Lord Kelvin: An account of his scientific life and work
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Title: Lord Kelvin: An account of his scientific life and work
Author: Andrew Gray
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ENGLISH MEN OF SCIENCE
EDITED BY
J. REYNOLDS GREEN, Sc.D.
LORD KELVIN
ENGLISH MEN
OF SCIENCE
EDITED BY
DR. J. REYNOLDS GREEN.
_With Photogravure Frontispiece._
_Small Cr. 8vo, 2s. 6d. net per vol._
SPENCER. By J. ARTHUR THOMPSON.
PRIESTLEY. By Dr. THORPE, C.B., F.R.S.
FLOWER. By Prof. R. LYDEKKER, F.R.S.
HUXLEY. By Prof. AINSWORTH DAVIS.
BENTHAM. By B. DAYDON JACKSON, F.L.S.
DALTON. By J. P. MILLINGTON, M.A.
_J. M. DENT & CO._
_All Rights Reserved_
[Illustration: Lord Kelvin]
LORD KELVIN
_AN ACCOUNT OF HIS SCIENTIFIC LIFE AND WORK_
BY
ANDREW GRAY
LL.D., F.R.S., V.-P.R.S.E.
PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW
PUBLISHED IN LONDON BY
J. M. DENT & CO., AND IN NEW
YORK BY E. P. DUTTON & CO.
1908
RICHARD CLAY & SONS, LIMITED,
BREAD STREET HILL, E.C., AND
BUNGAY, SUFFOLK.
PREFACE
This book makes no claim to be a biography of Lord Kelvin in the usual
sense. It is an extension of an article which appeared in the _Glasgow
Herald_ for December 19, 1907, and has been written at the suggestion
of various friends of Lord Kelvin, in the University of Glasgow and
elsewhere, who had read that article. The aim of the volume is to give
an account of Lord Kelvin's life of scientific activity, and to explain
to the student, and to the general reader who takes an interest in
physical science and its applications, the nature of his discoveries.
Only such a statement of biographical facts as seems in harmony with
this purpose is attempted. But I have ventured, as an old pupil and
assistant of Lord Kelvin, to sketch here and there the scene in his
class-room and laboratory, and to record some of the incidents of his
teaching and work.
I am under obligations to the proprietors of the _Glasgow Herald_ for
their freely accorded permission to make use of their article, and to
Messrs. Annan, photographers, Glasgow, and Messrs. James MacLehose &
Sons, Glasgow, for the illustrations which are given, and which I hope
may add to the interest of the book.
A. GRAY.
_The University_, _Glasgow_,
_May_ 20, 1908.
CONTENTS
CHAP. PAGE
I. PARENTAGE AND EARLY EDUCATION 1
II. CLASSES AT THE UNIVERSITY OF GLASGOW. FIRST SCIENTIFIC
PAPERS 13
III. UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS UNDERGRADUATE 23
IV. THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM.
ELECTRIC IMAGES. ELECTRIC INVERSION 33
V. THE CHAIR OF NATURAL PHILOSOPHY AT GLASGOW. ESTABLISHMENT
OF THE FIRST PHYSICAL LABORATORY 61
VI. FRIENDSHIP WITH STOKES AND JOULE. EARLY WORK AT GLASGOW 79
VII. THE 'ACCOUNT OF CARNOT'S THEORY OF THE MOTIVE POWER OF
HEAT'--TRANSITION TO THE DYNAMICAL THEORY OF HEAT 99
VIII. THERMODYNAMICS AND ABSOLUTE THERMOMETRY 114
IX. HYDRODYNAMICS--DYNAMICAL THEOREM OF MINIMUM
ENERGY--VORTEX MOTION 153
X. THE ENERGY THEORY OF ELECTROLYSIS--ELECTRICAL
UNITS--ELECTRICAL OSCILLATIONS 176
XI. THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'--GYROSTATIC
ACTION--'ELECTROSTATICS AND MAGNETISM' 194
XII. THE AGE OF THE EARTH 229
XIII. BRITISH ASSOCIATION COMMITTEE ON ELECTRICAL STANDARDS 244
XIV. THE BALTIMORE LECTURES 254
XV. SPEED OF TELEGRAPH SIGNALLING--LAYING OF SUBMARINE
CABLES--TELEGRAPH INSTRUMENTS--NAVIGATIONAL
INSTRUMENTS, COMPASS AND SOUNDING MACHINE 264
XVI. LORD KELVIN IN HIS CLASS-ROOM AND LABORATORY 279
XVII. PRACTICAL ACTIVITIES--HONOURS AND DISTINCTIONS--LAST
ILLNESS AND DEATH 299
CONCLUSION 305
INDEX 317
CORRIGENDUM
Page 105, line 9 from foot, for ∂e + O read ∂e + o
LIST OF ILLUSTRATIONS
_To face page_
LORD KELVIN (photogravure) _Frontispiece_
LORD KELVIN IN 1846 64
VIEW OF OLD COLLEGE 70
LORD KELVIN
CHAPTER I
PARENTAGE AND EARLY EDUCATION
Lord Kelvin came of a stock which has helped to give to the north of
Ireland its commercial and industrial supremacy over the rest of that
distressful country. His ancestors were county Down agriculturists of
Scottish extraction. His father was James Thomson, the well-known
Glasgow Professor of Mathematics, and author of mathematical text-books
which at one time were much valued, and are even now worth consulting.
James Thomson was born on November 13, 1786, near Ballynahinch, county
Down. Being the son of a small farmer he was probably unable to enter on
university studies at the usual age, for he did not matriculate in
Scotland until 1810. The class-lists of the time show that he
distinguished himself highly in mathematics, natural philosophy, and
classics.
An interesting incident of these student days of his father was related
by Lord Kelvin in his installation address as Chancellor of the
University in 1904, and is noteworthy as indicating how comparatively
recent are many of the characteristics of our present-day life and
commerce. James Thomson and some companions, walking from Greenock to
Glasgow, on their way to join the college classes at the commencement of
the session, "saw a prodigy--a black chimney moving rapidly beyond a
field on the left-hand side of their road. They jumped the fence, ran
across the field, and saw, to their astonishment, Henry Bell's 'Comet'
(then not a year old) travelling on the Clyde between Glasgow and
Greenock."[1] Sometimes then the passage from Belfast to Greenock took a
long time. Once James Thomson, crossing in an old lime-carrying smack,
was three or four days on the way, in the course of which the vessel,
becalmed, was carried three times by the tide round Ailsa Craig.
Mr. Thomson was elected in 1815 to the Professorship of Mathematics in
the Royal Academical Institution of Belfast, and held the post for
seventeen years, building up for himself an excellent reputation as a
teacher, and as a clear and accurate writer. Just then analytical
methods were beginning to supersede the processes of geometrical
demonstration which the form adopted by Newton for the Principia had
tended to perpetuate in this country. Laplace was at the height of his
fame in France, and was writing the great analytical Principia, his
_Mécanique Céleste_, applying the whole force of his genius, and all the
resources of the differential and integral calculus invented by Newton
and improved by the mathematicians of the intervening century, to the
elucidation and extension of the "system of the world," which had been
so boldly sketched by the founder of modern physical science.
In that period Fourier wrote his memoirs on the conduction of heat, and
gave to the world his immortal book to be an inspiration to the physical
philosophers of succeeding generations. Legendre had written memoirs
which were to lead, in the hands of Jacobi and his successors, to a new
province of mathematics, while, in Germany, Gauss had begun his stately
march of discovery.
The methods and results of this period of mathematical activity were at
first hardly known in this country: the slavish devotion of Cambridge to
the geometrical processes and the fluxional notation of Newton, an
exclusive partiality which Newton himself would have been the first to
condemn, led analytical methods, equally Newtonian, to be stigmatised as
innovations, because clothed in the unfamiliar garb of the continental
notation. A revolt against this was led by Sir John Herschel, Woodhouse,
Peacock, and some others at Cambridge, who wrote books which had a great
effect in bringing about a change of methods. Sir John thus described
the effect of the new movements:--"Students at our universities,
fettered by no prejudices, entangled by no habits, and excited by the
ardour and emulation of youth, had heard of the existence of masses of
knowledge from which they were debarred by the mere accident of
position. They required no more. The prestige which magnifies what is
unknown, and the attractions inherent in what is forbidden, coincided in
their impulse. The books were procured and read, and produced their
natural effects. The brows of many a Cambridge examiner were elevated,
half in ire, half in admiration, at the unusual answers which began to
appear in examination papers. Even moderators are not made of
impenetrable stuff, though fenced with sevenfold Jacquier, and tough
bull-hide of Vince and Wood."
The memoirs and treatises of the continental analysts were eagerly
procured and studied by James Thomson, and as he was bound by no
examination traditions, he freely adopted their methods, so far as these
came within the scope of his teaching, and made them known to the
English reading public in his text-books. Hence when the chair of
Mathematics at Glasgow became vacant in 1832 by the death of Mr. James
Millar, Mr. Thomson was at once chosen by the Faculty, which at that
time was the electing body.
The Faculty consisted of the Principal and the Professors of Divinity,
Church History, Oriental Languages, Natural Philosophy, Moral
Philosophy, Mathematics, Logic, Greek, Humanity, Civil Law, Practice of
Medicine, Anatomy, and Practical Astronomy. It administered the whole
revenues and property of the College, and possessed the patronage of the
above-named chairs with the exception of Church History, Civil Law,
Medicine, Anatomy, and Astronomy, so that Mr. Thomson became not only
Professor of Mathematics, but also, in virtue of his office, a member of
what was really the supreme governing body of the University. The
members of the Faculty, with the exception of the Professor of
Astronomy, who resided at the observatory, were provided with official
residences in the College. This arrangement is still adhered to; though
now the government is in the hands of a University Court, with the
Senate (which formerly only met to confer degrees or to manage the
library and some other matters) to regulate and superintend teaching and
discipline.
Professor Thomson was by no means the first or the only professor of the
name in the University of Glasgow, as the following passage quoted from
a letter of John Nichol, son of Dr. J. P. Nichol, and first Professor of
English at Glasgow, amusingly testifies:--
"Niebuhr, after examining a portion of the _Fasti Consulares_, arrived
at the conclusion that the _senatus populusque Romanus_ had made a
compact to elect every year a member of the Fabian house to one of the
highest offices of state, so thickly are the records studded with the
name of the Fabii. Some future Niebuhr of the New Zealand Macaulay
imagines, turning his attention to the annals of Glasgow College, will
undoubtedly arrive at the conclusion that the leaders of that
illustrious corporation had, during the period of which I am writing,
become bound in a similar manner to the name of Thomson. Members of that
great gens filled one-half of the chairs in the University. I will not
venture to say how many I have known. There was Tommy Thomson the
chemist; William Thomson of Materia Medica; Allen Thomson of Anatomy,
brother of the last; Dr. James Thomson of Mathematics; William, his son,
etc., etc. Old Dr. James was one of the best of Irishmen, a good
mathematician, an enthusiastic and successful teacher, the author of
several valuable school-books, a friend of my father's, and himself the
father of a large family, the members of which have been prosperous in
the world. They lived near us in the court, and we made a pretty close
acquaintanceship with them all."
A former Professor of Natural Philosophy, Dr. Anderson,[2] who appears
to have lived the closing years of his life in almost constant warfare
with his colleagues of the Faculty, and who established science classes
for workmen in Glasgow, bequeathed a sum of money to set up a college in
Glasgow in which such classes might be carried on. The result was the
foundation of what used to be called the "Andersonian University" in
George Street, the precursor of the magnificent Technical College of the
present day. This name, and the large number of Thomsons who had been
and were still connected with the University of Glasgow, caused the more
ancient institution to be not infrequently referred to as the
"Thomsonian University"!
The Thomas Thomson (no relative of the Belfast Thomsons) affectionately,
if a little irreverently, mentioned in the above quotation, was then the
Professor of Chemistry. He was the first to establish a chemical
laboratory for students in this country; indeed, his laboratory preceded
that of Liebig at Giessen by some years, and it is probable that as
regards experimental chemistry Glasgow was then in advance of the rest
of the world. His pupil and life-long admirer was destined to establish
the first physical laboratory for such students as were willing to spend
some time in the experimental investigation and verification of physical
principles, or to help the professor in his researches. The systematic
instruction of students in methods of experimenting by practical
exercises with apparatus was a much later idea, and this fact must be
taken account of when the laboratories of the present time are
contrasted with the much more meagre provision of those early days. The
laboratory is now, as much as the lecture-room, the place where classes
are held and instruction given in experimental science to crowds of
students, and it is a change for the better.
The arrival of James Thomson and his family at Glasgow College, in 1832,
was remarked at the time as an event which brought a large reinforcement
to the gens already inseparably associated with the place: how great
were to be its consequences not merely to the University but to the
world at large nobody can then have imagined. His family consisted of
four sons and two daughters: his wife, Margaret Gardner, daughter of
William Gardner, a merchant in Glasgow, had died shortly before, and the
care of the family was undertaken by her sister, Mrs. Gall. The eldest
son, James Thomson, long after to be Rankine's successor in the Chair of
Engineering, was ten years of age and even then an inveterate inventor;
William, the future Lord Kelvin (born June 26, 1824), was a child of
eight. Two younger sons were John (born in 1826)--who achieved
distinction in Medicine, became Resident Assistant in the Glasgow Royal
Infirmary, and died there of a fever caught in the discharge of his
duty--and Robert, who was born in 1829, and died in Australia in 1905.
Besides these four sons there were in all three daughters:--Elizabeth,
afterwards wife of the Rev. David King, D.D.; Anna, who was married to
Mr. William Bottomley of Belfast (these two were the eldest of the
family), and Margaret, the youngest, who died in childhood. Thus began
William Thomson's residence in and connection with the University of
Glasgow, a connection only terminated by the funeral ceremony in
Westminster Abbey on December 23, 1907.
Professor Thomson himself carefully superintended the education of his
sons, which was carried out at home. They were well grounded in the old
classical languages, and moreover received sound instruction in what
even now are called, but in a somewhat disparaging sense, modern
subjects. As John Nichol has said in his letters, "He was a stern
disciplinarian, and did not relax his discipline when he applied it to
his children, and yet the aim of his life was their advancement."
It would appear from John Nichol's recollections that even in childhood
and youth, young James Thomson was an enthusiastic experimentalist and
inventor, eager to describe his ideas and show his models to a
sympathetic listener.[3] And both then and in later years his charming
simplicity, his devouring passion for accuracy of verbal expression in
all his scientific writing and teaching, and his unaffected and
unconscious genius for the invention of mechanical appliances, all based
on true and intuitively perceived physical principles, showed that if he
had had the unrelenting power of ignoring accessories and unimportant
details which was possessed by his younger brother, he might have
accomplished far more than he did, considerable as that was. But William
had more rapid decision, and though careful and exact in expressing his
meaning, was less influenced by considerations of the errors that might
arise from the various connotations of such scientific terms as are also
words in common use; and he quickly completed work which his brother
would have pondered over for a long time, and perhaps never finished.
It is difficult for a stranger to Glasgow, or even for a resident in
Glasgow in these days of quick and frequent communication with England,
and for that matter with all parts of the world, to form a true idea of
life and work at the University of Glasgow seventy years ago. The
University had then its home in the old "tounis colledge" in the High
Street, where many could have wished it to remain, and, extending its
buildings on College Green, retain the old and include the new. Its fine
old gateway, and part of one of the courts, were still a quaint
adornment of the somewhat squalid street in 1871, after the University
had moved to its present situation on the windy top of Gilmorehill.
Deserted as it was, its old walls told something of the history of the
past, and reminded the passer-by that learning had flourished amid the
shops and booths of the townspeople, and that students and professors
had there lived and worked within sound of the shuttle and the forge.
The old associations of a town or a street or a building, linked as they
often are with the history of a nation, are a valuable possession, not
always placed in the account when the advantages or disadvantages of
proposed changes are discussed; but a University which for four hundred
years has seen the tide of human life flow round it in a great city, is
instinct with memories which even the demolition of its walls can only
partially destroy. Poets and statesmen, men of thought and men of
action, lords and commoners, rich men's sons and the children of
farmers, craftsmen and labourers, had mingled in its classes and sat
together on its benches; and so had been brought about a community of
thought and feeling which the practice of our modern and wealthy
cosmopolites, who affect to despise nationality, certainly does nothing
to encourage. In the eighteenth century the Provosts and the Bailies of
the time still dwelt among men and women in the High Street, and its
continuation the Saltmarket, or not far off in Virginia Street, the home
of the tobacco lords and the West India merchants. Their homely
hospitality, their cautious and at the same time splendid generosity,
their prudent courage, and their faithful and candid friendships are
depicted in the pages of Scott; and though a change in men and manners,
not altogether for the better, has been gradually brought about by sport
and fashion, those peculiarly Scottish virtues are still to be found in
the civic statesmen and merchant princes of the Glasgow of to-day.
Seventy years ago the great migration of the well-to-do towards the west
had commenced, but it had but little interfered with the life of the
High Street or of the College. Now many old slums besides the Vennel and
the Havannah have disappeared, much to the credit of the Corporation of
Glasgow; and, alas, so has every vestige of the Old College, much to the
regret of all who remember its quaint old courts. A railway company, it
is to be supposed, dare not possess an artistic soul to be saved; and
therefore, perhaps, it is that it builds huge and ugly caravanserais of
which no one, except perhaps the shareholders, would keenly regret the
disappearance. But both artists and antiquaries would have blessed the
directors--and such a blessing would have done them no harm--if they had
been ingenious and pious enough to leave some relic of the old buildings
as a memorial of the old days and the old life of the High Street.
A picture of the College in the High Street has recently been drawn by
one who lived and worked in it, though some thirty years after James
Thomson brought his family to live in its courts. Professor G. G. Ramsay
has thus portrayed some features of the place, which may interest those
who would like to imagine the environment in which Lord Kelvin grew up
from childhood, until, a youth of seventeen, he left Glasgow for
Cambridge.[4] "There was something in the very disamenities of the old
place that created a bond of fellowship among those who lived and worked
there, and that makes all old students, to this day, look back to it
with a sort of family pride and reverence. The grimy, dingy, low-roofed
rooms; the narrow, picturesque courts, buzzing with student-life; the
dismal, foggy mornings and the perpetual gas; the sudden passage from
the brawling, huckstering High Street into the academic quietude, or the
still more academic hubbub, of those quaint cloisters, into which the
policeman, so busy outside, was never permitted to penetrate; the
tinkling of the 'angry bell' that made the students hurry along to the
door which was closed the moment that it stopped; the roar and the flare
of the Saturday nights, with the cries of carouse or incipient murder
which would rise into our quiet rooms from the Vennel or the Havannah;
the exhausted lassitude of Sunday mornings, when poor slipshod creatures
might be seen, as soon as the street was clear of churchgoers, sneaking
over to the chemist's for a dose of laudanum to ease off the debauch of
yesterday; the conversations one would have after breakfast with the old
ladies on the other side of the Vennel, not twenty feet from one's
breakfast-table, who divided the day between smoking short cutty pipes
and drinking poisonous black tea--these sharp contrasts bound together
the College folk and the College students, making them feel at once part
of the veritable populace of the city, and also hedged off from it by
separate pursuits and interests."
The university removed in 1871 to larger and more airily situated
buildings in the western part of the city. Round these have grown up, in
the intervening thirty-eight years, new buildings for most of the great
departments of science, including a separate Institute of Natural
Philosophy, which was opened in April 1907, by the Prince and Princess
of Wales.
