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Title: James Clerk Maxwell and Modern Physics
Author: Glazebrook, Richard
Language: English
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[Illustration: (cover)]
_The Century Science Series_
Edited by Sir Henry E. Roscoe, D.C.L., LL.D., F.R.S.
JAMES CLERK MAXWELL AND MODERN PHYSICS
* * * * * *
The Century Science Series.
EDITED BY
SIR HENRY E. ROSCOE, D.C.L., F.R.S., M.P.
John Dalton and the Rise of Modern Chemistry.
By Sir HENRY E. ROSCOE, F.R.S.
Major Rennell, F.R.S., and the Rise of English Geography.
By CLEMENTS R. MARKHAM, C.B., F.R.S., President of the Royal
Geographical Society.
Justus von Liebig: his Life and Work (1803–1873).
By W. A. SHENSTONE, F.I.C., Lecturer on Chemistry in Clifton
College.
The Herschels and Modern Astronomy.
By AGNES M. CLERKE, Author of “A Popular History of Astronomy
during the 19th Century,” &c.
Charles Lyell and Modern Geology.
By Rev. Professor T. G. BONNEY, F.R.S.
James Clerk Maxwell and Modern Physics.
By R. T. GLAZEBROOK, F.R.S., Fellow of Trinity College,
Cambridge.
_In Preparation._
Michael Faraday: his Life and Work.
By Professor SILVANUS P. THOMPSON, F.R.S.
Humphry Davy.
By T. E. THORPE, F.R.S., Principal Chemist of the Government
Laboratories.
Pasteur: his Life and Work.
By M. ARMAND RUFFER, M.D., Director of the British Institute of
Preventive Medicine.
Charles Darwin and the Origin of Species.
By EDWARD B. POULTON, M.A., F.R.S., Hope Professor of Zoology
in the University of Oxford.
Hermann von Helmholtz.
By A. W. RÜCKER, F.R.S., Professor of Physics in the Royal
College of Science, London.
CASSELL & COMPANY, LIMITED, _London_; _Paris_ & _Melbourne_.
* * * * * *
[Illustration: J. Clerk Maxwell
(_From a Photograph of the Picture by G. Lowes Dickinson, Esq., in the
Hall of Trinity College, Cambridge._)]
The Century Science Series
JAMES CLERK MAXWELL AND MODERN PHYSICS
by
R. T. GLAZEBROOK, F.R.S.
Fellow of Trinity College, Cambridge
University Lecturer in Mathematics, and Assistant Director of the
Cavendish Laboratory
Cassell and Company, Limited
London, Paris & Melbourne
1896
All Rights Reserved
[Illustration]
PREFACE.
The task of giving some account of Maxwell’s work--of describing the
share that he has taken in the advance of Physical Science during the
latter half of this nineteenth century--has proved no light labour.
The problems which he attacked are of such magnitude and complexity,
that the attempt to explain them and their importance, satisfactorily,
without the aid of symbols, is almost foredoomed to failure. However,
the attempt has been made, in the belief that there are many who,
though they cannot follow the mathematical analysis of Maxwell’s work,
have sufficient general knowledge of physical ideas and principles to
make an account of Maxwell and of the development of the truths that he
discovered, subjects of intelligent interest.
Maxwell’s life was written in 1882 by two of those who were most
intimately connected with him, Professor Lewis Campbell and Dr.
Garnett. Many of the biographical details of the earlier part of this
book are taken from their work. My thanks are due to them and to their
publishers, Messrs. Macmillan, for permission to use any of the letters
which appear in their biography. I trust that my brief account may
be sufficient to induce many to read Professor Campbell’s “Life and
Letters,” with a view of learning more of the inner thoughts of one who
has left so strong an imprint on all he undertook, and was so deeply
loved by all who knew him.
R. T. G.
_Cambridge,
December, 1895._
CONTENTS.
CHAPTER PAGE
I. EARLY LIFE 9
II. UNDERGRADUATE LIFE AT CAMBRIDGE 28
III. EARLY RESEARCHES--PROFESSOR AT ABERDEEN 38
IV. PROFESSOR AT KING’S COLLEGE, LONDON--LIFE AT GLENLAIR 54
V. CAMBRIDGE--PROFESSOR OR PHYSICS 60
VI. CAMBRIDGE--THE CAVENDISH LABORATORY 73
VII. SCIENTIFIC WORK--COLOUR VISION 93
VIII. SCIENTIFIC WORK--MOLECULAR THEORY 108
IX. SCIENTIFIC WORK--ELECTRICAL THEORIES 148
X. DEVELOPMENT OF MAXWELL’S THEORY 202
JAMES CLERK MAXWELL
AND MODERN PHYSICS.
CHAPTER I.
EARLY LIFE.
“One who has enriched the inheritance left by Newton and has
consolidated the work of Faraday--one who impelled the mind of
Cambridge to a fresh course of real investigation--has clearly earned
his place in human memory.” It was thus that Professor Lewis Campbell
and Mr. Garnett began in 1882 their life of James Clerk Maxwell. The
years which have passed, since that date, have all tended to strengthen
the belief in the greatness of Maxwell’s work and in the fertility
of his genius, which has inspired the labours of those who, not in
Cambridge only, but throughout the world, have aided in developing the
seeds sown by him. My object in the following pages will be to give
some very brief account of his life and writings, in a form which may,
I hope, enable many to realise what Physical Science owes to one who
was to me a most kind friend as well as a revered master.
The Clerks of Penicuik, from whom Clerk Maxwell was descended, were
a distinguished family. Sir John Clerk, the great-great-grandfather
of Clerk Maxwell, was a Baron of the Exchequer in Scotland from
1707 to 1755; he was also one of the Commissioners of the Union,
and was in many ways an accomplished scholar. His second son George
married a first cousin, Dorothea Maxwell, the heiress of Middlebie in
Dumfriesshire, and took the name of Maxwell. By the death of his elder
brother James in 1782 George Clerk Maxwell succeeded to the baronetcy
and the property of Penicuik. Before this time he had become involved
in mining and manufacturing speculations, and most of the Middlebie
property had been sold to pay his debts.
The property of Sir George Clerk Maxwell descended in 1798 to his two
grandsons, Sir George Clerk and Mr. John Clerk Maxwell. It had been
arranged that the younger of the two was to take the remains of the
Middlebie property and to assume with it the name of Maxwell. Sir
George Clerk was member for Midlothian, and held office under Sir
Robert Peel. John Clerk Maxwell was the father of James Clerk Maxwell,
the subject of this sketch.[1]
John Clerk Maxwell lived with his widowed mother in Edinburgh until her
death in 1824. He was a lawyer, and from time to time did some little
business in the courts. At the same time he maintained an interest in
scientific pursuits, especially those of a practical nature. Professor
Campbell tells us of an endeavour to devise a bellows which would give
a continuous draught of air. In 1831 he contributed to the _Edinburgh
Medical and Philosophical Journal_ a paper entitled “Outlines of a Plan
for combining Machinery with the Manual Printing Press.”
In 1826 John Clerk Maxwell married Miss Frances Cay, of North Charlton,
Northumberland. For the first few years of their married life their
home was in Edinburgh. The old estate of Middlebie had been greatly
reduced in extent, and there was not a house on it in which the laird
could live. However, soon after his marriage, John Clerk Maxwell
purchased the adjoining property of Glenlair and built a mansion-house
for himself and his wife. Mr. Maxwell superintended the building work.
The actual working plans for some further additions made in 1843
were his handiwork. A garden was laid out and planted, and a dreary
stony waste was converted into a pleasant home. For some years after
he settled at Glenlair the house in Edinburgh was retained by Mr.
Maxwell, and here, on June 13, 1831, was born his only son, James Clerk
Maxwell. A daughter, born earlier, died in infancy. Glenlair, however,
was his parents’ home, and nearly all the reminiscences we have of
his childhood are connected with it. The laird devoted himself to his
estates and to the education of his son, taking, however, from time to
time his full share in such county business as fell to him. Glenlair in
1830 was very much in the wilds; the journey from Edinburgh occupied
two days. “Carriages in the modern sense were hardly known to the Vale
of Urr. A sort of double gig with a hood was the best apology for a
travelling coach, and the most active mode of locomotion was in a kind
of rough dog-cart known in the family speech as a hurly.”[2]
Mrs. Maxwell writes thus[3], when the boy was nearly three years old,
to her sister, Miss Jane Cay:--
“He is a very happy man, and has improved much since the
weather got moderate. He has great work with doors, locks,
keys, etc., and ‘Show me how it doos’ is never out of his
mouth. He also investigates the hidden course of streams and
bell-wires--the way the water gets from the pond through the
wall and a pend or small bridge and down a drain into Water
Orr, then past the smiddy and down to the sea, where Maggy’s
ships sail. As to the bells, they will not rust; he stands
sentry in the kitchen and Mag runs through the house ringing
them all by turns, or he rings and sends Bessy to see and shout
to let him know; and he drags papa all over to show him the
holes where the wires go through.”
To discover “how it doos” was thus early his aim. His cousin, Mrs.
Blackburn, tells us that throughout his childhood his constant question
was, “What’s the go of that? What does it do?” And if the answer were
too vague or inconclusive, he would add, “But what’s the _particular_
go of that?”
Professor Campbell’s most interesting account of these early years is
illustrated by a number of sketches of episodes in his life. In one
Maxwell is absorbed in watching the fiddler at a country dance; in
another he is teaching his dog some tricks; in a third he is helping a
smaller boy in his efforts to build a castle. Together with his cousin,
Miss Wedderburn, he devised a number of figures for a toy known as a
magic disc, which afterwards developed into the zoetrope or wheel of
life, and in which, by means of an ingenious contrivance of mirrors,
the impression of a continuous movement was produced.
This happy life went on until his mother’s death in December, 1839; she
died, at the age of forty-eight, of the painful disease to which her
son afterwards succumbed. When James, being then eight years old, was
told that she was now in heaven, he said: “Oh, I’m so glad! Now she’ll
have no more pain.”
After this his aunt, Miss Jane Cay, took a mother’s place. The problem
of his education had to be faced, and the first attempts were not
successful. A tutor had been engaged during Mrs. Maxwell’s last
illness, and he, it seems, tried to coerce Clerk Maxwell into learning;
but such treatment failed, and in 1841, when ten years old, he began
his school-life at the Edinburgh Academy.
School-life at first had its hardships. Maxwell’s appearance, his
first day at school, in Galloway home-spun and square-toed shoes with
buckles, was more than his fellows could stand. “Who made those shoes?”
they asked[4]; and the reply they received was--
“Div ye ken ’twas a man,
And he lived in a house,
In whilk was a mouse.”
He returned to Heriot Row that afternoon, says Professor Campbell,
“with his tunic in rags and wanting the skirt, his neat frill rumpled
and torn--himself excessively amused by his experiences and showing not
the slightest sign of irritation.”
No. 31, Heriot Row, was the house of his widowed aunt, Mrs. Wedderburn,
Mr. Maxwell’s sister; and this, with occasional intervals when he
was with Miss Cay, was his home for the next eight or nine years.
Mr. Maxwell himself, during this period, spent much of his time in
Edinburgh, living with his sister during most of the winter and
returning to Glenlair for the spring and summer.
Much of what we know of Clerk Maxwell’s life during this period comes
from the letters which passed between him and his father. They tell
us of the close intimacy and affection which existed between the two,
of the boy’s eager desire to please and amuse his father in the dull
solitude of Glenlair, and his father’s anxiety for his welfare and
progress.
Professor Campbell was his schoolfellow, and records events of those
years in which he shared, which bring clearly before us what Clerk
Maxwell was like. Thus he writes[5]:--
“He came to know Swift and Dryden, and after a while Hobbes,
and Butler’s ‘Hudibras.’ Then, if his father was in Edinburgh,
they walked together, especially on the Saturday half-holiday,
and ‘viewed’ Leith Fort, or the preparations for the Granton
railway, or the stratification of Salisbury Crags--always
learning something new, and winning ideas for imagination to
feed upon. One Saturday, February 12, 1842, he had a special
treat, being taken ‘to see electro-magnetic machines.’”
And again, speaking of his school-life:--
“But at school also he gradually made his way. He soon
discovered that Latin was worth learning, and the Greek
Delectus interested him when we got so far. And there were two
subjects in which he at once took the foremost place, when he
had a fair chance of doing so; these were Scripture Biography
and English. In arithmetic as well as in Latin his comparative
want of readiness kept him down.
“On the whole he attained a measure of success which helped
to secure for him a certain respect; and, however strange he
sometimes seemed to his companions, he had three qualities
which they could not fail to understand--agile strength
of limb, imperturbable courage, and profound good-nature.
Professor James Muirhead remembers him as ‘a friendly boy,
though never quite amalgamating with the rest.’ And another old
class-fellow, the Rev. W. Macfarlane of Lenzie, records the
following as his impression:--‘Clerk Maxwell, when he entered
the Academy, was somewhat rustic and somewhat eccentric. Boys
called him “Dafty,” and used to try to make fun of him. On one
occasion I remember he turned with tremendous vigour, with a
kind of demonic force, on his tormentors. I think he was let
alone after that, and gradually won the respect even of the
most thoughtless of his schoolfellows.’”
The first reference to mathematical studies occurs, says Professor
Campbell, in a letter to his father written soon after his thirteenth
birthday.[6]
“After describing the Virginian Minstrels, and betwixt
inquiries after various pets at Glenlair, he remarks, as if it
were an ordinary piece of news, ‘I have made a tetrahedron, a
dodecahedron, and two other hedrons, whose names I don’t know.’
We had not yet begun geometry, and he had certainly not at this
time learnt the definitions in Euclid; yet he had not merely
realised the nature of the five regular solids sufficiently to
construct them out of pasteboard with approximate accuracy, but
had further contrived other symmetrical polyhedra derived from
them, specimens of which (as improved in 1848) may be still
seen at the Cavendish Laboratory.
“Who first called his attention to the pyramid, cube, etc., I
do not know. He may have seen an account of them by chance in a
book. But the fact remains that at this early time his fancy,
like that of the old Greek geometers, was arrested by these
types of complete symmetry; and his imagination so thoroughly
mastered them that he proceeded to make them with his own
hand. That he himself attached more importance to this moment
than the letter indicates is proved by the care with which he
has preserved these perishable things, so that they (or those
which replaced them in 1848) are still in existence after
thirty-seven years.”
The summer holidays were spent at Glenlair. His cousin, Miss Jemima
Wedderburn, was with him, and shared his play. Her skilled pencil
has left us many amusing pictures of the time, some of which are
reproduced by Professor Campbell. There were expeditions and picnics of
all sorts, and a new toy known as “the devil on two sticks” afforded
infinite amusement. The winter holidays usually found him at Penicuik,
or occasionally at Glasgow, with Professor Blackburne or Professor
W. Thomson (now Lord Kelvin). In October, 1844, Maxwell was promoted
to the rector’s class-room. John Williams, afterwards Archdeacon of
Cardigan, a distinguished Baliol man, was rector, and the change was
in many ways an important one for Maxwell. He writes to his father: “I
like P---- better than B----. We have lots of jokes, and he speaks a
great deal, and we have not so much monotonous parsing. In the English
Milton is better than the History of Greece....”
P---- was the boys’ nickname for the rector; B---- for Mr. Carmichael,
the second master. This[7] is the account of Maxwell’s first interview
with the rector:--
_Rector_: “What part of Galloway do you come from?”
_J. C. M._: “From the Vale of Urr. Ye spell it o, err, err, or oo, err,
err.”
The study of geometry was begun, and in the mathematical master, Mr.
Gloag, Maxwell found a teacher with a real gift for his task. It was
here that Maxwell’s vast superiority to many who were his companions
at once showed itself. “He seemed,” says Professor Campbell, “to be in
the heart of the subject when they were only at the boundary; but the
boyish game of contesting point by point with such a mind was a most
wholesome stimulus, so that the mere exercise of faculty was a pure
joy. With Maxwell the first lessons of geometry branched out at once
into inquiries which became fruitful.”
In July, 1845, he writes:--
“I have got the 11th prize for Scholarship, the 1st for
English, the prize for English verses, and the Mathematical
Medal. I tried for Scripture knowledge, and Hamilton in the 7th
has got it. We tried for the Medal on Thursday. I had done them
all, and got home at half-past two; but Campbell stayed till
four. I was rather tired with writing exercises from nine till
half-past two.
“Campbell and I went ‘once more unto the b(r)each’ to-day at
Portobello. I can swim a little now. Campbell has got 6 prizes.
He got a letter written too soon, congratulating him upon _my_
medal; but there is no rivalry betwixt us, as B---- Carmichael
says.”
After a summer spent chiefly at Glenlair, he returned with his father
to Edinburgh for the winter, and began, at the age of fourteen, to go
to the meetings of the Royal Society of Edinburgh. At the Society of
Arts he met Mr. R. D. Hay, the decorative painter, who had interested
himself in the attempt to reduce beauty in form and colour to
mathematical principles. Clerk Maxwell was interested in the question
how to draw a perfect oval, and devised a method of drawing oval curves
which was referred by his father to Professor Forbes for his criticism
and suggestions. After discussing the matter with Professor Kelland,
Professor Forbes wrote as follows[8]:--
“MY DEAR SIR,--I am glad to find to-day, from Professor
Kelland, that his opinion of your son’s paper agrees with
mine, namely, that it is most ingenious, most creditable to
him, and, we believe, a new way of considering higher curves
with reference to foci. Unfortunately, these ovals appear
to be curves of a very high and intractable order, so that
possibly the elegant method of description may not lead to a
corresponding simplicity in investigating their properties. But
that is not the present point. If you wish it, I think that
the simplicity and elegance of the method would entitle it to
be brought before the Royal Society.--Believe me, my dear sir,
yours truly,
“JAMES D. FORBES.”
In consequence of this, Clerk Maxwell’s first published paper was
communicated to the Royal Society of Edinburgh on April 6th, 1846,
when its author was barely fifteen. Its title is as follows: “On the
Description of Oval Curves and those having a Plurality of Foci. By Mr.
Clerk Maxwell, Junior. With Remarks by Professor Forbes. Communicated
by Professor Forbes.”
The notice in his father’s diary runs: “M. 6 [Ap., 1846.] Royal Society
with Jas. Professor Forbes gave acct. of James’s Ovals. Met with very
great attention and approbation generally.”
This was the beginning of the lifelong friendship between Maxwell and
Forbes.
The curves investigated by Maxwell have the property that the sum found
by adding to the distance of any point on the curve from one focus a
constant multiple of the distance of the same point from a second focus
is always constant.
The curves are of great importance in the theory of light, for if this
constant factor expresses the refractive index of any medium, then
light diverging from one focus without the medium and refracted at a
surface bounding the medium, and having the form of one of Maxwell’s
ovals, will be refracted so as to converge to the second focus.
About the same time he was busy with some investigations on the
properties of jelly and gutta-percha, which seem to have been suggested
by Forbes’ “Theory of Glaciers.”
He failed to obtain the Mathematical Medal in 1846--possibly on account
of these researches--but he continued at school till 1847, when he
left, being then first in mathematics and in English, and nearly first
in Latin.
In 1847 he was working at magnetism and the polarisation of light.
Some time in that year he was taken by his uncle, Mr. John Cay, to see
William Nicol, the inventor of the polarising prism, who showed him the
colours exhibited by polarised light after passing through unannealed
glass. On his return, he made a polariscope with a glass reflector.
The framework of the first instrument was of cardboard, but a superior
article was afterwards constructed of wood. Small lenses mounted on
cardboard were employed when a conical pencil was needed. By means
of this instrument he examined the figures exhibited by pieces of
unannealed glass, which he prepared himself; and, with a camera lucida
and box of colours, he reproduced these figures on paper, taking care
to sketch no outlines, but to shade each coloured band imperceptibly
into the next. Some of these coloured drawings he forwarded to Nicol,
and was more than repaid by the receipt shortly afterwards of a pair of
prisms prepared by Nicol himself. These prisms were always very highly
prized by Maxwell. Once, when at Trinity, the little box containing
them was carried off by his bed-maker during a vacation, and destined
for destruction. The bed-maker died before term commenced, and it
was only by diligent search among her effects that the prisms were
recovered.[9] After this they were more carefully guarded, and they
are now, together with the wooden polariscope, the bits of unannealed
glass, and the water-colour drawings, in one of the showcases at the
Cavendish Laboratory.
About this time, Professor P. G. Tait and he were schoolfellows at the
Academy, acknowledged as the two best mathematicians in the school. It
was thought desirable, says Professor Campbell, that “we should have
lessons in physical science, so one of the classical masters gave them
out of a text-book.... The only thing I distinctly remember about these
hours is that Maxwell and P. G. Tait seemed to know much more about the
subject than our teacher did.”
An interesting account of these days is given by Professor Tait in an
obituary notice on Maxwell printed in the “Proceedings of the Royal
Society of Edinburgh, 1879–80,” from which the following is taken:--
“When I first made Clerk Maxwell’s acquaintance, about
thirty-five years ago, at the Edinburgh Academy, he was a year
before me, being in the fifth class, while I was in the fourth.
“At school he was at first regarded as shy and rather dull.
He made no friendships, and he spent his occasional holidays
in reading old ballads, drawing curious diagrams, and making
rude mechanical models. This absorption in such pursuits,
totally unintelligible to his schoolfellows (who were then
quite innocent of mathematics), of course procured him a not
very complimentary nickname, which I know is still remembered
by many Fellows of this Society. About the middle of his school
career, however, he surprised his companions by suddenly
becoming one of the most brilliant among them, gaining
high, and sometimes the highest, prizes for scholarships,
mathematics, and English verse composition. From this time
forward I became very intimate with him, and we discussed
together, with schoolboy enthusiasm, numerous curious
problems, among which I remember particularly the various plane
sections of a ring or tore, and the form of a cylindrical
mirror which should show one his own image unperverted. I
still possess some of the MSS. we exchanged in 1846 and early
in 1847. Those by Maxwell are on ‘The Conical Pendulum,’
‘Descartes’ Ovals,’ ‘Meloid and Apioid,’ and ‘Trifocal Curves.’
All are drawn up in strict geometrical form and divided into
consecutive propositions. The three latter are connected with
his first published paper, communicated by Forbes to this
society and printed in our ‘Proceedings,’ vol. ii., under the
title, ‘On the Description of Oval Curves and those having a
Plurality of Foci’ (1846). At the time when these papers were
written he had received no instruction in mathematics beyond a
few books of Euclid and the merest elements of algebra.”
In November, 1847, Clerk Maxwell entered the University of Edinburgh,
learning mathematics from Kelland, natural philosophy from J. D.
Forbes, and logic from Sir W. R. Hamilton. At this time, according to
Professor Campbell[10]--
“he still occasioned some concern to the more conventional
amongst his friends by the originality and simplicity of his
ways. His replies in ordinary conversation were indirect and
enigmatical, often uttered with hesitation and in a monotonous
key. While extremely neat in his person, he had a rooted
objection to the vanities of starch and gloves. He had a pious
horror of destroying anything, even a scrap of writing-paper.
He preferred travelling by the third class in railway journeys,
saying he liked a hard seat. When at table he often seemed
abstracted from what was going on, being absorbed in observing
the effects of refracted light in the finger-glasses, or in
trying some experiment with his eyes--seeing round a corner,
making invisible stereoscopes, and the like. Miss Cay used
to call his attention by crying, ‘Jamsie, you’re in a prop.’
He never tasted wine; and he spoke to gentle and simple in
exactly the same tone. On the other hand, his teachers--Forbes
above all--had formed the highest opinion of his intellectual
originality and force; and a few experienced observers, in
watching his devotion to his father, began to have some inkling
of his heroic singleness of heart. To his college companions,
whom he could now select at will, his quaint humour was an
endless delight. His chief associates, after I went to the
University of Glasgow, were my brother, Robert Campbell (still
at the Academy), P. G. Tait, and Allan Stewart. Tait went
to Peterhouse, Cambridge, in 1848, after one session of the
University of Edinburgh; Stewart to the same college in 1849;
Maxwell did not go up until 1850.”
During this period he wrote two important papers. The one, on “Rolling
Curves,” was read to the Royal Society of Edinburgh by Professor
Kelland--(“it was not thought proper for a boy in a round jacket to
mount the rostrum”)--in February, 1849; the other, on “The Equilibrium
of Elastic Solids,” appeared in the spring of 1850.
The vacations were spent at Glenlair, and we learn from letters to
Professor Campbell and others how the time was passed.
“On Saturday,” he writes[11]--April 26th, 1848, just after his
arrival home--“the natural philosophers ran up Arthur’s Seat with the
barometer. The Professor set it down at the top.... He did not set it
straight, and made the hill grow fifty feet; but we got it down again.”
In a letter of July in the same year he describes his laboratory:--
“I have regularly set up shop now above the wash-house at the
gate, in a garret. I have an old door set on two barrels, and
two chairs, of which one is safe, and a skylight above which
will slide up and down.
“On the door (or table) there is a lot of bowls, jugs, plates,
jam pigs, etc., containing water, salt, soda, sulphuric acid,
blue vitriol, plumbago ore; also broken glass, iron, and copper
wire, copper and zinc plate, bees’ wax, sealing wax, clay,
rosin, charcoal, a lens, a Smee’s galvanic apparatus, and a
countless variety of little beetles, spiders, and wood lice,
which fall into the different liquids and poison themselves. I
intend to get up some more galvanism in jam pigs; but I must
first copper the interiors of the pigs, so I am experimenting
on the best methods of electrotyping. So I am making copper
seals with the device of a beetle. First, I thought a beetle
was a good conductor, so I embedded one in wax (not at all
cruel, because I slew him in boiling water, in which he never
kicked), leaving his back out; but he would not do. Then I took
a cast of him in sealing wax, and pressed wax into the hollow,
and blackleaded it with a brush; but neither would that do. So
at last I took my fingers and rubbed it, which I find the best
way to use the blacklead. Then it coppered famously. I melt out
the wax with the lens, that being the cleanest way of getting a
strong heat, so I do most things with it that need heat. To-day
I astonished the natives as follows. I took a crystal of blue
vitriol and put the lens to it, and so drove off the water,
leaving a white powder. Then I did the same to some washing
soda, and mixed the two white powders together, and made a
small native spit on them, which turned them green by a mutual
exchange, thus:--1. Sulphate of copper and carbonate of soda.
2. Sulphate of soda and carbonate of copper (blue or green).”
Of his reading he says:--“I am reading Herodotus’ ‘Euterpe,’ having
taken the turn--that is to say that sometimes I can do props., read
Diff. and Int. Calc., Poisson, Hamilton’s dissertation, etc.”
In September he was busy with polarised light. “We were at Castle
Douglas yesterday, and got crystals of saltpetre, which I have been
cutting up into plates to-day in hopes to see rings.”
In July, 1849, he writes[12]:--
“I have set up the machine for showing the rings in crystals,
which I planned during your visit last year. It answers very
well. I also made some experiments on compressed jellies in
illustration of my props. on that subject. The principal one
was this:--The jelly is poured while hot into the annular space
contained between a paper cylinder and a cork; then, when cold,
the cork is twisted round and the jelly exposed to polarised
light, when a transverse cross, X, not +, appears, with rings
as the inverse square of the radius, all which is fully
verified. Hip! etc. _Q.E.D._”
And again on March 22nd, 1850:--
“At Practical Mechanics I have been turning Devils of sorts.
For private studies I have been reading Young’s ‘Lectures,’
Willis’s ‘Principles of Mechanism,’ Moseley’s ‘Engineering
and Mechanics,’ Dixon on ‘Heat,’ and Moigno’s ‘Répertoire
d’Optique.’ This last is a very complete analysis of all that
has been done in the optical way from Fresnel to the end of
1849, and there is another volume a-coming which will complete
the work. There is in it, besides common optics, all about the
other things which accompany light, as heat, chemical action,
photographic rays, action on vegetables, etc.
“My notions are rather few, as I do not _entertain_ them just
now. I have a notion for the torsion of wires and rods, not
to be made till the vacation; of experiments on the action of
compression on glass, jelly, etc., numerically done up; of
papers for the Physico-Mathematical Society (which is to revive
in earnest next session!); on the relations of optical and
mechanical constants, their desirableness, etc.; and suspension
bridges, and catenaries, and elastic curves. Alex. Campbell,
Agnew, and I are appointed to read up the subject of periodical
shooting stars, and to prepare a list of the phenomena to be
observed on the 9th August and 13th November. The society’s
barometer is to be taken up Arthur’s Seat at the end of the
session, when Forbes goes up, and All students are invited to
attend, so that the existence of the society may be recognised.”
It was at last settled that he was to go up to Cambridge. Tait had been
at Peterhouse for two years, while Allan Stewart had joined him there
in 1849, and after much discussion it was arranged that Maxwell should
enter at the same college.
Of this period of his life Tait writes as follows:--
“The winter of 1847 found us together in the classes of Forbes
and Kelland, where he highly distinguished himself. With the
former he was a particular favourite, being admitted to the
free use of the class apparatus for original experiments. He
lingered here behind most of his former associates, having
spent three years at the University of Edinburgh, working
(without any assistance or supervision) with physical and
chemical apparatus, and devouring all sorts of scientific
works in the library. During this period he wrote two valuable
papers, which are published in our ‘Transactions,’ on ‘The
Theory of Rolling Curves’ and on ‘The Equilibrium of Elastic
Solids.’ Thus he brought to Cambridge, in the autumn of 1850, a
mass of knowledge which was really immense for so young a man,
but in a state of disorder appalling to his methodical private
tutor. Though that tutor was William Hopkins, the pupil to a
great extent took his own way, and it may safely be said that
no high wrangler of recent years ever entered the Senate House
more imperfectly trained to produce ‘paying’ work than did
Clerk Maxwell. But by sheer strength of intellect, though with
the very minimum of knowledge how to use it to advantage under
the conditions of the examination, he obtained the position
of Second Wrangler, and was bracketed equal with the Senior
Wrangler in the higher ordeal of the Smith’s Prizes. His name
appears in the Cambridge ‘Calendar’ as Maxwell of Trinity,
but he was originally entered at Peterhouse, and kept his
first term there, in that small but most ancient foundation
which has of late furnished Scotland with the majority of the
professors of mathematics and natural philosophy in her four
universities.”
While W. D. Niven, in his preface to Maxwell’s collected works (p.
xii.), says:--
“It may readily be supposed that his preparatory training for
the Cambridge course was far removed from the ordinary type.
There had indeed for some time been practically no restraint
upon his plan of study, and his mind had been allowed to follow
its natural bent towards science, though not to an extent
so absorbing as to withdraw him from other pursuits. Though
he was not a sportsman--indeed, sport so-called was always
repugnant to him--he was yet exceedingly fond of a country
life. He was a good horseman and a good swimmer. Whence,
however, he derived his chief enjoyment may be gathered from
the account which Mr. Campbell gives of the zest with which he
quoted on one occasion the lines of Burns which describe the
poet finding inspiration while wandering along the banks of a
stream in the free indulgence of his fancies. Maxwell was not
only a lover of poetry, but himself a poet, as the fine pieces
gathered together by Mr. Campbell abundantly testify. He saw,
however, that his true calling was science, and never regarded
these poetical efforts as other than mere pastime. Devotion
to science, already stimulated by successful endeavour;
a tendency to ponder over philosophical problems; and an
attachment to English literature, particularly to English
poetry--these tastes, implanted in a mind of singular strength
and purity, may be said to have been the endowments with which
young Maxwell began his Cambridge career. Besides this, his
scientific reading, as we may gather from his papers to the
Royal Society of Edinburgh referred to above, was already
extensive and varied. He brought with him, says Professor Tait,
a mass of knowledge which was really immense for so young a
man, but in a state of disorder appalling to his methodical
private tutor.”
CHAPTER II.
UNDERGRADUATE LIFE AT CAMBRIDGE.
Maxwell did not remain long at Peterhouse; before the end of his
first term he migrated to Trinity, and was entered under Dr. Thompson
December 14th, 1850. He appeared to the tutor a shy and diffident
youth, but presently surprised Dr. Thompson by producing a bundle
of papers--copies, probably, of those he had already published--and
remarking, “Perhaps these may show that I am not unfit to enter at your
College.”
The change was pressed upon him by many friends, the grounds of the
advice being that, from the large number of high wranglers recently
at Peterhouse and the smallness of the foundation, the chances of a
Fellowship there for a mathematical man were less than at Trinity. It
was a step he never regretted; the prospect of a Fellowship had but
little influence on his mind. He found, however, at the larger college
ampler opportunities for self-improvement, and it was possible for him
to select his friends from among men whom he otherwise would never have
known.
The record of his undergraduate life is not very full; his letters to
his father have, unfortunately, been lost, but we have enough in the
recollections of friends still living to picture what it was like. At
first he lodged in King’s Parade with an old Edinburgh schoolfellow,
C. H. Robertson. He attended the College lectures on mathematics,
though they were somewhat elementary, and worked as a private pupil
with Porter, of Peterhouse. His father writes to him, November, 1850:
“Have you called on Professors Sedgwick, at Trin., and Stokes, at
Pembroke? If not, you should do both. Stokes will be most in your line,
if he takes you in hand at all. Sedgwick is also a great Don in his
line, and, if you were entered in geology, would be a most valuable
acquaintance.”
In his second year he became a pupil of Hopkins, the great coach; he
also attended Stokes’ lectures, and the friendship which lasted till
his death was thus begun. In April, 1852, he was elected a scholar,
and obtained rooms in College (G, Old Court). In June, 1852, he came
of age. “I trust you will be as discreet when major as you have been
while minor,” writes his father the day before. The next academic
year, October, 1852, to June, 1853, was a very busy one; hard grind
for the Tripos occupied his time, and he seems to have been thoroughly
overstrained. He was taken ill while staying near Lowestoft with the
Rev. C. B. Tayler, the uncle of a College friend. His own account of
the illness is given in a letter to Professor Campbell[13], dated July
14th, 1853.
“You wrote just in time for your letter to reach me as I
reached Cambridge. After examination, I went to visit the
Rev. C. B. Tayler (uncle to a Tayler whom I think you have
seen under the name of _Freshman_, etc., and author of many
tracts and other didactic works). We had little expedites and
walks, and things parochial and educational, and domesticity.
I intended to return on the 18th June, but on the 17th I felt
unwell, and took measures accordingly to be well again--_i.e._
went to bed, and made up my mind to recover. But it lasted more
than a fortnight, during which time I was taken care of beyond
expectation (not that I did not expect much before). When I
was perfectly useless and could not sit up without fainting,
Mr. Tayler did everything for me in such a way that I had no
fear of giving trouble. So did Mrs. Tayler; and the two nephews
did all they could. So they kept me in great happiness all the
time, and detained me till I was able to walk about and got
back strength. I returned on the 4th July.
“The consequence of all this is that I correspond with Mr.
Tayler, and have entered into bonds with the nephews, of all of
whom more hereafter. Since I came here I have been attending
Hop., but, with his approval, did not begin full swing. I
am getting on, though, and the work is not grinding on the
prepared brain.”
During this period he wrote some papers for the _Cambridge and Dublin
Mathematical Journal_ which will be referred to again later. He was
also a member of a discussion society known as the “Apostles,” and some
of the essays contributed by him are preserved by Professor Campbell.
Mr. Niven, in his preface to the collected edition of Maxwell’s works,
suggests that the composition of these essays laid the foundation of
that literary finish which is one of the characteristics of Maxwell’s
scientific writings.
Among his friends at the time were Tait, Charles Mackenzie of Caius,
the missionary bishop of Central Africa, Henry and Frank Mackenzie of
Trinity, Droop, third Wrangler in 1854; Gedge, Isaac Taylor, Blakiston,
F. W. Farrar,[14] H. M. Butler,[15] Hort, V. Lushington, Cecil Munro,
G. W. H. Tayler, and W. N. Lawson. Some of these who survived him have
given to Professor Campbell their recollections of these undergraduate
days, which are full of interest.
Thus Mr. Lawson writes[16]:--
“There must be many of his quaint verses about, if one could
lay hands on them, for Maxwell was constantly producing
something of the sort and bringing it round to his friends,
with a sly chuckle at the humour, which, though his own, no one
enjoyed more than himself.
“I remember Maxwell coming to me one morning with a copy of
verses beginning, ‘Gin a body meet a body going through the
air,’ in which he had twisted the well-known song into a
description of the laws of impact of solid bodies.
“There was also a description which Maxwell wrote of some
University ceremony--I forget what--in which somebody ‘went
before’ and somebody ‘followed after,’ and ‘in the midst were
the wranglers, playing with the symbols.’
“These last words, however meant, were, in fact, a description
of his own wonderful power. I remember, one day in lecture,
our lecturer had filled the black-board three times with
the investigation of some hard problem in Geometry of Three
Dimensions, and was not at the end of it, when Maxwell came up
with a question whether it would not come out geometrically,
and showed how, with a figure, and in a few lines, there was
the solution at once.
