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+The Project Gutenberg EBook of Lord Kelvin, by Andrew Gray
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+
+Title: Lord Kelvin
+       An account of his scientific life and work
+
+Author: Andrew Gray
+
+Release Date: April 4, 2012 [EBook #39373]
+
+Language: English
+
+Character set encoding: UTF-8
+
+*** START OF THIS PROJECT GUTENBERG EBOOK LORD KELVIN ***
+
+
+
+
+Produced by Laura Wisewell, Turgut Dincer, Tamise Totterdell
+and the Online Distributed Proofreading Team at
+http://www.pgdp.net (The original copy of this book was
+generously made available for scanning by the Department
+of Mathematics at the University of Glasgow.)
+
+
+
+
+
+
+
+
+  +--------------------------------------------------------------------+
+  |                                                                    |
+  |                                                                    |
+  | TRANSCRIBER'S NOTES:                                               |
+  |                                                                    |
+  | Mathematical expressions which are in italics in the original text |
+  | are displayed in non-italic characters to increase the readability |
+  | of the book.                                                       |
+  |                                                                    |
+  | For super and subscripts unicode characters have been used except  |
+  | a few cases where carat symbol and underline have been used.       |
+  |                                                                    |
+  | Mathematical notation in the body of text is duly conserved but    |
+  | parantheses have been added to expressions to avoid any kind of    |
+  | misenterpretation.                                                 |
+  |                                                                    |
+  +--------------------------------------------------------------------+
+
+
+
+  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.
+
+
+
+
+
+End of the Project Gutenberg EBook of Lord Kelvin, by Andrew Gray
+
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+<pre>
+
+The Project Gutenberg EBook of Lord Kelvin, by Andrew Gray
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org/license
+
+
+Title: Lord Kelvin
+       An account of his scientific life and work
+
+Author: Andrew Gray
+
+Release Date: April 4, 2012 [EBook #39373]
+
+Language: English
+
+Character set encoding: UTF-8
+
+*** START OF THIS PROJECT GUTENBERG EBOOK LORD KELVIN ***
+
+
+
+
+Produced by Laura Wisewell, Turgut Dincer, Tamise Totterdell
+and the Online Distributed Proofreading Team at
+http://www.pgdp.net (The original copy of this book was
+generously made available for scanning by the Department
+of Mathematics at the University of Glasgow.)
+
+
+
+
+
+
+</pre>
+
+
+<div class="figcenter" style="width: 640px;">
+<div class="figleft" style="width: 101px;">
+<img src="images/spine.jpg" width="101" height="570" alt="Spine" title="Spine" />
+</div>
+<div class="figright" style="width: 480px;">
+<img src="images/cover.jpg" width="480" height="572" alt="Cover" title="Cover" />
+</div>
+</div>
+<div style="clear: both;"></div>
+
+
+<p>&nbsp;</p>
+
+<p style='text-align:left; margin-left:35%'>ENGLISH<br />
+MEN OF SCIENCE</p>
+
+<p>&nbsp;</p>
+
+<p class="center"><small>EDITED BY</small><br />
+<big>J. REYNOLDS GREEN, Sc.D.</big></p>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+<p class="center"><big><b>LORD KELVIN</b></big></p>
+
+<p>&nbsp;</p>
+
+
+<div class="tn">
+<p><big><i>ENGLISH MEN<br />
+OF SCIENCE</i></big></p>
+
+<p><span class="smcap">Edited by</span></p>
+
+<p><span class="smcap">Dr. J. REYNOLDS GREEN.</span></p>
+
+<p>
+<i>With Photogravure Frontispiece.<br />
+Small Cr. 8vo, 2s. 6d. net per vol.</i><br />
+</p>
+
+<p>SPENCER. By <span class="smcap">J. Arthur
+Thompson</span>.</p>
+
+<p>PRIESTLEY. By Dr. <span class="smcap">Thorpe</span>,
+C.B., F.R.S.</p>
+
+<p>FLOWER. By Prof. <span class="smcap">R. Lydekker</span>,
+F.R.S.</p>
+
+<p>HUXLEY. By Prof. <span class="smcap">Ainsworth
+Davis</span>.</p>
+
+<p>BENTHAM. By <span class="smcap">B. Daydon
+Jackson</span>, F.L.S.</p>
+
+<p>DALTON. By <span class="smcap">J. P. Millington</span>,
+M.A.</p>
+
+<p><i>J. M. DENT &amp; CO.</i></p>
+</div>
+
+<p>&nbsp;</p>
+<h5><i>All Rights Reserved</i></h5>
+
+
+
+<div class="figcenter" style="width: 550px; position: relative;">
+<a name="frontis" id="frontis"></a>
+<img src="images/frontis.jpg" width="550" height="812" alt="Lord Kelvin" title="Lord Kelvin" />
+</div>
+
+
+
+<div class="figcenter" style="width: 550px; position: relative;">
+<img src="images/title.jpg" width="550" height="844" alt="Title" title="Title" />
+</div>
+
+
+<hr style='width: 70%'/>
+
+
+<h1>LORD KELVIN</h1>
+
+<h5><big><i>AN ACCOUNT OF HIS SCIENTIFIC<br />
+LIFE AND WORK</i></big></h5>
+
+<p>&nbsp;</p>
+<h6>BY</h6>
+
+<h3>ANDREW GRAY</h3>
+<h5>LL.D., F.R.S., V.-P.R.S.E.</h5>
+
+<h6>PROFESSOR OF NATURAL PHILOSOPHY IN THE<br />
+UNIVERSITY OF GLASGOW</h6>
+
+<div class="figcenter" style="width: 100px;">
+<a href="images/title.jpg">
+  <img src="images/mark.png" width="100" height="157" alt="printer's mark" title="Click for Original Title Page" />
+</a>
+</div>
+
+<p class="center"><small>
+PUBLISHED IN LONDON BY<br />
+J. M. DENT &amp; CO., AND IN NEW<br />
+YORK BY E. P. DUTTON &amp; CO.<br />
+</small></p>
+
+<h5>1908</h5>
+
+<p>&nbsp;</p>
+<hr style='width: 70%'/>
+<p>&nbsp;</p>
+
+<h6>
+<span class="smcap">Richard Clay &amp; Sons</span>, <span class="smcap">Limited</span>,<br />
+BREAD STREET HILL, E.C., AND<br />
+BUNGAY, SUFFOLK.</h6>
+
+<p>&nbsp;</p>
+<hr style='width: 70%'/>
+<p><span class='pagenum'><a name="Page_v" id="Page_v">v</a></span></p>
+
+<h3>PREFACE</h3>
+
+<p>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 <i>Glasgow Herald</i> 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.</p>
+
+<p>I am under obligations to the proprietors of the
+<i>Glasgow Herald</i> for their freely accorded permission to
+make use of their article, and to Messrs. Annan, photographers,
+Glasgow, and Messrs. James MacLehose &amp;
+Sons, Glasgow, for the illustrations which are given,
+and which I hope may add to the interest of the book.</p>
+
+<p class="quotsig"><span class="smcap">A. Gray</span>.</p>
+
+<p class="small"><i>The University, Glasgow,<br />
+&nbsp;&nbsp;&nbsp;&nbsp;May</i> 20, 1908.</p>
+
+<p><span class='pagenum'><a name="Page_vi" id="Page_vi">vi</a></span></p>
+
+<hr style='width: 70%'/>
+<p><span class='pagenum'><a name="Page_vii" id="Page_vii">vii</a></span></p>
+
+<h3>CONTENTS</h3>
+
+<table width="70%" summary="contents" cellpadding="10"  cellspacing="10">
+<tr>
+<td class="right10t"><small>CHAP.</small></td>
+<td class="left80">&nbsp;</td>
+<td class="right10"><small>PAGE</small></td>
+</tr>
+<tr>
+<td class="right10t">I.</td>
+<td class="left80">PARENTAGE AND EARLY EDUCATION</td>
+<td class="right10"><a href="#Page_1">1</a></td>
+</tr>
+<tr>
+<td class="right10t">II.</td>
+<td class="left80">CLASSES AT THE UNIVERSITY OF GLASGOW. FIRST SCIENTIFIC PAPERS</td>
+<td class="right10"><a href="#Page_13">13</a></td>
+</tr>
+<tr>
+<td class="right10t">III.</td>
+<td class="left80">UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS UNDERGRADUATE</td>
+<td class="right10"><a href="#Page_23">23</a></td>
+</tr>
+<tr>
+<td class="right10t">IV.</td>
+<td class="left80">THE MATHEMATICAL THEORY OF ELECTRICITY
+IN EQUILIBRIUM. ELECTRIC IMAGES. ELECTRIC INVERSION</td>
+<td class="right10"><a href="#Page_33">33</a></td>
+</tr>
+<tr>
+<td class="right10t">V.</td>
+<td class="left80">THE CHAIR OF NATURAL PHILOSOPHY AT
+GLASGOW. ESTABLISHMENT OF THE
+FIRST PHYSICAL LABORATORY</td>
+<td class="right10"><a href="#Page_61">61</a></td>
+</tr>
+<tr>
+<td class="right10t">VI.</td>
+<td class="left80">FRIENDSHIP WITH STOKES AND JOULE.
+EARLY WORK AT GLASGOW</td>
+<td class="right10"><a href="#Page_79">79</a></td>
+</tr>
+<tr>
+<td class="right10t">VII.</td>
+<td class="left80">THE 'ACCOUNT OF CARNOT'S THEORY OF
+THE MOTIVE POWER OF HEAT'&mdash;TRANSITION
+TO THE DYNAMICAL THEORY OF HEAT</td>
+<td class="right10"><a href="#Page_99">99</a></td>
+</tr>
+<tr>
+<td class="right10t">VIII.</td>
+<td class="left80">THERMODYNAMICS AND ABSOLUTE THERMOMETRY</td>
+<td class="right10"><a href="#Page_114">114</a></td>
+</tr>
+<tr>
+<td class="right10t">IX.</td>
+<td class="left80">HYDRODYNAMICS&mdash;DYNAMICAL THEOREM
+OF MINIMUM ENERGY&mdash;VORTEX MOTION</td>
+<td class="right10"><a href="#Page_153">153</a></td>
+</tr>
+<tr>
+<td class="right10t">X.</td>
+<td class="left80">THE ENERGY THEORY OF ELECTROLYSIS&mdash;ELECTRICAL
+UNITS&mdash;ELECTRICAL OSCILLATIONS</td>
+<td class="right10"><a href="#Page_176">176</a></td>
+</tr>
+<tr>
+<td class="right10t">XI.</td>
+<td class="left80">THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'&mdash;GYROSTATIC
+ACTION&mdash;'ELECTROSTATICS
+AND MAGNETISM'<span class='pagenum'><a name="Page_viii" id="Page_viii">viii</a></span></td>
+<td class="right10"><a href="#Page_194">194</a></td>
+</tr>
+<tr>
+<td class="right10t">XII.</td>
+<td class="left80">THE AGE OF THE EARTH</td>
+<td class="right10"><a href="#Page_229">229</a></td>
+</tr>
+<tr>
+<td class="right10t">XIII.</td>
+<td class="left80">BRITISH ASSOCIATION COMMITTEE ON
+ELECTRICAL STANDARDS</td>
+<td class="right10"><a href="#Page_244">244</a></td>
+</tr>
+<tr>
+<td class="right10t">XIV.</td>
+<td class="left80">THE BALTIMORE LECTURES</td>
+<td class="right10"><a href="#Page_254">254</a></td>
+</tr>
+<tr>
+<td class="right10t">XV.</td>
+<td class="left80">SPEED OF TELEGRAPH SIGNALLING&mdash;LAYING
+OF SUBMARINE CABLES&mdash;TELEGRAPH
+INSTRUMENTS&mdash;NAVIGATIONAL INSTRUMENTS,
+COMPASS AND SOUNDING MACHINE</td>
+<td class="right10"><a href="#Page_264">264</a></td>
+</tr>
+<tr>
+<td class="right10t">XVI.</td>
+<td class="left80">LORD KELVIN IN HIS CLASS-ROOM AND
+LABORATORY</td>
+<td class="right10"><a href="#Page_279">279</a></td>
+</tr>
+<tr>
+<td class="right10t">XVII.</td>
+<td class="left80">PRACTICAL ACTIVITIES&mdash;HONOURS AND
+DISTINCTIONS&mdash;LAST ILLNESS AND DEATH</td>
+<td class="right10"><a href="#Page_299">299</a></td>
+</tr>
+<tr>
+<td class="right10t">&nbsp;</td>
+<td class="left80">CONCLUSION</td>
+<td class="right10"><a href="#Page_305">305</a></td>
+</tr>
+<tr>
+<td class="right10t">&nbsp;</td>
+<td class="left80">INDEX</td>
+<td class="right10"><a href="#Page_317">317</a></td>
+</tr>
+</table>
+
+<hr style='width: 10%'/>
+
+<h4>CORRIGENDUM</h4>
+
+<p class="center"><a href="#Page_105">Page 105</a>, line 9 from foot, for
+<i>&#8706;</i><i>e</i>&nbsp;&#43;&nbsp;<i>O</i> read <i>&#8706;</i><i>e</i>&nbsp;&#43;&nbsp;o</p>
+<p><span class='pagenum'><a name="Page_ix" id="Page_ix">ix</a></span></p>
+
+<hr style='width: 70%'/>
+<p><span class='pagenum'><a name="Page_x" id="Page_x">x</a></span></p>
+
+<h3>LIST OF ILLUSTRATIONS</h3>
+
+<table width="70%" summary="illustrations" cellpadding="10"  cellspacing="10">
+<tr>
+<td class="right10t">&nbsp;</td>
+<td class="right10" colspan="2"><small><i>To face page</i></small></td>
+</tr>
+<tr>
+<td class="left80" colspan="2"><span class="smcap">Lord Kelvin</span> (<i>photogravure</i>)</td>
+<td class="right10"><a href="#frontis"><i>Frontispiece</i></a></td>
+</tr>
+<tr>
+<td class="left80" colspan="2"><span class="smcap">Lord Kelvin in 1846</span></td>
+<td class="right10"><a href="#thomson">64</a></td>
+</tr>
+<tr>
+<td class="left80" colspan="2"><span class="smcap">View of Old College</span></td>
+<td class="right10"><a href="#college">70</a></td>
+</tr>
+</table>
+
+
+<hr style='width: 70%'/>
+<p><span class='pagenum'><a name="Page_1" id="Page_1">1</a></span></p>
+
+<h1>LORD KELVIN</h1>
+
+<h3>CHAPTER I</h3>
+
+<h4>PARENTAGE AND EARLY EDUCATION</h4>
+
+<p>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.</p>
+
+<p>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
+<span class='pagenum'><a name="Page_2" id="Page_2">2</a></span>
+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&mdash;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."<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">1</a> 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.</p>
+
+<p>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 <i>Principia</i> had tended
+to perpetuate in this country. Laplace was at the
+height of his fame in France, and was writing the great
+analytical <i>Principia</i>, his <i>M&eacute;canique C&eacute;leste</i>, 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.</p>
+<p><span class='pagenum'><a name="Page_3" id="Page_3">3</a></span></p>
+<p>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.</p>
+
+<p>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:&mdash;"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<span class='pagenum'><a name="Page_4" id="Page_4">4</a></span>
+papers. Even moderators are not made of impenetrable
+stuff, though fenced with sevenfold Jacquier,
+and tough bull-hide of Vince and Wood."</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_5" id="Page_5">5</a></span>
+to manage the library and some other matters) to
+regulate and superintend teaching and discipline.</p>
+
+<p>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:&mdash;</p>
+
+<p>"Niebuhr, after examining a portion of the <i>Fasti
+Consulares</i>, arrived at the conclusion that the <i>senatus
+populusque Romanus</i> 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 <i>gens</i> 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."</p>
+
+<p>A former Professor of Natural Philosophy, Dr.<span class='pagenum'><a name="Page_6" id="Page_6">6</a></span>
+Anderson,<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">2</a> 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"!</p>
+
+<p>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
+<span class='pagenum'><a name="Page_7" id="Page_7">7</a></span>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.</p>
+
+<p>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 <i>gens</i> 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)&mdash;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&mdash;and
+Robert, who was born in 1829, and died in
+Australia in 1905. Besides these four sons there were
+in all three daughters:&mdash;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<span class='pagenum'><a name="Page_8" id="Page_8">8</a></span>
+Glasgow, a connection only terminated by the funeral
+ceremony in Westminster Abbey on December 23,
+1907.</p>
+
+<p>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."</p>
+
+<p>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.<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">3</a> 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,
+<span class='pagenum'><a name="Page_9" id="Page_9">9</a></span>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_10" id="Page_10">10</a></span>
+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<span class='pagenum'><a name="Page_11" id="Page_11">11</a></span>
+keenly regret the disappearance. But both artists
+and antiquaries would have blessed the directors&mdash;and
+such a blessing would have done them no harm&mdash;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.</p>
+
+<p>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.<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">4</a> "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
+<span class='pagenum'><a name="Page_12" id="Page_12">12</a></span>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&mdash;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."</p>
+
+<p>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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_13" id="Page_13">13</a></span></p>
+
+<h3>CHAPTER II</h3>
+
+<h4>CLASSES AT THE UNIVERSITY OF GLASGOW. FIRST
+SCIENTIFIC PAPERS</h4>
+
+<p><span class="smcap">In</span> 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&mdash;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&mdash;a translation of Lucian's <i>Dialogues of the Gods</i>
+"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<span class='pagenum'><a name="Page_14" id="Page_14">14</a></span>
+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.</p>
+
+<p>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:</p>
+
+<p>"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....</p>
+
+<p>"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....</p>
+
+<p>"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,<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">5</a> Dr. Nichol, added
+<span class='pagenum'><a name="Page_15" id="Page_15">15</a></span>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....</p>
+
+<p>"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. <i>That</i> was at a time
+when an imperfectly informed public used to regard
+the University of Glasgow as a stagnant survival of
+medi&aelig;valism, and used to call its professors the 'Monks
+of the Molendinar'!</p>
+
+<p>"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&mdash;a
+deserted wine-cellar of an old professorial house,
+enlarged a few years later by the annexation of a<span class='pagenum'><a name="Page_16" id="Page_16">16</a></span>
+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."</p>
+
+<p>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, <i>La
+Th&eacute;orie Analytique de la Chaleur</i>, 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 <i>Cambridge Mathematical Journal</i> (vol. ii, May
+1841). The object of the paper is to show in what
+cases a function <i>f</i>(<i>x</i>), which is to have certain arbitrary
+values between certain values of <i>x</i>, 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 <i>x</i>,
+between 0 and <i>a</i>, say, and for the assigned values of
+the function, <i>f</i>(<i>x</i>) 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 <i>x</i>, which
+lies outside the limits of that variable between which
+the values of the function are assigned, the values of
+<i>f</i>(<i>x</i>) there are to be found from the expansion adopted,
+by rules which are laid down in the paper.</p>
+
+<p>Fourier used sine-expansions or cosine-expansions
+as it suited him for the function between the limits,
+<span class='pagenum'><a name="Page_17" id="Page_17">17</a></span>
+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 <i>x</i> 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.</p>
+
+<p>The next paper is of the same final date, but
+appeared in the <i>Cambridge Mathematical Journal</i> 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
+<i>F</i>(<i>x</i>), which expresses the temperature at distance <i>x</i>
+from the centre, can be expanded in an infinite series
+of the form</p>
+
+<p class="center">
+<i>a</i><sub>1</sub> sin <i>n</i><sub>1</sub><i>x</i>&nbsp;&#43;&nbsp;<i>a</i><sub>2</sub> sin <i>n</i><sub>2</sub><i>x</i>&nbsp;&#43;&nbsp;...
+</p>
+
+<p>where <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ... are multipliers to be determined
+and <i>n</i><sub>1</sub>, <i>n</i><sub>2</sub>, ... are the roots, infinite in number, of
+the transcendental equation (<i>tan&nbsp;nX</i>)&nbsp;&frasl;&nbsp;<i>nX</i>&nbsp;=&nbsp;1&nbsp;&minus;&nbsp;<i>hX</i>.</p>
+
+<p>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<span class='pagenum'><a name="Page_18" id="Page_18">18</a></span>
+equation lies between 0 and <sup>1</sup>&frasl;<sub>2</sub>, the second between
+1 and <sup>3</sup>&frasl;<sub>2</sub>, the third between 2 and <sup>5</sup>&frasl;<sub>2</sub>, 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 <i>x</i> the same
+value as the series</p>
+
+<p class="center">
+<i>A</i><sub>1</sub> sin <small><sup>1</sup>&frasl;<sub>2</sub></small>&nbsp;<i>x</i>&nbsp;&#43;&nbsp;<i>A</i><sub>2</sub> sin <small><sup>3</sup>&frasl;<sub>2</sub></small>&nbsp;<i>x</i>&nbsp;&#43;&nbsp;...
+</p>
+
+<p>where <i>A</i><sub>1</sub>, <i>A</i><sub>2</sub>, ... are known in terms of <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ...
+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.</p>
+
+<p>The next ten papers, with two exceptions, are all
+on the motion of heat, and appeared in the <i>Cambridge
+Mathematical Journal</i> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_19" id="Page_19">19</a></span>
+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.</p>
+
+<p>Two other great works of his father's collection of
+mathematical books, Laplace's <i>M&eacute;canique C&eacute;leste</i> and
+Lagrange's <i>M&eacute;canique Analytique</i>, 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 <i>Natural Philosophy</i>.</p>
+
+<p>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<span class='pagenum'><a name="Page_20" id="Page_20">20</a></span>
+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.</p>
+
+<p>The <i>Cyclop&aelig;dia of Physical Science</i> 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 <i>Differential
+and Integral Calculus</i> 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.</p>
+
+<p>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 <i>Cambridge Mathematical Journal</i> 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<span class='pagenum'><a name="Page_21" id="Page_21">21</a></span>
+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.</p>
+
+<p>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 <i>Journal</i> (vols. iii and v), and in the celebrated
+memoir by Gauss "On General Theorems<span class='pagenum'><a name="Page_22" id="Page_22">22</a></span>
+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
+<i>Scientific Memoirs</i> in 1842. These anticipations are
+again referred to below.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_23" id="Page_23">23</a></span></p>
+
+<h3>CHAPTER III</h3>
+
+<h4>UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS
+UNDERGRADUATE</h4>
+
+<p><span class="smcap">Thomson</span> 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&mdash;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 <i>Cambridge Mathematical
+Journal</i>, 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<span class='pagenum'><a name="Page_24" id="Page_24">24</a></span>
+of the C.U.M.S., the Cambridge University Musical
+Society.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">6</a> The musical instruments he favoured were
+the cornet and especially the French horn&mdash;he was
+second horn in the original Peterhouse band&mdash;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.</p>
+
+<p>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.</p>
+
+<p>At Cambridge Thomson made the acquaintance of
+George Gabriel Stokes, who graduated as Senior
+<span class='pagenum'><a name="Page_25" id="Page_25">25</a></span>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.</p>
+
+<p>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 <i>Cambridge Mathematical Journal</i> between
+1842 and 1845; but three appeared in 1845 in
+Liouville's <i>Journal de Math&eacute;matiques</i>. 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.<span class='pagenum'><a name="Page_26" id="Page_26">26</a></span></p>
+
+<p>Another paper, published in the form of a letter, of
+date October 8, 1845, to M. Liouville, and published
+in the <i>Journal de Math&eacute;matiques</i> 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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_27" id="Page_27">27</a></span>
+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:&mdash;"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!</p>
+
+<p>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 <i>Cambridge Mathematical Journal</i>; 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.</p>
+
+<p>The Tripos Examination was held in the early part
+of January. On the 25th of that month Thomson<span class='pagenum'><a name="Page_28" id="Page_28">28</a></span>
+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. <a href="#Page_21">21</a>). 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.</p>
+
+<p>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
+<i>Cambridge Mathematical Journal</i> 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 <i>Journal</i>.</p>
+
+<p>Soon after the Tripos and Smith's Prize Examinations<span class='pagenum'><a name="Page_29" id="Page_29">29</a></span>
+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
+Acad&eacute;mie 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, "<i>Vous avez un M&egrave;moire
+de Green! M. Liouville me l'a dit.</i>" 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,
+"<i>Voila mon affaire!</i>"</p>
+
+<p>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.</p>
+
+<p>It is a striking testimony to Thomson's genius that,<span class='pagenum'><a name="Page_30" id="Page_30">30</a></span>
+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&mdash;Sturm
+and Gauss&mdash;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.</p>
+
+<p>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.<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">7</a> In 1850 he obtained the
+<span class='pagenum'><a name="Page_31" id="Page_31">31</a></span>republication of Green's 'Essay' in Crelle's <i>Journal</i>;
+in later years he frequently expressed regret that it had
+not been published in England.</p>
+
+<p>In the commencement of 1845 Thomson told
+Liouville of the method of <i>Electric Images</i> 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 <i>Philosophical Magazine</i>.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">8</a> This
+had been preceded by a series of papers in March,
+May, and November 1848, November 1849, and
+February 1850, in the <i>Cambridge and Dublin Mathematical
+Journal</i>, on various parts of the mathematical
+theory of electricity in equilibrium,<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">9</a> in which the
+theory of images is dealt with. The letters to Liouville
+promptly appeared in the <i>Journal</i>, and the veteran
+analyst wrote a long Note on their subject, which
+concludes as follows: "Mon but sera rempli, je le
+r&eacute;p&eacute;te, s'ils [ces d&eacute;veloppements] peuvent aider &agrave; bien
+faire comprendre la haute importance du travail de ce
+jeune g&eacute;om&egrave;tre, et si M. Thomson lui-m&ecirc;me veut bien
+y voir une preuve nouvelle de l'amiti&eacute; que je lui porte
+et de l'estime qui j'ai pour son talent."</p>
+
+<p><span class='pagenum'><a name="Page_32" id="Page_32">32</a></span></p><p>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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_33" id="Page_33">33</a></span></p>
+
+<h3>CHAPTER IV</h3>
+
+<h4>THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM.