CHAPTER II
CLASSES AT THE UNIVERSITY OF GLASGOW. FIRST SCIENTIFIC PAPERS
In 1834, that is at the age of ten, William Thomson entered the
University classes. Though small in stature, and youthful even for a
time when mere boys were University students, he soon made himself
conspicuous by his readiness in answering questions, and by his general
proficiency, especially in mathematical and physical studies. The
classes met at that time twice a day--in mathematics once for lecture
and once for oral examination and the working of unseen examples by
students of the class. It is still matter of tradition how, in his
father's class, William was conspicuous for the brilliancy of the work
he did in this second hour. His elder brother James and he seem to have
gone through their University course together. In 1834-5 they were
bracketed third in Latin Prose Composition. In 1835-6 William received a
prize for a vacation exercise--a translation of Lucian's _Dialogues of
the Gods_ "with full parsing of the first three Dialogues." In 1836-7
and 1837-8 the brothers were in the Junior and Senior Mathematical
Classes, and in each year the first and the second place in the
prize-list fell to William and James respectively. In the second of
these years, William appears as second prizeman in the Logic Class,
while James was third, and John Caird (afterwards Principal of the
University) was fifth. William and James Thomson took the first and
second prizes in the Natural Philosophy Class at the close of session
1838-9; and in that year William gained the Class Prize in Astronomy,
and a University Medal for an Essay on the Figure of the Earth. In
1840-1 he appears once more, this time as fifth prizeman in the Senior
Humanity Class.
In his inaugural address as Chancellor of the University, already quoted
above, Lord Kelvin refers to his teachers in Glasgow College in the
following words:
"To this day I look back to William Ramsay's lectures on Roman
Antiquities, and readings of Juvenal and Plautus, as more interesting
than many a good stage play that I have seen in the theatre....
"Greek under Sir Daniel Sandford and Lushington, Logic under Robert
Buchanan, Moral Philosophy under William Fleming, Natural Philosophy and
Astronomy under John Pringle Nichol, Chemistry under Thomas Thomson, a
very advanced teacher and investigator, Natural History under William
Cowper, were, as I can testify by my experience, all made interesting
and valuable to the students of Glasgow University in the thirties and
forties of the nineteenth century....
"My predecessor in the Natural Philosophy chair, Dr. Meikleham, taught
his students reverence for the great French mathematicians Legendre,
Lagrange, and Laplace. His immediate successor in the teaching of the
Natural Philosophy Class,[5] Dr. Nichol, added Fresnel and Fourier to
this list of scientific nobles: and by his own inspiring enthusiasm for
the great French school of mathematical physics, continually manifested
in his experimental and theoretical teaching of the wave theory of light
and of practical astronomy, he largely promoted scientific study and
thorough appreciation of science in the University of Glasgow....
"As far back as 1818 to 1830 Thomas Thomson, the first Professor of
Chemistry in the University of Glasgow, began the systematic teaching of
practical chemistry to students, and, aided by the Faculty of Glasgow
College, which gave the site and the money for the building, realised a
well-equipped laboratory, which preceded, I believe, by some years
Liebig's famous laboratory of Giessen, and was, I believe, the first
established of all the laboratories in the world for chemical research
and the practical instruction of University students in chemistry. That
was at a time when an imperfectly informed public used to regard the
University of Glasgow as a stagnant survival of mediævalism, and used to
call its professors the 'Monks of the Molendinar'!
"The University of Adam Smith, James Watt, and Thomas Reid was never
stagnant. For two centuries and a half it has been very progressive.
Nearly two centuries ago it had a laboratory of human anatomy.
Seventy-five years ago it had the first chemical students' laboratory.
Sixty-five years ago it had the first Professorship of Engineering of
the British Empire. Fifty years ago it had the first physical students'
laboratory--a deserted wine-cellar of an old professorial house,
enlarged a few years later by the annexation of a deserted
examination-room. Thirty-four years ago, when it migrated from its
four-hundred-years-old site off the High Street of Glasgow to this
brighter and airier hill-top, it acquired laboratories of physiology and
zoology; but too small and too meagrely equipped."
In the summer of 1840 Professor James Thomson and his two sons went for
a tour in Germany. It was stipulated that German should be the chief, if
not the only, subject of study during the holidays. But William had just
begun to study Fourier's famous book, _La Théorie Analytique de la
Chaleur_, and took it with him. He read that great work, full as it was
of new theorems and processes of mathematics, with the greatest delight,
and finished it in a fortnight. The result was his first original paper
"On Fourier's Expansions of Functions in Trigonometrical Series," which
is dated "Frankfort, July 1840, and Glasgow, April 1841," and was
published in the _Cambridge Mathematical Journal_ (vol. ii, May 1841).
The object of the paper is to show in what cases a function f(x), which
is to have certain arbitrary values between certain values of x, can be
expanded in a series of sines and when in a series of cosines. The
conclusion come to is that, for assigned limits of x, between 0 and a,
say, and for the assigned values of the function, f(x) can be expressed
either as a series of sines or as a series of cosines. If, however, the
function is to be calculated for any value of x, which lies outside the
limits of that variable between which the values of the function are
assigned, the values of f(x) there are to be found from the expansion
adopted, by rules which are laid down in the paper.
Fourier used sine-expansions or cosine-expansions as it suited him for
the function between the limits, and his results had been pronounced to
be "nearly all erroneous." From this charge of error, which was brought
by a distinguished and experienced mathematician, the young analyst
of sixteen successfully vindicated Fourier's work. Fourier was
incontestably right in holding, though he nowhere directly proved, that
a function given for any value of x between certain limits, could be
expressed either by a sine-series or by a cosine-series. The divergence
of the values of the two expressions takes place outside these limits,
as has been stated above.
The next paper is of the same final date, but appeared in the
_Cambridge Mathematical Journal_ of the following November. In his
treatment of the problem of the cooling of a sphere, given with an
arbitrary initial distribution of temperature symmetrical about the
centre, Fourier assumes that the arbitrary function F(x), which
expresses the temperature at distance x from the centre, can be
expanded in an infinite series of the form
a₁ sin n₁x + a₂ sin n₂x + ...
where a₁, a₂, ... are multipliers to be determined and n₁, n₂, ...
are the roots, infinite in number, of the transcendental equation
(tan nX)⧸nX = 1 － hX.
This equation expresses, according to a particular solution of the
differential equation of the flow of heat in the sphere, the condition
fulfilled at the surface, that the heat reaching the surface by
conduction from the interior in any time is radiated in that time to the
surroundings. Thomson dealt in this second paper with the possibility of
the expansion. He showed that, inasmuch as the first of the roots of the
transcendental equation lies between 0 and 1⧸2, the second between
1 and 3⧸2, the third between 2 and 5⧸2, and so on, with very close
approach to the upper limit as the roots become of high order, the
series assumed as possible has between the given limits of x the same
value as the series
A₁ sin (1⧸2)x + A₂ sin (3⧸2)x + ...
where A₁, A₂, ... are known in terms of a₁, a₂, ... Conversely, any
series of this form is capable of being replaced by a series of the
form assumed. Further, a series of the form just written can be made to
represent any arbitrary system of values between the given limits, and
so the possibility of the expansion is demonstrated.
The next ten papers, with two exceptions, are all on the motion of heat,
and appeared in the _Cambridge Mathematical Journal_ between 1841 and
1843, and deal with important topics suggested by Fourier's treatise. Of
the ideas contained in one or two of them some account will be given
presently.
Fourier's book was called by Clerk Maxwell, himself a man of much
spirituality of feeling, and no mean poet, a great mathematical poem.
Thomson often referred to it in similar terms. The idea of the
mathematician as poet may seem strange to some; but the genius of the
greatest mathematicians is akin to that of the true creative artist, who
is veritably inspired. For such a book was a work of the imagination as
well as of the reason. It contained a new method of analysis applied
with sublime success to the solution of the equations of heat
conduction, an analysis which has since been transferred to other
branches of physical mathematics, and has illuminated them with just
those rays which could reveal the texture and structure of the physical
phenomena. That method and its applications came from Fourier's mind in
full development; he trod unerringly in its use along an almost unknown
path, with pitfalls on every side; and he reached results which have
since been verified by a criticism searching and keen, and lasting from
Fourier's day to ours. The criticism has been minute and logical: it has
not, it is needless to say, been poetical.
Two other great works of his father's collection of mathematical books,
Laplace's _Mécanique Céleste_ and Lagrange's _Mécanique Analytique_,
seem also to have been read about this time, and to have made a deep
impression on the mind of the youthful philosopher. The effect of these
books can be easily traced in Thomson and Tail's _Natural Philosophy_.
The study of Fourier had a profound influence on Thomson's future work,
an influence which has extended to his latest writings on the theory of
certain kinds of waves. His treatment is founded on a strikingly
original use of a peculiar form of solution (given by Fourier) of a
certain fundamental differential equation in the theory of the flow of
heat. It is probable that William Thomson's earliest predilections as
regards study were in the direction of mathematics rather than of
physics. But the studies of the young mathematician, for such in a very
real and high sense he had become, were widened and deepened by the
interest in physical things and their explanation aroused by the
lectures of Meikleham, then Professor of Natural Philosophy, and
especially (as Lord Kelvin testified in his inaugural address as
Chancellor) by the teaching of J. P. Nichol, the Professor of Astronomy,
a man of poetical imagination and of great gifts of vivid and clear
exposition.
The _Cyclopædia of Physical Science_ which Dr. Nichol published is
little known now; but the first edition, published in 1857, to which
Thomson contributed several articles, including a sketch of
thermodynamics, contained much that was new and stimulating to the
student of natural philosophy, and some idea of the accomplishments of
its compiler and author can be gathered from its perusal. De Morgan's
_Differential and Integral Calculus_ was a favourite book in Thomson's
student days, and later when he was at Cambridge, and he delighted to
pore over its pages before the fire when the work of the day was over.
Long after, he paid a grateful tribute to De Morgan and his great work,
in the Presidential Address to the British Association at its Edinburgh
Meeting in 1870.
The next paper which Thomson published, after the two of which a sketch
has been given above, was entitled "The Uniform Motion of Heat in
Homogeneous Solid Bodies, and its Connection with the Mathematical
Theory of Electricity." It is dated "Lamlash, August 1841," so that it
followed the first two at an interval of only four months. It appeared
in the _Cambridge Mathematical Journal_ in February 1842, and is
republished in the "Reprint of Papers on Electrostatics and Magnetism."
It will always be a noteworthy paper in the history of physical
mathematics. For although, for the most part, only known theorems
regarding the conduction of heat were discussed, an analogy was pointed
out between the distribution of lines of flow and surfaces of equal
temperature in a solid and unequally heated body, with sources of heat
in its interior, and the arrangement of lines of forces and
equipotential surfaces in an insulating medium surrounding electrified
bodies, which correspond to the sources of heat in the thermal case. The
distribution of lines of force in a space filled with insulating media
of different inductive qualities was shown to be precisely analogous to
that of lines of flow of heat in a corresponding arrangement of media of
different heat-conducting powers. So the whole analysis and system of
solutions in the thermal case could be at once transferred to the
electrical one. The idea of the "conduction of lines of force," as
Faraday first and Thomson afterwards called it, was further developed in
subsequent papers, and threw light on the whole subject of electrostatic
force in the "field" surrounding an electric distribution. Moreover, it
made the subject definite and quantitative, and not only gave a guide to
the interpretation of unexplained facts, but opened a way to new
theorems and to further investigation.
This paper contains the extremely important theorem of the equivalence,
so far as external field is concerned, of any distribution of
electricity and a certain definite distribution, over any equipotential
surface, of a quantity equal to that contained within the surface. But
this general theorem and others contained in the paper had been
anticipated in Green's "Essay on the Application of Mathematical
Analysis to the Theories of Electricity and Magnetism," in memoirs by
Chasles in Liouville's Journal (vols. iii and v), and in the celebrated
memoir by Gauss "On General Theorems relating to Attractive and
Repulsive Forces varying inversely as the Square of the Distance,"
published in German in Leipzig in 1840, and in English in Taylor's
_Scientific Memoirs_ in 1842. These anticipations are again referred to
below.
CHAPTER III
UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS UNDERGRADUATE
Thomson entered at St. Peter's College, Cambridge, in October 1841, and
began the course of study then in vogue for mathematical honours. At
that time, as always down almost to the present day, everything depended
on the choice of a private tutor or "coach," and the devotion of the
pupil to his directions, and on adherence to the subjects of the
programme. His private tutor was William Hopkins, "best of all private
tutors," one of the most eminent of his pupils called him, a man of
great attainment and of distinction as an original investigator in a
subject which had always deeply interested Thomson--the internal
rigidity of the earth. But the curriculum for the tripos did not exhaust
Thomson's energy, nor was it possible to keep him entirely to the groove
of mastering and writing out book-work, and to the solution of problems
of the kind dear to the heart of the mathematical examiner. He wrote
original articles for the _Cambridge Mathematical Journal_, on points in
pure and in applied mathematics, and read mathematical books altogether
outside the scope of the tripos. Nor did he neglect athletic exercises
and amusements; he won the Colquhoun Sculls as an oarsman, and was an
active member, and later, during his residence at Cambridge, president
of the C.U.M.S., the Cambridge University Musical Society.[6] The
musical instruments he favoured were the cornet and especially the
French horn--he was second horn in the original Peterhouse band--but
nothing seems to be on record as to the difficulties or incidents of his
practice! Long afterwards, in a few extremely interesting lectures which
he gave annually on sound, he discoursed on the vibrations of columns of
air in wind instruments, and sometimes illustrated his remarks by
showing how notes were varied in pitch on the old-fashioned French horn,
played with the hand in the bell, a performance which always intensely
delighted the Natural Philosophy Class.
At the Jubilee commemoration of the society, 1893, Lord Kelvin recalled
that Mendelssohn, Weber and Beethoven were the "gods" of the infant
association. Those of his pupils who came more intimately in contact
with him will remember his keen admiration for these and other great
composers, especially Bach, Mozart, and Beethoven, and his delight in
hearing their works. The Waldstein sonata was a special favourite. It
has been remarked before now, and it seems to be true, that the music of
Bach and Beethoven has had special attractions for many great
mathematicians.
At Cambridge Thomson made the acquaintance of George Gabriel Stokes, who
graduated as Senior Wrangler and First Smith's Prizeman in 1841, and
eight years later became Lucasian Professor of Mathematics in the
University of Cambridge. Their acquaintance soon ripened into a close
friendship, which lasted until the death of Stokes in 1903. The Senior
Wrangler and the Peterhouse Undergraduate undertook the composition of a
series of notes and papers on points in pure and physical mathematics
which required clearing up, or putting in a new point of view; and so
began a life-long intercourse and correspondence which was of great
value to science.
Thomson's papers of this period are on a considerable variety of
subjects, including his favourite subject of the flux of heat. There are
sixteen in all that seem to have been written and published during his
undergraduate residence at Cambridge. Most of them appeared in the
_Cambridge Mathematical Journal_ between 1842 and 1845; but three
appeared in 1845 in Liouville's _Journal de Mathématiques_. Four are on
subjects of pure mathematics, such as Dupin's theorem regarding lines of
curvature of orthogonally intersecting surfaces, the reduction of the
general equation of surfaces of the second order (now called second
degree), six are on various subjects of the theory of heat, one is on
attractions, five are on electrical theory, and one is on the law of
gravity at the surface of a revolving homogeneous fluid. It is
impossible to give an account of all these papers here. Some of them are
new presentations or new proofs of known theorems, one or two are fresh
and clear statements of fundamental principles to be used later as the
foundation of more complete statements of mathematical theory; but all
are marked by clearness and vigour of treatment.
Another paper, published in the form of a letter, of date October 8,
1845, to M. Liouville, and published in the _Journal de Mathématiques_
in the same year, indicates that either before or shortly after taking
his degree, Thomson had invented his celebrated method of "Electric
Images" for the solution of problems of electric distribution. Of this
method, which is one of the most elegant in the whole range of physical
mathematics, and solves at a stroke some problems, otherwise almost
intractable, we shall give some account in the following chapter.
This record of work is prodigious for a student reading for the
mathematical tripos; and it is somewhat of an irony of fate that such
scientific activity is, on the whole, rather a hindrance than a help in
the preparation for that elaborate ordeal of examination. Great
expectations had been formed regarding Thomson's performance; hardly
ever before had a candidate appeared who had done so much and so
brilliant original work, and there was little doubt that he would be
easily first in any contest involving real mathematical power, that is,
ability to deal with new problems and to express new relations of facts
in mathematical language. But the tripos was not a test of power merely;
it was a test also of acquisition, and, to candidates fairly equal in
this respect, also of memory and of quickness of reproduction on paper
of acquired knowledge.
The moderators on the occasion were Robert Leslie Ellis and Harvey
Goodwin, both distinguished men. Ellis had been Senior Wrangler and
first Smith's Prizeman a few years before, and was a mathematician of
original power and promise, who had already written memoirs of great
merit. Goodwin had been Second Wrangler when Ellis was Senior, and
became known to a later generation as Bishop of Carlisle. In a life of
Ellis prefixed to a volume of his collected papers, Goodwin says:--"It
was in this year that Professor W. Thomson took his degree; great
expectations had been excited concerning him, and I remember Ellis
remarking to me, with a smile, 'You and I are just about fit to mend his
pens.'" Surely never was higher tribute paid to candidate by examiner!
Another story, which, however, does not seem capable of such complete
authentication, is told of the same examination, or it may be of the
Smith's Prize Examination which followed. A certain problem was solved,
so it is said, in practically identical terms by both the First and
Second Wranglers. The examiners remarked the coincidence, and were
curious as to its origin. On being asked regarding it, the Senior
Wrangler replied that he had seen the solution he gave in a paper which
had appeared in a recent number of the _Cambridge Mathematical Journal_;
Thomson's answer was that he was the author of the paper in question!
Thomson was Second Wrangler, and Parkinson, of St. John's College,
afterwards. Dr. Parkinson, tutor of St. John's and author of various
mathematical text-books, was Senior. These positions were reversed in
the examination for Smith's Prizes, which was very generally regarded as
a better test of original ability than the tripos, so that the temporary
disappointment of Thomson's friends was quickly forgotten in this higher
success.
The Tripos Examination was held in the early part of January. On the
25th of that month Thomson met his private tutor Hopkins in the "Senior
Wranglers' Walk" at Cambridge, and in the course of conversation
referred to his desire to obtain a copy of Green's 'Essay' (supra, p.
21). Hopkins at once took him to the rooms where he had attended almost
daily for a considerable time as a pupil, and produced no less than
three copies of the Essay, and gave him one of them. A hasty perusal
showed Thomson that all the general theorems of attractions contained in
his paper "On the Uniform Motion of Heat," etc., as well as those of
Gauss and Chasles, had been set forth by Green and were derivable from a
general theorem of analysis whereby a certain integral taken throughout
a space bounded by surfaces fulfilling a certain condition is expressed
as two integrals, one taken throughout the space, the other taken over
the bounding surface or surfaces.
It has been stated in the last chapter that Thomson had established, as
a deduction from the flow of heat in a uniform solid from sources
distributed within it, the remarkable theorem of the replacement,
without alteration of the external flow, of these sources by a certain
distribution over any surface of uniform temperature, and had pointed
out the analogue of this theorem in electricity. This method of proof
was perfectly original and had not been anticipated, though the theorem,
as has been stated, had already been given by Green and by Gauss. In the
paper entitled "Propositions in the Theory of Attraction," published in
the _Cambridge Mathematical Journal_ in November 1842, Thomson gave an
analytical proof of this great theorem, but afterwards found that this
had been done almost contemporaneously by Sturm in Liouville's Journal.