“Maxwell was, I daresay you remember, very fond of a talk upon
almost anything. He and I were pupils (at an enormous distance
apart) of Hopkins, and I well recollect how, when I had been
working the night before and all the morning at Hopkins’s
problems, with little or no result, Maxwell would come in for a
gossip, and talk on while I was wishing him far away, till at
last, about half an hour or so before our meeting at Hopkins’s,
he would say, ‘Well, I must go to old Hop.’s problems’; and, by
the time we met there, they were all done.
“I remember Hopkins telling me, when speaking of Maxwell,
either just before or just after his degree, ‘It is not
possible for that man to think incorrectly on physical
subjects’; and Hopkins, as you know, had had, perhaps, more
experience of mathematical minds than any man of his time.”
The last clause is part of a quotation from a diary kept by Mr. Lawson
at Cambridge, in which, under the date July 15th, 1853, he writes:--
“He (Hopkins) was talking to me this evening about Maxwell.
He says he is unquestionably the most extraordinary man he
has met with in the whole range of his experience; he says
it appears impossible for Maxwell to think incorrectly on
physical subjects; that in his analysis, however, he is far
more deficient. He looks upon him as a great genius with all
its eccentricities, and prophesies that one day he will shine
as a light in physical science--a prophecy in which all his
fellow-students strenuously unite.”
How many who have struggled through the “Electricity and Magnetism”
have realised the truth of the remark about the correctness of his
physical intuitions and the deficiency at times of his analysis!
Dr. Butler, a friend of these early days, preached the University
sermon on November 16th, 1879, ten days after Maxwell’s death, and
spoke thus:--
“It is a solemn thing--even the least thoughtful is touched
by it--when a great intellect passes away into the silence
and we see it no more. Such a loss, such a void, is present,
I feel certain, to many here to-day. It is not often, even
in this great home of thought and knowledge, that so bright
a light is extinguished as that which is now mourned by many
illustrious mourners, here chiefly, but also far beyond this
place. I shall be believed when I say in all simplicity that I
wish it had fallen to some more competent tongue to put into
words those feelings of reverent affection which are, I am
persuaded, uppermost in many hearts on this Sunday. My poor
words shall be few, but believe me they come from the heart.
You know, brethren, with what an eager pride we follow the
fortunes of those whom we have loved and reverenced in our
undergraduate days. We may see them but seldom, few letters may
pass between us, but their names are never common names. They
never become to us only what other men are. When I came up to
Trinity twenty-eight years ago, James Clerk Maxwell was just
beginning his second year. His position among us--I speak in
the presence of many who remember that time--was unique. He was
the one acknowledged man of genius among the undergraduates. We
understood even then that, though barely of age, he was in his
own line of inquiry not a beginner but a master. His name was
already a familiar name to men of science. If he lived, it was
certain that he was one of that small but sacred band to whom
it would be given to enlarge the bounds of human knowledge.
It was a position which might have turned the head of a
smaller man; but the friend of whom we were all so proud, and
who seemed, as it were, to link us thus early with the great
outside world of the pioneers of knowledge, had one of those
rich and lavish natures which no prosperity can impoverish,
and which make faith in goodness easy for others. I have often
thought that those who never knew the grand old Adam Sedgwick
and the then young and ever-youthful Clerk Maxwell had yet to
learn the largeness and fulness of the moulds in which some
choice natures are framed. Of the scientific greatness of our
friend we were most of us unable to judge; but anyone could
see and admire the boy-like glee, the joyous invention, the
wide reading, the eager thirst for truth, the subtle thought,
the perfect temper, the unfailing reverence, the singular
absence of any taint of the breath of worldliness in any of its
thousand forms.
“Brethren, you may know such men now among your college
friends, though there can be but few in any year, or indeed in
any century, that possess the rare genius of the man whom we
deplore. If it be so, then, if you will accept the counsel of
a stranger, thank God for His gift. Believe me when I tell you
that few such blessings will come to you in later life. There
are blessings that come once in a lifetime. One of these is the
reverence with which we look up to greatness and goodness in
a college friend--above us, beyond us, far out of our mental
or moral grasp, but still one of us, near to us, our own. You
know, in part at least, how in this case the promise of youth
was more than fulfilled, and how the man who, but a fortnight
ago, was the ornament of the University, and--shall I be
wrong in saying it?--almost the discoverer of a new world of
knowledge, was even more loved than he was admired, retaining
after twenty years of fame that mirth, that simplicity, that
child-like delight in all that is fresh and wonderful which we
rejoice to think of as some of the surest accompaniment of true
scientific genius.
“You know, also, that he was a devout as well as thoughtful
Christian. I do not note this in the triumphant spirit of a
controversialist. I will not for a moment assume that there is
any natural opposition between scientific genius and simple
Christian faith. I will not compare him with others who have
had the genius without the faith. Christianity, though she
thankfully welcomes and deeply prizes them, does not need
now, any more than when St. Paul first preached the Cross at
Corinth, the speculations of the subtle or the wisdom of the
wise. If I wished to show men, especially young men, the living
force of the Gospel, I would take them not so much to a learned
and devout Christian man to whom all stores of knowledge were
familiar, but to some country village where for fifty years
there had been devout traditions and devout practice. There
they would see the Gospel lived out; truths, which other men
spoke of, seen and known; a spirit not of this world, visibly,
hourly present; citizenship in heaven daily assumed and daily
realised. Such characters I believe to be the most convincing
preachers to those who ask whether Revelation is a fable
and God an unknowable. Yes, in most cases--not, I admit, in
all--simple faith, even peradventure more than devout genius,
is mighty for removing doubts and implanting fresh conviction.
But having said this, we may well give thanks to God that our
friend was what he was, a firm Christian believer, and that his
powerful mind, after ranging at will through the illimitable
spaces of Creation and almost handling what he called ‘the
foundation-stones of the material universe,’ found its true
rest and happiness in the love and the mercy of Him whom the
humblest Christian calls his Father. Of such a man it may be
truly said that he had his citizenship in heaven, and that he
looked for, as a Saviour, the Lord Jesus Christ, through whom
the unnumbered worlds were made, and in the likeness of whose
image our new and spiritual body will be fashioned.”
The Tripos came in January, 1854. “You will need to get muffetees for
the Senate Room. Take your plaid or rug to wrap round your feet and
legs,” was his father’s advice--advice which will appeal to many who
can remember the Senate House as it felt on a cold January morning.
Maxwell had been preparing carefully for this examination. Thus to
his aunt, Miss Cay, in June, 1853, he writes:--“If anyone asks how I
am getting on in mathematics, say that I am busy arranging everything
so as to be able to express all distinctly, so that examiner may be
satisfied now and pupils edified hereafter. It is pleasant work and
very strengthening, but not nearly finished.”
Still, the illness of July, 1853, had left some effect. Professor
Baynes states that he said that on entering the Senate House for the
first paper he felt his mind almost a blank, but by-and-by his mental
vision became preternaturally clear.
The moderators were Mackenzie of Caius, whose advice had been mainly
instrumental in leading him to migrate to Trinity, Wm. Walton of
Trinity, Wolstenholme of Christ’s, and Percival Frost of St. John’s.
When the lists were published, Routh of Peterhouse was senior, Maxwell
second. The examination for the Smith’s Prizes followed in a few days,
and then Routh and Maxwell were declared equal.
In a letter to Miss Cay[17] of January 13th, while waiting for the
three days’ list, he writes:--
“All my correspondents have been writing to me, which is kind,
and have not been writing questions, which is kinder. So I
answer you now, while I am slacking speed to get up steam,
leaving Lewis and Stewart, etc., till next week, when I will
give an account of the _five days_. There are a good many up
here at present, and we get on very jolly on the whole; but
some are not well, and some are going to be plucked or gulphed,
as the case may be, and others are reading so hard that they
are invisible. I go to-morrow to breakfast with shaky men, and
after food I am to go and hear the list read out, and whether
they are through, and bring them word. When the honour list
comes out the poll men act as messengers. Bob Campbell comes
in occasionally of an evening now, to discuss matters and vary
sports. During examination I have had men at night working with
gutta-percha, magnets, etc. It is much better than reading
novels or talking after 5½ hours’ hard writing.”
His father, on hearing the news, wrote from Edinburgh:--
“I heartily congratulate you on your place in the list. I
suppose it is higher than the speculators would have guessed,
and quite as high as Hopkins reckoned on. I wish you success
in the Smith’s Prizes; be sure to write me the result. I will
see Mrs. Morrieson, and I think I will call on Dr. Gloag to
congratulate him. He has at least three pupils gaining honours.”
His friends in Edinburgh were greatly pleased. “I get congratulations
on all hands,” his father writes,[18] “including Professor Kelland
and Sandy Fraser and all others competent.... To-night or on Monday
I shall expect to hear of the Smith’s Prizes.” And again, February
6th, 1854:--“George Wedderburn came into my room at 2 a.m. yesterday
morning, having seen the Saturday _Times_, received by the express
train.... As you are equal to the Senior in the champion trial, you are
very little behind him.”
Or again, March 5th, 1854:--
“Aunt Jane stirred me up to sit for my picture, as she said you
wished for it and were entitled to ask for it _qua_ Wrangler. I
have had four sittings to Sir John Watson Gordon, and it is now
far advanced; I think it is very like. It is kitcat size, to be
a companion to Dyce’s picture of your mother and self, which
Aunt Jane says she is to leave to you.”
And now the long years of preparation were nearly over. The cunning
craftsman was fitted with his tools; he could set to work to unlock the
secrets of Nature; he was free to employ his genius and his knowledge
on those tasks for which he felt most fitted.
CHAPTER III.
EARLY RESEARCHES.--PROFESSOR AT ABERDEEN.
From this time on Maxwell’s life becomes a record of his writings
and discoveries. It will, however, probably be clearest to separate
as far as possible biographical details from a detailed account of
his scientific work, leaving this for consecutive treatment in later
chapters, and only alluding to it so far as may prove necessary to
explain references in his letters.
He continued in Cambridge till the Long Vacation of 1854, reading
Mill’s “Logic.” “I am experiencing the effects of Mill,” he writes,
March 25th, 1854, “but I take him slowly. I do not think him the last
of his kind. I think more is wanted to bring the connexion of sensation
with science to light, and to show what it is not.” He also read
Berkeley on “The Theory of Vision” and “greatly admired it.”
About the same time he devised an ophthalmoscope.[19]
“I have made an instrument for seeing into the eye through
the pupil. The difficulty is to throw the light in at that
small hole and look in at the same time; but that difficulty
is overcome, and I can see a large part of the back of the eye
quite distinctly with the image of the candle on it. People
find no inconvenience in being examined, and I have got dogs
to sit quite still and keep their eyes steady. Dogs’ eyes are
very beautiful behind--a copper-coloured ground, with glorious
bright patches and networks of blue, yellow, and green, with
blood-vessels great and small.”
After the vacation he returned to Cambridge, and the letters refer to
the colour-top. Thus to Miss Cay, November 24th, 1854, p. 208:--
“I have been very busy of late with various things, and am just
beginning to make papers for the examination at Cheltenham,
which I have to conduct about the 11th of December. I have
also to make papers to polish off my pups. with. I have been
spinning colours a great deal, and have got most accurate
results, proving that ordinary people’s eyes are all made
alike, though some are better than others, and that other
people see two colours instead of three; but all those who do
so agree amongst themselves. I have made a triangle of colours
by which you may make out everything.
“If you can find out any people in Edinburgh who do not see
colours (I know the Dicksons don’t), pray drop a hint that
I would like to see them. I have put one here up to a dodge
by which he distinguishes colours without fail. I have also
constructed a pair of squinting spectacles, and am beginning
operations on a squinting man.”
A paper written for his own use originally some time in 1854, but
communicated as a parting gift to his friend Farrar, who was about to
become a master at Marlborough, gives us some insight into his view of
life at the age of twenty-three.
“He that would enjoy life and act with freedom must have the
work of the day continually before his eyes. Not yesterday’s
work, lest he fall into despair; nor to-morrow’s, lest he
become a visionary--not that which ends with the day, which is
a worldly work; nor yet that only which remains to eternity,
for by it he cannot shape his actions.
“Happy is the man who can recognise in the work of to-day a
connected portion of the work of life and an embodiment of
the work of Eternity. The foundations of his confidence are
unchangeable, for he has been made a partaker of Infinity. He
strenuously works out his daily enterprises because the present
is given him for a possession.
“Thus ought Man to be an impersonation of the divine process
of nature, and to show forth the union of the infinite with
the finite, not slighting his temporal existence, remembering
that in it only is individual action possible; nor yet shutting
out from his view that which is eternal, knowing that Time is
a mystery which man cannot endure to contemplate until eternal
Truth enlighten it.”
His father was unwell in the Christmas vacation of that year, and he
could not return to Cambridge at the beginning of the Lent term. “My
steps,” he writes[20] to C. J. Munro from Edinburgh, February 19th,
1855, “will be no more by the reedy and crooked till Easter term.... I
should like to know how many kept bacalaurean weeks go to each of these
terms, and when they begin and end. Overhaul the Calendar, and when
found make note of.”
He was back in Cambridge for the May term, working at the motion
of fluids and at his colour-top. A paper on “Experiments on Colour
as Perceived by the Eye” was communicated to the Royal Society of
Edinburgh on March 19th, 1855. The experiments were shown to the
Cambridge Philosophical Society in May following, and the results are
thus described in two letters[21] to his father, Saturday, May 5th,
1855:
“The Royal Society have been very considerate in sending me my
paper on ‘Colours’ just when I wanted it for the Philosophical
here. I am to let them see the tricks on Monday evening,
and I have been there preparing their experiments in the
gaslight. There is to be a meeting in my rooms to-night to
discuss Adam Smith’s ‘Theory of Moral Sentiments,’ so I must
clear up my litter presently. I am working away at electricity
again, and have been working my way into the views of heavy
German writers. It takes a long time to reduce to order all
the notions one gets from these men, but I hope to see my way
through the subject and arrive at something intelligible in the
way of a theory....
“The colour trick came off on Monday, 7th. I had the
proof-sheets of my paper, and was going to read; but I changed
my mind and talked instead, which was more to the purpose.
There were sundry men who thought that blue and yellow make
green, so I had to undeceive them. I have got Hay’s book of
colours out of the Univ. Library, and am working through the
specimens, matching them with the top. I have a new trick of
stretching the string horizontally above the top, so as to
touch the upper part of the axis. The motion of the axis sets
the string a-vibrating in the same time with the revolutions of
the top, and the colours are seen in the haze produced by the
vibration. Thomson has been spinning the top, and he finds my
diagram of colours agrees with his experiments, but he doubts
about browns, what is their composition. I have got colcothar
brown, and can make white with it, and blue and green; also,
by mixing red with a little blue and green and a great deal of
black, I can match colcothar exactly.
“I have been perfecting my instrument for looking into the eye.
Ware has a little beast like old Ask, which sits quite steady
and seems to like being looked at, and I have got several men
who have large pupils and do not wish to let me look in. I
have seen the image of the candle distinctly in all the eyes I
have tried, and the veins of the retina were visible in some;
but the dogs’ eyes showed all the ramifications of veins, with
glorious blue and green network, so that you might copy down
everything. I have shown lots of men the image in my own eye by
shutting off the light till the pupil dilated and then letting
it on.
“I am reading Electricity and working at Fluid Motion, and have
got out the condition of a fluid being able to flow the same
way for a length of time and not wriggle about.”
The British Association met at Glasgow in September, 1855, and Maxwell
was present, and showed his colour-top at Professor Ramsay’s house to
some of those interested. Letters[22] to his father about this time
describe some of the events of the meeting and his own plans for the
term.
“We had a paper from Brewster on ‘The theory of three colours
in the spectrum,’ in which he treated Whewell with philosophic
pity, commending him to the care of Prof. Wartman of Geneva,
who was considered the greatest authority in cases of his
kind--cases, in fact, of colour-blindness. Whewell was in the
room, but went out and avoided the quarrel; and Stokes made a
few remarks, stating the case not only clearly but courteously.
However, Brewster did not seem to see that Stokes admitted
his experiments to be correct, and the newspapers represented
Stokes as calling in question the accuracy of the experiments.
“I am getting my electrical mathematics into shape, and I see
through some parts which were rather hazy before; but I do not
find very much time for it at present, because I am reading
about heat and fluids, so as not to tell lies in my lectures.
I got a note from the Society of Arts about the platometer,
awarding thanks and offering to defray the expenses to the
extent of £10, on the machine being produced in working order.
When I have arranged it in my head, I intend to write to James
Bryson about it.
“I got a long letter from Thomson about colours and
electricity. He is beginning to believe in my theory about all
colours being capable of reference to three standard ones, and
he is very glad that I should poach on his electrical preserves.
“... It is difficult to keep up one’s interest in intellectual
matters when friends of the intellectual kind are scarce.
However, there are plenty friends not intellectual who serve
to bring out the active and practical habits of mind, which
overly-intellectual people seldom do. Wherefore, if I am to be
up this term, I intend to addict myself rather to the working
men who are getting up classes than to pups., who are in
the main a vexation. Meanwhile, there is the examination to
consider.
“You say Dr. Wilson has sent his book. I will write and thank
him. I suppose it is about colour-blindness. I intend to begin
Poisson’s papers on electricity and magnetism to-morrow. I have
got them out of the library. My reading hitherto has been of
novels--‘Shirley’ and ‘The Newcomes,’ and now ‘Westward Ho.’
“Macmillan proposes to get up a book of optics with my
assistance, and I feel inclined for the job. There is great
bother in making a mathematical book, especially on a subject
with which you are familiar, for in correcting it you do as
you would to pups.--look if the principle and result is right,
and forget to look out for small errors in the course of the
work. However, I expect the work will be salutary, as involving
hard work, and in the end much abuse from coaches and students,
and certainly no vain fame, except in Macmillan’s puffs. But,
if I have rightly conceived the plan of an educational book
on optics, it will be very different in manner, though not in
matter, from those now used.”
The examination referred to was that for a Fellowship at Trinity, and
Maxwell was elected on October 10th, 1855.
He was immediately asked to lecture for the College, on hydrostatics
and optics, to the upper division of the third year, and to set papers
for the questionists. In consequence, he declined to take pupils, in
order to have time for reading and doing private mathematics, and for
seeing the men who attended his lectures.
In November he writes: “I have been lecturing two weeks now, and the
class seems improving; and they come and ask questions, which is a good
sign. I have been making curves to show the relations of pressure and
volume in gases, and they make the subject easier.”
Still, he found time to attend Professor Willis’s lectures on mechanism
and to continue his reading. “I have been reading,” he writes, “old
books on optics, and find many things in them far better than what is
new. The foreign mathematicians are discovering for themselves methods
which were well known at Cambridge in 1720, but are now forgotten.”
The “Poisson” was read to help him with his own views on electricity,
which were rapidly maturing, and the first of that great series of
works which has revolutionised the science was published on December
10th, 1855, when his paper on “Faraday’s Lines of Force” was read to
the Cambridge Philosophical Society.
The next term found him back in Cambridge at work on his lectures, full
of plans for a new colour top and other matters. Early in February
he received a letter from Professor Forbes, telling him that the
Professorship of Natural Philosophy in Marischal College, Aberdeen, was
vacant, and suggesting that he should apply.
He decided to be a candidate if his father approved. “For my own part,”
he writes, “I think the sooner I get into regular work the better,
and that the best way of getting into such work is to profess one’s
readiness by applying for it.” On the 20th of February he writes:
“However, wisdom is of many kinds, and I do not know which dwells
with wise counsellors most, whether scientific, practical, political,
or ecclesiastical. I hear there are candidates of all kinds relying
on the predominance of one or other of these kinds of wisdom in the
constitution of the Government.”
The second part of the paper on “Faraday’s Lines of Force” was read
during the term. Writing on the 4th of March, he expresses the hope
soon to be able to write out fully the paper. “I have done nothing
in that way this term,” he says, “but am just beginning to feel the
electrical state come on again.”
His father was working at Edinburgh in support of his candidature for
Aberdeen, and when, in the middle of March, he returned North, he
found everything well prepared. The two returned to Glenlair together
after a few days in Edinburgh, and Maxwell was preparing to go back to
Cambridge, when, on the 2nd of April, his father died suddenly.
Writing to Mrs. Blackburn, he says: “My father died suddenly to-day at
twelve o’clock. He had been giving directions about the garden, and he
said he would sit down and rest a little, as usual. After a few minutes
I asked him to lie down on the sofa, and he did not seem inclined to do
so; and then I got him some ether, which had helped him before. Before
he could take any he had a slight struggle, and all was over. He hardly
breathed afterwards.”
Almost immediately after this, Maxwell was appointed to Aberdeen. His
father’s death had frustrated some at least of the intentions with
which he had applied for the post. He knew the old man would be glad
to see him the occupant of a Scotch chair. He hoped, too, to be able to
live with his father at Glenlair for one half the year; but this was
not to be. No doubt the laboratory and the freedom of the post, when
compared with the routine work of preparing men for the Tripos, had
their inducements; still, it may be doubted if the choice was a wise
one for him. The work of drilling classes, composed, for the most part,
of raw untrained lads, in the elements of physics and mechanics was, as
Niven says in his preface to the collected works, not that for which
he was best fitted; while at Cambridge, had he stayed, he must always
have had among his pupils some of the best mathematicians of the time;
and he might have founded some ten or fifteen years before he did that
Cambridge School of Physicists which looks back with so much pride to
him as their master.
Leave-taking at Trinity was a sad task. He writes[23] thus, June 4th,
to Mr. R. B. Litchfield:--
“On Thursday evening I take the North-Western route to the
North. I am busy looking over immense rubbish of papers, etc.,
for some things not to be burnt lie among much combustible
matter, and some is soft and good for packing.
“It is not pleasant to go down to live solitary, but it would
not be pleasant to stay up either, when all one had to do lay
elsewhere. The transition state from a man into a Don must come
at last, and it must be painful, like gradual outrooting of
nerves. When it is done there is no more pain, but occasional
reminders from some suckers, tap-roots, or other remnants of
the old nerves, just to show what was there and what might have
been.”
The summer of 1856 was spent at Glenlair, where various friends were
his guests--Lushington, MacLennan, the two cousins Cay, and others.
He continued to work at optics, electricity, and magnetism, and in
October was busy with “a solemn address or manifesto to the Natural
Philosophers of the North, which needed coffee and anchovies and a
roaring hot fire and spread coat-tails to make it natural.” This was
his inaugural lecture.
In November he was at Aberdeen. Letters[24] to Miss Cay, Professor
Campbell, and C. J. Munro tell of the work of the session. The last is
from Glenlair, dated May 20th, 1857, after work was over.
“The session went off smoothly enough. I had Sun, all the
beginning of optics, and worked off all the experimental part
up to Fraunhofer’s lines, which were glorious to see with a
water-prism I have set up in the form of a cubical box, five
inch side....
“I succeeded very well with heat. The experiments on latent
heat came out very accurate. That was my part, and the class
could explain and work out the results better than I expected.
Next year I intend to mix experimental physics with mechanics,
devoting Tuesday and THURSDAY (what would Stokes say?) to the
science of experimenting accurately....
“Last week I brewed chlorophyll (as the chemists word it), a
green liquor, which turns the invisible light red....
“My last grind was the reduction of equations of colour which I
made last year. The result was eminently satisfactory.”
Another letter,[25] June 5th, 1857, also to Munro, refers to the work
of the University Commission and the new statutes.
“I have not seen Article 7, but I agree with your dissent from
it entirely. On the vested interest principle, I think the
men who intended to keep their fellowships by celibacy and
ordination, and got them on that footing, should not be allowed
to desert the virgin choir or neglect the priestly office,
but on those principles should be allowed to live out their
days, provided the whole amount of souls cured annually does
not amount to £20 in the King’s Book. But my doctrine is that
the various grades of College officers should be set on such a
basis that, although chance lecturers might be sometimes chosen
from among fresh fellows who are going away soon, the reliable
assistant tutors, and those that have a plain calling that
way, should, after a few years, be elected permanent officers
of the College, and be tutors and deans in their time, and
seniors also, with leave to marry, or, rather, never prohibited
or asked any questions on that head, and with leave to retire
after so many years’ service as seniors. As for the men of the
world, we should have a limited term of existence, and that
independent of marriage or ‘parsonage.’”
It was more than twenty years before the scheme outlined in the above
letter came to anything; but, at the time of Maxwell’s death in 1879,
another Commission was sitting, and the plan suggested by Maxwell
became the basis of the statutes of nearly all the colleges.
For the winter session of 1857–58 he was again at Aberdeen.
The Adams Prize had been established in 1848 by some members of
St. John’s College, and connected by them with the name of Adams
“in testimony of their sense of the honour he had conferred upon
his College and the University by having been the first among the
mathematicians of Europe to determine from perturbations the unknown
place of a disturbing planet exterior to Uranus.” Professor Challis,
Dr. Parkinson, and Sir William Thomson, the examiners, had selected
as the subject for the prize to be awarded in 1857 the “Motions of
Saturn’s Rings.” For this Maxwell had decided to compete, and his
letters at the end of 1857 tell of the progress of the task. Thus,
writing[26] to Lewis Campbell from Glenlair on August 28th, he says:--
“I have been battering away at Saturn, returning to the charge
every now and then. I have effected several breaches in the
solid ring, and now I am splash into the fluid one, amid a
clash of symbols truly astounding. When I reappear it will be
in the dusky ring, which is something like the state of the
air supposing the siege of Sebastopol conducted from a forest
of guns 100 miles one way, and 30,000 miles the other, and the
shot never to stop, but go spinning away round a circle, radius
170,000 miles.”
And again[27] to Miss Cay on the 28th of November:--
“I have been pretty steady at work since I came. The class
is small and not bright, but I am going to give them plenty
to do from the first, and I find it a good plan. I have a
large attendance of my old pupils, who go on with the higher
subjects. This is not part of the College course, so they
come merely from choice, and I have begun with the least
amusing part of what I intend to give them. Many had been
reading in summer, for they did very good papers for me on
the old subjects at the beginning of the month. Most of my
spare time I have been doing Saturn’s rings, which is getting
on now, but lately I have had a great many long letters to
write--some to Glenlair, some to private friends, and some all
about science.... I have had letters from Thomson and Challis
about Saturn--from Hayward, of Durham University, about the
brass top, of which he wants one. He says that the earth has
been really found to change its axis regularly in the way I
supposed. Faraday has also been writing about his own subjects.
I have had also to write Forbes a long report on colours; so
that for every note I have got I have had to write a couple of
sheets in reply, and reporting progress takes a deal of writing
and spelling.”
He devised a model (now at the Cavendish Laboratory) to exhibit the
motions of the satellites in a disturbed ring, “for the edification of
sensible image-worshippers.”
The essay was awarded the prize, and secured for its author great
credit among scientific men.
In another letter, written during the same session, he says: “I find my
principal work here is teaching my men to avoid vague expressions, as
‘a certain force,’ meaning uncertain; _may_ instead of _must_; _will
be_ instead of _is_; _proportional_ instead of _equal_.”
The death, during the autumn, of his College friend Pomeroy, from fever
in India, was a great blow to him; his letters at the time show the
depth of his feelings and his beliefs.
The question of the fusion of the two Colleges at Aberdeen, King’s
College and the Marischal College, was coming to the fore. “Know
all men,” he says, in a letter to Professor Campbell, “that I am a
Fusionist.”
In February, 1858, he was still engaged on Saturn’s rings, while hard
at work during the same time with his classes. He had established a
voluntary class for his students of the previous year, and was reading
with them Newton’s “Lunar Theory and Astronomy.” This was followed by
“Electricity and Magnetism,” Faraday’s book being the backbone of
everything, “as he himself is the nucleus of everything electric since
1830.”
In February, 1858, he announced his engagement to Katherine Mary Dewar,
the daughter of the Principal of Marischal College.
“Dear Aunt” (he says,[28] February 18th, 1858), “this comes to
tell you that I am going to have a wife....
“Don’t be afraid; she is not mathematical, but there are other
things besides that, and she certainly won’t stop mathematics.
The only one that can speak as an eye-witness is Johnnie,
and he only saw her when we were both trying to act the
indifferent. We have been trying it since, but it would not do,
and it was not good for either.”
The wedding took place early in June. Professor Campbell has preserved
some of the letters written by Maxwell to Miss Dewar, and these
contain “the record of feelings which in the years that followed were
transfused in action and embodied in a married life which can only be
spoken of as one of unexampled devotion.”
The project for the fusion of the two Colleges, to which reference has
been made, went on, and the scheme was completed in 1860.
The two Colleges were united to form the University of Aberdeen, and
the new chair of Natural Philosophy thus created was filled by the
appointment of David Thomson, Professor of Natural Philosophy in King’s
College, and Maxwell’s senior. Mr. W. D. Niven, in his preface to
Maxwell’s works, when dealing with this appointment, writes:--
“Professor Thomson, though not comparable to Maxwell as
a physicist, was nevertheless a remarkable man. He was
distinguished by singular force of character and great
administrative faculty, and he had been prominent in bringing
about the fusion of the Colleges. He was also an admirable
lecturer and teacher, and had done much to raise the standard
of scientific education in the north of Scotland. Thus the
choice made by the Commissioners, though almost inevitable,
had the effect of making it appear that Maxwell failed as a
teacher. There seems, however, to be no evidence to support
such an inference. On the contrary, if we may judge from the
number of voluntary students attending his classes in his last
College session, he would seem to have been as popular as a
professor as he was personally estimable.”
The question whether Maxwell was a great teacher has sometimes been
discussed. I trust that the following pages will give an answer to
it. He was not a prominent lecturer. As Professor Campbell says,[29]
“Between his students’ ignorance and his vast knowledge it was
difficult to find a common measure. The advice which he once gave
to a friend whose duty it was to preach to a country congregation,
‘Why don’t you give it them thinner?’ must often have been applicable
to himself.... Illustrations of _ignotum per ignotius_, or of the
abstruse by some unobserved property of the familiar, were multiplied
with dazzling rapidity. Then the spirit of indirectness and paradox,
though he was aware of its dangers, would often take possession of him
against his will, and, either from shyness or momentary excitement, or
the despair of making himself understood, would land him in ‘chaotic
statements,’ breaking off with some quirk of ironical humour.”
But teaching is not all done by lecturing. His books and papers are
vast storehouses of suggestions and ideas which the ablest minds of the
past twenty years have been since developing. To talk with him for an
hour was to gain inspiration for a year’s work; to see his enthusiasm
and to win his praise or commendation were enough to compensate for
many weary struggles over some stubborn piece of apparatus which would
not go right, or some small source of error which threatened to prove
intractable and declined to submit itself to calculation. The sure
judgment of posterity will confirm the verdict that Clerk Maxwell was a
great teacher, though lecturing to a crowd of untrained undergraduates
was a task for which others were better fitted than he.
CHAPTER IV.
PROFESSOR AT KING’S COLLEGE, LONDON.--LIFE AT GLENLAIR.
In 1860 Forbes resigned the chair of Natural Philosophy at Edinburgh.
Maxwell and Tait were candidates, and Tait was appointed. In the
summer of the same year Maxwell obtained the vacant Professorship of
Natural Philosophy at King’s College, London. This he held to 1865,
and this period of his life is distinguished by the appearance of
some of his most important papers. The work was arduous; the College
course extended over nine months of the year; there were as well
evening lectures to artisans as part of his regular duties. His life in
London was useful to him in the opportunities it gave him for becoming
personally acquainted with Faraday and others. He also renewed his
intimacy with various Cambridge friends.
He was at the celebrated Oxford meeting of the British Association in
1860, where he exhibited his colour-box for mixing the colours of the
spectrum. In 1859, at the meeting at Aberdeen, he had read to Section
A his first paper on the “Dynamical Theory of Gases,” published in the
_Philosophical Magazine_ for January, 1860. The second part of the
paper, dealing with the conduction of heat and other phenomena in a
gas, was published in July, 1860, after the Oxford meeting.
A paper on the “Theory of Compound Colours” was communicated to
the Royal Society by Professor Stokes in January, 1860. It contains
the account of his colour-box in the form finally adopted (most of
the important parts of the apparatus are still at the Cavendish
Laboratory), and a number of observations by Mrs. Maxwell and himself,
which will be more fully described later.
In November, 1860, he received for this work the Rumford medal of the
Royal Society.
The next year, 1861, is of great importance in the history of
electrical science. The British Association met at Manchester, and a
Committee was appointed on Standards of Electrical Resistance. Maxwell
was not a member. The committee reported at the Cambridge meeting in
1862, and were reappointed with extended duties. Maxwell’s name, among
others, was added, and he took a prominent part in the deliberations
of the committee, which, as their Report[30] presented in 1863 states,
came to the opinion, “after mature consideration, that the system
of so-called absolute electrical units, based on purely mechanical
measurements, is not only the best system yet proposed, but is the
only one consistent with our present knowledge both of the relations
existing between the various electrical phenomena and of the connection
between these and the fundamental measurements of time, space, and
mass.”
Appendix C of this Report, “On the Elementary Relations between
Electrical Measurements,” bears the names of Clerk Maxwell and Fleeming
Jenkin, and is the foundation of everything that has been done in the
way of absolute electrical measurement since that date; while Appendix
D gives an account by the same two workers of the experiments on the
absolute unit of electrical resistance made in the laboratory of King’s
College by Maxwell, Fleeming Jenkin, and Balfour Stewart. Further
experiments are described in the report for 1864. The work thus begun
was consummated during the year 1894 by the legalisation throughout
the civilised world of a system of electrical units based on those
described in these reports.
Meanwhile, Maxwell’s views on electro-magnetic theory were quietly
developing. Papers on “Physical Lines of Force,” which appeared in the
_Philosophical Magazine_ during 1861 and 1862, contain the germs of
his theory--expressed at that time, it is true, in a somewhat material
form. In the paper published January, 1862, the now well-known relation
between the ratio of the electric units and the velocity of light was
established, and his correspondence with Fleeming Jenkin and C. J.
Munro about this time relates in part to the experimental verification
of this relation. His experiments on this matter were published in the
“Philosophical Transactions” for 1868.
This electrical theory occupied his mind mainly during 1863 and 1864.
In September of the latter year he writes[31] from Glenlair to C.
Hockin, who had taken Balfour Stewart’s place during the second series
of experiments on the measurement of resistance.
“I have been doing several electrical problems. I have got a
theory of ‘electric absorption,’ _i.e._, residual charge, etc.,
and I very much want determinations of the specific induction,
electric resistance, and absorption of good dielectrics, such
as glass, shell-lac, gutta-percha, ebonite, sulphur, etc.
“I have also cleared the electromagnetic theory of light from
all unwarrantable assumption, so that we may safely determine
the velocity of light by measuring the attraction between
bodies kept at a given difference of potential, the value of
which is known in electromagnetic measure.
“I hope there will be resistance coils at the British
Association.”
This work resulted in his greatest electrical paper, “A Dynamical
Theory of the Electromagnetic Field,” read to the Royal Society
December 8th, 1864.
But the molecular theory of gases was still prominently before his mind.
In 1862, writing[32] to H. R. Droop, he says:--
“Some time ago, when investigating Bernoulli’s theory of gases,
I was surprised to find that the internal friction of a gas (if
it depends on the collision of particles) should be independent
of the density.
“Stokes has been examining Graham’s experiments on the rate
of flow of gases through fine tubes, and he finds that the
friction, if independent of density, accounts for Graham’s
results; but, if taken proportional to density, differs from
those results very much. This seems rather a curious result,
and an additional phenomenon, explained by the ‘collision of
particles’ theory of gases. Still one phenomenon goes against
that theory--the relation between specific heat at constant
pressure and at constant volume, which is in air = 1·408, while
it ought to be 1·333.”
And again[33] in the same year, 21st April, 1862, to Lewis Campbell:--
“Herr Clausius of Zürich, one of the heat philosophers, has
been working at the theory of gases being little bodies flying
about, and has found some cases in which he and I don’t tally.
So I am working it out again. Several experimental results have
turned up lately rather confirmatory than otherwise of that
theory.
“I hope you enjoy the absence of pupils. I find the division of
them into smaller classes is a great help to me and to them;
but the total oblivion of them for definite intervals is a
necessary condition for doing them justice at the proper time.”
The experiments on the viscosity of gases, which formed the Bakerian
Lecture to the Royal Society read on February 8th, 1866, were the
outcome of this work. His house in 8, Palace Gardens, Kensington,
contained a large garret running the complete length.
“To maintain the proper temperature a large fire was for some days kept
up in the room in the midst of very hot weather. Kettles were kept on
the fire and large quantities of steam allowed to flow into the room.
Mrs. Maxwell acted as stoker, which was very exhausting work when
maintained for several consecutive hours. After this the room was kept
cool for subsequent experiments by the employment of a considerable
amount of ice.”
Next year, May, 1866, was read his paper on the “Dynamical Theory of
Gases,” in which errors in his former papers, which had been pointed
out by Clausius, were corrected.
Meanwhile he had resigned his London Professorship at the end of the
Session of 1865, and had been succeeded by Professor W. G. Adams.
