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Title: Lord Kelvin - An account of his scientific life and work
Author: Gray, Andrew, 1847-1925
Language: English
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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
the equivalent of the heat developed.

This result naturally suggests the formation of a new scale of
thermometry by the adoption of the defining relation T = 1⧸μ, where T
denotes temperature. A scale of temperature thus defined is proposed
in the paper by Joule and Thomson, "On the Thermal Effects of Fluids
in Motion," Part II, which was published in the _Philosophical
Transactions_ for June 1854, and is what is now universally known as
Thomson's scale of absolute thermodynamic temperature. It can, of
course, be made to give 100 as the numerical value of the temperature
difference between 0° C. and 100° C. by properly fixing the unit of T.
This scale was the natural successor, in the dynamical theory, of one
which Thomson had suggested in 1848, and which was founded, according
to Carnot's idea, on the condition that a unit of heat should do the
same amount of work in descending through each degree. This, as he
pointed out, might justly be called an absolute scale, since it would be
independent of the physical properties of any substance. In the same
sense the scale defined by T = 1⧸μ is truly an absolute scale.

The new scale gives a simple expression for the efficiency of a perfect
engine working between two physically given temperatures, and assigns
the numerical values of these temperatures; for the heat H taken in from
the source in the isothermal expansion which forms the first operation
of the cycle (p. 120) is Mdv, and, as we have seen, the work done in the
cycle is ∂p⧸∂t.dtdv, or μHdt. If we adopt the expression 1⧸T for
μ, we may put dT for dt; and we obtain for the work done the
expression HdT⧸T. The work done is thus the fraction dT⧸T of the heat
taken in, and this is what is properly called the efficiency of the
engine for the cycle.

If we suppose the difference of temperatures between source and
refrigerator to be finite, T - T', say, then since T is the temperature
of the source, we have for the efficiency (T - T')⧸T. If the heat taken
in be H, the heat rejected is HT'⧸T, so that the heat received by the
engine is to the heat rejected by it in the ratio of T' to T. Thus, as
was done by Thomson, we may define the temperatures of the source and
refrigerator as proportional to the heat taken in from the source and
the heat rejected to the refrigerator by a perfect engine, working
between those temperatures. The scale may be made to have 100 degrees
between the temperature of melting ice and the boiling point, as
already explained. We shall return to the comparison of this scale with
that of the air thermometer. At present we consider some of the
thermodynamic relations of the properties of bodies arrived at by
Thomson.

First we take the working substance of the engine as consisting of
matter in two states or phases; for example, ice and water, or water and
saturated steam. Let us apply equation (A) to this case. If v₁, v₂ be
the volume of unit of mass in the first and second states respectively,
the isothermal expansion of the first part of the cycle will take place
in consequence of the conversion of a mass dm from the first state to
the second. Thus dv, the change of volume, is dm(v₂ - v₁). Also if L
be the latent heat of the substance in the second state, _e.g._ the
latent heat of water, Mdv = Ldm; so that M(v₂ - v₁) = L. If dp be the
step of pressure corresponding to the step dT of temperature, equation
(A) becomes

  dT⧸T = dp(v₂ - v₁)⧸L                                     ... (B)

In the case of coexistence of the liquid and solid phases, this gives us
the very remarkable result that a change of pressure dp will raise or
lower the temperature of coexistence of the two phases, that is, the
melting point of the solid, by the difference of temperature, dT,
according as v₂ is greater or less than v₁ Thus a substance like
water, which expands in freezing, so that v₂ - v₁ is negative, has
its freezing point lowered by increase of pressure and raised by
diminution of pressure. This is the result predicted by Professor James
Thomson and verified experimentally by his brother (p. 113 above).
On the other hand, a substance like paraffin wax, which contracts in
solidifying, would have its melting point raised by increase of pressure
and lowered by a diminution of pressure.

The same conclusions would be applicable when the phases are liquid and
vapour of the same substance, if there were any case in which v₂ - v₁
is negative. As it is we see, what is well known to be the case, that
the temperature of equilibrium of a liquid with its vapour is raised by
increase of pressure.

Another important result of equation (B), as applied to the liquid and
vapour phases of a substance, is the information which it gives as to
the density of the saturated vapour. When the two phases coexist the
pressure is a function of the temperature only. Hence if the relation of
pressure to temperature is known, dp⧸dT can be calculated, or obtained
graphically from a curve; and the volume v₂ per unit mass of the vapour
will be given in terms of dp⧸dT, the temperature T, and the volume v per
unit mass of the liquid. The density of saturated steam at different
temperatures is very difficult to measure experimentally with any
approach of accuracy: but so far as experiment goes equation (B) is
confirmed. The theory here given is fully confirmed by other results,
and equation (B) is available for the calculation of v₂ for any
substance for which the relation between p and T is known. It is thus
that the density of saturated steam can best be found.

We can obtain another important result for the case of the working
substance in two phases from equation (B). The relation is

  ∂L⧸∂T + c - h = L⧸T                                      ... (C)

where c and h are the specific heats of the substance in the two phases
respectively, and L is the latent heat of the second phase at absolute
temperature T.

We shall obtain the relation in another way, which will illustrate
another mode of dealing with a cycle of operations which Thomson
employed. Any small step of change of a substance may be regarded as
made up of a step of volume, say, followed by a step of temperature,
that is, by an isothermal step followed by an adiabatic step. In this
way any cycle of operations whatever may be regarded as made up of a
series of Carnot cycles. But without regarding any cycle of a more
general kind than Carnot's as thus compounded, we can draw conclusions
from it by the dynamical theory provided only it is reversible. Suppose
a gramme, say, of the substance to be taken at a specified temperature T
in the lower phase, and to be changed to the other phase at that
temperature. The heat taken in will be L and the expansion will be
v₂ - v₁. Next, keeping the substance in the second phase, and in
equilibrium with the first phase (that is, for example, if the second
phase is saturated vapour, the saturation is to continue in the further
change), let the substance be lowered in temperature by dT. The heat
given out by the substance will be hdT, where h is the specific heat of
the substance in the second phase. Now at the new temperature T - dT let
the substance be wholly brought back to the second phase; the heat given
out will be L - ∂L⧸∂T.dT. Finally, let the substance, now again
all in the first phase, be brought to the original temperature: the heat
taken in will be cdt, where c is the specific heat in the first phase.
Thus the net excess of heat taken in over heat given out in the cycle is
(∂L⧸∂T + c - h)dT. This must, in the indicator diagram for the
changes specified, be the area of the cycle or (v₂ - v₁)∂p⧸∂T.dT.
But by equation (B) L⧸T(v₂ - v₁) = ∂p⧸∂T, and the area of the cycle is
(L⧸T)dT. Equating the two expressions thus found for the area we get
equation (C).

This relation was arrived at by Clausius in his paper referred to above,
and the priority of publication is his: it is here given in the form
which it takes when Thomson's scale of absolute temperature is used.

Regnault's experimental results for the heat required to raise unit mass
of water from the temperature of melting ice to any higher temperature
and evaporate it at that temperature enable the values of L⧸T and
∂L⧸∂T to be calculated, and therefore that of h to be found. It
appears that h is negative for all the temperatures to which Regnault's
experimental results can be held to apply. This, as was pointed out by
Thomson, means that if a mass of saturated vapour is made to expand so
as at the same time to fall in temperature, it must have heat given to
it, otherwise it will be partly condensed into liquid; and, on the other
hand, if the vapour be compressed and made to rise in temperature while
at the same time it is kept saturated, heat must be taken from it,
otherwise the vapour will become superheated and so cease to be
saturated.

It is convenient to notice here the article on Heat which Thomson wrote
for the ninth edition of the _Encyclopædia Britannica_. In that article
he gave a valuable discussion of ordinary thermometry, of thermometry by
means of the pressures of saturated vapour of different
substances--steam-pressure thermometers, he called them--of absolute
thermodynamic thermometry, all enriched with new experimental and
theoretical investigations, and appended to the whole a valuable
synopsis, with additions of his own, of the Fourier mathematics of heat
conduction.

First dealing with temperature as measured by the expansion of a liquid
in a less expansible vessel, he showed how it is in reality numerically
reckoned. This amounted to a discussion of the scale of an ordinary
mercury-in-glass thermometer, a subject concerning which erroneous
statements are not infrequently made in text-books. A sketch of
Thomson's treatment of it is given here.

Considering this thermometer as a vessel consisting of a glass bulb and
a long glass stem of fine and uniform bore, hermetically sealed and
containing only mercury and mercury vapour, he explained the numerical
relation between the temperature as shown by the instrument and the
volumes of the mercury and vessel. The scale is really defined by the
method of graduation adopted. Two points of reference are marked on the
stem at which the top of the mercury stands when the vessel is immersed
(1) in melting ice, (2) in saturated steam under standard atmospheric
pressure. The stem is divided into parts of equal volume of bore between
these two points and beyond each of them. For a centigrade thermometer
the bore-space between the two points is divided into 100 equal parts,
and the lower point of reference is marked 0 and the upper 100, and the
other dividing marks are numbered in accordance with this along the
stem. Each of these parts of the bore may be called a degree-space.

Now let the instrument contain in its bulb and stem, up to the mark 0, N
degree-spaces, and let v be the volume of a degree-space at that
temperature. The volume up to the mark 0 will be Nv, at that
temperature; and if the substance of the vessel be quite uniform in
quality and free from stress, N will be the same for all temperatures.
If v₀ be the volume of a degree-space at the temperature of melting ice
the volume of the mercury at that temperature will be Nv₀. If G be the
expansion of the glass when the volume of a degree-space is increased
from v₀ to v by the rise of temperature, then v = v₀(1 + G). The
volume of the mercury has been increased therefore to (N + n)v₀(1 + G)
by the same rise of temperature, if the top of the column is thereby
made to rise from the mark 0 so as to occupy n degree-spaces more than
before. But if E be the expansion of the mercury between the temperature
of melting ice and that which has now been attained, the volume of the
mercury is also Nv₀(1 + E). Hence N(1 + E) = (N + n)(1 + G). This gives
n = N(E - G)⧸(1 + G).

If we take, as is usual, n as measuring the temperature, and substitute
for it the symbol t, we have, since N = 100(1 + G₁₀₀)⧸(E₁₀₀ - G₁₀₀),

  t = 100 {(1 + G₁₀₀)⧸(1 + G)} {(E - G)⧸(E₁₀₀ - G₁₀₀)}     ... (D)

In this reckoning the definition of any temperature, let us say 37° C.,
is the temperature of the vessel and its contents when the top of the
mercury column stands at the mark 37 above 0, on the scale defined by
the graduation of the instrument; but the numerical signification with
relation to the volumes is given by equation (D). This shows that the
numerical measure of any temperature involves both the expansion of the
vessel and that of the glass vessel between the temperature of melting
ice and the temperature in question. This result may be contrasted with
the erroneous statement frequently made that equal increments of
temperature correspond to equal increments of the volume of the
thermometric substance. It also shows that different mercury-in-glass
thermometers, however accurately made and graduated, need not agree when
placed in a bath at any other temperature than 0° C. or 100° C. This
fact, and the results of the comparison of thermometers made with
different kinds of glass with the normal air thermometer, which was
carried out by Regnault, were always insisted on by Thomson in his
teaching when he dealt with the subject of heat. The scale of a
mercury-in-glass thermometer is too often in text-books, and even in
Acts of Parliament regarded as a perfectly definite thing, and the
expansion of a gas is not infrequently defined by this indefinite scale,
instead of being used as it ought to be, as the basis of definition of
the scale of the gas thermometer. The whole treatment of the so-called
gaseous laws is too often, from a logical point of view, a mass of
confusion.

In his article on Heat Thomson gave two definitions of the scale of
absolute temperature. One is that stated on p. 126 above, namely, that
the temperature of the source and refrigerator are in the ratio of the
heat taken in from the source to the heat given to the refrigerator,
when the engine describes a Carnot cycle consisting of two isothermal
and two adiabatic changes.

The other definition is better adapted for general use, as it applies to
any cycle whatever which is reversible. Let the working substance
expand under constant pressure by an amount dv (AB' in Fig. 12), and let
heat H be given to the substance at the same time. The external work
done is pdv. Thomson called pdv⧸H the work ratio. Now let the
temperature be raised by dT without giving heat to the substance or
taking heat from it, and let the corresponding pressure rise be dp; and
call dp⧸p the pressure ratio. The temperature ratio dT⧸T is equal to
the product of the work ratio and the pressure ratio, that is,

  dT⧸T = dvdp⧸H

This is clearly true; for dvdp is the area of a cycle like AB'C'D,
represented in Fig. 12, for which an amount of heat H is taken in,
though not in this case strictly at one temperature. And clearly, since
in Fig. 12 the change from B' to B is adiabatic, H is the heat which
would have to be taken in for the isothermal change AB in the Carnot
cycle ABCD, which has the same area as AB'C'D. Thus the efficiency of
the cycle is dvdp⧸H, and this by the former definition is dT⧸T.

Or we may regard the matter thus:--The amount of heat H which
corresponds to an infinitesimal expansion dv may be used in equation (A)
whether the expansion is isothermal or not, if we take T as the average
temperature of the expansion. Hence we have dp⧸dT = H⧸(dv.T), that is,
dT⧸T = dpdv⧸H. The theorem on p. 128 is obtained by what is virtually
this process.


COMPARISON OF ABSOLUTE SCALE WITH SCALE OF AIR THERMOMETER

The comparison which Joule and Thomson carried out of the absolute
thermodynamic scale with the scale of the constant pressure gas
thermometer has already been referred to, and it has been shown that the
two scales would exactly agree, that is, absolute temperature would be
simply proportional to the volume of the gas in a gas thermometer kept
at the temperature to be measured, if the internal energy of the gas
were not altered by an alteration of volume without alteration of
temperature, that is, if the de - ∂e of p. 107 above were zero. Joule
tested whether this was the case by immersing two vessels, connected by
a tube which could be opened or closed by a stopcock, in the water of a
calorimeter, ascertaining the temperature with a very sensitive
thermometer, and then allowing air which had already been compressed
into one of the vessels to flow into the other, which was initially
empty. It was found that no alteration of temperature of the water of
the calorimeter that could be observed was produced. But the volume of
the air had been doubled by the process, and if any sensible alteration
of internal energy had taken place it would have shown itself by an
elevation or a lowering of the temperature of the water, according as
the energy had been diminished or increased.

Thomson suggested that the gas to be examined should be forced through a
pipe ending in a fine nozzle, or, preferably, through a plug of porous
material placed in a pipe along which the gas was forced by a pump, and
observations made of the temperature in the steady stream on both sides
of the plug. The experiments were carried out with a plug of compressed
cotton-wool held between two metal disks pierced with holes, in a tube
of boxwood surrounded also by cotton-wool, and placed in a bath of water
closely surrounding the supply pipe. This was of metal, and formed the
end of a long spiral all immersed in the bath. Thus the temperature of
the gas approaching the plug was kept at a uniform temperature
determined by a delicate thermometer; another thermometer gave the
temperature in the steady stream beyond the plug.

In the case of hydrogen the experiments showed a slight heating effect
of passage through the plug; air, oxygen, nitrogen and carbonic acid
were cooled by the passage.

The theory of the matter is set forth in the original papers, and in a
very elegant manner in the article on Heat. The result of the analysis
shows that if ∂w be the positive or negative work-value of the heat
which will convert one gramme of the gas after passage to its original
temperature; and T be absolute temperature, and v volume of a gramme of
the gas at pressure p, and the difference of pressure on the two sides
of the plug be dp, the equation which holds is

  (1⧸T) (∂T⧸∂v) = 1⧸{v + (∂w⧸dp)}                      ... (E)[18]

It was found by Joule and Thomson that ∂w was proportional to dp for
values of dp up to five or six atmospheres. At different temperatures,
however, in the case of hydrogen the heating effect was found to
diminish with rise of temperature, being .100 of a degree centigrade at
4° or 5° centigrade, and .155 at temperatures of from 89° to 93°
centigrade for a difference of pressure due to 100 inches of mercury.

If there is neither heating nor cooling ∂w = 0, and we obtain by
integration T = Cv, where C is a constant.

Elaborate discussions of the theory of this experiment will be found in
modern treatises on thermodynamics, and in various recent memoirs, and
the differential equation has been modified in various ways, and
integrated on various suppositions, which it would be out of place to
discuss here.

The cooling effect of passing a gas such as air or oxygen through a
narrow orifice has been used to liquefy the gas. The stream of gas is
pumped along a pipe towards the opening, and that which has passed the
orifice and been slightly cooled is led on its way back to the pump
along the outside of the pipe by which more gas is approaching the
orifice, and so cools slightly the advancing current. The gas which
emerges later is thus cooler than that which emerged before, and the
process goes on until the issuing gas is liquefied and falls down into
the lower part of the pipe surrounding the orifice, whence it can be
drawn off into vessels constructed to receive and preserve it.

It is possible thus to liquefy hydrogen, which shows that at the low
temperature at which the process is usually started (an initial cooling
is applied) the passage through the orifice has a cooling effect as in
the other cases.

Another idea, that of _thermodynamic motivity_, on which Thomson
suggested might be founded a fruitful presentation of the subject of
thermodynamics, may be mentioned here. It was set forth in a letter
written to Professor Tait in May 1879. If a system of bodies be given,
all at different temperatures, it is possible to reduce them to a common
temperature, and by doing so to extract a certain amount of mechanical
energy from them. The temperatures must for this purpose be equalised
by perfect thermodynamic engines working between the final temperature
T₀, say, and the temperatures of the different parts of the system.
This process is one of the levelling up and the levelling down of
temperature; and the temperature T₀ is such that exactly the heat given
out at T₀ by certain engines, receiving heat from bodies of higher
temperature than T₀, is supplied to the engines which work between T₀
and bodies at lower temperatures. The whole useful work obtained in this
way was called by Thomson the motivity of the system. Of course
equalisation of temperature may be obtained by conduction, and in this
case the energy which might be utilised is lost. With two equal and
similar bodies at absolute temperatures T, T' the temperature to which
they are reduced when their motivity is extracted is √(TT'). If the
temperatures are equalised by conduction the resulting temperature is
higher, being ½(T + T'). Thus, if only the two bodies are available
for engines to work between, the motivity is the measure of the energy
lost when conduction brings about equalisation of temperature.

A very suggestive paper on the subject was published by Lord Kelvin in
the _Trans. R.S.E._, vol. 28, 1877-8.


DISSIPATION OF ENERGY

In connection with the theory of heat must be mentioned Thomson's great
generalisation, the theory of the dissipation of energy.[19] Most people
have some notion of the meaning of the physical doctrine of
conservation of energy, though in popular discourses it is usually
misstated. What is meant is that in a finite material system, which is
isolated in the sense that it is not acted on by force from without, the
total amount of energy--that is, energy of motion and energy of relative
position (including energy of chemical affinity) of the parts--remains
constant. The usual misstatement is that the energy of the universe is
constant. This may be true if the universe is finite; if the universe is
infinite in extent the statement has no meaning. In any case, we know
nothing about the universe as a whole, and therefore make no statements
regarding it.

But while there is thus conservation or constancy of amount of energy in
an isolated and finite material system, this energy may to residents on
the system become unavailable. For useful work within such a system is
done by conversion of energy from one form to another and the total
amount remains unchanged. But if this conversion is prevented all
processes which involve such conversion must cease, and among these are
vital processes.

The unavailable form which the energy of the system with which we are
directly and at present concerned, whatever may become of us ultimately,
is taking, according to Thomson's theory, is universally diffused heat.
How this comes about may be seen as follows. Even a perfect engine, if
the refrigerator be at the lowest available temperature, rejects a
quantity of heat which cannot be utilised for the performance of the
work. This heat is diffused by conduction and radiation to surrounding
bodies, and so to bodies more remote, and the general temperature of
the system is raised. Moreover, as heat engines are imperfect there is
heat rejected to the surroundings by conduction, and produced by work
done against friction, so that the heat thrown on the unavailable or
waste heap is still further increased.

Conduction of heat is the great agency by which energy is more and more
dispersed in this unavailable form throughout the totality of material
bodies. As has been seen, available motivity is continually wasted
through its agency; and in the flow of heat in the earth and in the sun
and other unequally heated bodies of our system the waste of energy is
prodigious. Aided by convection currents in the air and in the ocean it
continually equalises temperatures, but does so at an immense cost of
useful energy.

Then in our insanely wasteful methods of heating our houses by open
fires, of half burning the coal used in boiler furnaces, and allowing
unconsumed carbon to escape into the atmosphere in enormous quantities,
while a very large portion of the heat actually generated is allowed to
escape up chimneys with heated gases, the store of unavailable heat is
being added to at a rate which will entail great distress, if not ruin,
on humanity at no indefinitely distant future. It will be the height of
imprudence to trust to the prospect, not infrequently referred to at the
present time, of drawing on the energy locked up in the atomic structure
of matter. He would be a foolish man who would wastefully squander the
wealth he possesses, in the belief that he can recoup himself from mines
which all experience so far shows require an expenditure to work them
far beyond any return that has as yet been obtained.

It is not apart from our present theme to urge that it is high time the
question of the national economy of fuel, and the desirability of
utilising by afforestation the solar energy continually going to waste
on the surface of the earth, were dealt with by statesmen. If statesmen
would but make themselves acquainted with the results of physical
science in this magnificent region of cosmic economics there would be
some hope, but, alas! as a rule their education is one which inevitably
leads to neglect, if not to disdain of physical teaching.

From the causes which have been referred to, energy is continually being
dissipated, not destroyed, but locked up in greater and greater quantity
in the general heat of bodies. There is always friction, always heat
conduction and convection, so that as our stores of motional or
positional energy, whether of chemical substances uncombined, the
earth's motion, or what not, are drawn upon, the inevitable fraction,
too often a large proportion, is shed off and the general temperature
raised. After a large part of the whole existent energy has gone thus to
raise the dead level of things, no difference of temperature adequate
for heat engines to work between will be possible, and the inevitable
death of all things will approach with headlong rapidity.


THERMOELASTICITY AND THERMOELECTRICITY

In the second definition of the scale of absolute temperature just
discussed, stress of any type may be substituted for pressure, and the
corresponding displacement s for the change of volume. Thus for a piece
of elastic material put through a cycle of changes we may substitute dS
for dp and Ads for dv; where A is such a factor that AdSds is the work
done in the displacement ds by the stress dS. As an example consider a
wire subjected to simple longitudinal stress S. Longitudinal extension
is produced, but this is not the only change; there is at the same time
lateral contraction. However, s within certain limits is proportional to
S.

Let heat dH in dynamical measure be given to the wire while the stress S
is maintained constant, and let the extension increase from s to s + ds.
The stress S will do work ASds _on the wire_, and the work ratio will be
-ASds⧸dH. Now let the stress be increased to S + dS while the extension
is kept constant, and the absolute temperature raised from T to T + dT.
The stress ratio (as we may call it) is dS⧸S and the temperature ratio
dT⧸T. Thus we obtain (p. 134 above)

  -(dS⧸dT) = (1⧸TA) (dH⧸ds)

In his Heat article Thomson used the alteration e of strain under
constant stress (that is ds⧸l, where l is the length of the wire)
corresponding to an amount of heat sufficient to raise the temperature
under constant stress by 1°. Hence if K be the specific heat under
constant stress, and le be put for ds in the sense just stated, we have

  dT = -(TedS⧸Kρ)                                          ... (F)

where ρ is the density, since dH = KρlA.

The ratio of dH to the increase ds of the extension is positive or
negative, that is, the substance absorbs or evolves heat, when strained
under the condition of constant stress, according as dS⧸dT is negative
or positive. Or we may put the same thing in another way which is
frequently useful. If a wire subjected to constant stress has heat given
to it, ds is negative or positive, in other words the wire shortens or
lengthens, according as dS⧸dT is positive or negative, that is,
according as the stress for a given strain is increased or diminished by
increase of temperature.

It is known from experiment that a metal wire expands under constant
stress when heat is given to it, and thus we learn from the equation (F)
that the stress required for a given strain is diminished when the
temperature of the wire is raised. Again, a strip of india-rubber
stretched by a weight is shortened if its temperature is raised,
consequently the stress required for a given strain is increased by
rise of temperature.

These results, from a qualitative point of view, are self-evident. But
from what has been set forth it will be obvious that an equation exactly
similar to (F) holds whether the change ds of s is taken as before under
constant stress, or at uniform temperature, or whether the change dS of
S is effected adiabatically or at constant strain.

In all these cases the same equation

  dT = -T (edS⧸Kρ)                                         ... (G)

applies, with the change of meaning of dT involved.

This equation differs from that of Thomson as given in various places
(_e.g._ in the _Encyclopædia Britannica_ article on Elasticity which he
also wrote) in the negative sign on the right-hand side, but the
difference is only apparent. According to his specification a pressure
would be a positive stress, and an expansion a positive displacement,
and in applying the equation to numerical examples this must be borne in
mind so that the proper signs may be given to each numerical magnitude.
As an example of adiabatic change, a sudden extension of the wire
already referred to by an increase of stress dS may be considered. If
there is not time for the passage of heat from or to the surroundings of
the wire, the change of temperature will be given by equation (G).

This equation was applied by Thomson (article Elasticity) to find the
relation between what he called the kinetic modulus of elasticity and
the static modulus, that is, between the modulus for adiabatic strain
and the modulus for isothermal strain.

The augmentation of the strain produced by raising the temperature 1°
is e, and therefore edT, that is, -Te²dS⧸Kρ, is the increase of strain
due to the sudden rise of temperature dT. This added to the isothermal
strain produced by dS will give the whole adiabatic strain. Thus
if M be the static or isothermal modulus, the adiabatic strain
is dS⧸M - Te²dS⧸Kρ. If M' denote the kinetic or adiabatic modulus
its value is dS divided by the whole adiabatic strain, that is,
M' = M⧸(1 - MTe²⧸Kρ) and the ratio M'⧸M = 1⧸(1 - MTe²⧸Kρ).

It is well known and easy to prove, without the use of any theorem which
can be properly called thermodynamic, that this ratio of moduli is equal
to the ratio of the specific heat K of the substance, under the
condition of constant stress, to the specific heat N under the condition
of constant strain of the corresponding type. This, indeed, is
self-evident if two changes of stress, one isothermal the other
adiabatic, _which produce the same steps of displacement ds_, be
considered, and it be remembered that the step ∂T of temperature which
accompanies the adiabatic change may be regarded as made up of a step
-dT of temperature, accompanying a displacement ds effected at constant
stress, and then two successive steps dT and ∂T effected, at constant
strain, along with the steps of stress dS. The ratio M'⧸M is easily seen
to have the value (∂T + dT)⧸dt, and since -KdT + N(∂T + dT) = 0, by
the adiabatic condition, the theorem is proved.

Laplace's celebrated result for air, according to which the adiabatic
bulk-modulus is equal to the static bulk-modulus multiplied by the ratio
of the specific heat of air pressure constant to the specific heat of
air volume constant, is a particular example of this theory.

Thomson showed in the Elasticity article how, by the value of M'⧸M,
derived as above from thermodynamic theory, the value of K⧸N could be
obtained for different substances and for different types of stress, and
gave very interesting tables of results for solids, liquids, and gases
subjected to pressure-stress (bulk-modulus) and for solids subjected to
longitudinal stress (Young's modulus).

The discussion as to the relation of the adiabatic and isothermal moduli
of elasticity is part of a very important paper on "Thermoelastic,
Thermomagnetic, and Thermoelectric Properties of Matter," which he
published in the _Philosophical Magazine_ for January 1878. This was in
the main a reprint of an article entitled, "On the Thermoelastic and
Thermomagnetic Properties of Matter, Part I," which appeared in April
1855 in the first number of the _Quarterly Journal of Mathematics_. Only
thermoelasticity was considered in this article; the thermomagnetic
results had, however, been indicated in an article on "Thermomagnetism"
in the second edition of the _Cyclopædia of Physical Science_, edited
and in great part written by Professor J. P. Nichol, and published in
1860. For the same Cyclopædia Thomson also wrote an article entitled,
"Thermo-electric, Division I.--Pyro-Electricity, or Thermo-Electricity
of Non-conducting Crystals," and the enlarged _Phil. Mag._ article also
contained the application of thermodynamics to this kind of
thermoelectric action.

This great paper cannot be described without a good deal of mathematical
analysis; but the student who has read the earlier thermodynamical
papers of Thomson will have little difficulty in mastering it. It must
suffice to say here that it may be regarded as giving the keynote of
much of the general thermodynamic treatment of physical phenomena, which
forms so large a part of the physical mathematics of the present day,
and which we owe to Willard Gibbs Duhem, and other contemporary writers.

Thomson had, however, previous to the publication of this paper, applied
thermodynamic theory to thermoelectric phenomena. A long series of
papers containing experimental investigations, and entitled,
"Electrodynamic Qualities of Metals," are placed in the second volume of
his _Mathematical and Physical Papers_. This series begins with the
Bakerian Lecture (published in the _Transactions of the Royal Society_
for 1856) which includes an account of the remarkable experimental work
accomplished during the preceding four or five years by the volunteer
laboratory corps in the newly-established physical laboratory in the old
College. The subjects dealt with are the Electric Convection of Heat,
Thermoelectric Inversions, the Effects of Mechanical Strain and of
Magnetisation on the Thermoelectric Qualities of Metals, and the Effects
of Tension and Magnetisation on the Electric Conductivity of Metals. It
is only possible to give here a very short indication of the
thermodynamic treatment, and of the nature of Thomson's remarkable
discovery of the electric convection of heat.

It was found by Seebeck in 1822 that when a circuit is formed of two
different metals (without any cell or battery) a current flows round the
circuit if the two junctions are not at the same temperature. For
example, if the two metals be rods of antimony and bismuth, joined at
their extremities so as to form a complete circuit, and one junction be
warmed while the other is kept at the ordinary temperature, a current
flows across the hot junction in the direction from bismuth to antimony.
Similarly, if a circuit be made of a copper wire and an iron wire, a
current passes across the warmer junction from copper to iron. The
current strength--other things being the same--depends on the metals
used; for example, bismuth and antimony are more effective than other
metals.

It was found by Peltier that when a current, say from a battery, is sent
round such a circuit, that junction is cooled and that junction is
heated by the passage of the current, which, being respectively heated
and cooled, would without the cell have caused a current to flow in the
same direction. Thus the current produced by the difference of
temperature of the junctions causes an absorption of heat from the
warmer junction, and an evolution of heat at the colder junction.

This naturally suggested to Thomson the consideration of a circuit of
two metals, with the junctions at different temperatures, as a heat
engine, of which the hot junction was the source and the cold junction
the refrigerator, while the heat generated in the circuit by the current
and other work performed, if there was any, was the equivalent of the
difference between the heat absorbed and the heat evolved. Of course in
such an arrangement there is always irreversible loss of heat by
conduction; but when such losses are properly allowed for the circuit is
capable of being correctly regarded as a reversible engine.

Shortly after Seebeck's discovery it was found by Cumming that when the
hot junction was increased in temperature the electromotive force
increased more and more slowly, at a certain temperature of the hot
junction took its maximum value, and then as the temperature of the hot
junction was further increased began to diminish, and ultimately, at a
sufficiently high temperature, in most instances changed sign. The
temperature of maximum electromotive force was found to be independent
of the temperature of the colder junction. It is called the temperature
of the neutral point, from the fact that if the two junctions of a
thermoelectric circuit be kept at a constant small difference of
temperature, and be both raised in temperature until one is at a higher
temperature than the neutral point, and the other is at a lower, the
electromotive force will fall off, until finally, when this point is
reached, it has become zero.