Soon after the Tripos and Smith's Prize Examinations were over, Thomson
went to London, and visited Faraday in his laboratory in the Royal
Institution. Then he went on to Paris with his friend Hugh Blackburn,
and spent the summer working in Regnault's famous laboratory, making the
acquaintance of Liouville, Sturm, Chasles, and other French
mathematicians of the time, and attending meetings of the Académie des
Sciences. He made known to the mathematicians of Paris Green's 'Essay,'
and the treasures it contained, and frequently told in after years with
what astonishment its results were received. He used to relate that one
day, while he and Blackburn sat in their rooms, they heard some one come
panting up the stair. Sturm burst in upon them in great excitement, and
exclaimed, "_Vous avez un Mèmoire de Green! M. Liouville me l'a dit._"
He sat down and turned over the pages of the 'Essay,' looking at one
result after another, until he came to a complete anticipation of his
proof of the replacement theorem. He jumped up, pointed to the page, and
cried out, "_Voila mon affaire!_"
To this visit to Paris Thomson often referred in later life with
grateful recognition of Regnault's kindness, and admiration of his
wonderful experimental skill. The great experimentalist was then engaged
in his researches on the thermal constants of bodies, with the elaborate
apparatus which he designed for himself, and with which he was supplied
by the wise liberality of the French Government. This initiation into
laboratory work bore fruit not long after in the establishment of the
Glasgow Physical Laboratory, the first physical laboratory for students
in this country.
It is a striking testimony to Thomson's genius that, at the age of only
seventeen, he had arrived at such a fundamental and general theorem of
attractions, and had pointed out its applications to electrical theory.
And it is also very remarkable that the theorem should have been proved
within an interval of two or three years by three different authors, two
of them--Sturm and Gauss--already famous as mathematicians. Green's
treatment of the subject was, however, the most general and
far-reaching, for, as has been stated, the theorem of Gauss, Sturm, and
Thomson was merely a particular case of a general theorem of analysis
contained in Green's 'Essay.' It has been said in jest, but not without
truth, that physical mathematics is made up of continued applications of
Green's theorem. Of this enormously powerful relation, a more lately
discovered result, which is very fundamental in the theory of functions
of a complex variable, and which is generally quoted as Riemann's
theorem, is only a particular case.
Thomson had the greatest reverence for the genius of Green, and found in
his memoirs, and in those of Cauchy on wave propagation, the inspiration
for much of his own later work.[7] In 1850 he obtained the
republication of Green's 'Essay' in Crelle's Journal; in later years he
frequently expressed regret that it had not been published in England.
In the commencement of 1845 Thomson told Liouville of the method of
_Electric Images_ which he had discovered for the solution of problems
of electric distribution. On October 8, 1845, after his return to
Cambridge, he wrote to Liouville a short account of the results of the
method in a number of different cases, and in two letters written on
June 26 and September 16 of the following year, he stated some further
results, including the solution of the problem of the distribution upon
a spherical bowl (a segment of a spherical conducting shell made by a
plane section) insulated and electrified. This last very remarkable
result was given without proof, and remained unproved until Thomson
published his demonstration twenty-three years later in the
_Philosophical Magazine_.[8] This had been preceded by a series of
papers in March, May, and November 1848, November 1849, and February
1850, in the _Cambridge and Dublin Mathematical Journal_, on various
parts of the mathematical theory of electricity in equilibrium,[9] in
which the theory of images is dealt with. The letters to Liouville
promptly appeared in the Journal, and the veteran analyst wrote a long
Note on their subject, which concludes as follows: "Mon but sera rempli,
je le répéte, s'ils [ces développements] peuvent aider à bien faire
comprendre la haute importance du travail de ce jeune géomètre, et si M.
Thomson lui-même veut bien y voir une preuve nouvelle de l'amitié que je
lui porte et de l'estime qui j'ai pour son talent."
The method of images may be regarded as a development in a particular
direction of the paper "On the Uniform Motion of Heat" already referred
to, and, taken along with this latter paper, forms the most striking
indication afforded by the whole range of Thomson's earlier work of the
strength and originality of his mathematical genius. Accordingly a
chapter is here devoted to a more complete explanation of the first
paper and the developments which flowed from it. The general reader may
pass over the chapter, and return to it from time to time as he finds
opportunity, until it is completely understood.
CHAPTER IV
THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. ELECTRIC IMAGES.
ELECTRIC INVERSION
In describing Thomson's early electrical researches we shall not enter
into detailed calculations, but merely explain the methods employed. The
meaning of certain technical terms may be recalled in the first place.
The whole space in which a distribution of electricity produces any
action on electrified bodies is called the _electrical field_ of the
distribution. The force exerted on a very small insulated trial
conductor, on which is an electric charge of amount equal to that taken
as the unit quantity of electricity, measures the _field-intensity_ at
any point at which the conductor is placed. The direction of the
field-intensity at the point is that in which the small conductor is
there urged. If the charge on the small conductor were a negative unit,
instead of a positive, the direction of the force would be reversed; the
magnitude of the force would remain the same. To make the
field-intensity quite definite, a positive unit is chosen for its
specification. For a charge on the trial-conductor consisting of any
number of units, the force is that number of times the field-intensity.
The field-intensity is often specified by its components, X, Y, Z in
three chosen directions at right angles to one another.
Now in all cases in which the action, whether attraction or repulsion,
between two unit quantities of matter concentrated at points is
inversely as the square of the distance between the charges, the
field-intensity, or its components, can be found from a certain function
V of the charges forming the acting distribution [which is always
capable of being regarded for mathematical purposes as a system of small
charges existing at points of space, _point-charges_ we shall call
them], their positions, and the position of the point at which the
field-intensity is to be found. If q₁, q₂, ... be the point-charges, and
be positive when the charges are positive and negative when the charges
are negative, and r₁, r₂, ... be their distances from the point P,
V is q₁⧸r₁ + q₂⧸r₂ + ... The field-intensity is the rate of diminution
of the value of V at P, taken along the specified direction. The three
gradients parallel to the three chosen coordinate directions are
X, Y, Z; but for their calculation it is necessary to insert the values
of r₁, r₂, ... in terms of the coordinates which specify the positions
of the point-charges, and the coordinates x, y, z which specify the
position of P. Once this is done, X, Y, Z are obtained by a simple
systematic process of calculation, namely, differentiation of the
function V with respect to x, y, z.
This function V seems to have been first used by Laplace for
gravitational matter in the _Mécanique Céleste_; its importance for
electricity and magnetism was recognised by Green, who named it the
potential. It has an important physical signification. It represents the
work which would have to be done to bring a unit of positive
electricity, against the electrical repulsion of the distribution, up to
the point P from a point at an infinite distance from every part of the
distribution; or, in other words, what we now call the _potential
energy_ of a charge q situated at P is qV. The excess of the potential
at P, over the potential at any other point Q in the field, is the work
which must be spent in carrying a positive unit from Q to P against
electrical repulsion. Of course, if the force to be overcome from Q to P
is on the whole an attraction, work has not been spent in effecting the
transference, but gained by allowing it to take place. The difference of
potential is then negative, that is, the potential of Q is higher than
that of P.
The difference of potential depends only on the points P and Q, and not
at all on the path pursued between them. Thus, if a unit of electricity
be carried from P to Q by any path, and back by any other, no work is
done on the whole by the agent carrying the unit. This simple fact
precludes the possibility of obtaining a so-called perpetual motion (a
self-acting machine doing useful work) by means of electrical action.
The same thing is true _mutatis mutandis_ of gravitational action.
In the thermal analogy explained by Thomson in his first paper, the
positive point-charges are point-sources of heat, which is there poured
at constant rate into the medium (supposed of uniform quality) to be
drawn off in part from the medium at constant rate where there are sinks
(or negative sources),--the negative point-charges in the electrical
case,--while the remainder is conducted away to more and more distant
parts of the conducting medium supposed infinitely extended. Whenever a
point-source, or a point-sink, exists at a distance from other sources
or sinks, the flow in the vicinity is in straight lines from or to the
point, and these straight lines would be indefinitely extended if either
source or sink existed by itself. As it is, the direction and amount of
flow everywhere depends on the flow resulting from the whole arrangement
of sources and sinks. Lines can be drawn in the medium which show the
direction of the resultant flow from point to point, and these lines of
flow can be so spaced as to indicate, by their closeness together or
their distance apart, where the rate of flow is greater or smaller; and
such lines start from sources, and either end in sinks or continue their
course to infinity. In the electrical case these lines are the analogues
of the lines of electric force (or field-intensity) in the insulating
medium, which start from positive charges and end in negative, or are
prolonged to infinity.
Across such lines of flow can be drawn a family of surfaces, to each of
which the lines met by the surface are perpendicular. These surfaces are
the equitemperature surfaces, or, as they are usually called, the
isothermal surfaces. They can be drawn more closely crowded together, or
more widely separated, so as to indicate where the rate of falling off
of temperature (the "temperature slope") is greater or less, just as the
contour lines in a map show the slopes on a hill-side.
Instead of the thermal analogy might have been used equally well that of
steady flow in an indefinitely extended mass of homogeneous frictionless
and incompressible fluid, into which fluid is being poured at a constant
rate by sources and withdrawn by sinks. The isothermal surfaces are
replaced by surfaces of equal pressure, while lines of flow in one are
also lines of flow in the other.
Now let heat be poured into the medium at constant rate by a single
point-source P (Fig. 1), and drawn off at a smaller rate by a single
point-sink P', while the remainder flows to more and more remote parts
of the medium, supposed infinite in extent in every direction. After a
sufficient time from the beginning of the flow a definite system of
lines of flow and isothermal surfaces can be traced for this case in the
manner described above. One of the isothermal surfaces will be a sphere
S surrounding the sink, which, however, will not be at the centre of the
sphere, but so situated that the source, sink, and centre are in line,
and that the radius of the sphere is a mean proportional between the
distances of the source and sink from the centre. If a be the radius of
the sphere and f the distance of the source from the centre of the
sphere, the heat carried off by the sink is the fraction a⧸f of that
given out by the source.
[Illustration: FIG. 1.]
In the electrical analogue, the source and sink are respectively a
point-charge and what is called the "electric image" of that charge with
respect to the sphere, which is in this case an equipotential surface.
And just as the lines of flow of heat meet the spherical isothermal
surface at right angles, so the lines of force in the electrical case
meet the equipotential surface also at right angles. Now obviously in
the thermal case a spherical sink could be arranged coinciding with the
spherical surface so as to receive the flow there arriving and carry
off the heat from the medium, without in the least disturbing the flow
outside the sphere. The whole amount of heat arriving would be the same:
the amount received per unit area at any point on the sphere would
evidently be proportional to the gradient of temperature there towards
the surface. Of course the same thing could be done at any isothermal
surface, and the same proportionality would hold in that case.
Similarly the source could be replaced by a surface-distribution of
sources over any surrounding isothermal surface; and the condition to be
fulfilled in that case would be that the amount of heat given out per
unit area anywhere should be exactly that which flows out along the
lines of flow there in the actual case. Outside the surface the field of
flow would not be affected by this replacement. It is obvious that in
this case the outflow per unit area must be proportional to the
temperature slope outward from the surface.
The same statements hold for any complex system of sources and sinks.
There must be the same outflow from the isothermal surface or inflow
towards it, as there is in the actual case, and the proportionality to
temperature slope must hold.
This is exactly analogous to the replacement by a distribution on an
equipotential surface of the electrical charge or charges within the
surface, by a distribution over the surface, with fulfilment of
Coulomb's theorem (p. 43 below) at the surface. Thomson's paper on the
"Uniform Motion of Heat" gave an intuitive proof of this great theorem
of electrostatics, which the statements above may help to make clear to
those who have, or are willing to acquire, some elementary knowledge of
electricity.
Returning to the distribution on any isothermal surface surrounding the
sink (or sinks) we see that it represents a surface-sink in equilibrium
with the flow in the field. The distribution on a metal shell,
coinciding with the surface, which keeps the surface at a potential
which is the analogue of the temperature at the isothermal surface,
while the shell is under the influence of a point-charge of
electricity--the analogue of the thermal source--is the distribution as
affected by the induction of the point-charge. If the shell coincide
with the spherical equipotential surface referred to above, and the
distribution given by the theorem of replacement be made upon it, the
shell will be at zero potential, and the charge will be that which would
exist if the shell were uninsulated, that is, the "induced charge."
The consideration of the following simple problem will serve to make
clear the meaning of an electric image, and form a suitable introduction
to a description of the application of the method to the electrification
of spherical surfaces. Imagine a very large plane sheet of tinfoil
connected by a conducting wire with the earth. If there are no
electrified bodies near, the sheet will be unelectrified. But let a very
small metallic ball with a charge of positive electricity upon it be
brought moderately close to one face of the tinfoil. The tinfoil will be
electrified negatively by induction, and the distribution of the
negative charge will depend on the position of the ball. Now, it can be
shown that the field of electric force, on the same side of the tinfoil
as the ball, is precisely the same as would be produced if the foil (and
everything behind it) were removed, and an equal negative charge of
electricity placed behind the tinfoil on the prolonged perpendicular
from the ball to the foil, and as far from the foil behind as the ball
is from it in front. Such a negative charge behind the tinfoil sheet is
called an electric image of the positive charge in front. It is
situated, as will be seen at what would be, if the tinfoil were a
mirror, the optical image of the ball in the mirror.
[Illustration: FIG. 2.]
Now, suppose a second very large sheet of tinfoil to be placed parallel
to the first sheet, so that the small electrified sphere is between the
two sheets, and that this second sheet is also connected to the earth.
The charge on the ball induces negative electricity on both sheets, but
besides this each sheet by its charge influences the other. The problem
of distribution is much more complicated than in the case of a single
sheet, but its solution is capable of very simple statement. Let us call
the two sheets A and B (Fig. 2), and regard them for the moment as
mirrors. A first image of an object P between the two mirrors is
produced directly by each, but the image I₁ in A is virtually an object
in front of B, and the image J₁ in B an object in front of A, so that
a second image more remote from the mirror than the first is produced in
each case. These second images I₂ and J₂ in the same way produce third
images still more remote, and so on. The positions are determined just
as for an object and a single mirror. There is thus an infinite trail of
images behind each mirror, the places of which any one can assign.
[Illustration: FIG. 3.]
Every one may see the realisation of this arrangement in a shop window,
the two sides of which are covered by parallel sheets of mirror-glass.
An infinite succession of the objects in the window is apparently seen
on both sides. When the objects displayed are glittering new bicycles in
a row the effect is very striking; but what we are concerned with here
is a single small object like the little ball, and its two trails of
images. The electric force at any point between the two sheets of
tinfoil is exactly the same as if the sheets were removed and charges
alternately negative and positive were placed at the image-points,
negative at the first images, positive at the second images, and so on,
each charge being the same in amount as that on the ball. We have an
"electric kaleidoscope" with parallel mirrors. When the angle between
the conducting planes is an aliquot part of 360°, let us say 60°, the
electrified point and the images are situated, just as are the object
and its image in Brewster's kaleidoscope, namely at the angular points
of a hexagon, the sides of which are alternately (as shown in Fig. 3) of
lengths twice the distance of the electrified point from A and from B.
[Illustration: FIG. 4.]
Now consider the spherical surface referred to at p. 37, which is kept
at uniform potential by a charge at the external point P, and a charge
q' at the inverse point P' within the sphere. If E (Fig. 4) be any point
whatever on the surface, and r, r' be its distances from P and P', it is
easy to prove by geometry that the two triangles CPE and CEP' are
similar, and therefore r' = ra⧸f. [Here a⧸f is used to mean a divided
by f. The mark ⧸ is adopted instead of the usual bar of the fraction,
for convenience of printing.] Now, by the explanation given above, the
potential produced at any point by a charge q at another point, is equal
to the ratio of the charge q to the distance between the points. Thus
the potential at E due to the charge q at P is q⧸r, and that at E due to
a charge q' at P' is q'⧸r'. Thus if q' = －qa⧸f, q' at P' will produce
a potential at E = －qa⧸fr' = －q⧸r, by the value of r. Hence q at P
and -qa⧸f at P' coexisting will give potential q⧸r + －q⧸r or zero,
at E. Thus the charge －qa⧸f, at the internal point P' will in presence
of +q at P keep all points of the spherical surface at zero potential.
These two charges represent the source and sink in the thermal analogue
of p. 37 above.
Now replace S by a spherical shell of metal connected to the earth
by a long fine wire, and imagine all other conductors to be at a great
distance from it. If this be under the influence of the charge q at P
alone, a charge is induced upon it which, in presence of P, maintains
it at zero potential. The internal charge －qa⧸f, and the induced
distribution on the shell are thus equivalent as regards the potential
produced by either at the spherical surface; for each counteracts then
the potential produced by q at P. But it can be proved that if a
distribution over an equipotential surface can be made to produce the
same potential over that surface as a given internal distribution does,
they produce the same potentials at all external points, or, as it is
usually put, the external fields are the same. This is part of the
statement of what has been called the "theorem of replacement"
discovered by Green, Gauss, Thomson, and Chasles as described above.
Another part of the statement of the theorem may now be formulated.
Coulomb showed long ago that the surface-density of electricity at any
point on a conductor is proportional to the resultant field-intensity
just outside the surface at that point. Since the surface is throughout
at one potential this intensity is normal to the surface. Let it be
denoted by N, and s be the surface-density: then according to the
system of units usually adopted 4πs = N.
Let now the rate of diminution of potential per unit of distance
outwards (or downward gradient of potential) from the equipotential
surface be determined for every point of the surface, and let
electricity be distributed over the surface, so that the amount per unit
area at each point (the surface-density) is made numerically equal to
the gradient there divided by 4π. This, by Coulomb's law, stated
above, gives that field-intensity just outside the surface which exists
for the actual distribution, and therefore, as can be proved, gives the
same field everywhere else outside the surface. The external fields will
therefore be equivalent, and further, the amount of electricity on the
surface will be the same as that situated within it in the actual
distribution.
Thus it is only necessary to find for －qa⧸f at P' and q at P, the
falling off gradient N of potential outside the spherical surface at
any point E, and to take N⧸4π, to obtain s the surface-density at E.
Calculation of this gradient for the sphere gives 4πs = －q(f² － a²)⧸ar³.
The surface-density is thus inversely as the cube of the distance PE.
If the influencing point P be situated within the spherical shell, and
the shell be connected to earth as before, the induced distribution
will be on its interior surface. The corresponding point P will now
be outside, but given by the same relation. And a will now be greater
than f, and the density will be given by 4πs = －q(a² － f²)⧸ar³,
where, f and r have the same meanings with regard to E and P
as before.
P' is in each case called the image of P in the sphere S, and the
charge －qa⧸f there supposed situated is the _electric image_ of the
charge q at P. It will be seen that an electric image is a charge, or
system of charges, on one side of an electrified surface which produces
on the other side of that surface the same electrical field as is
produced by the actual electrification of the surface.
While by the theorem of replacement there is only one distribution over
a surface which produces at all points on one side of a surface the same
field as does a distribution D on the other side of the surface, this
surface distribution may be equivalent to several different arrangements
of D. Thus the point-charge at P' is only one of various
image-distributions equivalent to the surface-distribution in the sense
explained. For example, a uniform distribution over any spherical
surface with centre at P' (Fig. 4) would do as well, provided this
spherical surface were not large enough to extend beyond the surface S.
In order to find the potential of the sphere (Fig. 4) when insulated
with a charge Q upon it, in presence of the influencing charge q at the
external point P, it is only necessary to imagine uniformly distributed
over the sphere, already electrified in the manner just explained, the
charge Q + aq⧸f. Then the whole charge will be Q, and the uniformity of
distribution will be disturbed, as required by the action of the
influencing point-charge. The potential will be Q⧸a + q⧸f. For a
given potential V of the sphere, the total charge is aV － aq⧸f,
that is the charge is aV over and above the induced charge.
If instead of a single influencing point-charge at P there be a system
of influencing point-charges at different external points, each of these
has an image-charge to be found in amount and situation by the method
just described, and the induced distribution is that obtained by
superimposing all the surface distributions found for the different
influencing points.
The force of repulsion between the point-charge q and the sphere
(with total charge Q) can be found at once by calculating the sum
of the forces between q at P and the charges Q + aq⧸f at C
and －aq⧸f at P'.