For the next four years he lived chiefly at Glenlair, working at his
theory of electricity, occasionally, as we shall see, visiting London
and Cambridge, and taking an active interest in the affairs of his
own neighbourhood. In 1865 he had a serious illness, through which he
was nursed with great care by Mrs. Maxwell. His correspondence was
considerable, and absorbed much of his time. Much also was given to the
study of English literature; he was fond of reading Chaucer, Milton, or
Shakespeare aloud to Mrs. Maxwell.
He also read much theological and philosophical literature, and all he
read helped only to strengthen that firm faith in the fundamentals of
Christianity in which he lived and died.
In 1867 he and Mrs. Maxwell paid a visit to Italy, which was a source
of great pleasure to both.
His chief scientific work was the preparation of his “Electricity and
Magnetism,” which did not appear till 1873; the time was in the main
one of quiet thought and preparation for his next great task, the
foundation of the School of Physics in Cambridge.
In 1868 the principalship of the United College in the University of
St. Andrews was vacant by the resignation of Forbes, and Maxwell was
invited by several of the professors to stand. He, however, declined to
submit his name to the Crown.
CHAPTER V.
CAMBRIDGE.--PROFESSOR OF PHYSICS.
During his retirement at Glenlair from 1865 to 1870 Maxwell was
frequently at Cambridge. He examined in the Mathematical Tripos in 1866
and 1867, and again in 1869 and 1870.
The regulations for the Tripos had been in force practically unchanged
since 1848, and it was felt by many that the range of subjects included
was not sufficiently extensive, and that changes were urgently needed
if Cambridge were to retain its position as the centre of mathematical
teaching. Natural Philosophy was mentioned in the Schedule, but Natural
Philosophy included only Dynamics and Astronomy, Hydrostatics and
Physical Optics, with some simple Hydrodynamics and Sound.
The subjects of Heat, Electricity and Magnetism, the Theory of Elastic
Solids and Vibrations, Vortex-Motion in Hydrodynamics, and much else,
were practically new since 1848. Stokes, Thomson, and Maxwell in
England, and Helmholtz in Germany, had created them.
Accordingly in June, 1868, a new plan of examinations was sanctioned
by the Senate to come into force in January, 1873, and these various
subjects were explicitly included.
Mr. Niven, who was one of those examined by Maxwell in 1866, writes in
the preface to the collected works:--
“For some years previous to 1866, when Maxwell returned to
Cambridge as Moderator in the Mathematical Tripos, the studies
in the University had lost touch with the great scientific
movements going on outside her walls. It was said that some
of the subjects most in vogue had but little interest for the
present generation, and loud complaints began to be heard
that while such branches of knowledge as Heat, Electricity,
and Magnetism were left out of the Tripos examination, the
candidates were wasting their time and energy upon mathematical
trifles barren of scientific interest and of practical results.
Into the movement for reform Maxwell entered warmly. By his
questions in 1866, and subsequent years, he infused new life
into the examination; he took an active part in drafting the
new scheme introduced in 1873; but most of all by his writings
he exerted a powerful influence on the younger members of the
University, and was largely instrumental in bringing about the
change which has been now effected.”
But the University possessed no means of teaching these subjects, and a
Syndicate or Committee was appointed, November 25th, 1868, “to consider
the best means of giving instruction to students in Physics, especially
in Heat, Electricity and Magnetism, and the methods of providing
apparatus for this purpose.”
Dr. Cookson, Master of St. Peter’s College, took an active part in the
work of the Syndicate. Professor Stokes, Professor Liveing, Professor
Humphry, Dr. Phear, and Dr. Routh were among the members. Maxwell
himself was in Cambridge that winter, as Examiner for the Tripos, and
his work as Moderator and Examiner in the two previous years had done
much to show the necessity of alterations and to indicate the direction
which changes should take.
The Syndicate reported February 27th, 1869. They called attention to
the Report of the Royal Commission of 1850. The Commissioners had
“prominently urged the importance of cultivating a knowledge of the
great branches of Experimental Physics in the University”; and in
page 118 of their Report, after commending the manner in which the
subject of Physical Optics is studied in the University, and pointing
out that “there is, perhaps, no public institution where it is better
represented or prosecuted with more zeal and success in the way of
original research,” they had stated that “no reason can be assigned
why other great branches of Natural Science should not become equally
objects of attention, or why Cambridge should not become a great school
of physical and experimental, as it is already of mathematical and
classical, instruction.”
And again the Commissioners remark: “In a University so thoroughly
imbued with the mathematical spirit, physical study might be expected
to assume within its precincts its highest and severest tone, be
studied under more abstract forms, with more continual reference to
mathematical laws, and therefore with better hope of bringing them one
by one under the domain of mathematical investigation than elsewhere.”
After calling attention to these statements the Report of the Syndicate
then continues:--
“In the scheme of Examination for Honours in the Mathematical Tripos
approved by Grace of the Senate on the 2nd of June, 1868, Heat,
Electricity and Magnetism, if not introduced for the first time, had a
much greater degree of importance assigned to them than at any previous
period, and these subjects will henceforth demand a corresponding
amount of attention from the candidates for Mathematical Honours. The
Syndicate have limited their attention almost entirely to the question
of providing public instruction in Heat, Electricity and Magnetism.
They recognise the importance and advantage of tutorial instruction in
these subjects in the several colleges, but they are also alive to the
great impulse given to studies of this kind, and to the large amount of
additional training which students may receive through the instruction
of a public Professor, and by knowledge gained in a well-appointed
laboratory.”
“In accordance with these views, and at an early period in their
deliberations, they requested the Professors[34] of the University, who
are engaged in teaching Mathematical and Physical Science, to confer
together upon the present means of teaching Experimental Physics,
especially Heat, Electricity and Magnetism, and to inform them how the
increased requirements of the University in this respect could be met
by them.”
“The Professors, so consulted, favoured the Syndicate with a report
on the subject, which the Syndicate now beg leave to lay before the
Senate. It points out how the requirements of the University might
be “partially met,” but the Professors state distinctly that they
“do not think that they are able to meet the want of an extensive
course of lectures on Physics treated as such, and in great measure
experimentally. As Experimental Physics may fairly be considered
to come within the province of one or more of the above-mentioned
Professors, the Syndicate have considered whether now or at some
future time some arrangement might not be made to secure the effective
teaching of this branch of science, without having resort to the
services of an additional Professor. They are, however, of opinion that
such an arrangement cannot be made at the present time, and that the
exigencies of the case may be best met by founding a new professorship
which shall terminate with the tenure of office of the Professor first
elected. The services of a man of the highest attainments in science,
devoting his life to public teaching as such Professor, and engaged in
original research, would be of incalculable benefit to the University.”
The Report goes on to point out that a laboratory would be necessary,
and also apparatus. It is estimated that £5,000 would cover the cost
of the laboratory, and £1,300 the necessary apparatus. Provision is
also made for a demonstrator and a laboratory assistant, and the Report
closes with a recommendation that a special Syndicate of Finance should
be appointed to consider the means of raising the funds.
The Professors in their Report to the Syndicate point out that teaching
in Experimental Physics is needed for the Mathematical Tripos, the
Natural Sciences Tripos, certain Special examinations, and the first
examination for the degree of M.B. It appeared to them clear that there
was work for a new Professor.
In May, 1869, the Financial Syndicate recommended by the above Report
was appointed “to consider the means of raising the necessary funds for
establishing a professor and demonstrator of Experimental Physics, and
for providing buildings and apparatus required for that department of
science, and further to consider other wants of the University, and the
sources from which those wants may be supplied.”
The Syndicate endeavoured to meet the expenditure by inquiry from the
several Colleges whether they would be willing to make contributions
from their corporate funds, but without success.
“The answers of the Colleges indicated such a want of concurrence
in any proposal to raise contributions from the corporate funds of
Colleges by any kind of direct taxation that the Syndicate felt
obliged to abandon the notion of obtaining the necessary funds from
this source, and accordingly to limit the number of objects which they
should recommend the Senate to accomplish.”
External authority was necessary before the colleges would submit
to taxation for University purposes, and it was left to the Royal
Commission of 1877 to carry into effect many of the suggestions
made by the Syndicate. Meanwhile they contented themselves with
recommending means for raising an annual stipend of £660 for the
professor, demonstrator, and assistant, and a capital sum of £5,000, or
thereabouts, for the expenses of a building.
The Syndicate’s Report was issued in an amended form in the May term of
1870, and before any decision was taken on it the Vice-Chancellor, Dr.
Atkinson, on October 13th, 1870, published “the following munificent
offer of his grace the Duke of Devonshire, the Chancellor of the
University,” who had been chairman of the Commission on Scientific
Education.
“Holker Hall, Grange, Lancashire.
“MY DEAR MR. VICE-CHANCELLOR,--I have the honour to address you
for the purpose of making an offer to the University, which, if
you see no objection, I shall be much obliged to you to submit
in such manner as you may think fit for the consideration of
the Council and the University.
“I find in the report dated February 29th, 1869, of the
Physical Science Syndicate, recommending the establishment of
a Professor and Demonstrator of Experimental Physics, that the
buildings and apparatus required for this department of science
are estimated to cost £6,300.
“I am desirous to assist the University in carrying this
recommendation into effect, and shall accordingly be prepared
to provide the funds required for the building and apparatus as
soon as the University shall have in other respects completed
its arrangements for teaching Experimental Physics, and shall
have approved the plan of the building.
“I remain, my dear Mr. Vice-Chancellor,
“Yours very faithfully,
“DEVONSHIRE.”
By his generous action the University was relieved from all expense
connected with the building. A Grace establishing a Professorship of
Experimental Physics was confirmed by the Senate February 9th, 1871,
and March 8th was fixed for the election.
Meanwhile who was to be Professor? Sir W. Thomson’s name had been
mentioned, but he, it was known, would not accept the post. Maxwell
was then applied to, and at first he was unwilling to leave Glenlair.
Professor Stokes, the Hon. J. W. Strutt (Lord Rayleigh), Mr. Blore
of Trinity, and others wrote to him. Lord Rayleigh’s letter[35] is as
follows:
“Cambridge, 14th February, 1871.
“When I came here last Friday I found everyone talking about
the new professorship, and hoping that you would come. Thomson,
it seems, has definitely declined.... There is no one here in
the least fit for the post. What is wanted by most who know
anything about it is not so much a lecturer as a mathematician
who has actual experience in experimenting, and who might
direct the energies of the younger Fellows and bachelors into
a proper channel. There must be many who would be willing to
work under a competent man, and who, while learning themselves,
would materially assist him.... I hope you may be induced to
come; if not, I don’t know who it is to be. Do not trouble to
answer me about this, as I believe others have written to you
about it.”
On the 15th of February, Maxwell wrote to Mr. Blore:--
“I had no intention of applying for the post when I got your
letter, and I have none now, unless I come to see that I can do
some good by it.” The letter continues:--“The class of Physical
Investigations, which might be undertaken with the help of men
of Cambridge education, and which would be creditable to the
University, demand in general a considerable amount of dull
labour, which may or may not be attractive to the pupils.”
However, on the 24th of February, Mr. Blore wrote to the Electoral
Roll:--
“I am authorised to give notice that Mr. John (_sic_) Clerk Maxwell,
F.R.S., formerly Professor of Natural Philosophy at Aberdeen, and
at King’s College, London, is a candidate for the professorship of
Experimental Physics.”
Maxwell was elected without opposition. Writing[36] to his wife from
Cambridge, 20th March, 1871, he says:--
“There are two parties about the professorship. One wants
popular lectures, and the other cares more for experimental
work. I think there should be a gradation--popular lectures and
rough experiments for the masses; real experiments for real
students; and laborious experiments for first-rate men like
Trotter and Stuart and Strutt.”
While in a letter[37] from Glenlair to C. J. Munro, dated March 15th,
1871, he writes:--“The Experimental Physics at Cambridge is not built
yet, but we are going to try. The desideratum is to set a Don and a
Freshman to observe and register (say) the vibrations of a magnet
together, or the Don to turn a watch and the Freshman to observe and
govern him.”
In October he delivered his Introductory Lecture. A few quotations will
show the spirit in which he approached his task.
“In a course of Experimental Physics we may consider either
the Physics or the Experiments as the leading feature. We may
either employ the experiments to illustrate the phenomena of
a particular branch of Physics, or we may make some physical
research in order to exemplify a particular experimental
method. In the order of time, we should begin, in the Lecture
Room, with a course of lectures on some branch of Physics
aided by experiments of illustration, and conclude, in the
Laboratory, with a course of experiments of research.
“Let me say a few words on these two classes of
experiments--Experiments of Illustration and Experiments of
Research. The aim of an experiment of illustration is to throw
light upon some scientific idea so that the student may be
enabled to grasp it. The circumstances of the experiment are
so arranged that the phenomenon which we wish to observe or to
exhibit is brought into prominence, instead of being obscured
and entangled among other phenomena, as it is when it occurs
in the ordinary course of nature. To exhibit illustrative
experiments, to encourage others to make them, and to cultivate
in every way the ideas on which they throw light, forms an
important part of our duty. The simpler the materials of an
illustrative experiment, and the more familiar they are to the
student, the more thoroughly is he likely to acquire the idea
which it is meant to illustrate. The educational value of such
experiments is often inversely proportional to the complexity
of the apparatus. The student who uses home-made apparatus,
which is always going wrong, often learns more than one who has
the use of carefully adjusted instruments, to which he is apt
to trust, and which he dares not take to pieces.
“It is very necessary that those who are trying to learn from
books the facts of physical science should be enabled by the
help of a few illustrative experiments to recognise these facts
when they meet with them out of doors. Science appears to us
with a very different aspect after we have found out that it is
not in lecture-rooms only, and by means of the electric light
projected on a screen, that we may witness physical phenomena,
but that we may find illustrations of the highest doctrines of
science in games and gymnastics, in travelling by land and by
water, in storms of the air and of the sea, and wherever there
is matter in motion.
“If, therefore, we desire, for our own advantage and for the
honour of our University, that the Devonshire Laboratory should
be successful, we must endeavour to maintain it in living union
with the other organs and faculties of our learned body. We
shall therefore first consider the relation in which we stand
to those mathematical studies which have so long flourished
among us, which deal with our own subjects, and which differ
from our experimental studies only in the mode in which they
are presented to the mind.
“There is no more powerful method for introducing knowledge
into the mind than that of presenting it in as many different
ways as we can. When the ideas, after entering through
different gateways, effect a junction in the citadel of the
mind, the position they occupy becomes impregnable. Opticians
tell us that the mental combination of the views of an object
which we obtain from stations no further apart than our two
eyes is sufficient to produce in our minds an impression of the
solidity of the object seen; and we find that this impression
is produced even when we are aware that we are really looking
at two flat pictures placed in a stereoscope. It is therefore
natural to expect that the knowledge of physical science
obtained by the combined use of mathematical analysis and
experimental research will be of a more solid, available, and
enduring kind than that possessed by the mere mathematician or
the mere experimenter.
“But what will be the effect on the University if men pursuing
that course of reading which has produced so many distinguished
Wranglers turn aside to work experiments? Will not their
attendance at the Laboratory count not merely as time withdrawn
from their more legitimate studies, but as the introduction of
a disturbing element, tainting their mathematical conceptions
with material imagery, and sapping their faith in the formulæ
of the text-books? Besides this, we have already heard
complaints of the undue extension of our studies, and of the
strain put upon our questionists by the weight of learning
which they try to carry with them into the Senate-House. If we
now ask them to get up their subjects not only by books and
writing, but at the same time by observation and manipulation,
will they not break down altogether? The Physical Laboratory,
we are told, may perhaps be useful to those who are going out
in Natural Science, and who do not take in Mathematics, but
to attempt to combine both kinds of study during the time of
residence at the University is more than one mind can bear.
“No doubt there is some reason for this feeling. Many of us
have already overcome the initial difficulties of mathematical
training. When we now go on with our study, we feel that it
requires exertion and involves fatigue, but we are confident
that if we only work hard our progress will be certain.
“Some of us, on the other hand, may have had some experience
of the routine of experimental work. As soon as we can read
scales, observe times, focus telescopes, and so on, this kind
of work ceases to require any great mental effort. We may,
perhaps, tire our eyes and weary our backs, but we do not
greatly fatigue our minds.
“It is not till we attempt to bring the theoretical part of
our training into contact with the practical that we begin to
experience the full effect of what Faraday has called ‘mental
inertia’--not only the difficulty of recognising, among the
concrete objects before us, the abstract relation which we have
learned from books, but the distracting pain of wrenching the
mind away from the symbols to the objects, and from the objects
back to the symbols. This, however, is the price we have to pay
for new ideas.
“But when we have overcome these difficulties, and successfully
bridged over the gulph between the abstract and the concrete,
it is not a mere piece of knowledge that we have obtained; we
have acquired the rudiment of a permanent mental endowment.
When, by a repetition of efforts of this kind, we have more
fully developed the scientific faculty, the exercise of this
faculty in detecting scientific principles in nature, and in
directing practice by theory, is no longer irksome, but becomes
an unfailing source of enjoyment, to which we return so often
that at last even our careless thoughts begin to run in a
scientific channel.
“Our principal work, however, in the Laboratory must be to
acquaint ourselves with all kinds of scientific methods, to
compare them and to estimate their value. It will, I think,
be a result worthy of our University, and more likely to be
accomplished here than in any private laboratory, if, by the
free and full discussion of the relative value of different
scientific procedures, we succeed in forming a school of
scientific criticism and in assisting the development of the
doctrine of method.
“But admitting that a practical acquaintance with the methods
of Physical Science is an essential part of a mathematical
and scientific education, we may be asked whether we are not
attributing too much importance to science altogether as part
of a liberal education.
“Fortunately, there is no question here whether the University
should continue to be a place of liberal education, or
should devote itself to preparing young men for particular
professions. Hence, though some of us may, I hope, see reason
to make the pursuit of science the main business of our lives,
it must be one of our most constant aims to maintain a living
connexion between our work and the other liberal studies of
Cambridge, whether literary, philological, historical, or
philosophical.
“There is a narrow professional spirit which may grow up among
men of science just as it does among men who practise any other
special business. But surely a University is the very place
where we should be able to overcome this tendency of men to
become, as it were, granulated into small worlds, which are
all the more worldly for their very smallness? We lose the
advantage of having men of varied pursuits collected into one
body if we do not endeavour to imbibe some of the spirit even
of those whose special branch of learning is different from our
own.”
Another expression of his views on the position of Physics at the time
will be found in his address to Section A of the British Association,
when President at the Liverpool meeting of 1870.
CHAPTER VI.
CAMBRIDGE--THE CAVENDISH LABORATORY.
But the laboratory was not yet built. A Syndicate, of which Maxwell
was a member, was appointed to consider the question of a site, to
take professional advice, and to obtain plans and estimates. Professor
Maxwell and Mr. Trotter visited various laboratories at home and
abroad for the purpose of ascertaining the best arrangements. Mr. W.
M. Fawcett was appointed architect; the tender of Mr. John Loveday,
of Kebworth, for the building at a cost of £8,450, exclusive of gas,
water, and heating, was accepted in March, 1872, and the building[38]
was begun during the summer.
In the meantime Maxwell began to lecture, finding a home where he could.
“Lectures begin 24th,” he writes from Glenlair, October 19th,
1872. “Laboratory rising, I hear, but I have no place to erect
my chair, but move about like the cuckoo, depositing my notions
in the Chemical Lecture-room 1st term; in the Botanical in
Lent, and in Comparative Anatomy in Easter.”
It was not till June, 1874, that the building was complete, and on
the 16th the Chancellor formally presented his gift of the Cavendish
Laboratory to the University. In the correspondence previous to this
time it was spoken of as the Devonshire Laboratory. The name Cavendish
commemorated the work of the great physicist of a century earlier,
whose writings Maxwell was shortly to edit, as well as the generosity
of the Chancellor.
In their letter of thanks to the Duke of Devonshire the University
write:--
“Unde vero conventius poterat illis artibus succurri quam e tua domo
quæ in ipsis jam pridem inclaruerat. Notum est Henricum Cavendish quem
secutus est Coulombius primum ita docuisse, quæ sit vis electrica ut
eam numerorum modulis illustraret; adhibitis rationibus quas hodie
veras esse constat.” And they suggest the name as suitable for the
building. To this the Chancellor replied, after referring to the work
of Henry Cavendish: “Quod pono in officinâ ipsâ nuncupandâ nomen ejus
commemorare dignati sitis, id grato animo accepi.”
The building had cost far more than the original estimate, but the
Chancellor’s generosity was not limited, and on July 21st, 1874, he
wrote to the Vice-Chancellor:--
“It is my wish to provide all instruments for the Cavendish Laboratory
which Professor Maxwell may consider to be immediately required, either
in his lectures or otherwise.”
Maxwell prepared a list, but explained while doing it that time and
thought were necessary to secure the best form of instruments; and he
continues, writing to the Vice-Chancellor: “I think the Duke fully
understood from what I said to him that to furnish the Laboratory
will be a matter of several years’ duration. I shall consider myself,
however,” he says, “at liberty to contribute to the Laboratory any
instruments which I have had constructed in former years, and which
may be found still useful, and also from time to time to procure others
for special researches.”
In 1877 in his annual report Professor Maxwell announced that the
Chancellor[39] had now “completed his gift to the University by
furnishing the Cavendish Laboratory with apparatus suited to the
present state of science.”
The stock of apparatus, however, was still small, although Maxwell in
the most generous manner himself spent large sums in adding to it;
for the Professor was most particular in procuring only expensive
instruments by the best makers, with such additional improvements as he
could himself suggest.
In March, 1874, a Demonstratorship of Physics had been established, and
Mr. Garnett of St. John’s College was appointed.
Work began in the laboratory in October, 1874. At first the number of
students was small. Only seventeen names appear in the Natural Sciences
Tripos[40] list for 1874, and few of those did Physics.
The fear alluded to by the Professor in his introductory lecture,
that men reading for the Mathematical Tripos would not find
time for attendance at the laboratory, was justified. One of the
weaknesses of our Cambridge plan has been the divorce between
Mathematics and experimental work, encouraged by our system of
examinations. Experimental knowledge is supposed not to be needed for
the Mathematical Tripos; the Mathematics permitted in the Natural
Sciences Tripos are very simple; thus it came about that few men while
reading for the Mathematical Tripos attended the laboratory, and this
unfortunate result was intensified by the action of the University in
1877–78, when the regulations for the Mathematical Tripos were again
altered.[41]
Still there were pupils eager and willing to work, though they were
chiefly men who had already taken their B.A. degree, and who wished
to continue Physical reading and research, even though it involved “a
considerable amount of dull labour not altogether attractive.” My own
work there began in 1876, and it may be interesting if I recall my
reminiscences of that time.
The first experiments I can recollect related to the measurement
of electrical resistance. I well remember Maxwell explaining the
principle of Wheatstone’s bridge, and my own wish at the time that I
had come to the laboratory before the Tripos, instead of afterwards.
Lord Rayleigh had, during the examination, set an easy question which
I failed to do for want of some slight experimental knowledge, and the
first few words of Maxwell’s talk showed me the solution.
I did not attend his lectures regularly--they were given, I think, at
an hour which I was obliged to devote to teaching; besides, there was
his book, the “Electricity and Magnetism,” into which I had just dipped
before the Tripos, to work at.
Chrystal and Saunder were then busy at their verification of Ohm’s law.
They were using a number of the Thomson form of tray Daniell’s cells,
and Maxwell was anxious for tests of various kinds to be made on these
cells; these I undertook, and spent some time over various simple
measurements on them. He then set me to work at some of the properties
of a stratified dielectric, consisting, if I remember rightly, of
sheets of paraffin paper and mica. By this means I became acquainted
with various pieces of apparatus. There were no regular classes and no
set drill of demonstrations arranged for examination purposes; these
came later. In Maxwell’s time those who wished to work had the use of
the laboratory and assistance and help from him, but they were left
pretty much to themselves to find out about the apparatus and the best
methods of using it.
Rather later than this Schuster came and did some of his spectroscope
work. J. E. H. Gordon was busy with the preliminary observations
for his determination of Verdet’s constant, and Niven had various
electrical experiments on hand; while Fleming was at work on the B. A.
resistance coils.
My own tastes lay in the direction of optics. Maxwell was anxious that
I should investigate the properties of certain crystals. I think they
were the chlorate of potash crystals, about which Stokes and Rayleigh
have since written; but these crystals were to be grown, a slow process
which would, he supposed, take years; and as I wished to produce a
dissertation for the Trinity Fellowship examination in 1877, that work
had to be laid aside.
Eventually I selected as a subject the form of the wave surface in
a biaxial crystal, and set to work in a room assigned to me. The
Professor used to come in on most days to see how I was getting on.
Generally he brought his dog, which sometimes was shut up in the next
room while he went to college. Dogs were not allowed in college, and
Maxwell had an amusing way of describing how Toby once wandered into
Trinity, and by some doggish instinct discovered immediately, to his
intense amazement, that he was in a place where no dogs had been since
the college was. Toby was not always quiet in his master’s absence, and
his presence in the next room was somewhat disturbing.
When difficulties occurred Maxwell was always ready to listen. Often
the answer did not come at once, but it always did come after a little
time. I remember one day, when I was in a serious dilemma, I told him
my long tale, and he said:--
“Well, Chrystal has been talking to me, and Garnett and Schuster have
been asking questions, and all this has formed a good thick crust round
my brain. What you have said will take some time to soak through, but
we will see about it.” In a few days he came back with--“I have been
thinking over what you said the other day, and if you do so-and-so it
will be all right.”
My dissertation was referred to him, and on the day of the election,
when returning to Cambridge for the admission, I met him at Bletchley
station, and well remember his kind congratulations and words of warm
encouragement.
For the next year and a half I was working regularly at the laboratory
and saw him almost daily during term time.
Of these last years there really is but little to tell. His own
scientific work went on. The “Electricity and Magnetism” was written
mostly at Glenlair. About the time of his return to Cambridge, in
October, 1872, he writes[42] to Lewis Campbell:--
“I am continually engaged in stirring up the Clarendon Press,
but they have been tolerably regular for two months. I find
nine sheets in thirteen weeks is their average. Tait gives me
great help in detecting absurdities. I am getting converted to
quaternions, and have put some in my book.”
The book was published in 1873. The Text-book of Heat was written
during the same period, while “Matter and Motion,” “a small book on a
great subject,” was published in 1876.
In 1873 and 1874 he was one of the examiners for the Natural Sciences
Tripos, and in 1873 he was the first additional examiner for the
Mathematical Tripos, in accordance with the scheme which he had done so
much to promote in 1868.
Many of his shorter papers were written about the same time. The
ninth edition of the _Encyclopædia Britannica_ was being published,
and Professor Baynes had enlisted his aid in the work. The articles
“Atom,” “Attraction,” “Capillary Action,” “Constitution of Bodies,”
“Diffusion,” “Ether,” “Faraday,” and others are by him.
He also wrote a number of papers for _Nature_. Some of these are
reviews of books or accounts of scientific men, such as the notices
of Faraday and Helmholtz, which appeared with their portraits; others
again are original contributions to science. Among the latter many have
reference to the molecular constitution of bodies. Two lectures--the
first on “Molecules,” delivered before the British Association at
Bradford in 1873; the second on the “Dynamical Evidence of the
Molecular Constitution of Bodies,” delivered before the Chemical
Society in 1875--were of special importance. The closing sentences of
the first lecture have been often quoted. They run as follow:--
“In the heavens we discover by their light, and by their light
alone, stars so distant from each other that no material thing
can ever have passed from one to another; and yet this light,
which is to us the sole evidence of the existence of these
distant worlds, tells us also that each of them is built up of
molecules of the same kinds as those which we find on earth.
A molecule of hydrogen, for example, whether in Sirius or in
Arcturus, executes its vibrations in precisely the same time.
“Each molecule therefore throughout the universe bears
impressed upon it the stamp of a metric system, as distinctly
as does the metre of the Archives at Paris, or the double royal
cubit of the temple of Karnac.
“No theory of evolution can be formed to account for the
similarity of molecules, for evolution necessarily implies
continuous change, and the molecule is incapable of growth or
decay, of generation or destruction.
“None of the processes of Nature, since the time when Nature
began, have produced the slightest difference in the properties
of any molecule. We are therefore unable to ascribe either the
existence of the molecules or the identity of their properties
to any of the causes which we call natural.
“On the other hand, the exact equality of each molecule to all
others of the same kind gives it, as Sir John Herschel has well
said, the essential character of a manufactured article, and
precludes the idea of its being eternal and self-existent.
“Thus we have been led along a strictly scientific path,
very near to the point at which Science must stop--not that
Science is debarred from studying the internal mechanism of a
molecule which she cannot take to pieces any more than from
investigating an organism which she cannot put together. But in
tracing back the history of matter, Science is arrested when
she assures herself, on the one hand, that the molecule has
been made, and, on the other, that it has not been made by any
of the processes we call natural.
“Science is incompetent to reason upon the creation of matter
itself out of nothing. We have reached the utmost limits of our
thinking faculties when we have admitted that because matter
cannot be eternal and self-existent, it must have been created.
“It is only when we contemplate, not matter in itself, but the
form in which it actually exists, that our mind finds something
on which it can lay hold.
“That matter, as such, should have certain fundamental
properties, that it should exist in space and be capable of
motion, that its motion should be persistent, and so on, are
truths which may, for anything we know, be of the kind which
metaphysicians call necessary. We may use our knowledge of
such truths for purposes of deduction, but we have no data for
speculating as to their origin.
“But that there should be exactly so much matter and no more
in every molecule of hydrogen is a fact of a very different
order. We have here a particular distribution of matter--a
_collocation_, to use the expression of Dr. Chalmers, of things
which we have no difficulty in imagining to have been arranged
otherwise.
“The form and dimensions of the orbits of the planets, for
instance, are not determined by any law of nature, but depend
upon a particular collocation of matter. The same is the case
with respect to the size of the earth, from which the standard
of what is called the metrical system has been derived. But
these astronomical and terrestrial magnitudes are far inferior
in scientific importance to that most fundamental of all
standards which forms the base of the molecular system. Natural
causes, as we know, are at work which tend to modify, if they
do not at length destroy, all the arrangements and dimensions
of the earth and the whole solar system. But though in the
course of ages catastrophes have occurred and may yet occur in
the heavens, though ancient systems may be dissolved and new
systems evolved out of their ruins, the molecules out of which
these systems are built--the foundation stones of the material
universe--remain unbroken and unworn. They continue this day as
they were created--perfect in number and measure and weight;
and from the ineffaceable characters impressed on them we may
learn that those aspirations after accuracy in measurement, and
justice in action, which we reckon among our noblest attributes
as men, are ours because they are essential constituents of the
image of Him who in the beginning created, not only the heaven
and the earth, but the materials of which heaven and earth
consist.”
This was criticised in _Nature_ by Mr. C. J. Munro, and at a later time
by Clifford in one of his essays.
Some correspondence with the Bishop of Gloucester and Bristol on the
authority for the comparison of molecules to manufactured articles is
given by Professor Campbell, and in it Maxwell points out that the
latter part of the article “Atom” in the _Encyclopædia_ is intended to
meet Mr. Munro’s criticism.
In 1874 the British Association met at Belfast, under the presidency of
Tyndall. Maxwell was present, and published afterwards in _Blackwood’s
Magazine_ an amusing paraphrase of the president’s address. This, with
some other verses written at about the same time, may be quoted here.
Professor Campbell has collected a number of verses written by Maxwell
at various times, which illustrate in an admirable manner both the
grave and the gay side of his character.
BRITISH ASSOCIATION, 1874.
_Notes of the President’s Address._
In the very beginnings of science, the parsons, who managed
things then,
Being handy with hammer and chisel, made gods in the likeness
of men;
Till commerce arose, and at length some men of exceptional power
Supplanted both demons and gods by the atoms, which last
to this hour.
Yet they did not abolish the gods, but they sent them well
out of the way,
With the rarest of nectar to drink, and blue fields of
nothing to sway.
From nothing comes nothing, they told us--naught happens by
chance, but by fate;
There is nothing but atoms and void, all else is mere whims
out of date!
Then why should a man curry favour with beings who cannot exist,
To compass some petty promotion in nebulous kingdoms of mist?
But not by the rays of the sun, nor the glittering shafts of the
day,
Must the fear of the gods be dispelled, but by words, and their
wonderful play.
So treading a path all untrod, the poet-philosopher sings
Of the seeds of the mighty world--the first-beginnings of things;
How freely he scatters his atoms before the beginning of years;
How he clothes them with force as a garment, those small
incompressible spheres!
Nor yet does he leave them hard-hearted--he dowers them with love
and with hate,
Like spherical small British Asses in infinitesimal state;
Till just as that living Plato, whom foreigners nickname
Plateau,[43]
Drops oil in his whisky-and-water (for foreigners sweeten it so);
Each drop keeps apart from the other, enclosed in a flexible skin,
Till touched by the gentle emotion evolved by the prick of a pin:
Thus in atoms a simple collision excites a sensational thrill,
Evolved through all sorts of emotion, as sense, understanding,
and will
(For by laying their heads all together, the atoms, as
councillors do,
May combine to express an opinion to every one of them new).
There is nobody here, I should say, has felt true indignation at
all,
Till an indignation meeting is held in the Ulster Hall;
Then gathers the wave of emotion, then noble feelings arise,
Till you all pass a resolution which takes every man by surprise.
Thus the pure elementary atom, the unit of mass and of thought,
By force of mere juxtaposition to life and sensation is brought;
So, down through untold generations, transmission of structureless
gorms
Enables our race to inherit the thoughts of beasts, fishes, and
worms.
We honour our fathers and mothers, grandfathers and grandmothers
too;
But how shall we honour the vista of ancestors now in our view?
First, then, let us honour the atom, so lively, so wise,
and so small;
The atomists next let us praise, Epicurus, Lucretius, and all.
Let us damn with faint praise Bishop Butler, in whom many
atoms combined
To form that remarkable structure it pleased him to call--his mind.
Last, praise we the noble body to which, for the time, we belong,
Ere yet the swift whirl of the atoms has hurried us, ruthless,
along,
The British Association--like Leviathan worshipped by Hobbes,
The incarnation of wisdom, built up of our witless nobs,
Which will carry on endless discussions when I, and probably you,
Have melted in infinite azure--in English, till all is blue.
MOLECULAR EVOLUTION.
_Belfast, 1874._
At quite uncertain times and places,
The atoms left their heavenly path,
And by fortuitous embraces
Engendered all that being hath.
And though they seem to cling together,
And form “associations” here,
Yet, soon or late, they burst their tether,
And through the depths of space career.
So we who sat, oppressed with science,
As British Asses, wise and grave,
Are now transformed to wild Red Lions,[44]
As round our prey we ramp and rave.
Thus, by a swift metamorphōsis,
Wisdom turns wit, and science joke,
Nonsense is incense to our noses,
For when Red Lions speak they smoke.
Hail, Nonsense! dry nurse of Red Lions,[45]
From thee the wise their wisdom learn;
From thee they cull those truths of science,
Which into thee again they turn.
What combinations of ideas
Nonsense alone can wisely form!
What sage has half the power that she has,
To take the towers of Truth by storm?
Yield, then, ye rules of rigid reason!
Dissolve, thou too, too solid sense!
Melt into nonsense for a season,
Then in some nobler form condense.
Soon, all too soon, the chilly morning
This flow of soul will crystallise;
Then those who Nonsense now are scorning
May learn, too late, where wisdom lies.
TO THE COMMITTEE OF THE CAYLEY PORTRAIT FUND.
1874.
O wretched race of men, to space confined!
What honour can ye pay to him, whose mind
To that which lies beyond hath penetrated?
The symbols he hath formed shall sound his praise,
And lead him on through unimagined ways
To conquests new, in worlds not yet created.
First, ye Determinants! in ordered row
And massive column ranged, before him go,
To form a phalanx for his safe protection.
Ye powers of the _n^{th}_ roots of -1!
Around his head in ceaseless[46] cycles run,
As unembodied spirits of direction.
And you, ye undevelopable scrolls!
Above the host wave your emblazoned rolls,
Ruled for the record of his bright inventions.
Ye cubic surfaces! by threes and nines
Draw round his camp your seven-and-twenty lines--
The seal of Solomon in three dimensions.
March on, symbolic host! with step sublime,
Up to the flaming bounds of Space and Time!
There pause, until by Dickinson depicted,
In two dimensions, we the form may trace
Of him whose soul, too large for vulgar space,
In _n_ dimensions flourished unrestricted.
IN MEMORY OF EDWARD WILSON,
_Who repented of what was in his mind to write after section._
RIGID BODY (_sings_).
GIN a body meet a body
Flyin’ through the air,
Gin a body hit a body,
Will it fly? and where?
Ilka impact has its measure,
Ne’er a ane hae I;
Yet a’ the lads they measure me,
Or, at least, they try.
Gin a body meet a body
Altogether free,
How they travel afterwards
We do not always see.