Thus it was found that for every pair of metals there was at least one
such temperature of the hot junction, and it was assumed, with
consequences in agreement with experimental results, that when the
temperature was the neutral temperature there was neither absorption nor
evolution of heat at the junction. But then the source provided by the
thermodynamic view just stated had ceased to exist. The current still
flowed, there was evolution of heat at the cold junction, and likewise
Joulean evolution of heat in the wires of the circuit in consequence of
their resistance. Hence it was clear that energy must be obtained
elsewhere than at the junctions. Thomson solved the problem by showing
that (besides the Joulean evolution of heat) there is absorption (or
evolution) of heat when a current flows in a conductor along which there
is a gradient of temperature. For example, when an electric current
flows along an unequally heated copper wire, heat is evolved where the
current flows from the hot parts to the cold, and heat is absorbed where
the flow is from cold to hot. When the hot junction is at the
temperature of zero absorption or evolution of heat--the so-called
neutral temperature--the heat absorbed in the flow of the circuit along
the unequally heated conductors is greater than that evolved on the
whole, by an amount which is the equivalent of the energy electrically
expended in the circuit in the same time.

It was found, moreover, that the amount of heat absorbed by a given
current in ascending or descending through a given difference of
temperature is different in different metals. When the current was unit
current and the temperature difference also unity, Thomson called the
heat absorbed or evolved in a metal the specific heat of electricity in
the metal, a name which is convenient in some ways, but misleading in
others. The term rather conveys the notion that electricity has a
material existence. A substance such as copper, lead, water, or mercury
has a specific heat in a perfectly understood sense; electricity is not
a substance, hence there cannot be in the same proper sense a specific
heat of electricity.

However, this absorption and evolution of heat was investigated
experimentally and mathematically by Thomson, and is generally now
referred to in thermoelectric discussions as the "Thomson effect."

Part VI (_Trans. R.S._, 1875) of the investigations of the
electrodynamic qualities of metals dealt with the effects of stretching
and compressing force, and of torsion, on the magnetisation of iron and
steel and of nickel and cobalt.

One of the principal results was the discovery that the effect of
longitudinal pull is to increase the inductive magnetisation of soft
iron, and of transverse thrust to diminish it, so long as the
magnetising field does not exceed a certain value. When this value,
which depends on the specimen, is exceeded, the effect of stress is
reversed. The field-intensity at which the effect is reversed is called
the Villari critical intensity, from the fact, afterwards ascertained,
that the result had previously been established by Villari in Italy. No
such critical value of the field was found to exist for steel, or
nickel, or cobalt.

In some of the experiments the specimen was put through a cycle of
magnetic changes, and the results recorded by curves. These proved that
in going from one state to another and returning the material lagged in
its return path behind the corresponding states in the outward path.
This is the phenomenon called later "hysteresis," and studied in minute
detail by Ewing and others. Thomson's magnetic work was thus the
starting point of many more recent researches.



CHAPTER IX

HYDRODYNAMICS--DYNAMICAL THEOREM OF MINIMUM ENERGY--VORTEX MOTION


Thomson devoted great attention from time to time to the science of
hydrodynamics. This is perhaps the most abstruse subject in the domain
of applied mathematics, and when viscosity (the frictional resistance to
the relative motion of particles of the fluid) is taken into account,
passes beyond the resources of mathematical science in its present state
of development. But leaving viscosity entirely aside, and dealing only
with so-called perfect fluids, the difficulties are often overwhelming.
For a long time the only kind of fluid motion considered was, with the
exception of a few simple cases, that which is called irrotational
motion. This motion is characterised by the analytical peculiarity, that
the velocity of an element of the fluid in any direction is the rate of
variation per unit distance in that direction of a function of the
coordinates (the distances which specify the position) of the particle.
This condition very much simplifies the analysis; but when it does not
hold we have much more serious difficulties to overcome. Then the
elements of the fluid have what is generally, but quite improperly,
called molecular rotation. For we know little of the molecules of a
fluid; even when we deal with infinitesimal elements, in the analysis of
fluid motion, we are considering the fluid in mass. But what is meant
is elemental rotation, a rotation of the infinitesimal elements as they
move. We have an example of such motion in the air when a ring of smoke
escapes from the funnel of a locomotive or the lips of a tobacco-smoker,
in the motion of part of the liquid when a cup of tea is stirred by
drawing the spoon from one side to the other, or when the blade of an
oar is moving through the water. In these last two cases the depressions
seen in the surface are the ends of a vortex which extends between them
and terminates on the surface. In all these examples what have been
called vortices are formed, and hence the name vortex motion has been
given to all those cases in which the condition of irrotationality is
not satisfied.

The first great paper on vortex motion was published by von Helmholtz in
1858, and ten years later a memoir on the same subject by Thomson was
published in the _Transactions of the Royal Society of Edinburgh_. In
that memoir are given very much simpler proofs of von Helmholtz's main
theorems, and, moreover, some new theorems of wide application to the
motion of fluids. One of these is so comprehensive that it may be said
with truth to contain the whole of the dynamics of a perfect fluid. We
go on to indicate the contents of the principal papers, as far as that
can be done without the introduction of analysis of a difficult
description.

In Chapter VI reference has been made to the "Notes on Hydrodynamics"
published by Thomson in the _Cambridge and Dublin Mathematical Journal_
for 1848 and 1849. These Notes were not intended to be entirely
original, but were composed for the use of students, like Airy's Tracts
of fifteen years before.

The first Note dealt with the equation of continuity, that is to say,
the mathematical expression of the obvious fact that if any region of
space in a moving fluid be considered, the excess of rate of flow into
the space across the bounding surface, above the rate of flow out, is
equal to the rate of growth of the quantity of fluid within the space.
The proof given is that now usually repeated in text-books of
hydrodynamics.

The second Note discussed the condition fulfilled at the bounding
surface of a moving fluid. The chief mathematical result is the equation
which expresses the fact, also obvious without analysis, that there is
no flow of the fluid across the surface. In other words, the component
of the motion of a fluid particle in the immediate neighbourhood of the
surface at any instant, taken in the direction perpendicular to the
surface, must be equal to the motion of the surface in that direction at
the same instant.

The third Note, published a year later (February 1849), is of
considerable scientific importance. It is entitled, "On the Vis Viva of
a Liquid in Motion." What used to be called the "vis viva" of a body is
double what is now called the energy of motion, or kinetic energy, of
the body. The term liquid is merely a brief expression for a fluid, the
mass of which per unit volume is the same throughout, and suffers no
variation. The fluid, moreover, is supposed devoid of friction, that is,
the relative motions of its parts are unresisted by tangential force
between them. The chief theorem proved and discussed may be described as
follows.

The liquid is supposed to fill the space within a closed envelope, which
fulfils the condition of being "simply continuous." The condition will
be understood by imagining any two points A, B, within the space, to be
joined by two lines ACB, ADB both lying within the space. These two
lines will form a circuit ACBDA. If now this circuit, however it may be
drawn, can be contracted down to a point, without any part of the
circuit passing out of the space, the condition is fulfilled. Clearly
the space within the surface of an anchor-ring, or a curtain-ring, would
not fulfil this condition, for one part of the circuit might pass from A
to B round the ring one way, and the other from A to B the other way.
The circuit could not then be contracted towards a point without passing
out of the ring.

Now let the liquid given at rest in such a space be set in motion by any
arbitrarily specified variation of position of the envelope. The liquid
within will be set in motion in a manner depending entirely on the
motion of the envelope. It is possible to conceive of other motions of
the liquid than that taken, which all agree in having the specified
motion of the surface. Thomson's theorem asserts that the motion
actually taken has less kinetic energy than that of any of the other
motions which have the same motion of the bounding surface.

The motion produced has the property described by the word
"irrotational," that is, the elements of the fluid have no spinning
motion--they move without rotation. A small portion of a fluid may
describe any path--may go round in a circle, for example--and yet have
no rotation. The reader may imagine a ball carried round in a circle,
but in such a way that no line in the body ever changes its direction.
The body has translation, but no spin.

Irrotationality of a fluid is secured, as stated above, when the
velocity of each element in any direction is the rate of variation per
unit distance in that direction of a certain function of the
coordinates, the distances, taken parallel to three lines perpendicular
to one another and drawn from a point, which specify the position of the
particle. In fact, what is called a velocity-potential exists, similar
to the potential described in Chapter IV above, for an electric field.
This condition, together with the specified motion of the surface,
suffices to determine the motion of the fluid.

Two important particular consequences were pointed out by Thomson: (1)
that the motion of the fluid at any instant depends solely on the form
and motion of the bounding surface, and is therefore independent of the
previous motion; and (2) that if the bounding surface be instantaneously
brought to rest, the liquid throughout the vessel will also be instantly
brought to rest.

This theorem was afterwards generalised by Thomson (_Proc. R.S.E._,
1863), and applied to any material system of connected particles set
into motion by specified velocities simultaneously and suddenly imposed
at selected points of the system. It was already known that the kinetic
energy of a system of bodies connected in any manner, and set in motion
by impulses applied at specified points, was either a maximum or a
minimum, as compared with that for any other motion compatible with
these impulses, and with the connections of the system. This was proved
by Lagrange in the _Mécanique Analytique_ as a generalisation of a
theorem given by Euler for a rigid body set into rotation by an impulse.

Bertrand proved in 1842 that when the impulses applied are given in
amount, and are applied at specified points, the system starts off with
kinetic energy greater than that of any other motion which is consistent
with the given impulses and the connections of the system. This other
motion must be such as could be produced in the system by the given
impulses, together with any other set of impulses capable of doing no
work on the whole.

Thomson's theorem is curiously complementary to Bertrand's. Let the
system be acted on by impulses applied at certain specified points, and
by no other impulses of any kind; and let the impulses be such as to
start those selected points with any prescribed velocities. The system
will start off with kinetic energy which is less than that of any other
motion which the system could have consistently with the prescribed
velocities, and which it could be constrained to take by impulses which
do no work on the whole. In each case the difference of energies is the
energy of the motion which must be compounded with one motion to give
the other which is compared with it.

A simple example, such as might be taken of the particular case
considered by Euler, may help to make these theorems clear. Imagine a
straight uniform rod to lie on a horizontal table, between which and the
rod there is no friction. Let the rod be struck a blow at one end in a
horizontal direction at right angles to the length of the rod. If no
other impulse acts, the end of the rod will move off with a certain
definite velocity, and the other parts of the rod (which is supposed
perfectly unbending) will be started by the connections of the system.
It is obvious that any number of other motions of the rod can be
imagined, all of which give the same motion of the extremity struck. But
the actual motion taken is one of turning about that point of the rod
which is two-thirds of the length from the end struck. If the reader
will consider the kinetic energy for any other horizontal turning motion
consistent with the same motion of the end, he will find that the
kinetic energy is greater than that of the motion just specified. This
motion could be produced by applying at the point about which the rod
turns the impulse required to keep that point at rest. The impulse so
applied would do no work. The actual value is 1⧸8mv², where m denotes
the mass of the rod and v the velocity of the end. If the motion taken
were one of rotation about a point of the rod at distance x from the end
struck, the kinetic energy would be m(4l² - 6lx + 3x²)v²⧸6x², where
2l is the length of the rod, and this has its least value 1⧸8mv² for
x = 4l⧸3. For example, x = 2l gives 1⧸6mv², which is greater than the
value just found.

Bertrand's theorem applied to this case of motion is not quite so easy,
perhaps, to understand. The motion which is said to have maximum energy
is one given by a specified impulse at the end struck, and this, in the
absence of any other impulses, would be a motion of minimum energy. But
let the alternative motion, which is to be compared with that actually
taken, be one constrained by additional impulses such as can together
effect no work, and the existence of the maximum is accounted for.
The kinetic energy produced is one-half the product of the impulse
into the velocity of the point struck, that is ½Iv, and it has just
been seen that this is the product of (1⧸6)mv² by the factor
(4l² - 6lx + 3x²)⧸x². This factor is 3I⧸mv, and is a minimum when
x = 4l⧸3. Thus for a given I, v will have its maximum value when the
factor referred to is least, and ½Iv will then be a maximum.

The bar can be constrained to turn about another point by a fixed pivot
there situated. An impulse will be applied to the rod by the pivot,
simultaneously with the blow; and it is obvious that this impulse does
no work, since there is no displacement of the point to which it is
applied.

The two theorems are consequences of one principle. The constraint in
each case increases what may be called the effective inertia, which may
be taken as I⧸v. Thus when v is given, I is increased by any constraint
compelling the rod to rotate about a particular axis, and so ½Iv, or
the kinetic energy, is increased. On the other hand, when I is given the
same constraint diminishes v, and so ½Iv is diminished.

A short paper published in the B. A. Report for 1852 points out that the
lines of force near a small magnet, placed with its axis along the lines
of force in a uniform magnetic field, as it would rest under the action
of the field, are at corresponding points similar to those of the field
of an insulated spherical conductor, under the inductive influence of a
distant electric change. Further, the fact is noted that, if the magnet
be oppositely directed to the field, the lines of force are curved
outwards, just as the lines of flow of a uniform stream would be by a
spherical obstacle, at the surface of which no eddies were caused. This
is one of those instructive analogies between the theory of fluid motion
and other theories involving perfectly analogous fundamental ideas,
which Thomson was fond of pointing out, and which helped him in his
repeated attempts to imagine mechanical representations of physical
phenomena of different kinds.

With these may be placed another, which in lectures he frequently dwelt
on--a simple doublet, as it is called, consisting of a point-source of
fluid and an equal and closely adjacent point-sink. A short tube in an
infinite mass of liquid, which is continually flowing in at one end and
out at the other, may serve as a realisation of this arrangement. The
lines of flow outside the tube are exactly analogous to the lines of
force of a small magnet; and if at the same time there exist a uniform
flow of the liquid in the direction of the length of the tube, the field
of flow will be an exact picture of the field of force of the small
magnet, when it is placed with its length along the lines of a
previously existing uniform field. The flow in the doublet will be with
or against the general flow according as the magnet is directed with or
against the field.

The paper on vortex-motion has been referred to above, and an indication
given of the nature of the fluid-motion described by this title. There
are, however, two cases of fluid-motion which are referred to as
vortices, though the fundamental criterion of vortex-motion--the
non-existence of a velocity-potential--is satisfied in only one of them.
The exhibition of one of these was a favourite experiment in Thomson's
ordinary lectures, as his old students will remember. If water in a
large bowl is stirred rapidly with a teaspoon carried round and round in
a circle about the axis of the bowl, the surface will become concave,
and the form of the central part will be a paraboloid of revolution
about the vertical through the lowest point, that is to say, any section
of that part of the surface made by a vertical plane containing the axis
will be a parabola symmetrical about the axis. The motion can be better
produced by mounting the vessel on a whirling-table, and rotating it
about the vertical axis coinciding with its axis of figure; but the
phenomenon can be quite well seen without this machinery. In this case
the velocity of each particle of the water is proportional to its
distance from the axis, and the whole mass, when relative equilibrium is
set up, turns, as if it were rigid, about the axis of the vessel. Each
element of the fluid in this "forced vortex," as it is called, is in
rotation, and, like the moon, makes one turn in one revolution about the
centre of its path. This is, therefore, a true, though very simple, case
of vortex-motion.

On the other hand, what may be called a "free vortex" may exist, and is
approximated to sometimes when water in a vessel is allowed to run off
through an escape pipe at the bottom. The velocity of an element in this
"vortex" is inversely proportional to its distance from the centre, and
the form of the free surface is quite different from that in the other
case. The name "free vortex" is often given to this case of motion, but
there is no vortex-motion about it whatever.

Thomson's great paper on vortex-motion was read before the Royal Society
of Edinburgh in 1867, and was recast and augmented in the following
year. It will be possible to give here only a sketch of its scope and
main results.

The fluid is supposed contained in a closed fixed vessel which is either
simply or multiply continuous (see p. 156), and may contain immersed in
it simply or multiply continuous solids. When these solids exist their
surfaces are part of the boundary of the liquid; they are surrounded by
the liquid unless they are anywhere in contact with the containing
vessel, and their density is supposed to be the same as that of the
liquid. They may be acted on by forces from without, and they act on the
liquid with pressure-forces, and either directly or through the liquid
on one another.

The first result obtained is fairly obvious. The centre of mass of the
whole system must remain at rest whatever external forces act on the
solids, since the density is the same everywhere within the vessel, and
the vessel is fixed; that is to say, there is no momentum of the
contents of the vessel in any direction. For whatever motion of the
solids is set up by the external forces, must be accompanied by a motion
of the liquid, equal and opposite in the sense here indicated.

After a discussion of what he calls the impulse of the motion, which is
the system of impulsive forces on the movable solids which would
generate the motion from rest, Thomson proceeds to prove the important
proposition that the rotational motion of every portion of the liquid
mass, if it is zero at any one instant for every portion of the mass,
remains always zero. This is done by considering the angular momentum of
any small spherical portion of the liquid relatively to an axis through
the centre of the sphere, and proving that in order that it may vanish,
for every axis, the component velocities of the fluid at the centre
must be derivable from a velocity-potential. The angular momentum
of a particle about an axis is the product of the component of the
particle's momentum, at right angles to the plane through the particle
and the axis, by the distance of the particle from the axis. The sum of
all such products for the particles making up the body (when proper
account is taken of the signs according to the direction of turning
round the axis) is the angular momentum. The proof of this result
adopted is due to Stokes. The angular velocities of an element of
fluid at a point x, y, z, about the axes of x, y, z are shown to be
½(∂w⧸∂y - ∂v⧸∂z), etc.

The condition was therefore shown to be necessary; it remained to prove
that it was sufficient. This is obvious at once from the definition of
the velocity-potential, which must now be supposed to exist in order
that its sufficiency may be proved. If any diameter of the spherical
portion be taken as the axis, and any plane through that axis be
considered, the velocity of a particle at right angles to that plane can
be at once expressed as the rate at which the velocity-potential varies
per unit distance along the circle, symmetrical about the axis, on which
the particle lies. The integral of the velocity-potential round this
circle vanishes, and so the angular momentum for any thin uniform ring
of particles about the axis also vanishes, and as the sphere is made up
of such rings, the whole angular momentum is zero. Thus the condition is
sufficient.

Thomson then proves that if the angular momentum thus considered be
zero for every portion of the liquid at any one instant, it remains zero
at every subsequent instant; that is, no physical action whatsoever
could set up angular momentum within the fluid, which, it is to be
remembered, is supposed to be frictionless. The proof here given cannot
be sketched because it depends on the differential equation of
continuity satisfied by the velocity-potential throughout the fluid (the
same differential equation, in fact, that is satisfied by the
distribution of temperature in a uniform conducting medium in the
stationary state), and the consequent expression of this function for
any spherical space in the fluid as a series of spherical harmonic
functions. To a reader to whom the properties of these functions are
known the process can present no difficulty.

An entirely different proof of this proposition is given subsequently in
the paper, and depends on a new and very general theorem, which has been
described as containing almost the whole theory of the motion of a
fluid. This depends on what Thomson called the flow along any path
joining any two points P, Q in the fluid. Let q be the velocity of the
fluid at any element of length ds of such a path, and θ be the
angle between the direction of ds (taken positive in the sense from P
to Q) and the direction of q: q cos θ.ds is the flow along ds. If u,
v, w be the components of q at ds, parallel to the axes, and dx, dy, dz
be the projections of ds on the axes, udx + vdy + wdz is the same thing
as q cos θ.ds. The sum of the values of either of these expressions
for all the elements of the path between P and Q is the flow along the
path. The statement that u, v, w are the space-rates of variation of a
function φ (of x, y, z) parallel to the axes, or that q cos θ is
the space-rate of variation of φ along ds, merely means that this
sum is the same for whatever path may be drawn from P to Q. This,
however, is only the case when the paths are so taken that in each case
the value of φ returns after variation along a closed path to the
value which it had at the starting point, that is, the closed path must
be capable of being contracted to a point without passing out of space
occupied by irrotationally moving fluid.

Since the flow from P to Q is the same for any two paths which fulfil
this condition, the flow from P to Q by any one path and from Q to P by
any other must be zero. The flow round such a closed path is not zero if
the condition is not fulfilled, and its value was called by Thomson the
circulation round the path.

The general theorem which he established may now be stated. Consider
any path joining PQ, and moving with the fluid, so that the line
contains always the same fluid particles. Let u̇, v̇, ẇ be the
time-rates of change of u, v, w at an element ds of the path, at
any instant, and du, dv, dw the excesses of the values of u, v, w
at the terminal extremity of ds above the values at the other
extremity; then the time-rate of variation of udx + vdy + wdz
is u̇dx + v̇dy + ẇdz + udu + vdv + wdw or u̇dx + v̇dy + ẇdz + qdq,
where q has the meaning specified above. Thus if S be the flow for
the whole path PQ, and Ṡ its time-rate of variation, S' denote the
sum of u̇dx + v̇dy + ẇdz along the path from P to Q, and q₁, q₀ the
resultant fluid velocities at Q and P, we get Ṡ = S' + ½(q₁² - q₀²).
This is Thomson's theorem. If the curve be closed, that is, if P and Q
be coincident, q₁ = q₀ and Ṡ = S'. But in certain circumstances S' is
zero, and so therefore is also Ṡ. Thus in the circumstances referred to,
as the closed path moves with the fluid Ṡ is continually zero, and it
follows that if Ṡ is zero at any instant it remains zero ever after. But
Ṡ is only zero if u, v, w are derivable from a potential, single valued
in the space in which the closed path is drawn, so that the path could
be shrunk down to a point without ever passing out of such space. In a
perfect fluid if this condition is once fulfilled for a closed curve
moving with the fluid, it is fulfilled for this curve ever after.

The circumstances in which S' is zero are these:--the external force,
per unit mass, acting on the fluid at any point is to be derivable from
a potential-function, and the density of the fluid is to be a function
of the pressure (also a function of the coordinates); and these
functions must be such as to render S' always zero for the closed path.
This condition is manifestly fulfilled in many important cases; for
example, the forces are derivable from a potential due to actions, such
as gravity, the origin of which is external to the fluid; and the
density is a function of the pressure (in the present case it is a
constant), such that the part of S' which depends on pressure and
density vanishes for the circuit.

It is to be clearly understood that the motion of a fluid may be
irrotational although the value of S does not vanish for every closed
path that can be drawn in it. The fluid may occupy multiply continuous
space, and the path may or may not be drawn so that S shall be zero; but
what is necessary for irrotational motion within any space is that S
should vanish for all paths which are capable of being shrunk down to
zero without passing out of that space. S need not vanish for a path
which cannot be so shrunk down, but it must, if the condition just
stated is fulfilled, have the same value for any two paths, one of which
can be made to pass into the other by change of position without ever
passing in whole or in part out of the space. The potential is always
single valued in fluid filling a singly continuous space such as that
within a spherical shell, or between two concentric shells; within a
hollow anchor-ring the potential, though it exist, and the motion be
irrotational, is not single valued. In the latter case the motion is
said to be cyclic, in the former acyclic.

A number of consequences are deduced from this theorem; and from these
the properties of vortices, which had previously been discovered by von
Helmholtz, immediately follow. First take any surface whatever which has
for bounding edge a closed curve drawn in the fluid, and draw from any
element of this surface, of area dS, a line perpendicular to the surface
towards the side chosen as the positive side, and calculate the angular
velocity ω, say, of the fluid about that normal from the components of
angular velocity determined in the manner explained at p. 164. This
Thomson called the rotation of the element. Now take the product ωdS for
the surface element. It is easy to see that this is equal to half the
circulation round the bounding edge of the element. As the fluid
composing the element moves the area dS may change, but the circulation
round its edge by Thomson's theorem remains unaltered. Thus ω alters in
the inverse ratio of dS, and the line drawn at right angles to the
surface at dS, if kept of length proportional to ω, will lengthen or
shorten as dS contracts or expands.

Now sum the values of ωdS for the finite surface enclosed by the
bounding curve. It follows from the fact that ωdS is equal to half the
circulation round the edge of dS, that this sum, which is usually
denoted by ΣωdS, is equal to half the circulation round the closed
curve which forms the edge of the surface. Also as the fluid moves the
circulation round the edge remains unaltered, and therefore so does also
ΣωdS for the elements enclosed by it. It is important to notice
that this sum being determined by the circulation in the bounding curve
is the same for all surfaces which have the same boundary.

The equality of 2ΣωdS for the surface to the circulation round its
edge was expressed by Thomson as an analytical theorem of integration,
which was first given by Stokes in a Smith's Prize paper set in 1854. It
is here stated, apparently by an oversight, that it was first given in
Thomson and Tait's _Natural Philosophy_, § 190. In the second edition of
the _Natural Philosophy_ the theorem is attributed to Stokes. It is now
well known as Stokes's theorem connecting a certain surface integral
with a line integral, and has many applications both in physics and in
geometry.

Now consider the resultant angular velocity at any point of the fluid,
and draw a short line through that point in the direction of the axis of
rotation. That line may be continued from point to point, and will
coincide at every one of its points with the direction of the axis of
rotation there. Such an axial curve, as it may be called, it is clear
moves with the fluid. For take any infinitesimal area containing an
element of the line; the circulation round the edge of this area is
zero, since there is no rotation about a line perpendicular to the area.
Hence the circulation along the axial curve is zero, and the axial
curves move with the fluid.

Take now any small plane area dS moving with the fluid, and draw axial
lines through every point of its boundary. These will form an axial tube
enclosing dS. If θ be the angle between the direction of resultant
rotation and a perpendicular to dS, the cross-section of the tube at
right angles to the normal, and to the axial lines which bound it, is
dS.cosθ. Let these axial lines be continued in both directions from the
element dS. They will enclose a tube of varying normal cross-section;
but the product of rotation and area of normal cross-section has
everywhere the same value. A vortex-tube with the fluid within it is
called a vortex-filament.

It will be seen that this vortex-tube must be endless, that is, it must
either return into itself, or be infinitely long in one or both
directions. For if it were terminated anywhere within the fluid, it
would be possible to form a surface, starting from a closed circuit
round the tube, continued along the surface of the tube to the
termination, and then closed by a cap situated beyond the termination.
At no part of this surface would there be any rotation, and ΣωdS,
which is equal to the circulation, would be zero for it; and of course
this cannot be the case. Thus the tube cannot terminate within the
fluid. It can, however, have both of its ends on the surface, or one on
the bounding surface and the other at infinity, if the fluid is
infinitely extended in one direction, but in that case the termination
is only apparent. The section is widened out at the surface; some of the
bounding lines pass across to the other apparent termination, when it
also lies on the surface, while the other lines pass off to infinity
along the surface, and correspond to other lines coming in from
infinity to the other termination. Whether the surface is infinite or
not, the vortex is spread out into what is called a vortex-sheet, that
is, in a surface on the two sides of which the fluid moves with
different tangential velocities.

Through a vortex-ring or tube, the fluid circulates in closed lines of
flow, each one of which is laced through the tube. The circulation along
every line of flow which encloses the same system of vortex-tubes has
the same value.

If any surface be drawn cutting a vortex-tube, it is clear from the
definition of the tube that the value of ΣωdS for every such
surface must be the same. This Thomson calls the "rotation of the tube."

As was pointed out first by von Helmholtz, vortex-filaments correspond
to circuits carrying currents and the velocity in the surrounding fluid
to magnetic field-intensity. The "rotation of the tube" corresponds to
the strength of the current, and sources and sinks to positive and
negative magnetic poles. Thomson made great use of this analogy in his
papers on electromagnetism.

Examples of vortex-tubes are indicated on p. 154; and the reader may
experiment with vortices in liquids with water in a tea-cup, or in a
river or pond, at pleasure. Air vortices may be experimentally studied
by means of a simple apparatus devised by Professor Tait, which may be
constructed by anyone.

In one end of a packing-box, about 2ft. long by 18in. wide and 18in.
deep, a circular hole is cut, and the edges of the hole are thinned down
to a blunt edge. This can be closed at pleasure by a piece of board. The
opposite end is removed, and a sheet of canvas stretched tightly in its
place, and tacked to the ends of the sides. Through two holes bored in
one of the sides the mouths of two flasks with bent necks protrude into
the box. One of these flasks contains ammonia, the other hydrochloric
acid. When the hole at one end is closed up by a slip of tinplate, and
the liquids are heated with a spirit-lamp, the vapours form a cloud of
sal-ammoniac within the box, which is retained during its formation. The
hole is then opened, and the canvas struck smartly with the palm of the
open hand. Immediately a beautiful ring of smoke emerges, clear-cut and
definite as a solid, and moves across the room. (See Fig. 13.) Of
course, it is a ring of air, made visible by the smoke carried with it.
By varying the shape of the aperture--for example, by using instead of
the hole cut in the wood, a slide of tinplate with an elliptic hole cut
in it--the vortex-rings can be set in vibration as they are created, and
the vibrations studied as the vortex moves.

[Illustration: FIG. 13.]

Still more beautiful vortices can be formed in water by using a long
tank of clear water to replace the air in which the vortex moves, and a
compartment at one end filled with water coloured with aniline, instead
of the smoke-box. A hole in the dividing partition enables the vortex to
be formed, and a piston arrangement fitted to the opposite side enables
the impulse to the water to be given from without.

From the account of the nature of vortex-motion given above, it will be
clear that vortices in a perfect fluid once existent must be ever
existent. To create a vortex within a mass of irrotationally moving
perfect fluid is physically impossible. It occurred to Thomson,
therefore, that ordinary matter might be portions of a perfect fluid,
filling all space, differentiated from the surrounding fluid by the
rotation which they possess. Such matter would fulfil the law of
conservation, as it could neither be created nor destroyed by any
physical act.

The results of such experiments led Thomson to frame his famous
vortex-atom theory of matter, a theory, however, which he felt
ultimately was beset with so many difficulties as to be unworkable.

The paper on vortex-motion also deals with the modification of Green's
celebrated theorem of analysis, which, it was pointed out by Helmholtz,
was necessary to adapt it to a space which is multiply continuous. The
theorem connects a certain volume-integral taken throughout a closed
space with an integral taken over the bounding surface of the space.
This arises from the fact noticed above that in multiply continuous
space (for example, the space within an endless tube) the functions
which are the subject of integration may not be single valued. Such a
function would be the velocity-potential for fluid circulating round the
tube--cyclic motion, as it was called by Thomson. If a closed path of
any form be drawn in such a tube, starting from a point P, and doubling
back so as to return to P without making the circuit of the tube, the
velocity-potential will vary along the tube, but will finally return to
its original value when the starting point is reached. And the
circulation round this circuit will be zero. But if the closed path make
the circuit of the tube, the velocity-potential will continuously vary
along the path, until finally, when P is reached again, the value of
the function is greater (or less) than the value assumed for the
starting point, by a certain definite amount which is the same for every
circuit of the space. If the path be carried twice round in the same
direction, the change of the function will be twice this amount, and so
on. The space within a single endless tube such as an anchor-ring is
doubly continuous; but much more complicated cases can be imagined. For
example, an anchor-ring with a cross-connecting tube from one side to
the other would be triply continuous.

Thomson showed that the proper modification of the theorem is obtained
by imagining diaphragms placed across the space, which are not to be
crossed by any closed path drawn within the space, and the two surfaces
of each of which are to be reckoned as part of the bounding surface of
the space. One such diaphragm is sufficient to convert a hollow
anchor-ring into a singly continuous space, two would be required for
the hollow anchor-ring with cross-connection, and so on. The number of
diaphragms required is always one less than the degree of multiplicity
of the continuity.