This can be found also by calculating the energy of the system, which
will be found to consist of three terms, one representing the energy of
the sphere with charge Q uninfluenced by an external charge, one
representing the energy on a small conductor (not a point) at P existing
alone, and a third representing the mutual energy of the electrification
on the sphere and the charge q at P existing in presence of one another.
By a known theorem the energy of a system of conductors is one half of
the sum obtained by multiplying the potential of each conductor by its
charge and adding the products together. It is only necessary then to
find the variation of the last term caused by increasing f by a small
amount df. This will be the product F.df of the force F required and the
displacement.
Either method may be applied to find the forces of attraction and
repulsion for the systems of electrified spheres described below.
The problem of two mutually influencing non-intersecting spheres, S₁, S₂
(Fig. 5), insulated with given charges, q₁, q₂, may now be dealt with in
the following manner. Let each be supposed at first charged uniformly.
By the known theorem referred to above, the external field of each is
the same as if its whole charge were situated at the centre. Now if the
distribution on S₂, say, be kept unaltered, while that on S₁ is allowed
to change, the action of S₂ on S₁ is the same as if the charge q₂ were
at the centre C₂ of S₂. Thus if f be the distance between the centres
C₁, C₂, and a₁ be the radius of S₁, the distribution will be that
corresponding to q₁ + a₁q₂⧸f uniformly distributed on S₁ together with
the induced charge －a₁q₂⧸f, which corresponds to the image-charge at
the point I₁ (within S₁), the inverse of C₂ with respect to S₁. Now
let the charge on S₁ be fixed in the state just supposed while that
on S₂ is freed. The charge on S₂ will rearrange itself under the
influence of q₁ + a₁q₂⧸f ( = q') and －a₁q₂⧸f, considered as at C₁
and I₁ respectively. The former of these will give a distribution
equivalent to q₂ + a₂q'⧸f uniformly distributed over S₂, and an
induced distribution of amount a₂q'⧸f at J₁, the inverse point of C₁
with regard to S₂. The image-charge －a₁q₂⧸f at I₁ in S₁ will react
on S₂ and give an induced distribution －a₂(－a₁q₂⧸f)f', (I₁C₂ = f')
corresponding to an image-charge a₂a₁q₂⧸ff' at the inverse point J₂
of P₁ with respect to C₂S₂. Thus the distribution on S₂ is equivalent
to q₂ + a₂q'⧸f － a₂a₁q₂⧸ff' distributed uniformly over it, together with
the two induced distributions just described.
[Illustration: FIG. 5.]
In the same way these two induced distributions on S₂ may now be
regarded as reacting on the distribution on S₁ as would point-charges
－a₂q₁⧸f and a₂a₁q₂⧸ff', situated at J₁ and J₂ respectively, and would
give two induced distributions on S₁ corresponding to their images
in S₁.
Thus by partial influences in unending succession the equilibrium state
of the two spheres could be approximated to as nearly as may be desired.
An infinite trail of electric images within each of the two spheres is
thus obtained, and the final state of each conductor can be calculated
by summation of the effects of each set of images.
If the final potentials, V₁, V₂, say, of the spheres are given the
process is somewhat simpler. Let first the charges be supposed to
exist uniformly distributed over each sphere, and to be of amount a₁V₁,
a₂V₂ in the two cases. The uniform distribution on S₁ will raise the
potential of S₂ above V₂, and to bring the potential down to V₂ in
presence of this distribution we must place an induced distribution
over S₂, represented as regards the external field by the image-charge
－a₂a₁V₁⧸f (at the image of C₁ in S₂) where f is the distance
between the centres. The charge a₂V₂ on S₂ will similarly have an
action on S₁ to be compensated in the same way by an image-charge
－a₁a₂V₂⧸f at the image of C₂ in S₁. Now these two image-charges
will react on the spheres S₁ and S₂ respectively, and will have to be
balanced by induced distributions represented by second image-charges,
to be found in the manner just exemplified. These will again react on
the spheres and will have to be compensated as before, and so on
indefinitely. The charges diminish in amount, and their positions
approximate more and more, according to definite laws, and the final
state is to be found by summation as before.
The force of repulsion is to be found by summing the forces between all
the different pairs of charges which can be formed by taking one charge
of each system at its proper point: or it can be obtained by calculating
the energy of the system.
The method of successive influences was given originally by Murphy, but
the mode of representing the effects of the successive induced charges
by image-charges is due to Thomson. Quite another solution of this
problem is, however, possible by Thomson's method of electrical
inversion.
A similar process to that just explained for two charged and mutually
influencing spheres will give the distribution on two concentric
conducting spheres, under the influence of a point-charge q at P between
the inner surface of the outer and the outer surface of the inner, as
shown in Fig. 7. There the influence of q at P, and of the induced
distributions on one another, is represented by two series of images,
one within the inner sphere and one outside the outer. These charges and
positions can be calculated from the result for a single sphere and
point-charge.
Thomson's method of electrical inversion, referred to above, enabled the
solutions of unsolved problems to be inferred from known solutions of
simpler cases of distribution. We give here a brief account of the
method, and some of its results. First we have to recall the meaning of
geometrical inversion. In Fig. 6 the distances OP, OP, OQ, OQ' fulfil
the relation OP.OP' = OQ.OQ' = a². Thus P' is (see p. 37) the inverse
of the point P with respect to a sphere of radius a and centre O
(indicated by the dotted line in Fig. 6), and similarly Q' is the
inverse of Q with respect to the same sphere and centre. O is called the
centre of inversion, and the sphere of radius a is called the sphere of
inversion. Thus the sphere of Figs. 1 and 4 is the sphere of inversion
for the points P and P', which are inverse points of one another. For
any system of points P, Q, ..., another system P', Q', ... of inverse
points can be found, and if the first system form a definite locus, the
second will form a derived locus, which is called the inverse of the
former. Also if P', Q', ... be regarded as the direct system, P, Q, ...
will be the corresponding inverse system with regard to the same sphere
and centre. P' is the image of P, and P is the image of P', and so on,
with regard to the same sphere and centre of inversion.
[Illustration: FIG. 6.]
The inverse of a circle is another circle, and therefore the inverse of
a sphere is another sphere, and the inverse of a straight line is a
circle passing through the centre of inversion, and of an infinite plane
a sphere passing through the centre of inversion. Obviously the inverse
of a sphere concentric with the sphere of inversion is a concentric
sphere.
The line P'Q' is of course not the inverse of the line PQ, which has
for its inverse the circle passing through the three points O, P', Q',
as indicated in Fig. 6.
The following results are easily proved.
A locus and its inverse cut any line OP at the same angle.
To a system of point-charges q₁, q₂, ... at points P₁, P₂, ... on
one side of the surface of the sphere of inversion there is a system
of charges aq₁⧸f₁, aq₂⧸f₂, ... on the other side of the spherical
surface [OP₁ = f₁, OP₂ = f₂]. This inverse system, as we shall call
it, produces the same potential at any point of the sphere of inversion,
as does the direct system from which it is derived.
If V, V' be the potentials produced by the whole direct system at Q,
and by the whole inverse system at Q', V'⧸V = r⧸a = a⧸r', where OQ = r,
OQ' = r'.
Thus if V is constant over any surface S', V' is not a constant over the
inverse surface S', unless r is a constant, that is, unless the surface
S' is a sphere concentric with the sphere of inversion, in which case
the inverse surface is concentric with it and is an equipotential
surface of the inverse distribution.
Further, if q be distributed over an element dS of a surface, the
inverse charge aq⧸f will be distributed over the corresponding element
dS' of the inverse surface. But dS'⧸dS = a⁴⧸f⁴ = f'⁴⧸a⁴ where f, f'
are the distances of O from dS and dS'. Thus if s be the density on dS
and s' the inverse density on dS' we have s'⧸s = a³⧸f'³ = f³⧸a³.
When V is constant over the direct surface, while r has different
values for different directions of OQ, the different points of the
inverse surface may be brought to zero potential by placing at O a
charge －aV. For this will produce at Q' a potential －aV⧸r' which
with V' will give at Q' a potential zero. This shows that V' is the
potential of the induced distribution on S' due to a charge －aV at O,
or that －V' is the potential due to the induced charge on S' produced
by the charge aV at O.
Thus we have the conclusion that by the process of inversion we get from
a distribution in equilibrium, on a conductor of any form, an induced
distribution on the inverse surface supposed insulated and conducting;
and conversely we obtain from a given induced distribution on an
insulated conducting surface, a natural equilibrium distribution on the
inverse surface. In each case the inducing charge is situated at the
centre of inversion. The charges on the conductor (or conductors) after
inversion are always obtainable at once from the fact that they are the
inverses of the charges on the conductor (or conductors) in the direct
case, and the surface-densities or volume-densities can be found from
the relations stated above.
[Illustration: FIG. 7.]
Now take the case of two concentric spheres insulated and influenced by
a point-charge q placed at a point P between them as shown in Fig. 7. We
have seen at p. 49 how the induced distribution, and the amount of the
charge, on each sphere is obtained from the two convergent series of
images, one outside the outer sphere, the other inside the inner sphere.
We do not here calculate the density of distribution at any point, as
our object is only to explain the method; but the quantities on the
spheres S₁ and S₂, are respectively －q.OA.PB⧸(OP.AB), －q.OB.AP⧸(OP.AB).
It may be noticed that the sum of the induced charges is －q, and that
as the radii of the spheres are both made indefinitely great, while
the distance AB is kept finite, the ratios OA⧸OP, OB⧸OP approximate
to unity, and the charges to －q.PB⧸AB, －q.AP⧸AB, that is, the
charges are inversely as the distances of P from the nearest points of
the two surfaces. But when the radii are made indefinitely great we have
the case of two infinite plane conducting surfaces with a point-charge
between them, which we have described above.
Now let this induced distribution, on the two concentric spheres, be
inverted from P as centre of inversion. We obtain two non-intersecting
spheres, as in Fig. 5, for the inverse geometrical system, and for the
inverse electrical system an equilibrium distribution on these two
spheres in presence of one another, and charged with the charges which
are the inverses of the induced charges. These maintain the system of
two spheres at one potential. From this inversion it is possible to
proceed as shown by Maxwell in his _Electricity and Magnetism_, vol. i,
§ 173, to the distribution on two spheres at two different potentials;
but we have shown above how the problem may be dealt with directly by
the method of images.
[Illustration: FIG. 8.]
Again take the case of two parallel infinite planes under the influence
of a point-charge between them. This system inverted from P as centre
gives the equilibrium distribution on two charged insulated spheres in
contact (Fig. 8); for this system is the inverse of the planes and the
charges upon them. Another interesting case is that of the "electric
kaleidoscope" referred to above. Here the two infinite conducting planes
are inclined at an angle 360°⧸n, where n is a whole number, and are
therefore bounded in one direction by the straight line which is their
intersection. The image points I₁, J₁, ..., of P placed in the angle
between the planes are situated as shown in Fig. 3, and are n － 1 in
number. This system inverted from P as centre gives two spherical
surfaces which cut one another at the same angle as do the planes. This
system is one of electrical equilibrium in free space, and therefore the
problem of the distribution on two intersecting spheres is solved, for
the case at least in which the angle of intersection is an aliquot part
of 360°. When the planes are at right angles the result is that for two
perpendicularly intersecting planes, for which Fig. 9 gives a diagram.
[Illustration: FIG. 9.]
But the greatest achievement of the method was the determination of the
distribution on a segment of a thin spherical shell with edge in one
plane. The solution of this problem was communicated to M. Liouville in
the letter of date September 16, 1846, referred to above, but without
proof, which Thomson stated he had not time to write out owing to
preparation for the commencement of his duties as Professor of Natural
Philosophy at Glasgow on November 1, 1846. It was not supplied until
December 1868 and January 1869; and in the meantime the problem had not
been solved by any other mathematician.
As a starting point for this investigation the distribution on a thin
plane circular disk of radius a is required. This can be obtained by
considering the disk as a limiting case of an oblate ellipsoid of
revolution, charged to potential V, say. If Fig. 10 represent the disk
and P the point at which the density is sought, so that CP = r, and
CA = a, the density is V⧸{2π²√(a² － r²)}.
The ratio q⧸V, of charge to potential, which is called the electrostatic
capacity of the conductor, is thus 2a⧸π, that is a⧸1.571. It is, as
Thomson notes in his paper, very remarkable that the Hon. Henry
Cavendish should have found long ago by experiment with the rudest
apparatus the electrostatic capacity of a disk to be 1⧸1.57 of that
of a sphere of the same radius.
[Illustration: FIG. 10.]
[Illustration: FIG. 11.]
Now invert this disk distribution with any point Q as centre of
inversion, and with radius of inversion a. The geometrical inverse is
a segment of a spherical surface which passes through Q. The inverse
distribution is the induced distribution on a conducting shell
uninsulated and coincident with the segment, and under the influence of
a charge －aV situated at Q (Fig. 11). Call this conducting shell the
"bowl." If the surface-densities at corresponding points on the disk and
on the inverse, say points P and P', be s and s', then, as on page 51,
s' = sa³⧸QP'³. If we put in the value of s given above, that of s' can
be put in a form given by Thomson, which it is important to remark is
independent of the radius of the spherical surface. This expression is
applicable to the other side of the bowl, inasmuch as the densities at
near points on opposite sides of the plane disk are equal.
If v, v' be the potentials at any point R of space, due to the disk
and to its image respectively, －v' = av⧸QR. If then R be coincident
with a point P' on the spherical segment we have (since then v = V)
V' = aV⧸QP', which is the potential due to the induced distribution
caused by the charge －aV at Q as already stated.
The fact that the value of s' does not involve the radius makes it
possible to suppose the radius infinite, in which case we have the
solution for a circular disk uninsulated and under the influence of a
charge of electricity at a point Q in the same plane but outside the
bounding circle.
Now consider the two parts of the spherical surface, the bowl B, and the
remainder S of the spherical surface. Q with the charge －aV may be
regarded as situated on the latter part of the surface. Any other
influencing charges situated on S will give distributions on the bowl to
be found as described above, and the resulting induced electrification
can be found from these by summation. If S be uniformly electrified to
density s, and held so electrified, the inducing distribution will be
one given by integration over the whole of S, and the bowl B will be at
zero potential under the influence of this electrification of S, just as
if B were replaced by a shell of metal connected to the earth by a long
fine wire. The densities are equal at infinitely near points on the two
sides of B.
Let the bowl be a thin metal shell connected with the earth by a long
thin wire and be surrounded by a concentric and complete shell of
diameter f greater than that of the spherical surface, and let this
shell be rigidly electrified with surface density －s. There will be no
force within this shell due to its own electrification, and hence it
will produce no change of the distribution in the interior. But the
potential within will be －2πfs, for the charge is －πf²s, and the
capacity of the shell is ½f. The potential of the bowl will now be
zero, and its electrification will just neutralise the potential
-2πfs, that is, will be exactly the free electrification required to
produce potential 2πfs.
To find this electrification let the value of f be only infinitesimally
greater than the diameter of the spherical surface of which B is a part;
then the bowl is under the influence (1) of a uniform electrification of
density －s infinitely close to its outer surface, and (2) of a uniform
electrification of the same density, which may be regarded as upon the
surface which has been called S above. It is obvious that by (1) a
density s is produced on the outer surface of the bowl, and no other
effect; by (2) an equal density at infinitely near points on the
opposite sides of the bowl is produced which we have seen how to
calculate. Thus the distribution on the bowl freely electrified is
completely determined and the density can easily be calculated. The
value will be found in Thomson's paper.
Interesting results are obtained by diminishing S more and more until
the shell is a complete sphere with a circular hole in it. Tabulated
results for different relative dimensions of S will be found in
Thomson's paper, "Reprint of Papers," Articles V, XIV, XV. Also the
reader will there find full particulars of the mathematical calculations
indicated in this chapter, and an extension of the method to the case of
an influencing point not on the spherical surface of which the shell
forms part. Further developments of the problem have been worked out by
other writers, and further information with references will be found in
Maxwell's _Electricity and Magnetism_, loc. cit.
It is not quite clear whether Thomson discovered geometrical inversion
independently or not: very likely he did. His letter to Liouville of
date October 8, 1845, certainly reads as if he claimed the geometrical
transformation as well as the application to electricity. Liouville,
however, in his Note in which he dwells on the analytical theory of the
transformation says, "La transformation dont il s'agit est bien connue,
du reste, et des plus simples; c'est celle que M. Thomson lui-même a
jadis employée sous le nom de principe des images." In Thomson and
Tail's _Natural Philosophy_, § 513, the reference to the method is as
follows: "Irrespectively of the special electric application, the method
of images gives a remarkable kind of transformation which is often
useful. It suggests for mere geometry what has been called the
transformation by reciprocal radius-rectors, that is to say...." Then
Maxwell, in his review of the "Reprint of Papers" (Nature, vol. vii),
after referring to the fact that the solution of the problem of the
spherical bowl remained undemonstrated from 1846 to 1869, says that the
geometrical idea of inversion had probably been discovered and
rediscovered repeatedly, but that in his opinion most of these
discoveries were later than 1845, the date of Thomson's first paper.[10]
A very general method of finding the potential at any point of a region
of space enclosed by a given boundary was stated by Green in his 'Essay'
for the case in which the potential is known for every point of the
boundary. The success of the method depends on finding a certain
function, now called Green's function. When this is known the potential
at any point is at once obtained by an integration over the surface.
Thomson's method of images amounts to finding for the case of a region
bounded by one spherical surface or more the proper value of Green's
function. Green's method has been successfully employed in more
complicated cases, and is now a powerful method of attack for a large
range of problems in other departments of physical mathematics. Thomson
only obtained a copy of Green's paper in January 1845, and probably
worked out his solutions quite independently of any ideas derived from
Green's general theory.
CHAPTER V
THE CHAIR OF NATURAL PHILOSOPHY AT GLASGOW. ESTABLISHMENT OF THE FIRST
PHYSICAL LABORATORY
The incumbent of the Chair of Natural Philosophy in the University of
Glasgow, Professor Meikleham, had been in failing health for several
years, and from 1842 to 1845 his duties had been discharged by another
member of the Thomson gens, Mr. David Thomson, B.A., of Trinity College,
Cambridge, afterwards Professor of Natural Philosophy at Aberdeen. Dr.
Meikleham died in May 1846, and the Faculty thereafter proceeded on the
invitation of Dr. J. P. Nichol, the Professor of Astronomy, to consider
whether in consequence of the great advances of physical science during
the preceding quarter of a century it was not urgently necessary to
remodel the arrangements for the teaching of natural philosophy in the
University. The advance of science had indeed been very great. Oersted
and Ampère, Henry and Faraday and Regnault, Gauss and Weber, had made
discoveries and introduced quantitative ideas, which had changed the
whole aspect of experimental and mathematical physics. The electrical
discoveries of the time reacted on the other branches of natural
philosophy, and in no small degree on mathematics itself. As a result
the progress of that period has continued and has increased in
rapidity, until now the accumulated results, for the most part already
united in the grasp of rational theory, have gone far beyond the power
of any single man to follow, much less to master.
It is interesting to look into a course of lectures such as were usually
delivered in the universities a hundred years ago by the Professor of
Natural Philosophy. We find a little discussion of mechanics,
hydrostatics and pneumatics, a little heat, and a very little optics.
Electricity and magnetism, which in our day have a literature far
exceeding that of the whole of physics only sixty years ago, could
hardly be said to exist. The professor of the beginning of the
nineteenth century, when Lord Kelvin's predecessor was appointed,
apparently found himself quite free to devote a considerable part of
each lecture to reflections on the beauties of nature, and to rhetorical
flights fitter for the pulpit than for the physics lecture-table.