Ilka problem has its method
By analytics high;
For me, I ken na ane o’ them,
But what the waur am I?
Another task, which occupied much time, from 1874 to 1879, was the
edition of the works of Henry Cavendish. Cavendish, who was great-uncle
to the Chancellor, had published only two electrical papers, but he had
left some twenty packets of manuscript on Mathematical and Experimental
Electricity. These were placed in Maxwell’s hands in 1874 by the Duke
of Devonshire.
Niven, in his preface to the collected papers dealing with this book,
writes thus:--
“This work, published in 1879, has had the effect of increasing
the reputation of Cavendish, disclosing as it does the
unsuspected advances which that acute physicist had made in
the Theory of Electricity, especially in the measurement of
electrical quantities. The work is enriched by a variety of
valuable notes, in which Cavendish’s views and results are
examined by the light of modern theory and methods. Especially
valuable are the methods applied to the determination of the
electrical capacities of conductors and condensers, a subject
in which Cavendish himself showed considerable skill both of a
mathematical and experimental character.
“The importance of the task undertaken by Maxwell in connection
with Cavendish’s papers will be understood from the following
extract from his introduction to them:--
“‘It is somewhat difficult to account for the fact that
though Cavendish had prepared a complete description of his
experiments on the charges of bodies, and had even taken the
trouble to write out a fair copy, and though all this seems
to have been done before 1774, and he continued to make
experiments in electricity till 1781, and lived on till 1810,
he kept his manuscript by him and never published it.
“‘Cavendish cared more for investigation than for publication.
He would undertake the most laborious researches in order to
clear up a difficulty which no one but himself could appreciate
or was even aware of, and we cannot doubt that the result of
his enquiries, when successful, gave him a certain degree of
satisfaction. But it did not excite in him that desire to
communicate the discovery to others, which in the case of
ordinary men of science generally ensures the publication of
their results. How completely these researches of Cavendish
remained unknown to other men of science is shown by the
external history of electricity.’
“It will probably be thought a matter of some difficulty to
place oneself in the position of a physicist of a century
ago, and to ascertain the exact bearing of his experiments.
But Maxwell entered upon this undertaking with the utmost
enthusiasm, and succeeded in identifying himself with
Cavendish’s methods. He showed that Cavendish had really
anticipated several of the discoveries in electrical science
which have been made since his time. Cavendish was the first to
form the conception of and to measure Electrostatic Capacity
and Specific Inductive Capacity; he also anticipated Ohm’s law.”
During the last years of his life Mrs. Maxwell had a serious and
prolonged illness, and Maxwell’s work was much increased by his duties
as sick nurse. On one occasion he did not sleep in a bed for three
weeks, but conducted his lectures and experiments at the laboratory as
usual.
About this time some of those who had been “Apostles” in 1853–57
revived the habit of meeting together for discussion. The club, which
included Professors Lightfoot, Hort and Westcott, was christened
the “Eranus,” and three of Maxwell’s contributions to it have been
preserved and are printed by Professor Campbell.
After the Cavendish papers were finished, Maxwell had more time for his
own original researches, and two important papers were published in
1879. The one on “Stresses in Rarefied Gases arising from Inequalities
of Temperature” was printed in the Royal Society’s Transactions, and
deals with the Theory of the Radiometer; the other on “Boltzmann’s
Theorem” appears in the Transactions of the Cambridge Philosophical
Society. In the previous year he had delivered the Rede lecture on “The
Telephone.” He also began to prepare a second edition of “Electricity
and Magnetism.”
His health gave way during the Easter term of 1879; indeed for two
years previously he had been troubled with dyspeptic symptoms, but had
consulted no one on the subject. He left Cambridge as usual in June,
hoping that he would quickly recover at Glenlair, but he grew worse
instead. In October he was told by Dr. Sanders of Edinburgh that he had
not a month to live. He returned to Cambridge in order to be under the
care of Dr. Paget, who was able in some measure to relieve his most
severe suffering but the disease, of which his mother had died at the
same age, continued its progress, and he died on November 5th. His one
care during his last illness was for those whom he left behind. Mrs.
Maxwell was an invalid dependent on him for everything, and the thought
of her helplessness was the one thing which in these last days troubled
him.
A funeral service took place in the chapel at Trinity College, and
afterwards his remains were conveyed to Scotland and interred in the
family burying-place at Corsock, Kirkcudbright.
A memorial edition of his works was issued by the Cambridge University
Press in 1890. A portrait by Lowes Dickinson hangs in the hall of
Trinity College, and there is a bust by Boehm in the laboratory.
After his death Mrs. Maxwell gave his scientific library to the
Cavendish Laboratory, and on her death she left a sum of about £6,000
to found a scholarship in Physics, to be held at the laboratory.
* * * * *
The preceding pages contain some account of Clerk Maxwell’s life as
a man of science. His character had other sides, and any life of him
would be incomplete without some brief reference to these. His letters
to his wife and to other intimate friends show throughout his life
the depth of his religious convictions. The high purpose evidenced
in the paper given to the present Dean of Canterbury when leaving
Cambridge, animated him continually, and appears from time to time in
his writings. The student’s evening hymn, composed in 1853 when still
an undergraduate, expresses the same feelings--
Through the creatures Thou hast made
Show the brightness of Thy glory,
Be eternal truth displayed
In their substance transitory,
Till green earth and ocean hoary,
Massy rock and tender blade,
Tell the same unending story,
“We are Truth in form arrayed.”
Teach me so Thy works to read
That my faith, new strength accruing,
May from world to world proceed,
Wisdom’s fruitful search pursuing,
Till Thy breath my mind imbuing,
I proclaim the eternal creed,
Oft the glorious theme renewing,
God our Lord is God indeed.
His views on the relation of Science to Faith are given in his
letter[47] to Bishop Ellicott already referred to--
“But I should be very sorry if an interpretation founded
on a most conjectural scientific hypothesis were to get
fastened to the text in Genesis, even if by so doing it got
rid of the old statement of the commentators which has long
ceased to be intelligible. The rate of change of scientific
hypothesis is naturally much more rapid than that of Biblical
interpretations, so that if an interpretation is founded on
such an hypothesis, it may help to keep the hypothesis above
ground long after it ought to be buried and forgotten.
“At the same time I think that each individual man should do
all he can to impress his own mind with the extent, the order,
and the unity of the universe, and should carry these ideas
with him as he reads such passages as the 1st chapter of the
Epistle to Colossians (_see_ ‘Lightfoot on Colossians,’ p.
182), just as enlarged conceptions of the extent and unity of
the world of life may be of service to us in reading Psalm
viii., Heb. ii. 6, etc.”
And again in his letter[48] to the secretary of the Victoria Institute
giving his reasons for declining to become a member--
“I think men of science as well as other men need to learn from
Christ, and I think Christians whose minds are scientific are
bound to study science, that their view of the glory of God
may be as extensive as their being is capable of. But I think
that the results which each man arrives at in his attempts to
harmonise his science with his Christianity ought not to be
regarded as having any significance except to the man himself,
and to him only for a time, and should not receive the stamp of
a society.”
Professor Campbell and Mr. Garnett have given us the evidence of those
who were with him in his last days, as to the strength of his own
faith. On his death bed he said that he had been occupied in trying to
gain truth; that it is but little of truth that man can acquire, but it
is something to know in whom we have believed.
CHAPTER VII.
SCIENTIFIC WORK--COLOUR VISION.
Fifteen years only have passed since the death of Clerk Maxwell, and it
is almost too soon to hope to form a correct estimate of the value of
his work and its relation to that of others who have laboured in the
same field.
Thus Niven, at the close of his obituary notice in the Proceedings of
the Royal Society, says: “It is seldom that the faculties of invention
and exposition, the attachment to physical science and capability of
developing it mathematically, have been found existing in one mind
to the same degree. It would, however, require powers somewhat akin
to Maxwell’s own to describe the more delicate features of the works
resulting from this combination, every one of which is stamped with the
subtle but unmistakable impress of genius.” And again in the preface
to Maxwell’s works, issued in 1890, he wrote: “Nor does it appear to
the present editor that the time has yet arrived when the quickening
influence of Maxwell’s mind on modern scientific thought can be duly
estimated.”
It is, however, the object of the present series to attempt to give
some account of the work of men of science of the last hundred years,
and to show how each has contributed his share to our present stock
of knowledge. This task, then, remains to be done. While attempting
it I wish to express my indebtedness to others who have already
written about Maxwell’s scientific work, especially to Mr. W. D.
Niven, whose preface to the Maxwell papers has been so often referred
to; to Mr. Garnett, the author of Part II. of the “Life of Maxwell,”
which deals with his contributions to science; and to Professor Tait,
who in _Nature_ for February 5th, 1880, gave an account of Clerk
Maxwell’s work, “necessarily brief, but sufficient to let even the
non-mathematical reader see how very great were his contributions to
modern science”--an account all the more interesting because, again to
quote from Professor Tait, “I have been intimately acquainted with him
since we were schoolboys together.”
Maxwell’s main contributions to science may be classified under three
heads--“Colour Perception,” “Molecular Physics,” and “Electrical
Theories.” In addition to these there were other papers of the highest
interest and importance, such as the essay on “Saturn’s Rings,” the
paper on the “Equilibrium of Elastic Solids,” and various memoirs on
pure geometry and questions of mechanics, which would, if they stood
alone, have secured for their author a distinguished position as a
physicist and mathematician, but which are not the works by which his
name will be mostly remembered.
The work on “Colour Perception” was begun at an early date. We have
seen Maxwell while still at Edinburgh interested in the discussions
about Hay’s theories.
His first published paper on the subject was a letter to Dr. G.
Wilson, printed in the Transactions of the Royal Society of Arts for
1855; but he had been mixing colours by means of his top for some
little time previously, and the results of these experiments are given
in a paper entitled “Experiments on Colour,” communicated to the Royal
Society of Edinburgh by Dr. Gregory, and printed in their Transactions,
vol. xxi.
In the paper on “The Theory of Compound Colours,” printed in the
Philosophical Transactions for 1860, Maxwell gives a history of the
theory as it was known to him.
He points out first the distinction between the _optical_ properties
and the _chromatic_ properties of a beam of light. “The optical
properties are those which have reference to its origin and propagation
through media until it falls on the sensitive organ of vision;”
they depend on the periods and amplitudes of the ether vibrations
which compose the beam. “The chromatic properties are those which
have reference to its power of exciting certain sensations of colour
perceived through the organ of vision.” It is possible for two beams to
be optically very different and chromatically alike. The converse is
not true; two beams which are optically alike are also chromatically
alike.
The foundation of the theory of compound colours was laid by Newton.
He first shewed that “by the mixture of homogeneal light colours may
be produced which are like to the colours of homogeneal light as to
the appearance of colour, but not as to the immutability of colour and
constitution of light.” Two beams which differ optically may yet be
alike chromatically; it is possible by mixing red and yellow to obtain
an orange colour chromatically similar to the orange of the spectrum,
but optically different to that orange, for the compound orange can be
analysed by a prism into its component red and yellow; the spectrum
orange is incapable of further resolution.
Newton also solves the following problem:--
_In a mixture of primary colours, the quantity and quality of each
being given to know the colour of the compound_ (Optics, Book 1, Part
2, Prop. 6), and his solution is the following:--He arranges the seven
colours of the spectrum round the circumference of a circle, the length
occupied by each colour being proportional to the musical interval to
which, in Newton’s views, the colour corresponded. At the centre of
gravity of each of these arcs he supposes a weight placed proportional
to the number of rays of the corresponding colour which enter into the
mixture under consideration. The position of the centre of gravity of
these weights indicates the nature of the resultant colour. A radius
drawn through this centre of gravity points out the colour of the
spectrum which it most resembles; the distance of the centre of gravity
from the centre gives the fulness of the colour. The centre itself is
white. Newton gives no proof of this rule; he merely says, “This rule I
conceive to be accurate enough for practice, though not mathematically
accurate.”
Maxwell proved that Newton’s method of finding the centre of gravity of
the component colours was confirmed by his observations, and that it
involves mathematically the theory of three elements of colour; but
the disposition of the colours on the circle was only a provisional
arrangement; the true relations of the colours could only be determined
by direct experiment.
Thomas Young appears to have been the next, after Newton, to work
at the theory of colour sensation. He made observations by spinning
coloured discs much in the same way as that which was afterwards
adopted by Maxwell, and he developed the theory that three different
primary sensations may be excited in the eye by light, while the colour
of any beam depends on the proportions in which these three sensations
are excited. He supposes the three primary sensations to correspond
to red, green, and violet. A blue ray is capable of exciting both the
green and the violet; a yellow ray excites the red and the green. Any
colour, according to Young’s theory, may be matched by a mixture of
these three primary colours taken in proper proportion; the quality
of the colour depends on the proportion of the intensities of the
components; its brightness depends on the sum of these intensities.
Maxwell’s experiments were undertaken with the object of proving or
disproving the physical part of Young’s theory. He does not consider
the question whether there are three distinct sensations corresponding
to the three primary colours; that is a physiological inquiry, and one
to which no completely satisfactory answer has yet been given. He does
show that by a proper mixture of any three arbitrarily chosen standard
colours it is possible to match any other colour; the words “proper
mixture,” however, need, as will appear shortly, some development.
We may with advantage compare the problem with one in acoustics.
When a compound musical note consisting of a pure tone and its
overtones is sounded, the trained ear can distinguish the various
overtones and analyse the sound into its simple components. The same
sensation cannot be excited in two different ways. The eye has no such
corresponding power. A given yellow may be a pure spectral yellow,
corresponding to a pure tone in music, or it may be a mixture of a
number of other pure tones; in either case it can be matched by a
proper combination of three standard colours--this Maxwell proved.
It may be, as Young supposed, that if the three standard colours be
properly selected they correspond exactly to three primary sensations
of the brain. Maxwell’s experiments do not afford any light on this
point, which still remains more than doubtful.
When Maxwell began his work the theory of colours was exciting
considerable interest. Sir David Brewster had recently developed a
new theory of colour sensation which had formed the basis of some
discussions, and in 1852 von Helmholtz published his first paper
on the subject. According to Brewster, the three primitive colours
were red, yellow and blue, and he supposed that they corresponded to
three different kinds of objective light. Helmholtz pointed out that
experiments up to that date had been conducted by mixing pigments, with
the exception of those in which the rotating disc was used, and that
it is necessary to make them on the rays of the spectrum itself. He
then describes a method of mixing the light from two spectra so as to
obtain the combination of every two of the simple prismatic rays in all
degrees of relative strength.
From these experiments results, which at the time were unexpected, but
some of which must have been known to Young, were obtained. Among them
it was shown that a mixture of red and green made yellow, while one of
green and violet produced blue.
In a later paper (_Philosophical Magazine_, 1854) Helmholtz described
a method for ascertaining the various pairs of complementary
colours--colours, that is, which when mixed will give white--which had
been shown by Grassman to exist if Newton’s theory were true. He also
gave a provisional diagram of the curve formed by the spectrum, which
ought to take the place of the circle in Newton’s diagram; for this,
however, his experiments did not give the complete data.
Such was the state of the question when Maxwell began. His first
colour-box was made in 1852. Others were designed in 1855 and 1856,
and the final paper appeared in 1860. But before that time he had
established important results by means of his rotatory discs and colour
top. In his own description of this he says: “The coloured paper is
cut into the form of disc, each with a hole in the centre and divided
along a radius so as to admit of several of them being placed on the
same axis, so that part of each is exposed. By slipping one disc over
another we can expose any given portion of each colour. These discs
are placed on a top or teetotum, which is spun rapidly. The axis of the
top passes through the centre of the discs, and the quantity of each
colour exposed is measured by graduations on the rim of the top, which
is divided into 100 parts. When the top is spun sufficiently rapidly,
the impressions due to each colour separately follow each other in
quick succession at each point of the retina, and are blended together;
the strength of the impression due to each colour is, as can be shown
experimentally, the same as when the three kinds of light in the same
relative proportions enter the eye simultaneously. These relative
proportions are measured by the areas of the various discs which are
exposed. Two sets of discs of different radius are used; the largest
discs are put on first, then the smaller, so that the centre portion
of the top shows the colour arising from the mixture of those of the
smaller discs; the outer portion, that of the larger discs.”
In experimenting, six discs of each size are used, black, white, red,
green, yellow and blue. It is found by experiment that a match can be
arranged between any five of these. Thus three of the larger discs are
placed on the top--say black, yellow and blue--and two of the smaller
discs, red and green, are placed above these. Then it is found that it
is possible so to adjust the amount exposed of each disc that the two
parts of the top appear when it is spun to be of the same tint. In one
series of experiments the chromatic effect of 46·8 parts of black, 29·1
of yellow, and 24·1 of blue was found to be the same as that of 66·6
of red and 33·4 of green; each set of discs has a dirty yellow tinge.
Now, in this experiment, black is not a colour; practically no light
reaches the eye from a dead black. We have, however, to fill up
the circumference of the top in some way which will not affect the
impression on the retina arising from the mixture of the blue and
yellow; this we can do by using the black disc.
Thus we have shown that 66·6 parts of red and 33·4 parts of green
produce the same chromatic effect as 29·1 of yellow and 24·1 of
blue. Similarly in this manner a match can be arranged between any
four colours and black, the black being necessary to complete the
circumference of the discs.
Thus using A, B, C, D to denote the various colours, _a_, _b_, _c_,
_d_ the amounts of each colour taken, we can get a series of results
expressed as follows: _a_ parts of A together with _b_ parts of B match
_c_ parts of C together with _d_ parts of D; or we may write this as an
equation thus:--
_a_ A + _b_ B = _c_ C + _d_ D,
where the + stands for “combined with,” and the = for “matches in tint.”
We may also write the above--
_d_ D = _a_ A + _b_ B - _c_ C,
or _d_ parts of D can be matched by a _proper_ combination of colours
A, B, C. The sign - shows that in order to make the match we have to
combine the colour C with D; the combination then matches a mixture of
A and B.
In this way we can form a number of equations for all possible colours,
and if we like to take any three colours A, B, C as standards, we
obtain a result which may be written generally--
_x_ X = _a_ A + _b_ B + _c_ C,
or _x_ parts of X can be matched by _a_ parts of A, combined with _b_
parts of B and _c_ parts of C. If the sign of one of the quantities
_a_, _b_, or _c_ is negative, it indicates that that colour must be
combined with X to match the other two.
Now Maxwell was able to show that, if A, B, C be properly selected,
nearly every other colour can be matched by positive combinations of
these three. These three colours, then, are primary colours, and nearly
every other colour can be matched by a combination of the three primary
colours.
Experiments, however, with coloured discs, such as were undertaken by
Young, Forbes and Maxwell, were not capable of giving satisfactory
results. The colours of the discs were not pure spectrum colours, and
varied to some extent with the nature of the incident light. It was for
this reason that Helmholtz in 1852 experimented with the spectrum, and
that Maxwell about the same time invented his colour box.
The principle of the latter was very simple. Suppose we have a slit
S, and some arrangement for forming a pure spectrum on a screen. Let
there now be a slit R placed in the red part of the spectrum on the
screen. When light falls on the slit S, only the red rays can reach
R, and hence conversely, if the white source be placed at the other
end of the apparatus, so that R is illuminated with white light, only
red rays will reach S. Similarly, if another slit be placed in the
green at G, and this be illuminated by white light, only the green
rays will reach S, while from a third slit V in the violet, violet
light only can arrive at S. Thus by opening the three slits at V, G
and R simultaneously, and looking through S, the retina receives the
impression of the three different colours. The amount of light of each
colour will depend on the breadth to which the corresponding slit is
opened, and the relative intensities of the three different components
can be compared by comparing the breadths of the three slits. Any other
colour which is allowed by some suitable contrivance to enter the eye
simultaneously can now be matched, provided the red, green and violet
are primary colours.
By means of experiments with the colour box Maxwell showed conclusively
that a match could be obtained between any four colours; the
experiments could not be carried out in quite the simple manner
suggested by the above description of the principle of the box.
An account of the method will be found in Maxwell’s own paper. It
consisted in matching a standard white by various combinations of other
colours.
The main object of his research, however, was to examine the chromatic
properties of the different parts of the spectrum, and to determine the
form of the curve which ought to replace the circle in Newton’s diagram
of colour.
Maxwell adopted as his three standard colours: red, of about wave
length 6,302; green, wave length 5,281; and violet, 4,569 tenth metres.
On the scale of Maxwell’s instrument these are represented by the
numbers 24, 44 and 68.
Let us take three points A, B, C at the corners of an equilateral
triangle to represent on a diagram these three colours. The position
of any other colour on the diagram will be found by taking weights
proportional to the amounts of the colours A, B, C required to make the
match between A, B, C and the given colour; these weights are placed at
A, B, C respectively; the position of their centre of gravity is the
point required. Thus the position of white is given by the equation--
W = 18·6 (24) + 31·4 (44) + 30·5 (68)
which means that weights proportional to 18·6, 31·4 and 30·5 are to be
placed at A, B, C respectively, and their centre of gravity is to be
found. The point so found is the position of white. Any other colour is
given by the equation--
X = _a_ (24) + _b_ (44) + _c_ (68).
Again, the position on the diagram for all colours for which _a_,
_b_, _c_ are all positive lies within the triangle A B C. If one of
the coefficients, say _c_, is negative the same construction applies,
but the weight applied at C must be treated as acting in the opposite
direction to those at A and B. A mixture of the given colour and C
matches a mixture of A and B. It is clear that the point corresponding
to X will then lie outside the triangle A B C. Maxwell showed that,
with his standards, nearly all colours could be represented by points
inside the triangle. The colours he had selected as standards were very
close to primary colours.
Again, he proved that any spectrum colour between red and green, when
combined with a very slight admixture of violet, could be matched, in
the case of either Mrs. Maxwell or himself, by a proper mixture of
the red and green. The positions, therefore, of the spectrum colours
between red and green lie just outside the triangle A B C, being very
close to the line A B, while for the colours between green and violet
Maxwell obtained a curve lying rather further outside the side B C.
Any spectrum colour between green and violet, together with a slight
admixture of red, can be matched by a proper mixture of green and
violet.
Thus the circle of Newton’s diagram should be replaced by a curve,
which coincides very nearly with the two sides A B and B C of Maxwell’s
figure. Strictly, according to his observations, the curve lies just
outside these two sides. The purples of the spectrum lie nearly along
the third side, C A, of the triangle, being obtained approximately by
mixing the violet and the red.
To find the point on the diagram corresponding to the colour obtained
by mixing any two or more spectrum colours we must, in accordance
with Newton’s rule, place weights at the points corresponding to the
selected colours, and find the centre of gravity of these weights.
This, then, was the outcome of Maxwell’s work on colour. It verified
the essential part of Newton’s construction, and obtained for the first
time the true form of the spectrum curve on the diagram.
The form of this curve will of course depend on the eye of the
individual observer. Thus Maxwell and Mrs. Maxwell both made
observations, and distinct differences were found in their eyes. It
appears, however, that a large majority of persons have normal vision,
and that matches made by one such person are accepted by most others
as satisfactory. Some people, however, are colour blind, and Maxwell
examined a few such. In the case of those whom he examined it appeared
as though vision was dichromatic, the red sensation seemed to be
absent; nearly all colours could be matched by combinations of green
and violet. The colour diagram was reduced to the straight line B C.
Other forms of colour blindness have since been investigated.
In awarding to Maxwell the Rumford medal in 1860, Major-General
Sabine, vice-president of the Royal Society, after explaining the
theory of colour vision and the possible method of verifying it, said:
“Professor Maxwell has subjected the theory to this verification, and
thereby raised the composition of colours to the rank of a branch of
mathematical physics,” and he continues: “The researches for which
the Rumford medal is awarded lead to the remarkable result that to a
very near degree of approximation all the colours of the spectrum, and
therefore all colours in nature which are only mixtures of these, can
be perfectly imitated by mixtures of three actually attainable colours,
which are the red, green and blue belonging respectively to three
particular parts of the spectrum.”
It should be noticed in concluding our remarks on this part of
Maxwell’s work that his results are purely physical. They are not
inconsistent with the physiological part of Young’s theory, viz., that
there are three primary sensations of colour which can be transmitted
to the brain, and that the colour of any object depends on the relative
proportions in which these sensations are excited, but they do not
prove that theory. Any physiological theory which can be accepted as
true must explain Maxwell’s observations, and Young’s theory does this;
but it is, of course, possible that other theories may explain them
equally well, and be more in accordance with physiological observations
than Young’s. Maxwell has given us the physical facts which have to be
explained; it is for the physiologists to do the rest.
CHAPTER VIII.
SCIENTIFIC WORK--MOLECULAR THEORY.
Maxwell in his article “Atom,” in the ninth edition of the
_Encyclopædia Britannica_, has given some account of Modern Molecular
Science, and in particular of the molecular theory of gases. Of this
science, Clausius and Maxwell are the founders, though to their names
it has recently been shown that a third, that of Waterston, must be
added. In the present chapter it is intended to give an outline of
Maxwell’s contributions to molecular science, and to explain the
advances due to him.
The doctrine that bodies are composed of small particles in rapid
motion is very ancient. Democritus was its founder, Lucretius--de Rerum
Naturâ--explained its principles. The atoms do not fill space; there is
void between.
“Quapropter locus est intactus inane vacansque,
Quod si non esset, nullâ ratione moveri
Res possent; namque officium quod corporis extat
Officere atque obstare, id in omni tempore adesset
Omnibus. Haud igitur quicquam procedere posset
Principium quoniam cedendi nulla daret res.”
According to Boscovitch an atom is an indivisible point, having
position in space, capable of motion, and possessing mass. It is also
endowed with the power of exerting force, so that two atoms attract
or repel each other with a force depending on their distance apart.
It has no parts or dimensions: it is a mere geometrical point without
extension in space; it has not the property of impenetrability, for two
atoms can, it is supposed, exist at the same point.
In modern molecular science according to Maxwell, “we begin by assuming
that bodies are made up of parts each of which is capable of motion,
and that these parts act on each other in a manner consistent with the
principle of the conservation of energy. In making these assumptions
we are justified by the facts that bodies may be divided into
smaller parts, and that all bodies with which we are acquainted are
conservative systems, which would not be the case unless their parts
were also conservative systems.
“We may also assume that these small parts are in motion. This is the
most general assumption we can make, for it includes as a particular
case the theory that the small parts are at rest. The phenomena of the
diffusion of gases and liquids through each other show that there may
be a motion of the small parts of a body which is not perceptible to us.
“We make no assumption with respect to the nature of the small
parts--whether they are all of one magnitude. We do not even assume
them to have extension and figure. Each of them must be measured by its
mass, and any two of them must, like visible bodies, have the power
of acting on one another when they come near enough to do so. The
properties of the body or medium are determined by the configuration of
its parts.”
These small particles are called molecules, and a molecule in its
physical aspect was defined by Maxwell in the following terms:--
“A molecule of a substance is a small body, such that if, on
the one hand, a number of similar molecules were assembled
together, they would form a mass of that substance; while on
the other hand, if any portion of this molecule were removed,
it would no longer be able, along with an assemblage of other
molecules similarly treated, to make up a mass of the original
substance.”
We are to look upon a gas as an assemblage of molecules flying about
in all directions. The path of any molecule is a straight line, except
during the time when it is under the action of a neighbouring molecule;
this time is usually small compared with that during which it is free.
The simplest theory we could formulate would be that the molecules
behaved like elastic spheres, and that the action between any two was
a collision following the laws which we know apply to the collision of
elastic bodies. If the average distance between two molecules be great
compared with their dimensions, the time during which any molecule
is in collision will be small compared with the interval between the
collisions, and this is in accordance with the fundamental assumption
just mentioned. It is not, however, necessary to suppose an encounter
between two molecules to be a collision. One molecule may act on
another with a force, which depends on the distance between them, of
such a character that the force is insensible except when the molecules
are extremely close together.
It is not difficult to see how the pressure exerted by a gas on the
sides of a vessel which contains it may be accounted for on this
assumption. Each molecule as it strikes the side has its momentum
reversed--the molecules are here assumed to be perfectly elastic.
Thus each molecule of the gas is continually gaining momentum from
the sides of the vessel, while it gives up to the vessel the momentum
which it possessed before the impact. The rate at which this change of
momentum proceeds across a given area measures the force exerted on
that area; the pressure of the gas is the rate of change of momentum
per unit of area of the surface.
Again, it can be shown that this pressure is proportional to the
product of the mass of each molecule, the number of molecules in a unit
of volume, and the square of the velocity of the molecules.
Let us consider in the first instance the case of a jet of sand or
water of unit cross section which is playing against a surface. Suppose
for the present that all the molecules which strike the surface have
the same velocity.
Then the number of molecules which strike the surface per second, will
be proportional to this velocity. If the particles are moving quickly
they can reach the surface in one second from a greater distance than
is possible if they be moving slowly. Again, the number reaching the
surface will be proportional to the number of molecules per unit of
volume. Hence, if we call _v_ the velocity of each particle, and N
the number of particles per unit of volume, the number which strike
the surface in one second will be N _v_; if _m_ be the mass of each
molecule, the mass which strikes the surface per second is N _m_ _v_;
the velocity of each particle of this mass is _v_, therefore the
momentum destroyed per second by the impact is N _m_ _v_ × _v_, or N
_m_ _v_², and this measures the pressure.
Hence in this case if _p_ be the pressure
_p_ = N _m_ _v_².
In the above we assume that _all_ the molecules in the jet are moving
with velocity _v_ perpendicular to the surface. In the case of a crowd
of molecules flying about in a closed space this is clearly not true.
The molecules may strike the surface in any direction; they will not
all be moving normal to the surface. To simplify the case, consider a
cubical box filled with gas. The box has three pairs of equal faces at
right angles. We may suppose one-third of the particles to be moving at
right angles to each face, and in this case the number per unit volume
which we have to consider is not N, but ⅓ N. Hence the formula becomes
_p_ = ⅓ N _m_ _v_².
Moreover, if _ρ_ be the density of the gas--that is, the mass of
unit volume--then N_m_ is equal to _ρ_, for _m_ is the mass of each
particle, and there are N particles in a unit of volume.
Hence, finally, _p_ = ⅓ _ρ_ _v_².
Or, again, if V be the volume of unit mass of the gas, then _ρ_ V is
unity, or ρ is equal to 1/V.
Hence _p_V = ⅓_v_².
Formulæ equivalent to these appear first to have been obtained by
Herapath about the year 1816 (Thomson’s “Annals of Philosophy,” 1816).
The results only, however, were stated in that year. A paper which
attempted to establish them was presented to the Royal Society in 1820.
It gave rise to very considerable correspondence, and was withdrawn
by the author before being read. It is printed in full in Thomson’s
“Annals of Philosophy” for 1821, vol. i., pp. 273, 340, 401. The
arguments of the author are no doubt open to criticism, and are in many
points far from sound. Still, by considering the problem of the impact
of a large number of hard bodies, he arrived at a formula connecting
the pressure and volume of a given mass of gas equivalent to that just
given. These results are contained in Propositions viii. and ix. of
Herapath’s paper.
In his next step, however, Herapath, as we know now, was wrong. One
of his fundamental assumptions is that the temperature of a gas is
measured by the momentum of each of its particles. Hence, assuming
this, we have T = _m_ _v_, if T represents the temperature: and
_p_ = ⅓ N _m_ _v_² = ⅓ (N/_m_) (_m_ _v_)².
Or, again--
_p_ = ⅓ N·T·_v_ = ⅓·(N/_m_)·T².
These results are practically given in Proposition viii., Corr. (1)
and (2), and Proposition ix.[49] The temperature as thus defined by
Herapath is an absolute temperature, and he calculates the absolute
zero of temperature at which the gas would have no volume from the
above results. The actual calculation is of course wrong, for, as
we know now by experiment, the pressure is proportional to the
temperature, and not to its square, as Herapath supposed. It will be
seen, however, that Herapath’s formula gives Boyle’s law; for if the
temperature is constant, the formula is equivalent to
_p_ V = a constant.
Herapath somewhat extended his work in his “Mathematical Physics”
published in 1847, and applied his principles to explain diffusion, the
relation between specific heat and atomic weight, and other properties
of bodies. He still, however, retained his erroneous supposition
that temperature is to be measured by the momentum of the individual
particles.
The next step in the theory was made by Waterston. His paper was read
to the Royal Society on March 5th, 1846. It was most unfortunately
committed to the Archives of the Society, and was only disinterred by
Lord Rayleigh in 1892 and printed in the Transactions for that year.
In the account just given of the theory, it has been supposed that all
the particles move with the same velocity. This is clearly not the
case in a gas. If at starting all the particles had the same velocity,
the collisions would change this state of affairs. Some particles will
be moving quickly, some slowly. We may, however, still apply the
theory by splitting up the particles into groups, and, supposing that
each group has a constant velocity, the particles in this group will
contribute to the pressure an amount--_p_₁--equal to ⅓ N₁ _m_ _v_₁²,
where _v_₁ is the velocity of the group and N₁ the number of particles
having that velocity. The whole pressure will be found by adding that
due to the various groups, and will be given as before by _p_ = ⅓ N _m_
_v_², where _v_ is not now the actual velocity of the particles, but a
mean velocity given by the equation
N _v_² = N₁ _v_₁² + N₂ _v_₂² + .....,
which will produce the same pressure as arises from the actual impacts.
This quantity v² is known as the _mean square_ of the molecular
velocity, and is so used by Waterston.
In a paper in the _Philosophical Magazine_ for 1858 Waterston gives an
account of his own paper of 1846 in the following terms:--“Mr. Herapath
unfortunately assumed heat or temperature to be represented by the
simple ratio of the velocity instead of the square of the velocity,
being in this apparently led astray by the definition of motion
generally received, and thus was baffled in his attempts to reconcile
his theory with observation. If we make this change in Mr. Herapath’s
definition of heat or temperature--viz., that it is proportional
to the vis-viva or square velocity of the moving particle, not to
the momentum or simple ratio of the velocity--we can without much
difficulty deduce not only the primary laws of elastic fluids, but also
the other physical properties of gases enumerated above in the third
objection to Newton’s hypothesis. [The paper from which the quotation
is taken is on ‘The Theory of Sound.’] In the Archives of the Royal
Society for 1845–46 there is a paper on ‘The Physics of Media that
consist of perfectly “Elastic Molecules in a State of Motion,”’ which
contains the synthetical reasoning on which the demonstration of these
matters rests.... This theory does not take account of the size of the
molecules. It assumes that no time is lost at the impact, and that if
the impacts produce rotatory motion, the vis viva thus invested bears
a constant ratio to the rectilineal vis viva, so as not to require
separate consideration. It does, also, not take account of the probable
internal motion of composite molecules; yet the results so closely
accord with observation in every part of the subject as to leave no
doubt that Mr. Herapath’s idea of the physical constitution of gases
approximates closely to the truth.”
In his introduction to Waterston’s paper (Phil. Trans., 1892) Lord
Rayleigh writes:--“Impressed with the above passage, and with the
general ingenuity and soundness of Waterston’s views, I took the first
opportunity of consulting the Archives, and saw at once that the memoir
justified the large claims made for it, and that it marks an immense
advance in the direction of the now generally received theory.”
In the first section of the paper Waterston’s great advance consisted
in the statement that the mean square of the kinetic energy of each
molecule measures the temperature.
According to this we are thus to put in the pressure equation--½ _m_
_v_² = T, the temperature, and we have at once--_p_ V = ⅔ N · T.
Now this equation expresses, as we know, the laws of Boyle and Gay
Lussac.
The second section discusses the properties of media, consisting of
two or more gases, and arrives at the result that “in mixed media
the mean square molecular velocity is inversely proportional to the
specific weights of the molecules.” This was the great law rediscovered
by Maxwell fifteen years later. With modern notation it may be put
thus:--If _m_₁, _m_₂ be the masses of each molecule of two different
sets of molecules mixed together, then, when a steady state has been
reached, since the temperature is the same throughout, _m_₁ _v_₁² is
equal to _m_₂ _v_₂². The average kinetic energy of each molecule is the
same.
From this Avogadros’ law follows at once--for if _p_₁, _p_₂ be the
pressures, N₁, N₂ the numbers of molecules per unit volume--
_p_₁ = ⅓ N₁ _m_₁ _v_₁²,
_p_₂ = ⅓ N₂ _m_₂ _v_₂².
Hence, if _p_₁, is equal to _p_₂, since _m_₁ _v_₁² is equal to _m_₂
_v_₂², we must have N₁ equal to N₂, or the number of molecules in equal
volumes of two gases at the same pressure and temperature is the same.
The proof of this proposition given by Waterston is not satisfactory.
On this point, however, we shall have more to say. The third section of
the paper deals with adiabatic expansion, and in it there is an error
in calculation which prevented correct results from being attained.
At the meeting of the British Association at Ipswich, in 1851, a paper
by J. J. Waterston of Bombay, on “The General Theory of Gases,” was
read. The following is an extract from the Proceedings:--
The author “conceives that the atoms of a gas, being perfectly elastic,
are in continual motion in all directions, being constrained within
a limited space by their collisions with each other, and with the
particles of surrounding bodies.