+ELECTRIC IMAGES. ELECTRIC INVERSION</h4>
+
+<p><span class="smcap">In</span> 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.</p>
+
+<p>The whole space in which a distribution of electricity
+produces any action on electrified bodies is called
+the <i>electrical field</i> 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
+<i>field-intensity</i> 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, <i>X</i>, <i>Y</i>, <i>Z</i>
+in three chosen directions at right angles to one
+another.</p>
+
+<p><span class='pagenum'><a name="Page_34" id="Page_34">34</a></span>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
+<i>V</i> 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, <i>point-charges</i> we shall call them], their positions,
+and the position of the point at which the field-intensity
+is to be found. If <i>q</i><sub>1</sub>, <i>q</i><sub>2</sub>, ... be the point-charges,
+and be positive when the charges are positive
+and negative when the charges are negative, and
+<i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, ... be their distances from the point <i>P</i>, <i>V</i> is
+<i>q</i><sub>1</sub>&nbsp;&frasl;&nbsp;<i>r</i><sub>1</sub>&nbsp;&#43;&nbsp;<i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>r</i><sub>2</sub>&nbsp;&#43;&nbsp;...
+The field-intensity is the rate of diminution of the value of <i>V</i> at <i>P</i>, taken along
+the specified direction. The three gradients parallel to
+the three chosen coordinate directions are <i>X</i>, <i>Y</i>, <i>Z</i>; but
+for their calculation it is necessary to insert the values
+of <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, ... in terms of the coordinates which
+specify the positions of the point-charges, and the
+coordinates <i>x</i>, <i>y</i>, <i>z</i> which specify the position of <i>P</i>.
+Once this is done, <i>X</i>, <i>Y</i>, <i>Z</i> are obtained by a simple
+systematic process of calculation, namely, differentiation
+of the function <i>V</i> with respect to <i>x</i>, <i>y</i>, <i>z</i>.</p>
+
+<p>This function <i>V</i> seems to have been first used by
+Laplace for gravitational matter in the <i>M&eacute;canique
+C&eacute;leste</i>; its importance for electricity and magnetism
+was recognised by Green, who named it the <i>potential</i>.
+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 <i>P</i> from a point<span class='pagenum'><a name="Page_35" id="Page_35">35</a></span>
+at an infinite distance from every part of the distribution;
+or, in other words, what we now call the
+<i>potential energy</i> of a charge <i>q</i> situated at <i>P</i> is <i>qV</i>.
+The excess of the potential at <i>P</i>, over the potential at
+any other point <i>Q</i> in the field, is the work which
+must be spent in carrying a positive unit from <i>Q</i> to <i>P</i>
+against electrical repulsion. Of course, if the force to
+be overcome from <i>Q</i> to <i>P</i> 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 <i>Q</i> is higher than that of <i>P</i>.</p>
+
+<p>The difference of potential depends only on the
+points <i>P</i> and <i>Q</i>, and not at all on the path pursued
+between them. Thus, if a unit of electricity be
+carried from <i>P</i> to <i>Q</i> 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 <i>mutatis
+mutandis</i> of gravitational action.</p>
+
+<p>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 <i>sinks</i> (or negative sources),&mdash;the negative point-charges
+in the electrical case,&mdash;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<span class='pagenum'><a name="Page_36" id="Page_36">36</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.<span class='pagenum'><a name="Page_37" id="Page_37">37</a></span></p>
+
+<div class="figright" style="width: 300px; position: relative;">
+<a name="f1" id="f1"></a>
+<img src="images/fig01.png" width="300" height="179" alt="Fig. 1." title="" />
+<p class="caption"><span class="smcap">Fig. 1.</span></p>
+</div>
+
+<p>Now let heat be poured into the medium at constant
+rate by a single point-source <i>P</i> (Fig. <a href="#f1">1</a>), and drawn off
+at a smaller rate by a single point-sink <i>P'</i>, 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 <i>S</i> 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 <i>a</i> be the radius of the
+sphere and <i>f</i> the distance of the source from the centre
+of the sphere, the heat carried off by the sink is the
+fraction <i>a</i>&nbsp;&frasl;&nbsp;<i>f</i> of that given out by the source.</p>
+
+<p>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<span class='pagenum'><a name="Page_38" id="Page_38">38</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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. <a href="#Page_43">43</a> 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<span class='pagenum'><a name="Page_39" id="Page_39">39</a></span>
+are willing to acquire, some elementary knowledge of
+electricity.</p>
+
+<p>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&mdash;the analogue of the thermal
+source&mdash;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."</p>
+
+<p>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<span class='pagenum'><a name="Page_40" id="Page_40">40</a></span>
+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.</p>
+
+<div class="figcenter" style="width: 550px;">
+<a name="f2" id="f2"></a>
+<img src="images/fig02.png" width="550" height="276" alt="Fig. 2." title="" />
+<p class="caption"><span class="smcap">Fig. 2.</span></p>
+</div>
+
+
+<div class="figright" style="width: 300px; position: relative;">
+<a name="f3" id="f3"></a>
+<img src="images/fig03.png" width="300" height="224" alt="Fig. 3." title="" />
+<p class="caption"><span class="smcap">Fig. 3.</span></p>
+</div>
+
+<p>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 <i>A</i> and <i>B</i> (Fig. <a href="#f2">2</a>), and
+regard them for the moment as mirrors. A first image
+of an object <i>P</i> between the two mirrors is produced
+directly by each, but the image <i>I</i><sub>1</sub> in <i>A</i> is virtually an
+object in front of <i>B</i>, and the image <i>J</i><sub>1</sub> in <i>B</i> an object<span class='pagenum'><a name="Page_41" id="Page_41">41</a></span>
+in front of <i>A</i>, so that a second image more remote
+from the mirror than the first is produced in each case.
+These second images <i>I</i><sub>2</sub> and <i>J</i><sub>2</sub> 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.</p>
+
+<p>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&deg;, let us say 60&deg;, the electrified point and the
+images are situated, just as are the object and its image<span class='pagenum'><a name="Page_42" id="Page_42">42</a></span>
+in Brewster's kaleidoscope, namely at the angular points
+of a hexagon, the sides of which are alternately (as
+shown in Fig. <a href="#f3">3</a>) of lengths twice the distance of the
+electrified point from <i>A</i> and from <i>B</i>.</p>
+
+<div class="figcenter" style="width: 400px;">
+<a name="f4" id="f4"></a>
+<img src="images/fig04.png" width="400" height="195" alt="Fig. 4." title="" />
+<p class="caption"><span class="smcap">Fig. 4.</span></p>
+</div>
+
+<p>Now consider the spherical surface referred to at
+p. <a href="#Page_37">37</a>, which is kept at uniform potential by a charge
+at the external point <i>P</i>, and a charge <i>q'</i> at the inverse
+point <i>P'</i> within the sphere. If <i>E</i> (Fig. <a href="#f4">4</a>) be any
+point whatever on the surface, and <i>r</i>, <i>r'</i> be its distances
+from <i>P</i> and <i>P'</i>, it is easy to prove by geometry that
+the two triangles <i>CPE</i> and <i>CEP'</i> are similar, and
+therefore <i>r'</i>&nbsp;=&nbsp;<i>ra</i>&nbsp;&frasl;&nbsp;<i>f</i>. [Here <i>a</i>&nbsp;&frasl;&nbsp;<i>f</i> is used to mean <i>a</i>
+divided by <i>f</i>. The mark &nbsp;&frasl;&nbsp; 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 <i>q</i> at
+another point, is equal to the ratio of the charge <i>q</i> to
+the distance between the points. Thus the potential
+at <i>E</i> due to the charge <i>q</i> at <i>P</i> is
+<i>q</i>&nbsp;&frasl;&nbsp;<i>r</i>,
+and that at <i>E</i> due to a charge <i>q'</i> at <i>P'</i> is
+<i>q'</i>&nbsp;&frasl;&nbsp;<i>r'</i>.
+Thus if <i>q'</i>&nbsp;=&nbsp;&minus;&nbsp;<i>qa</i>&nbsp;&frasl;&nbsp;<i>f</i>,
+<i>q'</i> at <i>P'</i> will produce a potential at
+<i>E</i>&nbsp;=&nbsp;&minus;&nbsp;<i>qa</i>&nbsp;&frasl;&nbsp;<i>fr'</i>&nbsp;=&nbsp;&minus;&nbsp;<i>q</i>&nbsp;&frasl;&nbsp;<i>r</i>,
+by the value of <i>r</i>. Hence <i>q</i> at <i>P</i> and &minus;&nbsp;<i>qa</i>&nbsp;&frasl;&nbsp;<i>f</i> at <i>P'</i>
+coexisting will give potential <i>q</i>&nbsp;&frasl;&nbsp;<i>r</i>&nbsp;&#43;&nbsp;&minus;&nbsp;<i>q</i>&nbsp;&frasl;&nbsp;<i>r</i> or zero,
+at <i>E</i>. Thus the charge &minus;&nbsp;<i>qa</i>&nbsp;&frasl;&nbsp;<i>f</i>, at the internal point<span class='pagenum'><a name="Page_43" id="Page_43">43</a></span>
+<i>P'</i> will in presence of &#43;&nbsp;<i>q</i> at <i>P</i> keep all points of the
+spherical surface at zero potential. These two charges
+represent the source and sink in the thermal analogue
+of p. <a href="#Page_37">37</a> above.</p>
+
+<p>Now replace <i>S</i> 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 <i>q</i> at <i>P</i>
+alone, a charge is induced upon it which, in presence
+of <i>P</i>, maintains it at zero potential. The internal
+charge &minus;&nbsp;<i>qa</i>&nbsp;&frasl;&nbsp;<i>f</i>, 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 <i>q</i> at <i>P</i>.
+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
+<i>external</i> 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.</p>
+
+<p>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 <i>N</i>, and <i>s</i> be the
+surface-density: then according to the system of units
+usually adopted 4&#960;<i>s</i>&nbsp;=&nbsp;<i>N</i>.</p>
+
+<p>Let now the rate of diminution of potential per unit of<span class='pagenum'><a name="Page_44" id="Page_44">44</a></span>
+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 4&#960;. 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.</p>
+
+<p>Thus it is only necessary to find for &minus;&nbsp;<i>qa</i>&nbsp;&frasl;&nbsp;<i>f</i> at <i>P'</i>
+and <i>q</i> at <i>P</i>, the falling off gradient <i>N</i> of potential outside
+the spherical surface at any point <i>E</i>, and to take
+<i>N</i>&nbsp;&frasl;&nbsp;4<i>&#960;</i>,
+to obtain <i>s</i> the surface-density at <i>E</i>. Calculation of this
+gradient for the sphere gives
+4&#960;<i>s</i>&nbsp;=&nbsp;&minus;&nbsp;<i>q</i>&nbsp;(<i>f</i><sup>2</sup>&nbsp;&minus;&nbsp;<i>a</i><sup>2</sup>)&nbsp;&frasl;&nbsp;<i>ar</i><sup>3</sup>.
+The surface-density is thus inversely as the cube of the
+distance <i>PE</i>.</p>
+
+<p>If the influencing point <i>P</i> 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 <i>P'</i> will now be outside,
+but given by the same relation. And <i>a</i> will now
+be greater than <i>f</i>, and the density will be given by
+4&#960;<i>s</i>&nbsp;=&nbsp;&minus;&nbsp;<i>q</i>&nbsp;(<i>a</i><sup>2</sup>&nbsp;&minus;&nbsp;<i>f</i><sup>2</sup>)&nbsp;&frasl;&nbsp;<i>ar</i><sup>3</sup>, where, <i>f</i> and <i>r</i> have the same
+meanings with regard to <i>E</i> and <i>P</i> as before.</p>
+
+<p><i>P'</i> is in each case called the image of <i>P</i> in the
+sphere <i>S</i>, and the charge &minus;&nbsp;<i>qa</i>&nbsp;&frasl;&nbsp;<i>f</i> there supposed situated
+is the <i>electric image</i> of the charge <i>q</i> at <i>P</i>. It will be
+seen that an electric image is a charge, or system of<span class='pagenum'><a name="Page_45" id="Page_45">45</a></span>
+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.</p>
+
+<p>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 <i>D</i> on the other side of the surface, this
+surface distribution may be equivalent to several different
+arrangements of <i>D</i>. Thus the point-charge at <i>P'</i>
+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 <i>P'</i> (Fig. <a href="#f4">4</a>) would do as well,
+provided this spherical surface were not large enough
+to extend beyond the surface <i>S</i>.</p>
+
+<p>In order to find the potential of the sphere (Fig. <a href="#f4">4</a>)
+when insulated with a charge <i>Q</i> upon it, in presence
+of the influencing charge <i>q</i> at the external point <i>P</i>, it
+is only necessary to imagine uniformly distributed over
+the sphere, already electrified in the manner just
+explained, the charge <i>Q</i>&nbsp;&#43;&nbsp;<i>aq</i>&nbsp;&frasl;&nbsp;<i>f</i>. Then the whole
+charge will be <i>Q</i>, and the uniformity of distribution
+will be disturbed, as required by the action of the
+influencing point-charge. The potential will be
+<i>Q</i>&nbsp;&frasl;&nbsp;<i>a</i>&nbsp;&#43;&nbsp;<i>q</i>&nbsp;&frasl;&nbsp;<i>f</i>. For a given potential <i>V</i> of the sphere,
+the total charge is <i>aV</i>&nbsp;&minus;&nbsp;<i>aq</i>&nbsp;&frasl;&nbsp;<i>f</i>, that is the charge is <i>aV</i>
+over and above the induced charge.</p>
+
+<p>If instead of a single influencing point-charge at <i>P</i>
+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<span class='pagenum'><a name="Page_46" id="Page_46">46</a></span>
+that obtained by superimposing all the surface distributions
+found for the different influencing points.</p>
+
+<p>The force of repulsion between the point-charge <i>q</i>
+and the sphere (with total charge <i>Q</i>) can be found at
+once by calculating the sum of the forces between <i>q</i> at
+<i>P</i> and the charges <i>Q</i>&nbsp;&#43;&nbsp;<i>aq</i>&nbsp;&frasl;&nbsp;<i>f</i> at <i>C</i> and &minus;&nbsp;<i>aq</i>&nbsp;&frasl;&nbsp;<i>f</i> at <i>P'</i>.</p>
+
+<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 <i>Q</i> uninfluenced by an external charge, one
+representing the energy on a small conductor (not a
+point) at <i>P</i> existing alone, and a third representing the
+mutual energy of the electrification on the sphere and
+the charge <i>q</i> at <i>P</i> 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 <i>f</i> by a small amount <i>df</i>. This will be the
+product <i>F&nbsp;.&nbsp;df</i> of the force <i>F</i> required and the displacement.</p>
+
+<p>Either method may be applied to find the forces of
+attraction and repulsion for the systems of electrified
+spheres described below.</p>
+
+<p>The problem of two mutually influencing non-intersecting
+spheres, <i>S</i><sub>1</sub>, <i>S</i><sub>2</sub> (Fig. <a href="#f5">5</a>), insulated with
+given charges, <i>q</i><sub>1</sub>, <i>q</i><sub>2</sub>, 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 <i>S</i><sub>2</sub>, say, be kept unaltered,<span class='pagenum'><a name="Page_47" id="Page_47">47</a></span>
+while that on <i>S</i><sub>1</sub> is allowed to change, the action of
+<i>S</i><sub>2</sub> on <i>S</i><sub>1</sub> is the same as if the charge <i>q</i><sub>2</sub> were at the
+centre <i>C</i><sub>2</sub> of <i>S</i><sub>2</sub>. Thus if <i>f</i> be the distance between
+the centres <i>C</i><sub>1</sub>, <i>C</i><sub>2</sub>, and <i>a</i><sub>1</sub> be the radius of <i>S</i><sub>1</sub>, the
+distribution will be that corresponding to <i>q</i><sub>1</sub>&nbsp;&#43;&nbsp;<i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>f</i>
+uniformly distributed on <i>S</i><sub>1</sub> together with the induced
+charge &minus;&nbsp;<i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>f</i>, which corresponds to the image-charge
+at the point <i>I</i><sub>1</sub> (within <i>S</i><sub>1</sub>), the inverse of <i>C</i><sub>2</sub>
+with respect to <i>S</i><sub>1</sub>. Now let the charge on <i>S</i><sub>1</sub> be
+fixed in the state just supposed while that on <i>S</i><sub>2</sub> is
+freed. The charge on <i>S</i><sub>2</sub> will rearrange itself under
+the influence of <i>q</i><sub>1</sub>&nbsp;&#43;&nbsp;<i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>f</i>&nbsp;(&nbsp;=&nbsp;<i>q'</i>)
+and &minus;&nbsp;<i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>f</i>, considered
+as at <i>C</i><sub>1</sub> and <i>I</i><sub>1</sub> respectively. The former of
+these will give a distribution equivalent to <i>q</i><sub>2</sub>&nbsp;&#43;&nbsp;<i>a</i><sub>2</sub><i>q'</i>&nbsp;&frasl;&nbsp;<i>f</i>
+uniformly distributed over <i>S</i><sub>2</sub>, and an induced distribution
+of amount &minus;&nbsp;<i>a</i><sub>2</sub><i>q'</i>&nbsp;&frasl;&nbsp;<i>f</i> at <i>J</i><sub>1</sub>, the inverse point of <i>C</i><sub>1</sub>
+with regard to <i>S</i><sub>2</sub>. The image-charge &minus;&nbsp;<i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>f</i> at <i>I</i><sub>1</sub>
+in <i>S</i><sub>1</sub> will react on <i>S</i><sub>2</sub> and give an induced distribution
+&minus;&nbsp;<i>a</i><sub>2</sub>&nbsp;(&minus;&nbsp;<i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>f</i>&nbsp;)&nbsp;<i>f'</i>, (<i>I</i><sub>1</sub><i>C</i><sub>2</sub>&nbsp;=&nbsp;<i>f'</i>&nbsp;) corresponding to an
+image-charge <i>a</i><sub>2</sub><i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>ff'</i> at the inverse point <i>J</i><sub>2</sub> of <i>P</i><sub>1</sub>
+with respect to <i>C</i><sub>2</sub><i>S</i><sub>2</sub>. Thus the distribution on <i>S</i><sub>2</sub> is
+equivalent to <i>q</i><sub>2</sub>&nbsp;&#43;&nbsp;<i>a</i><sub>2</sub><i>q'</i>&nbsp;&frasl;&nbsp;<i>f</i>&nbsp;&minus;&nbsp;<i>a</i><sub>2</sub><i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>ff'</i>
+at the inverse point <i>J</i><sub>2</sub> of <i>P</i><sub>1</sub>
+distributed<span class='pagenum'><a name="Page_48" id="Page_48">48</a></span>
+uniformly over it, together with the two induced
+distributions just described.</p>
+
+<div class="figcenter" style="width: 500px; position: relative;"><a name="f5" id="f5"></a><img src="images/fig05.png" width="500" height="245" alt="Fig. 5." title="" />
+<p class="caption"><span class="smcap">Fig. 5.</span></p></div>
+
+<p>In the same way these two induced distributions on
+<i>S</i><sub>2</sub> may now be regarded as reacting on the distribution
+on <i>S</i><sub>1</sub> as would point-charges &minus;&nbsp;<i>a</i><sub>2</sub><i>q</i><sub>1</sub>&nbsp;&frasl;&nbsp;<i>f</i>
+and <i>a</i><sub>2</sub><i>a</i><sub>1</sub><i>q</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>ff'</i>,
+situated at <i>J</i><sub>1</sub> and <i>J</i><sub>2</sub> respectively, and would give two
+induced distributions on <i>S</i><sub>1</sub> corresponding to their
+images in <i>S</i><sub>1</sub>.</p>
+
+<p>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.</p>
+
+<p>If the final potentials, <i>V</i><sub>1</sub>, <i>V</i><sub>2</sub>, 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 <i>a</i><sub>1</sub><i>V</i><sub>1</sub>, <i>a</i><sub>2</sub><i>V</i><sub>2</sub> in the two
+cases. The uniform distribution on <i>S</i><sub>1</sub> will raise the
+potential of <i>S</i><sub>2</sub> above <i>V</i><sub>2</sub>, and to bring the potential
+down to <i>V</i><sub>2</sub> in presence of this distribution we must
+place an induced distribution over <i>S</i><sub>2</sub>, represented as
+regards the external field by the image-charge
+&minus;&nbsp;<i>a</i><sub>2</sub><i>a</i><sub>1</sub><i>V</i><sub>1</sub>&nbsp;&frasl;&nbsp;<i>f</i> (at the image of <i>C</i><sub>1</sub> in <i>S</i><sub>2</sub>) where <i>f</i> is the
+distance between the centres. The charge <i>a</i><sub>2</sub><i>V</i><sub>2</sub> on <i>S</i><sub>2</sub>
+will similarly have an action on <i>S</i><sub>1</sub> to be compensated
+in the same way by an image-charge &minus;&nbsp;<i>a</i><sub>1</sub><i>a</i><sub>2</sub><i>V</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>f</i> at
+the image of <i>C</i><sub>2</sub> in <i>S</i><sub>1</sub>. Now these two image-charges
+will react on the spheres <i>S</i><sub>1</sub> and <i>S</i><sub>2</sub> 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<span class='pagenum'><a name="Page_49" id="Page_49">49</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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 <i>q</i> at <i>P</i> between
+the inner surface of the outer and the outer surface
+of the inner, as shown in Fig. <a href="#f7">7</a>. There the influence
+of <i>q</i> at <i>P</i>, 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.</p>
+
+<p>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. <a href="#f6">6</a>
+<span class='pagenum'><a name="Page_50" id="Page_50">50</a></span>
+the distances <i>OP</i>, <i>OP'</i>, <i>OQ</i>, <i>OQ'</i> fulfil the relation
+<i>OP</i>.<i>OP'</i>&nbsp;=&nbsp;<i>OQ</i>.<i>OQ'</i>&nbsp;=&nbsp;<i>a</i><sup>2</sup>. Thus <i>P'</i> is (see p. <a href="#Page_37">37</a>)
+the inverse of the point <i>P</i> with respect to a sphere of
+radius <i>a</i> and centre <i>O</i> (indicated by the dotted line in
+Fig. <a href="#f6">6</a>), and similarly <i>Q'</i> is the inverse of <i>Q</i> with
+respect to the same sphere and centre. <i>O</i> is called the
+centre of inversion, and the sphere of radius <i>a</i> is called
+the sphere of inversion. Thus the sphere of Figs. <a href="#f1">1</a>
+and <a href="#f4">4</a> is the sphere of inversion for the points <i>P</i> and
+<i>P'</i>, which are inverse points of one another. For any
+system of points <i>P</i>, <i>Q</i>, ..., another system <i>P'</i>, <i>Q'</i>, ...
+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 <i>P'</i>, <i>Q'</i>, ... be regarded as the direct system,
+<i>P</i>, <i>Q</i>, ... will be the corresponding inverse system
+with regard to the same sphere and centre. <i>P'</i> is the
+image of <i>P</i>, and <i>P</i> is the image of <i>P'</i>, and so on, with
+regard to the same sphere and centre of inversion.</p>
+
+<div class="figcenter" style="width: 500px; position: relative;"><a name="f6" id="f6"></a><img src="images/fig06.png" width="500" height="236" alt="Fig. 6." title="" />
+<p class="caption"><span class="smcap">Fig. 6.</span></p></div>
+
+<p>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<span class='pagenum'><a name="Page_51" id="Page_51">51</a></span>
+passing through the centre of inversion. Obviously
+the inverse of a sphere concentric with the sphere of
+inversion is a concentric sphere.</p>
+
+<p>The line <i>P'Q'</i> is of course not the inverse of the line
+<i>PQ</i>, which has for its inverse the circle passing through
+the three points <i>O</i>, <i>P'</i>, <i>Q'</i>, as indicated in Fig. <a href="#f6">6</a>.</p>
+
+<p>The following results are easily proved.</p>
+
+<p>A locus and its inverse cut any line <i>OP</i> at the
+same angle.</p>
+
+<p>To a system of point-charges <i>q</i><sub>1</sub>, <i>q</i><sub>2</sub>, ... at points <i>P</i><sub>1</sub>,
+<i>P</i><sub>2</sub>, ... on one side of the surface of the sphere of inversion
+there is a system of charges <i>aq</i><sub>1</sub>&nbsp;&frasl;&nbsp;<i>f</i><sub>1</sub>, <i>aq</i><sub>2</sub>&nbsp;&frasl;&nbsp;<i>f</i><sub>2</sub>, ... on the
+other side of the spherical surface [<i>OP</i><sub>1</sub>&nbsp;=&nbsp;<i>f</i><sub>1</sub>, <i>OP</i><sub>2</sub>&nbsp;=&nbsp;<i>f</i><sub>2</sub>].