In the intervening time the form and fashion of scientific lectures has
entirely changed, and the change is a testimony to the progress of
science. It is visible even in the design of the apparatus. Microscopes,
for example, have a perfection and a power undreamed of by our
great-grandfathers, and they are supported on stands which lack the
ornamentation of that bygone time, but possess stability and
convenience. Everything and everybody--even the professor, if that be
possible--must be business-like; and each moment of time must be
utilised in experiments for demonstration, not for applause, and in
brief and cogent statements of theory and fact. To waste time in talk
that is not to the point is criminal. But withal there is need of grace
of expression and vividness of description, of clearness of exposition,
of imagination, even of poetical intuition: but the stern beauty of
modern science is only disfigured by the old artificial adornments and
irrelevancies.
This is the tone and temper of science at the present day: the task is
immense, the time is short. And sixty years since some tinge of the same
cast of thought was visible in scientific workers and teachers. The
Faculty agreed with Dr. Nichol that there was need to bring physical
teaching and equipment into line with the state of science at the time;
but they wisely decided to do nothing until they had appointed a
Professor of Natural Philosophy who would be able to advise them fully
and in detail. They determined, however, to make the appointment subject
to such alterations in the arrangements of the department as they might
afterwards find desirable.
On September 11, 1846, the Faculty met, and having considered the
resolutions which had been proposed by Dr. Nichol, resolved to the
effect that the appointment about to be made should not prejudice the
right of the Faculty to originate or support, during the incumbency of
the new professor, such changes in the arrangements for conducting
instruction in physical science as it might be expedient to adopt, and
that this resolution should be communicated to the candidate elected.
The minute then runs: "The Faculty having deliberated on the respective
qualifications of the gentlemen who have announced themselves candidates
for this chair, and the vote having been taken, it carried unanimously
in favour of Mr. William Thomson, B.A., Fellow of St. Peter's College,
Cambridge, and formerly a student of this University, who is
accordingly declared to be duly elected: and Mr. Thomson being within
call appeared in Faculty, and the whole of this minute having been read
to him he agreed to the resolution of Faculty above recorded and
accepted the office." It was also resolved as follows: "The Faculty
hereby prescribe Mr. Thomson an essay on the subject, _De caloris
distributione per terræ corpus_, and resolve that his admission be on
Tuesday the 13th October, provided that he shall be found qualified by
the Meeting and shall have taken the oath and made the subscriptions
which are required by law."
At that time, and down to within the last fifteen years, every
professor, before his induction to his chair, had to submit a Latin
essay on some prescribed subject. This was almost the last relic of the
customs of the days when university lectures were delivered in Latin, a
practice which appears to have been first broken through by Adam Smith
when Professor of Moral Philosophy. Whatever it may have been in the
eighteenth century, the Latin essay at the end of the nineteenth was
perhaps hardly an infallible criterion of the professor-elect's
Latinity, and it was just as well to discard it. But fifty years before,
and for long after, classical languages bulked largely in the curriculum
of every student of the Scottish Universities, and it is undoubtedly the
case that most of those who afterwards came to eminence in other
departments of learning had in their time acquitted themselves well in
the old _Litteræ Humaniores_. This was true, as we have seen, of
Thomson, and it is unlikely that the form of his inaugural dissertation
cost him much more effort than its matter.
[Illustration: PROFESSOR WILLIAM THOMSON, 1846]
The subject chosen had reference no doubt to the papers on the theory
of heat which Mr. Thomson had already published. The thesis was
presented to the Faculty on the day appointed, and approved, and Mr.
Thomson having produced a certificate of his having taken the oaths to
government, and promised to subscribe the formula of the Church of
Scotland as required by law, on the first convenient opportunity, "the
following oath was then administered to him, which he took and
subscribed: _Ego, Gulielmus Thomson, B.A., physicus professor in hac
Academia designatus, promitto sancteque polliceor me in munere mihi
demandato studiose fideliterque versaturum._" Professor Thomson was then
"solemnly admitted and received by all the Members present, and took his
seat as a Member of Faculty."
No translation of this essay was ever published, but its substance was
contained in various papers which appeared later. The following
reference to it is made in an introduction attached to Article XI of his
_Mathematical and Physical Papers_ (vol. i, 1882).
"An application to Terrestrial Temperature, of the principle set forth
in the first part of this paper relating to the age of thermal
distributions, was made the subject of the author's Inaugural
Dissertation on the occasion of his induction to the professorship of
Natural Philosophy in the University of Glasgow, in October 1846, '_De
Motu Caloris per Terræ Corpus_'[11]: which, more fully developed
afterwards, gave a very decisive limitation to the possible age of the
earth as a habitation for living creatures; and proved the untenability
of the enormous claims for TIME which, uncurbed by physical science,
geologists and biologists had begun to make and to regard as
unchallengeable. See 'Secular Cooling of the Earth, Geological Time,'
and several other Articles below." Some statement of the argument for
this limitation will be given later. [See Chap. XIV.]
Thomson thus entered at the age of twenty-five on what was to be his
life work as a teacher, investigator, and inventor. For he continued in
office fifty-three years, so that the united tenures of his predecessor
and himself amounted to only four years less than a century! He took up
his duties at the opening of the college session in November, and
promptly called the attention of the Faculty to the deficiencies of the
equipment of apparatus, which had been allowed to fall behind the times,
and required to have added to it many new instruments. A committee was
appointed to consider the question and report, and as a result of the
representations of this committee a sum of £100 was placed at Professor
Thomson's disposal to supply his most pressing needs. In the following
years repeated applications for further grants were made and various
sums were voted--not amounting to more than £500 or £600 in all--which
were apparently regarded as (and no doubt were, considering the times
and the funds at the disposal of the Faculty) a liberal provision for
the teaching of physical science. A minute of the Faculty, of date Nov.
26, 1847, is interesting.
After "emphatically deprecating" all idea that such large annual
expenditure for any one department was to be regularly contemplated, the
committee refer in their report to the "inadequate condition of the
department in question," and express their satisfaction "with the
reasonable manner in which the Professor of Natural Philosophy has on
all occasions readily modified his demands in accordance with the
economical suggestions of the committee." They conclude by saying that
they "view his ardour and anxiety in the prosecution of his profession
with the greatest pleasure," and "heartily concur in those anticipations
of his future celebrity which Monsr. Serville,[12] the French
mathematician, has recently thought fit to publish to the scientific
world."
Again, in April 1852, the Faculty agree to pay a sum of £137 6_s._
1½_d._ as the price of purchases of philosophical apparatus already
made, and approve of a suggestion of the committee that the expenditure
on this behalf during the next year should not exceed £50, and "they
desire that the purchases shall be made so far as is possible with the
previously obtained concurrence of the committee." It is easy to imagine
that the ardent young Professor of Natural Philosophy found the
leisurely methods of his older colleagues much too slow, and in his
enthusiasm anticipated consent to his demands by ordering his new
instruments without waiting for committees and meetings and reports.
In an address at the opening of the Physical and Chemical Laboratories
of the University College of North Wales, on February 2, 1885, Sir
William Thomson (as he was then) referred to his early equipment and
work as follows: "When I entered upon the professorship of Natural
Philosophy at Glasgow, I found apparatus of a very old-fashioned kind.
Much of it was more than a hundred years old, little of it less than
fifty years old, and most of it was worm-eaten. Still, with such
appliances, year after year, students of natural philosophy had been
brought together and taught as well as possible. The principles of
dynamics and electricity had been well illustrated and well taught, as
well taught as lectures and so imperfect apparatus--but apparatus merely
of the lecture-illustration kind--could teach. But there was absolutely
no provision of any kind for experimental investigation, still less
idea, even, for anything like students' practical work. Students'
laboratories for physical science were not then thought of."[13]
It appears that the class of Natural Philosophy (there was then as a
rule only one class in any subject, though supplementary work was done
in various ways) met for systematic lectures at 9 a.m., which is the
hour still adhered to, and for what was called "Experimental Physics" at
8 p.m.!
The _University Calendar_ for 1863-4 states that "the Natural Philosophy
Class meets two hours daily, 9 a.m. and 11 a.m. The first hour is
chiefly spent in statements of Principles, description of Results of
Observation, and Experimental Illustrations. The second hour is devoted
to Mathematical Demonstrations and Exercises, and Examinations on all
parts of the Course.
"The Text Books to be used are: 'Elements of Dynamics' (first part now
ready), Printed by George Richardson, University Printer. 'Elements of
Natural Philosophy,' by Professors W. Thomson and P. G. Tait (Two
Treatises to be published before November. Macmillan.[14])
"The shorter of the last mentioned Treatises will be used for the work
required of all students of Natural Philosophy in the regular
curriculum. The whole or specified parts of the larger Treatise will be
prescribed in connection with voluntary examinations and exercises in
the Class, and for candidates for the degree of M.A. with honours.
Students who desire to undertake these higher parts of the business of
the class, ought to be well prepared on all the subjects of the Senior
Mathematical Class.
"The Laboratory in connection with the class is open daily from 9 a.m.
to 4 p.m. for Experimental Exercises and Investigations, under the
direction of the Professor and his official assistant."
In 1847 the meetings for experimental physics were changed to 11 a.m.
The hour 9 a.m. is still (1908) retained for the regular meetings of the
ordinary class, and 11 a.m. for meetings held twice a week for exercises
and tutorial work, attendance at which is optional.
[A second graduating class has now been instituted and is very largely
attended. Each student attends three lectures and spends four hours in
the laboratory each week. A higher class, in two divisions, is also
held.]
At an early date in his career as a professor Thomson called in the aid
of his students for experimental research. In many directions the
properties of matter still lay unexplored, and it was necessary to
obtain exact data for the perfecting of the theories of elasticity,
electricity and heat, which had been based on the researches of the
first half of the nineteenth century. To the authors of these
theories--Gauss, Green, Cauchy and others--he was a fit successor. Not
knowing all that had been done by these men of genius, he reinvented,
as we have seen, some of their great theorems, and in somewhat later
work, notably in electricity and magnetism, set the theories on a new
basis cleared of all extraneous and unnecessary matter, and reduced the
hypotheses and assumptions to the smallest possible number, stated with
the most careful precautions against misunderstanding. As this work was
gradually accomplished the need for further experiment became more and
more clearly apparent. Accordingly he established at the old College in
the High Street, what he has justly claimed was the first physical
laboratory for students.[15] An old wine-cellar in the basement
adjoining the Natural Philosophy Class-room was first annexed, and was
the scene of early researches, which were to lead to much of the best
work of the present time. To this was added a little later the
Blackstone Examination-room, which, disused and "left unprotected," was
added to the wine-cellar, and gave space for the increasing corps of
enthusiastic workers who came under the influence of the new teacher,
and were eager to be associated with his work. A good many of the
researches which were carried out in this meagre accommodation in the
old College will be mentioned in what follows.
[Illustration: INNER COURT OF THE OLD COLLEGE
Showing Natural Philosophy Rooms]
[In the view of the inner court of the Old College given opposite, the
windows on the ground-floor to the right of the turret in front, are
those of the Blackstone Examination-room, which formed a large part of
the new Physical Laboratory. The windows above these, on the second
floor, are those of the Apparatus-room of the Natural Philosophy
Department. Between the turret on the right of the picture and the angle
of the court are the windows of the Natural Philosophy Class-room. The
attic above the Apparatus-room was at a later time occupied by the
Engineering Department, under Professor Macquorn Rankine.]
Here again we may quote from the Bangor address:
"Soon after I entered my present chair in the University of Glasgow in
1846 I had occasion to undertake some investigations of electrodynamic
qualities of matter, to answer questions suggested by the results of
mathematical theory, questions which could only be answered by direct
experiment. The labour of observing proved too heavy, much of it could
scarcely be carried on without two or more persons, working together. I
therefore invited students to aid in the work. They willingly accepted
the invitation, and lent me most cheerful and able help. Soon after,
other students, hearing that their class-fellows had got experimental
work to do, came to me and volunteered to assist in the investigation. I
could not give them all work in the particular investigation with which
I had commenced--'the electric convection of heat'--for want of means
and time and possibilities of arrangement, but I did all in my power to
find work for them on allied subjects (Electrodynamic Properties of
Metals, Moduluses of Elasticity of Metals, Elastic Fatigue, Atmospheric
Electricity, etc.). I then had an ordinary class of a hundred students,
of whom some attended lectures in natural philosophy two hours a day,
and had nothing more to do from morning till night. These were the balmy
days of natural philosophy in the University of Glasgow--the
pre-Commissional days. But the majority of the class really had very
hard work, and many of them worked after class-hours for self-support.
Some were engaged in teaching, some were city-missionaries, intending to
go into the Established Church of Scotland or some other religious
denomination of Scotland, or some of the denominations of Wales, for I
always had many Welsh students. In those days, as now, in the Scottish
Universities all intending theological students took a 'philosophical
curriculum'--'zuerst collegium logicum,' then moral philosophy, and
(generally last) natural philosophy. Three-fourths of my volunteer
experimentalists used to be students who entered the theological classes
immediately after the completion of the philosophical curriculum. I well
remember the surprise of a great German professor when he heard of this
rule and usage: 'What! do the theologians learn physics?' I said, 'Yes,
they all do; and many of them have made capital experiments. I believe
they do not find that their theology suffers at all from (their) having
learned something of mathematics and dynamics and experimental physics
before they enter upon it.'"
This statement, besides throwing an interesting light on the conditions
of university work sixty years ago, gives an illustration of the wide
interpretation in Scotland of the term Arts. Here it has meant, since
the Chair of Natural Philosophy was founded in 1577, and held by one of
the Regents of the University, _Artes Liberales_ in the widest sense,
that is, the study of _Litteræ Humaniores_ (including mental and moral
philosophy) and physical and mathematical science. These were all deemed
necessary for a liberal education at that time: in the scientific age in
which we live it is more imperative than ever that neither should be
excluded from the Arts curriculum of our Universities. The common
distinction between Arts and Science is a false one, and the product of
a narrow idea which is alien to the traditions of our northern
Universities.
It is to be noted, however, that the laboratory thus founded was
essentially a research laboratory; it was not designed for the
systematic instruction of students in methods of experimenting.
Laboratories for this purpose came later, and as a natural consequence.
But for the best students, ill prepared as, no doubt, some of them were
for the work of research, the experience gained in such a laboratory was
very valuable. They learned--and, indeed, had to learn--in an incidental
manner how to determine physical constants, such as specific gravities,
thermal capacities, electric resistances, and so forth. For, apart from
the _Relations des Expériences_ of Regnault, and the magnetic and
electric work of Gauss and Weber, there was no systematised body of
information available for the guidance of students. Good students could
branch out from the main line of inquiry, so as to acquire skill in
subsidiary determinations of this kind; to the more easily daunted
student such difficulties proved formidable, and often absolutely
deterrent.
It is not easy for a physicist of the present day to realise the state
of knowledge of the time, and so he often fails to recognise the full
importance of Thomson's work. The want of precise knowledge of physical
constants was to a considerable extent a consequence of the want of
exact definitions of quantities to be determined, and in a much greater
degree of the lack of any system of units of measurement. The study of
phenomena was in the main merely qualitative; where an attempt had been
made to obtain quantitative determinations, the units employed were
arbitrary and dependent on apparatus in the possession of the
experimenter, and therefore unavailable to others. In the department of
heat, as has been said, a great beginning had been made by Regnault, in
whose hands the exact determination of physical constants had become a
fine art.
In electricity and magnetism there were already the rudiments of
quantitative measurement. But it was only long after, when the actions
of magnets and of electric currents had been much further studied, that
the British Association entered on its great work of setting up a system
of absolute units for the measurement of such actions. Up till then the
resistance, for example, of a piece of wire, to the passage of an
electric current along it, was expressed by some such specification as
that it was equal to the resistance of a certain piece of copper wire in
the experimenter's possession. It was therefore practically impossible
for experimenters elsewhere to profit by the information. And so in
other cases. An example from Thomson's papers on the "Dynamical Theory
of Heat" may be cited here, though it refers to a time (1851) when some
progress towards obtaining a system of absolute units had been made. In
§ 118 (Art. XLVIII) he states that the electromotive force of a
thermoelectric couple of copper and bismuth, at temperatures 0° C. and
100° C. of its functions, might be estimated from a comparison made by
Pouillet of the strength of the current sent by this electromotive force
through a copper wire 20 metres long and 1 millimetre in diameter, with
the strength of a current decomposing water at a certain rate, were it
not that the specific resistances of different specimens of copper are
found to differ considerably from one another. Hence, though an estimate
is made, it is stated that, without experiments on the actual wire used
by Pouillet, it was impossible to arrive at an accurate result. Now if
it had been in Pouillet's power to determine accurately the resistance
of his circuit in absolute units, there would have been no difficulty in
the matter, and his result would have been immediately available for the
estimate required.
When submarine cables came to be manufactured and laid all this had to
be changed. For they were expensive; an Atlantic cable, for example,
cost half a million sterling. The state of the cable had to be
ascertained at short intervals during manufacture; a similar watch had
to be kept upon it during the process of laying, and afterwards during
its life of telegraphic use. The observations made by one observer had
therefore to be made available to all, so that, with other instruments
and at another place, equivalent observations could be made and their
results quantitatively compared with those of the former. To set up a
system of measurement for such purposes as these involved much
theoretical discussion and an enormous amount of experimental
investigation. This was undertaken by a special committee of the
Association, and a principal part in furnishing discussions of theory
and in devising experimental methods was taken by Thomson. The
committee's investigations took place at a date somewhat later in
Thomson's career than that with which we are here dealing, and some
account of them will be given in a later chapter; but much work,
preparatory for and leading up to the determination of electrical
standards, was done by the volunteer laboratory corps in the transformed
wine-cellar of the old College.
The selection and realisation of electrical standards was a work of
extraordinary importance to the world from every point of
view--political, commercial, and social. It not only rendered
applications of electricity possible in the arts and industries, but by
relieving experimental results from the vagueness of the specifications
formerly in use, made the further progress of pure electrical science a
matter in which every step forward, taken by an individual worker,
facilitated the advance of all. But like other toilsome services, the
nature of which is not clear to the general public, it has never
received proper acknowledgment from those who have profited by it. If
Thomson had done nothing more than the work he did in this connection,
first with his students and later with the British Association
Committee, he would have deserved well of his fellow-countrymen.
When Professor Thomson was entering on the duties of his chair, and
calling his students to his aid, the discoveries of Faraday on the
induction of currents by the motion of magnets in the neighbourhood of
closed circuits of wire, or, what comes to the same thing, the motion of
such circuits in the "fields" of magnets, had not been long given to the
world, and were being pondered deeply by natural philosophers. The time
was ripe for a quantitative investigation of current induction, like
that furnished by the genius of Ampère after the discovery by Oersted of
the deflection of a magnet by an electric current. Such an investigation
was immensely facilitated by Faraday's conception of lines of magnetic
force, the cutting of which by the wire of the circuit gave rise to the
induced current. Indeed, the mathematical ideas involved were indicated,
and not obscurely, by Faraday himself. But to render the mathematical
theory explicit, and to investigate and test its consequences, required
the highest genius. This work was accomplished in great measure by
Thomson, whose presentation of electrodynamic theory helped Maxwell to
the view that light was an affair of the propagation of electric and
magnetic vibrations in an insulating medium, the light-carrying ether.
Another investigation on which he had already entered in 1847 was of
great importance, not only for pure science but for the development and
proper economy of all industrial operations. The foundations on which a
dynamical theory of heat was to be raised had been partly laid by Carnot
and were being completed on the experimental side by James Prescott
Joule, whom Thomson met in 1847 at the meeting of the British
Association at Oxford. The meeting at Oxford in 1860 is memorable to the
public at large, mainly on account of the discussion which took place on
the Darwinian theory, and the famous dialectic encounter between Bishop
Wilberforce and Professor Huxley; the Oxford meeting of 1894 will always
be associated with the announcement of the discovery of argon by Lord
Rayleigh and Sir William Ramsay: the meeting of 1847 might quite as
worthily be remembered as that at which Joule laid down, with numerical
exactitude, the first law of thermodynamics. Joule brought his
experimental results before the Mathematical and Physical Section at
that meeting; and it appears probable that they would have received
scant attention had not their importance been forcibly pointed out by
Thomson. Communications thereafter passed frequently between the two
young physicists, and there soon began a collaboration of great value to
science, and a friendship which lasted till the death of Joule in 1884.