The paper also deals with the motion of solids in the fluid and the
analogous motions of vortex-rings and their attraction by ordinary
matter. These can be studied with vortex-rings in air produced by the
apparatus described above. Such a ring made to pass the re-entrant
corner of a wall--the edge of a window recess, for example--will appear
to be attracted. A large sphere such as a large terrestrial globe serves
also very well as an attracting body.

Two vortex-rings projected one after the other also act on one another
in a very curious manner. Their planes are perpendicular to the
direction of motion, and the fluid is moving round the circular core of
the ring. There is irrotational cyclic motion of the fluid through the
ring in one direction and back outside, as shown in Fig. 13, which can
be detected by placing a candle flame in the path of the centre. The
first ring, in consequence of the existence of that which follows it,
moves more slowly, and opens out more widely, the following ring hastens
its motion and diminishes in diameter, until finally it overtakes the
former and penetrates it. As soon as it has passed through it moves
ahead more and more slowly, until the one which has been left behind
begins to catch it up, and the changes which took place before are
repeated. The one penetrating becomes in its turn the penetrated, and so
on in alternation. Great care and skill are, however, necessary to make
this interesting experiment succeed.

We have not space to deal here with other hydrodynamical investigations,
such as the contributions which Thomson made to the discussion of the
many difficult problems of the motion of solids through a liquid, or to
his very numerous and important contributions to the theory of waves.
The number and importance of his hydrodynamical papers may be judged
from the fact that there are no less than fifty-two references to his
papers, and thirty-five to Thomson and Tait's _Natural Philosophy_ in
the latest edition of Lamb's Hydrodynamics, and that many of these are
concerned with general theorems and results of great value.



CHAPTER X

THE ENERGY THEORY OF ELECTROLYSIS--ELECTRICAL UNITS--ELECTRICAL
OSCILLATIONS


ELECTROLYSIS AND ELECTRICAL UNITS

In December 1851 Thomson communicated an important paper to the
_Philosophical Magazine_ on "The Mechanical Theory of Electrolysis," and
"Applications of Mechanical Effect to the Measurement of Electromotive
Forces, and of Galvanic Resistances, in Absolute Units."

In the first of these he supposed a machine of the kind imagined by
Faraday, consisting of a metal disk, rotating uniformly with its plane
at right angles to the lines of force of a uniform magnetic field, and
touched at its centre and its circumference by fixed wires, to send a
current through an electrochemical apparatus, to which the wires are
connected. A certain amount of work W was supposed to be spent in a
given time, during which a quantity of heat H was evolved in the
circuit, and a certain amount of work M spent in the chemical apparatus
in effecting chemical change. If H be taken in dynamical units, W = H +
M.

The work done in driving the disk, if the intensity of the field is I,
the current produced c, the radius of the disc r, and the angular
velocity of turning w, is ½Ir²cw.

Thomson assumed that the work done in the electrochemical apparatus was
equal to the heat of chemical combination of the substance or
substances which underwent the chemical action, taken with the proper
sign according to the change, if more compound substances than one were
acted on. Hence M represented this resultant heat of combination.

The electrochemical apparatus was a voltameter containing a definite
compound to be electrolysed, or a voltaic cell or battery. And by
Faraday's experiments on electrolysis it was known that the amount of
chemical action was proportional to the whole quantity of electricity
passed through the cell in a given time, so that the rate at which
energy was being spent in the cell was at any instant proportional to
the current at that instant.

The chemical change could be measured by considering only one of the
elements set free, or made to combine, by the passage of the current,
and considering the quantity of heat θ, say, for the whole chemical
change in the cell corresponding to the action on unit mass of that
element. Thus if E denote the whole quantity of that element operated on
the heat of combination in the vessel was θE. If E be taken for unit of
time, and ε denote the quantity set free by the passage of unit quantity
of electricity, then E = εc, since a current conveys c units of
electricity in one second. The number ε is a definite quantity of the
element, and is called its electrochemical equivalent. Again, from
Joule's experiments, H = Rc², if R denote the resistance of the current,
and so

  ½Ir²cw = Rc² + θεc

and

  c = (½Ir²w - θε)⧸R

The quantity ½Ir²w is the electromotive force due to the disk.

Thus c was positive or negative according as ½Ir²w was greater or less
than θε, and was zero when ½Ir²w = θε. Thus the electromotive force
of the disk was opposed by a back electromotive force θε due to the
chemical action in the voltameter or battery, to which the wires from
the disk were connected.

The conclusion arrived at therefore was that the electromotive force
(or, as it was then termed, the intensity) of the electrochemical action
was equal to the dynamical value of the whole chemical change effected
by a current of unit strength in unit of time.

From this result Thomson proceeded to calculate the electromotive forces
required to effect chemical changes of different kinds, and those of
various types of voltaic cell. Supposing a unit of electricity to be
carried by the current through the cell, he considered the chemical
changes which accompanied its passage, and from the known values of
heats of combination calculated their energy values. In some parts the
change was one of chemical combination, in others one of decomposition
of the materials, and regard had to be paid to the sign of the
heat-equivalent. By properly summing up the whole heat-equivalents a net
total was obtained which, according to Thomson, was the energy consumed
in the passage of unit current, and was therefore the electromotive
force. The theory was incomplete, and required to be supplemented by
thermodynamic theory, which shows that besides the electromotive force
there must be included in the quantity set against the sum of heats a
term represented by the product of the absolute temperature multiplied
by the rate of variation of electromotive force with alteration of
temperature. Thus the theory is only applicable when the electromotive
force is not affected by variation of temperature. The necessary
addition here indicated was made by Helmholtz.

In the next paper, which appeared in the same number (December 1851) of
the _Philosophical Magazine_, the principle of work is applied to the
measurement of electromotive forces and resistances in absolute units.
The advantages of such units are obvious. Nearly the whole of the
quantitative work of the older experimenters was useless except for
those who had actually made the observations: it was hardly possible for
one man to advance his researches by employing data obtained by others.
For the results were expressed by reference to apparatus and materials
in the possession of the observers, and to these others could obtain
access only with great difficulty and at great expense--to say nothing
of the uncertainty of comparisons made to enable the results of one man
to be linked on to those made elsewhere, and with other apparatus, by
another. It was imperative, therefore, to obtain absolute units--units
independent of accidents of place and apparatus--for the expression of
currents, electromotive forces, and resistances, so as to enable the
results of the work of experiments all over the world to be made
available to every one who read the published record. (See Chap. XIII.)

The magneto-electric machine imagined in the former paper gave a means
of estimating the electromotive force of a cell or battery in absolute
units. The same kind of machine is used here, in the simpler form of a
sliding conductor connecting a pair of insulated rails laid with their
plane perpendicular to the lines of force of a uniform magnetic field.
If the rails be connected by a wire, and the slider be moved so as to
cut across the lines of force, a current will be produced in the
circuit. The current can be measured in terms of the already known unit
of current, that current which flowing in a circle of radius unity
produces a magnetic field at the centre of 2π units. This current, c,
say, in strength, flowing in the circuit, renders a dynamical force cIl
necessary to move the slider of length l across the lines of force of
the field of intensity I, and if the speed of the slider required for
the current c be v, the rate at which work is done in moving the slider
is cIlv. This must be the rate at which work is done in the circuit by
the current, and if the only work done be in the heating of the
conductor, we have cIlv = Rc², or Ilv = Rc, so that Ilv is the
electromotive force. Any electromotive force otherwise produced, which
gave rise to the same current, must obviously be equal to Ilv, so that
the unit of electromotive force can thus be properly defined.

Thomson used a foot-grain-second system of units; but from this
arrangement are now obtained the C.G.S. units of electromotive force and
resistance. If I is one C.G.S. unit, l one centimetre, and v one
centimetre per second, we have unit electromotive force in the C.G.S.
system. Also in one C.G.S. unit of resistance if c be unity as well as
Ilv.

The idea of the determination of a resistance in absolute units on
correct principles was due to W. Weber, who also gave methods of
carrying out the measurement; and the first determination was made by
Kirchhoff in 1849. Thomson appears, however, to have been the first to
discuss the subject of units from the point of view of energy. This mode
of regarding the matter is important, as the absolute units are so
chosen as to enable work done by electric and magnetic forces to be
reckoned in the ordinary dynamical units. A vast amount of experimental
resource and skill has been spent since that time on the determination
of resistance, though not more than the importance of the subject
warranted. We shall have to return to the subject in dealing with the
work of the British Association on Electrical Standards, of which
Thomson was for long an active member.


ELECTRICAL OSCILLATIONS

In his famous tract on the conservation of energy, published in 1847,
von Helmholtz discussed some puzzling results obtained by Riess in the
magnetisation of iron wires by the current of a Leyden jar discharge
flowing in a coil surrounding them, and by the fact, observed by
Wollaston, that when water was decomposed by Leyden jar discharges a
mixture of oxygen and hydrogen appeared at each electrode, and suggested
that possibly the discharge was oscillatory in character.

In 1853 the subject was discussed mathematically by Thomson, in a paper
which was to prove fruitful in our own time in a manner then little
anticipated. The jar is given, let us say, with the interior coating
charged positively, and the exterior coating charged negatively. A coil
or helix of wire has its ends connected to the two coatings, and a
current immediately begins in the wire, and gradually (not slowly)
increases in strength. Accompanying the creation of the current is the
production of a magnetic field, that is, the surrounding space is made
the seat of magnetic action. The magnetic field, as we shall see from
another investigation of Thomson's, almost certainly involves motion in
or of a medium--the ether--filling the space where the magnetic action
is found to exist. The charge of the jar consists of a state of intense
and peculiar strain in the glass plate between the coatings. When the
plates are connected by the coil, this state of strain breaks down and
motion in the medium ensues, not merely between the plates, but also in
the surrounding space--in fact, in the whole field. This motion--which
is not to be confused with bodily displacement of finite parts of the
medium--is opposed by something akin to inertia of the medium (the
property that confers energy on matter when in motion), so that when the
motion is started it persists, until it is finally wiped out by
resistance of the nature of friction. The inertia here referred to
depends on the mode in which the coil is wound, or whether it contains
or not an iron core.

If the work done in charging a Leyden jar or electric condenser, by
bringing the charge to the condenser in successive small portions, is
considered, it is at once clear that it must be proportional to the
square of the whole quantity of electricity brought up. For whatever the
charge may be, let it be brought up from a great distance in a large
number N of equal instalments. The larger the whole amount the larger
must each instalment be, and therefore the greater the amount
accumulated on the condenser when any given number of instalments have
been deposited. But the greater any charge that is being brought up, and
also the greater the charge that has already arrived, the greater is
the repulsion that must be overcome in bringing up that instalment, in
simple proportion in each case, and therefore the greater the work done.
Thus the whole work done in bringing up the charge must be proportional
to Q². We suppose it to be ½Q²⧸C, where C is a constant depending on
the condenser and called its capacity.

The idea of the charge as a quantity of some kind of matter, brought up
and placed on the insulated plate of the condenser, has only a
correspondence to the fact, which is that the medium between the plates
is the seat, when the condenser is charged, of a store of energy, which
can only be made available by connecting the plates of the condenser by
a wire or other conductor. The charge is only a surface aspect of the
state of the medium, apparently a state of strain, to which the energy
belongs.

When a wire is used to connect the plates the state of strain
disappears; the energy comes out from the medium between the plates by
motion sideways of the tubes of strain (so that the insulating medium is
under longitudinal tension and lateral pressure) which, according to
Faraday's conception of lines of electric force connecting the charge on
a body with the opposite charges on other bodies, run from plate to
plate, when the condenser is in equilibrium in the changed state. These
tubes move out with their ends on the wire, carrying the energy with
them, and the ends run towards one another along the wire; the tube
shortens in the process, and energy is lost in the wire. The ends of a
tube thus moving represent portions of the charges which were on the
plates, and the oppositely-directed motions of the opposite charges
represent a current along the wire from one conductor to the other. The
motion of the tubes is accompanied by the development of a magnetic
field, the lines of force of which are endless, and the direction of
which at every point is perpendicular at once to the length of the tube
and to the direction in which it is there moving. In certain
circumstances the tube, by the time its ends have met, will have wholly
disappeared in the wire, and the whole energy will have gone to heat the
wire: in other circumstances the ends will meet before the tube has
disappeared, the ends will cross, and the tube will be carried back to
the condenser and reinserted in the opposite direction. At a certain
time this will have happened to all the tubes, though they will have
lost some of their energy in the process; and the condenser will again
be charged, though in the opposite way to that in which it was at first.
Then the tubes will move out again, and the same process will be
repeated: once more the condenser will be charged, but in the same
direction as at first, and once more with a certain loss of energy.
Again the process of discharge and charge will take place, and so on,
again and again, until the whole energy has disappeared. This process
represents, according to the modern theory of the flow of energy in the
electromagnetic field, with more or less accuracy, what takes place in
the oscillatory discharge of a condenser.

The motion of the tubes with their ends on the wire represents a certain
amount of energy, commonly regarded as kinetic, and styled
electrokinetic energy. If c denote the current, that is, the rate,
-dQ⧸dt, at which the charge of the condenser is being changed, and L a
quantity called self-inductance, depending mainly on the arrangement of
the connecting wire--whether it is wound in a coil or helix, with or
without an iron core, or not--the electrokinetic energy will be ½Lc².
This is analogous to the kinetic energy ½mv² of a body (say a pendulum
bob) of mass m and velocity v, so that L represents a quantity for the
conducting arrangement analogous to inertia, and c is the analogue of
the velocity of the body. The whole energy at any instant is thus

  ½Q²⧸C + ½Lc², or ½Q²⧸C + ½L(dQ⧸dt)².

The loss of energy due to heating of the conducting connection is not
completely understood, though its quantitative laws have been quite
fully ascertained and expressed in terms of magnitudes that are capable
of measurement. It was found by Joule to be proportional to the second
power, or square, of the current, and to a quantity R depending on the
conductor, and called its resistance. The generation of heat in the
conductor seems to be due to some kind of frictional action of particles
of the conductor set up by the penetration of the Faraday tubes into it.
A conductor is unable to bear any tangential action exerted upon it by
Faraday tubes, which, however, when they exist, begin and end at
material particles, except when they are endless, as they may be in the
radiation of energy. When the Faraday tubes are moving with any ordinary
speed they are not at their ends perpendicular to the conducting surface
from which they start or at which they terminate, but are there more or
less inclined to the surface, and consequently there is tangential
action which appears to displace the particles (not merely at the
surface, unless the alternation is very rapid) relatively to one
another and so cause frictional generation of heat.

The time rate of generation of heat is thus Rc², or R(dQ⧸dt)², when the
units in which R and c are expressed are such as to make this quantity a
rate of doing work in the true dynamical sense. This is the rate at
which the sum of energy already found is being diminished, and so the
equation

  ½d/dt{(Q²⧸C) + L(dQ⧸dt)²} = -R(dQ⧸dt)²

holds, or leaving out the common factor dQ⧸dt, the equation

  L(d²Q⧸dt²) + R(dQ⧸dt) + Q⧸C = 0

This last equation was established by Thomson, and is precisely that
which would be obtained for a pendulum bob of mass L, pulled back
towards the position of equilibrium with a force Q⧸C, where Q is the
displacement from the middle position, and having its motion damped out
by resisting force of amount R per unit of the velocity.

It is more instructive perhaps to take the oscillatory motion of a
spiral spring hung vertically with a weight on its lower end, as that
which has a differential equation equivalent to the equation just found.
When the stretch is of a certain amount, there is equilibrium--the
action of the spring just balances the weight,--and if the spring be
stretched further there will be a balance of pull developed tending to
bring the system back towards the equilibrium position. If left to
itself the system gets into motion, which, if the resistance is not too
great, is added to until the equilibrium position is reached; and the
motion, which is continued by the inertia of the mass, only begins to
fall off as that position is passed, and the pull of the spring becomes
insufficient to balance the weight. Thus the mass oscillates about the
position of equilibrium, and the oscillations are successively smaller
and smaller in extent, and die out as their energy is expended finally
in doing work against friction.

If the resisting force for finite motion is very great, as for example
when the vibrating mass of the pendulum or spring is immersed in a very
viscous fluid, like treacle, oscillation will not take place at all.
After displacement the mass will move at first fairly quickly, then more
and more slowly back to the position of equilibrium, which it will,
strictly speaking, only exactly reach after an infinite time. The
resisting force is here indefinitely small for an indefinitely small
speed, but it becomes so great when any motion ensues, that as the
restoring force falls off with the displacement, no work is finally done
by it, except to move the body through the resisting medium.

The differential equation is applicable to the spring if Q is again
taken as displacement from the equilibrium position, L as the inertia of
the vibrating body, 1⧸C as the pull exerted by the spring per unit of
its extension (that is, the stiffness of the spring), and R has the same
meaning as before.

In this case of motion, as well as in that of the pendulum, energy is
carried off by the production of waves in the medium in which the
vibrator is immersed. These are propagated out from the vibrator as
their source, but no account of them is taken in the differential
equation, which in that respect is imperfect. There is no difficulty,
only the addition of a little complication, in supplying the omission.

The formation of such waves by the spiral spring vibrator can be well
shown by immersing the vibrating body in a trough of water, and the much
greater rate of damping out of the motion in that case can then be
compared with the rate of damping in air.

It has been indicated that the differential equation does not represent
oscillatory motion if the value of R is too great. The exact condition
depends on the roots of the quadratic equation Lx² + Rx + 1⧸C = 0,
obtained by writing 1 for Q, and x for d⧸dt, and then treating x as a
quantity. These roots are -R⧸2L ± √(R²⧸4L² - 1⧸CL), and are
therefore real or imaginary according as 4L⧸C is less or greater than
R². If the roots are real, that is, if R² be greater than 4L⧸C, the
discharge will not be oscillatory; the Faraday tubes referred to above
will be absorbed in the wire without any return to the condenser. The
corresponding result happens with the vibrator when R is sufficiently
great, or L⧸C sufficiently small (a weak spring and a small mass, or
both), to enable the condition to be fulfilled.

If, however, the roots of the quadratic are imaginary, that is, if 4L⧸C
be greater than R² (a condition which will be fulfilled in the spring
analogue, by making the spring sufficiently stiff and the mass large
enough to prevent the friction from controlling the motion) the motion
is one in which Q disappears by oscillations about zero, of continually
diminishing amplitude. A complete discussion gives for the period of
oscillation 4πL⧸√(4L⧸C - R²), or if R be comparatively small, 2π√(LC).
The charge Q falls off by the fraction e^{-RT⧸2L} (where e is the
number 2.71828...) in each period T, and so gradually disappears.

Thus electric oscillations are produced, that is to say, the charged
state of the condenser subsides by oscillations, in which the charged
state undergoes successive reversals, with dissipation of energy in the
wire; and both the period and the rate of dissipation can be calculated
if L, C, and R are known, or can be found, for the system. These
quantities can be calculated and adjusted in certain definite cases, and
as the electric oscillations can be experimentally observed, the theory
can be verified. This has been done by various experimenters.

Returning to the pendulum illustration, it will be seen that the
pendulum held deflected is analogous to the charged jar, letting the
pendulum go corresponds to connecting the discharging coil to the
coatings, the motion of the pendulum is the analogue of that motion of
the medium in which consists the magnetic field, the friction of the air
answers to the resistance of the wire which finally damps out the
current. The inertia or mass of the bob is the analogue of what Thomson
called the electromagnetic inertia of the coil and connections; what is
now generally called the self-inductance of the conducting system. The
component of gravity along the path towards the lowest point, answers to
the reciprocal, 1⧸C, of the capacity of the condenser.

It appears from the analogy that just as the oscillations of a pendulum
can be prevented by immersing the bob in a more resisting medium, such
as treacle or oil, so that when released the pendulum slips down to the
vertical without passing it, so by properly proportioning the resistance
in the circuit to the electromagnetic inertia of the coil, oscillatory
discharge of the Leyden jar may also be rendered impossible.

All this was worked out in an exceedingly instructive manner in
Thomson's paper; the account of the matter by the motion of Faraday
tubes is more recent, and is valuable as suggesting how the inertia
effect of the coil arises. The analogy of the pendulum is a true one,
and enables the facts to be described; but it is to be remembered that
it becomes evident only as a consequence of the mathematical treatment
of the electrical problem. The paper was of great importance for the
investigation of the electric waves used in wireless telegraphy in our
own time. It enabled the period of oscillation of different systems to
be calculated, and so the rates of exciters and receivers of electric
waves to be found. For such vibrators are really Leyden jars, or
condensers, caused to discharge in an oscillatory manner.

This application was not foreseen by Thomson, and, indeed, could hardly
be, as the idea of electric waves in an insulating medium came a good
deal later in the work of Maxwell. Yet the analogy of the pendulum, if
it had then been examined, might have suggested such waves. As the bob
oscillates backwards and forwards the air in which it is immersed is
periodically disturbed, and waves radiate outwards from it through the
surrounding atmosphere. The energy of these waves is exceedingly small,
otherwise, as pointed out above, a term would have to be included in the
theory of the resisted motion of the pendulum to account for this energy
of radiation. So likewise when the electric vibrations proceed, and the
insulating medium is the seat of a periodically varying magnetic field,
electromagnetic waves are propagated outwards through the surrounding
medium--the ether--and the energy carried away by the waves is derived
from the initial energy of the charged condenser. In strictness also
Thomson's theory of electric oscillations requires an addition to
account for the energy lost by radiation. This is wanting, and the whole
decay of the amount of energy present at the oscillator is put down to
the action of resistance--that is, to something of the nature of
frictional retardation. Notwithstanding this defect of the theory, which
is after all not so serious as certain difficulties of exact calculation
of the self-inductance of the discharging conductor, the periods of
vibrators can be very accurately found. When these are known it is only
necessary to measure the length of an electrical wave to find its
velocity of propagation. When electromagnetic waves were discovered
experimentally in 1888 by Heinrich Hertz, it was thus that he was able
to demonstrate that they travelled with the velocity of light.

Thomson suggested that double, triple and quadruple flashes of lightning
might be successive flashes of an oscillatory discharge. He also pointed
out that if a spark-gap were included in a properly arranged condenser
and discharging wire, it might be possible, by means of Wheatstone's
revolving mirror, to see the sparks produced in the successive
oscillations, as "points or short lines of light separated by dark
intervals, instead of a single point of light, or of an unbroken line of
light, as it would be if the discharge were instantaneous, or were
continuous, or of appreciable duration."

This anticipation was verified by experiments made by Feddersen, and
published in 1859 (_Pogg. Ann._, 108, 1859). The subject was also
investigated in Helmholtz's laboratory at Berlin, by N. Schiller, who,
determining the period for condensers with different substances between
the plates, was able to deduce the inductive capacities of these
substances (_Pogg. Ann._, 152, 1874). [The specific inductive capacity
of an insulator is the ratio of the capacity of a condenser with the
substance between the plates to the capacity of an exactly similar
condenser with air between the plates.]

The particular case of non-oscillatory discharge obtained by supposing C
and Q both infinitely great and to have a finite ratio V (which will be
the potential, p. 34, of the charged plate), is considered in the paper.
The discharging conductor is thus subjected to a difference of potential
suddenly applied and maintained at one end, while the other end is kept
at potential zero. The solution of the differential equation for this
case will show how the current rises from zero in the wire to its final
steady value. If c be put as before for the current -dQ⧸dt, and the
constant value V for Q⧸C, the equation is

  L(dc⧸dt) + Rc = V

which gives, since c = 0 when t = 0,

  c = (V⧸R)[1 - e^{-(R⧸L)t}].

Thus, when an infinite time has elapsed the current has become V⧸R, the
steady value.

Thomson concludes by showing how, by measuring the non-oscillatory
discharge of a condenser (the capacity of which can be calculated) by
means of an electrodynamometer and an ordinary galvanometer arranged in
series, what W. Weber called the duration of the discharging current may
be determined. From this Thomson deduced a value for the ratio of the
electromagnetic unit of electricity to the electrostatic unit, and
indicated methods of determining this ratio experimentally. This ratio
is of fundamental importance in electromagnetic theory, and is
essentially of the nature of a speed. According to Maxwell it is the
speed of propagation of electromagnetic waves in an insulating medium
for which the units are defined. It was first determined in the Glasgow
laboratory by Mr. Dugald McKichan, and has been determined many times
since. It is practically identical with the speed of light as
ascertained by the best experiments.



CHAPTER XI

THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'--GYROSTATIC
ACTION--'ELECTROSTATICS AND MAGNETISM'


THE 'NATURAL PHILOSOPHY'

Professor Tait was appointed to the Chair of Natural Philosophy in the
University of Edinburgh in 1860, and came almost immediately into
frequent contact with Thomson. Both were Peterhouse men, trained by the
same private tutor--William Hopkins--both were enthusiastic
investigators in mathematical as well as in experimental physics, they
taught in the sister universities of Edinburgh and Glasgow, and had much
the same kind of classes to deal with and the same educational problems
to solve. Tait was an Edinburgh man--an old school-fellow of Clerk
Maxwell at the Edinburgh Academy--and had therefore been exposed to that
contact, in play and in work, with compeers of like age and
capabilities, which is one of the best preparations for the larger
school and more serious struggles of life. Thomson's early education,
under his father's anxious care, had no doubt certain advantages, and
his early entrance into college classes gave him to a great extent that
intercourse with others for which such advantages are never complete
compensation. The two men had much community of thought and experience,
and the literary partnership into which they entered was hailed as one
likely to do much for the progress of science.

In some ways, however, Thomson and Tait were very different
personalities. Thomson troubled himself little with metaphysical
subtleties, his conceptions were like those of Newton, absolutely clear
so far as they went; he never, in his teaching at least, showed any
disposition to discuss the "foundations of dynamics," or the conception
of motion in a straight line. These were taken for granted like the
fundamental ideas in a book on geometry; and the student was left to do
what every true dynamical student must do for himself sooner or
later--to compare the abstractions of dynamics with the products of his
experience in the world of matter and force. Perhaps a little guidance
now and then in the difficulties about conceptions, which beset every
beginner, might not have been amiss: but Thomson was so intent on the
concrete example in hand--pendulum or gyrostat, or what not--that he
left each man to form or correct his own ideas by the lessons which such
examples afford to every one who carefully examines them.

Tait, on the other hand, though he continually denounced metaphysical
discussion, was in reality much more metaphysical than Thomson, and
seemed to take pleasure in the somewhat transcendental arguments with
regard to matters of analysis which were put forward, especially in the
_Elements of Quaternions_, by Sir William Rowan Hamilton, of Dublin, a
master whom he much revered. But there is metaphysics and metaphysics!
and the pronouncements of professed metaphysicians were often
characterised as non-scientific and fruitless, which no doubt they were
from the physical point of view.

Then Tait was strongly convinced of the importance for physics of the
quaternion analysis: Thomson was not, to say the least; and this was
probably the main reason why the vectorial treatment of displacement,
velocities, and other directed quantities, has no place in the joint
writings of the two Scottish professors. In controversy Tait was a
formidable antagonist: when war was declared he gave no quarter and
asked for none, though he never fought an unchivalric battle. He admired
foreign investigators--and especially von Helmholtz--but he was always
ready to put on his armour and place lance in rest for the cause of
British science. Thomson was much less of a combatant, though he also
could bravely splinter a spear with an opponent on occasion, as in the
memorable discussion with Huxley on the Age of the Earth.

Tait's professorial lectures were always models of clear and logical
arrangement. Every statement bore on the business in hand; the
experimental illustrations, always carefully prepared beforehand, were
called for at the proper time and were invariably successful. With
Thomson it was otherwise: his digressions, though sometimes inspired and
inspiring, were fatal to the success of the utmost efforts of his
assistants to make his lectures successful systematic expositions of the
facts and principles of elementary physics.

As has been stated in Chapter IV, two books were announced in 1863 as in
course of preparation for the ensuing session of College. These were not
published until 1867 and 1873; the first issued was the famous _Treatise
on Natural Philosophy_, the second was entitled _Elements of Natural
Philosophy_, and consisted in the main of part of the non-mathematical
or large type portions of the Treatise. The scheme of the latter was
that of an articulated skeleton of statements of principles and results,
printed in ordinary type, with the mathematical deductions and proofs in
smaller type. As was to be expected, the Elements, to a student whose
mathematical reading was wide enough to tackle the Treatise, was the
more difficult book of the two to completely master. But the continued
large print narrative, as it may be called, is extremely valuable. It is
a memorial of a habit of mind which was characteristic of both authors.
They kept before them always the idea or thing rather than its symbol;
and thus the edifice which they built up seemed never obscured by the
scaffolding and machinery used in its erection. And as far as possible
in processes of deduction the ideas are emphasised throughout; there is
no mere putting in at one end and taking out at the other; the result is
examined and described at every stage. As in all else of Thomson's work,
physical interpretation is kept in view at every step, and made
available for correction and avoidance of errors, and the suggestion of
new inquiries.

The book as it stands consists of "Division I, Preliminary" and part of
"Division II, Abstract Dynamics." Division I includes the chapter on
Kinematics already referred to, a chapter on Dynamical Laws and
Principles, chapters on Experience and Measures and Instruments.
Division II is represented only by Chapter V, Introductory; Chapter VI,
Statics of a Particle and Attractions; and Chapter VII, Statics of
Solids and Fluids. Thus Abstract Dynamics is without the more complete
treatment of Kinetics to which, as well as to Statics, the discussion of
Dynamical Laws and Principles was intended to be an introduction. But to
a considerable extent, as we shall see, Kinetics is treated in this
introductory chapter: indeed, the discussion of the general theorems of
dynamics and their applications to kinetics is remarkably complete.

In Volume II it was intended to include chapters on the kinetics of a
particle and of solid and fluid bodies, on the vibrations of solid
bodies, and on wave-motion in general. It was expected also to contain a
chapter much referred to in Volume I, on "Properties of Matter." That
the work was not completed is a matter of keen regret to all physicists,
regret, however, now tempered by the fact that many of the subjects of
the unfulfilled programme are represented by such works as Lord
Rayleigh's _Theory of Sound_, Lamb's Hydrodynamics, and Routh's
_Dynamics of a System of Rigid Bodies_. But all deeply lament the loss
of the "Properties of Matter." No one can ever write it as Thomson would
have written it. His students obtained in his lectures glimpses of the
things it might have contained, and it was most eagerly looked for. If
that chapter only had been given, the loss caused by the discontinuance
of the book would not have been so irreparable.

The first edition of the book was published by the Clarendon Press,
Oxford. It was printed by Messrs. Constable, of Edinburgh, and is a
beautiful specimen of mathematical typography. In some ways the first
edition is exceedingly interesting, for it is not too much to say that
its issue had an influence on dynamical science, and its exposition in
this country, only second to that due to Newton's Principia. Three
other works, perhaps, have had the same degree and kind of influence on
mathematical thought--Laplace's _Mécanique Céleste_, Lagrange's
_Mécanique Analytique_, and Fourier's _Théorie Analytique de la
Chaleur_.

The second edition was issued by the Cambridge University Press as Parts
I and II in 1878 and 1883. Various younger mathematicians now of
eminence--Professor Chrystal, of Edinburgh, and Professor Burnside, of
Greenwich, may be mentioned--read the proofs, and it is on the whole
remarkably free from typographical and other errors. With the issue of
Part II, the continuation was definitely abandoned.

In the second edition many topics are more fully discussed, and the
contents include a very valuable account of cycloidal motion (or
oscillatory motion, as it is more usually called), and of a revised
version of the chapter on Statics which forms the concluding portion of
the book, and which discusses some of the great problems of terrestrial
and cosmical physics.

