the observationf 
that it is sufficient for the fulfilment of Fresnel's laws if the 
velocity of the irrotational waves in one of the two media is 
very small, without regard to the other medium. Following up 
this idea, Thomson assumed that in space void of ponderable 
matter the aether is practically incompressible by the forces 
concerned in light-waves, but that in the space occupied by 
liquids and solids it has a negatiye_CQmpres^biIiljy , so as to give 
zero velocity for longitudinal aether- waves in these bodies. 
This assumption was based on the conception that material 
atoms move through space without displacing the aether: a 
conception which, as Thomson remarked, contradicts the old 
scholastic axiom that two different portions of matter cannot 
simultaneously occupy the same space.J He supposed the 
aether to be attracted and repelled by the atoms, and thereby to 
be condensed or rarefied. 

The year 1839, which saw the publication of MacCullagh's 
dynamical theory of light and Cauchy's theory of the labile 
aether, was memorable also for the appearance of a memoir by 
Green on crystal-bptics.H This really contains two distinct 
theories, which respectively resemble Cauchy's First and Second 
Theories : in one of them, the stresses in the undisturbed state 

* Baltimore Lectures (edition 1904), p. 214. 

t Ibid. (ed. 1904), p. 411. 

^ Michell and Boscovich in the eighteenth century had taught the doctrine of 
the mutual penetration of matter, i.e. that two substances may be in the same 
place at the same time without excluding each other : cf. Priestley's History i., 
p. 392. 

6 Cf. Baltimore Lectures (ed. 1904), pp. 413-14, 463, and Appendices A and E. 

|j Cambridge Phil. Trans., 1839 ; Green's Math. Papers;?. 293. 

M 



162 The Aether as an Elastic Solid. 

of the aether are supposed to vanish, and the vibrations of the 
aether are supposed to be executed parallel to the plane of 
polarization of the light ; in the other theory, the initial stresses 
are not supposed to vanish, and the aether- vibrations are at 
right angles to the plane of polarization. The two investigations 
are generally known as Green's First and Second Theories of 
crystal-optics. 

The foundations of both theories are, however, the same. 
Green first of all determined the potential energy of a strained 
crystalline solid ; this in the most general case involves 27 
constants, or 21 if there is no initial stress.* If, however, as is 
here assumed, the medium possesses three planes of symmetry 
at right angles to each other, the number of constants reduces 
to. 12, or to 9 if there is no initial stress; if e denote the dis- 
placement, the potential energy per unit volume may be written 



fo \2 ("be \ 2 fr\f \ i \ (ffo \2 /f)p \2 /a/, N 

+ $} * () 1 + ** ft) + (I) + (I 



fty*. 

7 ty tz 9 

sf3e y 3g, 
* 4/1 ;? + s- 

2</ \dz ty 



* 

dx 

The usual variational equation 



= - [[f 



* For there are 21 terms in a homogeneous function of the second degree in six 
variables. 



The Aether as an Elastic Solid. 163 

then yields the differential equations of motion, namely : 



8 / de x de y fe,\ a / ae x a^ , a^\ 

+ a + h ^ + g + a + A + ], 
acVSaj ty y dzj dx\ fa ty y fa) 9 

and two similar equations. 

These differ from Cauchy's fundamental equations in having 
greater generality: for Cauchy's medium was supposed to be 
built up of point-centres of force attracting each other according 
to some function of the distance ; and, as we have seen, there 
are limitations in this method of construction, which render it 
incompetent to represent the most general type of elastic solid. 
Cauchy's equations for crystalline media are, in fact, exactly 
analogous to the equations originally found by Navier for 
isotropic media, which contain only one elastic constant instead 
of two. 

The number of constants in the above equations still exceeds 
the three which are required to specify the properties of a 
biaxal crystal : and Green now proceeds to consider how the 
number may be reduced. The condition which he imposes for 
this purpose is that for two of the three waves whose front is 
parallel to a given plane, the vibration of the aethereal molecules 
shall be accurately in the plane of the wave : in other words, 
that two of the three waves shall be purely distortional, the 
remaining one being consequently a normal vibration. This 
condition gives five relations,* which may be written : 

a. b = c = JJK; 

/'-j-2/ / = M -2<7; tf- M -2fc; 
where /z denotes a new constant, f 

* As Green showed, the hypothesis of transversality really involves the existence 
of planes of symmetry, so that it alone is capable of giving 14 relations between the 
21 constants : and 3 of the remaining 7 constants may be removed by change of 
axes, leaving only four. 

t It was afterwards shown by Barre de Saint- Venant (b. 1797, d. 1886), 
Journal de Math., vii (1863), p. 399, that if the initial stresses be supposed to 
vanish, the conditions which must be satisfied among the remaining nine constants 

M 2 



164 The Aether as an Elastic Solid. 

Thus the potential energy per unit volume may be written 

. n^* ^ rr de y^. T de * 

d> = 6r + L + -I ^~ 

ox oy oz 



i T ] Wx \ ^ y\ -L. oe * 
I 1 Ur + Ur- + Ur 



At this point Green's two theories of crystal-optics diverge 
from each other. According to the first theory, the initial 
stresses G-, H, I are zero, so that 






", *> >/>#>*>/' /, *' in order that the wave-surface may be Fresnel's, are the 
following : 

(34 _/) ( 3c -/)=(/ + /')* 



((3a- 



(Za - h) (3 - h) = (h + A') 



These reduce to Green's relations when the additional equation b = c is assumed. 

Saint-Venant disputed the validity of Green's relations, asserting that they?are 
compatible only with isotropy. On this controversy cf. E. T. Glazebrook, Brit. 
Assoc. Report, 1885, p. 171, and Karl Pearson in Todhunter and Pearson's History 
of Elasticity, ii, 147. 



The Aether as an Elastic Solid. 165 

This expression contains the correct number of constants, 
namely, four: three of them represent the optical constants 
of a biaxal crystal, and one (namely, ju) represents the square of 
the velocity of propagation of longitudinal waves. It is found 
that the two sheets of the wave-surface which correspond to the 
two distortional waves form a Fresnel's wave-surface, the third 
sheet, which corresponds to the longitudinal wave, being an 
ellipsoid. The directions of polarization and the wave- velocities 
of the distortional waves are identical with those assigned by 
Fresnel, provided it is assumed that the direction of vibration 
of the aether- particles is parallel to the plane of polarization ; 
but this last assumption is of course inconsistent with Green's 
theory of reflexion and refraction. 

In his Second Theory, Green, like Cauchy, used the condition 
that for the waves whose fronts are parallel to the coordinate 
planes, the wave- velocity depends only on the plane of polariza- 
tion, and not on the direction of propagation. He thus obtained 
the equations already found by Cauchy 

O-f-H-g-I-h. 

The wave-surface in this case also is Fresnel's, provided it 
is assumed that the vibrations of the aether are executed at 
right angles to the plane of polarization. 

The principle which underlies the Second Theories of Green 
and Cauchy is that the aether in a crystal resembles an elastic 
solid which is unequally pressed or pulled in different directions 
by the unmoved ponderable matter. This idea appealed strongly 
to W. Thomson (Kelvin), who long afterwards developed it 
further,* arriving at the following interesting result : Let an 
incompressible solid, isotropic when unstrained, be such that its 
potential energy per unit volume is 



P 7 
where q denotes its modulus of rigidity when unstrained, and 

* Proc. R. S. Edin. xv (1887), p. 21 : Phil. Mag. xxv (1888) p. 116 : Baltimore 
Lectures (ed. 1904), pp. 228-259. 



1 66 The Aether as an Elastic Solid. 

*> j3*> 7*> denote the proportions in which lines parallel to the 
axes of strain are altered ; then if the solid be initially strained 
in a way defined by given values of a, (3, y, by forces applied to 
its surface, and if waves of distortion be superposed on this 
initial strain, the transmission of these waves will follow exactly 
the laws of Fresnel's theory of crystal- optics, the wave-surface 

being 

* 




q q 

There is some difficulty in picturing the manner in which 
the molecules of ponderable matter act upon the aether so as to 
produce the initial strain required by this theory. Lord 
Kelvin utilized* the suggestion to which we have already 
referred, namely, that the aether may pervade the atoms of 
matter so as to occupy space jointly with them, and that its 
interaction with them may consist in attractions and repulsions 
exercised throughout the regions interior to the atoms. These 
forces may be supposed to be so large in comparison with those 
called into play in free aether that the resistance to compres- 
sion may be overcome, and the aether may be (say) condensed 
in the central region of an isolated atom, and rarefied in its 
outer parts. A crystal may be supposed to consist of a group 
of spherical atoms in which neighbouring spheres overlap each 
other ; in the central regions of the spheres the aether will be 
condensed, and within the lens-shaped regions of overlapping 
it will be still more rarefied than in the outer parts of a solitary 
atom, while in the interstices between the atoms its density 
will be unaffected. In consequence of these rarefactions and 
condensations, the reaction of the aether on the atoms tends 
to draw inwards the outermost atoms of the group, which, 
however, will be maintained in position by repulsions between 
the atoms themselves; and thus we can account for the pull 
which, according to the present hypothesis, is exerted on the 
aether by the ponderable molecules of crystals. 

* Baltimore Lectures (ed. 1904), p. 253. 



The Aether as an Elastic Solid. 167 

Analysis similar to that of Cauchy's and Green's Second 
Theory of crystal-optics may be applied to explain the doubly 
refracting property which is possessed by strained glass ; but 
in this case the formulae derived are found to conflict with 
the results of experiment. The discordance led Kelvin to 
doubt the truth of the whole theory. "After earnest and 
hopeful consideration of the stress theory of double refraction 
during fourteen years," he said,* " I am unable to see how it 
can give the true explanation either of the double refraction of 
natural crystals, or of double refraction induced in isotropic 
solids by the application of unequal pressures in different 
directions." 

It is impossible to avoid noticing throughout all Kelvin's 
work evidences of the deep impression which was made 
upon him by the writings of Green. The same may be said 
of Kelvin's friend and contemporary Stokes; and, indeed, it 
is no exaggeration to describe Green as the real founder of 
that " Cambridge school " of natural philosophers, of which 
Kelvin, Stokes, Lord Eayleigh, and Clerk Maxwell were the 
most illustrious members in the latter half of the nineteenth 
century, and which is now led by Sir Joseph Thomson and 
Sir Joseph Larrnor. In order to understand the peculiar 
position occupied by Green, it is necessary to recall some- 
thing of the history of mathematical studies at Cambridge. 

The century which elapsed between the death of Newton 
and the scientific activity of Green was the darkest in the 
history of the University. It is true that Cavendish and 
Young were educated at Cambridge; but they, after taking 
undergraduate courses, removed to London. In the entire 
period the only natural philosopher of distinction who lived 
and taught at Cambridge was Michell ; and for some reason 
which at this distance of time it is difficult to understand 
fully, Michell's researches seem to have attracted little or no 
attention among his collegiate contemporaries and successors, 

* Baltimore Lectures (ed. 1904), p. 258. 



168 The Aether as an Elastic Solid. 

who silently acquiesced when his discoveries were attributed to 
others, and allowed his name to perish entirely from Cambridge 
tradition. 

A few years before Green published his first paper, a 
notable revival of mathematical learning swept over the 
University ; the fluxional symbolism, which since the time of 
Newton had isolated Cambridge from the continental schools, 
was abandoned in favour of the differential notation, and the 
works of the great French analysts were introduced and 
eagerly read. Green undoubtedly received his own early 
inspiration from this source ; but in clearness of physical 
insight and conciseness of exposition he far excelled his 
masters ; and the slight volume of his collected papers has 
to this day a charm which is wanting to the voluminous 
writings of Cauchy and Poisson. It was natural that such an 
example should powerfully influence the youthful intellects of 
Stokes who was an undergraduate when Green read his memoir 
on double refraction to the Cambridge Philosophical Society 
and of William Thomson (Kelvin), who came into residence two 
years afterwards.* 

In spite of the advances which were made in the great 
memoirs of the year 1839, the fundamental question as to 
whether the aether-particles vibrate parallel or at right angles 
to the plane of polarization was still unanswered. More light 
was thrown on this problem ten years later by Stokes's inves- 
tigation of Diffraction.f Stokes showed that on almost any 
conceivable hypothesis regarding the aether, a disturbance in 
which the vibrations are executed at right angles to the plane 
of diffraction must be transmitted round the edge of an opaque 
body with less diminution of intensity than a disturbance whose 
vibrations are executed parallel to that plane. It follows that 
when light, of which the vibrations are oblique to the plane of 

*It was in the year Thomson took his degree (1845) that he bought, and read 
with delight, the electrical memoir which Green had published at Nottingham in 
1828. 

f Trans. Camb. Phil. Soc., ix (1849), p. 1. Stokes's Math, and Phys. Papers, 
ii, p. 243. 



The Aether as an Elastic Solid. 169 

diffraction, is so transmitted, the plane of vibration will be more 
nearly at right angles to the plane of diffraction in the diffracted 
than in the incident light. Stokes himself performed experi- 
ments to test the matter, using a grating in order to obtain 
strong light diffracted at a large angle, and found that when 
the plane of polarization of the incident light was oblique to the 
plane of diffraction, the plane of polarization of the diffracted 
light was more nearly parallel to the plane of diffraction. This 
result, which was afterwards confirmed by L. Lorenz,* appeared 
to confirm decisively the hypothesis of Fresnel, that the vibra- 
tions of the aethereal particles are executed at right angles to 
the plane of polarization. 

Three years afterwards Stokes indicatedf a second line of 
proof leading to the same conclusion. It had long been known 
that the blue light of the sky, which is due to the scattering of 
the sun's direct rays by small particles or molecules in the 
-atmosphere, is partly polarized. The polarization is most 
marked when the light comes from a part of the sky distant 90 
from the sun, in which case it must have been scattered in a 
direction perpendicular to that of the direct sunlight incident 
on the small particles ; and the polarization is in the plane 
through the sun. 

If, then, the axis of y be taken parallel to the light incident 
on a small particle at the origin, and the scattered light be 
observed along the axis of x, this scattered light is found to be 
polarized in the plane xy. Considering the matter from the 
dynamical point of view, we may suppose the material particle 
to possess so much inertia (compared to the aether) that it is 
practically at rest. Its motion relative to the aether, which is 
the cause of the disturbance it creates in the aether, will there- 
fore be in the same line as the incident aethereal vibration, 
but in the opposite direction. The disturbance must be 
transversal, and must therefore be zero in a polar direction and 

* Ann. d. Phj-s. exi (1860), p. 315. Phil. Mag. xxi (1861), p. 321. 
t Phil. Trans., 1852, p. 463. Stokes's Math, and Phys. Papers, iii, p. 267. 
f. the foot-note added on p. 361 oi the Math, and Phys. Paper*. 



170 The Aether as an Elastic Solia. 

a maximum in an equatorial direction, its amplitude being, in 
fact, proportional to the sine of the polar distance. The polar 
line must, by considerations of symmetry, be the line of the 
incident vibration. Thus we see that none of the light scattered 
in the ^-direction can come from that constituent of the incident 
light which vibrates parallel to the o>axis ; so the light observed 
in this direction must consist of vibrations parallel to the 2-axis. 
But we have seen that the plane of polarization of the scattered 
light is the plane of xy ; and therefore the vibration is at right 
angles to the plane of polarization.* 

The phenomena of diffraction and of polarization by scatter- 
ing thus agreed in confirming the result arrived at in Fresnel's 
and Green's theory of reflexion. The chief difficulty in accepting 
it arose in connexion with the optics of crystals. As we have 
seen, Green and Cauchy were unable to reconcile the hypothesis 
of aethereal vibrations at right angles to the plane of polariza- 
tion with the correct formulae of crystal-optics, at any rate so 
long as the aether within crystals was supposed to be free from 
initial stress. The underlying reason for this can be readily 
seen. In a crystal, where the elasticity is different in different 
directions, the resistance to distortion depends solely on the 
orientation of the plane of distortion, which in the case of light 
is the plane through the directions of propagation and vibration. 
Now it is known that for light propagated parallel to one of the 
axes of elasticity of a crystal, the velocity of propagation 
depends only on the plane of polarization of the light, being the 
same whichever of the two axes lying in that plane is the 
direction of propagation. Comparing these results, we see that 
the plane of polarization must be the plane of distortion, and 
therefore the vibrations of the aether-particles must be executed 
parallel to the plane of polarization.f 

* The theory of polarization by small particles was afterwards investigated by 
Lord Rayleigh, Phil. Mag. xli(187l). 

fin Fresnel's theory of crystal-optics, in which the aether-vibrations are at 
right angles to the plane of polarization, the velocity of propagation depends only 
on the direction of vibration, not on the plane through this and the direction of 
transmission. 



The Aether as an Elastic Solid. 171 

A way of escape from this conclusion suggested itself to 
Stokes,* and later to Eankinet and Lord Kayleigh.J; What if the 
aether in a crystal, instead of having its elasticity different in 
different directions, were to have its rigidity invariable and its 
inertia different in different directions ? This would bring the 
theory of crystal- op tics into complete agreement with Fresnel's 
and Green's theory of reflexion, in which the optical differences 
between media are attributed to differences of inertia of the 
aether contained within them. The only difficulty lies in 
conceiving how aelotropy of inertia can exist; and all three 
writers overcame this obstacle by pointing out that a solid 
which is immersed in a fluid may have its effective inertia 
different in different directions. For instance, a coin immersed 
in water moves much more readily in its own plane than in the 
direction at right angles to this. 

Suppose then that twice the kinetic energy per unit volume 
of the aether within a crystal is represented by the expression 



and that the potential energy per unit volume has the same 
value as in space void of ordinary matter. The aether is 
assumed to be incompressible, so that div e is zero : the potential 
energy per unit volume is therefore 



_ __ 

dz dx d 

where n denotes as usual the rigidity. 

* Stokes, in a letter to Lord Rayleigh, inserted in his Memoir and Scientific 
Correspondence, ii, p. 99, explains that the idea presented itself to him while he 
was writing the paper on Fluid Motion which appeared in Trans. Camh. Phil. 
Soc., via (1843), p. 105. He suggested the wave-surface to which this theory 
leads in Brit. Assoc. Rep., 1862, p. 269. 

t Phil. Mag. (4), i (1851), p. 441. J Phil. Mag. (4), xli (1871), p. 519. 



1 72 The Aether as an Elastic Solid. 

The variational equation of motion is 

pi ^r/ de x + p z % de y + p z 2 $e z [ dx dy dz 



where p denotes an undetermined function of (x, y, z) : the term 
in p being introduced on account of the kinematical constraint 
expressed by the equation 

div e = 0. 

The equations of motion which result from this variational 
equation are 

< h w = - +nV * e " '""' 

and two similar equations. It is evident that p resembles a 
hydrostatic pressure. 

Substituting in these equations the analytical expression 
for a plane wave, we readily find that the velocity F of the 
wave is connected with the direction-cosines (X, ^t, z/) of its 
normal by the equation 

A 2 u* v z 

n- PI V* r n-ptV" + n- p z V* = ' 

When this is compared with Fresnel's relation between the 
velocity and direction of a wave, it is seen that the new formula 
differs from his only in having the reciprocal of the velocity in 
place of the velocity. About 1867 Stokes carried out a series 
of experiments in order to determine which of the two theories 
was most nearly conformable to the facts : he found the con- 
struction of Huygens and Fresnel to be decidedly the more 
correct, the difference between the results of it and the rival 
construction being about 100 times the probable error of 
observation.* 

* Proc. R. S., June, 1872. After these experiments Stokes gave it as his opinion 
(Phil. Mag. xli (1871), p. 521) that the true theory of crystal-optics was yet to be 
found. On the accuracy of Fresnel's construction cf. Glazebrook, Phil. Trans, 
clxxi (1879) p. 421, and Hastings, Am. Journ. Sci. (3) xxxv (1887) p. 60. 



The Aether as an Elastic Solid. 173 

The hypothesis that in crystals the inertia depends on 
direction seemed therefore to be discredited when the theory 
based on it was compared with the results of observation. But 
when, in 1888, W. Thomson (Lord Kelvin) revived Cauchy's 
theory of the labile aether, the question naturally arose as to 
whether that theory could be extended so as to account for the 
optical properties of crystals : and it was shown by E. T. 
Glazebrook* that the correct formulae of crystal-optics ar& 
obtained when the Cauchy-Thomson hypothesis of zero velocity 
for the longitudinal wave is combined with the Stokes-Kankine- 
Rayleigh hypothesis of aelotropic inertia. 

For on reference to the formulae which have been already 
given, it is obvious that the equation of motion of an aether 
having these properties must be 

(pie*, p z e y , p 3 O = -n curl curl e, 

where e denotes the displacement, n the rigidity, and (p lt p 2 , /o 3 ) 
the inertia : and this equation leads by the usual analysis ta 
Fresnel's wave-surface. The displacement e of the aethereal 
particles is not, however, accurately in the wave-front, as in 
Fresnel's theory, but is at right angles to the direction of the 
ray, in the plane passing through the ray and the wave- 
normal, f 

Having now traced the progress of the elastic-solid theory 
so far as it is concerned with the propagation of light in 
ordinary isotropic media and in crystals, we must consider the 
attempts which were made about this time to account for the 
optical properties of a more peculiar class of substances. 

It was found by Arago in 181 IJ that the state of 
polarization of a beam of light is altered when the beam is 
passed through a plate of quartz along the optic axis. The 



* Phil. Mag. xxvi (1888), p. 521 ; xxviii (1889), p. 110. 

t This theory of crystal-optics may be assimilated to the electro-magnetic theory 
by interpreting the elastic displacement e as electric force, and the vector 
(pifx, p^y, ptfz) as electric displacement. 

+ Mem. de 1'Institut, 1811, Part I, p. 115, sqq. 



174 The Aether as an Elastic Solid. 

phenomenon was studied shortly afterwards by Biot,* who 
showed that the alteration consists in a rotation of the plane of 
polarization about the direction of propagation : the angle of 
rotation is proportional to the thickness of the plate and 
inversely proportional to the square of the wave-length. 

In some specimens of quartz the rotation is from left to 
right, in others from right to left. This distinction was shown 
by Sir John Herschelf (b. 1792, d. 1871) in 1820 to be 
associated with differences in the crystalline form of the 
specimens, the two types bearing the same relation to each 
other as a right-handed and left-handed helix respectively. 
FresnelJ and W. Thomsong proposed the term helical to 
denote the property of rotating the plane of polarization, 
exhibited by such bodies as quartz : the less appropriate term 
natural rotatory polarization is, however, generally used.|| 

Biot showed that many liquid organic bodies, e.g. turpentine 
and sugar solutions, possess the natural rotatory property : we 
might be led to infer the presence of a helical structure in 
the molecules of such substances ; and this inference is sup- 
ported by the study of their chemical constitution; for they 
are invariably of the "mirror-image" or "enantiomorphous" 
type, in which one of the atoms (generally carbon) is asym- 
metrically linked to other atoms. 

The next advance in the subject was due to Fresnel,1[ who 
showed that in naturally active bodies the velocity of propa- 
gation of circularly polarized light is different according as the 
polarization is right-handed or left-handed. From this 
property the rotation of the plane of polarization of a plane- 
polarized ray may be immediately deduced ; for the plane- 
polarized ray may be resolved into two rays circularly polarized 
in opposite senses, and these advance in phase by different 

* Mem. de 1'Institut, 1812, Part i, p. 218, sqq. ; Annales de Chim., ix (1818), 
p. 372; x (1819), p. 63. tCamb. Phil. Soc. Trans, i, p. 43. 

J Mem. de 1'Inst. vii, p. 73. Baltimore Lectures (ed. 1904), p. 31. 

|| The term rotatory may be applied with propriety to the property discovered 
by Faraday, which will be discussed later. 

H Annales de Chim. xxviii (1825), p. 147. 



The Aether as an Elastic Solid. 1 75 

amounts in passing through a given thickness of the substance-: 
at any stage they may be recompoundecl into a pkne-polarized 
ray, the azimuth of whose plane of polarization varies with the 
length of path traversed. 

It is readily seen from this that a ray of light incident on 
a crystal of quartz will in general bifurcate into two refracted 
rays, each of which will be elliptically polarized, i.e. will be 
capable of resolution into two plane-polarized components 
which differ in phase by a definite amount. The directions of 
these refracted rays may be determined by Huygens' con- 
struction, provided the wave-surface is supposed to consist of a 
sphere and spheroid which do not touch. 

The first attempt to frame a theory of naturally active 
bodies was made by MacCullagh in 1836.* Suppose a plane 
wave of light to be propagated within a crystal of quartz. Let 
(#, ?/, z) denote the coordinates of a vibrating molecule, when 
the axis of x is taken at right angles to the plane of the wave, 
and the axis of z at right angles to the axis of the 
crystal. Using Fand Zto denote the displacements parallel to 
the axes of y and z respectively at any time t, MacCullagh 
assumed that the differential equations which determine Y and 



__ _ 
w "" ** w ^'w 

where /* denotes a constant on which the natural rotatory 
property of the crystal depends. In order to avoid compli- 
cations arising from the ordinary crystalline properties of quartz, 
we shall suppose that the light is propagated parallel to the 
optic axis, so that we can take c, equal to c 2 . 

Assuming first that the beam is circularly polarized, let it 
be represented by 

(y f\ 

Y = A sin (Ix - ), Z = A cos (Ix - t), 

* Trans. Royal Irish Acad., xvii. ; MacCullagh's Coll. Works, p. 63. 



176 The Aether as an Elastic Solid. 

*he ambiguous sign being determined according as the circular 
polarization. ^ ri^ht-handed or left-handed. 

Substituting in ^ e above differential equations, we have 



or 



Since I// denotes the velocity of propagation, it is evident that 
the reciprocals of the velocities of propagation of a right-handed 
and left-handed beam differ by the quantity 



from which it is easily shown that the angle through which the 
plane of polarization of a plane-polarized beam rotates in unit 
length of path is 



rV 

If we neglect the variation of Ci with the period of the light, 
this expression satisfies Biot's law that the angle of rotation 
in unit length of path is proportional to the inverse square of 
the wave-length. 

MacCullagh's investigation can be scarcely called a theory, 
for it amounts only to a reduction of the phenomena to 
empirical, though mathematical, laws ; but it was on this 
foundation that later workers built the theory which is now 
accepted.* 

* The later developments of this theory will be discussed in a subsequent 
chapter ; hut mention may here he made of an attempt which was made in 1856 by 
Carl Neumann, then a very young man, to provide a rational basis for MacCullagh's 
equations. Neumann showed that the equations may be derived from the 
hypothesis that the relative displacement of one aethereal particle with respect to 
another acts on the latter according to the same law as an element of an electric 
current acts on a magnetic pole. Cf. the preface to C. Neumann's Die 
Drehung der Polarisationsebene des Lichtes, Halle, 1863. 



The Aether as an Elastic Solid. 177 

The great investigators who developed the theory of light 
after the death of Fresnel devoted considerable attention to 
the optical properties of metals. Their researches in this 
direction must now be reviewed. 

The most striking properties of metals are the power of 
brilliantly reflecting light at all angles of incidence, which is 
so well shown by the mirrors of reflecting telescopes, and the 
opacity, which causes a train of waves to be extinguished before 
it has proceeded many wave-lengths into a metallic medium. 
That these two attributes are connected appears probable 
from the fact that certain non- metallic bodies e.g., aniline 
dyes which strongly absorb the rays in certain parts of the 
spectrum, reflect those rays with almost metallic brilliance. 
A third quality in which metals differ from transparent bodies, 
and which, as we shall see, is again closely related to the other 
two, is in regard to the polarization of the light reflected from 
them. This was first noticed by Malus ; and in 1830 Sir David 
Brewster* showed that plane-polarized light incident on a 
metallic surface remains polarized in the same plane after 
reflexion if its polarization is either parallel or perpendicular 
to the plane of reflexion, but that in other cases the reflected 
light is polarized elliptically. 

It was this discovery of Brews ter's which suggested to the 
mathematicians a theory of metallic reflexion. For, as we have 
seen, elliptic polarization is obtained when plane-polarized 
light is totally reflected at the surface of a transparent body ; 
and this analogy between the effects of total reflexion and 
metallic reflexion led to the surmise that the latter pheno- 
menon might be treated in the same way as Fresnel had treated 
the former, namely, by introducing imaginary quantities into 
the formulae of ordinary reflexion. On these principles mathe- 
matical formulae were devised by MacCullaghf and Cauchy^ 

*Phil. Trans., 1830. 

+ Proc. Roy. Irish Acad., i (1836), p. 2 ; ii (1843), p. 376 : Trans. Roy. Irish 
Acad., xviii (1837), p. 71 : MacCullagh's Coll. Works, pp. 58, 132, 230. 

J Comptes Rendus, vii (1838), p. 953 ; riii (1839), pp. 553, 658, 961 ; xxvi 
(1848), p. 86. 

N 



178 The Aether as an Elastic Solid. 

To explain their method, we shall suppose the incident 
light to be polarized in the plane of incidence. According to 
Fresnel's sine-law, the amplitude of the light (polarized in this 
way) reflected from a transparent body is to the amplitude of 
the incident light in the ratio 

_ sin (i - r) 
sin (i + r)' 

where i denotes the angle of incidence and r is determined from 
the equation 

sin i = ft sin r. 

MacCullagh and Cauchy assumed that these equations hold good 
also for reflexion at a metallic surface, provided the refractive 
index /* is replaced by a complex quantity 

IJL = v(l *v/ 1) say, 

where v and K are to be regarded as two constants characteristic 
of the metal. We have therefore 

tan i - tan r (ju 2 - sin 2 i)% - cos i 

jj ,.-,.. - - - -- " 

tan i + tan r (fj 2 - sin 2 i)k + cos i 
If then we write 



so that equations defining U and v are obtained by equating 
separately the real and the imaginary parts of this equation, we 
have 

Ue^ ~ l - cos i 
J 



TT v \/ 1 

Ue v + cos 
and this may be written in the form 



where 



-=. 2 U* + cos 2 ^ - 2 U cos v cos i 
U" + cos 2 * + 2 U cos v cos i 
2 U cos i sin v 



tang = 



U* - cos 2 * 



The Aether as an Elastic Solid. 179 

The quantities J and S are interpreted in the same way as 

in Fresnel's theory of total reflexion : that is, we take J to 
mean the ratio of the intensities of the reflected and incident 
light, while 3 measures the change of phase experienced by 
the light in reflexion. 

The case of light polarized at right angles to the plane of 
incidence may be treated in the same way. 

When the incidence is perpendicular, U evidently reduces 
to v (1 + K 2 )*, and u reduces to - tan -1 K. For silver at perpen- 

2 

dicular incidence almost all the light is reflected, so J is nearly 
unity : this requires cos v to be small, and K to be very large. 
The extreme case in which K is indefinitely great but v indefinitely 
small, so that the quasi- index of refraction is a pure imaginary, 
is generally known as the case of ideal silver. 

The physical significance of the two constants v and K was 
more or less distinctly indicated by Cauchy; in fact, as the 
difference between metals and transparent bodies depends on 
the constant K, it is evident that K must in some way measure 
the opacity of the substance. This will be more clearly seen if 
we inquire how the elastic-solid theory of light can be extended 
so as to provide a physical basis for the formulae of MacCullagh 
and Cauchy.* The sine-formula of Fresnel, which was the 
starting-point of our investigation of metallic reflexion, is a 
consequence of Green's elastic-solid theory : and the differences 
between Green's results and those which we have derived arise 
solely from the complex value which we have assumed for yu. 
We have therefore to modify Green's theory in such a way as 
to obtain a complex value for the index of refraction. 

Take the plane of incidence as plane of xy, and the metallic 
surface as plane of yz. If the light is polarized in the plane of 
incidence, so that the light- vector is parallel to the axis of z, 
the incident light may be taken to be a function of the 
argument 

ax + by + ct, 

* This was done by Lord Rayleigh, Phil. Mag. xliii (1872), p. 321. 

N 2 



180 The Aether as an Elastic Solid. 

where 



a /p\l . b 

- = - - COS I, 

c \n c 



/p\* 

- - I sin ^ ; 
\nj 



here * denotes the angle of incidence, p the inertia of the aether,, 
and n its rigidity. 

Let the reflected light be a function of the argument 

OiX + by + ct, 

where, in order to secure continuity at the boundary, b and c 
must have the same values as before. Since Green's formulae 
are to be still applicable, we must have 



where sin i = ft sin r, but /j. has now a complex value. This- 
equation may be written in the form 



n 
Let the complex value of /u, z be written 



p 

the real part being written pi/p in order to exhibit the analogy 
with Green's theory of transparent media : then we have 

n n 

But an equation of this kind must (as in Green's theory) 
represent the condition to be satisfied in order that the 
quantity 

(a\x + by + ct) \/ - I 
t/ 

may satisfy the differential equation of motion of the aether ; 
from which we see that the equation of motion of the aether 
in the metallic medium is probably of the form 

d z e z A de z 



This equation of motion differs from that of a Greenian 



The Aether as an Elastic Solid. 181 

elastic solid by reason of the occurrence of the term in de z /dt. 
But this is evidently a " viscous " term, representing something 
like a frictional dissipation of the energy of luminous vibra- 
tions : a dissipation which, in fact, occasions the opacity of the 
metal. Thus the term which expresses opacity in the equation 
of motion of the luminiferous medium appears as the origin of the 
peculiarities of metallic reflexion.* It is curious to notice how 
closely this accords with the idea of Huygens, that metals are 
characterized by the presence of soft particles which damp the 
vibrations of light. 

There is, however, one great difficulty attending this 
explanation of metallic reflexion, which was first pointed out by 
Lord Rayleigh.f We have seen that for ideal silver ^ is real 
and negative : and therefore A must be zero and p negative ; 
that is to say, the inertia of the luminiferous medium in the 
metal must be negative. This seems to destroy entirely the 
physical intelligibility of the theory as applied to the case of 
ideal silver. 

The difficulty is a deep-seated one, and was not overcome 
for many years. The direction in which the true solution lies 
will suggest itself when we consider the resemblance which 
has already been noticed between metals and those substances 
which show "surface colour" e.g. the aniline dyes. In the 
case of the latter substances, the light which is so copiously 
reflected from them lies within a restricted part of the spectrum ; 
and it therefore seems probable that the phenomenon is not 
to be attributed to the existence of dissipative terms, but that 
it belongs rather to the same class of effects as dispersion, 
and is to be referred to the same causes. In fact, dispersion 
means that the value of the refractive index of a substance 
with respect to any kind of light depends on the period of 
the light ; and we have only to suppose that the physical 
causes which operate in dispersion cause the refractive index 

* It is easily seen that the amplitude is reduced by the factor e-* when light 
travels one wave-length in the metal : K is generally called the coefficient of 
absorption. 1" Loc. cit. 



182 The Aether as an Elastic Solid. 

to become imaginary for certain kinds of light, in order to 
explain satisfactorily both the surface colours of the aniline 
dyes and the strong reflecting powers of the metals. 

Dispersion was the subject of several memoirs by the 
founders of the elastic-solid theory. So early as 1830 Cauchy's 
attention was directed* to the possibility of constructing a 
mathematical theory of this phenomenon on the basis of 
Fresnel's " Hypothesis of Finite Impacts "f i.e. the assumption 
that the radius of action of one particle of the luminiferous 
medium on its neighbours is so large as to be comparable with 
the wave-length of light. Cauchy supposed the medium to 
be formed, as in Navier's theory of elastic solids, of a system 
of point-centres of force : the force between two of these 
point-centres, m at (x, y, z), and //, at (x + A#, y + Ay, z + Az), 
may be denoted by m^/(r), where r denotes the distance between 
m and p. When this medium is disturbed by light- waves pro- 
pagated parallel to the z-axis, the displacement being parallel 
to the #>axis, the equation of motion of m is evidently 



- 

+ p ) - rT , 

where denotes the displacement of m, ( -i A If) the displace- 
ment of p, and (r + p) the new value of r. Substituting for p its 
value, and retaining only terms of the first degree in A?, this 
equation becomes 



DT r dr 

Now, by Taylor's theorem, since depends only on z, we have 



Substituting, and remembering that summations which 
involve odd powers of Az must vanish when taken over all 

* Bull, des Sc. Math, xiv (1830), p. 9 : " Sur la dispersion de la lumiere," 
. Exevcwe* de Math., 1836. t Cf. p. 132. 



The Aether as an Elastic Solid. 183 

the point-centres within the sphere of influence of m, we obtain 
an equation of the form 

fft d'K o^E d'Z 

w = a &+Pw + y & + ---' 

where a, )3, 7 . . . denote constants. 

Each successive term on the right-hand side of this equation 
involves an additional factor (A^) 2 /X 8 as compared with the pre- 
ceding term, where X denotes the wave-length of the light : so 
if the radii of influence of the point-centres were indefinitely 
small in comparison with the wave-length of the light, the 
equation would reduce to 

8^_ cP 

ar- = "&*' 

which is the ordinary equation of wave-propagation in one 
dimension in non-dispersive media. But if the medium is so 
coarse-grained that A is not large compared with the radii of 
influence, we must retain the higher derivates of . Substi- 
tuting 



in the differential equation with these higher derivates retained, 
we have 

'2-irV 



which shows that c b the velocity of the light in the medium, 
depends on the wave-length A ; as it should do in order to 
explain dispersion. 

Dispersion is, then, according to the view of Fresnel and 
Cauchy, a consequence of the coarse-grainedness of the medium. 
Since the luminiferous medium was found to be dispersive only 
within material bodies, it seemed natural to suppose that in 
these bodies the aether is loaded by the molecules of matter, 
and that dispersion depends essentially on the ratio of the 
wave-length to the distance between adjacent material molecules. 



184 The Aether as an Elastic Solid. 

This theory, in one modification or another, held its ground 
until forty years later it was overthrown by the facts of 
anomalous dispersion. 

The distinction between aether and ponderable matter was 
more definitely drawn in memoirs which were published 
independently in 1841-2 by F. E. Neumann* and Matthew 
O'Brien.f These authors supposed the ponderable particles to 
remain sensibly at rest while the aether surges round them, and 
is acted on by them with forces which are proportional to its 
displacement. ThusJ the equation of motion of the aether 
becomes 

rP& 

p ~ 2 = - (k + ^n) grad div e - n curl curl e - Ce, 
ot 

where C denotes a constant on which the phenomena of dis- 
persion depend. For polarized plane waves propagated parallel 
to the axis of x, this equation becomes 

9 2 e 9 2 e 

fg^5r*5 

and substituting 



e = e 

where r denotes the period and V the velocity of the light, we 
have 

G T , 

772 - P 4^3 r ' 

an equation which expresses the dependence of the velocity on 
the period. 

The attempt to represent the properties of the aether by 
those of an elastic solid lost some of its interest after the 
rise of the electromagnetic theory of light. But in 1867, 

* Berlin Abhandlungen aus dem Jahre 1841, Zweiter Teil, p. 1 : Berlin, 1843. 
t Trans. Camb. Phil. Soc. vii (1842), p. 397. 
J O'Brien, loc. cit, 15, 28. 



The Aether as an Elastic Solid. 185 

before the electromagnetic hypothesis had attracted much 
attention, an elastic-solid theory in many respects preferable 
to its predecessors was presented to the French Academy* by 
Joseph Boussinesq (b. 1842). Until this time, as we have 
seen, investigators had been divided into two parties, according 
as they attributed the optical properties of different bodies to 
variations in the inertia of the luminiferous medium, or to 
variations in its elastic properties. Boussinesq, taking up a 
position apart from both these schools, assumed that the aether 
is exactly the same in all material bodies as in interplanetary 
space, in regard both to inertia and to rigidity, and that the 
optical properties of matter are due to interaction between the 
aether and the material particles, as had been imagined more or 
less by Neumann and O'Brien. These material particles he 
supposed to be disseminated in the aether, in much the same 
way as dust-particles floating in the air. 

If e denote the displacement at the point (x, y, z) in the 
aether, and e' the displacement of the ponderable particles 
at the same place, the equation of motion of the aether is 

rfie ?P&' 

P *jp = ~ ( k + ^ l ) g 11 " 1 div e + ^V 2 e - p, jp, (1) 

where p and p l denote the densities of the aether and matter 
respectively, and k and n denote as usual the elastic constants 
of the aether. This differs from the ordinary Cauchy-Green 
equation only in the presence of the term pi&*'/dP, which 
represents the effect of the inertia of the matter. To this 
equation we must adjoin another expressing the connexion 
between the displacements of the matter and of the aether: 
if we assume that these are simply proportional to each 
other say, 

e' = Ae, (2) 

* Journal de Math. (2) xiii (1868), pp. 313, 425 : cf. also Comptes Rendus, 
cxvii (1893), pp. 80, 139, 193. Equations kindred to some of those of Boussinesq 
M-ere afterwards deduced by Karl Pearson, Proc. Lond. Math. Soc , xx (1889), 
p. 297, from the hypothesis that the strain-energy involves the velocities. 



186 The Aether as an Elastic Solid. 

where the constant A depends on the nature of the ponderable 
body our equation becomes 

3 2 e 

(p + P1 A) ^ = - (k + Jw) grad div e + ^V 2 e, 
ot 

which is essentially the same equation as is obtained in those 
older theories which suppose the inertia of the luminiferous 
medium to vary from one medium to another. So far there 
would seem to be nothing very new in Boussinesq's work. But 
when we proceed to consider crystal-optics, dispersion, and 
rotatory polarization, the advantage of his method becomes 
evident: he retains equation (1) as a formula universally true 
at any rate for bodies at rest while equation (2) is varied 
to suit the circumstances of the case. Thus dispersion can be 
explained if, instead of equation (2), we take the relation 

e' = Ae - Z>V 2 e, 

where D is a constant which measures the dispersive power of 
the substance : the rotation of the plane of polarization of sugar 
solutions can be explained if we suppose that in these bodies 
equation (2) is replaced by 

e' = AQ + B curl e, 

where B is a constant which measures the rotatory power ; and 
the optical properties of crystals can be explained if we suppose 
that for them equation (2) is to be replaced by the equations 

e x ' = Atfx, ey = A z e yt e z ' = A 3 e, 

When these values for the components of e' are substituted 
in equation (1), we evidently obtain the same formulae as were 
derived from the Stokes-Eankine-Eayleigh hypothesis of inertia 
different in different directions in a crystal; to which Boussinesq's 
theory of crystal-optics is practically equivalent. 

The optical properties of bodies in motion may be accounted 
for by modifying equation (1), so that it takes the form 

a a a ay , 

- + W x + W y - + W~ C ,, 

ct cv oy ozj 



The Aether as an Elastic Solid. 187 

where w denotes the velocity of the ponderable body. If the 
body is an ordinary isotropic one, and if we consider light 
propagated parallel to the axis of z, in a medium moving in 
that direction, the light- vector being parallel to the axis of x, 
the equation reduces to 

d'e x d'e x id 9V 

O 7b ' Q]A. I -f- IV I 6 r i 

' O/2 ^W- \ ^/ ^i/v / 

C7t (j6 \(7 (72'/ 

substituting 

,-/(*- FO, 

where V denotes the velocity of propagation of light in the 
medium estimated with reference to the fixed aether, we obtain 

for V the value 

/ n \k o\A 



\p + pt p + 

The absolute velocity of light is therefore increased by the 
amount piAw/(p + piA) owing to the motion of the medium ; 
and this may be written (/** - 1) wjfjc, where ju denotes the 
refractive index ; so that Boussinesq's theory leads to the same 
formula as had been given half a century previously by Fresnel.* 
It is Boussinesq's merit to have clearly asserted that all 
space, both within and without ponderable bodies, is occupied 
by one identical aether, the same everywhere both in inertia 
and elasticity; and that all aethereal processes are to be re- 
presented by two kinds of equations, of which one kind expresses 
the invariable equations of motion of the aether, while the other 
kind expresses the interaction between aether and matter. 
Many years afterwards these ideas were revived in connexion 
with the electromagnetic theory, in the modern forms of which 
they are indeed of fundamental importance. 

* Cf. p. 115 sqq. 



( 188 ) 



CHAPTEK VI. 

FAKADAY. 

TOWARDS the end of the year 1812, Davy received a letter in 
which the writer, a bookbinder's journeyman named Michael 
Faraday, expressed a desire to escape from trade, and obtain 
employment in a scientific laboratory. With the letter was 
enclosed a neatly written copy of notes which the young man 
he was twenty-one years of age had made of Davy's own 
public lectures. The great chemist replied courteously, and 
arranged an interview ; at which he learnt that his correspon- 
dent had educated himself by reading the volumes which came 
into his hands for binding. "There were two," Faraday 
wrote later, "that especially helped me, the 'Encyclopaedia 
Britannica,' from which I gained my first notions of electricity, 
and Mrs. Marcet's ' Conversations on Chemistry/ which gave 
me my foundation in that science." Already, before his applica- 
tion to Davy, he had performed a number of chemical 
experiments, and had made for himself a voltaic pile, with 
which he had decomposed several compound bodies. 

At Davy's recommendation Faraday was in the following 
spring appointed to a post in the laboratory of the Koyal 
Institution, which had been established at the close of the 
eighteenth century under the auspices of Count Rumford ; and 
here he remained for the whole of his active life, first as 
assistant, then as director of the laboratory, and from 1833 
onwards as the occupant of a chair of chemistry which was 
founded for his benefit. 

For many years Faraday was directly under Davy's influence, 
and was occupied chiefly in chemical investigations. But in 
1821, when the new field of inquiry opened by Oersted's 



Faraday. 189 

discovery was attracting attention, he wrote an Historical 
Sketch of Electro- Magnetism* as a preparation for which he 
carefully repeated the experiments described by the writers he 
was reviewing ; and this seems to have been the beginning of 
the researches to which his fame is chiefly due. 

The memoir which stands first in the published volumes of 
Faraday's electrical workf was communicated to the Royal 
Society on November 24th, 1831. The investigation was 
inspired, as he tells us, by the hope of discovering analogies 
between the behaviour of electricity as observed in motion in 
currents, and the behaviour of electricity at rest on conductors. 
Static electricity was known to possess the power of " induction " 
i.e., of causing an opposite electrical state on bodies in its 
neighbourhood ; was it not possible that electric currents might 
show a similar property ? The idea at first was that if in any 
circuit a current were made to flow, any adjacent circuit would 
be traversed by an induced current, which would persist exactly 
as long as the inducing current. Faraday found that this was 
not the case ; a current was indeed induced, but it lasted only 
for an instant, being in fact perceived only when the primary 
current was started or stopped. It depended, as he soon 
convinced himself, not on the mere existence of the inducing 
current, but on its variation. 

Faraday now set himself to determine the laws of induction 
of currents, and for this purpose devised a new way of repre- 
senting the state of a magnetic field. Philosophers had been 
long accustomed? to illustrate magnetic power by strewing iron 
filings on a sheet of paper, and observing the curves in which 
they dispose themselves when a magnet is brought underneath. 

Published in Annals of Philosophy, ii (1821), pp. 195, 274; iii (1822), 
p. 107. 

t Experimental Researches in Electricity, by Michael Faraday : 3 vols. 

* The practice goes back at least as far as Niccolo Cabeo ; indeed the curves 
traced by Petrus Peregrinus on his globular lodestone (cf . p. 8) were projections 
of lines of force. Among eighteenth-century writers La Hire mentions the use of 
iron filings, Mem. de 1'Acad., 1717. Faraday had referred to them in his electro- 
magnetic paper of 1821, Exp. Res. ii, p. 127. 



190 Faraday. 

These curves suggested to Faraday* the idea of lines of magnetic 
force, or curves whose direction at every point coincides with 
the direction of the magnetic intensity at that point; the 
curves in which the iron filings arrange themselves on the 
paper resemble these curves so far as is possible subject to the 
condition of not leaving the plane of the paper. 

With these lines of magnetic force Faraday conceived all 
space to be filled. Every line of force is a closed curve, which 
in some part of its course passes through the magnet to which 
it belongs, f Hence if any small closed curve be taken in space, 
the lines of force which intersect this curve must form a 
tubular surface returning into itself ; such a surface is called a 
tiibe of force. From a tube of force we may derive information 
not only regarding the direction of the magnetic intensity, 
but also regarding its magnitude; for the product of this 
magnitude} and the cross-section of any tube is constant along 
the entire length of the tube. On the basis of this result, 
Faraday conceived the idea of partitioning all space into 
compartments by tubes, each tube being such that this product 
has the same definite value. For simplicity, each of these 
tubes may be called a " unit line of force " ; the strength of 
the field is then indicated by the separation or concentration of 
the unit lines of force,! I so that the number of them which 
intersect a unit area placed at right angles to their direction 

#They were first defined in Exp. Res., 114 : "By magnetic curves, I mean 
the lines of magnetic forces, however modified hy the juxtaposition of poles, 
which could be depicted by iron filings ; or those to which a very small magnetic 
needle would form a tangent." 

t Exp. Res. iii, p. 405. 

J Within the substance of magnetized bodies we must in this connexion under- 
stand the magnetic intensity to be that experienced in a crevice whose sides are 
perpendicular to the lines of magnetization : in other words, we must take it to be 
what since Maxwell's time has been called the magnetic induction. 

Exp. Res., 3073. This theorem was first proved by the French geometer 
Michel Chasles, in his memoir on the attraction of an ellipsoidal sheet, Journal 
de 1'Ecole Polyt. xv (1837), p. 266. 

|| Ibid., 3122. "The relative amount of force, or of lines of force, in a 
given space is indicated by their concentration or separation i.e., by their number 
in that space." 



Faraday. 191 

at any point measures the intensity of the magnetic field at 
that point. 

Faraday constantly thought in terms of lines of force. 
" I cannot refrain," he wrote, in 1851,* " from again expressing 
my conviction of the truthfulness of the representation, which 
the idea of lines of force affords in regard to magnetic action. 
All the points which are experimentally established in regard 
to that action i.e. all that is not hypothetical appear to be well 
and truly represented by it."f 

Faraday found that a current is induced in a circuit either 
when the strength of an adjacent current is altered, or when a 
magnet is brought near to the circuit, or when the circuit itself 
is moved about in presence of another current or a magnet. 
He saw from the firstj that in all cases the induction depends 
on the relative motion of the circuit and the lines of magnetic 
force in its vicinity. The precise nature of this dependence 
was the subject of long-continued further experiments. In 
1832 he found that the currents produced by induction under 
the same circumstances in different wires are proportional to 
the conducting powers of the wires a result which showed 
that the induction consists in the production of a definite 
electromotive force, independent of the nature of the wire, and 
dependent only on the intersections of the wire and the 
magnetic curves. This electromotive force is produced whether 
the wire forms a closed circuit (so that a current flows) or is 
open (so that electric tension results). 

All that now remained was to inquire in what way the 
electromotive force depends on the relative motion of the wire 
and the lines of force. The answer to this inquiry is, in 

* Exp. Res., 3174. 

t Some of Faraday's most distinguished contemporaries were far from sharing 
this conviction. " I declare," wrote Sir George Airy in 1855, " that I can hardly 
imagine anyone who practically and numerically knows this agreement " between 
observation and the results of calculation based on action at a distance, "to hesitate 
au instant in the choice between this simple and precise action, on the one hand, 
and anything so vague and varying as lines of force, on the other hand." Cf. 
Bence Jones's Life of Faraday, ii, p. 353. 

I Exp. Res., 116. Ibid., 213. 



192 Faraday. 

Faraday's own words,* that "whether the wire moves directly or 
obliquely across the lines of force, in one direction or another, it 
sums up the amount of the forces represented by the lines it 
has crossed," so that " the quantity of electricity thrown into a 
current is directly as the number of curves intersected."t The 
induced electromotive force is, in fact, simply proportional to 
the number of the unit lines of magnetic force intersected by 
the wire per second. 

This is the fundamental principle of the induction of 
currents. Faraday is undoubtedly entitled to the full honour 
of its discovery ; but for a right understanding of the progress 
of electrical theory at this period, it is necessary to remember 
that many years elapsed before all the conceptions involved in 
Faraday's principle became clear and familiar to his contem- 
poraries ; and that in the meantime the problem of formulating 
the laws of induced currents was approached with success from 
other points of view. There were indeed many obstacles to the 
direct appropriation of Faraday's work by the mathematical 
physicists of his own generation ; not being himself a mathe- 
matician, he was unable to address them in their own language ; 
and his favourite mode of representation by moving lines of 
force repelled analysts who had been trained in the school of 
Laplace and Poisson. Moreover, the idea of electromotive force 
itself, which had been applied to currents a few years previously 
in Ohm's memoir, was, as we have seen, still involved in 
obscurity and misapprehension. 

A curious question which arose out of Faraday's theory 
was whether a bar-magnet which is rotated on its own axis 
carries its lines of magnetic force in rotation with it. Faraday 
himself believed that the lines of force do not rotate J: on this 
view a revolving magnet like the earth is to be regarded as 
moving through its own lines of force, so that it must become 
charged at the equator and poles with electricity of opposite 
signs ; and if a wire not partaking in the earth's rotation were 
to have sliding contact with the earth at a pole and at the 

* Exp. Res., 3082. t Ibid., 3115. % Ibid., 3090. 



Faraday. 193 

equator, a current would steadily flow through it. Experiments 
confirmatory of these views were made by Faraday himself ;* 
but they do not strictly prove his hypothesis that the lines of 
force remain at rest ; for it is easily seenf that, if they were to 
rotate, that part of the electromotive force which would be 
produced by. their rotation would be derivable from a potential, 
and so would produce no effect in closed circuits such as Faraday 
used. 

Three years after the commencement of Faraday's researches 
on induced currents he was led to an important extension of 
them by an observation which was communicated to him by 
another worker. William Jenkin had noticed that an 
electric shock may be obtained with no more powerful source of 
electricity than a single cell, provided the wire through which 
the current passes is long and coiled ; the shock being felt when 
contact is broken. J As Jenkin did not choose to investigate 
the matter further, Faraday took it up, and showed that the 
powerful momentary current, which was observed when the 
circuit was interrupted, was really an induced current governed 
by the same laws as all other induced currents, but with this 
peculiarity, that the induced and inducing currents now flowed 
in the same circuit. In fact, the current in its steady state 
establishes in the surrounding region a magnetic field, whose 
lines of force are linked with the circuit ; and the removal of 
these lines of force when the circuit is broken originates an 
induced current, which greatly reinforces the primary current 
just before its final extinction. To this phenomenon the name 
of self-induction has been given. 

The circumstances attending the discovery of self-induction 

Exp. Res., $ 218, 3109, &c. 

t Cf. W. Weber, Ann. d. Phys. lii (1841) ; S. Tolver Preston, Phil. Mag. xix 
(1885), p. 131. In 1891 S. T. Preston, Phil. Mag. xxxi, p. 100, designed a crucial 
experiment to test the question ; but it was not tried for want of a sufficiently 
delicate electrometer. 

% A similar observation had been made by Henry, and published in the Amer. 
Jour. Sci. xxii (1832), p. 408. The spark at the rupture of a spirally-wound 
circuit had been often observed, e.g., by Pouillet and Nobili. 

Exp. Res., 1048. 

O 



1 94 Faraday. 

occasioned a comment from Faraday on the number of sugges- 
tions which were continually being laid before him. He re- 
marked that although at different times a large number of 
authors had presented him with their ideas, this case of 
Jenkin was the only one in which any result had followed. 
" The volunteers are serious embarrassments generally to the 
experienced philosopher."* 

The discoveries of Oersted, Ampere, and Faraday had shown 
the close connexion of magnetic with electric science. But the 
connexion of the different branches of electric science with 
each other was still not altogether clear. Although Wollaston's 
experiments of 1801 had in effect proved the identity in kind 
of the currents derived from frictional and voltaic sources, the 
question was still regarded as open thirty years afterwards,f no 
satisfactory explanation being forthcoming of the fact that 
frictional electricity appeared to be a surface-phenomenon, 
whereas voltaic electricity was conducted within the interior 
substance of bodies. To this question Faraday now applied him- 
self; and in 1833 he succeeded* in showing that every known 
effect of electricity physiological, magnetic, luminous, calorific, 
chemical, and mechanical may be obtained indifferently either 
with the electricity which is obtained by friction or with that 
obtained from a voltaic battery. Henceforth the identity of the 
two was beyond dispute. 

Some misapprehension, however, has existed among later 
writers as to the conclusions which may be drawn from this 
identification. What Faraday proved is that the process which 
goes on in a wire connecting the terminals of a voltaic cell is of 
the same nature as the process which for a short time goes on in 
a wire by which a condenser is discharged. He did not prove, 

* Bence Jones's Life of Faraday, ii, p. 45. 

t Cf. John Davy, Phil. Trans., 1832, p. 259 ; W. Ritchie, ibid., p. 279. Davy 
suggested that the electrical power, " according to the analogy of the solar ray," 
might be " not a simple power, but a combination of powers, which may occur 
variously associated, and produce all the varieties of electricity with which we are 
acquainted." 

J Exp. Jies. 9 Series iii. 






Faraday. 195 

and did not profess to have proved, that this process consists in 
the actual movement of a quasi-substance, electricity, from one 
plate of the condenser to the other, or of two quasi-substances, 
the resinous and vitreous electricities, in opposite directions. 
The process had been pictured in this way by many of his 
predecessors, notably by Volta; and it has since been so 
pictured by most of his successors : but from such assumptions 
Faraday himself carefully abstained. 

What is common to all theories, and is universally conceded, 
is that the rate of increase in the total quantity of electrostatic 
charge within any volume-element is equal to the excess of the 
influx over the efflux of current from it. This statement may 
be represented by the equation 

|+divi = 0, (1) 

where p denotes the volume-density of electrostatic charge, 
and i the current, at the place (x, y, z) at the time t. Volta's 
assumption is really one way of interpreting this equation 
physically: it presents itself when we compare equation (1) 
with the equation 



which is the equation of continuity for a fluid of density p and 
velocity v : we may identify the two equations by supposing i 
to be of the same physical nature as the product /v; and 
this is precisely what is done by those who accept Volta's 
assumption. 

But other assumptions might be made which would equally 
well furnish physical interpretations to equation (1). For 
instance, if we suppose p to be the convergence of any vector of 
which i is the time-flux,* equation (1) is satisfied automatically ; 

* In symbols, 

div 8 = - , 



where s denotes the vector in question. 

02 



196 Faraday. 

we can picture this vector as being of the nature of a displace- 
ment. By such an assumption we should avoid altogether the 
necessity for regarding the conduction-current as an actual 
flow of electric charges, or for speculating whether the drifting 
charges are positive or negative ; and there would be no longer 
anything surprising in the production of a null effect by the 
coalescence of electric charges of opposite signs. 

Faraday himself wished to leave the matter open, and to 
avoid any definite assumption.* Perhaps the best indication of 
his views is afforded by a laboratory notej- of date 1837 : 

"After much consideration of the manner in which the 
electric forces are arranged in the various phenomena generally,. 
I have come to certain conclusions which I will endeavour to 
note down without committing myself to any opinion as to the 
cause of electricity, i.e., as to the nature of the power. If 
electricity exist independently of matter, then I think that the 
hypothesis of one fluid will not stand against that of two fluids. 
There are, I think, evidently what I may call two elements of 
power, of equal force and acting toward each other. But these 
powers may be distinguished only by direction, and may be no 
more separate than the north and south forces in the elements 
of a magnetic needle. They may be the polar points of the 
forces originally placed in the particles of matter." 

It may be remarked that since the rise of the mathematical 
theory of electrostatics, the controversy between the supporters 
of the one-fluid and the two-fluid theories had become 
manifestly barren. The analytical equations, in which 
interest was now largely centred, could be interpreted equally 
well on either hypothesis; and there seemed to be little 
prospect of discriminating between them by any new experi- 
mental discovery. But a problem does not lose its fascination 

*"His principal aim," said Helmholtz in the Faraday Lecture of 1881, 
" was to express in his new conceptions only facts, with the least possible use of 
hypothetical substances and forces. This was really a progress in general 
scientific method, destined to purify science from the last remains of meta- 
physics." 

t Bence Jones's Life of Foradny^ ii, p. 77. 



Faraday. 197 

because it appears insoluble. " I said once to Faraday," wrote 
Stokes to his father-in-law in 1879, " as I sat beside him at a 
British Association dinner, that I thought a great step would 
be made when we should be able to say of electricity that 
which we say of light, in saying that it consists of undula- 
tions. He said to me he thought we were a long way off that 

yet."* 

For his next series of researches,! Faraday reverted to 
subjects which had been among the first to attract him as an 
apprentice attending Davy's lectures : the voltaic pile, and the 
relations of electricity to chemistry. 

It was at this time generally supposed that the decomposi- 
tion of a solution, through which an electric current is passed, 
is due primarily to attractive and repellent forces exercised on 
its molecules by the metallic terminals at which the current 
enters and leaves the solution. Such forces had been assumed 
both in the hypothesis of Grothuss and Davy, and in the rival 
hypothesis of De La Eive ;+ the chief difference between these 
being that whereas Grothuss and Davy supposed a chain 'of 
decompositions and recompositions in the liquid, De La Rive 
supposed the molecules adjacent to the terminals to be the 
only ones decomposed, and attributed to their fragments the 
power of travelling through the liquid from one terminal to the 
other. 

To test this doctrine of the influence of terminals, Faraday 
moistened a piece of paper in a saline solution, and supported 
it in the air on wax, so as to occupy part of the interval 
between two needle-points which were connected with an 
electric machine. When the machine was worked, the current 
was conveyed between the needle-points by way of the 
moistened paper and the two air-intervals on either side of it ; 
and under these circumstances it was found that the salt under- 
went decomposition. Since in this case no metallic terminals of 
.any kind were in contact with the solution, it was evident that 

* Stokes's Scientific Correspondence, vol. i, p. 353. 

t Exp. Res., 450 (1833). Cf. pp. 78-9. 



198 Faraday. 

all hypotheses which attributed decomposition to the action of 
the terminals were untenable. 

The ground being thus cleared by the demolition of previous 
theories, Faraday was at liberty to construct a theory of his 
own. He retained one of the ideas of Grothuss' and Davy's 
doctrine, namely, that a chain of decompositions and recombi- 
nations takes place in the liquid ; but these molecular processes 
he attributed not to any action of the terminals, but to a power 
possessed by the electric current itself, at all places in its 
course through the solution. If as an example we consider 
neighbouring molecules A, B, C, D, . . . of the compound say 
water, which was at that time believed to be directly decom- 
posed by the current Faraday supposed that before the 
passage of the current the hydrogen of A would be in close 
union with the oxygen of A, and also in a less close relation with 
the oxygen atoms of B, C, D, . . . : these latter relations being 
conjectured to be the cause of the attraction of aggregation in 
solids and fluids.* When an electric current is sent through the 
liquid, the affinity of the hydrogen of A for the oxygen of B is 
strengthened, if A and B lie along the direction of the current ; 
while the hydrogen of A withdraws some of its bonds from the 
oxygen of A, with which it is at the moment combined. So 
long as the hydrogen and oxygen of A remain in association, 
the state thus induced is merely one of polarization ; but the 
compound molecule is unable to stand the strain thus imposed 
on it, and the hydrogen and oxygen of A part company from 
each other. Thus decompositions take place, followed by 
recombinations : with the result that after each exchange an 
oxygen atom associates itself with a partner nearer to the 
positive terminal, while a hydrogen atom associates with a 
partner nearer to the negative terminal. 

This theory explains why, in all ordinary cases, the evolved 
substances appear only at the terminals ; for the terminals are 
the limiting surfaces of the decomposing substance ; and, except 
at them, every particle finds other particles having a contrary 

*Exp. lies., 523. 



Faraday. 199 

tendency with which it can combine. It also explains why, in 
numerous cases, the atoms of the evolved substances are not 
retained by the terminals (an obvious difficulty in the way of 
all theories which suppose the terminals to attract the atoms) : 
for the evolved substances v are expelled from the liquid, not 
drawn out by an attraction. 

Many of the perplexities which had harassed the older 
theories were at once removed when the phenomena were re- 
garded from Faraday's point of view. Thus, mere mixtures (as 
opposed to chemical compounds) are not separated into their 
constituents by the electric current ; although there would seem 
to be no reason why the Grothuss-Davy polar attraction should 
not operate as well on elements contained in mixtures as on 
elements contained in compounds. 

In the latter part of the same year (1833) Faraday took up 
the subject again.* It was at this time that he introduced the 
terms which have ever since been generally used to describe 
the phenomena of electro-chemical decomposition. To the 
terminals by which the electric current passes into or out of the 
decomposing body he gave the name electrodes. The electrode 
of high potential, at which oxygen, chlorine, acids, &c., are 
evolved, he called the anode, and the electrode of low potential, 
at which metals, alkalis, and bases are evolved, the cathode. 
Those bodies which are decomposed directly by the current 
he named electrolytes ; the parts into which they are decomposed, 
ions ; the acid ions, which travel to the anode, he named anions ; 
and the metallic ions, which pass to the cathode, cations. 

Faraday now proceeded to test the truth of a supposition 
which he had published rather more than a year previously ,f 
and which indeed had apparently been suspected by Gay-Lussac 
and ThenardJ so early as 1811; namely, that the rate at which 
an electrolyte is decomposed depends solely on the intensity of 
the electric current passing through it, and not at all on the 
size of the electrodes or the strength of the solution. Having 

* Exp. Res., 661. f /*'*., 377 (Dec. 1832). 

Recherches physico-chimiqucs faites sur la pile ; Paris, 1811, p. 12. 



200 Faraday. 

established the accuracy of this law,* he found by a comparison 
of different electrolytes that the mass of any ion liberated by 
a given quantity of electricity is proportional to its chemical 
equivalent, i.e. to the amount of it required to combine with 
some standard mass of some standard element. If an element 
is %-valent, so that one of its atoms can hold in combination 
n atoms of hydrogen, the chemical equivalent of this element 
may be taken to be 1/n of its atomic weight ; and therefore 
Faraday's result may be expressed by saying that an electric 
current will liberate exactly one atom of the element in 
question in the time which it would take to liberate n atoms 
of hydrogen.-)- 

The quantitative law seemed to Faraday:}: to indicate that 
" the atoms of matter are in some way endowed or associated 
with electrical powers, to which they owe their most striking 
qualities, and amongst them their mutual chemical affinity." 
Looking at the facts of electrolytic decomposition from this 
point of view, he showed how natural it is to suppose that 
the electricity which passes through the electrolyte is the exact 
equivalent of that which is possessed by the atoms separated at 
the electrodes ; which implies that there is a certain absolute 
quantity of the electric power associated with each atom of 
matter. 

The claims of this splendid speculation he advocated with 
conviction. " The harmony," he wrote, " which it introduces 
into the associated theories of definite proportions and electro- 
chemical affinity is very great. According to it, the equivalent 
weights of bodies are simply those quantities of them which 
contain equal quantities of electricity, or have naturally equal 
electric powers ; it being the ELECTRICITY which determines the 
equivalent number, because it determines the combining force. 
Or, if we adopt the atomic theory or phraseology, then the 

*Exp. Res., 713-821. 

t In the modern units, 96580 coulombs of electricity must pass round the 
circuit in order to liberate of each ion a number of grams equal to the quotient of 
the atomic weight by the valency. 

J Exp. Res., 852. Ibid., 869. 



Faraday. 201 

atoms of bodies which are equivalent to each other in their 
ordinary chemical action, have equal quantities of electricity 
naturally associated with them. " But," he added, " I must 
confess I am jealous of the term atom : for though it is very 
easy to talk of atoms, it is very difficult to form a clear idea 
of their nature, especially when compound bodies are under 
consideration." 

These discoveries and ideas tended to confirm Faraday in 
preferring, among the rival theories of the voltaic cell, that one 
to which all his antecedents and connexions predisposed him. 
The controversy between the supporters of Volta's contact 
hypothesis on the one hand, and the chemical hypothesis of 
Davy and Wollaston on the other, had now been carried on 
for a generation without any very decisive result. In Germany 
and Italy the contact explanation was generally accepted, under 
the influence of Christian Heiririch Pfaff, of Kiel (b. 1773, 
d. 1852), and of Ohm, and, among the younger men, of Gustav 
Theodor Fechner (b. 1801, d. 1887), of Leipzig,* and Stefano 
Marianini (b. 1790, d. 1866), of Modena. Among French writers 
De La Eive, of Geneva, was, as we have seen, active in support 
of the chemical hypothesis; and this side in the dispute had 
always been favoured by the English philosophers. 

There is no doubt that when two different metals are put 
in contact, a difference of potential is set up between them 
without any apparent chemical action ; but while the contact 
party regarded this as a direct manifestation of a "contact- 
force " distinct in kind from all other known forces of nature, 

* Johaim Christian Poggendorff (b. 1796, d. 1877), of Berlin, for long the editor 
of the Annalen der Physik, leaned originally to the chemical side, but in 1838 
became convinced of the truth of the contact theory, which he afterwards actively 
defended. Moritz Hermann Jacobi (b. 1801, d. 1874), of Dorpat, is also to be 
mentioned among its advocates. 

Faraday's first series of investigations on this subject were made in 1834 : 
Exp, es., series viii. In 1836 De La Kive followed on the same side with his 
Eccherches sur la Cause de V Electr. Voltaique. The views of Faraday and De La 
Rive were criticized by Pfaff, Revision der Lehre vom Galvanistntts, Kiel, 1837, and 
by Fechner, Ann. d. Phys., xlii (1837), p. 481, and xliii (1838), p. 433 : translated 
Phil. Mag., xiii (1838), pp. 205, 367. Faraday returned to the question in 1840, 
Exp, Jtes., series xvi and xvii. 



202 Faraday. 

the chemical party explained it as a consequence of chemical 
affinity or incipient chemical action between the metals and 
the surrounding air or moisture. There is also no doubt that 
the continued activity of a voltaic cell is always accompanied 
by chemical unions or decompositions ; but while the chemical 
party asserted that these constitute the efficient source of the- 
current, the contact party regarded them as secondary actions, 
and attributed the continual circulation of electricity to the 
perpetual tendency of the electromotive force of contact to 
transfer charge from one substance to another. 