+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.</p>
+
+<p>If <i>V</i>, <i>V'</i> be the potentials produced by the whole
+direct system at <i>Q</i>, and by the whole inverse system
+at <i>Q'</i>, <i>V'</i>&nbsp;&frasl;&nbsp;<i>V</i>&nbsp;=&nbsp;<i>r</i>&nbsp;&frasl;&nbsp;<i>a</i>&nbsp;=&nbsp;<i>a</i>&nbsp;&frasl;&nbsp;<i>r'</i>, where <i>OQ</i>&nbsp;=&nbsp;<i>r</i>, <i>OQ'</i>&nbsp;=&nbsp;<i>r'</i>.</p>
+
+<p>Thus if <i>V</i> is constant over any surface <i>S'</i>, <i>V'</i> is not
+a constant over the inverse surface <i>S'</i>, unless <i>r</i> is a
+constant, that is, unless the surface <i>S'</i> 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.</p>
+
+<p>Further, if <i>q</i> be distributed over an element <i>dS</i> of
+a surface, the inverse charge <i>aq</i>&nbsp;&frasl;&nbsp;<i>f</i> will be distributed
+over the corresponding element <i>dS'</i> of the inverse
+surface. But <i>dS'</i>&nbsp;&frasl;&nbsp;<i>dS</i>&nbsp;=&nbsp;<i>a</i><sup>4</sup>&nbsp;&frasl;&nbsp;<i>f</i><sup>4</sup>&nbsp;=&nbsp;<i>f'</i><sup>4</sup>&nbsp;&frasl;&nbsp;<i>a</i><sup>4</sup> where <i>f</i>, <i>f'</i>
+are the distances of <i>O</i> from <i>dS</i> and <i>dS'</i>. Thus if <i>s</i> be
+the density on <i>dS</i> and <i>s'</i> the inverse density on <i>dS'</i>
+we have <i>s'</i>&nbsp;&frasl;&nbsp;<i>s</i>&nbsp;=&nbsp;<i>a</i><sup>3</sup>&nbsp;&frasl;&nbsp;<i>f'</i><sup>3</sup>&nbsp;=&nbsp;<i>f</i><sup>3</sup>&nbsp;&frasl;&nbsp;<i>a</i><sup>3</sup>.</p>
+
+<p>When <i>V</i> is constant over the direct surface, while<span class='pagenum'><a name="Page_52" id="Page_52">52</a></span>
+<i>r</i> has different values for different directions of <i>OQ</i>,
+the different points of the inverse surface may be
+brought to zero potential by placing at <i>O</i> a charge
+&minus;&nbsp;<i>aV</i>. For this will produce at <i>Q'</i> a potential &minus;&nbsp;<i>aV</i>&nbsp;&frasl;&nbsp;<i>r'</i>
+which with <i>V'</i> will give at <i>Q'</i> a potential zero. This
+shows that <i>V'</i> is the potential of the induced distribution
+on <i>S'</i> due to a charge &minus;&nbsp;<i>aV</i> at <i>O</i>, or that &minus;&nbsp;<i>V'</i> is the
+potential due to the induced charge on <i>S'</i> produced
+by the charge <i>aV</i> at <i>O</i>.</p>
+
+<div class="figleft" style="width: 300px; position: relative;"><a name="f7" id="f7"></a><img src="images/fig07.png" width="300" height="283" alt="Fig. 7." title="" />
+<p class="caption"><span class="smcap">Fig. 7.</span></p>
+</div>
+
+<p>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.</p>
+
+<p>Now take the case of two concentric spheres
+insulated and influenced by a point-charge <i>q</i> placed
+at a point <i>P</i> between them as shown in Fig. <a href="#f7">7</a>. We
+have seen at p. <a href="#Page_49">49</a> how the induced distribution, and the<span class='pagenum'><a name="Page_53" id="Page_53">53</a></span>
+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 <i>S</i><sub>1</sub> and <i>S</i><sub>2</sub>, are respectively
+&minus;&nbsp;<i>q.OA.PB</i>&nbsp;&frasl;&nbsp;(<i>OP.AB</i>), &minus;&nbsp;<i>q.OB.AP</i>&nbsp;&frasl;&nbsp;(<i>OP.AB</i>).</p>
+
+<p>It may be noticed that the sum of the induced
+charges is &minus;&nbsp;<i>q</i>, and that as the radii of the spheres are
+both made indefinitely great, while the distance <i>AB</i> is
+kept finite, the ratios <i>OA</i>&nbsp;&frasl;&nbsp;<i>OP</i>, <i>OB</i>&nbsp;&frasl;&nbsp;<i>OP</i> approximate
+to unity, and the charges to &minus;&nbsp;<i>q.PB</i>&nbsp;&frasl;&nbsp;<i>AB</i>, &minus;&nbsp;<i>q.AP</i>&nbsp;&frasl;&nbsp;<i>AB</i>,
+that is, the charges are inversely as the distances of
+<i>P</i> 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.</p>
+
+<p>Now let this induced distribution, on the two
+concentric spheres, be inverted from <i>P</i> as centre of
+inversion. We obtain two non-intersecting spheres,
+as in Fig. <a href="#f5">5</a>, 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 <i>Electricity and Magnetism</i>, vol. i,
+&sect; 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.<span class='pagenum'><a name="Page_54" id="Page_54">54</a></span></p>
+
+<div class="figcenter" style="width: 250px; position: relative;"><a name="f8" id="f8"></a><img src="images/fig08.png" width="250" height="239" alt="Fig. 8." title="" />
+<p class="caption"><span class="smcap">Fig. 8.</span></p></div>
+
+<p>Again take the case of two parallel infinite planes
+under the influence of a point-charge between them.
+This system inverted from <i>P</i> as centre gives the
+equilibrium distribution on two charged insulated
+spheres in contact (Fig. <a href="#f8">8</a>); 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&deg;&nbsp;&frasl;&nbsp;<i>n</i>,
+where <i>n</i> is a whole number, and are therefore bounded
+in one direction by the straight line which is their
+intersection. The image points <i>I</i><sub>1</sub>, <i>J</i><sub>1</sub>, ..., of <i>P</i>
+placed in the angle between the planes are situated as
+shown in Fig. <a href="#f3">3</a>, and are <i>n</i>&nbsp;&minus;&nbsp;1 in number. This system
+inverted from <i>P</i> 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&deg;. When the planes
+are at right angles the result is that for two<span class='pagenum'><a name="Page_55" id="Page_55">55</a></span>
+perpendicularly intersecting planes, for which Fig. <a href="#f9">9</a>
+gives a diagram.</p>
+
+<div class="figcenter" style="width: 350px; position: relative;"><a name="f9" id="f9"></a><img src="images/fig09.png" width="350" height="419" alt="Fig. 9." title="" />
+<p class="caption"><span class="smcap">Fig. 9.</span></p></div>
+
+<p>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.</p>
+
+<p>As a starting point for this investigation the distribution
+on a thin plane circular disk of radius <i>a</i> is
+required. This can be obtained by considering the disk<span class='pagenum'><a name="Page_56" id="Page_56">56</a></span>
+as a limiting case of an oblate ellipsoid of revolution,
+charged to potential <i>V</i>, say. If Fig. <a href="#f10">10</a> represent the disk
+and <i>P</i> the point at which the density is sought, so that
+<i>CP</i>&nbsp;=&nbsp;<i>r</i>, and <i>CA</i>&nbsp;=&nbsp;<i>a</i>,
+the density is <i>V</i>&nbsp;&frasl;&nbsp;{2&#960;<sup>2</sup>&#8730;(<i>a</i><sup>2</sup>&nbsp;&minus;&nbsp;<i>r</i><sup>2</sup>)}.</p>
+
+<p>The ratio <i>q</i>&nbsp;&frasl;&nbsp;<i>V</i>, of charge to potential, which is
+called the electrostatic capacity of the conductor, is
+thus 2<i>a</i>&nbsp;&frasl;&nbsp;&#960;, that is <i>a</i>&nbsp;&frasl;&nbsp;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&nbsp;&frasl;&nbsp;1.57 of that of a sphere of the same
+radius.</p>
+
+<table width="100%" summary="figs. 10, 11" border="0">
+<tr>
+<td class="t50"><div class="figcenter" style="width: 250px; position: relative;"><a name="f10" id="f10"></a><img src="images/fig10.png" width="250" height="258" alt="Fig. 10." title="" />
+<p class="caption"><span class="smcap">Fig. 10.</span></p></div>
+</td>
+<td class="t50"><div class="figcenter" style="width: 250px; position: relative;"><a name="f11" id="f11"></a><img src="images/fig11.png" width="250" height="274" alt="Fig. 11." title="" />
+<p class="caption"><span class="smcap">Fig. 11.</span></p></div>
+</td>
+</tr>
+</table>
+
+<p>Now invert this disk distribution with any point <i>Q</i>
+as centre of inversion, and with radius of inversion <i>a</i>.
+The geometrical inverse is a segment of a spherical surface
+which passes through <i>Q</i>. The inverse distribution
+is the induced distribution on a conducting shell uninsulated
+and coincident with the segment, and under
+the influence of a charge &minus;&nbsp;<i>aV</i> situated at <i>Q</i> (Fig. <a href="#f11">11</a>).
+Call this conducting shell the "bowl." If the surface-densities
+at corresponding points on the disk and on
+the inverse, say points <i>P</i> and <i>P'</i>, be <i>s</i> and <i>s'</i>, then, as<span class='pagenum'><a name="Page_57" id="Page_57">57</a></span>
+on page <a href="#Page_51">51</a>, <i>s'</i>&nbsp;=&nbsp;<i>sa</i><sup>3</sup>&nbsp;&frasl;&nbsp;<i>QP'</i><sup>3</sup>. If we put in the value of
+<i>s</i> given above, that of <i>s'</i> 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.</p>
+
+<p>If <i>v</i>, <i>v'</i> be the potentials at any point <i>R</i> of space,
+due to the disk and to its image respectively,
+&minus;&nbsp;<i>v'</i>&nbsp;=&nbsp;<i>av</i>&nbsp;&frasl;&nbsp;<i>QR</i>. If then <i>R</i> be coincident with a point
+<i>P'</i> on the spherical segment we have (since then
+<i>v</i>&nbsp;=&nbsp;<i>V</i>) <i>V'</i>&nbsp;=&nbsp;<i>aV</i>&nbsp;&frasl;&nbsp;<i>QP'</i>, which is the potential due to
+the induced distribution caused by the charge &minus;&nbsp;<i>aV</i>
+at <i>Q</i> as already stated.</p>
+
+<p>The fact that the value of <i>s'</i> 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 <i>Q</i> in the same plane but outside
+the bounding circle.</p>
+
+<p>Now consider the two parts of the spherical surface,
+the bowl <i>B</i>, and the remainder <i>S</i> of the spherical
+surface. <i>Q</i> with the charge &minus;&nbsp;<i>aV</i>
+may be regarded
+as situated on the latter part of the surface. Any
+other influencing charges situated on <i>S</i> 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 <i>S</i> be uniformly electrified
+to density <i>s</i>, and held so electrified, the inducing
+distribution will be one given by <i>integration</i> over the
+whole of <i>S</i>, and the bowl <i>B</i> will be at zero potential
+under the influence of this electrification of <i>S</i>, just as
+if <i>B</i> were replaced by a shell of metal connected to<span class='pagenum'><a name="Page_58" id="Page_58">58</a></span>
+the earth by a long fine wire. The densities are equal
+at infinitely near points on the two sides of <i>B</i>.</p>
+
+<p>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 <i>f</i> greater
+than that of the spherical surface, and let this shell be
+rigidly electrified with surface density &minus;&nbsp;<i>s</i>.
+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 &minus;&nbsp;2&#960;<i>fs</i>,
+for the charge is &minus;&nbsp;&#960;<i>f</i><sup>2</sup><i>s</i>, and
+the capacity of the shell is &frac12;<i>f</i>. The potential of the
+bowl will now be zero, and its electrification will just
+neutralise the potential &minus;&nbsp;2&#960;<i>fs</i>, that is, will be exactly
+the free electrification required to produce potential 2&#960;<i>fs</i>.</p>
+
+<p>To find this electrification let the value of <i>f</i> be only
+infinitesimally greater than the diameter of the
+spherical surface of which <i>B</i> is a part; then the
+bowl is under the influence (1) of a uniform electrification
+of density &minus;&nbsp;<i>s</i> 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 <i>S</i> above. It is obvious that by (1) a density
+<i>s</i> 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.</p>
+
+<p>Interesting results are obtained by diminishing <i>S</i>
+more and more until the shell is a complete sphere
+with a circular hole in it. Tabulated results for<span class='pagenum'><a name="Page_59" id="Page_59">59</a></span>
+different relative dimensions of <i>S</i> 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 <i>Electricity and Magnetism</i>, loc. cit.</p>
+
+<p>It is not quite clear whether Thomson discovered
+<i>geometrical</i> 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-m&ecirc;me
+a jadis employ&eacute;e sous le nom de principe des
+<i>images</i>." In Thomson and Tail's <i>Natural Philosophy</i>,
+&sect; 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"
+(<i>Nature</i>, 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<span class='pagenum'><a name="Page_60" id="Page_60">60</a></span>
+opinion most of these discoveries were later than 1845,
+the date of Thomson's first paper.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">10</a></p>
+
+<p>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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_61" id="Page_61">61</a></span></p>
+
+<h3>CHAPTER V</h3>
+
+<h4>THE CHAIR OF NATURAL PHILOSOPHY AT GLASGOW.
+ESTABLISHMENT OF THE FIRST PHYSICAL
+LABORATORY</h4>
+
+<p><span class="smcap">The</span> 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 <i>gens</i>, 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 Amp&egrave;re, 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,<span class='pagenum'><a name="Page_62" id="Page_62">62</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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&mdash;even the professor, if that be
+possible&mdash;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<span class='pagenum'><a name="Page_63" id="Page_63">63</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_64" id="Page_64">64</a></span>
+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, <i>De caloris distributione per terr&aelig; corpus</i>, 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."</p>
+
+<p>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 <i>Litter&aelig; Humaniores</i>. 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.</p>
+
+<div class="figcenter" style="width: 550px; position: relative;"><a name="thomson" id="thomson"></a><img src="images/thomson.jpg" width="550" height="694" alt="Professor WILLIAM THOMSON" title="" />
+<p class="caption"><span class="smcap">Professor WILLIAM THOMSON</span>, 1846</p></div>
+
+<p>The subject chosen had reference no doubt to the<span class='pagenum'><a name="Page_65" id="Page_65">65</a></span>
+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: <i>Ego, Gulielmus
+Thomson, B.A., physicus professor in hac Academia designatus,
+promitto sancteque polliceor me in munere mihi demandato
+studiose fideliterque versaturum.</i>" Professor
+Thomson was then "solemnly admitted and received
+by all the Members present, and took his seat as a
+Member of Faculty."</p>
+
+<p>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
+<i>Mathematical and Physical Papers</i> (vol. i, 1882).</p>
+
+<p>"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, '<i>De Motu Caloris per Terr&aelig; Corpus</i>'<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">11</a>: 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
+<span class='pagenum'><a name="Page_66" id="Page_66">66</a></span>
+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. <a href="#Page_254">XIV.</a>]</p>
+
+<p>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 &pound;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&mdash;not amounting to more than
+&pound;500 or &pound;600 in all&mdash;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_67" id="Page_67">67</a></span>
+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,<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">12</a> the French mathematician, has
+recently thought fit to publish to the scientific world."</p>
+
+<p>Again, in April 1852, the Faculty agree to pay a sum
+of &pound;137 6<i>s.</i> 1&frac12;<i>d.</i> 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
+&pound;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.</p>
+
+<p>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
+<span class='pagenum'><a name="Page_68" id="Page_68">68</a></span>
+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&mdash;but apparatus merely of
+the lecture-illustration kind&mdash;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."<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">13</a></p>
+
+<p>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.!</p>
+
+<p>The <i>University Calendar</i> 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.</p>
+
+<p>"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.<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">14</a>)</p>
+
+<p><span class='pagenum'><a name="Page_69" id="Page_69">69</a></span></p><p>"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.</p>
+
+<p>"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."</p>
+
+<p>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.</p>
+
+<p>[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.]</p>
+
+<p>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&mdash;Gauss, Green,
+Cauchy and others&mdash;he was a fit successor. Not
+knowing all that had been done by these men of genius,<span class='pagenum'><a name="Page_70" id="Page_70">70</a></span>
+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.<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">15</a> 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.</p>
+
+<div class="figcenter" style="width: 550px; position: relative;"><a name="college" id="college"></a><img src="images/old_college.jpg" width="500" height="735" alt="INNER COURT OF THE OLD COLLEGE" title="" />
+<p class="caption"><span class="smcap">INNER COURT OF THE OLD COLLEGE</span><br />
+Showing Natural Philosophy Rooms</p></div>
+
+<p>[In the view of the inner court of the Old College
+given opposite, the windows on the ground-floor to
+<span class='pagenum'><a name="Page_71" id="Page_71">71</a></span>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.]</p>
+
+<p>Here again we may quote from the Bangor address:</p>
+
+<p>"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&mdash;'the electric convection
+of heat'&mdash;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<span class='pagenum'><a name="Page_72" id="Page_72">72</a></span>
+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&mdash;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'&mdash;'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.'"</p>
+
+<p>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 <i>Arts</i>. Here it has meant, since the Chair of
+Natural Philosophy was founded in 1577, and held by
+one of the Regents of the University, <i>Artes Liberales</i> in<span class='pagenum'><a name="Page_73" id="Page_73">73</a></span>
+the widest sense, that is, the study of <i>Litter&aelig; Humaniores</i>
+(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.</p>
+
+<p>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&mdash;and, indeed, had to learn&mdash;in
+an incidental manner how to determine physical constants,
+such as specific gravities, thermal capacities,
+electric resistances, and so forth. For, apart from the
+<i>Relations des Exp&eacute;riences</i> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_74" id="Page_74">74</a></span>
+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.</p>
+
+<p>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 &sect; 118 (Art. XLVIII) he states that the
+electromotive force of a thermoelectric couple of copper<span class='pagenum'><a name="Page_75" id="Page_75">75</a></span>
+and bismuth, at temperatures 0&deg;&nbsp;C. and 100&deg;&nbsp;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_76" id="Page_76">76</a></span>
+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.</p>
+
+<p>The selection and realisation of electrical standards
+was a work of extraordinary importance to the world
+from every point of view&mdash;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_77" id="Page_77">77</a></span>
+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
+Amp&egrave;re 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_78" id="Page_78">78</a></span>
+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. <a href="#Page_88">88</a> below.]</p>
+
+<p>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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_79" id="Page_79">79</a></span></p>
+
+<h3>CHAPTER VI</h3>
+
+<h4>FRIENDSHIP WITH STOKES AND JOULE.
+EARLY WORK AT GLASGOW</h4>
+
+<p><span class="smcap">During</span> 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 <i>Cambridge Mathematical Journal</i>. In
+1846 "Cambridge and Dublin" was substituted for
+"Cambridge" in the title of the <i>Journal</i>, 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.</p>
+
+<p>For the mathematical power and sure physical
+instinct of Stokes Thomson had always the greatest
+admiration. When asked on one occasion who was<span class='pagenum'><a name="Page_80" id="Page_80">80</a></span>
+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."</p>
+
+<p>After the death of Stokes in February 1902, Lord
+Kelvin again referred, in an enthusiastic tribute in
+<i>Nature</i> 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."</p>
+
+<p>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.<span class='pagenum'><a name="Page_81" id="Page_81">81</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_82" id="Page_82">82</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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
+<i>Electrodynamic Qualities of Metals</i>&mdash;the fact that up
+to a certain magnetising force the effect of pull,
+applied to a wire of soft iron, is to increase the<span class='pagenum'><a name="Page_83" id="Page_83">83</a></span>
+magnetisation produced, and for higher magnetising
+forces to diminish it&mdash;was telegraphed to him on the
+night on which the paper was read to the Royal
+Society.</p>
+
+<p>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.</p>
+
+<p>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 <i>Natural Philosophy</i>
+to bring back the current of teaching and thought in
+dynamical science to the ideas of the <i>Principia</i>, 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,<span class='pagenum'><a name="Page_84" id="Page_84">84</a></span>
+within the category of dynamical action between
+material systems.</p>
+
+<p>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.</p>
+
+<p>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 <i>Nature</i> article quoted above,
+p. <a href="#Page_80">80</a>, and the <i>Baltimore Lectures</i>, p. <a href="#Page_101">101</a>). 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<span class='pagenum'><a name="Page_85" id="Page_85">85</a></span>
+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.</p>
+
+<p>A letter of Stokes to Sir John Lubbock, printed in
+the <i>Scientific Correspondence of Sir George Gabriel Stokes</i>,
+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 a&euml;rial vibrations, as<span class='pagenum'><a name="Page_86" id="Page_86">86</a></span>
+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....</p>
+
+<p>"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."</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_87" id="Page_87">87</a></span>
+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
+<i>char-&agrave;-banc</i>, coming slowly up the hill behind him,
+lest it should get broken. But there, comfortably and
+safely seated in the <i>char-&agrave;-banc</i>, was his bride&mdash;the
+sympathetic companion and sharer in his work of after
+years. He had not told me in Section A, or in the<span class='pagenum'><a name="Page_88" id="Page_88">88</a></span>
+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."</p>
+
+<p>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&mdash;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<span class='pagenum'><a name="Page_89" id="Page_89">89</a></span>
+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.</p>
+
+<p>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&mdash;between those in which
+force is concerned and those into which enter only
+the idea of displacement in space and in time&mdash;is
+emphasised in Thomson and Tait's <i>Natural Philosophy</i>,
+which commences with a long chapter devoted entirely
+to kinematics.</p>
+
+<p>Whether Professor Thomson read the whole of the
+Introductory Lecture on the first occasion is uncertain&mdash;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<span class='pagenum'><a name="Page_90" id="Page_90">90</a></span>
+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&mdash;introductory or
+not&mdash;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<span class='pagenum'><a name="Page_91" id="Page_91">91</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_92" id="Page_92">92</a></span>
+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&mdash;of Humanity, Greek, Logic, Mathematics,
+Moral Philosophy and Natural Philosophy&mdash;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.</p>
+
+<p>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 <i>Outlines</i>, 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.</p>
+
+<p>Home exercises were not prescribed as part of the<span class='pagenum'><a name="Page_93" id="Page_93">93</a></span>
+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.</p>
+
+<p>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,<span class='pagenum'><a name="Page_94" id="Page_94">94</a></span>
+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<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">16</a> 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.</p>
+
+<p>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 <i>conf&eacute;rence</i>, 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
+<span class='pagenum'><a name="Page_95" id="Page_95">95</a></span>clearness and accuracy the knowledge he has gained,
+but they should be once a week to be beneficial."</p>
+
+<p>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&mdash;which in many cases must be on points
+which are sufficiently plain to the majority&mdash;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<span class='pagenum'><a name="Page_96" id="Page_96">96</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_97" id="Page_97">97</a></span>
+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&mdash;the professor
+certainly did not&mdash;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 &pound;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 <i>famulus</i> rather than the much<span class='pagenum'><a name="Page_98" id="Page_98">98</a></span>
+more measured respect paid by modern assistants to
+their chiefs.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_99" id="Page_99">99</a></span></p>
+
+<h3>CHAPTER VII</h3>
+
+<h4>THE "ACCOUNT OF CARNOT'S THEORY OF THE
+MOTIVE POWER OF HEAT"&mdash;TRANSITION TO THE
+DYNAMICAL THEORY OF HEAT</h4>
+
+<p><span class="smcap">The</span> 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 <i>Journal de l'&Eacute;cole Polytechnique</i>, 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
+<i>vice versa</i>. 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.<span class='pagenum'><a name="Page_100" id="Page_100">100</a></span>
+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<span class='pagenum'><a name="Page_101" id="Page_101">101</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_102" id="Page_102">102</a></span>
+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&mdash;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&deg;&nbsp;F.
+to 61&deg;&nbsp;F. An approximation to this conclusion was
+contained in the paper which he communicated to the
+British Association at Oxford in 1847.</p>
+
+<p>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.<span class='pagenum'><a name="Page_103" id="Page_103">103</a></span></p>
+
+<p>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.</p>
+
+<p>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&mdash;the specific heat at
+constant pressure and the specific heat at constant
+volume&mdash;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
+<i>p</i> and the increase of volume be <i>dv</i>, the work done
+against the external force is <i>pdv</i>. 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<span class='pagenum'><a name="Page_104" id="Page_104">104</a></span>
+is the specific heat <i>C<sub>p</sub></i> 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 <i>C<sub>v</sub></i>, the specific heat at constant volume. Now
+Mayer assumed that the excess of the specific heat <i>C<sub>p</sub></i>
+above <i>C<sub>v</sub></i> was the thermal equivalent of the work <i>pdv</i>
+done in the former case. Thus he obtained the equation
+<i>J</i>&nbsp;(<i>C<sub>p</sub></i>&nbsp;&minus;&nbsp;<i>C<sub>v</sub></i>)&nbsp;=&nbsp;<i>pdv</i>, where <i>J</i> denotes the dynamical
+equivalent of heat and <i>C<sub>p</sub></i>, <i>C<sub>v</sub></i> are taken in thermal
+units. But if a be the coefficient of expansion of the air
+under constant pressure (that is 1&nbsp;&frasl;&nbsp;273), and <i>v</i><sub>0</sub> be the
+volume of the air at 0&deg;&nbsp;C., we have dv&nbsp;=&nbsp;av<sub>0</sub>, so that
+<i>J</i>&nbsp;(<i>C<sub>p</sub></i>&nbsp;&minus;&nbsp;<i>C<sub>v</sub></i>)&nbsp;=&nbsp;<i>apv</i><sub>0</sub>. Now if <i>p</i> be one atmosphere, say
+1.014&thinsp;&times;&thinsp;10<sup>6</sup> dynes per square centimetre, and the
+temperature be the freezing point of water, the volume
+of a gramme of air is 1&nbsp;&frasl;&nbsp;.001293 in cubic centimetres.
+Hence</p>
+
+<div class="center"><img class="floatInsert22" src="images/f104.png" alt="" title="" />
+</div>
+
+<p>from which, if <i>C<sub>p</sub></i>&nbsp;&minus;&nbsp;<i>C<sub>v</sub></i> is known, the value of <i>J</i> can
+be found.</p>
+
+<p>In Mayer's time the difference of the specific heats
+of air was imperfectly known, and so <i>J</i> 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 <i>C<sub>p</sub></i>&nbsp;&minus;&nbsp;<i>C<sub>v</sub></i> may be taken as .0686.