Various speculations have been indulged in, from time to time, as to the
respective parts contributed to the work by the two authors, but these
are generally very wide of the mark. The mode of composition of the
sections on cycloidal (oscillatory) motion gives some idea of Thomson's
method of working. His proofs (of "T and T-dash" as the authors called
the book) were carried with him by rail and steamer, and he worked
incessantly (without, however, altogether withdrawing his attention from
what was going on around him!) at corrections and additions. He
corrected heavily on the proofs, and then overflowed into additional
manuscript. Thus, when he came to the short original § 343, he greatly
extended that in the first instance, and proceeded from section to
section until additions numbered from § 343a to § 343p, amounting in all
to some ten pages of small print, had been interpolated. Similarly § 345
was extended by the addition of §§ 345 (i) to 345 (xxviii), mainly on
gyrostatic domination. The method had the disadvantage of interrupting
the printers and keeping type long standing, but the matter was often
all the more inspiring through having been produced under pressure from
the printing office. Indeed, much was no doubt written in this way
which, to the great loss of dynamical science, would otherwise never
have been written at all.

The kinematical discussion begins with the consideration of motion along
a continuous line, curved or straight. This naturally suggests the ideas
of curvature and tortuosity, which are fully dealt with mathematically,
before the notion of velocity is introduced. When that is done, the
directional quality of velocity is not so much insisted on as is now the
case: for example, a point is spoken of as moving in a curve with a
uniform velocity; and of course in the language of the present time,
which has been rendered more precise by vector ideas, if not by
vector-analysis, the velocity of a point which is continually changing
the direction of its motion, cannot be uniform. The same remark may be
made regarding the treatment of acceleration: in both cases the
reference of the quantity to three Cartesian axes is immediate, and the
changes of the components, thus fixed in direction, are alone
considered.

There can be no doubt that greater clearness is obtained by the process
afterwards insisted on by Tait, of considering by a hodographic diagram
the changes of velocity in successive intervals of time, and from these
discovering the direction and magnitude of the rate of change at each
instant. This method is indeed indicated at § 37, but no diagram is
given, and the properties of the hodograph are investigated by means of
Cartesians. The subject is, however, treated in the Elements by the
method here indicated.

Remarkable features of this chapter are the very complete discussion of
simple harmonic or vibratory motion, the sections on rotation, and the
geometry of rolling and precessional motion, and on the curvature of
surfaces as investigated by kinematical methods. A remark made in § 96
should be borne in mind by all who essay to solve gyrostatic problems.
It is that just as acceleration, which is always at right angles to the
motion of a point, produces a change in the direction of the motion but
none in the speed of the point (it does influence the velocity), so an
action, tending always to produce rotation about an axis at right angles
to that about which a rigid body is already rotating, will change the
direction of the axis about which the body revolves, but will produce no
change in the rate of turning.[20]

A very full and clear account of the analysis of strains is given in
this chapter, in preparation for the treatment of elasticity which comes
later in the book; and a long appendix is added on Spherical Harmonics,
which are defined as homogeneous functions of the coordinates which
satisfy the differential equation of the distribution of temperature in
a medium in which there is steady flow of heat, or of distribution of
potential in an electrical field. This appendix is within its scope one
of the most masterly discussions of this subject ever written, though,
from the point of view of rigidity of proof, required by modern
function-theory, it may be open to objection.

In the next chapter, which is entitled "Dynamical Laws and Principles,"
the authors at the outset declare their intention of following the
Principia closely in the discussion of the general foundations of the
subject. Accordingly, after some definitions the laws of motion are
stated, and the opportunity is taken to adopt and enforce the Gaussian
system of absolute units for dynamical quantities. As has been indicated
above, the various difficulties more or less metaphysical which must
occur to every thoughtful student in considering Newton's laws of motion
are not discussed, and probably such a discussion was beyond the scheme
which the authors had in view. But metaphysics is not altogether
excluded. It is stated that "matter has an innate power of resisting
external influences, so that every body, as far as it can, remains at
rest, or moves uniformly in a straight line," and it is stated that this
property--inertia--is proportional to the quantity of matter in the
body. This statement is criticised by Maxwell in his review of the
_Natural Philosophy_ in Nature in 1879 (one of the last papers that
Maxwell wrote). He asks, "Is it a fact that 'matter has any power,
either innate or acquired, of resisting external influences'? Does not
every force which acts on a body always produce that change in the
motion of the body by which its value, as a force, is reckoned? Is a cup
of tea to be accused of resisting the sweetening influence of sugar,
because it persistently refuses to turn sweet unless the sugar is put
into it?"

This innate power of resisting is merely the _materiæ vis insita_ of
Newton's "Definitio III," given in the Principia, and the statement to
which Maxwell objects is only a free translation of that definition.
Moreover, when a body is drawn or pushed by other bodies, it reacts on
those bodies with an equal force, and this reaction is just as real as
the action: its existence is due to the inertia of the body. The
definition, from one point of view, is only a statement of the fact that
the acceleration produced in a body in certain circumstances depends
upon the body itself, as well as on the other bodies concerned, but from
another it may be regarded as accounting for the reaction. The mass or
inertia of the body is only such a number that, for different bodies in
the same circumstances as to the action of other bodies in giving them
acceleration, the product of the mass and the acceleration is the same
for all. It is, however, a very important property of the body, for it
is one factor of the quantum of kinetic energy which the body
contributes to the energy of the system, in consequence of its motion
relatively to the chosen axes of reference, which are taken as at rest.

The relativity of motion is not emphasised so greatly in the _Natural
Philosophy_ as in some more modern treatises, but it is not overlooked;
and whatever may be the view taken as to the importance of dwelling on
such considerations in a treatise on dynamics, there can be no doubt
that the return to Newton was on the whole a salutary change of the
manner of teaching the subject.

The treatment of force in the first and second laws of motion is frankly
causal. Force is there the cause of rate of change of momentum; and this
view Professor Tait in his own writings has always combated, it must be
admitted, in a very cogent manner. According to him, force is merely
rate of change of momentum. Hence the forces in equations of motion are
only expressions, the values of which as rates of change of momentum,
are to be made explicit by the solution of such equations in terms of
known quantities. And there does not seem to be any logical escape from
this conclusion, though, except as a way of speaking, the reference to
cause disappears.

The discussion of the third law of motion is particularly valuable, for,
as is well known, attention was therein called to the fact that in the
last sentences of the Scholium which Newton appended to his remarks on
the third law, the rates of working of the acting and reacting forces
between the bodies are equal and opposite. Thus the whole work done in
any time by the parts of a system on one another is zero, and the
doctrine of conservation of energy is virtually contained in Newton's
statement. The only point in which the theory was not complete so far as
ordinary dynamical actions are concerned, was in regard to work done
against friction, for which, when heat was left out of account, there
was no visible equivalent. Newton's statement of the equality of what
Thomson and Tait called "activity" and "counter-activity" is, however,
perfectly absolute. In the completion of the theory of energy on the
side of the conversion of heat into work, Thomson, as we have seen, took
a very prominent part.

After the introduction of the dynamical laws the most interesting part
of this chapter is the elaborate discussion which it contains of the
Lagrangian equations of motion, of the principle of Least Action, with
the large number of extremely important applications of these theories.
The originality and suggestiveness of this part of the book, taken
alone, would entitle it to rank with the great classics--the _Mécanique
Céleste_, the _Mécanique Analytique_, and the memoirs of Jacobi and
Hamilton--all of which were an outcome of the Principia, and from which,
with the Principia, the authors of the _Natural Philosophy_ drew their
inspiration.

It is perhaps the case, as Professor Tait himself suggested, that no one
has yet arisen who can bend to the fullest extent the bow which Hamilton
fashioned; but when this Ulysses appears it will be found that his
strength and skill have been nurtured by the study of the _Natural
Philosophy_. Lagrange's equations are now, thanks to the physical
reality which the expositions and examples of Thomson and Tait have
given to generalised forces, coordinates, and velocities, applied to all
kinds of systems which formerly seemed to be outside the range of
dynamical treatment. As Maxwell put it, "The credit of breaking up the
monopoly of the great masters of the spell, and making all their charms
familiar in our ears as household words, belongs in great measure to
Thomson and Tait. The two northern wizards were the first who, without
compunction or dread, uttered in their mother tongue the true and proper
names of those dynamical concepts, which the magicians of old were wont
to invoke only by the aid of muttered symbols and inarticulate
equations. And now the feeblest among us can repeat the words of power,
and take part in dynamical discussions which a few years ago we should
have left to our betters."

A very remarkable feature in this discussion is the use made of the idea
of "ignoration of coordinates." The variables made use of in the
Lagrangian equations must be such as to enable the positions of the
parts of the system which determine the motion to be expressed for any
instant of time. These parts, by their displacements, control those of
the other parts, through the connections of the system. They are called
the independent coordinates, and sometimes the "degrees of freedom," of
the system. Into the expressions of the kinetic and potential energies,
from which by a formal process the equations of motion, as many in
number as there are degrees of freedom, are derived, the value of these
variables and of the corresponding velocities enter in the general case.
But in certain cases some of the variables are represented by the
corresponding velocities only, and the variables themselves do not
appear in the equations of motion. For example, when fly-wheels form
part of the system, and are connected with the rest of the system only
by their bearings, the angle through which the wheel has turned from any
epoch of time is of no consequence, the only thing which affects the
energy of the system is the angular velocity or angular momentum of the
wheel. The system is said by Thomson and Tait in such a case to be under
gyrostatic domination. (See "Gyrostatic Action," p. 214 below.)

Moreover, since the force which is the rate of growth of the momentum
corresponding to any coordinate is numerically the rate of variation
with that coordinate of the difference of the kinetic and potential
energies, every force is zero for which the coordinate does not appear;
and therefore the corresponding momentum is constant. But that momentum
is expressed by means of the values of other coordinates which do appear
and their velocities, with the velocities for the absent coordinates;
and as many equations are furnished by the constant values of such
momenta as there are coordinates absent. The corresponding velocities
can be determined from these equations in terms of the constant momenta
and the coordinates which appear and their velocities. The values so
found, substituted in the expressions for the kinetic and potential
energies, remove from these expressions every reference to the absent
coordinates. Then from the new expression for the kinetic energy (in
which a function of the constant momenta now appears, and is taken as an
addition to the potential energy) the equations of motion are formed for
the coordinates actually present, and these are sufficient to determine
the motion. The other coordinates are thus in a certain sense ignored,
and the method is called that of "ignoration of coordinates."

Theorems of action of great importance for a general theory of optics
conclude this chapter; but of these it is impossible to give here any
account, without a discussion of technicalities beyond the reading of
ordinary students of dynamics.

In an Appendix to Part I an account is given of Continuous Calculating
Machines. Ordinary calculating machines, such as the "arithmometer" of
Thomas of Colmar, carry out calculations and exhibit the result as a row
of figures. But the machines here described are of a different
character: they exhibit their results by values of a continuously
varying quantity. The first is one for predicting the height of the
tides for future time, at any port for which data have been already
obtained regarding tidal heights, by means of a self-registering
tide-gauge. Two of these were made according to the ideas set forth in
this Appendix; one is in the South Kensington Museum, the other is at
the National Physical Laboratory at Bushy House, where it is used mainly
for drawing on paper curves of future tidal heights, for ports in the
Indian Ocean. From these curves tide-tables are compiled, and issued for
the use of mariners and others.

Another machine described in this Appendix was designed for the
mechanical solution of simultaneous linear equations. It is impossible
to explain here the interesting arrangement of six frames, carrying as
many pulleys, adjustable along slides (for the solution of equations
involving six unknown quantities), which Thomson constructed, and which
is now in the Natural Philosophy Department at Glasgow. The idea of
arranging the first practical machine for this number of variables, was
that it might be used for the calculation of the corrections on values
already found for the six elements of a comet or asteroid. The machine
was made, but some mechanical difficulties arose in applying it, and
the experiments with it were not at the time persevered with. Very
possibly, however, it may yet be brought into use.

[Illustration: FIG. 14.]

But the most wonderful of these mechanical arrangements is the machine
for analysing the curves drawn by a self-registering tide gauge, so as
to exhibit the constants of the harmonic curves, and thus enable the
prediction of tidal heights to be carried out either by the
tide-predicting machine, or by calculation. One day in 1876, Thomson
remarked to his brother, James Thomson, then Professor of Engineering at
Glasgow, that all he required for the construction of a tidal analyser
was a form of integrating machine more satisfactory for his purpose than
the usual type of integrator employed by surveyors and naval architects.
James Thomson at once replied that he had invented, a long time before,
what he called a disk-globe-cylinder-integrator. This consisted of a
brass disk, with its plane inclined to the horizontal, which could be
turned about its axis by a wheel gearing in teeth on the edge of the
disk, and driven by the operator in a manner which will presently
appear. Parallel and close to the disk, but not touching it, was placed
a horizontal cylinder of brass, about 2 inches in diameter (called the
registering cylinder), and between the disk and this cylinder was laid a
metal ball about 2½ inches in diameter. When the disk was kept at
rest, and the ball was rolled along between the cylinder and disk, the
trace of its rolling on the latter was a straight horizontal line
passing through the centre. Supposing then that the point of contact of
the ball with the disk was on one side, at a distance from the centre,
and that the disk was then turned, the ball was by the friction between
it and the disk made to roll, and so to turn the cylinder. The angular
velocity of rolling, and therefore the angular velocity of the cylinder,
was proportional to the speed of the part of the disk in contact with
it, that is, to y. It was also proportional to the speed of turning of
the disk.

The mode by which this machine effects an integration will now be
evident. Imagine the area to be found to lie between a curve and a
straight datum line, drawn on a band of paper. This is stretched on a
large cylinder, with the datum line round the cylinder. We call this the
paper-cylinder. The distances of the different points of the curve from
the datum line are values of y. A horizontal bar parallel to the
cylinder carries a fork at one end and a projecting style at the other.
The globe just fits between the prongs of the fork, and when the bar is
moved in the direction of its length carries the ball along the disk and
cylinder. When the style at the other end is on the datum line, the
centre of the ball is at the centre of the disk, and the turning of the
disk does not turn the cylinder. When the bar is displaced in the line
of its own length to bring the style from the datum line to a point on
the curve, the ball is displaced a distance y, and there is a
corresponding turning of the cylinder by the action of the ball. In the
use of the instrument the paper-cylinder is turned by the operator while
the style is kept on the curve, and the disk is turned by the gearing
already referred to, which is driven by a shaft geared with that of the
paper-cylinder. Thus the displacement of the ball is always y, the
ordinate of the curve, and for any displacement dx along the datum line,
the registering cylinder is turned through an angle proportional to ydx.
Thus any finite angle turned through is proportional to the integral of
ydx for the corresponding part of the curve: a scale round one end of
the registering cylinder gives that angle. Thomson immediately perceived
that this extremely ingenious integrating machine was just what he
required for his purpose. The curve of tidal heights drawn (on a reduced
scale, of course) by a tide-gauge, is really the resultant of a large
number of simple curves, represented by a series of harmonic terms, the
coefficients of which are certain integrals. The problem is the
evaluation of these integrals; and the method usually employed is to
obtain them by measurement of ordinates of the curve and an elaborate
process of calculation. But one of them is simply the integral area
between the curve and the datum line corresponding to the mean water
level, and the others are the integrals of quantities of the type
y sin nx.dx, where y is the ordinate of the curve, and n a number
inversely proportional to the period of the tidal constituent
represented by the term.

All that was necessary, in order to give the integral of a term
y sin nx.dx, was to make the disk oscillate about its axis as the
paper-cylinder was turned through an angle proportional to x. Thus one
disk, globe, and cylinder was arranged exactly as has been described for
the integral of ydx, and with this as many others as there were harmonic
terms to be evaluated from the curve were combined as follows. The disks
were placed all in one plane with their centres all on one horizontal
line, and the cylinders with their axes also in line, and a single
sliding bar, with a fork for each globe, gave in each case the
displacement y from the centre of the disk.

The requisite different speeds of oscillation were given to the disks by
shafts geared with the paper-cylinder, by trains of wheels cut with the
proper number of teeth for the speed required.

Thus the angles turned through by the registering cylinders when a curve
on the paper-cylinder was passed under the style were proportional to
the integrals required, and it was only necessary to calibrate the
graduation of the scales of these cylinders by means of known curves to
obtain the integrals in proper units.

One of these machines, which analyses four harmonic constituents, is in
the Natural Philosophy Department at Glasgow; a much larger machine, to
analyse a tidal curve containing five pairs of harmonic terms, or eleven
constituents in all, was made for the British Association Committee on
Tidal Observations, and is probably now in the South Kensington Museum.

But still more remarkable applications which Thomson made of his
brother's integrating machine were to the mechanical integration of
linear differential equations, with variable coefficients, to the
integration of the general linear differential equation of any order,
and, finally, to the integration of any differential equation of any
order.

These applications were all made in a few days, almost in a few hours,
after James Thomson first described the elementary machine, and papers
containing descriptions of the combinations required were at once
dictated by Thomson to his secretary, and despatched for publication.
Very possibly he had thought out the applications to some extent before;
but it is unlikely that he had done so in detail. But, even if it were
so, the connection of a series of machines by the single controlling
bar, and the production of the oscillations of the disks, all
controlled, as they were, by the motion of a simple point along the
curve, so as to give the required Fourier coefficients, were almost
instantaneous, and afford an example of invention amounting to
inspiration.

There should be noticed here also the geometrical slide for use in
safety-valves, cathetometers and other instruments, and the
hole-slot-and-plane mode of so supporting an instrument now used in all
laboratories. These were Thomson's inventions, and their importance is
insisted on in the _Natural Philosophy_.


In Part II, the principal subjects treated are attractions, elasticity,
such great hydrostatical examples as the equilibrium theory of the tides
and the equilibrium of rotating liquid spheroids, and such problems of
astronomical and terrestrial dynamics as the distribution of matter in
the earth, with the bearing on this subject of the precession of the
equinoxes, tidal friction, the earth's rigidity, the effects of elastic
tides, the secular cooling of the earth, the age of the earth, and the
"age of the sun's heat." Of these, with the exception of the age of the
earth, we shall not attempt to give any account. The importance of the
original contributions to elasticity contained in the book is indicated
by the large space devoted to the _Natural Philosophy_ in Professor Karl
Pearson's continuation of Todhunter's _History of Elasticity_. The heavy
task of editing Part II was performed mainly by Sir George Darwin, who
made many notable additions from his own researches to the matter
contained in the first edition.

In the next chapter an attempt will be made to present Thomson's views
on the subject of the age of the earth. These, when they were published,
attracted much attention, and received a good deal of hostile criticism
from geologists and biologists, whose processes they were deemed to
restrict to an entirely inadequate period of time.


GYROSTATIC ACTION

Thomson in his lectures and otherwise gave a great deal of attention to
the motion of gyrostats, and to the effect of the inclusion of gyrostats
in a system on its properties. Reference has been made to the treatment
of "gyrostatic domination" in "Thomson and Tait." A gyrostat consists of
a disk or wheel with a massive rim, which revolves within a case or
framework, by which the whole arrangement can be moved about, or
supported, without interfering with the wheel. The ordinary toy
consisting of wheel with a massive rim, and a light frame, is an
example. But much larger and more carefully made instruments, in which
the wheel is entirely enclosed, give the most interesting experiments.
The body seems to have its properties entirely altered by the rotation
of the wheel, and of course the case prevents any outward change from
being visible.

[Illustration: FIG. 15.]

Figure 15 shows one form of gyrostat mounted on a horizontal frame, held
in the hands of an experimenter. The axis of the fly-wheel is vertical
within the tubular part of the case; the fly-wheel is within the part on
which is engraved an arrow-head to show the direction of rotation. Round
the case in the plane of the wheel is a projecting rim sharpened to an
edge, on which the gyrostat can be supported in other experiments. To
the rim are screwed two projecting pivots, which can turn in bearings on
the two sides of the frame as shown. The centre of mass of the wheel is
on the level of these pivots, so that the instrument will remain with
either end of the axis up.

If the fly-wheel be not in rotation, the experimenter can carry the
arrangement about, and the fly-wheel and case move with it as if the
gyrostat were merely an ordinary rigid body. But now remove the
gyrostat from the frame, and set the wheel in rotation. This is done by
an endless cord wrapped round a small pulley fast on the axle (to which
access is obtained by a hole just opposite in the case) and passed also
round a larger pulley on the shaft of a motor. When the motor is started
the cord must be tightened only very gently at first, so that it slips
on the pulley, otherwise the motor would be retarded, and possibly
burned by the current. The fly-wheel gradually gets up speed, and then
the cord can be brought quite tight so that no slipping occurs. When the
speed is great enough the cord is cut with a stroke from a sharp knife
and runs out.

The gyrostat is now replaced on its pivots in the frame, with its axis
vertical, and moved about as it was before. If the experimenter, holding
the frame as shown, turns round in the direction of the arrow, which is
that of rotation, nothing happens. If, however, he turns round the other
way, the gyrostat immediately turns on its pivots so as to point the
other end of the axis up. If the experimenter continues his turning
motion, the gyrostat is now quiescent: for it is being carried round now
in the direction of rotation. Thus, with no gravitational stability at
all (since the centre is on a level with the pivots) the gyrostat is in
stable equilibrium when carried round in the direction of rotation, but
is in unstable equilibrium when carried round the opposite way.

Thus, if the observer knew nothing of the rotation of the fly-wheel, and
could see and feel only the outside of the case, the behaviour of the
instrument might well appear very astonishing.

This is a case of what Thomson and Tait call "gyrostatic domination,"
which is treated very fully in their Sections 345 (vi) to 345 (xxviii)
of Part I. It may be remarked here that this case of motion may be
easily treated mathematically in an exceedingly elementary manner, and
the instability of the one case, and the stability of the other, made
clear to the beginner who has only a notion of the composition of
angular momenta about different axes.

A year or two ago it was suggested by Professor Pickering, of Harvard,
that the fact that the outermost satellite of Saturn revolves in the
direction opposite to the planet's rotation, may be due to the fact that
originally Saturn rotated in the direction of the motion of this moon,
but inasmuch as his motion round the sun was opposite in direction to
his rotation, he was turned, so to speak, upside down, like the
gyrostat! The other satellites, it is suggested, were thrown off later,
as their revolution is direct. Professor Pickering refers to an
experiment (similar to that described above) which he gives as new.
Thomson had shown this experiment for many years, as an example of the
general discussion in "Thomson and Tait," and its theory had already
been explicitly published.[21]

Many other experiments with gyrostats used to be shown by Thomson to
visitors. Many of these are indicated in "Thomson and Tait." The earth's
precessional motion is a gyrostatic effect due to the differential
attraction of the sun, which tends to bring the plane of the equator
into coincidence with the ecliptic, and so alters the direction of the
axis of rotation. Old students will remember the balanced globe--with
inclined material axis rolling round a horizontal ring--by which the
kinematics of the motion could be studied, and the displacement of the
equinoxes on the ecliptic traced.

[Illustration: FIG. 16.]

Another example of the gyrostatic domination discussed in "Thomson and
Tait" is given in the very remarkable address entitled "A Kinetic Theory
of Matter," which Sir William Thomson delivered to Section A of the
British Association at Montreal, in 1884. Figure 16 shows an ordinary
double "coach spring," the upper and lower members of which carry two
hooked rods as shown. If the upper hook is attached to a fixed support,
and a weight is hung on the lower, the spring will be drawn out, and the
arrangement will be in equilibrium under a certain elongation. If the
weight be pulled down further and then left to itself, it will vibrate
up and down in a period depending upon the equilibrium elongation
produced by the weight. The same thing will happen if a spiral spring be
substituted for the coach spring. A spherical case, through which the
hooked rods pass freely, hides the internal parts from view.

[Illustration: FIG. 17.]

Figure 17 shows two hooked rods, as in the former case, attached by
swivels to two opposite corners of a frame formed of four rods jointed
together at their ends. Each of these is divided in the middle for the
insertion of a gyrostat, the axis of which is pivoted on the adjacent
ends of the two halves of the rod. A spherical case, indicated by the
circle, again hides the internal arrangement from inspection, but
permits the hooked rods to move freely up and down. The swivels allow
the frame, gyrostats and all, to be turned about the line of the hooks.

If now the gyrostats be not in rotation, the frame will be perfectly
limp, and will not in the least resist pull applied by a weight. But if
the gyrostats be rotated in the directions shown by the circles, with
arrow-heads drawn round the rods, there will be angular momentum of the
whole system about the line joining the hooks, and if a weight or a
force be applied to pull out the frame along that line, the pull will be
resisted just as it was in the other case by the spring. Moreover,
equilibrium will be obtained with an elongation proportional to the
weight hung on, and small oscillations will be performed just as if
there were a spring in the interior instead of the gyrostats.

According as the frame is pulled out, or shortened, the angular momentum
of the gyrostats about the line joining the hooks is increased or
diminished, and the frame, carrying the gyrostats with it, turns about
the swivels in one direction or the other, at the rate necessary to
maintain the angular momentum at a constant value. But this will not be
perceived from without.

The rotation of the fly-wheels thus gives to the otherwise limp frame
the elasticity which the spring possesses; without dissection of the
model the difference cannot be perceived. This illustrates Thomson's
idea that the elasticity of matter may be due to motion of molecules or
groups of molecules of the body, imbedded in a connecting framework,
deformed by applied forces as in this model, and producing displacements
which are resisted in consequence of the motion.

And here may be mentioned also Thomson's explanation of the phenomenon,
discovered by Faraday, of the rotation of the plane of a beam of
polarised light which is passed along the lines of force of a magnetic
field. This rotation is distinct altogether from that which is produced
when polarised light is passed along a tube filled with a solution of
sugar or tartaric acid. If the ray be reflected after passage, and made
to retraverse the medium, the rotation is annulled in the latter case,
it is doubled in the former. This led Thomson to the view that in sugar,
tartaric acid, quartz, etc., the turning is due to the structure of the
substance, and in the magnetic field to rotation already existing in the
medium. He used to say that a very large number of minute spiral
cavities all in the same direction, and all right-handed or all
left-handed, in the sugar or quartz, would give the effect; on the other
hand, the magnetic phenomenon could only be produced by some arrangement
analogous to a very large number of tops, or gyrostats, imbedded in the
medium with their axes all in one direction (or preponderatingly so) and
all turning the same way. The rotation of these tops or gyrostats
Thomson supposed to be caused by the magnetic field, and to be
essentially that which constitutes the magnetisation of the medium.

Let the frame of the gyrostatic spring-balance described above, turn
round the line joining the hooks so as to exactly compensate, by turning
in the opposite direction, the angular momentum about that line given by
the fly-wheels; then the arrangement will have no angular momentum on
the whole; and a large number of such balances, all very minute and
hooked together, will form a substance without angular momentum in any
part. But now by the equivalent of a magnetic force along the lines of
the hooks, let a different angular turning of the frames be produced;
the medium will possess a specific angular momentum in every part. If a
wave of transverse vibrations which are parallel to one direction (that
is, if the wave be plane-polarised) enter the medium in the direction of
the axes of the frames, the direction of vibration will be turned as the
wave proceeds, that is, the plane of polarisation will be turned round.

More recent research has shown an effect of a magnetic field on the
spectrum of light produced in the field, and viewed with a spectroscope
in a direction at right angles to the field--the Zeeman effect, as it is
called--and the explanation of this effect by equations of moving
electric charges, which are essentially gyrostatic equations, is
suggestive of an analogy or correspondence between the systems of moving
electrons which constitute these charges, and some such gyrostatic
molecules as Thomson imagined. It has been pointed out that the Zeeman
effect, in its simple forms at least, can be exactly imitated by the
motion of an ordinary pendulum having a gyrostat in its bob, with its
axis directed along the suspension rod.[22]


ELECTROSTATICS AND MAGNETISM

In the ten years from 1863 to 1873 Thomson was extremely busy with
literary work. In 1872, five years after the publication of the treatise
on _Natural Philosophy_, and just before the appearance of the Elements,
Messrs. Macmillan & Co. published for him a collection of memoirs
entitled _Reprint of Papers on Electrostatics and Magnetism_. The
volume contains 596 pages, and the subjects dealt with range from the
"Uniform Motion of Heat and its Connection with the Mathematical Theory
of Electricity" (the paper already described in Chapter II above) and
the discussion of Electrometers and Electrostatic Measuring Instruments,
to a complete mathematical theory of magnetism. The subject of
electrostatics led naturally to the consideration of electrical
measuring instruments as they existed forty years ago (about 1867), and
their replacement by others, the indications of which from day to day
should be directly comparable, and capable of being interpreted in
absolute units. Down to that time people had been obliged to content
themselves with gold-leaf electroscopes, and indeed it was impossible
for accurate measuring instruments to be invented until a system of
absolute units had been completely worked out. The task of fixing upon
definitions of units and of realising them in suitable standards had
been begun by the British Association, and it was as part of the Report
of that Committee to the Dundee Meeting in 1867 that Thomson's paper on
Electrometers first appeared.

It was there pointed out that an electrometer is essentially an
instrument for measuring differences of electric potential between
conductors, by means of effects of electrostatic force. Such a
difference is what a gold-leaf electroscope indicates for its gold
leaves and the walls surrounding the air-space in which they are
suspended. As electroscopes used to be constructed, these walls were
made of glass imperfectly covered, if at all, by conducting material,
and the electroscope was quite indefinite and uncertain in its action.
The instrument was also, as made, quite insensitive. Recently, however,
it has been rehabilitated in reputation, and brought into use as a very
sensitive indicator of effects of radio-activity.

Thomson described in this paper six species of electrometers of his own
devising. The best known of these are his quadrant electrometer and his
attracted-disk electrometers. The former is to be found in some form or
other in every laboratory nowadays, and need not be described in detail.
The action is of two conductors--the two pairs of opposite quadrants of
a shallow, horizontal, cylindrical box, made by dividing the box into
four by two slits at right angles--upon an electrified slip of aluminium
suspended by a two-thread suspension within the box, with its length
along one of the slits. The two pairs of opposite quadrants are at the
potential difference to be measured, and the slip of aluminium, or
"needle," has each end urged round from a quadrant at higher potential
towards one at a lower, and these actions conspire to turn the slip
against its tendency to return to the position in which the two threads
are in one plane. Thus the deflection (measured by the displacement of a
reflected ray of light used as index) gives an indication of the amount
of the potential difference.

The electrification of the "needle" was kept up by enclosing the
quadrantal box within an electrified Leyden jar, to the interior coating
of which contact is made by a platinum wire, depending from the needle
to sulphuric acid contained in the jar. The whole apparatus was enclosed
in a conducting case connected to earth. This made its action perfectly
definite. Variations of this electrification of the jar were shown by
an attached attracted-disk electrometer, the principle of which we shall
merely indicate.

The quadrant electrometer has now been vastly increased in sensibility
by the use of a single quartz fibre as suspension. By the invention of
this fibre, which is exceedingly strong and is, moreover, so definite in
its elastic properties that it comes back at once exactly to its former
zero state after twist, Mr. C. V. Boys has increased the delicacy of all
kinds of suspended indicators many fold. But it ought to be remembered
that a Dolezalek electrometer, with some hundred or more times the
sensibility of the bifilar instrument, was only made possible by its
predecessor.

Attracted-disk-electrometers simply measure, either by weighing or by
the deflection of a spring, the attractive force between two parallel
disks at different potentials. From the determination of this force, and
the measurement of the distance between the disks (or better, of an
alteration of the distance) a difference of potentials can be
determined, and a unit for it obtained, which is in direct and known
relation to ordinary dynamical units. Thomson's "Absolute Electrometer"
was designed specially for accurate determinations of this kind. Another
form, called the Long Range Electrometer, was devised for the
measurement of the potentials of the charged conductors in electric
machines and Leyden jars.