One of the most active supporters of the chemical theory 
among the English physicists immediately preceding Faraday 
was Peter Mark Eoget (b. 1779, d. 1869), to whom are due two- 
of the strongest arguments in its favour. In the first place, 
carefully distinguishing between the quantity of electricity put 
into circulation by a cell and the tension at which this electricity 
is furnished, he showed that the latter quantity depends on the 
" energy of the chemical action "* a fact which, when taken 
together with Faraday's discovery that the quantity of electricity 
put into circulation depends on the amount of chemicals con- 
sumed, places the origin of voltaic activity beyond all question. 
Koget's principle was afterwards verified by Faradayf and by 
De La EiveJ; " the electricity of the voltaic pile is proportionate 
in its intensity to the intensity of the affinities concerned in 
its production," said the former in 1834; while De La Kive 
wrote in 1836, " The intensity of the currents developed in 
combinations and in decompositions is exactly proportional to 
the degree of affinity which subsists between the atoms whose 
combination or separation has given rise to these currents." 

* " The absolute quantity of electricity which is thus developed, and made to 
circulate, will depend upon a variety of circumstances, such as the extent of the 
surfaces in chemical action, the facilities afforded to its transmission, &c. But 
its degree of intensity, or tension, as it is often termed, will be regulated by other 
causes, and more especially by the energy of the chemical action." Roget's 
Galvanism (1832), 70. 

t Exp. Res., 908, 909, 916, 988, 1958. 

I Annales de Chim., Ixi (1836), p. 38. 



Faraday. 203 

Not resting here, however, Koget brought up another argu- 
ment of far-reaching significance. " If," he wrote,* " there could 
exist a power having the property ascribed to it by the [contact] 
hypothesis, namely, that of giving continual impulse to a fluid 
in one constant direction, without being exhausted by its own 
action, it would differ essentially from all the other known 
powers in nature. All the powers and sources of motion, with 
the operation of which we are acquainted, when producing their 
peculiar effects, are expended in the same proportion as those 
effects are produced ; and hence arises the impossibility of 
obtaining by their agency a perpetual effect ; or, in other words, 
a perpetual motion. But the electro-motive force ascribed by 
Yolta to the metals when in contact is a force which, as long 
as a free course is allowed to the electricity it sets in motion, 
is never expended, and continues to be exerted with undi- 
minished power, in the production of a never-ceasing effect. 
Against the truth of such a supposition the probabilities are 
all but infinite." 

This principle, which is little less than the doctrine of 
conservation of energy applied to a voltaic cell, was reasserted 
by Faraday. The process imagined by the contact school 
" would," he wrote, "indeed be a creation of 'power -, like no other 
force in nature." In all known cases energy is not generated, 
but only transformed. There is no such thing in the world as 
"a pure creation of force; a production of power without a 
corresponding exhaustion of something to supply it."f 

As time went on, each of the rival theories of the cell 
became modified in the direction of the other. The contact 
party admitted the importance of the surfaces at which the 
metals are in contact with the liquid, where of course the chief 
chemical action takes place ; and the chemical party confessed 
their inability to explain the state of tension which subsists 
before the circuit is closed, without introducing hypotheses just 
as uncertain as that of contact force. 

*Roget's Galvanism (1832), 113. 
t Exp.Res., 2071 (1840). 



204 Faraday. 

Faraday's own view on this point* was that a plate of 
amalgamated zinc, when placed in dilute sulphuric acid, " has 
power so far to act, by its attraction for the oxygen of the 
particles'in contact with it, as to place the similar forces already 
active between these and the other particles of oxygen and 
the particles of hydrogen in the water, in a peculiar state of 
tension or polarity, and probably also at the same time to 
throw those of its own particles which are in contact with the 
water into a similar but opposed state. Whilst this state is 
retained, no further change occurs: but when it is relieved 
by completion of the circuit, in which case the forces determined 
in opposite directions, with respect to the zinc and the electro- 
lyte, are found exactly competent to neutralize each other, then 
a series of decompositions and recompositions takes place 
amongst the particles of oxygen and hydrogen which constitute 
the water, between the place of contact with the platina and 
the place where the zinc is active : these intervening particles 
being evidently in close dependence upon and relation to each 
other. The zinc forms a direct compound with those particles 
of oxygen which were, previously, in divided relation to both 
it and the hydrogen : the oxide is removed by the acid, and a 
fresh surface of zinc is presented to the water, to renew and 
repeat the action." 

These ideas were developed further by the later adherents 
of the chemical theory, especially by Faraday's friend Christian 
Friedrich Schonbein,f of Basle (6. 1799, d. 1868), the discoverer 
of ozone. Schonbein made the hypothesis more definite by 
assuming that when the circuit is open, the molecules of water 
adjacent to the zinc plate are electrically polarized, the oxygen 
side of each molecule being turned towards the zinc and being 
negatively charged, while the hydrogen side is turned away 
from the zinc and is positively charged. In the third quarter 

* Exp. &., 949. 

t Ann. d. Phys., Ixxviii (1849), p. 289, translated Archives des sc. phys., xiii 
(1850), p. 192. Faraday and Schonbein for many years carried on a correspondence, 
which has been edited by G. W. A. Kahlbaum and F. V. Darbishire : London, 
Williams and Norgate. 



Faraday. 205 

of the nineteenth century, the general opinion was in favour 
of some such conception as this. Helmholtz* attempted to- 
grasp the molecular processes more intimately by assuming 
that the different chemical elements have different attractive- 
powers (exerted only at small distances) for the vitreous and 
resinous electricities : thus potassium and zinc have strong 
attractions for positive charges, while oxygen, chlorine, and 
bromine have strong attractions for negative electricity. This 
differs from Volta's original hypothesis in little else but 
in assuming two electric fluids where Volta assumed only 
one. It is evident that the contact difference of potential; 
between two metals may be at once explained by Helmholtz's, 
hypothesis, as it was by Volta's ; and the activity of the voltaic 
cell may be referred to the same principles : for the two ions 
of which the liquid molecules are composed will also possess 
different attractive powers for the electricities, and may be 
supposed to be united respectively with vitreous and resinous, 
charges. Thus when two metals are immersed in the liquid,^ 
the circuit being open, the positive ions are attracted to the 
negative metal and the negative ions to the positive metal,, 
thereby causing a polarized arrangement of the liquid molecules 
near the metals. When the circuit is closed, the positively 
charged surface of the positive metal is dissolved into the fluid;, 
and as the atoms carry their charge with them, the positive 
charge on the immersed surface of this metal must be per- 
petually renewed by a current flowing in the outer circuit. 

It will be seen that Helmholtz did not adhere to Davy'ss 
doctrine of the electrical nature of chemical affinity quite as, 
simply or closely as Faraday, who preferred it in its most direct 
and uncompromising form. " All the facts show us," he wrote,f 
"that that power commonly called chemical affinity can be> 
communicated to a distance through the metals and certain 
forms of carbon ; that the electric current is only another form 
of the forces of chemical affinity ; that its power is in proportion. 

* In his celebrated memoir of 1847 on the Conservation, .o.Huergy. 
t Exp. Ties., 918. 



206 Faraday. 

to the chemical affinities producing it ; that when it is deficient 
in force it may be helped by calling in chemical aid, the want 
in the former being made up by an equivalent of the latter; 
that, in other words, the forces termed chemical affinity and 
electricity are one and the same." 

In the interval between Faraday's earlier and later papers 
on the cell, some important results on the same subject were 
published by Frederic Daniell (b. 1790, d. 1845), Professor of 
Chemistry in King's College, London.* Daniell showed that 
when a current is passed through a solution of a salt in water, 
the ions which carry the current are those derived from the salt, 
and not the oxygen and hydrogen ions derived from the water ; 
this follows since a current divides itself between different mixed 
electrolytes according to the difficulty of decomposing each, and 
it is known that pure water can be electrolysed only with great 
difficulty. Daniell further showed that the ions arising from 
(say) sodium sulphate are not represented by Na 2 and S0 3 , but 
by Na and S0 4 ; and that in such a case as this, sulphuric acid 
is formed at the anode and soda at the cathode by secondary 
action, giving rise to the observed evolution of oxygen and 
hydrogen respectively at these terminals. 

The researches of Faraday on the decomposition of chemical 
compounds placed between electrodes maintained at different 
potentials led him in 1837 to reflect on the behaviour of such 
substances as oil of turpentine or sulphur, when placed in the 
same situation. These bodies do not conduct electricity, and 
are not decomposed ; but if the metallic faces of a condenser 
are maintained at a definite potential difference, and if the 
space between them is occupied by one of these insulating 
substances, it is found that the charge on either face depends 
on the nature of the insulating substance. If for any particular 
insulator the charge has a value s times the value which it 
would have if the intervening body were air, the number f 
may be regarded as a measure of the influence which the 
insulator exerts on the propagation of electrostatic action 

* Phil. Trans., 1839, p. 97. 



Faraday. 207 

through it : it was called by Faraday the specific inductive 
capacity of the insulator.* 

The discovery of this property of insulating substances or 
dielectrics raised the question as to whether it could be 
harmonized with the old ideas of electrostatic action. Consider, 
for example, the force of attraction or repulsion between two 
small electrically- charged bodies. So long as they are in air, 
the force is proportional to the inverse square of the distance ; 
but if the medium in which they are immersed be partly 
changed e.g., if a globe of sulphur be inserted in the intervening 
space this law is no longer valid : the change in the dielectric 
affects the distribution of electric intensity throughout the 
entire field. 

The problem could be satisfactorily solved only by forming 
a physical conception of the action of dielectrics : and such a 
conception Faraday now put forward. 

The original idea had been promulgated long before by his 
master Davy. Davy, it will be remembered,f in his explanation 
of the voltaic pile, had supposed that at first, before chemical 
decompositions take place, the liquid plays a part analogous to 
that of the glass in a Leyden jar, and that in this is involved an 
electric polarization of the liquid molecules.^ This hypothesis 
was now developed by Faraday. Keferring first to his own work 
on electrolysis, he asserted that the behaviour of a dielectric is 
exactly the same as that of an electrolyte, up to the point at 
which the electrolyte breaks down under the electric stress ; a 
dielectric being, in fact, a body which is capable of sustaining 
the stress without suffering decomposition. 

" For," he argued,|| " let the electrolyte be water, a plate of 
ice being coated with platina foil on its two surfaces, and these 

* Exp. Res., 1252 (1837). Cavendish had discovered specific inductive capacity 
long before, but his papers were still unpublished. 

t Cf. p. 77. 

\ This is expressly stated in Davy's Elements of Chemical Philosophy (1812), 
Div. i, 7, where he lays it down that an essential " property of non-conductors" 
is "to receive electrical polarities." 

$ Exp. Res., 1164, 1338, 1343, 1621. 

|| Exp. Res., 1164. 



208 Faraday. 

coatings connected with any continued source of the two 
electrical powers, the ice will charge like a Leyden arrangement, 
presenting a case of common induction, but no current will pass. 
If the ice be liquefied, the induction will now fall to a certain 
degree, because a current can now pass ; but its passing is 
dependent upon a peculiar molecular arrangement of the particles 
consistent with the transfer of the elements of the electrolyte in 
opposite directions . . . As, therefore, in the electrolytic action, 
induction appeared to bethejfe step,and decomposition the second 
(the power of separating these steps from each other by giving 
the solid or fluid condition to the electrolyte being in our hands) ;: 
as the induction was the same in its nature as that through air, 
glass, wax, &c., produced by any of the ordinary means ; and as 
the whole effect in the electrolyte appeared to be an action of 
the particles thrown into a peculiar or polarized state, I was 
glad to suspect that common induction itself was in all cases an 
action of contiguous particles, and that electrical action at a 
distance (i.e., ordinary inductive action) never occurred except 
through the influence of the intervening matter." 

Thus at the root of Faraday's conception of electrostatic 
induction lay this idea that the whole of the insulating medium 
through which the action takes place is in a state of polarization 
similar to that which precedes decomposition in an electrolyte. 
" Insulators," he wrote,* " may be said to be bodies whose 
particles can retain the polarized state, whilst conductors are 
those whose particles cannot be permanently polarized." 

The conception which he at this time entertained of the 
polarization may be reconstructed from what he had already 
written concerning electrolytes. He supposedf that in the 
ordinary or unpolarized condition of a body, the molecules con- 
sist of atoms which are bound to each other by the forces of 
chemical affinity, these forces being really electrical in their 
nature ; and that the same forces are exerted, though to a less 

* Exp.Res., 1338. 

t This must not be taken to be more than an idea which Faraday mentioned as 
present to his mind. He declined as yet to formulate a definite hypothesis. 



Faraday. 209 

degree, between atoms which belong to different molecules, 
thus producing the phenomena of cohesion. When an electric 
field is set up, a change takes place in the distribution of these 
forces ; some are strengthened and some are weakened, the 
effect being symmetrical about the direction of the applied 
electric force. 

Such a polarized condition acquired by a dielectric when 
placed in an electric field presents an evident analogy to the 
condition of magnetic polarization which is acquired by a mass 
of soft iron when placed in a magnetic field ; and it was there- 
fore natural that in discussing the matter Faraday should 
introduce lines of electric force, similar to the lines of magnetic 
force which he had employed so successfully in his previous 
researches. A line of electric force he defined to be a curve 
whose tangent at every point has the same direction as the 
electric intensity. 

The changes which take place in an electric field when the 
dielectric is varied may be very simply described in terms of 
lines of force. Thus if a mass of sulphur, or other substance of 
high specific inductive capacity, is introduced into the field, 
the effect is as if the lines of force tend to crowd into it : as 
W. Thomson (Kelvin) showed later, they are altered in the 
same way as the lines of flow of heat, in a case of steady con- 
duction of heat, would be altered by introducing a body of 
greater conducting power for heat. By studying the figures of 
the lines of force in a great number of individual cases, Faraday 
was led to notice that they always dispose themselves as if they 
were subject to a mutual repulsion, or as if the tubes of force 
had an inherent tendency to dilate.* 

It is interesting to interpret by aid of these conceptions the 
law of Priestley and Coulomb regarding the attraction between 
two oppositely-charged spheres. In Faraday's view, the medium 
intervening between the spheres is the seat of a system of 
stresses, which may be represented by an attraction or tension 
along the lines of electric force at every point, together with a 

* Exp. Res., 1224, 1297 (1837). 
P 



210 Faraday. 

mutual repulsion of these lines, or pressure laterally. Where a 
line of force ends on one of the spheres, its tension is exercised 
on the sphere: in this way, every surface-element of each 
sphere is pulled outwards. If the spheres were entirely 
removed from each other's influence, the state of stress would be 
uniform round each sphere, and the pulls on its surf ace -elements 
would balance, giving no resultant force on the sphere. But 
when the two spheres are brought into each other's presence, 
the unit lines of force become somewhat more crowded together 
on the sides of the spheres which face than on the remote sides, 
and thus the resultant pull on either sphere tends to draw it 
toward the other. When the spheres are at distances great 
compared with their radii, the attraction is nearly proportional 
to the inverse square of the distance, which is Priestley's law. 

In the following year (1838) Faraday amplified* his theory 
of electrostatic induction, by making further use of the analogy 
with the induction of magnetism. Fourteen years previously 
Poisson had imaginedf an admirable model of the molecular 
processes which accompany magnetization; and this was now 
applied with very little change by Faraday to the case of induc- 
tion in dielectrics. " The particles of an insulating dielectric," 
he suggested, J " whilst under induction may be compared to a 
series of small magnetic needles, or, more correctly still, to a 
series of small insulated conductors. If the space round a 
charged globe were filled with a mixture of an insulating 
dielectric, as oil of turpentine or air, and small globular 
conductors, as shot, the latter being at a little distance from 
each other so as to be insulated, then these would in their 
condition and action exactly resemble what I consider to be 
the condition and action of the particles of the insulating 
dielectric itself. If the globe were charged, these little con- 
ductors would all be polar ; if the globe were discharged, they 
would all return to their normal state, to be polarized again 
upon the recharging of the globe/' 

That this explanation accounts for the phenomena of specific 

* Exp. Res., Series xiv. t Cf. p. 65. J Exp. Res., 1679. 



Faraday. 211 

inductive capacity may be seen by what follows, which is 
substantially a translation into electrostatical language of 
Poisson's theory of induced magnetism.* 

Let p denote volume-density of electric charge. For each 
of Faraday's " small shot " the integral 

JJJ pdx dy dz, 

integrated throughout the shot, will vanish, since the total 
charge of the shot is zero : but if r denote the vector (x, y, z), 

the integral 

J/J p r dx dy dz 

will not be zero, since it represents the electric polarization of 
the shot : if there are N shot per unit volume, the quantity 

P = &!!! P r dx dy dz 

will represent the total polarization per unit volume. If d 
denote the electric force, and E the average value of d, P will 
be proportional to E, say 

P - ( - 1) E. 

By integration by parts, assuming all the quantities concerned 
to vary continuously and to vanish at infinity, we have 

+ p * D * (x> y> z} ** dy ds = "Iff* ^ p ** dy dz> 

where ^ denotes an arbitrary function, and the volume-integrals 
are taken throughout infinite space. This equation shows that 
the polar-distribution of electric charge on the shot is equivalent 
to a volume- distribution throughout space, of density 

P = - div P. 

Now the fundamental equation of electrostatics may in 
suitable units be written, 

div d = p ; 

* W. Thomson (Kelvin), Camb. and Dub. Matb. Journal, November, 1845 ; 
"W. Thomson's Papers on Electrostatics and Magnetism, 43 sqq. ; F. 0. Mossotti, 
Arcb. des sc. phys. (Geneva) vi (1847), p. 193 ; Mem. della Soc. Ital. Modena, 
(2)xiv(1850), p. 49. 

P 2 



212 Faraday. 

and this gives on averaging 

div E = pi + jo, 

where pi denotes the volume-density of free electric charge, 
i.e. excluding that in the doublets ; or 

div (E + P) = Plt 
or div (* E) = p lt 

This is the fundamental equation of electrostatics, as modified 
in order to take into account the effect of the specific inductive 
capacity . 

The conception of action propagated step by step through a 
medium by the influence of contiguous particles had a firm hold 
on Faraday's mind, and was applied by him in almost every 
part of physics. " It appears to me possible," he wrote in 
1838,* " and even probable, that magnetic action may be 
communicated to a distance by the action of the intervening 
particles, in a manner having a relation to the way in which 
the inductive forces of static electricity are transferred to a 
distance ; the intervening particles assuming for the time more 
or less of a peculiar condition, which (though with a very 
imperfect idea) I have several times expressed by the term 
electro-tonic state."^ 

The same set of ideas sufficed to explain electric currents. 
Conduction, Faraday suggested,* might be " an action of 
contiguous particles, dependent on the forces developed in 
electrical excitement ; these forces bring the particles into a 
state of tension or polarity ; and being in this state the 
contiguous particles have a power or capability of communicating 
these forces, one to the other, by which they are lowered and 
discharge occurs." 

* Exp Res., 1729. 

f This name had been devised in 1831 to express the state of matter subject to 
magneto-electric induction ; cf. Exp.^Res., 60. 
J Exp. Res. iii, p. 513. 
As in electrostatic induction in dielectrics. 






Faraday. 213 

After working strenuously for the ten years which followed 
the discovery of induced currents, Faraday found in 1841 that 
his health was affected ; and for four years he rested. A second 
period of brilliant discoveries began in 1845. 

Many experiments had been made at different times by 
various investigators* with the purpose of discovering a 
connexion between magnetism and light. These had generally 
taken the form of attempts to magnetize bodies by exposure 
in particular ways to particular kinds of radiation ; and a 
successful issue had been more than once reported, only to be 
negatived on re-examination. 

The true path was first indicated by Sir John Herschel. 
After his discovery of the connexion between the outward form 
of quartz crystals and their property of rotating the plane of 
polarization of light, Herschel remarked that a rectilinear 
electric current, deflecting a needle to right and left all round 
it, possesses a helicoidal dissymmetry similar to that displayed 
by the crystals. " Therefore," he wrote,f " induction led me to 
conclude that a similar connexion exists, and must turn up 
somehow or other, between the electric current and polarized 
light, and that the plane of polarization would be deflected by 
magneto-electricity." 

The nature of this connexion was discovered by Faraday, 
who so far back as 1834J had transmitted polarized light 
through an electrolytic solution during the passage of the 
current, in the hope of observing a change of polarization. 
This early attempt failed ; but in September, 1845, he varied 
the experiment by placing a piece of heavy glass between the 
poles of an excited electro-magnet ; and found that the plane 
of polarization of a beam of light was rotated when the beam 
travelled through the glass parallel to the lines of force of the 
magnetic field. 

*e.g. by Morichini, of Rome, in 1813, Quart. Journ. Sci. xix, p. 338; by 
Samuel Hunter Christie, of Cambridge, in 1825, Phil. Trans., 1826, p. 219 ; and 
by Mary Somerville in the same year, Phil. Trans., 1826, p. 132. 

t Sir. J. Herschel in Bence Jones's Life of Faraday, p. 205. 

lExp. Res., 951. \Ib., 2152. 



214 Faraday. 

In the year following Faraday's discovery, Airy* suggested 
a way of representing the effect analytically; as might have been 
expected, this was by modifying the equations which had been 
already introduced by MacCullagh for the case of naturally 
active bodies. In Mac Cullagh's equations 
|8^F =c2 8 2 2 r + VZ 

d^Y 

the terms 8 3 ^/8# 3 and 8 3 F/8# 3 change sign with x, so that the 
rotation of the plane of polarization is always right-handed or 
always left-handed with respect to the direction of the beam. 
This is the case in naturally-active bodies ; but the rotation due 
to a magnetic field is in the same absolute direction whichever 
way the light is travelling, so that the derivations with respect 
to x must be of even order. Airy proposed the equations 
/8 2 F 2 8 2 F a 

I *-, * o 1 <*N 9 r^ *"\ / 



dx* dt' 

where p denotes a constant, proportional to the strength of the 
magnetic field which is used to produce the effect. He remarked, 
however, that instead of taking p dZ/dt and fj. 8 Y/dt as the additional 
terms, it would be possible to take 8 3 ^/8 3 and /u 8 3 F/8^ 3 , or 
im() 3 Z/dz?dt> and ju8 3 F/8^ 2 8^, or any other derivates in which the 
number of differentiations is odd with respect to t and even with 
respect to x. It may, in fact, be shown by the method pre- 
viously applied to Mac Cullagh's formulae that, if the equations 

are 

, 8 2 F 8 r+s ^ 



8 r+s F 



where (r + s) is an odd number, the angle through which the 

* Phil. Mag. xxviii (1846) p. 469. 



Faraday. 215 

plane of polarization rotates in unit length of path is a numerical 
multiple of 



where T denotes the period of the light. Now it was shown by 
Verdet* that the magnetic rotation is approximately proportional 
to the inverse square of the wave-length ; and hence we must 

have 

r + s= 3; 

so that the only equations capable of correctly representing 
Faraday's effect are either 



w ^dx-dt 

or 



dt* W * dt 

d'Z _ 2 d'Z 
W '" l ~W~^'W 

The former pair arise, as will appear later, in Maxwell's 
theory of rotatory polarization : the latter pair, which were 
suggested in 1868 by Boussinesq,f follow from that physical 
theory of the phenomenon which is generally accepted at the 
present time.* 

Airy's work on the magnetic rotation of light was limited 
in the same way as MacCullagh's work on the rotatory power 
of quartz ; it furnished only an analytical representation of the 
effect, without attempting to justify the equations. The earliest 
endeavour to provide a physical theory seems to have been 
made in 1858, in the inaugural dissertation of Carl Neumann, 

* Comptes Rendus, Ivi (1863), p. 630. 

t Journal de Math., xiii (1868), p. 430. 

J Fand Z being interpreted as components of electric force. 



216 Faraday. 

of Halle.* Neumann assumed that every element of an electric 
current exerts force on the particles of the aether ; and in parti- 
cular that this is true of the molecular currents which constitute 
magnetization, although in this case the force vanishes except 
when the aethereal particle is already in motion. If e denote the 
displacement of the aethereal particle ra, the force in question 
may be represented by the term 

km [ e. K] 

where K denotes the imposed magnetic field, and k denotes a 
magneto-optic constant characteristic of the body. When this 
term is introduced into the equations of motion of the aether, 
they take the form which had been suggested by Airy ; whence 
Neumann's hypothesis is seen to lead to the incorrect conclusion 
that the rotation is independent of the wave-length. 

The rotation of plane-polarized light depends, as Fresnel had 
shown,f on a difference between the velocities of propagation of 
the right-handed and left-handed circularly polarized waves into 
which plane-polarized light may be resolved. In the case of 
magnetic rotation, this difference was shown by Verdet to be 
proportional to the component of the magnetic force in the 
direction of propagation of the light; and CornuJ showed further 
that the mean of the velocities of the right-handed and left- 
handed waves is equal to the velocity of light in the medium 
when there is no magnetic field. From these data, by Fresnel's 
geometrical method, the wave- surf ace in the medium may be 
obtained; it is found to consist of two spheres (one relating 
to the right-handed and one to the left-handed light), each 
identical with the spherical wave-surface of the unmagnetized 
medium, displaced from each other along the lines of magnetic 
force. 

The discovery of the connexion between magnetism and 

* Explicare tentatur, quomodojiat, ut lucis planum polarisationis per vires el. vel 
mag. declinetur. Halis Saxonum, 1858. The results were republished in a tract 
Die magnetische Drehung der Polarisationsebene des Lichtes. Halle, 1863. 

t Cf. p. 174. J Comptes Rendus, xcii (1881), p. 1368. 

Cornu, Comptes Rendus, xcix (1884), p. 1045. 



Faraday. 217 

light gave interest to a short paper of a speculative character 
which Faraday published* in 1846, under the title " Thoughts 
on Kay- Vibrations." In this it is possible to trace the progress 
of Faraday's thought towards something like an electro-magnetic 
theory of light. 

Considering first the nature of ponderable matter, he suggests 
that an ultimate atom may be nothing else than a field of 
force electric, magnetic, and gravitational surrounding a point- 
centre ; on this view, which is substantially that of Michell and 
Boscovich, an atom would have no definite size, but ought 
rather to be conceived of as completely penetrable, and extend- 
ing throughout all space ; and the molecule of a chemical 
compound would consist not of atoms side by side, but of 
" spheres of power mutually penetrated, and the centres even 
coinciding."t 

All space being thus permeated by lines of force, Faraday 
suggested that light and radiant heat might be transverse 
vibrations propagated along these lines of force. In this way 
he proposed to " dismiss the aether," or rather to replace it by 
lines of force between centres, the centres together with their 
lines of force constituting the particles of material substances. 

If the existence of a luminiferous aether were to be admitted, 
Faraday suggested that it might be the vehicle of magnetic 
force ; " for," he wrote in 1851,{ "it is not at all unlikely that 
if there be an aether, it should have other uses than simply 
the conveyance of radiations." This sentence may be regarded 
as the origin of the electro-magnetic theory of light. 

At the time when the " Thoughts on Eay -Vibrations " were 
published, Faraday was evidently trying to comprehend every- 
thing in terms of lines of force ; his confidence in which had 
been recently justified by another discovery. A few weeks 
after the first observation of the magnetic rotation of light, he 
noticed that a bar of the heavy glass which had been used in 

* Phil. Mag. (3), xxviii (1846) : Exp. Res., iii, p. 447. 

t Cf. Bence Jones's Life of Faraday, ii, p. 178. 

J Exp. Res., $ 3075. Phil. Trans., 1846, p. 21 : Exp. Res., 2253. 



218 Faraday. 

this investigation, when suspended between the poles of an 
electro-magnet, set itself across the line joining the poles : thus 
behaving in the contrary way to a bar of an ordinary magnetic 
substance, which would tend to set itself along this line. A 
simpler manifestation of the effect was obtained when a cube 
or sphere of the substance was used ; in such forms it showed 
a disposition to move from the stronger to the weaker places 
of the magnetic field. The pointing of the bar was then seen 
to be merely the resultant of the tendencies of each of its 
particles to move outwards into the positions of weakest 
magnetic action. 

Many other bodies besides heavy glass were found to 
display the same property ; in particular, bismuth.* The name 
diamagnetic was given to them. 

" Theoretically," remarked Faraday, " an explanation of the 
movements of the diamagnetic bodies might be offered in the 
supposition that magnetic induction caused in them a contrary 
state to that which it produced in magnetic matter ; i.e. that if 
a particle of each kind of matter were placed in the magnetic 
field, both would become magnetic, and each would have its 
axis parallel to the resultant of magnetic force passing through 
it ; but the particle of magnetic matter would have its north 
and south poles opposite, or facing toward the contrary poles 
of the inducing magnet, whereas with the diamagnetic particles 
the reverse would be the case ; and hence would result approxi- 
mation in the one substance, recession in the other. Upon 
Ampere's theory, this view would be equivalent to the sup- 
position that, as currents are induced in iron and magnetics 
parallel to those existing in the inducing magnet or battery 
wire, so in bismuth, heavy glass, and diamagnetic bodies, the 
currents induced are in the contrary direction." 

This explanation became generally known as the " hypothesis 
of diamagnetic polarity " ; it represents diamagnetism as similar 

* The repulsion of bismuth in the magnetic field had been previously observed 
by A. Brugmans in 1778; Antonii Brugmans Magnetismus, Lugd. Bat., 1778. 
t Exp. Res., 2429. 



Faraday. 219 

to ordinary induced magnetism in all respects, except that the 
direction of the induced polarity is reversed. It was accepted 
by other investigators, notably by W. Weber, Pliicker, Eeich, 
and Tyndall ; but was afterwards displaced from the favour of 
its inventor by another conception, more agreeable to his peculiar 
views on the nature of the magnetic field. In this second 
hypothesis, Faraday supposed an ordinary magnetic or para- 
magnetic* body to be one which offers a specially easy passage 
to lines of magnetic force, so that they tend to crowd into 
it in preference to other bodies ; while he supposed a dia- 
magnetic body to have a low degree of conducting power for 
the lines of force, so that they tend to avoid it. " If, then," he 
reasoned,f "a medium having a certain conducting power occupy 
the magnetic field, and then a portion of another medium or 
substance be placed in the field having a greater conducting 
power, the latter will tend to draw up towards the place of 
greatest force, displacing the former." There is an electrostatic 
effect to which this is quite analogous ; a charged body attracts 
a body whose specific inductive capacity is greater than that of 
the surrounding medium, and repels a body whose specific 
inductive capacity is less; in either case the tendency is to 
afford the path of best conductance to the lines of force .J 

For some time the advocates of the "polarity" and 
" conduction " theories of diarnagnetism carried on a contro- 
versy which, indeed, like the controversy between the adherents 
of the one-fluid and two-fluid theories of electricity, persisted 
after it had been shown that the rival hypotheses were mathe- 
matically equivalent, and that no experiment could be suggested 
which would distinguish between them. 

Meanwhile new properties of magnetizable bodies were being 
discovered. In 1847 Julius Pliicker (b. 1801, d. 1868), Professor 
of Natural Philosophy in the University of Bonn, while 
repeating and extending Faraday's magnetic experiments, 

* This term was introduced by Faraday, Exp. Res., 2790. 
t Exp. es., 2798. 

J The mathematical theory of the motion of a magnetizable body in a non- 
uniform field of force was discussed by "W. Thomson (Kelvin) in 1847. 



220 Faraday. 

observed* that certain uniaxal crystals, when placed between 
the two poles of a magnet, tend to set themselves so that the 
optic axis has the equatorial position. At this time Faraday 
was continuing his researches ; and, while investigating the 
diamagnetic properties of bismuth, was frequently embarrassed 
by the occurrence of anomalous results. In 1848 he ascertained 
that these were in some way connected with the crystalline 
form of the substance, and showedf that when a crystal of 
bismuth is placed in a field of uniform magnetic force (so that 
no tendency to motion arises from its diamagnetism) it sets 
itself so as to have one of its crystalline axes directed along 
the lines of force. 

At first he supposed this effect to be distinct from that 
which had been discovered shortly before by Pliicker. " The 
results," he wrote,J " are altogether very different from those 
produced by diamagnetic action. They are equally distinct from 
those discovered and described by Pliicker, in his beautiful 
researches into the relation of the optic axis to magnetic action ; 
for there the force is equatorial, whereas here it is axial. So 
they appear to present to us a new force, or a new form of 
force, in the molecules of matter, which, for convenience sake, 
I will conventionally designate by a new word, as the magne- 
crystallic force." Later in the same year, however, he recognized^ 
that " the phaenomena discovered by Pliicker and those of which 
I have given an account have one common origin and cause." 

The idea of the " conduction " of lines of magnetic force by 
different substances, by which Faraday had so successfully 
explained the phenomena of diamagnetism, he now applied to 
the study of the magnetic behaviour of crystals. " If," he wrote,|| 
"the idea of conduction be applied to these magnecrystallic 
bodies, it would seem to satisfy all that requires explanation in 
their special results. A magnecrystallic substance would then 
be one which in the crystallized state could conduct onwards, or 

* Ann. d. Phys. Ixxii (1847), p. 315; Taylor's Scientific Memoirs, v, p. 353. 

t Phil. Trans., 1849, p. 1 ; Exp. Res., 2454. 

i Exp. Res., 2469. Ibid., 2605. || Ibid., 2837. 



Faraday. 221 

permit the exertion of the magnetic force with more facility in 
one direction than another ; and that direction would be the 
magnecrystallic axis. Hence, when in the magnetic field, the 
magnecrystallic axis would be urged into a position coincident 
with the magnetic axis, by a force correspondent to that 
difference, just as if two different bodies were taken, when the 
one with the greater conducting power displaces that which is 
weaker." 

This hypothesis led Faraday to predict the existence of 
another type of magnecrystallic effect, as yet unobserved. " If 
such a view were correct/' he wrote,* " it would appear to 
follow that a diamagnetic body like bismuth ought to be less 
diamagnetic when its magnecrystallic axis is parallel to the 
magnetic axis than when it is perpendicular to it. In the two 
positions it should be equivalent to two substances having 
different conducting powers for magnetism, and therefore if 
submitted to the differential balance ought to present 
differential phaenomena." This expectation was realized when 
the matter was subjected to the test of experiment. f 

The series of Faraday's " Experimental Researches in 
Electricity " end in the year 1855. The closing period of his 
life was quietly spent at Hampton Court, in a house placed at 
his disposal by the kindness of the Queen ; and here on August 
25th, 1867, he passed away. 

Among experimental philosophers Faraday holds by uni- 
versal consent the foremost place. The memoirs in which his 
discoveries are enshrined will never cease to be read with 
admiration and delight; and future generations will preserve 
with an affection not less enduring the personal records and 
familiar letters, which recall the memory of his humble and 
unselfish spirit. 

*Exp. Res., 2839. ^ Ibid., 2841. 



222 The Mathematical Electricians of the 



CHAPTEK VII. 

THE MATHEMATICAL ELECTRICIANS OF THE MIDDLE OF THE 
NINETEENTH CENTURY. 

WHILE Faraday was engaged in discovering the laws of induced 
currents in his own way, by use of the conception of lines of 
force, his contemporary Franz Neumann was attacking the 
same problem from a different point of view. Xeumann 
preferred to take Ampere as his model ; and in 1845 published 
a memoir,* in which the laws of induction of currents were 
deduced by the help of Ampere's analysis. 

Among the assumptions on which Neumann based his work 
was a rule which had been formulated, not long after Faraday's 
original discovery, by Emil Lenz,f and which may be enunciated 
as follows : when a conducting circuit is moved in a magnetic 
field, the induced current flows in such a direction that the 
ponderomotive forces on it tend to oppose the motion. 

Let ds denote an element of the circuit which is in motion, 
and let C ds denote the component, taken in the direction of 
motion, of the ponderomotive force exerted by the inducing 
current on d$, when the latter is carrying unit current ; so that 
the value of C is known from Ampere's theory. Then Lenz's 
rule requires that the product of C into the strength of the 
induced current should be negative. Xeumann assumed that 
this is because it consists of a negative coefficient multiplying 
the square of C\ that is, he assumed the induced electro- 
motive force to be proportional to C. He further assumed it to 
be proportional to the velocity v of the motion; and thus 
obtained for the electromotive force induced in ds the expression 

- ei-Cds, 
where e denotes a constant coefficient. By aid of this formula, 

Berlin Abhandlungen, 1845, p. 1 ; 1848, p. 1 ; reprinted as Xo. 10 and 
No. 36 of Ostwald's Klassiker-, translated Journal de Math, xiii (1848), p. 113. 
t Ann. d. Phys. xxxi (1834), p. 483. 



Middle of the Nineteenth Century. 223 

in the earlier part* of the memoir, he calculated the induced 
currents in various particular cases. 

But having arrived at the formulae in this way, Neumann 
noticedf a peculiarity in them which suggested a totally 
different method of treating the subject. In fact, on examining 
the expression for the current induced in a circuit which is in 
motion in the field due to a magnet, it appeared that this 
induced current depends only on the alteration caused by the 
motion in the value of a certain function ; and, moreover, that 
this function is no other than the potential of the ponderomotive 
forces which, according to Ampere's theory, act between the 
circuit, supposed traversed by unit current, and the magnet. 

Accordingly, Neumann now proposed to reconstruct his 
theory by taking this potential function as the foundation. 

The nature of Neumann's potential, and its connexion 
with Faraday's theory, will be understood from the following 
considerations : 

The potential energy of a magnetic molecule M in a field 
of magnetic intensity B is (B . M) ; and therefore the potential 
energy of a current i flowing in a circuit s in this field is 



where S denotes a diaphragm bounded by the circuit s ; as is 
seen at once on replacing the circuit by its equivalent magnetic 
shell S. If the field B be produced by a current i' flowing in a 
circuit s', we have, by the formula of Biot and Savart, 



1 *' 

curl 



* 1-8. It may be remarked that Neumann, in making use of Ohm's law, 
was (like everyone else at this time) unaware of the identity of electroscopic 
force with electrostatic potential. t 9. 



224 The Mathematical Electricians of the 

Hence, the mutual potential energy of the two currents is 

- . dS 



which hy Stokes's transformation may be written in the form 

(ds.ds') 



This expression represents the amount of mechanical work 
which must be performed against the electro-dynamic pondero- 
motive forces, in order to separate the two circuits to an infinite 
distance apart, when the current-strengths are maintained 
unaltered. 

The above potential function has been obtained by con- 
sidering the ponderomotive forces ; but it can now be connected 
with Faraday's theory of induction of currents. For by 
interpreting the expression 

(B . dS) 



If' 



in terms of lines of force, we see that the potential function 
represents the product of i into the number of unit-lines of 
magnetic force due to s' t which pass through the gap formed by 
the circuit s ; and since by Faraday's law the currents induced 
in s depend entirely on the variation in the number of these 
lines, it is evident that the potential function supplies all that 
is needed for the analytical treatment of the induced currents. 
This was Neumann's discovery. 

The electromotive force induced in a circuit s by the motion 
of other circuits s', carrying currents i' t is thus proportional to 
the time-rate of variation of the potential 

(ds.ds'). 



so that if we denote by a the vector 



Middle of the Nineteenth Century. 225 

which, of course, is a function of the position of the element ds 
from which r is measured, then the electromotive force induced 
in any circuit-element ds by any alteration in the currents 

which give rise to a is 

(a. ds). 

The induction of currents is therefore governed by the vector a ; 
this, which is generally known as the vector-potential, has from 
Neumann's time onwards played a great part in electrical theory. 
It may be readily interpreted in terms of Faraday's conceptions ; 
for (a . ds) represents the total number of unit lines of magnetic 
force which have passed across the line-element ds prior to the 
instant t. The vector-potential may in fact be regarded as the 
analytical measure of Faraday's electrotonic state* 

While Neumann was endeavouring to comprehend the laws 
of induced currents in an extended form of Ampere's theory, 
another investigator was attempting a still more ambitious 
project : no less than that of uniting electrodynamics into a 
coherent whole with electrostatics. 

Wilhelm Weber (6. 1804, d. 1890) was in the earlier part of 
his scientific career a friend and colleague of Gauss at Gottingen. 
In 1837, however, he became involved in political trouble. The 
union of Hanover with the British Empire, which had subsisted 
since the accession of the Hanoverian dynasty to the British 
throne, was in that year dissolved by the operation of the Salic 
law ; the Princess Victoria succeeded to the crown of England, 
and her uncle Ernest- Augustus to that of Hanover. The new 
king, who was a pronounced reactionary, revoked the free 
constitution which the Hanoverians had for some time enjoyed ; 
and Weber, who took a prominent part in opposing this action, 
was deprived of his professorship. From 1843 to 1849, when 
his principal theoretical researches in electricity were made, 
he occupied a chair in the University of Leipzig. 

The theory of Weber was in its origin closely connected 
with the work of another Leipzig professor, Fechner, who in 
1845f introduced certain assumptions regarding the nature of 

* Cf. pp. 212, 272. t Ann. d. Phys. Lxiv (1845), p. 337. 

Q 



226 The Mathematical Electricians of the 

electric currents. Fechner supposed every current to consist in 
a streaming of electric charges, the vitreous charges travelling 
in one direction, and the resinous charges, equal to them in 
magnitude and number, travelling in the opposite direction with 
equal velocity. He further supposed that like charges attract 
each other when they are moving parallel to the same direction, 
while unlike charges attract when they are moving in opposite 
directions. On these assumptions he succeeded in bringing 
Faraday's induction effects into connexion with Ampere's laws 
of electrodynamics. 

In 1846 Weber,* adopting the same assumptions as Fechner, 
analysed the phenomena in the following way : 

The formula of Ampere for the ponderomotive force between 
two elements ds, ds' of currents i t i ', may be written 



r ds ds r 2 ds ds' 

Suppose now that X units of vitreous electricity are contained 
in unit length of the wire s, and are moving with velocity u ; 
and that an equal quantity of resinous electricity is moving 
with velocity u in the opposite direction ; so that 



Let X', u', denote the corresponding quantities for the other 
current; and let the suffix ! be taken to refer to the action 
between the positive charges in the two wires, the suffix 2 to 
the action between the positive charge in s and the negative 
charge in s, the suffix 3 to the action between the negative 
charge in s and the positive charge in s', and the suffix 4 to the 
action between the negative charges in the two wires. Then 
we have 

'dr\ dr , dr 

= u + u ,, 
dtji ds ds 

* Elektrodynamische Maassbestimmungen, Leipzig Abhandl., 1846 : Ann. d. 
Phys. hcxiii (1848), p. 193: English translation in Taylor's Scientific Memoirs, 
v (1852), p. 489. 



Middle of the Nineteenth Century. 227 

and 

fd z r\ z d z r , c?r <Fr 

__ = u * __ + 2uu --?-, + u 2 -j-7- 
df <fo* dsds ds* 



By aid of these and the similar equations with the suffixes 3 , 3 , 4 , 
the equation for the ponderomotive force may be transformed 
into the equation 

d?r\ f dV 

I nt 

A A' tl&rtst' \ \ ///z /, \ ///* / 

F = 



But this is the equation which we should have obtained 
had we set out from the following assumptions : that the 
ponderomotive force between two current-elements is the 
resultant of the force between the positive charge in ds and the 
positive charge in ds', of the force between the positive charge 
in ds and the negative charge in ds t etc. ; and that any two 
electrified particles of charges e and e', whose distance apart 
is r, repel each other with a force of magnitude 



* l & 

Two such charges would, of course, also exert on each other an 
electrostatic repulsion, whose magnitude in these units would 
be eec'/r 2 , where c denotes a constant* of the dimensions of a 
velocity, whose value is approximately 3 x 10 10 cm./sec. So 
that on these assumptions the total repellent force would be 

ee'c z f rr r* 



* The units which have been adopted in the above investigation depend on the 
electrodynamic actions of currents ; i.e. they are such that two unit currents flowing 
in parallel circular circuits at a certain distance apart exert unit ponderomotive 
force on each other. The quantity of electricity conveyed in unit time by such a 
unit current is adopted as the unit~eharge. This unit charge is not identical with 
the electrostatic unit charge, which is definedHqbe such that two unit charges at 
unit distance apart repel each other with unit poniieiQmotive force. Hence the 
necessity for introducing the factor c. 

Q 



228 The Mathematical Electricians of the 

This expression for the force between two electric charges 
was taken by Weber as the basis of his theory. Weber's is the 
first of the electron-theories a name given to any theory which 
attributes the phenomena of electrodynamics to the agency 
of moving electric charges, the forces on which depend not 
only on the position of the charges (as in electrostatics), but 
also on their velocity. 

The latter feature of Weber's theory led its earliest critics 
to deny that his law of force could be reconciled with the 
principle of conservation of energy. They were, however, 
mistaken on this point, as may be seen from the following 
considerations. The above expression for the force between 
two charges may be written in the form 



where U denotes the expression 

ee'c~ 



Consider now two material particles at distance r apart, whose 
mechanical kinetic energy is T, and whose mechanical potential 
energy is F, and which carry charges e and e'. The equations 
of motion of these particles will be exactly the same as the 
equations of motion of a dynamical system for which the 
kinetic energy is 

ee'i* 






and the potential energy is 



To such a system the principle of conservation of energy may be 
applied : the equation of energy is, in fact, 

m -rr 1 > 6e ' G " 

T + V - ee r + - = constant. 
2r r 



Middle of the Nineteenth Century. 229 

The first objection made to Weber's theory is thus disposed 
of ; but another and more serious one now presents itself. The 
occurrence of the negative sign with the term - ee'r^/Zr implies 
that a charge behaves somewhat as if its mass were negative, so 
that in certain circumstances its velocity might increase indefi- 
nitely under the action of a force opposed to the motion. This 
is one of the vulnerable points of Weber's theory, and has been 
the object of much criticism. In fact,* suppose that one charged 
particle of mass /z. is free to move, and that the other charges 
are spread uniformly over the surface of a hollow spherical 
insulator in which the particle is enclosed. The equation of 
conservation of energy is 

^(fi-ep)v*+ V= constant, 

where e denotes the charge of the particle, v its velocity, V its 
potential energy with respect to the mechanical forces which act 
on it, and p denotes the quantity 

- cos-(v.r)dS, 

where the integration is taken over the sphere, and where o- 
denotes the surface-density ; p is independent of the position of 
the particle p within the sphere. If now the electric charge on 
the sphere is so great that ep is greate^-tbsciTT^ then v 2 and V 
must increase and diminish together; which is evidently absurd. 

Leaving this objection unanswered, we proceed to show how 
Weber's law of force between electrons leads to the formulae 
for the induction of currents. 

The mutual energy of two moving charges is 

~\ ~2cV' 

r ! * " L"v r ' Y ' ~1' 

r |_ * c r J 

where v and v' denote the velocities of the charges ; so that the 

* This example was given by Helmholtz, Journal fur Math. Ixxv (1873), p. 35 ; 
Phil. Mag. xliv (1872), p. 530. 



230 The Mathematical Electricians of the 

mutual energy of two current-elements containing charges e, e 
respectively of each kind of electricity, is 






r 3 



If ds, ds' denote the lengths of the elements, and i, i f the currents 
in them, we have 

ids = 2ev, i'ds' = 2V ; 

so the mutual energy of two current-elements is 

nf 

-(r.ds').(r.ds). 

The mutual energy of ids with all the other currents is therefore 

t(dt.a), 
where a denotes a vector-potential 



By reasoning similar to Neumann's, it may be shown that the 
electromotive force induced in ds by any alteration in the rest 
of the field is 

-(ds.a); 

and thus a complete theory of induced currents may be 
constructed. 

The necessity for induced currents may be inferred by 
general reasoning from the first principles of Weber's theory. 
When a circuit s moves in the field due to currents, the velocity 
of the vitreous charges in s is, owing to the motion of s, not 
equal and opposite to that of the resinous charges : this gives 
rise to a difference in the forces acting on the vitreous and 
resinous charges in s ; and hence the charges of opposite sign 
separate from each other and move in opposite directions. 

The assumption that positive and negative charges move 
with equal and opposite velocities relative to the matter of 



Middle of the Nineteenth Century. 231 

the conductor is one to which, for various reasons which will 
appear later, objection may be taken ; but it is an integral part 
of Weber's theory, and cannot be excised from it. In fact, 
if this condition were not satisfied, and if the law of force were 
Weber's, electric currents would exert forces on electrostatic 
charges at rest*; as may be seen by the following example. 
Let a current flow in a closed circuit formed by arcs of two 
concentric circles and the portions of the radii connecting their 
extremities; then, if Weber's law were true, and if only one 
kind of electricity were in motion, the current would evidently 
exert an electrostatic force on a charge placed at the centre of 
the circles. It has been shown,f indeed, that the assumption 
of opposite electricities moving with equal and opposite veloci- 
ties in a circuit is almost inevitable in any theory of the type 
of Weber's, so long as the mutual action of two charges is 
assumed to depend only on their relative (as opposed to their 
absolute) motion. 

The law of Weber is not the only one of its kind; an alterna- 
tive to it was suggested by Bernhard Eiemann (b. 1826, d. 1866), 
in a course of lectures which were delivered^ at Gottingen 
in 1861, and which were published after his death by 
K. Hattendorff. Kiemann proposed as the electrokinetic 
energy of two electrons e (x, y, z) and e\x f , y\ z') the expression 



this differs from the corresponding expression given by Weber 
only in that the relative velocity of the two electrons is 
substituted in place of the component of this velocity along 
the radius vector. Eventually, as will be seen later, the laws 

* This remark was first made by Clausius, Journal fur Math. Ixxxii (1877), 
p. 86: the simple proof given above is due to Grassmann, Journal fur Math. 
Ixxxiii (1877), p. 57. 

t H. Lorberg, Journal fur Math. Ixxxiv (1878), p. 305. 

J Schicere, Elektricitat und Magnetismus, nach den Vorlesungen von B. Riemann : 
Hannover, 1875, p. 326. Another alternative to Weber's law had been discovered 
by Gauss so far back as 1835, but was not published until after his death: cf. 
Gauss' Werke, v, p. 616. 






232 The Mathematical Electricians of the 

of Riemann and Weber were both abandoned in favour of a 
third alternative. 

At the time, however, Weber's discovery was felt to be a 
great advance ; and indeed it had, perhaps, the greatest share 
in awakening mathematical physicists to a sense of the possi- 
bilities latent in the theory of electricity. Beyond this, its 
influence was felt in general dynamics ; for Weber's electro- 
kinetic energy, which resembled kinetic energy in some respects 
and potential energy in others, could not be precisely classified 
under either head ; and its introduction, by helping to break 
down the distinction which had hitherto subsisted between the 
two parts of the kinetic potential, prepared the way for the 
modern transformation-theory of dynamics.* 

Another subject whose development was stimulated by the 
work of Weber was the theory of gravitation. That gravitation 
is propagated by the action of a medium, and consequently is a 
process requiring time for its accomplishment, had been an article 
of faith with many generations of physicists. Indeed, the 
dependence of the force on the distance between the attracting 
bodies seemed to suggest this idea ; for a propagation which is 
truly instantaneous would, perhaps, be more naturally conceived 
to be effected by some kind of rigid connexion between the 
bodies, which would be more likely to give a force independent 
of the mutual distance. 

It is obvious that, if the simple law of Newton is abandoned, 
there is a wide field of rival hypotheses from which to choose 
its successor. The first notable attempt to discuss the question 
was made by Laplace. f Laplace supposed gravity to be pro- 
duced by the impulsion on the attracted body of a " gravific 
fluid," which flows with a definite velocity toward the centre 
of attraction say, the sun. If the attracted body or planet 
is in motion, the velocity of the fluid relative to it will be 
compounded of the absolute velocity of the fluid and the 
reversed velocity of the planet, and the force of gravity will 

* Cf. "Whittaker, Analytical Dynamics, chapters ii, iii, xi. 
t Meeanique Celeste, Livre x, chap, vii, 22. 



Middle of the Nineteenth Century. 233 

act in the direction thus determined, its magnitude being 
unaltered by the planet's motion. This amounts to supposing 
that gravity is subject to an aberrational effect similar to that 
observed in the case of light. It is easily seen that the modi- 
fication thus introduced into Newton's law may be represented 
by an additional perturbing force, directed along the tangent 
to the orbit in the opposite sense to the motion, and pro- 
portional to the planet's velocity and to the inverse square of 
the distance from the sun. By considering the influence of 
this force on the secular equation of the moon's motion, Laplace 
found that the velocity of the gravific fluid must be at least a 
hundred million times greater than that of light. 

The assumptions made by Laplace are evidently in the 
highest degree questionable; but the generation immediately 
succeeding, overawed by his fame, seems to have found no way 
of improving on them. Under the influence of Weber's ideas, 
however, astronomers began to think of modifying Newton's 
law by^ adding a term involving the velocities of the bodies. 
Tisserand* in 1872 discussed the motion of the planets round 
the sun on the supposition that the law of gravitation is the 
same as Weber's law of electrodynamic action, so that the 
force is 



jp = / _^r n . -?.r J 

* . i x nv^* i i r ^i 



where / denotes the constant of gravitation, ra the mass of 
the planet, // the mass of the sun, r the distance of the planet 
from the sun, and h the velocity of propagation of gravitation. 
The equations of motion may be rigorously integrated by 
the aid of elliptic functions!; but the simplest procedure is 
to write 



* Comptes Rendus, Ixxv (1872), p. 760. Of. also Comptes Rendus, ex (1890), 
p. 313, and Holzmiiller, Zeitschrif t f iir Math. u. Phys., 1870, p. 69. 

t This had been done in an inaugural dissertation by Seegers, Gottingen, 1864. 



234 The Mathematical Electricians of the 

and, regarding F\ as a perturbing function, to find the variation 
of the constants of elliptic motion. Tisserand showed that the 
perturbations of all the elements are zero or periodic, and quite 
insensible, except that of the longitude of perihelion, which has 
a secular part. If A be assumed equal to the velocity of light, 
the effect would be to rotate the major axis of the orbit of 
Mercury in the direct sense 14" in a century. 

Now, as it happened, a discordance between theory and 
observation was known to exist in regard to the motion of 
Mercury's perihelion ; for Le Verrier had found that the attrac- 
tion of the planets might be expected to turn the perihelion 
527" in the direct sense in a century, whereas the motion 
actually observed was greater than this by 38". It is evident, 
however, that only f of the excess is explained by Tisserand's 
adoption of "Weber's law; and it seemed therefore that this 
suggestion would prove as unprofitable as Le Terrier's own 
hypothesis of an intra-mercurial planet. But it was found 
later* that f of the excess could be explained by substituting 
Eiemann's electrodynamic law for Weber's, and that a com- 
bination of the laws of Biemann and Weber would give exactly 
the amount desired.f 

After the publication of his memoir on the law of force 
between electrons, Weber turned his attention to the question 
of diamagnetism, and developed Faraday's idea regarding the 
explanation of diamagnetic phenomena by the effects of electric 
currents induced in the diamagnetic bodies.^ Weber remarked 
that if, with Ampere, we assume the existence of molecular 
circuits in which there is no ohmic resistance, so that currents 
can flow without dissipation of energy, it is quite natural to 
suppose that currents would be induced in these molecular 

* By Maurice Levy, Comptes Eendus, ex (1890), p. 545. 

t The consequences of adopting the electrodynamic law of Clausius (for which 
see later) were discussed by Oppenheim, Zur Frage nach der Fortpflanzungs- 
geschwindigJceit der Gravitation, Wien, 1895. 

I Leipzig Berichte, i (1847), p. 346 ; Ann. d. Phys. Ixxiii (1848), p. 241 ; 
translated Taylor's Scientific Memoirs, v, p. 477 ; Abhandl. der K. Sachs. Ges. i 
(1852), p. 483; Ann. d. Phys. Ixxxvii (1852), p. 145; trans. Tyndall and 
Francis' Scientific Memoirs, p. 163. 



Middle of the Nineteenth Century. 235 

circuits if they were situated in a varying magnetic field ; and 
he pointed out that such induced molecular currents would 
confer upon the substance the properties characteristic of 
dia magnetism. 

The difficulty with this hypothesis is to avoid explaining too 
much ; for, if it be accepted, the inference seems to be that all 
bodies, without exception, should be diamagnetic. Weber escaped 
from this conclusion by supposing that in iron and other 
magnetic substances there exist permanent molecular currents, 
which do not owe their origin to induction, and which, under 
the influence of the impressed magnetic force, set themselves in 
definite orientations. Since a magnetic field tends to give such 
a direction to a pre-existing current that its course becomes 
opposed to that of the current which would be induced by the 
increase of the magnetic force, it follows that a substance stored 
with such pre-existing currents would display the phenomena 
of paramagnetism: t The bodies ordinarily called paramagnetic 
are, according to this hypothesis, those bodies in which the 
paramagnetism is strong enough to mask the diamagnetism. 

The radical distinction which Weber postulated between the 
natures of paramagnetism and diamagnetism accords with many 
facts which have been discovered subsequently. Thus in 1895 
P. Curie showed* that the magnetic susceptibility per gramme- 
molecule is connected with the temperature by laws which are 
different for paramagnetic and diamagnetic bodies. For the 
former it varies in inverse proportion to the absolute tempe- 
rature, whereas for diamagnetic bodies it is independent of the 
temperature. 

The conclusions which followed from the work of Faraday 
and Weber were adverse to the hypothesis of magnetic fluids ; 
for according to that hypothesis the induced polarity would be 
in the same direction whether due to a change of orientation of 
pre-existing molecular magnets, or to a fresh separation of 
magnetic fluids in the molecules. " Through the discovery of 

* Annales de Chimie (7) v (1845), p. 289. 



236 The Mathematical Electricians of the 

diamagnetism," wrote Weber* in 1852, "the hypothesis of 
electric molecular currents in the interior of bodies is cor- 
roborated, and the hypothesis of magnetic fluids in the interior 
of bodies is refuted." The latter hypothesis is, moreover, unable 
to account for the phenomena shown by bodies which are 
strongly magnetic, like iron : for it is found that when the 
magnetizing force is gradually increased to a very large value, 
the magnetization induced in such bodies does not increase in 
proportion, but tends to a saturation value This effect cannot 
be explained on the assumptions of Poisson,but is easily deducible 
from those of Weber; for, according to Weber's theory, the 
magnetizing force merely orients existing magnets ; and when it 
has attained such a value that all of them are oriented in the 
same direction, there is nothing further to be done, 

Weber's theory in its original form is, however, open to 
some objection. If the elementary magnets are supposed to be 
free to orient themselves without encountering any resistance, 
it is evident that a very small magnetizing force would suffice 
to turn them all parallel to each other, and thus would produce 
immediately the greatest possible intensity of induced magnetism. 
To overcome this difficulty, Weber assumed that every displace- 
ment of a molecular circuit is resisted by a couple, which tends 
to restore the circuit to its original orientation. This assump- 
tion fails, however, to account for the fact that iron which 
has been placed in a strong magnetic field does not return 
to its original condition when it is removed from the field, 
but retains a certain amount of residual magnetization. 

Another alternative was to assume a frictional resistance 
to the rotation of the magnetic molecules ; but if such a 
resistance existed, it could be overcome only by a finite 
magnetizing force ; and this inference is inconsistent with the 
observation that some degree of magnetization is induced by 
every force, however feeble. 

The hypothesis which has ultimately gained acceptance is 
that the orientation is resisted by couples which arise from the 

* Ann. d. Phys. lxxxvii(1852), p. 145 ; Tyndall and Francis' Sci. Mem., p. 163. 



Middle of the Nineteenth Century. 237 

mutual action of the molecular magnets themselves. In the 
unmagnetized condition the molecules " arrange themselves so 
as to satisfy their mutual attraction by the shortest path, and 
thus form a complete closed circuit of attraction," as D. E. 
Hughes wrote* in 1883 ; when an external magnetizing force 
is applied, these small circuits are broken up ; and at any stage 
of the process a molecular magnet is in equilibrium under the 
joint influence of the external force and the forces due to the 
other molecules. 

This hypothesis was suggested by Maxwell,t and has been 
since developed by J. A. Ewing;J its consequences may be 
illustrated by the following simple example : 

Consider two magnetic molecules, each of magnetic moment 
m, whose centres are fixed at a distance c apart. When 
undisturbed, they dispose themselves in the position of stable 
equilibrium, in which they point in the same direction along 
the line c. Now let an increasing magnetic force H be made 
to act on them in a direction at right angles to the line c. 
The magnets turn towards the direction of H ; and when 
H attains the value Sm/c 3 , they become perpendicular to the 
line c, after which they remain in this position, when H is 
increased further. Thus they display the phenomena of induc- 
tion initially proportional to the magnetizing force, and of 
saturation. If the magnetizing force H be supposed to act 
parallel to the line c, in the direction in which the axes 
originally pointed, the magnets will remain at rest. But if H 
acts in the opposite direction, the equilibrium will be stable 
only so long as H is less than ra/c 3 ; when H increases 
beyond this limit, the equilibrium becomes unstable, and the 
magnets turn over so as to point in the direction of H\ when 
H is gradually decreased to zero, they remain in their new posi- 
tions, thus illustrating the phenomenon of residual magnetism. 

* Proc. Roy. Soc. xxxv (1883), p. 178. 
f Treatise on Elect. $ May., 443. 

I Phil. Mag. xxx (1890), p. 205 ; Magnetic Induction in Iron atid other Metals,. 
1891. 

E. G. Gallop, Messenger of Math, xxvii (1897), p. 6. 



238 The Mathematical Electricians of the 

By taking a large number of such pairs of magnetic molecules, 
originally oriented in all directions, and at such distances that 
the pairs do not sensibly influence each other, we may 
construct a model whose behaviour under the influence of 
an external magnetic field will closely resemble the actual 
behaviour of ferromagnetic bodies. 

In order that the magnets in the model may come to rest 
in their new positions after reversal, it will be necessary to 
suppose that they experience some kind of dissipative force 
which damps the oscillations ; to this would correspond in 
actual magnetic substances the electric currents which would 
be set up in the neighbouring mass when the molecular 
magnets are suddenly reversed ; in either case, the sudden 
reversals are attended by a transformation of magnetic energy 
into heat. 

The transformation of energy from one form to another is a 
subject which was first treated in a general fashion shortly 
before the middle of the nineteenth century. It had long been 
known that the energy of motion and the energy of position 
of a dynamical system are convertible into each other, and 
that the amount of their sum remains invariable when the 
system is self-contained. This principle of conservation of 
dynamical energy had been extended to optics by Fresnel, who 
had assumed* that the energy brought to an interface by 
incident light is equal to the energy carried away from the 
interface by the reflected and refracted beams. A similar 
conception was involved in Eoget's and Faraday's defencef of 
the chemical theory of the voltaic cell ; they argued that the 
work done by the current in the outer circuit must be provided 
at the expense of the chemical energy stored in the cell, and 
showed that the quantity of electricity sent round the circuit 
is proportional to the quantity of chemicals consumed, while 
its tension is proportional to the strength of the chemical 
affinities concerned in the reaction. This theory was extended 

*Cf. p. 133. tCf. p. 203. 



Middle of the Nineteenth Century. 239 

and completed by James Prescott Joule, of Manchester, in 1841. 
Joule, who believed* that heat is producible from mechanical 
work and convertible into it, measuredf the amount of heat 
evolved in unit time in a metallic wire, through which a 
current of known strength was passed; he found the amount 
to be proportional to the resistance of the wire multiplied by 
the square of the current- strength ; or (as follows from Ohm's 
law) to the current-strength multiplied by the difference of 
electric tensions at the extremities of the wire. 

The quantity of energy yielded up as heat in the outer 
circuit being thus known, it became possible to consider the 
transference of energy in the circuit as a whole. " When," 
wrote Joule, " any voltaic arrangement, whether simple or 
compound, passes a current of electricity through any substance, 
whether an electrolyte or not, the total voltaic heat which is 
generated in any time is proportional to the number of atoms 
which are electrolyzed in each cell of the circuit, multiplied 
by the virtual intensity of the battery : if a decomposing cell 
be in the circuit, the virtual intensity of the battery is reduced 
in proportion to its resistance to electrolyzation." In the same 
year hej enhanced the significance of this by showing that the 
quantities of heat which are evolved by the combustion of the 
equivalents of bodies are proportional to the intensities of their 
affinities for oxygen, as measured by the electromotive force 
of a battery required to decompose the oxide electrolytically. 

The theory of Koget and Faraday, thus perfected by Joule, 
enables us to trace quantitatively the transformations of energy 
in the voltaic cell and circuit. The primary source of energy 
is the chemical reaction : in a Daniell cell, ZnjZn SOJCu S0 4 |Cu, 
for instance, it is the substitution of zinc for copper as the 
partner of the sulphion. The strength of the chemical affinities 
concerned is in this case measured by the difference of the heats 
of formation of zinc sulphate and copper sulphate ; and it is 

*Cf. p. 33. 

t Phil. Mag. xix (1841), p. 260 ; Joule's Scientific Papers i, p. 60. 
I Phil. Mag. xx (1841), p. 98 : cf. also Phil. Mag. xxii (1843), p. 204. 



240 The Mathematical Electricians of the 

this which determines the electromotive force of the cell.* 
The amount of energy which is changed from the chemical to 
the electrical form in a given interval of time is measured by 
the product of the strength of the chemical affinity into the 
quantity of chemicals decomposed in that time, or (what is the 
same thing) by the product of the electromotive force of the 
cell into the quantity of electricity which is circulated. This 
energy may be either dissipated as heat in conformity to 
Joule's law, or otherwise utilized in the outer circuit. 

The importance of these principles was emphasized by 
Hermann von Helmholtz (b. 1821, d. 1894), in a memoir which 
was published in 1847, and which will be more fully noticed 
presently, and by W. Thomson (Lord Kelvin) in 1851f; the 
equations have subsequently received only one important 
modification, which is due to Helmholtz.:}: Helmholtz pointed 
out that the electrical energy furnished by a voltaic cell need 
not be derived exclusively from the energy of the chemical 
reactions : for the cell may also operate by abstracting heat- 
energy from neighbouring bodies, and converting this into 
electrical energy. The extent to which this takes place is 
determined by a law which was discovered in 1855 by Thomson. 
Thomson showed that if E denotes the " available energy," i.e., 
possible output of mechanical work, of a system maintained 
at the absolute temperature T, then a fraction 

TdE 
fidT 

of this work is obtained, not at the expense of the thermal or 

* The heat of formation of a gramme-molecule of ZnS04 is greater than the heat 
of formation of a gramme-molecule of CuSO* by about 50,000 calories ; and with 
divalent metals, 46,000 calories per gramme- molecule corresponds to ane.m.f. of one 
volt ; so the e.m.f. of a Daniell cell should be 50/46 volts, which is nearly the 
case. 

t Kelvin's Math, and Phys. Papers, i, pp. 472, 490. 

J Berlin Sitzungsber., 1882, pp. 22, 825 ; 1883, p. 647. 

Quart. Journ. Math., April, 1855 ; Kelvin's Math, and Phys. Papers, i, 
p. 297, eqn. (7). 



Middle of the Nineteenth Century. 241 

chemical energy of the system itself, but at the expense of the 
thermal energy of neighbouring bodies. Now in the case of 
the voltaic cell, the principle of Eoget, Faraday, and Joule is 
expressed by the equation 

^ = A, 

where E denotes the available or electrical energy, which is 
measured by the electromotive force of the cell, and where X 
denotes the heat of the chemical reaction which supplies this 
energy. In accordance with Thomson's principle, we must 
replace this equation by 

F \ 4- T dE 
^ =X + T dT' 

which is the correct relation between the electromotive force 
of a cell and the energy of the chemical reactions which occur 
in it. In general the term A is much larger than the term 
T dEjdT ; but in certain classes of cells e.g., concentration- 
cells A is zero; in which case the whole of the electrical 
energy is procured at the expense of the thermal energy of 
the cells' surroundings. 

Helmholtz's memoir of 1847, to which reference has already 
been made, bore the title, " On the Conservation of Force." It 
was originally read to the Physical Society of Berlin*; but 
though the younger physicists of the Society received it with 
enthusiasm, the prejudices of the older generation prevented 
its acceptance for the Annalen der Physik ; and it was eventually 
published as a separate treatise.f 

In this memoir it was asserted* that the conservation of 

* On July 23rd, 1847. 

t Berlin, G. A. Reimer. English Translation in Tyndall & Francis' Scientific 
Memoirs, p. 114. The publisher, to Helmholtz's "great surprise," gave him an 
honorarium. Cf. Hermann von Helmholtz, by Leo Koenigsbeiger ; English 
translation by F. A. Welby. 

j Helmholtz had been partly anticipated by "W. R. Grove, in his lectures on 
the Correlation of Physical Forces, which were delivered in 1843 and published in 
1846. Grove, after asserting that heat is " purely dynamical " in its nature, and 
that the various " physical forces " may be transformed into each other, remarked : 
" The great problem which remains to be solved, in regard to the correlation 
of physical forces, is the establishment of their equivalent of power, or their 
measurable relation to a given standard." 

P. 



242 The Mathematical Electricians of the 

energy is a universal principle of nature : that the kinetic and 
potential energy of dynamical systems may be converted into 
heat according to definite quantitative laws, as taught by 
Kumford, Joule, and Eobert Mayer* ; and that any of these 
forms of energy may be converted into the chemical, electro- 
static, voltaic, and magnetic forms. The latter Helmholtz 
examined systematically. 

Consider first the energy of an electrostatic field. It will 
be convenient to suppose that the system has been formed by 
continually bringing from a very great distance infinitesimal 
quantities of electricity, proportional to the quantities already 
present at the various points of the system ; so that the charge 
is always distributed proportionally to the final distribution. 
Let e typify the final charge at any point of space, and V the 
final potential at this point. Then at any stage of the process 
the charge and potential at this point will have the values \e 
and A F, where A denotes a proper fraction. At this stage let 
charges ed\ be brought from a great distance and added to the 
charges \e. The work required for this is 



so the total work required in order to bring the system from 
infinite dispersion to its final state is 

fi 

or 



By reasoning similar to that used in the case of electrostatic 
distributions, it may be shown that the energy of a magnetic 
field, which is due to permanent magnets and which also 
contains bodies susceptible to magnetic induction, is 

\ 
where p denotes the density of Poisson's equivalent magnetiza- 

* Julius Robert Mayer (b. 1814, d. 1878), who was a medical man in Heilbronn, 
asserted the equivalence of heat and work in 1842, Annal. d. Chemie, xlii, p. 233 ; 
his memoir, like that of Helmholtz, was first declined by the editors of the 
Annalen der Physik. An English translation of one of Mayer's memoirs was 
printed in Phil. Mag. xxv (1863), p. 493. 