[See p. 88 below.]
We shall devote the next few chapters to an account, as free from
technicalities as possible, of these great divisions of Thomson's
earlier original work as professor at Glasgow.
CHAPTER VI
FRIENDSHIP WITH STOKES AND JOULE. EARLY WORK AT GLASGOW
During his residence at Cambridge Thomson gained the friendship of
George Gabriel Stokes, who had graduated as Senior Wrangler and First
Smith's Prizeman in 1841. They discussed mathematical questions together
and contributed articles on various topics to the _Cambridge
Mathematical Journal_. In 1846 "Cambridge and Dublin" was substituted
for "Cambridge" in the title of the Journal, and a new series was begun
under the editorship of Thomson. A feature of the earlier volumes of the
new issue was a series of Notes on Hydrodynamics written by agreement
between Thomson and Stokes, and printed in vols. ii, iii, and v. The
first, second, and fifth of the series were written by Thomson, the
others by Stokes. The matter of these Notes was not altogether novel;
but many points were put in a new and more truly physical light, and the
series was no doubt of much service to students, for whose use the
articles were intended. Some account of these Notes will be given in a
later chapter on Thomson's hydrodynamical papers.
For the mathematical power and sure physical instinct of Stokes Thomson
had always the greatest admiration. When asked on one occasion who was
the most outstanding worker in physical science on the continent, he
replied, "I do not know, but whoever he is, I am certain that Stokes is
a match for him." In a report of an address which he delivered in June
1897, at the celebration of the Jubilee of Sir George Stokes as Lucasian
Professor of Mathematics, Lord Kelvin referred to their early
intercourse at Cambridge in terms which were reported as follows: "When
he reflected on his own early progress, he was led to recall the great
kindness shown to himself, and the great value which his intercourse
with Sir George Stokes had been to him through life. Whenever a
mathematical difficulty occurred he used to say to himself, 'Ask Stokes
what he thinks of it.' He got an answer if answer was possible; he was
told, at all events, if it was unanswerable. He felt that in his
undergraduate days, and he felt it more now."
After the death of Stokes in February 1902, Lord Kelvin again referred,
in an enthusiastic tribute in Nature for February 12, to these early
discussions. "Stokes's scientific work and scientific thought is but
partially represented by his published writings. He gave generously and
freely of his treasures to all who were fortunate enough to have an
opportunity of receiving from him. His teaching me the principles of
solar and stellar chemistry when we were walking about among the
colleges sometime prior to 1852 (when I vacated my Peterhouse Fellowship
to be no more in Cambridge for many years) is but one example."
The interchange of ideas between Stokes and Thomson which began in those
early days went on constantly and seems to have been stimulating to
both. The two men were in a sense complementary in nature and
temperament. Both had great power and great insight, but while Stokes
was uniformly calm, reflective, and judicial, Thomson's enthusiasm was
more outspokenly fervid, and he was apt to be at times vehement and
impetuous in his eagerness to push on an investigation; and though, as
became his nationality, he was cautious in committing himself to
conclusions, he exercised perhaps less reserve in placing his results
before the public of science.
A characteristic instance of Thomson's vehement pursuit of experimental
results may be given here, although the incidents occurred at a much
later date in his career than that with which we are at present
concerned. In 1880 the invention of the Faure Secondary Battery
attracted his attention. M. Faure brought from Paris some cells made up
and ready charged, and showed in the Physical Laboratory at Glasgow the
very powerful currents which, in consequence of their very low internal
resistance, they were capable of producing in a thick piece of copper
wire. The cells were of the original form, constructed by coating strips
of sheet lead on both sides with a paste of minium moistened with dilute
sulphuric acid, swathing them in woollen cloth sewed round them, and
then rolling two together to form the pair of plates for one cell.
A supply of sheet lead, minium, and woollen cloth was at once obtained,
and the whole laboratory corps of students and staff was set to work to
manufacture secondary batteries. A small Siemens-Halske dynamo was
telegraphed for to charge the cells, and the ventilating steam-engine of
the University was requisitioned to drive the dynamo during the night.
Thus the University stokers and engineer were put on double shifts; the
cells were charged during the night and the charging current and
battery-potential measured at intervals.
Then the cells were run down during the day, and their output measured
in the same way. Just as this began, Thomson was laid up with an ailment
which confined him to bed for a couple of weeks or so; but this led to
no cessation of the laboratory activity. On the contrary, the laboratory
corps was divided into two squads, one for the night, the other for the
day, and the work of charging and discharging, and of measurement of
expenditure and return of energy went on without intermission. The
results obtained during the day were taken to Thomson's bedside in the
evening, and early in the morning he was ready to review those which had
been obtained during the night, and to suggest further questions to be
answered without delay. This mode of working could not go on
indefinitely, but it continued until his assistants (some of whom had to
take both shifts!), to say nothing of the stokers and students, were
fairly well exhausted.
On other occasions, when he was from home, he found the post too slow to
convey his directions to his laboratory workers, and telegraphed from
day to day questions and instructions regarding the work on hand. Thus
one important result (anticipated, however, by Villari) of the series of
researches on the effects of stress on magnetisation which forms Part
VII of his _Electrodynamic Qualities of Metals_--the fact that up to a
certain magnetising force the effect of pull, applied to a wire of soft
iron, is to increase the magnetisation produced, and for higher
magnetising forces to diminish it--was telegraphed to him on the night
on which the paper was read to the Royal Society.
It will thus be seen that Thomson, whether confined to his room or on
holiday, kept his mind fixed upon his scientific or practical work, and
was almost impatient for its progress. Stokes worked mainly by himself;
but even if he had had a corps of workers and assistants, it is
improbable that such disturbances of hours of attendance and laboratory
and workshop routine would have occurred, as were not infrequent at
Glasgow when Thomson's work was, in the 'sixties and 'seventies, at its
intensest.
Stokes and Thomson were in succession presidents of the Royal Society,
Stokes from 1885 to 1890, and Thomson (from 1892 as Lord Kelvin) from
1890 to 1895. This is the highest distinction which any scientific man
in this country can achieve, and it is very remarkable that there should
have been in recent times two presidents in succession whose modes of
thought and mathematical power are so directly comparable with those of
the great founder of modern natural philosophy. Stokes had the
additional distinction of being the lineal successor of Newton as
Lucasian Professor of Mathematics at Cambridge. But it was reserved for
Thomson to do much by the publication of Thomson and Tait's _Natural
Philosophy_ to bring back the current of teaching and thought in
dynamical science to the ideas of the Principia, and to show how
completely the fundamental laws, as laid down in that great classic,
avail for the inclusion of the modern theory of energy, in all its
transformations, within the category of dynamical action between
material systems.
An exceedingly eminent politician, now deceased, said some years ago
that the present age was singularly deficient in minds of the first
quality. So far as scientific genius is concerned, the dictum was
singularly false: we have here a striking proof of the contrary. But
then few politicians know anything of science; indeed some of those who
guide, or aspire to guide, the destinies of the most scientific and
industrial empire the world has ever seen are almost boastful of their
ignorance. There are, of course, honourable exceptions.
It is convenient to refer here to the share which Stokes and Thomson
took in the physical explanation of the dark lines of the solar
spectrum, and to their prediction of the possibility of determining the
constitution of the stars and of terrestrial substances by what is now
known as spectrum analysis. Thomson used to give the physical theory of
these lines in his lectures, and say that he obtained the idea from
Stokes in a conversation which they had in the garden of Pembroke at
Cambridge, "some time prior to 1852" (see the quotation from his Nature
article quoted above, p. 80, and the _Baltimore Lectures_, p. 101). This
is confirmed by a student's note-book, of date 1854, which is now in the
Natural Philosophy Department. The statements therein recorded are
perfectly definite and clear, and show that at that early date the whole
affair of spectrum analysis was in his hands, and only required
confirmation by experiments on the reversal of the lines of terrestrial
substances by an atmosphere of the substance which produced the lines,
and a comparison of the positions of the bright lines of terrestrial
substances with those of the dark lines of the solar spectrum. Why
Thomson did not carry out all these experiments it would be difficult to
say. Some of them he did make, for Professor John Ferguson, who was a
student of Natural Philosophy in 1859-60, has recently told how he
witnessed Thomson make the experiment of reversing the lines of sodium
by passing the light from the salted flame of a spirit lamp through
vapour of sodium produced by heating the metal in an iron spoon. A few
days later, says Professor Ferguson, Thomson read a letter to his class
announcing Bunsen and Kirchhoff's discovery.
A letter of Stokes to Sir John Lubbock, printed in the _Scientific
Correspondence of Sir George Gabriel Stokes_, states his recollection of
the matter, and gives Thomson the credit of having inferred the method
of spectrum analysis, a method to which Stokes himself makes no claim.
He says, "I know, I think, what Sir William Thomson was alluding to. I
knew well, what was generally known, and is mentioned by Herschel in his
treatise on Light, that the bright D seen in flames is specially
produced when a salt of soda is introduced. I connected it in my own
mind with the presence of sodium, and I suppose others did so too. The
coincidence in position of the bright and dark D is too striking to
allow us to regard it as fortuitous. In conversation with Thomson I
explained the connection of the dark and bright line by the analogy of a
set of piano strings tuned to the same note, which, if struck, would
give out that note, and also would be ready to sound it, to take it up,
in fact, if it were sounded in air. This would imply absorption of the
aërial vibrations, as otherwise there would be a creation of energy.
Accordingly I accounted for the presence of the dark D in the solar
spectrum by supposing that there was sodium in the atmosphere, capable
of absorbing light of that particular refrangibility. He asked me if
there were any other instances of such coincidences of bright and dark
lines, and I said I thought there was one mentioned by Brewster. He was
much struck with this, and jumped to the conclusion that to find out
what substances were in the stars we must compare the positions of the
dark lines seen in their spectra with the spectra of metals, etc....
"I should have said that I thought Thomson was going too fast ahead, for
my notion at the time was that, though a few of the dark lines might be
traced to elementary substances, sodium for one, probably potassium for
another, yet the great bulk of them were probably due to compound
vapours, which, like peroxide of nitrogen and some other known compound
gases, have the character of selective absorption."
It will be remembered that the experimental establishment of the method
of spectrum analysis was published towards the end of 1859 by Bunsen and
Kirchhoff, to whom, therefore, the full credit of discoverers must be
given.
Lord Kelvin in the later years of his life used to tell the story of his
first meeting with Joule at Oxford, and of their second meeting a
fortnight later in Switzerland. He did so also in his address delivered
on the occasion of the unveiling of a statue of Joule, in Manchester
Town Hall, on December 7, 1893, and we quote the narrative on account of
its scientific and personal interest. "I can never forget the British
Association at Oxford in 1847, when in one of the sections I heard a
paper read by a very unassuming young man, who betrayed no consciousness
in his manner that he had a great idea to unfold. I was tremendously
struck with the paper. I at first thought it could not be true, because
it was different from Carnot's theory, and immediately after the reading
of the paper I had a few words with the author, James Joule, which was
the beginning of our forty years' acquaintance and friendship. On the
evening of the same day, that very valuable institution of the British
Association, its conversazione, gave us opportunity for a good hour's
talk and discussion over all that either of us knew of thermodynamics. I
gained ideas which had never entered my mind before, and I thought I,
too, suggested something worthy of Joule's consideration when I told him
of Carnot's theory. Then and there in the Radcliffe Library, Oxford, we
parted, both of us, I am sure, feeling that we had much more to say to
one another and much matter for reflection in what we had talked over
that evening. But ... a fortnight later, when walking down the valley of
Chamounix, I saw in the distance a young man walking up the road towards
me, and carrying in his hand something which looked like a stick, but
which he was using neither as an alpenstock nor as a walking-stick. It
was Joule with a long thermometer in his hand, which he would not trust
by itself in the _char-à-banc_, coming slowly up the hill behind him,
lest it should get broken. But there, comfortably and safely seated in
the _char-à-banc_, was his bride--the sympathetic companion and sharer
in his work of after years. He had not told me in Section A, or in the
Radcliffe Library, that he was going to be married in three days, but
now in the valley of Chamounix he introduced me to his young wife. We
appointed to meet again a fortnight later at Martigny to make
experiments on the heat of a waterfall (Sallanches) with that
thermometer: and afterwards we met again and again, and from that time,
indeed, remained close friends till the end of Joule's life. I had the
great pleasure and satisfaction for many years, beginning just forty
years ago, of making experiments along with Joule which led to some
important results in respect to the theory of thermodynamics. This is
one of the most valuable recollections of my life, and is indeed as
valuable a recollection as I can conceive in the possession of any man
interested in science."
At the beginning of his course of lectures each session, Professor
Thomson read, or rather attempted to read, an introductory address on
the scope and methods of physical science, which he had prepared for his
first session in 1846. It set forth the fact that in science there were
two stages of progress--a natural history stage and a natural philosophy
stage. In the first the discoverer or teacher is occupied with the
collection of facts, and their arrangement in classes according to their
nature; in the second he is concerned with the relations of facts
already discovered and classified, and endeavours to bring them within
the scope of general principles or causes. Once the philosophical stage
is reached, its methods and results are connected and enlarged by
continued research after facts, controlled and directed by the
conclusions of general theory. Thus the method is at first purely
inductive, but becomes in the second stage both inductive and
deductive; the general theory predicts by its deductions, and the
verification of these by experiment and observation give a validity to
the theory which no mere induction could afford. These stages of
scientific investigation are well illustrated by the laws of Kepler
arrived at by mere comparison of the motions of the planets, and the
deduction of these laws, with the remarkable correction of the third
law, given by the theory of universal gravitation. The prediction of the
existence and place of the planet Neptune from the perturbations of
Uranus is an excellent example of the predictive quality of a true
philosophical theory.
The lecture then proceeded to state the province of dynamics, to define
its different parts, and to insist on the importance of kinematics,
which was described as a purely geometrical subject, the geometry
of motion, considerations from which entered into every dynamical
problem. This distinction between dynamical and kinematical
considerations--between those in which force is concerned and those into
which enter only the idea of displacement in space and in time--is
emphasised in Thomson and Tait's _Natural Philosophy_, which commences
with a long chapter devoted entirely to kinematics.
Whether Professor Thomson read the whole of the Introductory Lecture on
the first occasion is uncertain--Clerk Maxwell is said to have asserted
that it was closely adhered to, for that one time only, and finished in
much less than the hour allotted to it. In later years he had never read
more than a couple of pages when some new illustration, or new fact of
science, which bore on his subject, led him to digress from the
manuscript, which was hardly ever returned to, and after a few minutes
was mechanically laid aside and forgotten. Once on beginning the session
he humorously informed the assembled class that he did not think he had
ever succeeded in reading the lecture through before, and added that he
had determined that they should hear the whole of it! But again occurred
the inevitable digression, in the professor's absorption in the new
topic the promise was forgotten, and the written lecture fared as
before! These digressions were exceedingly interesting to the best
students: whether they compensated for the want of a carefully prepared
presentation of the elements of the subject, suited to the wants of the
mass of the members of the class, is a matter which need not here be
discussed. All through his elementary lectures--introductory or not--new
ideas and new problems continually presented themselves. An eminent
physicist once remarked that Thomson was perhaps the only living man who
made discoveries while lecturing. That was hardly true; in the glow of
action and stress of expression the mind of every intense thinker often
sees new relations, and finds new points of view, which amount to
discoveries. But fecundity of mind has, of course, its disadvantages:
the unexpected cannot happen without causing distractions to all
concerned. A mind which can see a theory of the physical universe in a
smoke-ring is likely, unless kept under extraordinary and hampering
restraint, to be tempted to digress from what is strictly the subject in
hand, to the world of matters which that subject suggests. Professor
Thomson was, it must be admitted, too discursive for the ordinary
student, and perhaps did not study the art of boiling down physical
theories to the form most easily digestible. His eagerness of mind and
width of mental outlook gave his lectures a special value to the
advanced student, so that there was a compensating advantage.
The teacher of natural philosophy is really placed in a position of
extraordinary difficulty. The fabric of nature is woven without seam,
and to take it to pieces is in a manner to destroy it. It must, after
examination in detail, be reconstructed and considered as a whole, or
its meaning escapes us. And here lies the difficulty: every bit of
matter stands in relation to everything else, and both sides of every
relation must be considered. In other words, in the explanation of any
one phenomenon the explanation of all others is more or less involved.
This does not mean that investigation or exposition is impossible, or
that we cannot proceed step by step; but it shows the foolishness of
that criticism of science and scientific method which asks for complete
or ultimate knowledge, and of the popular demand for a simple form of
words to express what is in reality infinitely complex.
In the earlier years of his professorship Professor Thomson taught his
class entirely himself, and gathered round him, as he has told us in the
Bangor address, an enthusiastic band of workers who aided him in the
researches which he began on the electrodynamic qualities of metals, the
elastic properties of substances, the thermal and electrical
conductivities of metals, and at a later date in the electric and
magnetic work which he undertook as a member of the British Association
Committee on Electrical Standards. The class met, as has been stated,
twice a day, first for lectures, then for exercises and oral
examination. The changes which took place later in the curriculum, and
especially the introduction of honours classes in the different
subjects, rendered it difficult, if not impossible, for two hours'
attendance to be given daily on all subjects, and students were at first
excused attendance at the second hour, and finally such attendance
became practically optional. But so long as the old traditional
curriculum in Arts--of Humanity, Greek, Logic, Mathematics, Moral
Philosophy and Natural Philosophy--endured, a large number of students
found it profitable to attend at both hours, and it was possible to give
a large amount of excellent tutorial instruction by the working of
examples and oral examination.
Thomson always held that his commission included the subject of physical
astronomy, and though his lectures on that subject were, as a rule,
confined to a statement of Kepler's laws and Newton's deductions from
them, he took care that the written and oral examinations included
astronomical questions, for which the students were enjoined to prepare
by reading Herschel's Outlines, or some similar text-book. This
injunction not infrequently was disregarded, and discomfiture of the
student followed as a matter of course, if he was called on to answer.
Nor were the questions always easy to prepare for by reading. A man
might have a fair knowledge of elementary astronomy, and be unable to
answer offhand such a question as, "Why is the ecliptic called the
ecliptic?" or to say, when the lectures on Kepler had been omitted,
short and tersely just what was Newton's deduction from the third law of
the planetary motions.
Home exercises were not prescribed as part of the regular work except
from time to time in the "Higher Mathematical Class" which for thirty
years or more of Thomson's tenure of office was held in the department.
But the whole ordinary class met every Monday morning and spent the
usual lecture hour in answering a paper of dynamical and physical
questions. As many as ten, and sometimes eleven, questions were set in
these papers, some of them fairly difficult and involving novel ideas,
and by this weekly paper of problems the best students, a dozen or
more perhaps, were helped to acquire a faculty of prompt and brief
expression. It was not uncommon for a good man to score 80 or 90 or even
100 per cent. in the paper, no small feat to accomplish in a single
hour. But to a considerable majority of the class, it is doubtful
whether the weekly examination was of much advantage: they attempted
one or two of the more descriptive questions perhaps, but a good
many did next to nothing. The examinations came every week, and so
the preparation for one after another was neglected, and as much
procrastination of work ensued as there would have been if only four or
five papers a session had been prescribed. Then the work of looking over
so many papers was a heavy task to the professor's assistant, a task
which became impossible when, for a few years in the early 'eighties,
the students in the ordinary class numbered about 250.