Accurate determinations of the sparking resistance between parallel
plates charged to different potentials in air were made by means of
attracted-disk-electrometers in the course of some important experiments
described in the _Electrostatics and Magnetism_. These results have been
much referred to in later researches.

A small attracted-disk-electrometer was used as indicated above to keep
a watch on the electrification of the Leyden jar of the quadrant
instrument, and a small induction machine was added, by turning which
the operator could make good any loss of charge of the jar.

This electrical machine was an example of an apparatus on precisely the
same principle as the Voss or Wimshurst machines of the present day. In
it by a set of moving carriers, influenced by conductors, the charges of
the latter were increased according to a compound interest principle
only interfered with by leakage to the air or by the supports. Several
forms of this machine, on the same principle, were constructed by
Thomson, and described in 1868; but he afterwards found that he had been
anticipated by C. F. Varley in 1860. Still later it was discovered that
a similar instrument had been made a century before by Nicholson, and
called by him the "Revolving Doubler."

The experiments which Thomson made on atmospheric electricity at the old
College tower, and by means of portable electrometers in Arran and
elsewhere, can only be mentioned. They led no doubt to some improvements
on electrometers which he made, the method of bringing the nozzle of a
water-dropper, or a point on a portable electrometer to the potential of
the air, by the inductive action on a stream of water-drops in the one
case, or the particles of smoke from a burning match in the other. He
invented a self-acting machine, worked by a stream of water-drops, for
accumulating electric charges, on the principle of the revolving
doubler. It was this apparently that led to the machines with revolving
carriers, to which reference has been made above.

The mathematical theory of magnetism which Thomson gave in 1849, in the
_Phil. Trans. R.S._, was, when completed by various later papers, a
systematic discussion of the whole subject, including electromagnetism
and diamagnetism. To a large extent the ground covered by the 1849 paper
had been traversed before by Poisson, and partially by Murphy and Green;
but Thomson stated that one chief object of his memoir was to formally
construct the theory without reference to the two magnetic fluids, by
means of which the facts of experiment and conclusions of theory had so
far been expressed. He found it, however, convenient to introduce the
idea of positive and negative magnetic matter (attracting and repelling
as do charges of positive and negative electricity), which are to be
regarded as always present in equal amounts, not only in a magnet as a
whole, but in every portion of a magnet; and at first sight this might
appear like a return to the magnetic fluids. But it amounts on the whole
rather to a conception of a magnet as a conglomeration of doublets of
magnetic matter (that is, very close, equal and inseparable charges of
the two kinds of matter), the arrangement of which can be changed by the
action of magnetic force. This idea is set forth now in all the books on
magnetism and electricity. There can be no doubt that the systematic
presentment of the subject by Thomson, and the theorems and ideas of
magnetic force and magnetic permeability by which he rendered the clear,
and therefore mathematical, notions of Faraday explicitly quantitative,
had much influence in furthering the progress of electrical science,
and so leading on the one hand to the electromagnetic theories of
Maxwell, and on the other to modern research on the magnetic properties
of iron, and to the correct ideas which now prevail as to construction
of dynamo-electric machines and motors.



CHAPTER XII

THE AGE OF THE EARTH


From his student days throughout his life, Lord Kelvin took a keen
interest in geological questions. He was always an active member of the
Geological Society of Glasgow, and was its president for twenty-one
years (1872-1893). The distribution of heat in the substance of the
earth was the subject of his inaugural dissertation as Professor of
Natural Philosophy; and previously, as a student, he had written an
essay on "The Figure of the Earth," for which he had been awarded a
University Gold Medal. He never ceased to ponder over the problems of
terrestrial physics, and he wrote much on the subject. His papers are to
be found as Appendices to Thomson and Tait's _Natural Philosophy_, and
in vol. ii of his _Popular Lectures and Addresses_, which is devoted to
geology and general physics.

His conclusions regarding the age of the earth have been referred to in
the last chapter. The first allusion to the subject was contained (see
p. 65 above) in his inaugural dissertation "_De Caloris distributione in
Terræ Corpus_"; but he returned to it again in a communication made to
the Royal Society of Edinburgh in December, 1865, and entitled "The
Doctrine of Uniformity in Geology briefly refuted." On February 27,
1868, he delivered to the Geological Society of Glasgow an address
entitled "On Geological Time," in which the necessity for limiting
geological and other changes to an almost infinitesimal fraction of the
vast periods at that time demanded was insisted on, and which gave rise
to much discussion.

The address began with a protest against the old uniformitarian view of
geological changes as expressed by Playfair in his _Illustrations of the
Huttonian Theory_. The first objection taken to the idea that "in the
continuation of the different species of animals and vegetables that
inhabit the earth, we discern neither a beginning nor an end; in the
planetary motions where geometry has carried the eye so far, both into
the future and the past, we discover no mark either of the commencement
or the termination of the present order" is, that the stability of the
motions of the heavenly bodies, to which reference is made in this
statement, is founded upon what is essentially an approximate
calculation, which leaves out, by intention, the consideration of
frictional resistance.

He points out, for example, that the friction which accompanies the
relative motion of the waters of the earth and the land is attended by
the production of heat, and that, by the doctrine of the conservation of
energy, heat cannot be produced without a disappearance of an equivalent
quantity of energy, either of motion or of position. The chief source of
this energy is the earth's rotation. Since the earth turns under the
moon and the tidal spheroid--that is, the earth's shape as distorted by
the heaping up of the waters in the tides--remains on the whole
stationary with respect to the moon, the solid matter of the earth turns
under the distribution of the water, held more or less fixed by the
moon, as does a fly-wheel under a stationary friction band round its
rim. Then just as the band held fixed retards the fly-wheel, so the
earth must be retarded in its rotation by this water-brake. In the
earth's rotation there is a store of kinetic energy which, roughly
estimated, would not be exhausted in less than ten million million
years, although drawn upon continuously by friction, or other actions,
at the rate of one million horse-power; so that, no immediate
catastrophe, such as that we should be involved in by the stoppage or
considerable retardation of the spinning motion of the earth, is
possible. But it was pointed out by Thomson that the best results of
astronomical observation show that the earth would in one hundred years
fall behind a perfect time-keeper, with which its rotation kept pace at
the beginning of the time, by about twenty seconds. The tendency is to
make the earth turn slower, and the moon to increase its distance and
move more slowly in its orbit, but with a resultant effect towards
coincidence of the period of the earth's rotation with that of
revolution of the moon round the earth. After this coincidence has been
attained, however, the solar tides will tend to make the moon fall in
towards the earth.

If then the earth be rotating more and more slowly, as time goes on, at
present, it must have been rotating more rapidly in past time. A
thousand million years ago, at the present rate of retardation, the
earth must have been rotating one seventh part of its speed faster than
it is rotating at present, and this would give for centrifugal force at
the surface one thousand million years ago, greater than the centrifugal
force at present, in the ratio of 64 to 49. Apparently therefore the
earth must have solidified at a much later date than that epoch, a date
when it was rotating much more nearly with the angular speed which it
has now; otherwise the figure of the earth would have deviated much more
from the spherical form than it actually does. On the other hand, one
hundred million years ago centrifugal force would be only three per
cent. greater than it is at present, and consolidation of the earth at
that less remote period would give a shape to the earth not very
different from that which it now possesses. The argument therefore from
tidal retardation would cut down the time available for geological and
biological changes to something not much more than one hundred million
years, perhaps to less.

A second argument for limitation of the time available for such
processes is derived from the sun's heat. The sun cannot be regarded as
a miraculous body producing its light and heat from nothing. Changes of
the constitution of the sun must be continually proceeding, to account
for its enormous radiation of energy into space, a radiation of which
only an infinitesimal part is received by the bodies of the solar
system, and a still more minute portion by the earth. The effects of the
sun's light and heat on the earth show how enormous must be the quantity
of energy lost from the sun in a year. How is this loss of energy to be
accounted for? What is the physical change which gives rise to it? In
1854 Thomson put forward the theory that the sun's heat is kept up by
the falling in of meteors on the sun's surface, but he afterwards saw
reason to abandon that view. Helmholtz had advocated the theory that the
sun was a body heated by the coming together of the matter composing it
by its mutual attraction, a process which, although the sun is now a
continuous mass, is to be regarded as still going on. It is easy to
calculate the exhaustion of potential energy caused by the coming
together of the matter of the sun from universal dispersion through
infinite space to a sphere of uniform density of the present size of the
sun. The result is about as much energy as would be generated by burning
seven million million million million million tons of coal. The amount
radiated in each hour is about as much as would be generated by burning
something like nine tons of coal every hour on every square yard of the
sun's surface. It is certain that the sun must be still contracting, and
if it contracts sufficiently to just make good this expenditure by the
further exhaustion of potential energy involved in the closer
aggregation of the matter, it must diminish in radius in each year by as
much as 130 feet.

The amount of energy generated by the falling together of the matter of
the sun from universal diffusion to the dimensions which the sun has at
present, is only about 13,000,000 times the amount now radiated per
annum. In Thomson's paper Pouillet's estimate of the energy radiated per
second is used, and this number is raised to 20,000,000. Taking the
latter estimate, the whole potential energy exhausted by the
condensation of the sun's mass to uniform density would suffice for only
20,000,000 years' supply. But the sun is undoubtedly of much greater
density in the central parts than near the surface, and so the energy
exhausted must be much greater than that stated above. This will raise
the number of years provided for. On the other hand, a considerable
amount of energy would be dissipated during the process of
condensation, and this would reduce the period of radiation estimated.
Thomson suggests that 50,000,000, or 100,000,000, years is a possible
estimate.

It is not unlikely that the rate of radiation in past time, when the sun
had not nearly condensed to its present size, was so much less than it
is at present that the period suggested above may have to be
considerably augmented. Another source of radiation, which seems to be
regarded by some authorities as a probable, if not a certain, one, has
been suggested in recent years--the presence of radio-active substances
in the sun. So far as we know, Lord Kelvin did not admit that this
source of radiation was worthy of consideration; but of course, granted
its existence to an extent comparable with the energy derivable from
condensation of the sun's mass, the "age of the sun's heat" would have
to be very greatly extended. These are matters, however, on which
further light may be thrown as research in radio-activity progresses.
Lord Kelvin was engaged when seized with his last illness in discussing
the changes of energy in a gaseous, or partially gaseous, globe, slowly
cooling and shrinking in doing so; and a posthumous paper on the subject
will shortly be published which may possibly contain further information
on this question of solar physics.

But Thomson put forward a third argument in the paper on Geological
Time, which has always been regarded as the most important. It is
derived from the fact, established by abundant observations, that the
temperature in the earth's crust increases from the surface inwards; and
that therefore the earth must be continually losing heat by conduction
from within. If the earth be supposed to have been of uniform
temperature at some period of past time and in a molten state, and
certain assumptions as to the conductive power and melting point of its
material be made, the time of cooling until the gradient of temperature
at the surface acquired its present value can be calculated. This was
done by Thomson in a paper published in the _Transactions, R.S.E._, in
1862. We propose to give here a short sketch of his argument, which has
excited much interest, and been the cause of some controversy.

In order to understand this argument, the reader must bear in mind some
fundamental facts of the flow of heat in a solid. Let him imagine a slab
of any uniform material, say sandstone or marble, the two parallel faces
of which are continually maintained at two different temperatures,
uniform over each face. For example, steam may be continually blown
against one face, while ice-cold water is made to flow over the other.
Heat will flow across the slab from the hotter face to the colder. It
will be found that the rate of flow of heat per unit area of face, that
is per square centimetre, or per square inch, is proportional to the
difference of the temperatures in the slab at the two faces, and
inversely proportional to the thickness of the slab. In other words, it
is proportional to the fall of temperature from one face to the other
taken per unit of the thickness, that is, to the "gradient of
temperature" from one face to the other. Moreover, comparing the flow in
one substance with the flow in another, we find it different in
different substances for the same gradient of temperature. Thus we get
finally a flow of heat across unit area of the slab which is equal to
the gradient of temperature multiplied by a number which depends on the
material: that number is called the "conductivity" of the substance.

Now, borings made in the earth show that the temperature increases
inwards, and the same thing is shown by the higher temperatures found in
deeper coal mines. By means of thermometers sunk to different depths,
the rate of increase of temperature with depth has been determined.
Similar observations show that the daily and annual variations of
temperature caused by the succession of day and night, and summer and
winter, penetrate to only a comparatively small depth below the
surface--three or four feet in the former case, sixty or seventy in the
latter. Leaving these variations out of account, since the average of
their effects over a considerable interval of time must be nothing, we
have in the earth a body at every point of the crust of which there is a
gradient of increasing temperature inwards. The amount of this may be
taken as one degree of Fahrenheit's scale for every 50 feet of descent.
This gradient is not uniform, but diminishes at greater depths.
Supposing the material of uniform quality as regards heat-conducting
power, the mathematical theory of a cooling globe of solid material (or
of a straight bar which does not lose heat from its sides) gives on
certain suppositions the gradients at different depths. The surface
gradient of 1° F. in 50 feet may be taken as holding for 5000 feet or
6000 feet or more.

This gradient of diminution of temperature outwards leads inevitably to
the conclusion that heat must be constantly flowing from the interior of
the earth towards the surface. This is as certain as that heat flows
along a poker, one end of which is in the fire, from the heated end to
the other. The heat which arrives at the surface of the earth is
radiated to the atmosphere or carried off by convection currents; there
is no doubt that it is lost from the earth. Thus the earth must be
cooling at a rate which can be calculated on certain assumptions, and it
is possible on these assumptions to calculate backwards, and determine
the interval of time which must have elapsed since the earth was just
beginning to cool from a molten condition, when of course life cannot
have existed on its surface, and those geological changes which have
effected so much can hardly have began.

Considering a globe of uniform material, and of great radius, which was
initially at one temperature, and at a certain instant had its surface
suddenly brought to, let us say, the temperature of melting ice, at
which the surface was kept ever after, we can find, by Fourier's
mathematical theory of the flow of heat, the gradient of temperature at
any subsequent time for a point on the surface, or at any specified
distance within it. For a point on the surface this gradient is simply
proportional to the initial uniform temperature, and inversely
proportional to the square root of the product of the "diffusivity" of
the material (the ratio of the conductivity to the specific heat) by the
interval of time which has elapsed since the cooling was started. Taking
a foot as the unit of length, and a year as the unit of time, we find
the diffusivity of the surface strata to be 400. If we take the initial
temperature as 7000 degrees F.--which is high enough for melting
rock--and take the interval of time which has elapsed as 100,000,000
years, we obtain at the surface a gradient approximately equal to that
which now exists. A greater interval of time would give a lower
gradient, a smaller interval would give a higher gradient than that
which exists at present. A lower initial temperature would require a
smaller interval of time, a higher initial temperature a longer interval
for the present gradient.

With the initial temperature of 7,000 degrees F., an interval of
4,000,000 years would give a surface gradient of 1° F. in 10 ft. Thus,
on the assumption made, the surface gradient of temperature has
diminished from 1⧸10 to 1⧸50 in about 96,000,000 years. After 10,000
years from the beginning of the cooling the gradient of temperature
would be 2° F. per foot. But, as Thomson showed, such a large gradient
would not lead to any sensible augmentation of the surface temperature,
for "the radiation from earth and atmosphere into space would almost
certainly be so rapid" as to prevent this. Hence he inferred that
conducted heat, even at that early period, could not sensibly affect the
general climate.

Two objections (apart from the assumptions already indicated) will
readily occur to any one considering this theory, and these Thomson
answered by anticipation. The first is, that no natural action could
possibly bring the surface of a uniformly heated globe instantaneously
to a temperature 7000° lower, and keep it so ever after. In reply to
this Thomson urged "that a large mass of melted rock, exposed freely to
our earth and sky, will, after it once becomes crusted over, present in
a few hours, or a few days, or at most a few weeks, a surface so cool
that it can be walked over with impunity. Hence, after 10,000 years, or
indeed, I may say, after a single year, its condition will be sensibly
the same as if the actual lowering of temperature experienced by the
surface had been produced in an instant, and maintained constant ever
after." The other objection was, that the earth was probably never a
uniformly heated solid 7000° F. above the present surface temperature as
assumed for the purpose of calculation. This Thomson answers by giving
reasons for believing that "the earth, although once all melted, or
melted all round its surface, did, in all probability, really become a
solid at its melting temperature all through, or all through the outer
layer which has been melted; and not until the solidification was thus
complete, or nearly so, did the surface begin to cool."

Thomson was inclined to believe that a temperature of 7000° F. was
probably too high, and results of experiments on the melting of basalt
and other rocks led him to prefer a much reduced temperature. This, as
has already been pointed out, would give a smaller value for the age of
the earth. In a letter on the subject published in Nature (vol. 51,
1895) he states that he "is not led to differ much" from an estimate of
24,000,000 years founded by Mr. Clarence King (_American Journal of
Science_, January 1893) on experiments on the physical properties of
rocks at high temperatures.

It is to be observed that the assumptions made above that the physical
constants of the material are constant throughout the earth, and at all
temperatures, are confessedly far from the truth. Nevertheless Thomson
strongly held that the uncertainty of the data can at most extend the
earth's age to some value between 20,000,000 and 200,000,000 of years,
and that the enormously long periods which were wont to be asked for by
geologists and biologists for the changes of the earth's surface and the
development of its flora and fauna, cannot possibly be conceded.

In Nature for January 3, 1895, Professor John Perry suggested that very
possibly the conductivity of the material composing the interior of the
earth was considerably higher than that of the surface strata. If this
were so, then, as can be shown without difficulty, the attainment of the
present gradient would be very greatly retarded, and therefore the age
of the earth correspondingly increased. The question then arose, and was
discussed, as to whether the rocks and other materials at high
temperatures were more or less conducting than at low temperatures, and
experiments on the subject were instituted and carried out. On the
whole, the evidence seemed to show that the conductivity of most
substances is diminished, not increased, by the rise of temperature, and
so far as it went, therefore, the evidence was against Professor Perry's
suggestion. On the other hand, he contended that the inside of the earth
may be a mass of great rigidity, partly solid and partly fluid,
possessing a "quasi-conductivity" which might greatly increase the
period of cooling. The subject is a difficult one both from a
mathematical and from the physical point of view, and further
investigation is necessary, especially of the behaviour of materials
under the enormous stresses which they undoubtedly sustain in the
interior of the earth.

After the publication of the paper on Geological Time a reply to it was
made by Professor Huxley, in an address to the Geological Society of
London, delivered on February 19, 1869. He adopted the rôle of an
advocate retained for the defence of geology against what seems to have
been regarded as an unwarranted attack, made by one who had no right to
offer an opinion on a geological question. For, after a long and
eloquent "pleading," he concludes his address with the words: "My
functions, as your advocate, are at an end. I speak with more than the
sincerity of a mere advocate when I express the belief that the case
against us has entirely broken down. The cry for reform which has been
raised from without is superfluous, inasmuch as we have long been
reforming from within with all needful speed; and the critical
examination of the grounds upon which the very grave charge of
opposition to the principles of Natural Philosophy has been brought
against us, rather shows that we have exercised a wise discrimination in
declining to meddle with our foundations at the bidding of the first
passer-by who fancies our house is not so well built as it might be." To
this Thomson rejoined in an address entitled "Of Geological Dynamics,"
also delivered to the Geological Society of Glasgow on April 5, 1869;
and to this, with Professor Huxley's address, the reader must be
referred for the objection, brought against Thomson's arguments, and the
replies which were immediately forthcoming. This is not the place to
discuss the question, but reference may be made to an interesting paper
on the subject in the _Glasgow Herald_ for February 22, 1908, by
Professor J. W. Gregory, in which the suggestion of Professor Perry, of
a nearer approach to uniformity of temperature in the interior of the
earth than Thomson had thought possible, is welcomed as possibly
extending the interval of time available to a period sufficient for all
purposes. In Professor Gregory's opinion, "Lord Kelvin in one respect
showed a keener insight than Huxley, who, referring to possible changes
in the rate of rotation of the earth, or in the heat given forth from
the sun or in the cooling of the earth, declared that geologists are
Gallios, 'who care for none of these things.' An ever-increasing school
of geologists now cares greatly for these questions, and reveres Lord
Kelvin as one of the founders of the geology of the inner earth."

After all, the problem is not one to be dealt with by the geologist or
biologist alone, but to be solved, so far as it can be solved at all, by
a consideration of all relevant evidence, from whatsoever quarter it may
come. It will not do in these days for scientific men to shut themselves
up within their special departments and to say, with regard to branches
of science which deal with other aspects of nature and other problems of
the past, present and future of that same earth on which all dwell and
work, that they "care for none of these things." This is an echo of an
old spirit, not yet dead, that has done much harm to the progress of
science. The division of science into departments is unavoidable, for
specialisation is imperative; but it is all the more necessary to
remember that the divisions set up are more or less arbitrary, and that
there are absolutely no frontiers to be guarded and enforced. Chemistry,
physiology, and physics cannot be walled off from one another without
loss to all; and geology has suffered immensely through its having been
regarded as essentially a branch of natural history, the devotees of
which have no concern with considerations of natural philosophy. Lord
Kelvin's dignified questions were unanswerable. "Who are the occupants
of 'our house,' and who is the 'passer-by'? Is geology not a branch of
physical science? Are investigations, experimental and mathematical, of
underground temperature not to be regarded as an integral part of
geology?... For myself, I am anxious to be regarded by geologists not as
a mere passer-by, but as one constantly interested in their grand
subject, and anxious in any way, however slight, to assist them in their
search for truth."



CHAPTER XIII

BRITISH ASSOCIATION COMMITTEE ON ELECTRICAL STANDARDS


When Professor Thomson began his work as a teacher in the University of
Glasgow, there was, as has already been noticed, great vagueness of
specification of physical quantities. Few of the formal definitions of
units of measurement, now to be found in the pages of every elementary
text book, had been framed, and there was much confusion of quantities
essentially distinct, a confusion which is now, to some extent at least,
guarded against by the adoption of a definite unit, with a distinctive
name for each magnitude to be measured. Thus rate of working, or
activity, was confused with work done; the condition for maximum
activity in the circuit of a battery or dynamo was often quoted as the
condition of greatest efficiency, that is of greatest economy of energy,
although it was exactly that in which half the available energy was
wasted.

Partly as a consequence of this vagueness of specification, there was a
great want of knowledge of the values of physical constants; for without
exact definitions of quantities to be determined, such definitions as
would indicate units for their measurement, related to ordinary
dynamical units according to a consistent scheme, it was impossible to
devise satisfactory experimental methods to do for electricity and
magnetism what had been done by Regnault and others for heat.

The first steps towards the construction of a complete system of units
for the quantitative measurement of magnetic and electric quantities
were taken by Gauss, in his celebrated paper entitled _Intensitas vis
magneticæ terrestris ad mensuram absolutam revocata_, published in 1832.
In this he showed how magnetic forces could be expressed in absolute
units, and thus be connected with the absolute dynamical units which
Gauss, in the same paper, based on chosen fundamental units of length,
mass, and time. Thus the modern system of absolute units of dynamical
quantities, and its extension to magnetism, are due to the practical
insight of a great mathematician, not to the experimentalists or
"practicians" of the time.

Methods of measuring electric quantities in absolute units were
described by W. Weber, in Parts II and III of his _Elecktrodynamische
Maassbestimmungen_, published in 1852. These were great steps in
advance, and rendered further progress in the science of absolute
measurement comparatively easy. But they remained the only steps taken
until the British Association Committee began their work. We have
already (pp. 74-76) referred to the great importance of that work, not
only for practical applications but also for the advancement of science.
But it was not a task which struck the imagination or excited the wonder
of the multitude. For the realisation of standards of resistance, for
example, involved long and tedious investigations of the effects of
impurities on the resistance of metals, and the variation of resistance
caused by change of temperature and lapse of time. Then alloys had to
be sought which would have a temperature effect of small amount, and
which were stable and durable in all their properties.

The discoveries of the experimentalist who finds a new element of
hitherto undreamed-of properties attract world-wide attention, and the
glory of the achievement is deservedly great. But the patient, plodding
work which gives a universal system of units and related standards, and
which enables a great physical subject like electricity and magnetism to
rise from a mere enumeration of qualitative results to a science of the
most delicate and exact measurement, and to find its practical
applications in all the affairs of daily life and commerce, is equally
deserving of the admiration and gratitude of mankind. Yet it receives
little or no recognition.

The construction of a standard of resistance was the first task
undertaken by the committee; but other units, for example of quantity of
electricity, intensity of electric field and difference of potential,
had also to be defined, and methods of employing them in experimental
work devised. It would be out of place to endeavour to discuss these
units here, but some idea of the manner in which their definitions are
founded on dynamical conceptions may be obtained from one or two
examples. Therefore we shall describe two simple experiments, which will
illustrate this dynamical foundation. An account has been given in
Chapter XI of the series of electrometers which Thomson invented for the
measurement of differences of electric potential. These all act by the
evaluation in terms of ordinary dynamical units of the force urging an
electrified body from a place of higher towards a place of lower
potential.

Some indication of the meaning of electrical quantities has been given
in Chapter IV. Difference of electric potential between two points in an
electric field was there defined as the dynamical work done in carrying
a unit of positive electricity against the forces of the field from the
point of lower to the point of higher potential. Now by the definition
of unit quantity of electricity given in electrical theory--that
quantity which, concentrated at a point at unit distance from an equal
quantity also concentrated at a point, is repelled with unit force--we
can find, by the simple experiment of hanging two pith balls (or,
better, two hollow, gilded beads of equal size) by two fine fibres of
quartz, a metre long, say, electrifying the two balls as they hang in
contact, and observing the distance at which they then hang, the
numerical magnitude in absolute units of a charge of electricity, and
apply that to finding the charge on a large spherical conductor and the
potential at points in its field also in absolute units. If m be
the mass of a ball, g gravity in cm. sec. units, d the distance in
cms. of the centres of the balls apart, and l the length in cms. of
a thread, the charge q, say, on each ball is easily found to be
√[mgd³⧸√{4^(l² - d²)}]. Thus the charge is got in absolute
centimetre-gramme-second units in terms of the mass m obtained by
ordinary weighing, and l and d obtained by easy and exact measurements.

If one of the balls be now taken away without discharging the other, and
the latter be placed in the field of a large electrified spherical
conductor, the fibre will be deflected from the vertical by the force on
the ball. Let the two centres be now on the same level. That force is
got at once from the angle of deflection (which is easily observed),
the charge on the ball, and the value of m. The electric field-intensity
is obtained by dividing the value of the force by q. The field intensity
multiplied by D, the distance apart in cms. of the centres of the ball
and the conductor, gives the potential at the centre of the ball in
C.G.S. units. Multiplication again by D gives the charge on the
conductor.

When it made its first Report in 1862 (to the meeting at Cambridge) the
committee consisted of Professors A. Williamson, C. Wheatstone, W.
Thomson, W. H. Miller, Dr. A. Matthiessen, and Mr. F. Jenkin. At the
next meeting, at Newcastle, it had been augmented by the addition of
Messrs. Balfour Stewart, C. W. Siemens, Professor Clerk Maxwell, Dr.
Joule, Dr. Esselbach, and Sir Charles Bright. The duty with which the
committee had been charged was that of constructing a suitable standard
of resistance. A reference to the account given in Chapter X above, of
the derivation of what came to be called the electromagnetic unit of
difference of potential, or electromotive force, by means of a simple
magneto-electric machine--a disk turning on a uniform magnetic field, or
the simple rails and slider and magnetic field arrangement there
described--will show how from this unit and the electromagnetic unit of
current (there also defined) the unit of resistance is defined. It is
the resistance of the circuit of slider, rails, and connecting wire,
when with this electromagnetic unit of electromotive force the unit of
current is made to flow.

This was one clear and definite way of defining the unit of current, and
of attaining the important object of connecting the units in such a way
that the rate of working in a circuit, or the energy expended in any
time, should be expressed at once in ordinary dynamical units of
activity or energy. A considerable number of proposals were discussed by
the committee; but it was finally determined to take the basis here
indicated, and to realise a standard of resistance in material of
constant and durable properties, which should have some simple multiple
of the unit of resistance, in the system of dynamical units based on the
centimetre as unit of length, the gramme as unit of mass, and the second
as unit of time--the so-called C.G.S. system. The comparison of the
different metals and alloys available was a most important but
exceedingly laborious series of investigations, carried out mainly by
Dr. Matthiessen and Professor Williamson.

Professor Thomson suggested to the committee the celebrated method of
determining the resistance of a circuit by revolving a coil, which
formed the main part of the circuit about a vertical axis in the earth's
magnetic field. An account of the experiments made with this method is
contained in the Report of 1863. They were carried out at King's
College, London, where Maxwell was then Professor of Experimental
Physics, by Maxwell, Balfour Stewart, and Fleeming Jenkin. The
theoretical discussion and the description of the experiments was
written by Maxwell, the details of the apparatus were described by
Jenkin.

The principle of the method is essentially the same as that of the
simple magneto-electric machine, to which reference has just been made.
Two parallel coils of wire were wound in channels cut round rings of
brass, which, however, were cut across by slots filled with vulcanite,
to prevent induced currents from circulating in the brass. These coils
were mounted in a vertical position and could be driven as a rigid
system, at a constant measured speed, about a vertical axis passing
through the centre of the system. Between the coils at this centre was
hung, from a steady support, a small magnetic needle by a single fibre
of silk; and a surrounding screen prevented the needle and suspension
from being affected by currents of air.

The ends of the coil were connected together so that the whole revolved
as a closed circuit about the vertical axis. When the coil system was at
right angles to the magnetic meridian there was a magnetic induction
through it of amount AH, where A denotes the effective area of the
coils, and H the horizontal component of the earth's magnetic field. By
one half-turn the coil was reversed with reference to this magnetic
induction, and as the coil turned an induced current was generated,
which depended at any instant on the rate at which the magnetic
induction was varying at the instant, on the inductive electromotive
force due to the varying of the current in the coil itself, and on the
resistance of the circuit. A periodic current thus flowed in one
direction _relatively to the coil_ in one half-turn from a position
perpendicular to the magnetic meridian, and in the opposite direction in
the next half-turn. But as the position of the coil was reversed in
every half-turn as well as the current in it, the current flowed on the
whole in the same average direction relatively to the needle, and but
for self-induction would have had its maximum value always when the
plane of the coil was in the magnetic meridian.

The needle was deflected as it would have been by a certain average
current, and the deflection was opposed by the action of the earth's
horizontal magnetic field H. But this was the field cut by the coil as
it turned, and therefore (except for a small term depending on the
turning of the coil in the field of the needle) the value of H did not
appear in the result, and did not require to be known.

Full details of the theory of this method and of the experiments carried
out to test it will be found in various memoirs and treatises[23]; but
it must suffice here to state that the resistance of the coil was
determined in this way, by a large series of experiments, before and
after every one of which the resistance was compared with that of a
German-silver standard. The resistance of this standard therefore became
known in absolute units, and copies of it, or multiples or sub-multiples
of it, could be made.

A unit called the B.A. unit, which was intended to contain 10^9 C.G.S.
electromagnetic units of resistance, was constructed from these
experiments, and copies of it were soon after to be found in nearly all
the physical laboratories of the world. Resistance boxes were
constructed by various makers, in which the coils were various multiples
of the B.A. unit, so that any resistance within a certain range could be
obtained by connecting these coils in series (which was easily done by
removing short circuiting plugs), and thus the absolute units of current
electromotive force and resistance came into general use.