Middle of the Nineteenth Century. 243 

tion, for the permanent magnets only, and $ denotes the magnetic 
potential.* 

Helmholtz, moreover, applied the principle of energy to 
systems containing electric currents. For instance, when a 
magnet is moved in the vicinity of a current, the energy taken 
from the battery may be equated to the sum of that expended 
as Joulian heat, and that communicated to the magnet by the 
electromagnetic force : and this equation shows that the current 
is not proportional to the electromotive force of the battery, 
i.e. it reveals the existence of Faraday's magneto-electric 
induction. As, however, Helmholtz was at the time un- 
acquainted with the conception of the electrokinetic energy 
stored in connexion with a current, his equations were for the 
most part defective. But in the case of the mutual action of 
a current and a permanent magnet, he obtained the correct 
result that the time-integral of the induced electromotive 
force in the circuit is equal to the increase which takes 
place in the potential of the magnet towards a current of a 
certain strength in the circuit. 

The correct theory of the energy of magnetic and electro- 
magnetic fields is due mainly to W. Thomson (Lord Kelvin). 
Thomson's researches on this subject commenced with one or 
two short investigations regarding the ponderomotive forces 
which act on temporary magnets. In 1847 he discussed t the 
case of a small iron sphere placed in a magnetic field, showing 
that it is acted on by a ponderomotive force represented by 
- grad cR~, where c denotes a constant, and R denotes the magnetic 
force of the field ; such a sphere must evidently tend to move 
towards the places where E' is greatest. The same analysis 
may be applied to explain why diamagnetic bodies tend to 
move, as in Faraday's experiments, from the stronger to the 
weaker parts of the field. 

* We suppose all transitions to be continuous, so as to avoid the necessity for 
writing surf ace -integrals separately. 

tCamb. and Dub. Matb. Journal, ii (1847), p. 230; W. Thomson's Papers 
on Electrostatics and Magnetism, p. 499; cf. also Phil. Mag. xxxvii (1850), 
p. 241. 

R 2 



24:4 The Mathematical Electricians of the 

Two years later Thomson presented to the Koyal Society a 
memoir* in which the results of Poisson'a theory of magnetism 
were derived from experimental data, without making use of 
the hypothesis of magnetic fluids ; and this was followed in 
1850 by a second memoir,f in which Thomson drew attention 
to the fact previously noticed by Poisson,J that the magnetic 
intensity at a point within a magnetized body depends on the 
shape of the small cavity in which the exploring magnet is 
placed. Thomson distinguished two vectors ; one of these, by 
later writers generally denoted by B, represents the magnetic 
intensity at a point situated in a small crevice in the 
magnetized body, when the faces of the crevice are at right 
angles to the direction of magnetization ; the vector B is always 
circuital. The other vector, generally denoted by H, represents 
the magnetic intensity in a narrow tubular cavity tangential 
to the direction of magnetization ; it is an irrotational vector. 
The magnetic potential tends at any point to a limit which is 
independent of the shape of the cavity in which the point is 
situated ; and the space-gradient of this limit is identical with 
H. Thomson called B the " magnetic force according to the 
electro-magnetic definition," and H the " magnetic force accord- 
ing to the polar definition " ; but the names magnetic induction 
and magnetic force, proposed by Maxwell, have been generally 
used by later writers. 

It may be remarked that the vector to which Faraday 
applied the term " magnetic force," and which he represented 
by lines of force, is not H, but B ; for the number of unit lines 
of force passing through any gap must depend only on the gap, 
and not on the particular diaphragm filling up the gap, across 
which the flux is estimated ; and this can be the case only if the 
vector which is represented by the lines of force is a circuital 
vector. 



* Phil. Trans., 1851, p. 243 ; Thomson's Papers on Elect, and Mag., p. 345. 

t Phil. Trans., 1851, p. 269 ; Papers on Elect, and Nay., p. 382. 

I Of. p. 64. 

Loc. cit., 78 of the original paper, and 517 of the reprint^ 



Middle of the Nineteenth Century. 245 

Thomson introduced a number of new terms into magnetic 
science as indeed he did into every science in which he was 
interested. The ratio of the measure of the induced magnetiza- 
tion I,-, in a temporary magnet, to the magnetizing force H, 
he named the susceptibility ; it is positive for paramagnetic and 
negative for diamagnetic bodies, and is connected with Poisson's 
constant k p * by the relation 

3 if 
t\jp 

= SFTv 

where K denotes the susceptibility. By an easy extension of 
Poisson's analysis it is seen that the magnetic induction and 
magnetic force are connected by the equation 

B = H + 47rl, 

where I denotes the total intensity of magnetization : so if I 
denote the permanent magnetization, we have 

B = H + 47rl + 47rl,,, 
= )uH + 47rI , 

where //, denotes (1 + 4) : //, was called by Thomson the 
permeability. 

In 1851 Thomson extended his magnetic theory so as to 
include magnecrystallic phenomena. The mathematical founda- 
tions of the theory of magnecrystallic action had been laid by 
anticipation, long before the experimental discovery of the 
phenomenon, in a memoir read by Poisson to the Academy in 
February, 1824. Poisson, as will be remembered, had supposed 
temporary magnetism to be due to " magnetic fluids," movable 
within the infinitely small " magnetic elements " of which he 
assumed magnetizable matter to be constituted. He had not 
overlooked the possibility that in crystals these magnetic 
elements might be non-spherical (e.g. ellipsoidal), and symmetri- 
cally arranged ; and had remarked that a portion of such 
a crystal, when placed in a magnetic field, would act in a 
manner depending on its orientation. The relations connecting 

* Cf. p. 65. 



246 The Mathematical Electricians of the 

the induced magnetization I with the magnetizing force H he 
had given in a form equivalent to 

( I x = aH x + b'ffy + c"ff z , 
I y = a"H x + bH y + c'H Z) 
I z = a'H r + b"H y + cH z . 

Thomson now* showed that the nine coefficients a, b' ', c" . . ., 
introduced by Poisson, are not independent of each other. For 
a sphere composed of the magnecrystalline substance, if placed 
in a uniform field of force, would be acted on by a couple : and 
the work done by this couple when the sphere, supposed of 
unit volume, performs a complete revolution round the axis of x 
may be easily shown to be 7rH(l - H^j IP) (- &" + c). But this 
work must be zero, since the system is restored to its primitive 
condition ; and hence ~b" and c must be equal. Similarly e" = a, 
and a" = b f . By change of axes three more coefficients may be 
removed, so that the equations may be brought to the form 

777" T TT T TT 

JC ~ Kl/Z x , Iy = K-lJily, 1 Z = Ka/Zz, 

where KI, K Z , K 3 may be called the principal magnetic suscepti- 
bilities. 

In the same year (1851) Thomson investigated the energy 
which, as was evident from Faraday's work on self-induction, 
must be stored in connexion with every electric current. He 
showed that, in his own words, f " the value of a current in a 
closed conductor, left without electromotive force, is the 
quantity of work that would be got by letting all the infinitely 
small currents into which it may be divided along the lines of 
motion of the electricity come together from an infinite distance, 
and make it up. Each of these ' infinitely small currents ' is of 
course in a circuit which is generally of finite length ; it is the 
section of each partial conductor and the strength of the current 
in it that must be infinitely small." 

* Phil. Mag. (4) i (1851), p. 177: Papers on Electrostatics and Magnetism, 
p. 471. 

t Papers on Electrostatics and Magnetism, p. 446. 



Middle of the Nineteenth Century. 247 

Discussing next the mutual energy due to the approach of a 
permanent magnet and a circuit carrying a current, he arrived 
at the remarkable conclusion that in this case there is no 
electrokinetic energy which depends on the mutual action ; the 
energy is simply the sum of that due to the permanent magnets 
and that due to the currents. If a permanent magnet is 
caused to approach a circuit carrying a current, the electromotive 
force acting in the circuit is thereby temporarily increased ; the 
amount of energy dissipated as Joulian heat, and the speed of 
the chemical reactions in the cells, are temporarily increased also. 
But the increase in the Joulian heat is exactly equal to the 
increase in the energy derived from consumption of chemicals, 
together with the mechanical work done on the magnet by the 
operator who moves it ; so that the balance of energy is perfect, 
and none needs to be added to or taken from the electrokinetic 
form. It will now be evident why it was that Helmholtz 
escaped in this case the errors into which he was led in other 
cases by his neglect of electrokinetic energy ; for in this case 
there was no electrokinetic energy to neglect. 

Two years later, in 1853, Thomson* gave a new form to the 
expression for the energy of a system of permanent and 
temporary magnets. 

We have seen that the energy of such a system is represented 

by 



where p denotes the density of Poisson's equivalent magnetiza- 
tion for the permanent magnets, and <f> denotes the magnetic 
potential, and where the integration may be extended over the 
whole of space. Substituting for p n its value - div I ,f the 
expression may be written in the form 



- J 



< div Io dx dydz ; 



*Proc. Glasgow Phil. Soc. iii (1853), p. 281; Kelvin's Math, and Phys. 
Papers, i, p. 521. t Cf . p. fi4. 



248 The Mathematical Electricians of the 

or, integrating by parts, 

(!, . grad <) dx dy dz, or - J (H . I ) dx dy dz. 



Since B = yu,H + 47rI , this expression may be written in the 
form 



- (H. 
offJJJ 



but the former of these integrals is equivalent to 

>fff 
(B . grad <) dx dydz, or - < div B dx dy dz, 



which vanishes, since B is a circuital vector. The energy of the 
field, therefore, reduces to 
1 
BIT, 

integrated over all space; which is equivalent to Thomson's 
form.* 

In the same memoir Thomson returned to the question of the 
energy which is possessed by a circuit in virtue of an electric 
current circulating in it. As he remarked, the energy may 
be determined by calculating the amount of work which 
must be done in and on the circuit in order to double the 
circuit on itself while the current is sustained in it with 
constant strength; for Faraday's experiments show that a 
circuit doubled on itself has no stored energy. Thomson found 
that the amount of work required may be expressed in the form 
\Li*, where i denotes the current strength, and L, which is 
called the coefficient of self-induction^ depends only on the form of 
the circuit. 

It may be noticed that in the doubling process the inherent 

* The form actually given by Thomson was 

fff (*E? lA d-d 
Sir}}} \:-.*) 

which reduces to the above when we neglect that part of I 2 which is due to the 
permanent magnetism, over which we have no control. 



Middle of the Nineteenth Century. 249 

electrodynamic energy is being given up, and yet the operator is 
doing positive work. The explanation of this apparent paradox 
is that the energy derived from both these sources is being 
used to save the energy which would otherwise be furnished by 
the battery, and which is expended in Joulian heat. 

Thomson next proceeded* to show that the energy which is 
stored in connexion with a circuit in which a current is flowing 
may be expressed as a volume-integral extended over the whole 
of space, similar to the integral by which he had already 
represented the energy of a system of permanent and temporary 
magnets. The theorem, as originally stated by its author, 
applied only to the case of a single circuit; but it may be 
established for a system formed by any number of circuits in 
the following way : 

If N 8 denote the number of unit tubes of magnetic induction 
which are linked with the & h circuit, in which a current i s is 
flowing, the electrokinetic energy of the system is JSJV,^; which 

may be written |2/ r , where / r denotes the total current flowing 

through the gap formed by the r th unit tube of magnetic induc- 
tion. But if H denote the (vector) magnetic force, and H its 
numerical magnitude, it is known that (l/4?r) J Hds, integrated 
along a closed line of magnetic induction, measures the total 
current flowing through the gap formed by the line. The 
energy is therefore (l/8?r)S jffds, the summation being extended 
over all the unit tubes of magnetic induction, and the integra- 
tion being taken along them. But if dS denote the cross-section 
of one of these tubes, we have BdS = 1, where B denotes the 
numerical magnitude of the magnetic induction B : so the energy 
is (1 1 'Sir) SBdS / Hds ; and as the tubes fill all space, we may 
replace 'S.dSjds by ^dxdydz. Thus the energy takes the form 
(l/8?r) JJf BHdxdydz, where the integration is extended over the 
whole of space ; and since in the present case B = pH, the energy 
may also be represented by (Il8v)ffffjjrdxdydz. 

* Nichols* Cyclopaedia, 2nd ed., 1860, article " Magnetism, dynamical 
relations of; " reprinted in Thomson's Papers on Elect, and Mag., p. 447, and 
his Math, and Phys. Papers, p. 532. 



250 The Mathematical Electricians of the 

But this is identical with the form which was obtained for 
a field due to permanent and temporary magnets. It thus 
appears that in all cases the stored energy of a system of 
electric currents and permanent and temporary magnets is 

-' dxdydz, 

where the integration is extended over all space. 

It must, however, be remembered that this represents only 
what in thermodynamics is called the " available energy " ; and 
it must further be remembered that part even of this available 
energy may not be convertible into mechanical work within the 
limitations of the system : e.g., the electrokinetic energy of a 
current flowing in a single closed perfectly conducting circuit 
cannot be converted into any other form so long as the circuit 
is absolutely rigid. All that we can say is that the changes in 
this stored electrokinetic energy correspond to the work furnished 
by the system in any change. 

The above form suggests that the energy may not be localized 
in the substance of the circuits and magnets, but may be distri- 
buted over the whole of space, an amount (pH 2 /Sir) of energy 
being contained in each unit volume. This conception was 
afterwards adopted by Maxwell, in whose theory it is of 
fundamental importance. 

While Thomson was investigating the energy stored in 
connexion with electric currents, the equations of flow of the 
currents were being generalized by Gustav Kirchhoff (b. 1824, 
d. 1887). In 1848 Kirchhoff* extended Ohm's theory of linear 
conduction to the case of conduction in three dimensions ; this 
could be done without much difficulty by making use of the 
analogy with the flow of heat, which had proved so useful to 
Ohm. In Kirchhoff s memoir a system is supposed to be 
formed of three-dimensional conductors, through which steady 
currents are flowing. At any point let V denote the " tension " 
or " electroscopic force " a quantity the significance of which 

*Ann.d. Phys. Ixxv (1848), p. 189: Kirchhoff's Ges. AbhandL, p. 33. 



Middle of tke Nineteenth Century. 4 25l 

in electrostatics was not yet correctly known. Then, within 
the substance of any homogeneous conductor, the function V 
must satisfy Laplace's equation V- V= ; while at the air-surface 
of each conductor, the derivate of V taken along the normal 
must vanish. At the interface between two conductors formed 
of different materials, the function V has a discontinuity, 
which is measured by the value of Volta's contact force for the 
two conductors ; and, moreover, the condition that the current 
shall be continuous across such an interface requires that 
Jed VfoN shall be continuous, where k denotes the ohmic specific 
conductivity of the conductor, and 3/3^ denotes differentiation 
along the normal to the interface. The equations which have 
now been mentioned suffice to determine the flow of electricity 
in the system. 

Kirchhoff also showed that the currents distribute them- 
selves in the conductors in such a way as to generate the least 
possible amount of Joulian heat ; as is easily seen, since the 
quantity of Joulian heat generated in unit time is 



where k, as before, denotes the specific conductivity ; and this 
integral has a stationary value when V satisfies the equation 

a /ar\ a 



Kirchhoff next applied himself to establish harmony between 
electrostatical conceptions and the theory of Ohm. That 
theory had now been before the world for twenty years, and 
had been verified by numerous experimental researches ; in 
particular, a careful investigation was made at this time (1848) 
by Kudolph Kohlrausch (b. 1809, d. 1858), who showed* that 
the difference of the electric " tensions " at the extremities of a 
voltaic cell, measured electrostatically with the circuit open, 
was for different cells proportional to the electromotive force 

*Ann. d. Phys. Ixxv (1848), p. 220. 



252 The Mathematical Electricians of the 

measured by the electrodynamic effects of the cell with the 
circuit closed ; and, further,* that when the circuit was closed, 
the difference of the tensions, measured electrostatically, at any 
two points of the outer circuit was proportional to the ohmic 
resistance existing between them. But in spite of all that had 
been done, it was still uncertain how " tension," or " electro- 
scopic force," or " electromotive force " should be interpreted 
in the language of theoretical electrostatics ; it will be 
remembered that Ohm himself, perpetuating a confusion which 
had originated with Volta, had identified electroscopic force 
with density of electric charge, and had assumed that the 
electricity in a conductor is at rest when it is distributed 
uniformly throughout the substance of the conductor. 

The uncertainty was finally removed in 1849 by Kirchhoff,f 
who identified Ohm's electroscopic force with the electrostatic 
potential. That this identification is correct may be seen by 
comparing the different expressions which have been obtained 
for electric energy; Helmholtz's expression^ shows that the 
energy of a unit charge at any place is proportional to the 
value of the electrostatic potential at that place ; while Joule's 
result shows that the energy liberated by a unit charge in 
passing from one place in a circuit to another is proportional 
to the difference of the electric tensions at the two places. It 
follows that tension and potential are the same thing. 

The work of Kirchhoff was followed by several other 
investigations which belong to the borderland between electro- 
statics and electrodynamics. One of the first of these was the 
study of the Leyden jar discharge. 

Early in the century Wollaston, in the course of his experi- 
ments on the decomposition of water, had observed that when 
the decomposition is effected by a discharge of static electricity, 
the hydrogen and oxygen do not appear at separate electrodes ; 
but that at each electrode there is evolved a mixture of the 

* Ann. d. Phys, Ixxviii (1849), p. 1. 

f Ib. Ixxviii (1849), p. 506 ; Kirchhoff's Get. Abhandl, p. 49 ; Phil. Mag. (3), 
xxxvii (1850), p. 463. 

I Cf. p. 242. Cf. p. 239. 



Middle of the Nineteenth Century. 253 

gases, as if the current had passed through the water in both 
directions. After this F. Savary* had noticed that the 
discharge of a Ley den jar magnetizes needles in alternating 
layers, and had conjectured that " the electric motion during 
the discharge consists of a series of oscillations." A similar 
remark was made in connexion with a similar observation by 
Joseph Henry (ft. 1799, d. 1878), of Washington, in 1842.f 
" The phenomena," he wrote, " require us to admit the existence 
of a principal discharge in one direction, and then several reflex 
actions backward and forward, each more feeble than the 
preceding, until equilibrium is restored." Helmholtz had 
repeated the same suggestion in his essay on the conservation 
of energy : and in 1853 W. Thomson J verified it, by 
investigating the mathematical theory of the discharge, as 
follows : 

Let C denote the capacity of the jar, i.e., the measure of the 
charge when there is unit difference of potential between "the 
coatings ; let R denote the ohmic resistance of the discharging 
circuit, and L its coefficient of self-induction. Then if at 
any instant t the charge of the condenser be Q, and the 
current in the wire be i, we have i = dQ/dt ; while Ohm's law, 
modified by taking self-induction into account, gives the 
equation 



Eliminating i, we have 



an equation which shows that when IFC < 4Z, the subsidence 
of Q to zero is effected by oscillations of period 



27T 



(1- * 
\LC 4Z 



* Annales de Chiniie, xxxiv (1827), p. 5. 
tProc. Am. Phil. Soc. ii (1842), p. 193. 

J Phil. Mag. (4) v (1853), p. 400 ; Kelvin's Math, and Phys. Papers i, 
p. 540. 



254 The Mathematical Electricians of the 

This simple result may be regarded as the beginning of the 
theory of electric oscillations. 

Thomson was at this time much engaged in the problems 
of submarine telegraphy; and thus he was led to examine 
the vexed question of the " velocity of electricity " over long 
insulated wires and cables. Various workers had made 
experiments on this subject at different times, but with 
hopelessly discordant results. Their attempts had generally 
taken the form of measuring the interval of time between the 
appearance of sparks at two spark-gaps in the same circuit, 
between which a great length of wire intervened, but which 
were brought near each other in order that the discharges 
might be seen together. In one series of experiments, 
performed by Watson at Shooter's Hill in 1747-8,* the circuit 
was four miles in length, two miles through wire and two 
miles through the ground ; but the discharges appeared to be 
perfectly simultaneous; whence Watson concluded that the 
velocity of propagation of electric effects is too great to be 
measurable. 

In 1834 Charles Wheatstone,f Professor of Experimental 
Philosophy in King's College, London, by examining in a 
revolving mirror sparks formed a,t the extremities of a circuit, 
found the velocity of electricity in a copper wire to be about 
one and a half times the velocity of light. In 1850 H. Fizeau 
and E. GounelleJ experimenting with the telegraph lines from 
Paris to Eouen and to Amiens, obtained a velocity about one- 
third that of light for the propagation of electricity in an iron 
wire, and nearly two- thirds that of light for the propagation 
in a copper wire. 

The first step towards explaining these discrepancies was 
made by Faraday, who early in 1854 showed experimentally 
that a submarine cable, formed of copper wire covered with 

* Phil. Trans, xlv (1748), pp. 49, 491. 
t Phil. Trans., 1834, p. 583. 
; Comptes Rendus, xxx (1850), p. 437. 

Proc. Roy. Inst., Jan. 20, 1854: Phil. Mag-., June, 1854: Exp. Res. iii, 
pp. 508, 521. 



Middle of the Nineteenth Century. 255 

gutta-percha, " may be assimilated exactly to an immense 
Leyden battery ; the glass of the jars represents the gutta- 
percha ; the internal coating is the surface of the copper wire," 
while the outer cgating corresponds to the sea-water. It 
follows that in all calculations relating to the propagation of 
electric disturbances along submarine cables, the electrostatic 
capacity of the cable must be taken into account. 

The theory of signalling by cable originated in a corre- 
spondence between Stokes and Thomson in 1854. In the case 
of long submarine lines, the speed of signalling is so much 
limited by the electrostatic factor that electro-magnetic induc- 
tion has no sensible effect ; and it was accordingly neglected in 
the investigation. In view of other applications of the analysis ; 
however, we shall suppose that the cable has a self-induction L 
per unit length, and that E denotes the ohmic resistance, and 
C the capacity per unit length, Fthe electric potential at a 
distance x from one terminal, and i the current at this place. 
Ohm's law, as modified for inductance, is expressed by the 
equation 

9^ T di _>. 

- -^- = L + Ri ; 

dx dt 

moreover, since the rate of accumulation of charge in unit 
length at # is - di/dx, and since this increases the potential 
at the rate - (l/C^difix, we have 



- 

'dt dx 

Eliminating i between these two equations, we have 
1 8 2 F 



which is known as the equation of telegraphy* 

Thomson, in one of his letterst to Stokes in 1854, 
obtained this equation in the form which applies to Atlantic 
.cables, i.e., with the term in L neglected. In this form it is 

* "We have neglected leakage, which is beside our present purpose. 

t Proc. Roy. Soc., May, 1855 : Kelvin's Math, and Phys. Papers, ii, p. 61. 



256 The Mathematical Electricians of the 

the same as Fourier's equation for the linear propagation of 
heat : so that the known solutions of Fourier's theory may he 
used in a new interpretation. If we substitute 
v - /,2<V - i -j- \x 

y t/ > 

we obtain 

A, = (1 + -v/^l) (nCR)l J 

and therefore a typical elementary solution of the equation is 
V = e -( nCR ^ x sin \2nt - (nCR)^x}. 

The form of this solution shows that if a regular harmonic 
variation of potential is applied at one end of a cable, the phase 
is propagated with a velocity which is proportional to the 
square root of the frequency of the oscillations : since therefore 
the different harmonics are propagated with different velocities, 
it is evident that no definite " velocity of transmission " is to be 
expected for ordinary signals. If a potential is suddenly applied 
at one end of the cable, a certain time elapses before the current 
at the other end attains a definite percentage of its maximum 
value ; but it may easily be shown* that this retardation is 
proportional to the square of the length of the cable, so that 
the apparent velocity of propagation would be less, the greater 
the length of cable used. 

The case of a telegraph line insulated in the air on poles is 
different from that of a cable ; for here the capacity is small, 
and it is necessary to take into account the inductance. If in 
the general equation of telegraphy we write 

V = e nx ^~ l + P l , 
we obtain the equation 

R (R* n* \i 

2l f (L* ~ CL) ; 

as the capacity is small, we may replace the quantity under the 
radical by its second term : and thus we see that a typical 
elementary solution of the equation is 



F= e i siu n{x - (CL)- 1 * t}; 

* This result, indeed, follows at once from the theory of dimensions. 



Middle of the Nineteenth Century. 257 

this shows that any harmonic disturbance, and therefore any 
disturbance whatever, is propagated along the wire with 
velocity (CL}~\ The difference between propagation in an 
aerial wire and propagation in an oceanic cable is, as Thomson 
remarked, similar to the difference between the propagation 
of an impulsive pressure through a long column of fluid in a 
tube when the tube is rigid (case of the aerial wire) and when 
it is elastic, so as to be capable of local distension (case of the 
cable, the distension corresponding to the effect of capacity) : 
in the former case, as is well known, the impulse is propagated 
with a definite velocity, namely, the velocity of sound in the 
fluid. 

The work of Thomson on signalling along cables was followed 
in 1857 by a celebrated investigation* of Kirchhoff's, on the 
propagation of electric disturbance along an aerial wire of 
circular cross- section. 

Kirchhoff assumed that the electric charge is practically all 
resident on the surface of the wire, and that the current is 
uniformly distributed over its cross-section; his idea of the 
current was the same as that of Fechner and Weber, namely, 
that it consists of equal streams of vitreous and resinous elec- 
tricity flowing in opposite directions. Denoting the electric 
potential by V, the charge per unit length of wire by e, the 
length of the wire by I, and the radius of its cross-section by a, 
he showed that Fis determined approximately by the equationf 

V = 2e log (I/a). 

* Ann. d. Phys. c (1857), pp. 193, 251 : Kirchhoff's Ges. Abhandl., p. iai ; 
Phil. Mag. xiii (1857), p. 393. 

t His method of obtaining this equation was to calculate separately the effects of 
(1) the portion of the wire within a distance e on either side of the point con- 
sidered, where e denotes a length small compared with J, but large compared with o, 
and (2) the rest of the wire. He thus obtained the equation 



where the integration is to be taken over all the length of the wire except the 
portion 2e : the equation given in the text was then derived by an, approximation ,. 
which, however, is open to some objection. 

S 



258 The Mathematical Electricians of the 

The next factor to be considered is the mutual induction of 
the current-elements in different parts of the wire. Assuming 
with Weber that the electromotive force induced in an element 
ds due to another element ds' carrying a current i' is derivable 
from a vector-potential 



,.3 

Kirchhoff found for the vector-potential due to the entire wire 
the approximate value 

w = 2i log (//a), 

where i denotes the strength of the current ;* the vector- 
potential being directed parallel to the wire. Ohm's law then 
gives the equation 

ldw 



where k denotes the specific conductivity of the material of 
which the wire is composed; and finally the principle of 
conservation of electricity gives the equation 

di _ _de . 

dx~ ~di' 

Denoting log (I/a) by y, and eliminating e, i, w from these four 
equations, we have 

8 2 F 1 d*V 1 8F 



which is, as might have been expected, the equation of telegraphy. 
When the term in 3 V/dt is ignored, as we have seen is in certain 
cases permissible, the equation becomes 

8 2 F lF 



* This expression was derived in a similar way to that for F, by an intermediate 
formula 

2 c ci'ds' 

w = 2i log -- h cos 6 cos Q , 
& a J r 

where 6 and Q' denote respectively the angles made with r by ds and ds'. 



Middle of the Nineteenth Century. 259 

which shows that the electric disturbance is propagated along the 
wire with the velocity c* KirchhofF s procedure has, in fact, 
involved the calculation of the capacity and self-induction of 
the wire, and is thus able to supply the definite values of the 
quantities which were left undetermined in the general equation 
of telegraphy. 

The velocity c, whose importance was thus demonstrated, has 
already been noticed in connexion with Weber's law of force ; 
it is a factor of proportionality, which must be introduced when 
electrodynamic phenomena are described in terms of units which 
have been defined electrostatically ,f or conversely when units 
which have been defined electrodynamicallyj are used in the 
description of electrostatic phenomena. That the factor which 
is introduced on such occasions must be of the dimensions 
(length/time), may be easily seen : for the electrostatic re- 
pulsion between electric charges is a quantity of the same kind 
as the electrodynamic repulsion between two definite lengths of 
wire, carrying currents which may be specified by the amount 
of charge which travels past any point in unit time. 

Shortly before the publication of Kirchhoff s memoir, the 
value of c had been determined by Weber and Kohlrausch ; 
their determination rested on a comparison of the measures of the 
charge of a Leyden jar, as obtained by a method depending 
on electrostatic attraction, and by a method depending on the 

* In referring to the original memoirs of Weber and Kirchhoff, it must he 
remembered that the quantity which in the present work is denoted by e, and 
which represents the velocity of light in free aether, was by these writers denoted 
by c/V'2. Weber, in fact, denoted by c the relative velocity with which two charges 
must approach each other in order that the force between them, as calculated by 
his formula, should vanish. 

It must also be remembered that those writers who accepted the hypothesis 
that currents consist of equal and opposite streams of vitreous and resinous 
electricity, were accustomed to write 2t to denote the current-strength. 

f i.e., defining unit electric charge as that which exerts unit ponderomotive 
force on a conductor at unit distance which carries an equal charge ; and then 
defining unit current as that which conveys unit charge in unit time. 

% i.e., defining unit current by means of the ponderomotive force which it 
exerts on an equal current, when the two currents flow in circuits of specified 
form at a specified distance apart. 

Ann. d. Phys. xcix (1856), p. 10. 

S 2 



260 The Mathematical Electricians of the 

magnetic effects of the current produced by discharging the jar. 
The resulting value was nearly 

c = 3*1 x 10 10 cm./sec.; 

which was the same, within the limits of the errors of measure- 
ment, as the speed with which light travels in interplanetary 
space. The coincidence was noticed by Kirchhoff, who was thus 
the first to discover the important fact that the velocity with 
which an electric disturbance is propagated along a perfectly- 
conducting aerial wire is equal to the velocity of light. 

In a second memoir published in the same year, Kirchhoff* 
extended the equations of propagation of electric disturbance 
to the case of three-dimensional conductors. 

As in his earlier investigation, he divided the electromotive 
force at any point into two parts, of which one is the gradient 
of the electrostatic potential </>, and the other is the derivate 
with respect to the time (with sign reversed) of a vector- 
potential a ; so that if i denote the current and k the specific 
conductivity, Ohm's law is expressed by the equation 

i = k (c 2 grad < - a). 

Kirchhoff calculated the value of a by aid of Weber's formula 
for the inductive action of one current element on another; 
the result is 



where r denotes the vector from the point (x, y, z), at which a is 
measured, to any other point (x, y, z") of the conductor, at which 
the current is i' ; and the integration is extended over the whole 
volume of the conductor. The remaining general equations are 
the ordinary equation of the electrostatic potential 

V 2 < + 4irp = 

(where p denotes the density of electric charge), and the equation 
of conservation of electricity 

| + div i = 0. 
ot 

* Ann. d. Phys. cii (1857), p. 529 : Ges. AbhandL, p. 154. 



Middle of the Nineteenth Century. 261 

It will be seen that Kirchhoff's electrical researches were 
greatly influenced by those of Weber. The latter investiga- 
tions, however, did not enjoy unquestioned authority ; for there 
was still a question as to whether the expressions given by 
Weber for the mutual energy of two current elements, and for 
the mutual energy of two electrons, were to be preferred to the 
rival formulae of Neumann and Eiemann. The matter was 
examined in 1870 by Helmholtz, in a series of memoirs* to 
which reference has already been made.f Helmholtz remarked 
that, for two elements ds, ds', carrying currents i, i', the electro- 
dynamic energy is 

n'(ds.ds') 
r ' 
according to Neumann, and 

?V 
5-(r.ds)(r.ds'), 

according to Weber; and that these expressions differ from 
each other only by the quantity 

- cos (ds . ds') + cos (r . ds) cos (r . ds') ] , 
d z r 



or ^^ 



dsds 



since this vanishes when integrated round either circuit, the 
two formulae give the same result when applied to entire 
currents. A general formula including both that of Neumann 
and that of Weber is evidently 

n'(ds .ds') .., ffr 

+ ki^ -j-, dsds, 
r ds ds 

where k denotes an arbitrary constant.^ 

Helmholtz's result suggested to Clausius a new form for 
the law of force between electrons ; namely, that which is 

* Journal fur Math., Ixxii (1870), p. 57 : Ixxv (1873), p. 35: Ixxviii (1874), 
p. 273. t Cf. p. 229. 

+ Cf. H. Lamb, Proc. Lond. Math. Soc., xiv (1883), p. 301. 
Journal fiir Math. Ixxxii (1877), p. 85 : Phil. Mag., x (1880), p. 255. 



262 The Mathematical Electricians of the 

obtained by supposing that two electrons of charges e, e', and 
velocities v, v', possess electrokinetic energy of amount 

ee f (v .v') 7 , d~r , 

- - + kee - r =-> w . 

r dsds 

Subtracting from this the mutual electrostatic potential energy, 
which is ee'c'/r, we may write the mutual kinetic potential of 
the two electrons in the form 

(xx + ijy + zz f - c 2 ) + kee' > vv', 



where (x, y, z) denote the coordinates of e, and (X, y', z) 
those of e f . 

The unknown constant k has clearly no influence so long as 
closed circuits only are considered: if k be replaced by zero, 
the expression for the kinetic potential becomes 

ee' 

(xx + yy + zz - c 2 ), 

which, as will appear later, closely resembles the corresponding 
expression in the modern theory of electrons. 

Clausius' formula has the great advantage over Weber's, that 
it does not compel us to assume equal and opposite velocities 
for the vitreous and resinous charges in an electric current; 
on the other hand, Clausius' expression involves the absolute 
velocities of the electrons, while Weber's depends only on their 
relative motion; and therefore Clausius' theory requires the 
assumption of a fixed aether in space, to which the velocities 
v and V may be referred. 

When the behaviour of finite electrical systems is predicted 
from the formulae of Weber, Eiemann, and Clausius, the three 
laws do not always lead to concordant results. For instance, if 
a circular current be rotated with constant angular velocity 
round its axis, according to Weber's law there would be a 
development of free electricity on a stationary conductor in the 
neighbourhood ; whereas, according to Clausius' formula there 
would be no induction on a stationary body, but electrification 



Middle of the Nineteenth Century. 263 

would appear on a body turning with the circuit as if 
rigidly connected with it. Again,* let a magnet be suspended 
within a hollow metallic body, and let the hollow body be 
suddenly charged or discharged; then, according to Clausius' 
theory, the magnet is unaffected; but according to Weber's 
and Kiemann's theories it experiences an impulsive couple. 
And again, if an electrified disk be rotated in its own plane, 
under certain circumstances a steady current will be induced in 
a neighbouring circuit according to Weber's law, but not 
according to the other formulae. 

An interesting objection to Clausius' theory was brought 
forward in 1879 by Frohlichf namely, that when a charge of 
free electricity and a constant electric current are at rest 
relatively to each other, but partake together of the translatory 
motion of the earth in space, a force should act between them if 
Clausius' law were true. It was, however, shown by BuddeJ 
that the circuit itself acquires an electrostatic charge, partly 
as a result of the same action which causes the force on the 
external conductor, and partly as a result of electrostatic 
induction by the charge on the external conductor ; and that the 
total force between the circuit and external conductor is thus 
reduced to zero. 

We have seen that the discrimination between the different 
laws of electrodynamic force is closely connected with the 
question whether in an electric current there are two kinds of 
electricity moving in opposite directions, or only one kind 
moving in one direction. On the unitary hypothesis, that the 

* The two following crudal experiments, with others, were suggested by 
E. Budde, Ann. d. Phys. xxx (1887), p. 100. 

t Ann. d. Phys. ix (1880), p. 261. 

+ Ann. d. Phys. x (1880), p. 553. 

This case of a charge and current moving side hy side was afterwards 
examined by Fitz Gerald (Trans. Boy. Dub. Soc. i, 1882 ; Scient. Writings of 
G. F. Fitz Gerald, p. Ill) without reference to Clausius' formula, from the 
standpoint of Maxwell's theory. The result obtained was the same namely, 
that the electricity induced on the conductor carrying the current neutralizes the 
ponderomotive force between the current and the external charge. 



264 The Mathematical Electricians of the 

current consists in a transport of one kind of electricity with a 
definite velocity relative to the wire, it might be expected that 
a coil rotated rapidly about its own axis would generate a 
magnetic field different from that produced by the same coil 
at rest. Experiments to determine the matter were performed 
by A. Foppl* and by E. L. Nichols and W. S. Franklin,f but 
with negative results. The latter investigators found that the 
velocity of electricity must be such that the quantity conveyed 
past a specified point in a unit of time, when the direction of 
the current was that in which the coil was travelling, did not 
differ from that transferred when the current and coil were 
moving in opposite directions by as much as one part in ten 
million, even when the velocity of the wire was 9096 cm./sec. 
They considered that they would have been able to detect 
a change of deflexion due to the motion of the coil, even though 
the velocity of the current had been considerably greater than 
a thousand million metres per second. 

During the decades in the middle of the century consider- 
able progress was made in the science of thermo-electricity, 
whose beginnings we have already described. J In Faraday's 
laboratory note-book, under the date July 28th, 1836, we 
read : " Surely the converse of thermo-electricity ought to be 
obtained experimentally. Pass current through a circuit of 
antimony and bismuth." 

Unknown to Faraday, the experiment here indicated had 
already been made, although its author had arrived at it by a 
different train of ideas. In 1834 Jean Charles Peltier|| (b. 1785, 
d. 1845) attempted the task, which was afterwards performed 
with success by Joule,1J of measuring the heat evolved by the 
passage of an electric current through a conductor. He found 
that a current produces in a homogeneous conductor an elevation 

* Ann. d. Phys. xxvii (1886), p. 410. 

t Amer. Jour. Sci., xxxvii (1889), p. 103. 

J Cf. pp. 92, 93. Bence Jones's Life of Faraday, ii, p. 76. 

II Annales de Ciiimie, Ivi (1834), p. 371. If Cf. p. 239. 



Middle of the Nineteenth Century. 265 

of temperature, which is the same in all parts of the conductor 
where the cross-section is the same ; but he did not succeed in 
connecting the thermal phenomena quantitatively with the 
strength of .the current a failure which was due chiefly to the 
circumstance that his attention was fixed on the rise of 
temperature rather than on the amount of the heat evolved. 
But incidentally the investigation led to an important discovery 
namely, that when a current was passed in succession through 
two conductors made of dissimilar metals, there was an evolution 
of heat at the junction ; and that this depended on the direction of 
the current ; for if the junction was heated when the current 
flowed in one sense, it was cooled when the current flowed in the 
opposite sense. This Peltier effect, as it is called, is quite distinct 
from the ordinary Joulian liberation of heat, in which the 
amount of energy set free in the thermal form is unaffected by 
a reversal of the current ; the Joulian effect is, in fact, propor- 
tional to the square of the current-strength, while the Peltier 
effect is proportional to the current-strength directly. The 
Peltier heat which is absorbed from external sources when a 
current i flows for unit time through a junction from one metal 
B to another metal A may therefore be denoted by 



where T denotes the absolute temperature of the junction. The 
function n^ (T) is found to be expressible as the difference of 
two parts, of which one depends on the metal A only, and the 
other on the metal B only ; thus we can write 



In 1851 a general theory of thermo-electric phenomena was 
constructed on the foundation of Seebeck's* and Peltier's dis- 
coveries by W. Thomson.f Consider a circuit formed of two 

* Cf. pp. 92, 93. 

t Proc. R.S. Edinb. iii (1851), p. 91 ; Phil. Mag. iii (1852), p. 529 : Kelvin's 
Math, and Phys. Paper*, i, p. 316. Cf. also Trans. R. S. Edinb. xxi (1854), 
p. 123, reprinted in Papers, i, p. 232 : and Phil. Trans., 1856, reprinted in Papers, 
ii, p. 189. 



266 The Mathematical Electricians of the 

metals, A and B, and let one junction be maintained at a 
slightly higher temperature (T + $T) than the temperature T 
of the other junction. As Seebeck had shown, a thermo-electric 
current will be set up in the circuit. Thomson saw that such 
a system might be regarded as a heat-engine, which absorbs a 
certain quantity of heat at the hot junction, and converts part 
of this into electrical energy, liberating the rest in the form of 
heat at the cold junction. If the Joulian evolution of heat be 
neglected, the process is reversible, and must obey the second 
law of thermodynamics ; that is, the sum of the quantities of 
heat absorbed, each divided by the absolute temperature at 
which it is absorbed, must vanish. Thus we have 






T+ST 



so the Peltier effect H^(T) must be directly proportional to 
the absolute temperature T. This result, however, as Thomson 
well knew, was contradicted by the observations of Gumming, 
who had shown that when the temperature of the hot junction 
is gradually increased, the electromotive force rises to a maximum 
value and then decreases. The contradiction led Thomson to 
predict the existence of a hitherto unrecognized thermo-electric 
phenomenon namely, a reversible absorption of heat at places 
in the circuit other than the junctions. Suppose that a current 
flows along a wire which is of the same metal throughout, but 
varies in temperature from point to point. Thomson showed 
that heat must be liberated at some points and absorbed at 
others, so as either to accentuate or to diminish the differences 
of temperature at the different points of the wire. Suppose 
that the heat absorbed from external sources when unit 
electric charge passes from the absolute temperature T to the 
temperature (T + $T) in a metal A is denoted by S A (T).ST. 
The thermodynamical equation now takes the corrected form 



~ SA(T)} 



Middle of the Nineteenth Century. 267 

Since the metals A and B are quite independent, this gives 



This equation connects Thomson's " specific heat of electricity" 
S A (T) with the Peltier effect. 

In 1870 P. G. Tait* found experimentally that the specific 
heat of electricity in pure metals is proportional to the absolute 
temperature. We may therefore write S A (T) = a A T, where 
a A denotes a constant characteristic of the metal A. The 
thermodynamical equation then becomes 

_d \U A (T)) 
dT ( T ~ 

or 



where TT A denotes another constant characteristic of the metal. 
The chief part of the Peltier effect arises from the term ir A T. 

By the investigations which have been described in the 
present chapter, the theory of electric currents was considerably 
advanced in several directions. In all these researches, how- 
ever, attention was fixed on the conductor carrying the current 
as the seat of the phenomenon. In the following period, interest 
was centred not so much on the conductors which carry charges 
and currents, as on the processes which take place in the 
dielectric media .around them. 

* Proc. R. S. Edinb. vii (1870), p. 308. Cf. also Batelli, Atti delia R. Ace. di 
Torino, xxii (1886), p. 48, translated Phil. Mug. xxiv (1887), p. 295. 



( 268 ) 
CHAPTEE VIII. 

MAXWELL. 

SINCE the time of Descartes, natural philosophers have never 
ceased to speculate on the manner in which electric and 
magnetic influences are transmitted through space. About 
the middle of the nineteenth century, speculation assumed a 
definite form, and issued in a rational theory. 

Among those who thought much on the matter was Karl 
Friedrich Gauss (b. 1777, d. 1855). In a letter* to Weber of 
date March 19, 1845, Gauss remarked that he had long ago 
proposed to himself to supplement the known forces which act 
between electric charges by other forces, such as would cause 
electric actions to be propagated between the charges with a 
finite velocity. But he expressed himself as determined not 
to publish his researches until he should have devised a 
mechanism by which the transmission could be conceived to 
be effected ; and this he had not succeeded in doing. 

More than one attempt to realize Gauss's aspiration was 
made by his pupil Eiemann. In a fragmentary note,t which 
appears to have been written in 1853, but which was not 
published until after his death, Biemann proposed an aether 
whose elements should be endowed with the power of resisting 
compression, and also (like the elements of MacCullagh's 
aether) of resisting changes of orientation. The former pro- 
perty he conceived to be the cause of gravitational and 
electrostatic effects, and the latter to be the cause of optical 
and magnetic phenomena. The theory thus outlined was 
apparently not developed further by its author ; but in a short 
investigation^ which was published posthumously in 1867, he 

* Gauss' Werke, v, p. 629. t Riemann's Werke, 2 e Aufl., p. 526. 

J Ann. d. Phys. cxxxi (1867), p. 237 ; Riemann's Werke, 2 e Aufl., p. 288 ; 
Phil. Mag. xxxiv (1867), p. 368. 

It had been presented to the Gottingen Academy in 1858, but afterwards 
withdrawn. 

. 



Maxwell. 269 

returned to the question of the process by which electric action 
is propagated through space. In this memoir he proposed to 
replace Poisson's equation for the electrostatic potential, 
namely, 



by the equation 



according to which the changes of potential due to changing 
electrification would be propagated outwards from the charges 
with a velocity c. This, so far as it goes, is in agreement with 
the view which is now accepted as correct ; but Kiemann's 
hypothesis was too slight to serve as the basis of a complete 
theory. Success came only when the properties of the inter- 
vening medium were taken into account. 

In that power to which Gauss attached so much importance, 
of devising dynamical models and analogies for obscure physical 
phenomena, perhaps no one has ever excelled W. Thomson*; 
and to him, jointly with Faraday, is due the credit of having 
initiated the theory of the electric medium. In one of his 
earliest papers, written at the age of seventeen,! Thomson 
compared the distribution of electrostatic force, in a region 
containing electrified conductors, with the distribution of the 
flow of heat in an infinite solid : the equipotential surfaces in 
the one case correspond to the isothermal surfaces in the other, 
and an electric charge corresponds to a source of heat.J 

* As will appear from the present chapter, Maxwell had the same power in a 
very marked degree. It has always been cultivated hy the " Cambridge school " 
of natural philosophers. 

t Camb. Math. Journal, iii (Feb. 1842), p. 71 ; reprinted in Thomson's Papers 
<JH Electrostatics and Magnetism, p. 1. Also Camb. and Dub. Math. Journal, 
Nov., 1845 ; reprinted in Papers, p. 15. 

\ As regards this comparison, Thomson had been anticipated by Chasles, 
Journal de 1'Ec. Polyt. xv (1837), p. 266, who had shown that attraction accord- 
ing to Newton's law gives rise to the same fields as the steady conduction of heat, 
both depending on Laplace's equation v' V = 0. 

It will be remembered that Ohm had used an analogy between thermal conduction 
and galvanic phenomena. 



270 Maxwell. 

It may, perhaps, seem as if the value of such an analogy 
as this consisted merely in the prospect which it offered of 
comparing, and thereby extending, the mathematical theories 
of heat and electricity. But to the physicist its chief interest 
lay rather in the idea that formulae which relate to the electric 
field, and which had heen deduced from laws of action at a 
distance, were shown to be identical with formulae relating to 
the theory of heat, which had been deduced from hypotheses 
of action between contiguous particles. 

In 1846 the year after he had taken his degree as second 
wrangler at Cambridge Thomson investigated* the analogies 
of electric phenomena with those of elasticity. For this purpose 
he examined the equations of equilibrium of an incompressible 
elastic solid which is in a state of strain ; and showed that 
the distribution of the vector which represents the elastic 
displacement might be assimilated to the distribution of the 
electric force in an electrostatic system. This, however, as he 
went on to show, is not the only analogy which may be 
perceived with the equations of elasticity ; for the elastic 
displacement may equally well be identified with a vector a, 
defined in terms of the magnetic induction B by the relation 

curl a = B. 

The vector a is equivalent to the vector-potential which 
had been used in the memoirs of Neumann, Weber, and 
Kirchhoff, on the induction of currents ; but Thomson arrived 
at it independently by a different process, and without being at 
the time aware of the identification. 

The results of Thomson's memoir seemed to suggest a 
picture of the propagation of electric or magnetic force : might 
it not take place in somewhat the same way as changes in the 
elastic displacement are transmitted through an elastic solid ? 
These suggestions were not at the time pursued further 
by their author; but they helped to inspire another young 

* Camb. and Dub. Math. Journ. ii (1847), p. 61 : Thomson's Math, and Phys. 
Papers, i, p. 76. 



Maxwell. 271 

Cambridge man to take up the matter a few years later. 
James Clerk Maxwell, by whom the problem was eventually 
solved, was born in 1831, the son of a landed proprietor in 
Dumfriesshire. He was educated at Edinburgh, and at Trinity 
College, Cambridge, of which society he became in 1855 a 
Fellow; and not long after his election to Fellowship, he 
communicated to the Cambridge Philosophical Society the first 
of his endeavours* to form a mechanical conception of the 
electro-magnetic field. 

Maxwell had been reading Faraday's Experimental He- 
searches', and, gifted as he was with a physical imagination 
akin to Faraday's, he had been profoundly impressed by the 
theory of lines of force. At the same time, he was a trained 
mathematician ; and the distinguishing feature of almost all 
his researches was the union of the imaginative and the 
analytical faculties to produce results partaking of both 
natures. This first memoir may be regarded as an attempt to 
connect the ideas of Faraday with the mathematical analogies 
which had been devised by Thomson. 

Maxwell considered first the illustration of Faraday's lines 
of force which is afforded by the lines of flow of a liquid. The 
lines of force represent the direction of a vector; and the 
magnitude of this vector is everywhere inversely proportional 
to the cross-section of a narrow tube formed by such lines. 
This relation between magnitude and direction is possessed by 
any circuital vector ; and in particular by the vector which 
represents the velocity at any point in a fluid, if the fluid be 
incompressible. It is therefore possible to represent the 
magnetic induction B, which is the vector represented by 
Faraday's lines of magnetic force, as the velocity of an incom- 
pressible fluid. Such an analogy had been indicated some 
years previously by Faraday himself,f who had suggested that 
along the lines of magnetic force there may be a " dynamic 
condition," analogous to that of the electric current, and 

* Trans. Camb. Phil. Soc. x, p. 27; Maxwell's Scientific Papers, i, p. 155. 
t Exp. Res., 3269 (1852). 



272 Maxwell. 

that, in fact, " the physical lines of magnetic force are 
currents." 

The comparison with the lines of flow of a liquid is 
applicable to electric as well as to magnetic lines of force. In 
this case the vector which corresponds to the velocity of the 
fluid is, in free aether, the electric force E. But when different 
dielectrics are present in the field, the electric force is not a 
circuital vector, and, therefore cannot be represented by lines 
of force ; in fact, the equation 

div E = 
is now replaced by the equation 

div(eE) = 0, 

where g denotes the specific inductive capacity or dielectric 
constant at the place (x, y } z\ It is, however, evident from 
this equation that the vector cE is circuital ; this vector, 
which will be denoted by D, bears to E a relation similar to 
that which the magnetic induction B bears to the magnetic 
force H. It is the vector D which is represented by Faraday's 
lines of electric force, and which in the hydrodynamical 
analogy corresponds to the velocity of the incompressible fluid. 

In comparing fluid motion with electric fields it is necessary 
to introduce sources and sinks into the fluid to correspond to 
the electric charges ; for D is not circuital at places where there, 
is free charge. The magnetic analogy is therefore somewhat 
the simpler. 

In the latter half of his memoir Maxwell discussed how 
Faraday's "electrotonic state" might be represented in mathe- 
matical symbols. This problem he solved by borrowing from 
Thomson's investigation of 1847 the vector a, which is defined 
in terms of the magnetic induction by the equation 

curl a = B ; 

if, with Maxwell, we call a the electrotonic intensity, the. 
equation is equivalent to the statement that " the entire 
electrotonic intensity round the boundary of any surface 
measures the number of lines of magnetic force which pass, 



Maxwell. 273 

through that surface." The electromotive force of induction at 
the place (x, y, z) is - d&/dt : as Maxwell said, " the electromotive 
force on any element of a conductor is measured by the 
instantaneous rate of change of the electrotonic intensity on 
that element." From this it is evident that a is no other than 
the vector-potential which had been employed by Neumann, 
Weber, and Kirchhoff, in the calculation of induced currents ; 
and we may take* for the electrotonic intensity due to a 
current i r flowing in a circuit s' the value which results from 
Neumann's theory, namely, 



., f *s' 
= t' 

} r 



It may, however, be remarked that the equation 

curl a = B, 

taken alone, is insufficient to determine a uniquely ; for we can 
choose a so as to satisfy this, and also to satisfy the equation 

div a = ;//, 

where i// denotes any arbitrary scalar. There are, therefore, an 
infinite number of possible functions a. With the particular 
value of a which has been adopted, we have 



3 ., f dx' 8 f dy' 8 ., f dz 
div a = - i \ - + ^' -2- + - i' \ 
te I' r fy ) 8 , r dz J, r 



., 
* 



= 0; 
so the vector-potential a which we have chosen is circuital. 

In this memoir the physical importance of the operators 
curl and div first became evidentf ; for, in addition to those 
applications which have been mentioned, Maxwell showed that 

* Cf . p. 224. 

t These operators had, however, occurred frequently in the writings of Stokes 
especially in his memoir of 1849 on the Dynamical Theory of Diffraction. 

T 



274 Maxwell. 

he connexion between the strength i of a current and the 
magnetic field H, to which it gives rise, may be represented by 
the equation 

4?ri = curl H ; 

this equation is equivalent to the statement that " the entire 
magnetic intensity round the boundary of any surface measures 
the quantity of electric current which passes through that 
surface." 

In the same year (1856) in which Maxwell's investigation 
was published, Thomson* put forward an alternative inter- 
pretation of magnetism. He had now come to the conclusion, 
from a study of the magnetic rotation of the plane of polariza- 
tion of light, that magnetism possesses a rotatory character; 
and suggested that the resultant angular momentum of the 
thermal motions of a bodyf might be taken as the measure of 
the magnetic moment. " The explanation," he wrote, " of all 
phenomena of electromagnetic attraction or repulsion, or of 
electromagnetic induction, is to be looked for simply in the 
inertia or pressure of the matter of which the motions 
constitute heat. Whether this matter is or is not electricity, 
whether it is a continuous fluid interpermeating the spaces 
between molecular nuclei, or is itself molecularly grouped : or 
whether all matter is continuous, and molecular heterogeneous- 
ness consists in finite vortical or other relative motions of 
contiguous parts of a body: it is impossible to decide, and, 
perhaps, in vain to speculate, in the present state of science." 

The two interpretations of magnetism, in which the linear 
and rotatory characters respectively are attributed to it, occur 
frequently in the subsequent history of the subject. The 
former was amplified in 1858, when Helmholtz published his 
researches^ on vortex motion ; for Helmholtz showed that if a 

*Proc. Roy. Soc. viii (1856), p. 150 ; xi (1861), p. 327, foot-note: Phil. Mag. 
xiii (1857), p. 198; Baltimore Lectures, Appendix F. 

t This was written shortly before the kinetic theory of gases was developed 
by Clausius and Maxwell. 

+ Journal fur Math. Iv (1858), p. 25; Helmholtz's Wiss. Abh. i, p. 101; 
translated Phil. Mag. xxxiii (1867), p. 485. 



Harwell. 275 

magnetic field produced by electric currents is compared to the 
flow of an incompressible fluid, so that the magnetic vector is 
represented by the fluid velocity, then the electric currents 
correspond to the vortex-filaments in the fluid. This analogy 
correlates many theorems in hydrodynamics and electricity ; 
for instance, the theorem that a re-entrant vortex-filament is 
equivalent to a uniform distribution of doublets over any 
surface bounded by it, corresponds to Ampere's theorem of the 
equivalence of electric currents and magnetic shells. 

In his memoir of 1855, Maxwell had not attempted to 
construct a mechanical model of electrodynamic actions, but 
had expressed his intention of doing so. " By a careful study," 
he wrote,* " of the laws of elastic solids, and of the motions of 
viscous fluids, I hope to discover a method of forming a 
mechanical conception of this electrotonic state adapted to 
general reasoning " ; and in a foot-note he referred to the effort 
which Thomson had already made in this direction. Six years 
elapsed, however, before anything further on the subject was 
published. In the meantime, Maxwell became Professor of 
Natural Philosophy in King's College, London a position in 
which he had opportunities of personal contact with Faraday, 
whom he had long reverenced. Faraday had now concluded 
the Experimental Researches, and was living in retirement at 
Hampton Court ; but his thoughts frequently recurred to the 
great problem which he had brought so near to solution. It 
appears from his note-book that in 1857f he was speculating 
whether the velocity of propagation of magnetic action is of the 
same order as that of light, and whether it is affected by the 
susceptibility to induction of the bodies through which the 
action is transmitted. 

The answer to this question was furnished in 1861-2, 
when Maxwell fulfilled his promise of devising a mechanical 
conception of the electromagnetic field.* 

* Maxwell's Scientific Papers, i, p. 188. 
t Bence Jones's Life of Faraday ii, p. 379. 

I Phil. Mag. xxi (1861), pp. 161, 281, 338; xxiii (1862), pp. 12, 85; 
Maxwell's Scientific Papers, i, p. 451. 

T 2 



276 Maxwell. 

In the interval since the publication of his previous memoir 
Maxwell had become convinced by Thomson's arguments that 
magnetism is in its nature rotatory. "The transference of 
electrolytes in fixed directions by the electric current, and the 
rotation of polarized light in fixed directions by magnetic force, 
are," he wrote, "the facts the consideration of which has 
induced me to regard magnetism as a phenomenon of rotation, 
and electric currents as phenomena of translation." This con- 
ception of magnetism he brought into connexion with Faraday's 
idea, that tubes of force tend to contract longitudinally and to 
expand laterally. Such a tendency may be attributed to 
centrifugal force, if it be assumed that each tube of force 
contains fluid which is in rotation about the axis of the tube. 
Accordingly Maxwell supposed that, in any magnetic field, the 
medium whose vibrations constitute light is in rotation about 
the lines of magnetic force; each unit tube of force may for the 
present be pictured as an isolated vortex. 

The energy of the motion per unit volume is proportional 
to /jH 2 , where /j. denotes the density of the medium, and H 
denotes the linear velocity at the circumference of each vortex. 
But, as we have seen,* Thomson had already shown that the 
energy of any magnetic field, whether produced by magnets or 
by electric currents, is 



where the integration is taken over all space, and where it 
denotes the magnetic permeability, and H the magnetic force. 
It was therefore natural to identify the density of the medium 
at any place with the magnetic permeability, and the circum- 
ferential velocity of the vortices with the magnetic force. 

But an objection to the proposed analogy now presents 
itself. Since two neighbouring vortices rotate in the same 
direction, the particles in the circumference of one vortex must 
be moving in the opposite direction to the particles contiguous 

* Cf. pp. 248, 250. 



Maxwell. 277 

to them in the circumference of the adjacent vortex ; and it 
seems, therefore, as if the motion would be discontinuous. 
Maxwell escaped from this difficulty by imitating a well-known 
mechanical arrangement. When it is desired that two wheels 
should revolve in the same sense, an " idle " wheel is inserted 
between them so as to be in gear with both. The model of the 
electromagnetic field to which Maxwell arrived by the intro- 
duction of this device greatly resembles that proposed by 
Bernoulli in 1736.* He supposed a layer of particles, acting as 
idle wheels, to be interposed between each vortex and the next, 
and to roll without sliding on the vortices ; so that each vortex 
tends to make the neighbouring vortices revolve in the same 
direction as itself. The particles were supposed to be not other- 
wise constrained, so that the velocity of the centre of any 
particle would be the mean of the circumferential velocities of 
the vortices between which it is placed. This condition yields 
(in suitable units) the analytical equation 

47Ti = curl H, 

where the vector i denotes the flux of the particles, so that its 
^-component i x denotes the quantity of particles transferred 
in unit time across unit area perpendicular to the ^-direction. 
On comparing this equation with that which represents Oersted's 
discovery, it is seen that the flux i of the movable particles 
interposed between neighbouring vortices is the analogue of 
the electric current. 

It will be noticed that in Maxwell's model the relation 
between electric current and magnetic force is secured by a 
connexion which is not of a dynamical, but of a purely kine- 
matical character. The above equation simply expresses the 
existence of certain non-holonomic constraints within the 
system. 

If from any cause the rotatory velocity of some of the 
cellular vortices is altered, the disturbance will be propagated 
from that part of the model to all other parts, by the mutual 

* Cf. p. 100. 



278 Maxwell. 

action of the particles and vortices. This action is determined, 
as Maxwell showed, hy the relation 

fj$L = - curl E 

which connects E, the force exerted on a unit quantity of 
particles at any place in consequence of the tangential action 
of the vortices, with H, the rate of change of velocity of the 
neighbouring vortices. It will be observed that this equation 
is not kinematical but dynamical. On comparing it with the 
electromagnetic equations 

curl a = /*H, 

Induced electromotive force = - a, 
it is seen that E must be interpreted electromagnetically as the 
induced electromotive force. Thus the motion of the particles 
constitutes an electric current, the tangential force with which 
they are pressed by the matter of the vortex-cells constitutes 
electromotive force, and the pressure of the particles on each 
other may be taken to correspond to the tension or potential of 
the electricity. 

The mechanism must next be extended so as to take account 
of the phenomena of electrostatics. For this purpose Maxwell 
assumed that the particles, when they are displaced from their 
equilibrium position in any direction, exert a tangential action 
on the elastic substance of the cells ; and that this gives rise 
to a distortion of the cells, which in turn calls into play a 
force arising from their elasticity, equal and opposite to the 
force which urges the particles away from the equilibrium 
position. When the exciting force is removed, the cells recover 
their form, and the electricity returns to its former position. 
The state of the medium, in which the electric particles are 
displaced in a definite direction, is assumed to represent an 
electrostatic field. Such a displacement does not itself con- 
stitute a current, because when it has attained a certain value 
it remains constant ; but the variations of displacement are to 
be regarded as currents, in the positive or negative direction 
according as the displacement is increasing or diminishing. 



Maxwell. 279 

The conception of the electrostatic state as a displacement 
of something from its equilibrium position was not altogether 
new, although it had not been previously presented in this 
form. Thomson, as we have seen, had compared electric force 
to the displacement in an elastic solid ; and Faraday, who had 
likened the particles of a ponderable dielectric to small con- 
ductors embedded in an insulating medium,* had supposed that 
when the dielectric is subjected to an electrostatic field, there 
is a displacement of electric charge on each of the small 
conductors. The motion of these charges, when the field is 
varied, is equivalent to an electric current ; and it was from 
this precedent that Maxwell derived the principle, which became 
of cardinal importance in his theory, that variations of displace- 
ment are to be counted as currents. But in adopting the 
idea, he altogether transformed it ; for Faraday's conception of 
displacement was applicable only to ponderable dielectrics, and 
was in fact introduced solely in order to explain why the 
specific inductive capacity of such dielectrics is different from 
that of free aether; whereas according to Maxwell there is 
displacement wherever there is electric force, whether material 
bodies are present or not. 

The difference between the conceptions of Faraday and 
Maxwell in this respect may be illustrated by an analogy 
drawn from the theory of magnetism. When a piece of iron 
is placed in a magnetic field, there is induced in it a magnetic 
distribution, say of intensity I ; this induced magnetization 
exists only within the iron, being zero in the free aether 
outside. The vector I may be compared to the polarization 
or displacement, which according to Faraday is induced in 
dielectrics by an electric field; and the electric current con- 
stituted by the variation of this polarization is then analogous 
to dl/dt. But the entity which was called by Maxwell the 
electric displacement in the dielectric is analogous not to I, 
but to the magnetic induction B : the Maxwellian displace- 

* Cf. p. 210. 



280 Maxwell. 

merit-current corresponds to d'B/dt, and may therefore have a 
value different from zero even in free aether. 

It may be remarked in passing that the term displacement, 
which was thus introduced, and which has been retained in 
the later development of the theory, is perhaps not well chosen ; 
what in the early models of the aether was represented as an 
actual displacement, has in later investigations been conceived 
of as a change of structure rather than of position in the 
elements of the aether. 

Maxwell supposed the electromotive' force acting on the 
electric particles to be connected with the displacement D 
which accompanies it, by an equation of the form 



where c, denotes a constant which depends on the elastic 
properties of the cells. The displacement-current D must now 
be inserted in the relation which connects the current with 
the magnetic force ; and thus we obtain the equation 

curl H = 47rS, 

where the vector S, which is called the total current, is the 
sum of the convection-current i and the displacement-current 
D. By performing the operation div on both sides of this 
equation, it is seen that the total current is a circuital vector. 
In the model, the total current is represented by the total 
motion of the rolling particles ; and this is conditioned by the 
rotations of the vortices in such a way as to impose the 
kinematic relation 

div S = 0. 

Having obtained the equations of motion of his system 
of vortices and particles, Maxwell proceeded to determine the 
rate of propagation of disturbances through it. He considered 
in particular the case in which the substance represented is a 
dielectric, so that the conduction-current is zero. If, moreover, 



Maxwell. 281 

the constant fi be supposed to have the value unity, the 
equations may be written 

div H = 0, 

c, 2 curl H = E, 
- curl E = H. 
Eliminating E, we see* that H satisfies the equations 

jdivH = 0, 



But these are precisely the equations which the light- vector 
satisfies in a medium in which the velocity of propagation is c^ : 
it follows that disturbances are propagated through the model 
by waves which are similar to waves of light, the magnetic 
(and similarly the electric) vector being in the wave-front. 
For a plane-polarized wave propagated parallel to the axis of z, 
the equations reduce to 



2 y = x 2 *^y y 

" Cl dz '"' dt' Cl ~dz '' dt' dz dt' dz 

whence we have 

= E x - c\S x = E 



these equations show that the electric and magnetic vectors are 
at right angles to each other. 

The question now arises as to the magnitude of the constant 
Cj.f This may be determined by comparing different expressions 
for the energy of an electrostatic field. The work done by an 
electromotive force E in producing a displacement D is 

fD 

E . dD or JED 

o 

per unit volume, since E is proportional to D. But if it be 
assumed that the energy of an electrostatic field is resident in 
the dielectric, the amount of energy per unit volume may be 

* For if a denote any vector, we have identically 

V-a -f grad div a + curl curl a = 0. 

t For criticisms on the procedure by which Maxwell determined the velocity of 
propagation of disturbance, cf. P. Duhem, Les Theories Electriqv.es de J. Clerk 
Maxwell, Paris, 1902. 



282 Maxwell. 

calculated by considering the mechanical force required in 
order to increase the distance between the plates of a condenser, 
so as to enlarge the field comprised between them. The result 
is that the energy per unit volume of the dielectric is fE /2 /87r, 
where c denotes the specific inductive capacity of the dielectric 
and E' denotes the electric force, measured in terms of the 
electrostatic unit : if E denotes the electric force expressed in 
terms of the electrodynamic units used in the present investi- 
gation, we have E = cE', where c denotes the constant which* 
occurs in transformations of this kind. The energy is therefore 
fcE 2 /87TC 2 per unit volume. Comparing this with the expression 
for the energy in terms of E and D, we have 

D 



and therefore the constant Ci has the value ct*. Thus the 
result is obtained that the velocity of propagation of dis- 
turbances in Maxwell's medium is ce~, where denotes the 
specific inductive capacity and c denotes the velocity for which 
Kohlrausch and Weber had foundf the value 3*1 x 10 10 cm./sec. 
Now by this time the velocity of light was known, not only 
from the astronomical observations of aberration and of Jupiter's 
satellites, but also by direct terrestrial experiments. In 1849 
Hippolyte Louis FizeauJ had ' determined it by rotating a 
toothed wheel so rapidly that a beam of light transmitted 
through the gap between two teeth and reflected back from a 
mirror was eclipsed by one of the teeth on its return journey. 
The velocity of light was calculated from the dimensions and 
angular velocity of the wheel and the distance of the mirror ; 
the result being 3*15 x 10 10 cm. /sec. 

* Cf. pp. 227, 259. | Cf. p. 260. 

| Comptes Rendus, xxix (1849), p. 90. A determination made by Cornu in 
1874 was on this principle. 

A different experimental method was employed in 1862 hy Leon Foucault 
(Comptes Rendus, Iv, pp. 501, 792) ; in this a ray from an origin was reflected 
by a revolving mirror M to a fixed mirror, and so reflected back to J/, and again 
to O. It is evident that the returning ray ?dO must be deviated by twice the 
angle through which M turns while the light passes from M to the fixed mirror 
and back. The value thus obtained by Foucault for the velocity of light was 



Maxwell. 283 

Maxwell was impressed, as Kirchhoff had been before him, 
by the close agreement between the electric ratio c and the 
velocity of light* ; and having demonstrated that the propaga- 
tion of electric disturbance resembles that of light, he did not 
hesitate to assert the identity of the two phenomena. "We 
can scarcely avoid the inference," he said, " that light consists 
in the transverse undulations of the same medium which is the 
cause of electric and magnetic phenomena." Thus was answered 
the question which Priestley had asked almost exactly a hundred 
years before :f "Is there any electric fluid sui generis at all, 
distinct from the aether ? " 

The presence of the dielectric constant e in the expression 
ct -i, which Maxwell had obtained for the velocity of propaga- 
tion of electromagnetic disturbances, suggested a further test 
of the identity of these disturbances with light: for the velocity 
of light in a medium is known to be inversely proportional to 
the refractive index of the medium, and therefore the refractive 
index should be, according to the theory, proportional to the 
square root of the specific inductive capacity. At the time, 
however, Maxwell did not examine whether this relation 
was confirmed by experiment. 

In what has preceded, the magnetic permeability //, has been 
supposed to have the value unity. If this is not the case, the 

2-98 x 10 10 cm./sec. Subsequent determinations by Michelson in 187'.) (Ast. 
Papers of the Amer. Ephemeris, i), and by Newcomb in 1882 (ibid., ii) depended 
on the same principle. 

As was shown afterwards by Lord Rayleigh (Nature, xxiv, p. 382, xxv, p. 52) 
and by Gibbs (Nature, xxxiii, p. 582), the value obtained for the velocity of light 
by the methods of Fizeau and Foucault represents the group-velocity, not the wave- 
velocity ; the eclipses of Jupiter's satellites also give the group-velocity, while the 
value deduced from the coefficient of aberration is the wave- velocity. In a non- 
dispersive medium, the group- velocity coincides with the wave- velocity ; and the 
agreement of the values of the velocity of light obtained by the two astronomical 
methods seems to negative the possibility of any appreciable dispersion in free 
aether. 