“The vis viva of these motions in a given portion of a gas constitutes
the quantity of heat contained in it.
“He shows that the result of this state of motion must be to give the
gas an elasticity proportional to the mean square of the velocity of
the molecular motions, and to the total mass of the atoms contained in
unity of bulk” (unit of volume)--that is to say, to the density of the
medium.
“The elasticity in a given gas is the measure of temperature.
Equilibrium of pressure and heat between two gases takes place when the
number of atoms in unit of volume is equal and the vis viva of each
atom equal. Temperature, therefore, in all gases is proportional to the
mass of one atom multiplied by the mean square of the velocity of the
molecular motions, being measured from an absolute zero 491° below the
zero of Fahrenheit’s thermometer.”
It appears, therefore, from these extracts that the discovery of the
laws that temperature is measured by the mean kinetic energy of a
single molecule, and that in a mixture of gases the mean kinetic energy
of each molecule is the same for each gas, is due to Waterston. They
were contained in his paper of 1846, and published by him in 1851. Both
these papers, however, appear to have been unnoticed by all subsequent
writers until 1892.
Meanwhile, in 1848, Joule’s attention was called by his experiments
to the question, and he saw that Herapath’s result gave a means of
calculating the mean velocity of the molecules of a gas. For according
to the result given above, _p_ = ⅓ _ρ v_²; thus _v_² = 3 _p/ρ_, and _p_
and _ρ_ being known, we find _v_². Thus for hydrogen at freezing-point
and atmospheric pressure Joule obtains for _v_ the value 6,055 feet per
second, or, roughly, six times the velocity of sound in air.
Clausius was the next writer of importance on the subject. His first
paper is in “Poggendorff’s Annalen,” vol. c., 1857, “On the Kind
of Motion we call Heat.” It gives an exposition of the theory, and
establishes the fact that the kinetic energy of the translatory motion
of a molecule does not represent the whole of the heat it contains. If
we look upon a molecule as a small solid we must consider the energy it
possesses in consequence of its rotation about its centre of gravity,
as well as the energy due to the motion of translation of the whole.
Clausius’ second paper appeared in 1859. In it he considers the average
length of the path of a molecule during the interval between two
collisions. He determines this path in terms of the average distance
between the molecules and the distance between the centres of two
molecules at the time when a collision is taking place.
These two papers appear to have attracted Maxwell’s attention to the
matter, and his first paper, entitled “Illustrations of the Dynamical
Theory of Gases,” was read to the British Association at Aberdeen and
Oxford in 1859 and 1860, and appeared in the _Philosophical Magazine_,
January and July, 1860.
In the introduction to this paper Maxwell points out, while there was
then no means of measuring the quantities which occurred in Clausius’
expression for the mean free path, “the phenomena of the internal
friction of gases, the conduction of heat through a gas, and the
diffusion of one gas through another, seem to indicate the possibility
of determining accurately the mean length of path which a particle
describes between two collisions. In order, therefore, to lay the
foundation of such investigations on strict mechanical principles,” he
continues, “I shall demonstrate the laws of motion of an indefinite
number of small, hard and perfectly elastic spheres acting on one
another only during impact.”
Maxwell then proceeds to consider in the first case the impact of two
spheres.
But a gas consists of an indefinite number of molecules. Now it is
impossible to deal with each molecule individually, to trace its
history and follow its path. In order, therefore, to avoid this
difficulty Maxwell introduced the statistical method of dealing with
such problems, and this introduction is the first great step in
molecular theory with which his name is connected.
He was led to this method by his investigation into the theory of
Saturn’s rings, which had been completed in 1856, and in which he
had shown that the conditions of stability required the supposition
that the rings are composed of an indefinite number of free particles
revolving round the planet, with velocities depending on their
distances from the centre. These particles may either be arranged in
separate rings, or their motion may be such that they are continually
coming into collision with each other.
As an example of the statistical method, let us consider a crowd
of people moving along a street. Taken as a whole the crowd moves
steadily forwards. Any individual in the crowd, however, is jostled
backwards and forwards and from side to side; if a line were drawn
across the street we should find people crossing it in both directions.
In a considerable interval more people would cross it, going in the
direction in which the crowd is moving, than in the other, and the
velocity of the crowd might be estimated by counting the number which
crossed the line in a given interval. This velocity so found would
differ greatly from the velocity of any individual, which might have
any value within limits, and which is continually changing. If we knew
the velocity of each individual and the number of individuals we could
calculate the average velocity, and this would agree with the value
found by counting the resultant number of people who cross the line in
a given interval.
Again, the people in the crowd will naturally fall into groups
according to their velocities. At any moment there will be a certain
number of people whose velocities are all practically equal, or, to be
more accurate, do not differ among themselves by more than some small
quantity. The number of people at any moment in each of these groups
will be very different. The number in any group, which has a velocity
not differing greatly from the mean velocity of the whole, will be
large; comparatively few will have either a very large or a very small
velocity.
Again, at any moment, individuals are changing from one group to
another; a man is brought to a stop by some obstruction, and his
velocity is considerably altered--he passes from one group to a
different one; but while this is so, if the mean velocity remains
constant, and the size of the crowd be very great, the number of people
at any moment in a given group remains unchanged. People pass from that
group into others, but during any interval the same number pass back
again into that group.
It is clear that if this condition is satisfied the distribution is
a steady one, and the crowd will continue to move on with the same
uniform mean velocity.
Now, Maxwell applies these considerations to a crowd of perfectly
elastic spheres, moving anyhow in a closed space, acting upon each
other only when in contact. He shows that they may be divided into
groups according to their velocities, and that, when the steady state
is reached, the number in each group will remain the same, although the
individuals change. Moreover, it is shown that, if A and B represent
any two groups, the state will only be steady when the numbers which
pass from the group A to the group B are equal to the numbers which
pass back from the group B to the group A. This condition, combined
with the fact that the total kinetic energy of the motion remains
unchanged, enables him to calculate the number of particles in any
group in terms of the whole number of particles, the mean velocity, and
the actual velocity of the group.
From this an accurate expression can be found for the pressure of the
gas, and it is proved that the value found by others, on the assumption
that all the particles were moving with a common velocity, is correct.
Previous to this paper of Maxwell’s it had been realised that the
velocities could not be uniform throughout. There had been no attempt
to determine the distribution of velocity, or to submit the problem to
calculation, making allowance for the variations in velocity.
Maxwell’s mathematical methods are, in their generality and elegance,
far in advance of anything previously attempted in the subject.
So far it has been assumed that the particles in the vessel are all
alike. Maxwell next takes the case of a mixture of two kinds of
particles, and inquires what relation must exist between the average
velocities of these different particles, in order that the state may be
steady.
Now, it can be shown that when two elastic spheres impinge the effect
of the impact is always such as to reduce the difference between their
kinetic energies.
Hence, after a very large number of impacts the kinetic energies of the
two balls must be the same; the steady state, then, will be reached
when each ball has the same kinetic energy.
Thus if _m_₁, _m_₂ be the masses of the particles in the two sets
respectively, _v_₁, _v_₂ their mean velocities we must have finally--
½ _m_₁ _v_₁² = ½ _m_₂ _v_₂²
This is the second of the two great laws enunciated by Waterston in
1845 and 1851, but which, as we have seen, had remained unknown until
1859, when it was again given by Maxwell.
Now, when gases are mixed their temperatures become equal. Hence we
conclude, in Maxwell’s words, “that the physical condition which
determines that the temperature of two gases shall be the same, is that
the mean kinetic energy of agitation of the individual molecules of the
two gases are equal.”
Thus, as the result of Maxwell’s more exact researches on the motion of
a system of spherical particles, we find that we again can obtain the
equations--
T = ½ _mv_²
_p_ = ⅓ N _mv_² = ⅔ NT = ⅔ _ρ_ T/_m_
From these results we obtain as before the laws of Boyle, Charles and
Avrogadro.
Again if _σ_ be the specific heat of the gas at constant volume, the
quantity of heat required to raise a single molecule of mass _m_ one
degree will be _σ_ _m_.
Thus, when a molecule is heated, the kinetic energy must increase by
this amount. But the increase of temperature, which in this case is 1°,
is measured by the increase of kinetic energy of the single molecule.
Hence the amount of heat required to raise the temperature of a single
molecule of all gases 1° is the same. Thus the quantity _σ_ _m_ is the
same for all gases; or, in other words, the specific heat of a gas is
inversely proportional to the mass of its individual molecules. The
density of a gas--since the number of molecules per unit volume at
a given pressure and temperature is the same for all gases--is also
proportional to the mass of each individual molecule. Thus the specific
heats of all gases are inversely proportional to their densities.
This is the law discovered experimentally by Dulong and Petit to be
approximately true for a large number of substances.
* * * * *
In the next part of the paper Maxwell proceeded to determine the
average number of collisions in a given time, and hence, knowing the
velocities, to determine, in terms of the size of the particles and
their numbers, the mean free path of a particle; the result so found
differed somewhat from that already obtained by Clausius.
Having done this he showed how, by means of experiments on the
viscosity of gases, the length of the mean free path could be
determined.
An illustration due to Professor Balfour Stewart will perhaps make this
clear. Let us suppose we have two trains running with uniform speed in
opposite directions on parallel lines, and, further, that the engines
continue to work at the same rate, developing just sufficient energy to
overcome the resistance of the line, etc., and to maintain the speed
constant. Now suppose passengers commence to jump across from one train
to the other. Each man carries with him his own momentum, which is in
the opposite direction to that of the train into which he jumps; the
result is that the momentum of each train is reduced by the process;
the velocities of the two decrease; it appears as though a frictional
force were acting between the two. Maxwell suggests that a similar
process will account for the apparent viscosity of gases.
Consider two streams of gas, moving in opposite directions one over
the other; it is found that in each case the layers of gas near the
separating surface move more slowly than those in the interior of
the streams; there is apparently a frictional force between the two
streams along this surface, tending to reduce their relative velocity.
Maxwell’s explanation of this is that at the common surface particles
from the one stream enter the other, and carry with them their own
momentum; thus near this surface the momentum of each stream is
reduced, just as the momentum of the trains is reduced by the people
jumping across. Internal friction or viscosity is due to the diffusion
of momentum across this common surface. The effect does not penetrate
far into the gas, for the particles soon acquire the velocity of the
stream to which they have come.
Now, the rate at which the momentum is diffused will measure the
frictional force, and will depend on the mean free path of the
particles. If this is considerable, so that on the average a particle
can penetrate a considerable distance into the second gas before a
collision takes place and its motion is changed, the viscosity will be
considerable; if, on the other hand, the mean free path is small, the
reverse will be true. Thus it is possible to obtain a relation between
the mean free path and the coefficient of viscosity, and from this, if
the coefficient of viscosity be known, a value for the mean free path
can be found.
Maxwell, in the paper under discussion, was the first to do this,
and, using a value found by Professor Stokes for the coefficient of
viscosity, obtained as the length of the mean free path of molecules
of air 1/447000 of an inch, while the number of collisions per second
experienced by each molecule is found to be about 8,077,200,000.
Moreover, it appeared from his theory that the coefficient of viscosity
should be independent of the number of molecules of gas present, so
that it is not altered by varying the density. This result Maxwell
characterises as startling, and he instituted an elaborate series of
experiments a few years later with a view of testing it. The reason
for this result will appear if we remember that, when the density is
decreased, the mean free path is increased; relatively, then, to the
total number of molecules present, the number which cross the surface
in a given time is increased. And it appears from Maxwell’s result that
this relative increase is such that the total number crossing remains
unchanged. Hence the momentum conveyed across each unit area per second
remains the same, in spite of the decrease in density.
Another consequence of the same investigation is that the coefficient
of viscosity is proportional to the mean velocity of the molecules.
Since the absolute temperature is proportional to the square of the
velocity, it follows that the coefficient of viscosity is proportional
to the square root of the absolute temperature.
The second part of the paper deals with the process of diffusion of two
or more kinds of moving particles among one another.
If two different gases are placed in two vessels separated by a porous
diaphragm such as a piece of unglazed earthenware, or connected by
means of a narrow tube, Graham had shewn that, after sufficient time
has elapsed, the two are mixed together. The same process takes place
when two gases of different density are placed together in the same
vessel. At first the denser gas may be at the bottom, the less dense
above, but after a time the two are found to be uniformly distributed
throughout.
Maxwell attempted to calculate from his theory the rate at which
the diffusion takes place in these cases. The conditions of most of
Graham’s experiments were too complicated to admit of direct comparison
with the theory, from which it appeared that there is a relation
between the mean free path and the rate of diffusion. One experiment,
however, was found, the conditions of which could be made the subject
of calculation, and from it Maxwell obtained as the value of the mean
free path in air 1/389000 of an inch.
The number was close enough to that found from the viscosity to afford
some confirmation of his theory.
However, a few years later Clausius criticised the details of this
part of the paper, and Maxwell, in his memoir of 1866, admits the
calculation to have been erroneous. The main principles remained
unaffected, the molecules pass from one gas to the other, and this
constitutes diffusion.
Now, suppose we have two sets of particles in contact of such a nature
that the mean kinetic energy of the one set is different from that of
the other; the temperatures of the two will then be different. These
two sets will diffuse into each other, and the diffusing particles will
carry with them their kinetic energy, which will gradually pass from
those which have the greater energy to those which have the less, until
the average kinetic energy is equalised throughout. But the kinetic
energy of translation is the heat of the particles. This diffusion of
kinetic energy is a diffusion of heat by conduction, and we have here
the mechanical theory of the conduction of heat in a gas.
Maxwell obtained an expression, which, however, he afterwards modified,
for the conductivity of a gas in terms of the mean free path. It
followed from this that the conductivity of air was only about 1/7000
of that of copper.
Thus the diffusion of gases, the viscosity of gases, and the conduction
of heat in gases, are all connected with the diffusion of the particles
carrying with them their momenta and their energy; while values of the
mean free path can be obtained from observations on any one of these
properties.
In the third part of his paper Maxwell considers the consequences
of supposing the particles not to be spherical. In this case the
impacts would tend to set up a motion of rotation in the particles.
The direction of the force acting on any particle at impact would not
necessarily pass through its centre; thus by impact the velocity of its
centre would be changed, and in addition the particles would be made to
spin. Some part, therefore, of the energy of the particles will appear
in the form of the translational energy of their centres, while the
rest will take the form of rotational energy of each particle about its
centre.
It follows from Maxwell’s work that for each particle the average value
of these two portions of energy would be equal. The total energy will
be half translational and half rotational.
This theorem, in a more general form which was afterwards given to
it, has led to much discussion, and will be again considered later.
For the present we will assume it to be true. Clausius had already
called attention to the fact that some of the energy must be rotational
unless the molecules be smooth spheres, and had given some reasons
for supposing that the ratio of the whole energy to the energy of
translation is in a steady state a constant. Maxwell shows that for
rigid bodies this constant is 2. Let us denote it for the present by
the symbol β. Thus, if the translational energy of a molecule is ½ _m_
_v_², its whole energy is ½ β _m_ _v_².
The temperature is still measured by the translational energy, or ½ _m_
_v_²; the heat depends on the whole energy. Hence if H represent the
amount of heat--measured as energy--contained by a single molecule,
and T its temperature, we have--
H = βT
From this it can be shewn[50] that if γ represent the ratio of the
specific heat of a gas at constant pressure to the specific heat at
constant volume, then--
β = ⅔ 1/(γ-1)
For air and some other gases the value of γ has been shown to be 1·408.
From this it follows that β = 1·634. Now, Maxwell’s theory required
that for smooth hard particles, approximately spherical in shape, β
should be 2, and hence he concludes “we have shown that a system of
such particles could not possibly satisfy the known relation between
the two specific heats of all gases.”
Since this statement was made many more experiments on the value of γ
have been undertaken; it is not equal to 1·408 for _all_ gases. Hence
the value of β is different for various gases.
It is of some importance to notice that the value of β just found for
air is very approximately 1·66 or 5/3.
For mercury vapour the value of γ has been shown by Kundt to be 1·33
or 1⅓, and hence β is equal to 1. Thus all the energy of a particle of
mercury vapour is translational, and its behaviour in this respect is
consistent with the assumption that a particle of mercury vapour is a
smooth sphere.
The two results of this theory which seemed to lend themselves most
readily to experimental verification were (1) that the viscosity of
a gas is independent of its density, and (2) that it is proportional
to the square root of the absolute temperature. The next piece
of work connected with the theory was an attempt to test these
consequences, and a description of the experiments was published in the
“Philosophical Transactions” for 1865, in a paper on the “Viscosity
or Internal Friction of Air and other Gases,” and forms the Bakerian
lecture for that year.
The first result was completely proved. It is shewn that the value of
the coefficient[51] of viscosity “is the same for air at 0·5 inch and
at 30 inches pressure, provided that the temperature remains the same.”
It was clear also that the viscosity depended on the temperature,
and the results of the experiments seemed to show that it was nearly
proportional to the absolute temperature. Thus for two temperatures,
185° Fah. and 51° Fah., the ratio of the two coefficients found was
1·2624; the ratio of the two temperatures, each measured from absolute
zero, is 1·2605.
This result, then, does not agree with the hypothesis that a gas
consists of spherical molecules acting only on each other by a kind of
impact, for, if this were so, the coefficient would, as we have seen,
depend on the square root of the absolute temperature. But Maxwell’s
result, connecting viscosity with the first power of the absolute
temperature, has not been confirmed by other investigators. According
to it we should have as the relation between μ, the coefficient of
viscosity at t° and μ₀, that at zero the equation--
μ = μ₀ (1 + .00365 t).
The most recent results of Professor Holman (_Philosophical Magazine_,
Vol. xxi., p. 212) give--
μ = μ₀ (1 + .00275 t - .00000034 t²).
And results similar to this are given by O. E. Meyer, Puluj, and
Obermeyer. Maxwell’s coefficient ·00365 is too large, but ·00182, the
coefficient obtained by supposing the viscosity proportional to the
square root of the temperature, would be too small.
It still remains true, therefore, that the laws of the viscosity of
gases cannot be explained by the hypothesis of the impact of hard
spheres; but some deductions drawn by Maxwell in his next paper from
his supposed law of proportionality to the first power of the absolute
temperature require modification.
It was clear from his experiments just described that the simple
hypothesis of the impact of elastic bodies would not account for all
the phenomena observed. Accordingly, in 1866, Maxwell took up the
problem in a more general form in his paper on the “Dynamical Theory of
Gases,” Phil. Trans., 1866.
In it he considered the molecules of the gas not as elastic spheres
of definite radius, but as small bodies, or groups of smaller
molecules, repelling one another with a force whose direction always
passes very nearly through the centre of gravity of the molecules,
and whose magnitude is represented very nearly by some function of
the distance of the centres of gravity. “I have made,” he continues,
“this modification of the theory in consequence of the results of my
experiments on the viscosity of air at different temperatures, and I
have deduced from these experiments that the repulsion is inversely as
the fifth power of the distance.”
Since more recent observation has shown that the numerical results of
Maxwell’s work connecting viscosity and temperature are erroneous, this
last deduction does not hold; the inverse fifth power law of force
will not give the correct relation between viscosity and temperature.
Maxwell himself at a later date, “On the Stresses in Rarefied Gases,”
Phil. Trans., 1879, realised this; but even in this last paper he
adhered to the fifth power law because it leads to an important
simplification in the equations to be dealt with.
The paper of 1866 is chiefly important because it contains for the
first time the application of general dynamical methods to molecular
problems. The law of the distribution of velocities among the molecules
is again investigated, and a result practically identical with that
found for the elastic spheres is arrived at. In obtaining this
conclusion, however, it is assumed that the distribution of velocities
is uniform in all directions about any point, whatever actions may be
taking place in the gas. If, for example, the temperature is different
at different points, then, for a given velocity, all directions are not
equally probable. Maxwell’s expression, therefore, for the number of
molecules which at any moment have a given velocity only applies to the
permanent state in which the distribution of temperature is uniform.
When dealing, for example, with the conduction of heat, a modification
of the expression is necessary. This was pointed out by Boltzmann.[52]
In the paper of 1866, Maxwell applies his generalised results to the
final distribution of two gases under the action of gravity, the
equilibrium of temperature between two gases, and the distribution of
temperature in a vertical column. These results are, as he states,
independent of the law of force between the molecules. The dynamical
causes of diffusion viscosity and conduction of heat are dealt with,
and these involve the law of force.
It follows also from the investigation that, on the hypotheses assumed
as its basis, if two kinds of gases be mixed, the difference between
the average kinetic energies of translation of the gases of each kind
diminishes rapidly in consequence of the action between the two. The
average kinetic energy of translation, therefore, tends to become the
same for each kind of gas, and as before, it is this average energy of
translation which measures the temperature.
A molecule in the theory is a portion of a gas which moves about as a
single body. It may be a mere point, a centre of force having inertia,
capable of doing work while losing velocity. There may be also in each
molecule systems of several such centres of force bound together by
their mutual actions. Again, a molecule may be a small solid body of
determinate form; but in this case we must, as Maxwell points out,
introduce a new set of forces binding together the parts of each
molecule: we must have a molecular theory of the second order. In any
case, the most general supposition made is that a molecule consists of
a series of parts which stick together, but are capable of relative
motion among each other.
In this case the kinetic energy of the molecule consists of the energy
of its centre of gravity, together with the energy of its component
parts, relative to its centre of gravity.[53]
Now Clausius had, as we have seen, given reasons for believing that the
ratio of the whole energy of a molecule to the energy of translation of
its centre of gravity tends to become constant. We have already used β
to denote this constant. Thus, while the temperature is measured by the
average kinetic energy of translation of the centre of gravity of each
molecule, the heat contained in a molecule is its whole energy, and is
β times this quantity. Thus the conclusions as to specific heat, etc.,
already given on page 130, apply in this case, and in particular we
have the result that if γ be the ratio of the specific heat at constant
pressure to that at constant volume, then--
β = ⅔ 1/(γ-1)
Maxwell’s theorem of the distribution of kinetic energy among a system
of molecules applied, as he gave it in 1866, to the kinetic energy of
translation of the centre of gravity of each molecule. Two years later
Dr. Boltzmann, in the paper we have already referred to, extended
it (under certain limitations) to the parts of which a molecule is
composed. According to Maxwell the average kinetic energy of the centre
of gravity of each molecule tends to become the same. According to
Boltzmann the average kinetic energy of each part of the molecule tends
to become the same.
Maxwell, in the last paper he wrote on the subject (“On Boltzmann’s
Theorem on the Average Distribution of Energy in a System of
Material Points,” Camb. Phil. Trans., XII.), took up this problem.
Watson had given a proof of it in 1876 differing from Boltzmann’s,
but still limited by the stipulation that the time, during which a
particle is encountering other particles, is very small compared with
the time during which there is no sensible action between it and
other particles, and also that the time during which a particle is
simultaneously within the distance of more than one other particle may
be neglected.
Maxwell claims that his proof is free from any such limitation. The
material points may act on each other at all distances, and according
to any law which is consistent with the conservation of energy; they
may also be acted on by forces external to the system, provided these
are consistent with that law.
The only assumption which is necessary for the direct proof is that
the system, if left to itself in its actual state of motion, will
sooner or later pass through every phase which is consistent with the
conservation of energy.
In this paper Maxwell finds in a very general manner an expression for
the number of molecules which at any time have a given velocity, and
this, when simplified by the assumptions of the former papers, reduces
to the form already found. He also shows that the average kinetic
energy corresponding to any one of the variables which define his
system is the same for every one of the variables of his system.
Thus, according to this theorem, if each molecule be a single small
solid body, six variables will be required to determine the position
of each, three variables will give us the position of the centre of
gravity of the molecule, while three others will determine the position
of the body relative to its centre of gravity. If the six variables
be properly chosen, the kinetic energy can be expressed as a sum of
six squares, one square corresponding to each variable. According to
the theorem the part of the kinetic energy depending on each square is
the same. Thus, the whole energy is six times as great as that which
arises from any one of the variables. The kinetic energy of translation
is three times as great as that arising from each variable, for it
involves the three variables which determine the position of the centre
of gravity. Hence, if we denote by K the kinetic energy due to one
variable, the whole energy is 6 K, and the translational energy is 3 K;
thus, for this case--
β = 6K/3K = 2
Or, again, if we suppose that the molecule is such that _m_ variables
are required to determine its position relatively to its centre of
gravity, since 3 are needed to fix the centre of gravity, the total
number of variables defining the position of the molecule is _m_ + 3,
and it is said to have _m_ + 3 degrees of freedom. Hence, in this case,
its total energy is (_m_ + 3) K and its energy of translation is 3 K,
thus we find--
β = (_m_ + 3)/3
Hence γ = 1 + 2/(_m_ + 3) = 1 + 2/_n_
if _n_ be the number of degrees of freedom of the molecule.
Thus, if this Boltzmann-Maxwell theorem be true, the specific heat of a
gas will depend solely on the number of degrees of freedom of each of
its molecules. For hard rigid bodies we should have _n_ equal to 6, and
hence γ = 1·333. Now the fact that this is not the value of γ for any
of the known gases is a fundamental difficulty in the way of accepting
the complete theory.
Boltzmann has called attention to the fact that if _n_ be equal to
five, then γ has the value 1·40. And this agrees fairly with the value
found by experiment for air, oxygen, nitrogen, and various other gases.
We will, however, return to this point shortly.
There is, perhaps, no result in the domain of physical science in
recent years which has been more discussed than the two fundamental
theorems of the molecular theory which we owe to Maxwell and to
Boltzmann.
The two results in question are (1) the expression for the number of
molecules which at any moment will have a given velocity, and (2) the
proposition that the kinetic energy is ultimately equally divided
among all the variables which determine the system.
With regard to (1) Maxwell showed that his error law was one possible
condition of permanence. If at any moment the velocities are
distributed according to the error law, that distribution will be a
permanent one. He did not prove that such a distribution is the only
one which can satisfy all the conditions of the problem.
The proof that this law is a necessary, as well as a sufficient,
condition of permanence was first given by Boltzmann, for a single
monatomic gas in 1872, for a mixture of such gases in 1886, and for a
polyatomic gas in 1887. Other proofs have been given since by Watson
and Burbury. It would be quite beyond the limits of this book to go
into the question of the completeness or sufficiency of the proofs. The
discussion of the question is still in progress.
The British Association Report for 1894 contains an important
contribution to the question, in the shape of a report by Mr. G. H.
Bryan, and the discussion he started at Oxford by reading this report
has been continued in the pages of _Nature_ and elsewhere since that
time.
Mr. Bryan shows in the first place what may be the nature of the
systems of molecules to which the results will apply, and discusses
various points of difficulty in the proof.
The theorem in question, from which the result (1) follows as a simple
deduction, has been thus stated by Dr. Larmor.[54]
“There exists a positive function belonging to a group of molecules
which, as they settle themselves into a steady state--on the average
derived from a great number of configurations--maintains a steady
downward trend. The Maxwell-Boltzmann steady state is the one in which
this function has finally attained its minimum value, and is thus a
unique steady state, it still being borne in mind that this is only a
proposition of averages derived from a great number of instances in
which nothing is conserved in encounters, except the energy, and that
exceptional circumstances may exist, comparatively very few in number,
in which the trend is, at any rate, temporarily the other way.”
This theorem, when applied to cases of motion, such as that of a gas at
constant temperature enclosed in a rigid envelope impermeable to heat,
appears to be proved. For such a case, therefore, the Maxwell-Boltzmann
law is the only one possible.
But whether this be so or not, the law first introduced by Maxwell is
one of those possible, and the advance in molecular science due to its
introduction is enormous.
We come now to the second result, the equal partition of the energy
among all the degrees of freedom of each molecule. Lord Kelvin
has pointed out a flaw in Maxwell’s proof, but Boltzmann showed
(_Philosophical Magazine_, March, 1893) how this flaw can easily
be corrected, and it may be said that in all cases in which the
Boltzmann-Maxwell law of the distribution of velocities holds,
Maxwell’s law of the equal partition of energy holds also.
Three cases are considered by Mr. Bryan, in which the law of
distribution fails for rigid molecules: the first is when the molecules
have all, in addition to their velocities of agitation, a common
velocity of translation in a fixed direction; the second is when the
gas has a motion of uniform rotation about a fixed axis; while the
third is when each molecule has an axis of symmetry. In this last case
the forces acting during a collision necessarily pass through the
axis of symmetry, the angular velocity, therefore, of any molecule
about this axis remains constant, the number of molecules having a
given angular velocity will remain the same throughout the motion,
and the part of the kinetic energy which depends on this component of
the motion will remain fixed, and will not come into consideration
when dealing with the equal partition of the energy among the various
degrees of freedom.
Such a molecule has five, and not six, degrees of freedom; three
quantities are needed to determine the position of its centre of
gravity, and two to fix the position of the axis of symmetry.
In this case, then, as Boltzmann points out, in the expression for the
ratio of the specific heats, we must have _n_ equal to 5, and hence
γ = 1 + 2/_n_ = 1 + 2/5 = 1·4
agreeing fairly with the value found for air and various other
permanent gases.
For cases, then, in which we consider each atom as a single rigid body,
the Boltzmann-Maxwell theorem appears to give a unique solution,
and the Maxwell law of the distribution of the energy to be in fair
accordance with the results of observation.[55]
If we can never go further--and it must be admitted that the
difficulties in the way of further advance are enormous--it may,
I think, be claimed for Maxwell that the progress already made is
greatly due to him. Both these laws, for the case of elastic spheres,
are contained in his first paper of 1860; and while it is to the
genius of Boltzmann that we owe their earliest generalisation, and in
particular the proof of the uniqueness of the solution under proper
restrictions, Maxwell’s last paper contributed in no small degree to
the security of the position. Not merely the foundations, but much of
the superstructure of molecular science is his work.
The difficulties in the way of advance are, as we have said, enormous.
Boltzmann, in one of his papers, has considered the properties of a
complex molecule of a gas, consisting maybe of a number of atoms and
possibly of ether atoms bound with them, and he concludes that such a
molecule will behave in its progressive motion, and in its collisions
with other molecules, nearly like a rigid body. But to quote from Mr.
Bryan: “The case of a polyatomic molecule, whose atoms are capable of
vibrating relative to one another, affords an interesting field for
investigation and speculation. Is the Boltzmann distribution still
unique, or do other permanent distributions exist in which the kinetic
energy is unequally divided?”
Again, the spectroscope reveals to us vibrations of the ether, which
are connected in some way with the vibrations of the molecules of
gas, whose spectrum we are observing. It seems clear that the law of
equal partition does not apply to these, and yet, if we are to suppose
that the ether vibrations are due to actual vibrations of the atoms
which constitute a molecule, why does it not apply? Where does the
condition come in which leads to failure in the proof? Or, again,
is it, as has been suggested, the fact that the complex spectrum
of a gas represents the terms of a Fourier Series, into which some
elaborate vibration of the atoms is resolved by the ether? or is the
spectrum due simply to electro-magnetic vibrations on the surface of
the molecules--vibrations whose period is determined chiefly by the
size and shape of the molecule, but in which the atoms of which it is
composed take part? There are grave difficulties in the way of either
of these explanations, but we must not let our dread of the task which
remains to be done blind our eyes to the greatness of Maxwell’s work.
One other important paper, and a number of shorter articles, remain to
be mentioned.
The Boltzmann-Maxwell law applies only to cases in which the
temperature is uniform throughout. In a paper published in the
Philosophical Transactions for 1879, on “Stresses in Rarefied Gases
Arising from Inequalities of Temperature,” Maxwell deals, among other
matters, with the theory of the radiometer. He shows that the observed
motions will not take place unless gas, in contact with a solid, can
slide along the surface of the solid with a finite velocity between
places where the temperature is different; and in an appendix he proves
that, on certain assumptions regarding the nature of the contact of the
solid and the gas, there will be, even when the pressure is constant, a
flow of gas along the surface from the colder to the hotter parts.
Among his less important papers bearing on molecular theory must be
mentioned a lecture on “Molecules” to the British Association at its
Bradford meeting; “Scientific Papers of Clerk Maxwell,” vol. ii., p.
361; and another on “The Molecular Constitution of Bodies,” Scientific
Papers, vol. ii., p. 418.
In this latter, and also in a review in _Nature_ of Van der Waals’
book on “The Continuity of the Gaseous and Liquid States,”[56] he
explains and discusses Clausius’ virial equation, by means of which the
variations of the permanent gases from Boyle’s law are explained. The
lecture gives a clear account, in Maxwell’s own inimitable style, of
the advances made in the kinetic theory up to the date at which it was
delivered, and puts clearly the difficulties it has to meet. Maxwell
thought that those arising from the known values of the ratio of the
specific heats were the most serious.
In the articles, “Atomic Constitution of Bodies” and “Diffusion,” in
the ninth edition of the _Encyclopædia Britannica_, we have Maxwell’s
later views on the fundamental assumptions of the molecular theory.
The text-book on “Heat” contains some further developments of the
theory. In particular he shows how the conclusions of the second law
of thermo-dynamics are connected with the fact that the coarseness of
our faculties will not allow us to grapple with individual molecules.
The work described in the foregoing chapters would have been sufficient
to secure to Maxwell a distinguished place among those who have
advanced our knowledge; it remains still to describe his greatest work,
his theory of Electricity and Magnetism.
CHAPTER IX.
SCIENTIFIC WORK.--ELECTRICAL THEORIES.
Clerk Maxwell’s first electrical paper--that on Faraday’s “Lines of
Force”--was read to the Cambridge Philosophical Society on December
10th, 1855, and Part II. on February 11th, 1856. The author was then a
Bachelor of Arts, only twenty-three years in age, and of less than one
year’s standing from the time of taking his degree.
The opening words of the paper are as follows (Scientific Papers, vol.
i., p. 155):--
“The present state of electrical science seems peculiarly
unfavourable to speculation. The laws of the distribution of
electricity on the surface of conductors have been analytically
deduced from experiment; some parts of the mathematical
theory of magnetism are established, while in other parts the
experimental data are wanting; the theory of the conduction of
galvanism, and that of the mutual attraction of conductors,
have been reduced to mathematical formulæ, but have not
fallen into relation with the other parts of the science. No
electrical theory can now be put forth, unless it shows the
connection, not only between electricity at rest and current
electricity, but between the attractions and inductive effects
of electricity in both states. Such a theory must accurately
satisfy those laws, the mathematical form of which is known,
and must afford the means of calculating the effects in the
limiting cases where the known formulæ are inapplicable.
In order, therefore, to appreciate the requirements of the
science, the student must make himself familiar with a
considerable body of most intricate mathematics, the mere
retention of which in the memory materially interferes with
further progress. The first process, therefore, in the
effectual study of the science, must be one of simplification
and reduction of the results of previous investigation to a
form in which the mind can grasp them. The results of this
simplification may take the form of a purely mathematical
formula or of a physical hypothesis. In the first case we
entirely lose sight of the phenomena to be explained; and
though we may trace out the consequences of given laws, we can
never obtain more extended views of the connections of the
subject. If, on the other hand, we adopt a physical hypothesis,
we see the phenomena only through a medium, and are liable
to that blindness to facts and rashness in assumption which
a partial explanation encourages. We must therefore discover
some method of investigation which allows the mind at every
step to lay hold of a clear physical conception, without being
committed to any theory founded on the physical science from
which that conception is borrowed, so that it is neither drawn
aside from the subject in pursuit of analytical subtleties, nor
carried beyond the truth by a favourite hypothesis.
“In order to obtain physical ideas without adopting a physical
theory we must make ourselves familiar with the existence of
physical analogies. By a physical analogy I mean that partial
similarity between the laws of one science and those of another
which makes each of them illustrate the other. Thus all the
mathematical sciences are founded on relations between physical
laws and laws of numbers, so that the aim of exact science
is to reduce the problems of Nature to the determination of
quantities by operations with members. Passing from the most
universal of all analogies to a very partial one, we find the
same resemblance in mathematical form between two different
phenomena giving rise to a physical theory of light.
“The changes of direction which light undergoes in passing from
one medium to another are identical with the deviations of the
path of a particle in moving through a narrow space in which
intense forces act. This analogy, which extends only to the
direction, and not to the velocity of motion, was long believed
to be the true explanation of the refraction of light; and we
still find it useful in the solution of certain problems, in
which we employ it without danger as an artificial method. The
other analogy, between light and the vibrations of an elastic
medium, extends much farther, but, though its importance and
fruitfulness cannot be over-estimated, we must recollect that
it is founded only on a resemblance _in form_ between the
laws of light and those of vibrations. By stripping it of its
physical dress and reducing it to a theory of ‘transverse
alternations,’ we might obtain a system of truth strictly
founded on observation, but probably deficient both in the
vividness of its conceptions and the fertility of its method.
I have said thus much on the disputed questions of optics, as
a preparation for the discussion of the almost universally
admitted theory of attraction at a distance.