In 1881 Lord Rayleigh and Professor Schuster carried out a very careful
repetition of the British Association experiments with the same
apparatus at the Cavendish Laboratory, and obtained a somewhat different
result. They found that the former result was about 1.17 per cent. too
small. Lord Rayleigh next carried out an independent set of experiments
by the same method with improved apparatus, and found that this
percentage error must be increased to about 1.35.

It may be noticed here that the simple disk machine, of Thomson's
illustration of the absolute unit of electromotive force, has been used
by Lorenz to give a method of determining resistance which is now
recognised as the best of all. It is sketched here that the reader may
obtain some idea of later work on this very important subject; work
which is a continuation of that of the original British Association
Committee by their successors. A circuit is made up of a standard coil
of wire, the ends of which are made to touch at the circumference and
near the centre of the disk, which is placed symmetrically with respect
to a cylindrical coil, and within it. A current is sent round this coil
from a battery, and produces a magnetic field within the coil, the lines
of magnetic force of which pass across the plane of the disk. This
current, or a measured fraction of it, is also made to flow through the
standard coil. The disk is now turned at a measured speed about its
axis, so that the electromotive force due to the cutting of the field
tends to produce a current in the standard coil of wire. The
electromotive force of the disk is made to oppose the potential
difference between the ends of this coil due to the current, so that no
current flows along the disk or the wires connecting it with the
standard coil. The magnetic field within the coil can be calculated from
the form and dimensions of the coil and the current in it (supposed for
the moment to be known), and the electromotive force of the disk is
obtained in terms of its dimensions and its speed and the field
intensity. But this electromotive force, which is proportional to the
current in the coil, is equal to the product of the resistance of the
wire and the same current, or a known fraction of it. Thus the current
appears on both sides of the equation and goes out, and the value of the
resistance is found in absolute units.

Lord Rayleigh obtained, by this method, a result which showed that the
B.A. unit was 1.323 per cent. too small; and exact experiments have been
made by others with concordant results. Values of the units have been
agreed on by International Congresses as exact enough for general work,
and with these units all electrical researches, wherever made, are
available for use by other experimenters.

A vast amount of work has been done on this subject during the last
forty years, and though the value of the practical unit of
resistance--10^9 C.G.S. units, now called the "ohm"--is taken as
settled, and copies can now be had in resistance boxes, or separately,
adjusted with all needful accuracy, at the National Physical Laboratory
and at the Bureau of Standards at Washington, and elsewhere, experiments
are being made on the exact measurement of currents; while a careful
watch is kept on the standards laid up at these places to see whether
any perceptible variation of their resistance takes place with lapse of
time.

The British Association Committee also worked out a complete system of
units for all electrical and magnetic quantities, and gave the first
systematic statement of their relations, that is, of the so-called
dimensional equations of the quantities. This will be found in the works
to which reference has already been made (p. 251).



CHAPTER XIV

THE BALTIMORE LECTURES


The Baltimore Lectures were delivered in 1884 at Johns Hopkins
University, soon after the Montreal meeting of the British Association.
The subject chosen was the Wave Theory of Light; and the idea underlying
the course was to discuss the difficulties of this theory to
"Professorial fellow-students in physical science." A stenographic
report of the course was taken by Mr. A. S. Hathaway, and was published
soon after. The lectures were revised by Lord Kelvin, and the book now
known as _The Baltimore Lectures_ was published just twenty years later
(in 1904) at the Cambridge University Press. It is absolutely impossible
in such a memoir as the present to give any account of the discussions
contained in the lectures as now published. The difficulties dealt with
can for the most part only be understood by those who are acquainted
with the wave theory of light in its details, and such readers will
naturally go direct to the book itself.

Some of the difficulties, however, were frequently alluded to in Lord
Kelvin's ordinary lectures, and all his old students will remember the
animation with which he discussed the apparent anomaly of a medium like
the luminiferous ether, which is of such enormous rigidity that (on the
elastic solid theory) a wave of transverse oscillation is propagated
through it with a speed of 3 × 10^10 centimetres (186,000 miles) per
second, and yet appears to offer no impediment to the slow motion of the
heavenly bodies. For Lord Kelvin adopted the elastic solid theory of
propagation of light as "the only tenable foundation for the wave theory
of light in the present state of our knowledge," and dismissed the
electromagnetic theory (his words were spoken in 1884, it is to be
remembered) with the statement of his strong view that an electric
displacement perpendicular to the line of propagation, accompanied by a
magnetic disturbance at right angles to both, is inadmissible.

And he goes on to say that "when we have an electromagnetic theory of
light," electric displacement will be seen as in the direction of
propagation, with Fresnelian vibrations perpendicular to that direction.
In the preface, of date January 1904, the insufficiency of the elastic
solid theory is admitted, and the question of the electromagnetic theory
again referred to. He says there that the object of the Baltimore
Lectures was to ascertain how far the phenomena of light could be
explained within the limits of the elastic solid theory. And the answer
is "everything _non-magnetic; nothing magnetic_." But he adds, "The
so-called electromagnetic theory of light has not helped us hitherto,"
and that the problem is now fully before physicists of constructing a
"comprehensive dynamics of ether, electricity, and ponderable matter
which shall include electrostatic force, magnetostatic force,
electromagnetism, electrochemistry, and the wave theory of light."

All this is exceedingly interesting, for it seems to make clear Lord
Kelvin's attitude with respect to the electromagnetic theory of Maxwell,
which is now regarded by most physicists as affording on the whole a
satisfactory account, if not a dynamical theory in the sense understood
by Lord Kelvin, of light-propagation. That there is an electric
displacement perpendicular to the direction of propagation and a
magnetic displacement (or motion) perpendicular to both seems proved by
the experiments of Hertz, and the velocity of propagation of these
disturbances has been found to be that of light. Of course it remains to
be found out in what the electric and magnetic changes consist, and
whether the ether has or has not an atomic structure. Towards the answer
to this question on electromagnetic presuppositions some progress has
already been made, principally by Larmor. And, after all, while we may
imagine that we know something more definite of dynamical actions on
ponderable matter, it is not quite certain that we do: we are more
familiar with them, that is almost all. We know, for example, that at
every point in the gravitational field of the earth we may set up a
gravitation vector, or field-intensity; for a particle of matter there
is subjected to acceleration along that direction. But of the rationale
of the action we know nothing, or next to nothing. So we set up electric
and magnetic vectors in an insulating medium, corresponding to electric
and magnetic effects which we can observe; and it is not too much to say
that we know hardly less in this case than we do in the other, of the
inner mechanism of the action of which we see the effects.

Returning to the difficulty of the elastic solid theory, that while its
rigidity is enormous, it offers no obstacle to the planets and other
heavenly bodies which move through it, it may be interesting to recall
how Lord Kelvin used to deal with it in his elementary lectures. The
same discussion was given in the Introductory Lecture at Baltimore. The
difficulty is not got over by an explanation of what takes place: it is
turned by showing that a similar difficulty exists in reconciling
phenomena which can be observed every day with such ordinary materials
as pitch or shoemakers' wax. A piece of such wax can be moulded into a
tuning-fork or a bell, and will then, if struck, sound a musical note of
definite pitch. This indicates, for rapidly alternating deformations
started by a force of short duration, the existence of internal forces
of the kind called elastic, that is, depending on the amount of
deformation caused, not on the rate at which the deformation is
increasing or diminishing, as is the case for the so-called "viscous
forces" which are usually displayed by such material. But the
tuning-fork or bell, if left lying on the table, will gradually flatten
down into a thin sheet under only its own weight. Here the deformation
is opposed only by viscous forces, which, as the change is very slow,
are exceedingly small.

But let a large slab of it, three or four inches thick, be placed in a
glass jar ten or twelve inches in diameter, already partly filled with
water, and let some ordinary corks be imprisoned beneath, while some
lead bullets are laid on the upper surface. After a month or two it will
be found that the corks have disappeared from the water into the wax,
and that the orifices which they made in entering it have healed up
completely; similarly the bullets have sunk down into the slab, leaving
no trace behind. After two or three months more, the corks will be seen
to be bursting their way out through the upper surface of the slab, and
the bullets will be found in the water below. The very thing has taken
place that would have happened if water had been used instead of pitch,
only it has taken a very much longer time to bring it about. The corks
have floated up through the wax in consequence of hydrostatic upward
force exerted by the wax acting as a fluid; and the bullets have sunk
down in consequence of the excess of their weights above the upward
hydrostatic force exerted on them as on the corks. The motion in both
cases has been opposed by the viscous forces called into play.

The application of this to the luminiferous ether is immediate. Let the
ether be regarded as a substance which can perform vibrations only "when
times and forces are suitable," that is, when the forces producing
distortion act for only an infinitesimal time (as in the starting of the
tuning-fork by a small blow), and are not too great. Vibrations may be
set up locally, and the medium may have a true rigidity by which they
are propagated to more remote parts; that is to say, waves travel out
from the centre of disturbance. On the other hand, if the forces are
long continued, even if they be small, they produce continuously
increasing change of shape. Thus the planets move seemingly without
resistance.

The conclusion is that the apparently contradictory properties of the
ether are no more mysterious than the properties of pitch or shoemakers'
wax. And, after all, matter is still a profound mystery.

Dynamical illustrations, which old Glasgow students will recognise,
appear continually in the lectures. They will remember, almost with
affection, the system of three particles (7 lb. or 14 lb. weights!)
joined together in a vertical row by stout spiral springs of steel,
which were always to be taken as massless, and will recall Lord Kelvin's
experiments with them, demonstrating the three modes of vibration of a
system of three masses, each of which influenced those next it on the
two sides. Here they will find the problem solved for any number of
particles and intervening springs, and the solution applied to an
extension of the massive molecule which von Helmholtz imbedded in the
elastic ether, and used to explain anomalous dispersion. A highly
complex molecule is suggested, consisting of an outer shell embedded in
the ether as in the simpler case, a second shell within that connected
to the outer by a sufficient number of equal radial springs, a third
within and similarly connected to the second by radial springs, and so
on. This molecule will have as many modes of vibration as there are sets
of springs, and can therefore impart, if it is set into motion, a
complex disturbance to the ether in which it is imbedded.

The modification of this arrangement by which Lord Kelvin explained the
phosphorescence of such substances as luminous paint is also described,
and will be recognised by some as an old friend. A number, two dozen or
so, of straight rods of wood eighteen inches long are attached to a
steel wire four or five inches apart, like steps on a ladder made with a
single rope along the centres of the steps. The wire is so attached to
each rod that the rod must turn with the wire if the latter is twisted
round. Each rod is loaded with a piece of lead at each end to give it
more moment of inertia about the wire. The wire, with this "ladder"
attached to it, is rigidly attached to the centre of a cross-bar at the
top, which can be made to swing about the wire as an axis and so impart
twisting vibrations to the wire in a period depending on this driver.
Sliding weights attached to the bar enable its moment of inertia to be
changed at pleasure. The lower end of the wire carries a cross-bar with
two vanes, immersed in treacle in a vessel below. When the period of the
exciter was very long the waves of torsion did not travel down the
"ladder," but when the period was made sufficiently short the waves
travelled down and were absorbed in the treacle below. In the former
case the vibrations persisted; the case was analogous to that of
phosphorescence.

[Illustration: FIG. 18.]

Incidentally a full and very attractive account of the elastic solid
theory is given in these lectures, accompanied as it is by
characteristic digressions on points of interest which suggest
themselves, and on topics on which the lecturer held strong opinions,
such, for example, as the absurd British system of weights and measures.
The book reads in many places like a report of some of the higher
mathematical lectures which were given every session at Glasgow; and on
that account, if on no other, it will be read by the old students of the
higher class with affectionate interest. But the discussions of the
great fundamental difficulty presented at once by dispersion--the fact,
that is, that light of different wave lengths has different velocities
in ordinary transparent matter--the discussions of the various theories
of dispersion that have been put forward, the construction of the
molecules, gyrostatic and non-gyrostatic, with all their remarkable
properties, which Lord Kelvin invents in order to frame a dynamical
mechanism which will imitate the action of matter as displayed in the
complex manifestations of the optical phenomena, not only of isotropic
matter, but of crystals, will ever afford instruction to every
mathematician who has the courage to attack this subject, and remain as
a monument to the extraordinary genius of their author.

A subject is touched on in these lectures which has not been dealt with
in the present review of Lord Kelvin's work. By four lines of
argument--by the heat of combination of copper and zinc, together with
the difference of electric potential developed when these metals are put
in contact, from the thickness of a capillary film of soap and water
(measured by Rücker and Reinold) just before it gives way, and the work
spent in stretching it, from the kinetic theory of gases and the
estimated length of free path of a particle (given also by Loschmidt
and by Johnstone Stoney), and from the undulatory theory of light--Lord
Kelvin estimated superior and inferior limits to the "size of the atoms"
of bodies, or, more properly speaking, of the molecular structure of the
matter. We cannot discuss these arguments--and they can be read at
leisure by any one who will consult Volume I (Constitution of Matter) of
Lord Kelvin's _Popular Lectures and Addresses_, for his Royal
Institution Lecture on the subject, there given in full--but we may
state his conclusion. Let a drop of water, a rain drop, for example, be
magnified to the size of the earth, that is, from a sphere a quarter of
an inch, or less, in diameter to a sphere 8000 miles in diameter, and
let the dimensions of the molecular structure be magnified in the same
proportion. "The magnified structure would be more coarse-grained than a
heap of small shot, but probably less coarse-grained than a heap of
cricket-balls."

Of course, it is not intended here to convey the idea that the molecules
are spheres like shot or cricket-balls; they undoubtedly have a
structure of their own. And no pronouncement is made as to the
divisibility or non-divisibility of the molecules. All that is alleged
is that if the division be carried to a minuteness near to or beyond
that of the dimensions of the structure, portions of the substance will
be obtained which have not the physical properties of the substance in
bulk.

The recent interesting researches of chemists and physicists into
phenomena which seem to demonstrate the disintegration, not merely of
molecules, but even of the atomic structure of matter, attracted Lord
Kelvin's attention in his last years, and _suo more_ he endeavoured to
frame dynamical explanations of electronic (or, as he preferred to call
it, "electrionic") action. But though keenly interested in all kinds of
research, he turned again and again to the older theories of light, and
his dynamical representations of the ether and of crystals, with renewed
vigour and enthusiasm.



CHAPTER XV

SPEED OF TELEGRAPH SIGNALLING--LAYING OF SUBMARINE CABLES--TELEGRAPH
INSTRUMENTS--NAVIGATIONAL INSTRUMENTS, COMPASS AND SOUNDING MACHINE


THEORY OF SIGNALLING

When the question of laying an Atlantic cable began to be debated in the
middle of the nineteenth century, Professor Thomson undertook the
discussion of the theory of signalling through such a cable. It was not
generally understood by practical telegraphists that the conditions of
working would be very different from those to which they were accustomed
on land lines, and that the instruments employed on such lines would be
useless for a cable. Such a cable consists of a copper conductor
separated from the sea-water by a coating of gutta-percha; it forms an
elongated Leyden jar of very great capacity, which, when a battery is
connected to one end of the conducting core, is gradually charged up,
first at that end, and later and later at greater distances from it, and
then is gradually discharged again when the battery is withdrawn and the
end of the conductor connected to earth. Here, again, an application of
Fourier's analysis solved the problem, which, with certain
modifications, and on the supposition that the working is slow, is
essentially the same problem as the diffusion of heat along a
conducting bar, or the diffusion of a salt solution along a column of
water. The signals are retarded (and this was one of the results of the
investigation) in such a manner "that the time required to reach a
stated fraction of the maximum strength of current at the remote end,"
when a given potential difference is applied at the other, or home end,
is proportional to the product of the capacity and resistance of the
cable, each taken per unit of the length, and also proportional to the
square of the length of cable. In other words, the retardation is
proportional to the product of the resistance of the copper conductor
and the total capacity of the cable. This gave a practical rule of great
importance for guidance in the manufacture of submarine cables. The
conductor should have the highest conductivity obtainable, and should
therefore be of pure copper; the insulating covering should, while
forming a nearly absolutely non-conducting sheath, have as low a
specific inductive capacity as possible. The first of these conditions
ran counter to some views that had been put forward, to the effect that
it was only necessary to have the internal conductor highly conducting
on its surface; and some controversy on the subject ensued. The inverse
square law, as it was called, was vehemently called in question, from a
mistaken interpretation of some experiments that were made to test it.
For if the potential at the home end be regularly altered, according to
the simple harmonic law, so that the number of periods of oscillation in
a second is n, the changes of potential are propagated with velocity
2√(πn⧸cr), where c and r are the capacity and resistance of the cable,
each taken per unit length. In this case, for a long cable, there is a
velocity of propagation independent of the length; and this fact seems
to have misled the experimenters. Thomson's view prevailed, and the
result was the establishment, first by Thomas Bolton & Sons,
Stoke-on-Trent, of mills for the manufacture of high conductivity
copper, which is now a great industry.

The Fourier mathematics of the conduction of heat along a bar suffices
to solve the problem, so long as the signalling is so slow as not to
bring into play electromagnetic induction to any serious extent. For
rapid signalling in which very quick changes of current are concerned
the electromotive forces due to the growth or dying out of the current
would be serious, and the theory of diffusion would not apply. But
ordinary cable working is quite slow enough to enable such electromotive
forces to be disregarded.


LAYING OF FIRST AMERICAN CABLES

The first cable of 1858 was laid by the U.S. frigate Niagara and H.M.S.
Agamemnon, after having been manufactured with all the precautions
suggested by Professor Thomson's researches. It is hard to realise how
difficult such an enterprise was at the time. The manufacture of a huge
cable, the stowage of it in cable tanks on board the vessels, the
invention of laying and controlling and picking-up machinery had to be
faced with but little experience to guide the engineers. Here again
Thomson, by his knowledge of dynamics and true engineering instinct, was
of great assistance. In 1865 he read a very valuable paper on the forces
concerned in the laying and lifting of deep-sea cables, showing how the
strains could be minimised in various practical cases of
importance--for example, in the lifting of a cable for repairs.

A first Atlantic cable had been partly laid in 1857 by the Niagara, when
it broke in 2000 fathoms of water, about 330 miles from Valentia, where
the laying had begun. An additional length of 900 miles was made, and
the enterprise was resumed. This time it was decided that the two
vessels, each with half of the cable on board, should meet and splice
the cable in mid-ocean, and then steam in opposite directions, the
Agamemnon towards Valentia, the Niagara towards Newfoundland. Professor
Thomson was engineer in charge of the electrical testing on board of the
Agamemnon. After various mishaps the cable was at last safely laid on
August 6, 1858, and congratulations were shortly after exchanged between
Great Britain and the United States. On September 6 it was announced
that signals had ceased to pass, and an investigation of the cause of
the stoppage was undertaken by Professor Thomson and the other
engineers. The report stated that the cable had been too hastily made,
that, in fact, it was not good enough, and that the strains in laying it
had been too great and unequal. It was found impossible to repair it, so
that there was no option but to abandon it.

This cable probably suffered seriously from the violent means which seem
to have been employed to force signals through it. Now only a very
moderate difference of potential is applied to a cable at the sending
end, and speed of signalling is obtained by the use of instruments, the
moving parts of which have little inertia, and readily respond to only
an exceedingly feeble current.

A second cable was made and laid in 1865 by the Great Eastern, which
could take on board the whole at once and steam from shore to shore. It
was also well adapted for cable work through having both screw and
paddles. As Thomson points out, "steerage way" could be got on the
vessel by driving the screw ahead, so as to send a stream of water
astern towards the rudder, while the paddles were driven astern to
prevent the ship from going ahead. This was of great advantage in
manœuvring on many occasions.

This cable also broke, but a third was laid successfully in 1866 by the
same vessel, and the second was recovered and repaired, so that two good
cables were secured for commercial working. On both expeditions
Professor Thomson acted as electrical engineer, and received the honour
of knighthood and the thanks of the Anglo-American Telegraph Company on
his return home, when he was also presented with the freedom of the city
of Glasgow.

He afterwards acted as engineer for the French Atlantic Cable, for the
Brazilian and River Plate Company, and for the Commercial Company, whose
two new Atlantic cables were laid in 1882-4.


MIRROR GALVANOMETER AND SIPHON RECORDER

Since whatever the potential applied at the sending end of the cable
might be (and, of course, as has been stated, this potential had to be
kept to as low a value as possible) the current at the receiving end
only rose gradually, it was necessary to have as delicate a receiving
instrument as possible, so that it would quickly respond to the growing
and still feeble current. For unless the cable could be worked at a
rate which would permit of charges per word transmitted which were
within the reach of commercial people, it was obvious that the
enterprise would fail of its object. And as a cable could not cost less
than half a million sterling, the revenue to be aimed at was very
considerable. This problem Thomson also solved by the invention of his
mirror galvanometer. The suspended magnet was made of small pieces of
watch-spring cemented to a small mirror, so that the whole moving part
weighed only a grain or two. Its inertia, or resistance to being set
into motion, was thus very small, and it was hung by a single fibre of
silk within a closed chamber at the centre of the galvanometer coil. A
ray of light from a lamp was reflected to a white paper scale in front
of the mirror, which as it turned caused a spot of illumination to move
along the paper. A motion of this long massless index to the left was
regarded as a dot, a motion to the right as a dash, and the Morse
alphabet could therefore be employed. This instrument was used in the
1858 cable expedition, and a special form of suspension was invented for
it by Thomson, to enable it to be used on board ship. The suspension
thread, instead of being held at one end only, was stretched from top to
bottom of the chamber in which the needle hung, and kept tight by being
secured at both ends. Thus the minimum of disturbance was caused to the
mirror by the rolling or pitching of the ship.

The galvanometer was also enclosed in a thick iron case to guard it
against the magnetic field due to the iron of the ship. The "iron-clad
galvanometer" first used in submarine telegraphy (on the 1858
expedition in the U.S. frigate Niagara) is in the collection of
historical apparatus in the Natural Philosophy Department of the
University of Glasgow.

The mirror galvanometer then invented has become one of the most useful
instruments of the laboratory. Mirror deflection is now used also for
the indicators of many kinds of instruments.

The galvanometer was replaced later by another invention of Professor
Thomson--the siphon recorder. Here a small and delicate pen was formed
by a piece of very fine glass tube (vaccination tubing, in fact) in the
form of a siphon, of which the shorter end dipped into an ink-bottle,
while the other end wrote the message in little zig-zag notches on a
ribbon of paper drawn past it by machinery. The siphon was moved to and
fro by the signalling currents, which flowed in a small coil hung
between the poles of an electromagnet, excited by a local battery, and
the ink was spirted in a succession of fine drops from the pen to the
paper. This was accomplished by electrifying the ink-bottle and ink by a
local electrical machine, and keeping the paper in contact with an
uninsulated metal roller. Electric attraction between the electrified
ink and the unelectrified paper thus drew the ink-drops out, and the
pen, which never touched the paper, was quite unretarded by friction.
Both these instruments had the inestimable advantage that the to and fro
motions of the spot of light or the pen took place independently of
ordinary earth-currents through the cable.

The arrangement of magnet and suspended coil in this instrument has
become widely known as that of the "d'Arsonval galvanometer." This
application was anticipated by Thomson, and is distinctly mentioned in
his recorder patent, long before such galvanometers were ever used. It
was later proposed by several experimenters before M. d'Arsonval.

It is not too much to say that, by his discussion of the speed of
signalling, his services as an electrical engineer, and especially by
his invention of instruments capable of responding to very feeble
currents, Thomson made submarine telegraphy commercially possible. Later
he entered into partnership with Mr. C. F. Varley and Professor Fleeming
Jenkin. A combination of inventions was made by the firm: Varley had
patented a method of signalling by condensers, and Jenkin later
suggested and patented an automatic key for "curb-sending" on a
cable--that is, signalling by placing one pole of the battery for an
interval a little shorter than the usual one to the line, and then
reversing the battery for the remainder. This gave sharper signals, as
the reversal helped to discharge the cable more rapidly than it would
have been by the mere connection to earth between two signals. The firm
of Thomson, Varley & Jenkin took a prominent part in cable work; and
Thomson and Jenkin acted as engineers for many large undertakings. They
employed a staff of young electricians at the cable-works at Millwall
and elsewhere, keeping watch over the cable during manufacture, and sent
them to sea as representatives and assistants to perform similar duties
during the process of cable-laying. On their staff were many men who
have come to eminence in electrical and engineering pursuits in later
life.


MARINERS' COMPASS AND SOUNDING MACHINE

After the earlier Atlantic expeditions Sir William Thomson turned his
attention to the construction of navigational instruments, and invented
the mariner's compass and wire-sounding apparatus which are now so well
known. He had come to the conclusion that the compasses in use had much
too large needles (some of them bar-magnets seven or eight inches long!)
to respond quickly and certainly to changes of course, and, what was
still more serious, to admit of the application of correcting magnets,
and of masses of soft-iron to annul the action of the magnetism of the
ship.

The compass card consists of a paper ring, on which the "points" and
degrees are engraved in the ordinary way, and is kept circular by a
light ring of aluminium. Threads of silk extend radially from the rim to
a central boss of aluminium in which is a cap of aluminium. In the top
of the cap is a sapphire bearing, which rests on an iridium point
projecting upward from the compass bowl. Eight magnets of glass-hard
steel, from 3¼ inches to 2 inches long, and about the thickness of a
knitting-needle, which form the compass needle, are strung like the
steps of a rope ladder, on two silk threads attached to four of the
radial threads.

The weight of the card is extremely small--only 170½ grains; that is
less than ⅖ of an ounce. But the matter is not merely made small in
amount; it is distributed on the whole at a great distance from the
axis; consequently the period of free vibration is long, and the card is
very steady. The great lightness of the card also causes the error due
to friction on the point of support to be very small.

The errors of the compass in an iron ship are mainly the semicircular
error and the quadrantal error. We can only briefly indicate how these
arise and how they are corrected. The ship's magnetism may be considered
as partly permanent, and partly inductive. The former changes only very
slowly, the latter alters as the ship changes course and position. For
the ship is a combination of longitudinal, transverse, and vertical
girders and beams. As a whole it is a great iron or steel girder, but
its structure gives it longitudinal, transverse, and vertical
magnetisation. This disturbs the compass, which is also affected by the
magnetisation of the iron or steel masts and spars, or of iron or steel
carried as cargo.

The semicircular error is due to a great extent to permanent magnetism,
but also in part to induced magnetism. It is so called because when the
ship's head is turned through 360°, the error attains a maximum on two
courses 180° apart. It may amount to over 20° in an ordinary iron
vessel, and to 30° or 40° in an armour-clad. It is corrected by two sets
of steel magnets placed with their centres under the needle in the
binnacle. One set have their lengths fore and aft, the others in the
thwart-ship direction. These magnets annul the error on the north and
south and on the east and west courses, due to the two horizontal
components of magnetic force produced mainly by the permanent magnetism
of the ship. A regular routine of swinging the ship when marks on the
shore (the true bearings of which from the ship are known) are
available, is followed for the adjustment.

The quadrantal error is so called because its maxima are found on four
compass courses successively a quadrant, or 90°, from one another. It
amounts in general to from 5° to 10° at most. It is due to induced
magnetism, and is corrected by a pair of soft-iron spheres, placed on
the two sides of the compass with their centres in a line transverse to
the ship, through the centre of the compass needle. There are, however,
exceptional cases in which they are placed in the fore and aft line one
afore, the other abaft, the needle. When the quadrantal error has once
been annulled it is always zero, for as the induced magnetism changes,
so does that of the spheres, and the adjustment remains good. In a new
ship the permanent magnetism slowly alters, and so the semicircular
correction has to be improved from time to time by changing the magnets.

These adjustments are not quite all that have to be made; but enough has
been stated to show how the process of compensation can be carried out
with the Thomson compass. The immensely-too-large magnets used formerly
as compass needles, through a mistaken notion, apparently, that more
directive force would be got by their means, rendered the quadrantal
adjustment an impossibility. The card swinging round brought the large
needles into different positions relatively to the iron balls, when
these were used, and exerted an inductive action on them which reacted
on the needles, producing more error, perhaps, than was corrected.

Thomson invented also an instrument called a "deflector," by which it is
possible to adjust a compass when sights of sun or stars, or bearings of
terrestrial objects, cannot be obtained. By means of it the directive
forces on the needles on different courses can be compared. Then the
adjustment is made by placing the correctors so that the directive force
is as nearly as may be the same on all courses. The compass is then
quite correct.

The theory of deviations of the compass, it is right to say, was
discussed first partially by Poisson, but afterwards very completely and
elegantly by the late Mr. Archibald Smith of Jordanhill, whose memoirs,
now incorporated in the _Admiralty Manual of Deviations of the Compass_,
led to Lord Kelvin's inventions.

Lord Kelvin's compass is now almost universally in use in the merchant
service of this country, and in most of the navies of the world. It has
added greatly to the certainty and safety of navigation.

The sounding machine is also well known. At first pianoforte wire was
used for deep-sea sounding by Commodore Belknap of the U.S. Navy, and by
others, on Sir William Thomson's recommendation. Finally, a form of
machine was made by which a sinker could be lowered to the bottom of the
sea and brought up again in a few minutes; so that it was possible to
take a sounding without the long delay involved in the old method with a
reel of hemp-rope, which often tempted shipmasters to run risks of going
ashore rather than stop the ship for the purpose. The wire offered
little resistance to motion through the water, and by a proper winding
machine, with brake to prevent the wire from running out too fast and
kinking, when it was almost certain to break, one man could quickly
sound and heave up again, while another attended to the wire and sinker.
A gauge consisting of a long quill-tube closed at the upper end, and
coated inside with chromate of silver, showed by the action of the
sea-water on the coating how far the water had passed up the tube,
compressing the air above it; and from this, by placing the tube along a
wooden rule properly graduated, the depth was read off at once. With the
improved machine a ship approaching the shore in thick weather could
take soundings at short intervals without stopping, and discover at once
any beginning of shallowing of the water, and so avoid danger.

The single wire is not now used, as a thin stranded wire is found safer
and quite as effective. The gauge also has been improved. The apparatus
can be seen in any well-found sea-going vessel; though there are still,
or were until not very long ago, steam vessels without this apparatus,
though crossing the English Channel with passengers. These depended for
soundings on the obsolete hemp-rope, wrapped round an iron spindle held
vertically on the deck by members of the ship's company, while the cord
was unwound by the descent of the sinker.[24]

Sir William Thomson's electrical and other inventions are too numerous
to specify here, and they are in constant use wherever precision of
measurement is aimed at or required. Long ago he invented electrometers
for absolute measurements of electrical potential ("electric pressure");
more recently his current-balances have given the same precision to
electrodynamic measurement of currents. All his early instruments were
made by Mr. James White, Glasgow. The business founded by Mr. White,
and latterly carried on at Cambridge Street, has developed immensely,
and is now owned by a limited liability company--Messrs. Kelvin and
James White (Limited).

For many years Sir William Thomson was a keen yachtsman, and his
schooner yacht, the _Lalla Rookh_, was well known on the Clyde and in
the Solent. An expert navigator, he delighted to take deep-sea voyages
in his yacht, and went more than once as far as Madeira. Many
navigational and hydrodynamical problems were worked out on these
expeditions. For a good many years, however, he had given up sea-faring
during his times of relaxation, and lived in Glasgow and London and in
Largs, Ayrshire, where he built, in 1875, a large and comfortable house,
looking out towards the Firth and the Argyleshire lochs he knew and
loved so well.