The velocity of light in dispersive media was directly investigated by Michelson 
in 1883-4, with results in accordance with theory. 

* He had "worked out the formulae in the country, before seeing Weber's 
result." Cf. Campbell and Garnett's Life of Maxwell, p. 244. 

f Priestley's Eistory, p. 488. 



284 Maxwell. 

velocity of propagation of disturbance may be shown, by the 
same analysis, to be ct~i^~i ; so that it is diminished when /u is 
greater than unity, i.e., in paramagnetic bodies. This inference 
had been anticipated by Faraday : " Nor is it likely," he wrote,* 
" that the paramagnetic body oxygen can exist in the air and 
not retard the transmission of the magnetism." 

It was inevitable that a theory so novel and so capacious as 
that of Maxwell should involve conceptions which his contempo- 
raries understood with difficulty and accepted with reluctance. 
Of these the most difficult and unacceptable was the principle 
that the total current is always a circuital vector ; or, as it is 
generally expressed, that " all currents are closed." According 
to the older electricians, a current which is employed in charging 
a condenser is not closed, but terminates at the coatings of the 
condenser, where charges are accumulating. Maxwell, on the 
other hand, taught that the dielectric between the coatings 
is the seat of a process the displacement-current which is 
proportional to the rate of increase of the electric force in the 
dielectric ; and that this process produces the same magnetic 
effects as a true current, and forms, so to speak, a continuation, 
through the dielectric, of the charging current, so that the 
latter may be regarded as flowing in a closed circuit. 

Another characteristic feature of Maxwell's theory is the 
conception for which, as we have seen, he was largely indebted 
to Faraday and Thomson that magnetic energy is the kinetic 
energy of a medium occupying the whole of space, and that 
electric energy is the energy of strain of the same medium. 
By this conception electromagnetic theory was brought into 
such close parallelism with the elastic- solid theories of the 
aether, that it was bound to issue in an electromagnetic theory 
of light. 

Maxwell's views were presented in a more developed form 
in a memoir entitled "A Dynamical Theory of the Electro- 
magnetic Field," which was read to the Koyal Society in 1864 ;f 

* Faraday's laboratory note-book for 1857 : of. Bence Jones's Life of Faraday, 
ii, p. 380. 

t Phil. Trans, civ (1865), p. 459 : Maxwell's Scient. Papers, i, p. 526 



Maxwell. 285 

in this the architecture of his system was displayed, stripped of 
the scaffolding by aid of which it had been first erected. 

As the equations employed were for the most part the same 
as had been set forth in the previous investigation, they need 
only be briefly recapitulated. The magnetic induction juH, being 
a circuital vector, may be expressed in terms of a vector-potential 

A by the equation 

luiK = curl A. 

The electric displacement D is connected with the volume- 
density p of free electric charge by the electrostatic equation 

div D = p. 

The principle of conservation of electricity yields the equation 
div i = - dp/dt, 

where i denotes the conduction-current. 

The law of induction of currents namely, that the total 
electromotive force in any circuit is proportional to the rate of 
decrease of the number of lines of magnetic induction which 
pass through it may be written 

- curl E = /LtH ; 

from which it follows that the electric force E must be expressible 

in the form 

E = - A + grad i//, 

where ^ denotes some scalar function. The quantities A and ;// 
which occur in this equation are not as yet completely deter- 
minate ; for the equation by which A is defined in terms of the 
magnetic induction specifies only the circuital part of A ; and as 
the irrotational part of A is thus indeterminate, it is evident 
that \p also must be indeterminate. Maxwell decided the matter 
by assuming* A to be a circuital vector ; thus 

divA = 0, 
and therefore div E = - 



* This is the effect of the introduction of (F 1 , G', H'} in 98 of the memoir ; 
cf. also Maxwell's Treitise on Electricity and Magnetism, 616. 



286 Maxwell. 

from which equation it is evident that ^ represents the electro- 
static potential. 

The principle which is peculiar to Maxwell's theory must 
now be introduced. Currents of conduction are not the only 
kind of currents ; even in the older theory of Faraday, Thomson, 
and Mossotti, it had been assumed that electric charges 
are set in motion in the particles of a dielectric when the 
dielectric is subjected to an electric field ; and the prede- 
cessors of Maxwell would not have refused to admit that the 
motion of these charges is in some sense a current. Suppose, 
then, that S denotes the total current which is capable of 
generating a magnetic field : since the integral of the magnetic 
force round any curve is proportional to the electric current 
which flows through the gap enclosed by the curve, we have in 
suitable units 

curl H = 4;rS. 

In order to determine S, we may consider the case of a con- 
denser whose coatings are supplied with electricity by a 
conduction-current i per unit-area of coating. If o- denote 
the surface-density of electric charge on the coatings, we have 

i = d(r/dt t and o- = D, 

where D denotes the magnitude of the electric displacement D 
in the dielectric between the coatings ; so i = D. But since the 
total current is to be circuital, its value in the dielectric must 
be the same as the value i which it has in the rest of the 
circuit ; that is, the current in the dielectric has the value D. 
We shall assume that the current in dielectrics always has this 
value, so that in the general equations the total current must 
be understood to be i + D. 

The above equations, together with those which express the 
proportionality of E to D in insulators, and to i in conductors, 
constituted Maxwell's system for a field formed by isotropic 
bodies which are not in motion. When the magnetic field is 
.due entirely to currents (including both conduction-currents 






Maxwell. 287 

and displacement-currents), so that there is no magnetization, 
we have 

V 2 A = - curl curl A = - curl H 

= - 47TS, 

so that the vector-potential is connected with the total current 
by an equation of the same form as that which connects the 
scalar potential with the density of electric charge. To these 
potentials Maxwell inclined to attribute a physical significance ; 
he supposed i// to be analogous to a pressure subsisting in the 
mass of particles in his model, and A to be the measure of 
the electrotonic state. The two functions are, however, of 
merely analytical interest, and do not correspond to physical 
entities. For let two oppositely-charged conductors, placed 
close to each other, give rise to an electrostatic field throughout 
all space. In such a field the vector-potential A is everywhere 
zero, while the scalar potential $ has a definite value at every 
point. Now let these conductors discharge each other ; the 
electrostatic force at any point of space remains unchanged 
until the point in question is reached by a wave of disturbance, 
which is propagated outwards from the conductors with the 
velocity of light, and which annihilates the field as it passes 
over it. But this order of events is not reflected in the 
behaviour of Maxwell's functions ;// and A ; for at the instant 
of discharge, ^ is everywhere annihilated, and A suddenly 
acquires a finite value throughout all space. 

As the potentials do not possess any physical significance, 
it is desirable to remove them from the equations. This was 
afterwards done by Maxwell himself, who* in 1868- proposed 
to base the electromagnetic theory of light solely on the 
equations 

curl H = 47rS, 

- curl E = B, 

together with the equations which define S in terms of E, and B 
in terms of H. 

* Phil. Trans, clviii (1868), p. 643 : Maxwell's Scient. Papers, ii, p. 125. 



288 Maxwell. 

The memoir of 1864 contained an extension of the equations 
to the case of bodies in motion ; the consideration of which 
naturally revives the question as to whether the aether is in 
any degree carried along with a body which moves through it. 
Maxwell did not formulate any express doctrine on this subject ; 
but his custom was to treat matter as if it were merely a 
modification of the aether, distinguished only by altered 
values of such constants as the magnetic permeability and 
the specific inductive capacity ; so that his theory may be 
said to involve the assumption that matter and aether move 
together. In deriving the equations which are applicable to 
moving bodies, he made use of Faraday's principle that the 
electromotive force induced in a body depends only on the 
relative motion of the body and the lines of magnetic force, 
whether one or the other is in motion absolutely. From this 
principle it may be inferred that the equation which determines 
the electric force* in terms of the potentials, in the case of a 
body which is moving with velocity w, is 

E = [w . /zH] - A + grad ^. 

Maxwell thought that the scalar quantity -fy in this equation 
represented the electrostatic potential; but the researches of 
other investigators-)- have indicated that it represents the sum 
of the electrostatic potential and the quantity (A . w). 

The electromagnetic theory of light was moreover extended 
in this memoir so as to account for the optical properties of 
crystals. For this purpose Maxwell assumed that in crystals 
the values of the coefficients of electric and magnetic induction 
depend on direction, so that the equation 

fjbK = curl A 
is replaced by 

= curl A ; 



* It may be here remarked that later writers have distinguished between the 
electric force in a moving body and the electric force in the aether through which 
the body is moving, and that E in the present equation corresponds to the former 
of these vectors. 

t Helmholtz, Journ. fiir Math., Ixxviii (1874), p. 309; H. W. Watson, Phil. 
Mag. (5), xxv (1888), p. 271. 



Maxwell. 289 

and similarly the equation 

E = 47rcO>/6 

is replaced by 

E = 4;r (c?D. xt c?D y , cjD z \ 

The other equations are the same as in isotropic media ; so that 

the propagation of disturbance is readily seen to depend on the 

equation 

(/i J? ft.ffy, H Z H Z } = - curl [c, 2 (curl 5),, tf(cuilH} y , Ca 2 ( curl -#)*) 

Now, if jui, ju 2 , A*3 are supposed equal to each other, this 
equation is the same as the equation of motion of MacCullagh's 
aether in crystalline media,* the magnetic force H corresponding 
to MacCullagh's elastic displacement ; and we may therefore 
immediately infer that Maxwell's electromagnetic equations 
yield a satisfactory theory of the propagation of light in 
crystals, provided it is assumed that the magnetic permeability 
is (for optical purposes) the same in all directions, and pro- 
vided the plane of polarization is identified with the plane 
which contains the magnetic vector. It is readily shown that 
the direction of the ray is at right angles to the magnetic 
vector and the electric force, and that the wave-front is the 
plane of the magnetic vector and the electric displacement.f 

After this Maxwell proceeded to investigate the propagation 
of light in metals. The difference between metals and dielectrics, 
so far as electricity is concerned, is that the former are con- 
ductors ; and it was therefore natural to seek the cause of the 
optical properties of metals in their ohmic conductivity. This 
idea at once suggested a physical reason for the opacity of 
metals namely, that within a metal the energy of the light 
vibrations is converted into Joulian heat in the same way as 
the energy of ordinary electric currents. 

* Cf. pp. 154 et sqq. 

f In the memoir of 1864 Maxwell left open the choice between the above theory 
and that which is obtained by assuming that in crystals the specific inductive 
rapacity is (for optical purposes) the same in all directions, while the magnetic 
permeability is aeolotropic. In the latcer case the plane of polarization must be 
identified with the plane which contains the electric displacement. Nine years 
later, in his Treatise ( 794), Maxwell definitely adopted the former alternative. 

U 



290 Maxwell. 

The equations of the electromagnetic field in the metal may 
be written 

curl H = 47rS, 

- curl E = H, 

S = i + D = K E + 



where K denotes the ohmic conductivity ; whence it is seen that 
the electric force satisfies the equation 

=c 2 V 2 E. 



This is of the same form as the corresponding equation in 
the elastic-solid theory* ; and, like it, furnishes a satisfactory 
general explanation of metallic reflexion. It is indeed correct 
in all details, so long as the period of the disturbance is not too 
short i.e., so long as the light- waves considered belong to the 
extreme infra-red region of the spectrum ; but if we attempt to 
apply the theory to the case of ordinary light, we are confronted 
by the difficulty which Lord Eayleigh indicated in the elastic- 
solid theory,f and which attends all attempts to explain the 
peculiar properties of metals by inserting a viscous term in 
the equation. The difficulty is that, in order to account for the 
properties of ideal silver, we must suppose the coefficient of 
E negative that is, the dielectric constant of the metal must 
be negative, which would imply instability of electrical 
equilibrium in the metal. The problem, as we have already 
remarked,:}: was solved only when its relation to the theory of 
dispersion was rightly understood. 

At this time important developments were in progress in 
the last-named subject. Since the time of Fresnel, theories of 
dispersion had proceeded! from the assumption that the radii 
of action of the particles of luminiferous media are so large 
as to be comparable with the wave-length of light. It was 
generally supposed that the aether is loaded by the molecules 

* Cf. p. iso. 

t Cf. p. 181. Cf. also Rayleigh, Phil. Mag. (5) xii (1881), p. 81, and 
H. A. Lorentz, Over de Theorie de Terugkaatsing, Arnhem, 1875. 
+ Cf. p. 181. Cf. p. 182. 



Maxwell. 291 

of ponderable matter, and that the amount of dispersion 
depends on the ratio of the wave-length to the distance 
between adjacent molecules. This hypothesis was, however, 
seen to be inadequate, when, in 1862, F. P. Leroux* found that 
a prism filled with the vapour of iodine refracted the red rays 
to a greater degree than the blue rays; for in all theories 
which depend on the assumption of a coarse-grained lumini- 
f erous medium, the refractive index increases with the frequency 
of the light. 

Leroux's phenomenon, to which the name anomalous dis- 
persion was given, was shown by later investigators-)- to be 
generally associated with " surface-colour." i.e., the property of 
brilliantly reflecting incident light of some particular frequency. 
Such an association seemed to indicate that the dispersive 
property of a substance is intimately connected with a certain 
frequency of vibration which is peculiar to that substance, and 
which, when it happens to fall within the limits of the visible 
spectrum, is apparent in the surface-colour. This idea of a 
frequency of vibration peculiar to each kind of ponderable 
matter is found in the writings of Stokes as far back as the 
year 1852 ; when, discussing fluorescence, he remarked: 
" Nothing seems more natural than to suppose that the incident 
vibrations of the luminiferous aether produce vibratory move- 
ments among the ultimate molecules of sensitive substances, 
and that the molecules in turn, swinging on their own account, 
produce vibrations in the luminiferous aether, and thus cause 
the sensation of light. The periodic times of these vibrations 
depend on the periods in which the molecules are disposed to 
swing, not upon the periodic time of the incident vibrations." 

The principle here introduced, of considering the molecules 
as dynamical systems which possess natural free periods, and 
which interact with the incident vibrations, lies at the basis of 

* Comptes Rendus, Iv (1862), p. 126. In 1870 C. Christiansen (Ann. d. Phys. 
cxli, p. 479 ; cxliii, p. 250) observed a similar effect in a solution of fuchsin. 

r Especially by Kundt, in a series of papers in the Annalen d. Phys., from 
vol. cxlii (1871) onwards. 

j Phil. Trans., 1852, p. 463. Stokes's Coll. Papers, iii., p. 267. 

U 2 



292 Maxwell. 

all modern theories of dispersion. The earliest of these was 
devised by Maxwell, who, in the Cambridge Mathematical Tripos 
for 1869,* published the results of the following investigation : 

A model of a dispersive medium may be constituted by 
embedding systems which represent the atoms of ponderable 
matter in a medium which represents the aether. We may 
picture each atomj- as composed of a single massive particle 
supported symmetrically by springs from the interior face of 
a massless spherical shell : if the shell be fixed, the particle 
will be capable of executing vibrations about the centre of the 
sphere, the effect of the springs being equivalent to a force on 
the particle proportional to its distance from the centre. The 
atoms thus constituted may be supposed to occupy small 
spherical cavities in the aether, the outer shell of each atom 
being in contact with the aether at all points and partaking 
of its motion. An immense number of atoms is supposed to 
exist in each unit volume of the dispersive medium, so that 
the medium as a whole is fine-grained. 

Suppose that the potential energy of strain of free aether 
per unit volume is 



where j denotes the displacement and E an elastic constant ; 
so that the equation of wave-propagation in free aether is 

3*1 a 2 ,, 

''a? = K & 

where p denotes the aethereal density. 

Then if <r denote the mass of the atomic particles in unit 
volume, (TJ + ) the total displacement of an atomic particle at 
the place x at time t, and <rp 2 the attractive force, it is evident 
that for the compound medium the kinetic energy per unit 
volume is 



* Cambridge Calendar, 1869 ; republished by Lord Kayleigh, Phil. Mag. xlviii 
(1899), p. 151. t This illustration is due to "W. Thomson. 



Maxwell. 293 



and the potential energy per unit volume is 

+ 



The equations of motion, derived by the process usual in 
dynamics, are 



Consider the propagation, through the medium thus constituted, 
of vibrations whose frequency is n, and whose velocity of pro- 
pagation in the medium is v ; so that r\ and are harmonic 
functions of n(t - x/v). Substituting these values in the 
differential equations, we obtain 

1 o oil? 2 



Now, p/^ 7 has the value 1/c 2 , where c denotes the velocity of 
light in free aether; and c/v is the refractive index ju of the 
medium for vibrations of frequency n. So the equation, which 
may be written 



determines the refractive index of the substance for vibrations 
of any frequency n. The same formula was independently 
obtained from similar considerations three years later by 
W. Sellmeier * 

If the oscillations are very slow, the incident light being in 
the extreme infra-red part of the spectrum, n is small, and the 
equation gives approximately ju 2 = (p + a)jp : for such oscilla- 
tions, each atomic particle and its shell move together as a 
rigid body, so that the effect is the same as if the aether were 
simply loaded by the masses of the atomic particles, its rigidity 
remaining unaltered. 

* Ann. d. Phys. oxlv (1872), pp. 399, 520 : cxlvii (1872), pp. 386, 525. 
Cf. also Helmholtz, Ann. d. Phys. cliv (1875), p. 582. 



294 Maxwell. 

The dispersion of light within the limits of the visible 
spectrum is for most substances controlled by a natural 
frequency p which corresponds to a vibration beyond the violet 
end of the visible spectrum : so that, n being smaller than p, 
we may expand the fraction in the formula of dispersion, and 
obtain the equation 

(T I n z n* 

fJL 2 = 1 + - (1 + - + -+... 
f>\ P* P* 

which resembles the formula of dispersion in Cauchy's theory* ; 
indeed, we may say that Cauchy's formula is the expansion of 
Maxwell's formula in a series which, as it converges only when 
n has values within a limited range, fails to represent the 
phenomena outside that range. 

The theory as given above is defective in that it becomes 
meaningless when the frequency n of the incident light is 
equal to the frequency p of the free vibrations of the atoms. 
This defect may be remedied by supposing that the motion of 
an atomic particle relative to the shell in which it is contained 
is opposed by a dissipative force varying as the relative 
velocity ; such a force suffices to prevent the forced vibration 
from becoming indefinitely great as the period of the incident 
light approaches the period of free vibration of the atoms ; its 
introduction is justified by the fact that vibrations in this 
part of the spectrum suffer absorption in passing through the 
medium. When the incident vibration is not in the same 
region of the spectrum as the free vibration, the absorption is 
not of much importance, and may be neglected. 

It is shown by the spectroscope that the atomic systems 
which emit and absorb radiation in actual bodies possess more 
than one distinct free period. The theory already given may, 
however, readily be extended-)- to the case in which the atoms 
have several natural frequencies of vibration ; we have only to 
suppose that the external massless rigid shell is connected by 
springs to an interior massive rigid shell, and that this again 
* Cf. p. 183. 

t This subject was developed by Lord Kelvin in the' Baltimore Lectures. 



Maxwell. 295 

is connected by springs to another massive shell inside it, and 
so on. The corresponding extension of the equation for the 
refractive index is 



where p^ p 2 , . . . denote the frequencies of the natural periods 
of vibration of the atom. 

The validity of the Maxwell- Sellmeier formula of disper- 
sion was strikingly confirmed by experimental researches in 
the closing years of the nineteenth century. In 1897 Rubens* 
showed that the formula represents closely the refractive 
indices of sylvin (potassium chloride) and rock-salt, with 
respect to light and radiant heat of wave-lengths between 
4,240 A.U. and 223,000 A.U. The constants in the formula 
being known from this comparison, it was possible to predict 
the dispersion for radiations of still lower frequency ; and it 
was found that the square of the refractive index should have 
a negative value (indicating complete reflexion) for wave- 
lengths 370,000 A.U. to 550,000 A.U. in the case of rock-salt, 
and for wave-lengths 450,000 ^to 670,000 A.U. in the case of 
sylvin. This inference was verified experimentally in the 
following year.f 

It may seem strange that Maxwell, having successfully 
employed his electromagnetic theory to explain the propagation 
of light in isotropic media, in crystals, and in metals, should 
have omitted to apply it to the problem of reflexion and refrac- 
tion. This is all the more surprising, as the study of the optics 
of crystals had already revealed a close analogy between the 
electromagnetic theory and MacCullagh's elastic-solid theory; 
and in order to explain reflexion and refraction electro- 
magnetically, nothing more was necessary than to transcribe 
MacCullagh's investigation of the same problem, interpreting e 
(the time-flux of the displacement of MacCullagh's aether) as 
the magnetic force, and curl e as the electric displacement. As 

* Ann. d. Phys. Ix (1897), p. 454. 

t Rubens and Aschkinass, Ann. d. Phys. Ixiv (1898). 



296 Maxwell. 

in MacCullagh's theory the difference between the contiguous 
media is represented by a difference of their elastic constants, 
so in the electromagnetic theory it may be represented by a 
difference in their specific inductive capacities. From a letter 
which Maxwell wrote to Stokes in 1864, and which has been 
preserved,* it appears that the problem of reflexion and refrac- 
tion was engaging Maxwell's attention at the time when he was 
preparing his Eoyal Society memoir on the electromagnetic 
field; but he was not able to satisfy himself regarding the 
conditions which should be satisfied at the interface between 
the media. He seems to have been in doubt which of the rival 
elastic-solid theories to take as a pattern ; and it is not unlikely 
that he was led astray by relying too much on the analogy 
between the electric displacement and an elastic displacement. t 
For in the elastic-solid theory all three components of the dis- 
placement must be continuous across the interface between two 
contiguous media ; but Maxwell found that it was impossible to 
explain reflexion and refraction if all three components of the 
electric displacement were supposed to be continuous across the 
interface ; and, unwilling to give up the analogy which had 
hitherto guided him aright, yet unable to disprove^ the Greenian 
conditions at bounding surfaces, he seems to have laid aside the 
problem until some new light should dawn upon it. 

This was not the only difficulty which beset the electro- 
magnetic theory. The theoretical conclusion, that the specific 
inductive capacity of a medium should be equal to the square of 
its refractive index with respect to waves of long period, was 
not as yet substantiated by experiment; and the theory of 
displacement-currents, on which everything else depended, was 

* Stokes's Scientific Correspondence, ii, pp. 25, 26. 

t It must be remembered tbat Maxwell pictured tbe electric displacement as a 
real displacement of a medium. "My theory of electrical forces," he \vrote, " is 
that they are called into play in insulating media by slight electric displacements, 
which put certain small portions of the medium into a state of distortion, which, 
being resisted by the elasticity of the medium, produces an electromotive force." 
Campbell and Garnett's Life of Maxwell, p. 244. 

| The letter to Stokes already mentioned appears to indicate that Maxwell for 
a time doubted the correctness of Green's conditions. 






Maxwell. 297 



unfavourably received by the most distinguished of Maxwell's 
contemporaries. Helmholtz indeed ultimately accepted it, but 
only after many years ; and W. Thomson (Kelvin) seems never 
to have thoroughly believed it to the end of his long life. In 
1888 he referred to it as a "curious and ingenious, but not 
wholly tenable hypothesis,"* and proposedf to replace it by an 
extension of the older potential theories. In 1896 he had some 
inclination? to speculate that alterations of electrostatic force 
due to rapidly-changing electrification are propagated by con- 
densational waves in the luminiferous aether. In 1904 he 
admittedg that a bar-magnet rotating about an axis at right 
angles to its length is equivalent to a lamp emitting light of 
period equal to the period of the rotation, but gave his final 
judgment in the sentence|| : " The so-called electromagnetic 
theory of light has not helped us hitherto." 

Thomson appears to have based his ideas of the propagation 
of electric disturbance on the case which had first become 
familiar to him that of the transmission of signals along a 
wire. He clung to the older view that in such a disturbance 
the wire is the actual medium of transmission ; whereas in 
\ Maxwell's theory the function of the wire is merely to guide 
the disturbance, which is resident in the surrounding dielectric. 

This opinion that conductors are the media of propagation 
of electric disturbance was entertained also by Ludwig Lorenz 
(&. 1829, d. 1891), of Copenhagen, who independently developed 
an electromagnetic theory of lightH a few years after the 
publication of Maxwell's memoirs. The procedure which 
Lorenz followed was that which Kiemann had suggested** in 
1858 namely, to modify the accepted formulae of electro- 
dynamics by introducing terms which, though too small to be 

* Nature, xxxviii (1888) p. 571. t Brit. Assoc. Report, 1888, p. 567. 

J Cf. Bottouiley, in Nature, liii (1896), p. 268 ; Kelvin, ib., p. 316 ; J. Willard 
Gibbs, ib., p. 509. 

Baltimore Lectures (ed. 1904), p. 376. || Ibid., preface, p. 7. 

H Oversiyt over det K. danske Vid. Selskaps Forhandliiiger, 1867, p. 26; Annul, 
der Phys. cxxxi (1867), p. 243 ; Phil. Mag., xxxiv (1867), p. 287. 

** Cf. p. 268. Riemann's memoir was, however, published only in the same 
year (1867) as Lorenz's. 



298 Maxwell. 

appreciable in ordinary laboratory experiments, would be 
capable of accounting for the propagation of electrical effects 
through space with a finite velocity. We have seen that in 
Neumann's theory the electric force E was determined by the 
equation 

-a, (1) 



where < denotes the electrostatic potential defined by the 
equation 

4>-{\\(p'lr) dx'dy'dz', 



p being the density of electric charge at the point (x, y, z'), and 
where a denotes the vector-potential, defined by the equation 

a={\[(i'lr)dx'dy'dz, 

J J J 

i' being the conduction-current at (x', y\ z'). We suppose the 
specific inductive capacity and the magnetic permeability to 
be everywhere unity. 

Lorenz proposed to replace these by the equations 



= \\\{p(t-r/c)/r\dx'dy'dz', 
{i'(t-r/c)/r}dx'dy'd3f' 9 



the change consists in replacing the values which p and i' have 
at the instant t by those which they have at the instant (t - r/c], 
which is the instant at which a disturbance travelling with 
velocity c must leave the place (x', y, z) in order to arrive at 
the place (x, y, z) at the instant t. Thus the values of the 
potentials at (x, y, z] at any instant t would, according to 
Lorenz's theory, depend on the electric state at the point 
(x', y', z') at the previous instant (t - r/c) : as if the potentials 
were propagated outwards from the charges and currents with 
velocity c. The functions <f> and a formed in this way are 
generally known as the retarded potentials. 



Maxwell. 299 

The equations by which (f> and a have been defined are 
equivalent to the equations 

V 2 </> - $1* = - 4^, (2) 

V 2 a - a/c 2 = - 47ri, (3) 

while the equation of conservation of electricity, 

div i + p = 
gives 

div a + <f> = 0. (4) 

From equations (1), (2), (4), we may readily derive the equation 

divE = 47rcV; (I) 

and from (1), (3), (4), we have 

curl H = E/c 2 + 47rt, (II) 

where H or curl a denotes the magnetic force : while from (1) 
we have 

curl E = - H. (Ill) 

The equations (I), (II), (III) are, however, the fundamental 
equations of Maxwell's theory; and therefore the theory of 
L. Lorenz is practically equivalent to that of Maxwell, so far 
as concerns the propagation of electromagnetic disturbances 
through free aether. Lorenz himself, however, does not appear 
to have clearly perceived this ; for in his memoir he postulated 
the presence of conducting matter throughout space, and was 
consequently led to equations resembling those which Maxwell 
had given for the propagation of light in metals. Observing 
that his equations represented periodic electric currents at 
right angles to the direction of propagation of the disturbance, 
he suggested that all luminous vibrations might be constituted 
by electric currents, and hence that there was " no longer any 
reason for maintaining the hypothesis of an aether, since we 
can admit that space contains sufficient ponderable matter to 
enable the disturbance to be propagated." 

Lorenz was unable to derive from his equations any explana- 
tion of the existence of refractive indices, and his theory lacks 



300 Maxwell. 

the rich physical suggestiveness of Maxwell's ; the value of 
his memoir lies chiefly in the introduction of the retarded 
potentials. It may be remarked in passing that Lorenz's 
retarded potentials are not identical with Maxwell's scalar 
and vector potentials ; for Lorenz's a is not a circuital vector, 
and Lorenz's < is not, like Maxwell's, the electrostatic potential, 
but depends on the positions occupied by the charges at certain 
previous instants. 

For some years no progress was made either with Maxwell's 
theory or with Lorenz's. Meanwhile, Maxwell had in 1865 
resigned his chair at King's College, and had retired to his 
estate in Dumfriesshire, where he occupied himself in writing 
a connected account of electrical theory. In 1871 he returned 
to Cambridge as Professor of Experimental Physics; and two 
years later published his Treatise on Electricity and Magnetism. 

In this celebrated work is comprehended almost every 
branch of electric and magnetic theory; but the intention of 
the writer was to discuss the whole as far as possible from a 
single point of view, namely, that of Faraday; so that little 
or no account was given of the hypotheses which had been pro- 
pounded in the two preceding decades by the great German 
electricians. So far as Maxwell's purpose was to disseminate 
the ideas of Faraday, it was undoubtedly fulfilled ; but the 
Treatise was less successful when considered as the exposition 
of its author's own views. The doctrines peculiar to Maxwell 
the existence of displacement-currents, and of electromagnetic 
vibrations identical with light were not introduced in the first 
'volume, or in the first half of the second volume ; and the 
account which was given of them was scarcely more complete, 
and was perhaps less attractive, than that which had been 
furnished in the original memoirs. 

Some matters were, however, discussed more fully in the 
Treatise than in Maxwell's previous writings ; and among these 
was the question of stress in the electromagnetic field. 

It will be remembered* that Faraday, when studying the 

* Cf. p. 209. 



Maxwell. 301 

curvature of lines of force in electrostatic fields, had noticed 
an apparent tendency of adjacent lines to repel each other, as 
if each tube of force were inherently disposed to distend 
laterally ; and that in addition to this repellent or diverging 
force in the transverse direction, he supposed an attractive or 
contractile force to be exerted at right angles to it, that is to 
say, in the direction of the lines of force. 

Of the existence of these pressures and tensions Maxwell 
was fully persuaded ; and he determined analytical expressions 
suitable to represent them. The tension along the lines of 
force must be supposed to maintain the ponderomotive force 
which acts on the conductor on which the lines of force 
terminate ; and it may therefore be measured by the force 
which is exerted on unit area of the conductor, i.e., *E 2 /87rc 2 or 
iDE. The pressure at right angles to the lines of force must 
then be determined so as to satisfy the condition that the aether 
is to be in equilibrium. 

For this purpose, consider a thin shell of aether included 
between two equipotential surfaces. The equilibrium of the, 
portion of this shell which is intercepted by a tube of force- 
requires (as in the theory of the equilibrium of liquid films), 
that the resultant force per unit area due to the above- 
mentioned normal tensions on its two faces shall have the- 
value T(l/pi + l//o 2 ), where pi and p z denote the principal radii 
of curvature of the shell at the place, and where T denotes, 
the lateral stress across unit length of the surface of the shell,, 
T being analogous to the surface-tension of a liquid film. 

Now, if t denote the thickness of the shell, the area inter- 
cepted on the second face by the tube of force bears to the 
area intercepted on the first face the ratio (pi + t) (p z + t)/p!p 2 > 
and by the fundamental property of tubes of force, D and E 
vary inversely as the cross-section of the tube, so the total force 
on the second face will bear to that on the first face the ratio 

piptKpi + 1} (p z + 1), 

or approximately 



302 Maxwell. 

the resultant force per unit area along the outward normal is 
therefore 

- IDE . t . (l//t>i + I//*), 
and so we have 

T = - IDE . t ; 

or the pressure at right angles to the lines of force is |DE per 
unit area that is, it is numerically equal to the tension along 
the lines of force. 

The principal stresses in the medium being thus determined, 
it readily follows that the stress across any plane, to which the 
unit vector N is normal, is 

(D.N)E-i( D - E ) N - 

Maxwell obtained* a similar formula for the case of magnetic 
fields ; the ponderornotive forces on magnetized matter and on 
conductors carrying currents may be accounted for by assuming 
a stress in the medium, the stress across the plane N" being 
represented by the vector 

1(B.K).H-1( B .H).N. ;j 

This, like the corresponding electrostatic formula, represents a 
tension across planes perpendicular to the lines of force, and a 
pressure across planes parallel to them. 

It may be remarked that Maxwell made no distinction 
between stress in the material dielectric and stress in the 
aether : indeed, so long as it was supposed that material bodies 
when displaced carry the contained aether along with them, 
no distinction was possible. In the modifications of Maxwell's 
theory which were developed many years afterwards by his 
followers, stresses corresponding to those introduced by Maxwell 
were assigned to the aether, as distinct from ponderable matter ; 
and it was assumed that the only stresses set up in material 
bodies by the electromagnetic field are produced indirectly: 
they may be calculated by the methods of the theory of 
elasticity, from a knowledge of the ponderomotive forces 
exerted on the electric charges connected with the bodies. 

* Maxwell's Treatise on Electricity and Magnetism, 643. 



Maxwell. 303 

Another remark suggested by Maxwell's theory of stress 
in the medium is that he considered the question from the 
purely statical point of view. He determined the stress so that 
it might produce the required forces on ponderable bodies, and 
be self-equilibrating in free aether. But* if the electric and 
magnetic phenomena are not really statical, but are kinetic in 
their nature, the stress or pressure need not be self-equilibrating. 
This may be illustrated by reference to the hydrodynamical 
models of the aether shortly to be described, in which perforated 
solids are immersed in a moving liquid : the ponderomotive 
forces exerted on the solids by the liquid correspond to those 
which act on conductors carrying currents in a magnetic field, 
and yet there is no stress in the medium beyond the pressure 
of the liquid. 

Among the problems to which Maxwell applied his theory 
of stress in the medium was one which had engaged the 
attention of many generations of his predecessors. The ad- 
herents of the corpuscular theory of light in the eighteenth 
century believed that their hypothesis would be decisively con- 
firmed if it could be shown that rays of light possess momentum : 
to determine the matter, several investigators directed powerful 
beams of light on delicately-suspended bodies, and looked for 
evidences of a pressure due to the impulse of the corpuscles. 
Such an experiment was performed in 1708 by Homberg,f who 
imagined that he actually obtained the effect in question ; but 
Mairan and Du Fay in the middle of the century, having 
repeated his operations, failed to confirm his conclusion.* 

The subject was afterwards taken up by Michell, who "some 
years ago," wrote Priestley in 1772, " endeavoured to ascertain 
the momentum of light in a much more accurate manner than 
those in which M. Homberg and M. Mairan had attempted it." 
He exposed a very thin and delicately-suspended copper plate 

* Cf. V. Bjeiknes, Phil. Mag. ix (1905), p. 491. 
t Histoire de 1'Acad., 1708, p. 21. 
% J. J. (ie Mairan, Traite de V A urore boreale, p. 370. 
History of Vision, i, p. 387. 



304 Maxwell. 

to the rays of the sun concentrated by a mirror, and observed 
a deflexion. He was not satisfied that the effect of the heating 
of the air had been altogether excluded, but " there seems to 
be no doubt," in Priestley's opinion, " but that the motion above 
mentioned is to be ascribed to the impulse of the rays of light." 
A similar experiment was made by A. Bennet,* who directed 
the light from the focus of a large lens on writing-paper 
delicately suspended in an exhausted receiver, but " could not 
perceive any motion distinguishable from the effects of heat." 
" Perhaps," he concluded, " sensible heat and light may not be 
caused by the influx or rectilineal projections of fine particles, 
but by the vibrations made in the universally diffused caloric 
or matter of heat, or fluid of light." Thus Bennet, and after 
him Young, f regarded the non-appearance of light- repulsion in 
this experiment as an argument in favour of the undulatory 
system of light. " For," wrote Young, " granting the utmost 
imaginable subtility of the corpuscles of light, their effects 
might naturally be expected to bear some proportion to the 
effects of the much less rapid motions of the electrical fluid, 
which are so very easily perceptible, even in their weakest 
states." 

This attitude is all the more remarkable, because Euler 
many years before had expressed the opinion that light-pressure 
might be expected just as reasonably on the undulatory 'as on 
the corpuscular hypothesis. "Just as," he wrote, J "a vehement 
sound excites not only a vibratory motion in the particles of 
the air, but there is also observed a real movement of the small 
particles of dust which are suspended therein, it is not to be 
doubted but that the vibratory motion set up by the light 
causes a similar effect." Euler not only inferred the existence 
of light-pressure, but even (adopting a suggestion of Kepler's) 
accounted for the tails of comets by supposing that the solar 
rays, impinging on the atmosphere of a comet, drive off from 
it the more subtle of its particles. 

* Phil. Trans., 1792, p. 81. + Ibid., 1802, p. 46. 

J Histoire de /' Acad. de Berlin, ii (1748), p. 117. 



Maxwell. 305 

The question was examined by Maxwell* from the point 
of view of the electromagnetic theory of light ; which readily 
furnishes reasons for the existence of light-pressure. For 
suppose that light falls on a metallic reflecting surface at 
perpendicular incidence. The light may be regarded as con- 
stituted of a rapidly-alternating magnetic field ; and this must 
induce electric currents in the surface layers of the metal. But 
a metal carrying currents in a magnetic field is acted on by a 
ponderomotive force, which is at right angles to both the 
magnetic force and the direction of the current, and is there- 
fore, in the present case, normal to the reflecting surface : 
this ponderomotive force is the light-pressure. Thus, according 
to Maxwell's theory, light-pressure is only an extended case of 
effects which may readily be produced in the laboratory. 

The magnitude of the light-pressure was deduced by 
Maxwell from his theory of stresses in the medium. We have 
seen that the stress across a plane whose unit-normal is N is 
represented by the vector 

(D . N) . E - J (D . E) . N + (B . N) . H - ~ (B . H) . N. 

47T O7T 

Now, suppose that a plane wave is incident perpendicularly on 
a perfectly reflecting metallic sheet: this sheet must support 
the mechanical stress which exists at its boundary in the 
aether. Owing to the presence of the reflected wave, D is zero 
at the surface ; and B is perpendicular to N, so (B . N) vanishes. 
Thus the stress is a pressure of magnitude (l/8?r) (B . H) 
normal to the surface : that is, the light-pressure is equal to 
the density of the aethereal energy in the region immediately 
outside the metal. This was Maxwell's result. 

This conclusion has been reached on the assumption that 
the light is incident normally to the reflecting surface. If, on 
the other hand, the surface is placed in an enclosure completely 
surrounded by a radiating shell, so that radiation falls on it 
from all directions, it may be shown that the light-pressure is 
measured by one-third of the density of aethereal energy. 

* Maxwell's Treatise on Electricity and Magnetism, 792. 
X 



306 Maxwell. 

A different way of inferring the necessity for light-pressure 
was indicated in 1876 by A. Bartoli,* who showed that, when 
radiant energy is transported from a cold body to a hot one by 
means of a moving mirror, the second law of thermodynamics 
would be violated unless a pressure were exerted on the mirror 
by the light. 

The thermodynamical ideas introduced into the subject by 
Bartoli have proved very fruitful. If a hollow vessel be at a 
definite temperature, the aether within the vessel must be full 
of radiation crossing from one side to the other : and hence the 
aether, when in radiative equilibrium with matter at a given 
temperature, is the seat of a definite quantity of energy per 
unit volume. 

If U denote this energy per unit volume, and P the light- 
pressure on unit area of a surface exposed to the radiation, we 
may applyf the equation of available energy! 



U-T dF P 
~ 1 dT 



Since, as we have seen, 



this equation gives . dU 

dT' 

and therefore U must be proportional to T*. From this it may 
be inferred that the intensity of emission of radiant energy by 
a body at temperature T is proportional to the fourth power of 
the absolute temperature a law which was first discovered 
experimentally by Stefan in 1879. 

In the year in which Maxwell's treatise was published, 
Sir William Crookes|| obtained experimental evidence of a 
pressure accompanying the incidence of light; but this was 

* Bartoli, Sopra i movimenti prodotti dalla luce e dal calore e sopra il radiometro 
di Crookes. Firenze, 1876. Also Nuovo Cimento (3) xv (1884), p. 193 ; and 
Exner's Rep., xxi (1885), p. 198. 

t Boltzmann, Ann. d. Phys. xxii (1884), p. 31. Cf. also B. Galitzine, Ann. d. 
Phys. xlvii (1892), p. 479. 

| Cf. p. 240. Wien. Ber. Ixxix (1879), p. 391. 

|| Phil. Trans, clxiv (1874), p. 501. The radiometer was discovered in 1875. 



Maxwell. 307 

soon found to be due to thermal effects ; and the existence of 
a true light-pressure was not confirmed experimentally* until 
1899. Since then the subject has been considerably developed, 
especially in regard to the part played by the pressure of radiation 
in cosmical physics. 

Another matter which received attention in Maxwell's 
Treatise was the influence of a magnetic field on the propagation 
of light in material substances. We have already seenf that 
the theory of magnetic vortices had its origin in Thomson's 
speculations on this phenomenon ; and Maxwell in his memoir 
of 1861-2 had attempted by the help of that theory to arrive 
at some explanation of it. The more complete investigation 
which is given in the Treatise is based on the same general 
assumptions, namely, that in a medium subjected to a magnetic 
field there exist concealed vortical motions, the axes of the 
vortices being in the direction of the lines of magnetic force ; 
and that waves of light passing through the medium disturb 
the vortices, which thereupon react dynamically on the luminous 
motion, and so affect its velocity of propagation. 

The manner of this dynamical interaction must now be 
more closely examined. Maxwell supposed that the magnetic 
vortices are affected by the light-waves in the same way as 
vortex-filaments in a liquid would be affected by any other 
coexisting motion in the liquid. The latter problem had been 
already discussed in Helrnholtz'js great memoir on vortex- 
motion ; adopting Helmholtz's results, Maxwell assumed for the 
additional term introduced into the magnetic force by the dis- 
placement of the vortices the value 9e/B0, where e denotes the 
displacement of the medium (i.e. the light vector), and the 
operator d/dO denotes H^/dx + Hy'dj^y + H z d/dz, H denoting the 
imposed magnetic field. Thus the luminous motion, by dis- 
turbing the vortices, gives rise to an electric current in the 
medium, proportional to curl 



*P. Lebedew, Archives des Sciences Phys. et Nat. (4) viii (1899), p. 184. 
Ann. d. Phys. vi (1901), p. 433. E. F. Nichols and G. F. Hull, Phys. Rev. 
xiii (1901), p. 293 ; Astrophys. Jour., xvii (1903), p. 315. t Cf. p. 274. 

X 2 



308 Maxwell. 

Maxwell further assumed that the current thus produced 
interacts dynamically with the luminous motion in such a 
manner that the kinetic energy of the medium contains a 
term proportional to the scalar product of e and curl de/30. 
The total kinetic energy of the medium may therefore be 
written 

\p& + Jcr (e . curl 9e/a0), 

where p denotes the density of the medium, and cr denotes a 
constant which measures the capacity of the medium to rotate 
the plane of polarization of light in a magnetic field. 

The equation of motion may now be derived as in the 
elastic- solid theories of light : it is 

32 

pe = % V 2 e - o- r - curl e. 
ot cu 

When the light is transmitted in the direction of the lines 
of force, and the axis of x is taken parallel to this direction, 
the equation reduces to 




and these equations, as we have seen,* furnish an explanation 
of Faraday's phenomenon. 

It may be remarked that the term 

J(T (e . curl 9e/80) 

in the kinetic energy may by partial integration be transformed 
into a term 

Jo- (curie. 9e/90),t 

together with surface-terms ; or, again, into 

- Jo- (curl e . 8e/80), 

together with surf ace- terms. These different forms all yield 
* Cf. p. 215. 

f This form was suggested by Fitz Gerald six years later, Phil. Trans., 1880, 
p. 691 : Fitz Gerald's Scientific Writings, p. 45. 



Maxwell. 309 

the same equation of motion for the medium; but, owing to 
the differences in the surface-terms, they yield different con- 
ditions at the boundary of the medium, and consequently give 
rise to different theories of reflexion. 

The assumptions involved in Maxwell's treatment of the 
magnetic rotation of light were such as might scarcely be 
justified in themselves ; but since the discussion as a whole 
proceeded from sound dynamical principles, and its conclu- 
sions were in harmony with experimental results, it was fitted 
to lead to the more perfect explanations which were afterwards 
devised by his successors. At the time of Maxwell's death, 
which happened in 1879, before he had completed his forty- 
ninth year, much yet remained to be done both in this and in 
the other investigations with which his name is associated; 
and the energies of the next generation were largely spent in 
extending and refining that conception of electrical and optical 
phenomena whose origin is correctly indicated in its name of 
Maxwell's Theory. 



( 310 ) 



CHAPTEK IX. 

MODELS OF THE AETHER. 

THE early attempts of Thomson and Maxwell to represent the 
electric medium by mechanical models opened up a new field of 
research, to which investigators were attracted as much by its 
intrinsic fascination as by the importance of the services which 
it promised to render to electric theory. 

Of the models to which reference has already been made, 
some such as those described in Thomson's memoir* of 1847 
and Maxwell's memoirf of 1861-2 attribute a linear character 
to electric force and electric current, and a rotatory character 
to magnetism; others such as that devised by Maxwell in 
1855J and afterwards amplified by Helmholtz regard mag- 
netic force as a linear and electric current as a rotatory 
phenomenon. This distinction furnishes a natural classification 
of models into two principal groups. 

Even within the limits of the former group diversity has 
already become apparent ; for in Maxwell's analogy of 1861-2, 
a continuous vortical motion is supposed to be in progress about 
the lines of magnetic induction ; whereas in Thomson's analogy 
the vector-potential was likened to the displacement in an 
elastic solid, so that the magnetic induction at any point would 
be represented by the twist of an element of volume of the 
solid from its equilibrium position ; or, in symbols, 

a = e, E = - e, B = curl e, 

where a denotes the vector-potential, E the electric force, B the 
magnetic induction, and e the elastic displacement. 

Thomson's original memoir concluded with a notice of his 
intention to resume the discussion in another communication 
His purpose was fulfilled only in 1890, when|| he showed tha 

* Cf. p. 270. t Of. p. 276. % Cf. p. 271. Cf. p. 274. 

|| Kelvin's Math, and Phys. Papers, iii, p. 436. 



Models of the Aether. 311 

in his model a linear current could be represented by a piece 
of endless, cord, of the same quality as the solid and embedded 
in it, if a tangential force were applied to the cord uniformly 
all round the circuit. The forces so applied tangentially pro- 
duce a tangential drag on the surrounding solid ; and the 
rotatory displacement thus caused is everywhere proportional 
to the magnetic vector. 

In order to represent the effect of varying permeability, 
Thomson abandoned the ordinary type of elastic solid, and 
replaced it by an aether of Mac Cullagh^s type; that is to say, 
an ideal incompressible substance, having no rigidity of the 
ordinary kind (i.e. elastic resistance to change of shape), but 
capable of resisting absolute rotation a property to which the 
name gyrostatic rigidity was given. The rotation of the solid 
representing the magnetic induction, and the coefficient of 
gyrostatic rigidity being inversely proportional to the permea- 
bility, the normal component of magnetic induction will be 
continuous across an interface, as it should be.* 

We have seen above that in models of this kind the electric 
force is represented by the translatory velocity of the medium. 
It might therefore be expected that a strong electric field would 
perceptibly affect the velocity of propagation of light ; and that 
this does not appear to be the case,f is an argument against the 
validity of the scheme. 

We now turn to the alternative conception, in which electric 
phenomena are regarded as rotatory, and magnetic force is 
represented by the linear velocity of the medium; in symbols, 

4-TrD = curl e, 
H = e, 

where D denotes the electric displacement, H the magnetic 
force, and e the displacement of the medium. In Maxwell's 
memoir of 1855, and in most of the succeeding writings for 

* Thomson inclined to believe (Papers, iii, p. I&5) that light might he correctly 
represented by the vibratory motion of such a solid. 

t Wilberforce, Trans. Camb. Phil. Soc. xiv (1887), p. 170 ; Lodge, Phil. Trans, 
clxxxix (1897), p. 149. 



312 Models of the Aether. 

many years, attention was directed chiefly to magnetic fields of 
a steady, or at any rate non-oscillatory, character ; in such fields, 
the motion of the particles of the medium is continuously 
progressive ; and it was consequently natural to suppose the 
medium to be fluid. 

Maxwell himself, as we have seen,* afterwards abandoned 
this conception in favour of that which represents magnetic 
phenomena as rotatory. "According to Ampere and all his 
followers," he wrote in 1870,f " electric currents are regarded 
as a species of translation, and magnetic force as depending on 
rotation. I am constrained to agree with this view, because 
the electric current is associated with electrolysis, and other 
undoubted instances of translation, while magnetism is asso- 
ciated with the rotation of the plane of polarization of light." 
But the other analogy was felt to be too valuable to be 
altogether discarded, especially when in 1858 Helmholtz 
extended itj by showing that if magnetic induction is com- 
pared to fluid velocity, then electric currents correspond to 
vortex-filaments in the fluid. Two years afterwards Kirchhoff 
developed it further. If the analogy has any dynamical (as 
distinguished from a merely kinematical) value, it is evident that 
the ponderomotive forces between metallic rings carrying electric 
currents should be similar to the ponderomotive forces between 
the same rings when they are immersed in an infinite incom- 
pressible fluid; the motion of the fluid being such that its 
circulation through the aperture of each ring is proportional to 
the strength of the electric current in the corresponding ring. 
In order to decide the question, Kirchhoff attempted, and solved, 
the hydrodynamical problem of the motion of two thin, rigid 
rings in an incompressible frictionless fluid, the fluid motion 
being irrotational ; and found that the forces between the rings 
are numerically equal to those which the rings would exert on 

* Cf. p. 276. 

t Proc. Lond. Math. Soc. iii (1870), p. 224 ; Maxwell's Sclent. Papers, ii, p. 263. 
J Cf. p. 274. 

Journnl fur Math. Ixxi (1869) ; Kirchhoff's Ge*amm. AbhandL, p. 404. Cf. 
also C. Neumann, Leipzig Berichte, xliv (1892), p. 86. 



Models of the Aether. 3 13 

each other if they were traversed by electric currents pro- 
portional to the circulations. 

There is, however, an important difference between the two 
cases, which was subsequently discussed by W. Thomson, who 
pursued the analogy in several memoirs.* In order to represent 
the magnetic field by a conservative dynamical system, we shall 
suppose that it is produced by a number of rings of perfectly 
conducting material, in which electric currents are circulating ; 
the surrounding medium being free aether. Now any perfectly 
conducting body acts as an impenetrable barrier to lines of 
magnetic force ; for, as Maxwell showed,f when a perfect con- 
ductor is placed in a magnetic field, electric currents are induced 
on its surface in such a way as to make the total magnetic force 
zero throughout the interior of the conductor.^ Lines of force 
are thus deflected by the body in the same way as the lines of 
flow of an incompressible fluid would be deflected by an obstacle 
of the same form, or as the lines of flow of electric current in a 
uniform conducting mass would be deflected by the introduction 
of a body of this form and of infinite resistance. If, then, for 
simplicity we consider two perfectly conducting rings carrying 
currents, those lines of force which are initially linked with a 
ring cannot escape from their entanglement, and new lines 
cannot become involved in it. This implies that the total 
number of lines of magnetic force which pass through the 
aperture of each ring is invariable. If the coefficients of self 
and mutual induction of the rings are denoted by Z,, Z 2 , Z 12 , 
the electrokinetic energy of the system may be represented by 

T = J (Z,*V + 2Z 12 ^ + Z 2 v), 

where i\, i> denote the strengths of the currents; and the 
condition that the number of lines of force linked with each 
circuit is to be invariable gives the equations 
Liii + Z 12 i 2 = constant, 
L z iz = constant. 



* Thomson's Reprint of Papers in Elect, and Mag., 573, 733, 751 (1870- 
1872). t Maxwell's Treatise on Elect, and Mag., 654. 

% For this reason "W. Thomson called a perfect conductor nn ideal extreme 
diamagnetic. 



314 Models of the Aether. 

It is evident that, when the system is considered from the 
point of view of general dynamics, the electric currents must be 
regarded as generalized velocities, and the quantities 

(L 1 i 1 + Z, 2 i 2 ) and (Z 12 ^ + L 9 i 2 ) 

as momenta. The electromagnetic ponderomotive force on the 
rings tending to increase any coordinate x is dT/dv. In the 
analogous hydrodynamical system, the fluid velocity corresponds 
to the magnetic force: and therefore the circulation through 
each ring (which is defined to be the integral fvds, taken round 
a path linked once with the ring) corresponds kinematically to 
the electric current ; and the flux of fluid through each ring 
corresponds to the number of lines of magnetic force which 
pass through the aperture of the ring. But in the hydro- 
dynamical problem the circulations play the part of generalized 
momenta ; while the fluxes of fluid through the rings play the 
part of generalized velocities. The kinetic energy may indeed 
be expressed in the form 



where KI, c 2 , denote the circulations (so that KI and c 2 are 
proportional respectively to ^ and 4), and NI, N n , N 2 , depend 
on the positions of the rings ; but this is the Hamiltonian (as 
opposed to the Lagrangian) form of the energy-function,* and 
the ponderomotive force on the rings tending to increase 
any coordinate x is - dK/dx. Since dK/dx is equal to dT/dx, 
we see that the ponderomotive forces on the rings in any 
position in the hydrodynamical system are equal, but opposite, 
to the ponderomotive forces on the rings in the electric 
system. 

The reason for the difference between the two cases may 
readily be understood. The rings cannot cut through the lines 
of magnetic force in the one system, but they can cut through 
the stream-lines in the other : consequently the flux of fluid 
through the rings is not invariable when the rings are moved, the 
invariants in the hydrodynamical system being the circulations. 

* Cf. Whittaker, Analytical Dynamics, 109. 



Modds of the Aethtr. 315 

If a thin ring, for which the circulation is zero, is introduced 
into the fluid, it will experience no ponderomotive forces ; but 
if a ring initially carrying no current is introduced into a 
magnetic field, it will experience ponderomotive forces, owing 
to the electric currents induced in it by its motion, 

Imperfect though the analogy is, it is not without interest. 
A bar-magnet, being equivalent to a current circulating in a wire 
wound round it, may be compared (as W. Thomson remarked) 
to a straight tube immersed in a perfect fluid, the fluid entering 
at one end and flowing out by the other, so that the particles 
of fluid follow the lines of magnetic force. If two such tubes 
are presented with like ends to each other, they attract ; with 
unlike ends, they repeL The forces are thus diametrically 
opposite in direction to those of magnets ; but in other respects 
the laws of mutual action between these tubes and between 
magnets are precisely the same.* 

* The mathematical analysis in this ease is very simple. A narrow rube through 
which water is flowing may be regarded as equivalent to a source at one end of the 
tube and a sink at the other; and the problem may therefore be reduced to the 
consideration of sinks in an unlimited fluid, If there are two sinks in sneh a fluid, 
of strengths m and */, the Telocity-potential is 

at/r + m*//, 

where r and i" denote distance from the sinks. The kinetic energy per unit 
of the fluid is 



thedensiryof the fluid; whence it is easily seen that the total 
energy of the fluid, when the two sinks are at a dtBtance I apart, exceeds the total 
cneigy when they are at an infinite distance apart by an amount 



0*i*ae*i*+^ld^&m^Mmt1to+k&lm*at1i*> 
small spheres *, /, surrounding the sinks. By Green's 
reduces at once to 



where the integration is taken over * and ", and m 

or '. The integral taken over *' vanishe 

hare 



of the fluid is therefore greater when sinks of strengths at, at* are at a 



3 1 6 Models of the Aether. 

Thomson, moreover, investigated* the ponderomotive forces 
which act between two solid bodies immersed in a fluid, when 
one of the bodies is constrained to perform small oscillations. 
If, for example, a small sphere immersed in an incompressible 
fluid is compelled to oscillate along the line which joins its 
centre to that of a much larger sphere, which is free, the free 
sphere will be attracted if it is denser than the fluid ; while 
if it is less dense than the fluid, it will be repelled or attracted 
according as the ratio of its distance from the vibrator to its 
radius is greater or less than a certain quantity depending on 
the ratio of its density to the density of the fluid. Systems 
of this kind were afterwards extensively investigated by 
C. A. Bjerknes.f Bjerknes showed that two spheres which 
are immersed in an incompressible fluid, and which pulsate 
(i.e., change in volume) regularly, exert on each other (by the 
mediation of the fluid) an attraction, determined by the inverse 
square law, if the pulsations are concordant ; and exert on 
each other a repulsion, determined likewise by the inverse 
square law, if the phases of the pulsations differ by half a 
period. It is necessary to suppose that the medium is incom- 
pressible, so that all pulsations are propagated instantaneously : 
otherwise attractions would change to repulsions and vice versa 
at distances greater than a quarter wave-length.^ If the 
spheres, instead of pulsating, oscillate to and fro in straight 
lines about their mean positions, the forces between them are 
proportional in magnitude and the same in direction, but 

mutual distance I than when sinks of the same strengths are at infinite distance 
apart by an amount lirpmm'/l. Since, in the case of the tubes, the quantities m 
correspond to the fluxes of fluid, this expression corresponds to the Lagrangian 
form of the kinetic energy ; and therefore the force tending to increase the coordi- 
nate x of one of the sinks is (3/9#) (4ny> ww'/Z). "Whence it is seen that the like ends 
of two tubes attract, and the unlike ends repel, according to the inverse square la\\ r . 

* Phil. Mag. xli (1870), p. 427. 

t Repertorium d. Mathematik von Konisberger und Zeuner (1876), p. 268. 
Gottinger Nachrichten, 1876, p. 245. Comptes Rendus, Ixxxiv (1877), p. 1377. 
Cf. Nature, xxiv (1881), p. 360. 

J On the mathematical theory of the force between two pulsating spheres in 
a fluid, cf. W. M. Hicks, Proc. Camb. Phil. Soc. iii (1879), p. 276 ; iv (1880), 
p. 29. 



Models of the Aether. 3 1 7 

opposite in sign, to those which act between two magnets 
oriented along the directions of oscillation.* 

The results obtained by Bjerknes were extended by 
A. H. Leahyf to the case of two spheres pulsating in an 
elastic medium ; the wave-length of the disturbance being 
supposed large in comparison with the distance between the 
spheres. For this system Bjerknes' results are reversed, the 
law being now that of attraction in the case of unlike phases, 
and of repulsion in the case of like phases : the intensity is as 
before proportional to the inverse square of the distance. 

The same author afterwards discussed \ the oscillations 
which may be produced in an elastic medium by the 
displacement, in the direction of the tangent to the cross- 
section, of the surfaces of tubes of small sectional area : 
the tubes either forming closed curves, or extending inde- 
finitely in both directions. The direction and circumstances 
of the motion are in general analogous to ordinary vortex- 
motions in an incompressible fluid ; and it was shown by Leahy 
that, if the period of the oscillation be such that the waves 
produced are long compared with ordinary finite distances, the 
displacement due to the tangential disturbances is proportional 
to the velocity due to vortex-rings of the same form as the 
tubular surfaces. One of these " oscillatory twists," as the 
tubular surfaces may be called, produces a displacement which 
is analogous to the magnetic force due to a current flowing in 
a curve coincident with the tube ; the strength of the current 
being proportional to b'w sin pt, where b denotes the radius of 
the twist, and t sin pt its angular displacement. If the field 
of vibration is explored by a rectilineal twist of the same 
period as that of the vibration, the twist will experience a force 

* A theory of gravitation has heen hased by Korn on the assumption that 
gravitating particles resemhle slightly compressible spheres immersed in an incom- 
pressible perfect fluid : the spheres execute pulsations, whose intensity corresponds 
to the mass of the gravitating particles, and thus forces of the Newtonian kind are 
produced between them. Cf. Korn, Eine Theorie der Gravitation und der elect. 
Etscheinungen, Berlin, 1898. 

t Trans. Camb. Phil. Soc. xiv (1884), p. 45. 

J Trans. Camb. Phil. Soc. xiv (1885), p. 188. 



3 1 8 Models of the Aether. 

at right angles to the plane containing the twist and the 
direction of the displacement which would exist if the twist 
were removed ; if the displacement of the medium be repre- 
sented by F sin pt, and the angular displacement of the twist 
by w sin pt, the magnitude of the force is proportional to the 
vector-product of V (in the direction of the displacement) and 
w (in the direction of the axis of the twist). 

A model of magnetic action may evidently be constructed 
on the basis of these results. A bar-magnet must be regarded 
as vibrating tangentially, the direction of vibration being 
parallel to the axis of the body. A cylindrical body carrying 
a current will have its surface also vibrating tangentially ; but 
in this case the direction of vibration will be perpendicular to 
the axis of the cylinder. A statically electrified body, on the 
other hand, may, as follows from the same author's earlier work, 
be regarded as analogous to a body whose surface vibrates in 
the normal direction. 

We have now discussed models in which the magnetic force 
is represented as the velocity in a liquid, and others in which 
it is represented as the displacement in an elastic solid. Some 
years before the date of Leahy's memoir, George Francis 
Fitz Gerald (b. 1851, d. 1901)* had instituted a comparison 
between magnetic force and the velocity in a quasi-elastic 
solid of the type first devised by MacCullagh.f An analogy 
is at once evident when it is noticed that the electromagnetic 
equation 

4?rD = curl H 
is satisfied identically by the values 

4?rD = curl e, 
H = e, 

where e denotes any vector; and that, on substituting these 
values in the other electromagnetic equation, 

- curl (4irc s D/e) - H, 

* Phil. Trans., 1880, p. 691 (presented October, 1878). Fitz Gerald's Scientific 
Writings, p. 45. t Cf. p. 155. 



Moaeh of the Aether. 319 

we obtain the equation 

ee + c 2 curl curl e = 0, 

which is no other than the equation of motion of MacCullagh's 
aether,* the specific inductive capacity corresponding to the 
reciprocal of MacCullagh's constant of elasticity. In the 
analogy thus constituted, electric displacement corresponds to 
the twist of the elements of volume of the aether ; and electric 
charge must evidently be represented as an intrinsic rotational 
strain. Mechanical models of the electromagnetic field, based on 
Fitz Gerald's analogy, were afterwards studied by A. Sommerfeld,f 
by K. Keiff,J and by Sir J. Larmor. The last-named authorll 
supposed the electric charge to exist in the form of discrete 
electrons, for the creation of which he suggested the following 
ideal processIF : A filament of aether, terminating at two 
nuclei, is supposed to be removed, and circulatory motion is 
imparted to the walls of the channel so formed, at each point 
of its length, so as to produce throughout the medium a 
rotational strain. When this has been accomplished, the 
channel is to be filled up again with aether, which is to be 
made continuous with its walls. When the constraint is 
removed from the walls of the channel, the circulation imposed 
on them proceeds to undo itself, until this tendency is balanced 
by the elastic resistance of the aether with which the channel 
has been filled up ; thus finally the system assumes a state of 
equilibrium in which the nuclei, which correspond to a positive 
and a negative electron, are surrounded by intrinsic rotational 
strain. 

Models in which magnetic force is represented by the 
velocity of an aether are not, however, secure from objection. 
It is necessary to suppose that the aether is capable of flowing 
like a perfect fluid in irrotational motion (which would corre- 

* Cf. p. 155. t Ann. d. Phys. xlvi (1892), p. 139. 

I Reiff, Elasticitat und Elektricitdt, Freiburg, 1893. 
Phil. Trans, clxxxv (1893), p. 719. 

|| In a supplement, of date August, 1894, to his above-cited memoir of 1893. 
H Phil. Trans, clxxxv (1894), p. 810; cxc (1897), p. 210; Larmor, Aether 
.and Matter (1900), p. 326. 



320 Models of the Aether. 

spond to a steady magnetic field), and that it is at the same 
time endowed with the power (which is requisite for the 
explanation of electric phenomena) of resisting the rotation of 
any element of volume.* But when the aether moves irrota- 
tionally in the fashion which corresponds to a steady magnetic 
field, each element of volume acquires after a finite time a 
rotatory displacement from its original orientation, in con- 
sequence of the motion ; and it might therefore be expected that 
the quasi-elastic power of resisting rotation would be called 
into play i.e., that a steady magnetic field would develop 
electric phenomena.f 

A further objection to all models in which magnetic force 
corresponds to velocity is that a strong magnetic field, being in 
such models represented by a steady drift of the aether, might 
be expected to influence the velocity of propagation of light. 
The existence of such an effect appears, however, to be disproved 
by the experiments of Sir Oliver Lodge ; J at any rate, unless it 
is assumed that the aether has an inertia at least of the same 
order of magnitude as that of ponderable matter, in which case 
the motion might be too slow to be measurable. 

Again, the evidence in favour of the rotatory as opposed to 
the linear character of magnetic phenomena has perhaps, on the 
whole, been strengthened since Thomson originally based his 
conclusion on the magnetic rotation of light. This brings us 
to the consideration of an experimental discovery. 

In 1879 E. H. Hall, at that time a student at Baltimore, 

* Larmor (loc. cit.) suggested the analogy of a liquid filled with magnetic 
molecules under the action of an external magnetic field. 

It has often heen objected to the mathematical conception of a perfect fluid 
that it contains no safeguard against slipping between adjacent layers, so that 
there is no justification for the usual assumption that the motion of <i perfect fluid 
is continuous. Larmor remarked that a rotational elasticity, such as is attributed 
to the medium above considered, furnishes precisely such a safeguard ; and that 
without some property of this kind a continuous frictionless fluid cannot be imagined. 

t Larmor proposed to avoid this by assuming that the rotation which is resisted 
by an element of volume of the aether is the vector sum of the series of differential 
rotations which it has experienced. J Phil. Trans, clxxxix (1897), p. 149. 

Am. Jour. Math, ii, p. 287 ; Am. J. Sci. xix, p. 200, and xx, p. 161 ; Phil. 
Mag. ix, p. 225, and x, p. 301. 



Models of the Aether. 321 

repeating an experiment which had been previously suggested 
by H. A. Kowland, obtained a new action of a magnetic field 
on electric currents. A strip of gold leaf mounted on glass, 
forming part of an electric circuit through which a current 
was passing, was placed between the poles of an electro- 
magnet, the plane of the strip being perpendicular to the 
lines of magnetic force. The two poles of a sensitive galvano- 
meter were then placed in connexion with different parts of the 
strip, until two points at the same potential were found. When 
the magnetic field was created or destroyed, a deflection of the 
galvanometer needle was observed, indicating a change in the 
relative potential of the two poles. It was thus shown that 
the magnetic field produces in the strip of gold leaf a new 
electromotive force, at right angles to the primary electromotive 
force and to the magnetic force, and proportional to the product 
of these forces. 

From the physical point of view we may therefore regard 
Hall's effect as an additional electromotive force generated by 
the action of the magnetic field on the current ; or alternatively 
we may regard it as a modification of the ohmic resistance of 
the metal, such as would be produced if the molecules of the 
metal assumed a helicoidal structure about the lines of magnetic 
force. From the latter point of view, all that is needed is 
to modify Ohm's law 

S = E 

(where S denotes electric current, k specific conductivity, and E 
electric force) so that it takes the form 

S = KE + h [E . H] 

where H denotes the imposed magnetic force, and h denotes a 
constant on which the magnitude of Hall's phenomenon 
depends. It is a curious circumstance that the occurrence, in 
the case of magnetized bodies, of an additional term in Ohm's 
law, formed from a vector-product of E, had been expressly 
suggested in Maxwell's Treatise*: although Maxwell had not- 
indicated the possibility of realizing it by Hall's experiment. 

* Elect, and Mag., 303. Cf. Hopkinson, Phil. Mag. x (1880), p. 430. 

Y 



322 Models of the Aether. 

An interesting application of Hall's discovery was made in 
the same year by Boltzmann,* who remarked that it offered a 
prospect of determining the absolute velocity of the electric 
charges which carry the current in the strip. For if it is 
supposed that only one kind (vitreous or resinous) of electricity 
is in motion, the force on one of the charges tending to drive it 
to one side of the strip will be proportional to the vector- 
product of its velocity and the magnetic intensity. Assuming 
that Hall's phenomenon is a consequence of this tendency of 
charges to move to one side of the strip, it is evident that the 
velocity in question must be proportional to the magnitude of 
the Hall electromotive force due to a unit magnetic field. On 
the basis of this reasoning, A. von Ettingshausenf found for the 
current sent by one or two Daniell's cells through a gold strip 
a velocity of the order of 0*1 cm. per second. It is clear, however, 
that, if the current consists of both vitreous and resinous charges 
in motion in opposite directions, Boltzmann's argument fails ; 
for the two kinds of electricity would give opposite directions 
to the current in Hall's phenomenon. 

In the year following his discovery, Hall} extended his 
researches in another direction, by investigating whether a 
magnetic field disturbs the distribution of equipotential lines in 
a dielectric which is in an electric field ; but no effect could be 
observed. Such an effect, indeed,|| was not to be expected on 
theoretical grounds; for when, in a material system, all the 
velocities are reversed, the motion is reversed, it being 
understood that, in the application of this theorem to electrical 
theory, an electrostatic state is to be regarded as one of rest, and 
a current as a phenomenon of motion ; and if such a reversal be 



* Wien Anz., 1880, p. 12. Phil. Mag. ix (1880), p. 307. 

t Ann. d. Phys. xi (1880), pp. 432, 1044. 

1 Am. Jour. Sci. xx (1880), p. 164. 

In 1885-6 E. van Aubel, Bull, de 1'Acad. Roy. de Belgique (3) x, p. 609 ; 
xii, p. 280, repeated the investigation in an improved form, and confirmed the 
result that a magnetic field has no influence on the electrostatic polarization of 
dielectrics. 

|| H. A. Lorentz, Arch. Neerl. xix (1884), p. 123. 



Models of the Aether. 323 

performed in the present system, the poles of the electro- 
magnet are exchanged, while in the dielectric no change takes 
place. 