The subject of natural philosophy had become so extensive in 1846 that
Professor J. P. Nichol called attention to the necessity for special
arrangements for its adequate teaching. What would he say if he could
survey its dimensions at the present time! To give even a brief outline
of the principal topics in dynamics, heat, acoustics, light, magnetism,
and electricity is more than can be accomplished in any course of
university lectures; and the only way to teach well and economically the
large numbers of students[16] who now throng the physics classes is to
give each week, say, three lectures as well considered and arranged as
possible, without any interruption from oral examination, and assemble
the students in smaller classes two or three times a week for exercises
and oral examination.
Thomson stated his views as to examinations and lectures in the Bangor
address. "The object of a university is teaching, not testing, ... in
respect to the teaching of a university the object of examination is to
promote the teaching. The examination should be, in the first place,
daily. No professor should meet his class without talking to them. He
should talk to them and they to him. The French call a lecture a
conférence, and I admire that idea. Every lecture should be a conference
of teachers and students. It is the true ideal of a professorial
lecture. I have found that many students are afflicted when they come up
to college with the disease called 'aphasia.' They will not answer when
questioned, even when the very words of the answer are put in their
mouths, or when the answer is simply 'yes' or 'no.' That disease wears
off in a few weeks, but the great cure for it is in repeated and careful
and very free interchange of question and answer between teacher and
student.... Written examinations are very important, as training the
student to express with clearness and accuracy the knowledge he has
gained, but they should be once a week to be beneficial."
The great difficulty now, when both classes and subject have grown
enormously, is to have free conversation between professor and student,
and yet give an adequate account of the subject. To examine orally in a
thorough way two students in each class-hour is about as much as can be
done if there is to be any systematic exposition by lecture at all; and
thus the conference between teacher and individual student can occur
only twice a year at most. Nevertheless Lord Kelvin was undoubtedly
right: oral examination and the training of individual students in the
art of clear and ready expression are very desirable. The real
difficulties of the subject are those which occur to the best students,
and a discussion of them in the presence of others is good for all. This
is difficult nowadays, for large classes cannot afford to wait while two
or three backward students grope after answers to questions--which in
many cases must be on points which are sufficiently plain to the
majority--to say nothing of the temptation to disorder which the display
of personal peculiarities or oddities of expression generally affords to
an assembly of students. But time will be economised and many advantages
added, if large classes are split up into sections for tutorial work, to
supplement the careful presentation of the subject made in the
systematic lectures delivered to the whole class in each case. The
introduction of a tutorial system will, however, do far more harm than
good, unless the method of instruction is such as to foster the
self-reliance of the student, who must not be, so to speak, spoon-fed:
such a method, and the advantages of the weekly examination on paper
may be secured, by setting the tutorial class to work out on the spot
exercises prescribed by the lecturer. But the danger, which is a very
real one, can only be fully avoided by the precautions of a skilful
teacher, who in those small classes will draw out and direct the ideas
of his students, rather than impart knowledge directly.
After a few years Thomson found it necessary to appoint an assistant,
and Mr. Donald McFarlane, who had distinguished himself in the
Mathematics and Natural Philosophy classes, was chosen. Mr. McFarlane
was originally a block-printer, and seems to have been an apprentice at
Alexandria in the Vale of Leven, at the time of the passing of the first
Reform Bill. After some time spent in the cotton industry of the
district, he became a teacher in a village school in the Vale of Leven,
and afterwards entered the University as a student. He discharged his
duties in the most faithful and self-abnegating manner until his
retirement in 1880, when he had become advanced in years. He had charge
of the instruments of the department, got ready the lecture
illustrations and attended during lecture to assist in the experiments
and supply numerical data when required, prepared the weekly class
examination paper and read the answers handed in, and assisted in the
original investigations which the professor was always enthusiastically
pursuing. A kind of universal physical genius was McFarlane; an expert
calculator and an exact and careful experimentalist. Many a long and
involved arithmetical research he carried out, much apparatus he made in
a homely way, and much he repaired and adjusted. Then, always when the
professor was out of the way and calm had descended on the
apparatus-room, if not on the laboratory, McFarlane sat down to reduce
his pile of examination papers, lest Monday should arrive with a new
deluge of crude answers and queer mistakes, ere the former had
disappeared. On Friday afternoons at 3 o'clock he gave solutions of the
previous Monday's questions to any members of the class who cared to
attend; and his clear and deliberate explanations were much appreciated.
An unfailing tribute was rendered to him every year by the students, and
often took the form of a valuable gift for which one and all had
subscribed. A recluse he was in his way, hardly anybody knew where he
lived--the professor certainly did not--and a man of the highest ability
and of the most absolute unselfishness. An hour in the evening with one
or two special friends, and the study of German, were the only
recreations of McFarlane's solitary life. He was full of humour, and
told with keen enjoyment stories of the University worthies of a bygone
age. For thirty years he worked on for a meagre salary, for during the
earlier part of that time no provision for assistants was made in the
Government grant to the Scottish Universities. By an ordinance issued in
1861 by the University Commissioners, appointed under the Act of 1858, a
grant of £100 a year was made from the Consolidated Fund for an
assistant in each of the departments of Humanity, Greek, Mathematics,
and Natural Philosophy, and for two in the department of Chemistry; and
McFarlane's position was somewhat improved. His veneration for Thomson
was such as few students or assistants have had for a master: his
devotion resembled that of the old famulus rather than the much more
measured respect paid by modern assistants to their chiefs.
After his retirement McFarlane lived on in Glasgow, and amused himself
reading out-of-the-way Latin literature and with the calculation of
eclipses! He finally returned to Alexandria, where he died in February
1897. "Old McFarlane" will be held in affectionate remembrance so long
as students of the Natural Philosophy Class in the 'fifties and 'sixties
and 'seventies, now, alas! a fast vanishing band, survive.
Soon after taking his degree of B.A. at Cambridge in 1845, Thomson had
been elected a Fellow of St. Peter's College. In 1852 he vacated his
Fellowship on his marriage to Miss Margaret Crum, daughter of Mr. Walter
Crum of Thornliebank, near Glasgow, but was re-elected in 1871, and
remained thereafter a Fellow of Peterhouse throughout his life.
CHAPTER VII
THE "ACCOUNT OF CARNOT'S THEORY OF THE MOTIVE POWER OF HEAT"--TRANSITION
TO THE DYNAMICAL THEORY OF HEAT
The meeting of Thomson and Joule at Oxford in 1847 was fraught with
important results to the theory of heat. Thomson had previously become
acquainted with Carnot's essay, most probably through Clapeyron's
account of it in the _Journal de l'École Polytechnique_, 1834, and had
adopted Carnot's view that when work was done by a heat engine heat was
merely let down from a body at one temperature to a body at a lower
temperature. Joule apparently knew nothing of Carnot's theory, and had
therefore come to the consideration of the subject without any
preconceived opinions. He had thus been led to form a clear notion of
heat as something which could be transformed into work, and _vice
versa_. This was the root idea of his attempt to find the dynamical
equivalent of heat. It was obvious that a heat engine took heat from a
source and gave heat to a refrigerator, and Joule naturally concluded
that the appearance of the work done by the engine must be accompanied
by the disappearance of a quantity of heat of which the work done was
the equivalent. He carried this idea consistently through all his work
upon energy-changes, not merely in heat engines but in what might be
called electric engines. For he pointed out that the heat produced in
the circuit of a voltaic battery was the equivalent of the
energy-changes within the battery, and that, moreover, when an
electromagnetic engine was driven by the current, or when
electrochemical decomposition was effected in a voltameter in the
circuit, the heat evolved in the circuit for a given expenditure of the
materials of the battery was less than it would otherwise have been, by
the equivalent of the work done by the engine, or of the chemical
changes effected in the voltameter. Thus Joule was in possession at an
earlier date than Thomson of the fundamental notion upon which the true
dynamical theory of heat engines is founded. Thomson, on the other hand,
as soon as he had received this idea, was able to add to it the
conception, derived from Carnot, of a reversible engine as the engine of
greatest efficiency, and to deduce in a highly original manner all the
consequences of these doctrines which go to make up the ordinary
thermodynamics even of the present time. Though Clausius was the first,
as we shall see, to deduce various important theorems, yet Thomson's
discussion of the question had a quality peculiarly its own. It was
marked by that freedom from unstated assumptions, from extraneous
considerations, from vagueness of statement and of thought, which
characterises all his applications of mathematics to physics. The
physical ideas are always set forth clearly and in such a manner that
their quantitative representation is immediate: we shall have an example
of this in the doctrine of absolute temperature. In most of the
thermodynamical discussions which take the great memoir of Clausius as
their starting point, temperature is supposed to be given by a
hypothetical something which is called a perfect gas, and it is very
difficult, if not impossible, to gather a precise notion of the
properties of such a gas and of the temperature scale thereon founded.
Thomson's scale enables a perfect gas to be defined, and the deviations
of the properties of ordinary gases from those of such a gas to be
observed and measured.
The idea, then, which Joule had communicated to Section A, when Thomson
interposed to call attention to its importance, was that work spent in
overcoming friction had its equivalent in the heat produced, that, in
fact, the amount of heat generated in such a case was proportional to
the work spent, quite irrespective of the materials used in the process,
provided no change of the internal energy of any of them took place so
as to affect the resulting quantity of heat. This forced upon physicists
the view pointed to by the doctrine of the immateriality of heat,
established by the experiments of Rumford and Davy, that heat itself was
a form of energy; and thus the principle of conservation of energy was
freed from its one defect, its apparent failure when work was done
against friction.
Rumford had noted the very great evolution of heat when gun-metal was
rubbed by a blunt borer, and had come to the reasonable conclusion that
what was evolved in apparently unlimited quantity by the abrasion or
cutting down of a negligible quantity of materials could not be a
material substance. He had also made a rough estimate of the relation
between the work spent in driving the borer by horse-power and the heat
generated. Joule's method of determining the work-equivalent of heat was
a refinement of Rumford's, but differed in the all-important respect
that accurate means were employed for measuring the expenditure of work
and the gain of heat. He stirred a liquid, such as water or mercury, in
a kind of churn driven by a falling weight. The range of descent of the
weight enabled the work consumed to be exactly estimated, and a
sensitive thermometer in the liquid measured the rise of temperature;
thus the heat produced was accurately determined. The rise of
temperature was very slight, and the change of state of the liquid, and
therefore any possible change in its internal energy, was infinitesimal.
The experiments were carried out with great care, and included very
exact measurements of the various corrections--for example, the amount
of work spent at pulleys and pivots without affecting the liquid, and
the loss of heat by radiation. The experiments proved that the work
spent on the liquid and the heat produced were in direct proportion to
one another. He found, finally, in 1850, that 772 foot-pounds of work at
Manchester generated one British thermal unit, that is, as much heat as
sufficed to raise a pound of water from 60° F. to 61° F. An
approximation to this conclusion was contained in the paper which he
communicated to the British Association at Oxford in 1847.
The results of a later determination made with an improved apparatus,
and completed in 1878, gave a very slightly higher result. When
corrected to the corresponding Fahrenheit degree on the air thermometer
it must be increased by somewhat less than one per cent. The exact
relation has been the subject during the last twenty years of much
refined experimental work, but without any serious alteration of the
number indicated above.
It is probable that in consequence of the conference which he had with
Joule at Oxford Thomson had his thoughts turned for some time almost
exclusively to the dynamical theory of heat engines. He worked at the
subject almost continuously for a long time, sending paper after paper
to the Edinburgh Royal Society. As we have seen, he had given Joule a
description of Carnot's essay on the Motive Power of Heat and the
conclusions, or some of them, therein contained. Joule's result, and the
thermodynamic law which it established, gave the key to the correction
of Carnot's theory necessary to bring it into line with a complete
doctrine of energy, which should take account of work done against
frictional resistances.
Mayer of Heilbronn had endeavoured to determine the dynamical equivalent
of heat in 1842, by calculating from the knowledge available at the time
of the two specific heats of air--the specific heat at constant pressure
and the specific heat at constant volume--the heat value of the work
spent in compressing air from a given volume to a smaller one. The
principle of this determination is easily understood, but it involves an
assumption that is not always clearly perceived. Let the air be imagined
confined in a cylinder closed by a frictionless piston, which is kept
from moving out under the air pressure by force applied from without.
Let heat be given to the air so as to raise its temperature, while the
piston moves out so as to keep the pressure constant. If the pressure be
p and the increase of volume be dv, the work done against the external
force is pdv. Let the rise of temperature be one degree of the
Centigrade scale, and the mass of air be one gramme, the heat given to
the gas is the specific heat Cp of the gas at constant pressure, for
there is only slight variation of specific heat with temperature. But if
the piston had been fixed the heat required for the same rise of
temperature would have been Cv, the specific heat at constant volume.
Now Mayer assumed that the excess of the specific heat Cp above Cv was
the thermal equivalent of the work pdv done in the former case. Thus he
obtained the equation J(Cp － Cv) = pdv, where J denotes the dynamical
equivalent of heat and Cp, Cv are taken in thermal units. But if a be
the coefficient of expansion of the air under constant pressure (that is
1⧸273), and v₀ be the volume of the air at 0° C., we have dv = av₀,
so that J(Cp － Cv) = apv₀. Now if p be one atmosphere, say 1.014 × 10^6
dynes per square centimetre, and the temperature be the freezing point
of water, the volume of a gramme of air is 1⧸.001293 in cubic
centimetres. Hence
J(Cp － Cv) = (1.014 × 10^6)⧸(273 × .001293)
from which, if Cp － Cv is known, the value of J can be found.
In Mayer's time the difference of the specific heats of air was
imperfectly known, and so J could not be found with anything like
accuracy. From Regnault's experiments on the specific heat at constant
pressure, and from the known ratio of the specific heats as deduced
from the velocity of sound combined with Regnault's result, the value of
Cp － Cv may be taken as .0686. Thus J works out to 42.2 × 10^6,
in ergs per calorie, which is not far from the true value. Mayer
obtained a result equivalent to 36.5 × 10^6 ergs per calorie.
The assumption on which this calculation is founded is that there is no
alteration of the internal energy of the gas in consequence of
expansion. If the air when raised in temperature, and at the same time
increased in volume, contained less internal energy than when simply
heated without alteration of volume, the energy evolved would be
available to aid the performance of the work done against external
forces, and less heat would be required, or, in the contrary case, more
heat would be required, than would be necessary if the internal energy
remained unaltered. Thus putting dW for pdv, the work done, e for the
internal energy before expansion, and dH for the heat given to the gas,
we have obviously the equation
JdH = de + dW
where de is the change of internal energy due to the alteration of
volume, together with the alteration of temperature. If now the
temperature be altered without expansion, no external work is done and
dW for that case is zero. Let ∂e and ∂H be the energy change and the
heat supplied, then in this case
J∂H = ∂e + O
Thus
J(dH － ∂H) = de － ∂e + dW
and the assumption is that de = ∂e, so that dW = J(dH － ∂H); that
is, dW = J(Cp － Cv), when the rise of temperature is 1° C. and the
mass of air is one gramme. This assumption requires justification, and
by an experiment of Joule's, which was repeated in a more sensitive form
devised by Thomson, it was shown to be a very close approximation to
the truth. Joule's experiment is well known: the explanation given
above may serve to make clear the nature of the research undertaken
later by Thomson and Joule conjointly.
The inverse process, the conversion of heat into work, required
investigation, and it is this that constitutes the science of
thermodynamics. It was the subject of the celebrated _Réflexions sur la
Puissance Motrice du Feu, et sur les Machines Propres à Développer cette
Puissance_, published in 1824 by Sadi Carnot, an uncle of the late
President of the French Republic. Only a few copies of this essay were
issued, and its text was known to very few persons twenty-four years
later, when it was reprinted by the Academy of Sciences. Its methods and
conclusions were set forth by Thomson in 1849 in a memoir which he
entitled, "An Account of Carnot's Theory of the Motive Power of Heat."
Numerical results deduced from Regnault's experiments on steam were
included; and the memoir as a whole led naturally in Thomson's hands to
a corrected theory of heat engines, which he published in 1852. Carnot's
view of the working of a heat engine was founded on the analogy of the
performance of work by a stream of water descending from a higher level
to a lower. The same quantity of water flows away in a given time from a
water wheel in the tail-race as is received in that time by the wheel
from the supply stream. Now a heat engine receives heat from a supplying
body, or source, at one temperature and parts with heat to another body
(for example, the condenser of a steam engine) at a lower temperature.
This body is usually called the refrigerator. According to Carnot these
temperatures corresponded to the two levels in the case of the water
wheel; the heat was what flowed through the engine. Thus in his theory
as much heat was given up by a heat engine to the body at the lower
temperature as was received by it from the source. The heat was simply
transferred from the body at the higher temperature to the body at the
lower; and this transference was supposed to be the source of the
work.[17]
The first law of thermodynamics based on Joule's proportionality of heat
produced to work expended, and the converse assumed and verified _a
posteriori_, showed that this view is erroneous, and that the heat
delivered to the refrigerator must be less in amount than that received
from the source, by exactly the amount which is converted into work,
together with the heat which, in an imperfect engine, is lost by
conduction, etc., from the cylinder or other working chamber. This
change was made by Thomson in his second paper: but he found the ideas
of Carnot of direct and fruitful application in the new theory. These
were the cycle of operations and the ideal reversible engine.
In the Carnot cycle the working substance--which might be a gas or a
vapour, or a liquid, or a vapour and its liquid in contact: it did not
matter what for the result--was supposed to be put through a succession
of changes in which the final state coincided with the initial. Thus the
substance having been brought back to the same physical condition as it
had when the cycle began, has the same internal energy as it had at the
beginning, and in the reckoning of the work done by or against external
forces, nothing requires to be set to the account of the working
substance. This is the first great advantage of the method of reasoning
which Carnot introduced.
The ideal engine was a very simple affair: but the notion of
reversibility is difficult to express in a form sufficiently definite
and precise. Carnot does not attempt this; he merely contents himself
with describing certain cycles of operations which obviously can be
carried through in the reverse order. Nor does Thomson go further in his
"Account of Carnot's Theory," though he states the criterion of a
perfect engine in the words, "A perfect thermodynamic engine is such
that, whatever amount of mechanical effect it can derive from a certain
thermal agency, if an equal amount be spent in working it backwards, an
equal reverse thermal effect will be produced." This proposition was
proved by Carnot: and the following formal statement in the essay is
made: "La puissance motrice de la chaleur est independante des agents
mis en œuvre pour la réaliser: sa quantité est fixée uniquement par
les temperatures des corps entre lesquels se fait, en dernier résultat,
le transport du calorique." The result involved in each, that the work
done in a cycle by an ideal engine depends on the temperatures between
which it works and not at all on the working substance, is, as we shall
see, of the greatest importance. The proof of the proposition, by
supposing a more efficient engine than the ideal one to exist, and to be
coupled with the latter, so that the more efficient would perform the
cycle forwards and the ideal engine the same cycle backwards, is well
known. In Carnot's view the former would do more work by letting down a
given quantity of heat from the higher to the lower temperature than was
spent on the latter in transferring the same quantity of heat from the
lower to the higher temperature, so that no heat would be taken from or
given to source or refrigerator, while there would be a gain of work on
the whole. This would be equivalent to admitting that useful work could
be continually performed without any resulting thermal or other change
in the agents performing the work. Even at that time this could not be
admitted as possible, and hence the supposition that a more efficient
engine than the reversible one could exist was untenable.
Carnot showed that the work done by an ideal engine, in transferring
heat from one temperature to another, was to be found by means of a
certain function of the temperature, hence called "Carnot's function."
The corresponding function in the true dynamical theory is always called
Carnot's. A certain assignment of value to it gave, as we shall see,
Thomson's famous absolute thermodynamic scale of temperature.