In the course of his deep-sea expeditions in his yacht he became
impressed with the utility of Sumner's method of determining the
position of a ship. Let us suppose that at a given instant the altitude
of the sun is determined from the ship. The Greenwich meantime, and
therefore the longitude at which the sun is vertical, is known by
chronometer, and the declination of the sun is known from the Nautical
Almanac. The point on the earth vertically under the sun can be marked
on the chart, and a circle (or rather, what would be a circle on a
terrestrial globe) drawn round it from every point of which the sun
would have the observed altitude. The ship is at a point on this circle.
Some time after the altitude of the sun is observed again, and a new
"circle" is drawn. If the first "circle" be bodily shifted on the chart
along the distance run in the interval, it will intersect the second in
two points, one of which will be the position of the ship, and it is
generally possible to tell which, without danger of mistake.

Sir William Thomson printed tables for facilitating the calculations in
the use of Sumner's method, and continually used them in his own
voyages. He was well versed in seamanship of all kinds, and used his
experience habitually to throw light on abstruse problems of dynamics.
Some of these will be found in "Thomson and Tait"; for instance, in Part
I, § 325, where a number of nautical phenomena are cited in illustration
of an important principle of hydrodynamics. The fifth example stated is
as follows: "In a smooth sea, with moderate wind blowing parallel to the
shore, a sailing ship heading towards the shore, with not enough of sail
set, can only be saved from creeping ashore by setting more sail, and
sailing rapidly towards the shore, or the danger that is to be avoided,
so as to allow her to be steered away from it. The risk of going ashore
in fulfilment of Lagrange's equations is a frequent incident of 'getting
under way' while lifting anchor or even after slipping from moorings."
His seamanship was well known to shipmasters, with whom he had much
intercourse, and whose intelligence and practical skill he held in very
high regard.



CHAPTER XVI

LORD KELVIN IN HIS CLASS-ROOM AND LABORATORY


It is impossible to convey to those who never studied at Glasgow any
clear conception of Thomson as he appeared to students whom he met daily
during the session. His appearance at meetings of the British
Association, and his vivacious questionings of the various authors of
papers, his absorption in his subject and oblivion to the flight of time
when he read a paper himself, will long be remembered by scientific men:
but though they suffice to suggest what he was like in his own
lecture-room, the picture lacks the setting of furniture, apparatus,
assistants, and students, which all contributed to the unique impression
made by his personality on his pupils. The lecture-table--with long
straight front and ends refracted inward, flanked by higher small round
tables supported on cylindrical pillars--laden with instruments; the
painted diagrams of the solar spectrum and of the paths of coloured rays
through a prism, hung round the walls; the long wire with the
cylindrical vibrator attached, for experiments on torsion, and the
triple spiral spring vibrator, which hung at the two ends of the long
blackboard; the pendulum thirty feet long, consisting of a steel wire
and a twelve-pound cannon-ball as bob, suspended from the apex of the
dome-roof above the lecture-table; the large iron wheel in the
beautiful oriel window on the right of the lecturer, and the collection
of optical instruments on the table in front of the central window
spaces, from which the small iron-framed panes--dear to the heart of the
architect--had been removed; the clock on either side of the room, one
motionless, the other indicating the time, and having attached to it the
alarm which showed when the "angry bell" outside had ceased to toll; the
ten benches of eager and merry students, which filled the auditorium;
all these combined to form a scene which every student fondly recalls,
and which cannot be adequately described. A similar scene, with some
differences of arrangement and having its own particular associations,
will occur to every student who attended in the Old College.

The writer will never forget the lecture-room when he first beheld it,
from his place on Bench VIII, a few days after the beginning of session
1874-5. Sir William Thomson, with activity emphasised rather than
otherwise by his lameness, came in with the students, passed behind the
table, and, putting up his eye-glass, surveyed the apparatus set out.
Then, as the students poured in, an increasing stream, the alarm weight
was released by the bell-ringer, and fell slowly some four or five feet,
from the top of the clock to a platform below. By the time the weight
had descended the students were in their places, and then, as Thomson
advanced to the table, all rose to their feet, and he recited the third
Collect from the Morning Service of the Church of England. It was the
custom then, and it is still one better honoured in the observance than
in the breach (which has become rather common) to open all the first and
second classes of the day with prayer; and the selection of the prayers
was left to the discretion of the professors. Next came the roll-call by
the assistant; each name was called in its English, or Scottish (for the
clans were always well represented) form, and the answer "adsum" was
returned.

Then the Professor began his lecture, generally with the examination of
one of the students, who rose in his place when his name was called.
Thomson, as the quotation in Chapter VI from the Bangor Address shows,
was fond of oral examination, and after the second hour had begun to
decline as one of regular attendance, habitually devoted ten or fifteen
minutes to asking questions and criticising the answers. The names of
the students to be questioned were selected at random from the class
register, or by a kind of lottery, carried out by placing a small card
for each student in a box on the table, and drawing a name whenever a
member of the class was to be examined. The interest in the drawing each
day was intense, for there was a glorious uncertainty as to what might
be the line of examination adopted. Sometimes, in the midst of a
criticism of an answer, an idea would suddenly occur to the Professor,
and he would enlarge upon it, until the forgotten examinee slipped
quietly back into his seat, to be no more disturbed at least for that
day! And how great the relief if the ordeal was well passed and the card
was placed in that receptacle of the blessed, the compartment reserved
for those who had been called and duly passed the assize! But there was
a third compartment reserved for the cards of those unfortunates who
failed to satisfy the judge! The reader may have anticipated the fact
that the three divisions of this fateful box were commonly known to
students by the names of the three great habitations of spirits
described in the _Divina Commedia_ of Dante.

As has been stated, the oral examination with which the lectures opened
was the cause of a good deal of excitement, which was added to by the
element of chance introduced by drawing the names from the purgatorial
compartment of the box. The ordeal was dreaded by backward students,
whom Thomson found, as he said, aphasic, when called on to answer in
examination, but who certainly were anything but aphasic in more
congenial circumstances. Occasionally they abstained from responding
to their names, modestly seeking the seclusion of the crowd, and
some little time would be spent in ascertaining whether the
examinee-designate was present. When at last he was discovered, he
generally rose with a fervent appeal to his fellows on either side to
help him in his need.

McFarlane used to tell of an incident which illustrated the ingenuity
with which it was sometimes attempted to evade the ordeal of the _viva
voce_ examination. One afternoon, when he was busily preparing the
lecture-illustrations for next day, a student came into the class-room,
and engaging him in conversation on some point of dynamics, regarding
which he professed to have a difficulty, hovered round the box which
contained the three compartments popularly known as Purgatory, Heaven,
and Hell! Always when McFarlane left the room to bring something from
the adjoining cabinet of apparatus, he found, when he returned, his
inquiring friend hurriedly quitting the immediate vicinity of the box.
At last the student took leave, with many apologies for giving so much
trouble. As McFarlane suspected would be the case, the ticket bearing
the name of that student was no longer to be found! He used to conclude
the story as follows: "I just made a new ticket for him, and placed it
on the top of the other tickets, and next day Sir William called him,
the very first time." What were his feelings, who had fondly thought
himself safe for the session, and now found himself subjected to a
"heckling" which he probably expected would be repeated indefinitely,
may be imagined.

The subject of the first lecture which the writer attended was simple
harmonic motion, and was illustrated by means of pendulums, spiral
springs with weights, a long vertical rod of steel tipped with an ivory
ball and fastened to a heavy base, tuning-forks, etc.

The motion was defined as that of a particle moving along the diameter
of a circle--the "auxiliary circle," Thomson called it--so as always to
keep pace, as regards displacement in the direction along that diameter,
with a particle moving with uniform speed in the circle. Then the
velocity and acceleration were found, and it was shown that the particle
was continually accelerated towards the centre in proportion to the
distance of the particle from that point. The constant ratio of
acceleration to displacement was proved to be equal to the square of the
angular velocity in the auxiliary circle, and from this fact, and the
particular value of the acceleration when the particle was at either end
of its range of motion, an expression for the period in terms of the
speed and radius of the auxiliary circle was deduced. Then the ordinary
simple pendulum formula was obtained.

This mode of treatment of an elementary matter, so entirely different
from anything in the ordinary text-books, arrested the attention at
once, and conveyed, to some at least of those present, an idea of simple
harmonic motion which was directly applicable to all kinds of cases,
such as the motion of the air in a sound wave, or of the medium which
conveys the waves of light.

The subject of Kepler's laws was dealt with in the early lectures of
every course, and Newton's deductions were insisted on as containing the
philosophy of the whole question, leading, as they did, to the single
principle from which the laws could be deduced, and the third law
corrected when the mass of the planet was comparable with that of the
sun. Sometimes Thomson would read the remarkable passage in Hegel's
Logik, in which he refers to the Newtonian theory of gravitation and
says, "The planets are not pulled this way and that, they move along in
their orbits like the blessed gods," and remark upon it. On one occasion
his remark was, "Well, gentlemen, if these be his physics, what must his
metaphysics be?" And certainly that a philosopher should deny, as Hegel
seemed to do, all merit to the philosophical setting in which Newton
placed the empirical results of Kepler, is a very remarkable phenomenon.

The vivacity and enthusiasm of the Professor at that time were very
great. The animation of his countenance as he looked at a gyrostat
spinning, standing on a knife-edge on the glass plate in front of him,
and leaning over so that its centre of gravity was on one side of the
point of support; the delight with which he showed that hurrying of the
precessional motion caused the gyrostat to rise, and retarding the
precessional motion caused the gyrostat to fall, so that the freedom to
"precess" was the secret of its not falling; the immediate application
of the study of the gyrostat to the explanation of the precession of the
equinoxes, and illustration by a model of a terrestrial globe, arranged
so that the centre should be a fixed point, while its axis--a material
spike of brass--rolled round a horizontal circle, the centre of which
represented the pole of the ecliptic, and the diameter of which
subtended an angle at the centre of the globe of twice the obliquity of
the ecliptic; the pleasure with which he pointed to the motion of the
equinoctial points along a circle surrounding the globe on a level with
its centre, and representing the plane of the ecliptic, and the smile
with which he announced, when the axis had rolled once round the circle,
that 26,000 years had elapsed--all these delighted his hearers, and made
the lecture memorable.

Then the gyrostat, mounted with its axis vertical on trunnions on a
level with the fly-wheel, and resting on a wooden frame carried about by
the professor! The delight of the students with the quiescence of the
gyrostat when the frame, gyrostat and all, was carried round in the
direction of the spin of the fly-wheel, and its sudden turning upside
down when the frame was carried round the other way, was extreme, and
when he suggested that a gyrostat might be concealed on a tray of
glasses carried by a waiter, their appreciation of what would happen was
shown by laughter and a tumult of applause.

Some would have liked to follow the motions of spinning bodies a little
more closely, and to have made out clearly why they behaved as they did.
Apparently Thomson imagined the whole affair was self-evident, for he
never gave more than the simple parallelogram diagram showing the
composition, with the already existing angular momentum about the axis
of the top, of that generated about another axis, in any short time, by
the action of gravity.

As a matter of fact, the stability and instability of the gyrostat on
the tray give the best possible illustration of the two different forms
of solution of the differential equation, [:θ] + μθ = 0, according as μ
is positive or negative; though it is also possible to explain the
inversion very simply from first principles. All this was no doubt
regarded by Thomson as obvious; but it was far from being self-evident
to even good students of the ordinary class, who, without exception,
were beginning the study of dynamics.

Thomson's absorption in the work of the moment was often very great, and
on these occasions he much disliked to be brought down to sublunary
things by any slight mischance or inconvenience. Examples will occur to
every old pupil of the great emphasis with which he commanded that
precautions should be taken to prevent the like from happening again.
Copies of Thomson and Tait's _Natural Philosophy_--"T and T'" was its
familiar title--and of other books, including Barlow's Tables and other
collections of numerical data, were always kept on the lecture-table.
But occasionally a laboratory student would stray in after everything
had been prepared for the morning lecture, and carry off Barlow to make
some calculation, and of course forget to return it. Next morning some
number would be wanted from Barlow in a hurry, and the book would be
missing. Then Thomson would order that Barlow should be chained to the
lecture-table, and enjoin his assistant to see that that was done
without an hour's delay!

On one occasion, after working out part of a calculation on the long
fixed blackboard on the wall behind the table, his chalk gave out, and
he dropped his hand down to the long ledge which projected from the
bottom of the board to find another piece. None was just there; and he
had to walk a step or two to obtain one. So he enjoined McFarlane, his
assistant, who was always in attendance, to have a sufficient number of
pieces on the ledge in future, to enable him to find one handy wherever
he might need it. McFarlane forgot the injunction, or could not obtain
more chalk at the time, and the same thing happened next day. So the
command was issued, "McFarlane, I told you to get plenty of chalk, and
you haven't done it. Now have a hundred pieces of chalk on this ledge
to-morrow; remember, a hundred pieces; I will count them!" McFarlane,
afraid to be caught napping again, sent that afternoon for several boxes
of chalk, and carefully laid the new shining white sticks on the shelf,
all neatly parallel at an angle to the edge. The shelf was about sixteen
feet long, so that there was one piece of chalk for every two inches,
and the effect was very fine. The class next morning was delighted, and
very appreciative of McFarlane's diligence. Thomson came in, put up his
eye-glass, looked at the display, smiled sweetly, and, turning to the
applauding students, began his lecture.

From time to time there were special experiments, which excited the
interest of the class to an extraordinary degree. One was the
determination of the velocity of a bullet fired from a rifle into a
Robins ballistic pendulum. The pendulum, consisting of a massive bob of
lead attached to a rigid frame of iron bars turning about knife-edges,
was set up behind the lecture-table, and the bullet was fired by Thomson
from a Jacob rifle into the bob of the pendulum. The velocity was
deduced from the deflection of the pendulum, its known moment of inertia
about the line of the knife-edges, the distance of the line of fire from
that line, and the mass of the bullet.

In some of the notices of Lord Kelvin that have appeared in the
newspapers, the imagination of the writers has converted the Jacob rifle
into one which Professor Thomson carried in the early years of the
volunteer movement, as a member of a Glasgow corps. It is still used in
the Natural Philosophy Department for the same experiment, and is a
muzzle-loading rifle of large calibre, which throws an ounce bullet. It
was invented by the well-known Indian sportsman, Colonel Jacob, for
big-game shooting in India. Thomson held a commission as captain in the
K (or University) Company of rifle volunteers, and so did not shoulder a
rifle, except when he may have indulged in target practice.

The front bench students were always in a state of excitement, mingled
in some cases perhaps with a little trepidation. For the target was very
near them, and though danger was averted by placing a large wooden
screen in front of the bob, to prevent splinters of the bullet from
flying about in the event of its missing the target and striking the
iron casing of the bob, there was a slight amount of nervousness as to
what might happen. The rifle, loaded by McFarlane, who had weighed out
the charge of powder (so many drams) from a prescription kept in a
cavity of the stock, was placed on the table, and two rests, provided
with V notches to receive the rifle, were placed in the proper position
to enable a bull's eye to be obtained. Thomson generally produced a
small box of cotton wool, and inserted a little in each of his ears to
prevent injury to the tympanum from the report, and advised the
spectators to do the same. Then, adjusting his eye-glass, he bent down,
placed the rifle in position, and fired, and the solemn stillness with
which the aiming and adjustments had been witnessed was succeeded by
vociferous applause. The length of tape drawn out under a light spring
was read off by McFarlane, who had already placed on the blackboard the
formula for calculation of the velocity, with the factor by which the
length of tape had to be multiplied to give the velocity in feet per
second. Then, with the intimation that a question involving numerical
calculation would be set on the subject, in the ensuing Monday morning
examination paper, the lecture generally closed, or was rounded off with
some further observations on angular (or, as Thomson always preferred to
call it, moment of) momentum.

Long after in the course of a debate in the House of Lords on a proposal
to make the use of the metric system of weights and measures compulsory,
Lord Kelvin told their lordships how he had weighed out the powder to
charge this rifle, and, mistaking the weights, had loaded the rifle with
an amount of powder which would have been almost certain to burst the
piece, but had happily paused before firing it off.

He often interrupted the course of a lecture with a denunciation of the
British "no-system of weights and measures"--"insane," "brain-wasting,"
"dangerous," were among the mildest epithets he applied to it, and he
would deeply sympathise with the student whose recollection of
avoirdupois weight, troy weight, apothecaries' weight, etc., was
somewhat hazy. The danger of the system consisted mainly in the fact
that the apothecaries' dram is 60 grains, while the avoirdupois dram is
27⅓ grains. Thus so many drams of powder required to charge a rifle
is a very much larger quantity when reckoned in apothecaries' drams than
when reckoned in avoirdupois. As a rule he left the loading of the
rifle, like all the other lecture-room experiments, to his assistants.

Another experiment which caused a great sensation was that known as the
"dewdrop"! A funnel of brass, composed of a tube about 30 inches long
and an inch wide, and a conical mouth about ten inches wide, had a piece
of stout sheet India-rubber stretched, as tightly as it could be by
hand, across its mouth, and made water-tight by a serving of twine and
cement round the edge. A wire soldered round the outside of the lip gave
a good hold for this serving and made all perfectly secure. On the plane
surface of the sheet geometrical figures were drawn in ink, so that
their distortion could be afterwards studied. The funnel was then hung
by a strong support in an inverted position behind the table, and water
poured gently into it from a rubber supply pipe connected with the
water-main. As the water was allowed to accumulate--very slowly at
first--the sheet of rubber gradually stretched and bulged out, at first
to a flat lens-shape, and gradually more and more, till an immense
water-drop had been formed, 15 or 18 inches in horizontal diameter, and
of still greater vertical dimensions. The rubber film was now, at the
place of greatest tension, quite thin and transparent, and its giving
way was anticipated by the students with keen enjoyment. A large tub had
been placed below to receive the water, but the deluge always extended
over the whole floor space behind the table, and was greeted with
rapturous applause.

Before the drop burst, and while it was forming, Thomson discoursed on
surface tension, emphasising the essential difference between the
tension in the rubber-film and the surface-film of a dewdrop, and
pointing out how the geometrical figures had changed in shape. Then he
would poke it with the pointer he held in his hand, and, turning to the
class, as the mass quivered, remark, "The trembling of the dewdrop,
gentlemen!"

Vibrations of elastic solids were illustrated in various ways,
frequently by means of a symmetrical shape of calves'-foot jelly, at the
top of which a coloured marble had been imbedded as a molecule, the
motions of which could be followed. And then he would discourse on the
Poisson-Navier theory of isotropic solids, and the impossibility of the
fixed relation which that theory imposed between the modulus of rigidity
and the modulus of compression; and refer with approval to the series of
examples of "perfectly uniform, homogeneous, isotropic solids," which
Stokes had shown could be obtained by making jellies of different
degrees of stiffness. Another example, frequently adduced as indicating
the falsity of the theory, was the entirely different behaviour of
blocks of India-rubber and cork, under compression applied by a Bramah
press. The cork diminished in thickness without spreading out laterally;
the rubber, being very little compressible, bulged out all round as its
thickness was diminished.

The lectures on acoustics, which came late in the course, were also
exceedingly popular. Two French horns, with all their crooks and
accessories, were displayed, and sometimes, to the great delight of the
class, Thomson would essay to show how the pitch of a note could be
modified by means of the keys, or by the hand inserted in the bell. The
determination by the siren of the pitch of the notes of tuning-forks
excited by a 'cello bow, and the tuning of a major third by sounding at
the same time the perfect fifth of the lower note, were often exhibited,
and commented on with acute remarks, of which it is a pity no statement
was ever published.[25]

The closing lecture of the ordinary course was usually on light, and the
subject which was generally the last to be taken up--for as the days
lengthened in spring, it was possible sometimes to obtain sunlight for
the experiments--was often relegated to the last day or two of the
session. So after an hour's lecture Thomson would say, "As this is the
last day of the session, I will go on for a little longer, after those
who have to leave have gone to their classes." Then he would resume
after ten o'clock, and go on to eleven, when another opportunity would
be given for students to leave, and the lecture would be again resumed.
Messengers would be sent from his house, where he was wanted for
business of different sorts, to find out what had become of him, and the
answer brought would be, hour after hour, "He is still lecturing." At
last he would conclude about one o'clock, and gently thank the small and
devoted band who had remained to the end, for their kind and prolonged
attention.

In the course of his lectures Thomson continually called on his
assistants for data of all kinds. In the busiest time of his life--the
fifteen years from 1870 to 1885--he trusted to his assistants for the
preparation of his class illustrations, and it was sometimes a little
difficult to anticipate his wishes, for without careful rehearsal it is
almost impossible to make sure that in an experimental lecture
everything will go without a hitch. The digressions, generally most
interesting and instructive, in which he frequently indulged, almost
always rendered it necessary to bring some experiment before the class
which had not been anticipated, and all kinds of things were kept in
readiness, lest they should be wanted suddenly.

It has often been asserted that Thomson appealed to his assistant for
information contained in the multiplication-table, and could not perform
the ordinary operations of arithmetic. His active mind, working on ahead
of the statements he was making at the moment, often could not be
brought back to the consideration of the value of 9 times 6, and the
like; but it was quite untrue that he was incapable of making
calculations. His memory was good, and though he never could be, for
example, sure whether the aqueous humour was before or behind the
crystalline in the eye, he was generally able at once to tell when a
misstatement had been made as to any numerical question regarding the
subject under discussion.

In the higher mathematical class, to which he lectured on Wednesdays, at
noon, Thomson was exceedingly interesting. There he seemed to work at
the subject as he lectured; new points to be investigated continually
presented themselves, and the students were encouraged to work them out
in the week-long intervals between his lectures. Always the physical
interpretation of results was aimed at, even intermediate steps were
discussed. Thus the meaning of the mathematical processes was ever kept
in view, and the men who could follow were made to think while they
worked, and to regard the mathematical analysis as merely an aid, not an
end in itself. "A little expenditure of chalk is a saving of brains;"
"the art of reading mathematical books is judicious skipping," were
remarks he sometimes made, and illustrate his view of the relative
importance of mathematical work when he regarded it as the handmaid of
the physical thinker. Yet he valued mathematics for its own sake, and
was keenly alive to elegance of form and method, as readers of such
great mathematical discussions as the "Appendix on Spherical Harmonics,"
in Thomson and Tait, will observe. He spoke with unqualified admiration
of the work of Green and Stokes, of Cauchy's great memoir on Waves, and
of Hamilton's papers on Dynamics. But no form of vector-analysis,
neither the Quaternions of Hamilton nor the Vectors of Willard Gibbs and
Heaviside, appealed to him, and the example of his friend and co-worker,
Tait, had no effect in modifying his adverse verdict regarding this
department of mathematics, a verdict which in later years became only
more emphatic.

One session he began the first lecture of the higher class by writing
dx⧸dt in the middle of the blackboard, and demanding of each of the ten
or a dozen students present, some of them distinguished graduates, what
it meant! One student described it as the limiting value of the ratio of
the increment of the dependent variable x to the increment of the
independent variable t, when the latter increment is made indefinitely
small. He retorted, "That's what Todhunter would say!" The others gave
various slightly different versions of the same definition. At last he
impatiently remarked, "Does nobody know that dx⧸dt means velocity?" Here
the physical idea as a whole was before his mind; and he did not reflect
that if t denoted time and x distance in any direction, the explanation
given by the student did describe velocity with fair accuracy.

An embarrassing peculiarity of his mathematical discussions was his
tendency, when a difficulty of symbolisation occurred, to completely
change the notation. Also he was not uniformly accurate in analytical
work; but he more than made up for this by the faculty he had of
devising a test of the accuracy of the result and of divining the error
which had crept in, if the test was not satisfied.

The subjects he treated were always such great branches of mathematics
as the theory of the tides--he discussed the tidal phenomena of the
English Channel in one course--the general theory of vibrations, Fourier
analysis, the theory of waves in water, etc., etc. A very good idea of
the manner and matter of his mathematical prelections can be obtained
from a perusal of the _Baltimore Lectures_.

In the physical laboratory he was both inspiring and distracting. He
continually thought of new things to be tried, and interrupted the
course of the work with interpolated experiments which often robbed the
preceding sequence of operations of their final result. His ideas were
on the whole better worked out by a really good corps of students when
he was from home, and could only communicate by letter his views on the
work set forth in the daily reports which were forwarded to him.

He insisted with emphasis that a student who found that a quadrant
electrometer would not work well should take it to pieces to ascertain
what was the matter. This of course generally resulted in the return of
the instrument to White's shop to be put together again and adjusted.
But, as he said, there was a cause for every trouble of that kind, and
the great thing was to find out at once what it was.

Thomson's concentration on the work in hand, and his power of simply
taking possession of men, even mere spectators, and converting them into
assistants, was often shown in the laboratory. Several men who have
since become eminent were among the assistants enrolled from the
laboratory students. Professor W. E. Ayrton and, later, Professor John
Perry, were students at Glasgow for a time, and rendered the most able
and willing help in the researches which were then proceeding. This
power was, no doubt, the secret of his success in gathering round him an
enthusiastic corps of laboratory workers in the early years of his
professorship, and it was shown also by the ease with which he annexed
the Blackstone examination-room and, later, various spaces in the new
University buildings. There, after a time, the Natural Philosophy rooms
were found by the senatus to include not only the original class-room,
laboratory, etc., but also all the spare attics and corridors in the
neighbourhood, and even the University tower itself! One of his
colleagues, who venerated him highly, remarked recently, "He had a great
faculty for annexation!"

The incident referred to occurred while he was preparing the article on
Heat for the ninth edition of the _Encyclopædia Britannica_. It seemed
at first a pity that Thomson should undertake to write such articles;
but in the course of their preparation he came upon so many points on
which experimental information was wanting, and instituted so many
researches to answer his questions, that the essays took very much the
character of original papers. In the article on Heat (he also wrote
Elasticity), will be found a long account of "Steam Thermometry," that
is, of thermometers in which the indicating substance was to be the
saturated vapours of different substances, water, sulphurous acid, etc.,
etc., for he did not limit the term "steam" to water-vapour. For some
time every one in the laboratory was employed in making sulphurous acid,
by heating copper in sulphuric acid in the usual way, and condensing the
gas in tubes immersed in freezing mixtures; and the atmosphere of the
room was of a sort which, however noxious to germs of different kinds,
it was a little difficult to breathe. One morning, when all were thus
occupied, an eminent chemist, who had just come home from the south for
a vacation, called to pay his respects. After a word or two of inquiry
as to how his young friend was prospering in his new post, Thomson said,
"We are all very busy brewing liquid sulphurous acid, for use in
sulphurous acid steam thermometers; we want a large quantity of the
liquid; would you mind helping us?" So, desiring an assistant to find a
flask and materials, he enrolled this new and excellent recruit on the
spot; and what was intended to be a mere call, was prolonged into a long
day of ungrudging work at an elementary chemical exercise!



CHAPTER XVII

PRACTICAL ACTIVITIES--HONOURS AND DISTINCTIONS--LAST ILLNESS AND DEATH


It remains to say something of Lord Kelvin's public and practical
activities. All over the world he came ultimately to be recognised as
the greatest living scientific authority in almost all branches of
physics. Every existing learned society sought to make him a Fellow,
honorary degrees were showered on him from all quarters. A list of some
of the most important of these distinctions is given in the Royal
Society Year-Book for 1907; it is doubtful if a complete list could be
compiled. He was awarded the Keith Medal and the Victoria Jubilee Medal
by the Royal Society of Edinburgh, and received in succession the Copley
and Royal Medals of the Royal Society of London, of which he was elected
a Fellow in 1851, and was President from 1890 to 1895. For several
periods of years he was President of the Royal Society of Edinburgh, to
which he communicated his papers on heat, dissipation of energy, vortex
motion, and many other memoirs.

He was President of the British Association at the Edinburgh meeting in
1871, when he delivered a presidential address, noteworthy in many
respects, but chiefly remarkable in the popular mind on account of his
suggestion that life was conveyed to the earth by a seed, a germ
enclosed in a crevice of a meteorite. This was understood at the time by
many people as an attempt to explain the origin of life itself, instead
of what it was intended to be, an explanation of the beginning of the
existence of living things on a planet which was originally, on the
completion of its formation by the condensation of nebular matter, red
hot even at its surface. On several occasions he was president of
Section A, and he was constant in attendance at the Association
meetings, and an eager listener and participator in the discussions and
debates. His scientific curiosity was never at rest, and he dearly liked
to meet and converse with scientific workers.

Lady Thomson, who had been long an invalid, died in 1870, and in 1874
Sir William Thomson was married to Miss Frances Anna Blandy (daughter of
Mr. Charles R. Blandy of Madeira) who survives him as Lady Kelvin. To
her tender solicitude he owed much of his constant and long-continued
activity in all kinds of work. She accompanied him on all public
occasions, and he relied greatly on her helpfulness and ever watchful
care.

In 1892 Sir William Thomson, while President of the Royal Society, was
raised to the Peerage, with the title of Baron Kelvin of Netherhall,
Largs; and more lately he was created a member of the Order of Merit and
a G.C.V.O. His foreign distinctions were very numerous. He was a Knight
of the Order _Pour le Mèrite_ of Prussia, a Foreign Associate of the
Institute of France, and a Grand Officer of the Legion of Honour. But no
public honour or mark of royal favour could raise him in the estimation
of all who know anything of science or of the labours of the scientific
men to whom we owe the necessities and luxuries of our present
civilisation.

In 1896 the City and University of Glasgow celebrated the jubilee of his
Professorship of Natural Philosophy. The rejoicings on that occasion
will never be forgotten by those whose privilege it was to take part in
them. Delegates came from every country in the world, and kings and
princes, universities and learned societies, colleges and scholastic
institutions of every kind, vied with each other in doing honour to the
veteran who had fought for truth and light for so many years, and won so
many victories. A memorial volume of the proceedings was published,
including a review of Lord Kelvin's work by the late Professor
FitzGerald, and a full report appeared in Nature and other journals at
the time, so that it is unnecessary to give particulars here. And indeed
it is impossible by any verbal description to convey an idea of the
enthusiasm with which the scientific world acclaimed its leader, and of
the dignity and state of the ceremonies.

In 1899, at the age of seventy-five, Lord Kelvin resigned the Chair of
Natural Philosophy, and retired, not to rest, but to investigate more
vigorously than ever the properties of matter. One remarkable fruit of
his leisure we have in his great book, the _Baltimore Lectures_, in
which theories of light are discussed with a power which excites the
reverence of all engaged in the new researches and which recent
discoveries have called into existence. And it is not too much to say
that the means of discussing and extending these discoveries are in
great measure due to Lord Kelvin.

During the year 1907 Lord Kelvin performed many University duties and
seemed to be in unusually good health. He presided as Chancellor at the
installation of Mr. Asquith as Lord Rector on January 11, and in the
same capacity attended a few days later the funeral of Principal Story,
the Vice-Chancellor, who died on January 13. On April 23 he presided at
the long and arduous ceremonies of honorary graduation, and the public
opening of the new Natural Philosophy Institute and the new Medical
Buildings, by the Prince of Wales. As Chancellor he conferred the degree
of Doctor of Laws on the Prince and Princess, and took the chair at the
luncheon which followed the proceedings, when he proposed in a short and
graceful speech the health of the Princess.

He was able to take part also in various political and social meetings,
and to give attention to the work in progress at the factories of his
firm in Cambridge Street. Lady Kelvin and he left Netherhall, Largs, for
Aix les Bains, at the end of July, but visited the British Association
at Leicester in passing. There he heard the presidential address of his
old friend, Sir David Gill, to whom he moved a vote of thanks in his
usual vivacious manner.