We must now consider the bearing of Hall's effect on 
the question as to whether magnetism is a rotatory or a 
linear phenomenon.* If magnetism be linear, electric currents 
must be rotatory; and if Hall's phenomenon be supposed to take 
place in a horizontal strip of metal, the magnetic force being 
directed vertically upwards, and the primary current flowing 
horizontally from north to south, the only geometrical entities 
involved are the vertical direction and a rotation in the east- 
and-west vertical plane ; and these are indifferent with respect 
to a rotation in the nor th-and- south vertical plane, so that there 
is nothing in the physical circumstances of the system to 
determine in which direction the secondary current shall flow. 
The hypothesis that magnetism is linear appears therefore 
to be inconsistent with the existence of Hall's effect, f There 
are, however, some considerations which may be urged on the 
other side. Hall's effect, like the magnetic rotation of light, 
takes place only in ponderable bodies, not in free aether ; and 
its direction is sometimes in one sense, sometimes in the other, 
according to the nature of the substance. It may therefore be 
doubted whether these phenomena are not of a secondary 
character, and the argument based on them invalid. Moreover, 
as Fitz Gerald remarked,^ the magnetic lines of force associated 
with a system of currents are circuital and have no open ends, 
making it difficult to imagine how alteration of rotation inside 
them could be produced. 

Of the various attempts to represent electric and magnetic 
phenomena by the motions and strains of a continuous medium, 
none of those hitherto considered has been found free from 

* Of. F. Kol&cek, Ann. d. Phys. Iv (1895), p. 503. 

t Further evidence in favour of the hypothesis that it is the electric phenomena 
which are linear is furnished by the fact that pyro-electric effects (the production of 
electric polarization by warming) occur in acentric crystals, and only in such. Cf. 
M. Abraham, Encyklopiidie der rnrith. Wiss. iv (2), p. 43. 

I Cf. Larmor, Phil. Trans, clxxxv, p. 780. 

Y 2 



324 Models of the Aether. 

objection.* Before proceeding to consider models which are not 
constituted by a continuous medium, mention must be made of 
a suggestion offered by Biemann in his lecturesf of 1861. Rie- 
mann remarked that the scalar-potential and vector-potential 
a, corresponding to his own law of force between electrons, 
satisfy the equation 

+ div a = ; 

an equation which, as we have seen, is satisfied also by the 
potentials of L. Lorenz.j This appeared to Riemann to indicate 
that <j> might represent the density of an aether, of which a 
represents the velocity. It will be observed that on this 
hypothesis the electric and magnetic forces correspond to second 
derivates of the displacement a circumstance which makes it 
somewhat difficult to assimilate the energy possessed by the 
electromagnetic field to the energy of the model. 

We must now proceed to consider those models in which 
the aether is represented as composed of more than one kind of 
constituent : of these Maxwell's model of 1861-2, formed of 
vortices and rolling particles, may be taken as the type. Another 
device of the same class was described in 1885 by Fitz Gerald ; 
this was constituted of a number of wheels, free to rotate on 
axes fixed perpendicularly in a plane board ; the axes were fixed 
at the intersections of two systems of perpendicular lines ; and 
each wheel was geared to each of its four neighbours by an 
indiarubber band. Thus all the wheels could rotate without 
any straining of the system, provided they all had the same 
angular velocity; but if some of the wheels were revolving 
faster than others, the indiarubber bands would become strained. 
It is evident that the wheels in this model play the same part 
as the vortices in Maxwell's model of 1861-2 : their rotation is 

* Cf. H. "Witte, Ueber den gegenwdrtigen Stand der Frage nach einer mecha- 
nischen Erkldrung der elektrischen Erscheinungen ; Berlin, 1906. 

t Edited after his death by K. Hattendorff, under the title Schwere, Elektricitiit, 
und Magnetismus, 1875, p. 330. 

I Cf. p. 299. 

Scient. Proc. Koy. Dublin Soc., 1885; Phil. Mag. June, 1885; Fitz Gerald's 
Seient. Writings, pp. 142, 157. 






^B Models of the Aether. 325 

the analogue of magnetic force ; and a region in which the masses 
of the wheels are large corresponds to a region of high magnetic 
permeability. The indiarubber bands of Fitz Gerald's model 
correspond to the medium in which Maxwell's vortices were 
embedded ; and a strain on the bands represents dielectric polari- 
zation, the line joining the tight and slack sides of any band 
being the direction of displacement. A body whose specific 
inductive capacity is large would be represented by a region 
in which the elasticity of the bands is feeble. Lastly, 
conduction may be represented by a slipping of the bands 
on the wheels. 

Such a model is capable of transmitting vibrations analogous 
to those of light. For if any group of wheels be suddenly set 
in rotation, those in the neighbourhood will be prevented by 
their inertia from immediately sharing in the motion; but 
presently the rotation will be communicated to the adjacent 
wheels, which will transmit it to their neighbours; and so a 
wave of motion will be propagated through the medium. The 
motion constituting the wave is readily seen to be directed in 
the plane of the wave, i.e. the vibration is transverse. The axes 
of rotation of the wheels are at right angles to the direction 
of propagation of the wave, and the direction of polarization of 
the bands is at right angles to both these directions. 

The elastic bands may be replaced by lines of governor 
balls :* if this be done, the energy of the system is entirely of 
the kinetic type.f 

Models of types different from the foregoing have been 
suggested by the researches of Helmholtz and W. Thomson on 
vortex-motion. The earliest attempts in this direction, however, 
were intended to illustrate the properties of ponderable matter 
rather than of the luminiferous medium. A vortex existing in 
a perfect fluid preserves its individuality throughout all changes, 

* Fitz Gerald's Scient. Writings, p. 271. 

t It is of course possible to devise models of this class in which the rotation may 
be interpreted as having the electric instead of the magnetic character. Such a 
model was proposed by Boltzinann, Vorlesungen iiber Maxwell's Theorie, ii. 



326 Models of the Aetker. 

and cannot be destroyed ; so that if, as Thomson* suggested in 
1867, the atoms of matter are constituted of vortex-rings in a 
perfect fluid, the conservation of matter may be immediately 
explained. The mutual interactions of atoms may be illustrated 
by the behaviour of smoke-rings, which after approaching each 
other closely are observed to rebound : and the spectroscopic 
properties of matter may be referred to the possession by 
vortex-rings of free periods of vibration. f 

There are, however, objections to the hypothesis of vortex- 
atoms. It is not easy to understand how the large density of 
ponderable matter as compared with aether is to be explained ; 
and further, the virtual inertia of a vortex-ring increases as its 
energy increases ; whereas the inertia of a ponderable body is, 
so far as is known, unaffected by changes of temperature. It 
is, moreover, doubtful whether vortex-atoms would be stable. 
" It now seems to me certain," wrote W. Thomson^ (Kelvin) in 
1905, " that if any motion be given within a finite portion of 
an infinite incompressible liquid, originally at rest, its fate is 
necessarily dissipation to infinite distances with infinitely small 
velocities everywhere; while the total kinetic energy remains 
constant. After many years of failure to prove that the motion 
in the ordinary Helmholtz circular ring is stable, I came to the 
conclusion that it is essentially unstable, and that its fate must 
be to become dissipated as now described." 

The vortex-atom hypothesis is not the only way in which 
the theory of vortex-motion has been applied to the construc- 
tion of models of the aether. It was shown in 1880 by 
W. Thomson that in certain circumstances a mass of fluid can 
exist in a state in which portions in rotational and irrotational 

* Phil. Mag. xxxiv(1867), p. 15; Proc. R.S. Edinb. vi, p. 94. 

t An attempt was made in 1883 by J. J. Thomson, Phil. Mag. xv (1883), 
p. 427, to explain the phenomena of the electric discharge through gases in terms 
of tho theory of vortex-atoms. The electric field was supposed to consist in a 
distribution of velocity in the medium whose vortex-motion constituted the atoms 
of the gas ; and Thomson considered the effect of this field on the dissociation and 
recoupling of vortex-rings. 

J Proc. Roy. Soc. Edinb., xxv (1905), p. 565. 

Brit. Assoc. Rep., 1880, p. 473. 



Models of the Aether. 327 

motion are finely mixed together, so that on a large scale the 
mass is homogeneous, having within any sensible volume an 
equal amount of vortex-motion in all directions. To a fluid 
having such a type of motion he gave the name vortex-sponge. 

TiveTyears later, Fitzgerald*" discussed the suitability of the 
vortex-sponge as a model of the aether. Since vorticity in a 
perfect fluid cannot be created or destroyed, the modification 
of the system which is to be analogous to an electric field must 
be a polarized state of the vortex motion, and light must be 
represented by a communication of this polarized motion from 
one part of the medium to another. Many distinct types of 
polarization may readily be imagined : for instance, if the 
turbulent motion were constituted of vortex-rings, these might 
be in motion parallel to definite lines or planes ; or if it were 
constituted of long vortex filaments, the filaments might be 
bent spirally about axes parallel to a given direction. The 
energy of any polarized state of vortex-motion would be greater 
than that of the unpolarized state; so that if the motion of 
matter had the effect of reducing the polarization, there would 
be forces tending to produce that motion. Since the forces due 
to a small vortex vary inversely as a high power of the distance 
from it, it seems probable that in the case of two infinite 
planes, separated by a region of polarized vortex-motion, the 
forces due to the polarization between the planes would depend 
on the polarization, but not on the mutual distance of the 
planes a property which is characteristic of plane distributions 
whose elements attract according to the Newtonian law. 

It is possible to conceive polarized forms of vortex- motion 
which are steady so far as the interior of the medium is 
concerned, but which tend to yield up their energy in producing 
motion of its boundary a property parallel to that of the 
aether, which, though itself in equilibrium, tends to move 
objects immersed in it. 

In the same year Hicksf discussed the possibility of trans- 

* Scient. Proc. Roy. Dublin Soc., 188o Scientific Wntings of FitzGerakl, 
p. 154. + Brit. Assoc. Rep., 1885, p. 930. 



328 Models of the Aether. 

mitting waves through a medium consisting of an incompressible 
fluid in which small vortex-rings are closely packed together. 
The wave-length of the disturbance was supposed large in com- 
parison with the dimensions and mutual distances of the rings ; 
and the translatory motion of the latter was supposed to be so 
slow that very many waves can pass over any one before it has 
much changed its position. Such a medium would probably 
act as a fluid for larger motions. The vibration in the wave- 
front might be either swinging oscillations of a ring about a 
diameter, or transverse vibrations of the ring, or apertural 
vibrations ; vibrations normal to the plane of the ring appear 
to be impossible. Hicks determined in each case the velocity 
of translation, in terms of the radius of the rings, the distance 
of their planes, and their cyclic constant. 

The greatest advance in the vortex-sponge theory of the 
aether was made in 1887, when W. Thomson* showed that the 
equation of propagation of laminar disturbances in a vortex- 
sponge is the same as the equation of propagation of luminous 
vibrations in the aether. The demonstration, which in the 
circumstances can scarcely be expected to be either very simple 
or very rigorous, is as follows : 

Let (u, v, w) denote the components of velocity, and p the 
pressure, at the point (x, y, z) in an incompressible fluid. Let 
the initial motion be supposed to consist of a laminar motion 
{/(?/), 0, Oj, superposed on a homogeneous, isotropic, and fine- 
grained distribution (u' 0t v , w ) : so that at the origin of time 
the velocity is {/ (y) + u' , v , w n \ : it is desired to find a 
function / (y, t) such that at any time t the velocity shall 
be \f(y, t) + u', v, w), where u', v, w, are quantities of which 
every average taken over a sufficiently large space is zero. 

Substituting these values of the components of velocity in 
the equation of motion 

du _ du du du dp 

dt dx ~ dy~ dz ~ dx' 

* Phil. Mug. xxiv (1887), p. 342 : Kelvin's Math, and Phys. Papers, iv, p. 308. 



Models of the Aether. 329 

there results 



W dp 

- w - . 

dz dx 

Take now the #2-averages of both members. The quantities 
du'/dt, du'/dx, v, dp/dx have zero averages; so the equation 
takes the form 



df(y*t) ( ,W M 

- = - A . [u -- + v + w 
dt \ dx dy 

if the symbol A is used to indicate that the xz- average is to be 
taken of the quantity following. Moreover, the incompressi- 
bility of the fluid is expressed by the equation 



whence 



du' dv dw 

+ ~ + = ' 



f\ A I / *** / t/* 1 ' ^ \JWJ 

1 aaT" 1 * ^ + l 9z 



When this is added to the preceding equation, the first and 
third pairs of terms of the second member vanish, since the 
^-average of any derivate dQ/dx vanishes if Q is finite for 
infinitely great values of x ; and the equation thus becomes 

a) 



From this it is seen that if the turbulent motion were to remain 
continually isotropic as at the beginning,/ (T/, t) would constantly 
retain its critical value /(y). In order to examine the deviation 
from isotropy, we shall determine Ad (u'v)/dt, which may be 
done in the following way : Multiplying the u- and ^-equations 
of motion by v, u' respectively, and adding, we have 



-. 

' fa ty dx 

d (u'v) d (u'v) dp , dp 

-V- - ~ w ~V -v^--u f ^- 

dy dz dx ty 



330 Models of the Aether. 

Taking the ^-average of this, we observe that the first term of 
the first member disappears, since A . v is zero, and the first 
term of the second member disappears, since A . 3 (u'v]fix is 
zero. Denoting by %R Z the average value of u z , v z , or w 1 , so 
that R may be called the average velocity of the turbulent 
motion, the equation becomes 



It ^ (- 


V * # 2 y^ 


' y O 


9 )j m 3-"' ^ 


Vj 


where 






n. i l a > *'*>,, 


9(^) d(u 


y) 9p , 3p 






i ^ _ f u 
dx dy 



Let p be written (jp' + TO), where y denotes the value which p 
would have if / were zero. The equations of motion immediately 
give 



and on subtracting the forms which this equation takes in the 
two cases, we have 



which, when the turbulent motion is fine-grained, so that 
f(y t t) is sensibly constant over ranges within which u' t v, w 
pass through all their values, may be written 



Moreover, we have 

. , , tyu'v) d(uv) 




for positive and negative values of u, v, w are equally probable ; 
and therefore the value of the second member of this equation 
is doubled by adding to itself what it becomes when for u', v, w 
we substitute - u', -v, -w, which (as may be seen by inspection 
of the above equation in V 2 ^>) does not change the value of p'. 



Models of the Aether. 331 

Comparing this equation with that which determines the value 
of Q, we have 

' d^ 



or substituting for CT, 

The isotropy with respect to x and z gives the equation 



, 8 a\ 8 _ 

-+ ^0- h" V ^ 



But by integration by parts we obtain the equation 

' U. v -^ o= _ 



and by the condition of incompressibility the second member 
may be written 

A . (tojty) . (d/ty) . V- 2 Vo, or - A . v . (d/ z df) . V-^o ; 
so we have 



On account of the isotropy, we may write J for 



and, therefore, 



The deviation from isotropy shown by this equation is very 
small, because of the smallness of df(y, t)/dy. The equation is 
therefore not restricted to the initial values of the two members, 



332 Models of the Aether. 

for we may neglect an infinitesimal deviation from (2/9) IP in 
the first factor of the second member, in consideration of the 
smallness of the second factor. Hence for all values of t we 
have the equation 



which, in combination with (1), yields the result 



the form of this equation shows that laminar disturbances are 
propagated through the vortex-sponge in the same manner as waves 
of distortion in a homogeneous elastic solid. 

The question of the stability of the turbulent motion remained 
undecided ; and at the time Thomson seems to have thought it 
likely that the motion would suffer diffusion. But two years 
later* he showed that stability was ensured at any rate when 
space is filled with a set of approximately straight hollow vortex 
filaments. Fitz Geraldf subsequently determined the energy per 
unit-volume in a turbulent liquid which is transmitting laminar 
waves. Writing for brevity 

(2/9) R* - V\ f(y, t) = P, and A (u'v) = 7, 
the equations are 

s ?.--h and h-.y*^ 
dt dy' ft " 8y 

If the quantity 

p-f jVP2S 

is integrated throughout space, and the variations of the 
integral with respect to time are determined, it is found that 






JIM- 



* Proc. Roy. Irish Acad. (3) i (1889), p. 340 ; Kelvin's Math, and Phys. Papers, 
iv, p. 202. 

t Brit. Assoc. Rep., 1899. Fitz Gerald's Scientific Writings, p. 484. 



Models of the Aether. 333 

Integrating the second term under the integral by parts, and 
omitting the superficial terms (which may be at infinity, or 
wherever energy enters the space under consideration), we have 

0fa***.JJJp(*+g 

Hence it appears that the quantity S, which is of the dimensions 
of energy, must be proportional to the energy per unit- volume 
of the medium a result which shows that there is a pronounced 
similarity between the dynamics of a vortex- sponge and of 
Maxwell's elastic aether. 

A definite vortex-sponge model of the aether was described 
by Hicks in his Presidential Address to the mathematical 
section of the British Association in 1895.* In this the small 
motions whose function is to confer the quasi-rigidity were not 
completely chaotic, but were disposed systematically. The 
medium was supposed to be constituted of cubical elements of 
fluid, each containing a rotational circulation complete in itself : 
in any element, the motion close to the central vertical diameter 
of the element is vertically upwards : the fluid which is thus 
carried to the upper part of the element flows outwards over 
the top, down the sides, and up the centre again. In each of 
the six adjoining elements the motion is similar to this, but in 
the reverse direction. The rotational motion in the elements 
confers on them the power of resisting distortion, so that waves 
may be propagated through the medium as through an elastic 
solid ; but the rotations are without effect on irrotational 
motions of the fluid, provided the velocities in the irrotational 
motion are slow compared with the velocity of propagation of 
distortional vibrations. 

A different model was described four years later by 
Fitz Gerald, f Since the distribution of velocity of a fluid in the 



* Brit. Assoc. Rep., 1895, p. 595. 

t Proc. Roy. Dublin Soc., December 12, 1899; Fitz Gerald's Scientific 
Writings, p. 472. 



334 Models of the Aether. 

neighbourhood of a vortex filament is the same as the distribu- 
tion of magnetic force around a wire of identical form carrying 
an electric current, it is evident that the fluid has more energy 
when the filament has the form of a helix than when it is 
straight ; so if space were filled with vortices, whose axes 
were all parallel to a given direction, there would be an 
increase in the energy per unit volume when the vortices 
were bent into a spiral form ; and this could be measured by 
the square of a vector say, E which may be supposed parallel 
to this direction. 

If now a single spiral vortex is surrounded by parallel 
straight ones, the latter will not remain straight, but will be 
bent by the action of their spiral neighbour. The transference 
of spirality may be specified by a vector H, which will be dis- 
tributed in circles round the spiral vortex ; its magnitude will 
depend on the rate at which spirality is being lost by the 
original spiral, and can be taken such that its square is equal 
to the mean energy of this new motion. The vectors E and H 
will then represent the electric and magnetic vectors; the 
vortex spirals representing tubes of electric force. 

Fitz Gerald's spirality is essentially similar to the laminar 
motion investigated by Lord Kelvin, since it involves a flow in 
the direction of the axis of the spiral, and such a flow cannot 
take place along the direction of a vortex filament without a 
spiral deformation of a filament. 

Other vortex analogues have been devised for electro- 
statical systems. One such, which was described in 1888 by 
W. M. Hicks,* depends on the circumstance that if two bodies 
in contact in an infinite fluid are separated from each other, and 
if there be a vortex filament which terminates on the bodies, 
there will be formed at the point where they separate a hollow 
vortex filamentf stretching from one to the other, with rotation 

* Brit. Assoc. Rep., 1888, p. 577. 

} A hollow vortex is a cyclic motion existing in a fluid without the presence of 
any actual rotational filaments. On the general theory cf. Hicks, Phil. Trans, 
clxxv (1883), p. 161 ; clxxvi (1885), p. 725 ; cxcii (1898), p. 33. 



Models of the Aether. 335 

equal and opposite to that of the original filament. As the 
bodies are moved apart, the hollow vortex may, through failure 
of stability, dissociate into a number of smaller ones ; and if 
the resulting number be very large, they will ultimately take 
up a position of stable equilibrium. The two sets of filaments 
the original filaments and their hollow companions will be 
intermingled, and each will distribute itself according to the 
same law as the lines of force between the two bodies which are 
equally and oppositely electrified. 

Since the pressure inside a hollow vortex is zero, the portion 
of the surface on which it abuts experiences a diminution of 
pressure ; the two bodies are therefore attracted. Moreover, as 
the two bodies separate further, the distribution of the filaments 
being the same as that of lines of electric force, the diminution 
of pressure for each line is the same at all distances, and there- 
fore the force between the two bodies follows the same law as 
the force between two bodies equally and oppositely electrified. 
It may be shown that the effect of the original filaments is 
similar, the diminution of pressure being half as large again as 
for the hollow vortices. 

If another surface were brought into the presence of the 
others, those of the filaments which encounter it would break 
off and rearrange themselves so that each part of a broken 
filament terminates on the new body. This analogy thus gives 
a complete account of electrostatic actions both quantitatively 
and qualitatively : the electric charge on a body corresponds 
to the number of ends of filaments abutting on it, the sign 
being determined by the direction of rotation of the filament 
as viewed from the body. 

A magnetic field may be supposed to be produced by the 
motion of the vortex filaments through the stationary aether, 
the magnetic force being at right angles to the filament and to 
its direction of motion. Electrostatic and magnetic fields thus 
correspond to states of motion in the medium, in which, how- 
ever, there is no bodily flow; for the two kinds of filament 
produce circulation in opposite directions. 



336 Models of the Aether. 

It is possible that hollow vortices are better adapted than 
ordinary vortex-filaments for the construction of models of the 
aether. Such, at any rate, was the opinion of Thomson (Kelvin) 
in his later years.* The analytical difficulties of the subject are 
formidable, and progress is consequently slow ; but among the 
many mechanical schemes which have been devised to represent 
electrical and optical phenomena, none possesses greater interest 
than that which pictures the aether as a vortex-sponge. 

* Proc. Roy. Irish Acad., November 30, 1889 ; Kelvin's Math, and Phys. 
Papers, iv, p. 202. "Rotational vortex-cores," he wrote, "must he absolutely 
discarded ; and we must have nothing hut irrotational revolution and vacuous 
cores." 



( 337 ) 



CHAPTEE X. 

THE FOLLOWERS OF MAXWELL. 

THE most notable imperfection in the electromagnetic theory 
of light, as presented in Maxwell's original memoirs, was the 
absence of any explanation of reflexion and refraction. Before 
the publication of Maxwell's Treatise, however, a method of 
supplying the omission was indicated by Helmholtz.* The 
principles on which the explanation depends are that the 
normal component of the electric displacement D, the tangential 
components of the electric force E, and the magnetic vector B 
or H, are to be continuous across the interface at which the 
reflexion takes place; the optical difference between the con- 
tiguous bodies being represented by a difference in their 
dielectric constants, and the electric vector being assumed to 
be at right angles to the plane of polarization.-)- The analysis 
required is a mere transcription of MacCullagh's theory of 
reflexion,| if the derivate of MacCullagh's displacement e with 
respect to the time be interpreted as the magnetic force, 
fi curl e as the electric force, and curl e as the electric displace- 
ment. The mathematical details of the solution were not given 
by Helmholtz himself, but were supplied a few years later in 
the inaugural dissertation of H. A. Lorentz. 

In the years immediately following the publication of 
Maxwell's Treatise, a certain amount of evidence in favour of 

* Journal fur Math. Ixxii (1870), p. 68, note. 

t Helmholtz (loc. cit.) pointed out that if the optical difference between the 
media were assumed to be due to a difference in their magnetic permeabilities, it 
would be necessary to suppose the magnetic vector at right angles to the plane of 
polarization in order to obtain Fresnel's sine and tangent formulae of reflexion. 

I Cf. pp. 148, 149, 154-156. 

Zeitschrift fiir Math. u. Phys. xxii (1877), pp. 1, 205 : Over de theorie der 
terugkaatsing en breking van het licht, Arnhem, 1875. Lorentz's work was based 
on Helmholtz's equations, but remains substantially unchanged when Maxwell's 
formulae are substituted. 

Z 



338 The Followers of Maxwell. 

his theory was furnished by experiment. That an electric field 
is closely concerned with the propagation of light was demon- 
strated in 1875, when John Kerr* showed that dielectrics 
subjected to powerful electrostatic force acquire the property 
of double refraction, their optical behaviour being similar to 
that of uniaxal crystals whose axes are directed along the lines 
of force. 

Other researches undertaken at this time had a more direct 
bearing on the questions at issue between the hypothesis of 
Maxwell and the older potential theories. In 1875-6 Helmholtzf 
and his pupil Schiller^ attempted to discriminate between the 
various doctrines and formulae relative to unclosed circuits by 
performing a crucial experiment. 

It was agreed in all theories that a ring-shaped magnet, 
which returns into itself so as to have no poles, can exert no 
ponderomotive force on other magnets or 011 closed electric 
currents. Helmholtz had, however, shown in 1873 that accord- 
ing to the potential-theories such a magnet would exert a 
ponderomotive force on an unclosed current. The matter was 
tested by suspending a magnetized steel ring by a long fibre 
in a closed metallic case, near which was placed a terminal of 
a Holtz machine. No ponderomotive force could be observed 
when the machine was put in action so as to produce a brush 
discharge from the terminal : from which it was inferred that 
the potential-theories do not correctly represent the phenomena, 
at least when displacement-currents and convection -currents 
(such as that of the electricity carried by the electrically repelled 
air from the terminal) are not taken into account. 

The researches of Helmholtz and Schiller brought into 
prominence the question as to the effects produced by the 

* Phil. Mag. (4) 1 (1875), pp. 337, 446 ; (5) viii (1879), pp. 85, 229 ; xiii (1882), 
pp. 153, 248. 

t Monatsberichte d. Acad. d. Berlin/1875, p. 400. Ann. d. Phys., clviii (1876), 
p. 87. t Ann. d. Phys. clix (1876), pp. 456, 537 ; clx (1877), p. 333. 

\ The valuable memoirs by Helmholtz in Journal fiir Math. Ixxii (1870), 
p. 57 ; Ixxv (1873), p. 35 ; Ixxviii (1874), p. 273, to which reference has already 
been made, contain a full discussion of the various possibilities of the potential- 
theories. 






The Followers of Maxwell. 339 



translatory motion of electric charges. That the convection 
of electricity is equivalent to a current had been suggested 
long before by Faraday.* "If," he wrote in 1838, "a baU 
be electrified positively in the middle of a room and be then 
moved in any direction, effects will be produced as if a current 
in the same direction had existed." To decide the matter 
a new experiment inspired by Helmholtz was performed by 
H. A. Kowlandf in 1876. The electrified body in Kowland's 
disposition was a disk of ebonite, coated with gold leaf and 
capable of turning rapidly round a vertical axis between two 
fixed plates of glass, each gilt on one side. The gilt faces 
of the plates could be earthed, while the ebonite disk received 
electricity from a point placed near its edge ; each coating of 
the disk thus formed a condenser with the plate nearest to it. 
An astatic needle was placed above the upper condenser-plate, 
nearly over the edge of the disk; and when the disk was rotated 
a magnetic field was found to be produced. This experiment, 
which has since been repeated under improved conditions by 
Kowland and Hutchinson,J H. Fender , and Eichenwald,|| shows 
that the " convection-current " produced by the rotation of a 
charged disk, when the other ends of the lines of force are on an 
earthed stationary plate parallel to it, produces the same mag- 
netic field as an ordinary conduction-current flowing in a circuit 
which coincides with the path of the convection-current. When 
two disks forming a condenser are rotated together, the 
magnetic action is the sum of the magnetic actions of each of 
the disks separately. It appears, therefore, that electric charges 
cling to the matter of a conductor and move with it, so far as 
Rowland's phenomenon is concerned. 

The first examination of the matter from the point of view 
of Maxwell's theory was undertaken by J. J. Thomson,1[ in 1881. 
If an electrostatically charged body is in motion, the change in 

* Exper.Re*., 1644. 

t Monatsberichte d. Akad. d. Berlin, 1876, p. 211 : Ann. d. Phys. clviii (1876), 
p. 487 : Annales de Chim. et de Phys. xii (1877) p. 119. 

i Phil. Mag. xxvii (1889), p. 445. \ Ibid, ii (1901), p. 179 : v (1903), p. 34. 
|| Ann. d. Phys. xi (1901), p. 1. H Phil. Mag. xi (1881), p. 229. 

Z 2 



340 The Followers of Maxwell. 

the location of the charge must produce a continuous alteration 
of the electric field at any point in the surrounding medium ; or, 
in the language of Maxwell's theory, there must be displacement- 
currents in the medium. It was to these displacement-currents 
that Thomson, in his original investigation, attributed the 
magnetic effects of moving charges. The particular system 
which he considered was that formed by a charged spherical 
conductor, moving uniformly in a straight line. It was assumed 
that the distribution of electricity remains uniform over the 
surface during the motion, and that the electric field in any 
position of the sphere is the same as if the sphere were at 
rest ; these assumptions are true so long as quantities of order 
(V/c) 2 are neglected, where v denotes the velocity of the sphere 
and c the velocity of light. 

Thomson's method was to determine the displacement- 
currents in the space outside the sphere from the known 
values of the electric field, and then to calculate the vector- 
potential due to these displacement-currents by means of the 
formula 



where S' denotes the displacement-current at (x'y'z f ). The 
magnetic field was then determined by the equation 

H = curl A. 

A defect in this investigation was pointed out by Fitz Gerald, 
who, in a short but most valuable note,* published a few months 
afterwards, observed that the displacement-currents of Thomson 
do not satisfy the circuital condition. This is most simply seen 
by considering the case in which the system consists of two 
parallel plates forming a condenser; if one of the plates is 
fixed, and the other plate is moved towards it, the electric field 
is annihilated in the space over which the moving plate travels : 
this destruction of electric displacement constitutes a displace- 
ment-current, which, considered alone, is evidently not a closed 

* Proc. Roy. Dublin Soc., November, 1881 ; Fitz Gerald's Scientific Writings, 
p. 102. 



The Followers #/ Maxwell. 341 

current. The defect, as Fitz Gerald showed, may be immediately 
removed by assuming that a moving charge itself is to be counted 
as a current-element : the total current, thus composed of the 
displacement- currents and the convection-current, is circuital. 
Making this correction, Fitz Gerald found that the magnetic 
force due to a sphere of charge e moving with velocity v along 
the axis of z is curl (0, 0, ev/r) a formula which shows that the 
displacement-currents have no resultant magnetic effect, since 
the term ev/r would be obtained from the convection-current 
alone. 

The expressions obtained by Thomson and Fitz Gerald were 
correct only to the first order of the small quantity v/c. The 
effect of including terms of higher order was considered in 1889 
by Oliver Heaviside,* whose solution may be derived in the 
following manner : 

Suppose that a charged system is in motion with uniform 
velocity v parallel to the axis of z ; the total current consists of 
the displacement- cur rent E/4?rc 2 where E denotes the electric 
force, and the convection-current pv where p denotes the 
volume-density of electricity. So the equation which connects 
magnetic force with electric current may be written 

E/c 2 = curl H - 4:irpv. 
Eliminating E between this and the equation 

curl E = - H, 
and remembering that H is here circuital, we have 

H/c 2 - V 2 H = 4?r curl pv. 
If, therefore, a vector-potential a be defined by the equation 

a/c 3 - V 2 a = 4?rpv, 

the magnetic force will be the curl of a ; and from the equation 
for a it is evident that the components a x and a y are zero, and 
that a z is to be determined from the equation 
a z /c~ - V"a z = 4npv. 

* Phil. Mag. xxvii (1889), p. 324. 



342 The Followers of Maxwell. 

Now, let (x, y, ) denote coordinates relative to axes which 
are parallel to the axes (a;, y, z) , and which move with the 
charged bodies ; then a z is a function of (x, y, ) only ; so we 
have 

a a , a a 

5 -IT and *' "Vr 

and the preceding equation is readily seen to be equivalent to 



where 1 denotes (1 - v'/c 2 )'^. But this is simply Poisson's 
equation, with & substituted for z; so the solution may be 
transcribed from the known solution of Poisson's equation : it is 

/LV dx' dy d%i' 



the integrations being taken over all the space in which there 
are moving charges ; or 



_rrr 

jJJ 



If the moving system consists of a single charge e at the point 
5 = 0, this gives 

ev 

%(1 - tf sin 8 0/c")* ' 
where sin 2 = (a 8 + y 2 )/r 2 . 

It is readily seen that the lines of magnetic force due to the 
moving point-charge are circles whose centres are on the line of 
motion, the magnitude of the magnetic force being 

ev (1 - v 2 /c 2 ) sin 8 



The electric force is radial, its magnitude being 



r 2 (l - v 2 sin 2 0/c 2 )f 

The fact that the electric vector due to a moving point- 
charge is everywhere radial led Heaviside to conclude that the 
same solution is applicable when the charge is distributed over 






The Followers of Maxwell. 343 

a perfectly conducting sphere whose centre is at the point, the 
only chaftge being that E and H would now vanish inside the 
sphere. This inference was subsequently found* to be incorrect : 
a distribution of electric charge on a moving sphere could in 
fact not be in equilibrium if the electric force were radial, since 
there would then be nothing to balance the mechanical force 
exerted on the moving charge (which is equivalent to a current) 
by the magnetic field. The moving system which gives rise to 
the same field as a moving point-charge is not a sphere, but an 
oblate spheroid whose polar axis (which is in the direction of 
motion) bears to its equatorial axis the ratio (1 - tf/c*)^ : !.) 

The energy of the field surrounding a charged sphere is 
greater when the sphere is in motion than when it is at rest. 
To determine the additional energy quantitatively (retaining 
only the lowest significant powers of v/c), we have only to 
integrate, throughout the space outside the sphere, the expression 
H 2 /87r, which represents the electrokinetic energy per unit 
volume : the result is e z v~/3a, where e denotes the charge, v the 
velocity, and a the radius of the sphere. 

It is evident from this result that the work required to be 
done in order to communicate a given velocity to the sphere 
is greater when the sphere is charged than when it is uncharged ; 
that is to say, the virtual mass of the sphere is increased by an 
amount 2e 2 /3a, owing to the presence of the charge. This may 
be regarded as arising from the self-induction of the convection- 
current which is formed when the charge is set in motion. It 
was suggested by J. LarmorJ and by W. Wien that the inertia 
of ordinary ponderable matter may ultimately prove to be of 
this nature, the atoms being constituted of systems of electrons. || 

By G. F. C. Searle. 

t Cf. Searle, Phil. Trans, clxxxvii (1896), p. 675, and Phil. Ma ? . xliv (1897), 
p. 329. On the theory of the moving electrified sphere, cf. also J. J. Thomson, 
Recent Researches in Elect, and Mag., p. 16; 0. Heaviside, Electrical Papers, ii, 
p. 514; Electromag. Theory, i, p. 269; W. B. Morton, Phil. Mag, xli (1896), 
p. 488 ; A. Schuster, Phil. Mag. xliii (1897), p. 1. 

+ Phil. Trans, clxxxvi (1895), p. 697. Arch. Neerl (3) v (1900), p. 96. 

|| Experimental evidence that the inertia of electrons is purely electromagnetic 
was afterwards furnished hy W. Kaufmann, Gott. Nach., 1901, p. 143 ; 1902, p 291. 



344 The Followers of Maxwell. 

It may, however, be remarked that this view of th>rigin of 
mass is not altogether consistent with the principle^that the 
electron is an indivisible entity. For the so-called self-induction 
of the spherical electron is really the mutual induction of the 
convection-currents produced by the elements of electric charge 
which are distributed over its surface ; and the calculation of 
this quantity presupposes the divisibility of the total charge into 
elements capable of acting severally in all respects as ordinary 
electric charges ; a property which appears scarcely consistent 
with the supposed fundamental nature of the electron. 

After the first attempt of J. J. Thomson to determine the 
field produced by a moving electrified sphere, the mathematical 
development of Maxwell's theory proceeded rapidly. The 
problems which admit of solution in terms of known functions 
are naturally those in which the conducting surfaces involved 
have simple geometrical forms planes, spheres, and cylinders.* 

A result which was obtained by Horace Lamb,f when 
investigating electrical motions in a spherical conductor, led 
to interesting consequences. Lamb found that if a spherical 
conductor is placed in a rapidly alternating field, the induced 
currents are almost entirely confined to a superficial layer ; and 
his result was shortly afterwards generalized by Oliver Heavi- 
side,| who showed that whatever be the form of a conductor 
rapidly alternating currents do not penetrate far into its sub- 
stance. The reason for this may be readily understood : it is 
virtually an application of the principle|| that a perfect conductor 
is impenetrable to magnetic lines of force. No perfect conductor 
is known to exist ; butU if the alternations of magnetic force to 
which a good conductor such as copper is exposed are very 

* Cf., e.g., C. Niven, Phil. Trans, clxxii (1881), p. 307 ; H. Lamb, Phil. Trans, 
clxxiv (1883), p. 519 ; J. J. Thomson, Proc. Lond. Math. Soc. xv (1884), p. 197 : 
H. A. Rowland, Phil. Mag. xvii (1884), p. 413 ; J. J. Thomson, Proc. Lond. Math. 
Soc. xvii (1886), p. 310; xix (1888), p. 520; and many investigations of Oliver 
Heaviside, collected in his Electrical Papers. 

t Loc. cit. % Electrician, Jan. 1885. 

The mathematical theory was given hy Lord Rayleigh, Phil. Mag. xxi. (18S6), 
p. 381. Cf. Maxwell's Treatise, 689. || Cf. p. 313. 

H As was first remarked by Lord Rayleigh, Phil. Mag. xiii (1882), p. 344. 



The Followers of Maxwell. 345 

rapid, the. conductor has not time (so to speak) to display 
the impSfection of its conductivity, and the magnetic field 
is therefore unable to extend far below the surface. 

The same conclusion may be reached by different reasoning.* 
When the alternations of the current are very rapid, the ohmic 
resistance ceases to play a dominant part, and the ordinary 
equations connecting electromotive force, induction, and current 
are equivalent to the conditions that the currents shall be so 
distributed as to make the electrokinetic or magnetic energy a 
minimum. Consider now the case of a single straight wire of 
circular cross-section. The magnetic energy in the space outside 
the wire is the same whatever be the distribution of current in 
the cross-section (so long as it is symmetrical about the centre), 
since it is the same as if the current were flowing along the 
central axis ; so the condition is that the magnetic energy in 
the wire shall be a minimum ; and this is obviously satisfied 
when the current is concentrated in the superficial layer, since 
then the magnetic force is zero in the substance of the wire. 

In spite of the advances which were effected by Maxwell 
and his earliest followers in the theory of electric oscillations, 
the gulf between the classical electrodynamics and the theory 
of light was not yet completely bridged. For in all the cases 
considered in the former science, energy is merely exchanged 
between one body and another, remaining within the limits of a 
given system ; while in optics the energy travels freely through 
space, unattached to any material body. The first discovery of 
a more complete connexion between the two theories was made 
by Fitz Gerald, who argued that if the unification which had 
been indicated by Maxwell is valid, it ought to be possible to 
generate radiant energy by purely electrical means; and in 
1883f he described methods by which this could be done. 

Fitz Gerald's system is what has since become known as 
the magnetic oscillator : it consists of a small circuit, in which 

* Of. J. Stefan, Wiener, Situungsber. xcix (1890), p. 319 ; Ann. d. Phys. xli 
(1890), p. 400. 

t Trans. Roy. Dublin Soc. iii (1883) ; Fitz Gerald's Scient. Writings, p. 122. 



346 The Followers of Maxwell. 

the strength of the current is varied according to the simple 
periodic law. The circuit will be supposed to be ft circle of 
small area S, whose centre is the origin and whose plane is the 
plane of xy ; and the surrounding medium will be supposed 
to be free aether. The current may be taken to be of strength 
A cos (2ni?/jF), so that the moment of the equivalent magnet 
is SA cos (2irt/T). Now in the older electrodynamics, the 
vector-potential due to a magnetic molecule of (vector) moment 
M at the origin is (l/47r) curl (M/r), where r denotes distance 
from the origin. The vector-potential due to Fitz Gerald's 
magnetic oscillator would therefore be (l/47r) curl K, where K 
denotes a vector parallel to the axis of z, and of magnitude 
(1/r) SA cos (2-n-t/ T). The change which is involved in replacing 
the assumptions of the older electrodynamics by those of 
Maxwell's theory is in the present case equivalent* to retarding 
the potential ; so that the vector-potential a due to the oscillator 
is (l/47r) curl K where K is still directed parallel to the axis of 
z, and is of magnitude 

SA 27T/ r 
K = - cos [ t 



The electric force E at any point of space is - a, and the 
magnetic force H is curl a : so that these quantities may be 
calculated without difficulty. The electric energy per unit 
volume is E 2 /8?rc 2 : performing the calculations, it is found that 
the value of this quantity averaged over a period of the 
oscillation and also averaged over the surface of a sphere of 
radius r is 



The part of this which is radiated is evidently that which 
is proportional to the inverse square of the distance,! so the 

* Cf. pp. 298, 299. 

tThe other term, which is neglected, is very small compared to the term 
retained, at great distances from the origin ; it is what would be obtained if the 
effects of induction of the displacement-currents were neglected : i.e. it is the 
energy of the forced displacement-currents which are produced directly by the 
variation of the primary current, and which originate the radiating displacement- 
currents. 



The Followers of Maxwell. 347 

average value of the radiant energy of electric type at distance 
r from the oscillator is 2iT z A*S 2 /3c*r i T* per unit volume. The 
radiant energy of magnetic type ma}' be calculated in a similar 
way, and is found to have the same value ; so the total radiant 
energy at distance r is 47r 3 ^4 2 /S^/3cVT 4 per unit volume; 
and therefore the energy radiated in unit time is 16ir 4t A' i S 2 /3c s T*. 
This is small, unless the frequency is very high ; so that 
ordinary alternating currents would give no appreciable radia- 
tion. Fitz Gerald, however, in the same year* indicated a 
method by which the difficulty of obtaining currents of 
sufficiently high frequency might be overcome: this was, to 
employ the alternating currents which are produced when 
a condenser is discharged. 

The Fitz Gerald radiator constructed on this principle is 
closely akin to the radiator afterwards developed with such 
success by Hertz : the only difference is that in Fitz Gerald's 
arrangement the condenser is used merely as the store of 
energy (its plates being so close together that the electrostatic 
field due to the charges is practically confined to the space 
between them), and the actual source of radiation is the 
alternating magnetic field due to the circular loop of wire: 
while in Hertz's arrangement the loop of wire is abolished, 
the condenser plates are at some distance apart, and the source 
of radiation is the alternating electrostatic field due to their 
charges. 

In the study of electrical radiation, valuable help is afforded 
by a general theorem on the transfer of energy in the electro- 
magnetic field, which was discovered in 1884 by John Henry 
Poynting.-)- We have seen that the older writers on electric 
currents recognized that an electric current is associated with 
the transport of energy from one place (e.g. the voltaic cell 
which maintains the current) to another (e.g. an electromotor 
which is worked by the current) ; but they supposed the energy 
to be conveyed by the current itself within the wire, in much 

* Brit. Assoc. Rep., 1883 ; FitzGerald's Scientific Writings, p. 129. 
tPhil. Trans, clxxv (1884), p. 343. 



348 The Followers of Maxwell. 

the same way as dynamical energy is carried by water flowing 
in a pipe; whereas in Maxwell's theory, the storehouse and 
vehicle of energy is the dielectric medium surrounding the wire. 
What Poynting achieved was to show that the flux of energy at 
any place might be expressed by a simple formula in terms of 
the electric and magnetic forces at the place. 

Denoting as usual by E the electric force, by D the electric 
displacement, by H the magnetic force, and by B the magnetic 
induction, the energy stored in unit volume of the medium is* 
l ED + (1/8*) BH ; 

so the increase of this in unit time is (since in isotropic media 
D is proportional to E, and B is proportional to H) 

ED + (1/4*) HB 

or E (S - i) + (1/4*) HB, 

where S denotes the total current, and i the current of 

conduction ; or (in virtue of the fundamental electromagnetic 

equations) 

- (E . i) 4. (1/4*) (E . curl H) - (1/4*) (H . curl E}, 
or - (E . i) - (1/4*) div [E . H]. 

Now (E . i) is the amount of electric energy transformed into 
heat per unit volume per second; and therefore the quantity 
- (1/4*) div [E . H] must represent the deposit of energy in unit 
volume per second due to the streaming of energy; which 
shows that the flux of energy is represented by the vector 
(1/4*) [E.HJ.f This is Poynting's theorem: that the flux of 
energy at any place is represented by the vector-product of the 
electric and magnetic forces, divided by 4*.* 

* Cf. pp. 248, 250, 282. 

t Of course any circuital vector may be added. II. M. Macdonald, Electric Waves, 
p. 72, propounded a form which differs from Poynting's by a non-circuital vector. 

J The analogue of Poynting's theorem in the theory of the vibrations of an 
isotropic elastic solid may be easily obtained ; for from the equation of motion of 
an elastic solid, 

p& = - (k + 4/3) gnid div e n curl curl e, 
it follows that 

tot* + i (* + $) (div e) + in (curl e)'} = - div W, 



The Followers of Maxwell. 349 

In the special case of the field which surrounds a straight 
wire carrying a continuous current, the lines of magnetic force 
are circles round the axis of the wire, while the lines of electric 
force are directed along the wire ; hence energy must be flowing 
in the medium in a direction at right angles to the axis of the 
wire. A current in any conductor may therefore be regarded 
as consisting essentially of a convergence of electric and magnetic 
energy from the medium upon the conductor, and its trans- 
formation there into other forms. 

This association of a current with motions at right angles to 
the wire in which it flows doubtless suggested to Poynting the 
conceptions of a memoir which he published* in the following 
year. When an electric current flowing in a straight wire is 
gradually increased in strength from zero, the surrounding space 
becomes filled with lines of magnetic force, which have the form 
of circles round the axis of the wire. Poynting, adopting 
Faraday's idea of the physical reality of lines of force, assumed 
that these lines of force arrive at their places by moving out- 
wards from the wire ; so that the magnetic field grows by a con- 
tinual emission from the wire of lines of force, which enlarge 
and spread out like the circular ripples from the place where a. 
stone is dropped into a pond. The electromotive force which is- 
associated with a changing magnetic field was now attributed 
directly to the motion of the lines of force, so that wherever 
electromotive force is produced by change in the magnetic field,, 
or by motion of matter through the field, the electric intensity 
is equal to the number of tubes of magnetic force intersected 
by unit length in unit time. 

A similar conception was introduced in regard to lines of 
electric force. It was assumed that any change in the total 

where W denotes the vector 

- (k + 4w/3) div e . e + n [curl e . e] ; 

and since the expression which is differentiated with respect to t represents the 
sum of the kinetic and potential energies per unit volume of the solid (save for 
terms which give only surface-integrals), it is seen that W is the analogue of the 
Poynting vector. Cf. L. Donati, Bologna Mem. (5) vii (1899), p. 633. 
* Phil. Trans, clxxvi (1885), p. 277. 



350 The Followers of Maxwell. 

electric induction through a curve is caused by the passage of 
tubes of force in or out across the boundary ; so that whenever 
magnetomotive force is produced by change in the electric field, 
or by motion of matter through the field, the magnetomotive 
force is proportional to the number of tubes of electric force 
intersected by unit length in unit time. 

Poynting, moreover, assumed that when a steady current C 
flows in a straight wire, C tubes of electric force close in upon 
the wire in unit time, and are there dissolved, their energy 
appearing as heat. If E denote the magnitude of the electric 
force, the energy of each tube per unit length is \E, so 
the amount of energy brought to the wire is \CE per unit 
length per unit time. This is, however, only half the energy 
actually transformed into heat in the wire : so Poynting further 
assumed that E tubes of magnetic force also move in per unit 
length per unit time, and finally disappear by contraction to 
infinitely small rings. This motion accounts for the existence 
of- the electric field ; and since each tube (which is a closed ring) 
contains energy of amount J(7, the disappearance of the tubes 
accounts for the remaining \GE units of energy dissipated in 
the wire. 

The theory of moving tubes of force has been extensively 
developed by Sir Joseph Thomson.* Of the two kinds of tubes 
magnetic and electric which had been introduced by Faraday 
and used by Poynting, Thomson resolved to discard the former 
and employ only the latter. This was a distinct departure 
from Faraday's conceptions, in which, as we have seen, great 
significance was attached to the physical reality of the magnetic 
lines ; but Thomson justified his choice by inferences drawn 
from the phenomena of electric conduction in liquids and gases. 
As will appear subsequently, these phenomena indicate that 
molecular structure is closely connected with tubes of electro- 
static force perhaps much more closely than with tubes of 
magnetic force ; and Thomson therefore decided to regard 

* Phil. Mag. xxxi (1891), p. 149; Thomson's Recent Researches in Elect, and 
Mag. (1893), chapter i. 



The Followers of Maxwell. 351 

magnetism as the secondary effect, and to ascribe magnetic 
fields, not to the presence of magnetic tubes, but to the motion 
of electric tubes. In order to account for the fact that magnetic 
fields may occur without any manifestation of electric force, he 
assumed that tubes exist in great numbers everywhere in space, 
either in the form of closed circuits or else terminating on atoms, 
and that electric force is only perceived when the tubes have a 
greater tendency to lie in one direction than in another. In a 
steady magnetic field the positive and negative tubes might be 
conceived to be moving in opposite directions with equal 
velocities. 

A beam of light might, from this point of view, be regarded 
simply as a group of tubes of force which are moving with the 
velocity of light at right angles to their own length. Such a 
conception almost amounts to a return to the corpuscular 
theory ; but since the tubes have definite directions per- 
pendicular to the direction of propagation, there would now 
be no difficulty in explaining polarization. 

The energy accompanying all electric and magnetic pheno- 
mena was supposed by Thomson to be ultimately kinetic energy 
of the aether ; the electric part of it being represented by rota- 
tion of the aether inside and about the tubes, and the magnetic 
part being the energy of the additional disturbance set up in 
the aether by the movement of the tubes. The inertia of this 
latter motion he regarded as the cause of induced electromotive 
force. 

There was, however, one phenomenon of the electromagnetic 
field as yet unexplained in terms of these conceptions namely, 
the ponderomotive force which is exerted by the field on a 
conductor carrying an electric current. Now any pondero- 
motive force consists in a transfer of mechanical momentum 
from the agent which exerts the force to the body which 
experiences it ; and it occurred to Thomson that the pondero- 
motive forces of the electromagnetic field might be explained if 
the moving tubes of force, which enter a conductor carrying a 
current and are there dissolved, were supposed to possess 



352 The Followers of Maxwell. 

mechanical momentum, which could be yielded up to the 
conductor. It is readily seen that such momentum must be 
directed at right angles to the tube and to the magnetic 
induction a result which suggests that the momentum stored 
in unit volume of the aether may be proportional to the vector- 
product of the electric and magnetic vectors. 

For this conjecture reasons of a more definite kind may be 
given.* We have already seenf that the ponderomotive forces 
on material bodies in the electromagnetic field may be accounted 
for by Maxwell's supposition that across any plane in the aether 
whose unit normal is N, there is a stress represented by 

P N = (D . N) E - J (D .E)N + (l/47r) (B . H)H - (I/Sir) (B. H) N. 

So long as the field is steady (i.e. electrostatic or magnetostatic) 
the resultant of the stresses acting on any element of volume of 
the aether is zero, so that the element is in equilibrium. But 
when the field is variable, this is no longer the case. The 
resultant stress on the aether contained within a surface S is 

JJ PN . dS 

integrated over the surface : transforming this into a volume- 
integral, the term (D . N) E gives a term div D . E + (D . V) E, 
where V denotes the vector operator (9/9a?, d/dy, d/dz) ; and the 
first of these terms vanishes, since D is a circuital vector; 
the term - J (D . E) N gives in the volume-integral a term 
J grad (D . E) ; and the magnetic terms give similar results. 
So the resultant force on unit- volume of the aether is 

(D . V) E + J grad (D . E) + (l/4ir) (B . V) E + (I /Sir) grad (B . H), 
which may be written 

[curl E . D] + (l/47r) [curl H . B] ; 

* The hypothesis that the aether is a storehouse of mechanical momentum, 
which was first advanced by ,T. J. Thomson (Recent Researches in Elect, and Mag. 
(1893), p. 13), was afterwards developed by H. Poincare, Archives Neerl. (2) v 
(1900), p. 252, and by M. Abraham, Gott, Nach., 1902, p. 20. 

tCf. p. 302. 



The Followers of Maxwell. 353 

or, by virtue of the fundamental equations for dielectrics, 
[- B . D] + [D . B] , or (a/ft) [D . B]. 

This result compels us to adopt one of three alternatives: 
either to modify the theory so as to reduce to zero the resultant 
force on an element of free aether ; this expedient has not met 
with general favour ;* or to assume that the force in question 
sets the aether in motion: this alternative was chosen by 
Helmholtz,f but is inconsistent with the theory of the aether 
which was generally received in the closing years of the century; 
or lastly, with Thomson^ to accept the principle that the aether 
is itself the vehicle of mechanical momentum, of amount [D . B] 
per unit volume. 

Maxwell's theory was now being developed in ways which 
could scarcely have been anticipated by its author. But although 
every year added something to the superstructure, the founda- 
tions remained much as Maxwell had laid them ; the doubtful 
argument by which he had sought to justify the introduction 
of displacement- currents was still all that was offered in their 
defence. In 1884, however, the theory was established on a 
different basis by a pupil of Helmholtz', Heinrich Hertz 
(b. 1857, d. 1894). 

The train of Hertz' ideas resembles that by which Ampere, 
on hearing of Oersted's discovery of the magnetic field produced 
by electric currents, inferred that electric currents should exert 
ponderomotive forces on each other. Ampere argued that a 
current, being competent to originate a magnetic field, must be 
equivalent to a magnet in other respects ; and therefore that 
currents, like magnets, should exhibit forces of mutual attraction 
and repulsion. 

* It was, however, adopted by G. T. "Walker, Aberration and the Electromagnetic 
Field, Camb., 1900. 

t Berlin Sitzungsberichte, 1893, p. 649; Ann. d. Phys. liii (1894), p. 135. 
Helmholtz supposed the aether to behave as a frictionless incompressible fluid. 

+ Loc. cit. 

Ann. d. Phys. xxiii (1884), p. 84: English version in Hertz's Miscellaneous 
Papers, translated by D. E. Jones and G. A. Schott, p. 273. 

2 A 



354 The Followers of Maxwell. 

Ampere's reasoning rests on the assumption that the mag- 
netic field produced by a current is in all respects of the same 
nature as that produced by a magnet ; in other words, that only 
one land of magnetic force exists. This principle of the " unity 
of magnetic force" Hertz now proposed to supplement by assert- 
ing that the electric force generated by a changing magnetic 
field is identical in nature with the electric force due to electro- 
static charges; this second principle he called the "unity of 
electric force." Suppose, then, that a system of electric currents 
i exists in otherwise empty space. According to the older 
theory, these currents give rise to a vector-potential a, , equal 
to Pot i ;* and the magnetic force H t is the curl of a t : while 
the electric force E! at any point in the field, produced by the 
variation of the currents, is ai. 

It is now assumed that the electric force so produced is 
indistinguishable from the electric force which would be set 
up by electrostatic charges, and therefore that the system of 
varying currents exerts ponderomobive forces on electrostatic 
charges ; the principle of action and reaction then requires that 
electrostatic charges should exert ponderomotive forces on a 
system of varying currents, and consequently (again appealing 
to the principle of the unity of electric force) that two systems 
of varying currents should exert on each other ponderomotive 
forces due to the variations. 

But just as Helmholtz,f by aid of the principle of conser- 
vation of energy, deduced the existence of an electromotive 
force of induction from the existence of the ponderomotive 
forces between electric currents (Le. variable electric systems), 
so from the existence of ponderomotive forces between variable 
systems of currents (i.e. variable magnetic systems) we may 
infer that variations in the rate of change of a variable magnetic 
system give rise to induced magnetic forces in the surrounding 
space. The analytical formulae which determine these forces 

* a = Pot /3 is used to denote the solution of the equation V'a + 47r = 0. 
fCf. p. 243. 



The Followers of Maxwell. 355 

will be of the same kind as in the electric case ; so that the 
induced magnetic force H' is given by an equation of the form 



where c denotes some constant, and bi, which is analogous to 
the vector-potential in the electric case, is a circuital vector 
whose curl is the electric force E! of the variable magnetic 
system. The value of bi is therefore (l/47r) curl Pot E t : so 
we have 

H' = - J-. |, curl Pot a, 

47TC" (jt~ 

This must be added to Hi. Writing H 2 for the sum, Hi + H', we 
see that H 2 is the curl of a 2 , where 



and the electric force E 2 will then be - a 2 . 

This system is not, however, final ; for we must now perform 
the process again with these improved values of the electric 
and magnetic forces and the vector-potential ; and so we obtain 
for the magnetic force the value curl a 3 , and for the electric 
force the value - a 3 , where 



1 r) z 1 ^* 

= a x - - Pot ax + -- Pot Pot 

4rrc 2 fit* 



This process must again be repeated indefinitely ; so finally we 
obtain for the magnetic force H the value curl a, and for the 
electric force E the value - a, where 



1 }*> 

- Pot Pot Pot a! + 



(47TC 2 ) 3 
2A2 



356 The Followers of Maxwell. 

It is evident that the quantity a thus defined satisfies the 
equation 



or v*a - - a = - 47ri. 

c 2 dt' 

This equation may be written 



while the equations H = curl a, E = - a give 

curl E = - H. 

These are, however, the fundamental equations of Maxwell's 
theory in the form given in his memoir of 1868,* 

That Hertz's deduction is ingenious and interesting will 
readily be admitted. That it is conclusive may scarcely be 
claimed : for the argument of Helmholtz regarding the induc- 
tion of currents is not altogether satisfactory; and Hertz, in 
following his master, is on no surer ground. 

In the course of a discussion^ on the validity of Hertz's 
assumptions, which followed the publication of his paper, 
E. AulingerJ brought to light a contradiction between the 
principles of the unity of electric and of magnetic force and 
the electrodynamics of Weber. Consider an electrostatically 
charged hollow sphere, in the interior of which is a wire 
carrying a variable current. According to Weber's theory, 
the sphere would exert a turning couple on the wire; but 
according to Hertz's principles, no action would be exerted, 
since charging the sphere makes no difference to either the 
electric or the magnetic force in its interior. The experiment 
thus suggested would be a crucial test of the correctness of 
Weber's theory ; it has the advantage of requiring nothing 
but closed currents and electrostatic charges at rest ; but 
the quantities to be observed would be on the limits of 
observational accuracy. 
Cf. p. 287. 

f Lorberg, Ann. d. Phys. xxvii (1886), p. 666; xxxi (1887), p. 131. 
Boltzmann, ibid, xxix (1886), p. 598. + Ann. d. Phys. xxvii (1886), p. 119. 



The Followers of Maxwell. 357 

After his attempt to justify the Maxwellian equations on 
theoretical grounds, Hertz turned his attention to the possibility 
of verifying them by direct experiment. His interest in the 
matter had first been aroused some years previously, when the 
Berlin Academy proposed as a prize subject " To establish 
experimentally a relation between electromagnetic actions and 
the polarization of dielectrics." Helmholtz suggested to Hertz 
that he should attempt the solution ; but at the time he saw 
no way of bringing phenomena of this kind within the limits of 
observation. From this time forward, however, the idea of electric 
oscillations was continually present to his mind ; and in the 
spring of 1886 he noticed an effect* which formed the starting- 
point of his later researches. When an open circuit was formed 
of a piece of copper wire, bent into the form of a rectangle, 
so that the ends of the wire were separated only by a short air- 
gap, and when this open circuit was connected by a wire with 
any point of a circuit through which the spark -discharge of an 
induction-coil was taking place, it was found that a spark 
passed in the air-gap of the open circuit. This was explained 
by supposing that the change of potential, which is propagated 
along the connecting wire from the induction-coil, reaches one 
end of the open circuit before it reaches the other, so that a 
spark passes between them; and the phenomenon therefore 
was regarded as indicating a finite velocity of propagation of 
electric potential along wires.! 

* Ann. d. Phys. xxxi (1887), p. 421. Hertz's Electric Waves, translated by 
D. E. Jones, p. 29. 

t Unknown to Hertz, the transmission of electric waves along wires had been 
observed in 1870 by Wilhelm von Bezold, Miinchen Sitzungsbericlite, i (1870), 
p. 113 ; Phil. Mag. xl (1870), p. 42. * If," he wrote at the conclusion of a series 
of experiments, "electrical waves be sent into a wire insulated at the end, they 
will be reflected at that end. The phenomena which accompany this process in 
alternating discharges appear to owe their origin to the interference of the 
advancing and reflected waves," and, "an electric discharge travels with the 
atne rapidity in wires of equal length, without reference to the materials of 
which these wires are made." 

The subject was investigated by 0. J. Lodge and A. P. Chattock at almost the 
same time as Hertz's experiments were being carried out: mention was made of 
their researches at the meeting of the British Association in 1888. 



358 The Followers of Maxwell. 

Continuing his experiments, Hertz* found that a spark 
could be induced in the open or secondary circuit even when it 
was not in metallic connexion with the primary circuit in which 
the electric oscillations were generated; and he rightly inter- 
preted the phenomenon by showing that the secondary circuit 
was of such dimensions as to make the free period of electric 
oscillations in it nearly equal to the period of the oscillations 
in the primary circuit ; the disturbance which passed from one 
circuit to the other by induction would consequently be greatly 
intensified in the secondary circuit by resonance. 

The discovery that sparks may be produced in the air-gap 
of a secondary circuit, provided it has the dimensions proper 
for resonance, was of great importance : for it supplied a method 
of detecting electrical effects in air at a distance from the primary 
disturbance ; a suitable detector was in fact all that was needed 
in order to observe the propagation of electric waves in free 
space, and thereby decisively test the Maxwellian theory. To 
this work Hertz now addressed himself.f 

The radiator or primary source of the disturbances studied 
by Hertz may be constructed of two sheets of metal in the 
same plane, each sheet carrying a stiff wire which projects 
towards the other sheet and terminates in a knob ; the sheets 
are to be excited by connecting them to the terminals of an 
induction coil. The sheets may be regarded as the two coatings 
of a modified Leyden jar, with air as the dielectric between 
them ; the electric field is extended throughout the air, instead 
of being confined to the narrow space between the coatings, as 
in the ordinary Leyden jar. Such a disposition ensures that 
the system shall lose a large part of its energy by radiation 
at each oscillation. 

* Loc. cit. 

t Sir Oliver Lodge was about this time independently studying electric oscilla- 
tions in air in connexion with the theory of lightning-conductors : cf. Lodge, 
Phil. Mag. xxvi (1888), p. 217. So long before as 1842, Joseph Henry, of 
Washington, had noticed that the inductive effects of the Leyden jar discharge 
could be observed at considerable distances, and had even suggested a comparison 
with " a spark from flint and steel in the case of light." 



The Followers of Maxwell. 359 

As in the jar discharge,* the electricity surges from one 
sheet to the other, with a period proportional to (CL)l, where 
denotes the electrostatic capacity of the system formed by 
the two sheets, and L denotes the self-induction of the 
connexion. The capacity and induction should be made as 
small as possible in order to make the period small. The 
detector used by Hertz was that already described, namely, 
a wire bent into an incompletely closed curve, and of such 
dimensions that its free period of oscillation was the same 
as that of the primary oscillation, so that resonance might take 
place. 

Towards the end of the year 1887, when studying the sparks 
induced in the resonating circuit by the primary disturbance, 
Hertz noticedf that the phenomena were distinctly modified 
when a large mass of an insulating substance was brought 
into the neighbourhood of the apparatus ; thus confirming the 
principle that the changing electric polarization which is pro- 
duced when an alternating electric force acts on a dielectric 
is capable of displaying electromagnetic effects. 

Early in the following year (1888) Hertz determined to 
verify Maxwell's theory directly by showing that electro- 
magnetic actions are propagated in air with a finite velocity .{ 
For this purpose he transmitted the disturbance from the 
primary oscillator by two different paths, viz., through the air 
and along a wire ; and having exposed the detector to the joint 
influence of the two partial disturbances, he observed inter- 
ference between them. In this way he found the ratio of the 
velocity of electric waves in air to their velocity when conducted 
by wires ; and the latter velocity he determined by observing 
the distance between the nodes of stationary waves in the wire, 
and calculating the period of the primary oscillation. The 
velocity of propagation of electric disturbances in air was in 



* Cf. p. 253. 

t Ann. d. Phys. xxxiv, p. 373. Electric Waves (English edition), p. 95. 

J Ann. d. Phys. xxxiv (1888), p. 551. Electric Waves (English edition) p. 107. 



360 The Followers of Maxwell. 

this way shown to be finite and of the same order as the 
velocity of light.* 

Later in 1888 Hertzf showed that electric waves in air are 
reflected at the surface of a wall ; stationary waves may thus 
be produced, and interference may be obtained between direct 
and reflected beams travelling in the same direction. 

The theoretical analysis of the disturbance emitted by a 
Hertzian radiator according to Maxwell's theory was given by 
Hertz in the following year.J 

The effects of the radiator are chiefly determined by the 
free electric charges which, alternately appearing at the two 
sides, generate an electric field by their presence and a magnetic 
field by their motion. In each oscillation, as the charges on 
the poles of the radiator increase from zero, lines of electric 
force, having their ends on these poles, move outwards into 
the surrounding space. When the charges on the poles attain 
their greatest values, the lines cease to issue outwards, and the 
existing lines begin to retreat inwards towards the poles; but 
the outer lines of force contract in such a way that their upper 
and lower parts touch each other at some distance from the 
radiator, and the remoter portion of each of these lines thus 
takes the form of a loop ; and when the rest of the line of 
force retreats inwards towards the radiator, this loop becomes 
detached and is propagated outwards as radiation. In this 
way the radiator emits a series of whirl-rings, which as they 
move grow thinner and wider; at a distance, the disturbance 

* Hertz's experiments gave the value 45/28 for the ratio of the velocity of 
electric waves in air to the velocity of electric waves conducted by the wires, and 
2 x 10 10 cms. per sec. for the latter velocity. These numbers were afterwards 
found to be open to objection: Poincare (Comptes Rendus, cxi (1890), p. 322) 
showed that the period calculated by Hertz was V2 x the true period, which would 
make the velocity of propagation in air equal to that of light x v'2. Ernst Lecher 
(Wiener Berichte, May 8, 1890; Phil. Mag. xxx (1890), p. 128), experimenting 
on the velocity of propagation of electric vibrations in wires, found instead of 
Hertz's 2 x 10 10 cms. per sec., a value within two per cent, of the velocity of 
light. E. Sarasin and L. De La Rive at Geneva (Archives des Sc. Phys. xxix (1893)) 
finally proved that the velocities of propagation in air and along wires are equal. 

t Ann. d. Phys. xxxiv (1888), p. 610. Electric Waves (English edition), p. 124. 

J Ibid., xxxvi (1889), p. 1. Electric Waves (English edition), p. 137. 



The Followers of Maxwell. 361 

is approximately a plane wave, the opposite sides of the ring 
representing the two phases of the wave. When one of these 
rings has become detached from the radiator, the energy con- 
tained may subsequently be regarded as travelling outwards 
with it. 

To discuss the problem analytically* we take the axis of 
the radiator as axis of z, and the centre of the spark-gap as 
origin. The field may be regarded as due to an electric doublet 
formed of a positive and an equal negative charge, displaced 
from each other along the axis of the vibrator, and of 

moment 

Ae~ p ^ sin (2irct/\), 

the factor e~ p ^ being inserted to represent the damping. 

The simplest method of proceeding, which was suggested by 
Fitz Gerald,f is to form the retarded potentials < and a of 
L. Lorenz.J These are determined in terms of the charges and 
their velocities by the equations 

I. * = a <^, o.-s^, 

whence it is readily shown that in the present case 



4> = - dF/dz, a = (0, 0, 
where 

Ae'KC-'P , 2;r . , 
F = sin (ct - r). 

T A* 

The electric and magnetic forces are then determined by the 
equations 

E = c 2 grad < - a, H = curl a. 

It is found that the electric force may be regarded as com- 
pounded of a force < 2 , parallel to the axis of the vibrator and 
depending at any instant only on the distance from the vibrator, 
together with a force fa sin acting in the meridian plane 

* Cf. Karl Pearson and A. Lee, Phil. Trans, cxciii (1899), p. 165. 
+ Brit. Assoc. Rep., Leeds (1890), p. 755. 

J Cf. p. 298. The use of retarded potentials was also recommended in the 
following year by Poincare, Comptes Rendus, cxiii (1891), p. 515. 



362 The Followers of Maxwell. 

perpendicular to the radius from the centre, where $1 depends at 
any instant only on the distance from the vibrator, and 
denotes the angle which the radius makes with the axis of the 
oscillator. At points on the axis, and in the equatorial plane, 
the electric force is parallel to the axis. At a great distance 
from the oscillator, 2 is small compared with 0,, so the wave is 
purely transverse. The magnetic force is directed along circles 
whose centres are on the axis of the radiator ; and its magnitude 
may be represented in the form 3 sin 9, where 3 depends 
only on r and t ; at great distances from the radiator, c< 3 is 
approximately equal to 0,. 

If the activity of the oscillator be supposed to be continually 
maintained, so that there is no damping, we may replace p { by 
zero, and may proceed as in the case of the magnetic oscillator* 
to determine the amount of energy radiated. The mean out- 
ward flow of energy per unit time is found to be Jc 3 ^ 2 (27T/X) 4 ; 
from which it is seen that the rate of loss of energy by radiation 
increases greatly as the wave-length decreases. 

The action of an electrical vibrator may be studied by the 
aid of mechanical models. In one of these, devised by Larmor,f 
the aether is represented by an incompressible elastic solid, in 
which are two cavities, corresponding to the conductors of the 
vibrator, filled with incompressible fluid of negligible inertia. 
The electric force is represented by the displacement of the 
solid. For such rapid alternations as are here considered, 
the metallic poles behave as perfect conductors; and the 
tangential components of electric force at their surfaces' are 
zero. This condition may be satisfied in the model by suppos- 
ing the lining of each cavity to be of flexible sheet-metal, so as 
to be incapable of tangential displacement ; the normal displace- 
ment of the lining then corresponds to the surface-density of 
electric charge on the conductor. 