In the light of the facts and theories which now exist, and are almost
the commonplaces of physical text books, it is very interesting to
review the ideas and difficulties which occurred to the founders of the
science of heat sixty years ago. For example, Thomson asks, in his
"Account of Carnot's Theory," what becomes of the mechanical effect
which might be produced by heat which is transferred from one body to
another by conduction. The heat leaves one body and enters another and
no mechanical effect results: if it passed from one to the other through
a heat engine, mechanical effect would be produced: what is produced in
place of the mechanical effect which is lost? This he calls a very
"perplexing question," and hopes that it will, before long, be cleared
up. He states, further, that the difficulty would be entirely avoided by
abandoning Carnot's principle that mechanical effect is obtained by "the
transference of heat from one body to another at a lower temperate."
Joule urges precisely this solution of the difficulty in his paper, "On
the Changes of Temperature produced by the Rarefaction and Condensation
of Air" (_Phil. Mag._, May 1845). Thomson notes this, but adds, "If we
do so, however, we meet with innumerable other difficulties--insuperable
without further experimental investigation, and an entire reconstruction
of the theory of heat from its foundation. It is in reality to
experiment that we must look, either for a verification of Carnot's
axiom, and an explanation of the difficulty we have been considering, or
for an entirely new basis of the Theory of Heat."
The experiments here asked for had already, as was soon after perceived
by Thomson, been made by Joule, not merely in his determinations of the
dynamical equivalent of heat, but in his exceedingly important
investigation of the energy changes in the circuit of a voltaic cell, or
of a magneto-electric machine. Moreover, the answer to this "very
perplexing question" was afterwards to be given by Thomson himself in
his paper, "On a Universal Tendency in Nature to the Dissipation of
Mechanical Energy," published in the Edinburgh Proceedings in 1852.
Again, we find, a page or two earlier in the "Account of Carnot's
Theory," the question asked with respect to the heat evolved in the
circuit of a magneto-electric machine, "Is the heat which is evolved in
one part of the closed conductor merely transferred from those parts
which are subject to the inducing influence?" and the statement made
that Joule had examined this question, and decided that it must be
answered in the negative. But Thomson goes on to say, "Before we can
finally conclude that heat is absolutely generated in such operations,
it would be necessary to prove that the inducing magnet does not become
lower in temperature and thus compensate for the heat evolved in the
conductor."
Here, apparently, the idea of work done in moving the magnet, or the
conductor in the magnetic field, is not present to Thomson's mind; for
if it had been, the idea that the work thus spent might have its
equivalent, in part, at least, in heat generated in the circuit, would
no doubt have occurred to him and been stated. This idea had been used
just a year before by Helmholtz, in his essay "Die Erhaltung der Kraft,"
to account for the heat produced in the circuit by the induced current,
that is, to answer the first question put above in the sense in which
Joule answered it. The subject, however, was fully worked out by Thomson
in a paper published in the _Philosophical Magazine_ for December 1851,
to which we shall refer later.
Tables of the work performed by various steam engines working between
different stated temperatures were given at the close of the "Account,"
and compared with the theoretical "duty" as calculated for Carnot's
ideal perfect engine. Of course the theoretical duty was calculated from
the temperatures of the boiler and condenser; the much greater fall of
temperature from the furnace to the boiler was neglected as inevitable,
so that the loss involved in that fall is not taken account of. Carnot's
theory gave for the theoretical duty of one heat unit (equivalent to
1390 foot-pounds of work) 440 foot-pounds for boiler at 140° C. and
condenser at 30° C.; and the best performance recorded was 253
foot-pounds, giving a percentage of 57.5 per cent. The worst was that of
common engines consuming 12 lb. of coal per horse-power per hour, and
gave 38.1 foot-pounds, or a percentage of 8.6 per cent. These
percentages become on the dynamical theory 68 and 10.3, since the true
theoretical duty for the heat unit is only 371 foot-pounds.
It is worthy of notice that the indicator-diagram method of graphically
representing the changes in a cycle of operations is adopted in
Thomson's "Account," but does not occur in Carnot's essay. The cycles
consist of two isothermal changes and two adiabatic changes; that is,
two changes at the temperatures of the source and refrigerator
respectively, and two changes--from the higher to the lower temperature,
and from the lower to the higher. These changes are made subject to the
condition in each case that the substance neither gains nor loses energy
in the form of heat, but is cooled in the one case by expansion and
heated in the other by compression. The indicator diagram was due not to
Thomson but to Clapeyron (see p. 99 above), who used it to illustrate an
account of Carnot's theory.
There appeared in the issue of the Edinburgh _Philosophical
Transactions_ for January 2, 1849, along with the "Account of Carnot's
Theory," a paper by James Thomson, entitled, "Theoretical Considerations
on the Effect of Pressure in Lowering the Freezing Point of Water." The
author predicted that, unless the principle of conservation of energy
was at fault, the effect of increase of pressure on water in the act of
freezing would be to lower the freezing point; and he calculated from
Carnot's theory the amount of lowering which would be produced by a
given increment of pressure. The prediction thus made was tested by
experiments carried out in the Physical Laboratory by Thomson, and the
results obtained completely confirmed the conclusions arrived at by
theory. This prediction and its verification have been justly regarded
as of great importance in the history of the dynamical theory of heat;
and they afford an excellent example of the predictive character of a
true scientific theory. The theory of the matter will be referred to in
the next chapter.
CHAPTER VIII
THERMODYNAMICS AND ABSOLUTE THERMOMETRY
The first statement of the true dynamical theory of heat, based on the
fundamental idea that the work done in a Carnot cycle is to be accounted
for by an excess of the heat received from the source over the heat
delivered to the refrigerator, was given by Clausius in a paper which
appeared in _Poggendorff's Annalen_ in March and April 1850, and in the
_Philosophical Magazine_ for July 1850, under a title which is a German
translation of that of Carnot's essay. In that paper the First Law of
Thermodynamics is explicitly stated as follows: "In all cases in which
work is produced by the agency of heat, a quantity of heat proportional
to the amount of work produced is expended, and, inversely, by the
expenditure of that amount of work exactly the same amount of heat is
generated." Modern thermodynamics is based on this principle and on the
so-called Second Law of Thermodynamics; which is, however, variously
stated by different authors. According to Clausius, who used in his
paper an argument like that of Carnot based on the transference of heat
from the source to the refrigerator, the foundation of the second law
was the fact that heat tends to pass from hotter to colder bodies. In
1854 (_Pogg. Ann._, Dec. 1854) he stated his fundamental principle
explicitly in the form: "Heat can never pass from a colder to a hotter
body, unless some other change, connected therewith, take place at the
same time," and gives in a note the shorter statement, which he regards
as equivalent: "Heat cannot of itself pass from a colder to a hotter
body."
We shall not here discuss the manner in which Clausius applied this
principle: but he arrived at and described in his paper many important
results, of which he must therefore be regarded as the primary
discoverer. His theory as originally set forth was lacking in clearness
and simplicity, and was much improved by additions made to it on its
republication, in 1864, with other memoirs on the Theory of Heat.
In the _Transactions R.S.E._, for March 1851, Thomson published his
great paper, "On the Dynamical Theory of Heat." The object of the paper
was stated to be threefold: (1) To show what modifications of Carnot's
conclusions are required, when the dynamical theory is adopted: (2) To
indicate the significance in this theory of the numerical results
deduced from Regnault's observations on steam: (3) To point out certain
remarkable relations connecting the physical properties of all
substances established by reasoning analogous to that of Carnot, but
founded on the dynamical theory.
This paper, though subsequent to that of Clausius, is very different in
character. Many of the results are identical with those previously
obtained by Clausius, but they are reached by a process which is
preceded by a clear statement of fundamental principles. These
principles have since been the subject of discussion, and are not free
from difficulty even now; but a great step in advance was made by their
careful formulation in Thomson's paper, as a preliminary to the
erection of the theory and the deduction of its consequences. Two
propositions are stated which may be taken as the First and Second Laws
of Thermodynamics. One is equivalent to the First Law as stated in p.
116, the other enunciates the principle of Reversibility as a criterion
of "perfection" of a heat engine. We quote these propositions.
"Prop. I (Joule).--When equal quantities of mechanical effect are
produced by any means whatever from purely thermal sources, or lost in
purely thermal effects, equal quantities of heat are put out of
existence or are generated."
"Prop. II (Carnot and Clausius).--If an engine be such that when worked
backwards, the physical and mechanical agencies in every part of its
motions are all reversed, it produces as much mechanical effect as can
be produced by any thermodynamic engine, with the same temperatures of
source and refrigerator, from a given quantity of heat."
Prop. I was proved by assuming that heat is a form of energy and
considering always the work effected by causing a working substance to
pass through a closed cycle of changes, so that there was no change of
internal energy to be reckoned with.
Prop. II was proved by the following "axiom": "It is impossible, by
means of inanimate material agency, to derive mechanical effect from any
portion of matter by cooling it below the temperature of the coldest of
the surrounding objects." This is rather a postulate than an axiom; for
it can hardly be contended that it commands assent as soon as it is
stated, even from a mind which is conversant with thermal phenomena. It
sets forth clearly, however, and with sufficient guardedness of
statement, a principle which, when the process by which work is done is
always a cyclical one, is not found contradicted by experience, and one,
moreover, which can be at once explicitly applied to demonstrate that no
engine can be more efficient than a reversible one, and that therefore
the efficiency of a reversible engine is independent of the nature of
the working substance.
It has been suggested by Clerk Maxwell that this "axiom" is contradicted
by the behaviour of a gas. According to the kinetic theory of gases an
elevation of temperature consists in an increase of the kinetic energy
of the translatory motion of the gaseous particles; and no doubt there
actually is, from time to time, a passage of some more quickly moving
particles from a portion of a gas in which the average kinetic energy is
low, to a region in which the average kinetic energy is high, and thus a
transference of heat from a region of low temperature to one of higher
temperature. Maxwell imagined a space filled with gas to be divided into
two compartments A and B by a partition in which were small massless
trapdoors, to open and shut which required no expenditure of energy. At
each of these doors was stationed a "sorting demon," whose duty it was
to allow every particle having a velocity greater than the average to
pass through from A to B, and to stop all those of smaller velocity than
the average. Similarly, the demons were to prevent all quickly moving
particles from going across from B to A, and to pass all slowly moving
particles. In this way, without the expenditure of work, all the quickly
moving particles could be assembled in one compartment, and all the
slowly moving particles in the other; and thus a difference of
temperatures between the two compartments could be brought about, or a
previously existing one increased by transference of heat from a colder
to a hotter mass of gas.
Contrary to a not uncommon belief, this process does not invalidate
Thomson's axiom as he intended it to be understood. For the gas referred
to here is what he would have regarded as the working substance of the
engine, by the cycles of which all the mechanical effect was derived;
and it is not, at the end of the process, in the state as regards
average kinetic energy of the particles in which it was at first. That
this was his answer to the implied criticism of his axiom contained in
Maxwell's illustration, those who have heard him refer to the matter in
his lectures are well aware. But of course it is to be understood that
the substance returns to the same state only in a statistical sense.
Thomson's demonstration that a reversible engine is the most efficient
is well known, and need not here be repeated in detail. The reversible
engine may be worked backwards, and the working substance will take in
heat where in the direct action it gave it out, and _vice versa_: the
substance will do work against external forces where in the direct
action it had work done upon it, and _vice versa_: in short, all the
physical and mechanical changes will be of the same amount, but merely
reversed, at every stage of the backward process. Thus if an engine A be
more efficient than a reversible one B, it will convert a larger
percentage of an amount of heat H taken in at the source into work than
would the reversible one working between the same temperatures. Thus if
h be the heat given to the refrigerator by A, and h' that given by B
when both work directly and take in H; h must be less than h'. Then
couple the engines together so that B works backwards while A works
directly. A will take in H and deliver h, and do work equivalent to
H － h. B will take h' from the refrigerator and deliver H to the source,
and have work equivalent to H － h' spent upon it. There will be no heat
on the whole given to or taken from the source; but heat h' － h will be
taken from the refrigerator, and work equivalent to this will be done.
Thus _by a cyclical process_, which leaves the working substance as it
was, work is done at the expense of heat taken from the refrigerator,
which Thomson's postulate affirms to be impossible. Therefore the
assumption that an engine more efficient than the reversible engine
exists must be abandoned; and we have the conclusion that all reversible
engines are equally efficient.
Thomson acknowledged in his paper the priority of Clausius in his proof
of this proposition, but stated that this demonstration had occurred to
him before he was aware that Clausius had dealt with the matter. He now
cited, as examples of the First Law of Thermodynamics, the results of
Joule's experiments regarding the heat produced in the circuits of
magneto-electric machines, and the fact that when an electric current
produced by a thermal agency or by a battery drives a motor, the heat
evolved in the circuit by the passage of the current is lessened by the
equivalent of the work done on the motor.
[Illustration: FIG. 12.]
In the Carnot cycle, the first operation is an isothermal expansion (AB
in Fig. 12), in which the substance increases in volume by dv, and takes
in from the source heat of amount Mdv. The second operation is an
adiabatic expansion, BC, in which the volume is further increased and
the temperature sinks by dt to the temperature of the refrigerator. The
third operation is an isothermal compression, CD, until the volume and
pressure are such that an adiabatic compression DA will just bring the
substance back to the original state. If ∂p⧸∂t be the rate of
increase of pressure with temperature when the volume is constant, the
step of pressure from one isothermal to the other is ∂p⧸∂t.dt; and
thus the area of the closed cycle in the diagram which measures the
external work done in the succession of changes is ∂p⧸∂t.dtdv. Now,
by the second law, the work done must be a certain fraction of the
work-equivalent of the heat, Mdv, taken in from the source. This
fraction is independent of the nature of the working substance, but
varies with the temperature, and is therefore a function of the
temperature. Its ratio to the difference of temperature dt between
source and refrigerator was called "Carnot's function," and the
determination of this function by experiment was at first perhaps the
most important problem of thermodynamics. Denoting it by μ, we have
the equation
∂p⧸∂t = μM ... (A)
which may be taken as expressing in mathematical language the second law
of thermodynamics. M is here so chosen that Mdv is the heat expressed in
units of work, so that μ does not involve Joule's equivalent of heat.
This equation was given by Carnot: it is here obtained by the dynamical
theory which regards the work done as accounted for by disappearance,
not transference merely, of heat.
The work done in the cycle becomes now μMdtdv, or if H denote Mdv, it
is μHdt. The fraction of the heat utilised is thus μdt. This is
called the efficiency of the engine for the cycle.
From the first law Thomson obtained another fundamental equation. For
every substance there is a relation connecting the pressure p (or more
generally the stress of some type), the volume v (or the configuration
according to the specified stress), and the temperature. We may
therefore take arbitrary changes of any two of these quantities: the
relation referred to will give the corresponding change of the third.
Thomson chose v and t as the quantities to be varied, and supposed them
to sustain arbitrary small changes dv and dt in consequence of the
passage of heat to the substance from without. The amount of heat taken
in is Mdv + Ndt, where Mdv and Ndt are heats required for the changes
taken separately. But the substance expanding through dv does external
work pdv. Thus the net amount of energy given to the substance from
without is Mdv + Ndt － pdv or (M － p)dv + Ndt; and if the substance
is made to pass through a cycle of changes so that it returns to the
physical state from which it started, the whole energy received in the
cycle must be zero. From this it follows that the rate of variation of
M － p when the temperature but not the volume varies, is equal to the
rate of variation of N when the volume but not the temperature varies.
To see that this relation holds, the reader unacquainted with the
properties of perfect differentials may proceed thus. Let the substance
be subjected to the infinitesimal closed cycle of changes defined by (1)
a variation consisting of the simultaneous changes dv, dt of volume and
temperature, (2) a variation －dv of volume only, (3) a variation －dt of
temperature only. M － p and N vary so as to have definite values for
the beginning and end of each step, and the proper mean values can
be written down for each step at once, and therefore the value of
(M － p)dv + Ndt obtained. Adding together these values for the three
steps we get the integral for the cycle. The condition that this should
vanish is at once seen to be the relation stated above.
This result combined with the equation A derived from the second law,
gives an important expression for Carnot's function.
We shall not pursue this discussion further: so much is given to make
clear how certain results as to the physical properties of substances
were obtained, and to explain Thomson's scale of absolute thermodynamic
temperature, which is by far the most important discovery within the
range of theoretical thermodynamics.
There are several scales of temperature: in point of fact the scale of a
mercury-in-glass thermometer is defined by the process of graduation,
and therefore there are as many such scales as there are thermometers,
since no two specimens of glass expand in precisely the same way. Equal
differences of temperature do not correspond to equal increments of
volume of the mercury: for the glass envelope expands also and in its
own way. On the scale of a constant pressure gas thermometer changes of
temperature are measured by variations of volume of the gas, while the
pressure is maintained constant; on a constant volume gas thermometer
changes of temperature are measured by alterations of pressure while the
volume of the gas is kept constant. Each scale has its own independent
definition, thus if the pressure of the gas be kept constant, and the
volume at temperature 0° C. be v₀ and that at any other temperature be
v₁ we define the numerical value t, this latter temperature, by the
equation v = v₀(1 + Et), where E is 1⧸100 of the increase of volume
sustained by the gas in being raised from 0° C. to 100° C. These are
the temperatures of reference on an ordinary centigrade thermometer,
that is, the temperature of melting ice and of saturated steam
under standard atmospheric pressure, respectively. Thus t has
the value (v⧸v₀ － 1)⧸E, and is the temperature (on the constant
pressure scale of the gas thermometer) corresponding to the volume v.
Equal differences of temperature are such as correspond to equal
increments of the volume at 0° C.
Similarly, on the constant volume scale we obtain a definition of
temperature from the pressure p, by the equation t = (p⧸p₀ － 1)⧸E',
where p₀ is the pressure at 0° C., and E' is 1⧸100 of the change of
pressure produced by raising the temperature from 0° C. to 100° C.
For air E is approximately 1⧸273, and thus t = 273(v － v₀)⧸v₀.
If we take the case of v = 0, we get t = －273. Now, although this
temperature may be inaccessible, we may take it as zero, and the
temperature denoted by t is, when reckoned from this zero, 273 + t.
This zero is called the absolute zero on the constant pressure air
thermometer. The value of E' is very nearly the same as that of E; and
we get in a similar manner an absolute zero for the constant volume
scale. If the gas obeyed Boyle's law exactly at all temperatures, E
would not differ from E'.
It was suggested to Thomson by Joule, in a letter dated December 9,
1848, that the value of μ might be given by the equation
μ = JE⧸(1 + Et). Here we take heat in dynamical units, and therefore
the factor J is not required. With these units Joule's suggestion is
that μ = E⧸(1 + Et), or with E = 1⧸273 μ = 1⧸(273 + t), that is,
μ = 1⧸T where T is the temperature reckoned in centigrade degrees from
the absolute zero of the constant pressure air thermometer.
The possibility of adopting this value of μ was shown by Thomson to
depend on whether or not the heat absorbed by a given mass of gas in
expanding without alteration of temperature is the equivalent of the
work done by the expanding gas against external pressure. The heat H
absorbed by the air in expanding from volume V to another volume V' at
constant temperature is the integral of Mdv taken from the former volume
to the latter. But by the value of M given on p. 121, if W be the
integral of pdv, that is the work done by the air in the expansion,
∂W⧸∂t = μH. The equation fulfilled by the gas at constant pressure
(the defining equation for t), v = v₀(1 + Et), gives for the integral
of pdv, that is W, the equation W = pv₀(1 + Et)log(V'⧸V), so that
∂W⧸∂t = EW⧸(1 + Et). Thus μH = EW⧸(1 + Et).
Hence it follows that if μ = E⧸(1 + Et), the value of H will be simply
W. Thus Joule's suggested value of μ is only admissible if the work
done by the gas in expanding from a given volume to any other is the
equivalent of the heat absorbed; or, which is the same thing, if the
external work done in compressing the gas from one volume to another is
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