Lord Kelvin had been accustomed for a good many years to spend a month
or six weeks in summer or early autumn at the famous French
watering-place, from which he seemed always to receive much benefit. For
a long time he had suffered from an intermittent and painful form of
facial neuralgia, which, except during its attacks, which came and
passed suddenly, did not incapacitate him from work. With the exception
of a rather serious illness in 1906, this was the only ailment from
which he had suffered for many years, and his general health was
otherwise uniformly good.

Lord and Lady Kelvin returned to Netherhall on September 14, with the
intention of going in a day or two to Belfast, to open the new
scientific buildings of Queen's College. But, unfortunately, on the day
of their arrival Lady Kelvin became very seriously ill, and the visit to
Ireland had to be abandoned. His address was, however, read by his
nephew, James Thomson, son of his elder brother, and was a tribute to
the city of his birth, and the memory of his father.

The illness of Lady Kelvin caused much anxiety for many weeks, and this,
and perhaps some incautious exposure, led to the impairment of Lord
Kelvin's health. A chill caught on November 23 caused him to be confined
to bed; and though he managed for a week or two still to do some work on
a paper with which he had been occupied for a considerable time, he
became worse, and gradually sank, until his death at a quarter-past ten
o'clock on the evening of December 18.

The keen sorrow which was universally felt for Lord Kelvin's death was
manifested by all classes of the community. In Glasgow every one mourned
as for the greatest of the land, and the testimony to the affection in
which he was held, and the reverence for his character and scientific
achievements, was extraordinary. And this feeling was universal; from
all parts of the world poured in telegrams of respectful sympathy with
Lady Kelvin and with the University of Glasgow in their bereavement.

The view was immediately and strongly expressed, both privately and by
the press, that the most illustrious natural philosopher since Newton
should rest beside the great founder of physical science in Westminster
Abbey, and a requisition was immediately prepared and forwarded by the
Royal Society of London to the Dean of Westminster. The wish of the
whole scientific world was at once acceded to, and on December 23, at
noon, the interment took place, with a state and yet a simplicity which
will never be forgotten by those who were present.

Nearly all the scientific notabilities of the country were present, and
the coffin, preceded by the choristers and the clergy, while the hymn,
"Brief life is here our portion," was sung, was followed round the
cloistered aisles from St. Faith's chapel to the choir, by the
relatives, representatives of His Majesty the King and the Prince of
Wales, by the Royal Society, by delegates from the Institute of France,
representatives of the Universities of Cambridge, Oxford, Glasgow, and
other universities, of the Royal Society of Edinburgh (of which Lord
Kelvin was president when he died), and of most of the learned societies
of the kingdom. Then, after a short service, the body was followed to
the grave in the cloisters by the same company of mourners, and to the
solemn words of the Burial Service was laid close by where rests all
that was mortal of Isaac Newton. There he sleeps well who toiled during
a long life for the cause of natural knowledge, and served nobly, as a
hero of peace, his country and the world.



CONCLUSION


The imperfect sketch of Lord Kelvin's scientific life and work which
this book contains can only give a faint notion of the great
achievements of the long life that has now ended. Beyond the researches
which he carried out and the discoveries he made, there is the
inspiration which his work and example gave to others. Inspired himself
by Lagrange, Laplace, Ampère, and Fourier, and led to experimental
research by the necessity for answers to the questions which his
mathematical expression of the discoveries of the twenty-five years
which preceded the establishment of his laboratory had suggested--the
theories of electricity and magnetism, of heat, of elasticity, his
discoveries in general dynamics and in fluid motion, the publication of
"Thomson and Tait," all made him the inspirer of others; and there was
no one, however eminent, who was not proud to acknowledge his
obligations to his genius. Clerk Maxwell, before he wrote the most
original treatise on electricity that has ever appeared, gave himself to
the study of Faraday's Experimental Researches and to the papers of
Thomson. And if some, like FitzGerald and others, have regretted that
the electromagnetic theory of light to which Maxwell was led by Faraday,
and, indeed, by Thomson himself, did not meet with a more sympathetic
reception at his hands, they have not been unmindful of the source from
which much of their illumination has come.

He has founded a school of thought in mathematical physics, of men in
whose minds the symbol is always the servant of the ideas, whose motto
is interpretation by dynamical processes and models as far as that is
possible, who shirk no mathematical difficulties when they have to be
encountered, but are never led away from the straight road to the goal
which they seek to reach--the systematic and clear formulation of the
course of physical action.

And in Lord Kelvin's mind there was blended with a clear physical
instinct which put aside all that was extraneous and unessential to the
main issue an extraordinary power of concentration on the problem in
hand, and a determination that was never daunted by failure, which
consented to postponement but never to relinquishment, and which led
often after long intervals of time to success in the end. He believed
that light would come at last on the most baffling of problems, if only
it were looked at from every point of view and its conditions were
completely formulated; but he could put what was for the time impossible
aside, and devote himself to the immediately possible and realisable.
And as often happens with every thinker, his mind, released from the
task, returned to it of itself, and what before appeared shrouded in
impenetrable mist stood out suddenly sharp and distinct like a
mountain-top before a climber who has at last risen above the clouds.

With the great mathematical power and sure instinct which led him to
success in physical research was combined a keen perception of the
importance of practical applications. Sometimes the practical question
suggested the theoretical and experimental research, as when the needs
of submarine telegraphy led to the discussion of the speed of signalling
and the evolution of the reflecting galvanometer and the siphon
recorder. On the other hand, the mathematical theory of electricity and
magnetism had led to quantitative measurement and absolute units at an
earlier time, when the need for these was beginning to be felt clearly
by scientific workers and dimly by those far-sighted practical men who
dreamed--for a dream it was thought at the time--of linking the Old
World with the New by a submarine cable. But the quantitative study of
electricity in the laboratory threw light on economic conditions, and
the mass of data already obtained, mainly as a mere matter of
experimental investigation of the properties of matter, became at once a
valuable asset of the race of submarine cable engineers which suddenly
sprang into existence.

And so it has been with the more recent applications of electricity. The
induction of currents discovered by Faraday could not become of
practical importance until its laws had been quantitatively discussed, a
much longer process than that of discovery; and we have seen how the
British Association Committee, led by Thomson and Maxwell, brought the
ideas and quantities of this new branch of science into numerical
relation with the units of already existing practical enterprise. The
electrical measuring instruments--first the electrometers, and more
recently the electric current balances and other beautiful instruments
for the dynamo-room and the workshop--which Lord Kelvin invented have
brought the precision of the laboratory into the everyday duties of the
secondary battery attendant and the wireman.

And as to methods of measurement, those who remember the haziness of
even telegraph engineers as to the measurement of the efficiency of
electrical currents and electromotive forces in the circuits of lamps
and dynamos, in the early days of electric lighting, know how much the
world is indebted to Thomson.[26] He it was who showed at first how
cables were to be tested, as well as how they were to be worked; it was
his task, again, to show how instruments were to be calibrated for
practical measurement of current and energy supplied by the early
contractors to consumers. He had in the quiet of his laboratory long
before elaborated methods of comparing resistances, and given the
Wheatstone balance its secondary conductors for the comparison of low
resistances; he now showed how the same principles could be applied to
measure the efficiencies of dynamos and to make up the account of charge
and discharge for a secondary battery.

And if the siphon-recorder and the mariners' compass and the sounding
machine proved pecuniarily profitable, the reward was that of the
inventor, who has an indefeasible right to the fruit of his brain and
his hand. But Lord Kelvin's activity was not confined merely to those
practical things which have, to use the ordinary phrase, "money in
them"; he gave his time and energies freely to the perfecting of the
harmonic analysis of the tides, undertook again, for a Committee of the
British Association, the investigation of the tides for different parts
of the world, superintended the analysis of tidal records, and invented
tide-predicting machines and improved tide-gauges.

Lord Kelvin's work in the theory of heat and in the science of energy
generally would have given him a title to immortality even if it had
stood alone; and there can be no doubt, even in the mind of the most
determined practical contemner of the Carnot cycle, of the enormous
importance of these achievements. Here he was a pioneer, and yet his
papers, theoretical and yet practical, written one after another in
pencil and despatched, rough as they were, to be printed by the Royal
Society of Edinburgh, form, as they are collected in volume i of his
_Mathematical and Physical Papers_, in some respects the best treatise
on thermodynamics at the present time! There are treatises written from
a more general standpoint, which deal with complex problems of chemical
and physical change of means of thermodynamic potentials, and processes
which are not to be found set forth in this volume of papers; but even
these are to a great extent an outcome of his "Thermoelastic,
Thermomagnetic and Thermoelectric Properties of Matter."

In hydrodynamics also Lord Kelvin never lost sight of practical
applications, even while pursuing the most intensely theoretical
researches into the action of vortices or the propagation of waves. In
his later years he worked out the theory of ship-waves with a power
which has made more than one skilful and successful cultivator of this
branch of science say that he was no mere mathematician, but a man who,
like the prophets of old, could divine what is hid from the eyes of
ordinary mortals. Of the ultimate importance of these for practical
questions of the construction of ships, and the economy of fuel in their
propulsion, there can be little doubt. Unhappily, the applications will
have now to be made by others.

It is interesting to note that the investigation of waves in canals with
which Lord Kelvin recently enriched the _Proceedings of the Royal
Society of Edinburgh_ have been carried out by a strikingly ingenious
adaptation of the Fourier solution of the differential equation of the
diffusion of heat along a bar, or of electricity along a slowly worked
cable. Thus, beginning with Fourier mathematics in his earliest
researches, he has in some of his last work applied the special
exponential form of Fourier solution of the diffusion equation to a
case, that of wave propagation, essentially different in physical
nature, and distinct in mathematical signification, from that for which
it was originally given.

Lord Kelvin's written work consists of the _Electrostatics and
Magnetism_, three volumes of _Collected Mathematical and Physical
Papers_, three of _Popular Lectures and Addresses_, the _Baltimore
Lectures_, a very considerable number of papers as yet uncollected, and
the _Natural Philosophy_. But this, great as it was, represented only a
relatively small part of his activities. He advised public companies on
special engineering and electrical questions, served on Royal
Commissions, acted as consulting engineer to cable companies and other
corporations, was employed as arbiter in disputes when scientific
questions were involved, advocated distinctive signalling for
lighthouses and devised apparatus for this purpose, and he was, above
all, a great inventor. His patents are many and important. One of them
was for a water-tap warranted not to drip, another, for electrical
generating machines, meters, etc., was perhaps the patent of largest
extent ever granted.

To Lord Kelvin's class teaching reference has been made in an earlier
chapter. He was certainly inspiring to the best students. At meetings of
the British Association his luminous remarks in discussion helped and
encouraged younger workers, and his enthusiasm was infectious. But with
the ordinary student who cannot receive or retain his mental nutriment
except by a carefully studied mode of presentation, he was not so
successful. He saw too much while he spoke; new ideas or novel modes of
viewing old ones presented themselves unexpectedly, associations crowded
upon his mind, and he was apt to be discursive, to the perplexity of all
except those whose minds were endued also with something of the same
kind of physical instinct or perception. Then he was so busy with many
things that he did not find time to ponder over and arrange the matter
of his elementary lectures, from the point of view of the presentment
most suitable to the capacity of his hearers. To the suggestion which
has lately been made, that he should not have been obliged to lecture to
elementary students, he would have been the first to object. As a matter
of fact, in his later years he lectured to the ordinary class only twice
a week, and to the higher class once. The remainder of the lectures were
given by his nephew, Dr. J. T. Bottomley, who for nearly thirty years
acted as his deputy as regards a great part of the routine work of the
chair.

It is hardly worth while to refute the statement often made that Lord
Kelvin could not perform the operations of simple arithmetic. The truth
is, that in the class-room he was too eager in the anticipation of the
results of a calculation, or too busy with thoughts of what lay beyond,
to be troubled with the multiplication table, and so he often appealed
to his assistants for elementary information which at the moment his
rapidly working mind could not be made to supply for itself.

To sum up, Lord Kelvin's scientific activity had lasted for nearly
seventy years. He was born four years after Oersted made his famous
discovery of the action of an electric current on a magnet, and two
years before Ampère, founding on this experiment, brought forth the
first great memoir on electromagnetism. Thus his life had seen the
growth of modern electrical science from its real infancy to its now
vigorous youth. The discoveries of Faraday in electrical induction were
given to the world when Lord Kelvin was a boy, and one of the great
tasks which he accomplished was to weave these discoveries together in a
uniform web of mathematical theory. This theory suggested, as we have
seen, new problems to be solved by experiment, which he attacked with
the aid of his students in the small and meagrely equipped laboratory
established sixty years ago in the Old College in the High Street. It
was his lot to live to see his presentations of theory lead to new
developments in his own hands and the hands of other men of
genius--Helmholtz and Clerk Maxwell, for example--and to survive until
these developments had led to practical applications throughout our
industries, and in all the affairs of present-day life and work. His
true monument will be his work and its results, and to only a few men
in the world's history has such a massive and majestic memorial been
reared.

He was a tireless worker. In every day of his life he was occupied with
many things, but he was never cumbered. The problems of nature were ever
in his mind, but he could put them aside in the press of affairs, and
take them up again immediately to push them forward another stage
towards solution. His "green book" was at hand on his table or in his
pocket; and whenever a moment's leisure occurred he had pencil in hand,
and was deep in triple integrals and applications of Green's Theorem,
that unfailing resource of physical mathematicians.

  Saepe stilum vertas quae digna legi sint
  Scripturus,

the motto which Horace recommends, was his, and he would playfully quote
it, pointing to the eraser-pad in the top of his gold pencil-case. He
erased, corrected, amended, and rewrote with unceasing diligence, to the
dismay of his shorthand-writing secretary.

The theories and facts of electricity and magnetism, the production and
propagation of waves in water or in the luminiferous ether, the
structure and density of the ether itself, the relations of heat and
work, the motions of the heavenly bodies, the constitution of crystals,
the theory of music, the practical problems of navigation, of
telegraphing under the sea, and of the electric lighting of cities--all
these and more came before his mind in turn, and sometimes most of them
in the course of a single day. He could turn from one thing to another,
and find mental rest in diversity of mental occupation.

He would lecture from nine to ten o'clock in the morning to his ordinary
class, though generally this was by no means the first scientific work
of the day. At ten o'clock he passed through his laboratory and spoke to
his laboratory students or to any one who might be waiting to consult
him, answered some urgent letter, or gave directions to his secretary;
then he walked or drove to White's workshop to immerse himself in the
details of instrument construction until he was again due at the
university for luncheon, or to lecture to his higher mathematical class
on some such subject as the theory of the tides or the Fourier analysis.

As scientific adviser to submarine telegraph companies and other public
bodies, and more recently as President of the Royal Society of London,
he made frequent journeys to London. These were arranged so as to
involve the minimum expenditure of time. He travelled by night when
alone, and could do so with comfort, for he possessed the gift of being
able to sleep well in almost any circumstances. Thus he would go to
London one night, spend a busy day in all kinds of business--scientific,
practical, or political--and return the next night to Glasgow, fresh and
eager for work on his arrival. Here may be noticed his power of
detaching himself from his environment, and of putting aside things
which might well have been anxieties, and of becoming again absorbed in
the problem which circumstances had made him temporarily abandon.

Genius has been said to be the power of taking infinite pains: it is
that indeed, but it is also far more. Genius means ideas, intuition, a
faculty of seizing by thought the hidden relations of things, and
withal the power of proceeding step by step to their clear and full
expression, whether in the language of mathematical analysis or in the
diction of daily life. Such was the genius of Lord Kelvin; it was lofty
and it was practical. He understood--for he had felt--the fascination of
knowledge apart from its application to mechanical devices; he did not
disdain to devote his great powers to the service of mankind. His
objects of daily contemplation were the play of forces, the actions of
bodies in all their varied manifestations, or, as he preferred to sum up
the realm of physics, the observation and discussion of properties of
matter. But his eyes were ever open to the bearing of all that he saw or
discovered on the improvement of industrial appliances, to the
possibility of using it to increase the comfort and safety of men, and
so to augment the sum total of human happiness.

His statement, which has been so often quoted, that after fifty-five
years of constant study he knew little more of electricity and magnetism
than he did at the beginning of his career, is not to be taken as a
confession of failure. It was, like Newton's famous declaration, an
indication of his sense of the vastness of the ocean of truth and the
manifoldness of the treasures which still lie within its "deep
unfathomed caves." Like Newton, he had merely wandered along the shore
of that great ocean, and here and there sounded its accessible depths,
while its infinite expanse lay unexplored. And also like Newton--indeed
like all great men--he stood with deep reverence before the great
problems of the soul and destiny of man. He believed that Nature, which
he had sought all his life to know and understand, showed everywhere
the handiwork of an infinite and beneficent intelligence, and he had
faith that in the end all that appeared dark and perplexing would stand
forth in fulness of light.



FOOTNOTES:


  [1] Lord Kelvin's address on his installation as Chancellor of
  the University of Glasgow, November 29, 1904.

  [2] Successor of Dr. Dick, the Professor of Natural Philosophy
  who induced the Faculty to grant a workshop to James Watt when
  the Corporation of Hammermen prevented him from starting
  business in Glasgow, and for whom Watt was repairing the
  Newcomen engine when he invented the separate condenser.

  [3] A model steam-engine which he made in his youth was
  carefully preserved by his brother in the Natural Philosophy
  Department. It was homely but accurate in construction: the
  beam was of wood, and the piston was an old thick copper penny!

  [4] Proceedings on the occasion of the Presentation to the
  University of Glasgow of the Portrait of Emeritus Professor G.
  G. Ramsay. November 6, 1907.

  [5] Apparently for a short time in 1841, when Dr. Meikleham was
  laid aside by illness.

  [6] The C.U.M.S. began as a Peterhouse society in 1843, and
  after a first concert, which was followed by a supper, and that
  by "certain operations on the chapel roof," the Master would
  only give permission to hold a second concert in the Red Lion
  at Cambridge, there being no room in College, on condition that
  the society called itself the University Musical Society. The
  new society was formed in May 1844; the first president was G.
  E. Smith, of Peterhouse, the second was Blow, also of
  Peterhouse, a violin player and 'cellist, and the third was
  Thomson. [See _Cambridge Chronicle_, July 10, 1903, and _The
  Cambridge Review_, Feb. 20, 1908.]

  [7] It is rather strange that the ninth edition of the
  _Encyclopædia Britannica_ contains no biography of Green. Born
  in the year 1793 at Nottingham, the son of a baker, he assisted
  his father, who latterly acquired a miller's business at the
  neighbouring village of Sneinton. In 1829 his father died, and
  he disposed of the business in order that he might have leisure
  to give to mathematics, in which, though entirely self-taught,
  he had begun to make original researches. His famous 'Essay'
  was published by subscription in 1828, and attracted but little
  attention. In 1833, at forty years of age, Green entered at
  Gonville and Caius College, and obtained the fourth place in
  the mathematical tripos of 1837, the year of Griffin,
  Sylvester, and Gregory. His university career, whatever else it
  may have done, apparently did not tend to make his earlier work
  much better known to the general scientific public, and he died
  in 1841 without the scientific recognition which was his due.
  That came later when, as stated below, Thomson discovered him
  to the French mathematicians and republished his 'Essay.'

  [8] January 1869, Reprint, etc., Article XV.

  [9] Reprint, Article V.

  [10] The geometrical idea was, however, given and applied at
  least as early as 1836 by Bellavitis, for a paper entitled
  "Teoria delle figure inversa" appears in the _Annali delle
  Scienze del Regno Lombardo-Veneto_ for that year. It was also
  described as an independent discovery by Mr. John Wm. Stubbs,
  in a paper in the _Philosophical Magazine_ for November 1843.
  In a note on the history of the transformation in Taylor's
  _Geometry of Conics_ the date (without reference) of Bellavitis
  is given, and it is stated that the method of inversion was
  given afresh by Messrs. Ingram and Stubbs (Dublin, _Phil. Soc.
  Trans._ I). The note also mentions that inversion was "applied
  by Dr. Hirst to attractions," but contains no reference to
  Thomson's papers!

  [11] "_De Caloris distributione per Terræ Corpus_" in the
  Faculty minute, as stated above.

  [12] _Sic._ Without doubt a mistake of the scribe for
  "Liouville."

  [13] _North Wales Chronicle_, Report, Feb. 7, 1885.

  [14] Published: _Treatise on Natural Philosophy_, vol. i in
  1867; _Elements of Natural Philosophy_ in 1873.

  [15] The exact date at which this was done cannot be determined
  from the Minutes of the Faculty, as they contain no reference
  to the appropriation of space for the purpose. In his _Oration
  on James Watt_, delivered at the Ninth Jubilee of the
  University of Glasgow, in 1901, Lord Kelvin referred to the
  Glasgow Physical Laboratory as having grown up between 1846 and
  1856; and elsewhere he has referred to it as having been
  "incipient" in 1851.

  [16] There are now in Glasgow in the winter session alone about
  360 elementary students and 80 advanced students, and about 250
  taking practical laboratory work.

  [17] Before his death (in 1832) Carnot had obtained a clear
  perception of the true state of the case, and of the complete
  doctrine of the conservatism of energy. [See extracts from
  Carnot's unpublished writings appended, with a biography, to
  the reprinted Memoir, by his younger brother, Hippolyte
  Carnot.]

  [18] This equation for the porous plug experiment may be
  established in the following manner, which forms a good example
  of Thomson's second definition of absolute temperature. Take
  pressure and volume of the gas on the supply side of the plug
  as p + dp and v, and on the delivery side as p and v + dv, so
  that dp and dv are positive. The net work done in forcing the
  gas through the plug = (p + dp)v - p(v + dv) = - pdv + vdp.
  Let a heating effect result so that temperature is changed from
  T to T + ∂T. Let this be annulled by abstraction of heat
  Cp∂T at constant pressure. (Cp = sp. heat press. const.)
  [It is to be understood that dv is the total expansion
  existing, after this abstraction of heat.] The energy e of the
  fluid has been increased by de = - pdv + vdp - Cp∂T.

  Now, since the original temperature has been restored, the
  same expansion dv if imposed isothermally would involve the
  same energy change de; but in that case heat dH (dynamical)
  would be absorbed, and work pdv would be done by the gas.
  Hence de = dH - pdv. This, with the former value of de, gives
  dH = vdp - Cp∂T. Thomson's work-ratio is thus pdv⧸(vdp - Cp∂T).
  Now suppose dp imposed without change of volume, and dT to be the
  resulting temperature change. The temperature and pressure ratios
  are dT⧸T, dp⧸p. Thus dT⧸T = dp dv⧸(vdp - Cp∂T), or

    (v⧸T)(dT⧸dv) = 1⧸[1 - (Cp⧸v)(∂T⧸dp)]

  which is Thomson's equation. The minus sign on the right arises
  from a heating effect having been taken here as the normal
  case.

  If the temperature T is restored by removing the heat at
  constant volume, a similar process gives the equation

    (v⧸T)(dT⧸dv) = [1 + (∂T⧸∂p)(∂T⧸dp)]⧸[1 - (Cv⧸v)(∂T⧸dp)]

  where dp is the change of pressure before the restoration of
  the temperature T, and ∂T⧸∂p is the rate of variation of T
  with p, volume constant.

  [19] "On a Universal Tendency in Nature to Dissipation of
  Energy," _Proc. R.S.E._, 1852, and _Phil. Mag._, Oct., 1852.

  [20] To this may be added the extremely useful theorem for such
  problems, that if any directed quantity L, say, characteristic
  of the motion of a body, be associated with a line or axis Ol,
  which is changing in direction, it causes a rate of production
  of the same quantity for a line or axis instantaneously at
  right angles to Ol, towards which Ol is turning with angular
  velocity ω, of amount ωL. If M be the amount of the
  quantity already existing for this latter line or axis, the
  total rate of growth of the quantity is there M + ωL. For
  example, a particle moving with uniform speed v in a circle of
  radius r, has momentum mv along the tangent. But the tangent is
  turning round as the particle moves with angular speed v⧸r,
  towards the radius. The rate of growth of momentum towards the
  centre is therefore

                mv × v⧸r = mv²⧸r.

  [21] See Gray's _Lehrbuch der Physik_, s. 278. Vieweg u. Sohn,
  1904.

  [22] Gray, Royal Institution, Friday Evening Discourse,
  February 1898.

  [23] See the _Reports of the Committee on Electrical Standards_,
  edited by Prof. Fleeming Jenkin, F.R.S., Maxwell's _Electricity
  and Magnetism_, and Gray's _Theory and Practice of Absolute
  Measurements in Electricity and Magnetism_, Vol. II, Part II.

  [24] The writer once, on a thick night, in a passenger steamer
  in the Race of Alderney, when the engines were stopped and
  soundings were being taken, saw the reel and cord go overboard,
  nearly taking one of the men with it. A new hank of cord had to
  be got and bent on a new reel; an operation that took a long
  time, during which the exact locality of the ship was a matter
  of uncertainty. Comment is needless!

  [25] The tuning of a major third, in this way, is described in
  the paper entitled "Beats on Imperfect Harmonies," published in
  _Popular Lectures and Addresses_, vol. ii.

  [26] The writer well remembers meeting a man of some experience
  in cable work who was on his way to measure the alternating
  currents in a Jablochkoff candle installation by the aid of an
  Ayrton and Perry galvanometer with steel needle!



INDEX


  Atlantic cables, 267, 268

  Atmospheric electricity, 226

  Atoms, size of, 261

  Ayrton, W. E., 296


  Baltimore lectures, 254-263

  Bertrand's theorem of maximum kinetic energy, 158

  Bottomley, James Thomson, 311

  Bottomley, William, 7

  British Association, electrical standards, 244-253


  Cambridge University Musical Society, 24

  _Cambridge and Dublin Mathematical Journal_, 25, 31, 78

  Carnot, Sadi, 77, 101

  Carnot's _Théorie Motrice du Feu_, 87, 101, 108 _et seq._

  Cauchy, 294

  Chasles, 28, 43

  Clapeyron, 101, 112

  Clausius, 114 _et seq._

  College, the old, of Glasgow, 10

  Compass, errors of, 273


  "Dew-drop," artificial, 290

  Dynamical theorems, Thomson's and Bertrand's, 158 _et seq._


  Earth, the age of, 196, 229-243

  Earth, tidal retardation of, 230

  Elasticity, Poisson-Navier theory of, 291;
    encyclopædia article on, 297

  Electrical oscillations, 181 _et seq._

  Electricity, mathematical theory of, 33

  Electrolysis, mechanical theory of, 176

  Electrometers, 223 _et seq._

  Electromotive forces, estimation of, by heats of combination, 178

  Electromotive forces, measurement of, 179

  _Electrostatics and Magnetism_, 222 _et seq._

  Ellis, Robert Leslie, 26

  Energy, dissipation of, 139


  Faculty, the, of the University of Glasgow, 4, 63-67

  Faraday, 61

  Faure, M., 81

  FitzGerald, G. F., 301, 305

  Fourier, _Théorie Analytique de la Chaleur_, 16 _et seq._


  Gauss, 28

  Gauss and Weber, 245

  Green, George, of Nottingham, 21, 30, 294

  Gregory, J. W., 241

  Goodwin, Harvey, 26

  Gyrostats and gyrostatic action, 214, 284-286


  Hamilton, Sir William Rowan, 196, 294

  Heat, encyclopædia article on, 297

  Heaviside, Oliver, 294

  Helmholtz, von, 113

  Hertz, 191, 256

  Hopkins, William, 23

  Huxley, 77, 196, 242

  Hydrodynamics, 153-175


  Images, electric, 31, 38-59

  Inversion, electrical, 49 _et seq._

  Inversion, geometrical, 59, 60


  Joule, James Prescott, 77, 86 _et seq._, 101 _et seq._


  Larmor, Joseph, 256

  Lectures on Natural Philosophy at Glasgow, 279 _et seq._

  Liouville, 31

  Liouville's _Journal de Mathématiques_, 25, 26, 31

  Loschmidt, 262

  Lubbock, Sir John (Lord Avebury), 85

  Luminiferous ether, motion of planets through, 256


  Magnetism, theory of, 227

  Mariners' compass, 272 _et seq._

  Maxwell, 117, 193, 305

  Mayer, of Heilbronn, 105

  McFarlane, Donald, 96, 287, 289

  McKichan, Dugald, 193

  _Mécanique Analytique_ of Lagrange, 199, 205

  _Mécanique Céleste_ of Laplace, 199, 205

  Meikleham, William, 61

  Mirror galvanometer, 268

  Motivity, thermodynamic, 138


  Natural Philosophy, Chair of, at Glasgow, 63

  _Natural Philosophy_, Thomson and Tait's, 196 _et seq._

  Navigational sounding machine, 272

  Newton, 195, 202

  Nichol, John, Professor of English Language and Literature, 5

  Nichol, John Pringle, Professor of Astronomy, 5, 20, 61, 63


  Oersted, 61

  Oscillations, electrical, 181 _et seq._


  Parkinson, Stephen, 27

  Peltier, 148

  Pendulum, ballistic, 288

  Perry, John, 240, 296

  Phosphorescence, dynamical theory of, 259

  Physical laboratory, first, 70

  Pickering, 217

  Polarised light, rotation of plane of, 220

  Principia, Newton's, 195, 202


  Ramsay, George Gilbert, Professor of Humanity, 11

  Regnault, 29

  Royal Society of Edinburgh, presidency of, 299

  Royal Society of London, presidency of, 299

  Rumford, Count, 103


  Seebeck, 148

  Signalling, theory of telegraphic, 264

  Siphon recorder, 268, 270

  Smith, Archibald, 275

  Spectrum analysis, dynamical theory of, 84

  Stokes, Sir George Gabriel, 24, 79, 80, 81, 85, 291, 294

  Stoney, Dr. Johnstone, 262

  Sun's heat, age of, 232


  Tait, Peter Guthrie, 194 _et seq._

  Temperature, absolute, 125 _et seq._;
    comparison of, with scale of air thermometer, 135

  Thermodynamics, 99-152

  Thermoelasticity, 142 _et seq._

  Thermoelectricity, 147 _et seq._

  Thermometry, absolute, 114-152

  Thomson, David, 61

  Thomson, James, Professor of Mathematics, 1-4, 7

  Thomson, James, Professor of Engineering, 113, 209;
    integrating machine, 209, 303

  Thomson and Tait's Natural Philosophy, 68, 196 _et seq._, 218

  Thomson's theorem of minimum kinetic energy, 158

  Thomson, Thomas, Professor of Chemistry, 6



  Thomson, prevalence of name at Glasgow College, 5

  Thomson, William, Lord Kelvin:--
    Parentage and early education, 1-12
    Career at Universities of Glasgow and Cambridge, 13-32
    Early researches, 16, 18, 31
    Election to Chair of Natural Philosophy at Glasgow, 64
    Scientific researches, passim;
      Jubilee of, 301;
      Chancellor of University of Glasgow, 302
    In class-room and laboratory, 279-298
    Practical activities, honours and distinctions, last illness and
        death, 299-304;
      funeral in Westminster Abbey, 304

  Tidal Analyser, 211

  Tide Predicter, 208


  Vortex-Motion, 161-175


  Waldstein sonata, 24

  Weber, W., 193

  Weights and measures, British, 289, 290

  White, James, 276

  Willard Gibbs, 294



  RICHARD CLAY & SONS, LIMITED,
  BREAD STREET HILL, E.C., AND
  BUNGAY, SUFFOLK.





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