In order to obtain oscillations in the solid resembling those 
of an electric vibrator, we may suppose that the two cavities 
* Cf. p. 346. 

7 Proc. Camb. Phil. Soc. vii (1891), p. 165. 



The Followers of Maxwell. 36li 

have the form of semicircular tubes forming the two halves 
of a complete circle. Each tube is enlarged at each of 
its ends, so as to present a front of considerable area to the 
corresponding front at the end of the other tube. Thus at each 
end of one diameter of the circle there is a pair of opposing 
fronts, which are separated from each other by a thin sheet 
of the elastic solid. 

The disturbance may be originated by forcing an excess of 
liquid into one of the enlarged ends of one of the cavities. This 
involves displacing the thin sheet of elastic solid, which 
separates it from the opposing front of the other cavity, and 
thus causing a corresponding deficiency of liquid in the enlarged 
end behind this front. The liquid will then surge backwards 
and forwards in each cavity between its enlarged ends ; and, 
the motion being communicated to the elastic solid, vibrations 
will be generated resembling those which are produced in the 
aether by a Hertzian oscillator. 

In the latter part of the year 1888 the researches of Hertz* 
yielded more complete evidence of the similarity of electric 
waves to light. It was shown that the part of the radiation 
from an oscillator which was transmitted through an opening in 
a screen was propagated in a straight line, with diffraction effects. 
Of the other properties of light, polarization existed in the 
original radiation, as was evident from the manner in which it 
was produced ; and polarization in other directions was obtained 
by passing the waves through a grating of parallel metallic wires ; 
the component of the electric force parallel to the wires was 
absorbed, so that in the transmitted beam the electric vibration 
was at right angles to the wires. This effect obviously resembled 
the polarization of ordinary light by a plate of tourmaline. 
Refraction was obtained by passing the radiation through 
prisms of hard pitch. j- 

* Ann. d. Phys.xxxvi (1889), p. 769; Electric Waves (Englished.), p. 172. 

I 0. J. Lodge and J. L. Ho ward in the same year showed that electric radiation 
might be refracted and concentrated hy means of large lenses. Cf. Phil. Mag. 
xxvii (1889), p. 48. 



364 The Followers of Maxwell. 

The old question as to whether the light-vector is in, or at 
right angles to, the plane of polarization* now presented itself 
in a new aspect. The wave-front of an electric wave contains 
two vectors, the electric and magnetic, which are at right angles 
to each other. Which of these is in the plane of polarization ? 
The answer was furnished by Fitz Gerald and Trouton,f who 
found on reflecting Hertzian waves from a wall of masonry that 
no reflexion was obtained at the polarizing angle when the 
vibrator was in the plane of reflexion. The inference from this 
is that the magnetic vector is in the plane of polarization of the 
electric wave, and the electric vector is at right angles to the 
plane of polarization. An interesting development followed in 
1890, when 0. Wiener^ succeeded in photographing stationary 
waves of light. The stationary waves were obtained by the 
composition of a beam incident on a mirror with the reflected 
beam, and were photographed on a thin film of transparent 
collodion, placed close to the mirror and slightly inclined to it. 
If the beam used in such an experiment is plane-polarized, and 
is incident at an angle of 45, the stationary vector is evidently 
that perpendicular to the plane of incidence; but Wiener 
found that under these conditions the effect was obtained only 
when the light was polarized in the plane of incidence ; so 
that the chemical activity must be associated with the vector 
perpendicular to the plane of polarization i.e., the electric 
vector. 

In 1890 and the years immediately following appeared 
several memoirs relating to the fundamental equations of 
electro-magnetic theory. Hertz, after presenting the general 



* Cf. pp. 168 et sqq. 

f Nature, xxxix (1889), p. 391. 

\ Ann. d. Phys. xl (1890), p. 203. Cf. a controversy regarding the results ; 
Comptes Rendus, cxii (1891), pp. 186, 325, 329, 365, 383, 456 ; and Ann. d. Phys. 
xli (1890), p. 154 ; xliii(1891), p. 177; xlviii (1893), p. 119. 

Gott. Nach. 1890, p. 106; Aim. d. Phys. xl (1890), p. 577; Electric Waves 
(English ed.), p. 195. In this memoir Hertz advocated the form of the equations 
which Maxwell had used in his paper of 1868 (cf. supra, p. 287) in preference to 
the earlier form, which involved the scalar and vector potentials. 



The Followers of Maxwell. 365 

content of Maxwell's theory for bodies at rest, proceeded* to 
extend the equations to the case in which material bodies are 
in motion in the field. 

In a really comprehensive and correct theory, as Hertz 
remarked, a distinction should be drawn between the quantities 
which specify the state of the aether at every point, and those 
which specify the state of the ponderable matter entangled with 
it. This anticipation has been fulfilled by later investigators ; 
but Hertz considered that the time was not ripe for such a 
complete theory, and preferred, like Maxwell, to assume that 
the state of the compound system matter plus aether can be 
specified in the same way when the matter moves as when it is 
at rest ; or, as Hertz himself expressed it, that " the aether 
contained within ponderable bodies moves with them." 

Maxwell's own hypothesis with regard to moving systemsf 
amounted merely to a modification in the equation 

B = - curl E, 

which represents the law that the electromotive force in a 
closed circuit is measured by the rate of decrease in the number 
of lines of magnetic induction which pass through the circuit. 
This law is true whether the circuit is at rest or in motion ; but 
in the latter case, the E in the equation must be taken to be the 
electromotive force in a stationary circuit whose position 
momentarily coincides with that of the moving circuit; and 
since an electromotive force [w . B] is generated in matter by 
its motion with velocity w in a magnetic field B, we see that E 
is connected with the electromotive force E' in the moving 
ponderable body by the equation 

E' = E + [w . B], 

so that the equation of electromagnetic induction in the moving 
body is 

B = - curl E' + curl [w . B]. 

* Ann. d. Phys. xli (1890), p. 369 ; Electric Waves (English ed.), p. 241. 

The propagation of light through a moving dielectric had been discussed 
previously, on the basis of Maxwell's equations for moving bodies, by J. J. Thomson, 
Phil. Mag. ix (1880), p. 284 ; Proc. Camb. Phil. Soc. v (1885), p. 250. 

tCf. p. 288. 



366 The Followers of Maxwell. 

Maxwell made no change in the other electromagnetic 
equations, which therefore retained the customary forms 

D = f E'/47rc 2 , div D = 0, 47r(i 4 D) = curl H, 
Hertz, however, impressed by the duality of electric and 
magnetic phenomena, modified the last of these equations by 
assuming that a magnetic force 4?r [D . w] is generated in a 
dielectric which moves with velocity w in an electric field ; such 
a force would be the magnetic analogue of the electromotive 
force of induction. A term involving curl |D . w] is then 
introduced into the last equation. 

The theory of Hertz resembles in many respects that of 
Heaviside,* who likewise insisted much on the duplex nature 
of the electromagnetic field, and was in consequence disposed 
to accept the term involving curl [D . w] in the equations of 
moving media. Heaviside recognized more clearly than his 
predecessors the distinction between the force E', which 
determines the flux D, and the force E, whose curl represents 
the electric current ; and, in conformity with his principle of 
duality, he made a similar distinction between the magnetic 
force H', which determines the flux B, and the force H, whose 
curl represents the " magnetic current." This distinction, as 
Heaviside showed, is of importance when the system is 
acted on by " impressed forces," such as voltaic electromotive 
forces, or permanent magnetization; these latter must be 
included in E' and H', since they help to give rise to the fluxes 
D and B ; but they must not be included in E and H, since their 
curls are not electric or magnetic currents ; so that in general 

we have 

E' = E + e, H' = + h, 

where e and h denote the impressed forces. 

Developing the theory by the aid of these conceptions, 
Heaviside was led to make a further modification. An im- 

* Heaviside's general theory was published in a series of papers in the 
Electrician, from 1885 onwards. His earlier work was republished in his 
Electrical Papers (2 vols., 1892), and his Electromagnetic Theory (2 vols., 1894). 
Mention may be specially made of a memoir in Phil. Trans, clxxxiii (1892), 
p. 423. 



The Followers of Maxwell. 367 

pressed force is best defined in terms of the energy which it 
communicates to the system ; thus, if e be an impressed electric 
force, the energy communicated to unit volume of the electro- 
magnetic system in unit time is e x the electric current. 
In order that this equation may be true, it is necessary to 
regard the electric current in a moving medium as composed 
of the conduction-current, displacement-current, convection- 
current, and also of the term curl [D . w] , whose presence in 
the equation we have already noticed. This may be called 
the current of dielectric convection. Thus the total current is 

S = D + i + pw + curl [D . w] , 

where pw denotes the conduction-current ; and the equation 
connecting current with magnetic force is 
curl (H' - h ) = 4?rS, 

where h denotes the impressed magnetic forces other than that 
induced by motion of the medium. 

We must now consider the advances which were effected 
during the period following the publication of Maxwell's 
Treatise in some of the special problems of electricity and 
optics. 

We have seen* that Maxwell accounted for the rotation of 
the plane of polarization of light in a medium subjected to a 
magnetic field K by adding to the kinetic energy of the aether, 
which is represented by Jpe*, a term J<r (e . curl 9e/90), where 
cr is a magneto-optic constant characteristic of the substance 
through which the light is transmitted, and d/dO stands for 
Kxdl'dx + Kydldy + K-d/dz. This theory was developed further 
in 1879 by Fitz Gerald,f who brought it into closer connexion 
with the electromagnetic theory of light by identifying the curl 
of the displacement e of the aethereal particles with the electric 
displacement ; the derivate of e with respect to the time then 
corresponds to the magnetic force. Being thus in possession of 
a definitely electromagnetic theory of the magnetic rotation of 

* Cf . p. 308. 

t Phil. Trans., 1879, p. 691. Fitz Gerald's Scient. Writings, p. 45. 



368 The Followers of Maxwell. 

light, Fitz Gerald proceeded to extend it so as to take account 
of a closely related phenomenon. In 1876 J. Kerr* had shown 
experimentally that when plane-polarized light is regularly 
reflected from either pole of an iron electromagnet, the reflected 
ray has a component polarized in a plane at right angles to the 
ordinary reflected ray. Shortly after this discovery had been 
made known, Fitz Geraldf had proposed to explain it by means 
of the same term in the equations which accounts for the mag- 
netic rotation of light in transparent bodies. His argument was 
that if the incident plane-polarized ray be resolved into two 
rays circularly polarized in opposite senses, the refractive index 
will have different values for these two rays, and hence the 
intensities after reflexion will be different; so that on re- 
compounding them, two plane-polarized rays will be obtained 
one polarized in the plane of incidence, and the other polarized 
at right angles to it. 

The analytical discussion of Kerr's phenomenon, which was 
given by Fitz Gerald in his memoir of 1879, was based on these 
ideas ; the most essential features of the phenomenon were 
explained, but the investigation was in some respects imperfect.} 

Anew and fruitful conception was introduced in 1879-1880, 
when H. A. Eowland suggested a connexion between the 
magnetic rotation of light and the phenomenon which had been 
discovered by his pupil Hall.|| Hall's effect may be regarded 

* Phil. Mag. (5) iii (1877), p. 321. 

t Proc. 11. S. xxv (1877), p. 447 ; Fitz Gerald's Sclent. Writings, p. 9. 

J Cf. Larmor's remarks in his Report on the Action oj Magnetism on Light, 
Brit. Assoc. Kep., 1893 ; and his editorial comments in Fitz Gerald's Scientific 
Writings. Larmor traced to its source an inconsistency in the equations hy which 
Fitz Gerald had represented the boundary-conditions at an interface between the 
media. Fitz Gerald had indeed made the mistake, similar to that which was so often 
made hy the earlier writers on the elastic-solid theory of light, of forgetting that when 
a medium is assumed to be incompressible, the condition of in compressibility must 
be introduced into the variational equation of motion (as was done supra, p. 172). 
Larmor showed that when this correction was made, new terms (resembling the 
terms in p, supra, p. 172) made their appearance; and the inconsistency in the 
equations was thus removed. 

Amer. Jour. Math, ii, p. 354, iii, p. 89; Phil. Mag. xi (1881), p. 254. 

|| Cf. p. 321. 



The Followers of Maxwell. 369 

as a rotation of conduction-currents under the influence of a 
magnetic field ; and if it be assumed that displacement-currents 
in dielectrics are rotated in the same way, the Faraday effect 
may evidently be explained. Considering the matter from the 
analytical point of view, the Hall effect may be represented by 
the addition of a term k [K . S] to the electromotive force, 
where K denotes the impressed magnetic force, and S denotes 
the current : so Kowland assumed that in dielectrics there is an 
additional term in the electric force, proportional to [K . D], i.e. 
proportional to the rate of increase of [K . D]. Now it is 
universally true that the total electric force round a circuit is 
proportional to the rate of decrease of the total magnetic 
induction through the circuit : so the total magnetic induction 
through the circuit must contain a term proportional to the 
integral of [K . D] taken round the circuit : and therefore the 
magnetic induction at any point must contain a term proportional 
to curl [K . D]. We may therefore write 

B = H + a curl [K . D], 

where <r denotes a constant. But if this be combined with the 
customary electromagnetic equations 

curl H = 47rD, curl E = - B, D = eE/47rc 3 , 

and all the vectors except B be eliminated (K being treated 
as a constant), we obtain the equation 



B - (c 7 /0 V 2 B + O/47r) curl 

where 3/80 stands for (K x d/fa + K y d/dy + K z d/dz) ; and this is 
identical with the equation which Maxwell had given* for the 
motion of the aether in magnetized media. It follows that the 
assumptions of Maxwell and of Eowland, different though they 
are physically, lead to the same analytical equations at any 
rate so far as concerns propagation through a homogeneous 
medium. 

The connexions of Hall's phenomenon with the magnetic 
rotation of light, and with the reflexion of light from magnetized 

* Cf. p. 308. 

2 B 



370 The Followers of Maxwell. 

metals, were extensively studied* in the years following the 
publication of Kowland's memoir: but it was not until the 
modern theory of electrons had been developed that a satisfactory 
representation of the molecular processes involved in magneto- 
optic phenomena was attained. 

The allied phenomenon of rotary polarization in naturally 
active bodies was investigated in 1892 by Goldhammer.f It 

* The theory of Basset (Phil. Trans, clxxxii (1891), p. 371) was, like Rowland's, 
based on the idea of extending Hall's phenomenon to dielectric media. An objec- 
tion to this theory was that the tangential component of the electromotive force 
was not continuous across the interface between a magnetized and an unmagnetized 
medium ; but Basset subsequently overcame this difficulty (Nature, Hi (1 895), p. 618 ; 
liii (1895), p. 130; Amer. Jour. Math, xix (1897), p. 60) the effect analogous to 
Hall's being introduced into the equation connecting electric displacement with 
electric force, so that the equation took the form 

E = (47rc 2 / ) D + ff [K . D]. 

Basset, in 1893 (Proc. Camb. Phil. Soc. viii, p. 68), derived analytical 
expressions which represent Kerr's magneto-optic phenomenon by substituting u 
complex quantity for the refractive index in the formulae applicable to transparent 
magnetized substances. 

The magnetic rotation of light and Kerr's phenomenon have been investigated 
also by R. T. Glazebrook, Phil. Mag. xi (1881), p. 397 ; by J. J. Thomson, 
Recent Researches, p. 482 : by D. A. Goldhammer, Ann. d. Phys. xlvi (1892), 
p. 71 ; xlvii (1892), p. 345; xlviii (1893), p. 740; 1 (1893), p. 772 : by P. Drude, 
Ann. d. Phys. xlvi (1892), p. 353; xlviii (1893), p. 122; xlix (1893), p. 690; 
lii (1894)) p. 496 : by C. H. Wind, Verslagen Kon. Akad. Amsterdam, 29th Sept., 
1894 : by Reiff, Ann. d. Phys. Ivii (1896), p. 281 : by J. G. Leathern, Phil. 
Trans, cxc (1897), p. 89; Trans. Camb. Phil. Soc. xvii (1898), p. 16: and by 
W. Voigt in many memoirs, and in his treatise, Magneto- und Elektro-optik. 
Larmor's report presented to the British Association in 1893 has been already 
mentioned. 

In most of the later theories the equations of propagation of light in magnetized 
metals are derived from the two fundamental electromagnetic equations 

curl H = 4?rS, - curl E = H ; 

the total current S being assumed to consist of a part (the displacement-current) 
proportional to E, a part (the conduction -current) proportional to E, and a part 
proportional to the vector-product of E and the magnetization. 

Various mechanical models of media in which magneto-optic phenomena take 
place have been devised at different times. W. Thomson (Proc. Lond. Math. Soc. 
vi (1875)) investigated the propagation of waves of displacement along a stretched 
chain whose links contain rotating fly-wheels : cf . also Larmor, Proc. Lond. Math. 
Soc. xxi. (1890), p. 423 ; xxiii (1891), p. 127 ; F. Hasenohrl, Wien Sitzungsberichte 
cvii, 2a (189S), p. 1015 ; W. Thomson (Kelvin), Phil. Mag. xlviii (1899), p. 236, 
and Baltimore Lectures ; and Fitz Gerald, Electrician, Aug. 4, 1899, Fitz Gerald's 
Scientific Writings, p. 481. t Journal de Physique (3) i, pp. 205, 345. 



The Followers of Maxwell. 371 

will be remembered* that in the elastic-solid theory of 
Boussinesq, the rotation of the plane of polarization of 
saccharine solutions had been represented by substituting the 

equation 

e' = Ae + B curl e 

in place of the usual equation 

e' = Ae. 

Goldhammer now proposed to represent rotatory power in the 
electromagnetic theory by substituting the equation 

E = (4ircVO D + k curl D, 
in place of the customary equation 

E = (4ircVO D : 

the constant k being a measure of the natural rotatory power 
of the substance concerned. The remaining equations are as 

usual. 

curl H = 47rD, - curl E = H 

Eliminating H and E, we have 

fi = (c 2 / ) V 2 D + (k/4w) V 2 curl D. 

For a plane wave which is propagated parallel to the axis of x, 
this equation reduces to 

k_ 

47T 

k 

47T ~& ' 

and, as MacCullagh had shown in 1836,f these equations are 
competent to represent the rotation of the plane of polarization. 
In the closing years of the nineteenth century, the general 
theory of aether and electricity assumed a new form. But 
before discussing the memoirs in which the new conception was 
unfolded, we shall consider the progress which had been made 
since the middle of the century in the study of conduction in 
liquid and gaseous media. 

*Cf. p. 186. tcf. p. 175. 

2B2 




( 372 ) 



CHAPTEE XI. 

CONDUCTION IN SOLUTIONS AND GASES, FROM FARADAY TO 
J. J. THOMSON. 

THE hypothesis which Grothuss and Davy had advanced* to 
explain the decomposition of electrolytes was open to serious 
objection in more than one respect. Since the electric force 
was supposed first to dissociate the molecules of the electrolyte 
into ions, and afterwards to set them in motion toward the 
electrodes, it would seem reasonable to expect that doubling 
the electric force would double both the dissociation of the 
molecules and the velocity of the ions, and would therefore 
quadruple the electrolysis an inference which is not verified 
by observation. Moreover it might be expected, on Grothuss' 
theory, that some definite magnitude of electromotive force 
would be requisite for the dissociation, and that no electrolysis 
at all would take place when the electromotive force was below 
this value, which again is contrary to experience. 

A way of escape from these difficulties was first indicated, in 
1850, by Alex. Williamson,-)- who suggested that in compound 
liquids decompositions and recombinations of the molecules are 
continually taking place throughout the whole mass of the liquid, 
quite independently of the application of an external electric 
force. An atom of one element in the compound is thus paired 
now with one and now with another atom of another element, 
and in the intervals between these alliances the atom may be 
regarded as entirely free. In 1857 this idea was made by 

* Cf. p. 78. 

f Phil. Mag. xxxvii (1850), p. 350 ; Liebig's Annulen d. Chem. u. Pharni. 
Ixxvii (1851) p. 37. 



Conduction in Solutions and Gase*, etc. 373 

K. Clausius,* of Zurich, the basis of a theory of electrolysis. 
According to it, the electromotive force emanating from the 
electrodes does not effect the dissociation of the electrolyte 
into ions, since a degree of dissociation sufficient for the purpose 
already exists in consequence of the perpetual mutability of the 
molecules of the electrolyte. Clausius assumed that these ions 
are in opposite electric conditions; the applied electric force 
therefore causes a general drift of all the ions of one kind 
towards the anode, and of all the ions of the other kind towards 
the cathode. These opposite motions of the two kinds of ions 
constitute the galvanic current in the liquid. 

The merits of the Williamson-Clausius hypothesis were not 
fully recognized for many years ; but it became the foundation 
of that theory of electrolysis which was generally accepted at 
the end of the century. 

Meanwhile another aspect of electrolysis was receiving 
attention. It had long been known that the passage of a 
current through an electrolytic solution is attended not only 
by the appearance of the products of decomposition at the 
electrodes, but also by changes of relative strength in different 
parts of the solution itself. Thus in the electrolysis of a solution 
of copper sulphate, with copper electrodes, in which copper is 
dissolved off the anode and deposited on the cathode, it is found 
that the concentration of the solution diminishes near the 
cathode, and increases near the anode. Some experiments on 
the subject were made by Faradayf in 1835 ; and in 1844 it 
was further investigated by Frederic Daniell and W. A. Miller, J 
who explained it by asserting that the cation and anion have 
not (as had previously been supposed) the same facility of 
moving to their respective electrodes ; but that in many cases 
the cation appears to move but little, while the transport is 
effected chiefly by the anion. 

* Ann. d. Phys. ci (1857), p. 338 ; Phil. Mag. xv (1858), p. 94. 
t Exper. Res. 525-53C. 

* Phil. Trans., 1844, p. 1. Cf. also Pouillet, Comptes Rendus xx (1845), 
p. 1544. 



374 Conduction in Solutions and Gases, 

This idea was adopted by W. Hittorf, of Minister, who, in the 
years 1853 to 1859, published* a series of memoirs on the 
migration of the ions. Let the velocity of the anions in the 
solution be to the velocity of the cations in the ratio v : u. 
Then it is easily seen that if (u + v) molecules of the electrolyte 
are decomposed by the current, and yielded up as ions at the 
electrodes, v of these molecules will have been taken from the 
fluid on the side of the cathode, and u of them from the fluid 
on the side of the anode. By measuring the concentration of 
the liquid round the electrodes after the passage of a current, 
Hittorf determined the ratio v/u in a large number of cases of 
electrolysis.! 

The theory of ionic movements was advanced a further 
stage by F. W. KohlrauschJ (I. 1840, d. 1910), of Wurzburg. 
Kohlrausch showed that although the ohmic specific conduc- 
tivity k of a solution diminishes indefinitely as the strength 
of the solution is reduced, yet the ratio k/m, where m denotes 
the number of gramme-equivalents of salt per unit volume, tends 
to a definite limit, when the solution is indefinitely dilute. This 
limiting value may be denoted by A. He further showed that 
A may be expressed as the sum of two parts, one of which 
depends on the cation, but is independent of the nature of the 
anion; while the other depends on the anion, but not on the 
cation a fact which may be explained by supposing that, in 
very dilute solutions, the twos ions move independently under 
the influence of the electric force. Let u and v denote the 
velocities of the cation and anion respectively, when the 
potential difference per cm. in the solution is unity : then the 
total current carried through a cube of unit volume is mE(u + v), 
where E denotes the electric charge carried by one gramme- 

# Ann. d. Phys. Ixxxix (1853), p. 177 ; xcviii (1856), p. 1 ; ciii (1858), p. 1 ; 
cvi (1859), pp. 337, 513. 

t The ratio v/(u + v) was termed by Hittorf the transport, number of the anion. 
J Ann d. Phys. vi (1879), pp. 1, 145. The chief results had been communicated 
to the Academy of Gottingen in 1876 and 1877. 

A gramme-equivalent means a muss of the salt whose weight in grammes 
is the molecular weight divided by the valency of the ions. 



from Faraday to J. J. Thomson. 375 

equivalent of ion.* Thus mE (u + v) = total current = k = raA, 
or A = E (u + v). The determination of v/u by the method of 
Hittorf, and of (u + v) by the method of Kohlrausch, made it 
possible to calculate the absolute velocities of drift of the ions 
from experimental data. 

Meanwhile, important advances in voltaic theory were 
being effected in connexion with a different class of investi- 
gations. 

Suppose that two mercury electrodes are placed in a solution 
of acidulated water, and that a difference of potential, insufficient 
to produce continuous decomposition of the water, is set up 
between the electrodes by an external agency. Initially a 
slight electric current the polarizing current,f as it is called 
is observed; but after a short time it ceases; and after its cessation 
the state of the system is one of electrical equilibrium. It is 
evident that the polarizing current must in some way have set 
up in the cell an electromotive force equal and opposite to the 
external difference of potential ; and it is also evident that the 
seat of this electromotive force must be at the electrodes, which 
are now said to be polarized. 

An abrupt fall of electric potential at an interface between 
two media, such as the mercury and the solution in the present 
case, requires that there should be a field of electric force, of 
considerable intensity, within a thin stratum at the interface > 
and this must owe its existence to the presence of electric 
charges. Since there is no electric field outside the thin stratum, 
there must be as much vitreous as resinous electricity present ; 
but the vitreous charges must preponderate on one side of the 
stratum, and the resinous charges on the other side ; so that 
the system as a whole resembles the two coatings of a con- 
denser with the intervening dielectric. In the case of the 

* i.e. E is 96580 coulombs. 

t The phenomenon of voltaic polarization was discovered by Hitter in 1803. 
Hitter explained it by comparing the action of the polarizing current to that of a 
current which is used to charge a condenser. Volta in 1805 put forward the 
alternative explanation, that the products of decomposition set tip a reverse 
electromotive force. 



376 Conduction in Solutions and Gases, 

polarized mercury cathode in acidulated water, there must be 
on the electrode itself a negative charge : the surface of this 
electrode in the polarized state may be supposed to be either 
mercury, or mercury covered with a layer of hydrogen. In 
the solution adjacent to the electrode, there must be an excess 
of cations and a deficiency of anions, so as to constitute the 
other layer of the condenser : these cations may be either 
mercury cations dissolved from the electrode, or the hydrogen 
cations of the'solution. 

It was shown in 1870 by Cromwell Fleetwood Varley* that 
a mercury cathode, thus polarized in acidulated water, shows a 
tendency to adopt a definite superficial form, as if the surface- 
tension at the interface between the mercury and the solution 
were in some way dependent on the electric conditions. The 
matter was more fully investigated in 1873 by a young 
French physicist, then preparing for his inaugural thesis, 
Gabriel Lippmann.f In Lippmann's instrumental disposition, 
which is called a capillary electrometer, mercury electrodes are 
immersed in acidulated water : the anode H Q has a large 
surface, wkile^the cathode H has a variable surface S small in 
comparison. When the external electromotive force is applied, 
it is easily seen that the fall of potential at the large electrode 
is only slightly affected, while the fall of potential at the small 
electrode is altered by polarization by an amount practically 
equal to the external electromotive force. Lippmann found 
that the constant of capillarity of the interface at the small 
electrode was a function of the external electromotive force, and 
therefore of the difference of potential between the mercury 
and the electrolyte. 

Let V denote the external electromotive force: we may, 
without loss of generality, assume the potential of [ to be zero, 
so that the potential of H is - V. The state of the system may 
be varied by altering either V or /S; we assume that these 

* Phil. Trans, clxi (1871), p. 129. 

f Comptes Rendus Ixxvi (1873), p. 1407. Phil. Mag. xlvii (1874), p. 281. 
Ann. de Chim. et de Phys. v (1875), p. 494, xii (1877), p. 265. 



from Faraday to J . J . Thomson. 377 

alterations may be performed independently, reversibly, and 
isothermally, and that the state of the large electrode H, } is not 
altered thereby. Let de denote the quantity of electricity which 
passes through the cell from 5" to H, when the state of the 
system is thus varied : then if E denote the available energy of 
the system, and y the surface-tension at H, we have 

dE = ydS + Vde, 

y being measured by the work required to increase the surface 
when no electricity flows through the circuit. 

In order that equilibrium may be re-established between the 
electrode and the solution when the fall of potential at the 
cathode is altered, it will be necessary not only that some 
hydrogen cations should come out of the solution and be 
deposited on the electrode, yielding up their charges, but also 
that there should be changes in the clustering of the charged 
ions of hydrogen, mercury, and sulphion in the layer of the 
solution immediately adjacent to the electrode. Each of these 
circumstances necessitates a flow of electricity in the outer 
circuit : in the one case to neutralize the charges of the cations 
deposited, and in the other case to increase the surface-density 
of electric charge on the electrode, which forms the opposite 
sheet of the quasi-condenser. Let Sf (V) denote the total 
quantity of electricity which has thus flowed in the circuit 
when the external electromotive force has attained the value V. 
Then evidently 



so 

dE= {y+ Vf(V)\dS + VSf (V}dV. 

Since this expression must be an exact differential, we have 



so that - dy/d V is equal to that flux of electricity per unit of 
new surface formed, which will maintain the surface in a 



378 Conduction in Solutions and Gases, 

constant condition (V being constant) when it is extended. 
Integrating the previous equation, we have 



Lippmann found that when the external electromotive force 
was applied, the surface-tension increased at first, until, when 
the external electromotive force amounted to about one volt, 
the surface-tension attained a maximum value, after which it 
diminished. He found that d-y/d F 2 was sensibly independent 
of F, so that the curve which represents the relation between 
7 and F is a parabola.* 

The theory so far is more or less independent of assumptions 
as to what actually takes place at the electrode : on this latter 
question many conflicting views have been put forward. In 
1878 Josiah Willard Gibbs,t of Yale (b. 1839, d. 1903), discussed 
the problem on the supposition that the polarizing current is 
simply an ordinary electrolytic conduction-current, which 
causes a liberation of hydrogen from the ionic form at the 
cathode. If this be so, the amount of electricity which passes 
through the cell in any displacement must be proportional to 
the quantity of hydrogen which is yielded up to the electrode 
in the displacement; so that dy/dV must be proportional to 
the amount of hydrogen deposited per unit area of the 
electrode.:}: 

A different view of the physical conditions at the polarized 
electrode was taken by Helmholtz, who assumed that the ions 
of hydrogen which are brought to the cathode by the polarizing 
current do not give up their charges there, but remain in the 
vicinity of the electrode, and form one face of a quasi-condenser 



* Lippman, Coniptes Eendus, xcv (1882), p. 686. 

t Trans. Conn. Acad. iii (1876-1878), pp. 108, 343; Gibbs' Scientific Papers, 
i, p. 55. 

J This is embodied in equation (690) of Gibbs' memoir. 

Berlin Monatsber., 1881, p. 945 ; Wiss. Abh. i, p. 925 ; Ann. d. Phys. xvi. 
(1882), p. 31. Cf. also Planck, Ann. d. Phys. xliv (1891), p. 385. 



from Faraday to J . J. Thomson. 379 

of which the other face is the electrode itself.* If a denote 
the surface-density of electricity on either face of this quasi- 
condenser, we have, therefore, 

de = - d(Sa) ; so a = dyfd V. 

This equation shows that when dyldV is zero i.e., when 
the surface-tension is a maximum a must be zero ; that is to 
say, there must be no difference of potential between the 
mercury and the electrolyte. The external electromotive force 
is then balanced entirely by the discontinuity of potential at 
the other electrode J7" ; and thus a method is suggested of 
measuring the latter discontinuity of potential. All previous 
measurements of differences of potential had involved the 
employment of more than one interface ; and it was not known 
how the measured difference of potential should be distributed 
among these interfaces ; so that the suggestion of a means of 
measuring single differences of potential was a distinct advance, 
even though the hypotheses on which the method was based 
were somewhat insecure. 

A further consequence deduced by Helmholtz from this 
theory leads to a second method of determining the difference 
of potential between mercury and an electrolyte. If a mercury 
surface is rapidly extending, and electricity is not rapidly 
transferred through the electrolyte, the electric surface-density 
in the double layer must rapidly decrease, since the same 
quantity of electricity is being distributed over an increasing 
area. Thus it may be inferred that a rapidly extending 
mercury-surface in an electrolyte is at the same potential as 
the electrolyte. 

This conception is realized in the dropping-electrode, in 

* The conception of double layers of electricity at the surface of separation of 
two bodies had been already applied by Helmholtz to explain various other 
phenomena e.g., the Volta contact-difference of potential of two metals, fiictional 
electricity, and *' electric endosmose," or the transport of fluid which occurs when 
an electric current is passed through two conducting liquids separated by a porous 
barrier. Cf. Helmholtz, Berlin Monatsberichte, February 27, 1879 ; -Ann. d. Phys. 
vii (1879), p. 337 ; Helmholtz, Wiss. Abh. i, p. 855. 



380 Conduction in Solutions and Gases , 

which a jet of mercury, falling from a reservoir into an electro- 
lytic solution, is so adjusted that it breaks into drops when 
the jet touches the solution. According to Helmholtz's 
conclusion there is no difference of potential between the 
drops and the electrolyte ; and therefore the difference of 
potential between the electrolyte and a layer of mercury 
underlying it in the same vessel is equal to the difference of 
potential between this layer of mercury and the mercury 
in the upper reservoir, which difference is a measurable 
quantity. 

It will be seen that according to the theories both of Gibbs 
and of Helmholtz, and indeed according to all other theories on 
the subject,* d^ldV is zero for an electrode whose surface is 



* E.g., that of Warburg, Ann. d. Phys. xli (1890), p. 1. In this it is assumed 
that the electrolytic solution near the electrodes originally contains a salt of 
mercury in solution. When the external electromotive force is applied, a conduc- 
tion-current passes through the electrolyte, which in the hody of the electrolyte 
is carried by the acid and hydrogen ions. Warburg supposed that at the 
cathode the hydrogen ions react with the salt of mercury, reducing it to metallic 
mercury, which is deposited on the electrode. Thus a considerable change in 
concentration of the salt of mercury is caused at the cathode. At the anode, the 
acid ions carrying the current attack the mercury of the electrode, and thus 
increase the local concentration of the mercuric salt ; but on account of the size of 
the anode this increase is trivial and may be neglected. 

Warburg thus supposed that the electromotive force of the polarized cell is 
really that of a concentration cell, depending on the different concentrations of 
mercuric salt at the electrodes. He found dy/dV to be equal to the amount of 
mercuric salt at the cathode per unit area of cathode, divided by the electro- 
chemical equivalent of mercury. The equation previously obtained is thus 
presented in a new physical interpretation. 

Warburg connected the increase of the surface-tension with the fact that the 
surface-tension between mercury and a solution always increases when the con- 
centration of the solution is diminished. His theory, of course, leads to no 
conclusion regarding the absolute potential difference between the mercury and the 
solution, as Helmholtz' does. 

Alan electrode whose surface is rapidly increasing e.g., a dropping electrode 
Warburg supposed that the surface-density of mercuric salt tends to zero, so 
dyldV is zero. 

The explanation of dropping electrodes favoured by Nernst, Beilage zu den 
Ann. d. Phys. Iviii (1896), is that the difference of potential corresponding to the 
equilibrium between the mercury and the electrolyte is instantaneously 
established ; but that ions are withdrawn from the solution in order to form the 
double layer necessary for this, and that these ions are carried down with the drops 



from Faraday to J. J . Thomson. 381 

rapidly increasing e.g., a dropping electrode; that is to say, 
the difference of potential between an ordinary mercury 
electrode and the electrolyte, when the surface-tension has its 
maximum value, is equal to the difference of potential between 
a dropping-electrode and the same electrolyte. This result has 
been experimentally verified by various investigators, who have 
shown that the applied electromotive force when the surface- 
tension has its maximum value in the capillary electrometer, is 
equal to the electromotive force of a cell having as electrodes a 
large mercury electrode and a dropping electrode. 

Another memoir which belongs to the same period of 
Helmholtz' career, and which has led to important develop- 
ments, was concerned with a special class of voltaic cells. The 
most usual type of cell is that in which the positive electrode 
is composed of a different metal from the negative electrode, 
and the evolution of energy depends on the difference in the 
chemical affinities of these metals for the liquids in the cell. 
But in the class of cells now considered* by Helmholtz, the 
two electrodes are composed of the same metal (say, copper) ; 
and the liquid (say, solution of copper sulphate) is more con- 
centrated in the neighbourhood of one electrode than in the 
neighbourhood of the other. When the cell is in operation, the 
salt passes from the places of high concentration to the places 
of low concentration, so as to equalize its distribution ; and this 
process is accompanied by the flow of a current in the outer 
circuit between the electrodes. Such cells had been studied 
experimentally by James Moser a short time previously! to 
Helmholtz' investigation. 

The activity of the cell is due to the fact that the available 
energy of a solution depends on its concentration ; the molecules 

of mercury, until the upper layer of the solution is so much impoverished that the 
double layer can no longer be formed. The impoverishment of the upper layer of 
the solution has actually been observed by Palniaer, Zeitsch. Phys. Chem. xxv 
(1898), p. 265 ; xxviii (1899), p. 257 ; xxxvi (1901), p. 664. 

* Berlin Monatsber., 1877, p. 713 ; Phil. Mag. (5) v (1878), p. 348; reprinted 
with additions in Ann. d. Phys. iii (1878), p. 201. 

t Ann. d. Phys. iii (1878), p. 216. 



382 Conduction in Solutions and Gases , 

of salt, in passing from a high to a low concentration, are 
therefore capable of supplying energy, just as a compressed gas 
is capable of supplying energy when its degree of compression 
is reduced. To examine the matter quantitatively, let nf(nf V) 
denote the term in the available energy of a solution, which is 
due to the dissolution of n gramme-molecules of salt in a volume 
V of pure solvent ; the function / will of course depend also on 
the temperature. Then when dn gramme-molecules of solvent 
are evaporated from the solution, the decrease in the available 
energy of the system is evidently equal to the available energy of 
dn gramme-molecules of liquid solvent, less the available energy 
of dn gramme-molecules of the vapour of the solvent, together 
with nf(n/ V) less nf{n/(V-v dn) } , where v denotes the volume 
of one gramme-molecule of the liquid. But this decrease in 
available energy must be equal to the mechanical work supplied 
to the external world, which is dn . p (v - v), if p l denote the 
vapour-pressure of the solution at the temperature in question, 
and v denote the volume of one gramme-molecule of vapour. 
We have therefore 

dn . pi (v' - v) = available energy of dn gramme-molecules of 

solvent vapour 
+ available energy of dn gramme-molecules of 

liquid solvent 
+ nf(n/ V) - nf {n/( V-v dn) \ . 

Subtracting from this the equation obtained by making n zero, 
we have 

dn . (Pi - p ) (v - v) = nf(n/ V) - nf( n/( V - v dn) } , 

where p Q denotes the vapour-pressure of the pure solvent at the 
temperature in question ; so that 

(Pi -Po) <>' - v) = - (n'/V*)f(n/V)v. 

Now, it is known that when a salt is dissolved in water, the 
vapour-pressure is lowered in proportion to the concentration 
of the salt at any rate when the concentration is small : in 



from Faraday to J . J. Thomson. 383 

fact, by the law of Kaoult, (p -pi)/po is approximately equal to 
nv/ V ; so that the previous equation becomes 

p. V(v' -f.) -*/(/ F). 

Neglecting v in comparison with v', and making use of the 
equation of state of perfect gases (namely, 

pjt = ST. 

where T denotes the absolute temperature, and R denotes the 
constant of the equation of state), we have 



and therefore 



Thus in the available energy of one gramme-molecule of a 
dissolved salt, the term which depends on the concentration is 
proportional to the logarithm of the concentration ; and hence, 
if in a concentration-cell one gramme-molecule of the salt 
passes from a high concentration c 2 at one electrode to a low 
concentration GI at the other electrode, its available energy is 
thereby diminished by an amount proportional to log (c 2 /c,). 
The energy which thus disappears is given up by the system in 
the form of electrical work; and therefore the electromotive 
force of the concentration-cell must be proportional to log (Cz/cJ.. 
The theory of solutions and their vapour-pressure was 
not at the time sufficiently developed to enable Helmholtz 
to determine precisely the coefficient of log (c 2 /Ci) in the 
expression.* 

An important advance in the theory of solutions was effected 
in 1887, by a young Swedish physicist, Svante Arrhenius.f 

* The formula given by Helmholtz was that the electromotive force of the cell 
is equal to b(l - ri) v log (czjc\), where ci and c\ denote the concentrations of the solu- 
tion at the electrodes, v denotes the volume of one gramme of vapour in equilibrium 
with the water at the temperature in question, n denotes the transport number for 
the cation (Hittorfs 1/w), and b denotes q x the lowering of vapour- pressure when 
one gramme-equivalent of salt is dissolved in q grammes of water, where q denotes 
a large number. 

t Zeitschrift fur phys. Chem. i (1887), p. 631. Previous investigations, in 
which the theory was to some extent foreshadowed, were published in Bihang 
till Svenska Vet. Ak. Forh. viii (1884), Nos. 13 and 14. 



384 Conduction in Solutions and Gases, 

Interpreting the properties discovered by Kohlrausch* in the 
light of the ideas of Williamson and Clausius regarding the 
spontaneous dissociation of electrolytes, Arrhenius inferred that 
in very dilute solutions the electrolyte is completely dissociated 
into ions, but that in more concentrated solutions the salt is 
less completely dissociated; and that as in all solutions the 
transport of electricity in the solution is effected solely by the 
movement of ions, the equivalent conductivityf must be pro- 
portional to the fraction which expresses the degree of ionization. 
By aid of these conceptions it became possible to estimate the 
dissociation quantitatively, and to construct a general theory 
of electrolytes. 

Contemporary physicists and chemists found it difficult 
at first to believe that a salt exists in dilute solution only 
in the form of ions, e.g. that the sodium and chlorine exist 
separately and independently in a solution of common salt. 
But there is a certain amount of chemical evidence in favour 
of Arrhenius' conception. For instance, the tests in chemical 
analysis are really tests for the ions ; iron in the form of a fer- 
rocyanide, and chlorine in the form of a chlorate, do not respond 
to the characteristic tests for iron and chlorine respectively, 
which are really the tests for the iron and chlorine ions. 

The general acceptance of Arrhenius' views was hastened 
by the advocacy of Ostwald, who brought to light further 
evidence in their favour. For instance, all permanganates 
in dilute solution show the same purple colour; and 
Ostwald considered their absorption-spectra to be identical ;J 
this identity is easily accounted for on Arrhenius' theory, by 
supposing that the spectrum in question is that of the anion 
which corresponds to the acid radicle. The blue colour 
which is observed in dilute solutions of copper salts, even 
when the strong solution is not blue, may in the same way be 

* Cf. p. 374. 

t I.e. the ohmic specific conductivity of the solution divided by the number of 
gramme-equivalents of salt per unit volume. 

J Examination of the spectra with higher dispersion does not altogether 
confirm this conclusion. 



from Faraday to J .J. Thomson. 385 

ascribed to a blue copper cation. A striking instance of the 
same kind is afforded by ferric sulphocyanide ; here the strong 
solution shows a deep red colour, due to the salt itself ; but on 
dilution the colour disappears, the ions being colourless. 

If it be granted that ions can have any kind of permanent 
existence in a salt solution, it may be shown from thermo- 
dynamical considerations that the degree of dissociation must 
increase as the dilution increases, and that at infinite dilution 
there must be complete dissociation. For the available energy 
of a dilute solution of volume V, containing j gramme-molecules 
of one substance, >/ 2 gramme-molecules of another, and so on, is 
(as may be shown by an obvious extension of the reasoning 
already employed in connexion with concentration-cells)* 

r (T) + RT^n r log (UT! V) + the available energy 



possessed by the solvent before the introduction of the solutes, 
where r (T) depends on T and on the nature of the r th solute, 
but not on V, and R denotes the constant which occurs in the 
equation of state of perfect gases. When the system is in 
equilibrium, the proportions of the reacting substances will 
be so adjusted that the available energy has a stationary 
value for small virtual alterations Swj, &^, ...... of the 

proportions ; and therefore 



- SSn r .<t> r (T) + RT2$n r .log (n r jV) 



Applying this to the case of an electrolyte in which the 
disappearance of one molecule of salt (indicated by the suffix ,) 
gives rise to one cation (indicated by the suffix 2 ) and one anion 
(indicated by the suffix 3 ), we have B^ = - 7^ = - Sn* ; so the 
equation becomes 

= 0, (T) - 2 (T) - 03 (T) + RT log (n, V/n.n,) - RT, 

or 

= a function of T only. 

* Cf. pp. 382-383. 
2 C 



386 Conduction in Solutions and Gases, 

Since in a neutral solution the number of anions is equal to the 
number of cations, this equation may be written 

nf = Fw-i x a function of T only ; 

it shows that when V is very large (so that the solution is very 
dilute), n 2 is very large compared with n^ ; that is to say, the 
salt tends towards a state of complete dissociation. 

The ideas of Arrhenius contributed to the success of Walther 
Nernst* in perfecting Helmholtz' theory of concentration-cells, 
and representing their mechanism in a much more definite 
fashion than had been done heretofore. 

In an electrolytic solution let the drift-velocity of the 
cations under unit electric force be u, and that of the anions 
be v t so that the fraction uj(u + v} of the current is transported 
by the cations, and the fraction v/(u + v) by the anions. If the 
concentration of the solution be Cj at one electrode, and c 2 at the 
other, it follows from the formula previously found for the 
available energy that one gramme - ion of cations, in moving 
from one electrode to the other, is capable of yielding up an 
amountf RT log (c 2 /c,) of energy; while one gramme - ion 
of anions going in the opposite direction must absorb the same 
amount of energy. The total quantity of work furnished when 
one gramme-molecule of salt is transferred from concentration 
c t to concentration c { is therefore 



u + v 



The quantity of electric charge which passes in the circuit 
when one gramme-molecule of the salt is transferred is pro- 
portional to the valency v of the ions, and the work furnished 
is proportional to the product of this charge and the electro- 

*Zeitschr. fur phys. Chem. ii (1888), p. 613; iv (1889), p. 129; Berlin 
Sitzungsberichte, 1889, p. 83 ; Ann. d. Phys. xlv (1892), p. 360. Cf. also 
Max Planck, Ann. d. Phys. xxxix (1890), p. 161 ; xl (1890), p. 561. 

t The correct law of dependence of the available energy on the temperature was 
by this time known. 



from Faraday to J . J . Thomson. 387 

motive force E of the cell ; so that in suitable units we have 

-, RTu-v. c, 
E = -- - log -. 

v u + v Ci 

A typical concentration-cell to which this formula may be 
applied may be constituted in the following way : Let a 
quantity of zinc amalgam, in which the concentration of zinc 
is d, be in contact with a dilute solution of zinc sulphate, and 
let this in turn be in contact with a quantity of zinc amalgam 
of concentration c z . When the two masses of amalgam are con- 
nected by a conducting wire outside the cell, an electric current 
flows in the wire from the weak to the strong amalgam,* while 
zinc cations pass through the solution from the strong amalgam 
to the weak. The electromotive force of such a cell, in which 
the current may be supposed to be carried solely by cations, is 

RT. c, 

lo- 



V 



Not content with the derivation of the electromotive force 
from considerations of energy, Nernst proceeded to supply a 
definite mechanical conception of the process of conduction in 
electrolytes. The ions are impelled by the electric force asso- 
ciated with the gradient of potential in the electrolyte. But 
this is not the only force which acts on them ; for, since their 
available energy decreases as the concentration decreases, there 
must be a force assisting every process by which the concentra- 
tion is decreased. The matter may be illustrated by the analogy 
of a gas compressed in a cylinder fitted with a piston; the 
available energy of the gas decreases as its degree of compression 
decreases; and therefore that movement of the piston which 
tends to decrease the compression is assisted by a force the 
"pressure" of the gas on the piston. Similarly, if a solution 
were contained within a cylinder fitted with a piston which is 
permeable to the pure solvent but not to the solute, and if the 
whole were immersed in pure solvent, the available energy of 

* It will hardly be necessary to remark that this supposed direction of the 
.current is purely conventional. 

2 C 2 



388 Conduction in Solutions and Gases, 

the system would be decreased if the piston were to move 
outwards so as to admit more solvent into the solution; and 
therefore this movement of the piston would be assisted by a 
force the "osmotic pressure of the solution," as it is called.* 

Consider, then, the case of a single electrolyte supposed to 
be perfectly dissociated ; its state will be supposed to be the 
same at all points of any plane at right angles to the axis of x. 
Let v denote the valency of the ions, and V the electric potential 
at any point. Sincef the available energy of a given quantity of 
a substance in very dilute solution depends on the concentration 
in exactly the same way as the available energy of a given 
quantity of a perfect gas depends on its density, it follows that 
the osmotic pressure p for each ion is determined in terms of 
the concentration and temperature by the equation of state 
of perfect gases 

Mp = ETc, 

where M denotes the molecular weight of the salt, and c the 
mass of salt per unit volume. 

Consider the cations contained in a parallelepiped at the 
place x, whose cross-section is of unit area and whose length 
is dx. The mechanical force acting on them due to the electric 
field is - (vc/M) d Vfdx . dx, and the mechanical force on them 
due to the osmotic pressure is - dp/dx . dx. If u denote the 
velocity of drift of the cations in a field of unit electric force, 
the total amount of charge which would be transferred by 
cations across unit area in unit time under the influence of the 
electric forces alone would be - (uvc/M) d V/dx ; so, under the 
influence of both forces, it is 

_uvcidV_ ET dc\ 
M\dx cv dx) 

Similarly, if v denote the velocity of drift of the anions in a 

* Cf . van't Hoff, Svenska Vet.-Ak. Handlingar xxi (1886), No. 17; Zeitschrift 
fiir Phys. Chem. i (1887), p. 481. 

t As follows from the expression obtained, supra, p. 383. 



from Faraday to jf . J . Thomson. 389 

unit electric field, the charge transferred across unit area in 
unit time by the anions is 

vvcfdV^ RT dc\ 
M\dx cv dx) 

We have therefore, if the total current be denoted by i, 
. vc dV RT do 



- u+ M^- u - v >^Tdx> 

or 

dV 7 Mdx u-v RT dc , 

- - T - dx = - - ^ + -- dx. 
dx (u + v)vc u + v vc dx 

The first term on the right evidently represents the product of 
the current into the ohmic resistance of the parallelepiped dx, 
while the second term represents the internal electromotive 
force of the parallelepiped. It follows that if r denote the 
specific resistance, we must have 

u + v = Mjrvc, 

in agreement with Kohlrausch's equation ;* while by integrating 
the expression for the internal electromotive force of the 
parallelepiped dx, we obtain for the electromotive force of a 
cell whose activity depends on the transference of electrolyte 
between the concentrations c, and c z , the value 

u-v RT fl dc . 

-- - T- <te> 
u + v v c dx 



u-v RT , c, 

or log-, 

u + v v GI 

in agreement with the result already obtained. 

It may be remarked that although the current arising from 
a concentration cell which is kept at a constant temperature is 
capable of performing work, yet this work is provided, not by 
any diminution in the total internal energy of the cell, but by 
the abstraction of thermal energy from neighbouring bodies. 
This indeed (as may be seen by reference to W. Thomson's general 

* Cf. P . 374. 



390 Conduction in Solutions and Gases, 

equation of available energy)* must be the case with any 
system whose available energy is exactly proportional to the 
absolute temperature. 

The advances which were effected in the last quarter of the 
nineteenth century in regard to the conduction of electricity 
through liquids, considerable though these advances were, may 
be regarded as the natural development of a theory which had 
long been before the world. It was otherwise with the kindred 
problem of the conduction of electricity through gases : for 
although many generations of philosophers had studied the 
remarkable effects which are presented by the passage of a 
current through a rarefied gas, it was not until recent times 
that a satisfactory theory of the phenomena was discovered. 

Some of the electricians of the earlier part of the eighteenth 
century performed experiments in vacuous spaces ; in particular, 
Hauksbeef in 1705 observed a luminosity when glass is rubbed 
in rarefied air. But the first investigator of the continuous 
discharge through a Tarefied gas seems to have been Watson,! 
who, by means of an electrical machine, sent a current through 
an exhausted glass tube three feet long and three inches in 
diameter. " It was," he wrote, " a most delightful spectacle, 
when the room was darkened, to see the electricity in its 
passage : to be able to observe not, as in the open air, its 
brushes or pencils of rays an inch or two in length, but here 
the coruscations were of the whole length of the tube between 
the plates, that is to say, thirty-two inches." Its appearance 
he described as being on different occasions " of a bright silver 
hue," " resembling very much the most lively coruscations of 
the aurora borealis," and " forming a continued arch of lambent 
flame." His theoretical explanation was that the electricity " is 
seen, without any preternatural force, pushing itself on through 
the vacuum by its own elasticity, in order to maintain the 

* Cf. p. 241. 

t Phil. Trans, xxiv (1705), p. 2165. Fra. Hauksbee, Physico- Mechanical 
Experiments, London, 1709. 

I Phil. Trans, xlv (1748), p. 93, xlvii (1752), p. 362. 



from Faraday to J . J . Thomson. 391 

equilibrium in the machine " a conception which follows 
naturally from the combination of Watson's one-fluid theory 
with the prevalent doctrine of electrical atmospheres.* 

A different explanation was put forward by Nollet, who 
performed electrical experiments in rarefied air at about the 
same time as Watson,f and saw in them a striking confirmation 
of his own hypothesis of efflux and afflux of electric matter.J 
According to Nollet, the particles of the effluent stream collide 
with those of the affluent stream which is moving in the 
opposite direction ; and being thus violently shaken, are excited 
to the point of emitting light. 

Almost a century elapsed before anything more was dis- 
covered regarding the discharge in vacuous spaces. But in 
1838 Faraday, while passing a current from the electrical 
machine between two brass rods in rarefied air, noticed that 
the purple haze or stream of light which proceeded from the 
positive pole stopped short before it arrived at the negative 
rod. The negative rod, which was itself covered with a con- 
tinuous glow, was thus separated from the purple column by 
a narrow dark space: to this, in honour of its discoverer, 
the name Faraday's dark space has generally been given by 
subsequent writers. 

That vitreous and resinous electricity give rise to different 
types of discharge had long been known; and indeed, as we 
have seen,) | it was the study of these differences that led 
Franklin to identify the electricity of glass with the superfluity of 
fluid, and the electricity of amber with the deficiency of it. But 
phenomena of this class are in general much more complex 
than might be supposed from the appearance which they 
present at a first examination ; and the value of Faraday's 
discovery of the negative glow and dark space lay chiefly in 
the simple and definite character of these features of the 
discharge, which indicated them as promising subjects for 
further research. Faraday himself felt the importance of 

* Cf. ch. ii. f Nollet, Recherches sur FElectricite, 1749, troisiemediscours. 
t Cf. p. 40. Phil. Trans., 1838 ; Exper. Res. i, 1526. || Cf. p. 44. 



392 Conduction in Solutions and Gases, 

investigations in this direction. " The results connected with 
the different conditions of positive and negative discharge," he 
wrote,* " will have a far greater influence on the philosophy of 
electrical science than we at present imagine." 

Twenty more years, however, passed before another notable 
advance was made. That a subject so full of promise should 
progress so slowly may appear strange ; but one reason at any 
rate is to be found in the incapacity of the air-pumps then in 
use to rarefy gases to the degree required for effective study 
of the negative glow. The invention of Geissler's mercurial 
air-pump in 1855 did much to remove this difficulty; and it 
was in Geissler's exhausted tubes that Julius Plticker,t of Bonn, 
studied the discharge three years later. 

It had been shown by Sir Humphrey Davy in 1821 J that 
one form of electric discharge namely, the arc between carbon 
poles is deflected when a magnet is brought near to it. 
Pliicker now performed a similar experiment with the vacuum 
discharge, and observed a similar deflexion. But the most 
interesting of his results were obtained by examining the 
behaviour of the negative glow in the magnetic field ; when 
the negative electrode was reduced to a single point, the whole 
of the negative light became concentrated along the line of 
magnetic force passing through this point. In other words, 
the negative glow disposed itself as if it were constituted of 
flexible chains of iron filings attached at one end to the 
cathode. 

Pliicker noticed that when the cathode was of platinum, 
small particles were torn off it and deposited on the walls of 
the glass bulb. " It is most natural," he wrote, " to imagine 
that the magnetic light is formed by the incandescence of these 
platinum particles as they are torn from the negative electrode." 
He likewise observed that during the discharge the walls of 

* Exper. Res., 1523. 

| Ann. d. Phys. ciii (1858), pp. 88, 151 ; civ (1858), pp. 113, 622 ; cv (1858), 
p. 67; cvii (1859), p. 77. Phil. Mag. xvi (1858), pp. 119, 408; xviii (1859), 
pp. 1, 7. 

J Phil. Trans., 1821, p. 425. 



from Faraday to J . J. Thomson. 393 

the tube, near the cathode, glowed with a phosphorescent light, 
and remarked that the position of this light was altered when 
the magnetic field was changed. This led to another discovery ; 
for in 1869 Plucker's pupil, W. Hittorf,* having placed a solid 
body between a point-cathode and the phosphorescent light, was 
surprised to find that a shadow was cast. He rightly inferred 
from this that the negative glow is formed of rays which 
proceed from the cathode in straight lines, and which cause the 
phosphorescence when they strike the walls of the tube. 

Hittorf's observation was amplified in 1876 by Eugen 
Goldstein,f who found that distinct shadows were cast, not 
only when the cathode was a single point, but also when it 
formed an extended surface, provided the shadow-throwing 
object was placed close to it. This clearly showed that the 
cathode rays (a term now for the first time introduced) are not 
emitted indiscriminately in all directions, but that each portion 
of the cathode surface emits rays which are practically confined 
to a single direction ; and Goldstein found this direction to be 
normal to the surface. In this respect his discovery established 
an important distinction between the manner in which cathode 
rays are emitted from an electrode and that in which light is 
emitted from an incandescent surface. 

The question as to the nature of the cathode rays attracted 
much attention during the next two decades. In the year 
following Hittorf's investigation, Cromwell VarleyJ put forward 
the hypothesis that the rays are composed of " attenuated par- 
ticles of matter, projected from the negative pole by electricity" ; 
and that it is in virtue of their negative charges that these 
particles are influenced by a magnetic field. 

During some years following this, the properties of highly 

* Ann. <1. Phys. cxxxvi (1869), pp. 1, 197; translated, Annales de Cbimie, xvi 
(1869), p. 487. 

t Berlin Monatsberichte, 1876, p. 279. 

J Proc. Roy. Soc. xix (1871), p. 236. 

Priestley in 1766 had shown that a current of electri6ed air flows from the 
points of hodies which are electrified either vitreously or resinously : cf. Priestley's 
History of Electricity, p. 591. 



394 Conduction in Solutions and Gases, 

rarefied gases were investigated by Sir William Crookes. 
Influenced, doubtless, by the ideas which were developed in 
connexion with his discovery of the radiometer, Crookes,* like 
Varley, proposed to regard the cathode rays as a molecular 
torrent : he supposed the molecules of the residual gas, coming 
into contact with the cathode, to acquire from it a resinous charge, 
and immediately to fly off normally to the surface, by reason of 
the mutual repulsion exerted by similarly electrified bodies. 
Carrying the exhaustion to a higher degree, Crookes was enabled 
to study a dark space which under such circumstances appears 
between the cathode and the cathode glow ; and to show that at 
the highest rarefactions this dark space (which has since been gene- 
rally known by his name) enlarges until the whole tube is occupied 
by it. He suggested that the thickness of the dark space may 
be a measure of the mean length of free path of the molecules. 
" The extra velocity," he wrote, " with which the molecules 
rebound from the excited negative pole keeps back the more 
slowly moving molecules which are advancing towards that pole. 
The conflict occurs at the boundary of the dark space, where the 
luminous margin bears witness to the energy of the collisions."f 
Thus according to Crookes the dark space is dark and the 
glow bright because there are collisions in the latter and not in 
the former. The fluorescence or phosphorescence on the walls 
of the tube he attributed to the impact of the particles on 
the glass. 

Crookes spoke of the cathode rays as an " ultra-gaseous " or 
" fourth state " of matter. These expressions have led some 
later writers to ascribe to him the enunciation or prediction of 
a hypothesis regarding the nature of the particles projected from 
the cathode, which arose some years afterwards, and which we 
shall presently describe ; but it is clear from Crookes' memoirs 
that he conceived the particles of the cathode rays to be 
ordinary gaseous molecules, carrying electric charges ; and by 



* Phil. Trans, clxx (1879), pp. 135, 641 ; Phil. Mag. vii (1879), p. 57. 
t Phil. Mag. vii (1879), p. 57. 



from Faraday to J . J . 71wmson. 395 

" a new state of matter " he understood simply a state in which 
the free path is so long that collisions may be disregarded. 

Crookes found that two adjacent pencils of cathode rays 
appeared to repel each other. At the time this was regarded as 
a direct confirmation of the hypothesis that the rays are streams 
of electrically charged particles ; but it was shown later that 
the deflexion of the rays must be assigned to causes other than 
mutual repulsion. 

How admirably the molecular- torrent theory accounts for 
the deviation of the cathode rays by a magnetic field was shown 
by the calculations of Eduard Riecke in 1881.* If the axis of 
z be taken parallel to the magnetic force H t the equations of 
motion of a particle of mass ra, charge e, and velocity (u, v, w) 

are 

mdu/dt = evH, mdvjdt = - euH, mdw/dt = 0. 

The last equation shows that the component of velocity of the 
particle parallel to the magnetic force is constant; the other 
equations give 

u = A sin (eHt/m), v = A cos (eHi/m), 

showing that the projection of the path on a plane at right 
angles to the magnetic force is a circle. Thus, in a magnetic 
field the particles of the molecular torrent describe spiral paths 
whose axes are the lines of magnetic force. 

But the hypothesis of Varley and Crookes was before long 
involved in difficulties. Taitf in 1880 remarked that if the 
particles are moving with great velocities, the periods of the 
luminous vibrations received from them should be affected to a 
measurable extent in accordance with Doppler's principle. 
Tait tried to obtain this effect, but without success. It may, 
however, be argued that if, as Crookes supposed, the particles 
become luminous only when they have collided with other 
particles, and have thereby lost part of their velocity, the 
phenomenon in question is not to be expected. 

* Gott. Nach., 2 February, 1881; reprinted, Ann. d. Phys. xiii (1881), p. 191. 
t Proc. Roy. Soc. Edinb. x (1880). p. 430. 



396 Conduction in Solutions and Gases , 

The alternative to the molecular-torrent theory is to suppose 
that the cathode radiation is a disturbance of the aether. This 
view was maintained by several physicists,* and notably by 
Hertz,f who rejected Varley's hypothesis when he found 
experimentally that the rays did not appear to produce any 
external electric or magnetic force, and were apparently not 
affected by an electrostatic field. It was, however, pointed 
out by Fitz Gerald* that external space is probably screened 
from the effects of the rays by other electric actions which 
take place in the discharge tube. 

It was further urged against the charged-particle theory 
that cathode rays are capable of passing through films of metal 
which are so thick as to be quite opaque to ordinary light ; 
it seemed inconceivable that particles of matter should not be 
stopped by even the thinnest gold-leaf. At the time of Hertz's 
experiments on the subject, an attempt to obviate this difficulty 
was made by J.-J. Thomson,! | who suggested that the metallic 
film when bombarded by the rays might itself acquire the 
property of emitting charged particles, so that the rays which 
were observed on the further side need not have passed through 
the film. It was Thomson who ultimately found the true 
explanation ; but this depended in part on another order of 
ideas, whose introduction and development must now be 
traced. 

The tendency, which was now general, to abandon the 
electron-theory of Weber in favour of Maxwell's theory 
involved certain changes in the conceptions of electric charge. 

* E.g. E. Wiedemann, Ann. d. Phys. x (1880), p. 202; translated, Phil. Mag. 
x (1880), p. 357. E. Goldstein, Ann. d. Phys. xii (1881), p. 249. 

t Ann. d. Phys. xix (1883), p. 782. 

J Nature, November 5, 1896 ; Fitz Gerald's Scientific Writings, p. 433. 

The penetrating power of the rays had been noticed by Hittorf, and by 
E. "Wiedemann and Ebert, Sitzber. d. phys.-med. Soc. zu Erlangen, llth December, 
1891. It was investigated more thoroughly by Hertz, Ann. d. Phys. xlv (1892), 
p. 28, and by Philipp Lenard, of Bonn, Ann. d. Phys. li (1894), p. 225 ; lii (1894), 
p. 23, who conducted a series of experiments on cathode rays which had passed out 
of the discharge tube through a thin window of aluminium. 

|| J. J. Thomson, Recent Researches, p. 126. 



from Faraday to J . J . Thomson. 397 



In the theory of Weber, electric phenomena were attributed 
to the agency of stationary or moving charges, which could 
most readily be pictured as having a discrete and atom-like 
existence. The conception of displacement, on the other hand, 
which lay at the root of the Maxwellian theory, was more in 
harmony with the representation of electricity as something 
of a continuous nature; and as Maxwell's views met with 
increasing acceptance, the atomistic hypothesis seemed to have 
entered on a period of decay. Its revival was due largely to the 
advocacy of Helmholtz,* who, in a lecture delivered to the 
Chemical Society of London in 1881, pointed outf that it was 
thoroughly in accord with the ideas of Faraday,J on which 
Maxwell's theory was founded. " If," he said, " we accept the 
hypothesis that the elementary substances are composed of 
atoms, we cannot avoid concluding that electricity also, positive 
as well as negative, is divided into definite elementary portions 
which behave like atoms of electricity." 

When the conduction of electricity is considered in the 
light of this hypothesis, it seems almost inevitable to conclude 
that the process is of much the same character in gases as in 
electrolytes ; and before long this view was actively maintained. 
It had indeed long been known that a compound gas might be 
decomposed by the electric discharge ; and that in some cases 
the constituents are liberated at the electrodes in such a way 
as to suggest an analogy with electrolysis. The question had 
been studied in 1861 by Adolphe Perrot, who examined the 
gases liberated by the passage of the electric spark through 
steam. He found that while the product of this action was 
a detonating mixture of hydrogen and oxygen, there was a 
decided preponderance of hydrogen at one pole and of oxygen 
at the other. 

The analogy of gaseous conduction to electrolysis was 
applied by W. Giese,|| of Berlin, in 1882, in order to explain 

* Cf. also G. Johnstone Stoney, Phil. Mag., May, 1881. 

f Journ. Chem. Soc. xxxix (1881), p. 277. \ Cf. p. 200. 

Annales de Chimie (3), Ixi, p. 161. 

|| Ann. d. Phys. xvii (1882), pp. 1, 236, 519. 



398 Conduction in Solutions and Gases , 

the conductivity of the hot gases of flames. " It is assumed," 
he wrote, " that in electrolytes, even before the application of 
an external electromotive force, there are present atoms or 
atomic groups the ions, as they are called which originate 
when the molecules dissociate ; hy these the passage of electri- 
city through the liquid is effected, for they are set in motion 
by the electric field and carry their charges with them. We 
shall now extend this hypothesis by assuming that in gases 
also the property of conductivity is due to the presence of 
ions. Such ions may be supposed to exist in small numbers 
in all gases at the ordinary temperature and pressure ; and as 
the temperature rises their numbers will increase." 

Ideas similar to this were presented in a general theory of 
the discharge in rarefied gases, which was devised two years 
later by Arthur Schuster, of Manchester.* Schuster remarked 
that when hot liquids are maintained at a high potential, the 
vapours which rise from them are found to be entirely free 
from electrification ; from which he inferred that a molecule 
striking an electrified surface in its rapid motion cannot carry 
away any part of the charge, and that one molecule cannot 
communicate electricity to another in an encounter in which 
both molecules remain intact. Thus he was led to the con- 
clusion that dissociation of the gaseous molecules is necessary 
for the passage of electricity through gases. f 

Schuster advocated the charged-particle theory of cathode 
rays, and by extending and interpreting an experiment of 
Hittorf s was able to adduce strong evidence in its favour. 
He placed the positive and negative electrodes so close to each 
other that at very low pressures the Crookes' dark space 
extended from the cathode to beyond the anode. In these 
circumstances it was found that the discharge from the positive 
electrode always passed to the nearest point of the inner 
boundary of the Crookes' dark space which, of course, was in 

* Proc. Roy. Soc. xxxvii (1884), p. 317. 

t In the case of an elementary gas, this would imply dissociation of the molecule 
into two atoms chemically alike, but oppositely charged ; in electrolysis the 
.dissociation is into two chemically unlike ions. 



Jrom Faraday to J . J . Thomson* 399 

the opposite direction to the ijathode. Thus, in the neighbour- 
hood of the positive discharge, the current was flowing in two 
opposite directions at closely adjoining places ; which could 
scarcely happen unless the current in one direction were 
carried by particles moving against the lines of force by 
virtue of their inertia. 

Continuing his researches, Schuster* showed in 1887 that 
a steady electric current may be obtained in air between 
electrodes whose difference of potential is but small, provided 
that an independent current is maintained in the same 
vessel ; that is to say, a continuous discharge produces in 
the air such a condition that conduction occurs with the 
smallest electromotive forces. This effect he explained by 
aid of the hypothesis previously advanced ; the ions produced 
by the main discharge become diffused throughout the vessel, 
and, coming under the influence of the field set up by the 
auxiliary electrodes, drift so as to carry a current between 
the latter. 