“We have all acquired the mathematical conception of these
attractions. We can reason about them and determine their
appropriate forms or formulæ. These formulæ have a distinct
mathematical significance, and their results are found to be
in accordance with natural phenomena. There is no formula
in applied mathematics more consistent with Nature than the
formula of attractions, and no theory better established in
the minds of men than that of the action of bodies on one
another at a distance. The laws of the conduction of heat in
uniform media appear at first sight among the most different in
their physical relations from those relating to attractions.
The quantities which enter into them are _temperature_, _flow
of heat_, _conductivity_. The word _force_ is foreign to the
subject. Yet we find that the mathematical laws of the uniform
motion of heat in homogeneous media are identical in form
with those of attractions varying inversely as the square of
the distance. We have only to substitute _source of heat_ for
_centre of attraction_, _flow of heat_ for _accelerating effect
of attraction_ at any point, and _temperature_ for _potential_,
and the solution of a problem in attractions is transformed
into that of a problem in heat.
“This analogy between the formulæ of heat and attraction was, I
believe, first pointed out by Professor William Thomson in the
_Cambridge Mathematical Journal_, Vol. III.
“Now the conduction of heat is supposed to proceed by an
action between contiguous parts of a medium, while the force
of attraction is a relation between distant bodies, and yet,
if we knew nothing more than is expressed in the mathematical
formulæ, there would be nothing to distinguish between the one
set of phenomena and the other.
“It is true that, if we introduce other considerations and
observe additional facts, the two subjects will assume very
different aspects, but the mathematical resemblance of some
of their laws will remain, and may still be made useful in
exciting appropriate mathematical ideas.
“It is by the use of analogies of this kind that I have
attempted to bring before the mind, in a convenient and
manageable form, those mathematical ideas which are necessary
to the study of the phenomena of electricity. The methods are
generally those suggested by the processes of reasoning which
are found in the researches of Faraday, and which, though they
have been interpreted mathematically by Professor Thomson and
others, are very generally supposed to be of an indefinite and
unmathematical character, when compared with those employed by
the professed mathematicians. By the method which I adopt, I
hope to render it evident that I am not attempting to establish
any physical theory of a science in which I have hardly made
a single experiment, and that the limit of my design is to
show how, by a strict application of the ideas and methods
of Faraday, the connection of the very different orders of
phenomena which he has discovered may be clearly placed before
the mathematical mind. I shall therefore avoid as much as I can
the introduction of anything which does not serve as a direct
illustration of Faraday’s methods, or of the mathematical
deductions which may be made from them. In treating the simpler
parts of the subject I shall use Faraday’s mathematical methods
as well as his ideas. When the complexity of the subject
requires it, I shall use analytical notation, still confining
myself to the development of ideas originated by the same
philosopher.
“I have in the first place to explain and illustrate the idea
of ‘lines of force.’
“When a body is electrified in any manner, a small body
charged with positive electricity, and placed in any given
position, will experience a force urging it in a certain
direction. If the small body be now negatively electrified, it
will be urged by an equal force in a direction exactly opposite.
“The same relations hold between a magnetic body and the north
or south poles of a small magnet. If the north pole is urged
in one direction, the south pole is urged in the opposite
direction.
“In this way we might find a line passing through any point
of space, such that it represents the direction of the
force acting on a positively electrified particle, or on an
elementary north pole, and the reverse direction of the force
on a negatively electrified particle or an elementary south
pole. Since at every point of space such a direction may be
found, if we commence at any point and draw a line so that,
as we go along it, its direction at any point shall always
coincide with that of the resultant force at that point, this
curve will indicate the direction of that force for every point
through which it passes, and might be called on that account a
_line of force_. We might in the same way draw other lines of
force, till we had filled all space with curves indicating by
their direction that of the force at any assigned point.
“We should thus obtain a geometrical model of the physical
phenomena, which would tell us the _direction_ of the force,
but we should still require some method of indicating the
_intensity_ of the force at any point. If we consider these
curves not as mere lines, but as fine tubes of variable section
carrying an incompressible fluid, then, since the velocity of
the fluid is inversely as the section of the tube, we may make
the velocity vary according to any given law, by regulating the
section of the tube, and in this way we might represent the
intensity of the force as well as its direction by the motion
of the fluid in these tubes. This method of representing the
intensity of a force by the velocity of an imaginary fluid in
a tube is applicable to any conceivable system of forces, but
it is capable of great simplification in the case in which
the forces are such as can be explained by the hypothesis of
attractions varying inversely as the square of the distance,
such as those observed in electrical and magnetic phenomena.
In the case of a perfectly arbitrary system of forces, there
will generally be interstices between the tubes; but in the
case of electric and magnetic forces it is possible to arrange
the tubes so as to leave no interstices. The tubes will then be
mere surfaces, directing the motion of a fluid filling up the
whole space. It has been usual to commence the investigation of
the laws of these forces by at once assuming that the phenomena
are due to attractive or repulsive forces acting between
certain points. We may, however, obtain a different view of the
subject, and one more suited to our more difficult inquiries,
by adopting for the definition of the forces of which we treat,
that they may be represented in magnitude and direction by the
uniform motion of an incompressible fluid.
“I propose, then, first to describe a method by which the
motion of such a fluid can be clearly conceived; secondly
to trace the consequences of assuming certain conditions of
motion, and to point out the application of the method to some
of the less complicated phenomena of electricity, magnetism,
and galvanism; and lastly, to show how by an extension of these
methods, and the introduction of another idea due to Faraday,
the laws of the attractions and inductive actions of magnets
and currents may be clearly conceived, without making any
assumptions as to the physical nature of electricity, or adding
anything to that which has been already proved by experiment.
“By referring everything to the purely geometrical idea of the
motion of an imaginary fluid, I hope to attain generality and
precision, and to avoid the dangers arising from a premature
theory professing to explain the cause of the phenomena.
If the results of mere speculation which I have collected
are found to be of any use to experimental philosophers, in
arranging and interpreting their results, they will have served
their purpose, and a mature theory, in which physical facts
will be physically explained, will be formed by those who by
interrogating Nature herself can obtain the only true solution
of the questions which the mathematical theory suggests.”
The idea was a bold one: for a youth of twenty-three to explain, by
means of the motions of an incompressible fluid, some of the less
complicated phenomena of electricity and magnetism, to show how
the laws of the attractions of magnets and currents may be clearly
conceived without making any assumption as to the physical nature of
electricity, or adding anything to that which has already been proved
by experiment.
It may be useful to review in a very few words the position of
electrical theory[57] in 1855.
Coulomb’s experiments had established the fundamental facts of
electrostatic attraction and repulsion, and Coulomb himself, about
1785, had stated a theory based on these experiments which could “only
be attacked by proving his experimental results to be inaccurate.”[58]
Coulomb supposes the existence of two electric fluids, the theory
developed previously by Franklin, but says--
“Je préviens pour mettre la théorie qui va suivre à l’abri de
toute dispute systématique, que dans la supposition de deux
fluides électriques, je n’ai autre intention que de présenter
avec le moins d’éléments possible les résultats du calcul et
de l’expérience, et non d’indiquer les véritables causes de
l’électricité.”
Cavendish was working in England about the same time as Coulomb, but
he published very little, and the value and importance of his work
was not recognised until the appearance in 1879 of the “Electrical
Researches of Henry Cavendish,” edited by Clerk Maxwell.
Early in the present century the application of mathematical analysis
to electrical problems was begun by Laplace, who investigated the
distribution of electricity on spheroids, and about 1811 Poisson’s
great work on the distribution of electricity on two spheres placed
at any given distance apart was published. Meanwhile the properties
of the electric current were being investigated. Galvani’s discovery
of the muscular contraction in a frog’s leg, caused by the contact of
dissimilar metals, was made in 1790. Volta invented the voltaic pile in
1800, and Oersted in 1820 discovered that an electric current produced
magnetic force in its neighbourhood. On this Ampère laid the foundation
of his theory of electro-dynamics, in which he showed how to calculate
the forces between circuits carrying currents from an assumed law of
force between each pair of elements of the circuits. His experiments
proved that the consequences which follow from this law are consistent
with all the observed facts. They do not prove that Ampère’s law alone
can explain the facts.
Maxwell, writing on this subject in the “Electricity an Magnetism,”
vol. ii., p. 162, says--
“The experimental investigation by which Ampère established the
laws of the mechanical action between electric currents is one
of the most brilliant achievements in science.
“The whole, 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 electro-dynamics.
“The method of Ampère, however, though cast into an inductive
form, does not allow us to trace the formation of the ideas
which guided it. We can scarcely believe that Ampère really
discovered the law of action by means of the experiments which
he describes. We are led to suspect, what, indeed, he tells us
himself, that he discovered the law by some process which he
has not shown us, and that when he had afterwards built up a
perfect demonstration, he removed all traces of the scaffolding
by which he had built it.”
The experimental evidence for Ampère’s theory, so far, at least, as
it was possible to obtain it from experiments on closed circuits, was
rendered unimpeachable by W. Weber about 1846, while in the previous
year Grassman and F. E. Neumann both published laws for the attraction
between two elements of current which differ from that of Ampère, but
lead to the same result for closed circuits. In a paper published in
1846 Weber announced his hypothesis connecting together electrostatic
and electro-dynamic action. In this paper he supposed that the force
between two particles of electricity depends on the motion of the
particles as well as on their distance apart. A somewhat similar
theory was proposed by Gauss and published after his death in his
collected works. It has been shown, however, that Gauss’ theory is
inconsistent with the conservation of energy. Weber’s theory avoids
this inconsistency and leads, for closed circuits, to the same results
as Ampère. It has been proved, however, by Von Helmholtz, that, under
certain circumstances, according to it, a body would behave as though
its mass were negative--it would move in a direction opposite to that
of the force.[59]
Since 1846 many other theories have been proposed to explain Ampère’s
laws. Meanwhile, in 1821, Faraday observed that under certain
circumstances a wire carrying a current could be kept in continuous
rotation in a magnetic field by the action between the magnets and
the current. In 1824 Arago observed the motion of a magnet caused by
rotating a copper disc in its neighbourhood, while in 1831 Faraday
began his experimental researches into electro-magnetic induction.
About the same period Joseph Henry, of Washington, was making,
independently of Faraday, experiments of fundamental importance on
electro-magnetic induction, but sufficient attention was not called to
his work until comparatively recent years.
In 1833 Lenz made some important researches, which led him to discover
the connection between the direction of the induced currents and
Ampère’s laws, summed up in his rule that the direction of the induced
current is always such as to oppose by its electro-magnetic action the
motion which induces it.
In 1845 F. E. Neumann developed from this law the mathematical theory
of electro-magnetic induction, and about the same time W. Weber showed
how it might be deduced from his elementary law of electrical action.
The great name of Von Helmholtz first appears in connection with this
subject in 1851, but of his writings we shall have more to say at a
later stage.
Meanwhile, during the same period, various writers, Murphy, Plana,
Charles, Sturm, and Gauss, extended Poisson’s work on electrostatics,
treating the questions which arose as problems in the distribution of
an attracting fluid, attracting or repelling according to Newton’s law,
though here again the greatest advances were made by a self-taught
Nottingham shoemaker, George Green by name, in his paper “On the
Application of Mathematical Analysis to the Theories of Electricity and
Magnetism,” 1828.
Green’s researches, Lord Kelvin writes, “have led to the elementary
proposition which must constitute the legitimate foundation of every
perfect mathematical structure that is to be made from the materials
furnished by the experimental laws of Coulomb.”
Green, it may be remarked, was the inventor of the term Potential.
His essay, however, lay neglected from 1828, until Lord Kelvin called
attention to it in 1845. Meanwhile, some of its most important results
had been re-discovered by Gauss and Charles and Thomson himself.
Until about 1845, the experimental work on which these mathematical
researches in electrostatics were based was that of Coulomb. An
electrified body is supposed to have a charge of some imponderable
fluid “electricity.” Particles of electricity repel each other
according to a certain law, and the fluid distributes itself in
equilibrium over the surface of any charged conductor in accordance
with this law. There are on this theory two opposite kinds of electric
fluid, positive and negative, two charges of the same kind repel, two
charges of opposite kinds attract; the repulsion or attraction is
proportional to the product of the charges, and inversely proportional
to the square of the distance between them.
The action between two charges is action at a distance taking place
across the space which separates the two.
Faraday, in 1837, in the eleventh series of his “Experimental
Researches,” published his first paper on “Electrostatic Induction.”
He showed--as indeed Cavendish had proved long previously, though the
result remained unpublished--that the force between two charged bodies
will depend on the insulating medium which surrounds them, not merely
on their shape and position. Induction, as he expresses it, takes place
along curved lines, and is an action of contiguous particles; these
curved lines he calls the “lines of force.”
Discussing these researches in 1845, Lord Kelvin writes[60]:--
“Mr. Faraday’s researches ... were undertaken with a view to
test an idea which he had long possessed that the forces of
attraction and repulsion exercised by free electricity are not
the resultants of actions exercised at a distance, but are
propagated by means of molecular action among the contiguous
particles of the insulating medium surrounding the electrified
bodies, which he therefore calls the dielectric. By this idea
he has been led to some very remarkable views upon induction,
or, in fact, upon electrical action in general. As it is
impossible that the phenomena observed by Faraday can be
incompatible with the results of experiment which constitute
Coulomb’s theory, it is to be expected that the difference
of his ideas from those of Coulomb must arise solely from a
different method of stating and interpreting physically the
same laws; and further, it may, I think, be shown that either
method of viewing this subject, when carried sufficiently
far, may be made the foundation of a mathematical theory
which would lead to the elementary principles of the other as
consequences. This theory would, accordingly, be the expression
of the ultimate law of the phenomena, independently of any
physical hypothesis we might from other circumstances be led
to adopt. That there are necessarily two distinct elementary
ways of viewing the theory of electricity may be seen from the
following considerations....”
In the pages which follow, Lord Kelvin develops the consequences of an
analogy between the conduction of heat and electrostatic action, which
he had pointed out three years earlier (1842), in his paper on “The
Uniform Motion of Heat in Homogeneous Solid Bodies,” and discusses its
connection with the mathematical theory of electricity.
The problem of distributing sources of heat in a given homogeneous
conductor of heat, so as to produce a definite steady temperature at
each point on the conductor is shewn to be _mathematically_ identical
with that of distributing electricity in equilibrium, so as to produce
at each point an electrical potential having the same value as the
temperature.
Thus the fundamental laws of the conduction of heat may be made the
basis of the mathematical theory of electricity, but the physical
idea which they suggest is that of the propagation of some effect by
means of the mutual action of contiguous particles, rather than that
of material particles attracting or repelling at a distance, which
naturally follows from the statement of Coulomb’s law.
Lord Kelvin continues:--
“All the views which Faraday has brought forward and
illustrated, as demonstrated by experiment, lead to this method
of establishing the mathematical theory, and, as far as the
analysis is concerned, it would in most _general_ propositions
be more simple, if possible, than that of Coulomb. Of course
the analysis of _particular_ problems would be identical in the
two methods. It is thus that Faraday arrives at a knowledge of
some of the most important of the mathematical theorems which
from their nature seemed destined never to be perceived except
as mathematical truths.”
Lord Kelvin’s papers on “The Mathematical Theory of Electricity,”
published from 1848 to 1850, his “Propositions on the Theory of
Attraction” (1842), his “Theory of Electrical Images” (1847), and his
paper on “The Mathematical Theory of Magnetism” (1849), contain a
statement of the most important results achieved in the mathematical
sciences of Electrostatics and Magnetism up to the time of Maxwell’s
first paper.
The opening sentences of that paper have already been quoted. In the
preface to the “Electricity and Magnetism” Maxwell writes thus:--
“Before I began the study of electricity I resolved to read
no mathematics on the subject till I had first read through
‘Experimental Researches on Electricity.’ I was aware that
there was supposed to be a difference between Faraday’s way of
conceiving phenomena and that of the mathematicians, so that
neither he nor they were satisfied with each other’s language.
I had also the conviction that this discrepancy did not arise
from either party being wrong. I was first convinced of this by
Sir William Thomson, to whose advice and assistance, as well as
to his published papers, I owe most of what I have learned on
the subject.
“As I proceeded with the study of Faraday, I perceived that his
method of conceiving the phenomena was also a mathematical
one, though not exhibited in the conventional form of
mathematical symbols. I also found that these methods were
capable of being expressed in the ordinary mathematical forms,
and thus compared with those of the professed mathematicians.
“For instance, Faraday, in his mind’s eye, saw lines of force
traversing all space where the mathematicians saw centres of
force attracting at a distance. Faraday saw a medium where
they saw nothing but distance. Faraday sought the seat of the
phenomena in real actions going on in the medium. They were
satisfied that they had found it in a power of action at a
distance impressed on the electric fluids.”
Now, Maxwell saw an analogy between electrostatics and the steady
motion of an incompressible fluid like water, and it is this analogy
which he develops in the first part of his paper. The water flows along
definite lines; a surface which consists wholly of such lines of flow
will have the property that no water ever crosses it. In any stream
of water we can imagine a number of such surfaces drawn, dividing it
up into a series of tubes; each of these will be a tube of flow, each
of these tubes remain always filled with water. Hence, the quantity
of water which crosses per second any section of a tube of flow
perpendicular to its length is always the same. Thus, from the form of
the tube, we can obtain information as to the direction and strength of
the flow, for where the tube is wide the flow will be proportionately
small, and _vice versâ_.
Again, we can draw in the fluid a number of surfaces, over each of
which the pressure is the same; these surfaces will cut the tubes
of flow at right angles. Let us suppose they are drawn so that the
difference of pressure between any two consecutive surfaces is unity,
then the surfaces will be close together at points at which the
pressure changes rapidly; where the variation of pressure is slow, the
distance between two consecutive surfaces will be considerable.
If, then, in any case of motion, we can draw the pressure surfaces,
and the tubes of flow, we can determine the motion of the fluid
completely. Now, the same mathematical expressions which appear in
the hydro-dynamical theory occur also in the theory of electricity,
the meaning only of the symbols is changed. For velocity of fluid we
have to write electrical force. For difference of fluid pressure we
substitute work done, or difference of electrical potential or pressure.
The surfaces and tubes, drawn as the solution of any hydro-dynamical
problem, give us also the solution of an electrical problem; the
tubes of flow are Faraday’s tubes of force, or tubes of induction,
the surfaces of constant pressure are surfaces of equal electrical
potential. Induction may take place in curved lines just as the tubes
of flow may be bent and curved; the analogy between the two is a
complete one.
But, as Maxwell shows, the analogy reaches further still. An electric
current flowing along a wire had been recognised as having many
properties similar to those of a current of liquid in a tube. When a
steady current is passing through any solid conductor, there are formed
in the conductor tubes of electrical flow and surfaces of constant
pressure. These tubes and surfaces are the same as those formed by the
flow of liquid through a solid whose boundary surface is the same
as that of the conductor, provided the flow of liquid is properly
proportioned to the flow of electricity.
These analogies refer to steady currents in which, therefore, the flow
at any point of the conductor does not depend on the time. In Part
II. of his paper Maxwell deals with Faraday’s electro-tonic state.
Faraday had found that when _changes_ are produced in the magnetic
phenomena surrounding a conductor, an electric current is set up in
the conductor, which continues so long as the magnetic changes are in
progress, but which ceases when the magnetic state becomes steady.
“Considerations of this kind led Professor Faraday to connect
with his discovery of the induction of electric currents the
conception of a state into which all bodies are thrown by the
presence of magnets and currents. This state does not manifest
itself by any known phenomena as long as it is undisturbed,
but any change in this state is indicated by a current or
tendency towards a current. To this state he gave the name of
the ‘Electro-tonic State,’ and although he afterwards succeeded
in explaining the phenomena which suggested it by means of less
hypothetical conceptions, he has on several occasions hinted at
the probability that some phenomena might be discovered which
would render the electro-tonic state an object of legitimate
induction. These speculations, into which Faraday had been
led by the study of laws which he has well established, and
which he abandoned only for want of experimental data for the
direct proof of the unknown state, have not, I think, been
made the subject of mathematical investigation. Perhaps it
may be thought that the quantitative determinations of the
various phenomena are not sufficiently rigorous to be made
the basis of a mathematical theory. Faraday, however, has not
contented himself with simply stating the numerical results
of his experiments and leaving the law to be discovered by
calculation. Where he has perceived a law he has at once stated
it, in terms as unambiguous as those of pure mathematics,
and if the mathematician, receiving this as a physical
truth, deduces from it other laws capable of being tested by
experiment, he has merely assisted the physicist in arranging
his own ideas, which is confessedly a necessary step in
scientific induction.
“In the following investigation, therefore, the laws
established by Faraday will be assumed as true, and it will
be shown that by following out his speculations other and
more general laws can be deduced from them. If it should,
then, appear that these laws, originally devised to include
one set of phenomena, may be generalised so as to extend to
phenomena of a different class, these mathematical connections
may suggest to physicists the means of establishing physical
connections, and thus mere speculation may be turned to account
in experimental science.”
Maxwell shows how to obtain a mathematical expression for Faraday’s
electro-tonic state. In his “Electricity and Magnetism,” this
electro-tonic state receives a new name. It is known as the Vector
Potential,[61] and the paper under consideration contains, though
in an incomplete form, his first statement of those equations of the
electric field which are so indissolubly bound up with Maxwell’s name.
The great advance in theory made in the paper is the distinct
recognition of certain mathematical functions as representing Faraday’s
electrotonic-state, and their use in solving electro-magnetic problems.
The paper contains no new physical theory of electricity, but in a
few years one appeared. In his later writings Maxwell adopted a more
general view of the electro-magnetic field than that contained in his
early papers on “Physical Lines of Force.” It must, therefore, not be
supposed that the somewhat gross conception of cog-wheels and pulleys,
which we are about to describe, were anything more to their author than
a model, which enabled him to realise how the changes, which occur when
a current of electricity passes through a wire, might be represented by
the motion of actual material particles.
The problem before him was to devise a physical theory of electricity,
which would explain the forces exerted on electrified bodies by means
of action between the contiguous parts of the medium in the space
surrounding these bodies, rather than by direct action across the
distance which separates them. A similar question, still unanswered,
had arisen in the case of gravitation. Astronomers have determined the
forces between attracting bodies; they do not know how those forces
arise.
Maxwell’s fondness for models has already been alluded to; it had led
him to construct his top to illustrate the dynamics of a rigid body
rotating about a fixed point, and his model of Saturn’s rings (now in
the Cavendish Laboratory) to illustrate the motion of the satellites
in the rings. He had explained many of the gaseous laws by means of
the impact of molecules, and now his fertile ingenuity was to imagine
a mechanical model of the state of the electro-magnetic field near a
system of conductors carrying currents.
Faraday, as we have seen, looked upon electrostatic and magnetic
induction as taking place along curved lines of force. He pictures
these lines as ropes of molecules starting from a charged conductor, or
a magnet, as the case may be, and acting on other bodies near. These
ropes of molecules tend to shorten, and at the same time to swell
outwards laterally. Thus the charged conductor tends to draw other
bodies to itself, there is a tension along the lines of force, while
at the same time each tube of molecules pushes its neighbours aside; a
pressure at right angles to the lines of force is combined with this
tension. Assuming for a moment this pressure and tension to exist, can
we devise a mechanism to account for it? Maxwell himself has likened
the lines of force to the fibres of a muscle. As the fibres contract,
causing the limb to which they are attached to move, they swell
outwards, and the muscle thickens.
Again, from another point of view, we might consider a line of force
as consisting of a string of small cells of some flexible material
each filled with fluid. If we then suppose this series of cells caused
to rotate rapidly about the direction of the line of force, the cells
will expand laterally and contract longitudinally; there will again be
tension along the lines of force and pressure at right angles to them.
It was this last idea, as we shall see shortly, of which Maxwell made
use--
“I propose now” [he writes (“On Physical Lines of Force,”
_Phil. Mag._, vol. xxi.)] “to examine magnetic phenomena from
a mechanical point of view, and to determine what tensions in,
or motions of, a medium are capable of producing the mechanical
phenomena observed. If by the same hypothesis we can connect
the phenomena of magnetic attraction with electro-magnetic
phenomena, and with those of induced currents, we shall have
found a theory which, if not true, can only be proved to be
erroneous by experiments, which will greatly enlarge our
knowledge of this part of physics.”
Lord Kelvin had in 1847 given a mechanical representation of electric,
magnetic and galvanic forces by means of the displacements of an
elastic solid in a state of strain. The angular displacement at each
point of the solid was taken as proportional to the magnetic force, and
from this the relation between the various other electric quantities
and the motion of the solid was developed. But Lord Kelvin did not
attempt to explain the origin of the observed forces by the effects due
to these strains, but merely made use of the mathematical analogy to
assist the imagination in the study of both.
Maxwell considered magnetic action as existing in the form of pressure
or tension, or more generally, of some stress in some medium. The
existence of a medium capable of exerting force on material bodies and
of withstanding considerable stress, both pressure and tension, is
thus a fundamental hypothesis with him; this medium is to be capable
of motion, and electro-magnetic forces arise from its motion and its
stresses.
Now, Maxwell’s fundamental supposition is that, in a magnetic field,
there is a rotation of the molecules continually in progress about the
lines of magnetic force. Consider now the case of a uniform magnetic
field, whose direction is perpendicular to the paper; we are to look
upon the lines of force as parallel strings of molecules, the axes of
these strings being perpendicular to the paper. Each string is supposed
to be rotating in the same direction about its axis, and the angular
velocity of rotation is a measure of the magnetic force. In consequence
of this rotation there will be differences of pressure in different
directions in the medium; the pressure along the axes of the strings
will be less than it would be if the medium were at rest, that in the
directions at right angles to the axes will be greater, the medium will
behave as though it were under tension along the axes of the molecules
under pressure at right angles to them. Moreover, it can be shown that
the pressure and the tension are both proportional to the square of the
angular velocity--the square, that is, of the magnetic force--and this
result is in accordance with the consequences of experiment.
More elaborate calculation shows that this statement is true generally.
If we draw the lines of force in any magnetic field, and then suppose
the molecules of the medium set in rotation about these lines of force
as axes, with velocities which at each point are proportional to the
magnetic force, the distribution of pressure throughout is that which
we know actually to exist in the magnetic field.
According to this hypothesis, then, a permanent bar magnet has the
power of setting the medium round it into continuous molecular rotation
about the lines of force as axes. The molecules which are set in
rotation we may consider as spherical, or nearly spherical, cells
filled with a fluid, or an elastic solid substance, and surrounded by a
kind of membrane, or sack, holding the contents together.
So far the model does not give any account of electrical actions which
go on in the magnetic field.
The energy is wholly rotational, and the forces wholly magnetic.
Consider, however, any two contiguous strings of molecules. Let them
cut the paper as shown in the two circles in Fig. 1:--
[Illustration: Fig. 1.
Fig. 2.]
Then these cells are both rotating in the same direction, hence at C,
where they touch, their points of contact will be moving in opposite
directions, as shown by the arrow heads, and it is difficult to imagine
how such motion can continue; it would require the surfaces of the
cells to be perfectly smooth, and if this were so they would lose the
power of transmitting action from one cell to the next.
The cells A and B may be compared to two cog-wheels placed close
together, which we wish to turn in the same direction. If the cogs can
interlock, as in Fig. 2, this is impossible: consecutive wheels in the
train must move in opposite directions.
[Illustration: Fig. 3.]
But in many machines the desired end is attained by inserting between
the two wheels A and B a third idle wheel C, as shewn in Fig. 3. This
may be very small, its only function is to transmit the motion of A to
B in such a way that A and B may both turn in the same direction. It is
not necessary that there should be cogs on the wheels; if the surfaces
be perfectly rough, so that no slipping can take place, the same result
follows without the cogs.
Guided by this analogy Maxwell extended his model by supposing each
cell coated with a number of small particles which roll on its surface.
These particles play the part of the idle wheels in the machine, and by
their rolling merely enable the adjacent parts of two cells to move in
opposite directions.
Consider now a number of such cells and their idle wheels lying in a
plane, that of the paper, and suppose each cell is rotating with the
same uniform angular velocity about an axis at right angles to that
plane, each idle wheel will be acted on by two equal and opposite
forces at the ends of the diameter in which it is touched by the
adjacent cells; it will therefore be set in rotation, but there will be
no force tending to drive it onwards; it does not matter whether the
axis on which it rotates is free to move or fixed, in either case the
idle wheel simply rotates. But suppose now the adjacent cells are not
rotating at the same rate. In addition to its rotation the idle wheel
will be urged onward with a velocity which depends on the difference
between the rotations, and, if it can move freely, it will move on from
between the two cells. Imagine now that the interstices between the
cells are fitted with a string of idle wheels. So long as the adjacent
cells move with different velocity there will be a continual stream of
rolling particles or idle wheels between them. Maxwell in the paper
considered these rolling particles to be particles of electricity.
Their motion constitutes an electric current. In a uniform magnetic
field there is no electric current; if the strength of the field
varies, the idle wheels are set in motion and there may be a current.
These particles are very small compared with the magnetic vortices.
The mass of all the particles is inappreciable compared with the mass
of the vortices, and a great many vortices with their surrounding
particles are contained in a molecule of the medium; the particles
roll on the vortices without touching each other, so that so long as
they remain within the same molecule there is no loss of energy by
resistance. When, however, there is a current or general transference
of particles in one direction they must pass from one molecule to
another, and in doing so may experience resistance and generate heat.
Maxwell states that the conception of a particle, having its motion
connected with that of a vortex by perfect rolling contact, may appear
somewhat awkward. “I do not bring it forward,” he writes, “as a mode of
connection existing in Nature, or even as that which I would willingly
assent to as an electrical hypothesis. It is, however, a mode of
connection which is mechanically conceivable and easily investigated,
and it serves to bring out the actual mechanical connections between
the known electro-magnetic phenomena, so that I venture to say that
anyone who understands the provisional and temporary character of this
hypothesis will find himself rather helped than hindered by it in his
search after the true interpretation of the phenomena.”
The first part of the paper deals with the theory of magnetism; in the
second part the hypothesis is applied to the phenomena of electric
currents, and it is shown how the known laws of steady currents and
of electro-magnetic induction can be deduced from it. In Part III.,
published January and February, 1862, the theory of molecular vortices
is applied to statical electricity.
The distinction between a conductor and an insulator or dielectric
is supposed to be that in the former the particles of electricity
can pass with more or less freedom from molecule to molecule. In the
latter such transference is impossible, the particles can only be
displaced within the molecule with which they are connected; the cells
or vortices of the medium are supposed to be elastic, and to resist by
their elasticity the displacement of the particles within them. When
electrical force acts on the medium this displacement of the particles
within each molecule takes place until the stresses due to the elastic
reaction of the vortices balance the electrical force; the medium
behaves like an elastic body yielding to pressure until the pressure is
balanced by the elastic stress. When the electric force is removed the
cells or vortices recover their form, the electricity returns to its
former position.
In a medium such as this waves of periodic displacement could be
set up, and would travel with a velocity depending on its electric
properties. The value for this velocity can be obtained from electrical
observations, and Maxwell showed that this velocity, so found, was,
within the limits of experimental error, the same as that of light.
Moreover, the electrical oscillations take place, like those of light,
in the front of the wave. Hence, he concludes, “the elasticity of the
magnetic medium in air is the same as that of the luminiferous medium,
if these two coexistent, coextensive, and equally elastic media are not
rather one medium.”
The paper thus contains the first germs of the electro-magnetic theory
of light. Moreover, it is shown that the attraction between two small
bodies charged with given quantities of electricity depends on the
medium in which they are placed, while the specific inductive capacity
is found to be proportional to the square of the refractive index.
The fourth and final part of the paper investigates the propagation of
light in a magnetic field.
Faraday had shown that the direction of vibration in a wave of
polarised light travelling parallel to the lines of force in a magnetic
field is rotated by its passage through the field. The numerical laws
of this relation had been investigated by Verdet, and Maxwell showed
how his hypothesis of molecular vortices led to laws which agree in the
main with those found by Verdet.
He points out that the connection between magnetism and electricity
has the same mathematical form as that between certain other pairs
of phenomena, one of which has a _linear_ and the other a _rotatory_
character; and, further, that an analogy may be worked out assuming
either the linear character for magnetism and the rotatory character
for electricity, or the reverse. He alludes to Prof. Challis’ theory,
according to which magnetism is to consist in currents in a fluid
whose directions correspond with the lines of magnetic force, while
electric currents are supposed to be accompanied by, if not dependent
upon, a rotatory motion of the fluid about the axis of the current;
and to Von Helmholtz’s theory of a somewhat similar character. He then
gives his own reasons--agreeing with those of Sir W. Thomson (Lord
Kelvin)--for supposing that there must be a real rotation going on in
a magnetic field in order to account for the rotation of the plane of
polarisation, and, accepting these reasons as valid, he develops the
consequences of his theory with the results stated above.
His own verdict on the theory is given in the “Electricity and
Magnetism” (vol. ii., § 831, first edition, p. 416):--
“A theory of molecular vortices, which I worked out at
considerable length, was published in the _Phil. Mag._ for
March, April, and May, 1861; Jan. and Feb., 1862.
“I think we have good evidence for the opinion that some
phenomenon of rotation is going on in the magnetic field, that
this rotation is performed by a great number of very small
portions of matter, each rotating on its own axis, this axis
being parallel to the direction of the magnetic force, and that
the rotations of these different vortices are made to depend on
one another by means of some kind of mechanism connecting them.
“The attempt which I then made to imagine a working model of
this mechanism must be taken for no more than it really is,
a demonstration that mechanism may be imagined capable of
producing a connection mechanically equivalent to the actual
connection of the parts of the electro-magnetic field. The
problem of determining the mechanism required to establish a
given species of connection between the motions of the parts of
a system always admits of an infinite number of solutions. Of
these, some may be more clumsy or more complex than others, but
all must satisfy the conditions of mechanism in general.
“The following results of the theory, however, are of higher
value:--
“(1) Magnetic force is the effect of the centrifugal force of
the vortices.
“(2) Electro-magnetic induction of currents is the effect of
the forces called into play when the velocity of the vortices
is changing.
“(3) Electromotive force arises from the stress on the
connecting mechanism.
“(4) Electric displacement arises from the elastic yielding of
the connecting mechanism.”
In studying this part of Maxwell’s work, it must clearly be remembered
that he did not look upon the ether as a series of cog-wheels with
idle wheels between, or anything of the kind. He devised a mechanical
model of such cogs and idle wheels, the properties of which would in
some respects closely resemble those of the ether; from this model he
deduced, among other things, the important fact that electric waves
would travel outwards with the velocity of light. Other such models
have been devised since his time to illustrate the same laws. Prof.
Fitzgerald has actually constructed one of wheels connected together by
elastic bands, which shows clearly the kind of processes which Maxwell
supposed to go on in a dielectric when under electric force. Professor
Lodge, in his book, “Modern Views of Electricity,” has very fully
developed a somewhat different arrangement of cog-wheels to attain the
same result.
Maxwell’s predictions as to the propagation of electric waves have
in recent days received their full verification in the brilliant
experiments of Hertz and his followers; it remains for us, before
dealing with these, to trace their final development in his hands.
The papers we have been discussing were perhaps too material to receive
the full attention they deserved; the ether is not a series of cogs,
and electricity is something different from material idle wheels. In
his paper on “The Dynamical Theory of the Electro-magnetic Field,”
_Phil. Trans._, 1864, Maxwell treats the same questions in a more
general manner. On a former occasion he says, “I have attempted to
describe a particular kind of motion and a particular kind of strain
so arranged as to account for the phenomena. In the present paper I
avoid any hypothesis of this kind; and in using such words as electric
momentum and electric elasticity in reference to the known phenomena of
the induction of currents and the polarisation of dielectrics, I wish
merely to direct the mind of the reader to mechanical phenomena, which
will assist him in understanding the electrical ones. All such phrases
in the present paper are to be considered as illustrative and not as
explanatory.” He then continues:--
“In speaking of the energy of the field, however, I wish to
be understood literally. All energy is the same as mechanical
energy, whether it exists in the form of motion or in that of
elasticity, or in any other form.
“The energy in electro-magnetic phenomena is mechanical energy.
The only question is, Where does it reside?
“On the old theories it resides in the electrified bodies,
conducting circuits, and magnets, in the form of an unknown
quality called potential energy, or the power of producing
certain effects at a distance. On our theory it resides in
the electro-magnetic field, in the space surrounding the
electrified and magnetic bodies, as well as in those bodies
themselves, and is in two different forms, which may be
described without hypothesis as magnetic polarisation and
electric polarisation, or, according to a very probable
hypothesis, as the motion and the strain of one and the same
medium.
“The conclusions arrived at in the present paper are
independent of this hypothesis, being deduced from experimental
facts of three kinds:--
“(1) The induction of electric currents by the increase or
diminution of neighbouring currents according to the changes in
the lines of force passing through the circuit.
“(2) The distribution of magnetic intensity according to the
variations of a magnetic potential.
“(3) The induction (or influence) of statical electricity
through dielectrics.