+Thus <i>J</i> works out to 42.2&nbsp;&times;&nbsp;10<sup>6</sup>, in ergs per calorie,
+which is not far from the true value. Mayer obtained
+a result equivalent to 36.5&nbsp;&times;&nbsp;10<sup>6</sup> ergs per calorie.</p>
+
+<p><span class='pagenum'><a name="Page_105" id="Page_105">105</a></span>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 <i>dW</i> for <i>pdv</i>, the work done, <i>e</i> for the
+internal energy before expansion, and <i>dH</i> for the heat
+given to the gas, we have obviously the equation</p>
+
+<div class="center">
+<i>JdH</i>&nbsp;=&nbsp;<i>de</i>&nbsp;&#43;&nbsp;<i>dW</i>
+</div>
+
+<p>where <i>de</i> 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 <i>dW</i>
+for that case is zero. Let <i>&#8706;</i><i>e</i> and <i>&#8706;</i><i>H</i> be the energy
+change and the heat supplied, then in this case</p>
+
+<div class="center">
+<i>J</i><i>&#8706;</i><i>H</i>&nbsp;=&nbsp;<i>&#8706;</i><i>e</i>&nbsp;&#43;&nbsp;<i>O</i>
+</div>
+
+<p>Thus</p>
+
+<div class="center">
+<i>J</i>&nbsp;(<i>dH</i>&nbsp;&minus;&nbsp;<i>&#8706;</i><i>H</i>)&nbsp;=&nbsp;<i>de</i>&nbsp;&minus;&nbsp;<i>&#8706;</i><i>e</i>&nbsp;&#43;&nbsp;<i>dW</i><br />
+</div>
+
+<p>and the assumption is that <i>de</i>&nbsp;=&nbsp;<i>&#8706;</i><i>e</i>, so that
+<i>dW</i>&nbsp;=&nbsp;<i>J</i>&nbsp;(<i>dH</i>&nbsp;&minus;&nbsp;<i>&#8706;</i><i>H</i>); that is, <i>dW</i>&nbsp;=&nbsp;<i>J</i>&nbsp;(<i>C<sub>p</sub></i>&nbsp;&minus;&nbsp;<i>C<sub>v</sub></i>), when
+the rise of temperature is 1&deg;&nbsp;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<span class='pagenum'><a name="Page_106" id="Page_106">106</a></span>
+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.</p>
+
+<p>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 <i>R&eacute;flexions sur la Puissance
+Motrice du Feu, et sur les Machines Propres &agrave;
+D&eacute;velopper cette Puissance</i>, 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<span class='pagenum'><a name="Page_107" id="Page_107">107</a></span>
+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.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">17</a></p>
+
+<p>The first law of thermodynamics based on Joule's
+proportionality of heat produced to work expended,
+and the converse assumed and verified <i>a posteriori</i>,
+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.</p>
+
+<p>In the Carnot cycle the working substance&mdash;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&mdash;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
+<span class='pagenum'><a name="Page_108" id="Page_108">108</a></span>
+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.</p>
+
+<p>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 &oelig;uvre pour la r&eacute;aliser: sa quantit&eacute; est fix&eacute;e
+uniquement par les temperatures des corps entre
+lesquels se fait, en dernier r&eacute;sultat, 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<span class='pagenum'><a name="Page_109" id="Page_109">109</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_110" id="Page_110">110</a></span>
+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" (<i>Phil. Mag.</i>, May
+1845). Thomson notes this, but adds, "If we do so,
+however, we meet with innumerable other difficulties&mdash;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."</p>
+
+<p>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 <i>Proceedings</i> in 1852.<span class='pagenum'><a name="Page_111" id="Page_111">111</a></span></p>
+
+<p>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."</p>
+
+<p>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 <i>Philosophical
+Magazine</i> for December 1851, to which we shall refer
+later.</p>
+
+<p>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<span class='pagenum'><a name="Page_112" id="Page_112">112</a></span>
+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&deg;&nbsp;C. and condenser at 30&deg;&nbsp;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.</p>
+
+<p>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&mdash;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. <a href="#Page_99">99</a> above), who used it to illustrate an account
+of Carnot's theory.</p>
+
+<p>There appeared in the issue of the Edinburgh<span class='pagenum'><a name="Page_113" id="Page_113">113</a></span>
+<i>Philosophical Transactions</i> 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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_114" id="Page_114">114</a></span></p>
+
+<h3>CHAPTER VIII</h3>
+
+<h4>THERMODYNAMICS AND ABSOLUTE THERMOMETRY</h4>
+
+<p><span class="smcap">The</span> 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 <i>Poggendorff's Annalen</i> in
+March and April 1850, and in the <i>Philosophical
+Magazine</i> 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
+(<i>Pogg. Ann.</i>, Dec. 1854) he stated his fundamental
+principle explicitly in the form: "Heat can never<span class='pagenum'><a name="Page_115" id="Page_115">115</a></span>
+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."</p>
+
+<p>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.</p>
+
+<p>In the <i>Transactions R.S.E.</i>, 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_116" id="Page_116">116</a></span>
+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. <a href="#Page_116">116</a>, the
+other enunciates the principle of Reversibility as a
+criterion of "perfection" of a heat engine. We quote
+these propositions.</p>
+
+<p>"Prop. I (Joule).&mdash;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."</p>
+
+<p>"Prop. II (Carnot and Clausius).&mdash;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."</p>
+
+<p>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.</p>
+
+<p>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.<span class='pagenum'><a name="Page_117" id="Page_117">117</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_118" id="Page_118">118</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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 <i>vice versa</i>: the substance will do work against
+external forces where in the direct action it had work
+done upon it, and <i>vice versa</i>: 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 <i>H</i> taken in at the source into
+work than would the reversible one working between
+the same temperatures. Thus if <i>h</i> be the heat given<span class='pagenum'><a name="Page_119" id="Page_119">119</a></span>
+to the refrigerator by A, and <i>h'</i> that given by B when
+both work directly and take in <i>H</i>; <i>h</i> must be less than
+<i>h'</i>. Then couple the engines together so that B works
+backwards while A works directly. A will take in <i>H</i>
+and deliver <i>h</i>, and do work equivalent to <i>H-h</i>. B
+will take <i>h'</i> from the refrigerator and deliver <i>H</i> to the
+source, and have work equivalent to <i>H-h'</i> spent upon
+it. There will be no heat on the whole given to or
+taken from the source; but heat <i>h'-h</i> will be taken
+from the refrigerator, and work equivalent to this will
+be done. Thus <i>by a cyclical process</i>, 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.</p>
+
+<p>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.</p>
+
+<div class="figcenter" style="width: 450px; position: relative;"><a name="f12" id="f12"></a><img src="images/fig12.png" width="450" height="417" alt="Fig. 12." title="" />
+<p class="caption"><span class="smcap">Fig. 12.</span></p></div>
+
+<p>In the Carnot cycle, the first operation is an isothermal
+expansion (<i>AB</i> in Fig. <a href="#f12">12</a>), in which the substance
+increases in volume by <i>dv</i>, and takes in from<span class='pagenum'><a name="Page_120" id="Page_120">120</a></span>
+the source heat of amount <i>Mdv</i>. The second
+operation is an adiabatic expansion, <i>BC</i>, in which
+the volume is further increased and the temperature
+sinks by <i>dt</i> to the temperature of the refrigerator.
+The third operation is an isothermal compression,
+<i>CD</i>, until the volume and pressure are such that
+an adiabatic compression <i>DA</i> will just bring the
+substance back to the original state. If &#8706;<i>p</i>&nbsp;&frasl;&nbsp;&#8706;<i>t</i> 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 <i>&#8706;p&nbsp;&frasl;&nbsp;&#8706;t&nbsp;.&nbsp;dt</i>; and thus the area
+of the closed cycle in the diagram which measures the
+external work done in the succession of changes is
+<i>&#8706;p&nbsp;&frasl;&nbsp;&#8706;t&nbsp;.&nbsp;dtdv</i>. Now, by the second law, the work done
+must be a certain fraction of the work-equivalent of
+the heat, <i>Mdv</i>, taken in from the source. This
+fraction is independent of the nature of the working
+substance, but varies with the temperature, and is<span class='pagenum'><a name="Page_121" id="Page_121">121</a></span>
+therefore a function of the temperature. Its ratio to
+the difference of temperature <i>dt</i> 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 <i>&#956;</i>, we have the equation</p>
+
+<div class="center"><img class="floatInsert22" src="images/f121.png" alt="" title="" />
+</div>
+
+<p>which may be taken as expressing in mathematical
+language the second law of thermodynamics. <i>M</i> is here
+so chosen that <i>Mdv</i> is the heat expressed in units of
+work, so that &#956; 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.</p>
+
+<p>The work done in the cycle becomes now <i>&#956;Mdtdv</i>,
+or if <i>H</i> denote <i>Mdv</i>, it is <i>&#956;Hdt</i>. The fraction of the
+heat utilised is thus <i>&#956;dt</i>. This is called the <i>efficiency</i> of
+the engine for the cycle.</p>
+
+<p>From the first law Thomson obtained another fundamental
+equation. For every substance there is a relation
+connecting the pressure <i>p</i> (or more generally
+the stress of some type), the volume <i>v</i> (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 <i>v</i> and <i>t</i> as the quantities to be varied,
+and supposed them to sustain arbitrary small changes
+<i>dv</i> and <i>dt</i> in consequence of the passage of heat to the<span class='pagenum'><a name="Page_122" id="Page_122">122</a></span>
+substance from without. The amount of heat taken
+in is <i>Mdv</i>&nbsp;&#43;&nbsp;<i>Ndt</i>, where <i>Mdv</i> and <i>Ndt</i> are heats
+required for the changes taken separately. But the
+substance expanding through <i>dv</i> does external work
+pdv. Thus the net amount of energy given to the
+substance from without is <i>Mdv</i>&nbsp;&#43;&nbsp;<i>Ndt</i>&nbsp;&minus;&nbsp;<i>pdv</i> or
+(<i>M</i>&nbsp;&minus;&nbsp;<i>p</i>)&nbsp;<i>dv</i>&nbsp;&#43;&nbsp;<i>Ndt</i>; 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 <i>M</i>&nbsp;&minus;&nbsp;<i>p</i> when the
+temperature but not the volume varies, is equal to
+the rate of variation of <i>N</i> 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 <i>dv</i>, <i>dt</i> of volume and
+temperature, (2) a variation &minus;&nbsp;<i>dv</i> of volume only,
+(3) a variation &minus;&nbsp;<i>dt</i> of temperature only. <i>M</i>&nbsp;&minus;&nbsp;<i>p</i> and
+<i>N</i> 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 (<i>M</i>&nbsp;&minus;&nbsp;<i>p</i>)&nbsp;<i>dv</i>&nbsp;&#43;&nbsp;<i>Ndt</i> 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.</p>
+
+<p>This result combined with the equation <i>A</i> derived
+from the second law, gives an important expression
+for Carnot's function.</p>
+
+<p>We shall not pursue this discussion further: so
+much is given to make clear how certain results as to<span class='pagenum'><a name="Page_123" id="Page_123">123</a></span>
+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.</p>
+
+<p>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&deg;&nbsp;C. be <i>v</i><sub>0</sub>
+and that at any other temperature be <i>v</i><sub>1</sub> we define the
+numerical value <i>t</i>, this latter temperature, by the equation
+<i>v</i>&nbsp;=&nbsp;<i>v</i><sub>0</sub>&nbsp;(1&nbsp;&#43;&nbsp;<i>Et</i>), where <i>E</i> is 1&nbsp;&frasl;&nbsp;100 of the increase
+of volume sustained by the gas in being raised from
+0&deg;&nbsp;C. to 100&deg;&nbsp;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 <i>t</i> has the value (<i>v</i>&nbsp;&frasl;&nbsp;<i>v</i><sub>0</sub>&nbsp;&minus;&nbsp;1)&nbsp;&frasl;&nbsp;<i>E</i>, and is the temperature
+(on the constant pressure scale of the gas thermometer)
+corresponding to the volume <i>v</i>. Equal<span class='pagenum'><a name="Page_124" id="Page_124">124</a></span>
+differences of temperature are such as correspond to
+equal increments of the volume at 0&deg;&nbsp;C.</p>
+
+<p>Similarly, on the constant volume scale we obtain a
+definition of temperature from the pressure <i>p</i>, by the
+equation <i>t</i>&nbsp;=&nbsp;(<i>p</i>&nbsp;&frasl;&nbsp;<i>p</i><sub>0</sub>&nbsp;&minus;&nbsp;1)&nbsp;&frasl;&nbsp;<i>E'</i>, where <i>p</i><sub>0</sub> is the pressure
+at 0&deg;&nbsp;C., and <i>E'</i> is 1&nbsp;&frasl;&nbsp;100 of the change of pressure
+produced by raising the temperature from 0&deg;&nbsp;C. to
+100&deg;&nbsp;C.</p>
+
+<p>For air <i>E</i> is approximately 1&nbsp;&frasl;&nbsp;273, and thus
+<i>t</i>&nbsp;=&nbsp;273&nbsp;(<i>v</i>&nbsp;&minus;&nbsp;<i>v</i><sub>0</sub>)&nbsp;&frasl;&nbsp;<i>v</i><sub>0</sub>. If we take the case of <i>v</i>&nbsp;=&nbsp;0,
+we get <i>t</i>&nbsp;=&nbsp;&minus;&nbsp;273. Now, although this temperature
+may be inaccessible, we may take it as zero, and the
+temperature denoted by <i>t</i> is, when reckoned from this
+zero, 273&nbsp;&#43;&nbsp;<i>t</i>. This zero is called the absolute zero
+on the constant pressure air thermometer. The value
+of <i>E'</i> is very nearly the same as that of <i>E</i>; 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, <i>E</i> would not differ from <i>E'</i>.</p>
+
+<p>It was suggested to Thomson by Joule, in a letter
+dated December 9, 1848, that the value of <i>&#956;</i> might be
+given by the equation <i>&#956;</i>&nbsp;=&nbsp;<i>JE</i>&nbsp;&frasl;&nbsp;(1&nbsp;&#43;&nbsp;<i>Et</i>). Here we
+take heat in dynamical units, and therefore the factor
+<i>J</i> is not required. With these units Joule's suggestion
+is that <i>&#956;</i>&nbsp;=&nbsp;<i>E</i>&nbsp;&frasl;&nbsp;(1&nbsp;&#43;&nbsp;<i>Et</i>), or with <i>E</i>&nbsp;=&nbsp;1&nbsp;&frasl;&nbsp;273
+<i>&#956;</i>&nbsp;=&nbsp;1&nbsp;&frasl;&nbsp;(273&nbsp;&#43;&nbsp;<i>t</i>), that is, <i>&#956;</i>&nbsp;=&nbsp;1&nbsp;&frasl;&nbsp;<i>T</i> where <i>T</i> is the
+temperature reckoned in centigrade degrees from the
+absolute zero of the constant pressure air thermometer.</p>
+
+<p>The possibility of adopting this value of &#956; 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.
+<span class='pagenum'><a name="Page_125" id="Page_125">125</a></span>
+The heat <i>H</i> absorbed by the air in expanding from
+volume <i>V</i> to another volume <i>V'</i> at constant temperature
+is the integral of <i>Mdv</i> taken from the former
+volume to the latter. But by the value of <i>M</i> given
+on p. <a href="#Page_121">121</a>, if <i>W</i> be the integral of <i>pdv</i>, that is the work
+done by the air in the expansion, &#8706;<i>W</i>&nbsp;&frasl;&nbsp;&#8706;<i>t</i>&nbsp;=&nbsp;&#956;<i>H</i>.
+The equation fulfilled by the gas at constant pressure
+(the defining equation for <i>t</i>), <i>v</i>&nbsp;=&nbsp;<i>v</i><sub>0</sub>&nbsp;(1&nbsp;&#43;&nbsp;<i>Et</i>),
+gives for the integral of <i>pdv</i>, that is <i>W</i>, the
+equation <i>W</i>&nbsp;=&nbsp;<i>pv</i><sub>0</sub>&nbsp;(1&nbsp;&#43;&nbsp;<i>Et</i>)&nbsp;log&nbsp;(<i>V'</i>&nbsp;&frasl;&nbsp;<i>V</i>), so that
+&#8706;<i>W</i>&nbsp;&frasl;&nbsp;&#8706;<i>t</i>&nbsp;=&nbsp;<i>EW</i>&nbsp;&frasl;&nbsp;(1&nbsp;&#43;&nbsp;<i>Et</i>). Thus &#956;<i>H</i>&nbsp;=&nbsp;<i>EW</i>&nbsp;&frasl;&nbsp;(1&nbsp;&#43;&nbsp;<i>Et</i>).</p>
+
+<p>Hence it follows that if &#956;&nbsp;=&nbsp;<i>E</i>&nbsp;&frasl;&nbsp;(1&nbsp;&#43;&nbsp;<i>Et</i>), the value
+of <i>H</i> will be simply <i>W</i>. Thus Joule's suggested value
+of &#956; 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.</p>
+
+<p>This result naturally suggests the formation of a
+new scale of thermometry by the adoption of the
+defining relation <i>T</i>&nbsp;=&nbsp;1&nbsp;&frasl;&nbsp;&#956;, where <i>T</i> 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 <i>Philosophical Transactions</i> 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&deg;&nbsp;C. and 100&deg;&nbsp;C.
+by properly fixing the unit of <i>T</i>. This scale was the
+natural successor, in the dynamical theory, of one
+which Thomson had suggested in 1848, and which<span class='pagenum'><a name="Page_126" id="Page_126">126</a></span>
+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 <i>absolute</i> scale,
+since it would be independent of the physical properties
+of any substance. In the same sense the scale defined
+by <i>T</i>&nbsp;=&nbsp;1&nbsp;&frasl;&nbsp;&#956; is truly an absolute scale.</p>
+
+<p>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 <i>H</i> taken
+in from the source in the isothermal expansion which
+forms the first operation of the cycle (p. <a href="#Page_120">120</a>) is <i>Mdv</i>,
+and, as we have seen, the work done in the cycle is
+<i>&#8706;</i><i>p</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>t</i>&nbsp;.&nbsp;<i>dtdv</i>, or &#956;<i>Hdt</i>. If we adopt the expression 1&nbsp;&frasl;&nbsp;<i>T</i>
+for &#956;, we may put <i>dT</i> for <i>dt</i>; and we obtain for the
+work done the expression <i>HdT</i>&nbsp;&frasl;&nbsp;<i>T</i>. The work done
+is thus the fraction <i>dT</i>&nbsp;&frasl;&nbsp;<i>T</i> of the heat taken in, and
+this is what is properly called the efficiency of the
+engine for the cycle.</p>
+
+<p>If we suppose the difference of temperatures between
+source and refrigerator to be finite, <i>T</i>&nbsp;&minus;&nbsp;<i>T'</i>, say, then
+since <i>T</i> is the temperature of the source, we have for
+the efficiency (<i>T</i>&nbsp;&minus;&nbsp;<i>T'</i>)&nbsp;&frasl;&nbsp;<i>T</i>. If the heat taken in be
+<i>H</i>, the heat rejected is <i>HT'</i>&nbsp;&frasl;&nbsp;<i>T</i>, so that the heat
+received by the engine is to the heat rejected by it in
+the ratio of <i>T'</i> to <i>T</i>. 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,<span class='pagenum'><a name="Page_127" id="Page_127">127</a></span>
+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.</p>
+
+<p>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 <i>v</i><sub>1</sub>, <i>v</i><sub>2</sub> 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 <i>dm</i> from the first state to the
+second. Thus <i>dv</i>, the change of volume, is <i>dm</i>&nbsp;(<i>v</i><sub>2</sub>&nbsp;&minus;&nbsp;<i>v</i><sub>1</sub>).
+Also if <i>L</i> be the latent heat of the substance in the
+second state, <i>e.g.</i> the latent heat of water, <i>Mdv</i>&nbsp;=&nbsp;<i>Ldm</i>;
+so that <i>M</i>&nbsp;(<i>v</i><sub>2</sub>&nbsp;&minus;&nbsp;<i>v</i><sub>1</sub>)&nbsp;=&nbsp;<i>L</i>. If <i>dp</i> be the step of pressure
+corresponding to the step <i>dT</i> of temperature, equation
+(A) becomes</p>
+
+<div class="center"><img class="floatInsert22" src="images/f127.png" alt="" title="" />
+</div>
+
+<p>In the case of coexistence of the liquid and solid
+phases, this gives us the very remarkable result that a
+change of pressure <i>dp</i> 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, <i>dT</i>,
+according as <i>v</i><sub>2</sub> is greater or less than <i>v</i><sub>1</sub> Thus a
+substance like water, which expands in freezing, so that
+<i>v</i><sub>2</sub>&nbsp;&minus;&nbsp;<i>v</i><sub>1</sub> 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. <a href="#Page_113">113</a> above).
+On the other hand, a substance like paraffin wax,<span class='pagenum'><a name="Page_128" id="Page_128">128</a></span>
+which contracts in solidifying, would have its melting
+point raised by increase of pressure and lowered by a
+diminution of pressure.</p>
+
+<p>The same conclusions would be applicable when the
+phases are liquid and vapour of the same substance, if
+there were any case in which <i>v</i><sub>2</sub>&nbsp;&minus;&nbsp;<i>v</i><sub>1</sub> 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.</p>
+
+<p>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,
+<i>dp</i>&nbsp;&frasl;&nbsp;<i>dT</i> can be calculated, or obtained graphically from
+a curve; and the volume <i>v</i><sub>2</sub> per unit mass of the
+vapour will be given in terms of <i>dp</i>&nbsp;&frasl;&nbsp;<i>dT</i>, the temperature
+<i>T</i>, and the volume <i>v</i> 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 <i>v</i><sub>2</sub> for any substance for
+which the relation between <i>p</i> and <i>T</i> is known. It is
+thus that the density of saturated steam can best be found.</p>
+
+<p>We can obtain another important result for the case
+of the working substance in two phases from equation
+(B). The relation is</p>
+
+<div class="center"><img class="floatInsert22" src="images/f128.png" alt="" title="" />
+</div>
+
+<p><span class='pagenum'><a name="Page_129" id="Page_129">129</a></span></p>
+
+<p>where <i>c</i> and <i>h</i> are the specific heats of the substance in
+the two phases respectively, and <i>L</i> is the latent heat of
+the second phase at absolute temperature <i>T</i>.</p>
+
+<p>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 <i>T</i> in the lower phase, and to be changed
+to the other phase at that temperature. The heat
+taken in will be <i>L</i> and the expansion will be <i>v</i><sub>2</sub>&nbsp;&minus;&nbsp;<i>v</i><sub>1</sub>.
+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 <i>dT</i>. The heat given
+out by the substance will be <i>hdT</i>, where <i>h</i> is the
+specific heat of the substance in the second phase.
+Now at the new temperature <i>T</i>&nbsp;&minus;&nbsp;<i>dT</i> let the substance
+be wholly brought back to the second phase; the heat
+given out will be <i>L</i>&nbsp;&minus;&nbsp;<i>&#8706;</i><i>L</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>T</i>&nbsp;.&nbsp;<i>dT</i>. Finally, let the
+substance, now again all in the first phase, be brought
+to the original temperature: the heat taken in will be
+<i>cdt</i>, where <i>c</i> is the specific heat in the first phase.
+Thus the net excess of heat taken in over heat given
+<span class='pagenum'><a name="Page_130" id="Page_130">130</a></span>
+out in the cycle is (<i>&#8706;</i><i>L</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>T</i>&nbsp;&#43;&nbsp;<i>c</i>&nbsp;&minus;&nbsp;<i>h</i>)&nbsp;<i>dT</i>. This must,
+in the indicator diagram for the changes specified, be
+the area of the cycle or (<i>v</i><sub>2</sub>&nbsp;&minus;&nbsp;<i>v</i><sub>1</sub>)&nbsp;<i>&#8706;</i><i>p</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>T</i>&nbsp;.&nbsp;<i>dT</i>. But by
+equation (B) <i>L</i>&nbsp;&frasl;&nbsp;<i>T</i>&nbsp;(<i>v</i><sub>2</sub>&nbsp;&minus;&nbsp;<i>v</i><sub>1</sub>)&nbsp;=&nbsp;<i>&#8706;</i><i>p</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>T</i>, and the area
+of the cycle is (<i>L</i>&nbsp;&frasl;&nbsp;<i>T</i>)&nbsp;<i>dT</i>. Equating the two
+expressions thus found for the area we get equation (C).</p>
+
+<p>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.</p>
+
+<p>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 <i>L</i>&nbsp;&frasl;&nbsp;<i>T</i> and <i>&#8706;</i><i>L</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>T</i>
+to be calculated, and therefore that of <i>h</i> to be found.
+It appears that <i>h</i> 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.</p>
+
+<p>It is convenient to notice here the article on <i>Heat</i>
+which Thomson wrote for the ninth edition of the
+<i>Encyclop&aelig;dia Britannica</i>. In that article he gave a
+valuable discussion of ordinary thermometry, of thermometry
+by means of the pressures of saturated vapour of
+different substances&mdash;steam-pressure thermometers, he
+called them&mdash;of absolute thermodynamic thermometry,<span class='pagenum'><a name="Page_131" id="Page_131">131</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Now let the instrument contain in its bulb and
+stem, up to the mark 0, <i>N</i> degree-spaces, and let <i>v</i> be<span class='pagenum'><a name="Page_132" id="Page_132">132</a></span>
+the volume of a degree-space at that temperature. The
+volume up to the mark 0 will be <i>Nv</i>, at that temperature;
+and if the substance of the vessel be quite uniform
+in quality and free from stress, <i>N</i> will be the same for
+all temperatures. If <i>v</i><sub>0</sub> be the volume of a degree-space
+at the temperature of melting ice the volume of the
+mercury at that temperature will be <i>Nv</i><sub>0</sub>. If <i>G</i> be
+the expansion of the glass when the volume of a
+degree-space is increased from <i>v</i><sub>0</sub> to <i>v</i> by the rise
+of temperature, then <i>v</i>&nbsp;=&nbsp;<i>v</i><sub>0</sub>&nbsp;(1&nbsp;&#43;&nbsp;<i>G</i>). The volume
+of the mercury has been increased therefore to
+(<i>N</i>&nbsp;&#43;&nbsp;<i>n</i>)&nbsp;<i>v</i><sub>0</sub>&nbsp;(1&nbsp;&#43;&nbsp;<i>G</i>) 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 <i>n</i> degree-spaces more than before.
+But if <i>E</i> 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
+<i>Nv</i><sub>0</sub>&nbsp;(1&nbsp;&#43;&nbsp;<i>E</i>). Hence <i>N</i>&nbsp;(1&nbsp;&#43;&nbsp;<i>E</i>)&nbsp;=&nbsp;(<i>N</i>&nbsp;&#43;&nbsp;<i>n</i>)&nbsp;(1&nbsp;&#43;&nbsp;<i>G</i>).
+This gives <i>n</i>&nbsp;=&nbsp;<i>N</i>&nbsp;(<i>E</i>&nbsp;&minus;&nbsp;<i>G</i>)&nbsp;&frasl;&nbsp;(1&nbsp;&#43;&nbsp;<i>G</i>).</p>
+
+<p>If we take, as is usual, <i>n</i> as measuring the temperature,
+and substitute for it the symbol <i>t</i>, we have, since
+<i>N</i>&nbsp;=&nbsp;100&nbsp;(1&nbsp;&#43;&nbsp;<i>G</i><sub>100</sub>)&nbsp;&frasl;&nbsp;(<i>E</i><sub>100</sub>&nbsp;&minus;&nbsp;<i>G</i><sub>100</sub>),</p>
+
+<div class="center">
+<img class="floatInsert22" src="images/f132.png" alt="" title="" />
+</div>
+
+<p>In this reckoning the definition of any temperature, let
+us say 37&deg;&nbsp;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<span class='pagenum'><a name="Page_133" id="Page_133">133</a></span>
+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&deg;&nbsp;C.