A discovery related to this was made in the same year by 
Hertz,f in the course of the celebrated researches? which have 
been already mentioned. Happening to notice that the passage 
of one spark is facilitated by the passage of another spark in 
its neighbourhood, he followed up the observation, and found 
the phenomenon to be due to the agency of ultra-violet light 
emitted by the latter spark. It appeared in fact that the 
distance across which an electric spark can pass in air is 
greatly increased when light of very short wave-length is 
allowed to fall on the spark-gap. It was soon found that the 
effective light is that which falls on the negative electrode 
of the gap ; and Wilhelm Hallwachs|| extended the discovery 

* Proc. Roy. Soc. xlii (1887), p. 371. Hittorf had discovered that very small 
electromotive forces are sufficient to cause a discharge across a space through which 
the cathode radiation is passing. 

t Berlin Ber., 1887, p. 487 ; Ann. d. Phys. xxxi (1887), p. 983 ; Electric Waves 
(English ed.), p. 63. 

I Cf.p. 357. 

By E. Wiedemann and Ebert, Ann. d. Phys. xxxiii (1888), p. 241. 

|| Ann. d. Phys. xxxiii (1888), p. 301. 



400 Conduction in Solutions and Gases, 

by showing that when a sheet of metal is negatively electrified 
and exposed to ultra-violet light, the adjacent air is thrown 
into a state which permits the charge to leak rapidly away. 

Interest was now thoroughly aroused in the problem of 
conductivity in gases ; and it was generally felt that the best 
hope of divining the nature of the process lay in studying the 
discharge at high rarefactions. " If a first step towards under- 
standing the relations between aether and ponderable matter is 
to be made," said Lord Kelvin in 1893,* " it seems to me that 
the most hopeful foundation for it is knowledge derived from 
experiments on electricity in high vacuum." 

Within the two following years considerable progress was 
effected in this direction. J. J. Thomson,^ by a rotating-mirror 
method, succeeded in measuring the velocity of the cathode rays, 
finding it to be| 1*9 x 10 7 cm./sec. ; a value so much smaller than 
that of the velocity of light that it was scarcely possible to 
conceive of the rays as vibrations of the aether. A further 
blow was dealt at the latter hypothesis when Jean Perrin, 
having received the rays in a metallic cylinder, found that 
the cylinder became charged with resinous electricity. When 
the rays were deviated by a magnet in such a way that they 
could no longer enter the cylinder, it no longer acquired a 
charge. This appeared to demonstrate that the rays transport 
negative electricity. 

With cathode rays is closely connected another type 
of radiation, which was discovered in December, 1895, by 
W. C. K6ntgen.ll The discovery seems to have originated 
in an accident : a photographic plate which, protected in the 
usual way, had been kept in a room in which vacuum-tube 
experiments were carried on, was found on development to show 
distinct markings. Experiments suggested by this showed 

* Proc. Roy. Soc. liv (1893), p. 389. 
t Phil. Mag. xxxviii (1894), p. 358. 

The value found by the same investigator in 1897 was much larger than this. 
$ Cornptes Rendus, cxxi (1895), p. 1130. 

|| Sitzungsber. der Wiirzburger Physikal. -Medic. Gesellschaft, 1895 ; reprinted, 
Ann. d. Phys. Ixiv (1898), pp. 1, 12; translated, Nature, liii (1896), p. 274. 



Jrom Faraday to J. J . Thomson. 401 

that radiation, capable of affecting sensitive plates and of 
causing fluorescence in certain substances, is emitted by tubes 
in which the electric discharge is passing; and that the radia- 
tion proceeds from the place where the cathode rays strike 
the glass walls of the tube. The X-rays, as they were called 
by their discoverer, are propagated in straight lines, and can 
neither be refracted by any of the substances which refract 
light, nor deviated from -their course by a magnetic field ; 
they are moreover able to pass with little absorption through 
many substances which are opaque to ordinary and ultra-violet 
light a property of which considerable use has been made 
in surgery. 

The nature of the new radiation was the subject of much 
speculation. Its discoverer suggested that it might prove to 
represent the long-sought-for longitudinal vibrations of the 
aether ; while other writers advocated the rival claims of 
aethereal vortices, infra-red light, and "sifted" cathode rays. 
The hypothesis which subsequently obtained general acceptance 
was first propounded by Schuster* in the month following the 
publication of Kontgen's researches. It is, that the X-rays are 
transverse vibrations of the aether, of exceedingly small wave- 
length. A suggestion which was put forward later in the year 
by E. Wiechertf and Sir George StokesJ to the effect that the 
rays are pulses generated in the aether when the glass of the 
discharge tube is bombarded by the cathode particles, is not 
really distinct from Schuster's hypothesis ; for ordinary white 
light likewise consists of pulses, as Gouy had shown, and the 
essential feature which distinguishes the Eontgen pulses is that 
the harmonic vibrations into which they can be resolved by 
Fourier's analysis are of very short period. 



* Nature, January 23, 1896, p. 268. Fitz Gerald independently made the same 
suggestion in a letter to O. J. Lodge, printed in the Electrician xxxvii, p. 372. 

t Ann. d. Phys. lix (1896), p. 321. 

I Xature, September 3, 1896, p. 427 : Proc. Canib. Phil. Soc. ix (1896), p. 215 ; 
Mem. Manchester Lit. & Phil. Soc. xli (1896-7). 

$ Journ. de Phys. v (1886), p. 354. 

2 D 



402 Conduction in Solutions and Gases, 

The rapidity of the vibrations explains the failure of all 
attempts to refract the X-rays. For in the formula 

= !- "** 



of the Maxwell-Sellmeier theory,* n denotes the frequency, and 
so is in this case extremely large ; whence we have 

/*' = !, 

i.e., the refractive index of all substances for the X-rays is unity. 
In fact, the vibrations alternate too rapidly to have an effect 
on the sluggish systems which are concerned in refraction. 
Some years afterwards H. Haga and C. H. Wind,f having 
measured the diffraction-patterns produced by X-rays, concluded 
that the wave-length of the vibrations concerned was of the 

o 

order of one Angstrom unit, that is about 1/6000 of the wave- 
length of the yellow light of sodium. 

One of the most important properties of X-rays was 
discovered, shortly after the rays themselves had become known, 
by J. J. Thomson,]: who announced that when they pass through 
any substance, whether solid, liquid, or gaseous, they render it 
conducting. This he attributed, in accordance with the ionic 
theory of conduction, to " a kind of electrolysis, the molecule of 
the non-conductor being split up, or nearly split up, by the 
Kb'ntgen rays." 

The conductivity produced in gases by this means was at 
once investigated! more closely. It was found that a gas which 
had acquired conducting power by exposure to X-rays lost this 
quality when forced through a plug of glass-wool; whence 
it was inferred that the structure in virtue of which the 
gas conducts is of so coarse a character that it is unable to 
survive the passage through the fine pores of the plug. The 

* Cf. p. 293. 

t Proceedings of the Amsterdam Acad., March 25th, 1899 (English edition, i, 
p. 420), and September 27th, 1902 (English edition, v, p. 247). 
% Nature, February 27, 1896, p. 391. 
J. J. Thomson and E. Rutherford, Phil. Mag. xlii (1896), p. 392. 



from Faraday to J. J. Thomson. 403 

conductivity was also found to be destroyed when an electric 
current was passed through the gas a phenomenon for which 
a parallel may be found in electrolysis. For if the ions were 
removed from an electrolytic solution by the passage of a 
current, the solution would cease to conduct as soon as 
sufficient electricity had passed to remove them all ; and it 
may be supposed that the conducting agents which are produced 
in a gas by exposure to X-rays are likewise abstracted from it 
when they are employed to transport charges. 

The same idea may be applied to explain another property 
of gases exposed to X-rays. The strength of the current 
through the gas depends both on the intensity of the radiation 
and also on the electromotive force ; but if the former factor be 
constant, and the electromotive force be increased, the current 
does not increase indefinitely, but tends to attain a certain 
" saturation " value. The existence of this saturation value is 
evidently due to the inability of the electromotive force to do 
more than to remove the ions as fast as they are produced by 
the rays. 

Meanwhile other evidence was accumulating to show that 
the conductivity produced in gases by X-rays is of the same 
nature as the conductivity of the gases from flames and from 
the path of a discharge, to which the theory of Giese and 
Schuster had already been applied. One proof of this identity 
was supplied by observations of the condensation of water- 
vapour into clouds. It had been noticed long before by 
John Aitken* that gases rising from flames cause precipita- 
tion of the aqueous vapour from a saturated gas; and 
E. von Helmholtzf had found that gases through which an 
electric discharge has been passed possess the same property. 
It was now shown by C. T. E. Wilson, % working in the 
Cavendish Laboratory at Cambridge, that the same is true of 
gases which have been exposed to X-rays. The explanation 

* Trans. R. S. Edinb. xxx (1880), p. 337. 

t Ann. d. Phys. xxxii (1887), p. 1. 

% Proc. Roy. Soc., March 19, 1896 ; Phil. Trans., 1897, p. 265. 

2 T> 2 



404 Conduction in Solutions and Gases, 

furnished by the ionic theory is that in all three cases the gas 
contains ions which act as centres of condensation for the 
vapour. 

During the year which followed their discovery, the X-rays 
were so thoroughly examined that at the end of that period 
they were almost better understood than the cathode rays 
from which they derived their origin. But the obscurity in 
which this subject had been so long involved was now to be 
dispelled. 

Lecturing at the Eoyal Institution on April 30th, 1897, 
J. J. Thomson advanced a new suggestion to reconcile the 
molecular- torrent hypothesis with Lenard's observations of the 
passage of cathode rays through material bodies. " We see 
from Lenard's table," he said, " that a cathode ray can travel 
through air at atmospheric pressure a distance of about half a 
centimetre before the brightness of the phosphorescence falls to 
about half its original value. Now the mean free path of 
the molecule of air at this pressure is about 10~ 5 cm., and if a 
molecule of air were projected it would lose half its momentum 
in a space comparable with the mean free path. Even if we 
suppose that it is not the same molecule that is carried, the 
effect of the obliquity of the collisions would reduce the 
momentum to half in a short multiple of that path. 

" Thus, from Lenard's experiments on the absorption of the 
rays outside the tube, it follows on the hypothesis that the 
cathode rays are charged particles moving with high velocities 
that the size of the carriers must be small compared with the 
dimensions of ordinary atoms or molecules.* The assumption 
of a state of matter more finely subdivided than the atom of 
an element is a somewhat startling one ; but a hypothesis that 
would involve somewhat similar consequences viz. that the 
so-called elements are compounds of some primordial element 
has been put forward from time to time by various 
chemists." 

* A similar suggestion was made by E. Wiechert, Verhandl. d. physik.-ocon. 
Gesellscb. in Konigsberg, Jan. 1897. 



from Faraday tcr J. J . Thomson. 405 

Thomson's lecture drew from Fitz Gerald* the suggestion 
that " we are dealing with free electrons in these cathode rays " 
a remark the point of which will become more evident when 
we come to consider the direction in which the Maxwellian 
theory was being developed at this time. 

Shortly afterwards Thomson himself published an accountf of 
experiments in which the only outstanding objections to the 
charged-particle theory were removed. The chief of these was 
Hertz' failure to deflect the cathode rays by an electrostatic 
field. Hertz had caused the rays to travel between parallel 
plates of metal maintained at different potentials ; but Thomson 
now showed that in these circumstances the rays generate 
ions in the rarefied gas, which settle on the plates, and annul 
the electric force in the intervening space. By carrying the 
exhaustion to a much higher degree, he removed this source of 
confusion, and obtained the expected deflexion of the rays. 

The electrostatic and magnetic deflexions taken together 
suffice to determine the ratio of the mass of a cathode particle 
to the charge which it carries. For the equation of motion of 
the particle is 

rar = eE .+ e[v. H], 

where r denotes the vector from the origin to the position of 
the particle ; E and H denote the electric and magnetic forces ; 
e the charge, m the mass, and v the velocity of the particle. 
By observing the circumstances in which the force #E, due to 
the electric field, exactly balances the force e [v . H], due to the 
magnetic field, it is possible to determine v ; and it is readily 
seen from the above equation that a measurement of the 
deflexion in the magnetic field supplies a relation between v 
and m/e ; so both v and m/e may be determined. Thomson 
found the value of m/e to be independent of the nature of the 
rarefied gas : its amount was 10~ 7 (grammes/electromagnetic units 
of charge), which is only about the thousandth part of the value 
of m/e for the hydrogen atom in electrolysis. If the charge 

* Electrician, May 21, 1897. t Phil. Mag. xliv (1897), p. 298. 



406 Conduction in Solutions and Gases, 

were supposed to be of the same order of magnitude as that on 
an electrolytic ion, it would be necessary to conclude that the 
particle whose mass was thus measured is much smaller than 
the atom, and the conjecture might be entertained that it is the 
primordial unit or corpuscle of which all atoms are ultimately 
composed.* 

The nature of the resinously charged corpuscles which 
constitute cathode rays being thus far determined, it became of 
interest to inquire whether corresponding bodies existed carrying 
charges of vitreous electricity a question to which at any rate 
a provisional answer was given by W. Wienf of Aachen in the 
same year. More than a decade previously E. Goldstein^ had 
shown that when the cathode of a discharge-tube is perforated, 
radiation of a certain type passes outward through the per- 
forations into the part of the tube behind the cathode. To 
this radiation he had given the name canal rays. Wien now 
showed that the canal rays are formed of positively charged 
particles, obtaining a value of m/e immensely larger than 
Thomson had obtained for the cathode rays, and indeed of 
the same order of magnitude as the corresponding ratio in 
electrolysis. 

The disparity thus revealed between the corpuscles of 
cathode rays and the positive ions of Goldstein's rays excited 
great interest ; it seemed to offer a prospect of explaining the 
curious differences between the relations of vitreous and of 
resinous electricity to ponderable matter. These phenomena 
had been studied by many previous investigators ; in particular 
Schuster, in the Bakerian lecture of 1890, had remarked that 
" if the law of impact is different between the molecules of the 
gas and the positive and negative ions respectively, it follows 
that the rate of diffusion of the two sets of ions will in general 
be different," and had inferred from his theory of the discharge 

* The value of m/e for cathode rays was determined also in the same year by 
W. Kaufmaim, Ann. d. Phys. Ixi, p. 544. 

t Verh;ndl. der physik. Gesells. zu Berlin, xvi (1897), p. 165; Ann. d. Phys. 
Ixv (1898), {>. 440. 

J Berlin Sitzungsber., 1886, p. 691. Proc. R.S. xlvii (1890), p. 526. 



Jrom Faraday to J . J . Thomson. 407 

that " the negative ions diffuse more rapidly." This inference 
was confirmed in 1898 by John Zeleny,* who showed that of 
the ions produced in air by exposure to X-rays, the positive 
are decidedly less mobile than the negative. 

The magnitude of the electric charge on the ions of gases 
was not known with certainty until 1898, when a plan for 
determining it was successfully executed by J. J. Thomson.f 
The principles on which this celebrated investigation was based 
are very ingenious. By measuring the current in a gas which 
is exposed to Rontgen rays and subjected to a known electro- 
motive force, it is possible to determine the value of the product 
nev, where n denotes the number of ions in unit volume of the 
gas, e the charge on an ion, and v the mean velocity of the 
positive and negative ions under the electromotive force. As 
v had been already determined ,J the experiment led to a 
determination of ne ; so if n could be found, the value of e 
might be deduced. 

The method employed by Thomson to determine n was 
founded on the discovery, to which we have already referred, 
that when X-rays pass through dust-free air, saturated with 
aqueous vapour, the ions act as nuclei around which the water 
condenses, so that a cloud is produced by such a degree of 
saturation as would ordinarily be incapable of producing con- 
densation. The size of the drops was calculated from measure- 
ments of the rate at which the cloud sank ; and, by comparing 
this estimate with the measurement of the mass of water 
deposited, the number of drops was determined, and hence the 
number n of ions. The value of e consequently deduced was 
found to be independent of the nature of the gas in which the 
ions were produced, being approximately the same in hydrogen 
as in air, and being apparently in both cases the same as for 
the charge carried by the hydrogen ion in electrolysis. 

Since the publication of Thomson's papers his general 
conclusions regarding the magnitudes of e and m/e for gaseous 

* Phil. Mag. xlvi (1898;, p. 120. t Phil. Mag. xlvi (1898), p. 528. 

+ By E. Rutherford, Phil. Mag. xliv (1897), p. 422. 



408 Conduction in Solutions and Gases, 

ions have been abundantly confirmed. It appears certain that 
electric charge exists in discrete units, vitreous and resinous, 
each of magnitude 1*5 x 10~ 19 coulombs approximately. Each 
ion, whether in an electrolytic liquid or in a gas, carries one 
(or an integral number) of these charges. An electrolytic ion 
also contains one or more atoms of matter; and a positive 
gaseous ion has a mass of the same order of magnitude as that 
of an atom of matter. But it is possible in many ways to 
produce in a gas negative ions which are not attached to atoms 
of matter ; for these the inertia is only about one- thousandth 
of the inertia of an atom; and there is reason for believing 
that even this apparent mass is in its origin purely electrical.* 

The closing years of the nineteenth century saw the founda- 
tion of another branch of experimental science which is closely 
related to the study of conduction in gases. When Rontgen 
announced his discovery of the X-rays, and described their 
power of exciting phosphorescence, a number of other workers 
commenced to investigate this property more completely. In 
particular, Henri Becquerel resolved to examine the radiations 
which are emitted by the phosphorescent double sulphate of 
uranium and potassium after exposure to the sun. The result 
was communicated to the French Academy on February 24th, 
1896.f " Let a photographic plate," he said, " be wrapped in 
two sheets of very thick black paper, such that the plate is not 
affected by exposure to the sun for a day. Outside the paper 
place a quantity of the phosphorescent substance, and expose 
the whole to the sun for several hours. When the plate is 
developed, it displays a silhouette of the phosphorescent 
substance. So the latter must emit radiations which are 
capable of passing through paper opaque to ordinary light, and 
of reducing salts of silver." 

At this time Becquerel supposed the radiation to have been 
excited by the exposure of the phosphorescent substance to the 
sun ; but a week later he announced^ that it persisted for an 

* Cf. p. 343. f Comptes Rendus, cxxii (1890), p. 420. 

I Ibid., cxxii (March 2nd, 1896), p. 501. 






from Faraday to J'. J. Thomson. 409 

indefinite time after the substance had been removed from the 
sunlight, and after the luminosity which properly constitutes 
phosphorescence had died away ; and he was thus led to con- 
clude that the activity was spontaneous and permanent. It 
was soon found that those salts of uranium which do not 
phosphoresce e.g., the uranous salts, and the metal itself, all 
emit the rays ; and it became evident that what Becquerel had 
discovered was a radically new physical property, possessed by 
the element uranium in all its chemical compounds. 

Attempts were now made to trace this activity in other 
substances. In 1898 it was recognized in thorium and its 
compounds;* and in the same year P. Curie and Madame 
Sklodowska Curie announced to the French Academy the 
separation from the mineral pitchblende of two new highly 
active elements, to which they gave the names of poloniumf and 
radium.}: A host of workers was soon engaged in studying the 
properties of the Becquerel rays. The discoverer himself had 
shown in 1896 that these rays, like the X- and cathode rays, 
impart conductivity to gases. It was found in 1899 by 
Kutherfordll that the rays from uranium are not all of the same 
kind, biit that at least two distinct types are present ; one of 
these, to which he gave the name a-rays, is readily absorbed ; 
while another, which he named /3-radiation, has a greater 
penetrating power. It was then shown by Giesel, Becquerel, and 
others, that part of the radiation is deflected by a magnetic field,1T 
and part is not.** After this Monsieur and Madame Curieft 
found that the deviable rays carry negative electric charges, 

* By Schmidt, Ann. d. Phys., Ixv (1898), p. 141 ; and by Ma.iame Curie, 
Comptes Rendus, cxxvi (1898), p. 1101. 

t Comptes Rendus, cxxvii (1898), p. 175. % Ibid., cxxvii (1898), p. 1215. 

Ibid., cxxii (1896), p. 559. || Phil. Mag. (5), xlvii (1899), p. 109. 

H Giesel, Ann. d. Phys. Ixix (1899), p. 834 (working with polonium); 
Becquerel, Comptes Rendus, cxxix (1899), p. 996 (working with radium) ; 
Meyer andv. Schweidler, Phys. Zeitschr. i (1899), p. 113 (working with polonium 
and radium). 

** Bc-cquerel, Comptes Rendus, cxxix (1889), p. 1205); cx\x (1900), pp. 206, 
372. Curie, ibid, exxx (1900), p. 73. 

ft Comptes Rendus, cxxx (1900), p. 647. 



410 Conduction in Solutions and Gases. 

and Becquerel* succeeded in deviating them by an electrostatic 
field. The deviable or j3- rays were thus clearly of the same 
nature as cathode rays ; and when measurements of the electric 
and magnetic deviations gave for the ratio m/e a value of the 
order 10~ 7 , the identity of the /3-particles with the cathode-ray 
corpuscles was fully established. 

The subsequent history of the new branch of physics thus 
created falls outside the limits of the present work. We must 
now consider the progress which was achieved in the general 
theory of aether and electricity in the last decade of the 
nineteenth century. 

* Comptes Eendus, c>xx (1900), p. 809. 



CHAPTEE XII. 

THE THEORY OF AETHER AND ELECTRONS IN THE CLOSING 
YEARS OF THE NINETEENTH CENTURY. 

THE attempts of Maxwell* and of Hertzf to extend the theory 
of the electromagnetic field to the case in which ponderable 
bodies are in motion had not been altogether successful. 
Neither writer had taken account of any motion of the material 
particles relative to the aether entangled with them, so that in 
both investigations the moving bodies were regarded simply 
as homogeneous portions of the medium which fills all space, 
distinguished only by special values of the electric and 
magnetic constants. Such an assumption is evidently incon- 
sistent with the admirable theory by which FresnelJ had 
explained the optical behaviour of moving transparent bodies ; 
it was therefore not surprising that writers subsequent to Hertz 
should have proposed to replace his equations by others 
designed to agree with Fresnel's formulae. Before discussing 
these, however, it may be well to review briefly the evidence 
for and against the motion of the aether in and adjacent 
to moving ponderable bodies, as it appeared in the last decade 
of the nineteenth century. 

The phenomena of aberration had been explained by Young 
on the assumption that the aether around bodies is unaffected 
by their motion. But it was shown by Stokes|| in 1845 that 
this is not the only possible explanation. For suppose that 
the motion of the earth communicates motion to the neighbour- 
ing portions of the aether ; this may be regarded as superposed 
on the vibratory motion which the aethereal particles have 

* Cf p. 288. t Cf. p. 365. I Cf. p. 116. $ Cf. p. 115. 

|| Phil. Mag. xxvii (1845), p. 9 ; xxviii (1846), p. 76; xxix (1846), p. 6. 



-412 The Theory of Aether and Electrons in the 

when transmitting light : the orientation of the wave-fronts of 
the light will consequently in general be altered ; and the direc- 
tion in which a heavenly body is seen, being normal to the wave- 
fronts will thereby be affected. But if the aethereal motion 
is irrotational, so that the elements of the aether do not 
rotate, it is easily seen that the direction of propagation of the 
light in space is unaffected ; the luminous disturbance is still 
propagated in straight lines from the star, while the normal 
to the wave-front at any point deviates from this line of 
propagation by the small angle ujc, where u denotes the 
component of the aethereal velocity at the point, resolved at 
right angles to the line of propagation, and c denotes the 
velocity of light. If it be supposed that the aether near the 
earth is at rest relatively to the earth's surface, the star will 
appear to be displaced towards the direction in which the 
earth is moving, through an angle measured by the ratio of 
the velocity of the earth to the velocity of light, multiplied by 
the sine of the angle between the direction of the earth's 
motion and the line joining the earth and star. This is 
precisely the law of aberration. 

An objection to Stokes's theory has been pointed out by 
several writers, amongst others by H. A. Lorentz.* This is, 
that the irrotational motion of an incompressible fluid is 
completely determinate when the normal component of the 
velocity at its boundary is given : so that if the aether were 
supposed to have the same normal component of velocity as 
the earth, it would not have the same tangential component of 
velocity. It follows that no motion will in general exist which 
satisfies Stokes's conditions ; and the difficulty is not solved in 
any very satisfactory fashion by either of the suggestions which, 
have been proposed to meet it. One of these is to suppose that 
the moving earth does generate a rotational disturbance, which, 
however, being radiated away with the velocity of light, does not 
affect the steadier irrotational motion ; the other, which was 

* Archives Neerl, xxi (1896), p. 103. 



Closing Years of the Nineteenth Century. 413- 

advanced by Planck,* is that the two conditions of Stokes's 
theory namely, that the motion of the aether is to be 
irrotational and that at the earth's surface its velocity is to be 
the same as that of the earth may both be satisfied if the 
aether is supposed to be compressible in accordance with 
Boyle's law, and subject to gravity, so that round the earth it 
is compressed like the atmosphere ; the velocity of light being 
supposed independent of the condensation of the aether. 

Lorentz,f in calling attention to the defects of Stokes's 
theory, proposed to combine the ideas of Stokes and Fresnel, by 
assuming that the aether near the earth is moving irrotationally 
(as in Stokes's theory), but that at the surface of the earth the 
aethereal velocity is not necessarily the same as that of ponder- 
able matter, and that (as in Fresiiel's theory) a material body 
imparts the fraction (ju 2 - l}/ju 2 of its own motion to the aether 
within it. Fresnel's theory is a particular case of this new 
theory, being derived from it by supposing the velocity -potential 
to be zero. 

Aberration is by no means the only astronomical phenomenon 
which depends on the velocity of propagation of light ; we have 
indeed seent that this velocity was originally determined by 
observing the retardation of the eclipses of Jupiter's satellites. 
It was remarked by Maxwell in 1879 that these eclipses 
furnish, theoretically at least, a means of determining the 
velocity of the solar system relative to the aether. For if the 
distance from the eclipsed satellite to the earth be divided by 
the observed retardation in time of the eclipse, the quotient 
represents the velocity of propagation of light in this direction, 
relative to the solar system; and this will differ from the velocity 
of propagation of light relative to the aether by the component, 
in this direction, of the sun's velocity relative to the aether. 
By taking observations when Jupiter is in different signs of the 

* Of. Lorentz, Proc. Amsterdam Acad. (English ed.), i (1899), p. 443. 
t Archives Neerl. xxi (1886), p. 103 : cf. also Zittinsgsversl. Kon. Ak. Amster- 
dam, 1897-98, p. 266. 

J Cf. p. 22. Proc. R. S. xxx (1880), p. 108. 



414 The Theory of Aether and Electrons in the 

zodiac, it should therefore be possible to determine the sun's 
velocity relative to the aether, or at least that component of it 
which lies in the ecliptic. 

The same principles may be applied to the discussion of 
other astronomical phenomena. Thus the minimum of a 
variable star of the Algol type will be retarded or accelerated 
by an interval of time which is found by dividing the projection 
of the radius from the sun to the earth on the direction from 
the sun to the Algol variable by the velocity, relative to the 
solar system, of propagation of light from the variable ; and thus 
the latter quantity may be deduced from observations of the 
retardation.* 

Another instance in which the time taken by light to cross 
an orbit influences an observable quantity is afforded by the 
astronomy of double stars. Savaryf long ago remarked that 
when the plane of the orbit of a double star is not at right 
angles to the line of sight, an inequality in the apparent motion 
must be caused by the circumstance that the light from the 
remoter star has the longer journey to make. Yvon VillarceauJ 
showed that the effect might be represented by a constant 
alteration of the elliptic elements of the orbit (which alteration 
is of course beyond detection), together with a periodic 
inequality, which may be completely specified by the following 
statement : the apparent coordinates of one star relative to the 
other have the values which in the absence of this effect they 
would have at an earlier or later instant, differing from the 
actual time by the amount 

m, - >)i z z 
m } + m z ' c' 

where m l and m 2 denote the masses of the stars, c the velocity 
of light, and z the actual distance of the two stars from each 

* The velocity of light was found from observations of Algol, by C. V. L. 
Charlier, Of versigt af K. Vet.-Ak. Forhandl. xivi (1889), p. 523. 

t Conn, des Temps, 1830. 

J Additions a la Connaissance des Temps, 1878 : an improved deduction was 
given by H. Seeliger, Sitzungsberichte d. K. Ak. zu Miinchen, xix (1889), p. 19. 



Closing Years of the Nineteenth Century. 415 

other at the time when the light was emitted, resolved along 
the line of sight. In the existing state of double-star astronomy, 
this effect would be masked by errors of observation. 

Villarceau also examined the consequences of supposing 
that the velocity of light depends on the velocity of the source 
by which it is emitted. If, for instance, the velocity of light 
from a star occulted by the moon were less than the velocity of 
light reflected by the moon, then the apparent position of the 
lunar disk would be more advanced in its movement than that 
of the star, so that at emersion the star would first appear at 
some distance outside the lunar disk, and at immersion the star 
would be projected on the interior of the disk at the instant of 
its disappearance. The amount by which the image of the star 
could encroach on that of the disk on this account could not be 
so much as 0"'71 ; encroachment to the extent of more than 
1" has been observed, but is evidently to be attributed for the 
most part to other causes. 

Among the consequences of the finite velocity of propagation 
of light which are of importance in astronomy, a leading place 
must be assigned to the principle enunciated in 1842 by Christian 
Doppler,* that the motion of a source of light relative to an 
observer modifies the period of the disturbance which is 
received by him. The phenomenon resembles the depression 
of the pitch of a note when the source of sound is receding from 
the observer. In either case, the period of the vibrations 
perceived by the observer is (c + v) / c x the natural period, 
where v denotes the velocity of separation of the source and 
observer, and c denotes the velocity of propagation of the 
disturbance. If, e.g., the velocity of separation is equal to 
the orbital velocity of the earth, the D lines of sodium in the 
spectrum of the source will be displaced towards the red, as 
compared with lines derived from a terrestrial sodium flame, bs 
about one-tenth of the distance between them. The application 
of this principle to the determination of the relative velocity of 

* Abhandl. der K. Hohm. Ges. der Wissensch. (5) ii (1842), p. 465. 



416 The Theory of Aether and Electrons in the 

stars in the line of sight, which has proved of great service in 
astrophysical research, was suggested by Fizeau in 1848.* 

Passing now from the astronomical observatory, we must 
examine the information which has been gained in the physical 
laboratory regarding the effect of the earth's motion on optical 
phenomena. We have alreadyf referred to the investigations 
by which the truth of Fresnel's formula was tested. An 
experiment of a different type was suggested in 1852 by 
FizeauJ who remarked that, unless the aether is carried along 
by the earth, the radiation emitted by a terrestrial source should 
have different intensities in different directions. It was, how- 
ever, shown long afterwards by Lorentz that such an experiment 
would not be expected on theoretical grounds to yield a positive 
result ; the amount of radiant energy imparted to an absorbing 
body is independent of the earth's motion. A few years later 
Fizeau investigatedll another possible effect. If a beam of 
polarized light is sent obliquely through a glass plate, the 
azimuth of polarization is altered to an extent which depends, 
amongst other things, on the refractive index of the glass. 
Fizeau performed this experiment with sunlight, the light 
being sent through the glass in the direction of the terrestrial 
motion, and in the opposite direction ; the readings seemed to 
differ in the two cases, but on account of experimental difficulties 
the result was indecisive. 

Some years later, the effect of the earth's motion on the 
rotation of the plane of polarization of light propagated along 
the axis of a quartz crystal was investigated by Mascart.^f The 
result was negative, Mascart stating that the rotation could 
not have been altered by more than the (l/40,000)th part when 
the orientation of the apparatus was reversed from that of 

* An apparatus for demonstrating the Doppler-Fizeau effect in the laboratory 
was constructed by Belopolsky, Astrophys. Journal xiii (1901), p. 15. 

t Of. pp. 117-120. + Ann. d. Phys. xcii (1854), p. 652. 

\ Proc. Amsterdam Acad. (English edition), iv (1902) p. 678. 

|| Annales de Chim. (3) Ixviii (1860), p. 129; Ann. d. Phys. cxiv (1861), 
p. 554. 

H Annales de 1'Ec. Norm. (2) i (1872), p. 157. 



Closing Years of the Nineteenth Century. 417 

the terrestrial motion to the opposite direction. This was 
afterwards confirmed by Lord Kayleigh,* who found that the 
alteration, if it existed, could not amount to (l/100,000)th 
part. 

In terrestrial methods of determining the velocity of light 
the ray is made to retrace its path, so that any velocity which 
the earth might possess with respect to the luminiferous medium 
would affect the time of the double passage only by an amount 
proportional to the square of the constant of aberration.f In 
1881, however, A. A. MichelsonJ remarked that the effect, 
though of the second order, should be manifested by a measur- 
able difference between the times for rays describing equal 
paths parallel and perpendicular respectively to the direction of 
the earth's motion. He produced interference-fringes between 
two pencils of light which had traversed paths perpendicular 
to each other ; but when the apparatus was rotated through a 
right angle, so that the difference would be reversed, the expected 
displacement of the fringes could not be perceived. This result 
was regarded by Michelson himself as a vindication of Stokes's 
theory^ in which the aether in the neighbourhood of the 
dearth is supposed to be set in motion. Lorentzj), however, 
showed that the quantity to be measured had only half the 
value supposed by Michelson, and suggested that the negative 
result of the experiment might be explained by that combina- 
tion of Fresnel's and Stokes's theories which was developed in 
his own memoirIF ; since, if the velocity of the aether near the 
earth were (say) half the earth's velocity, the displacement of 
Michelson's fringes would be insensible. 

* Phil. Mag. iv. (1902), p. 215. 

t The constant of aberration is the ratio of the earth's orbital velocity to the 
velocity of light ; cf. supra, p. 100. 

Amer. Journ. Sci. xxii (1881), p. 20. His method was afterwards improved : 
cf. Michelson and Morley, Amer. Journ. Sci. xxxiv (1887), p. 333; Phil. Mag. 
xxiv (1887), p. 449. 

Cf. p. 411. 

|| Arch. Xeerl. xxi (1886), p. 103. On the Micbelson-Morley experiment cf. 
also Hicks, Phil. Mag. iii (1902), p. 9. 

U Cf. p. 413. 

2 E 



418 The Theory of Aether and Electrons in the 

A sequel to the experiment of Michelson and Morley was 
performed in 1897, when Michelson* attempted to determine 
by experiment whether the relative motion of earth and aether 
varies with the vertical height above the terrestrial surface. 
No result, however, could be obtained to indicate that the 
velocity of light depends on the distance from the centre of 
the earth ; and Michelson concluded that if there were no choice 
<"but between the theories of Fresnel and Stokes, it would be 
necessary to adopt the latter, and to suppose that the earth's 
influence on the aether exends to many thousand kilometres 
above its surface. By this time, however, as will subsequently 
appear, a different explanation was at hand. 

Meanwhile the perplexity of the subject was increased by 
experimental results which pointed in the opposite direction 
to that of Michelson. In 1892 Sir Oliver Lodgef observed the 
interference between the two portions of a bifurcated beam of 
light, which were made to travel in opposite directions round 
a closed path in the space * between two rapidly rotating steel 
disks. The observations showed that the velocity of light is 
not affected by the motion of adjacent matter to the extent of 
(l/200)th part of the velocity of the matter. Continuing his 
investigations, Lodge} strongly magnetized the moving matter 
(iron in this experiment), so that the light was propagated 
across a moving magnetic field ; and electrified it so that the 
path of the beams lay in a moving electrostatic field ; but in 
no case was the velocity of the light appreciably affected. 

We must now trace the steps by which theoretical physicists 
not only arrived at a solution of the apparent contradictions 
furnished by experiments with moving bodies, but so extended 
the domain of electrical science that it became necessary to 
enlarge the boundaries of space and time to contain it. 

The first memoir in which the new conceptions were 
unfolded-j was published by H. A. Lorentzg in 1892. The 

* Amer. Journ. Sci. (4) iii (1897), p. 475. 

t Phil. Trans, clxxxiv (1893), p. 727. J Ibid., clxxxix (1897), p. 149. 

Archives Neerl. xxv (1892), p. 363 : the theory is given in eh. iv, pp. 432 
et sqq. 



Closing Years of the Nineteenth Century. 419 

theory of Lorentz was, like those of Weber, Kiemann, and 
Clausius,* a theory of electrons ; that is to say, all electro- 
dynamical phenomena were ascribed to the agency of moving 
electric charges, which were supposed in a magnetic field to 
experience forces proportional to their velocities, and to com- 
municate these forces to the ponderable matter with which 
they might be associated.t 

In spite of the fact that the earlier theories of electrons 
had failed to fulfil the expectations of their authors, the 
assumption that all electric and magnetic phenomena are due 
to the presence or motion of individual electric charges was 
one to which physicists were at this time disposed to give a 
favourable consideration ; for, as we have seen,* evidence of 
the atomic nature of electricity was now contributed by the 
study of the conduction of electricity through liquids and gases. 
Moreover, the discoveries of Hertz had shown that a molecule 
which is emitting light must contain some system resembling 
a Hertzian vibrator; and the essential process in a Hertzian 
vibrator is the oscillation of electricity to and fro. Lorentz 
himself from the outset of his career! | had supposed the inter- 
action of ponderable matter with the electric field to be effected 
by the agency of electric charges associated with the material 
atoms. 

The principal difference by which the theory now advanced 
by Lorentz is distinguished from the theories of Weber, 

* Cf. pp. 226, 231, 262. 

+ Some writers have inclined to use the term ' electron-theory ' as if it were 
specially connected with Sir Joseph Thomson's justly celebrated discovery (cf . p. 407, 
supra) that all negative electrons have equal charges. But Thomson's discovery, 
though undoubtedly of the greatest importance as a guide to the structure of the 
universe, has hitherto exercised hut little influence on general electromagnetic 
theory. The reason for this is that in theoretical investigations it is customary 
to denote the changes of electrons by symbols, e\, e-z, . . . ; and the equality or 
non-equality of these makes no difference to the equations. To take an illustration 
from Celestial Mechanics, it would clearly make no difference in the general 
equations of the planetary theory if the masses of the planets happened to be 
all equal. 

* Cf. chapter xi. 

Cf. pp. 357-363. 

|| Verb. d. Ak. v. Wetenschappen, Amsterdam, Deel xviii (1878). 

2 E 2 



420 The Theory of Aether and Electrons in the 

Kiemann, and Clausing, and from Lorentz' own earlier work, 
lies in the conception which is entertained of the propagation 
of influence from one electron to another. In the older writ- 
ings, the electrons were assumed to be capable of acting on 
each other at a distance, with forces depending on their 
charges, mutual distances, and velocities ; in the present 
memoir, on the other hand, the electrons were supposed to 
interact not directly with each other, but with the medium in 
which they were embedded. To this medium were ascribed the 
properties characteristic of the aether in Maxwell's theory. 

The only respect in which Lorentz' medium differed from 
Maxwell's was in regard to the effects of the motion of bodies. 
Impressed by the success of Fresnel's beautiful theory of 
the propagation of light in moving transparent substances,* 
Lorentz designed his equations so as to accord with that 
theory, and showed that this might be done by drawing a 
distinction between matter and aether, and assuming that a 
moving ponderable body cannot communicate its motion to 
the aether which surrounds it, or even to the aether which 
is entangled in its own particles ; so that no part of the aether 
can be in motion relative to any other part. Such an aether 
simply space endowed with certain dynamical properties. 

The general plan of Lorentz' investigation was to reduce all 
the complicated cases of electromagnetic action to one simple 
and fundamental case, in which the field contains only free 
aether with solitary electrons dispersed in it ; the theory which 
he adopted in this fundamental case was a combination of 
Clausius' theory of electricity with Maxwell's theory of the 
aether. 

Suppose that e (x, y, z) and e(x, y', z) are two electrons. 
In the theory of Clausius,f the kinetic potential of their mutual 
action is 

ee' 

(xx + yy + ss' - c 2 ) ; 

so when any number of electrons are present, the part of the 

*Cf. pp. 116 etxqq. t Cf. p. 262. 






Closing Years of the Nineteenth Century. 421 

kinetic potential which concerns any one of them say, e may 
be written 

L e = e (a x x + a y y + a z z - c 2 <), 

where a and c denote potential functions, defined by the 

equations 

/ f c r r f 

dxdy'dz, </> = \\\ p - dx'dy'dz' ; 

p denoting the volume-density of electric charge, and v its 
velocity, and the integration being taken over all space. 

We shall now reject Clausius' assumption that electrons act 
instantaneously at a distance, and replace it by the assumption 
that they act on each other only through the mediation of an 
aether which fills all space, and satisfies Maxwell's equations. 
This modification may be effected in Clausius' theory without 
difficulty ; for, as we have seen,* if the state of Maxwell's 
aether at any point is defined by the electric vector d and 
magnetic vector h,f these vectors may be expressed in terms 
of potentials a and ^ by the equations 

d = c" grad < - a, h = curl a ; 

and the functions a and < may in turn be expressed in terms of 
the electric charges by the equations 

a - JTJ \((**)'lr\ dx'dy'dz', </> = J/J |(J5) dxdtfdsf, 

where the bars indicate that the values of (pv r )' and (p)' refer 
to the instant (t - r/c). Comparing these formulae with those 
given above for Clausius' potentials, we see that the only change 
which it is necessary to make in Clausius' theory is that of 
retarding the potentials in the way indicated by L. Lorenz.J 
The electric and magnetic forces, thus defined in terms of the 

* Cf. pp. 298, 299. 

t We shall use the small letters d and h. in place of E and H, when MC are 
concerned with Lorentz' fundamental case, in which the system consists solely of 
free aether and isolated electrons. 

% Cf. p. 298. 



422 The Theory of Aether and Electrons in the 

position and motion of the charges, satisfy the Maxwellian 
equations 

div d = 47rc 2 / o, 

div h * 0, 

curl d = - K, 

curl h = d/c 2 + 47r/ov. 

The theory of Lorentz is based on these four aethereal 
equations of Maxwell, together with the equation which deter- 
mines the ponderomotive force on a charged particle ; this, 
which we shall now derive, is the contribution furnished by 
Clausius' theory. 

The Lagrangian equations of motion of the electron e are 

^-0 

fa- 

and two similar equations, where L denotes the total kinetic 
otential due to all causes, electric and mechanical. The 
ponderomotive force exerted on the electron by the electro- 
magnetic field has for its ^-component 

dx ~ dt\ dx 
or 

fda x . da v . da z . d<t>\ da x 

e{ x + - it H z c*} e '- 

\dx dx* dx dxj dt 

which, since 



reduces to 

- e I & -^ 



or ed x + e (yh z - z 

so that the force in question is 

ed + e [v . h]. 
This was Lorentz' expression for the ponderomotive force on an 



Closing Years of the Nineteenth Century. 423 

electrified corpuscle of charge e moving with velocity v in a field 
defined by the electric force d and magnetic force h. 

In Lorentz' fundamental case, which has thus been examined, 
account has been taken only of the ultimate constituents of 
which the universe is supposed to be composed, namely, cor- 
puscles and the aether. We must now see how to build up 
from these the more complex systems which are directly 
presented to our experience. 

The electromagnetic field in ponderable bodies, which to our 
senses appears in general to vary continuously, would present a 
different aspect if we were able to discern molecular structure ; 
we should then perceive the individual electrons by which the 
field is produced, and the rapid fluctuations of electric and 
magnetic force between them. As it is, the values furnished 
by our instruments represent averages taken over volumes 
which, though they appear small to us, are large compared 
with molecular dimensions.* We shall denote an average 
value of this kind by a bar placed over the corresponding symbol. 

Lorentz supposed that the phenomena of electrostatic charge 
and of conduction-currents are due to the presence or motion of 
simple electrons such as have been considered above. The part 
of p arising from these is the measurable density of electrostatic 
charge ; this we shall denote by pi. If w denote the velocity 
of the ponderable matter, and if the velocity v of the electrons 
be written w + u, then the quantity pv, so far as it arises from 
electrons of this type, may be written ^ w + pu. The former of 
these terms represents the convection-current, and the latter 
the conduction-current. 

Consider next the phenomena of dielectrics. Following 
Faraday, Thomson, and Mossotti,f Lorentz supposed that each 
dielectric molecule contains corpuscles charged vitreously and 
also corpuscles charged resinously. These in the absence of an 

* These principles had been enunciated, and to some extent developed, by 
J. Willard Gibbs in 1882-3 : Amer. Journ. Sci. xxiii, pp. 262, 460, xxv, p. 107 ; 
Gibbs' Scientific Papei-s, ii, pp. 182, 195, 211. 

t Cf. pp. 210, 211. 



424 The Theory of Aether and Electrons in the 

external field are so arranged as to neutralize each other's electric 
fields outside the molecule. For simplicity we may suppose 
that in each molecule only one corpuscle, of charge e, is capable 
of being displaced from its position ; it follows from what has 
been assumed that the other corpuscles in the molecule exert 
the same electrostatic action as a charge - e situated at the 
original position of this corpuscle. Thus if e is displaced to an 
adjacent position, the entire molecule becomes equivalent to an 
electric doublet, whose moment is measured by the- product of e 
and the displacement of e. The molecules in unit volume, taken 
together, will in this way give rise to a (vector) electric moment 
per unit volume, P, which may be compared to the (vector) 
intensity of magnetization in Poisson's theory of magnetism.* 
As in that theory, we may replace the doublet -distribution P 
of the scalar quantity p by a volume-distribution of p, determined 
by the equationf 

p = - div P. 

This represents the part of jo due to the dielectric molecules. 

Moreover, the scalar quantity pw x has also a doublet-distri- 
bution, to which the same theorem may be applied ; the average 
value of the part of pw x , due to dielectric molecules, is therefore 
determined by the equation 

pwx = - div (W.J.?) = - w x div P - (P . V) w x , 
or 

/ow = - div P . w - (P . V) w. 

We have now to find that part of ju which is due to dielectric 
molecules. For a single doublet of moment p we have, by 
differentiation, 

f JJ pM dx dy dz = dp/dt, 

where the integration is taken throughout the molecule; so 
that 

/// P M dxdydz = (d/dt) ( FP), 

where the integration is taken throughout a volume V, which 
*Cf. p. 64. 

t We assume all transitions gradual, so as to avoid surface-distributions. 



Closing Years of the Nineteenth Century. 425 

encloses a large number of molecules, but which is small com- 
pared with measurable quantities; and this equation may be 
written 



Now, if P refers to differentiation at a fixed point of space (as 
opposed to a differentiation which accompanies the moving body), 

we have 

(/&)*-? + (w.V)P, 

and (d/dt) V = Fdiv w; 
so that 

/ou = P + (w . V) P + div w . P 

= P + curl [P . w] + div P . w + (P . V) w, 
and therefore 

pu + pw = P + curl [P . w]. 

This equation determines the part of f>v which arises from the 
dielectric molecules. 

The general equations of the aether thus become, when the 
averaging process is performed, 

div d = 4>!r<?pi ~ 4-Trc 2 div P, div h = 0, 
curl d = - h, 

curl h =- (1/c 2 ) d + 47r , 

( + P + curl [P . w] I 

In order to assimilate these to the ordinary electromagnetic 
equations, we must evidently write 

d = E, the electric force; 
(1/4-7TC 2 ) E + P = D, the electric induction ; 

h = H, the magnetic vector. 

The equations then become (writing p for p lt as there is no 
longer any need to use the subscript), 

div D = p, - curl E = H, 

where div H = ' curl H = 4lrS ' 

S = conduction-current + convection-current + D + curl [P . w]. 



convection-current + conduction-current . 



426 The Theory of Aether and Electrons in the 

The term D in S evidently represents the displacement- 
current of Maxwell ; and the term curl [P . w] will be 
recognized as a modified form of the term curl [D . w], which 
was first introduced into the equations by Hertz.* It will 
be remembered that Hertz supposed this term to repre- 
sent the generation of a magnetic force within a dielectric 
which is in motion in an electric field ; and that Heaviside,f by 
adducing considerations relative to the energy, showed that the 
term ought to be regarded as part of the total current, and 
inferred from its existence that a dielectric which moves in an 
electric field is the seat of an electric current, which produces 
a magnetic field in the surrounding space. The modification 
introduced by Lorentz consisted in replacing D by P in the 
vector-product ; this implied that the moving dielectric does 
not carry along the aethereal displacement, which is represented 
by the term E/4?rc 2 in D, but only carries along the charges 
which exist at opposite ends of the molecules of the ponderable 
dielectric, and which are represented by the term P. The part 
of the total current represented by the term curl [P . w] is 
generally called the current of dielectric convection. 

That a magnetic field is produced when an uncharged 
dielectric is in motion at right angles to the lines of force of a 
constant electrostatic field had been shown experimentally in 
1888 by Rontgen.J His experiment consisted in rotating a 
dielectric disk between the plates of a condenser ; a magnetic 
field was produced, equivalent to that which would be produced 
by the rotation of the " fictitious charges " on the two faces of 
the dielectric, i.e., charges which bear the same relation to 
the dielectric polarization that Poisson's equivalent surface- 
density of magnetism bears to magnetic polarization. If U 
denote the difference of potential between the opposite coatings 
of the condenser, and * the specific inductive capacity of the 
dielectric, the surf ace -density of electric charge on the coatings 

* Cf. p. 366. t Cf. p. 367. 

I Ann. d. Phys. xxxv (1888), p. 264 ; xl (1890), p. 93. 
Cf. p. 64. 



Closing Years of the Nineteenth Century. 427 

is proportional to t7, and the fictitious charge on the sur- 
faces of the dielectric is proportional to + (a - 1) U. It is evident 
from this that if a plane condenser is charged to a given 
difference of potential, and is rotated in its own plane, the 
magnetic field produced is proportional to * if (as in Kowland's 
experiment*) the coatings are rotated while the dielectric 
remains at rest, but is in the opposite direction, and is propor- 
tional to (c - 1) if (as in Kontgen's experiment) the dielectric is 
rotated while the coatings remain at rest. If the coatings and 
dielectric are rotated together, the magnetic action (being the 
sum of these) should be independent of f a conclusion which 
was verified later by Eichenwald.f 

Hitherto we have taken no account of the possible mag- 
netization of the ponderable body. This would modify the 
equations in the usual manner,:}: so that they finally take the 
form 

div D = p, (I) 

div B = 0, (II) 

curl H = 47rS, (III) 

-curl.E = B, (IV), 

where S denotes the total current formed of the displacement - 
current, the convection-current, the conduction-current, and the 
current of dielectric convection. Moreover, since 

S =pv + d'/47rc 2 , 
we have 

div S = div pv + (l/4;rc 2 ) div (ad/80 

= div v 



*Cf. p. 339. 

t Ann. d. Piiys. xi (1903), p. 421 ; xiii (1904), p. 919. Eichenwald performed 
other experiments of a similar character, e.g. he observed the magnetic field due 
to the changes of polarization in a dielectric which was moved in a non- 
homogeneous electric field. 

J It is possible to construct a purely electronic theory of magnetization, a 
magnetic molecule being supposed to contain electrons in orbital revolution. It 
then appears that the vector which represents the average value of h. is not H, 
but B. 



428 The Theory of Aether and Electrons in the 
which vanishes by virtue of the principle of conservation of 

div 8 = 0, (V) 



electricity. Thus 



or the total current is a circuital vector. Equations (I) to (Y) 
are the fundamental equations of Lorentz' theory of electrons. 

We have now to consider the relation by which the polari- 
zation P of dielectrics is determined. If the dielectric is 
moving with velocity w, the ponderomotive force on unit 
electric charge moving with it is (as in all theories)* 

E' = E + [w . B ]. (1) 

In order to connect P with E', it is necessary to consider the 
motion of the corpuscles. Let e denote the charge and m the 
mass of a corpuscle, (, *?, ) its displacement from its position of 
equilibrium, k* (, 77, ) the restitutive force which retains it in 
the vicinity of this point ; then the equations of motion of the 
corpuscle are 

ra + A- 2 = eE x ' t 

and similar equations in 17 and . When the corpuscle is set in 
motion by light of frequency n passing through the medium, 
the displacements and forces will be periodic functions of nt 
say, 



Substituting these values in the equations of motion, we obtain 
A(Jc* - mn z ) -= eE, and therefore ? (k z - tun*) = eE' x . 

Thus, if N denote the number of polarizable molecules per unit 
volume, the polarization is determined by the equation 

* = Ne (g, TJ, ?) = JVVE7(& 2 - m?i 2 ). 

In the particular case in which the dielectric is at rest, this 
equatio^ gives 

= (l/47rc 2 )E + P = (l/47rc 2 )E + Ne*E/(k 2 - mw\ 
But, as we have seen,f D bears to E the ratio ^u 2 /47rc 2 , where ^ 

*Cf. p. 365. tCf. p. 281. 



Closing Years of the Nineteenth Century. 429 

denotes the refractive index of the dielectric ; and therefore the 
refractive index is determined in terms of the frequency by the 
equation 

- mn z ). 



This formula is equivalent to that which Maxwell and 
Sellmeier* had derived from the elastic-solid theory. Though 
superficially different, the derivations are alike in their 
essential feature, which is the assumption that the molecules 
of the dielectric contain systems which possess free periods 
of vibration, and which respond to the oscillations of the 
incident light. The formula may be derived on electro- 
magnetic principles without any explicit reference to electrons ; 
all that is necessary is to assume that the dielectric polarization 
has a free period of vibration.f 

When the luminous vibrations are very slow, so that n is 
small, fjr reduces to the dielectric constant ej ; so that the. 
theory of Lorentz leads to the expression 



e <= 

* Cf. p. 293. 

t A theory of dispersion, which, so far as its physical assumptions and results 
are concerned, resembles that described above, was published in the same year 
(1892) by Helmholtz, Berl. Ber., 1892, p. 1093, Ann d. Phys. xlviii (1893), 
pp. 389, 723. In this, as in Lorentz' theory, the incident light is supposed to 
excite sympathetic vibrations in the electric doublets which exist in the molecules 
of transparent bodies. Helmholtz' equations were, however, derived in a different 
way from those of Lorentz, being deduced from the Principle of Least Action. 
The final result is, as in Lorentz' theory, represented (when the effect of damping 
is neglected) by the Maxwell-Sellmeier formula. Helmholtz' theory was developed 
further by Reiff, Ann. d. Phys. Iv (1895), p. 82. 

In a theory .of dispersion given by Planck, Berl. Ber., 1902, p. 470, the 
damping of the oscillations is assumed to be due to the loss of energy by radiation : 
so that no new constant is required in order to express it. 

Lorentz, in his lectures on the Iheory of Electrons (Leipzig, 1909), p. 141, 
suggested that the dissipative term in the equations of motion of dielectric electrons 
might be ascribed to the destruction of the regular vibrations of tit^electrons 
within a molecule by the collisions of the molecule with other molecule 

Some interesting references to the ideas of Hertz on the elet.:t)magnetic 
explanation of dispersion will be found in a memoir by Drude, Ann. d. Phys. (6) i 
(1900), p. 437. 

+ Cf. p. 283. 



430 The Theory of Aether and Electrons in the 

for the specific inductive capacity in terms of the number and 
circumstances of the electrons.* 

Eeturning now to the case in which the dielectric is sup- 
posed to be in motion, the equation for the polarization may be 
written 

from this equation, Fresnel's formula for the velocity of light in 
a moving dielectric may be deduced. For, let the axis of z be 
taken parallel to the direction of motion of the dielectric, which 
is supposed to be also the direction of propagation of the light ; 
and, considering a plane -polarized wave, take the axis of x 
parallel to the electric vector, so that the magnetic vector 
must be parallel to the axis of y. Then equation (III) above 
becomes 

equation (IV) becomes (assuming B equal to H, as is always 
the case in optics), 

The equation which defines the electric induction gives 

IV* (1/4**)** + P.; 

and equations (1) and (2) give 

4arc*P x = (ft - 1) (E x - wH y ). 
Eliminating D x , P x , and H y , we have 

-iV- + ' 

. A 

or, neglecting w~/c 2 , 

~dz*~ = 7 ~W ' ~~? dtfc '* 

Substituting E x = e n ^ , so that V denotes the velocity 

of light in the moving dielectric with respect to the fixed aether 
we have 



* Cf. p. 211. 

t This equation was first given as a result of the theory of electrons by Lorentz 
in the last chapter of his memoir of 1892, Arch. Neeii. xxv, p. 525. It was also 
given by Larmor, Phil. Trans., clxxxv (1894), p. 821. 



Closing Years of the Nineteenth Century. 431 
or (neglecting 



C fjC - 1 

y - _ + - ^, 
P M" 

which is the formula of Fresnel.* The hypothesis of Fresnel, 
that a ponderable body in motion carries with it the excess of 
aether which it contains as compared with space free from 
matter, is thus seen to be transformed in Lorentz' theory 
into the supposition that the polarized molecules of the 
dielectric, like so many small condensers, increase the dielectric 
constant, and that it is (so to speak) this augmentation of the 
dielectric constant which travels with the moving matter. One 
evident objection to Fresnel's theory, namely, that it required 
the relative velocity of aether and matter to be different for 
light of different colours, is thus removed ; for the theory of 
Lorentz only requires that the dielectric constant should have 
different values for light of different colours, and of this 
a satisfactory explanation is provided by the theory of 
dispersion. 

The correctness of Lorentz' hypothesis, as opposed to that of { 
Hertz (in which the whole of the contained aether was supposed to 
be transported with the moving body), was afterwards confirmed 
by various experiments. In 1901 E. Blondlotf drove a current 
of air through a magnetic field, at right angles to the lines of 
magnetic force. The air-current was made to pass between the 
faces of a condenser, which were connected by a wire, so as to be 
at the same potential. An electromotive force E' would be 
produced in the air by its motion in the magnetic field ; and, 
according to the theory of Hertz, this should produce an 
electric induction D of amount (e/47rc 2 ) E' (where t denotes the 
specific inductive capacity of the air, which is practically 
unity) ; so that, according to Hertz, the faces of the condenser 
should become charged. According to Lorentz' theory, on the 
other hand, the electric induction D is determined by the 
equation 

47rc 2 D = E + (e - 1) E' 

* Cf. p. 117. t Comptes Rendus cxxxiii (1901), p. 778. 



432 The Theory of Aether and Electrons in the 

where E denotes the electric force on a charge at rest, which is 
zero in the present case. Thus, according to Lorentz' theory, 
the charges on the faces would have only (e - l)/e of the values 
which they would have in Hertz' theory ; that is, they would be 
practically zero. The result of Blondlot's experiment was in 
favour of the theory of Lorentz. 

An experiment of a similar character was performed in 
1905 by H. A. Wilson.* In this, the space between the inner 
and outer coatings of a cylindrical condenser was filled with 
the dielectric ebonite. When the coatings of such a con- 
denser are maintained at a definite difference of potential, 
charges are induced on them ; and if the condenser be rotated 
on its axis in a magnetic field whose lines of force are parallel 
to the axis, these charges will be altered, owing to the 
additional polarization which is produced in the dielectric 
molecules by their motion in the magnetic field. As before, 
the value of the additional charge according to the theory of 
Lorentz is (e - l)/e times its value as calculated by the theory 
of Hertz. The result of Wilson's experiments was, like that of 
Blondlot's, in favour of Lorentz. 

The reconciliation of the electromagnetic theory with 
Fresnel's law of the propagation of light in moving bodies was 
a distinct advance. But the theory of the motionless aether 
was hampered by one difficulty : it was, in its original form, 
incompetent to explain the negative result of the experiment 
of Michelson and Morley.f The adjustment of theory to 
observation in this particular was achieved by means of a 
remarkable hypothesis which must now be introduced. 

In the issue of " Nature" for June 16th, 1892,J Lodge 

mentioned that Fitz Gerald had communicated to him a new 



suggestion for overcoming the difficulty. This was, to suppose 
that the dimensions of material bodies are slightly altered 
when they are in motion relative to the aether. Five months 
afterwards, this hypothesis of Fitz Gerald's was adopted by 

*Phii. Trans, cciv (1905), p. 121. 

t Cf. p 417. J Nature, xlvi (1892), p. 165, 



Closing Years of the Nineteenth Century. 433 

Lorentz, in a communication to the Amsterdam Academy;* 
after which it won favour in a gradually widening circle, until 
eventually it came to be generally taken as the basis of all 
theoretical investigations on the motion of ponderable bodies 
through the aether. 

Let us first see how it explains Michelson's result. 
On the supposition that the aether is motionless, one of the 
two portions into which the original beam of light is divided 
should accomplish its journey in a time less than the other by 
ur*l/c?, where w denotes the velocity of the earth, c the velocity 
of light, and I the length of each arm. This would be exactly 
compensated if the arm which is pointed in the direction of the 
terrestrial motion were shorter than the other by an amount 
w 2 //2c 3 ; as would be the case if the linear dimensions of 
moving bodies were always contracted in the direction of 
their motion in the ratio of (1 - w' /2c~) to unity. This is 
Fitz Gerald's hypothesis of contraction. Since for the earth the 
ratio w/c is only 

30 km. /sec. 
300,000 km./sec.' 

the fraction w^jc- is only one hundred-millionth. 

Several further contributions to the theory of electrons in a 
motionless aether were made in a short treatisef which was 
published by Lorentz in 1895. One of these related to the 
explanation of an experimental result obtained some years 
previously by Th. des Coudres,J of Leipzig. Des Coudres had 
observed the mutual inductance of coils in different circum- 
stances of inclination of their common axis to the direction of 
the earth's motion, but had been unable to detect any effect 
depending on the orientation. Lorentz now showed that this 
could be explained by considerations similar to those which 

* Verslagen d. Kon. Ak. van Wetenschappen, 1892-3, p. 74 (November 26th 
1892). 

t Versuch einer Theorie Jer electrischen und optischen Erscheinungen in bewegten 
Eorpern, von H. A. Lorentz ; Leiden, E. J. Brill. It was reprinted by Teubner, 
of Leipzig, in 1906. 

+ Ann. d. Phys. xxxviii (1889), p. 73. 

2 F 



434 The Theory of Aether and Electrons in the 

Budde and Fitz Gerald* had advanced in a similar case ; a 
conductor carrying a constant electric current and moving with 
the earth would exert a force on electric charges at relative 
rest in its vicinity, were it not that this force induces on the 
surface of the conductor itself a compensating electrostatic 
charge, whose action annuls the expected effect. 

The most satisfactory method of discussing the influence of 
the terrestrial motion on electrical phenomena is to transform 
the fundamental equations of the aether and electrons to axes 
moving with the earth. Taking the axis of x parallel to the 
direction of the earth's motion, and denoting the velocity of the 
earth by w, we write 

x = #1 + wt t y = 2/1, z = Zi, 

so that (x-i, yi t Zj) denote coordinates referred to axes moving 
with the earth. Lorentz completed the change of coordinates 
by introducing in place of the variable t a "local time" t l} 
defined by the equation 

t = t l + m^/c 2 . 

It is also necessary to introduce, in place of d and h, the electric 
and magnetic forces relative to the moving axes : these aret 

d 1 = d + [w.h] 
h 1 = h + (l/c 2 ) [d.w]; 

and in place of the velocity v of an electron referred to^the 
original fixed axes, we must introduce its velocity Vi relative to 
the moving axes, which is given by the equation 

V, = V - W. 

The fundamental equations of the aether and electrons, 
referred to the original axes, are 

div d = 47re 2 , curl d = - h, 



div h = 0, curl h = (1/c 2 ) d + 

F = d + [v . h], 

where F denotes the ponderomotive force on a particle carrying 
a unit charge. 

* Cf . p. 263. t Cf. pp. 365, 366. 



Closing Years of the Nineteenth Century. 435 

By direct transformation from the original to the new 
variables it is found that, when quantities of order iv*/c* and 
wv/c* are neglected, these equations take the form 

divj d! = 4?rc 2 p, curl a di = - Sh^B^, 

div t H! = 0, curl, hi = (1/c 2 ) ddj/fy 

F = d! + [v lt hj, 
where div, d, stands for 



Since these have the same form as the original equations, 
it follows that when terms depending on the square of the 
constant of aberration are neglected, all electrical phenomena 
may be expressed with reference to axes moving with the earth 
by the same equations as if the axes were at rest relative to the 
aether. 

In the last chapter of the Versuch Lorentz discussed those 
experimental results which were as yet unexplained by the 
theory of the motionless aether. That the terrestrial motion 
exerts no influence on the rotation of the plane of polariza- 
tion in quartz* might be explained by supposing that two 
independent effects, which are both due to the earth's motion, 
cancel each other; but Lorentz left the question undecided. 
Five years later Larmorf criticized this investigation, and 
arrived at the conclusion that there should be no first-order 
effect ; but LorentzJ afterwards maintained his position against 
Larmor's criticism. 

Although the physical conceptions of Lorentz had from 
the beginning included that of atomic electric charges, the 
analytical equations had hitherto involved p, the volume-density 
of electric charge; that is, they had been conformed to the 
hypothesis of a continuous distribution of electricity in space. 
It might hastily be supposed that in order to obtain an 

* Cf. p. 416. t Larmor, Aether and Matter, 1900. 

J Proc. Amsterdam Acad. (English ed.), iv (1902), p. 669. 
2 F 2 



436 The Theory of Aether and Electrons in the 

analytical theory of electrons, nothing more would be required 
than to modify the formulae by writing e (the charge of an 
electron) in place of pdxdydz. That this is not the case was 
shown* a few years after the publication of the Versuch. 

Consider, for example, the formula for the scalar potential 
at any point in the aether, 



where the bar indicates that the quantity underneath it is to 
have its retarded value, f 

This integral, in which the integration is extended over all 
elements of space, must be transformed before the integration 
can be taken to extend over moving elements of charge. Let 
de denote the sum of the electric charges which are accounted 
for under the heading of the volume- element dx'dy'dz in 
the above integral. This quantity de is not identical with 
~p'dx'dy'dz'. For, to take the simplest case, suppose that it is 
required to compute the value of the potential-function for the 
origin at the time t, and that the charge is receding from the 
origin along the axis of x with velocity u. The charge which 
is to be ascribed to any position x is the charge which occupies 
that position at the instant t - x/c; so that when the reckoning 
is made according to intervals of space, it is necessary to 
reckon within a segment (x 2 - a?i) not the electricity which at 
any one instant occupies that segment, but the electricity which 
at the instant (t - xjc) occupies a segment (x* - x\\ where x\ 
denotes the point from which the electricity streams to x l in the 
interval between the instants (t - x z ,'c) and (t - x^/c). We have 
evidently 

&'i - %'\ = u (%2 - %i)/c, or x z - x\ = (x 2 - #0 (1 + u/c). 

For this case we should therefore have 



I^ ?' dx'dy'dz' = (l + ^'dx'dy'dz'. 

Xi \ cj 



* E. Wiechert, Arch. Neerl. (2) v (1900), p. 549. Cf. also A. Lienard, 
L' Eclairage elect, xvi (1898), pp. 5, 53, 106. 
t Cf. p. 298. 



Closing Years of the Nineteenth Century. 437 

In the general case, it is only necessary to replace u by the 
component of velocity of the electric charge in the direction of 
the radius vector from the point at which the potential is to be 
computed. This component may be written v cos (v . r), where r 
is measured positively from the point in question to the charge, 
and v denotes the velocity of the charge. Thus 

cde' = {c + v cos (v . r) } ~p dx'dy'dz', 
and therefore 



f de' 
}cr + (r.v)' 



where the integration is extended over all the charges in the 
field, and the bars over the letters imply that the position of 
the charge considered is that which it occupied at the instant 
t - r/c. In the same way the vector- potential may be shown to 
have the value 



r vde' 
J cr + (r . v 



Meanwhile the unsettled problem of the relative motion of 
earth and aether was provoking a fresh series of experimental 
investigations. The most interesting of these was due to 
Fitz Gerald,* who shortly before his death in February, 1901, 
commenced to examine the phenomena manifested by a 
charged electrical condenser, as it is carried through space in 
consequence of the terrestrial motion. On the assumption 
that a moving charge develops a magnetic field, there will be 
associated with the condenser a magnetic force at right angles 
to the lines of electric force and to the direction of the 
motion: magnetic energy must therefore be stored in the 
medium, when the plane of the condenser includes the direc- 
tion of the drift; but when the plane of the condenser is at 
right angles to the terrestrial motion, the effects of the 
opposite charges neutralize each other. Fitz Gerald's original 
idea was that, in order to supply the magnetic energy, there 
must be a mechanical drag on the condenser at the moment of 

* Fitz Gerald's Scientific Writings, p. 557. 



438 The Theory of Aether and Electrons in the 

charging, similar to that which would be produced if the mass 
of a body at the surface of the earth were suddenly to become 
greater. Moreover, it was conjectured that the condenser, 
when freely suspended, would tend to move so as to assume the 
longitudinal orientation, which is that of maximum kinetic 
energy* : the transverse position would therefore be one of 
unstable equilibrium. 

For both effects a search was made by Fitz Gerald's pupil 
Trouton :f in the experiments designed to observe the turning 
couple, a condenser was suspended in a vertical plane by a 
fine wire, and charged. If the plane of the condenser were 
that of the meridian, about noon there should be no couple 
tending to alter the orientation, because the drift of aether due 
to the earth's motion would be at right angles to this plane; 
at any other hour, a couple should act. The effect to be 
detected was extremely small ; for the magnetic force due to 
the motion of the charges would be of order w/c, where w 
denotes the velocity of the earth ; so the magnetic energy of 
the system, which depends on the square of the force, would be 
of order (w/c)' ; and the couple, which depends on the derivate 
of this with respect to the azimuth, would therefore be likewise 
of the second order in (w/c). 

No couple could be detected. As the energy of the magnetic 
field must be derived from some source, there seems to be no 
escape from the conclusion that the electrostatic energy of a 
charged condenser is diminished by the fraction (w/c)" of its 
amount when the condenser is moving with velocity w at 
right angles to its lines of electrostatic force. To explain this 
diminution, it is necessary to admit Fitz Gerald's hypothesis 
of contraction. The negative result of the experiment may be 
taken to indicate^ that the kinetic potential of the system, 
when the Fitz Gerald contraction is taken into account as a 

* Larmor, in Fitz Gerald's Scientific Papers, p 566. 

t F. T. Trouton. Trans. Roy. Dub. Soc., April, 1902; F. T. Trouton and 
H. R. Noble, Phil. Trans, ccii (1903), p. 165. 

J Cf. P. Langevin, Comptes Rendus, cxl (1905), p. 1171. 