“We may now proceed to demonstrate from these principles the
existence and laws of the mechanical forces, which act upon
electric currents, magnets, and electrified bodies placed in
the electro-magnetic field.”
In his introduction to the paper, he discusses in a general way the
various explanations of electric phenomena which had been given, and
points out that--
“It appears, therefore, that certain phenomena in electricity
and magnetism lead to the same conclusion as those of optics,
namely, that there is an ætherial medium pervading all bodies,
and modified only in degree by their presence; that the parts
of this medium are capable of being set in motion by electric
currents and magnets; that this motion is communicated from
one part of the medium to another by forces arising from the
connection of those parts; that under the action of these
forces there is a certain yielding depending on the elasticity
of these connections; and that, therefore, energy in two
different forms may exist in the medium, the one form being
the actual energy of motion of its parts, and the other being
the potential energy stored up in the connections in virtue of
their elasticity.
“Thus, then, we are led to the conception of a complicated
mechanism capable of a vast variety of motion, but at the
same time so connected that the motion of one part depends,
according to definite relations, on the motion of other parts,
these motions being communicated by forces arising from the
relative displacement of the connected parts, in virtue of
their elasticity. Such a mechanism must be subject to the
general laws of dynamics, and we ought to be able to work out
all the consequences of its motion, provided we know the form
of the relation between the motions of the parts.”
These general laws of dynamics, applicable to the motion of any
connected system, had been developed by Lagrange, and are expressed
in his generalised equations of motion. It is one of Maxwell’s chief
claims to fame that he saw in the electric field a connected system to
which Lagrange’s equations could be applied, and that he was able to
deduce the mechanical and electrical actions which take place by means
of fundamental propositions of dynamics.
The methods of the paper now under discussion were developed further
in the “Treatise on Electricity and Magnetism,” published in 1873; in
endeavouring to give some slight account of Maxwell’s work, we shall
describe it in the form it ultimately took.
The task which Maxwell set himself was a double one; he had first to
express in symbols, in as general a form as possible, the fundamental
laws of electro-magnetism as deduced from experiments, chiefly
the experiments of Faraday, and the relations between the various
quantities involved; when this was done he had to show how these laws
could be deduced from the general dynamical laws applicable to any
system of moving bodies.
There are two classes of phenomena, electric and magnetic, which have
been known from very early times, and which are connected together.
When a piece of sealing-wax is rubbed it is found to attract other
bodies, it is said to exert electric force throughout the space
surrounding it; when two different metals are dipped in slightly
acidulated water and connected by a wire, certain changes take place
in the plates, the water, the wire, and the space round the wire,
electric force is again exerted and a current of electricity is said
to flow in the wire. Again, certain bodies, such as the lodestone, or
pieces of iron and steel which have been treated in a certain manner,
exhibit phenomena of action at a distance: they are said to exert
magnetic force, and it is found that this magnetic force exists in the
neighbourhood of an electric current and is connected with the current.
Again, when electric force is applied to a body, the effects may be in
part electrical, in part mechanical; the electrical state of the body
is in general changed, while in addition, mechanical forces tending to
move the body are set up. Experiment must teach us how the electrical
state depends on the electric force, and what is the connection
between this electric force and the magnetic forces which may, under
certain circumstances, be observed. Now, in specifying the electric
and magnetic conditions of the system, various other quantities, in
addition to the electric force, will have to be introduced; the first
step is to formulate the necessary quantities, and to determine the
relations between them and the electric force.
Consider now a wire connecting the two poles of an electric battery--in
its simplest form, a piece of zinc and a piece of copper in a vessel
of dilute acid--electric force is produced at each point of the wire.
Let us suppose this force known; an electric current depending on the
material and the size of the wire flows along it, its value can be
determined at each point of the wire in terms of the electric force
by Ohm’s law. If we take either this current or the electric force
as known, we can determine by known laws the electric and magnetic
conditions elsewhere. If we suppose the wire to be straight and very
long, then, so long as the current is steady and we neglect the small
effect due to the electrostatic charge on the wire, there is no
electric force outside the wire. There is, however, magnetic force,
and it is found that the lines of magnetic force are circles round the
wire. It is found also that the work done in travelling once completely
round the wire against the magnetic force is measured by the current
flowing through the wire, and is obtained in the system of units
usually adopted by multiplying the current by 4π. This last result then
gives us one of the necessary relations, that between the magnetic
force due to a current and the strength of the current.
Again, consider a steady current flowing in a conductor of any form or
shape, the total flow of current across any section of the conductor
can be measured in various ways, and it is found that at any time this
total flow is the same for each section of the conductor. In this
respect the flow of a current resembles that of an incompressible
fluid through a pipe; where the pipe is narrow the velocity of flow
is greater than it is where the pipe is broad, but the total quantity
crossing each section at any given instant is the same.
Consider now two conducting bodies, two spheres, or two flat plates
placed near together but insulated. Let each conductor be connected
to one of the poles of the battery by a conducting wire. Then, for a
very short interval after the contact is made, it is found that there
is a current in each wire which rapidly dies away to zero. In the
neighbourhood of the balls there is electric force; the balls are said
to be charged with electricity, and the lines of force are curved lines
running from one ball to the other. It is found that the balls slightly
attract each other, and the space between them is now in a different
condition from what it was before the balls were charged. According
to Maxwell, _Electric Displacement_ has been produced in this space,
and the electric displacement at each point is proportional to the
electric force at that point.
Thus, (i) when electric force acts on a conductor, it produces a
current, the current being by Ohm’s law proportional to the force:
(ii) when it acts on an insulator it produces electric displacement,
and the displacement is proportional to the force; while (iii) there
is magnetic force in the neighbourhood of the current, and the work
done in carrying a magnetic pole round any complete circuit linked
with the current is proportional to the current. The first two of
these principles give us two sets of equations connecting together the
electric force and the current in a conductor or the displacement in a
dielectric respectively; the third connects the magnetic force and the
current.
Now let us go back to the variable period when the current is flowing
in the wires; and to make ideas precise, let the two conductors be two
equal large flat plates placed with their faces parallel, and at some
small distance apart. In this case, when the plates are charged, and
the current has ceased, the electric displacement and the force are
confined almost entirely to the space between the plates. During the
variable period the total flow at any instant across each section of
the wire is the same, but in the ordinary sense of the word there is no
flow of electricity across the insulating medium between the plates.
In this space, however, the electric displacement is continuously
changing, rising from zero initially to its final steady value when the
current ceases. It is a fundamental part of Maxwell’s theory that this
variation of electric displacement is equivalent in all respects to a
current. The current at any point in a dielectric is measured by the
rate of change of displacement at that point.
Moreover, it is also an essential point that if we consider any section
of the dielectric between the two plates, the rate of change of the
total displacement across this section is at each moment equal to the
total flow of current across each section of the conducting wire.
Currents of electricity, therefore, including displacement
currents, always flow in closed circuits, and obey the laws of an
incompressible fluid in that the total flow across each section of the
circuit--conducting or dielectric--is at any moment the same.
It should be clearly remembered that this fundamental hypothesis of
Maxwell’s theory is an assumption only to be justified by experiment.
Von Helmholtz, in his paper on “The Equations of Motion of Electricity
for Bodies at Rest,” formed his equations in an entirely different
manner from Maxwell, and arrived at results of a more general
character, which do not require us to suppose that currents flow always
in closed circuits, but permit of the condensation of electricity at
points in the circuit where the conductors end and the non-conducting
part of the circuit begins. We leave for the present the question which
of the two theories, if either, represents the facts.
We have obtained above three fundamental relations--(i) that between
electric force and electric current in a conductor; (ii) that between
electric force and electric displacement in a dielectric; (iii) that
between magnetic force and the current which gives rise to it. And
we have seen that an electric current--_i.e._ in a dielectric the
variation of the strength of an electric field of force--gives rise
to magnetic force. Now, magnetic force acting on a medium produces
“magnetic displacement,” or magnetic induction, as it is called. In
all media except iron, nickel, cobalt, and a few other substances, the
magnetic induction is proportional to the magnetic force, and the ratio
between the magnetic induction produced by a given force and the force
is found to be very nearly the same for all such media. This ratio is
known as the permeability, and is generally denoted by the symbol μ.
A relation reciprocal to that given in (iii) above might be
anticipated, and was, in fact, discovered by Faraday. Changes in a
field of magnetic induction give rise to electric force, and hence to
displacement currents in a dielectric or to conduction currents in a
conductor. In considering the relation between these changes and the
electric force, it is simplest at first not to deal with magnetic
matter such as iron, nickel, or cobalt; and then we may say that (iv)
the work which at any instant would be done in carrying a unit quantity
of electricity round a closed circuit in a magnetic field against the
electric forces due to the field is equal to the rate at which the
total magnetic induction which threads the circuit is being decreased.
This law, summing up Faraday’s experiments on electro-magnetic
induction, gives a fourth principle, leading to a fourth series of
equations connecting together the electric and magnetic quantities
involved.
The equations deduced from the above four principles, together with the
condition implied in the continuity of an electric current, constitute
Maxwell’s equations of the electro-magnetic field.
If we are dealing only with a dielectric medium, the reciprocal
relation between the third and fourth principle may be made more clear
by the following statement:--
(A) The work done at any moment in carrying a unit quantity of
magnetism round a closed circuit in a field in which electric
displacement is varying, is equal to the rate of change of the total
electric displacement through the circuit multiplied by 4 π.[62]
(B) The work done at any moment in carrying a unit quantity of
electricity round a circuit in a field in which the magnetic induction
is varying, is equal to the rate of change of the total magnetic
induction through the circuit.
From these two principles, combined with the laws connecting electric
force and displacement, magnetic force and induction, and with the
condition of continuity, Maxwell obtained his equations of the field.
Faraday’s experiments on electro-magnetic induction afford the proof of
the truth of the fourth principle. It follows from those experiments
that when the number of lines of magnetic induction which are linked
with any closed circuit are made to vary, an induced electromotive
force is brought into play round that circuit. This electromotive force
is, according to Faraday’s results, measured by the rate of decrease
in the number of lines of magnetic induction which thread the circuit.
Maxwell applies this principle to all circuits, whether conducting or
not.
In obtaining equations to express in symbols the results of the fourth
principle just enunciated, Maxwell introduces a new quantity, to which
he gives the name of the “vector potential.” This quantity appears in
his analysis, and its physical meaning is not at first quite clear.
Professor Poynting has, however, put Maxwell’s principles in a slightly
different form, which enables us to see definitely the meaning of the
vector potential, and to deduce Maxwell’s equations more readily from
the fundamental statements.
We are dealing with a circuit with which lines of magnetic induction
are linked, while the number of such lines linked with the circuit is
varying. Now, let us suppose the variation to take place in consequence
of the lines of induction moving outwards or inwards, as the case may
be, so as to cut the circuit. Originally there are none linked with
the circuit. As the magnetic field has grown to its present strength
lines of magnetic induction have moved inwards. Each little element of
the circuit has been cut by some, and the total number linked with the
circuit can be found by adding together those cut by each element. Now,
Professor Poynting’s statement of Maxwell’s fourth principle is that
the electrical force in the direction of any element of the circuit is
found by dividing by the length of the element the number of lines of
magnetic induction which are cut in one second by it.
Moreover, the total number of lines of magnetic induction which have
been cut by an element of unit length is defined as the component
of the vector potential in the direction of the element; hence the
electrical force in any direction is the rate of decrease of the
component of the vector potential in that direction. We have thus a
physical meaning for the vector potential, and shall find that in the
dynamical theory this quantity is of great importance.
Professor Poynting has modified Maxwell’s third principle in a similar
manner; he looks upon the variation in the electric displacement as
due to the motion of tubes of electric induction,[63] and the magnetic
force along any circuit is equal to the number of tubes of electric
induction cutting or cut by unit length of the circuit per second,
multiplied by 4π.
From the equations of the field, as found by Maxwell, it is possible
to derive two sets of symmetrical equations. The one set connects the
rate of change of the electric force with quantities depending on the
magnetic force; the other set connects in a similar manner the rate of
change of the magnetic force with quantities depending on the electric
force. Several writers in recent years adopt these equations as the
fundamental relations of the field, establishing them by the argument
that they lead to consequences which are found to be in accordance with
experiment.
We have endeavoured to give some account of Maxwell’s historical
method, according to which the equations are deduced from the laws of
electric currents and of electro-magnetic induction derived directly
from experiment.
While the manner in which Maxwell obtained his equations is all his
own, he was not alone in stating and discussing general equations
of the electro-magnetic field. The next steps which we are about to
consider are, however, in a special manner due to him. An electrical
or magnetic system is the seat of energy; this energy is partly
electrical, partly magnetic, and various expressions can be found for
it. In Maxwell’s theory it is a fundamental assumption that energy has
position. “The electric and magnetic energies of any electro-magnetic
system,” says Professor Poynting, “reside, therefore, somewhere in the
field.” It follows from this that they are present wherever electric
and magnetic force can be shown to exist. Maxwell showed that all the
electric energy is accounted for by supposing that in the neighbourhood
of a point at which the electric force is R there is an amount of
energy per unit of volume equal to KR²/8π, K being the inductive
capacity of the medium, while in the neighbourhood of a point at which
the magnetic force is H, the magnetic energy per unit of volume is
μH²/8π, μ being the permeability. He supposes, then, that at each point
of an electro-magnetic system energy is stored according to these
laws. It follows, then, that the electro-magnetic field resembles a
dynamical system in which energy is stored. Can we discover more of
the mechanism by which the actions in the field are maintained? Now
the motion of any point of a connected system depends on that of other
points of the system; there are generally, in any machine, a certain
number of points called driving-points, the motion of which controls
the motion of all other parts of the machine; if the motion of the
driving-points be known, that of any other point can be determined.
Thus in a steam engine the motion of a point on the fly-wheel can be
found if the motion of the piston and the connections between the
piston and the wheel be known.
In order to determine the force which is acting on any part of the
machine we must find its momentum, and then calculate the rate at
which this momentum is being changed. This rate of change will give us
the force. The method of calculation which it is necessary to employ
was first given by Lagrange, and afterwards developed, with some
modifications, by Hamilton. It is usually referred to as Hamilton’s
principle; when the equations in the original form are used they are
known as Lagrange’s equations.
Now Maxwell showed how these methods of calculation could be applied
to the electro-magnetic field. The energy of a dynamical system is
partly kinetic, partly potential. Maxwell supposes that the magnetic
energy of the field is kinetic energy, the electric energy potential.
When the kinetic energy of a system is known, the momentum of any
part of the system can be calculated by recognised processes. Thus if
we consider a circuit in an electro-magnetic field we can calculate
the energy of the field, and hence obtain the momentum corresponding
to this circuit. If we deal with a simple case in which the conducting
circuits are fixed in position, and only the current in each circuit is
allowed to vary, the rate of change of momentum corresponding to any
circuit will give the force in that circuit. The momentum in question
is electric momentum, and the force is electric force. Now we have
already seen that the electric force at any point of a conducting
circuit is given by the rate of change of the vector potential in the
direction considered. Hence we are led to identify the vector potential
with the electric momentum of our dynamical system; and, referring to
the original definition of vector potential, we see that the electric
momentum of a circuit is measured by the number of lines of magnetic
induction which are interlinked with it.
Again, the kinetic energy of a dynamical system can be expressed in
terms of the squares and products of the velocities of its several
parts. It can also be expressed by multiplying the velocity of each
driving-point by the momentum corresponding to that driving-point, and
taking half the sum of the products. Suppose, now, we are dealing with
a system consisting of a number of wire circuits in which currents are
running, and let us suppose that we may represent the current in each
wire as the velocity of a driving-point in our dynamical system. We can
also express in terms of these currents the electric momentum of each
wire circuit; let this be done, and let half the sum of the products of
the corresponding velocities and momenta be formed.
In maintaining the currents in the wires energy is needed to supply
the heat which is produced in each wire; but in starting the currents
it is found that more energy is needed than is requisite for the
supply of this heat. This excess of energy can be calculated, and when
the calculation is made it is found that the excess is equal to half
the sum of the products of the currents and corresponding momenta.
Moreover, if this sum be expressed in terms of the magnetic force, it
is found to be equal to μ H²/8 π, which is the magnetic energy of the
field. Now, when a dynamical system is set in motion against known
forces, more energy is supplied than is needed to do the work against
the forces; this excess of energy measures the kinetic energy acquired
by the system.
Hence, Maxwell was justified in taking the magnetic energy of the field
as the kinetic energy of the mechanical system, and if the strengths
of the currents in the wires be taken to represent the velocities of
the driving-points, this energy is measured in terms of the electrical
velocities and momenta in exactly the same way as the energy of a
mechanical system is measured in terms of the velocities and momenta of
its driving-points.
The mechanical system in which, according to Maxwell, the energy is
stored is the ether. A state of motion or of strain is set up in the
ether of the field. The electric forces which drive the currents, and
also the mechanical forces acting on the conductors carrying the
currents, are due to this state of motion, or it may be of strain, in
the ether. It must not be supposed that the term electric displacement
in Maxwell’s mind meant an actual bodily displacement of the particles
of the ether; it is in some way connected with such a material
displacement. In his view, without motion of the ether particles
there would be no electric action, but he does not identify electric
displacement and the displacement of an ether particle.
His mechanical theory, however, does account for the electro-magnetic
forces between conductors carrying currents. The energy of the system
depends on the relative positions of the currents which form part of
it. Now, any conservative mechanical system tends to set itself in
such a position that its potential energy is least, its kinetic energy
greatest. The circuits of the system, then, will tend to set themselves
so that the electro-kinetic energy of the system may be as large as
possible; forces will be needed to hold them in any position in which
this condition is not satisfied.
We have another proof of the correctness of the value found for the
energy of the field in that the forces calculated from this value agree
with those which are determined by direct experiment.
Again, the forces applied at the various driving-points are transmitted
to other points by the connections of the machine; the connections
are thrown into a state of strain; stress exists throughout their
substance. When we see the piston-rod and the shaft of an engine
connected by the crank and the connecting-rod, we recognise that the
work done on the piston is transmitted thus to the shaft. So, too, in
the electro-magnetic field, the ether forms the connection between the
various circuits in the field; the forces with which those circuits
act on each other are transmitted from one circuit to another by the
stresses set up in the ether.
To take another instance, consider the electrostatic attraction between
two charged bodies. Let us suppose the bodies charged by connecting
each to the opposite pole of a battery; a current flows from the
battery setting up electric displacement in the space between the
bodies, and throwing the ether into a state of strain. As the strain
increases the current gets less; the reaction resulting from the strain
tends to stop it, until at last this reaction is so great that the
current is stopped. When this is the case the wires to the battery may
be removed, provided this is done without destroying the insulation of
the bodies; the state of strain will remain and shows itself in the
attraction between the balls.
Looking at the problem in this manner, we are face to face with two
great questions--the one, What is the state of strain in the ether
which will enable it to produce the observed electrostatic attractions
and repulsions between charged bodies? and the other, What is the
mechanical structure of the ether which would give rise to such a state
of strain as will account for the observed forces? Maxwell gives one
answer to the first question; it is not the only answer which could
be given, but it does account for the facts. He failed to answer the
second. He says (“Electricity and Magnetism,” vol. i. p. 132):--
“It must be carefully borne in mind that we have made only
one step in the theory of the action of the medium. We have
supposed it to be in a state of stress, but have not in
any way accounted for this stress, or explained how it is
maintained.... I have not been able to make the next step,
namely, to account by mechanical considerations for these
stresses in the dielectric.”
Faraday had pointed out that the inductive action between two bodies
takes place along the lines of force, which tend to shorten along their
length and to spread outwards in other directions. Maxwell compares
them to the fibres of a muscle, which contracts and at the same time
thickens when exerting force. In the electric field there is, on
Maxwell’s theory, a tension along the lines of electric force and a
pressure at right angles to those lines. Maxwell proved that a tension
K R²/8 π along the lines of force, combined with an equal pressure
in perpendicular directions, would maintain the equilibrium of the
field, and would give rise to the observed attractions or repulsions
between electrified bodies. Other distributions of stress might be
found which would lead to the same result. The one just stated will
always be connected with Maxwell’s name. It will be noticed that the
tension along the lines of force and the pressure at right angles to
them are each numerically equal to the potential energy stored per unit
of volume in the field. The value of each of the three quantities is K
R²/8 π.
In the same way, in a magnetic field, there is a state of stress, and
on Maxwell’s theory this, too, consists of a tension along the lines
of force and an equal pressure at right angles to them, the values of
the tension and the pressure being each equal to that of the magnetic
energy per unit of volume, or μH²/8π.
In a case in which both electric and magnetic force exists, these two
states of stress are superposed. The total energy per unit of volume
is KR²/8π + μH²/8π; the total stress is made up of tensions KR²/8π and
μH²/8π along the lines of electric and magnetic force respectively, and
equal pressures at right angles to these lines.
We see, then, from Maxwell’s theory, that electric force produced at
any given point in space is transmitted from that point by the action
of the ether. The question suggests itself, Does the transmission take
time, and if so, does it proceed with a definite velocity depending on
the nature of the medium through which the change is proceeding?
According to the molecular-vortex theory, we have seen that waves of
electric force are transmitted with a definite velocity. The more
general theory developed in the “Electricity and Magnetism” leads to
the same result. Electric force produced at any point travels outwards
from that point with a velocity given by 1/√(Kμ). At a distant point
the force is zero, until the disturbance reaches it. If the disturbance
last only for a limited interval, its effects will at any future time
be confined to the space within a spherical shell of constant thickness
depending on the interval; the radii of this shell increase with
uniform speed 1/√(Kμ).
If the initial disturbance be periodic, periodic waves of electric
force will travel out from the centre, just as waves of sound travel
out from a bell, or waves of light from a candle flame. A wire carrying
an alternating current may be such a source of periodic disturbance,
and from the wire waves travel outwards into space.
Now, it is known that in a sound wave the displacements of the air
particles take place in the direction in which the wave is travelling;
they lie at right angles to the wave front, and are spoken of as
longitudinal. In light waves, on the other hand, the displacements are,
as Fresnel proved, in the wave front, at right angles, that is, to the
direction of propagation; they are transverse.
Theory shows that in general both these waves may exist in an elastic
solid body, and that they travel with different velocities. Of which
nature are the waves of electric displacement in a dielectric? It
can be shewn to follow as a necessary consequence of Maxwell’s views
as to the closed character of all electric currents, that waves of
electric displacement are transverse. Electric vibrations, like those
of light, are in the wave front and at right angles to the direction
of propagation; they depend on the rigidity or quasi-rigidity of the
medium through which they travel, not on its resistance to compression.
Again, an electric current, whether due to variation of displacement
in a dielectric or to conduction in a conductor, is accompanied by
magnetic force. A wave of periodic electric displacement, then, will be
also a wave of periodic magnetic force travelling at the same rate;
and Maxwell shewed that the direction of this magnetic force also
lies in the wave front, and is always at right angles to the electric
displacement. In the ordinary theory of light the wave of linear
displacement is accompanied by a wave of periodic angular twist about
a direction lying in the wave front and perpendicular to the linear
displacement.
In many respects, then, waves of electric displacement resemble waves
of light, and, indeed, as we proceed we shall find closer connections
still. Hence comes Maxwell’s electro-magnetic theory of light.
It is only in dielectric media that electric force is propagated by
wave motion. In conductors, although the third and fourth of Maxwell’s
principles given on page 185 still are true, the relation between
the electric force and the electric current differs from that which
holds in a dielectric. Hence the equations satisfied by the force are
different. The laws of its propagation resemble those of the conduction
of heat rather than those of the transmission of light.
Again, light travels with different velocities in different transparent
media. The velocity of electric waves, as has been stated, is equal to
1/√(μK); but in making this statement it is assumed that the simple
laws which hold where there is no gross matter--or, rather, where
air is the only dielectric with which we are concerned--hold also in
solid or liquid dielectrics. In a solid or a liquid, as in vacuo, the
waves are propagated by the ether. We assume, as a first step towards
a complete theory, that so far as the electric waves are concerned
the sole effect produced by the matter shews itself in a change of
inductive capacity or of permeability. It is not likely that such a
supposition should be the whole truth, and we may, therefore, expect
results deduced from it to be only approximation to the true result.
Now, electro-magnetic experiments show that, excluding magnetic
substances, the permeability of all bodies is very nearly the same,
and differs very slightly from that of air. The inductive capacity,
however, of different bodies is different, and hence the velocity with
which electro-magnetic waves travel differs in different bodies.
But the refraction of waves of light depends on the fact that light
travels with different velocities in different media; hence we should
expect to have waves of electric displacement reflected and refracted
when they pass from one dielectric, such as air, to another, such as
glass or gutta-percha; moreover, for light the refractive index of
a medium such as glass is the ratio of the velocity in air to the
velocity in the glass.
Thus the electrical refractive index of glass is the ratio of the
velocity of electric waves in air to their velocity in glass.
Now let K₀ be the inductive capacity of air, K₁ that of glass, taking
the permeability of air and glass to be the same, we have the result
that--
Electrical refractive index = √(K₁/K₀).
But the ratio of the inductive capacity of glass to that of air is
known as the specific inductive capacity of glass.
Hence, the specific inductive capacity of any medium is equal to the
square of the electrical refractive index of that medium.
Since Maxwell’s time the mathematical laws of the reflexion and
refraction of electric waves have been investigated by various writers,
and it has been shewn that they agree exactly with those enunciated by
Fresnel for light.
Hitherto we have been discussing the propagation of electric waves
in an isotropic medium, one which has identical properties in all
directions about a point. Let us now consider how these laws are
modified if the dielectric be crystalline in structure.
Maxwell assumes that the crystalline character of the dielectric can
be sufficiently represented by supposing the inductive capacity to
be different in different directions; experiments have since shewn
that this is true for crystals such as Iceland Spar and Aragonite;
he assumes also, and this, too, is justified by experiment, that the
magnetic permeability does not depend on the direction. It follows
from these assumptions that a crystal will produce double refraction
and polarisation of electric waves which fall upon it, and, further,
that the laws of double refraction will be those given by Fresnel for
light waves in a doubly refracting medium. There will be two waves in
the crystal. The disturbance in each of these will be plane polarised;
their velocity and the position of their plane of polarisation can be
found from the direction in which they are travelling by Fresnel’s
construction exactly.
Maxwell’s theory, then, would appear to indicate some close connection
between electric waves and those of light. Faraday’s experiments on
the rotation of the plane of polarisation by magnetic force shew one
phenomenon in which the two are connected, and Maxwell endeavoured to
apply his theory to explain this. Here, however, it became necessary
to introduce an additional hypothesis--there must be some connection
between the motion of the ether to which magnetic force is due and that
which constitutes light. It is impossible to give a mechanical account
of the rotation of the plane of polarisation without some assumption as
to the relation between these two kinds of motion. Maxwell, therefore,
supposes the linear displacements of a point in the ether to be those
which give rise to light, while the components of the magnetic force
are connected with these in the same way as the components of a vortex
in a liquid in vortex motion are connected with the displacements of
the liquid. He further assumes the existence of a term of special form
in the expression for the kinetic energy, and from these assumptions he
deduces the laws of the propagation of polarised light in a magnetic
field. These laws agree in the main with the results of Verdet’s
experiments.
CHAPTER X.
DEVELOPMENT OF MAXWELL’S THEORY.
We have endeavoured in the preceding pages to give some account of
Maxwell’s contributions to electrical theory and the physics of the
ether. We must now consider very briefly what evidence there is to
support these views. At Maxwell’s death such evidence, though strong,
was indirect. His supporters were limited to some few English-speaking
pupils, young and enthusiastic, who were convinced, it may be, in no
small measure, by the affection and reverence with which they regarded
their master. Abroad his views had made very little way.
In the last words of his book he writes, speaking of various
distinguished workers--
“There appears to be in the minds of these eminent men some
prejudice, or _à priori_ objection, against the hypothesis
of a medium in which the phenomena of radiation of light and
heat, and the electric actions at a distance, take place. It
is true that, at one time, those who speculated as to the
causes of physical phenomena were in the habit of accounting
for each kind of action at a distance by means of a special
ætherial fluid, whose function and property it was to produce
these actions. They filled all space three and four times over
with æthers of different kinds, the properties of which were
invented merely to ‘save appearances,’ so that more rational
enquirers were willing rather to accept not only Newton’s
definite law of attraction at a distance, but even the dogma
of Cotes,[64] that action at a distance is one of the primary
properties of matter, and that no explanation can be more
intelligible than this fact. Hence the undulatory theory of
light has met with much opposition, directed not against its
failure to explain the phenomena, but against its assumption of
the existence of a medium in which light is propagated.
“We have seen that the mathematical expression for
electro-dynamic action led, in the mind of Gauss, to the
conviction that a theory of the propagation of electric
action in time would be found to be the very key-stone of
electro-dynamics. Now we are unable to conceive of propagation
in time, except either as the flight of a material substance
through space, or as the propagation of a condition of motion,
or stress, in a medium already existing in space.
“In the theory of Neumann, the mathematical conception called
potential, which we are unable to conceive as a material
substance, is supposed to be projected from one particle to
another in a manner which is quite independent of a medium,
and which, as Neumann has himself pointed out, is extremely
different from that of the propagation of light.
“In the theories of Riemann and Betti it would appear that the
action is supposed to be propagated in a manner somewhat more
similar to that of light.
“But in all of these theories the question naturally
occurs:--If something is transmitted from one particle to
another at a distance, what is its condition after it has
left one particle and before it has reached the other? If
this something is the potential energy of the two particles,
as in Neumann’s theory, how are we to conceive this energy
as existing in a point of space, coinciding neither with the
one particle nor with the other? In fact, whenever energy is
transmitted from one body to another in time, there must be
a medium or substance in which the energy exists after it
leaves one body and before it reaches the other, for energy,
as Torricelli[65] remarked, ‘is a quintessence of so subtle a
nature that it cannot be contained in any vessel except the
inmost substance of material things.’ Hence all these theories
lead to a conception of a medium in which the propagation takes
place, and if we admit this medium as an hypothesis, I think
it ought to occupy a prominent place in our investigations,
and that we ought to endeavour to construct a mental
representation of all the details of its action, and this has
been my constant aim in this treatise.”
Let us see, then, what were the experimental grounds in Maxwell’s day
for accepting as true his views on electrical action, and how since
then, by the genius of Heinrich Hertz and the labours of his followers,
those grounds have been rendered so sure that nearly the whole progress
of electrical science during the last twenty years has consisted in
the development of ideas which are to be found in the “Treatise on
Electricity and Magnetism.”
The purely electrical consequences of Maxwell’s theory were of course
in accord with all known electrical observations. The equations of the
field accounted for the electro-magnetic forces observed in various
experiments, and from them the laws of electro-magnetic induction
could be correctly deduced; but there was nothing very special in
this. Similar equations had been obtained from the theory of action at
a distance by various writers; in fact, Helmholtz’s theory, based on
the most general form of expression for the force between two elements
of current consistent with certain experiments of Ampère’s, was more
general in its character than Maxwell’s. The destructive features of
Maxwell’s theory were:
(1) The assumption that all currents flow in closed circuits.
(2) The idea of energy residing throughout the electro-magnetic
field in consequence of the strains and stresses set up in the
electro-magnetic medium by the actions to which it was subject.
(3) The identification of this electro-magnetic medium with the
luminiferous ether, and the consequent view that light is an
electro-magnetic phenomena.
(4) The view that electro-magnetic forces arise entirely from strains
and stresses set up in the ether; the electrostatic charge of an
insulated conductor being one of the forms in which the ether strain is
manifested to us.
(5) A dielectric under the action of electric force is said to
become polarised, and, according to Maxwell (vol. i. p. 133), all
electrification is the residual effect of the polarisation of the
dielectric.
Now it must, I think, be admitted that in Maxwell’s day there was
direct proof of very few of these propositions. No one has even yet so
measured the displacement currents in a dielectric as to show that the
total flow across every section of a circuit is at any given moment
the same, though there are other experiments of an indirect character
which have now completely justified Maxwell’s hypothesis. Experiments
by Schiller and Von Helmholtz prove it is true that some action in
the dielectric must be taken into consideration in any satisfactory
theory; they therefore upset various theories based on direct action at
a distance, “but they tell us nothing as to whether any special form
of the dielectric theory, such as Maxwell’s or Helmholtz’s, is true or
not.” (J. J. Thomson, “Report on Electrical Theories,” B.A. Report,
1885, p. 149.)
When Maxwell died there had been little if any experimental evidence
as to the stresses set up in a body by electric force. Fontana, Govi,
and Duter had all observed that changes take place in the volume of
the dielectric of a condenser when it is charged. Quincke had taken
up the work, and the first of his classic papers on this subject was
published in 1880, the year following Maxwell’s death. Maxwell himself
was fond of shewing an experiment in which a charged insulated sphere
was brought near to the surface of paraffin; the stress on the surface
causes a heaping up of the paraffin under the sphere.
Kerr had shewn in 1875 that many substances become doubly refracting
under electric stress; his complete determination of the laws of this
action was published at a later date.
As to direct measurements on electric waves, there were none; the value
of the velocity with which, if Maxwell’s theory were true, they must
travel had been determined from electrical observations of quite a
different character. Weber and Kohlrausch had measured the value of K
for air, for which μ is unity, and from their observations it follows
that the value of the wave velocity for electro-magnetic waves is about
31 × 10⁹ centimetres per second. The velocity of light was known, from
the experiments of Fizeau and Foucault, to have about this value, and
it was the near coincidence of these two values which led Maxwell to
write in 1864:--
“The agreement of the results seems to show that light and magnetism
are affections of the same substance, and that light is an
electro-magnetic disturbance propagated through the field according to
electro-magnetic laws.”
By the time the first edition of the “Electricity and Magnetism”
was published, Maxwell and Thomson (Lord Kelvin) had both made
determinations of K, and had shewn that for air at least the resulting
value for the velocity of electro-magnetic waves was very nearly that
of light.
For other substances at that date the observations were fewer still.
Gibson and Barclay had determined the specific inductive capacity
of paraffin, and found that its square root was 1·405, while its
refractive index for long waves is 1·422. Maxwell himself thought
that if a similar agreement could be shewn to hold for a number of
substances, we should be warranted in concluding that “the square root
of K, though it may not be the complete expression for the index of
refraction, is at least the most important term in it.”
Between this time and Maxwell’s death enough had been done to more
than justify this statement. It was clear from the observations of
Boltzmann, Silow, Hopkinson, and others that there were many substances
for which the square root of the specific inductive capacity was very
nearly indeed equal to the refractive index, and good reason had been
given why in some cases there should be a considerable difference
between the two.
Hopkinson found that in the case of glass the differences were very
large, and they have since been found to be considerable for most
solids examined, with the exception of paraffin and sulphur. For
petroleum oil, benzine, toluene, carbon-bisulphide, and some other
liquids the agreement between Maxwell’s theory and experiment is
close. For the fatty oils, such as castor oil, olive oil, sperm oil,
neatsfoot oil, and also for ether, the differences are considerable.
It seems probable that the reason for this difference lies in the
fact that, in the light waves, we are dealing with the wave velocity
of a disturbance of an extremely short period. Now, we know that the
substances mentioned shew optical dispersion, and we have at present
no completely satisfactory theory from which we can calculate, from
experiments on very short waves, what the velocity for very long
waves will be. In most cases Cauchy’s formula has been used to obtain
the numbers given. The value of K, however, as found by experiment,
corresponds to these infinitely long waves, and to quote Professor
J. J. Thomson’s words, “the marvel is not that there should not be
substances for which the relation K = μ² does not hold, but that there
should be any for which it does.”[66]
It has been shewn, moreover, both by Professor J. J. Thomson himself
and by Blondlot, that when the value of K is measured under very
rapidly varying electrifications, changing at the rate of about
25,000,000 to the second, the value of the inductive capacity for glass
is reduced from about 6·8 or 7 to about 2·7; the square root of this is
1·6, which does not differ much from its refractive index. The values
of the inductive capacity of paraffin and sulphur, which it will be
remembered agree fairly with Maxwell’s theory, were found to be not
greatly different in the steady and in the rapidly varying field.
On the other hand, some experiments of Arons and Rubens in rapidly
varying fields lead to values which do not differ greatly from those
given by other methods. The theory, however, of these experiments seems
open to criticism.
To attempt anything like a complete account of modern verifications
of Maxwell’s views and modern developments of his theory is a task
beyond our limits, but an account of Maxwell written in 1895 would be
incomplete without a reference to the work of Heinrich Hertz.
Maxwell told us what the properties of electro-magnetic waves in air
must be. Hertz[67] in 1887 enabled us to measure those properties, and
the measurements have verified completely Maxwell’s views.
The method of producing electrical oscillations in a conductor had
long been known. Thomson and Von Helmholtz had both pointed it out.
Schiller had examined such oscillations in 1874, and had determined the
inductive capacity of glass by their means, using oscillations whose
period varied from ·000056 to ·00012 of a second.