+or 100&deg;&nbsp;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.</p>
+
+<p>In his article on <i>Heat</i> Thomson gave two definitions
+of the scale of absolute temperature. One is that
+stated on p. <a href="#Page_126">126</a> 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.</p>
+
+<p>The other definition is better adapted for general use,
+as it applies to any cycle whatever which is reversible.<span class='pagenum'><a name="Page_134" id="Page_134">134</a></span>
+Let the working substance expand under constant
+pressure by an amount <i>dv</i> (<i>AB'</i> in Fig. <a href="#f12">12</a>), and let
+heat <i>H</i> be given to the substance at the same time.
+The external work done is <i>pdv</i>. Thomson called
+<i>pdv</i>&nbsp;&frasl;&nbsp;<i>H</i> the work ratio. Now let the temperature be
+raised by <i>dT</i> without giving heat to the substance or
+taking heat from it, and let the corresponding pressure
+rise be <i>dp</i>; and call <i>dp</i>&nbsp;&frasl;&nbsp;<i>p</i> the pressure ratio. The
+temperature ratio <i>dT</i>&nbsp;&frasl;&nbsp;<i>T</i> is equal to the product of the
+work ratio and the pressure ratio, that is,</p>
+
+<div class="center"><img class="floatInsert22" src="images/f134b.png" alt="" title="" />
+</div>
+
+<p>This is clearly true; for <i>dvdp</i> is the area of a cycle
+like <i>AB'C'D</i>, represented in Fig. <a href="#f12">12</a>, for which an
+amount of heat <i>H</i> is taken in, though not in this case
+strictly at one temperature. And clearly, since in
+Fig. <a href="#f12">12</a> the change from <i>B'</i> to <i>B</i> is adiabatic, <i>H</i> is the
+heat which would have to be taken in for the isothermal
+change <i>AB</i> in the Carnot cycle <i>ABCD</i>, which has the
+same area as AB'C'D. Thus the efficiency of the
+cycle is <i>dvdp</i>&nbsp;&frasl;&nbsp;<i>H</i>, and this by the former definition
+is <i>dT</i>&nbsp;&frasl;&nbsp;<i>T</i>.</p>
+
+<p>Or we may regard the matter thus:&mdash;The amount
+of heat <i>H</i> which corresponds to an infinitesimal expansion
+<i>dv</i> may be used in equation (A) whether the
+expansion is isothermal or not, if we take <i>T</i> as the
+average temperature of the expansion. Hence we
+have <i>dp</i>&nbsp;&frasl;&nbsp;<i>dT</i>&nbsp;=&nbsp;<i>H</i>&nbsp;&frasl;&nbsp;(<i>dv</i>.<i>T</i>), that is, <i>dT</i>&nbsp;&frasl;&nbsp;<i>T</i>&nbsp;=&nbsp;<i>dpdv</i>&nbsp;&frasl;&nbsp;<i>H</i>.
+The theorem on p. <a href="#Page_128">128</a> is obtained by what is virtually
+this process.<span class='pagenum'><a name="Page_135" id="Page_135">135</a></span></p>
+
+<h3><span class="smcap">Comparison of Absolute Scale with Scale of
+Air Thermometer</span></h3>
+
+<p>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 <i>de</i>&nbsp;&minus;&nbsp;<i>&#8706;</i><i>e</i> of p. <a href="#Page_107">107</a> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_136" id="Page_136">136</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>The theory of the matter is set forth in the original
+papers, and in a very elegant manner in the article on
+<i>Heat</i>. The result of the analysis shows that if <i>&#8706;</i><i>w</i> 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 <i>T</i> be absolute temperature,
+and <i>v</i> volume of a gramme of the gas at pressure <i>p</i>, and
+the difference of pressure on the two sides of the plug
+be <i>dp</i>, the equation which holds is</p>
+
+<div class="center">
+<a name="FNanchor_18_18" id="FNanchor_18_18"></a>
+  <a href="#Footnote_18_18" class="fnanchor">
+    <img class="floatInsert35" src="images/136.png" alt="" title="" />
+  </a>
+</div>
+
+<p  class="after">It was found by Joule and Thomson that <i>&#8706;</i><i>w</i> was
+proportional to <i>dp</i> for values of <i>dp</i> 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&deg; or 5&deg; centigrade, and .155 at
+temperatures of from 89&deg; to 93&deg; centigrade for a
+difference of pressure due to 100 inches of mercury.</p>
+
+<p>If there is neither heating nor cooling <i>&#8706;</i><i>w</i>&nbsp;=&nbsp;0, and
+we obtain by integration <i>T</i>&nbsp;=&nbsp;<i>Cv</i>, where <i>C</i> is a
+constant.</p>
+
+<p><span class='pagenum'><a name="Page_138" id="Page_138">138</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Another idea, that of <i>thermodynamic motivity</i>, 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
+<span class='pagenum'><a name="Page_139" id="Page_139">139</a></span>
+be equalised by perfect thermodynamic engines working
+between the final temperature <i>T</i><sub>0</sub>, 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 <i>T</i><sub>0</sub> is such
+that exactly the heat given out at <i>T</i><sub>0</sub> by certain engines,
+receiving heat from bodies of higher temperature than
+<i>T</i><sub>0</sub>, is supplied to the engines which work between <i>T</i><sub>0</sub>
+and bodies at lower temperatures. The whole useful
+work obtained in this way was called by Thomson the
+<i>motivity</i> 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
+<i>T</i>, <i>T'</i> the temperature to which they are reduced
+when their motivity is extracted is &#8730;(<i>TT'</i>). If the
+temperatures are equalised by conduction the resulting
+temperature is higher, being &frac12;(<i>T</i>&nbsp;&#43;&nbsp;<i>T'</i>). 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.</p>
+
+<p>A very suggestive paper on the subject was published
+by Lord Kelvin in the <i>Trans. R.S.E.</i>, vol. 28, 1877-8.</p>
+
+<h3><span class="smcap">Dissipation of Energy</span></h3>
+
+<p>In connection with the theory of heat must be
+mentioned Thomson's great generalisation, the theory
+of the dissipation of energy.<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">19</a> Most people have some
+<span class='pagenum'><a name="Page_140" id="Page_140">140</a></span>
+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&mdash;that is, energy of motion and energy
+of relative position (including energy of chemical affinity)
+of the parts&mdash;remains constant. The usual misstatement
+is that the energy of the universe is constant.
+This may be true if the <i>universe</i> 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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_141" id="Page_141">141</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p><span class='pagenum'><a name="Page_142" id="Page_142">142</a></span>
+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.</p>
+
+<p>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.</p>
+
+<h3><span class="smcap">Thermoelasticity and Thermoelectricity</span></h3>
+
+<p>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<span class='pagenum'><a name="Page_143" id="Page_143">143</a></span>
+may substitute <i>dS</i> for <i>dp</i> and <i>Ads</i> for <i>dv</i>; where <i>A</i> is
+such a factor that <i>AdSds</i> is the work done in the displacement
+<i>ds</i> by the stress <i>dS</i>. As an example consider a
+wire subjected to simple longitudinal stress <i>S</i>. Longitudinal
+extension is produced, but this is not the only
+change; there is at the same time lateral contraction.
+However, <i>s</i> within certain limits is proportional to <i>S</i>.</p>
+
+<p>Let heat <i>dH</i> in dynamical measure be given to the
+wire while the stress <i>S</i> is maintained constant, and let
+the extension increase from <i>s</i> to <i>s</i>&nbsp;&#43;&nbsp;<i>ds</i>. The stress <i>S</i>
+will do work <i>ASds</i> <i>on the wire</i>, and the work ratio will
+be &minus;&nbsp;<i>ASds</i>&nbsp;&frasl;&nbsp;<i>dH</i>. Now let the stress be increased to
+<i>S</i>&nbsp;&#43;&nbsp;<i>dS</i> while the extension is kept constant, and the
+absolute temperature raised from <i>T</i> to <i>T</i>&nbsp;&#43;&nbsp;<i>dT</i>. The
+stress ratio (as we may call it) is <i>dS</i>&nbsp;&frasl;&nbsp;<i>S</i> and the temperature
+ratio <i>dT</i>&nbsp;&frasl;&nbsp;<i>T</i>. Thus we obtain (p. <a href="#Page_134">134</a> above)</p>
+
+<div class="center"><img class="floatInsert22" src="images/f143a.png" alt="" title="" />
+</div>
+
+<p>In his <i>Heat</i> article Thomson used the alteration <i>e</i>
+of strain under constant stress (that is <i>ds</i>&nbsp;&frasl;&nbsp;<i>l</i>, where <i>l</i> is
+the length of the wire) corresponding to an amount of
+heat sufficient to raise the temperature under constant
+stress by 1&deg;. Hence if <i>K</i> be the specific heat under
+constant stress, and <i>le</i> be put for <i>ds</i> in the sense just
+stated, we have</p>
+
+<div class="center"><img class="floatInsert25" src="images/f143b.png" alt="" title="" />
+</div>
+
+<p>where &#961; is the density, since <i>dH</i>&nbsp;=&nbsp;<i>K</i>&#961;<i>lA</i>.</p>
+
+<p>The ratio of <i>dH</i> to the increase <i>ds</i> of the extension
+is positive or negative, that is, the substance absorbs
+or evolves heat, when strained under the condition of<span class='pagenum'><a name="Page_144" id="Page_144">144</a></span>
+constant stress, according as <i>dS</i>&nbsp;&frasl;&nbsp;<i>dT</i> 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, <i>ds</i> is negative or
+positive, in other words the wire shortens or lengthens,
+according as <i>dS</i>&nbsp;&frasl;&nbsp;<i>dT</i> is positive or negative, that is,
+according as the stress for a given strain is increased
+or diminished by increase of temperature.</p>
+
+<p>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.</p>
+
+<p>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 <i>ds</i> of <i>s</i> is taken as before
+under constant stress, or at uniform temperature, or
+whether the change <i>dS</i> of <i>S</i> is effected adiabatically or
+at constant strain.</p>
+
+<p>In all these cases the same equation</p>
+
+<div class="center"><img class="floatInsert25" src="images/f144.png" alt="" title="" />
+</div>
+
+<p>applies, with the change of meaning of <i>dT</i> involved.</p>
+
+<p>This equation differs from that of Thomson as
+given in various places (<i>e.g.</i> in the <i>Encyclop&aelig;dia
+Britannica</i> article on <i>Elasticity</i> which he also wrote)
+in the negative sign on the right-hand side, but the<span class='pagenum'><a name="Page_145" id="Page_145">145</a></span>
+difference is only apparent. According to his specification
+a <i>pressure</i> would be a positive stress, and an
+<i>expansion</i> 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 <i>dS</i> 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).</p>
+
+<p>This equation was applied by Thomson (article
+<i>Elasticity</i>) 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.</p>
+
+<p>The augmentation of the strain produced by raising
+the temperature 1&deg; is <i>e</i>, and therefore <i>edT</i>, that is,
+&minus;&nbsp;<i>Te</i><sup>2</sup><i>dS</i>&nbsp;&frasl;&nbsp;<i>K</i>&#961;, is the increase of strain due to the sudden
+rise of temperature <i>dT</i>. This added to the isothermal
+strain produced by <i>dS</i> will give the whole adiabatic
+strain. Thus if <i>M</i> be the static or isothermal modulus,
+the adiabatic strain is <i>dS</i>&nbsp;&frasl;&nbsp;<i>M</i>&nbsp;&minus;&nbsp;<i>Te</i><sup>2</sup><i>dS</i>&nbsp;&frasl;&nbsp;<i>K</i>&#961;. If <i>M'</i>
+denote the kinetic or adiabatic modulus its value
+is <i>dS</i> divided by the whole adiabatic strain, that
+is, <i>M'</i>&nbsp;=&nbsp;<i>M</i>&nbsp;&frasl;&nbsp;(1&nbsp;&minus;&nbsp;<i>MTe</i><sup>2</sup>&nbsp;&frasl;&nbsp;<i>K</i>&#961;) and the ratio
+<i>M'</i>&nbsp;&frasl;&nbsp;<i>M</i>&nbsp;=&nbsp;1&nbsp;&frasl;&nbsp;(1&nbsp;&minus;&nbsp;<i>MTe</i><sup>2</sup>&nbsp;&frasl;&nbsp;<i>K</i>&#961;).</p>
+
+<p>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 <i>K</i> of the substance, under the condition
+of constant stress, to the specific heat <i>N</i> under
+the condition of constant strain of the corresponding<span class='pagenum'><a name="Page_146" id="Page_146">146</a></span>
+type. This, indeed, is self-evident if two changes of
+stress, one isothermal the other adiabatic, <i>which produce
+the same steps of displacement ds</i>, be considered, and it
+be remembered that the step <i>&#8706;</i><i>T</i> of temperature which
+accompanies the adiabatic change may be regarded as
+made up of a step &minus;&nbsp;<i>dT</i> of temperature, accompanying
+a displacement ds effected at constant stress, and then
+two successive steps <i>dT</i> and <i>&#8706;</i><i>T</i> effected, at constant
+strain, along with the steps of stress <i>dS</i>. The ratio
+<i>M'</i>&nbsp;&frasl;&nbsp;<i>M</i> is easily seen to have the value (<i>&#8706;</i><i>T</i>&nbsp;&#43;&nbsp;<i>dT</i>)&nbsp;&frasl;&nbsp;<i>dt</i>,
+and since &minus;&nbsp;<i>KdT</i>&nbsp;&#43;&nbsp;<i>N</i>&nbsp;(<i>&#8706;</i><i>T</i>&nbsp;&#43;&nbsp;<i>dT</i>)&nbsp;=&nbsp;0, by the adiabatic
+condition, the theorem is proved.</p>
+
+<p>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.</p>
+
+<p>Thomson showed in the <i>Elasticity</i> article how, by
+the value of <i>M'</i>&nbsp;&frasl;&nbsp;<i>M</i>, derived as above from thermodynamic
+theory, the value of <i>K</i>&nbsp;&frasl;&nbsp;<i>N</i> 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).</p>
+
+<p>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 <i>Philosophical Magazine</i> for January
+1878. This was in the main a reprint of an article
+entitled, "On the Thermoelastic and Thermomagnetic<span class='pagenum'><a name="Page_147" id="Page_147">147</a></span>
+Properties of Matter, Part I," which appeared in
+April 1855 in the first number of the <i>Quarterly
+Journal of Mathematics</i>. 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 <i>Cyclop&aelig;dia of
+Physical Science</i>, edited and in great part written by
+Professor J. P. Nichol, and published in 1860. For
+the same <i>Cyclop&aelig;dia</i> Thomson also wrote an article entitled,
+"Thermo-electric, Division I.&mdash;Pyro-Electricity,
+or Thermo-Electricity of Non-conducting Crystals,"
+and the enlarged <i>Phil. Mag.</i> article also contained the
+application of thermodynamics to this kind of thermoelectric
+action.</p>
+
+<p>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.</p>
+
+<p>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 <i>Mathematical and Physical Papers</i>.
+This series begins with the Bakerian Lecture (published
+in the <i>Transactions of the Royal Society</i> for
+1856) which includes an account of the remarkable<span class='pagenum'><a name="Page_148" id="Page_148">148</a></span>
+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.</p>
+
+<p>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&mdash;other things being the same&mdash;depends
+on the metals used; for example, bismuth and antimony
+are more effective than other metals.</p>
+
+<p>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<span class='pagenum'><a name="Page_149" id="Page_149">149</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.<span class='pagenum'><a name="Page_150" id="Page_150">150</a></span></p>
+
+<p>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&mdash;the so-called neutral
+temperature&mdash;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.</p>
+
+<p>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,<span class='pagenum'><a name="Page_151" id="Page_151">151</a></span>
+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.</p>
+
+<p>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."</p>
+
+<p>Part VI (<i>Trans. R.S.</i>, 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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_152" id="Page_152">152</a></span>
+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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_153" id="Page_153">153</a></span></p>
+
+<h3>CHAPTER IX</h3>
+
+<h4>HYDRODYNAMICS&mdash;DYNAMICAL THEOREM OF
+MINIMUM ENERGY&mdash;VORTEX MOTION</h4>
+
+<p>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
+<i>molecules</i> of a fluid; even when we deal with infinitesimal
+elements, in the analysis of fluid motion, we are<span class='pagenum'><a name="Page_154" id="Page_154">154</a></span>
+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
+<i>vortices</i> are formed, and hence the name vortex motion
+has been given to all those cases in which the condition
+of irrotationality is not satisfied.</p>
+
+<p>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 <i>Transactions of the Royal Society of
+Edinburgh</i>. 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.</p>
+
+<p>In Chapter VI reference has been made to the
+"Notes on Hydrodynamics" published by Thomson
+in the <i>Cambridge and Dublin Mathematical Journal</i>
+for 1848 and 1849. These Notes were not intended
+to be entirely original, but were composed for the<span class='pagenum'><a name="Page_155" id="Page_155">155</a></span>
+use of students, like Airy's <i>Tracts</i> of fifteen years
+before.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p><span class='pagenum'><a name="Page_156" id="Page_156">156</a></span>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 <i>A</i>, <i>B</i>, within the
+space, to be joined by two lines <i>ACB</i>, <i>ADB</i> both lying
+within the space. These two lines will form a circuit
+<i>ACBDA</i>. 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 <i>A</i> to <i>B</i> round the ring one way, and the
+other from <i>A</i> to <i>B</i> the other way. The circuit could
+not then be contracted towards a point without passing
+out of the ring.</p>
+
+<p>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.</p>
+
+<p>The motion produced has the property described by
+the word "irrotational," that is, the elements of the
+fluid have no spinning motion&mdash;they move without
+rotation. A small portion of a fluid may describe any
+path&mdash;may go round in a circle, for example&mdash;and yet
+have no rotation. The reader may imagine a ball
+carried round in a circle, but in such a way that no<span class='pagenum'><a name="Page_157" id="Page_157">157</a></span>
+line in the body ever changes its direction. The body
+has translation, but no spin.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>This theorem was afterwards generalised by Thomson
+(<i>Proc. R.S.E.</i>, 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<span class='pagenum'><a name="Page_158" id="Page_158">158</a></span>
+by Lagrange in the <i>M&eacute;canique Analytique</i> as a generalisation
+of a theorem given by Euler for a rigid body
+set into rotation by an impulse.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_159" id="Page_159">159</a></span>
+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 <small><sup>1</sup>&frasl;<sub>8</sub></small><i>mv</i><sup>2</sup>, where <i>m</i> denotes the mass of the
+rod and <i>v</i> the velocity of the end. If the motion
+taken were one of rotation about a point of the rod at
+distance <i>x</i> from the end struck, the kinetic energy would
+be <i>m</i>&nbsp;(4<i>l</i><sup>2</sup>&nbsp;&minus;&nbsp;6<i>lx</i>&nbsp;&#43;&nbsp;3<i>x</i><sup>2</sup>)&nbsp;<i>v</i><sup>2</sup>&nbsp;&frasl;&nbsp;6<i>x</i><sup>2</sup>,
+where 2<i>l</i> is the length of
+the rod, and this has its least value <small><sup>1</sup>&frasl;<sub>8</sub></small><i>mv</i><sup>2</sup> for <i>x</i>&nbsp;=&nbsp;4<i>l</i>&nbsp;&frasl;&nbsp;3.
+For example, <i>x</i>&nbsp;=&nbsp;2<i>l</i> gives <small><sup>1</sup>&frasl;<sub>6</sub></small><i>mv</i><sup>2</sup>, which is greater than
+the value just found.</p>
+
+<p>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<span class='pagenum'><a name="Page_160" id="Page_160">160</a></span>
+produced is one-half the product of the impulse into
+the velocity of the point struck, that is &frac12;<i>Iv</i>, and it
+has just been seen that this is the product of <small><sup>1</sup>&frasl;<sub>6</sub></small><i>mv</i><sup>2</sup>
+by the factor (4<i>l</i><sup>2</sup>&nbsp;&minus;&nbsp;6<i>lx</i>&nbsp;&#43;&nbsp;3<i>x</i><sup>2</sup>)&nbsp;&frasl;&nbsp;<i>x</i><sup>2</sup>. This factor is
+3<i>I</i>&nbsp;&frasl;&nbsp;<i>mv</i>, and is a minimum when <i>x</i>&nbsp;=&nbsp;4<i>l</i>&nbsp;&frasl;&nbsp;3. Thus
+for a given <i>I</i>, <i>v</i> will have its maximum value when
+the factor referred to is least, and &frac12;<i>Iv</i> will then be a
+maximum.</p>
+
+<p>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.</p>
+
+<p>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>I</i>&nbsp;&frasl;&nbsp;<i>v</i>. Thus when <i>v</i> is given, <i>I</i> is increased by any
+constraint compelling the rod to rotate about a
+particular axis, and so &frac12;<i>Iv</i>, or the kinetic energy, is
+increased. On the other hand, when <i>I</i> is given the
+same constraint diminishes <i>v</i>, and so &frac12;<i>Iv</i> is diminished.</p>
+
+<p>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<span class='pagenum'><a name="Page_161" id="Page_161">161</a></span>
+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.</p>
+
+<p>With these may be placed another, which in lectures
+he frequently dwelt on&mdash;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.</p>
+
+<p>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&mdash;the
+non-existence of a velocity-potential&mdash;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.<span class='pagenum'><a name="Page_162" id="Page_162">162</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>Thomson's great paper on vortex-motion was read
+before the Royal Society of Edinburgh in 1867, and<span class='pagenum'><a name="Page_163" id="Page_163">163</a></span>
+was recast and augmented in the following year. It
+will be possible to give here only a sketch of its scope
+and main results.</p>
+
+<p>The fluid is supposed contained in a closed fixed
+vessel which is either simply or multiply continuous
+(see p. <a href="#Page_156">156</a>), 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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_164" id="Page_164">164</a></span>
+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 <i>x</i>, <i>y</i>, <i>z</i>, about the axes
+of <i>x</i>, <i>y</i>, <i>z</i> are shown to be &frac12;&nbsp;(<i>&#8706;</i><i>w</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>y</i>&nbsp;&minus;&nbsp;<i>&#8706;</i><i>v</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>z</i>), etc.</p>
+
+<p>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.</p>
+
+<p>Thomson then proves that if the angular momentum<span class='pagenum'><a name="Page_165" id="Page_165">165</a></span>
+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.</p>
+
+<p>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 <i>P</i>, <i>Q</i> in the
+fluid. Let <i>q</i> be the velocity of the fluid at any element
+of length <i>ds</i> of such a path, and <i>&#952;</i> be the angle between
+the direction of <i>ds</i> (taken positive in the sense from <i>P</i> to
+<i>Q</i>) and the direction of <i>q</i>: <i>q</i> cos <i>&#952;</i>&nbsp;.&nbsp;<i>ds</i> is the flow along <i>ds</i>.
+If <i>u</i>, <i>v</i>, <i>w</i> be the components of <i>q</i> at <i>ds</i>, parallel to the
+axes, and <i>dx</i>, <i>dy</i>, <i>dz</i> be the projections of <i>ds</i> on the axes,
+<i>udx</i>&nbsp;&#43;&nbsp;<i>vdy</i>&nbsp;&#43;&nbsp;<i>wdz</i> is the same thing as <i>q</i> cos <i>&#952;</i>&nbsp;.&nbsp;<i>ds</i>.
+The sum of the values of either of these expressions for
+all the elements of the path between <i>P</i> and <i>Q</i> is the
+flow along the path. The statement that <i>u</i>, <i>v</i>, <i>w</i> are
+the space-rates of variation of a function <i>&#966;</i> (of <i>x</i>, <i>y</i>, <i>z</i>)
+parallel to the axes, or that <i>q</i> cos <i>&#952;</i> is the space-rate of<span class='pagenum'><a name="Page_166" id="Page_166">166</a></span>
+variation of <i>&#966;</i> along <i>ds</i>, merely means that this sum is
+the same for whatever path may be drawn from <i>P</i> to <i>Q</i>.