Closing Years of the Nineteenth Century. 439 

constraint, is independent of the orientation of the plates with 
respect to the direction of the terrestrial motion. 

It may be remarked that the existence of the couple, had 
it been observed, would have demonstrated the possibility of 
drawing on the energy of the earth's motion for purposes of 
terrestrial utility. 

The Fitz Gerald contraction of matter as it moves through 
the aether might conceivably be supposed to affect in some 
way the optical properties of the moving matter; for in- 
stance, transparent substances might become doubly refracting. 
Experiments designed to test this supposition were per- 
formed by Lord Eayieigh in 1902,* and by D. B. Brace in 
1904t ; but no double refraction comparable with the propor- 
tion (w/cf of the single refraction could be detected. The 
Fitz Gerald contraction of a material body cannot therefore be 
of the same nature as the contraction which would be produced 
in the body by pressure, but must be accompanied by such 
concomitant changes in the relations of the molecules to the 
aether that an isotropic substance does not lose its simply 
refracting character. 

By this time, indeed, the hypothesis of contraction, which 
originally had no direct connexion with electric theory, had 
assumed a new aspect. Lorentz, as we have seen,J had 
obtained the equations of a moving electric system by 
applying a transformation to the fundamental equations of 
the aether. In the original form of this transformation, 
quantities of higher order than the first in w/c were neglected. 
But in 1900 Larmorg extended the analysis so as to include 
small quantities of the second order, and thereby discovered a 
remarkable connexion between the equations of transforma- 
tion and the equations which represent Fitz Gerald's con- 

Phil. Mag. iv (1902), p. 678. 
t Phil. Mag. vii (1904), p. 317. 

+ Cf. p. 434. Cf. also Lorentz, Proc. Amsterdam Acad. (English ed.), i (1899), 
p. 427. 

Larmor, Aether and Matter, p. 173. 



440 The Theory of Aether and Electrons in the 

traction. After this Lorentz* went further still, and obtained 
the transformation in a form which is exact to all orders of the 
small quantity w/c. In this form we shall now consider it. 
The fundamental equations of the aether are 

div d = 4irc*p, curl d = - h, 

div h = 0, curl h = d/c 2 + 47r/ov. 

It is desired to find a transformation from the variables 
x, y, z, t, p, d, h, v, to new variables x lt y^ z ly t h p lt d,, hi, YI, such 
that the equations in terms of these new variables may take 
the same form as the original equations, namely : 

divi d x = 47rc 2 jOi, curl, di = - dh^d^, 
d^ h! = 0, curlj hi = (1/c 2 ) Bdj/9^ 



Evidently one particular class of such transformations is 
that which corresponds to rotations of the axes of coordinates 
about the origin : these may be described as the linear homo- 
geneous transformations of determinant unity which transform 
the expression (x 2 + if + z z ) into itself. 

These particular transformations are, however, of little 
interest, since they do not change the variable t. But in place 
of them consider the more general class formed of all those 
linear homogeneous transformations of determinant unity in 
the variables x, y, z, ct, which transform the expression 
(x- + y* + z - c't") into itself : we shall show that these trans- 
formations have the property of transforming the differential 
equations into themselves. 

All transformations of this class may be obtained by the 
combination and repetition (with interchange of letters) of one 
of them, in which two of the variables say, y and z are 
unchanged. The equations of this typical transformation may 

* Proc. Amsterdam Acad. (English ed.), vi, p. 809. Lorentz' work was 
completed in respect to the formulae which connect pi, vi, with p, v, by Einstein, 
Ann. d. Phys., xvii (1905), p. 891. It should be added that the transformation 
in question had been applied to the equation of vibratory motions many years 
before by Voigt, Gott. Nach. 1887, p. 41. 



Closing Years of the Nineteenth Century. 441 

easily be derived by considering that the equation of the 

rectangular hyperbola 

x 2 - (cty = 1 

(in the plane of the variables x, ct) is unaltered when any pair 
of conjugate diameters are taken as new axes, and a new unit 
of length is taken proportional to the length of either of these 
diameters. The equations of transformation are thus found to be 

x = Xi cosh a + cti sinh a, y = y\ y 

t = ti cosh a + (x } /c) sinh a, z = z,, 

where a denotes a constant. The simpler equations previously 
given by Lorentz* may evidently be derived from these by 
writing w/c for tanh a, and neglecting powers of w/c above the 
first. By an obvious extension of the equations given by 
Lorentz for the electric and magnetic forces, it is seen that the 
corresponding equations in the present transformation are 

= d Xlt h x = 



d y = d yi cosh a + ch zi sinh a, 
d z = d zi cosh a - ch v . sinh a, 



h y = h yi cosh a - (l/c)d zi sinh a, 
h z = h zi cosh a + (l/c)d yi sinh a. 

The connexion between p and p l may be obtained in the 
following way. It is assumed that if a charge e is attached to 
a particle which occupies the position (f, 77, J) at the instant t y 
an equal charge will be attached to the corresponding point 
(f i, 77 1} \) at the corresponding instant ti in the transformed 
system ; so that a charge e attached to an adjacent particle 
(f + A TI + Arj, %+ A) at the instant t will give rise in the 
derived system to a charge e at the place 



V %1 A . Ofrl A %l 

fi +^Af + ^-AT/4 ^\ 

at the instant 

* Cf. P . 434. 



442 The Theory of Aether and Electrons in the 

that is to say, at the place 

(f ! + Af cosh a, 77, + AT?, + A?) 

at the instant (^ - sinh a . Af/c). Thus at the instant ^, this 
charge will occupy the position 

(f i + Af cosh a + sinh a . Af . 0^/e, 771 + AT; + sinh a . Af . v yi /c, 

1 + Af + sinh a . A f . v ai /c). 

The charges corresponding to those in the original system which 
were at the instant t contained in a volume A| AT? A will 
therefore in the derived system at the instant t l occupy a volume 



cosh a + sinh a . v x jc . 



AS, 



sinh a .Vyjc 1 

sinh a . v z jc 1 
or, 

(cosh a + sinh a . v x jc) A? Aj A. 

Thus if jOi denote the volume-density of electric charge in the 
transformed system, we shall have 

pi (cosh a + sinh a . v x jc) = p ; 

this equation expresses the connexion between pi and p. We 
have moreover 

dx fix fix dx 

~ ($_ dt dt dt 

v x , sech a 

= c tann a + - , 

cosh a + v Xl c^sinha 

and similarly 

v yi 

cosh a + v Xl c~ l sinh a 
and 



cosh a + v Xl c l sinh a 



Closi?ig Years of the Nineteenth Century. 443 

When the original variables are by direct substitution replaced 
by the new variables in the differential equations, the latter 
take the form 



div! hi = 0, cur^ h t 

that is to say, the fundamental equations of the aether retain 
their form unaltered, when the variables are subjected to the 
transformation which has been specified. 

We are now in a position to show the connexion of this 
transformation with Fitz Gerald's hypothesis of contraction. 
Suppose that two material particles are moving along the axis 
of x with velocity w - c tanh a. From the relation 

v x . sech a 
v x = c tanh a + = - 



cosh a + v Xi c~ l sinh a ' 

it follows that v Xl is zero for each of the particles, which implies 
that they are at rest relative to the new axes. Let %i and x\ 
denote their coordinates with respect to this latter system ; then 
the coordinates of one particle at the instant ti, referred to the 
original axes, will be given by the equations 

x = Xi cosh a + ct l sinh o, t t\ cosh a + Xi c~ l sinh a ; 
and the coordinates of the other particle will be given by 
x f - x\ cosh o f cti sinh a, t f = t l cosh a + x\ c' 1 sinh a ; 

so that at time t the latter particle will have the coordinate x", 
where 

x" = of + w (t - t') 

= x\ cosh a + ct l sinh a + (x - x\) sinh 2 a sech a, 
which gives 

x" x = (a?'i Xi) (1 w'/c 2 )^. 

This equation shows that the distance between the par- 
ticles in the system of measurement furnished by the original 
axes, with reference to which the particles were moving with 
velocity w, bears the ratio (1 - w-/c'-)^ : 1 to their distance in the 



444 The Theory of Aether and Electrons in the 

system of measurement furnished by the transformed axes, 
with reference to which the particles are at rest. But accord- 
ing to FitzGerald's hypothesis of contraction, when a material 
body is in motion relative to the aether, in a direction parallel 
to the axis of x, its dimensions parallel to this direction 
contract in precisely this ratio; so that the equation of the 
body, in terms of the coordinates x }) y^ z ly which move with 
it, is unaltered. Thus the hypothesis of Fitz Gerald may be 
expressed by the statement .that the equations of the figures 
of ponderable bodies are covariant with respect to those trans- 
formations for which the fundamental equations of the aether 
are covariant. 

The covariance holds with respect to all linear homogeneous 
transformations in the variables (x, y, z, t), of determinant 
unity, which transform the expression (x" 2 + y 2 - + z~ - c 2 f) into 
itself. This group comprises an infinite number of transforma- 
tions ; so that there are an infinite number of sets of variables 
resembling (x }) y lt c,, ,), of which any one set (x r , y r , z r , t r ) can 
be derived from any other set (.r s , y s , z s , t s ) by a transformation 
of the group ; among the sets we must of course include the 
original set of coordinates (x, y, z, t). But hitherto we have 
proceeded on the assumption that the original set (x, y, z, t) is 
entitled to a primacy among all the other sets, since the axes 
(x, y, z) have been supposed to possess the special property of 
having no motion relative to the aether, and the time repre- 
sented by the variable t has been understood to be a definite 
physical quantity. The other sets of variables (.r r , y r , s r , t r ) 
have been regarded merely as symbols convenient for use in 
problems relating to moving bodies, but not as corresponding 
to physical entities in the same degree as (x, y, z, t). "We 
must now inquire whether this view is justified. 

The question amounts to asking whether absolute position 
in space, or at any rate absolute fixity relative to the aether, is 
something which can be brought within the bounds of human 
knowledge. 

It is well known that the science of dynamics, as founded 



Closing Years of the Nineteenth Century. 445 

on Newton's laws of motion, does not supply any criterion by 
which rest may be distinguished from uniform motion ; for if 
the laws of motion are applicable when the position of bodies 
is referred to any particular set of axes, they will be equally 
applicable when position is referred to any other set of axes 
which have a uniform motion of translation relative to these. 

The older theories of electrostatics, magnetism, and electro- 
dynamics, which are based on the conception of action at a 
distance, are concerned only with relative configurations and 
motions, and are therefore useless in the search for a basis of 
absolute reckoning. 

But the existence of an aether, which is postulated in the 
undulatory theory of light, seems at first sight to involve the 
conceptions of rest and motion relative to it, and thus to afford 
a means of specifying absolute position. Suppose, for instance, 
that a disturbance is generated at any point in free aether; 
this disturbance will spread outwards in the form of a sphere ; 
and the centre of this sphere will for all subsequent time 
occupy an unchanged position relative to the aether. In this 
way, or in many other ways, we might hope to determine, by 
electrical or optical experiments, the velocity of the earth 
relative to the aether. 

The failure of such experiments as had been tried led 
Fitz Gerald* to suggest that the dimensions of material bodies 
undergo contraction when the bodies are in motion relative 
to the aether. By the transformation of Lorentz and Larmor, 
as we have seen, this hypothesis came to be expressed in a new 
form ; namely that the equation of the figure of the body, 
referred to a frame of reference moving with it, is always the 
same, but that frames of reference which are in motion relative to 
each other are based on different standards of length and time. 
This way of regarding the matter brings into prominence the 
fundamental questions involved. Before speaking of lengths 
and velocities, it is necessary to examine the nature of systems 
of measurement of space and time. 

* Cf . p. 432. 



446 The Theory of Aether and Electrons in the 

Of the events with which Natural Philosophy is concerned, 
each is perceived to happen at some definite location at some 
definite moment. When a material object has been observed 
to occupy a certain position at a certain instant, the same 
object may again be observed at a subsequent instant ; but it 
is impossible to determine whether the object is or is not in 
the same position, since there is no obvious means of preserving 
the identity of any location from one moment to another. 
The physicist, however, finds it convenient to construct a 
framework of axes in space and time for the purpose of fitting 
his experiences into an orderly arrangement ; and the ques- 
tion at issue is whether experience furnishes the means of 
determining a framework completely and uniquely by 
absolute properties, or whether the selection inevitably rests 
on arbitrary choice and accidental circumstance. 

In attempting to answer this question, it may first be 
observed that the choice is always made so as to simplify 
the description of natural phenomena as much as possible ; 
thus, the variable which is to measure time is so chosen that 
its increment in the interval between any two consecutive 
beats of a pendulum is the same as its increment in the interval 
between any other two consecutive beats. If the selection of 
the four variables (x, y, z y t) is well made, it should be possible 
to express the laws of nature by statements of a simple character, 
e.g., that a body isolated from the influence of external agents 
moves through equal intervals of space in equal intervals of 
time. 

Accepting, then, the principle that the framework of axes 
is to be chosen so as to furnish the simplest possible expression 
of the natural laws, it becomes of importance to determine 
which of the natural laws are entitled, by reason of their 
primary importance, to receive the greatest consideration. 

Now many indications point to the probability that the 
various types of forces which are observed in ponderable 
bodies forces of cohesion, of chemical union, and so forth 
are ultimately electric in their nature. Such an assumption 



Closing Years of the Nineteenth Century. 447 

would have the great advantage of explaining the contraction 
postulated by Fitz Gerald, since it would represent the con- 
traction as actually produced by the motion. But if this 
assumption be correct, the theory of electricity and aether is 
without doubt the fundamental theory of Natural Philosophy ; 
and the framework of space and time should be chosen with 
a view chiefly to the expression of electrical phenomena. This 
may most naturally be done by stipulating that the wave- 
fronts of disturbances generated in free aether shall, in the 
system of length and time adopted, be accounted spheres whose 
centres are at the origins of disturbance and whose radii are 
proportional to the times elapsed since their initiation. Eeferred 
to axes of (#,y,z,) which satisfy these conditions, the fundamental 
equations of the electric field assume the form which has been 
taken as the basis of all our theoretical investigations. 

Imagine now a distant star which is moving with a uniform 
velocity w or c tanh a relative to this framework (x, y, z, t). The 
theorem of transformation shows that there exists another 
framework (a?,, y\,z\, t^, with respect to which the star is at rest, 
and in which moreover the condition laid down regarding the 
wave-surface is satisfied. This framework is peculiarly fitted 
for the representation of the phenomena which happen on the 
star ; whose inhabitants would therefore naturally adopt it as 
their system of space and time. Beings, on the other hand, who 
dwell on a body which is at rest with respect to the axes 
(x, y, z, t) would prefer to use the latter system ; and from the 
point of view of the universe at large, either of these systems 
is as good as the other. The equations of motion of the aether 
are the same with respect to both sets of coordinates, and 
therefore neither can claim to possess the only property which 
could confer a primacy namely, an absolute relation to the 
aether.* 

To sum up, we may say that the phenomena whose study 
is the object of Natural Philosophy take place each at a definite 

* This was first clearly expressed by Einstein, Ann. d. Phys. xvii (1905), 
p. 891. 



448 The Theory of Aether a?id Electrons in the 

location at a definite moment ; the whole constituting a four- 
dimensional world of space and time. To construct a set of 
axes of space and time is equivalent to projecting this four- 
dimensional world into a three-dimensional world of space and 
a one-dimensional world of time ; and this projection may be 
performed in an infinite number of ways, each of which is 
distinguished from the others only by characteristics merely 
arbitrary and accidental.* 

In order to represent natural phenomena without introducing 
this contingent element, it would be necessary to abandon the 
customary three-dimensional system of coordinates, and to 
operate in four dimensions. Analysis of this kind has been 
devised, and has been applied to the theory of the aether ; 
but its development belongs to the twentieth century, and 
consequently falls outside the scope of the present work. 

From what has been said, it will be evident that, in the 
closing years of the nineteenth century, electrical investigation 
was chiefly concerned with systems in motion. The theory of 
electrons was, however, applied with success in other directions, 
and notably to the explanation of a new experimental discovery. 

The last recorded observation of Faradayf was an attempt 
to detect changes in the period, or in the state of polarization, 
of the light emitted by a sodium flame, when the flame was 
placed in a strong magnetic field. No result was obtained; 
but the conviction that an effect of this nature remained to be 
discovered was felt by many of his successors. TaitJ examined 
the influence of a magnetic field on the selective absorption of 
light ; impelled thereto, as he explained, by theoretical considera- 
tions. For from the phenomenon of magnetic rotation it may be 
inferred that rays circularly polarized in opposite senses are 
propagated with different velocities in the magnetized medium ; 
and therefore if only those rays are absorbed which have a 

* Cf. H. Minkowski, Raum und Zeit. : Leipzig, 1909. 
t Bence Jones' Life of Faraday, ii, p. 449. 
J Proc. R.S. Edinb. ix (1875), p. 118. 
Cf. pp. 174, 216. 



Closing Years of the Nineteenth Century. 449' 

certain definite wave-length in the medium, the period of the 
ray absorbed from a beam of circularly polarized white light 
will not be the same when the polarization is right-handed 
as when it is left-handed. "Thus," wrote Tait, "what was 
originally a single dark absorption-line might become a double 
line." 

The effect anticipated under different forms by Faraday and 
Tait was discovered, towards the end of 1896, by P. Zeeman.* 
Eepeating Faraday's procedure, he placed a sodium flame 
between the poles of an electromagnet, and observed a widen- 
ing of the D -lines in the spectrum when the magnetizing 
current was applied. 

A theoretical explanation of the phenomenon was imme- 
diately furnished to Zeeman by Lorentz.f The radiation i& 
supposed to be emitted by electrons which describe orbits 
within the sodium atoms. If e denote the charge of an electron 
of mass ra, the ponderomotive force which acts on it by virtue 
of the external magnetic field is e [r . K], where K denotes the 
magnetic force and r denotes the displacement of the electron 
from its position of equilibrium; and therefore, if the force 
which restrains the electron in its orbit be &, the equation of 
motion of the electron is 

mi? -t- K 2 r = e [f . K]. 

The motion of the electron may (as is shown in treatises 
on dynamics) be represented by the superposition of certain 
particular solutions called principal oscillations, whose distin- 
guishing property is that they are periodic in the time. In order 
to determine the principal oscillations, we write T^e nt ^- * for r, 
where r denotes a vector which is independent of the time, and 
n denotes the frequency of the principal oscillation : substitut- 
ing in the equation, we have 

( K - - mn*) r c = en^/^~l [r, E]. 

* Zittingsverslagen der Akad. v. "Wet. te Amsterdam v (1896), pp. 181, 242 ; 
vi (1897), pp. 13, 99 ; Phil. Mag. (5) xliii (1897), p. 226. 
t Phil. Mag. xliii (1897), p. 232. 

2 G 



450 The Theory of Aether and Electrons in the 

This equation may be satisfied either (1) if r is parallel to K, 

in which case it reduces to 

K* - mri* = 0, 

so that n has the value Km"*, or (2) if r is at right angles to K, 
in which case by squaring both sides of the equation we obtain 

the result 

(V - mn*) z = #tfK\ 

which gives for n the approximate values KW~* el/2m. 

When there is no external magnetic field, so that K is zero, 
the three values of n which have been obtained all reduce to 
icw~V which represents the frequency of vibration of the 
emitted light before the magnetic field is applied. When the 
field is applied, this single frequency is replaced by the three 
frequencies cm~, cm"i + eK/2m, icm'i - eK/2m ; that is to say, 
the single line in the spectrum is replaced by three lines close 
together. The apparatus used by Zeeman in his earliest experi- 
ments was not of sufficient power to exhibit this triplication 
distinctly, and the effect was therefore described at first as a 
widening of the spectral lines.* 

We have seen above that the principal oscillation of the 
electron corresponding to the frequency *cra~ is performed in a 
direction parallel to the magnetic force K. It will therefore 
give rise to radiation resembling that of a Hertzian vibrator, 
and the electric vector of the radiation will be parallel to the 
lines of force of the external magnetic field. It follows that 
when the light received in the spectroscope is that which has 
been emitted in a direction at right angles to the magnetic 
field, this constituent (which is represented by the middle line 
of the triplet in the spectrum) will appear polarized in a plane 
at right angles to the field ; but when the light received in the 
spectroscope is that which has been emitted in the direction of 
the magnetic force, this constituent will be absent. 

We have also seen that the principal oscillations of the 
electron corresponding to the frequencies Km-* eJ/2m are 

* Later observations, with more powerful apparatus, have shown that the 
primitive spectral line is frequently replaced by more than three components. 



Closing Years of the Nineteenth Century. 45 1 

performed in a plane at right angles to the magnetic field K. 
In order to determine the nature of these two principal oscilla- 
tions, we observe that it is possible for the electron to describe 
a circular orbit in 'this plane, if the radius of the orbit be 
suitably chosen ; for in a circular motion the forces * 2 r and 
.-e[r . K] would be directed towards the centre of the circle ; and 
it would therefore be necessary only to adjust the radius so that 
these furnish the exact amount of centripetal force required. 
Such a motion, being periodic, would be a principal oscillation. 
Moreover, since the force e [r . K] changes sign when the 
sense of the movement in the circle is reversed, it is evident 
that there are two such [circular orbits, corresponding to the 
two senses in which the electron may circulate; these must, 
therefore, be no other than the two principal oscillations of 
frequencies K.m~^ el/2m. When the light received in the 
spectroscope is that which has been emitted in a direction at 
right angles to the external magnetic field, the circles, are seen 
edgewise, and the light appears polarized in a plane parallel to 
the field ; but when the light examined is that which has been 
emitted in a direction parallel to the external magnetic force, 
,the radiations of frequencies Km~i eK/2m are seen to be 
circularly polarized in opposite senses. All these theoretical 
conclusions have been verified by observation. 

It was found by Cornu* and by C. G-. W. Konigf that the 
more refrangible component (i.e., the one whose period is shorter 
than that of the original radiation) has its circular vibration 
in the same sense as the current in the electromagnet. From 
this it may be inferred that the vibration must be due to a 
resinously charged electron; for let the magnetizing current 
and the electron be supposed to circulate round the axis of z in 
the direction in which a right-handed screw must turn in order 
to progress along the positive direction of the axis of z ; then 
the magnetic force is directed positively along the axis of z, 
jand, in order that the force on the electron may be directed 

* Comptes Rendus, cxxv (1897), p. 555. 

* Ann. d. Phys. Ixii (1897), p. 240. 

2G2 



452 The Theory of Aether and Electrons in the 

inward to the axis of z (so as to shorten the period), the charge 
on the electron must be negative. 

The value of e/m for this negative electron may be determined 
by measurement of the separation between the components of 
the triplet in a magnetic field of known strength ; for, as we 
have seen, the difference of the frequencies of the outer com- 
ponents is eKjm. The values of e/m thus determined agree well 
with the estimations* of e/m for the corpuscles of cathode rays. 

The phenomenon discovered by Zeeman is closely related to 
the magnetic rotation of the plane of polarization of light. f 
Both effects may be explained by supposing that the molecules 
of material bodies contain electric systems which possess 
natural periods of vibration, the simplest example of such a 
system being an electron which is attracted to a fixed centre 
with a force proportional to the distance. Zeeman's effect 
represents the influence of an external magnetic field on the 
free oscillations of these electric systems, while Faraday's effect 
represents the influence of the external magnetic field on the 
forced oscillations which the systems perform under the stimulus 
of incident light. The latter phenomenon may be analysed 
without difficulty on these principles, the equation of motion of 
one of the electrons being taken in the form 

mr + K 2 r = eE + e[r.H], 

where m denotes the mass and e the charge of the electron,, 
r its distance from the centre of force, K 2 r the restitutive force, 
E and H the electric and magnetic forces. When the electron 
performs forced oscillations under the influence of light of 
frequency n, this equation becomes 

( K 2 -m?i 2 )r = eE + e[r.H]. 

The influence of the magnetic force on the motion of the 
electron is small compared with the influence of the electric 
force, i.e. the second term on the right is small compared with 
the first term ; so in the second term we may replace r by its 

* Cf. p. 405. t Cf. pp. 213-216, 307-309, 367-370. 



Closing Years of the Nineteenth Century. 453 

value as found from the first term, namely, eE/(ic z - run*). The 
equation thus becomes 

r = K 2 - mn* + (i'-wm 1 ) 1 ^'^' 
If P denote* the electric moment" per unit volume, we have 

P = ei x the number of such systems in unit volume of the 
medium ; 

so P must be of the form 



where e evidently represents the dielectric constant of the 
medium, and o- is the coefficient which measures the magnetic 
rotatory power. In the magneto-optic term we may replace 
H by K, the external magnetic force, since this is large com- 
pared with the magnetic force of the luminous vibrations. 
Thus if D denote the electric induction, we have 

D = fE/47rc 2 + <r [E . K]. 

'Combining this with the usual electromagnetic equations, 

curl H = 47ri>, 

curl E = - H, 
we have 

- curl curl E = *E/c 2 + 4:r<r [E . K]. 

When a plane wave of light is propagated through the 
medium in the direction of the lines of magnetic force, and 
the axis of x is taken parallel to this direction, the equation 
gives 

(VEy 



.and these equations, as we have seen,f are competent to explain 
the rotation of the plane of polarization. 

*Cf. p. 428. t Cf. p 215. 



454 The Theory of Aether and Electrons in the 

From the occurrence of the factor (KT - mw) in the denomi- 
nator of the expression for the magneto-optic constant <r, it 
may be inferred that the magnetic rotation will be very large 
for light whose period is nearly the same as a free period of 
vibration of the electrons. A large rotation is in fact observed* 
when plane-polarized light, whose frequency differs but little 
from the frequencies of the D-lines. is passed through sodium 
vapour in a direction parallel to the lines of magnetic force. 

The optical properties of metals may be explained, according 
to the theory of electrons, by a slight extension of the analysis 
which applies to the propagation of light in transparent sub- 
stances. It is, in fact, only necessary to suppose that some of 
the electrons in metals are free instead of being bound to the 
molecules : a supposition which may be embodied in the equations 
by assuming that an electric force E gives rise to a polarization. 
P, where 

E = a P + /3P + 7 P ; 

the term in a represents the effect of the inertia of the electrons ; 
the term in ]3 represents their ohmic drift ; and the term in y 
represents the effect of the restitutive forces where these exist. 
This equation is to be combined with the customary electro- 
magnetic equations 

curl H = E/c 2 + 47rP, - curl E = H. 

In discussing the propagation of light through the metal, we 
may for convenience suppose that the beam is plane-polarized 

* The phenomenon was first observed by D. Macaluso and 0. M. Corbino,. 
Comptes Rendus, cxxvii (1898), p. 548, Rend. Lincei (5) vii (2) (1898), p. 293. The 
theoretical explanation was supplied by AV. Voigt, Gott. Nach., 1898, p. 349, 
Ann. d. Phys. Ixvii (1899), p. 345. Cf. also P. Zeeman, Proc. Amst. Acad. 
v (1902), p. 41, and J. J. Hallo, Arch. N6erl. (2) x (1905), p. 148. 

Voigt also predicted that if plane-polarized light, of period nearly tbe same as 
that of the D radiation, were passed through sodium vapour in a magnetic field, 
in a direction perpendicular to the lines of magnetic force, the velocity of propa- 
gation would be found to depend on the orientation of the plane of polarization, 
so that the sodium vapour would behave as a uniaxal crystal. This prediction was 
confirmed experimentally by Voigt and Wiechert : cf . Voigt, Gott. Nach., 1898, 
p. 355: Ann. d. Phys. Ixvii. (1899), p. 345. Cf. also A. Cotton, Cornpte* 
Rendus, cxxviii (1899), p. 294, and J. Geest, Arch. Neerl. (2), x (1905), p. 291. 



Closing Years of the Nineteenth Century. 455 

and propagated parallel to the axis of 2, the electric vector being- 
parallel to the axis of x. Thus the equations of motion reduce 
to 

= -r -=-^- + 4n- 



For E x and P* we may substitute exponential functions of 



where n denotes the frequency of the light, and /* the quasi-index 
of refraction of the metal : the equations then give at once 

<y _ i) (_ a?l t + pnS~^i + y ) = 47TC 2 . 

Writing v (1 - K v/ - 1) for p, so that v is inversely proportional 
to the velocity of light in the medium, and denotes the 
coefficient of absorption, and equating separately the real and 
imaginary parts of the equation, we obtain 

4*0 (y- an*) 



f?n z + (y - aw 2 ) 2 

When the wave-length of the light is very large, the inertia 
represented by the constant a has but little influence, and the 
equations reduce to those of Maxwell's original theory* of the 
propagation of light in metals. The formulae were experi- 
mentally confirmed for this case by the researches of E. Hagen 
and H. Kubensf with infra-red light ; a relation being thus 
established between the ohmic conductivity of a metal .and 
its optical properties with respect to light of great wave- 
length. 

When, however, the luminous vibrations are performed 
more rapidly, the effect of the inertia becomes predominant; and 

* Cf. p. 290. 

t Berlin Sitzungsber., 1903, pp. 269, 410; Ann. d. Phys. xi (1903), p. 873 ; 
Phil. Mag. vii (1904), p. 157. 



456 The Theory of Aether and Electrons in 'the 

If the constants of the metal are such that, for a certain range 
of values of n, V Z K is small, while v~ (I - K 2 ) is negative, it is evident 
that, for this range of values of n, v will be small and K large, 
i.e., the properties of the metal will approach those of ideal 
.silver.* Finally, for indefinitely great values of n, V~K is small 
.and v 2 (1 - K 2 ) is nearly unity, so that v tends to unity and K 
to zero : an approximation to these conditions is realized in 
the X-rays.f 

In the last years of the nineteenth century, attempts were 
made to form more definite conceptions regarding the behaviour 
of electrons within metals. It will be remembered that the 
original theory of electrons had been proposed by WeberJ for 
the purpose of explaining the phenomena of electric currents 
in metallic wires. Weber, however, made but little progress 
towards an electric theory of metals ; for being concerned 
chiefly with magneto-electric induction and electromagnetic 
ponder omotive force, he scarcely brought the metal into the 
discussion at all, except in the assumption that electrons of 
opposite signs travel with equal and opposite velocities relative 
to its substance. The more comprehensive scheme of his 
successors half a century afterwards aimed at connecting in 
a unified theory all the known electrical properties of metals, 
such as the conduction of currents according to Ohm's law, the 
thermo-electric effects of Seebeck, Peltier, and W. Thomson, 
the gal vano- magnetic effect of Hall, and other phenomena which 
will be mentioned subsequently. 

The later investigators, indeed, ranged beyond the group 
of purely electrical properties, and sought by aid of the theory of 
electrons to explain the conduction of heat. The principal ground 
on which this extension was justified was an experimental result 
obtained in 1853 by G. Wiedemann and K. FranzJ who found 

* Cf. p. 179. 

t Models illustrating the selective reflexion and absorption of light by metallic 
bodies and by gases were discussed by H. Lamb, Mem. and Proc. Manchester Lit. 
.and Phil. Soc. xlii (1898), p. 1 ; Proc. Lond. Math. Soc. xxxii (1900), p.. 11 ; Trans. 
'Camb. Phil. Soc. xviii (1900), p. 348. 

+ Cf. p. 226. Ann. d. Phys. Ixxxix (1853), p. 497. 



Closing Years of the Nineteenth Century. 457 

that at any temperature the ratio of the thermal conductivity 
of a body to its ohmic conductivity is approximately the same 
for all metals, and that the value of this ratio is proportional 
to the absolute temperature. In fact, the conductivity of a 
pure metal for heat is almost independent of the temperature; 
while the electric conductivity varies in inverse proportion to 
the absolute temperature, so that a pure metal as it approaches 
the absolute zero of temperature tends to assume the character 
of a perfect conductor. That the two conductivities are closely 
related was shown to be highly probable by the experiments 
of Tait^ in which pieces of the same metal were found to exhibit 
variations in ohmic conductivity exactly parallel to variations 
in their thermal conductivity. 

The attempt to explain the electrical and thermal properties 
of metals by aid of the theory of electrons rests on the assump- 
tion that conduction in metals is more or less similar to 
conduction in electrolytes ; at any rate, that positive and 
negative charges drift in opposite directions through the sub- 
stance of the conductor under the influence of an electric 
field. It was remarked in 1888 by J. J. Thomson,* who must 
be regarded as the founder of the modern theory, that the 
differences which are perceived between metallic and electro- 
lytic conduction may be referred to special features in the two 
cases, which do not affect their general resemblance. In 
electrolytes the carriers are provided only by the salt, which 
is dispersed throughout a large inert mass of solvent ; whereas 
in metals it may be supposed that every molecule is capable 
of furnishing carriers. Thomson, therefore, proposed to regard 
the current in metals as a series of intermittent discharges, 
caused by the rearrangement of the constituents of molecular 
systems a conception similar to that by which Grothussf had 
pictured conduction in electrolytes. This view would, as he 
showed, lead to a general explanation of the connexion between 
thermal and electrical conductivities. 

* J. J. Thomson, Applications of Dynamics to Physics and Chemistry, 1888, 
p. 296. Cf. also Giese, Ann. d. Phys. xxxvii (1889), p. 576. t Cf. p. 78. 



458 The Theory of Aether and Electrons in the 

Most of the later writers on metallic conduction have pre- 
ferred to take the hypothesis of Arrhenius* rather than that of 
Grothuss as a pattern ; and have therefore supposed the 
interstices between the molecules of the metal to be at all 
times swarming with electric charges in rapid motion. In 
1898 E. Eieckef effected an important advance by examining 
the consequences of the assumption that the average velocity of 
this random motion of the charges is nearly proportional to the 
square root of the absolute temperature T. P. DrudeJ in 1900 
replaced this by the more definite assumption that the kinetic 
energy of each moving charge is equal to the average kinetic 
energy of a molecule of a perfect gas at the same temperature , 
and may therefore be expressed in the form qT, where q denotes 
a universal constant. 

In the same year J. J. Thomson remarked that it would 
accord with the conclusions drawn from the study of ionization 
in gases to suppose that the vitreous and resinous charges play 
different parts in the process of conduction : the resinous 
charges may be conceived of as carried by simple negative 
corpuscles or electrons, such as constitute the cathode rays : 
they may be supposed to move about freely in the interstices 
between the atoms of the metal. The vitreous charges, on the 
other hand, may be regarded as more or less fixed in attachment 
to the metallic atoms. According to this view the transport of 
electricity is due almost entirely to the motion of the negative 
charges. 

An experiment which was performed at this time by Eiecke|| 
lent some support to Thomson's hypothesis. A cylinder of 
aluminium was inserted between two cylinders of copper in 
a circuit, and a current was passed for such a time that the 
amount of copper deposited in an electrolytic arrangement 

* Cf. p. 384. 

t G-ott. Nach., 1898, pp. 48, 137. Ann. d. Phys. Ixvi (1898), pp. 353, 545, 
1199; ii. (1900), p. 835. 

I Ann. d. Phys. (4) i (1900), p. 566 ; iii (1900), p. 369 ; vii (1902), p. 687. 
Rapports pres. au Congres de Physique, Paris, 1900, iii, p. 138. 
|| Phys. Zeitsch. iii (1901), p. 639.' 



Closing Years of the Nineteenth Centwy. 459* 

would have amounted to over a kilogramme. The weight of 
each of the three cylinders, however, showed no measurable 
change; from which it appeared unlikely that metallic con- 
duction is accompanied by the transport of metallic ions. 

The ideas of Thomson, Kiecke, and Drude were combined by 
Lorentz* in an investigation which, as it is the most complete r 
will here be given as the representative of all of them. 

It is supposed that the atoms of the metal are fixed, and 
that in the interstices between them a large number of resinous- 
electrons are in rapid motion. The mutual collisions of the 
electrons are disregarded, so that their collisions with the 
fixed atoms alone come under consideration ; these are 
regarded as analogous to collisions between moving and fixed 
elastic spheres. 

The flow of heat and electricity in the metal is 
supposed to take place in a direction parallel to the axis of 
x t so that the metal is in the same condition at all points of 
any plane perpendicular to this direction ; and the flow is 
supposed to be steady, so that the state of the system is 
independent of the time. 

Consider a slab of thickness dx and of unit area ; and suppose 
that the number of electrons in this slab whose ^-components 
of velocity lie between u and u + du, whose ^-components of 
velocity lie between v and v + dv, and whose ^-components of 
velocity lie between w and w + dw, is 

/ (u t v, w, x) dx du dv dw. 

One of these electrons, supposing it to escape collision, 
will in the interval of time dt travel from (x, y, z) to (x + u dt, 
y + vdt, z + wdt) : and its ^-component of velocity will at the 
end of the interval be increased by an amount e Edtjm^ if ra and 
e denote its mass and charge, and E denotes the electric force. 
Suppose that the number of electrons lost to this group by 
collisions in the interval dt is a dx du dv dw dt, and that the 

* Amsterdam Proceedings (English edition) vii (1904-1905), pp. 438, 585, 684 



460 The Theory of Aether and Electrons in the 

number added to the group by collisions in the same interval is 
b dx du dv dw dt. Then w^e have 

f (u, v, w, x) + (b a) dt = f (u + eE dt/m, v, w, x + u dt), 
.and therefore 

7JT ^\ J? r\ J? 

i &fi of oT 

b - a = + u 

m du dx 

Now, the law of distribution of velocities which Maxwell 
postulated for the molecules of a perfect gas at rest is expressed 
by the equation 

r z 
/= TT~^ a' 3 Ne"*, 

where N denotes the number of moving corpuscles in unit 
volume, r denotes the resultant velocity of a corpuscle (so that 
r 2 = u* + v~ + w*), and a denotes a constant which specifies the 
-average intensity of agitation, and consequently the temperature. 
It is assumed that the law of distribution of velocities 
among the electrons in a metal is nearly of this form; but a 
term must be added in order to represent the general drifting of 
the electrons parallel to the axis of x. The simplest assumption 
that can be made regarding this term is that it is of the form 

u x a function of r only ; 
we shall, therefore, write 



a 



/ = NTT~* a' 3 e 2 + u^ (r). 
The value of ^ (r) may now be determined from the equation 

eE'df df 
b - a = ~- + u -; 
m du dx 

for on the left-hand side^ the Maxwellian term 



would give a zero result, since b is equal to a in Maxwell's 
.system ; thus b - a must depend solely on the term u-% (r) ; and 



Closing Years cf the^ Nineteenth Century. 46 T 

an examination of the circumstances of a collision, in the manner 
of the kinetic theory of gases, shows that (b - a) must have the 
form - ur^ (r)/l, where I denotes a constant which is closely 
related to the mean free path of the electrons. In the terms 
on the right-hand side of the equation, on the other hand, 
Maxwell's term gives a result different from zero; and in 
comparison with this we may neglect the terms which arise 
from u\ (r). Thus we have 

urv(r) leE d 8\ N --, 



/ \m 

or 

lu --, fieNE d (N\ 2M* da) . 



and thus the law of distribution of velocities is determined. 
The electric current i is determined by the equation 

i = e Jj'J uf (it, v, w) du dv dw, 

where the integration is extended over all possible values of the 
components of velocity of the electrons. The Maxwellian term 
in f (u, v, w) furnishes no contribution to this integral, so we 

have 

i = e JJJ v? x (r) du dv dw. 

When the integration is performed, this formula becomes 






mu dx ' dxf 

or 

STT^W a . m /a 2 dN da\ 

'~&N l + 2~e(N~fa'* a dx)' 

The coefficient of i in this equation must evidently represent 
the ohmic specific resistance of the metal ; so if y denote the 
specific conductivity, we have 

4/r N 



Let the equation be next applied to the case of two metals 
A and B in contact at the . same . temperature T, forming an 



462 The Theory of Aether and Electrons in the 

open circuit in which there is no conduction of heat or electricity 
.(so that i and da/dx are zero). Integrating the equation 

= m a 2 clN 
" 2eNdx 

.across the junction of the metals, we have 

Discontinuity of potential at junction = -^ log - ; 

or since f ma 2 , which represents the average kinetic energy of an 
electron, is by Drude's assumption equal to q/T, where gr denotes 
a universal constant, we have 

2 q N 

Discontinuity of potential at junction = ^ - T log ~- 

O > J\ A 

This may be interpreted as the difference of potential con- 
nected with the Peltier* effect at the junction of two metals ; 
the product of the difference of potential and the current 
measures the evolution of heat at the junction. The Peltier 
discontinuity of potential is of the order of a thousandth of a 
volt, and must be distinguished from Volta's contact-difference 
of potential, which is generally much larger, and which, as it 
presumably depends on the relation of the metals to the medium 
in which they are immersed, is beyond the scope of the present 
investigation. 

Eeturning to the general equations, we observe that the flux 
of energy JFis parallel to the axis of x, and is given by the 
equation 

W = \m HI ur i f(u, v, iv) du dv dw, 

where the integration is again extended over all possible values 
of the components of velocity ; performing the integration, we 
have 



or, substituting for E from the equation already found, 

TT _ ma 2 . 4ml da 

W = ^ - -rr Na z -7- 

e 871-2 dx 

* Cf. p. 264. 



Closing Years of the Nineteenth Century. 463 

Consider now the case in which there is conduction of heat 
without conduction of electricity. The flux of energy will in this 
case be given by the equation 



W--*** 

K 



where K denotes the thermal conductivity of the metal expressed 
in suitable units ; or 

3ma da 
W = - K.-^ -j-- 

2q dx 

If it be assumed that the conduction of heat in metals is 
effected by motion of the electrons, this expression may be 
compared with the preceding; thus we have 



and comparing this with the formula already found for the 
electric conductivity, we have 



7 W ' 

an equation which shows that the ratio of the thermal to the 
electric conductivity is of the form T x a constant which is the 
same for all metals. This result accords with the law of 
Wiedernann and Franz. 

Moreover, the value of q is known from the kinetic theory of 
gases; and the value of e has been determined by J. J. Thomson* 
and his followers ; substituting these values in the formula for K/y, 
a fair agreement is obtained with the values of K/y determined 
experimentally. 

It was remarked by J. J. Thomson that if, as is postulated 
in the above theory, a metal contains a great number of free 
electrons in temperature equilibrium with the atoms, the 
specific heat of the metal must depend largely on the energy 
required in order to raise the temperature of the electrons. 
Thomson considered that the observed specific heats of metals 
are smaller than is compatible with the theory, and was thus 

* Cf. p. 407. 



464 The Theory of Aether and Electrons in the 

led to investigate* the consequences of his original hypothesis^ 
regarding the motion of the electrons, which differs from the 
one just described in much the same way as Grothuss' theory. of 
electrolysis differs from Arrhenius'. Each electron was now 
supposed to be free only for a very short time, from the moment 
when it is liberated by the dissociation of an atom to the moment 
when it collides with, and is absorbed by, a different atom. The 
atoms were conceived to be paired in doublets, one pole of each 
doublet being negatively, and the other positively, electrified. 
Under the influence of an external electric field the doublets 
orient themselves parallel to the electric force, and the electrons 
which are ejected from their negative poles give rise to a current 
predominantly in this direction. The electric conductivity of 
the metal may thus be calculated. In order to comprise the 
conduction of heat in his theory, Thomson assumed that the 
kinetic energy with which an electron leaves an atom is pro- 
portional to the absolute temperature ; so that if one part of the 
metal is hotter than another, the temperature will be equalized 
by the interchange of corpuscles. This theory, like the other, leads 
to a rational explanation of the law of Wiedemann and Franz. 

The theory of electrons in metals has received support 
from the study of another phenomenon. It was known to 
the philosophers of the eighteenth century that the air near 
an incandescent metal acquires the power of conducting elec- 
tricity. "Let the end of a poker," wrote Canton,J "when 
red-hot, be brought but for a moment within three or four 
inches of a small electrified body, and its electrical power will 
be almost, if not entirely, destroyed." 

The subject continued to attract attention at intervals ; 

* J. J. Thomson, The Corpuscular Theory of Matter ; London, 1907. 

f Cf. p. 457. \ Phil. Trans, lii (1762), p. 457. 

Cf. E. Becquerel, Annales de Chimie xxxix (1853), p. 355 ; Guthrie, Phil. 
Mag. xlvi (1873), p. 254; also various memoirs by Elster and Geitel in the 
A'nnaleri d. Phys. from 1882 onwards. The phenomenon is very noticeable, as 
Edison showed (Engineering, December 12, 1884, p. 553), when a filament of 
carbon is hearted to incandescence in a rarefied gas. In recent years it has been 
found that ions are emitted when magnesia, or any of the oxides of the alkaline 
earth metals, is heated to a dull red heal. ; , 



Closing Years of the Nineteenth Century. 465 

and as the process of conduction in gases came to be better 
understood, the conductivity produced in the neighbourhood of 
incandescent metals was attributed to the emission of electrically 
charged particles by the metals. But it was not until the develop- 
ment of J. J. Thomson's theory of ionization in gases that notable 
advances were made. In 1899, Thomson* determined the ratio 
of the charge to the mass of the resinously charged ions emitted 
by a hot filament of carbon in rarefied hydrogen, by observing 
their deflexion in a magnetic field. The value obtained for 
the ratio was nearly the same as that which he had found for 
the corpuscles of cathode rays ; whence he concluded that 
the negative ions emitted by the hot carbon were negative 
electrons. 

The corresponding investigation-)- for the positive leak from 
hot bodies yielded the information that the mass of the positive 
ions is of the same order of magnitude as the mass of material 
atoms. There are reasons for believing that these ions are 
produced from gas which has been absorbed by the superficial 
layer of the metal.J 

If, when a hot metal is emitting ions in a rarefied gas, an 
electromotive force be established between the metal and a 
neighbouring electrode, either the positive or the negative ions 
are urged towards the electrode by the electric field, and a current 
is thus transmitted through the intervening space. When the 
metal is at a higher potential than the electrode, the current is 
carried by the vitreously charged ions : when the electrode is 
at the higher potential, by those with resinous charges. In 
either case, it is found that when the electromotive force is 
increased indefinitely, the current does not increase indefinitely 
likewise, but acquires a certain " saturation " value. The 
obvious explanation of this is that the supply of ions available 
for carrying the current is limited. 

* Phil. Mag. xlviii (1899), p. 547. 

t J. J. Thomson, Proc. Camb. Phil. Soc. xv (1909), p. 64 ; 0. W. Richardson, 
Phil. Mag. xvi. (1908), p. 740. 

+ Cf. Richardson, Phil. Trans, ccvii (1906), p. 1. 

2 H 



466 The Theory of Aether and Electrons in the 

When the temperature of the metal is high, the ions 
emitted are mainly negative; and it is found* that in these 
circumstances, when the surrounding gas is rarefied, the satura- 
tion-current is almost independent of the nature of the gas or 
of its pressure. The leak of resinous electricity from a metallic 
surface in a rarefied gas must therefore depend only on the 
temperature and on the nature of the metal ; and it was shown 
by 0. W. Richardsonf that the dependence on the temperature 
may be expressed by an equation of the form 

b 



where i denotes the saturation-current per unit area of 
surface (which is proportional to the number of ions emitted in 
unit time), T denotes the absolute temperature, and A and b 
are constants.! 

In order to account for these phenomena, Eichardson 
adopted the hypothesis which had previously been proposed || 
for the explanation of metallic conductivity ; namely, that 
a metal is to be regarded as a sponge- like structure of 
comparatively large fixed positive ions and molecules, in the 
interstices of which negative electrons are in rapid motion. 
Since the electrons do not all escape freely at the surface, he 
postulated a superficial discontinuity of potential, sufficient to 
restrain most of them. Thus, let N denote the number of free 
electrons in unit volume of the metal ; then in a parallelepiped 
whose height measured at right angles to the surface is dx, 
and whose base is of unit area, the number of electrons whose 

* Cf. J. A. McClelland, Proe. Camb. Phil. Soc. x (1899), p. 241; xi (1901), 
p. 296. On the results obtained when the gas is hydrogen, cf. H. A. Wilson, 
Phil. Trans, ccii (1903), p. 243; ccviii (1908), p. 247; and 0. W. Richardson, 
Phil. Trans, ccvii (1906), p. 1. 

fProc. Camb. Phil. Soc. xi (1902), p. 286; Phil. Trans, cci (1903), p. 497. 
Cf. also H. A. Wilson, Phil. Trans, ccii (1903), p. 243. 

J The same law applies to the emission from other bodies, e.g. heated 
alkaline earths, and to the emission of positive ions at any rate when a steady 
state of emission has been reached in a gas which is at a definite pressure. 

Phil. Trans, cci (1903), p. 497. 

|| Cf. pp. 457 et sqq. 



Closing Years of the^ Nineteenth Century. 467 

^-components of velocity are comprised between u and u + du is 

^ 
TT * a" 1 JVe * du dx, where | ma 2 = qT, 

m denoting the mass of an electron, T the absolute temperature, 
and q the universal constant previously introduced. 

Now, an electron whose ^-component of velocity is u will 
arrive at the interface within an interval dt of time, provided 
that at the beginning of this interval it is within a distance u dt 
of the interface. So the number of electrons whose ^-com- 
ponents of velocity are comprised between u and u + du which 
arrive at unit area of the interface in the interval dt is 



If the work which an electron must perform in order to escape 
through the surface layer be denoted by </>, the number of 
electrons emitted by unit area of metal in unit time is 
therefore 



c **udu, or 
*** 

The current issuing from unit area of the hot metal is thus 

20 30 

JirWeae'^, or N 

where t denotes the charge on an electron. This expression, 
being of the form 



agrees with the experimental measures ; and the comparison 
furnishes the value of the superficial discontinuity of potential 
which is implied in the existence of 0.* 

A few years after the date of this investigation, a plan was 

* This discontinuity of potential was found to be 2*45 volts for sodium, 4-1 
volts for platinum, and 6-1 volts for carbon. 



468 The Theory of Aether and Electrons in the 

devised and successfully carried out* for determining experi- 
mentally the kinetic energy possessed by the ions after 
emission. The mean kinetic energy of both negative and 
positive ions was found to be the same for various metals 
(platinum, gold, silver, etc.), and to be directly proportional to 
the absolute temperature; and the distribution of velocities 
among the ions proved to be that expressed by Maxwell's law. 
The ions may therefore be regarded as kinetically equivalent 
to the molecules of a gas whose temperature is the same as that 
of the metal. 

By the investigations which have been recorded, the hypo- 
thesis of atomic electric charges has been, to all appearances, 
decisively established. But all the parts of the theory of 
electrons do not enjoy an equal degree of security; and in 
particular, it is possible that the future may bring important 
changes in the conception of the aether. The hope was 
formerly entertained of discovering an aether by reference to 
which motion might be estimated .absolutely ; but such a hope 
has been destroyed by the researches which have sprung from 
Fitz Gerald's hypothesis of contraction; and in some recent 
writings it is possible to recognize a tendency to replace the 
classical aether by other conceptions, which, however, have 
been as yet but indistinctly outlined. 

In any event, the close of the nineteenth century brought to 
an end a well-marked era in the history of natural philosophy ; 
and this is true not only with respect to the discoveries them- 
selves, but also in regard to the conditions of scientific organiza- 
tion and endeavour, which in the last decades of that period 
became profoundly changed. The investigators who advanced 
the theories of aether and electricity, from the time of Descartes 
to that of Lord Kelvin, were, with very few exceptions, 
congregated within a narrow territory : from Dublin to the 
western provinces of Russia, and from Stockholm to the north 
of Italy, may be circumscribed by a circle of no more than six 

* 0. W. Richardson and F. C. Brown, Phil. Mag. xvi (1908), pp. 353, 890 ; 
F. C. Brown, Phil. Mag. xvii (1909), p. 355 ; xviii (1909), p. 649 



Closing Years of the Nineteenth Century. 469 

hundred miles radius. But throughout the whole of Kelvin's 
long life, the domain of culture was rapidly extending : the 
learning of the Germanic and Latin peoples was carried to the 
furthest regions of the earth : new universities were founded, 
and inquiries into the secrets of nature were instituted in 
every quarter of the globe. Let this record close with the 
anticipation that fellowship in the pursuit of knowledge will 
increase in the nations the spirit of generous emulation and 
mutual respect. 



INDEX OF AUTHORS CITED. 



Abraham, M., 323, 352. 

Aepinus, F. U. T., 47-52, 55. 

Airy, Sir G. B., 120, 191, 214, 215. 

Aitken, J., 403. 

Ampere, A. M., 87-92, 312. 

Ango, P., 24. 

Arago, F., 86, 114, 116, 121, 122, 136, 

173. 

Arrhenius, S., 383, 384. 
Aschkinass, E., 295. 
Aubel, E. van, 322. 
Anlinger, E., 356. 

Bacon, Sir F., Lord Verulam, 2, 3, 33. 

Banks, Sir J., 75. 

Bartholin, E., 25. 

Bartoli, A., 306. 

Basset, A. B., 370. 

liatelli, A., 267. 

Becearia, G. B., 49, 53, 67, 75. 

Becher, J. J., 36. 

Becquerel, A. C., 93, 94. 

Becquerel, E., 464. 

Bec 4 uerel, H., 408, 409, 410. 

Belopolsky, A., 416. 

Bennet, A., 73, 304. 

Bernoulli, D., 9, 50. 

Bernoulli, John (the elder), 101.. 

Bernoulli, John (the younger), 9, 100- 
102. 

Berthollet, A., 112. 

Berzelius, J. J., 80-83. 

Betti, E., 65. 

Bezold, W. v., 357. 

Biot, J. B., 86, 114, 174. 

Bjerknes, C. A., 316, 317. 

Bjeiknes, V., 303. 

Blondlot, R., 431, 432. 

Boerhaave, H., 35. 

Boltzmann, L., 206, 322, 325, 356. 

Boscovich, R. G., 33, 161, 217. 



Bottomley, J. T., 297. 

Boussinesq, J., 185-187, 215. 

Boyle, U., 11, 17, 31-33, 35. 

Brace, D. B., 439. 

Bradley, J., 99, 100. 

Brewster, Sir D., Ill, 113, 134, 177. 

Brougham, H., Lord, 108. 

Brouncker, Viscount, 10. 

Brown, F. C., 467. 

Brugmans, A., 56, 218. 

Budde, E., 263. 

Buffon, G. L. L., Comte de, 48. 

Cabeo, N., 31, 189. 

Campbell, L., 283, 296. 

Canton, J., 50, 464. 

Carlisle, Sir A., 75, 78. 

Cascariolo, V., 19, 20. 

Cassini, G. D., 22. 

Cauchy, A. L M 132, 139, 142-150, 158, 

159, 161, 163, 165, 167, 170, 177- 

179, 182, 183, 294. 
Cavendish, Lord C., 51. 
Cavendish, Hon. H., 51-54, 75. 94, 

167, 207. 

Chandler, S. C., 100. 
Charlier, C. V. L., 190, 269. 
Chasles, M. 190, 269. 
Chattock, A. P., 357. 
Chladni, E. F. F., 110. 
Christiansen, C., 291. 
Christie, S. H., 213. 
Clausius, R., 231, 234, 261-263, 274, 

373, 420-422. 
ColUnson, P., 43, 46. 
Corbino, 0. M., 454. 
Cornu, M. A., 216, 282, 451. 
Cotton, A., 454. 
Coulomb, C. A., 56-59. 
Courtivron, G., Marquis de, 104. 
Crookes, Sir W., 306, 394, 395, 



472 



Index. 



Cruickshank, W., 75, 76. 
Gumming, J., 93, 266. 
Cunaeus, 41. 
Curie, P., 235, 409. 
Curie, Mme. S., 409. 

Daniell, F., 206, 373. 

Darbishire, F. V., 204. 

Davy, J., 194. 

Davy, SirH., 76-78, 80, 94, 95, 188, 

197, 372, 392. 
De La Hire, nee La Hire. 
Delambre, J. B. J., 22. 
De la Eive, A., 79, 80. 
De la Rive, L., 197, 201, 202, 360. 
Desaguliers, J. T., 37-39. 
Descartes, R., 2-9, 38, 85. 
Des Coudres, T., 433. 
Desormes, C. B., 84. 
Digby, K., 31. 
Donati, L., 349. 
Doppler, C., 415. 
Drude, P., 370, 429, 458, 459. 
Du Fay, C. F., 39, 40, 44, 303. 
Duhem, P., 281. 
Dulong, P. L., 132. 

Ebert, 396, 399. 
Edison, T., 464. 
Eichenwald, A., 339, 427. 
Einstein, A., 440, 447. 
Elster, J., 464. 
Ettingshausen, A. v., 322. 
Euler, L., 9, 66, 103, 104, 304. 
Ewing, J. A., 237. 

Fabroni, G., 71, 76. 

Faraday, M., 45, 58, 82, 85, 188-221, 
244, 248, 254, 264, 269, 271, 272, 
275, 276, 279, 284, 286, 288, 300, 
339, 349, 350, 373, 391, 448. 

Fechner, G. T., 98, 201, 225, 226. 

Fermat, P. de, 9, 10, 102, 103. 

Fitz Gerald, G. F., 157, '263, 308, 318, 
319, 323, 324, 325, 327, 332, 333, 
334, 340, 341, 345-347, 361, 364, 
367, 368, 370, 396, 401, 405, 432, 
433, 437, 438. 

Fi/eau, H. L., 117, 136, 254, 282, 283, 
416. 

Fiippl, A., 264. 



Foucault, L., 136, 282, 283. 

Fourcroy, A. F. de, 93. 

Fourier, J., Baron, 95, 132, 139, 256. 

Franklin, B., 41-51,84, 103. 

Franklin, W. S., 264. 

Franz, R., 456, 457. 

Fresnel, A., 24, 28, 113-136, 148, 174. 

Frohlich, I., 263. 

Galileo, G., 21. 

Galitzine, B., 306. 

Gallop, E. G., 237, 238. 

Galvani, L., 67-71. 

Garnett, W., 283, 296. 

Gauss, K. F., 58 S 225-231, 268, 269. 

Gautherot, N., 94. 

Guy-Lussac, L. J., 199. 

Geest, J., 454. 

Geissler, H., 392. 

Geitel, H., 464. 

Gibbs, J. Willard, 283, 297, 378, 380, 
423. 

Giese, W., 397, 398, 457. 

Giesel, F., 409. 

Gilberd or Gilbert, W., 8, 29-31. 

Glasenapp, S. von, 22. 

Glazebrook, R. T., 131, 160, 164, 172, 

173, 370. 

Goldhammer, D. A., 370, 371. 
Goldstein, E., 393, 396, 406. 
Gounelle, E., 254. 
Gouy, G., 401. 
Grassnmnn, H., 91, 231. 
Gray, S., 37, 38, 49. 
s'Gravesande, W. J., 32, 34, 36, 108. 
Green, G., 64-66, 150-154, 158, 161- 

165, 167, 168, 170, 179, 296. 
Gren, F. A. C., 70, 74. 
Grimaldi, F. M., 11. 
Grothuss, T., Freiherr v., 78-81. 
Grove, Sir W. R., 241. 
Guericke, 0. v., 37- 
Guthrie, F., 464. 

Hachette, J. N. P., 84. 
Haga, H., 402. 
Hagen, E., 455. 
Hall, E. H., 320-323. 
Halley, E., 99, 106. 
Hallo, J. J., 454. 



Index. 



473 



Hallwachs, W., 399, 400. 
Hamilton, Sir W. R., 131, 139. 
Hansteen, C., 84. 
Hasenohrl, F., 370. 
Hastings, C. S., 131, 172. 
Hattendorf, K., 231. 
Hauksbee, F., 39, 390. 
Heaviside, 0., 341-344, 366, 367. 
Heliodorus of Larissa, 10. 
Helmholtz, H. v., 196, 205, 229, 240- 
243, 247, 253, 261, 274, 275, 288, 
293, 297, 307, 312, 325, 337-339, 
353, 357, 378-382, 386, 397, 429. 
Helmholtz, R. v., 403. 
Henry, J., 193, 253, 358. 

Hero of Alexandria, 10. 

Herschel, Sir J., 174, 213. 

Herschel, Sir W., 54. 

Hertz, H., 347, 353-366, 396, 399, 405, 
411, 429, 431, 432. 

Hicks, W. M., 316, 327, 328, 333-336, 
417. 

Hittorf, W., 374, 375, 393, 396, 398, 
399. 

Hoek, M., 118, 120. 

Holzmiiller, G., 233. 

Homberg, W., 34, 35, 303. 

Hooke, R., 11-17, 33, 36, 122. 

Hopkinson, J., 321. 

Horsley, S., 17. 

Howard, J. L., 363. 

Hughes, D. E., 237. 

Hull, G. F., 307. 

Hutchinson, C. T., 339. 

Huygens, C., 6, 17, 22-28, 99, 145, 
181. 

Jacobi, M. H., 201. 
Jaequier, F., 54. 
Jenkin, W., 193, 194. 
Joule, J. P., 239, 240, 242. 

Kahlbaum, G. W. A., 204. 

Kaufmann, W., 343, 406. 

Kelvin, see Thomson, W. 

Kepler, J., 304. 

Kerr, J., 338, 368, 370. 

Kirchhoff, G., 250-252, 257-259, 260- 

261, 312. 
Kleist, E. G. v., 41. 



Koenigsberger, L., 241. 
Kohlrausch, F. W., 374. 
Kohlrausch, R., 251, 252, 259. 
Kolacek, F., 323. 
Konig, C. G. W., 451. 
Korn, A, 317. 
Korteweg, D. J., 91. 
Kundt, A., 291. 
Kiistner, F., 100. 

Lagrange, J. L., 60, 103, 139. 

La Hire, P. de, 22, 189. 

Lamb, H., 261, 344, 456. 

Lambert, J. H., 55. 

Langevin, P., 438. 

Laplace, P. S., Marquis de, 60, 61, 109, 

110, 112, 114, 132, 139, 232, 233. 
Larmor, Sir J., 118, 167, 319, 323, 343, 

362, 363, 368, 370, 430, 435, 438, 

439. 

Lavoisier, A. L., 33, 35, 36. 
Leahy, A. H., 317, 318. 
Leathern, J. G., 370. 
Lebedew, P., 307. 
Lecher, E., 360, 
Lee, A., 361. 
Legendre, A.M., 60. 
Lenard, P., 396, 404. 
Lenz, E., 222. 
Leroux, F. P., 291. 
Le Seur, T., 54. 
Le Verrier, U. J. J., 234. 
Levy, M., 234. 
Lienard, H., 436. 
Lippmann, G., 375-378. 
Lloyd, H., 131. 
Lodge, Sir 0. J., 311, 320, 357, 358, 

363, 401, 418, 432. 
Lorberg, H., 231, 356. 
Lorentz, H. A., 290, 322, 337, 412, 

413, 416-449, 459-463. 
Lorenz, L., 169, 297-300, 324, 361. 

Macaluso, D., 454. 

Macaulay, Lord, 108. 

McClelland, J. A., 466. 

MacCullagh, J., 130, 148-150, 154-157, 

175-179, 289, 295, 296. 
Macdonald, H. M., 348. 



2 I 



474 



Index. 



Mairan, J. J. de, 303. 

Malus, E. L., Ill, 112, 177. 

Marcet, M., 188. 

Marianini, S., 201. 

Mascart, E., 121, 416. 

Maupertuis, P. L. M. de, 102, 103. 

Maxwell, J. Clerk, 52, 65, 92, 102, 167, 

190, 215, 237, 250, 263, 268-313, 

321, 333, 337, 348, 365, 397, 411, 

413, 460. 
Mayer, E., 242. 
Mayer, T., 55. 
Melvill, T., 104. 
Meyer, S., 409. 
Michel), J., 54, 55, 116, 161, 167, 217, 

303. 

Michelson, A. A., 117, 283, 417, 418. 
Miller, W. A., 373. 
Minkowski, H., 448. 
Morichini, D. P., 213. 
Morley, E. W., 117, 417,418. 
Morton, W. B., 343. 
Moser, J., 381. 
Mossotti, F. 0., 211, 286. 
Mottelay, P. F., 8. 
Musschenbroek, P. van, 41, 55. 

Navier, C. L. M. H., 138-140. 
Nernst, W., 380, 386-389. 
Neumann, C., 176, 215, 216, 312. 
Neumann, F. E., 143, 148, 149, 184, 

222--22S, 261. 
Newcomb, S., 283. 
Newton, Sir I., 9, 15-21, 28, 31-34, 

53, 106, 107. 
Nichols, E. F., 307. 
Nichols, E. L., 264. 
Nicholson, W., 75, 77, 78. 
Niven, C., 344. 
Nobili, L., 193. 
Noble, H. E., 438. 
Nollet, J. A., 40-42, 47, 48, 391. 
Nyren, M., 100. 

O'Brien, M., 142, 184. 
Oersted, H. C., 84-87. 
Ohm, G. S., 95-98, 201,. 
Oppenheim, S., 234. 
Ostwald, W., 384. 



Palmaer, W., 381. 

Pardies, I. G., 24. 

Peacock, G., 108, 125. 

Pearson, K., 140, 164, 185, 361. 

Peltier, J. C., 264-267. 

Pender, H., 339. 

Peregrinus, P., 7, 8, 189. 

Pen-in, J., 400. 

Perrot, A., 397. 

Pfaff, C. H., 76, 201. 

Planck, M., 378, 386, 413, 429. 

Pliicker, J., 219, 220, 392, 393. 

Poggendorff, J. C., 201. 

Poincare, H., 352, 360, 361. 

Poisson, S. D., 59-65, 114, 115, 134 r 

139-141, 245, 246. 
Pouillet, C. S. M., 193, 373. 
Poynting, J. H., 347-350. 
Preston, S. T., 193. 
Priestley, J., 36, 50-54, 75, 161, 283, 

303, 304, 393. 

Eankine, W. J. M., 140, 171. 

Eaoult, F., 383. 

Eayleigh, J. W. Strutt, Lord, 167, 170 r 

171, 179, 181, 283, 290, 292, 344, 

417, 439. 
Eeich, F., 219. 
Eeiff, E., 319, 370, 429. 
Eespighi, L., 120. 
Eichardson, 0. W., 465, 466, 467. 
Eiecke, E., 395, 458, 459. 
Eiemann, B., 231, 234, 261-263, 268, 

269, 297, 324. 
Eitchie, W., 194. 
Eitter, J. W., 75, 375. 
Eobison, J., 51, 116. 
Eoemer, 0., 22, 99. 
Eoget, P. M., 78, 202, 203. 
Eontgen, W. C., 400, 401, 426, 427. 
Eowlaml, H. A., 321, 339, 344, 368, 

369, 370, 427. 
Eubens, H., 295, 455. 
Eumford, B. Thompson, Count, 35, 188, 

242. 
Eutherford, E., 402, 407, 409. 

Saint-Venant, B. de, 163, 164. 
Sampson, E.A., 22. 



Index. 



475 



Sarasin, E., 360. 

Savart, F., 86. 

Savary, F., 253, 414. 

Scheele, K. W., 35, 36. 

Schiller, N., 338. 

Schmidt, G. C., 409. 

Schonbein, C. F., 204. 

Schuster, A., 343, 398, 399, 401, 406. 

Schweidler, E. v., 409. 

Searle, G. F. C., 343. 

Seebeck, T. J., 92, 93, 265, 266. 

Seegers, 233. 

Seeliger, H., 100, 414. 

Sellmeier, W., 293, 295. 

Snell, W., 6. 

Socin, A., 50. 

Somerville, M., 213. 

Sommerfeld, A., 319. 

Spence, 43. 

Stahl, G. E., 36. 

Stefan, J., 306, 345. 

Stokes, Sir G. G., 117, 131, 132, 137, 
141, 167-169, 171, 172, 197, 255, 
273, 291, 296, 401, 411, 412. 

Stoney, G. Johnstone, 397. 

Struve, W., 100. 

Sulzer, J. G., 67. 

Symmer, R., 56. 

Tait, P. G., 91, 267, 395, 448, 449, 
457. 

Talcott, 100. 

Taylor, B., 35. 

Thenard, L. J., 93, 199. 

Thomson, Sir J. J., 167, 326, 339, 340, 
343, 344, 350-353, 365, 370, 396, 
400, 402-407, 419, 457, 458, 459, 
463, 464, 465. 

Thomson, W. (Lord Kelvin), 52, 101, 
140, 157-161, 165-168, 173, 174, 
209, 211, 219, 240-250, 253-257, 
265-267, 269, 270, 274-276, 279, 
284, 286, 292, 294, 297, 310, 311, 
313, 315, 316, 325, 326, 328-332, 
336, 370, 400. 

Tisserand, F., 233, 234. 



Todhunter, I., 140. 
Torricelli, E., 23. 
Trouton, F. T., 364, 438. 
Tyndall, J., 219. 

Van Marum, M., 57, 76, 84. 
Van 't Hoff, J. H., 388. 
Varley, C. F., 376, 393. 
Vauquelin, L. N., 93. 
Verdet, E., 125, 215, 216. 
Villarceau, Y., 414, 415. 
Voigt, W., 370, 440, 454. 
Volta, A., 57, 70-76, 195, 252, 375. 

Walker, G. T., 353. 

Wangerin, A., 143. 

Warburg, E., 380. 

Watson, Sir W., 42, 43, 48, 51, 254, 

390, 391. 

Watson, H. W., 288. 
Weber, W., 193, 2J9, 225-236, 259, 

261-263, 268, 282, 283, 356, 456. 
Welby, F. A., 241. 
Wheatstone, Sir C., 98, 254. 
Whiston, W., 9. 

Wiechert, E., 401, 404, 436, 454. 
Wiedemann, E., 396, 399. 
Wiedemann, G., 456, 457. 
Wien, W., 343, 406. 
Wiener, 0., 364. 
Wilberforce, L. R., 311. 
Wilcke, J. K., 48, 50, 56. 
Williams, A., 37. 
Williamson, A., 372, 373. 
Wilson, C. T. R., 403. 
Wilson, H. A., 432, 466. 
Wilson, P., 116. 
Wind, C. H., 370, 402. 
Witte, H., 324. 
Wollaston, W. H., 76, 77, 109, 252. 

Young, T., -28, 105-111, 115, 121-123, 
125, 132, 134, 136, 167, 304. 

Zeeman, P., 449, 450, 451. 
Zeleny, J., 407. 



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