These oscillations were produced by discharging a condenser through a
coil of wire having self-induction. If the electrical resistance of the
coil be not too great, the charge oscillates backwards and forwards
between the plates of the condenser until its energy is dissipated in
the heat produced in the wire, and in the electro-magnetic radiations
which leave it.
The period of these oscillations under proper conditions is given by
the formula T = 2π√(CL) where L, the coefficient of self induction,
and _C_ the capacity of the condenser. These quantities can be
calculated, and hence the time of an oscillation is known. From such
an arrangement waves radiate out into space. If we could measure
by any method the length of such a wave we could determine its
velocity by dividing the wave length by the period. But it is clear
that since the velocity is comparable with that of light the wave
length will be enormous, unless the period is very short. Thus, a
wave, travelling with the velocity of light, whose period was ·0001
second, such as the waves Schiller worked with, would have a length of
·0001 × 30,000,000,000 or 3,000,000 centimetres, and would be quite
unmeasurable. Before measurements on electric waves could be made it
was necessary (1) to produce waves of sufficiently rapid period, (2) to
devise means to detect them. This is what Hertz did.
The wave length of the electrical oscillations can be reduced by
reducing either the electrical capacity of the system, or the
coefficient of self-induction of the wire. Hertz adopted both these
expedients. His vibrator, in some of his more important experiments,
consisted of two square brass plates 40 cm. in the side. To each of
these is attached a piece of copper wire about 30 cm. in length, and
each wire ends in a small highly-polished brass ball. The plates are
placed so that the wires lie in the same straight line, the brass
balls being separated by a very small air gap. The two plates are then
charged, the one positively the other negatively, until the insulation
resistance of the air gap breaks down and a discharge passes across.
Under these conditions the discharge is oscillatory. It does not
consist of a single spark, but of a series of sparks, which pass
and repass in opposite directions, until the energy of the original
charge is radiated into space or dissipated as heat; the plates are
then recharged and the process repeated. In Hertz’s experiments the
oscillator was charged by being connected to the secondary terminals of
an induction coil.
In 1883 Professor Fitzgerald had called attention to this method of
producing electric waves in air, and had given two metres as the
minimum wave length which might be attained. In 1870 Herr von Bezold
had actually made observations on the propagation and reflection of
electrical oscillations, but his work, published as a preliminary
communication, had attracted little notice. Hertz was the first to
undertake in 1887 in a systematic manner the investigation of the
electric waves in air which proceed from such an oscillator with a view
to testing various theories of electro-magnetic action.
It remained, however, necessary to devise an apparatus for detecting
the waves. When the waves are incident on a conductor, electric
surgings are set up in the conductor, and may, under proper conditions,
be observed as tiny sparks. Hertz used as his detector a loop of wire,
the ends of which terminated in two small brass balls. The wire was
bent so that the balls were very close together, and the sparks could
be seen passing across the tiny air gap which separated them. Such
a wire will have a definite period of its own for oscillations of
electricity with which it may be charged, and if the frequency of the
electric waves which fall on it agrees with that of the waves which
it can itself emit, the oscillations which are set up in the wire will
be stronger than under other conditions, the sparks seen will be more
brilliant.[68] Hertz’s resonator was a circle of wire thirty-five
centimetres in radius, the period for such a resonator would, he
calculated, be the same as that of his vibrator.
There is, however, very considerable difficulty in determining the
period of an electric oscillator from its dimensions, and the value
obtained from calculation for that of Hertz’s radiator is not very
trustworthy. The complete period is, however, comparable with two
one hundredth millionths of a second; in his original papers, Hertz,
through an error, gave a value greater than this.
With these arrangements Hertz was able to detect the presence of
electrical radiation at considerable distances from the radiator; he
was also able to measure its wave length. In the case of sound waves
the existence of nodes and loops formed under proper conditions is
well known. When waves are directly reflected from a flat surface,
interference takes place between the incident and reflected waves,
stationary vibrations are set up, and nodes and loops--places, that
is, of minimum and of maximum motion respectively--are formed. The
position of these nodes and loops can be determined by the aid of
suitable apparatus, and it can be shewn that the distance between two
consecutive nodes is half the wave length.
Similarly when electrical vibrations fall on a reflector, a large
flat surface of metal, for example, stationary vibrations due to the
interference between the incident and reflected waves are produced, and
these give rise to electrical nodes and loops. The position of such
nodes and loops can be found by the use of Hertz’s apparatus, or in
other ways, and hence the length of the electrical waves can be found.
The existence of the nodes and loops shews that the electric effects
are propagated by wave motion. The length of the waves is found to be
definite, since the nodes and loops recur at equal intervals apart.
If it be assumed that the frequency is known, the velocity of wave
propagation can be determined. Hertz found from his experiments that
in air the waves travelled with the velocity of light. It appears,
however, that there were two errors in the calculation which happened
to correct each other, so that neither the value of the frequency given
in Hertz’s paper nor the wave length observed is correct.
By modifying the apparatus it was possible to measure the wave length
of the waves transmitted along a copper wire, and hence, again
assuming the period of oscillation, to calculate the velocity of wave
propagation along the wire. Hertz made the experiment, and found from
his first observations that the waves were propagated along the wire
with a finite velocity, but that the velocity differed from that in
air. The half-wave length in the wire was only about 2·8 metres; that
in air was about 4·5 metres.
Now, this experiment afforded a crucial test between the theories of
Maxwell and Von Helmholtz. According to the former, the waves do not
travel in the wire at all; they travel through the air alongside the
wire, and the wave length observed by Hertz ought to have been the same
as in air. According to Von Helmholtz, the two velocities observed
by Hertz should have been different, as, indeed, they were, and the
experiment appeared to prove that Maxwell’s theory was insufficient and
that a more general one, such as that of Von Helmholtz, was necessary.
But other experiments have not led to the same result. Hertz himself,
using more rapid oscillations in some later measurements, found that
the wave length of the electric waves from a given oscillator was the
same whether they were transmitted through free space or conducted
along a wire.[69] Lecher and J. J. Thomson have arrived at the same
result; but the most complete experiments on this point are those of
Sarasin and De la Rive.
It may be taken, then, as established that Maxwell’s theory is
sufficient, and that the greater generality of Von Helmholtz is
unnecessary.
In a later paper Hertz showed that electric waves could be reflected
and refracted, polarised and analysed, just like light waves. In his
introduction to his “Collected Papers” he writes (p. 19):--
“Casting now a glance backwards, we see that by the experiments
above sketched the propagation in time of a supposed action
at a distance is for the first time proved. This fact forms
the philosophic result of the experiments, and indeed, in a
certain sense, the most important result. The proof includes
a recognition of the fact that the electric forces can
disentangle themselves from material bodies, and can continue
to subsist as conditions or changes in the state of space. The
details of the experiments further prove that the particular
manner in which the electric force is propagated exhibits the
closest analogy[70] with the propagation of light; indeed, that
it corresponds almost completely to it. The hypothesis that
light is an electrical phenomenon is thus made highly probable.
To give a strict proof of this hypothesis would logically
require experiments upon light itself.
“What we here indicate as having been accomplished by the
experiments is accomplished independently of the correctness
of particular theories. Nevertheless, there is an obvious
connection between the experiments and the theory in connection
with which they were really undertaken. Since the year 1861
science has been in possession of a theory which Maxwell
constructed upon Faraday’s views, and which we therefore call
the Faraday-Maxwell theory. This theory affirms the possibility
of the class of phenomena here discovered just as positively
as the remaining electrical theories are compelled to deny
it. From the outset Maxwell’s theory excelled all others in
elegance and in the abundance of the relations between the
various phenomena which it included.
“The probability of this theory, and therefore the number of
its adherents, increased from year to year. But as long as
Maxwell’s theory depended solely upon the probability of its
results, and not on the certainty of its hypotheses, it could
not completely displace the theories which were opposed to it.
“The fundamental hypotheses of Maxwell’s theory contradicted
the usual views, and did not rest upon the evidence of decisive
experiments. In this connection we can best characterise the
object and the result of our experiments by saying: The object
of these experiments was to test the fundamental hypotheses of
the Faraday-Maxwell theory, and the result of the experiments
is to confirm the fundamental hypotheses of the theory.”
Since Maxwell’s death volumes have been written on electrical
questions, which have all been inspired by his work. The standpoint
from which electrical theory is regarded has been entirely changed. The
greatest masters of mathematical physics have found, in the development
of Maxwell’s views, a task that called for all their powers, and the
harvest of new truths which has been garnered has proved most rich. But
while this is so, the question is still often asked, What is Maxwell’s
theory? Hertz himself concludes the introduction just referred to with
his most interesting answer to this question. Prof. Boltzmann has made
the theory the subject of an important course of lectures. Poincaré,
in the introduction to his “Lectures on Maxwell’s Theories and the
Electro-magnetic Theory of Light,” expresses the difficulty, which many
feel, in understanding what the theory is. “The first time,” he says,
“that a French reader opens Maxwell’s book a feeling of uneasiness,
often even of distrust, is mingled with his admiration. It is only
after prolonged study, and at the cost of many efforts, that this
feeling is dissipated. Some great minds retain it always.” And again
he writes: “A French _savant_, one of those who have most completely
fathomed Maxwell’s meaning, said to me once, ‘I understand everything
in the book except what is meant by a body charged with electricity.’”
In considering this question, Poincaré’s own remark--“Maxwell does
not give a mechanical explanation of electricity and magnetism, he is
only concerned to show that such an explanation is possible”--is most
important.
We cannot find in the “Electricity” an answer to the question--What is
an electric charge? Maxwell did not pretend to know, and the attempt to
give too great definiteness to his views on this point is apt to lead
to a misconception of what those views were.
On the old theories of action at a distance and of electric and
magnetic fluids attracting according to known laws, it was easy to be
mechanical. It was only necessary to investigate the manner in which
such fluids could distribute themselves so as to be in equilibrium,
and to calculate the forces arising from the distribution. The problem
of assigning such a mechanical structure to the ether as will permit
of its exerting the action which occurs in an electro-magnetic field
is a harder one to solve, and till it is solved the question--What is
an electric charge?--must remain unanswered. Still, in order to grasp
Maxwell’s theory this knowledge is not necessary.
The properties of ether in dielectrics and in conductors must be quite
different. In a dielectric the ether has the power of storing energy by
some change in its configuration or its structure; in a conductor this
power is absent, owing probably to the action of the matter of which
the conductor is composed.
When we are said to charge an insulated conductor we really act on the
ether in the neighbourhood of the body so as to store it with energy;
if there be another conductor in the field we cannot store energy in
the ether it contains. As, then, we pass from the outside of this
conductor to its interior there is a sudden change in some mechanical
quantity connected with the ether, and this change shows itself as a
force of attraction between the two conductors. Maxwell called the
change in structure, or in property, which occurs when a dielectric
is thus stored with electrostatic energy, _Electric Displacement_; if
we denote it by D, then the electric force R is equal to 4πD/K, and
hence the energy in a unit of volume is 2πD²/K, where K is a quantity
depending on the insulator.
Now, D, the electric displacement, is a quantity which has direction
as well as magnitude. Its value, therefore, at any point can be
represented by a straight line in the usual way; inside a conductor it
is zero. The total change in D, which takes place all over the surface
of a conductor as we enter it from the outside measures, according
to Maxwell, the total charge on the conductor. At points at which
the lines representing D enter the conductor the charge is negative;
at points at which they leave it the charge is positive; along the
lines of the displacement there exists throughout the ether a tension
measured by 2πD²/K; at right angles to these lines there is a pressure
of the same amount.
In addition to the above the components of the displacement D must
satisfy certain relations which can only be expressed in mathematical
form, the physical meaning of which it is difficult to state in
non-mathematical language.
When these relations are so expressed the problem of finding the value
of the displacement at all points of space becomes determinate, and
the forces acting on the conductors can be obtained. Moreover, the
total change of displacement on entering or leaving a conductor can be
calculated, and this gives the quantity which is known as the total
electrical charge on the conductor. The forces obtained by the above
method are exactly the same as those which would exist if we supposed
each conductor to be charged in the ordinary sense with the quantities
just found, and to attract or repel according to the ordinary laws.
If, then, we define electric displacement as that change which takes
place in a dielectric when it becomes the seat of electrostatic
energy, and if, further, we suppose that the change, whatever it
be mechanically, satisfies certain well-known laws, and that in
consequence certain pressures and tensions exist in the dielectric,
electrostatic problems can be solved without reference to a charge of
electricity residing on the conductors.
Something such as this, it appears to me, is Maxwell’s theory of
electricity as applied to electrostatics. It is not necessary, in order
to understand it, to know what change in the ether constitutes electric
displacement, or what is an electric charge, though, of course, such
knowledge would render our views more definite, and would make the
theory a mechanical one.
When we turn to magnetism and electro-magnetism, Maxwell’s theory
develops itself naturally. Experiment proves that magnetic induction is
connected with the rate of change of electric displacement, according
to the laws already given. If, then, we knew the nature of the change
to which the name “electric displacement” has been given, the nature
of magnetic induction would be known. The difficulties in the way of
any mechanical explanation are, it is true, very great; assuming,
however, that some mechanical conception of “electric displacement”
is possible, Maxwell’s theory gives a consistent account of the other
phenomena of electro-magnetism.
Again, we have, it is true, an electro-magnetic theory of light, but
we do not know the nature of the change in the ether which affects
our eyes with the sensation of light. Is it the same as electric
displacement, or as magnetic induction, or since, when electric
displacement is varying, magnetic induction always accompanies it, is
the sensation of light due to the combined effect of the two?
These questions remain unanswered. It may be that light is neither
electric displacement nor magnetic induction, but some quite different
periodic change of structure of the ether, which travels through the
ether at the same rate as these quantities, and obeys many of the same
laws.
In this respect there is a material difference between the ordinary
theory of light and the electro-magnetic theory. The former is a
mechanical theory; it starts from the assumption that the periodic
change which constitutes light is the ordinary linear displacement of a
medium--the ether--having certain mechanical properties, and from those
properties it deduces the laws of optics with more or less success.
Lord Kelvin, in his labile ether, has devised a medium which could
exist and which has the necessary mechanical properties. The periodic
linear displacements of the labile ether would obey the laws of
light, and from the fundamental hypotheses of the theory, a mechanical
explanation, reasonably satisfactory in its main features, can be given
of most purely optical phenomena. The relations between light and
electricity, or light and magnetism, are not, however, touched by this
theory; indeed, they cannot be touched without making some assumption
as to what electric displacement is.
In recent years various suggestions have been made as to the nature
of the change which constitutes electric displacement. One theory,
due to Von Helmholtz, supposes that the electro-kinetic momentum, or
vector potential of Maxwell, is actually the momentum of the moving
ether; according to another, suggested, it would appear originally
in a crude form by Challis, and developed within the last few months
in very satisfactory detail by Larmor, the velocity of the ether is
magnetic force; others have been devised, but we are still waiting for
a second Newton to give us a theory of the ether which shall include
the facts of electricity and magnetism, luminous radiation, and it may
be gravitation.[71]
Meanwhile we believe that Maxwell has taken the first steps towards
this discovery, and has pointed out the lines along which the future
discoverer must direct his search, and hence we claim for him a
foremost place among the leaders of this century of science.
FOOTNOTES
[1] A full biographical account of the Clerk and Maxwell families is
given in a note by Miss Isabella Clerk in the “Life of James Clerk
Maxwell,” and from this the above brief statement has been taken.
[2] “Life of J. C. Maxwell,” p. 26.
[3] “Life of J. C. Maxwell,” p. 27.
[4] “Life of J. C. Maxwell,” p. 49.
[5] “Life of J. C. Maxwell,” p. 52.
[6] “Life of J. C. Maxwell,” p. 56.
[7] “Life of J. C. Maxwell,” p. 67.
[8] “Life of J. C. Maxwell,” p. 75.
[9] Professor Garnett in _Nature_, November 13th, 1879.
[10] “Life of J. C. Maxwell,” p. 105.
[11] “Life of J. C. Maxwell,” p. 116.
[12] “Life of J. C. Maxwell,” pp. 123–129.
[13] “Life of J. C. Maxwell,” p. 190.
[14] Dean of Canterbury.
[15] Master of Trinity.
[16] “Life of J. C. Maxwell,” p. 174.
[17] “Life of J. C. Maxwell,” p. 195.
[18] “Life of J. C. Maxwell,” p. 207.
[19] “Life of J. C. Maxwell,” p. 208.
[20] “Life of J. C. Maxwell,” p. 210.
[21] “Life of J. C. Maxwell,” p. 211.
[22] “Life of J. C. Maxwell,” p. 216.
[23] “Life of J. C. Maxwell,” p. 256.
[24] “Life of J. C. Maxwell,” p. 267.
[25] “Life of J. C. Maxwell,” p. 269.
[26] “Life of J. C. Maxwell,” p. 278.
[27] “Life of J. C. Maxwell,” p. 292.
[28] “Life of J. C. Maxwell,” p. 303.
[29] “Life of J. C. Maxwell,” p. 259.
[30] B.A. Report, Newcastle, 1863.
[31] “Life of J. C. Maxwell,” p. 340.
[32] “Life of J. C. Maxwell,” p. 332.
[33] “Life of J. C. Maxwell,” p. 336.
[34] The Professors who were consulted were Challis, Willis, Stokes,
Cayley, Adams, and Liveing.
[35] “Life of J. C. Maxwell,” p. 349.
[36] “Life of J. C. Maxwell,” p. 381.
[37] “Life of J. C. Maxwell,” p. 379.
[38] An account of the laboratory is given in _Nature_, vol. x., p. 139.
[39] The Chancellor continued to take to the end of his life a warm
interest in the work at the laboratory. In 1887, the Jubilee year, as
Proctor--at the same time I held the office of Demonstrator--it was
my duty to accompany the Chancellor and other officers to Windsor to
present an address from the University to Her Majesty. I was introduced
to the Chancellor at Paddington, and he at once began to question me
closely about the progress of the laboratory, the number of students,
and the work being done there, showing himself fully acquainted with
recent progress.
[40] In 1894 the list contained, in Part II., sixteen names, and in
Part I., one hundred and three names.
[41] Under the new regulations Physics was removed from the first part
of the Tripos and formed, with the more advanced parts of Astronomy
and Pure Mathematics, a part by itself, to which only the Wranglers
were admitted. Thus the number of men encouraged to read Physics was
very limited. This pernicious system was altered in the regulations
at present in force, which came into action in 1892. Part I. of the
Mathematical Tripos now contains Heat, Elementary Hydrodynamics
and Sound, and the simpler parts of Electricity and Magnetism, and
candidates for this examination do come to the laboratory, though not
in very large numbers. The more advanced parts both of Mathematics and
Physics are included in Part II.
[42] “Life of J. C. Maxwell,” p. 383.
[43] “Statique Expérimentale et Théorique des Liquides soumis aux
seules Forces Moléculaires.” Par J. Plateau, Professeur à l’Université
de Gaud.
[44] The “Red Lions” are a club formed by Members of the British
Association to meet for relaxation after the graver labours of the day.
[45] “Leonum arida nutrix.”--_Horace._
[46] _v.r._, endless.
[47] “Life of J. C. Maxwell,” p. 394.
[48] “Life of J. C. Maxwell,” p. 404.
[49] In his “Hydrodynamics,” published in 1738, Daniel Bernouilli
had discussed the constitution of a gas, and had proved from general
considerations that the pressure, if it arose from the impact of a
number of moving particles, must be proportional to the square of their
velocity. (_See_ “Pogg. Ann.,” Bd. 107, 1859, p. 490.)
[50] The proof is as follows:--
If σ be the specific heat at constant volume, σ′ at constant pressure,
and consider a unit of mass of gas at pressure p and volume v, let the
volume increase by an amount dv, while the temperature dy.
Thus σ′dT = σdT + pdv
But pv = ⅔T/m
Hence p being constant,
pdv = ⅔ dT/m
Therefore σ′ = σ + ⅔ 1/m
Now suppose an amount of heat, dH, is given to a single molecule and
that its temperature is T. Its specific heat is σ, and
dH = σmdT
But dH = βdT
Therefore β = σm
Hence 1/m = σ/β
Thus σ′ = σ(1 + 2/(3β))
And σ′/σ = γ
Therefore γ = 1 + 2/(3β)
Or β = 2/(3(γ-1))
[51] Owing to an error of calculation the actual value obtained by
Maxwell from these observations for the coefficient of viscosity is too
great. More recent observers have found lower values than those given
by him; the difference is thus explained.
[52] Studien über das Gleichgewicht der lebendigen Kraft zwischen
bewegten materiellen Punkten Sitz d. k. Akad Wien, Band LVIII., 1868.
[53] Another supposition which might be made, and which is necessary
in order to explain various actions observed in a compound gas under
electric force, is that the parts of which a molecule is composed are
continually changing. Thus a molecule of steam consists of two parts of
hydrogen, one of oxygen, but a given molecule of oxygen is not always
combined with the same two molecules of hydrogen; the particles are
continually changed. In Maxwell’s paper an hypothesis of this kind is
not dealt with.
[54] _Nature_, vol. 1., p. 152 (December 13th, 1894).
[55] See papers by Mr. Capstick, _Phil. Trans._, vols. 185–186.
[56] _Nature_, vol. x.
[57] An historical account of the development of the science of
electricity will be found in the article “Electricity” in the
_Encyclopædia Britannica_, ninth edition, by Professor Chrystal.
[58] Thomson (Lord Kelvin), “Papers on Electrostatics and Magnetism,”
p. 15.
[59] J. J. Thomson, B.A., Report, 1885, pp. 109, 113, Report on
Electrical Theories.
[60] Papers on “Electrostatics,” etc., p. 26.
[61] It is difficult to explain without analysis exactly what is
measured by Maxwell’s Vector Potential. Its rate of change at any
point of space measures the electromotive force at that point, so far
as it is due to variations of the electric current in neighbouring
conductors; the magnetic induction depends on the first differential
coefficients of the components of the electro-tonic state; the
electric current is related to their second differential coefficients
in the same manner as the density of attracting matter is related
to the potential it produces. In language which is now frequently
used in mathematical physics, the electromotive force at a point
due to magnetic induction is proportioned to the rate of change of
the Vector Potential, the magnetic induction depends on the “curl”
of the Vector Potential, while the electric current is measured by
the “concentration” of the Vector Potential. From a knowledge of the
Vector Potential these other quantities can be obtained by processes of
differentiation.
[62] The 4 π is introduced because of the system of units usually
employed to measure electrical quantities. If we adopted Mr. Oliver
Heaviside’s “rational units,” it would disappear, as it does in (B).
[63] For an exact statement as to the relation between the directions
of the lines of electric displacement and of the magnetic force,
reference must be made to Professor Poynting’s paper, _Phil. Trans._,
1885, Part II., pp. 280, 281. The ideas are further developed in a
series of articles in the _Electrician_, September, 1895. Reference
should also be made to J. J. Thomson’s “Recent Researches in
Electricity and Magnetism.”
[64] Preface to Newton’s “Principia,” 2nd edition.
[65] “Lezioni Accademiche” (Firenze, 1715), p. 25.
[66] In his sentence μ stands for the refractive index.
[67] Hertz’s papers have been translated into English by D. E. Jones,
and are published under the title of _Electric Waves_.
[68] Some of the consequences of this electrical resonance have been
very strikingly shown by Professor Oliver Lodge. _See_ _Nature_,
February 20th, 1890.
[69] Hertz’s original results were no doubt affected by waves reflected
from the walls and floor of the room in which he worked. An iron
stove also, which was near his apparatus, may have had a disturbing
influence; but for all this, it is to his genius and his brilliant
achievements that the complete establishment of Maxwell’s theory is due.
[70] The analogy does not consist only in the agreement between the
more or less accurately measured velocities. The approximately equal
velocity is only one element among many others.
[71] For a very suggestive account of some possible theories,
reference should be made to the presidential address of Professor W. M.
Hicks to Section A of the British Association at Ipswich in 1895.
INDEX.
Aberdeen, Maxwell elected Professor at, 45;
formation of University of, 51
Adams, W. G., succeeds Maxwell as Professor at King’s College,
London, 58
Adams Prize, The, 48;
gained by Maxwell, 50
Ampère, 155, 204
Ampère’s Law, 155, 156
_Annals of Philosophy_, Thomson’s, 112, 113
“Apostles,” club so called, 30, 89
Arago, 157
Aragonite, 200
Atom, article by Maxwell in _Encyclopædia Britannica_, 108
Avogadros’ Law, 117, 124
Bakerian Lecture, delivered by Maxwell, 58
Berkeley on the Theory of Vision, 38
Bernouilli, D., 113
Blackburne, Professor, 16
Blore, Rev. E. W., 67
Boehm, Bust of Maxwell by, 90
Boltzmann, Dr., 135, 137, 138, 144, 216
Boltzmann-Maxwell Theory, The, 140, 145
Boscovitch on Atoms, 108, 109
Boyle’s Law, 114, 117, 124
Brewster, Sir David, on Colour Sensation, 99
British Association, Maxwell and, 42,54;
Lecture before, 80–82;
Lines on President’s address, 83, 84
Butler, Dr. H. M., extract from sermon on Maxwell, 32–35
Bryan, G. H., 141, 143
Cambridge, Maxwell at, 28–46;
Mathematical Tripos at, 60;
Foundation of Professorship of Experimental Physics at, 66
_Cambridge and Dublin Mathematical Journal_, Papers by Maxwell in, 30
Campbell, Professor L., 9, 10, 12, 14, 22, 52, 57, 79
Cauchy’s Formula, 208
Cavendish, Henry, 73, 74;
Works of, edited by Maxwell, 87, 154, 155
Cavendish Laboratory, built and presented to University of
Cambridge, 73, 74
Cay, Miss Frances, 11
Cayley Portrait Fund, lines to Committee, 86
Challis, Professor, 49
Charles’ Law, 124
Chemical Society, Maxwell’s lecture before, 80–82
Clausius, on kinetic theory of gases, 119, 129, 130, 137
Clerks of Penicuik, The, 9, 10
Colour Perception, 94
Colour Sensation, Young on, 97, 98;
Sir D. Brewster on, 99
Colours, paper by Maxwell, on, 40, 41;
Helmholtz on, 99
Conductors and Insulators, Distinction between, 173
Cookson, Dr., 61
Corsock, Maxwell buried at, 90
Cotes, 202
Coulomb, 154
Curves, investigated by Maxwell, 19
Daniell’s cells, 77
Democritus, 108
Demonstrator of Physics, W. Garnett appointed, 75
Description of Oval Curves, first paper by Maxwell, 19
Devonshire, Duke of, Cavendish Laboratory built by, 73, 74;
Letter of Thanks from University of Cambridge, 74
Dewar, Miss K. M., her marriage to Maxwell, 51
Dickinson, Lowes; Portrait of Maxwell by, 90
Diffusion of gases, 128
Discs for colour experiments, 99–101
Droop, H. R., 57
Dynamical Theory of the Electro-magnetic Field, Maxwell on, 57, 177
Dynamical Theory of Gases, Maxwell on, 58, 134
Edinburgh Academy, Maxwell’s school-life at, 13–18
Edinburgh, Royal Society of, Maxwell at meetings of, 18
Edinburgh, University of, Maxwell at, 22
Elastic Spheres, 144
Electric Displacement, 218, 219, 220
Electrical Theories, 94, 154, 155
Electricity and Magnetism, Maxwell’s book on, 59, 77, 79, 147, 155,
156, 176, 180–201;
papers by Lord Kelvin on, 161–2;
Application of Mathematical Analysis to, paper by G. Green, 158
Electricity, Modern Views of, by Professor Lodge, 177
Electro-kinetic Momentum, 221
Electro-magnetic Field, Dynamical Theory of, Maxwell on, 57, 177
Electro-magnetic Induction, 157
Electro-magnetic Theory of Light, 174
Electro-tonic State, 164
Electrostatic Induction, Faraday on, 159
_Encyclopædia Britannica_, articles by Maxwell in, 80, 108, 146
Ether, labile, 220
Experimental Physics, foundation of Professorship at Cambridge, 66;
Election of Maxwell, 68
Faraday on electrical science, 157;
on electrostatic induction, 159
Faraday’s Lines of Force, paper by Maxwell on, 44, 45, 148–153
Fawcett, W. M., architect of Cavendish Laboratory, 73
Fitzgerald, Professor, 177, 211
Forbes, Professor J. D., 18, 44, 54;
friendship with Maxwell, 19;
paper on Theory of Glaciers, 19;
resigns Professorship at Edinburgh, 54
Galvani, 155
Garnett, W., appointed Demonstrator of Physics at Cambridge, 75;
Life of Maxwell by, 94
Gases, Molecular theory of, 57, 108;
Waterston on general theory of, 118;
Clausius on, 119;
diffusion of, 128
Gauss’ Theory, 156
Gay Lussac’s Law, 117
General Theory of Gases, Waterston on, 118;
Clausius on, 119
Glenlair, home of Maxwell, 11, 23;
laboratory at, 24;
Maxwell’s life at, 58, 59;
“Electricity and Magnetism” written at, 79
Gordon, J. E. H., 77, 78
Green, G., of Nottingham, paper on electricity and magnetism, 158;
inventor of term “Potential,” 158
Hamilton, Sir W. R., 22
Hamilton’s Principle, 190
Heat, Text-book on, by Maxwell, 79
Helmholtz, 99, 156, 157, 175, 221
Henry, J., of Washington, on electro-magnetic induction, 157
Herapath on molecules, 112–116
Hertz, Heinrich, 204, 209–213
Hicks, W. M., 221
Hockin, C., 56
Holman, Professor, 133
Iceland Spar, 200
Insulators and Conductors, Distinction between, 173
Jenkin, Fleeming, 55, 56
Kelland, Professor, 22
Kelvin, Lord, 16, 142, 158, 159, 160, 168;
on the Uniform Motion of Heat, 160;
papers on Electricity and Magnetism, 161, 162
Kinetic energy, 124, 129, 136, 139, 191
King’s College, London, Maxwell elected Professor at, 54
Kohlrausch, 206
Kundt, 132
Labile Ether, 220
Laboratory at Glenlair, 24
Lagrange, 179
Lagrange’s Equations, 179, 190
Laplace, 155
Larmor, J., 141, 142
Lecher, 214
Lenz, 157
Litchfield, R. B., 46
Light, Electro-magnetic Theory of, 174;
Waves of, 198, 199
Lodge, Professor, book on Modern Views of Electricity, 177
Lucretius, 108
Luminous Radiation, 221
Mathematical Tripos at Cambridge, subjects, 60;
Maxwell an examiner for, 60, 80;
experimental work in, 76
Matter and Motion, Maxwell on, 79
Maxwell, James Clerk, parentage and birthplace, 10, 11;
childhood and school-days, 12–18;
his mother’s death, 13;
first lessons in geometry, 17;
attends meetings of Royal Society of Edinburgh, 18;
his first published paper, 19;
friendship with Professor Forbes, 19;
his polariscope, 20;
enters the University of Edinburgh, 22;
papers on Rolling Curves and Elastic Solids, 23;
vacations at Glenlair, 23;
laboratory at Glenlair, 24;
undergraduate life at Cambridge, 28–36;
elected scholar of Trinity, 29;
illness at Lowestoft, 29;
his friends at Cambridge, 30;
Tripos and degree, 35–37;
early researches, 38–44;
paper on Colours, 40, 41;
elected Fellow of Trinity, 43;
Lecturer at Trinity, 43;
Professor at Aberdeen, 45;
his father’s death, 45;
gains the Adams Prize, 50;
marriage, 51;
powers as teacher and lecturer, 52, 53;
Professor at King’s College, London, 54;
gains the Rumford Medal, 55;
delivers Bakerian lecture, 58;
resigns Professorship at King’s College, London, 58;
life at Glenlair, 58, 59;
visit to Italy, 59;
Examiner for Mathematical Tripos, 60, 80;
elected Professor of Experimental Physics at Cambridge, 68;
Introductory Lecture, 68–72;
Examiner for Natural Sciences Tripos, 79;
articles in _Encyclopædia Britannica_, 80, 118, 146;
papers in Nature, 80;
lectures before British Association and Chemical Society, 80–82;
humorous poems, 83–87;
delivers Rede Lecture on the Telephone, 89;
last illness and death, 89, 90;
buried at Corsock, 90;
bust and portrait, 90;
religious views, 91, 92
Maxwell, John Clerk, 10, 11
Meyer, O. E., 133
Mill’s Logic, 38
Molecular Evolution, Lines on, 85
---- Physics, 94
---- Constitution of Bodies, Maxwell on, 146
---- Theory of Gases, 57, 108
Molecules, 109, 110;
Herapath on, 112–116;
lecture by Maxwell on, 146
Motion of Saturn’s Rings, subject for Adams Prize, 49
Munro, J. C., 40, 56, 68, 82
Natural Sciences Tripos, Maxwell Examiner for, 79
_Nature_, papers by Maxwell in, 80
Neumann, F. E., 156, 157
Newton’s Lunar Theory and Astronomy, 50
---- Principia, 202
Nicol, Wm., inventor of the polarising prism, 20
Niven, W. D., 27, 46, 51, 52, 60, 78, 87, 88, 93
Obermeyer, 134
Ohm’s Law, 77
Ophthalmoscope devised by Maxwell, 83
Oval Curves, Description of, Maxwell’s first paper, 19
Parkinson, Dr., 49
_Philosophical Magazine_, 56, 99, 115, 120, 133, 142
_Philosophical Transactions_, 56, 89, 132, 145
Physical Lines of Force, Maxwell on, 56, 158
Physics, Instruction in, at Cambridge, 61;
Report of Syndicate on, 62–64;
Demonstrator appointed, 75
Poincaré, 216
Poisson, 44;
on distribution of electricity, 155
Polariscope, made by Maxwell, 20
“Potential,” term invented by G. Green, 158;
the Vector, 165, 221
Poynting, Professor, 187–189
Puluj, 134
Quincke, 206
Radiation, Luminous, 221
Rarefied Gases, Stresses in, paper by Maxwell, 135, 145
Rayleigh, Lord, 67, 77
Rede Lecture on the Telephone, delivered by Maxwell, 89
Report on Electrical Theories, J. J. Thomson, 204
---- of Syndicate as to instruction in Physics at Cambridge, 62–64
Robertson, C. H., 28
Rolling Curves, Maxwell on, 23
Royal Society, The, Maxwell and, 55;
Transactions of, 89
Rumford Medal gained by Maxwell, 55, 106
Sabine, Major-General, Vice-President of Royal Society, 106
Smith’s Prizes, 36
Standards of Electrical Resistance, Committee on, 55
Stewart, Balfour, 56, 125
Stresses in Rarefied Gases, Maxwell on, 135, 155
Tait, Professor P. G., 21, 26, 94
Tayler, Rev. C. B., 29
Telephone, Rede Lecture by Maxwell on, 89
Theory of Glaciers, Prof. Forbes on, 19
Thomson, J. J., 157, 208;
Report on Electrical Theories, 205
Thomson’s _Annals of Philosophy_, 112, 113
Uniform Motion of Heat in Homogeneous Solid Bodies, paper by Lord
Kelvin, 160, 161
University Commission, 47, 48, 62
Urr, Vale of, 11
Vector Potential, The, 165, 221
Viscosity of Gases, Experiments on, 58, 125, 132
Volta, Inventor of voltaic pile, 155
Waterston, J. J., on molecular theory of gases, 114, 115;
on general theory of gases, 118
Waves of Light, 198, 199
Weber, W., 156, 206
Wedderburn, Mrs., 14
Wheatstone’s Bridge, 77
Williams, J., Archdeacon of Cardigan, 16
Willis, Professor, 44
Wilson, E., lines in memory of, 86, 87
Young, T., on colour sensation, 97, 98
PRINTED BY CASSELL & COMPANY, LIMITED, LA BELLE SAUVAGE, LONDON, E.C.
* * * * * *
Transcriber’s note:
Punctuation, hyphenation, and spelling were made consistent when a
predominant preference was found in the original book; otherwise they
were not changed.
Simple typographical errors were corrected; unpaired quotation
marks were remedied when the change was obvious, and otherwise left
unpaired.
Illustrations in this eBook have been positioned between paragraphs
and outside quotations.
Footnotes, originally at the bottoms of pages, have been collected,
renumbered, and placed just before the Index.
The Index was not checked for proper alphabetization or correct page
references.
Some values in the original book are known today to be incorrect, but
have not been changed here.
Page 133: The last equation on the page,
μ = μ₀ (1 + .00275 t - .00000034 t²)
was misprinted as
μ = μ₀ {1 + .00275 t .00000034 t²}.
It is shown here with corrections based on its cited source:
https://archive.org/details/s05philosophicalmag21londuoft/page/212
Page 144: “possibly of ether atoms bound with them” was printed that
way, but “ether” may be a misprint for “other”.
Page 170: “hence at C, where they touch” was printed as “A”, but Figure
1 at that point is labelled “C”.
*** End of this LibraryBlog Digital Book "James Clerk Maxwell and Modern Physics" ***
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