+This, however, is only the case when the paths are
+so taken that in each case the value of <i>&#966;</i> 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.</p>
+
+<p>Since the flow from <i>P</i> to <i>Q</i> is the same for any two
+paths which fulfil this condition, the flow from <i>P</i> to <i>Q</i>
+by any one path and from <i>Q</i> to <i>P</i> 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.</p>
+
+<p>The general theorem which he established may now
+be stated. Consider any path joining <i>PQ</i>, and moving
+with the fluid, so that the line contains always the
+same fluid particles. Let <i>u</i>&#775;, <i>v</i>&#775;, <i>w</i>&#775; be the time-rates of
+change of <i>u</i>, <i>v</i>, <i>w</i> at an element <i>ds</i> of the path, at any
+instant, and <i>du</i>, <i>dv</i>, <i>dw</i> the excesses of the values of <i>u</i>, <i>v</i>,
+<i>w</i> at the terminal extremity of <i>ds</i> above the values at the
+other extremity; then the time-rate of variation of
+<i>udx</i>&nbsp;&#43;&nbsp;<i>vdy</i>&nbsp;&#43;&nbsp;<i>wdz</i> is <i>u</i>&#775;<i>dx</i>&nbsp;&#43;&nbsp;<i>v</i>&#775;<i>dy</i>&nbsp;&#43;&nbsp;<i>w</i>&#775;<i>dz</i>&nbsp;&#43;&nbsp;<i>udu</i>&nbsp;&#43;&nbsp;<i>vdv</i>&nbsp;&#43;&nbsp;<i>wdw</i>
+or <i>u</i>&#775;<i>dx</i>&nbsp;&#43;&nbsp;<i>v</i>&#775;<i>dy</i>&nbsp;&#43;&nbsp;<i>w</i>&#775;<i>dz</i>&nbsp;&#43;&nbsp;<i>qdq</i>, where <i>q</i> has the
+meaning specified above. Thus if <i>S</i> be the flow for
+the whole path <i>PQ</i>, and <i>&#7776;</i> its time-rate of variation, <i>S'</i>
+denote the sum of <i>u</i>&#775;<i>dx</i>&nbsp;&#43;&nbsp;<i>v</i>&#775;<i>dy</i>&nbsp;&#43;&nbsp;<i>w</i>&#775;<i>dz</i> along the path
+from <i>P</i> to <i>Q</i>, and <i>q</i><sub>1</sub>, <i>q</i><sub>0</sub> the resultant fluid velocities at
+<i>Q</i> and <i>P</i>, we get <i>&#7776;</i>&nbsp;=&nbsp;<i>S'</i>&nbsp;&#43;&nbsp;&frac12;(<i>q</i><sub>1</sub><sup>2</sup>&nbsp;&minus;&nbsp;<i>q</i><sub>0</sub><sup>2</sup>). This is
+Thomson's theorem. If the curve be closed, that is, if
+<i>P</i> and <i>Q</i> be coincident, <i>q</i><sub>1</sub>&nbsp;=&nbsp;<i>q</i><sub>0</sub> and <i>&#7776;</i>&nbsp;=&nbsp;<i>S'</i>. But in
+certain circumstances <i>S'</i> is zero, and so therefore is
+also <i>&#7776;</i>. Thus in the circumstances referred to, as the<span class='pagenum'><a name="Page_167" id="Page_167">167</a></span>
+closed path moves with the fluid <i>&#7776;</i> is continually zero,
+and it follows that if <i>&#7776;</i> is zero at any instant it remains
+zero ever after. But <i>&#7776;</i> is only zero if <i>u</i>, <i>v</i>, <i>w</i> 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.</p>
+
+<p>The circumstances in which <i>S'</i> is zero are these:&mdash;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 <i>S'</i> 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 <i>S'</i>
+which depends on pressure and density vanishes for the
+circuit.</p>
+
+<p>It is to be clearly understood that the motion of a
+fluid may be irrotational although the value of <i>S</i> does
+not vanish for <i>every</i> 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 <i>S</i> shall
+be zero; but what is necessary for irrotational motion
+within any space is that <i>S</i> should vanish for all paths
+which are capable of being shrunk down to zero without
+passing out of that space. <i>S</i> need not vanish for a
+path which cannot be so shrunk down, but it must, if the<span class='pagenum'><a name="Page_168" id="Page_168">168</a></span>
+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 <i>cyclic</i>, in the former <i>acyclic</i>.</p>
+
+<p>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 <i>dS</i>, a line perpendicular to the surface towards
+the side chosen as the positive side, and calculate the
+angular velocity <i>&#969;</i>, say, of the fluid about that normal
+from the components of angular velocity determined in
+the manner explained at p. <a href="#Page_164">164</a>. This Thomson
+called the <i>rotation</i> of the element. Now take the product
+<i>&#969;dS</i> 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 <i>dS</i> may change, but the circulation
+round its edge by Thomson's theorem remains
+unaltered. Thus <i>&#969;</i> alters in the inverse ratio of <i>dS</i>,
+and the line drawn at right angles to the surface at <i>dS</i>,
+if kept of length proportional to <i>&#969;</i>, will lengthen or
+shorten as <i>dS</i> contracts or expands.</p>
+
+<p>Now sum the values of <i>&#969;dS</i> for the finite surface
+enclosed by the bounding curve. It follows from the<span class='pagenum'><a name="Page_169" id="Page_169">169</a></span>
+fact that <i>&#969;dS</i> is equal to half the circulation round the
+edge of <i>dS</i>, that this sum, which is usually denoted by
+&#931;<i>&#969;dS</i>, 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 &#931;<i>&#969;dS</i> 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.</p>
+
+<p>The equality of 2&#931;<i>&#969;dS</i> 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 <i>Natural Philosophy</i>, &sect; 190.
+In the second edition of the <i>Natural Philosophy</i> 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.</p>
+
+<p>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.</p>
+
+<p><span class='pagenum'><a name="Page_170" id="Page_170">170</a></span>Take now any small plane area <i>dS</i> moving with the
+fluid, and draw axial lines through every point of its
+boundary. These will form an axial tube enclosing
+<i>dS</i>. If <i>&#952;</i> be the angle between the direction of resultant
+rotation and a perpendicular to <i>dS</i>, the cross-section
+of the tube at right angles to the normal, and to the
+axial lines which bound it, is <i>dS</i>.cos<i>&#952;</i>. Let these
+axial lines be continued in both directions from the
+element <i>dS</i>. 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.</p>
+
+<p>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 &#931;<i>&#969;dS</i>, 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<span class='pagenum'><a name="Page_171" id="Page_171">171</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>If any surface be drawn cutting a vortex-tube, it is
+clear from the definition of the tube that the value of
+&#931;<i>&#969;dS</i> for every such surface must be the same. This
+Thomson calls the "rotation of the tube."</p>
+
+<p>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.</p>
+
+<p>Examples of vortex-tubes are indicated on p. <a href="#Page_154">154</a>;
+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.</p>
+
+<div class="figleft" style="width: 150px; position: relative;"><a name="f13" id="f13"></a><img src="images/fig13.png" width="150" height="165" alt="Fig. 13." title="" />
+<p class="caption"><span class="smcap">Fig. 13.</span></p></div>
+
+<p>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<span class='pagenum'><a name="Page_172" id="Page_172">172</a></span>
+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. <a href="#f13">13</a>.) Of course, it is a ring
+of air, made visible by the smoke carried
+with it. By varying the shape of the
+aperture&mdash;for example, by using instead
+of the hole cut in the wood, a slide of
+tinplate with an elliptic hole cut in it&mdash;the vortex-rings
+can be set in vibration as they are created, and the
+vibrations studied as the vortex moves.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_173" id="Page_173">173</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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&mdash;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 <i>P</i>, and doubling back so as to return
+to <i>P</i> 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 <i>P</i> is<span class='pagenum'><a name="Page_174" id="Page_174">174</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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&mdash;the edge of a window
+recess, for example&mdash;will appear to be attracted. A
+large sphere such as a large terrestrial globe serves also
+very well as an attracting body.</p>
+
+<p>Two vortex-rings projected one after the other also<span class='pagenum'><a name="Page_175" id="Page_175">175</a></span>
+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. <a href="#f13">13</a>, 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.</p>
+
+<p>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 <i>Natural Philosophy</i> in the
+latest edition of Lamb's <i>Hydrodynamics</i>, and that
+many of these are concerned with general theorems
+and results of great value.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_176" id="Page_176">176</a></span></p>
+
+<h3>CHAPTER X</h3>
+
+<h4>THE ENERGY THEORY OF ELECTROLYSIS&mdash;ELECTRICAL
+UNITS&mdash;ELECTRICAL OSCILLATIONS</h4>
+
+<h3><span class="smcap">Electrolysis and Electrical Units</span></h3>
+
+<p><span class="smcap">In</span> December 1851 Thomson communicated an
+important paper to the <i>Philosophical Magazine</i> on
+"The Mechanical Theory of Electrolysis," and
+"Applications of Mechanical Effect to the Measurement
+of Electromotive Forces, and of Galvanic
+Resistances, in Absolute Units."</p>
+
+<p>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 <i>W</i> was supposed to be spent in a given time,
+during which a quantity of heat <i>H</i> was evolved in the
+circuit, and a certain amount of work <i>M</i> spent in the
+chemical apparatus in effecting chemical change. If
+<i>H</i> be taken in dynamical units, <i>W</i>&nbsp;=&nbsp;<i>H</i>&nbsp;&#43;&nbsp;<i>M</i>.</p>
+
+<p>The work done in driving the disk, if the intensity
+of the field is <i>I</i>, the current produced <i>c</i>, the radius of the
+disc <i>r</i>, and the angular velocity of turning <i>w</i>, is &frac12;<i>Ir</i><sup>2</sup><i>cw</i>.</p>
+
+<p>Thomson assumed that the work done in the electrochemical
+apparatus was equal to the heat of chemical<span class='pagenum'><a name="Page_177" id="Page_177">177</a></span>
+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 <i>M</i> represented
+this resultant heat of combination.</p>
+
+<p>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.</p>
+
+<p>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 <i>&#952;</i>, say, for the whole chemical
+change in the cell corresponding to the action on unit
+mass of that element. Thus if <i>E</i> denote the whole
+quantity of that element operated on the heat of combination
+in the vessel was <i>&#952;</i><i>E</i>. If <i>E</i> be taken for
+unit of time, and &#949; denote the quantity set free by the
+passage of unit quantity of electricity, then <i>E</i>&nbsp;=&nbsp;<i>&#949;c</i>,
+since a current conveys <i>c</i> units of electricity in one
+second. The number <i>&#949;</i> is a definite quantity of the
+element, and is called its electrochemical equivalent.
+Again, from Joule's experiments, <i>H</i>&nbsp;=&nbsp;<i>Rc</i><sup>2</sup>, if <i>R</i> denote
+the resistance of the current, and so</p>
+
+<div class="center"><img class="floatInsert13" src="images/f177a.png" alt="" title="" />
+</div>
+
+<p>and</p>
+
+<div class="center"><img class="floatInsert25" src="images/f177b.png" alt="" title="" />
+</div>
+
+<p><span class='pagenum'><a name="Page_178" id="Page_178">178</a></span></p>
+
+<p>The quantity &frac12;<i>Ir</i><sup>2</sup><i>w</i> is the electromotive force due to
+the disk.</p>
+
+<p>Thus <i>c</i> was positive or negative according as &frac12;<i>Ir</i><sup>2</sup><i>w</i>
+was greater or less than <i>&#952;&#949;</i>, and was zero when
+&frac12;<i>Ir</i><sup>2</sup><i>w</i>&nbsp;=&nbsp;<i>&#952;&#949;</i>. Thus the electromotive force of the disk
+was opposed by a back electromotive force <i>&#952;&#949;</i> due to
+the chemical action in the voltameter or battery, to
+which the wires from the disk were connected.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_179" id="Page_179">179</a></span>
+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.</p>
+
+<p>In the next paper, which appeared in the same
+number (December 1851) of the <i>Philosophical Magazine</i>,
+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&mdash;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&mdash;units
+independent of accidents of place and apparatus&mdash;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.
+<a href="#Page_244">XIII</a>.)</p>
+
+<p>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<span class='pagenum'><a name="Page_180" id="Page_180">180</a></span>
+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 2&#960; units. This current,
+<i>c</i>, say, in strength, flowing in the circuit, renders a
+dynamical force <i>cIl</i> necessary to move the slider of
+length <i>l</i> across the lines of force of the field of intensity
+<i>I</i>, and if the speed of the slider required for the current
+<i>c</i> be <i>v</i>, the rate at which work is done in moving the
+slider is <i>cIlv</i>. 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
+<i>cIlv</i>&nbsp;=&nbsp;<i>Rc</i><sup>2</sup>, or <i>Ilv</i>&nbsp;=&nbsp;<i>Rc</i>, so that <i>Ilv</i> is the electromotive
+force. Any electromotive force otherwise
+produced, which gave rise to the same current, must
+obviously be equal to <i>Ilv</i>, so that the unit of electromotive
+force can thus be properly defined.</p>
+
+<p>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>I</i> is one
+C.G.S. unit, <i>l</i> one centimetre, and <i>v</i> 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 <i>c</i> be unity as well as <i>Ilv</i>.</p>
+
+<p>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,<span class='pagenum'><a name="Page_181" id="Page_181">181</a></span>
+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.</p>
+
+<h3><span class="smcap">Electrical Oscillations</span></h3>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_182" id="Page_182">182</a></span>
+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&mdash;the ether&mdash;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&mdash;in fact, in the whole field. This
+motion&mdash;which is not to be confused with bodily displacement
+of finite parts of the medium&mdash;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.</p>
+
+<p>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 <i>N</i> 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,<span class='pagenum'><a name="Page_183" id="Page_183">183</a></span>
+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 <i>Q</i><sup>2</sup>. We suppose it to be
+&frac12;<i>Q</i><sup>2</sup>&nbsp;&frasl;&nbsp;<i>C</i>, where <i>C</i> is a constant depending on the
+condenser and called its capacity.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_184" id="Page_184">184</a></span>
+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.</p>
+
+<p>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
+<i>c</i> denote the current, that is, the rate, &minus;&nbsp;<i>dQ</i>/<i>dt</i>, at
+which the charge of the condenser is being changed,
+and <i>L</i> a quantity called self-inductance, depending<span class='pagenum'><a name="Page_185" id="Page_185">185</a></span>
+mainly on the arrangement of the connecting wire&mdash;whether
+it is wound in a coil or helix, with or without
+an iron core, or not&mdash;the electrokinetic energy will be
+&frac12;<i>Lc</i><sup>2</sup>. This is analogous to the kinetic energy &frac12;<i>mv</i><sup>2</sup>
+of a body (say a pendulum bob) of mass <i>m</i> and velocity
+<i>v</i>, so that <i>L</i> represents a quantity for the conducting
+arrangement analogous to inertia, and <i>c</i> is the analogue
+of the velocity of the body. The whole energy at
+any instant is thus</p>
+
+<div class="center">
+&frac12;<i>Q</i><sup>2</sup>&nbsp;&frasl;&nbsp;<i>C</i>&nbsp;&#43;&nbsp;&frac12;<i>Lc</i><sup>2</sup>, or &frac12;<i>Q</i><sup>2</sup>&nbsp;&frasl;&nbsp;<i>C</i>&nbsp;&#43;&nbsp;&frac12;<i>L</i>&nbsp;(<i>dQ</i>&nbsp;&frasl;&nbsp;<i>dt</i>)<sup>2</sup>.<br />
+</div>
+
+<p>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 <i>R</i> 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)
+<span class='pagenum'><a name="Page_186" id="Page_186">186</a></span>
+relatively to one another and so cause frictional
+generation of heat.</p>
+
+<p>The time rate of generation of heat is thus <i>Rc</i><sup>2</sup>, or
+<i>R</i>&nbsp;(<i>dQ</i>&nbsp;&frasl;&nbsp;<i>dt</i>)<sup>2</sup>, when the units in which <i>R</i> and <i>c</i> 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</p>
+
+<div class="center"><img class="floatInsert30" src="images/f186a.png" alt="" title="" />
+</div>
+
+<p>holds, or leaving out the common factor <i>dQ</i>&nbsp;&frasl;&nbsp;<i>dt</i>, the
+equation</p>
+
+<div class="center"><img class="floatInsert25" src="images/f186b.png" alt="" title="" />
+</div>
+
+<p>This last equation was established by Thomson, and
+is precisely that which would be obtained for a pendulum
+bob of mass <i>L</i>, pulled back towards the position
+of equilibrium with a force <i>Q</i>&nbsp;&frasl;&nbsp;<i>C</i>, where <i>Q</i> is the displacement
+from the middle position, and having its
+motion damped out by resisting force of amount <i>R</i> per
+unit of the velocity.</p>
+
+<p>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&mdash;the
+action of the spring just balances the weight,&mdash;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<span class='pagenum'><a name="Page_187" id="Page_187">187</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>The differential equation is applicable to the spring
+if <i>Q</i> is again taken as displacement from the equilibrium
+position, <i>L</i> as the inertia of the vibrating body, 1&nbsp;&frasl;&nbsp;<i>C</i>
+as the pull exerted by the spring per unit of its
+extension (that is, the stiffness of the spring), and <i>R</i> has
+the same meaning as before.</p>
+
+<p>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<span class='pagenum'><a name="Page_188" id="Page_188">188</a></span>
+equation, which in that respect is imperfect. There
+is no difficulty, only the addition of a little complication,
+in supplying the omission.</p>
+
+<p>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.</p>
+
+<p>It has been indicated that the differential equation
+does not represent oscillatory motion if the value of
+<i>R</i> is too great. The exact condition depends on the
+roots of the quadratic equation <i>Lx</i><sup>2</sup>&nbsp;&#43;&nbsp;<i>Rx</i>&nbsp;&#43;&nbsp;1&nbsp;&frasl;&nbsp;<i>C</i>&nbsp;=&nbsp;0,
+obtained by writing 1 for <i>Q</i>, and <i>x</i> for <i>d</i>&nbsp;&frasl;&nbsp;<i>dt</i>, and
+then treating <i>x</i> as a quantity. These roots are
+&minus;&nbsp;<i>R</i>&nbsp;&frasl;&nbsp;2<i>L</i>&nbsp;&plusmn;&nbsp;&#8730;(<i>R</i><sup>2</sup>&nbsp;&frasl;&nbsp;4<i>L</i><sup>2</sup>&nbsp;&minus;&nbsp;1&nbsp;&frasl;&nbsp;<i>CL</i>), and are therefore real
+or imaginary according as 4<i>L</i>&nbsp;&frasl;&nbsp;<i>C</i> is less or greater
+than <i>R</i><sup>2</sup>. If the roots are real, that is, if <i>R</i><sup>2</sup> be greater
+than 4<i>L</i>&nbsp;&frasl;&nbsp;<i>C</i>, 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
+<i>R</i> is sufficiently great, or <i>L</i>&nbsp;&frasl;&nbsp;<i>C</i> sufficiently small (a
+weak spring and a small mass, or both), to enable the
+condition to be fulfilled.</p>
+
+<p>If, however, the roots of the quadratic are imaginary,
+that is, if 4<i>L</i>&nbsp;&frasl;&nbsp;<i>C</i> be greater than <i>R</i><sup>2</sup> (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 <i>Q</i> disappears by oscillations
+about zero, of continually diminishing amplitude. A
+complete discussion gives for the period of oscillation
+<span class='pagenum'><a name="Page_189" id="Page_189">189</a></span>
+4&#960;<i>L</i>&nbsp;&frasl;&nbsp;&#8730;(4<i>L</i>&nbsp;&frasl;&nbsp;<i>C</i>&nbsp;&minus;&nbsp;<i>R</i><sup>2</sup>), or if <i>R</i> be comparatively small,
+2&#960;&#8730;(<i>LC</i>). The charge <i>Q</i> falls off by the fraction
+<i>e</i><sup>&nbsp;&minus;&nbsp;<i>RT</i>&frasl;2<i>L</i></sup> (where <i>e</i> is the number 2.71828...) in each
+period <i>T</i>, and so gradually disappears.</p>
+
+<p>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 <i>L</i>, <i>C</i>, and <i>R</i> 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.</p>
+
+<p>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&nbsp;&frasl;&nbsp;<i>C</i>, of the capacity of the
+condenser.</p>
+
+<p>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<span class='pagenum'><a name="Page_190" id="Page_190">190</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_191" id="Page_191">191</a></span>
+vibrations proceed, and the insulating medium is the
+seat of a periodically varying magnetic field, electromagnetic
+waves are propagated outwards through the
+surrounding medium&mdash;the ether&mdash;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&mdash;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.</p>
+
+<p>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."</p>
+
+<p><span class='pagenum'><a name="Page_192" id="Page_192">192</a></span>This anticipation was verified by experiments made
+by Feddersen, and published in 1859 (<i>Pogg. Ann.</i>,
+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 (<i>Pogg. Ann.</i>,
+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.]</p>
+
+<p>The particular case of non-oscillatory discharge
+obtained by supposing <i>C</i> and <i>Q</i> both infinitely great
+and to have a finite ratio <i>V</i> (which will be the potential,
+p. <a href="#Page_34">34</a>, 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 <i>c</i> be put as before
+for the current &minus;&nbsp;<i>dQ</i>&nbsp;&frasl;&nbsp;<i>dt</i>, and the constant value <i>V</i> for
+<i>Q</i>&nbsp;&frasl;&nbsp;<i>C</i>, the equation is</p>
+
+<div class="center"><img class="floatInsert22" src="images/f192a.png" alt="" title="" />
+</div>
+
+<p>which gives, since <i>c</i>&nbsp;=&nbsp;0 when <i>t</i>&nbsp;=&nbsp;0,</p>
+
+<div class="center"><img class="floatInsert22" src="images/f192b.png" alt="" title="" />
+</div>
+
+<p>Thus, when an infinite time has elapsed the current
+has become <i>V</i>&nbsp;&frasl;&nbsp;<i>R</i>, the steady value.</p>
+
+<p><span class='pagenum'><a name="Page_193" id="Page_193">193</a></span>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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_194" id="Page_194">194</a></span></p>
+
+<h3>CHAPTER XI</h3>
+
+<h4>THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'&mdash;GYROSTATIC
+ACTION&mdash;'ELECTROSTATICS AND MAGNETISM'</h4>
+
+<h3><span class="smcap">The 'Natural Philosophy'</span></h3>
+
+<p><span class="smcap">Professor Tait</span> 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&mdash;William
+Hopkins&mdash;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&mdash;an old school-fellow of
+Clerk Maxwell at the Edinburgh Academy&mdash;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<span class='pagenum'><a name="Page_195" id="Page_195">195</a></span>
+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.</p>
+
+<p>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&mdash;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&mdash;pendulum or gyrostat, or what not&mdash;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.</p>
+
+<p>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 <i>Elements of Quaternions</i>, 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<span class='pagenum'><a name="Page_196" id="Page_196">196</a></span>
+as non-scientific and fruitless, which no doubt they
+were from the physical point of view.</p>
+
+<p>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&mdash;and especially
+von Helmholtz&mdash;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.</p>
+
+<p>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.</p>
+
+<p>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
+<i>Treatise on Natural Philosophy</i>, the second was entitled<span class='pagenum'><a name="Page_197" id="Page_197">197</a></span>
+<i>Elements of Natural Philosophy</i>, and consisted in the
+main of part of the non-mathematical or large type
+portions of the <i>Treatise</i>. 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 <i>Elements</i>, to a student
+whose mathematical reading was wide enough to
+tackle the <i>Treatise</i>, 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_198" id="Page_198">198</a></span>
+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.</p>
+
+<p>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 <i>Theory of Sound</i>, Lamb's
+<i>Hydrodynamics</i>, and Routh's <i>Dynamics of a System of
+Rigid Bodies</i>. 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_199" id="Page_199">199</a></span>
+to that due to Newton's <i>Principia</i>. Three other works,
+perhaps, have had the same degree and kind of influence
+on mathematical thought&mdash;Laplace's <i>M&eacute;canique C&eacute;leste</i>,
+Lagrange's <i>M&eacute;canique Analytique</i>, and Fourier's <i>Th&eacute;orie
+Analytique de la Chaleur</i>.</p>
+
+<p>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&mdash;Professor
+Chrystal, of Edinburgh, and Professor
+Burnside, of Greenwich, may be mentioned&mdash;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.</p>
+
+<p>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.</p>
+
+<p>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-<i>dash</i>" 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<span class='pagenum'><a name="Page_200" id="Page_200">200</a></span>
+short original &sect; 343, he greatly extended that in the
+first instance, and proceeded from section to section
+until additions numbered from &sect; 343<i>a</i> to &sect; 343<i>p</i>,
+amounting in all to some ten pages of small print,
+had been interpolated. Similarly &sect; 345 was extended
+by the addition of &sect;&sect; 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.</p>
+
+<p>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.</p>
+
+<p>There can be no doubt that greater clearness is
+obtained by the process afterwards insisted on by Tait,<span class='pagenum'><a name="Page_201" id="Page_201">201</a></span>
+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 &sect; 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
+<i>Elements</i> by the method here indicated.</p>
+
+<p>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 &sect; 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 <i>speed</i> of the point (it
+does influence the <i>velocity</i>), 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.<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">20</a></p>
+
+<p><span class='pagenum'><a name="Page_202" id="Page_202">202</a></span>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.</p>
+
+<p>In the next chapter, which is entitled "Dynamical
+Laws and Principles," the authors at the outset declare
+their intention of following the <i>Principia</i> 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&mdash;inertia&mdash;is proportional to the quantity of
+matter in the body. This statement is criticised by
+Maxwell in his review of the <i>Natural Philosophy</i> in<span class='pagenum'><a name="Page_203" id="Page_203">203</a></span>
+<i>Nature</i> 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?"</p>
+
+<p>This innate power of resisting is merely the <i>materi&aelig;
+vis insita</i> of Newton's "Definitio III," given in the
+<i>Principia</i>, 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.</p>
+
+<p>The relativity of motion is not emphasised so greatly<span class='pagenum'><a name="Page_204" id="Page_204">204</a></span>
+in the <i>Natural Philosophy</i> 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.</p>
+
+<p>The treatment of force in the first and second laws
+of motion is frankly causal. Force is there the <i>cause</i>
+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.</p>
+
+<p>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 <i>Scholium</i> 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<span class='pagenum'><a name="Page_205" id="Page_205">205</a></span>
+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.</p>
+
+<p>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&mdash;the <i>M&eacute;canique C&eacute;leste</i>,
+the <i>M&eacute;canique Analytique</i>, and the memoirs of Jacobi
+and Hamilton&mdash;all of which were an outcome of the
+<i>Principia</i>, and from which, with the <i>Principia</i>, the
+authors of the <i>Natural Philosophy</i> drew their inspiration.</p>
+
+<p>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 <i>Natural Philosophy</i>. 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<span class='pagenum'><a name="Page_206" id="Page_206">206</a></span>
+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."</p>
+
+<p>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<span class='pagenum'><a name="Page_207" id="Page_207">207</a></span>
+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. <a href="#Page_214">214</a> below.)</p>
+
+<p>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."</p>
+
+<p>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<span class='pagenum'><a name="Page_208" id="Page_208">208</a></span>
+a discussion of technicalities beyond the reading of
+ordinary students of dynamics.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_209" id="Page_209">209</a></span>
+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.</p>
+
+<div class="figcenter" style="width: 550px; position: relative;"><a name="f14" id="f14"></a><img src="images/fig14.png" width="550" height="225" alt="Fig.14." title="" />
+<p class="caption"><span class="smcap">Fig. 14.</span></p></div>
+
+<p>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<span class='pagenum'><a name="Page_210" id="Page_210">210</a></span>
+of brass, about 2 inches in diameter (called the registering
+cylinder), and between the disk and this cylinder
+was laid a metal ball about 2&frac12; 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 <i>y</i>. It was also proportional to
+the speed of turning of the disk.</p>
+
+<p>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 <i>y</i>. 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<span class='pagenum'><a name="Page_211" id="Page_211">211</a></span>
+a distance <i>y</i>, 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 <i>y</i>, the
+ordinate of the curve, and for any displacement <i>dx</i>
+along the datum line, the registering cylinder is turned
+through an angle proportional to <i>ydx</i>. Thus any finite
+angle turned through is proportional to the integral of
+<i>ydx</i> 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 <i>y</i> sin <i>nx</i>&nbsp;.&nbsp;<i>dx</i>,
+where <i>y</i> is the ordinate of the curve, and <i>n</i> a number
+inversely proportional to the period of the tidal constituent
+represented by the term.</p>
+
+<p>All that was necessary, in order to give the integral
+of a term <i>y</i> sin <i>nx</i>&nbsp;.&nbsp;<i>dx</i>, was to make the disk oscillate
+about its axis as the paper-cylinder was turned through<span class='pagenum'><a name="Page_212" id="Page_212">212</a></span>
+an angle proportional to <i>x</i>. Thus one disk, globe, and
+cylinder was arranged exactly as has been described for
+the integral of <i>ydx</i>, 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 <i>y</i> from the centre
+of the disk.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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,<span class='pagenum'><a name="Page_213" id="Page_213">213</a></span>
+and, finally, to the integration of any differential
+equation of any order.</p>
+
+<p>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.</p>
+
+<p class="tb">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 <i>Natural Philosophy</i>.</p>
+
+<p>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<span class='pagenum'><a name="Page_214" id="Page_214">214</a></span>
+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 <i>Natural
+Philosophy</i> in Professor Karl Pearson's continuation of
+Todhunter's <i>History of Elasticity</i>. 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.</p>
+
+<p>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.</p>
+
+<h3><span class="smcap">Gyrostatic Action</span></h3>
+
+<p>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<span class='pagenum'><a name="Page_215" id="Page_215">215</a></span>
+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.</p>
+
+<div class="figcenter" style="width: 550px; position: relative;"><a name="f15" id="f15"></a><img src="images/fig15.png" width="550" height="283" alt="Fig. 15." title="" />
+<p class="caption"><span class="smcap">Fig. 15.</span></p></div>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_216" id="Page_216">216</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.<span class='pagenum'><a name="Page_217" id="Page_217">217</a></span></p>
+
+<p>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.</p>
+
+<p>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.<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">21</a></p>
+
+<p>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
+<span class='pagenum'><a name="Page_218" id="Page_218">218</a></span>globe&mdash;with inclined material axis rolling round a
+horizontal ring&mdash;by which the kinematics of the
+motion could be studied, and the displacement of the
+equinoxes on the ecliptic traced.</p>
+
+<div class="figcenter" style="width: 300px; position: relative;"><a name="f16" id="f16"></a><img src="images/fig16.png" width="300" height="455" alt="Fig. 16." title="" />
+<p class="caption"><span class="smcap">Fig. 16.</span></p></div>
+
+<p>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<span class='pagenum'><a name="Page_219" id="Page_219">219</a></span>
+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.</p>
+
+<div class="figcenter" style="width: 300px; position: relative;"><a name="f17" id="f17"></a><img src="images/fig17.png" width="300" height="445" alt="Fig. 17." title="" />
+<p class="caption"><span class="smcap">Fig. 17.</span></p></div>
+
+<p>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.<span class='pagenum'><a name="Page_220" id="Page_220">220</a></span></p>
+
+<p>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 arrowheads
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_221" id="Page_221">221</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_222" id="Page_222">222</a></span>
+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.</p>
+
+<p>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&mdash;the Zeeman effect,
+as it is called&mdash;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.<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">22</a></p>
+
+<h3><span class="smcap">Electrostatics and Magnetism</span></h3>
+
+<p>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 <i>Natural
+Philosophy</i>, and just before the appearance of the
+<i>Elements</i>, Messrs. Macmillan &amp; Co. published for him
+a collection of memoirs entitled <i>Reprint of Papers on</i>
+<span class='pagenum'><a name="Page_223" id="Page_223">223</a></span>
+<i>Electrostatics and Magnetism</i>. 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 <i>measuring</i> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_224" id="Page_224">224</a></span>
+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.</p>
+
+<p>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&mdash;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&mdash;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_225" id="Page_225">225</a></span>
+by an attached attracted-disk electrometer, the principle
+of which we shall merely indicate.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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 <i>Electrostatics and Magnetism</i>. These
+results have been much referred to in later researches.</p>
+
+<p><span class='pagenum'><a name="Page_226" id="Page_226">226</a></span>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.</p>
+
+<p>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."</p>
+
+<p>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<span class='pagenum'><a name="Page_227" id="Page_227">227</a></span>
+led to the machines with revolving carriers, to which
+reference has been made above.</p>
+
+<p>The mathematical theory of magnetism which
+Thomson gave in 1849, in the <i>Phil. Trans. R.S.</i>, 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<span class='pagenum'><a name="Page_228" id="Page_228">228</a></span>
+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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_229" id="Page_229">229</a></span></p>
+
+<h3>CHAPTER XII</h3>
+
+<h4>THE AGE OF THE EARTH</h4>
+
+<p><span class="smcap">From</span> 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 <i>Natural Philosophy</i>, and in vol. ii of his <i>Popular
+Lectures and Addresses</i>, which is devoted to geology
+and general physics.</p>
+
+<p>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. <a href="#Page_65">65</a> above) in his
+inaugural dissertation "<i>De Caloris distributione in Terr&aelig;
+Corpus</i>"; 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<span class='pagenum'><a name="Page_230" id="Page_230">230</a></span>
+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.</p>
+
+<p>The address began with a protest against the old
+uniformitarian view of geological changes as expressed
+by Playfair in his <i>Illustrations of the Huttonian
+Theory</i>. 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.</p>
+
+<p>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&mdash;that is, the
+earth's shape as distorted by the heaping up of the
+waters in the tides&mdash;remains on the whole stationary
+with respect to the moon, the solid matter of the
+earth turns under the distribution of the water, held<span class='pagenum'><a name="Page_231" id="Page_231">231</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_232" id="Page_232">232</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_233" id="Page_233">233</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_234" id="Page_234">234</a></span>
+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.</p>
+
+<p>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&mdash;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_235" id="Page_235">235</a></span>
+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
+<i>Transactions, R.S.E.</i>, 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_236" id="Page_236">236</a></span>
+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.</p>
+
+<p>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&mdash;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&deg;&nbsp;F. in 50 feet may be taken as holding for 5000
+feet or 6000 feet or more.</p>
+
+<p>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<span class='pagenum'><a name="Page_237" id="Page_237">237</a></span>
+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.</p>
+
+<p>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.&mdash;which is high enough for melting
+rock&mdash;and take the interval of time which has elapsed
+as 100,000,000 years, we obtain at the surface a<span class='pagenum'><a name="Page_238" id="Page_238">238</a></span>
+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.</p>
+
+<p>With the initial temperature of 7,000 degrees F.,
+an interval of 4,000,000 years would give a surface
+gradient of 1&deg;&nbsp;F. in 10 ft. Thus, on the assumption
+made, the surface gradient of temperature has diminished
+from <sup>1</sup>&frasl;<sub>10</sub> to <sup>1</sup>&frasl;<sub>50</sub> in about 96,000,000 years.
+After 10,000 years from the beginning of the cooling
+the gradient of temperature would be 2&deg;&nbsp;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.</p>
+
+<p>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&deg; 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<span class='pagenum'><a name="Page_239" id="Page_239">239</a></span>
+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&deg;&nbsp;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."</p>
+
+<p>Thomson was inclined to believe that a temperature
+of 7000&deg;&nbsp;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 <i>Nature</i> (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
+(<i>American Journal of Science</i>, January 1893) on experiments
+on the physical properties of rocks at high
+temperatures.</p>
+
+<p>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<span class='pagenum'><a name="Page_240" id="Page_240">240</a></span>
+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.</p>
+
+<p>In <i>Nature</i> 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.</p>
+
+<p>After the publication of the paper on Geological
+Time a reply to it was made by Professor Huxley, in<span class='pagenum'><a name="Page_241" id="Page_241">241</a></span>
+an address to the Geological Society of London,
+delivered on February 19, 1869. He adopted the <i>r&ocirc;le</i>
+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
+<i>Glasgow Herald</i> 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<span class='pagenum'><a name="Page_242" id="Page_242">242</a></span>
+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."</p>
+
+<p>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<span class='pagenum'><a name="Page_243" id="Page_243">243</a></span>
+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."</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_244" id="Page_244">244</a></span></p>
+
+<h3>CHAPTER XIII</h3>
+
+<h4>BRITISH ASSOCIATION COMMITTEE ON ELECTRICAL
+STANDARDS</h4>
+
+<p><span class="smcap">When</span> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_245" id="Page_245">245</a></span>
+to do for electricity and magnetism what had been
+done by Regnault and others for heat.</p>
+
+<p>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 <i>Intensitas vis
+magnetic&aelig; terrestris ad mensuram absolutam revocata</i>,
+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.</p>
+
+<p>Methods of measuring electric quantities in absolute
+units were described by W. Weber, in Parts II and
+III of his <i>Elecktrodynamische Maassbestimmungen</i>, 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. <a href="#Page_74">74-76</a>) 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<span class='pagenum'><a name="Page_246" id="Page_246">246</a></span>
+be sought which would have a temperature effect of
+small amount, and which were stable and durable in all
+their properties.</p>
+
+<p>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.</p>
+
+<p>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.<span class='pagenum'><a name="Page_247" id="Page_247">247</a></span></p>
+
+<p>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&mdash;that
+quantity which, concentrated at a point at unit distance
+from an equal quantity also concentrated at a point, is
+repelled with unit force&mdash;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 <i>m</i> be the mass of a ball, <i>g</i> gravity in cm.
+sec. units, <i>d</i> the distance in cms. of the centres of
+the balls apart, and <i>l</i> the length in cms. of a thread,
+the charge <i>q</i>, say, on each ball is easily found to be
+<img class="floatInsert18" src="images/f247.png" alt="" title="" />
+Thus the charge is got in
+absolute centimetre-gramme-second units in terms of
+the mass <i>m</i> obtained by ordinary weighing, and <i>l</i> and <i>d</i>
+obtained by easy and exact measurements.</p>
+
+<p>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<span class='pagenum'><a name="Page_248" id="Page_248">248</a></span>
+easily observed), the charge on the ball, and the value
+of <i>m</i>. The electric field-intensity is obtained by
+dividing the value of the force by <i>q</i>. The field intensity
+multiplied by <i>D</i>, 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 <i>D</i> gives the charge on the conductor.</p>
+
+<p>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&mdash;a
+disk turning on a uniform magnetic field, or the simple
+rails and slider and magnetic field arrangement there
+described&mdash;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_249" id="Page_249">249</a></span>
+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&mdash;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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_250" id="Page_250">250</a></span>
+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.</p>
+
+<p>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 <i>AH</i>, where <i>A</i> denotes
+the effective area of the coils, and <i>H</i> 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 <i>relatively to the coil</i> 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.</p>
+
+<p>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 <i>H</i>. But this was the field cut by the coil
+as it turned, and therefore (except for a small term<span class='pagenum'><a name="Page_251" id="Page_251">251</a></span>
+depending on the turning of the coil in the field of
+the needle) the value of <i>H</i> did not appear in the result,
+and did not require to be known.</p>
+
+<p>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<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">23</a>; 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.</p>
+
+<p>A unit called the B.A. unit, which was intended to
+contain 10<sup>9</sup> 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.</p>
+
+<p>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
+<span class='pagenum'><a name="Page_252" id="Page_252">252</a></span>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_253" id="Page_253">253</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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&mdash;10<sup>9</sup> C.G.S.
+units, now called the "ohm"&mdash;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.</p>
+
+<p>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. <a href="#Page_251">251</a>).</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_254" id="Page_254">254</a></span></p>
+
+<h3>CHAPTER XIV</h3>
+
+<h4>THE BALTIMORE LECTURES</h4>
+
+<p><span class="smcap">The</span> 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 <i>The Baltimore Lectures</i> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_255" id="Page_255">255</a></span>
+3&nbsp;&times;&nbsp;10<sup>10</sup> 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.</p>
+
+<p>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 <i>non-magnetic; nothing
+magnetic</i>." 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."</p>
+
+<p>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<span class='pagenum'><a name="Page_256" id="Page_256">256</a></span>
+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 <i>rationale</i> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_257" id="Page_257">257</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_258" id="Page_258">258</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>Dynamical illustrations, which old Glasgow students
+will recognise, appear continually in the lectures.
+They will remember, almost with affection, the system
+of three <i>particles</i> (7 lb. or 14 lb. weights!) joined<span class='pagenum'><a name="Page_259" id="Page_259">259</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_260" id="Page_260">260</a></span>
+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.</p>
+
+<div class="figcenter" style="width: 300px; position: relative;"><a name="f18" id="f18"></a><img src="images/fig18.png" width="300" height="473" alt="Fig. 18." title="" />
+<p class="caption"><span class="smcap">Fig. 18.</span></p></div>
+
+<p>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<span class='pagenum'><a name="Page_261" id="Page_261">261</a></span>
+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&mdash;the fact,
+that is, that light of different wave lengths has different
+velocities in ordinary transparent matter&mdash;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.</p>
+
+<p>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&mdash;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 R&uuml;cker 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<span class='pagenum'><a name="Page_262" id="Page_262">262</a></span>
+Loschmidt and by Johnstone Stoney), and from the
+undulatory theory of light&mdash;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&mdash;and
+they can be read at leisure by any one who
+will consult Volume I (Constitution of Matter) of Lord
+Kelvin's <i>Popular Lectures and Addresses</i>, for his Royal
+Institution Lecture on the subject, there given in full&mdash;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."</p>
+
+<p>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.</p>
+
+<p>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 <i>suo more</i> he endeavoured
+to frame dynamical explanations of electronic (or, as he<span class='pagenum'><a name="Page_263" id="Page_263">263</a></span>
+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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_264" id="Page_264">264</a></span></p>
+
+<h3>CHAPTER XV</h3>
+
+<h4>SPEED OF TELEGRAPH SIGNALLING&mdash;LAYING OF SUBMARINE
+CABLES&mdash;TELEGRAPH INSTRUMENTS&mdash;NAVIGATIONAL
+INSTRUMENTS, COMPASS AND
+SOUNDING MACHINE</h4>
+
+<h3><span class="smcap">Theory of Signalling</span></h3>
+
+<p><span class="smcap">When</span> 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<span class='pagenum'><a name="Page_265" id="Page_265">265</a></span>
+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 <i>n</i>, the changes of potential are
+propagated with velocity 2&#8730;(&#960;<i>n</i>&frasl;<i>cr</i>), where <i>c</i> and <i>r</i> are
+the capacity and resistance of the cable, each taken
+per unit length. In this case, for a long cable, there<span class='pagenum'><a name="Page_266" id="Page_266">266</a></span>
+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 &amp; Sons, Stoke-on-Trent,
+of mills for the manufacture of high
+conductivity copper, which is now a great industry.</p>
+
+<p>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.</p>
+
+<h3><span class="smcap">Laying of First American Cables</span></h3>
+
+<p>The first cable of 1858 was laid by the U.S. frigate
+<i>Niagara</i> and H.M.S. <i>Agamemnon</i>, 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<span class='pagenum'><a name="Page_267" id="Page_267">267</a></span>
+practical cases of importance&mdash;for example, in the lifting
+of a cable for repairs.</p>
+
+<p>A first Atlantic cable had been partly laid in 1857
+by the <i>Niagara</i>, 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
+<i>Agamemnon</i> towards Valentia, the <i>Niagara</i> towards
+Newfoundland. Professor Thomson was engineer in
+charge of the electrical testing on board of the <i>Agamemnon</i>.
+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.</p>
+
+<p>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.<span class='pagenum'><a name="Page_268" id="Page_268">268</a></span></p>
+
+<p>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 man&oelig;uvring on many occasions.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<h3><span class="smcap">Mirror Galvanometer and Siphon Recorder</span></h3>
+
+<p>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<span class='pagenum'><a name="Page_269" id="Page_269">269</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_270" id="Page_270">270</a></span>
+in the U.S. frigate <i>Niagara</i>) is in the collection of historical
+apparatus in the Natural Philosophy Department
+of the University of Glasgow.</p>
+
+<p>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.</p>
+
+<p>The galvanometer was replaced later by another
+invention of Professor Thomson&mdash;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.</p>
+
+<p>The arrangement of magnet and suspended coil in
+this instrument has become widely known as that of
+the "d'Arsonval galvanometer." This application was<span class='pagenum'><a name="Page_271" id="Page_271">271</a></span>
+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.</p>
+
+<p>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&mdash;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 &amp; 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.<span class='pagenum'><a name="Page_272" id="Page_272">272</a></span></p>
+
+<h3><span class="smcap">Mariners' Compass and Sounding Machine</span></h3>
+
+<p>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.</p>
+
+<p>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 3&frac14; 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.</p>
+
+<p>The weight of the card is extremely small&mdash;only
+170&frac12; grains; that is less than <sup>2</sup>&frasl;<sub>5</sub> 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<span class='pagenum'><a name="Page_273" id="Page_273">273</a></span>
+the card also causes the error due to friction on the
+point of support to be very small.</p>
+
+<p>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.</p>
+
+<p>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&deg;, the error attains a
+maximum on two courses 180&deg; apart. It may amount
+to over 20&deg; in an ordinary iron vessel, and to 30&deg; or
+40&deg; 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.<span class='pagenum'><a name="Page_274" id="Page_274">274</a></span></p>
+
+<p>The quadrantal error is so called because its maxima
+are found on four compass courses successively a
+quadrant, or 90&deg;, from one another. It amounts in
+general to from 5&deg; to 10&deg; 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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_275" id="Page_275">275</a></span>
+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.</p>
+
+<p>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 <i>Admiralty Manual of Deviations
+of the Compass</i>, led to Lord Kelvin's inventions.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_276" id="Page_276">276</a></span>
+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.</p>
+
+<p>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.<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">24</a></p>
+
+<p>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
+<span class='pagenum'><a name="Page_277" id="Page_277">277</a></span>business founded by Mr. White, and latterly carried
+on at Cambridge Street, has developed immensely, and
+is now owned by a limited liability company&mdash;Messrs.
+Kelvin and James White (Limited).</p>
+
+<p>For many years Sir William Thomson was a keen
+yachtsman, and his schooner yacht, the <i>Lalla Rookh</i>,
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_278" id="Page_278">278</a></span>
+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.</p>
+
+<p>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, &sect; 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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_279" id="Page_279">279</a></span></p>
+
+<h3>CHAPTER XVI</h3>
+
+<h4>LORD KELVIN IN HIS CLASS-ROOM AND LABORATORY</h4>
+
+<p><span class="smcap">It</span> 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&mdash;with
+long straight front and ends refracted inward,
+flanked by higher small round tables supported on
+cylindrical pillars&mdash;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<span class='pagenum'><a name="Page_280" id="Page_280">280</a></span>
+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&mdash;dear to the heart of the architect&mdash;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_281" id="Page_281">281</a></span>
+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.</p>
+
+<p>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<span class='pagenum'><a name="Page_282" id="Page_282">282</a></span>
+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 <i>Divina Commedia</i> of Dante.</p>
+
+<p>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.</p>
+
+<p>McFarlane used to tell of an incident which illustrated
+the ingenuity with which it was sometimes
+attempted to evade the ordeal of the <i>viva voce</i>
+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<span class='pagenum'><a name="Page_283" id="Page_283">283</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>The motion was defined as that of a particle moving
+along the diameter of a circle&mdash;the "auxiliary circle,"
+Thomson called it&mdash;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.<span class='pagenum'><a name="Page_284" id="Page_284">284</a></span></p>
+
+<p>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.</p>
+
+<p>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 <i>Logik</i>, 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 <i>philosopher</i> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_285" id="Page_285">285</a></span>
+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&mdash;a
+material spike of brass&mdash;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&mdash;all
+these delighted his hearers, and made the lecture
+memorable.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_286" id="Page_286">286</a></span>
+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.</p>
+
+<p>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,
+&#1258;&nbsp;&#43;&nbsp;&#956;&#1256;&nbsp;=&nbsp;0, according as &#956; 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.</p>
+
+<p>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 <i>Natural Philosophy</i>&mdash;"T and
+T'" was its familiar title&mdash;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 <i>Barlow</i> to make some calculation,
+and of course forget to return it. Next morning some
+number would be wanted from <i>Barlow</i> in a hurry, and
+the book would be missing. Then Thomson would<span class='pagenum'><a name="Page_287" id="Page_287">287</a></span>
+order that <i>Barlow</i> should be chained to the lecture-table,
+and enjoin his assistant to see that that was
+done without an hour's delay!</p>
+
+<p>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 <i>hundred</i> pieces of chalk
+on this ledge to-morrow; remember, a <i>hundred</i> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_288" id="Page_288">288</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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,<span class='pagenum'><a name="Page_289" id="Page_289">289</a></span>
+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 <big>V</big> 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.</p>
+
+<p>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.</p>
+
+<p>He often interrupted the course of a lecture with a<span class='pagenum'><a name="Page_290" id="Page_290">290</a></span>
+denunciation of the British "no-system of weights and
+measures"&mdash;"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 27<small><sup>1</sup>&frasl;<sub>3</sub></small> 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.</p>
+
+<p>Another experiment which caused a great sensation
+was that known as the "dew-drop"! 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&mdash;very
+slowly at first&mdash;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<span class='pagenum'><a name="Page_291" id="Page_291">291</a></span>-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.</p>
+
+<p>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!"</p>
+
+<p>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<span class='pagenum'><a name="Page_292" id="Page_292">292</a></span>
+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.</p>
+
+<p>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.<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">25</a></p>
+
+<p>The closing lecture of the ordinary course was
+usually on light, and the subject which was generally
+the last to be taken up&mdash;for as the days lengthened in
+spring, it was possible sometimes to obtain sunlight for
+the experiments&mdash;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
+<span class='pagenum'><a name="Page_293" id="Page_293">293</a></span>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.</p>
+
+<p>In the course of his lectures Thomson continually
+called on his assistants for data of all kinds. In the
+busiest time of his life&mdash;the fifteen years from 1870 to
+1885&mdash;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_294" id="Page_294">294</a></span>
+been made as to any numerical question regarding
+the subject under discussion.</p>
+
+<p>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,<span class='pagenum'><a name="Page_295" id="Page_295">295</a></span>
+a verdict which in later years became only more
+emphatic.</p>
+
+<p>One session he began the first lecture of the higher
+class by writing <i>dx</i>&nbsp;&frasl;&nbsp;<i>dt</i> 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
+<i>x</i> to the increment of the independent variable <i>t</i>, 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 <i>dx</i>&nbsp;&frasl;&nbsp;<i>dt</i> means velocity?" Here
+the physical idea as a whole was before his mind; and
+he did not reflect that if <i>t</i> denoted time and <i>x</i> distance
+in any direction, the explanation given by the student
+did describe velocity with fair accuracy.</p>
+
+<p>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.</p>
+
+<p>The subjects he treated were always such great
+branches of mathematics as the theory of the tides&mdash;he
+discussed the tidal phenomena of the English
+Channel in one course&mdash;the general theory of vibrations,
+Fourier analysis, the theory of waves in water,
+etc., etc. A very good idea of the manner and matter<span class='pagenum'><a name="Page_296" id="Page_296">296</a></span>
+of his mathematical prelections can be obtained from a
+perusal of the <i>Baltimore Lectures</i>.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_297" id="Page_297">297</a></span>
+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!"</p>
+
+<p>The incident referred to occurred while he was
+preparing the article on <i>Heat</i> for the ninth edition of
+the <i>Encyclop&aelig;dia Britannica</i>. 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 <i>Heat</i> (he also wrote <i>Elasticity</i>), 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<span class='pagenum'><a name="Page_298" id="Page_298">298</a></span>
+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!</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_299" id="Page_299">299</a></span></p>
+
+<h3>CHAPTER XVII</h3>
+
+<h4>PRACTICAL ACTIVITIES&mdash;HONOURS AND DISTINCTIONS&mdash;LAST
+ILLNESS AND DEATH</h4>
+
+<p><span class="smcap">It</span> 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.</p>
+
+<p>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<span class='pagenum'><a name="Page_300" id="Page_300">300</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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 <i>Pour le
+M&egrave;rite</i> 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.<span class='pagenum'><a name="Page_301" id="Page_301">301</a></span></p>
+
+<p>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
+<i>Nature</i> 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.</p>
+
+<p>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 <i>Baltimore
+Lectures</i>, 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.</p>
+
+<p>During the year 1907 Lord Kelvin performed
+many University duties and seemed to be in unusually<span class='pagenum'><a name="Page_302" id="Page_302">302</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.<span class='pagenum'><a name="Page_303" id="Page_303">303</a></span></p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_304" id="Page_304">304</a></span>
+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.</p>
+
+<p>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.</p>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_305" id="Page_305">305</a></span></p>
+
+<h3>CONCLUSION</h3>
+
+<p><span class="smcap">The</span> 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,
+Amp&egrave;re, 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&mdash;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<span class='pagenum'><a name="Page_306" id="Page_306">306</a></span>
+unmindful of the source from which much of their
+illumination has come.</p>
+
+<p>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&mdash;the systematic and clear formulation of the
+course of physical action.</p>
+
+<p>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.</p>
+
+<p>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<span class='pagenum'><a name="Page_307" id="Page_307">307</a></span>
+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&mdash;for
+a dream it was thought at the time&mdash;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.</p>
+
+<p>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&mdash;first
+the electrometers, and more recently the electric
+current balances and other beautiful instruments for
+the dynamo-room and the workshop&mdash;which Lord
+Kelvin invented have brought the precision of the
+laboratory into the everyday duties of the secondary
+battery attendant and the wireman.<span class='pagenum'><a name="Page_308" id="Page_308">308</a></span></p>
+
+<p>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.<a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">26</a> 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.</p>
+
+<p>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
+<span class='pagenum'><a name="Page_309" id="Page_309">309</a></span>for different parts of the world, superintended the
+analysis of tidal records, and invented tide-predicting
+machines and improved tide-gauges.</p>
+
+<p>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 <i>Mathematical and Physical
+Papers</i>, 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."</p>
+
+<p>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<span class='pagenum'><a name="Page_310" id="Page_310">310</a></span>
+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.</p>
+
+<p>It is interesting to note that the investigation of waves
+in canals with which Lord Kelvin recently enriched
+the <i>Proceedings of the Royal Society of Edinburgh</i> 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.</p>
+
+<p>Lord Kelvin's written work consists of the <i>Electrostatics
+and Magnetism</i>, three volumes of <i>Collected
+Mathematical and Physical Papers</i>, three of <i>Popular
+Lectures and Addresses</i>, the <i>Baltimore Lectures</i>, a very
+considerable number of papers as yet uncollected,
+and the <i>Natural Philosophy</i>. 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<span class='pagenum'><a name="Page_311" id="Page_311">311</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>It is hardly worth while to refute the statement<span class='pagenum'><a name="Page_312" id="Page_312">312</a></span>
+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.</p>
+
+<p>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 Amp&egrave;re, 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&mdash;Helmholtz and Clerk Maxwell, for example&mdash;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<span class='pagenum'><a name="Page_313" id="Page_313">313</a></span>
+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.</p>
+
+<p>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.</p>
+
+<div class="poem">
+<div class="stanza">
+<span class="i0">Saepe stilum vertas quae digna legi sint</span>
+<span class="i0">Scripturus,</span>
+</div>
+</div>
+
+<p>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.</p>
+
+<p>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&mdash;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.<span class='pagenum'><a name="Page_314" id="Page_314">314</a></span></p>
+
+<p>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.</p>
+
+<p>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&mdash;scientific, practical, or political&mdash;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.</p>
+
+<p>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<span class='pagenum'><a name="Page_315" id="Page_315">315</a></span>
+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&mdash;for he had felt&mdash;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.</p>
+
+<p>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&mdash;indeed like all
+great men&mdash;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<span class='pagenum'><a name="Page_316" id="Page_316">316</a></span>
+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.</p>
+
+<hr />
+<h3>FOOTNOTES</h3>
+<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">1</span></a> Lord Kelvin's address on his installation as Chancellor of the
+University of Glasgow, November 29, 1904.</p></div>
+
+<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">2</span></a> 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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">3</span></a> 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!</p></div>
+
+<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">4</span></a> Proceedings on the occasion of the Presentation to the University
+of Glasgow of the Portrait of Emeritus Professor G. G. Ramsay.
+November 6, 1907.</p></div>
+
+<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">5</span></a> Apparently for a short time in 1841, when Dr. Meikleham was
+laid aside by illness.</p></div>
+
+<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">6</span></a> 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 <i>Cambridge Chronicle</i>, July 10, 1903, and <i>The Cambridge
+Review</i>, Feb. 20, 1908.]</p></div>
+
+<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">7</span></a> It is rather strange that the ninth edition of the <i>Encyclop&aelig;dia
+Britannica</i> 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.'</p></div>
+
+<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">8</span></a> January 1869, <i>Reprint</i>, etc., Article XV.</p></div>
+
+<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">9</span></a> <i>Reprint</i>, Article V.</p></div>
+
+<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">10</span></a> 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 <i>Annali delle Scienze del Regno Lombardo-Veneto</i>
+for that year. It was also described as an independent discovery
+by Mr. John Wm. Stubbs, in a paper in the <i>Philosophical Magazine</i>
+for November 1843. In a note on the history of the transformation
+in Taylor's <i>Geometry of Conics</i> 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, <i>Phil. Soc.
+Trans.</i> I). The note also mentions that inversion was "applied by
+Dr. Hirst to attractions," but contains no reference to Thomson's
+papers!</p></div>
+
+<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">11</span></a> "<i>De Caloris distributione per Terr&aelig; Corpus</i>" in the Faculty minute,
+as stated above.</p></div>
+
+<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">12</span></a> <i>Sic.</i> Without doubt a mistake of the scribe for "Liouville."</p></div>
+
+<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">13</span></a> <i>North Wales Chronicle</i>, Report, Feb. 7, 1885.</p></div>
+
+<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">14</span></a> Published: <i>Treatise on Natural Philosophy</i>, vol. i in 1867; <i>Elements
+of Natural Philosophy</i> in 1873.</p></div>
+
+<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">15</span></a> 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 <i>Oration on James Watt</i>, 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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">16</span></a> 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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">17</span></a> 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.]</p></div>
+
+<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">18</span></a> 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 <i>p</i>&nbsp;&#43;&nbsp;<i>dp</i>
+and <i>v</i>, and on the delivery side as <i>p</i> and <i>v</i>&nbsp;&#43;&nbsp;<i>dv</i>, so that <i>dp</i> and <i>dv</i> are
+positive. The net work done in forcing the gas through the plug
+= (<i>p</i>&nbsp;&#43;&nbsp;<i>dp</i>)&nbsp;<i>v</i>&nbsp;&minus;&nbsp;<i>p</i> (<i>v</i>&nbsp;&#43;&nbsp;<i>dv</i>)&nbsp;=&nbsp;&minus;&nbsp;<i>pdv</i>&nbsp;&#43;&nbsp;<i>vdp</i>. Let a heating effect result
+so that temperature is changed from <i>T</i> to <i>T</i>&nbsp;&#43;&nbsp;<i>&#8706;</i><i>T</i>. Let this be annulled
+by abstraction of heat <i>C<sub>p</sub></i><i>&#8706;</i><i>T</i> at constant pressure. (<i>C<sub>p</sub></i>&nbsp;=&nbsp;sp. heat press.
+const.) [It is to be understood that <i>dv</i> is the total expansion existing,
+<i>after</i> this abstraction of heat.] The energy <i>e</i> of the fluid has been
+increased by <i>de</i>&nbsp;=&nbsp;&minus;&nbsp;<i>pdv</i>&nbsp;&#43;&nbsp;<i>vdp</i>&nbsp;&minus;&nbsp;<i>C<sub>p</sub></i><i>&#8706;</i><i>T</i>.
+</p><p>
+Now, since the original temperature has been restored, the same
+expansion <i>dv</i> if imposed isothermally would involve the same energy
+change <i>de</i>; but in that case heat <i>dH</i> (dynamical) would be absorbed,
+and work <i>pdv</i> would be done by the gas. Hence <i>de</i>&nbsp;=&nbsp;<i>dH</i>&nbsp;&minus;&nbsp;<i>pdv</i>.
+This, with the former value of <i>de</i>, gives <i>dH</i>&nbsp;=&nbsp;<i>vdp</i>&nbsp;&minus;&nbsp;<i>C<sub>p</sub></i><i>&#8706;</i><i>T</i>. Thomson's
+work-ratio is thus <i>pdv</i>&nbsp;&frasl;&nbsp;(<i>vdp</i>&nbsp;&minus;&nbsp;<i>C<sub>p</sub></i><i>&#8706;</i><i>T</i>). Now suppose <i>dp</i> imposed
+without change of volume, and <i>dT</i> to be the resulting temperature
+change. The temperature and pressure ratios are <i>dT</i>&nbsp;&frasl;&nbsp;<i>T</i>, <i>dp</i>&nbsp;&frasl;&nbsp;<i>p</i>. Thus
+<i>dT</i>&nbsp;&frasl;&nbsp;<i>T</i>&nbsp;=&nbsp;<i>dp</i> <i>dv</i>&nbsp;&frasl;&nbsp;(<i>vdp</i>&nbsp;&minus;&nbsp;<i>C<sub>p</sub></i><i>&#8706;</i><i>T</i>), or
+</p>
+<div class="center">
+<img class="floatInsert35" src="images/ftn18a.png" alt="" title="" />
+</div>
+<p>
+which is Thomson's equation. The <i>minus</i> sign on the right arises from
+a heating effect having been taken here as the normal case.
+</p><p>
+If the temperature <i>T</i> is restored by removing the heat at constant
+volume, a similar process gives the equation
+</p>
+<div class="center">
+<img class="floatInsert45" src="images/ftn18b.png" alt="" title="" />
+</div>
+<p>
+where <i>dp</i> is the change of pressure <i>before</i> the restoration of the
+temperature <i>T</i>, and <i>&#8706;</i><i>T</i>&nbsp;&frasl;&nbsp;<i>&#8706;</i><i>p</i> is the rate of variation of <i>T</i> with <i>p</i>, volume
+constant.</p></div>
+
+<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">19</span></a> "On a Universal Tendency in Nature to Dissipation of Energy,"
+<i>Proc. R.S.E.</i>, 1852, and <i>Phil. Mag.</i>, Oct., 1852.</p></div>
+
+<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">20</span></a> To this may be added the extremely useful theorem for such
+problems, that if any directed quantity <i>L</i>, say, characteristic of the motion
+of a body, be associated with a line or axis <i>Ol</i>, 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 <i>Ol</i>, towards which <i>Ol</i> is turning
+with angular velocity <i>&#969;</i>, of amount <i>&#969;L</i>. If <i>M</i> be the amount of the
+quantity already existing for this latter line or axis, the total rate of
+growth of the quantity is there <i>M</i>&nbsp;&#43;&nbsp;<i>&#969;L</i>. For example, a particle
+moving with uniform speed <i>v</i> in a circle of radius <i>r</i>, has momentum
+<i>mv</i> along the tangent. But the tangent is turning round as the particle
+moves with angular speed <i>v</i>&nbsp;&frasl;&nbsp;<i>r</i>, towards the radius. The rate of growth
+of momentum towards the centre is therefore</p>
+
+<p class="center"><i>mv</i> &times; <i>v</i>&nbsp;&frasl;&nbsp;<i>r</i> = <i>mv</i><sup>2</sup>&nbsp;&frasl;&nbsp;<i>r</i>.</p></div>
+
+<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">21</span></a> See Gray's <i>Lehrbuch der Physik</i>, s. 278. Vieweg u. Sohn, 1904.</p></div>
+
+<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">22</span></a> Gray, Royal Institution, Friday Evening Discourse, February 1898.</p></div>
+
+<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">23</span></a> See the <i>Reports of the Committee on Electrical Standards</i>, edited by
+Prof. Fleeming Jenkin, F.R.S., Maxwell's <i>Electricity and Magnetism</i>, and
+Gray's <i>Theory and Practice of Absolute Measurements in Electricity and
+Magnetism</i>, Vol. II, Part II.</p></div>
+
+<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">24</span></a> 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!</p></div>
+
+<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">25</span></a> The tuning of a major third, in this way, is described in the paper
+entitled "Beats on Imperfect Harmonies," published in <i>Popular Lectures
+and Addresses</i>, vol. ii.</p></div>
+
+<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">26</span></a> 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!</p></div>
+
+<hr />
+
+<p><span class='pagenum'><a name="Page_317" id="Page_317">317</a></span></p>
+
+<h3>INDEX</h3>
+
+<ul class="IX">
+<li>
+Atlantic cables, <a href="#Page_267">267</a>, <a href="#Page_268">268</a>
+</li><li>
+Atmospheric electricity, <a href="#Page_226">226</a>
+</li><li>
+Atoms, size of, <a href="#Page_261">261</a>
+</li><li>
+Ayrton, W. E., <a href="#Page_296">296</a>
+</li><li>&nbsp;
+</li><li>
+Baltimore lectures, <a href="#Page_254">254</a>-<a href="#Page_263">263</a>
+</li><li>
+Bertrand's theorem of maximum kinetic energy, <a href="#Page_158">158</a>
+</li><li>
+Bottomley, James Thomson, <a href="#Page_311">311</a>
+</li><li>
+Bottomley, William, <a href="#Page_7">7</a>
+</li><li>
+British Association, electrical standards, <a href="#Page_244">244</a>-<a href="#Page_253">253</a>
+</li><li>&nbsp;
+</li><li>
+Cambridge University Musical Society, <a href="#Page_24">24</a>
+</li><li>
+<i>Cambridge and Dublin Mathematical Journal</i>, <a href="#Page_25">25</a>, <a href="#Page_31">31</a>, <a href="#Page_78">78</a>
+</li><li>
+Carnot, Sadi, <a href="#Page_77">77</a>, <a href="#Page_101">101</a>
+</li><li>
+Carnot's <i>Th&eacute;orie Motrice du Feu</i>, <a href="#Page_87">87</a>, <a href="#Page_101">101</a>, <a href="#Page_108">108</a> <i>et seq.</i>
+</li><li>
+Cauchy, <a href="#Page_294">294</a>
+</li><li>
+Chasles, <a href="#Page_28">28</a>, <a href="#Page_43">43</a>
+</li><li>
+Clapeyron, <a href="#Page_101">101</a>, <a href="#Page_112">112</a>
+</li><li>
+Clausius, <a href="#Page_114">114</a> <i>et seq.</i>
+</li><li>
+College, the old, of Glasgow, <a href="#Page_10">10</a>
+</li><li>
+Compass, errors of, <a href="#Page_273">273</a>
+</li><li>&nbsp;
+</li><li>
+"Dew-drop," artificial, <a href="#Page_290">290</a>
+</li><li>
+Dynamical theorems, Thomson's and Bertrand's, <a href="#Page_158">158</a> <i>et seq.</i>
+</li><li>&nbsp;
+</li><li>
+Earth, the age of, <a href="#Page_196">196</a>, <a href="#Page_229">229</a>-<a href="#Page_243">243</a>
+</li><li>
+Earth, tidal retardation of, <a href="#Page_230">230</a>
+</li><li>
+Elasticity, Poisson-Navier theory of, <a href="#Page_291">291</a>;
+<ul><li>
+encyclop&aelig;dia article on, <a href="#Page_297">297</a>
+</li></ul></li><li>
+Electrical oscillations, <a href="#Page_181">181</a> <i>et seq.</i>
+</li><li>
+Electricity, mathematical theory of, <a href="#Page_33">33</a>
+</li><li>
+Electrolysis, mechanical theory of, <a href="#Page_176">176</a>
+</li><li>
+Electrometers, <a href="#Page_223">223</a> <i>et seq.</i>
+</li><li>
+Electromotive forces, estimation of, by heats of combination, <a href="#Page_178">178</a>
+</li><li>
+Electromotive forces, measurement of, <a href="#Page_179">179</a>
+</li><li>
+<i>Electrostatics and Magnetism</i>, <a href="#Page_222">222</a> <i>et seq.</i>
+</li><li>
+Ellis, Robert Leslie, <a href="#Page_26">26</a>
+</li><li>
+Energy, dissipation of, <a href="#Page_139">139</a>
+</li><li>&nbsp;
+</li><li>
+Faculty, the, of the University of Glasgow, <a href="#Page_4">4</a>, <a href="#Page_63">63</a>-<a href="#Page_67">67</a>
+</li><li>
+Faraday, <a href="#Page_61">61</a>
+</li><li>
+Faure, M., <a href="#Page_81">81</a>
+</li><li>
+FitzGerald, G. F., <a href="#Page_301">301</a>, <a href="#Page_305">305</a>
+</li><li>
+Fourier, <i>Th&eacute;orie Analytique de la Chaleur</i>, <a href="#Page_16">16</a> <i>et seq.</i>
+</li><li>&nbsp;
+</li><li>
+Gauss, <a href="#Page_28">28</a>
+</li><li>
+Gauss and Weber, <a href="#Page_245">245</a>
+</li><li>
+Green, George, of Nottingham, <a href="#Page_21">21</a>, <a href="#Page_30">30</a>, <a href="#Page_294">294</a>
+</li><li>
+Gregory, J. W., <a href="#Page_241">241</a>
+</li><li>
+Goodwin, Harvey, <a href="#Page_26">26</a>
+</li><li>
+Gyrostats and gyrostatic action, <a href="#Page_214">214</a>, <a href="#Page_284">284</a>-<a href="#Page_286">286</a>
+</li><li>&nbsp;
+</li><li>
+Hamilton, Sir William Rowan, <a href="#Page_196">196</a>, <a href="#Page_294">294</a>
+</li><li>
+Heat, encyclop&aelig;dia article on, <a href="#Page_297">297</a>
+</li><li>
+Heaviside, Oliver, <a href="#Page_294">294</a>
+</li><li>
+Helmholtz, von, <a href="#Page_113">113</a>
+</li><li>
+Hertz, <a href="#Page_191">191</a>, <a href="#Page_256">256</a>
+</li><li>
+Hopkins, William, <a href="#Page_23">23</a>
+</li><li>
+Huxley, <a href="#Page_77">77</a>, <a href="#Page_196">196</a>, <a href="#Page_242">242</a>
+</li><li>
+Hydrodynamics, <a href="#Page_153">153</a>-<a href="#Page_175">175</a>
+</li><li>&nbsp;
+</li><li>
+Images, electric, <a href="#Page_31">31</a>, <a href="#Page_38">38</a>-<a href="#Page_59">59</a>
+</li><li>
+Inversion, electrical, <a href="#Page_49">49</a> <i>et seq.</i>
+<span class='pagenum'><a name="Page_318" id="Page_318">318</a></span>
+</li><li>
+Inversion, geometrical, <a href="#Page_59">59</a>, <a href="#Page_60">60</a>
+</li><li>&nbsp;
+</li><li>
+Joule, James Prescott, <a href="#Page_77">77</a>, <a href="#Page_86">86</a> <i>et seq.</i>, <a href="#Page_101">101</a> <i>et seq.</i>
+</li><li>&nbsp;
+</li><li>
+Larmor, Joseph, <a href="#Page_256">256</a>
+</li><li>
+Lectures on Natural Philosophy at Glasgow, <a href="#Page_279">279</a> <i>et seq.</i>
+</li><li>
+Liouville, <a href="#Page_31">31</a>
+</li><li>
+Liouville's <i>Journal de Math&eacute;matiques</i>, <a href="#Page_25">25</a>, <a href="#Page_26">26</a>, <a href="#Page_31">31</a>
+</li><li>
+Loschmidt, <a href="#Page_262">262</a>
+</li><li>
+Lubbock, Sir John (Lord Avebury), <a href="#Page_85">85</a>
+</li><li>
+Luminiferous ether, motion of planets through, <a href="#Page_256">256</a>
+</li><li>&nbsp;
+</li><li>
+Magnetism, theory of, <a href="#Page_227">227</a>
+</li><li>
+Mariners' compass, <a href="#Page_272">272</a> <i>et seq.</i>
+</li><li>
+Maxwell, <a href="#Page_117">117</a>, <a href="#Page_193">193</a>, <a href="#Page_305">305</a>
+</li><li>
+Mayer, of Heilbronn, <a href="#Page_105">105</a>
+</li><li>
+McFarlane, Donald, <a href="#Page_96">96</a>, <a href="#Page_287">287</a>, <a href="#Page_289">289</a>
+</li><li>
+McKichan, Dugald, <a href="#Page_193">193</a>
+</li><li>
+<i>M&eacute;canique Analytique</i> of Lagrange, <a href="#Page_199">199</a>, <a href="#Page_205">205</a>
+</li><li>
+<i>M&eacute;canique C&eacute;leste</i> of Laplace, <a href="#Page_199">199</a>, <a href="#Page_205">205</a>
+</li><li>
+Meikleham, William, <a href="#Page_61">61</a>
+</li><li>
+Mirror galvanometer, <a href="#Page_268">268</a>
+</li><li>
+Motivity, thermodynamic, <a href="#Page_138">138</a>
+</li><li>&nbsp;
+</li><li>
+Natural Philosophy, Chair of, at Glasgow, <a href="#Page_63">63</a>
+</li><li>
+<i>Natural Philosophy</i>, Thomson and Tait's, <a href="#Page_196">196</a> <i>et seq.</i>
+</li><li>
+Navigational sounding machine, <a href="#Page_272">272</a>
+</li><li>
+Newton, <a href="#Page_195">195</a>, <a href="#Page_202">202</a>
+</li><li>
+Nichol, John, Professor of English Language and Literature, <a href="#Page_5">5</a>
+</li><li>
+Nichol, John Pringle, Professor of Astronomy, <a href="#Page_5">5</a>, <a href="#Page_20">20</a>, <a href="#Page_61">61</a>, <a href="#Page_63">63</a>
+</li><li>&nbsp;
+</li><li>
+Oersted, <a href="#Page_61">61</a>
+</li><li>
+Oscillations, electrical, <a href="#Page_181">181</a> <i>et seq.</i>
+</li><li>&nbsp;
+</li><li>
+Parkinson, Stephen, <a href="#Page_27">27</a>
+</li><li>
+Peltier, <a href="#Page_148">148</a>
+</li><li>
+Pendulum, ballistic, <a href="#Page_288">288</a>
+</li><li>
+Perry, John, <a href="#Page_240">240</a>, <a href="#Page_296">296</a>
+</li><li>
+Phosphorescence, dynamical theory of, <a href="#Page_259">259</a>
+</li><li>
+Physical laboratory, first, <a href="#Page_70">70</a>
+</li><li>
+Pickering, <a href="#Page_217">217</a>
+</li><li>
+Polarised light, rotation of plane of, <a href="#Page_220">220</a>
+</li><li>
+<i>Principia</i>, Newton's, <a href="#Page_195">195</a>, <a href="#Page_202">202</a>
+</li><li>&nbsp;
+</li><li>
+Ramsay, George Gilbert, Professor of Humanity, <a href="#Page_11">11</a>
+</li><li>
+Regnault, <a href="#Page_29">29</a>
+</li><li>
+Royal Society of Edinburgh, presidency of, <a href="#Page_299">299</a>
+</li><li>
+Royal Society of London, presidency of, <a href="#Page_299">299</a>
+</li><li>
+Rumford, Count, <a href="#Page_103">103</a>
+</li><li>&nbsp;
+</li><li>
+Seebeck, <a href="#Page_148">148</a>
+</li><li>
+Signalling, theory of telegraphic, <a href="#Page_264">264</a>
+</li><li>
+Siphon recorder, <a href="#Page_268">268</a>, <a href="#Page_270">270</a>
+</li><li>
+Smith, Archibald, <a href="#Page_275">275</a>
+</li><li>
+Spectrum analysis, dynamical theory of, <a href="#Page_84">84</a>
+</li><li>
+Stokes, Sir George Gabriel, <a href="#Page_24">24</a>, <a href="#Page_79">79</a>, <a href="#Page_80">80</a>, <a href="#Page_81">81</a>, <a href="#Page_85">85</a>, <a href="#Page_291">291</a>, <a href="#Page_294">294</a>
+</li><li>
+Stoney, Dr. Johnstone, <a href="#Page_262">262</a>
+</li><li>
+Sun's heat, age of, <a href="#Page_232">232</a>
+</li><li>&nbsp;
+</li><li>
+Tait, Peter Guthrie, <a href="#Page_194">194</a> <i>et seq.</i>
+</li><li>
+Temperature, absolute, <a href="#Page_125">125</a> <i>et seq.</i>;
+<ul><li>
+comparison of, with scale of air thermometer, <a href="#Page_135">135</a>
+</li></ul></li><li>
+Thermodynamics, <a href="#Page_99">99</a>-<a href="#Page_152">152</a>
+</li><li>
+Thermoelasticity, <a href="#Page_142">142</a> <i>et seq.</i>
+</li><li>
+Thermoelectricity, <a href="#Page_147">147</a> <i>et seq.</i>
+</li><li>
+Thermometry, absolute, <a href="#Page_114">114</a>-<a href="#Page_152">152</a>
+</li><li>
+Thomson, David, <a href="#Page_61">61</a>
+</li><li>
+Thomson, James, Professor of Mathematics, <a href="#Page_1">1</a>-<a href="#Page_4">4</a>, <a href="#Page_7">7</a>
+</li><li>
+Thomson, James, Professor of Engineering, <a href="#Page_113">113</a>, <a href="#Page_209">209</a>;
+<ul><li>
+integrating machine, <a href="#Page_209">209</a>, <a href="#Page_303">303</a>
+</li></ul></li><li>
+Thomson and Tait's Natural Philosophy, <a href="#Page_68">68</a>, <a href="#Page_196">196</a> <i>et seq.</i>, <a href="#Page_218">218</a>
+</li><li>
+Thomson's theorem of minimum kinetic energy, <a href="#Page_158">158</a>
+</li><li>
+Thomson, Thomas, Professor of Chemistry, <a href="#Page_6">6</a>
+<span class='pagenum'><a name="Page_319" id="Page_319">319</a></span>
+</li><li>
+<i>Thomson</i>, prevalence of name at Glasgow College, <a href="#Page_5">5</a>
+</li><li>
+Thomson, William, Lord Kelvin:&mdash;
+<ul><li>
+  Parentage and early education, <a href="#Page_1">1</a>-<a href="#Page_12">12</a>
+  </li><li>
+  Career at Universities of Glasgow and Cambridge, <a href="#Page_13">13</a>-<a href="#Page_32">32</a>
+  </li><li>
+  Early researches, <a href="#Page_16">16</a>, <a href="#Page_18">18</a>, <a href="#Page_31">31</a>
+  </li><li>
+  Election to Chair of Natural Philosophy at Glasgow, <a href="#Page_64">64</a>
+  </li><li>
+  Scientific researches, <i>passim</i>;
+    <ul><li>
+    Jubilee of, <a href="#Page_301">301</a>;
+    </li><li>
+    Chancellor of University of Glasgow, <a href="#Page_302">302</a>
+    </li></ul></li><li>
+  In class-room and laboratory, <a href="#Page_279">279</a>-<a href="#Page_298">298</a>
+  </li><li>
+  Practical activities, honours and distinctions, last illness and death, <a href="#Page_299">299</a>-<a href="#Page_304">304</a>;
+    <ul><li>
+    funeral in Westminster Abbey, <a href="#Page_304">304</a>
+    </li></ul></li></ul></li><li>
+
+Tidal Analyser, <a href="#Page_211">211</a>
+</li><li>
+Tide Predicter, <a href="#Page_208">208</a>
+</li><li>&nbsp;
+</li><li>
+Vortex-Motion, <a href="#Page_161">161</a>-<a href="#Page_175">175</a>
+</li><li>&nbsp;
+</li><li>
+Waldstein sonata, <a href="#Page_24">24</a>
+</li><li>
+Weber, W., <a href="#Page_193">193</a>
+</li><li>
+Weights and measures, British, <a href="#Page_289">289</a>, <a href="#Page_290">290</a>
+</li><li>
+White, James, <a href="#Page_276">276</a>
+</li><li>
+Willard Gibbs, <a href="#Page_294">294</a>
+</li></ul>
+
+<hr />
+<p class="center">
+<span class="smcap">Richard Clay &amp; Sons, Limited</span>,<br />
+<small>BREAD STREET HILL, E.C., AND<br />
+BUNGAY, SUFFOLK.</small><br />
+</p>
+
+
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
+End of the Project Gutenberg EBook of Lord Kelvin, by Andrew Gray
+
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diff --git a/LICENSE.txt b/LICENSE.txt
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@@ -0,0 +1,11 @@
+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
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diff --git a/README.md b/README.md
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #39373 (https://www.gutenberg.org/ebooks/39373)
author	Roger Frank <rfrank@pglaf.org>	2025-10-14 20:12:36 -0700
committer	Roger Frank <rfrank@pglaf.org>	2025-10-14 20:12:36 -0700
commit	c137749e541d5946db129883038780083a0bcd59 (patch)
tree	c9de17f82160587c2c3acfae2cd810778376d45a